(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 8.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 835, 17] NotebookDataLength[ 376966, 8484] NotebookOptionsPosition[ 294889, 6647] NotebookOutlinePosition[ 363251, 8038] CellTagsIndexPosition[ 363147, 8032] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJzt3V9sVOeZx/FIKyHFHmzPeBiDApEsERLZNEQEO8GkJEQkaLsKf2LUNsRZ KJFIIFQrF4JK1RCllqIVVSWjJm16gUTCHUjJXSxxV1nlGiGtVlqhvdiLvZi5 2Iu92IuVVvtwHp/nfc7fOWM7IeDvp1N3OPOe97znOe/xhX99z4ye/Kc3T/7d Y4899g/y3xOPP/bY/fcjAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAFZJq9VaxsZW Uk+H61X1YXc9dNeBbajGD696+/IjZttUrIN24ttUr2F23/KadG1Zcrgi1Y/S 015VhrS8YVcs7zLaV+ywqGX2QEV6bVyl/+qnX95D1w6rl6vKCHsdHgAAAAAA AAAAD7VWq/XrX/96fHxc8xHduHPnTtm4ZXTUb9y9e7dstBhF/3nhwoUzZ87I 7s1m0/6EfvLkyf379/t9U4fTHVPGI6mPpHM5kHSe6i132KL80NJb0afa4Hqj Uf01PDwsA6vY+FLU3h9a3hxrNP7caHxdr9+Nfsp72fJko6HF1DaX4h6Oue1F w7Y2fsfZej116PJ9/afPNJunok9vRYOUN5fr9YOZc+naVQmtg3QrO96NDiRv Trk65PYjA0tVL7vXQVf/gwVDetW1OVVt2NJg175Xps+ekdebH5z+yYnjW8fG slPUt5cG0kzby2v7xK6i9tLz0V9+IA2KejOv/fyn0lJ6tpbyU7bYwEpeupf8 1PbljfcdnbaSFvUvZ/f8y3uL6iBDrT4qLe+yi+DLLn586KAdWk7kic2bsyOs ODw5wa7jAQAAAAAAAADgodBqte7cuXPmzBlLfDRHu3fv3szMjG7UZnNzcwsL C41GY8vo6NWrV6WB7Hjjxo17EWssbt68eeHChdxMSv4pjaWfGxkTExP79++X T/1GaSlbrly5kurt5MmTsl2OIuPx+YgcWgeZDXq089QuqbHdjSKqiq+BgYGh oaGKjb+q16WxnIVW+NU4ccu+bkdhlhXzerx9Nuohe15+2NbG7ygvOZyedckp p/qXn6eiweQO8ut6fTIeZNeuSkzGAV9uHY7Fw04FkQeLB2Z7ydGfdM2+dvX3 XfkqPd3Xl1sl3/7Hhw7OfXPzj3/7a+r1q7/8aXTbU9nsafvELvko2372i8+f f3lvNjI++ssP5FM5xBObN6dGmxqJdmud6MbsgXJfO17aI3vt2vdKlcYyVK2n HKVrY6uDP6ncChSNSo4yffbMsovgK/n7W99mj/LWh+c0JbR5XnF4b35wuuhX BwAAAAAAAAAAD5cNGzZcjfj0SuO/+fl5v3FhYWFubk62yPY7d+7MzMwMRLaO jZ0/f17aHzlyRJfUaT6YmxDJ4aTlvn37dN+hJI3wBmK6URrr4Sy70REuRnzo Iz81r5RdZHsqL7hy5Yrmg9mcyMbWUz5Yq9Wq54NfDg3JGekpHKvQ/qBbn6hb ztZq0kN25H7Y1sbveDfKIjf29VXcVy/35YLYzidxPiIs6qpk7k0Wx3z2utRo pFLLKtW7HKeBs+4spvr6UrNi0u3y6fr1ekGL8kE59FsfntOo6MTHH02+/tqP dr/47NTuvUcOz37xuWz8/a1vZaMPqrZP7NKIShpIM2ksL2mj7eW17+h0Ixlw azSmMVZJxmqp1rN7pqyZ5ncXr1+TgelLj5h+7Zlqbdokx33+5b16LuXttz23 Qw8hdGy57Q+9f0qTUzll2cXflTrUiqOSo6ykCLr93blPdCTSg56C/Dzwzts6 QvlpI5SiaSfyadnwpnaPbntKpwf5IAAAAAAAAADgYaerBe/cuTM0NKRRhUZ4 GsBZpKUbDx069MTmzbpaMBXwzc/Pa8go7W/cuFEUw/l8UBsbaezzQftUO5c+ rcMto6PaifzcuXOnZTryU5rJufi4U487Pj6uCx7PnTunJ5WbD87W6/I6W6vp 69P1631+ZNv1lcoHrw0Ophr413R/vx73mWbTh2Kyl3w01dcnP+X93Xix28Yo zJJT8PmgyI48Fcxpm1Q+KK/fRCVNRS25+8rGg8l95dwPRIM8Xqv5ZY8+dpSe s12VJClSh1RX0nmqDvqSQ1suM5kcWG71dMA6f/wSQo1ofW7lM1A5kfJ88LWf /1Qjpx/tfjE6v5rOUp2uB955W7M2K/LWsTENB+Uja2/2Hjlsi+YaLvh+84PT tmDt2andReOxVEsGY1Nd88HZLz5//PHHbXhFZBfNBw+9f6prex2h5YO57fW8 pAI6Bj8qHWqVUWkKuZIiWIx78fo1+UVhZdcRNjeO6Ag1edTfbNrJ+IsvyAi7 Fq1keSkAAAAAAAAAAA+LVvRNgvfu3ZuYmNC/z2tIt3PnTvmpS2Zs4xObN2sq J2/0D+m6VE1+WsgoquSDQy6ONJYP2p/ihbyR3jQf1F3kWIuLi7Va7erVq7rI 0dZPSTNdzGiLfUbcw1Hl05J8ULbIsTRH0EBnqq/Prz5bt26dZhw+5vChmG+Q YiflM6lP169P7TLd36/hoA5yFfPB3AV0RfmgT+5kSH6QMravk59arFM9H9SH l5bXwT6Vw2muJwP7s9srW3DdS3qTjVrtoiWEreirFf0ANBsqWvMoGzXs03DQ 8ms9hKbYO17aM+BCWF3C9taH54raa6R48fo1n21pNPazc7O6zK21aVPRWleL xmzYlg/qEe1wKXqxpLHlgxqMlrTXivl80Oazp3eErtHTVYc+gPN3QcmoxEqK sHVsTGNcCwf9EXWE+45O25XynWgdcofn61D2+xQAAAAAAAAAgIeB/tlfv4JQ /zauYVytVltcXPQbFxYWBgYGNB+0v7rrX8tT0V7FfNAyEVvP5TvRkEWHJyPR R5tq8iUjkf6l2czMTGrloxxaOpcGPjfcMjqqTyjtmg/qIilNcBrR1/b5aMli lEb88E/56eMqy00aGZaA2HK2W/W6xluaWWgzW4dlodsq5oPZp4zm7juZSe78 icubjS42/Wxw0J5o2lM+aMO7Ha3dS5VO3vgg9el4NaVtuTY4mBqY7jUZPffV pz+5Swhle0+LB/Wr+i5ev2bjtPlps9QutE5FjdIaG5rac7a9RWkyPy211Ghs /MUX3rv8z5qRDRU8p7c8H7QOc+klsHzQlgmXtBc+H8yekQ5ARqKncOCdt1MB XG4dskdZYRH02aSyi4Wk/rx0hDpV7EplOykfIQAAAAAAAAAAj4Bms2lfQSgs fZubm/OPDJV/pvJB2VF7aCWjva754JkzZ6Sf/c6W0VELGfdFdPuRI0f0kaFb x8a0Q13tOD4+Lsd6YvNmDf4sidB8ULboIseGW28o7S0fzF2OZOdiUUIqH8zG DdlQrDyXeSbZ3tas2S6aOWrcpmNYxXxQG/ioJXffg8mzToVBsou0+SoO127V 67bqqno+6I/72eBgbh1Sw9Dsz7Ycj6qXSpk1pLMc2TbOZqLGXhcPaupkS+2y WZUVR99sn9jll/Jle9aB/eJ3l+zxmDqBNRqTLY0NTV2xqF+ulwouK+aDucW3 AaTywa75l4WeJf3bKWi3OpN9Ptj1+ZyWDy6jCLLl9B8u+8WA2cNpoVTLPf7U V7J8hAAAAAAAAAAAPAIsQRsaGtKvF5yYmBgYGDh06JA9MlS/fPD+o/lWIx/M kt2H4+8f9GQA8/PzOh7tUJ8Uao/3tO891CTi5s2b+uWGcjo2Bn2v+eD58+dz B5YlDbL5YGrH3HytpGff4YH+/twIwy9WWpV88Fa97r/xcDJK0DbE39iYzTdn kwvrsuHRhuiLGq2N5nSNZApZPR88Gyd9vg6p4uu3N2aXczbib+7zpbOQzo7l lxB+un69dNXT4kHLBydff61ilKbrDU//4XKVKE2X2mm8ZdGY7KgPIM19wOYP Nh9sxV//t/fI4ZXng7KL1Lx6EWzLtud2VAz7yAcBAAAAAAAAAGtTK/4Kwq1j Y5oJavqmWaGu5tPleLLR54D2p/5e80HpUL8wzp6lqY21E/tIlwGObnvKf9eh PvXUnkV55MgRWyooDeTQ0ol8pN+HKNttLaHmg0UDyy3LMvLB8vWDXTtMyeaD ttDM5C7c8/ngtcHB47WatbGv82sVrH/02d+6det8EGyjWt18UI/rG7ei74JM tem1etZVdgmhjwvLFw+OJBOrKgdNRW8lUVpqqZ3lg89O7dZbY/aLz/VLDFOL Fn+w+aBstIemrjAf1CLILroksEoRss8yLT9WbifkgwAAAAAAAACAtaAVfwXh zMyMPlPU0jd91uj58+dtyV7qKwKtB83pqueDFgvqgTRHsM7t08XFRRuPNNCj yDj940n1gaX2HFTNB+3Ro7Jlfn6+68Byy9JTPnhtcPBsrTZbr2dfz0SJ3grz wVv1+pdDQ1/V67LFv75yyVdRPrhu3Tr56ZvZ9xuW54P2cNFUZVY9H8w2zo6t qHq2bDDLPvJLCP1qyq6LB0eSiVWVyVMxekvlg7oWMhWNjW57KvcBmxXzwdyQ OneQXb+vcMTlg0XfP7hldPTduU/8GLLfP1jyBX/Zatv/UaFiEVaeDzaKv4ex a1cAAAAAAAAAADxc9CsI5+fnFxcXT548qX8nHx4evnDhwo2Ifvng0NDQxMSE PoA0lQ9KS8sQq+SDGv+llsL58FEHoKv/tL00vnLliozwRpJs8YfWfFDIGO7c ueNXIH6n+WDJS/rR3GEl+WCVV0k+6BfN3Y2fMppdo/cw5oNFJXq1+FsI71Ze PDjygPJBuwX+/h/fyT5gsyQfvHj9mmzc8dIeGUP2tXVszPIvHeTsF5/LAA6f fk8OnX1Z+xGXD46/+MKze6Z8tz85cfytD8/pysHf3/pWFw9qlmdDLTmK9GBH SVVbi5D7lNHVzQePX/qtjDB3ePuOTme/cRIAAAAAAAAAgIedfQWhpWn6x3b7 QsA33njDQkNd06d/ute/6m8ZHZWN2WV62cU4zWbT8r5UztJKPqRU/yCvjwwV 9rzTmZkZffqoLT/0kaXlg3Lo8fFx2S5bNCxoRKsLH0g+KPvqGT2ofFBTMB+Q 6VNGZVRV8sHsNyR+D/lgtk1u9UpKpGXXGZhaQng3el9l8eDIg8sH9exke/Yp oyX5YPlL+tdubZDlL2nj87iu7U98/FFjQ9PiOT/UiqPKVluLkH3KaNd8sOuz TEdcPljykgb2q6xrhwAAAAAAAAAAPCxa8VcQLi4u+ocHDg8P6xI8aWPfJqYP +dTvAdy/f/+VK1ekzcLCgn5BoYZ08/Pz/hGganx8XHqomA/q3/Yto5Rd/DcJ 2sKiRrQOTgajKxwtH9Shyhvf+Ht4vmju6+nVyAdL+q+SD+ql+Tr5MFIZwCOf DxYtIZQ+/VQvnwkPKh/UYWefMlqSD859c1M6LHrteGmPZaY2yMcjtfj7QD1L 6EZcPrh9Ytf9JYRTu2U88pIe9LiNDU1Non2VbKjlR2m4/y9BtgjyUfYpo6ub D8oZlQyvyjwBAAAAAAAAAODh0oq/gnB+ft7/aV1+Xr16dWFhwS8IGh4e3r9/ /40bN3Rpoe5lsZ3GcPfyaDbnnxeamw9aWKAbbQmhPgHVZxZKn4O6uLho+aAG ZPKR/ByOaMubN29+d/ng2Vpt3bp1GjF4Pv5YST5Y1H+tcj7YiB4r6hO0p/v6 quSDP+Tni+oEkPe1KO+bcmeUqnC2/hVTpAebD+otkHrKqBSn5PsHNYnL5lyp JM4GmVqQ6+mxUusHLUfTNvJGnyy6a98r/heFXWvLB/XTrEb8JNiialcswgrz waI66P+7gPWDAAAAAAAAAIBHjz7Ez0IB+1u9vLc/ktujGnVj7hKb3E+zf3W3 BDB3DLZkKTUwG57f1w/SN2hlyPAakSpLgZaRD9riqUaSjcd3eKC/v9d8MLf/ ocwzQovyQU1LU08ZLc8HN8ZL8FKV+R7yQV+r6f7+onzQT0hRkg9WuaBFM+EB 5oMj8S3gnzIq/yzPB3PnoZ+KqXxwKJm5ez7ps3zQYnpdw7vv6LQmd9lgLpXZ 6WnmHqUkH6xYhA0r+P5B7WQo86WoqeEBAAAAAAAAAPAosZxF/zxuf6u3VXjZ jRbbae5gD3LUflKhnrbRZv4Q2TGkPrUObffUvn6QvXZeXpBe80ELR7LRpHom uSau13wwt3/poWI+qEnHUPIpo9nxH8x7RGfRqG6vOB/8dP36bGN5nxqGtPGL H4+76tmsk3EeS66OlDZ+YdoK88G9Rw73lA/OfvF5lXxQui3PB1t5TxnVpCw3 H/SL+HKNVA4xU1fN8kHfvyZ3uoRQS+QnTK9r+orywSpF8GHf8vLBKvMBAAAA AAAAAIBHSSpBKNk4EsdtuetrLDUoWoaT7S17uOzGkoU8uQlIxRMsqcYy8sHy nKXZbN52a/dyExM/wmw+mO0/dwxF+aCGLKmnjKbyQf/pb6K1n6mzfjKKBVOd 95oP+oRRVyluSK4JvexCTE36/EpJOW5uqpgNLi0kqnJBi2bCT04c/+Pf/vqz c7NVojT5dOvYmLS/eP1aSSiWTaaK8sGReJGsPWCzuXGkaz5YnsStJB/M9i9F tiWEqd5WKx+sUgTp/Be/uyRbJl9/rUqSm3sVyAcBAAAAAAAAAFizlp0Plqwf lI987HU8fq6jNTgW5YZPxk9AXfV8cCQOWWYzSwhtX/n0lvv0QH9/arHYn5Pr +OzplNVLIdtPuU4+Xb/e8h2rg30qg7EI0h96OhqYr16qT3/WFS9o0UzYPrHL wq+SkKvlVrrpMrcto6NFC9nkI43bmhtH7PGeJQ8y9Q/Y/Nm52R9UPij7Fi0h XMV8sGsR5FNNck98/FH5SWXjS/JBAAAAAAAAAADQaz746fr10kZ2yX1Nxl+7 5h8xKq8vo+d5SoODLvyyiNDHfKuVD+ou2aeM+n19PCevy/W6nsWxKK3zyd3G uCw+H+xaCjmEX4Sop3wsrsPl5MAO9Pfr+OUQryYH9tngoFXvutuuaxJT5VpJ PigDvnj9mqZpuQvT5J9vfXhu+8QuW6SpIVfRI0bln+/OfaJJlj0OtDwfbCUf sKk/x1984YeQD45ESwgnX38tu4RwdfPB8iJI509s3rz0ANJo9+w3jW4ZHZXK y88N8bepkg8CAAAAAAAAAADVaz5Y/vqqXtfoTXo4VqH9gXhx3HeRD2rIksra /KI/Oe7lvO8o9K/b0WM/7RmezWTuWV4KXQw4mYwIc1+6DNDSn4rVm44jRR8P LTsf1CJvn9ilwZNFhBYwbR0b04xp7pubWnwNqjRSnP3i89FtT2l73UUjKo23 5L19+WB5NDaSfMCmvlYrHyz/3kwdhpxXST64tC41WtCnJbI1sD4f9Os9i6yk CPakU6ntgXfetuBVRyIXUS/KL353ya6Izwcbpet/iQ4BAAAAAAAAAHi0rW4+ +OXQkKYPzWazEa13K4rGbkfxloUp30U+OBKHLJcK8kFdYJh9Bqm9vq7XNRz0 uUxPpbCI8FbBUW5HDy+VQ1gapRFP1+rpekPLenq6oCWTQVpue26HRoRz39x8 68Nzb35wWl76hXf6bYP6pFDNxeSnRYQaSGn7d+c+sU6kQ1s8WCUaG0k+YHNV 8kFpf+j9U4dPv6fDy772HZ22MyrJB3VsO17ao9mclMLmkgZw5UeR19axMW2/ kiLoldKIUIts/Z/+w2XdeOLjj1qbNll0qMM7fum3MsKS4Um5dBJ2nS0AAAAA AAAAAOAh1Wq1JqPFbl8ODcnr6b6+7OMK9fsB5dNrg4Plr98MDFhipenGxr6+ 47WafHQ7DrbkvWzZ6HI3OdyleAzT/f25eZaNwbfxO8qhLfhL7fhk9IhOHb/v X8OgoeisZXf5VDO1W/X6Z1FLi+10EdbySqFRqbyXDqVbCwqlzdlazfJHX3aL CLtWzxaC9XRBS+jqxdamTSc+/kgDPnvNfXNz75HDUhA/Wg2q5J+HT7+nX8xn L9n90PunmhtH/JI6HeH02TP3o7E9U0VjayUfsDn+4gt+pZ490bRKPrhr3yt+ VEWvX/3lT1rMrvmgX0IoZ61tLIDr+nr+5b06dZfywWUVwWbItud2WCBoLxnb s1O79Uql4suuLxmVLRqtMmEAAAAAAAAAAMDDSIOGgShfy65HG4kDEW3weKnU wxU1bBqKVtL5fS1jsocfWrNUUpY7BmuT3TH3OY0aAqb2taxK8zvZmDq71CCX XYrUCWaPomsMszFfxeoVrTsruaBd54Mdd9tzO56d2i2vLaOj2lv2AvkKSzNt r2sGi2qo7e3Ec4dhyaMd1CJavV5Fy+6yp2M1tFPI0g51Uln/RcOzNjYwK1r5 UXw1tP3yijDiQmT9VMsuLw1k/Rn5GLcWKxledjYCAAAAAAAAAIBHjOY1w8PD jUZDVzZlIzYfZ5TQHny4YJ2nWtqBfJLYiBStpcq2SW0sysv8CWb7104sJUwN siS266kURTs24i/yy41julZveRe0Kx+bdq1JyTjtBLOrQbV9+dhsGL4Tq2TF ZW4l51J0vbr279ukpmLXiWG7rKQIfhhVrlRP83YZEwYAAAAAAAAAADx0LODQ NyUNusr2kLtvqlkrqfogWxmreILfRSm6tqk4sIp7dW3ZtYemU1ST8vbZXbpe r/KW1Xe39tWvV5XpNJI3Y3udGCssQurstOCaNuZeqV6HV6WwAAAAAAAAAAAA eCQtIy0iYHogKDsAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5NoJAAAAAAAAAAAAYM14 AwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAfO/+BQAAAAAAAAAAAMCa8a8AAAAAAAAAAAAA 1ox/AwAAAAAAAAAAALBm/DsAAAAAAAAAAACANeM/AAAAAAAAAAAAAKwZ/wkA AAAAAAAAAABgzei0O51Ou925/x/93+h/7v/Qf9x/q/9Z+nd7qYG9iXde2pzc 3o47bEf/CE2s9VKj6Hjxgd0+rsHSfzt+VB2/59Ln4eB2bu2278+104/iAdgB ls6+3YmP1w5vlz4M9QmnEXfrxtF2R0zVzZW9k+jSjSiuWXupEKHfxPHCj7BT +iKE6xafi52sbV7q0q5bXG+7JHHp7ZK33TCtvKGOcSM3l8I+S8N1F91qZpWz UoSDWYXc9bTZGy5UouyutPa5v35+9sQzOTGtXVXDJAnzLpx+Ym64qRjfA6na 2blbm0Tv/p5qu6O58XWS7xOjCsXsxLMtDCtxCf1Jtv3phnNzt0coqh9V8l4I V87fxPF2G4ifvskzDL8QbCrHc68d/tdNA3/4RCE7bovNJrsAnXCaHWtqZ2s/ /fQPN2DyctsYE0e1yof6JoYZ5pSfKe6Ojo8TJpi/Vdp2tRJ3qrWw0zLWS3yT +BHZzWaVbYfP3G+BThhBuE06bkDWh7vY4Y7thHslvI3P282tMK/TR+i0XX9h 1OEXRHzwMAHbbuShUPFMcTdhfDw/i8K8CBchDC7c4HYeoS5uNsSn4i+dXf70 Tex+CdmRE9MjzGR/3ds2Yn8Lhg9tqO5ODW3cLHIT3y5QmJZ+xtrUCaOz6+NH mfhtZGdtfVi5rKyh8omZZpfU3dyJS+amnV3TcIKulu4/jk2/+CqHWRsmZvIg 1qv/tRNmok12O2Y4fPqqhyP5m8OOELfv2K7hX3F1wkUKd6KvZrgl3PXrhL3s t4ex0wx3WrudbuFOyCa0n3m2m5+lyYvvjutmTWJ4dj3ju8eK6G7H1NYwncOV 80f2k9qmRmpSd+ydm1OJK9QOEzWeCW6KhpnrrmOYxW1fAz9qf+ptP1y7TxM1 C7d5Yg7YlY1ngJ+V7v5I1MD2CkMJNQ/3XPJOizuzoVsP8Zm3M0O23kPzcD3t EMlZZlciHnTivd1h6donbvH4ZrTpZO3iGy5MFDdjwpj9JLRZGiacDSVZNruT 3fxt+18INpkSY7QauUvsmoYrlyx+x43CX/VOONtw69g1sKtlF9UdONwZYSra lXI3iV3xcEOF2d4JVU1UxS5SmG3u0iQmhl3txOGXKmZzO/wW8PenmxTtdmqs Np64dn5rKGuYF223SzxfO2GANm9SN6K7gOGHlTj03rHTdCcZ5mM7jM0mYVwU d1HdYPw8Skwhq6KvTaiAnzvJ6sYlj29rG1m4ccPo/cRzg0/MSDf57aKGLeFk fdnstN1gQoUTdbIS2z3nr2aY1HZ+1j78OxzeDdj9Imi7Htwg/OzuuGOFKxzu Xnfp4/3DfZiYWnap3Mm6qWxTIdxVndB54ldDO3GyNmg7ms3+9O0SrqzrKlTL Kmr1sqH5OyVZRHfDJiri5mYn7ivUJdw3dvBQqMRst10TZxE3T064MLfDRQqN QnXa/jTjy9q2EVn5fWHi1jZnXDO7E23+2BlZQcKtGCanm4mh4p3ESfipYYNN HDzcgXbIjp1vuJOtTZgXnfiudneWv9385Ai3ie+ynTyrUDZX+cRFsWLbvWCj SN2tHd+x3bjWyF1MmzM22f2daOfqz8hqnLolE1fAdxp6dvepu1MTu4TfB/EN 5foMdfJ1DXMhzGebHMkZ6I4YCpP5PRBvT9yA7maN53C8KXzkLlQYs90z7vra mMKFa4cm7p5uhwOEqeDnfjvRkZt2NkXcpEn8OrBr4npxw/QXNRzCRu/vTnfL ujG7EYY92uF/QqN4loSr0Q7n4k7apmsYqLvqYXKE//EXqG3nljihMJNsGG6A 4bZyt4hNAz8X/P2SOEO7T+L37qpaVf2+4Qq76sTN7dYI1zbcTpkDpu6AduKu DFPA3Y3xvZi4WslboGMF8fVqu9478X0cqmTV9xfXjhVOwK51OE64P21axjM8 rqa7Y0Jjdx5WXHc7hA3uKrjBdJLHS81B17FdUJtGfo7GRwi/R9zUs0KEKx6m WTzz3EXuWD1CUd2d6IYaZmoolTsJq087jMeqGgpjEy7T1g4S/uUnT2Ki2Hji EvgZHW6jeA7ZNUgMNuxlbUM9E3e4u45WTptX+ua/AAAAAAAAAAAAAKwZ/w0A AAAAAAAAAABgzfgfAAAAAAAAAAAAAGvG/wIAAAAAAAAAAABYM/4PAAAAAAAA AAAAwJrx/1oIIGU= "], {{0, 75}, {2400, 0}}, {0, 255}, ColorFunction->RGBColor], BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable->False], BaseStyle->"ImageGraphics", ImageSize->Magnification[1], ImageSizeRaw->{2400, 75}, PlotRange->{{0, 2400}, {0, 75}}]], "hidefromslideshowgraphic", CellChangeTimes->{{3.4483017593296423`*^9, 3.448301769562791*^9}, { 3.4487297016867533`*^9, 3.448729710294153*^9}, {3.449486136735977*^9, 3.449486146926845*^9}, {3.449490735677544*^9, 3.4494907482045183`*^9}, { 3.473785056790244*^9, 3.4737850735465307`*^9}, {3.485608891427413*^9, 3.485608902078108*^9}, {3.516534228793694*^9, 3.516534238460294*^9}, { 3.517925345960129*^9, 3.5179253474520397`*^9}, {3.518187873028657*^9, 3.518187875044894*^9}, {3.5181990234752483`*^9, 3.5181990242399282`*^9}, { 3.518200170676401*^9, 3.5182001710513353`*^9}}, Background->None], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Analysis and Design of Feedback Control Systems Using ", FontSize->36], StyleBox["Mathematica", FontSize->36, FontSlant->"Italic"] }], "Title", ShowGroupOpener->False, CellChangeTimes->{{3.448190122176695*^9, 3.44819012385317*^9}, { 3.485609127497636*^9, 3.485609133015955*^9}, {3.514308355088097*^9, 3.514308374696691*^9}, 3.5146594348020153`*^9, 3.5146601881617107`*^9, 3.5280393276552057`*^9, 3.528039389335505*^9}], Cell["Pradipto Ghosh", "Subtitle", CellChangeTimes->{{3.485609136120798*^9, 3.4856091511532907`*^9}, { 3.4856091945334663`*^9, 3.485609199379443*^9}, {3.4951031489375*^9, 3.49510314984375*^9}, {3.495106455296875*^9, 3.495106455453125*^9}, { 3.5143083846926413`*^9, 3.514308395249558*^9}, {3.528039452580358*^9, 3.52803946812458*^9}, {3.528074019943356*^9, 3.528074029554685*^9}, 3.528134279820133*^9}], Cell["Wolfram Research, Inc.", "Subsubtitle", CellChangeTimes->{ 3.483202458953512*^9, {3.495105345328125*^9, 3.495105347890625*^9}, { 3.49510644571875*^9, 3.495106448390625*^9}, {3.5143083980990458`*^9, 3.514308409442589*^9}, {3.528039471775152*^9, 3.528039506473639*^9}, { 3.528074033317417*^9, 3.528074034625581*^9}, 3.528134293916136*^9}], Cell[BoxData[""], "Input", CellChangeTimes->{{3.528138586111092*^9, 3.528138588026973*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Talk Outline", "Section", CellChangeTimes->{{3.528135934583589*^9, 3.528135936444151*^9}, { 3.528136443446286*^9, 3.528136444626192*^9}}], Cell["\<\ System Model Representations Operations on Models Analysis of Model Properties Design of Control Systems\ \>", "Text", CellChangeTimes->{{3.5281361026195498`*^9, 3.5281361093280153`*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Model Representations", "Section", CellChangeTimes->{ 3.483202458955147*^9, {3.514308340990994*^9, 3.514308352103572*^9}, 3.5280399780628977`*^9, {3.528052587781603*^9, 3.528052596878368*^9}}], Cell["Transfer function models and state-space models", "Text", CellChangeTimes->{{3.495209008234375*^9, 3.49520915653125*^9}, 3.495209919765625*^9, 3.4952106014375*^9, {3.4952106824375*^9, 3.495210832234375*^9}, 3.514307848543872*^9, {3.514308058576482*^9, 3.514308065607885*^9}, {3.51430841745117*^9, 3.514308419642997*^9}, { 3.5149152616687326`*^9, 3.514915280523456*^9}, {3.514915328702818*^9, 3.5149153375415287`*^9}, 3.514915444638068*^9, 3.528039961512518*^9, { 3.528040001740385*^9, 3.5280400446568947`*^9}, {3.528074059318348*^9, 3.528074084452113*^9}, 3.5281387807871723`*^9, {3.528138827338632*^9, 3.528138858988262*^9}}], Cell[CellGroupData[{ Cell["Tachometer", "Subsection", CellChangeTimes->{ 3.483202458955147*^9, {3.495918529775271*^9, 3.4959185355106363`*^9}, { 3.495920008158016*^9, 3.495920032708687*^9}, {3.52804071533072*^9, 3.52804071753509*^9}}], Cell["\<\ A very simple system : A tachometer, which is a device used for measuring \ angular speed in terms of electric voltage \ \>", "Text", CellChangeTimes->{{3.5280407377333384`*^9, 3.5280407435141907`*^9}}], Cell[BoxData[ GraphicsBox[{ {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{194.6226, 329.}, {204.7006, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{203.6802, 329.}, {201.49020000000002`, 332.603}, {201.6122, 332.685}, {207.4872, 330.351}, {209.5342, 329.901}, {211.58120000000002`, 329.451}, {213.62820000000002`, 329.}, { 211.58120000000002`, 328.55}, {209.5342, 328.1}, {207.4872, 327.649}, { 201.6122, 325.315}, {201.49020000000002`, 325.377}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{274.92769999999996`, 329.}, { 294.96369999999996`, 329.}}}]}, { {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{331.2451, 329.}, {345.6441, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{372.0752687617261, 327.7522150562854}, { 369.8852687617261, 331.35521505628543`}, {370.0072687617261, 331.4372150562854}, {375.8822687617261, 329.1032150562854}, { 377.92926876172606`, 328.65321505628543`}, {379.9762687617261, 328.20321505628544`}, {382.02326876172606`, 327.7522150562854}, { 379.9762687617261, 327.30221505628543`}, {377.92926876172606`, 326.85221505628544`}, {375.8822687617261, 326.4012150562854}, { 370.0072687617261, 324.0672150562854}, {369.8852687617261, 324.12921505628543`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{482.8262, 329.}, {500.03119999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{499.01169999999996`, 329.}, { 496.82269999999994`, 332.603}, {496.9437, 332.685}, {502.8187, 330.351}, {504.86569999999995`, 329.901}, {506.91369999999995`, 329.451}, {508.9607, 329.}, {506.91369999999995`, 328.55}, { 504.86569999999995`, 328.1}, {502.8187, 327.649}, {496.9437, 325.315}, { 496.82269999999994`, 325.377}}}]}}, {Thickness[0.0012626262626262627`], {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{664.623, 208.01949999999997`}, { 668.2270000000001, 210.21049999999997`}, {668.307, 210.08749999999998`}, {665.975, 204.21249999999998`}, {665.523, 202.16549999999995`}, {665.0740000000001, 200.11949999999996`}, { 664.623, 198.07249999999996`}, {664.172, 200.11949999999996`}, { 663.7230000000001, 202.16549999999995`}, {663.2710000000001, 204.21249999999998`}, {660.937, 210.08749999999998`}, {661., 210.21049999999997`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{179.6226, 303.1777}, {176.0196, 300.98670000000004`}, {175.9376, 301.10970000000003`}, {178.2716, 306.98470000000003`}, {178.7216, 309.0317}, {179.1716, 311.0787}, { 179.6226, 313.1257}, {180.0726, 311.0787}, {180.5226, 309.0317}, { 180.9736, 306.98470000000003`}, {183.3076, 301.10970000000003`}, { 183.2456, 300.98670000000004`}}}]}}, { InsetBox["", {227.29126922539376`, 329.68859319586625`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontWeight->Bold, FontColor->GrayLevel[0.], FontOpacity->1., Background->GrayLevel[1.]], {245.6199527883859, 329.68859319586625`}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {313.69792030807093`, 231.49921696555123`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {405.3413381230317, 391.2206023001969}, {Left, Baseline}, Alignment->{Left, Top}]}, {AbsoluteThickness[1.6218100973589302`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{256.3299665404043, 289.19907691239325`}, {468.208269570708, 290.3662750961539}}]}, InsetBox["", {339.5087534722222, 337.0542024465812}, {Left, Baseline}, Alignment->{Left, Top}], {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{488.7785273202445, 330.2175859559195}, {488.6969608392103, 366.9006507073905}}]}, {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 507.81418328998126`, 293.6010039058056}, {507.81418328998126`, 283.632716571212}, {499.21197535047287`, 275.55185456743123`}, { 488.60057270397004`, 275.55185456743123`}, {477.9891700574671, 275.55185456743123`}, {469.3869621179584, 283.632716571212}, { 469.3869621179584, 293.6010039058056}, {469.3869621179584, 303.56929124039914`}, {477.9891700574671, 311.6501532441799}, { 488.60057270397004`, 311.6501532441799}, {499.21197535047287`, 311.6501532441799}, {507.81418328998126`, 303.56929124039914`}, { 507.81418328998126`, 293.6010039058056}}}, CurveClosed->{1}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{487.61132913648527`, 222.13584319416768`}, {487.5297626554513, 258.3135949018067}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["T", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->12, Background->GrayLevel[1.]], {485.40852644230813`, 290.3662750961538}, { Left, Baseline}, Alignment->{Left, Top}], {EdgeForm[{GrayLevel[0.], Opacity[0.62], AbsoluteThickness[ 1.8836490894898006`], JoinForm[{"Miter", 4.}]}], EdgeForm[None], RectangleBox[{484.24132825854724`, 327.71661697649574`}, \ {494.7461119123934, 310.2086442200855}]}, {EdgeForm[{GrayLevel[0.], Opacity[0.62], AbsoluteThickness[ 1.8836490894898006`], JoinForm[{"Miter", 4.}]}], EdgeForm[None], RectangleBox[{483.657729166667, 276.3598968910255}, \ {494.162512820513, 258.8519241346153}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{472.5693464209402, 219.16718588675212`}, {501.7493010149575, 219.16718588675212`}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{477.8217382478639, 212.16399678418816`}, {494.1625128205139, 212.16399678418816`}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{481.32333279914667`, 205.1608076816242}, {490.66091826923275`, 205.1608076816242}}]}, {AbsoluteThickness[1.6218100973589302`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{489.64011542210494`, 366.81775613247885`}, {525.7292769272666, 367.9849543162395}}]}, {EdgeForm[{GrayLevel[0.], Opacity[0.62], AbsoluteThickness[ 1.6218100973589302`], JoinForm[{"Miter", 4.}]}], EdgeForm[None], DiskBox[{530.9292556089746, 367.401355224359}, \ {5.835990918803532, 7.003189102564079}]}, InsetBox[ GraphicsBox[ {Dashing[{Small, Small}], CircleBox[{0, 0}, 0.1, NCache[{Rational[-1, 6] Pi, Rational[1, 6] Pi}, {-0.5235987755982988, 0.5235987755982988}]]}, ImageSize->{Automatic, 44.815750202128896`}], Scaled[{0.3034188034188035, 0.4916600841896388}], Center, {64.19590010683766, 47.53491962820512}, {{1., 0.}, {0., 1.}}], {Arrowheads[0.027039583641088438`], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], ArrowBox[{{327.9028394679598, 274.5467386824394}, {323.8093812235013, 260.9499877313238}}]}, InsetBox[ StyleBox[ TagBox["\[Theta]", HoldForm], FontSize->13, Background->GrayLevel[1.]], {304.49280795940183`, 267.0223114209401}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[ TagBox["+", HoldForm], FontSize->13, Background->GrayLevel[1.]], {546.1028319978633, 365.0669588568375}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ StyleBox["e", FontSize->16], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {535.5980483440171, 341.72299518162384`}, { Left, Baseline}, Alignment->{Left, Top}]}, ImagePadding->{{0., 1.}, {1., 3.1072}}, ImageSize->{469., Automatic}, PlotRange->{{165.012625, 711.261375}, {187.584524, 401.708578}}, PlotRangePadding->Automatic]], "Input", CellChangeTimes->{{3.5280417768054247`*^9, 3.528042037463716*^9}, { 3.52804207937269*^9, 3.528042149304494*^9}, {3.5280422765383472`*^9, 3.528042287722774*^9}, {3.528042324640727*^9, 3.528042423395144*^9}, { 3.52804246503131*^9, 3.528042507590324*^9}}], Cell[BoxData[ RowBox[{"tfTacho", " ", "=", " ", RowBox[{"TransferFunctionModel", "[", RowBox[{"s", ",", "s"}], "]"}]}]], "Input", CellChangeTimes->{{3.494946982054118*^9, 3.494947000142241*^9}, 3.495919335502689*^9, {3.4959194441070538`*^9, 3.495919476086821*^9}, 3.4959195547984943`*^9, 3.528040769914503*^9}] }, Closed]], Cell[CellGroupData[{ Cell["Electric Network", "Subsection", CellChangeTimes->{ 3.483202458955147*^9, {3.495918529775271*^9, 3.4959185355106363`*^9}, { 3.495920008158016*^9, 3.495920032708687*^9}, {3.528040841563778*^9, 3.528040848311912*^9}}], Cell[TextData[{ "A slightly more complicated system, the state-space model of an electrical \ network. The ", Cell[BoxData[ ButtonBox["SystemsModelLabels", BaseStyle->"Link"]], "InlineFormula"], " option labels the input, output and state variables:" }], "Text", CellChangeTimes->{{3.5280408904566193`*^9, 3.528040930773987*^9}}], Cell[BoxData[ GraphicsBox[{ {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{194.6226, 329.}, {204.7006, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{203.6802, 329.}, {201.49020000000002`, 332.603}, {201.6122, 332.685}, {207.4872, 330.351}, {209.5342, 329.901}, {211.58120000000002`, 329.451}, {213.62820000000002`, 329.}, { 211.58120000000002`, 328.55}, {209.5342, 328.1}, {207.4872, 327.649}, { 201.6122, 325.315}, {201.49020000000002`, 325.377}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{274.92769999999996`, 329.}, { 294.96369999999996`, 329.}}}]}, { {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{331.2451, 329.}, {345.6441, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{372.0752687617261, 327.7522150562854}, { 369.8852687617261, 331.35521505628543`}, {370.0072687617261, 331.4372150562854}, {375.8822687617261, 329.1032150562854}, { 377.92926876172606`, 328.65321505628543`}, {379.9762687617261, 328.20321505628544`}, {382.02326876172606`, 327.7522150562854}, { 379.9762687617261, 327.30221505628543`}, {377.92926876172606`, 326.85221505628544`}, {375.8822687617261, 326.4012150562854}, { 370.0072687617261, 324.0672150562854}, {369.8852687617261, 324.12921505628543`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{482.8262, 329.}, {500.03119999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{499.01169999999996`, 329.}, { 496.82269999999994`, 332.603}, {496.9437, 332.685}, {502.8187, 330.351}, {504.86569999999995`, 329.901}, {506.91369999999995`, 329.451}, {508.9607, 329.}, {506.91369999999995`, 328.55}, { 504.86569999999995`, 328.1}, {502.8187, 327.649}, {496.9437, 325.315}, { 496.82269999999994`, 325.377}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{624.2067928178433, 352.4443582066745}, { 622.0177928178433, 356.0473582066745}, {622.1387928178433, 356.1293582066745}, {628.0137928178433, 353.7953582066745}, { 630.0607928178433, 353.3453582066745}, {632.1087928178433, 352.89535820667453`}, {634.1557928178433, 352.4443582066745}, { 632.1087928178433, 351.9943582066745}, {630.0607928178433, 351.54435820667453`}, {628.0137928178433, 351.0933582066745}, { 622.1387928178433, 348.7593582066745}, {622.0177928178433, 348.8213582066745}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{648.1934000000001, 329.}, { 700.3364000000001, 329.}}}]}}, {Thickness[0.0012626262626262627`], {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{664.623, 329.}, {664.623, 207.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{664.623, 208.01949999999997`}, { 668.2270000000001, 210.21049999999997`}, {668.307, 210.08749999999998`}, {665.975, 204.21249999999998`}, {665.523, 202.16549999999995`}, {665.0740000000001, 200.11949999999996`}, { 664.623, 198.07249999999996`}, {664.172, 200.11949999999996`}, { 663.7230000000001, 202.16549999999995`}, {663.2710000000001, 204.21249999999998`}, {660.937, 210.08749999999998`}, {661., 210.21049999999997`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{179.6226, 303.1777}, {176.0196, 300.98670000000004`}, {175.9376, 301.10970000000003`}, {178.2716, 306.98470000000003`}, {178.7216, 309.0317}, {179.1716, 311.0787}, { 179.6226, 313.1257}, {180.0726, 311.0787}, {180.5226, 309.0317}, { 180.9736, 306.98470000000003`}, {183.3076, 301.10970000000003`}, { 183.2456, 300.98670000000004`}}}]}}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{205.1493686663968, 216.7427301193011}, {205.0678021853626, 274.06001879347446`}}]}, { InsetBox["", {227.29126922539376`, 329.68859319586625`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontWeight->Bold, FontColor->GrayLevel[0.], FontOpacity->1., Background->GrayLevel[1.]], {245.6199527883859, 329.68859319586625`}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {313.69792030807093`, 231.49921696555123`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {405.3413381230317, 391.2206023001969}, {Left, Baseline}, Alignment->{Left, Top}], { {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 218.1028244906338, 288.11626822284865`}, {218.1028244906338, 280.7204421358921}, {212.10734622976423`, 274.7249638750225}, { 204.71152014280773`, 274.7249638750225}, {197.3156940558511, 274.7249638750225}, {191.3202157949815, 280.7204421358921}, { 191.3202157949815, 288.11626822284865`}, {191.3202157949815, 295.51209430980515`}, {197.3156940558511, 301.5075725706747}, { 204.71152014280773`, 301.5075725706747}, {212.10734622976423`, 301.5075725706747}, {218.1028244906338, 295.51209430980515`}, { 218.1028244906338, 288.11626822284865`}}}, CurveClosed->{1}]}, FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, { 0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 184.3965780525378, 301.987506080743}, {184.3965780525378, 303.391506080743}, {187.25057805253783`, 303.391506080743}, { 187.25057805253783`, 306.223506080743}, {188.63257805253778`, 306.223506080743}, {188.63257805253778`, 303.391506080743}, { 191.48557805253785`, 303.391506080743}, {191.48557805253785`, 301.987506080743}, {188.63257805253778`, 301.987506080743}, { 188.63257805253778`, 299.135506080743}, {187.25057805253783`, 299.135506080743}, {187.25057805253783`, 301.987506080743}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 187.55211834693006`, 266.9001004785577}, {194.54811834693015`, 266.9001004785577}, {194.54811834693015`, 265.6721004785577}, { 187.55211834693006`, 265.6721004785577}}}]}, StyleBox[ InsetBox["", {179.4707287315495, 284.44689558569735`}, {Left, Baseline}, Alignment->{Left, Top}], FontSize->13]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{339.9289196260143, 348.3757255411389}, { 337.73991962601445`, 351.9787255411389}, {337.8609196260143, 352.0607255411389}, {343.7359196260143, 349.7267255411389}, { 345.78291962601435`, 349.2767255411389}, {347.83091962601435`, 348.8267255411389}, {349.8779196260144, 348.3757255411389}, { 347.83091962601435`, 347.9257255411389}, {345.78291962601435`, 347.4757255411389}, {343.7359196260143, 347.0247255411389}, { 337.8609196260143, 344.6907255411389}, {337.73991962601445`, 344.7527255411389}}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{205.1493686663972, 303.04630016071854`}, {205.067802185363, 352.33916842050763`}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{205.705070534898, 352.1941128022442}, {254.99621900504698`, 352.6138709010023}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{256.60696668353637`, 352.0054553718966}, {263.24156688596554`, 368.67971513990636`}, {275.40500059041904`, 339.4997605458893}, { 286.4626675944676, 368.67971513990636`}, {296.35313641475557`, 345.9841949001153}}]}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], LineBox[{{296.32242067307703`, 342.89019336538456`}, {307.994402510684, 368.56855340811967`}}]}, {AbsoluteThickness[0.8222426499470712], JoinForm[{"Miter", 4.}], LineBox[{{306.9105756257633, 368.45183358974344`}, {314.9142203144081, 348.842904102564}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{315.42169980840276`, 348.10891915908195`}, {364.71284827855175`, 348.52867725784006`}}]}, {AbsoluteThickness[1.3963683610559372`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[CompressedData[" 1:eJyN0k9I03EcxvExTap1KKEQy6IskNEhLLQUaqmEKJUahGVICUJumv9yOgsS Zx6GQWpESlKIRTgMslCxnH9SM81000JnGiQJmQdRULGS6vD+fWC/Qx1EXpf9 vp/neXanZCakajUaje3P39//90+/aq1rMBvK9MbY2Kk8Ay5dTrevBZkVd9dn WO3hYm3ylaTEU+LjWzIPrrskLurO1DXmih35WdPJpWaDW9tWEd+h9k999ktd ldng39RmXHOpHTaVXdliNxsS0xwR9q/iuzva/RNX1LaU55hSHWbD6FD7grcu X7GvtaP/WYA4IaSzNvmA+Pa3zkJdpHhT3Gt96mVxjFe31vea2NbU7XbcEvel 9TSaHop9Anptfs/FUcO9KT094mLrm7CcMXFXSJ/vru9qcx8mH0yemtm+2YFf +Ur+mH4w/WH6xfSPo6PSR+K3iTeUm37Uecn3Pd0/aQxcmc9T7sPcj8kHk5+n +X3y9jTvox9Pcw99ci8mD0xemDwxeWP6wPTF9/G/9oJbonIjfZ2yH0x+mHwx +S8t525vn5a+8NGat10FmwsUX4/rr94XKF5tGoi5ES0+bHy3Z3+S2BwwuDqW IX4xPOi6WSResL6vD64UB4cOFX9+JM6aHTpf1qI275+Lc26cmRTrvV1fKuYL lHyeGEcq5rZaVCbfmYBRY1WQ2vSz1zkacSJcbfrE9I3ZQ0rJB//FkxZlL5g9 YfaG2SNmv/y+p9k/78fci43NrtZjXmLywuTpab6Hed/TmuHgIwNqcw/mXvoj D0xe9O9p8sXkj+kH0yemf8xe/ndvh+xXF01Lcj8mP0y+mPwx/WD6w/SNa0M/ Ljy4KL4QP1b4+J5FeR/mnmrv8YQzdrG7eVyvcahNHn4mt7ZhWG3yO7tzwn1u Wm3yvuOcaPRZUpu+MH1i+sfsA7MfzL4we3SVfLI9X1+o+Df+uTfJ "]]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{416.5438336004273, 349.89338246794864`}, {525.0932646901714, 352.22777883546996`}}]}, {AbsoluteThickness[1.6218100973589302`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{204.11376415598312`, 217.99998770299146`}, {652.3178667200864, 219.16718588675212`}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{456.2285718482906, 349.89338246794864`}, {456.2285718482908, 297.3694641987179}}]}, {AbsoluteThickness[1.3335214321633237`], JoinForm[{"Miter", 4.}], LineBox[{{439.887797275641, 296.2022660149572}, {469.0677518696583, 296.2022660149572}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{441.6385945512825, 282.19588780982906`}, {470.8185491452998, 282.19588780982906`}}]}, {AbsoluteThickness[1.4962356560944332`], JoinForm[{"Miter", 4.}], LineBox[{{456.22857184829127`, 281.6122887179487}, {456.2285718482915, 217.41638861111107`}}]}, {AbsoluteThickness[1.3963683610559372`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[CompressedData[" 1:eJyN01tIk3EYx3ETIUkxEmwipUUHpajQgoWBLTCCpINFLRJKTLM8Ts0JpaZm WAqbm4dNp5PMSLqwrEQp0lBwoEVGkmFkBywjKw8QkcKsLr57YO9FXby8fG7e ///5/Z53dULmoSRPDw+P8j/P37e/06BOSS/SpKhTjfvncjU40StjatFuvcvN nRmONq34XUpmU1yKOCRYl+ddID7xXHegwyhuKM0KTWjWa+5ojdcf65UeVWcv +HXoNQfbjUtVRUoHTmaPPHToNdNLKi+kXxUbEisn+sxKa+05t8+M6jWbuysP BzWIn6hMPbob4rQs00ZHm9h30GRZ2SXeW2jOHBgQfx4xv141LL4SXrUn7404 rKLq3tNP4v7xqpC10+LTUdUV53+JvazVP4c8i11umak+FeqrNPNh8sHkGR1T 86wgoNiVP6YfTH+YfjH94/Wx6aojEeKJxrSdrSFyvrtbv6Qmz/vKfJj5Mflg 8nM33ydvd3M/+nE389An82LywOSFyROTN6YPTF+cj/+1L7g29lxZwDfJB5Mf Jl9M/sNeuSd7F6QvPN5Ss2M4WFzirLm5IUzc2157cSRSHO9jmdwULfZIsmhL 94nt3Za+0aPiqEDrlvB48ViW1VZ2Vpw/aF08lq00939QWPe2/LL4+Ku6mPcG +R+sFfVrDNeUJt/tH+uN47eUpp+XUbb5yPtK0yemb8w+6K22ZNMj+X8w+4TZ N8w+YvaX77ub/ef+mHnxXHh9p9oqJi9Mnu7mPMz9Vqyry9mWrzTzYOalP/LA 5EX/7iZfTP6YfjB9YvrH7Mv/7pt/V6460z/PNT8mP0y+mPwx/WD6w/SNl8/a Xkz0i2edDarJD3I/zDzmY42Xdn0VR9xtnLL8UJo8hnzscd+dSpOfLsnuiPYu UZi8/XrsW23LlKYvTJ+Y/jH7gdkfzH5h9rEtsKlpJkj8GzJ7G3g= "]]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{578.2007820512824, 352.2277788354701}, {660.1902459594937, 353.39497701923074`}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{659.3210558226502, 351.28105141975306`}, {659.3210558226505, 315.0979077231719}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{652.9014658119657, 274.82467103302724`}, {652.901465811966, 218.79915821251439`}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{657.70889171191, 313.64431656701936`}, {674.639075313804, 307.6930358759997}, {645.977704351385, 294.35344502222404`}, { 675.5830500653763, 284.4911301399877}, {653.3083536640795, 273.686227888236}}]}, StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["e", TraditionalForm]], FontSize->12]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {176.10100774572652`, 282.19588780982906`}, {Left, Baseline}, Alignment->{Left, Top}], FontSize->12], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["y", TraditionalForm]], FontSize->12]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {686.1666140491453, 293.8678696474359}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->12], InsetBox["", {339.5087534722222, 337.0542024465812}, {Left, Baseline}, Alignment->{Left, Top}], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["i", "1"], TraditionalForm]], FontSize->12]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {340.6759516559829, 328.8838151602564}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->12], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["e", "c"], TraditionalForm]], FontSize->12]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {478.4053373397436, 289.19907691239314`}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->12], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["i", "2"], TraditionalForm]], FontSize->12]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {613.8003266559829, 337.0542024465812}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->12], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["R", "1"], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {277.647249732906, 382.5749316132478}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["L", "1"], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {385.0294826388889, 380.24053524572645`}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["C", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {418.8782299679492, 284.5302841773505}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["L", "2"], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {554.273219284188, 381.40773342948717`}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["R", "2"], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {625.4723084935897, 291.5334732799145}, { Left, Baseline}, Alignment->{Left, Top}]}, ImagePadding->{{0., 1.}, {1., 3.1072}}, ImageSize->{469., Automatic}, PlotRange->{{165.012625, 711.261375}, {187.584524, 401.708578}}, PlotRangePadding->Automatic]], "Input", CellChangeTimes->{{3.5279726781804857`*^9, 3.527973162539296*^9}, { 3.527973205402227*^9, 3.527973241598978*^9}, {3.527973274932774*^9, 3.527973660208012*^9}, {3.527973708059843*^9, 3.527973756998856*^9}, { 3.527974001735463*^9, 3.5279741021553183`*^9}, {3.527974234218172*^9, 3.527974234998461*^9}, {3.527974527779599*^9, 3.527974675404333*^9}, { 3.527974708811222*^9, 3.5279747653381977`*^9}, {3.527974833921845*^9, 3.527974838841164*^9}, {3.5279748933901*^9, 3.527974979105378*^9}}], Cell[BoxData[ RowBox[{"ssNtwrk", " ", "=", " ", RowBox[{"StateSpaceModel", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"(", GridBox[{ { RowBox[{"-", FractionBox[ SubscriptBox["R", "1"], SubscriptBox["L", "1"]]}], "0", RowBox[{"-", FractionBox["1", SubscriptBox["L", "1"]]}]}, {"0", RowBox[{"-", FractionBox[ SubscriptBox["R", "2"], SubscriptBox["L", "2"]]}], FractionBox["1", SubscriptBox["L", "2"]]}, { FractionBox["1", "C"], RowBox[{"-", FractionBox["1", "C"]}], "0"} }], ")"}], ",", RowBox[{"(", GridBox[{ { FractionBox["1", SubscriptBox["L", "1"]]}, {"0"}, {"0"} }], ")"}], ",", RowBox[{"(", GridBox[{ {"0", SubscriptBox["R", "2"], "0"} }], ")"}]}], "}"}], ",", RowBox[{"SystemsModelLabels", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", "\"\\"", "}"}], ",", RowBox[{"{", "\"\\"", "}"}], ",", RowBox[{"{", RowBox[{ "\"\<\!\(\*SubscriptBox[\(i\), \(1\)]\)\>\"", ",", "\"\<\!\(\*SubscriptBox[\(i\), \(2\)]\)\>\"", ",", "\"\<\!\(\*SubscriptBox[\(e\), \(c\)]\)\>\""}], "}"}]}], "}"}]}]}], "]"}]}]], "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Model Representations", "Section", CellChangeTimes->{ 3.483202458955147*^9, {3.51430857274755*^9, 3.514308578875259*^9}, 3.528042620896772*^9, 3.528052628955443*^9}], Cell[CellGroupData[{ Cell["DC Motor", "Subsection", CellChangeTimes->{ 3.483202458955147*^9, {3.495918529775271*^9, 3.4959185355106363`*^9}, { 3.495920008158016*^9, 3.495920032708687*^9}, {3.528040841563778*^9, 3.528040848311912*^9}, {3.52804107049142*^9, 3.528041075328792*^9}}], Cell["\<\ The state-space model of a separately excited DC motor\ \>", "Text", CellChangeTimes->{{3.5280408904566193`*^9, 3.528040930773987*^9}, 3.528041108295559*^9, 3.528041142105851*^9}], Cell[BoxData[ GraphicsBox[{ {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{194.6226, 329.}, {204.7006, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{203.6802, 329.}, {201.49020000000002`, 332.603}, {201.6122, 332.685}, {207.4872, 330.351}, {209.5342, 329.901}, {211.58120000000002`, 329.451}, {213.62820000000002`, 329.}, { 211.58120000000002`, 328.55}, {209.5342, 328.1}, {207.4872, 327.649}, { 201.6122, 325.315}, {201.49020000000002`, 325.377}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{274.92769999999996`, 329.}, { 294.96369999999996`, 329.}}}]}, { {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{331.2451, 329.}, {345.6441, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{372.0752687617261, 327.7522150562854}, { 369.8852687617261, 331.35521505628543`}, {370.0072687617261, 331.4372150562854}, {375.8822687617261, 329.1032150562854}, { 377.92926876172606`, 328.65321505628543`}, {379.9762687617261, 328.20321505628544`}, {382.02326876172606`, 327.7522150562854}, { 379.9762687617261, 327.30221505628543`}, {377.92926876172606`, 326.85221505628544`}, {375.8822687617261, 326.4012150562854}, { 370.0072687617261, 324.0672150562854}, {369.8852687617261, 324.12921505628543`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{482.8262, 329.}, {500.03119999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{499.01169999999996`, 329.}, { 496.82269999999994`, 332.603}, {496.9437, 332.685}, {502.8187, 330.351}, {504.86569999999995`, 329.901}, {506.91369999999995`, 329.451}, {508.9607, 329.}, {506.91369999999995`, 328.55}, { 504.86569999999995`, 328.1}, {502.8187, 327.649}, {496.9437, 325.315}, { 496.82269999999994`, 325.377}}}]}}, {Thickness[0.0012626262626262627`], {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{664.623, 208.01949999999997`}, { 668.2270000000001, 210.21049999999997`}, {668.307, 210.08749999999998`}, {665.975, 204.21249999999998`}, {665.523, 202.16549999999995`}, {665.0740000000001, 200.11949999999996`}, { 664.623, 198.07249999999996`}, {664.172, 200.11949999999996`}, { 663.7230000000001, 202.16549999999995`}, {663.2710000000001, 204.21249999999998`}, {660.937, 210.08749999999998`}, {661., 210.21049999999997`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{179.6226, 303.1777}, {176.0196, 300.98670000000004`}, {175.9376, 301.10970000000003`}, {178.2716, 306.98470000000003`}, {178.7216, 309.0317}, {179.1716, 311.0787}, { 179.6226, 313.1257}, {180.0726, 311.0787}, {180.5226, 309.0317}, { 180.9736, 306.98470000000003`}, {183.3076, 301.10970000000003`}, { 183.2456, 300.98670000000004`}}}]}}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{205.1493686663968, 216.7427301193011}, {205.0678021853626, 274.06001879347446`}}]}, { InsetBox["", {227.29126922539376`, 329.68859319586625`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontWeight->Bold, FontColor->GrayLevel[0.], FontOpacity->1., Background->GrayLevel[1.]], {245.6199527883859, 329.68859319586625`}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {313.69792030807093`, 231.49921696555123`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {405.3413381230317, 391.2206023001969}, {Left, Baseline}, Alignment->{Left, Top}], { {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 218.1028244906338, 288.11626822284865`}, {218.1028244906338, 280.7204421358921}, {212.10734622976423`, 274.7249638750225}, { 204.71152014280773`, 274.7249638750225}, {197.3156940558511, 274.7249638750225}, {191.3202157949815, 280.7204421358921}, { 191.3202157949815, 288.11626822284865`}, {191.3202157949815, 295.51209430980515`}, {197.3156940558511, 301.5075725706747}, { 204.71152014280773`, 301.5075725706747}, {212.10734622976423`, 301.5075725706747}, {218.1028244906338, 295.51209430980515`}, { 218.1028244906338, 288.11626822284865`}}}, CurveClosed->{1}]}, FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, { 0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 184.3965780525378, 301.987506080743}, {184.3965780525378, 303.391506080743}, {187.25057805253783`, 303.391506080743}, { 187.25057805253783`, 306.223506080743}, {188.63257805253778`, 306.223506080743}, {188.63257805253778`, 303.391506080743}, { 191.48557805253785`, 303.391506080743}, {191.48557805253785`, 301.987506080743}, {188.63257805253778`, 301.987506080743}, { 188.63257805253778`, 299.135506080743}, {187.25057805253783`, 299.135506080743}, {187.25057805253783`, 301.987506080743}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 187.55211834693006`, 266.9001004785577}, {194.54811834693015`, 266.9001004785577}, {194.54811834693015`, 265.6721004785577}, { 187.55211834693006`, 265.6721004785577}}}]}, StyleBox[ InsetBox["", {179.4707287315495, 284.44689558569735`}, {Left, Baseline}, Alignment->{Left, Top}], FontSize->13]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{339.9289196260143, 348.3757255411389}, { 337.73991962601445`, 351.9787255411389}, {337.8609196260143, 352.0607255411389}, {343.7359196260143, 349.7267255411389}, { 345.78291962601435`, 349.2767255411389}, {347.83091962601435`, 348.8267255411389}, {349.8779196260144, 348.3757255411389}, { 347.83091962601435`, 347.9257255411389}, {345.78291962601435`, 347.4757255411389}, {343.7359196260143, 347.0247255411389}, { 337.8609196260143, 344.6907255411389}, {337.73991962601445`, 344.7527255411389}}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{205.1493686663972, 303.04630016071854`}, {205.067802185363, 352.33916842050763`}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{205.705070534898, 352.1941128022442}, {254.99621900504698`, 352.6138709010023}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{256.60696668353637`, 352.0054553718966}, {263.24156688596554`, 368.67971513990636`}, {275.40500059041904`, 339.4997605458893}, { 286.4626675944676, 368.67971513990636`}, {296.35313641475557`, 345.9841949001153}}]}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], LineBox[{{296.32242067307703`, 342.89019336538456`}, {307.994402510684, 368.56855340811967`}}]}, {AbsoluteThickness[0.8222426499470712], JoinForm[{"Miter", 4.}], LineBox[{{306.9105756257633, 368.45183358974344`}, {314.9142203144081, 348.842904102564}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{315.42169980840276`, 348.10891915908195`}, {364.71284827855175`, 348.52867725784006`}}]}, {AbsoluteThickness[1.3963683610559372`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[CompressedData[" 1:eJyN0k9I03EcxvExTap1KKEQy6IskNEhLLQUaqmEKJUahGVICUJumv9yOgsS Zx6GQWpESlKIRTgMslCxnH9SM81000JnGiQJmQdRULGS6vD+fWC/Qx1EXpf9 vp/neXanZCakajUaje3P39//90+/aq1rMBvK9MbY2Kk8Ay5dTrevBZkVd9dn WO3hYm3ylaTEU+LjWzIPrrskLurO1DXmih35WdPJpWaDW9tWEd+h9k999ktd ldng39RmXHOpHTaVXdliNxsS0xwR9q/iuzva/RNX1LaU55hSHWbD6FD7grcu X7GvtaP/WYA4IaSzNvmA+Pa3zkJdpHhT3Gt96mVxjFe31vea2NbU7XbcEvel 9TSaHop9Anptfs/FUcO9KT094mLrm7CcMXFXSJ/vru9qcx8mH0yemtm+2YFf +Ur+mH4w/WH6xfSPo6PSR+K3iTeUm37Uecn3Pd0/aQxcmc9T7sPcj8kHk5+n +X3y9jTvox9Pcw99ci8mD0xemDwxeWP6wPTF9/G/9oJbonIjfZ2yH0x+mHwx +S8t525vn5a+8NGat10FmwsUX4/rr94XKF5tGoi5ES0+bHy3Z3+S2BwwuDqW IX4xPOi6WSResL6vD64UB4cOFX9+JM6aHTpf1qI275+Lc26cmRTrvV1fKuYL lHyeGEcq5rZaVCbfmYBRY1WQ2vSz1zkacSJcbfrE9I3ZQ0rJB//FkxZlL5g9 YfaG2SNmv/y+p9k/78fci43NrtZjXmLywuTpab6Hed/TmuHgIwNqcw/mXvoj D0xe9O9p8sXkj+kH0yemf8xe/ndvh+xXF01Lcj8mP0y+mPwx/WD6w/SNa0M/ Ljy4KL4QP1b4+J5FeR/mnmrv8YQzdrG7eVyvcahNHn4mt7ZhWG3yO7tzwn1u Wm3yvuOcaPRZUpu+MH1i+sfsA7MfzL4we3SVfLI9X1+o+Df+uTfJ "]]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{416.3389530681717, 349.89338246794864`}, {486.78060515832476`, 352.22777883546996`}}]}, {AbsoluteThickness[1.6218100973589302`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{203.90003046259324`, 217.99998770299146`}, {487.9566565032199, 219.16718588675212`}}]}, StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["e", TraditionalForm]], FontSize->12]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {176.10100774572652`, 282.19588780982906`}, {Left, Baseline}, Alignment->{Left, Top}], FontSize->12], InsetBox["", {339.5087534722222, 337.0542024465812}, {Left, Baseline}, Alignment->{Left, Top}], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["i", TraditionalForm]], FontSize->12]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {340.6759516559829, 328.8838151602564}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->12], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["R", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {277.647249732906, 382.5749316132478}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["L", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {385.0294826388889, 380.24053524572645`}, { Left, Baseline}, Alignment->{Left, Top}], {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{486.44413095272273`, 313.8768113832698}, {486.36256447168853`, 350.5598761347408}}]}, {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 507.81418328998126`, 293.6010039058056}, {507.81418328998126`, 283.632716571212}, {499.21197535047287`, 275.55185456743123`}, { 488.60057270397004`, 275.55185456743123`}, {477.9891700574671, 275.55185456743123`}, {469.3869621179584, 283.632716571212}, { 469.3869621179584, 293.6010039058056}, {469.3869621179584, 303.56929124039914`}, {477.9891700574671, 311.6501532441799}, { 488.60057270397004`, 311.6501532441799}, {499.21197535047287`, 311.6501532441799}, {507.81418328998126`, 303.56929124039914`}, { 507.81418328998126`, 293.6010039058056}}}, CurveClosed->{1}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{487.61132913648436`, 218.66366519979692`}, {487.5297626554504, 275.8478533828392}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["M", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->12, Background->GrayLevel[1.]], {481.90693189102586`, 289.19907691239314`}, { Left, Baseline}, Alignment->{Left, Top}], {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{509.59119002359887`, 293.91551333684913`}, {556.9631656411603, 293.8195318356938}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], CircleBox[{557.7748138354707, 293.867869647436}, \ {9.337585470085628, 21.009567307692237`}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{558.5609230802377, 316.4495702174197}, {596.6807017785941, 317.97401929883387`}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{557.3937248964776, 273.26323741827446`}, {595.513503594834, 274.7876864996886}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], CircleBox[{595.1251557158123, 296.7858651068376}, \ {9.337585470085628, 21.009567307692237`}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{605.22677743904, 295.6663106124902}, {638.7337845831689, 295.57032911133484`}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{640.5142912091354, 281.0077193120255}, {640.4327247281013, 308.5200178756288}}]}, {AbsoluteThickness[21.134890398366476`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.91]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{673.3274340277777, 362.7325624893162}, {673.3274340277779, 225.0031768055556}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{629.9362206963781, 328.3002160683761}, {663.0275309061893, 328.30021606837585`}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{627.601824328857, 267.60591051282046`}, {659.5259363549071, 267.60591051282046`}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{629.1262770256716, 309.10226238851055`}, {629.6213275645265, 328.4205442444848}}]}, {AbsoluteThickness[1.8836490894898006`], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{627.9590788419114, 267.66672686500635`}, {628.4541293807663, 286.9850087209806}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["J", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {579.9515793269235, 253.01593321581203`}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["f", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {617.3019212072657, 330.0510133440171}, { Left, Baseline}, Alignment->{Left, Top}]}, ImagePadding->{{0., 1.}, {1., 3.1072}}, ImageSize->{469., Automatic}, PlotRange->{{165.012625, 711.261375}, {187.584524, 401.708578}}, PlotRangePadding->Automatic]], "Input", CellChangeTimes->{{3.52798226528192*^9, 3.5279823228278112`*^9}, { 3.527982362714779*^9, 3.527982557535479*^9}, {3.527982605708246*^9, 3.527982679876608*^9}, {3.527982715368225*^9, 3.5279827592113943`*^9}, { 3.527982799154055*^9, 3.527982847859202*^9}, {3.527982923422763*^9, 3.527983069280219*^9}, {3.5279834200985727`*^9, 3.5279836664688063`*^9}, { 3.5280504356669064`*^9, 3.528050436342203*^9}}], Cell[BoxData[ RowBox[{"ssMotor", " ", "=", " ", RowBox[{"StateSpaceModel", "[", RowBox[{"{", RowBox[{ RowBox[{"(", GridBox[{ {"0", "1", "0"}, {"0", RowBox[{"-", FractionBox["f", "J"]}], FractionBox[ SubscriptBox["k", "1"], "J"]}, {"0", RowBox[{"-", FractionBox[ SubscriptBox["k", "2"], "L"]}], RowBox[{"-", FractionBox["R", "L"]}]} }], ")"}], ",", RowBox[{"(", GridBox[{ {"0"}, {"0"}, { FractionBox["1", "L"]} }], ")"}], ",", RowBox[{"(", GridBox[{ {"1", "0", "0"} }], ")"}]}], "}"}], "]"}]}]], "Input"], Cell["\<\ The corresponding transfer-function description of the same system can be \ obtained by converting the state-space model object to a transfer-function \ object\ \>", "Text", CellChangeTimes->{{3.528041420205476*^9, 3.528041424551833*^9}, { 3.528042629655366*^9, 3.528042643897427*^9}, {3.528059863145752*^9, 3.528059867473868*^9}}], Cell[BoxData[ RowBox[{"tfMotor", " ", "=", " ", RowBox[{ RowBox[{"TransferFunctionModel", "[", "ssMotor", "]"}], "//", "Simplify"}]}]], "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Operations on Models", "Section", CellChangeTimes->{ 3.483202458955147*^9, {3.514308863196991*^9, 3.5143088633311243`*^9}, 3.528043368065051*^9, {3.5280526340898447`*^9, 3.528052645284409*^9}, { 3.528134566352951*^9, 3.52813457826093*^9}}], Cell[CellGroupData[{ Cell["B-747 Longitudinal Dynamics", "Subsection", CellChangeTimes->{ 3.483202458955147*^9, {3.495918529775271*^9, 3.4959185355106363`*^9}, { 3.495920008158016*^9, 3.495920032708687*^9}, {3.528040841563778*^9, 3.528040848311912*^9}, {3.52804107049142*^9, 3.528041075328792*^9}, { 3.528042731806477*^9, 3.528042754715267*^9}, {3.528045386077153*^9, 3.528045412733033*^9}}], Cell["\<\ Linearized longitudinal equations of motion flying near sea level. Elevator \ and thrust are control inputs, wind velocity is the disturbance input, pitch \ rate is the measured output.\ \>", "Text", CellChangeTimes->{{3.5280408904566193`*^9, 3.528040930773987*^9}, 3.528041108295559*^9, 3.528041142105851*^9, {3.5280428062614803`*^9, 3.528042839715294*^9}, {3.5280455936644773`*^9, 3.52804564868721*^9}, 3.5280456970255404`*^9, {3.528046448686235*^9, 3.528046506330749*^9}, { 3.528046706991976*^9, 3.5280467203264112`*^9}}], Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{ StyleBox[ RowBox[{ OverscriptBox[ StyleBox["x", "TB"], "."], "(", "t", ")"}], "DisplayFormula"], StyleBox["=", "DisplayFormula"], RowBox[{ RowBox[{ StyleBox["a", "TB"], StyleBox[" ", "TB"], RowBox[{ StyleBox["x", "TB"], StyleBox["(", "DisplayFormula"], StyleBox["t", "DisplayFormula"], StyleBox[")", "DisplayFormula"]}]}], StyleBox["+", "DisplayFormula"], RowBox[{ SubscriptBox[ StyleBox["b", "TB"], "u"], RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["\[Delta]", "e"], "(", "t", ")"}], "}"}], ",", RowBox[{"{", RowBox[{ SubscriptBox["\[Delta]", "T"], "(", "t", ")"}], "}"}]}], "}"}]}], " ", "+", RowBox[{ SubscriptBox[ StyleBox["b", "TB"], "w"], RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["u", "w"], "(", "t", ")"}], "}"}], ",", RowBox[{"{", RowBox[{ SubscriptBox["w", "w"], "(", "t", ")"}], "}"}]}], "}"}]}]}]}], " "}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{ StyleBox["y", "TB"], StyleBox["(", "DisplayFormula"], StyleBox["t", "DisplayFormula"], StyleBox[")", "DisplayFormula"]}], StyleBox["=", "DisplayFormula"], RowBox[{ StyleBox["c", "TB"], StyleBox[" ", "TB"], RowBox[{ StyleBox["x", "TB"], StyleBox["(", "DisplayFormula"], StyleBox["t", "DisplayFormula"], StyleBox[")", "DisplayFormula"]}]}]}], TraditionalForm]}], "DisplayFormula", CellChangeTimes->{{3.495921379934836*^9, 3.495921383427203*^9}, { 3.495921729714572*^9, 3.49592172999667*^9}, {3.5280457462048073`*^9, 3.528045781742552*^9}, {3.528045831440888*^9, 3.5280460230386257`*^9}, { 3.528046171985634*^9, 3.5280462341544237`*^9}, 3.52804642894932*^9}, CounterAssignments->{{"Section", 1}, {"NumberedEquation", 2}}, CellTags->"3.2"], Cell[BoxData[{ RowBox[{ RowBox[{"a", "=", RowBox[{"(", GridBox[{ { RowBox[{"-", "0.021"}], "0.122", "0", RowBox[{"-", "0.322"}], "1"}, { RowBox[{"-", "0.209"}], RowBox[{"-", "0.530"}], "2.21", "0", RowBox[{"-", "0.044"}]}, {"0.017", RowBox[{"-", "0.164"}], RowBox[{"-", "0.412"}], "0", "0.544"}, {"0", "0", "1", "0", "0"}, {"0", "0", "0", "0", RowBox[{"-", "0.25"}]} }], ")"}]}], ";", RowBox[{ SubscriptBox["b", "u"], "=", RowBox[{"(", GridBox[{ {"0.01", "0"}, { RowBox[{"-", "0.064"}], "0"}, { RowBox[{"-", "0.378"}], "0"}, {"0", "0"}, {"0", "0.25"} }], ")"}]}], ";", RowBox[{ SubscriptBox["b", "w"], "=", RowBox[{"(", GridBox[{ {"0.021", RowBox[{"-", "0.122"}]}, {"0.209", "0.530"}, { RowBox[{"-", "0.017"}], "0.164"}, {"0", "0"}, {"0", "0"} }], ")"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"c", "=", RowBox[{"(", GridBox[{ {"0", "0", "1", "0", "0"} }], ")"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"ssb", "=", " ", RowBox[{"StateSpaceModel", "[", RowBox[{"{", RowBox[{"a", ",", RowBox[{"ArrayFlatten", "[", RowBox[{"{", RowBox[{"{", RowBox[{ SubscriptBox["b", "u"], ",", SubscriptBox["b", "w"]}], "}"}], "}"}], "]"}], ",", "c"}], "}"}], "]"}]}]}], "Input", CellChangeTimes->{{3.528046303701735*^9, 3.528046417160079*^9}, { 3.5280465115111313`*^9, 3.5280466835831423`*^9}, {3.528046725842333*^9, 3.528046771895629*^9}}], Cell[CellGroupData[{ Cell["Discretization", "Subsubsection", CellChangeTimes->{{3.5280468085589457`*^9, 3.528046869897873*^9}, { 3.528050334338785*^9, 3.528050349453521*^9}, {3.5281133525034027`*^9, 3.528113354869666*^9}}], Cell["\<\ Discretize the model with a sampling period of \[Tau] = 0.1 using method \ \"ZeroOrderHold\"\ \>", "Text", CellChangeTimes->{{3.5281133402790623`*^9, 3.528113346459859*^9}}], Cell[BoxData[ RowBox[{"ToDiscreteTimeModel", "[", RowBox[{"ssb", ",", "0.1", ",", RowBox[{"Method", "\[Rule]", " ", "\"\\""}]}], "]"}]], "Input", CellChangeTimes->{{3.5280503151171827`*^9, 3.528050360617659*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Transfer-function representation\ \>", "Subsubsection", CellChangeTimes->{{3.5280503709261017`*^9, 3.528050393454832*^9}, { 3.5280526762790537`*^9, 3.528052677562377*^9}, {3.52811339779311*^9, 3.528113414161963*^9}}], Cell["Obtain the transer-function matrix of the discrete model", "Text", CellChangeTimes->{{3.528113377896427*^9, 3.528113390027738*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"TransferFunctionModel", "[", "%", "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.5280504042681417`*^9, 3.528050414480063*^9}, { 3.528050788768917*^9, 3.52805079869354*^9}, {3.528050829020368*^9, 3.528050834289752*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Model Extraction", "Subsubsection", CellChangeTimes->{{3.528058626631716*^9, 3.5280586778245697`*^9}, { 3.5281139371910563`*^9, 3.528113942979045*^9}}], Cell["Extract the subsystem associated with the first two inputs", "Text", CellChangeTimes->{{3.528113426997793*^9, 3.528113433976997*^9}}], Cell[BoxData[ RowBox[{"SystemsModelExtract", "[", RowBox[{"%", ",", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.528058684028675*^9, 3.528058695864015*^9}, { 3.5280587508199472`*^9, 3.528058782985837*^9}}] }, Closed]] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Model Connections", "Section", CellChangeTimes->{ 3.483202458955147*^9, {3.514308863196991*^9, 3.5143088633311243`*^9}, 3.5280508660463963`*^9, 3.528052797473154*^9, {3.52811402410958*^9, 3.528114026137493*^9}}], Cell[CellGroupData[{ Cell["Thyroid Hormone Control System", "Subsection", CellChangeTimes->{ 3.483202458955147*^9, {3.495918529775271*^9, 3.4959185355106363`*^9}, { 3.495920008158016*^9, 3.495920032708687*^9}, {3.528040841563778*^9, 3.528040848311912*^9}, {3.52804107049142*^9, 3.528041075328792*^9}, { 3.528042731806477*^9, 3.528042754715267*^9}}], Cell["\<\ Model built up from simpler TransferFunctionModel subsystems\ \>", "Text", CellChangeTimes->{{3.5280408904566193`*^9, 3.528040930773987*^9}, 3.528041108295559*^9, 3.528041142105851*^9, {3.5280428062614803`*^9, 3.528042839715294*^9}}], Cell[BoxData[ GraphicsBox[{ {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{194.6226, 329.}, {204.7006, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{203.6802, 329.}, {201.49020000000002`, 332.603}, {201.6122, 332.685}, {207.4872, 330.351}, {209.5342, 329.901}, {211.58120000000002`, 329.451}, {213.62820000000002`, 329.}, { 211.58120000000002`, 328.55}, {209.5342, 328.1}, {207.4872, 327.649}, { 201.6122, 325.315}, {201.49020000000002`, 325.377}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{274.92769999999996`, 329.}, { 294.96369999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{293.9438, 329.}, {291.7538, 332.603}, { 291.8758, 332.685}, {297.7508, 330.351}, {299.7978, 329.901}, {301.8448, 329.451}, {303.8918, 329.}, {301.8448, 328.55}, {299.7978, 328.1}, { 297.7508, 327.649}, {291.8758, 325.315}, {291.7538, 325.377}}}]}, { {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{317.63669999999996`, 353.749}, { 321.23969999999997`, 355.939}, {321.32169999999996`, 355.817}, { 318.98769999999996`, 349.942}, {318.5377, 347.89500000000004`}, { 318.0877, 345.848}, {317.63669999999996`, 343.80100000000004`}, { 317.1867, 345.848}, {316.7367, 347.89500000000004`}, { 316.28569999999996`, 349.942}, {313.95169999999996`, 355.817}, { 314.0137, 355.939}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{331.2451, 329.}, {345.6441, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{372.0752687617261, 327.7522150562854}, { 369.8852687617261, 331.35521505628543`}, {370.0072687617261, 331.4372150562854}, {375.8822687617261, 329.1032150562854}, { 377.92926876172606`, 328.65321505628543`}, {379.9762687617261, 328.20321505628544`}, {382.02326876172606`, 327.7522150562854}, { 379.9762687617261, 327.30221505628543`}, {377.92926876172606`, 326.85221505628544`}, {375.8822687617261, 326.4012150562854}, { 370.0072687617261, 324.0672150562854}, {369.8852687617261, 324.12921505628543`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{482.8262, 329.}, {500.03119999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{499.01169999999996`, 329.}, { 496.82269999999994`, 332.603}, {496.9437, 332.685}, {502.8187, 330.351}, {504.86569999999995`, 329.901}, {506.91369999999995`, 329.451}, {508.9607, 329.}, {506.91369999999995`, 328.55}, { 504.86569999999995`, 328.1}, {502.8187, 327.649}, {496.9437, 325.315}, { 496.82269999999994`, 325.377}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{445.7907144902713, 330.20726561808283`}, { 443.6017144902713, 333.81026561808284`}, {443.7227144902713, 333.89226561808283`}, {449.5977144902713, 331.5582656180828}, { 451.6447144902713, 331.10826561808284`}, {453.6927144902713, 330.65826561808285`}, {455.73971449027135`, 330.20726561808283`}, { 453.6927144902713, 329.75726561808284`}, {451.6447144902713, 329.30726561808285`}, {449.5977144902713, 328.85626561808283`}, { 443.7227144902713, 326.5222656180828}, {443.6017144902713, 326.58426561808284`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{648.1934000000001, 329.}, { 700.3364000000001, 329.}}}]}}, { {Thickness[0.0012626262626262627`], {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{664.623, 329.}, {664.623, 207.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{664.623, 208.01949999999997`}, { 668.2270000000001, 210.21049999999997`}, {668.307, 210.08749999999998`}, {665.975, 204.21249999999998`}, {665.523, 202.16549999999995`}, {665.0740000000001, 200.11949999999996`}, { 664.623, 198.07249999999996`}, {664.172, 200.11949999999996`}, { 663.7230000000001, 202.16549999999995`}, {663.2710000000001, 204.21249999999998`}, {660.937, 210.08749999999998`}, {661., 210.21049999999997`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{179.6226, 303.1777}, {176.0196, 300.98670000000004`}, {175.9376, 301.10970000000003`}, {178.2716, 306.98470000000003`}, {178.7216, 309.0317}, {179.1716, 311.0787}, { 179.6226, 313.1257}, {180.0726, 311.0787}, {180.5226, 309.0317}, { 180.9736, 306.98470000000003`}, {183.3076, 301.10970000000003`}, { 183.2456, 300.98670000000004`}}}]}}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{0, 2, 0}, {0, 1, 0}}}, {{{673.6843213703485, 215.77772302958124`}, {311.9001163248639, 215.77772302958124`}, { 311.9001163248639, 315.8441961238991}}}, CurveClosed->{0}]}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{415.6071882034252, 329.6085244466593}, {443.51406827482856`, 330.71663150221}}]}, {AbsoluteThickness[1], StrokeForm[{GrayLevel[0.], Opacity[0.64]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{316.3147046983335, 330.78655481789576`}, {347.4053462125612, 330.786554817896}}]}}, {AbsoluteThickness[1.5667510701081488`], StrokeForm[{GrayLevel[0.], Opacity[0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{672.814877114072, 330.892517103107}, {674.0626620577868, 215.83983535734086`}}]}, { InsetBox["", {227.29126922539376`, 329.68859319586625`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontWeight->Bold, FontColor->GrayLevel[0.], FontOpacity->1., Background->GrayLevel[1.]], {245.6199527883859, 329.68859319586625`}, { Left, Baseline}, Alignment->{Left, Top}], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["y", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {655.7236151982551, 341.7163175500202}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], InsetBox["", {313.69792030807093`, 231.49921696555123`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {405.3413381230317, 391.2206023001969}, {Left, Baseline}, Alignment->{Left, Top}], {Thickness[0.0012626262626262627`], EdgeForm[{AbsoluteThickness[ 1.023292992280754], JoinForm[{"Miter", 4.}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{517.4469609725278, 304.3966027199046}, {455.4412507993677, 304.3966027199046}, { 455.4412507993677, 357.1034094302431}, {517.4469609725278, 357.1034094302431}}}]}, {{ {FaceForm[{RGBColor[0.8630000000000001, 0.894, 0.89], Opacity[1.]}], FilledCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 326.6901564341327, 329.49126303719174`}, {326.6901564341327, 321.7592630371917}, {320.42215643413266`, 315.49126303719174`}, { 312.6901564341327, 315.49126303719174`}, {304.9581564341326, 315.49126303719174`}, {298.6901564341327, 321.7592630371917}, { 298.6901564341327, 329.49126303719174`}, {298.6901564341327, 337.22326303719177`}, {304.9581564341326, 343.49126303719174`}, { 312.6901564341327, 343.49126303719174`}, {320.42215643413266`, 343.49126303719174`}, {326.6901564341327, 337.22326303719177`}, { 326.6901564341327, 329.49126303719174`}}}]}, {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 326.6522555803771, 331.3026010219939}, {326.6522555803771, 323.90677493503733`}, {320.6567773195075, 317.9112966741678}, { 313.260951232551, 317.9112966741678}, {305.8651251455944, 317.9112966741678}, {299.8696468847249, 323.90677493503733`}, { 299.8696468847249, 331.3026010219939}, {299.8696468847249, 338.6984271089504}, {305.8651251455944, 344.69390536981996`}, { 313.260951232551, 344.69390536981996`}, {320.6567773195075, 344.69390536981996`}, {326.6522555803771, 338.6984271089504}, { 326.6522555803771, 331.3026010219939}}}, CurveClosed->{1}]}}, FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, { 0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 294.1132073260418, 345.17383887988825`}, {294.1132073260418, 346.57783887988825`}, {296.96720732604183`, 346.57783887988825`}, { 296.96720732604183`, 349.40983887988824`}, {298.3492073260418, 349.40983887988824`}, {298.3492073260418, 346.57783887988825`}, { 301.20220732604184`, 346.57783887988825`}, {301.20220732604184`, 345.17383887988825`}, {298.3492073260418, 345.17383887988825`}, { 298.3492073260418, 342.3218388798883}, {296.96720732604183`, 342.3218388798883}, {296.96720732604183`, 345.17383887988825`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 293.76715306915196`, 319.42401874778835`}, {300.76315306915205`, 319.42401874778835`}, {300.76315306915205`, 318.19601874778834`}, { 293.76715306915196`, 318.19601874778834`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{312.072789392625, 307.3729291397307}, { 308.3463368040363, 305.39919128859634`}, {308.27176639394435`, 305.5268324024223}, {310.94966013297375`, 311.2532681282877}, { 311.52011924062214`, 313.2700193292297}, {312.09051911563, 315.2857722859659}, {312.66197646748407`, 317.3024642542674}, { 312.9899371450054, 315.232403676893}, {313.31795705516726`, 313.16334134372437`}, {313.6469159768944, 311.0932215337094}, { 315.6288261902785, 305.09028784208135`}, {315.5596494347432, 304.9711762284846}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{289.95152908215715`, 330.7187883180905}, { 287.7605290821571, 334.3217883180905}, {287.88352908215717`, 334.4037883180905}, {293.75852908215717`, 332.0697883180905}, { 295.8055290821572, 331.6197883180905}, {297.8515290821571, 331.1697883180905}, {299.89852908215715`, 330.7187883180905}, { 297.8515290821571, 330.2687883180905}, {295.8055290821572, 329.8187883180905}, {293.75852908215717`, 329.3677883180905}, { 287.88352908215717`, 327.0337883180905}, {287.7605290821571, 327.0957883180905}}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{235.63894698516998`, 330.7527856724211}, {289.8083336524264, 330.75278567242134`}}]}, StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["r", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {216.9810685651708, 330.8331874504331}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{350.6444937510316, 330.8618614596187}, { 348.4554937510316, 334.4648614596187}, {348.5764937510316, 334.5468614596187}, {354.4514937510316, 332.2128614596187}, { 356.4984937510316, 331.7628614596187}, {358.5464937510316, 331.3128614596187}, {360.59349375103164`, 330.8618614596187}, { 358.5464937510316, 330.4118614596187}, {356.4984937510316, 329.9618614596187}, {354.4514937510316, 329.5108614596187}, { 348.5764937510316, 327.1768614596187}, {348.4554937510316, 327.2388614596187}}}]}}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], StrokeForm[ Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{650.2741496990091, 330.851936453746}, {695.663175246591, 332.0997213974613}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{695.2424071836341, 332.1556051627651}, { 693.0534071836342, 335.7586051627651}, {693.1744071836341, 335.8406051627651}, {699.0494071836341, 333.5066051627651}, { 701.0964071836341, 333.0566051627651}, {703.1444071836341, 332.6066051627651}, {705.1914071836342, 332.1556051627651}, { 703.1444071836341, 331.7056051627651}, {701.0964071836341, 331.2556051627651}, {699.0494071836341, 330.8046051627651}, { 693.1744071836341, 328.4706051627651}, {693.0534071836342, 328.5326051627651}}}]}, {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{415.3508615106698, 304.85206023642985`}, {360.3034314899709, 304.85206023642985`}, { 360.3034314899709, 353.78960234021645`}, {415.3508615106698, 353.78960234021645`}}}]}, {FaceForm[{RGBColor[0.8630000000000001, 0.894, 0.89], Opacity[1.]}], FilledCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 572.8089752543574, 331.94631359898835`}, {572.8089752543574, 324.2143135989883}, {566.5409752543574, 317.94631359898835`}, { 558.8089752543574, 317.94631359898835`}, {551.0769752543573, 317.94631359898835`}, {544.8089752543574, 324.2143135989883}, { 544.8089752543574, 331.94631359898835`}, {544.8089752543574, 339.6783135989884}, {551.0769752543573, 345.94631359898835`}, { 558.8089752543574, 345.94631359898835`}, {566.5409752543574, 345.94631359898835`}, {572.8089752543574, 339.6783135989884}, { 572.8089752543574, 331.94631359898835`}}}]}, {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 572.1573117601523, 331.9163636624423}, {572.1573117601523, 324.5205375754857}, {566.1618334992827, 318.52505931461616`}, { 558.7660074123262, 318.52505931461616`}, {551.3701813253696, 318.52505931461616`}, {545.3747030645001, 324.5205375754857}, { 545.3747030645001, 331.9163636624423}, {545.3747030645001, 339.3121897493988}, {551.3701813253696, 345.30766801026834`}, { 558.7660074123262, 345.30766801026834`}, {566.1618334992827, 345.30766801026834`}, {572.1573117601523, 339.3121897493988}, { 572.1573117601523, 331.9163636624423}}}, CurveClosed->{1}]}, {AbsoluteThickness[1], StrokeForm[{GrayLevel[0.], Opacity[0.64]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{517.6288507657492, 332.014080098794}, {536.4442394709882, 332.014080098794}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{537.2283364476609, 332.08938674051683`}, { 535.0393364476608, 335.69238674051684`}, {535.1603364476609, 335.77438674051683`}, {541.0353364476609, 333.44038674051683`}, { 543.082336447661, 332.99038674051684`}, {545.130336447661, 332.54038674051685`}, {547.177336447661, 332.08938674051683`}, { 545.130336447661, 331.63938674051684`}, {543.082336447661, 331.18938674051685`}, {541.0353364476609, 330.73838674051683`}, { 535.1603364476609, 328.4043867405168}, {535.0393364476608, 328.46638674051684`}}}]}, {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{647.9669022410068, 309.14839871957514`}, {592.9194722203079, 309.14839871957514`}, { 592.9194722203079, 358.08594082336174`}, {647.9669022410068, 358.08594082336174`}}}]}, {AbsoluteThickness[1], StrokeForm[{GrayLevel[0.], Opacity[0.64]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{571.6399631252998, 331.40031745834426`}, {590.4553518305388, 331.40031745834426`}}]}, {AbsoluteThickness[1.5667510701081488`], StrokeForm[{GrayLevel[0.], Opacity[0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{558.1163012706986, 377.39168314459164`}, {557.9961776133671, 346.8222735487837}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{558.6977223274746, 354.30982726715365`}, { 562.3293525264748, 356.4509933274371}, {562.4097453713285, 356.3289196181613}, {559.9982641552983, 350.4852957985838}, { 559.5212359653303, 348.444425125078}, {559.0441945523373, 346.4025545390002}, {558.5661664497975, 344.3616970885196}, { 558.1432733249828, 346.414468484702}, {557.7203934231936, 348.4682397934564}, {557.296500385807, 350.52102441266396`}, { 555.0403897158322, 356.42637331385737`}, {555.1039842813439, 356.54654290750494`}}}]}, FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 539.618263505817, 316.32699477876423`}, {539.618263505817, 317.7309947787642}, {542.4722635058171, 317.7309947787642}, { 542.4722635058171, 320.5629947787642}, {543.854263505817, 320.5629947787642}, {543.854263505817, 317.7309947787642}, { 546.7072635058171, 317.7309947787642}, {546.7072635058171, 316.32699477876423`}, {543.854263505817, 316.32699477876423`}, { 543.854263505817, 313.47499477876426`}, {542.4722635058171, 313.47499477876426`}, {542.4722635058171, 316.32699477876423`}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 539.618263505817, 350.6977026439328}, {539.618263505817, 352.1017026439328}, {542.4722635058171, 352.1017026439328}, { 542.4722635058171, 354.9337026439328}, {543.854263505817, 354.9337026439328}, {543.854263505817, 352.1017026439328}, { 546.7072635058171, 352.1017026439328}, {546.7072635058171, 350.6977026439328}, {543.854263505817, 350.6977026439328}, { 543.854263505817, 347.84570264393284`}, {542.4722635058171, 347.84570264393284`}, {542.4722635058171, 350.6977026439328}}}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ FractionBox[ StyleBox["2", FontSize->12], StyleBox[ RowBox[{"s", "+", "2"}], FontSize->12]], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {376.76073595505625`, 326.34736393483115`}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->12], StyleBox[ RowBox[{"s", "+", "a"}], FontSize->12]], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {476.1902837078652, 327.57488921572997`}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["10", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {611.2180646067416, 330.0299397775277}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["reference", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {211.04482303370793`, 301.7968583168538}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["d", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {536.3390224719101, 381.58600157528076`}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {581.7574578651686, 387.72362797977513`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["medication", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {554.7519016853932, 384.0410521370785}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["Hyp/Pit", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {360.8029073033709, 283.3839791033703}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["Thyroid", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {462.68750561797754`, 283.3839791033705}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["Metabolism", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {585.4400337078652, 287.0665549460671}, { Left, Baseline}, Alignment->{Left, Top}]}, ImagePadding->{{0., 1.}, {1., 3.1072}}, ImageSize->{446., Automatic}, PlotRange->{{165.012625, 711.261375}, {187.584524, 401.708578}}, PlotRangePadding->Automatic]], "Input", CellChangeTimes->{{3.527976189604295*^9, 3.527976373970902*^9}, { 3.527976435400079*^9, 3.527976741796983*^9}, {3.527976779976487*^9, 3.5279768657866993`*^9}, {3.52797690458046*^9, 3.527977116885365*^9}, { 3.527977327982197*^9, 3.5279773744726343`*^9}, {3.527977415160507*^9, 3.5279775741296463`*^9}, {3.527979987494904*^9, 3.527980048264749*^9}}], Cell[BoxData[ RowBox[{"Quit", "[", "]"}]], "Input", CellChangeTimes->{{3.528056280380464*^9, 3.5280562820214043`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"tfHypPit", "=", " ", RowBox[{"TransferFunctionModel", "[", RowBox[{ FractionBox["2", RowBox[{"s", "+", "2"}]], ",", "s"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"tfThyroid", " ", "=", " ", RowBox[{"TransferFunctionModel", "[", RowBox[{ FractionBox["1", RowBox[{"s", "+", "a"}]], ",", "s"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"tfMetabolism", " ", "=", " ", RowBox[{"TransferFunctionModel", "[", RowBox[{"10", ",", "s"}], "]"}]}], ";"}]}], "Input", CellChangeTimes->{{3.528042900527746*^9, 3.528042934716194*^9}}], Cell[CellGroupData[{ Cell["Interconnection", "Subsubsection", CellChangeTimes->{{3.528042992371563*^9, 3.52804302058673*^9}, { 3.528113249877809*^9, 3.528113257223853*^9}, {3.528114007354463*^9, 3.5281140153272533`*^9}}], Cell["\<\ Equivalent system from the reference and the medication inputs to the output\ \>", "Text", CellChangeTimes->{{3.528113992227886*^9, 3.5281139986153097`*^9}}], Cell[BoxData[ RowBox[{"tfry", "=", " ", RowBox[{"SystemsModelFeedbackConnect", "[", RowBox[{"SystemsModelSeriesConnect", "[", RowBox[{"tfHypPit", ",", RowBox[{"SystemsModelSeriesConnect", "[", RowBox[{"tfThyroid", ",", "tfMetabolism"}], "]"}]}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.528043049152657*^9, 3.528043059808455*^9}}], Cell[BoxData[ RowBox[{"tfdy", "=", RowBox[{"SystemsModelFeedbackConnect", "[", RowBox[{"tfMetabolism", ",", RowBox[{"SystemsModelSeriesConnect", "[", RowBox[{"tfHypPit", ",", "tfThyroid"}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.528043102706972*^9, 3.528043111533369*^9}}] }, Closed]] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Model Connections", "Section", CellChangeTimes->{ 3.528051191553873*^9, {3.528052844534576*^9, 3.5280528554406567`*^9}, 3.52811404173918*^9}], Cell[CellGroupData[{ Cell["A More Complicated Model", "Subsection", CellChangeTimes->{ 3.483202458955147*^9, {3.495918529775271*^9, 3.4959185355106363`*^9}, { 3.495920008158016*^9, 3.495920032708687*^9}, {3.528040841563778*^9, 3.528040848311912*^9}, {3.52804107049142*^9, 3.528041075328792*^9}, { 3.528042731806477*^9, 3.528042754715267*^9}, {3.5280513795034113`*^9, 3.528051387954815*^9}, {3.5282065903003407`*^9, 3.5282065970384407`*^9}}], Cell["\<\ Simplify an interconnected system of blocks constituting a motor control \ system:\ \>", "Text", CellChangeTimes->{{3.5280408904566193`*^9, 3.528040930773987*^9}, 3.528041108295559*^9, 3.528041142105851*^9, {3.5280428062614803`*^9, 3.528042839715294*^9}, {3.528051397342969*^9, 3.528051432866795*^9}, { 3.5280514732348137`*^9, 3.528051507780052*^9}, {3.528051826207505*^9, 3.5280518400432167`*^9}, {3.5280522398305387`*^9, 3.528052268052758*^9}}], Cell[BoxData[ GraphicsBox[{ {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{194.6226, 329.}, {204.7006, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{203.6802, 329.}, {201.49020000000002`, 332.603}, {201.6122, 332.685}, {207.4872, 330.351}, {209.5342, 329.901}, {211.58120000000002`, 329.451}, {213.62820000000002`, 329.}, { 211.58120000000002`, 328.55}, {209.5342, 328.1}, {207.4872, 327.649}, { 201.6122, 325.315}, {201.49020000000002`, 325.377}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{274.92769999999996`, 329.}, { 294.96369999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{293.9438, 329.}, {291.7538, 332.603}, { 291.8758, 332.685}, {297.7508, 330.351}, {299.7978, 329.901}, {301.8448, 329.451}, {303.8918, 329.}, {301.8448, 328.55}, {299.7978, 328.1}, { 297.7508, 327.649}, {291.8758, 325.315}, {291.7538, 325.377}}}]}, { {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{317.63669999999996`, 353.749}, { 321.23969999999997`, 355.939}, {321.32169999999996`, 355.817}, { 318.98769999999996`, 349.942}, {318.5377, 347.89500000000004`}, { 318.0877, 345.848}, {317.63669999999996`, 343.80100000000004`}, { 317.1867, 345.848}, {316.7367, 347.89500000000004`}, { 316.28569999999996`, 349.942}, {313.95169999999996`, 355.817}, { 314.0137, 355.939}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{482.8262, 329.}, {500.03119999999996`, 329.}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{648.1934000000001, 329.}, { 700.3364000000001, 329.}}}]}}, { {Thickness[0.0012626262626262627`], {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{664.623, 329.}, {664.623, 207.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{664.623, 208.01949999999997`}, { 668.2270000000001, 210.21049999999997`}, {668.307, 210.08749999999998`}, {665.975, 204.21249999999998`}, {665.523, 202.16549999999995`}, {665.0740000000001, 200.11949999999996`}, { 664.623, 198.07249999999996`}, {664.172, 200.11949999999996`}, { 663.7230000000001, 202.16549999999995`}, {663.2710000000001, 204.21249999999998`}, {660.937, 210.08749999999998`}, {661., 210.21049999999997`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{179.6226, 303.1777}, {176.0196, 300.98670000000004`}, {175.9376, 301.10970000000003`}, {178.2716, 306.98470000000003`}, {178.7216, 309.0317}, {179.1716, 311.0787}, { 179.6226, 313.1257}, {180.0726, 311.0787}, {180.5226, 309.0317}, { 180.9736, 306.98470000000003`}, {183.3076, 301.10970000000003`}, { 183.2456, 300.98670000000004`}}}]}}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{0, 2, 0}, {0, 1, 0}}}, {{{444.79673727726276`, 219.25941437532012`}, {256.4506762459639, 219.25941437532012`}, { 256.4506762459639, 310.2289353701543}}}, CurveClosed->{0}]}}, { InsetBox["", {227.29126922539376`, 329.68859319586625`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontWeight->Bold, FontColor->GrayLevel[0.], FontOpacity->1., Background->GrayLevel[1.]], {245.6199527883859, 329.68859319586625`}, { Left, Baseline}, Alignment->{Left, Top}], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["y", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {685.4358903028576, 345.52558358907174`}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], InsetBox["", {313.69792030807093`, 231.49921696555123`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {405.3413381230317, 391.2206023001969}, {Left, Baseline}, Alignment->{Left, Top}], {AbsoluteThickness[1.023292992280754], StrokeForm[{GrayLevel[0.], Opacity[ 0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{676.457045472355, 219.45207968863912`}, {483.0621998657426, 218.6902264808288}}]}, {{ {FaceForm[{RGBColor[0.8630000000000001, 0.894, 0.89], Opacity[1.]}], FilledCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 266.45035257008664`, 327.84377399475983`}, {266.45035257008664`, 322.2015037244895}, {262.21521743495146`, 317.62755777854363`}, { 256.9908931106272, 317.62755777854363`}, {251.7665687863028, 317.62755777854363`}, {247.53143365116767`, 322.2015037244895}, { 247.53143365116767`, 327.84377399475983`}, {247.53143365116767`, 333.4860442650301}, {251.7665687863028, 338.059990210976}, { 256.9908931106272, 338.059990210976}, {262.21521743495146`, 338.059990210976}, {266.45035257008664`, 333.4860442650301}, { 266.45035257008664`, 327.84377399475983`}}}]}, {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 267.42176452437843`, 328.19440474333965`}, {267.42176452437843`, 322.0312163375425}, {263.2582379543302, 317.0349844534845}, { 258.1222476161659, 317.0349844534845}, {252.98625727800163`, 317.0349844534845}, {248.82273070795333`, 322.0312163375425}, { 248.82273070795333`, 328.19440474333965`}, {248.82273070795333`, 334.35759314913673`}, {252.98625727800163`, 339.3538250331947}, { 258.1222476161659, 339.3538250331947}, {263.2582379543302, 339.3538250331947}, {267.42176452437843`, 334.35759314913673`}, { 267.42176452437843`, 328.19440474333965`}}}, CurveClosed->{1}]}}, FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, { 0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 237.59711386563902`, 341.4124273220417}, {237.59711386563902`, 342.5356273220417}, {239.30951386563902`, 342.5356273220417}, { 239.30951386563902`, 344.80122732204165`}, {240.13871386563898`, 344.80122732204165`}, {240.13871386563898`, 342.5356273220417}, { 241.85051386563902`, 342.5356273220417}, {241.85051386563902`, 341.4124273220417}, {240.13871386563898`, 341.4124273220417}, { 240.13871386563898`, 339.1308273220417}, {239.30951386563902`, 339.1308273220417}, {239.30951386563902`, 341.4124273220417}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 237.39001569118824`, 318.6621655399778}, {244.38601569118833`, 318.6621655399778}, {244.38601569118833`, 317.4341655399778}, { 237.39001569118824`, 317.4341655399778}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{256.5885989161184, 309.1114491615899}, { 253.60743684524743`, 307.48601799006747`}, {253.54778051717386`, 307.5911342014536}, {255.6900955083974, 312.3070224462839}, { 256.14646279451614`, 313.96787637647145`}, {256.60278269452243`, 315.6279082231954}, {257.05994857600564`, 317.28871337356134`}, { 257.32231711802274`, 315.5839576039589}, {257.5847330461522, 313.88002391782}, {257.8479001835339, 312.1752193683959}, { 259.43342835424124`, 307.2316269164669}, {259.378086949813, 307.1335349993872}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{241.38464258393304`, 329.4131861498596}, { 239.74139258393302`, 331.9352861498596}, {239.83364258393306`, 331.9926861498596}, {244.23989258393306`, 330.3588861498596}, { 245.77514258393308`, 330.0438861498596}, {247.309642583933, 329.7288861498596}, {248.84489258393302`, 329.4131861498596}, { 247.309642583933, 329.0981861498596}, {245.77514258393308`, 328.7831861498596}, {244.23989258393306`, 328.4674861498596}, { 239.83364258393306`, 326.83368614985955`}, {239.74139258393302`, 326.87708614985957`}}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{229.8009257747583, 329.2290792568004}, {240.28787514475556`, 329.2290792568003}}]}, StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["r", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {181.595127099482, 334.6852542314971}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{481.42187369444014`, 219.60380656625946`}, { 483.58092781874234`, 215.98157309941413`}, {483.4572075271061, 215.9006639563124}, {477.60307606336505`, 218.28652350901814`}, { 475.56013530630855`, 218.7546069023972}, {473.5181945101547, 219.2226814530859}, {471.47526259578854`, 219.69176480736766`}, { 473.5261617741779, 220.12364622637284`}, {475.57606099166446`, 220.5555364880684}, {477.6269690127442, 220.9884178679763}, { 483.5223781551314, 223.2703758088371}, {483.6448250993544, 223.20729058195735`}}}]}}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], StrokeForm[ Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{666.6140314434665, 331.72710614403735`}, {686.1376030167526, 332.14270402104995`}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{687.3259819411286, 332.62127723585417`}, { 685.1369819411287, 336.2242772358542}, {685.2579819411286, 336.30627723585417`}, {691.1329819411286, 333.97227723585416`}, { 693.1799819411286, 333.5222772358542}, {695.2279819411286, 333.0722772358542}, {697.2749819411287, 332.62127723585417`}, { 695.2279819411286, 332.1712772358542}, {693.1799819411286, 331.7212772358542}, {691.1329819411286, 331.27027723585417`}, { 685.2579819411286, 328.93627723585416`}, {685.1369819411287, 328.9982772358542}}}]}, {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 228.90748156296524`, 317.4433232578872}, {203.26895251222874`, 317.4433232578872}, {203.26895251222874`, 342.288536941348}, { 228.90748156296524`, 342.288536941348}}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{169.88245500273692`, 330.13902303197165`}, {192.64465718172295`, 330.13902303197153`}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{193.86596950371114`, 330.40513567296375`}, { 192.08740700371123`, 333.28753567296377`}, {192.18571950371114`, 333.3531356729638}, {196.95915700371114`, 331.48593567296376`}, { 198.62234450371113`, 331.1259356729638}, {200.28634450371118`, 330.7659356729638}, {201.94953200371117`, 330.40513567296375`}, { 200.28634450371118`, 330.0451356729638}, {198.62234450371113`, 329.6851356729638}, {196.95915700371114`, 329.32433567296374`}, { 192.18571950371114`, 327.4571356729638}, {192.08740700371123`, 327.50673567296377`}}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["c", "1"], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {212.3596618283109, 328.4634326198955}, { Left, Baseline}, Alignment->{Left, Top}], {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 471.17680164664716`, 206.9746081253906}, {445.5382725959107, 206.9746081253906}, {445.5382725959107, 231.81982180885143`}, { 471.17680164664716`, 231.81982180885143`}}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["c", "2"], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {453.3740641562063, 217.42177237377967`}, { Left, Baseline}, Alignment->{Left, Top}], {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{665.4493696382789, 318.9670296735075}, {639.8108405875424, 318.9670296735075}, { 639.8108405875424, 343.81224335696834`}, {665.4493696382789, 343.81224335696834`}}}]}, InsetBox[ StyleBox[ TagBox["", HoldForm], FontSize->13, Background->GrayLevel[1.]], {652.9796046025103, 329.795120525802}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->12], StyleBox["s", FontSize->12]], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {650.6940449790793, 329.03326731799154`}, { Left, Baseline}, Alignment->{Left, Top}], {AbsoluteThickness[1.5667510701081488`], StrokeForm[{GrayLevel[0.], Opacity[0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{677.7309973837205, 331.7980168756683}, {677.107104911863, 219.95399303013505`}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{268.6554393730846, 329.2290792568003}, {279.1423887430819, 329.22907925680016`}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{280.23915618225936`, 329.4131861498595}, { 278.59590618225934`, 331.9352861498595}, {278.6881561822594, 331.9926861498595}, {283.0944061822594, 330.3588861498595}, { 284.6296561822594, 330.0438861498595}, {286.1641561822593, 329.7288861498595}, {287.69940618225934`, 329.4131861498595}, { 286.1641561822593, 329.0981861498595}, {284.6296561822594, 328.7831861498595}, {283.0944061822594, 328.4674861498595}, { 278.6881561822594, 326.83368614985943`}, {278.59590618225934`, 326.87708614985945`}}}]}, {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{313.4731876299108, 316.68147005007677`}, {287.8346585791743, 316.68147005007677`}, { 287.8346585791743, 341.5266837335376}, {313.4731876299108, 341.5266837335376}}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["k", "a"], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {297.1941565550906, 327.50956090237094`}, { Left, Baseline}, Alignment->{Left, Top}], {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{314.36663184170385`, 329.22907925680016`}, {324.85358121170106`, 329.22907925680005`}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{325.9503486508786, 329.41318614985937`}, { 324.3070986508785, 331.9352861498594}, {324.39934865087855`, 331.99268614985937`}, {328.80559865087855`, 330.35888614985936`}, { 330.34084865087857`, 330.04388614985936`}, {331.87534865087855`, 329.72888614985936`}, {333.41059865087857`, 329.41318614985937`}, { 331.87534865087855`, 329.0981861498594}, {330.34084865087857`, 328.7831861498594}, {328.80559865087855`, 328.4674861498594}, { 324.39934865087855`, 326.8336861498593}, {324.3070986508785, 326.87708614985934`}}}]}, {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{357.6606736829093, 315.91961684226635`}, {332.02214463217285`, 315.91961684226635`}, { 332.02214463217285`, 340.7648305257272}, {357.6606736829093, 340.7648305257272}}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["k", "r"], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {339.8579361924685, 325.6049278828451}, { Left, Baseline}, Alignment->{Left, Top}], {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 392.7466172091762, 328.57533134724474`}, {392.7466172091762, 322.4121429414476}, {388.58309063912793`, 317.4159110573896}, { 383.4471003009636, 317.4159110573896}, {378.3111099627994, 317.4159110573896}, {374.147583392751, 322.4121429414476}, { 374.147583392751, 328.57533134724474`}, {374.147583392751, 334.7385197530418}, {378.3111099627994, 339.7347516370998}, { 383.4471003009636, 339.7347516370998}, {388.58309063912793`, 339.7347516370998}, {392.7466172091762, 334.7385197530418}, { 392.7466172091762, 328.57533134724474`}}}, CurveClosed->{1}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{358.5541178947025, 328.46722604898974`}, {369.0410672646997, 328.46722604898963`}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{367.09042187263594`, 328.65133294204895`}, { 365.44717187263586`, 331.173432942049}, {365.5394218726359, 331.23083294204895`}, {369.9456718726359, 329.59703294204894`}, { 371.4809218726359, 329.28203294204894`}, {373.0154218726359, 328.96703294204895`}, {374.5506718726359, 328.65133294204895`}, { 373.0154218726359, 328.33633294204895`}, {371.4809218726359, 328.02133294204896`}, {369.9456718726359, 327.70563294204896`}, { 365.5394218726359, 326.0718329420489}, {365.44717187263586`, 326.1152329420489}}}]}, {FaceForm[{RGBColor[0.8630000000000001, 0.894, 0.89], Opacity[1.]}], FilledCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 392.5173713612859, 328.6303837279319}, {392.5173713612859, 322.5701675117156}, {388.4516416315561, 317.65741075495896`}, { 383.4362902802048, 317.65741075495896`}, {378.4209389288534, 317.65741075495896`}, {374.35520919912364`, 322.5701675117156}, { 374.35520919912364`, 328.6303837279319}, {374.35520919912364`, 334.6905999441481}, {378.4209389288534, 339.60335670090484`}, { 383.4362902802048, 339.60335670090484`}, {388.4516416315561, 339.60335670090484`}, {392.5173713612859, 334.6905999441481}, { 392.5173713612859, 328.6303837279319}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{365.3813546033221, 320.5667985595035}, {372.3773546033222, 320.5667985595035}, { 372.3773546033222, 319.3387985595035}, {365.3813546033221, 319.3387985595035}}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{393.5993654539772, 329.22907925680016`}, {404.0863148239744, 329.22907925680005`}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{404.4212290553416, 329.41318614985926`}, { 402.7779790553415, 331.9352861498593}, {402.87022905534155`, 331.99268614985925`}, {407.27647905534155`, 330.35888614985925`}, { 408.8117290553416, 330.04388614985925`}, {410.34622905534155`, 329.72888614985925`}, {411.8814790553416, 329.41318614985926`}, { 410.34622905534155`, 329.09818614985926`}, {408.8117290553416, 328.78318614985926`}, {407.27647905534155`, 328.46748614985927`}, { 402.87022905534155`, 326.8336861498592}, {402.7779790553415, 326.8770861498592}}}]}, {AbsoluteThickness[1.5667510701081488`], StrokeForm[{GrayLevel[0.], Opacity[0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{383.2747325650315, 315.93681937506085`}, {382.650840093174, 261.15607300173855`}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{382.9593446999612, 309.4310885263554}, { 379.2532304494605, 307.89442025421323`}, {379.17726921222743`, 307.9899163452584}, {381.7968594672925, 312.38952973435664`}, { 382.3468043244054, 313.93610265240045`}, {382.8967982716, 315.4834393543629}, {383.4477419230682, 317.0299747329324}, { 383.7967119855887, 315.4496162880526}, {384.1456329580275, 313.8684940592542}, {384.495601814903, 312.2880980749002}, { 386.53838360966995`, 307.7132504199459}, {386.4705184597617, 307.6231600131945}}}]}, {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{468.0107854117566, 314.4356887286373}, {412.20928100721267`, 314.4356887286373}, { 412.20928100721267`, 343.7982139909092}, {468.0107854117566, 343.7982139909092}}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ StyleBox["1", FontSize->12], StyleBox["/", FontSize->12], SubscriptBox[ StyleBox["R", FontSize->12], "a"]}], RowBox[{ RowBox[{ SubscriptBox[ StyleBox["T", FontSize->12], "1"], StyleBox["s", FontSize->12]}], StyleBox[" ", FontSize->12], StyleBox["+", FontSize->12], StyleBox[" ", FontSize->12], StyleBox["1", FontSize->12]}]], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {423.6617890516039, 329.79512052580196`}, { Left, Baseline}, Alignment->{Left, Top}], {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{513.840581284025, 316.68147005007654`}, {488.20205223328867`, 316.68147005007654`}, { 488.20205223328867`, 341.52668373353737`}, {513.840581284025, 341.52668373353737`}}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["k", "m"], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {496.03784379358433`, 326.3667810906553}, { Left, Baseline}, Alignment->{Left, Top}], {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{469.02283302719894`, 329.22907925680005`}, {479.50978239719615`, 329.22907925679993`}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{480.60654983637363`, 329.41318614985914`}, { 478.96329983637355`, 331.93528614985917`}, {479.0555498363736, 331.99268614985914`}, {483.4617998363736, 330.35888614985913`}, { 484.9970498363736, 330.04388614985913`}, {486.5315498363736, 329.72888614985914`}, {488.0667998363736, 329.41318614985914`}, { 486.5315498363736, 329.09818614985915`}, {484.9970498363736, 328.78318614985915`}, {483.4617998363736, 328.46748614985916`}, { 479.0555498363736, 326.8336861498591}, {478.96329983637355`, 326.8770861498591}}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{514.7340254958182, 329.22907925679993`}, {525.2209748658154, 329.2290792567998}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{526.3177423049929, 329.41318614985903`}, { 524.6744923049928, 331.93528614985905`}, {524.7667423049928, 331.992686149859}, {529.1729923049928, 330.358886149859}, { 530.7082423049928, 330.043886149859}, {532.2427423049928, 329.728886149859}, {533.7779923049928, 329.41318614985903`}, { 532.2427423049928, 329.09818614985903`}, {530.7082423049928, 328.78318614985903`}, {529.1729923049928, 328.46748614985904`}, { 524.7667423049928, 326.833686149859}, {524.6744923049928, 326.877086149859}}}]}, {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{590.6691518692182, 314.43568872863716`}, {534.8676474646743, 314.43568872863716`}, { 534.8676474646743, 343.79821399090906`}, {590.6691518692182, 343.79821399090906`}}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ FractionBox[ StyleBox[ RowBox[{"1", "/", "c"}], FontSize->12], RowBox[{ RowBox[{ SubscriptBox[ StyleBox["T", FontSize->12], "2"], StyleBox["s", FontSize->12]}], StyleBox[" ", FontSize->12], StyleBox["+", FontSize->12], StyleBox[" ", FontSize->12], StyleBox["1", FontSize->12]}]], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {544.0345958856345, 328.27141411018124`}, { Left, Baseline}, Alignment->{Left, Top}], {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{592.2577999107255, 330.9008762397817}, {638.5409443413303, 330.9008762397816}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{632.9771913984379, 330.9368925654801}, { 631.3339413984379, 333.4589925654801}, {631.4261913984378, 333.5163925654801}, {635.8324413984378, 331.8825925654801}, { 637.367691398438, 331.5675925654801}, {638.9021913984378, 331.2525925654801}, {640.4374413984378, 330.9368925654801}, { 638.9021913984378, 330.6218925654801}, {637.367691398438, 330.3068925654801}, {635.8324413984378, 329.9911925654801}, { 631.4261913984378, 328.35739256548004`}, {631.3339413984379, 328.40079256548006`}}}]}, {AbsoluteThickness[1.5667510701081488`], StrokeForm[{GrayLevel[0.], Opacity[0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{620.5920067979464, 331.08386719548184`}, {619.9681143260889, 263.3687779284586}}]}, {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{508.5076088293527, 251.16209417838877`}, {482.86907977861637`, 251.16209417838877`}, { 482.86907977861637`, 276.0073078618496}, {508.5076088293527, 276.0073078618496}}}]}, {AbsoluteThickness[1.023292992280754], StrokeForm[{GrayLevel[0.], Opacity[ 0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{619.9629631792056, 262.87771253382675`}, {517.9342580228181, 262.11585932601633`}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{517.9908276693354, 262.6485128075425}, { 520.1498817936376, 259.0262793406971}, {520.0261615020015, 258.94537019759537`}, {514.1720300382603, 261.33122975030113`}, { 512.129089281204, 261.7993131436802}, {510.08714848505, 262.2673876943689}, {508.0442165706839, 262.7364710486506}, { 510.0951157490734, 263.16835246765584`}, {512.1450149665598, 263.6002427293514}, {514.1959229876395, 264.0331241092593}, { 520.0913321300268, 266.31508205012005`}, {520.2137790742497, 266.25199682324035`}}}]}, {AbsoluteThickness[1.023292992280754], StrokeForm[{GrayLevel[0.], Opacity[ 0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{482.0507463419967, 260.97307951430093`}, {381.5376742914116, 260.9730795143008}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["k", "b"], TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {492.2285777545327, 260.08555201115746`}, { Left, Baseline}, Alignment->{Left, Top}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 364.0647463621522, 340.2696475103261}, {364.0647463621522, 341.3928475103261}, {365.7771463621522, 341.3928475103261}, { 365.7771463621522, 343.658447510326}, {366.6063463621522, 343.658447510326}, {366.6063463621522, 341.3928475103261}, { 368.3181463621522, 341.3928475103261}, {368.3181463621522, 340.2696475103261}, {366.6063463621522, 340.2696475103261}, { 366.6063463621522, 337.98804751032606`}, {365.7771463621522, 337.98804751032606`}, {365.7771463621522, 340.2696475103261}}}]}, ImagePadding->{{0., 1.}, {1., 3.1072}}, ImageSize->{718., Automatic}, PlotRange->{{165.012625, 711.261375}, {187.584524, 401.708578}}, PlotRangePadding->Automatic]], "Input", CellChangeTimes->{{3.527963403996524*^9, 3.527964024854034*^9}, { 3.527964061802137*^9, 3.527964441081657*^9}, {3.5279645298359127`*^9, 3.527964612409268*^9}, {3.527964668999148*^9, 3.5279648192815657`*^9}, { 3.527964857459429*^9, 3.52796541774369*^9}, {3.5279654496089487`*^9, 3.527965538688401*^9}, {3.527966056201812*^9, 3.527966061213402*^9}, { 3.5279671894607058`*^9, 3.5279671943603363`*^9}, {3.528051632060562*^9, 3.528051634719224*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Quit", "[", "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{"SystemsModelSeriesConnect", "[", RowBox[{ RowBox[{"SystemsModelFeedbackConnect", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"SystemsModelSeriesConnect", "[", RowBox[{ RowBox[{"SystemsModelSeriesConnect", "[", " ", RowBox[{ RowBox[{"TransferFunctionModel", "[", RowBox[{ SubscriptBox["k", "a"], ",", "s"}], "]"}], ",", RowBox[{"SystemsModelSeriesConnect", "[", " ", RowBox[{ RowBox[{"TransferFunctionModel", "[", RowBox[{ SubscriptBox["k", "r"], ",", "s"}], "]"}], ",", RowBox[{"SystemsModelFeedbackConnect", "[", RowBox[{ RowBox[{"SystemsModelSeriesConnect", "[", RowBox[{ RowBox[{"TransferFunctionModel", "[", RowBox[{ FractionBox[ RowBox[{"1", "/", SubscriptBox["R", "a"]}], RowBox[{ RowBox[{ SubscriptBox["T", "1"], "s"}], "+", "1"}]], ",", "s"}], "]"}], ",", RowBox[{"SystemsModelSeriesConnect", "[", RowBox[{ RowBox[{"TransferFunctionModel", "[", RowBox[{ SubscriptBox["k", "m"], ",", "s"}], "]"}], ",", RowBox[{"TransferFunctionModel", "[", RowBox[{ FractionBox[ RowBox[{"1", "/", "c"}], RowBox[{ RowBox[{ SubscriptBox["T", "2"], "s"}], "+", "1"}]], ",", "s"}], "]"}]}], "]"}]}], "]"}], ",", RowBox[{"TransferFunctionModel", "[", RowBox[{ SubscriptBox["k", "b"], ",", "s"}], "]"}]}], "]"}]}], "]"}]}], "]"}], ",", " ", RowBox[{"TransferFunctionModel", "[", RowBox[{ FractionBox["1", "s"], ",", "s"}], "]"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"TransferFunctionModel", "[", RowBox[{ SubscriptBox["c", "2"], ",", "s"}], "]"}]}], "]"}], ",", RowBox[{"TransferFunctionModel", "[", RowBox[{ SubscriptBox["c", "1"], ",", "s"}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.5235010128161373`*^9, 3.523501163541617*^9}, { 3.5280522789795218`*^9, 3.5280523028341208`*^9}}] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Linearization", "Section", CellChangeTimes->{3.5280525283605423`*^9, 3.528052917715294*^9, 3.528114375586277*^9}], Cell["\<\ Start with non-linear systems, and find linear approximations before \ proceeding with linear control analysis and design. \ \>", "Text", CellChangeTimes->{{3.5280408904566193`*^9, 3.528040930773987*^9}, 3.528041108295559*^9, 3.528041142105851*^9, {3.5280428062614803`*^9, 3.528042839715294*^9}, {3.528051397342969*^9, 3.528051432866795*^9}, { 3.5280514732348137`*^9, 3.528051507780052*^9}, {3.528051826207505*^9, 3.5280518400432167`*^9}, {3.5280522398305387`*^9, 3.528052268052758*^9}, { 3.528055455777595*^9, 3.528055494809483*^9}}], Cell[BoxData[ RowBox[{"\t\t\t\t\t\t\t", GraphicsBox[{ {RGBColor[1, 0.5, 0], EdgeForm[GrayLevel[0]], FaceForm[{RGBColor[0, 1, 1], Opacity[0.38]}], DiskBox[{0, 0}]}, {Thickness[0.004], Dashing[{Small, Small}], LineBox[{{0, 0}, {2, 0}}]}, {Thickness[0.004], ArrowBox[{{0, 0}, {1.5, 1.5}}]}, {PointSize[Large], PointBox[{1.5, 1.5}]}, {Thickness[0.004], ArrowBox[{{1.505, 1.505}, {1.85, 1.85}}]}, {Thickness[0.004], ArrowBox[{{1.505, 1.505}, {1.55, 2.2}}]}, {Thickness[0.004], ArrowBox[{{1.505, 1.505}, {1.2, 1.8}}]}, {Thickness[0.004], Dashing[{Small, Small}], ArrowBox[{{1.505, 1.505}, {0.8, 2.2}}]}, {Thickness[0.004], Dashing[{Small, Small}], CircleBox[{0, 0}, 1.2, NCache[{0, Rational[1, 4] Pi}, {0, 0.7853981633974483}]]}, {Thickness[0.004], Dashing[{Small, Small}], CircleBox[{1.5, 1.5}, 0.4, NCache[{Rational[1, 2] Pi, 2.3876104167282426`}, {1.5707963267948966`, 2.3876104167282426`}]]}, InsetBox[ StyleBox["\<\"\[Theta]\"\>", StripOnInput->False, FontSize->16], {1, 0.4}], InsetBox[ StyleBox["\<\"\\!\\(\\*SubscriptBox[\\(v\\), \\(r\\)]\\)\"\>", StripOnInput->False, FontSize->16], {1.9, 1.9}], InsetBox[ StyleBox["\<\"\\!\\(\\*SubscriptBox[\\(v\\), \\(t\\)]\\)\"\>", StripOnInput->False, FontSize->16], {0.8, 2.1}], InsetBox[ StyleBox["\[Phi]", StripOnInput->False, FontSize->16], {1.32, 2}], InsetBox[ StyleBox["\<\"a\"\>", StripOnInput->False, FontSize->16], {1.6, 2.25}], InsetBox[ StyleBox["\<\"spacecraft\"\>", StripOnInput->False, FontSize->16], {2, 1.5}], InsetBox[ StyleBox["\<\" Earth\"\>", StripOnInput->False, FontSize->16], {0, -0.1}], InsetBox[ StyleBox["\<\"reference line\"\>", StripOnInput->False, FontSize->16], {1.6, -0.1}], InsetBox[ StyleBox["\<\"r\"\>", StripOnInput->False, FontSize->16], {1, 1.2}]}, ImageSize->{303., Automatic}]}]], "Input", CellChangeTimes->{{3.528055593053981*^9, 3.52805560647519*^9}, { 3.528058256970532*^9, 3.528058324530101*^9}, {3.5280711867608843`*^9, 3.528071187077248*^9}}], Cell["\<\ Obtain a state-space model by linearizing the two-body equations of motion \ for a low-thrust spacecraft orbit transfer:\ \>", "Text", CellChangeTimes->{{3.5280557161411963`*^9, 3.5280557743988647`*^9}, { 3.528055804933606*^9, 3.528055817720752*^9}, {3.528071114108581*^9, 3.528071118040118*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"r", "'"}], "=", SubscriptBox["v", "r"]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Theta]", "'"}], "=", FractionBox[ SubscriptBox["v", "t"], "r"]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ SubscriptBox["v", "r"], "'"}], "=", RowBox[{ FractionBox[ SuperscriptBox[ SubscriptBox["v", "t"], "2"], "r"], "-", FractionBox["1", SuperscriptBox["r", "2"]], "+", RowBox[{"a", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ SubscriptBox["v", "t"], "'"}], "=", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{ SubscriptBox["v", "r"], " ", SubscriptBox["v", "t"]}], "r"]}], "+", " ", RowBox[{"a", " ", RowBox[{ "Cos", "[", "\[Phi]", "]"}]}]}]}], "\[LineSeparator]"}], "DisplayFormula",\ CellChangeTimes->{{3.528055828557571*^9, 3.5280560277555237`*^9}, 3.5280565890859833`*^9, {3.528071197284628*^9, 3.528071201872499*^9}, { 3.528194093139471*^9, 3.5281940952978907`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"StateSpaceModel", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"r", "'"}], "[", "t", "]"}], "\[Equal]", RowBox[{ SubscriptBox["v", "r"], "[", "t", "]"}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "t", "]"}], "\[Equal]", " ", RowBox[{ RowBox[{ SubscriptBox["v", "t"], "[", "t", "]"}], "/", RowBox[{"r", "[", "t", "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["v", "r"], "'"}], "[", "t", "]"}], "\[Equal]", " ", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["v", "t"], "[", "t", "]"}], "*", RowBox[{ RowBox[{ SubscriptBox["v", "t"], "[", "t", "]"}], "/", RowBox[{"r", "[", "t", "]"}]}]}], "-", RowBox[{"1", "/", RowBox[{ RowBox[{"(", RowBox[{"r", "[", "t", "]"}], ")"}], "^", "2"}]}], " ", "+", " ", RowBox[{"a", " ", RowBox[{"Sin", "[", RowBox[{"\[Phi]", "[", "t", "]"}], "]"}]}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["v", "t"], "'"}], "[", "t", "]"}], "\[Equal]", " ", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{ SubscriptBox["v", "r"], "[", "t", "]"}]}], " ", RowBox[{ RowBox[{ SubscriptBox["v", "t"], "[", "t", "]"}], "/", RowBox[{"r", "[", "t", "]"}]}]}], "+", " ", RowBox[{"a", " ", RowBox[{"Cos", "[", RowBox[{"\[Phi]", "[", "t", "]"}], "]"}]}]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"r", "[", "t", "]"}], ",", "0.75"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"\[Theta]", "[", "t", "]"}], ",", RowBox[{"Pi", "/", "3"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["v", "r"], "[", "t", "]"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["v", "t"], "[", "t", "]"}], ",", "0.05"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"\[Phi]", "[", "t", "]"}], ",", RowBox[{"Pi", "/", "4"}]}], "}"}], "}"}], ",", RowBox[{"{", RowBox[{"r", "[", "t", "]"}], "}"}], ",", "t"}], "]"}], "//", "Chop"}]], "Input", CellChangeTimes->{{3.528056491422389*^9, 3.528056523011256*^9}, { 3.528056610540135*^9, 3.528056619196653*^9}, {3.528071214127562*^9, 3.5280712277911053`*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Analysis", "Section", CellChangeTimes->{3.528056786872114*^9}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " provides tools for both state-space and analysis and classical analysis. \ With a model, verify controllablity, observability, absolute and relative \ stability..." }], "Text", CellChangeTimes->{{3.5280568142957163`*^9, 3.52805682046609*^9}, { 3.528056859142531*^9, 3.528056895639441*^9}, {3.528057909690545*^9, 3.528058086636138*^9}, {3.5280584499175863`*^9, 3.528058470683444*^9}, { 3.5280589214421997`*^9, 3.528058931212862*^9}, {3.52805923644221*^9, 3.528059286917594*^9}, 3.5280593733935957`*^9, {3.528115516242867*^9, 3.5281155289352493`*^9}}], Cell["\<\ Functions to compute state-space transformations and decompositions, analyze \ frequency-domain behavior, visualize system responses...\ \>", "Text", CellChangeTimes->{{3.528058973808673*^9, 3.528059081564186*^9}, { 3.528059128164311*^9, 3.528059227940619*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ State-space Analysis\ \>", "Section", CellChangeTimes->{ 3.5280593212301607`*^9, {3.528060465323888*^9, 3.52806046856448*^9}, { 3.528063583915557*^9, 3.528063584310898*^9}}], Cell["A state-space model", "Text", CellChangeTimes->{{3.528059621793222*^9, 3.5280596393574467`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"ss", "=", " ", RowBox[{"StateSpaceModel", "[", RowBox[{"{", RowBox[{ RowBox[{"(", GridBox[{ {"0", "1", "0", "0"}, {"0", "0", "1", "0"}, {"0", "0", "0", "1"}, { RowBox[{"-", "99.65"}], RowBox[{"-", "153.7"}], RowBox[{"-", "74.96"}], RowBox[{"-", "14.6"}]} }], ")"}], ",", RowBox[{"(", GridBox[{ {"0"}, {"0"}, {"0"}, {"1"} }], ")"}], ",", RowBox[{"(", GridBox[{ {"26", "36", "11", "1"} }], ")"}]}], "}"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.528060488102303*^9, 3.528060509363229*^9}}], Cell[CellGroupData[{ Cell["Controllable and Observable?", "Subsubsection", CellChangeTimes->{ 3.483202458955147*^9, {3.495918529775271*^9, 3.4959185355106363`*^9}, { 3.495920008158016*^9, 3.495920032708687*^9}, {3.495922447933578*^9, 3.49592247695435*^9}, {3.4959264782249413`*^9, 3.495926505621825*^9}, 3.496006573894264*^9, {3.5280597522209263`*^9, 3.528059769143961*^9}}], Cell[BoxData[ RowBox[{"Through", "@", RowBox[{ RowBox[{"{", RowBox[{"ControllableModelQ", ",", "ObservableModelQ"}], "}"}], "@", "ss"}]}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Gramians ", "Subsubsection", CellChangeTimes->{{3.528060037166987*^9, 3.528060056940626*^9}, 3.5281144496818943`*^9}], Cell["Compute the controllability and observability gramians ", "Text", CellChangeTimes->{{3.528114445046245*^9, 3.528114457158304*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"wc", ",", "wo"}], "}"}], "=", RowBox[{"Through", "@", RowBox[{ RowBox[{"{", RowBox[{"ControllabilityGramian", ",", "ObservabilityGramian"}], "}"}], "@", "ss"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"MatrixForm", "/@", RowBox[{"{", RowBox[{"wc", ",", "wo"}], "}"}]}], "//", "Chop"}]}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Balanced realization", "Subsubsection", CellChangeTimes->{{3.52811447238237*^9, 3.5281144850881577`*^9}}], Cell["\<\ The realization is not balanced. Compute a balanced realization:\ \>", "Text", CellChangeTimes->{{3.5280602963553877`*^9, 3.528060353018664*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"s1", ",", "ib"}], "}"}], "=", RowBox[{"InternallyBalancedDecomposition", "[", "ss", "]"}]}]], "Input", CellChangeTimes->{3.52806038309879*^9}] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["State-space Analysis", "Section", CellChangeTimes->{ 3.528060539481229*^9, {3.528063589434031*^9, 3.528063589788954*^9}}], Cell[CellGroupData[{ Cell["Jordan Decomposition", "Subsubsection", CellChangeTimes->{{3.5281145910768337`*^9, 3.528114615204001*^9}}], Cell[BoxData[ RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"MatrixForm", "@", RowBox[{ RowBox[{"Normal", "[", RowBox[{ RowBox[{"JordanModelDecomposition", "[", "#", "]"}], "[", RowBox[{"[", "2", "]"}], "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], "&"}], ",", RowBox[{"{", RowBox[{"ss", ",", "ib"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.528060827312921*^9, 3.528060844211741*^9}}], Cell["\<\ InternallyBalancedDecomposition has effected a similarity transformation : \ Jordan decompositions of the original and transformed models are identical\ \>", "Text", CellChangeTimes->{{3.52811462580929*^9, 3.528114633470931*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["New gramians", "Subsubsection", CellChangeTimes->{{3.528060884963244*^9, 3.5280609131912527`*^9}, { 3.5281147000206213`*^9, 3.528114702639024*^9}}], Cell["After balancing, the new gramians are also identical", "Text", CellChangeTimes->{{3.528114684722168*^9, 3.5281146896558*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"wC", ",", "wO"}], "}"}], "=", RowBox[{"Through", "@", RowBox[{ RowBox[{"{", RowBox[{"ControllabilityGramian", ",", "ObservabilityGramian"}], "}"}], "@", "ib"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"MatrixForm", "/@", RowBox[{"{", RowBox[{"wC", ",", "wO"}], "}"}]}], "//", "Chop"}]}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Lower Order Model", "Subsubsection", CellChangeTimes->{{3.528114721885806*^9, 3.52811475073383*^9}}], Cell["\<\ A model of reduced order by retaining the two most controllable and \ observable states\ \>", "Text", CellChangeTimes->{{3.528061049074724*^9, 3.528061090829834*^9}}], Cell[BoxData[ RowBox[{"ssred", "=", RowBox[{"SystemsModelDelete", "[", RowBox[{"ib", ",", "None", ",", "None", ",", RowBox[{"{", RowBox[{"3", ",", "4"}], "}"}]}], "]"}]}]], "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["State-space Analysis", "Section", CellChangeTimes->{ 3.528061210181376*^9, {3.52806359432196*^9, 3.5280635946448383`*^9}}], Cell[CellGroupData[{ Cell["Controllability and Observability", "Subsubsection", CellChangeTimes->{{3.528114804582966*^9, 3.528114830326255*^9}}], Cell["The reduced system is controllable and observable", "Text", CellChangeTimes->{{3.528061271613521*^9, 3.5280612734431963`*^9}, { 3.528061360497589*^9, 3.52806138492754*^9}}], Cell[BoxData[ RowBox[{"Through", "@", RowBox[{ RowBox[{"{", RowBox[{"ControllableModelQ", ",", "ObservableModelQ"}], "}"}], "@", "ssred"}]}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Bode Plot", "Subsubsection", CellChangeTimes->{{3.528114835865966*^9, 3.5281148441243896`*^9}}], Cell["\<\ The Bode plots of the actual and reduced systems are almost coincident in \ this scale \ \>", "Text", CellChangeTimes->{{3.528061463810968*^9, 3.5280614795150833`*^9}}], Cell[BoxData[ RowBox[{"BodePlot", "[", RowBox[{ RowBox[{"{", RowBox[{"ss", ",", "ssred"}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", " ", "250"}]}], "]"}]], "Input", CellChangeTimes->{{3.5280614992870913`*^9, 3.52806151767654*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["State space Transformation", "Subsubsection", CellChangeTimes->{{3.528114893492135*^9, 3.5281149136573687`*^9}}], Cell["\<\ The original system can be retieved from the balanced system by a state-space \ transformation\ \>", "Text", CellChangeTimes->{{3.528061544367999*^9, 3.528061556911147*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"ss", "==", RowBox[{"StateSpaceTransform", "[", RowBox[{"ib", ",", RowBox[{"Inverse", "@", "s1"}]}], "]"}]}], "//", "Chop"}]], "Input", CellChangeTimes->{{3.52806157223484*^9, 3.528061635927229*^9}}] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["State-space Analysis", "Section", CellChangeTimes->{ 3.5280626783038588`*^9, {3.528063598910928*^9, 3.528063599164928*^9}}], Cell[CellGroupData[{ Cell["Non-linear system", "Subsubsection", CellChangeTimes->{{3.528114946441771*^9, 3.5281149609022007`*^9}, { 3.528115107083301*^9, 3.5281151143909817`*^9}}], Cell["\<\ Objective: analyze stability of a non-linear system based on local \ linearization.\ \>", "Text", CellChangeTimes->{{3.528062702440034*^9, 3.528062728728478*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["x", "1"], "'"}], "[", "t", "]"}], "\[Equal]", " ", RowBox[{ RowBox[{ RowBox[{"-", "1.1"}], " ", RowBox[{ SubscriptBox["x", "1"], "[", "t", "]"}]}], "+", " ", RowBox[{ FractionBox["2", "\[Pi]"], RowBox[{"ArcTan", "[", RowBox[{"0.7", " ", "\[Pi]", " ", RowBox[{ SubscriptBox["x", "2"], "[", "t", "]"}]}], "]"}]}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["x", "2"], "'"}], "[", "t", "]"}], "\[Equal]", " ", RowBox[{ RowBox[{ RowBox[{"-", "1.1"}], " ", RowBox[{ SubscriptBox["x", "2"], "[", "t", "]"}]}], "+", " ", RowBox[{ FractionBox["2", "\[Pi]"], RowBox[{"ArcTan", "[", RowBox[{"0.7", " ", "\[Pi]", " ", RowBox[{ SubscriptBox["x", "1"], "[", "t", "]"}]}], "]"}]}]}]}]}], "DisplayFormula", CellChangeTimes->{{3.528035461122913*^9, 3.52803560895012*^9}, 3.528035653002511*^9}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell["Equilibrium points", "Subsubsection", CellChangeTimes->{{3.5280628846557083`*^9, 3.528062898781172*^9}, { 3.52811497745107*^9, 3.528114978086358*^9}}], Cell[BoxData[ RowBox[{"eqpt", "=", RowBox[{"FindInstance", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "1.1"}], " ", SubscriptBox["x", "1"]}], "+", " ", RowBox[{ RowBox[{"(", RowBox[{"2", "/", "Pi"}], ")"}], RowBox[{"ArcTan", "[", RowBox[{"0.7", " ", "Pi", " ", SubscriptBox["x", "2"]}], "]"}]}]}], "\[Equal]", "0"}], ",", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"2", "/", "Pi"}], ")"}], RowBox[{"ArcTan", "[", RowBox[{"0.7", " ", "Pi", " ", SubscriptBox["x", "1"]}], "]"}]}], "-", RowBox[{"1.1", SubscriptBox["x", "2"]}]}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"{", RowBox[{ SubscriptBox["x", "1"], ",", SubscriptBox["x", "2"]}], "}"}], ",", "Reals", ",", "3"}], "]"}]}]], "Input", CellChangeTimes->{{3.528062917936084*^9, 3.528062920505563*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Linear Model", "Subsubsection", CellChangeTimes->{{3.528114998909855*^9, 3.528115011888507*^9}}], Cell["Obtain linear models about equilibrium points :", "Text", CellChangeTimes->{{3.528062946488715*^9, 3.5280629628615093`*^9}, { 3.5280629946923923`*^9, 3.528063010686384*^9}}], Cell[BoxData[ RowBox[{"MapThread", "[", RowBox[{ RowBox[{ RowBox[{"StateSpaceModel", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["x", "1"], "'"}], "[", "t", "]"}], "\[Equal]", RowBox[{ RowBox[{ RowBox[{"-", "1.1"}], " ", RowBox[{ SubscriptBox["x", "1"], "[", "t", "]"}]}], "+", " ", RowBox[{ RowBox[{"(", RowBox[{"2", "/", "Pi"}], ")"}], RowBox[{"ArcTan", "[", RowBox[{"0.7", " ", "Pi", " ", RowBox[{ SubscriptBox["x", "2"], "[", "t", "]"}]}], "]"}]}]}]}], ",", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["x", "2"], "'"}], "[", "t", "]"}], "\[Equal]", " ", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"2", "/", "Pi"}], ")"}], RowBox[{"ArcTan", "[", RowBox[{"0.7", " ", "Pi", " ", RowBox[{ SubscriptBox["x", "1"], "[", "t", "]"}]}], "]"}]}], "-", RowBox[{"1.1", " ", RowBox[{ SubscriptBox["x", "2"], "[", "t", "]"}]}]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["x", "1"], "[", "t", "]"}], ",", "#1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["x", "2"], "[", "t", "]"}], ",", "#2"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"u", "[", "t", "]"}], ",", "0"}], "}"}], "}"}], ",", RowBox[{"{", "0", "}"}], ",", "t"}], "]"}], "&"}], ",", RowBox[{"Transpose", "[", RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["x", "1"], ",", SubscriptBox["x", "2"]}], "}"}], "/.", "eqpt"}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.528063046795642*^9, 3.528063053407756*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Lyapunov equation", "Subsubsection", CellChangeTimes->{{3.5281150220632772`*^9, 3.528115047639944*^9}, { 3.528115084696147*^9, 3.5281150914720383`*^9}}], Cell["\<\ Verify asymptotic stability of the equilibrium points by checking \ positive-definiteness of the Lyapunov equation solutions :\ \>", "Text", CellChangeTimes->{{3.528063146756068*^9, 3.528063154575418*^9}, { 3.5280631880867*^9, 3.528063222548221*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["a", "i"], ".", SubscriptBox["p", "i"]}], " ", "+", " ", RowBox[{ SubscriptBox["p", "i"], ".", RowBox[{ SubscriptBox["a", "i"], "\[ConjugateTranspose]"}]}]}], "\[Equal]", "c"}]], "DisplayFormula", CellChangeTimes->{{3.528063306650928*^9, 3.528063372480197*^9}}], Cell[BoxData[ RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"PositiveDefiniteMatrixQ", "[", RowBox[{"LyapunovSolve", "[", RowBox[{ RowBox[{ RowBox[{"Normal", "[", "#", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"DiagonalMatrix", "[", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", RowBox[{"-", "2"}]}], "}"}], "]"}]}], "]"}], "]"}], "&"}], ",", "%"}], "]"}]], "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Classical Analysis", "Section", CellChangeTimes->{ 3.528063572445952*^9, {3.528063604080558*^9, 3.528063604380383*^9}}], Cell[CellGroupData[{ Cell["System Model", "Subsubsection", CellChangeTimes->{{3.5280639451535673`*^9, 3.5280639768204107`*^9}, { 3.528115266139327*^9, 3.528115271314509*^9}, {3.5281162163791428`*^9, 3.528116218774898*^9}}], Cell[BoxData[ RowBox[{"tfol", " ", "=", " ", RowBox[{"TransferFunctionModel", "[", RowBox[{ FractionBox[ RowBox[{"2.7778", RowBox[{"(", RowBox[{ RowBox[{"s", "^", "2"}], "+", RowBox[{"0.192", "s"}], "+", "1.92"}], ")"}]}], RowBox[{"s", RowBox[{ RowBox[{"(", RowBox[{"s", "+", "1"}], ")"}], "^", "2"}], RowBox[{"(", RowBox[{ RowBox[{"s", "^", "2"}], "+", RowBox[{"0.384", "s"}], "+", "2.56"}], ")"}]}]], ",", "s"}], "]"}]}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Open-loop Stability", "Subsubsection", CellChangeTimes->{{3.5281152929415283`*^9, 3.528115294538246*^9}, 3.5281156167064962`*^9, {3.5281161976775084`*^9, 3.5281162221793013`*^9}}], Cell["\<\ Open loop system is marginally stable, not BIBO stable\ \>", "Text", CellChangeTimes->{{3.528064411578314*^9, 3.528064454601282*^9}, { 3.528115621515959*^9, 3.528115670102899*^9}, {3.528142841453095*^9, 3.528142842672906*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Flatten", "[", RowBox[{"TransferFunctionPoles", "[", "tfol", "]"}], "]"}], "//", "TableForm"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Closed-loop Stability", "Subsubsection", CellChangeTimes->{{3.528116228099503*^9, 3.528116263411998*^9}}], Cell["\<\ Closed-loop system stable under unity negative feedback since no net \ encirclement of critical point\ \>", "Text", CellChangeTimes->{{3.528064474562072*^9, 3.528064507273224*^9}, { 3.52806454174802*^9, 3.528064561605749*^9}, 3.528064592298052*^9, { 3.528067176770578*^9, 3.528067178276842*^9}}], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Graphics", "[", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "}"}], "]"}], "]"}], ",", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", "Medium", "]"}], ",", RowBox[{"Point", "[", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0"}], "}"}], "]"}]}], "}"}], "]"}], ",", RowBox[{"NyquistPlot", "[", "tfol", "]"}]}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", " ", "300"}], ",", RowBox[{"Axes", "\[Rule]", " ", "True"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "2.5"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.528070174655938*^9, 3.5280701862689447`*^9}}] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Classical Analysis", "Section", CellChangeTimes->{3.5280646111756563`*^9}], Cell["\<\ Step response of closed-loop system is highly oscillatory:\ \>", "Text", CellChangeTimes->{{3.528064635388146*^9, 3.528064676303048*^9}}], Cell[CellGroupData[{ Cell["Closed-loop Performance", "Subsubsection", CellChangeTimes->{{3.52811631296634*^9, 3.5281163315040693`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"tfcl", " ", "=", " ", RowBox[{"SystemsModelFeedbackConnect", "[", "tfol", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"OutputResponse", "[", RowBox[{"tfcl", ",", "1", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "500"}], "}"}]}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{"%", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "500"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", "All"}], ",", RowBox[{"Frame", "\[Rule]", " ", "True"}]}], "]"}]}], "Input", CellChangeTimes->{{3.528070201579647*^9, 3.5280702052993*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Verify Stability Margins", "Subsubsection", CellChangeTimes->{{3.528116435345717*^9, 3.5281164573084917`*^9}}], Cell["Bode plot shows poor stability margins", "Text", CellChangeTimes->{{3.52806471631686*^9, 3.528064734077729*^9}}], Cell[BoxData[ RowBox[{"BodePlot", "[", RowBox[{"tfol", ",", RowBox[{"StabilityMargins", "\[Rule]", " ", "True"}], ",", RowBox[{"ImageSize", "\[Rule]", " ", "300"}], ",", RowBox[{"StabilityMarginsStyle", "\[Rule]", " ", RowBox[{"Directive", "[", RowBox[{"Red", ",", "Thick"}], "]"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.528064759938352*^9, 3.52806477350406*^9}}], Cell["Numerical values:", "Text", CellChangeTimes->{{3.5280648689740047`*^9, 3.5280648795323668`*^9}, 3.528067238303629*^9}], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"gm", ",", "pm"}], "}"}], "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"GainMargins", "[", "tfol", "]"}], "[", RowBox[{"[", RowBox[{"1", ",", "2"}], "]"}], "]"}], ",", RowBox[{ RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"#", "[", RowBox[{"[", "1", "]"}], "]"}], ",", FractionBox[ RowBox[{"#", "[", RowBox[{"[", "2", "]"}], "]"}], "Degree"]}], "}"}], "&"}], ",", RowBox[{"PhaseMargins", "[", "tfol", "]"}]}], "]"}], "[", RowBox[{"[", RowBox[{"1", ",", "2"}], "]"}], "]"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.528064887687705*^9, 3.5280649657014847`*^9}}] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Classical Analysis", "Section", CellChangeTimes->{3.5280648441227913`*^9}], Cell[CellGroupData[{ Cell["Proportional Control ", "Subsubsection", CellChangeTimes->{{3.5281168647321577`*^9, 3.5281168740362597`*^9}, { 3.528116979510298*^9, 3.528116996355503*^9}, 3.528196280217875*^9, 3.528202408064889*^9}], Cell["\<\ Attempt to improve performance by proportional control.\ \>", "Text", CellChangeTimes->{{3.5280650455783243`*^9, 3.528065101425419*^9}, 3.5280657263116083`*^9}], Cell[BoxData[ GraphicsBox[{ {EdgeForm[{AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}]}], FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{567.8275822417065, 290.30825401430843`}, {339.4742760615837, 290.30825401430843`}, { 339.4742760615837, 361.6895654277628}, {567.8275822417065, 361.6895654277628}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{194.6226, 329.}, {204.7006, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{203.6802, 329.}, {201.49020000000002`, 332.603}, {201.6122, 332.685}, {207.4872, 330.351}, {209.5342, 329.901}, {211.58120000000002`, 329.451}, {213.62820000000002`, 329.}, { 211.58120000000002`, 328.55}, {209.5342, 328.1}, {207.4872, 327.649}, { 201.6122, 325.315}, {201.49020000000002`, 325.377}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{274.92769999999996`, 329.}, { 294.96369999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{293.9438, 329.}, {291.7538, 332.603}, { 291.8758, 332.685}, {297.7508, 330.351}, {299.7978, 329.901}, {301.8448, 329.451}, {303.8918, 329.}, {301.8448, 328.55}, {299.7978, 328.1}, { 297.7508, 327.649}, {291.8758, 325.315}, {291.7538, 325.377}}}]}, { {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{317.63669999999996`, 353.749}, { 321.23969999999997`, 355.939}, {321.32169999999996`, 355.817}, { 318.98769999999996`, 349.942}, {318.5377, 347.89500000000004`}, { 318.0877, 345.848}, {317.63669999999996`, 343.80100000000004`}, { 317.1867, 345.848}, {316.7367, 347.89500000000004`}, { 316.28569999999996`, 349.942}, {313.95169999999996`, 355.817}, { 314.0137, 355.939}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{331.2451, 329.}, {345.6441, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{372.0752687617261, 327.7522150562854}, { 369.8852687617261, 331.35521505628543`}, {370.0072687617261, 331.4372150562854}, {375.8822687617261, 329.1032150562854}, { 377.92926876172606`, 328.65321505628543`}, {379.9762687617261, 328.20321505628544`}, {382.02326876172606`, 327.7522150562854}, { 379.9762687617261, 327.30221505628543`}, {377.92926876172606`, 326.85221505628544`}, {375.8822687617261, 326.4012150562854}, { 370.0072687617261, 324.0672150562854}, {369.8852687617261, 324.12921505628543`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{482.8262, 329.}, {500.03119999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{499.01169999999996`, 329.}, { 496.82269999999994`, 332.603}, {496.9437, 332.685}, {502.8187, 330.351}, {504.86569999999995`, 329.901}, {506.91369999999995`, 329.451}, {508.9607, 329.}, {506.91369999999995`, 328.55}, { 504.86569999999995`, 328.1}, {502.8187, 327.649}, {496.9437, 325.315}, { 496.82269999999994`, 325.377}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{647.1273016048697, 323.56695975669345`}, { 644.9383016048697, 327.16995975669346`}, {645.0593016048697, 327.25195975669345`}, {650.9343016048697, 324.91795975669345`}, { 652.9813016048697, 324.46795975669346`}, {655.0293016048697, 324.0179597566935}, {657.0763016048697, 323.56695975669345`}, { 655.0293016048697, 323.11695975669346`}, {652.9813016048697, 322.6669597566935}, {650.9343016048697, 322.21595975669345`}, { 645.0593016048697, 319.88195975669345`}, {644.9383016048697, 319.94395975669346`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{648.1934000000001, 329.}, { 700.3364000000001, 329.}}}]}}, { {Thickness[0.0012626262626262627`], {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{664.623, 329.}, {664.623, 207.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{700.5735575868373, 300.8917737659963}, { 704.1775575868373, 303.08277376599625`}, {704.2575575868373, 302.95977376599626`}, {701.9255575868374, 297.08477376599626`}, { 701.4735575868374, 295.03777376599623`}, {701.0245575868373, 292.99177376599624`}, {700.5735575868373, 290.9447737659963}, { 700.1225575868373, 292.99177376599624`}, {699.6735575868374, 295.03777376599623`}, {699.2215575868374, 297.08477376599626`}, { 696.8875575868374, 302.95977376599626`}, {696.9505575868372, 303.08277376599625`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{179.6226, 303.1777}, {176.0196, 300.98670000000004`}, {175.9376, 301.10970000000003`}, {178.2716, 306.98470000000003`}, {178.7216, 309.0317}, {179.1716, 311.0787}, { 179.6226, 313.1257}, {180.0726, 311.0787}, {180.5226, 309.0317}, { 180.9736, 306.98470000000003`}, {183.3076, 301.10970000000003`}, { 183.2456, 300.98670000000004`}}}]}}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{0, 2, 0}, {0, 1, 0}}}, {{{625.2663963862341, 222.90132515231497`}, {242.39293680752434`, 221.95213551305866`}, { 242.39293680752434`, 317.06482281062813`}}}, CurveClosed->{0}]}, {AbsoluteThickness[1], StrokeForm[{GrayLevel[0.], Opacity[0.64]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{254.0668412874445, 331.21966702852296`}, {276.8216286070817, 331.2196670285232}}]}}, {AbsoluteThickness[1.5667510701081488`], StrokeForm[{GrayLevel[0.], Opacity[0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{624.025796433958, 322.15969575643487`}, {625.2735813776728, 222.54096787339074`}}]}, { InsetBox["", {227.29126922539376`, 329.68859319586625`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontWeight->Bold, FontColor->GrayLevel[0.], FontOpacity->1., Background->GrayLevel[1.]], {245.6199527883859, 329.68859319586625`}, { Left, Baseline}, Alignment->{Left, Top}], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["u", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {265.0598831806616, 340.36946292682865`}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], InsetBox["", {313.69792030807093`, 231.49921696555123`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {405.3413381230317, 391.2206023001969}, {Left, Baseline}, Alignment->{Left, Top}], {{ {FaceForm[{RGBColor[0.8630000000000001, 0.894, 0.89], Opacity[1.]}], FilledCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 258.7835476589956, 330.489889636826}, {258.7835476589956, 322.75788963682595`}, {252.51554765899556`, 316.489889636826}, { 244.7835476589956, 316.489889636826}, {237.0515476589955, 316.489889636826}, {230.7835476589956, 322.75788963682595`}, { 230.7835476589956, 330.489889636826}, {230.7835476589956, 338.221889636826}, {237.0515476589955, 344.489889636826}, { 244.7835476589956, 344.489889636826}, {252.51554765899556`, 344.489889636826}, {258.7835476589956, 338.221889636826}, { 258.7835476589956, 330.489889636826}}}]}, {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 256.7483936059712, 330.3039744223591}, {256.7483936059712, 322.90814833540253`}, {250.75291534510166`, 316.91267007453297`}, { 243.35708925814515`, 316.91267007453297`}, {235.96126317118853`, 316.91267007453297`}, {229.96578491031903`, 322.90814833540253`}, { 229.96578491031903`, 330.3039744223591}, {229.96578491031903`, 337.6998005093156}, {235.96126317118853`, 343.69527877018515`}, { 243.35708925814515`, 343.69527877018515`}, {250.75291534510166`, 343.69527877018515`}, {256.7483936059712, 337.6998005093156}, { 256.7483936059712, 330.3039744223591}}}, CurveClosed->{1}]}}, FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, { 0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 219.2162123534641, 349.1683452784252}, {219.2162123534641, 350.5723452784252}, {222.07021235346414`, 350.5723452784252}, { 222.07021235346414`, 353.4043452784252}, {223.4522123534641, 353.4043452784252}, {223.4522123534641, 350.5723452784252}, { 226.30521235346416`, 350.5723452784252}, {226.30521235346416`, 349.1683452784252}, {223.4522123534641, 349.1683452784252}, { 223.4522123534641, 346.3163452784252}, {222.07021235346414`, 346.3163452784252}, {222.07021235346414`, 349.1683452784252}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 220.86741129584303`, 316.4281389488846}, {227.86341129584312`, 316.4281389488846}, {227.86341129584312`, 315.2001389488846}, { 220.86741129584303`, 315.2001389488846}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{242.62672478074813`, 307.3729291397307}, { 238.90027219215943`, 305.39919128859634`}, {238.82570178206748`, 305.5268324024223}, {241.50359552109688`, 311.2532681282877}, { 242.07405462874527`, 313.2700193292297}, {242.64445450375314`, 315.2857722859659}, {243.2159118556072, 317.3024642542674}, { 243.54387253312854`, 315.232403676893}, {243.8718924432904, 313.16334134372437`}, {244.2008513650175, 311.0932215337094}, { 246.18276157840165`, 305.09028784208135`}, {246.11358482286636`, 304.9711762284846}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{221.0462937073857, 330.71878831809005`}, { 218.85529370738567`, 334.32178831809006`}, {218.97829370738572`, 334.40378831809005`}, {224.85329370738572`, 332.06978831809005`}, { 226.90029370738574`, 331.61978831809006`}, {228.94629370738568`, 331.16978831809007`}, {230.9932937073857, 330.71878831809005`}, { 228.94629370738568`, 330.26878831809006`}, {226.90029370738574`, 329.81878831809007`}, {224.85329370738572`, 329.36778831809005`}, { 218.97829370738572`, 327.03378831809005`}, {218.85529370738567`, 327.09578831809006`}}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{182.6915402620839, 330.75278567242066`}, {220.90309827765498`, 330.75278567242066`}}]}, StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["r", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {198.09243915141235`, 340.1955923350955}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], StyleBox[InsetBox[ StyleBox[Cell[BoxData[ FormBox["y", TraditionalForm]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {633.7447955836466, 334.59469875049354`}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{277.1590406462517, 331.06607498898177`}, { 274.97004064625173`, 334.6690749889818}, {275.0910406462517, 334.75107498898177`}, {280.9660406462517, 332.41707498898177`}, { 283.01304064625174`, 331.9670749889818}, {285.06104064625174`, 331.5170749889818}, {287.10804064625177`, 331.06607498898177`}, { 285.06104064625174`, 330.6160749889818}, {283.01304064625174`, 330.1660749889818}, {280.9660406462517, 329.71507498898177`}, { 275.0910406462517, 327.38107498898177`}, {274.97004064625173`, 327.4430749889818}}}]}}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{567.6261553996312, 323.2630861751625}, {645.9129122890446, 323.2630861751623}}]}, {EdgeForm[{AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}]}], FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{320.7789432147263, 311.35966805559997`}, {286.87495889103997`, 311.35966805559997`}, { 286.87495889103997`, 347.0503237623271}, {320.7789432147263, 347.0503237623271}}}]}, InsetBox["\:f350", Scaled[{0.5, 0.5}], Center], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"2.778", RowBox[{"(", RowBox[{ SuperscriptBox["s", "2"], "+", RowBox[{"0.192", "s"}], " ", "+", " ", "1.92"}], ")"}]}], RowBox[{ SuperscriptBox[ RowBox[{"s", "(", RowBox[{"s", "+", "1"}], ")"}], "2"], RowBox[{"(", RowBox[{ SuperscriptBox["s", "2"], "+", RowBox[{"0.384", "s"}], " ", "+", " ", "2.56"}], ")"}]}]], TraditionalForm]], FormatType->"TraditionalForm", FontSize->16]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->12, Background->GrayLevel[1.]], {368.2331380255942, 322.42481610054836`}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ StyleBox["k", FontSize->14], TraditionalForm]], FormatType->"TraditionalForm"]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->12, Background->GrayLevel[1.]], {303.3224090493602, 326.4193224990858}, { Left, Baseline}, Alignment->{Left, Top}], {AbsoluteThickness[1], StrokeForm[{GrayLevel[0.], Opacity[0.64]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{321.3859938645293, 329.22241382925415`}, {341.17276544682255`, 329.2224138292544}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{330.58556372669045`, 329.5681350895302}, { 328.3965637266905, 333.1711350895302}, {328.51756372669047`, 333.2531350895302}, {334.39256372669047`, 330.9191350895302}, { 336.4395637266905, 330.4691350895302}, {338.4875637266905, 330.0191350895302}, {340.5345637266905, 329.5681350895302}, { 338.4875637266905, 329.1181350895302}, {336.4395637266905, 328.6681350895302}, {334.39256372669047`, 328.2171350895302}, { 328.51756372669047`, 325.8831350895302}, {328.3965637266905, 325.9451350895302}}}]}}, ImagePadding->{{0., 1.}, {1., 3.1072}}, ImageSize->{548., Automatic}, PlotRange->{{165.012625, 711.261375}, {187.584524, 401.708578}}, PlotRangePadding->Automatic]], "Input", CellChangeTimes->{{3.528065171408169*^9, 3.5280652380043573`*^9}, { 3.528065287592812*^9, 3.528065354929263*^9}, {3.5280653899309177`*^9, 3.5280656805588818`*^9}, {3.5280660075472507`*^9, 3.5280660085839243`*^9}}], Cell["\<\ Conditionally stable system. Gain reduction moves dominant poles into LHP, \ improving stability margins.\ \>", "Text", CellChangeTimes->{{3.5280658382907333`*^9, 3.528065945165195*^9}, { 3.528066039317*^9, 3.528066054622809*^9}}], Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"RootLocusPlot", "[", RowBox[{ RowBox[{"k", FractionBox[ RowBox[{"2.7778", RowBox[{"(", RowBox[{ RowBox[{"s", "^", "2"}], "+", RowBox[{"0.192", "s"}], "+", "1.92"}], ")"}]}], RowBox[{"s", RowBox[{ RowBox[{"(", RowBox[{"s", "+", "1"}], ")"}], "^", "2"}], RowBox[{"(", RowBox[{ RowBox[{"s", "^", "2"}], "+", RowBox[{"0.384", "s"}], "+", "2.56"}], ")"}]}]]}], ",", RowBox[{"{", RowBox[{"k", ",", "0", ",", "5"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1.2"}], ",", "0.2"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1.5"}], ",", "1.5"}], "}"}]}], "}"}]}], ",", RowBox[{"AspectRatio", "\[Rule]", "1"}], ",", RowBox[{"PlotLabel", "\[Rule]", " ", "kValue"}], ",", RowBox[{"PoleZeroMarkers", "\[Rule]", RowBox[{"{", RowBox[{"Automatic", ",", RowBox[{"\"\\"", "\[Rule]", "kValue"}]}], "}"}]}]}], "]"}], ",", RowBox[{"{", RowBox[{"kValue", ",", "0", ",", "5", ",", "0.001"}], "}"}]}], "]"}]], "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Classical Analysis", "Section", CellChangeTimes->{3.528066073121125*^9}], Cell[CellGroupData[{ Cell["Closed-loop Simulation", "Subsubsection", CellChangeTimes->{{3.528117071927143*^9, 3.528117090476362*^9}, { 3.528206334921595*^9, 3.528206335549573*^9}}], Cell["Performance with gain = 0.27 :", "Text", CellChangeTimes->{{3.528066088469788*^9, 3.528066127253386*^9}, 3.528074548770336*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"tfnew", "=", RowBox[{"TransferFunctionModel", "[", RowBox[{ RowBox[{"0.27", FractionBox[ RowBox[{"2.7778", RowBox[{"(", RowBox[{ RowBox[{"s", "^", "2"}], "+", RowBox[{"0.192", "s"}], "+", "1.92"}], ")"}]}], RowBox[{"s", RowBox[{ RowBox[{"(", RowBox[{"s", "+", "1"}], ")"}], "^", "2"}], RowBox[{"(", RowBox[{ RowBox[{"s", "^", "2"}], "+", RowBox[{"0.384", "s"}], "+", "2.56"}], ")"}]}]]}], ",", "s"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"OutputResponse", "[", RowBox[{ RowBox[{"SystemsModelFeedbackConnect", "[", "tfnew", "]"}], ",", "1", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "500"}], "}"}]}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{"%", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "60"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", "All"}], ",", RowBox[{"Frame", "\[Rule]", " ", "True"}]}], "]"}]}], "Input", CellChangeTimes->{{3.528066366357209*^9, 3.528066396676746*^9}, { 3.528066432248612*^9, 3.5280664327943077`*^9}, {3.528066473911262*^9, 3.528066538015565*^9}, {3.528066657897266*^9, 3.528066663864408*^9}, { 3.5280702382844877`*^9, 3.528070241763608*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["New Stability Margins", "Subsubsection", CellChangeTimes->{{3.52811711254084*^9, 3.528117148715715*^9}}], Cell["\<\ The gain and phase margins have also improved to about 4 and 40 degrees \ respectively:\ \>", "Text", CellChangeTimes->{{3.528066558828498*^9, 3.528066573483316*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"gmnew", ",", "pmnew"}], "}"}], "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"GainMargins", "[", "tfnew", "]"}], "[", RowBox[{"[", RowBox[{"1", ",", "2"}], "]"}], "]"}], ",", RowBox[{ RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"#", "[", RowBox[{"[", "1", "]"}], "]"}], ",", FractionBox[ RowBox[{"#", "[", RowBox[{"[", "2", "]"}], "]"}], "Degree"]}], "}"}], "&"}], ",", RowBox[{"PhaseMargins", "[", "tfnew", "]"}]}], "]"}], "[", RowBox[{"[", RowBox[{"1", ",", "2"}], "]"}], "]"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.528066587727481*^9, 3.52806663127109*^9}, { 3.52806666973394*^9, 3.5280666835948687`*^9}}] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Classical Analysis", "Section", CellChangeTimes->{3.528066987324098*^9}], Cell[CellGroupData[{ Cell["Sensitivity", "Subsubsection", CellChangeTimes->{{3.528117963480217*^9, 3.528117975547476*^9}}], Cell["\<\ Alternately, search for a gain which gives a maximum sensitivity of about 2, \ signifying robust stability against plant parameter variations.\ \>", "Text", CellChangeTimes->{{3.5280670086825523`*^9, 3.5280670178645697`*^9}, { 3.5281459139338503`*^9, 3.528145916602726*^9}}], Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Show", "[", RowBox[{"{", RowBox[{ RowBox[{"NyquistPlot", "[", RowBox[{ RowBox[{"k", FractionBox[ RowBox[{"2.7778", RowBox[{"(", RowBox[{ RowBox[{"s", "^", "2"}], "+", RowBox[{"0.192", "s"}], "+", "1.92"}], ")"}]}], RowBox[{"s", RowBox[{ RowBox[{"(", RowBox[{"s", "+", "1"}], ")"}], "^", "2"}], RowBox[{"(", RowBox[{ RowBox[{"s", "^", "2"}], "+", RowBox[{"0.384", "s"}], "+", "2.56"}], ")"}]}]]}], ",", RowBox[{"{", RowBox[{"0.01", ",", "100"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "0.2"}], "}"}]}], "}"}]}], ",", RowBox[{"PlotLabel", "\[Rule]", " ", RowBox[{"StringJoin", "[", RowBox[{"\"\\"", ",", RowBox[{"ToString", "[", "k", "]"}], ",", "\"\<, |S(j\[Omega])\!\(\*SubscriptBox[\(|\), \(max\)]\)= \>\"", ",", RowBox[{"ToString", "[", "2", "]"}]}], "]"}]}]}], "]"}], ",", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"Red", ",", RowBox[{"Circle", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0"}], "}"}], ",", "0.5"}], "]"}]}], "}"}], "]"}]}], "}"}], "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "0.001", ",", "1"}], "}"}]}], "]"}]], "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Design of a 2-DOF DC Servo Motor Controller\ \>", "Section", CellChangeTimes->{{3.5280679989897757`*^9, 3.5280680102554207`*^9}, { 3.528068459438525*^9, 3.528068462696151*^9}, {3.528068555374709*^9, 3.5280685961139517`*^9}, {3.5280703471727953`*^9, 3.528070387893551*^9}, { 3.5281462219282427`*^9, 3.528146235790579*^9}}], Cell[BoxData[ RowBox[{"\t\t\t", GraphicsBox[{ {EdgeForm[{AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}]}], FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 476.2234973439403, 194.98562665931468`}, {396.71054286959736`, 194.98562665931468`}, {396.71054286959736`, 239.02941455272267`}, { 476.2234973439403, 239.02941455272267`}}}]}, {EdgeForm[{AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}]}], FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 521.5383208564447, 290.5820347899397}, {359.53557223285736`, 290.5820347899397}, {359.53557223285736`, 361.96334620339405`}, { 521.5383208564447, 361.96334620339405`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{194.6226, 329.}, {204.7006, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{203.6802, 329.}, {201.49020000000002`, 332.603}, {201.6122, 332.685}, {207.4872, 330.351}, {209.5342, 329.901}, {211.58120000000002`, 329.451}, {213.62820000000002`, 329.}, { 211.58120000000002`, 328.55}, {209.5342, 328.1}, {207.4872, 327.649}, { 201.6122, 325.315}, {201.49020000000002`, 325.377}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{274.92769999999996`, 329.}, { 294.96369999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{293.9438, 329.}, {291.7538, 332.603}, { 291.8758, 332.685}, {297.7508, 330.351}, {299.7978, 329.901}, {301.8448, 329.451}, {303.8918, 329.}, {301.8448, 328.55}, {299.7978, 328.1}, { 297.7508, 327.649}, {291.8758, 325.315}, {291.7538, 325.377}}}]}, { {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{317.63669999999996`, 353.749}, { 321.23969999999997`, 355.939}, {321.32169999999996`, 355.817}, { 318.98769999999996`, 349.942}, {318.5377, 347.89500000000004`}, { 318.0877, 345.848}, {317.63669999999996`, 343.80100000000004`}, { 317.1867, 345.848}, {316.7367, 347.89500000000004`}, { 316.28569999999996`, 349.942}, {313.95169999999996`, 355.817}, { 314.0137, 355.939}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{331.2451, 329.}, {345.6441, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{372.0752687617261, 327.7522150562854}, { 369.8852687617261, 331.35521505628543`}, {370.0072687617261, 331.4372150562854}, {375.8822687617261, 329.1032150562854}, { 377.92926876172606`, 328.65321505628543`}, {379.9762687617261, 328.20321505628544`}, {382.02326876172606`, 327.7522150562854}, { 379.9762687617261, 327.30221505628543`}, {377.92926876172606`, 326.85221505628544`}, {375.8822687617261, 326.4012150562854}, { 370.0072687617261, 324.0672150562854}, {369.8852687617261, 324.12921505628543`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{482.8262, 329.}, {500.03119999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{499.01169999999996`, 329.}, { 496.82269999999994`, 332.603}, {496.9437, 332.685}, {502.8187, 330.351}, {504.86569999999995`, 329.901}, {506.91369999999995`, 329.451}, {508.9607, 329.}, {506.91369999999995`, 328.55}, { 504.86569999999995`, 328.1}, {502.8187, 327.649}, {496.9437, 325.315}, {496.82269999999994`, 325.377}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{561.1780908947657, 331.4347908989821}, { 558.9890908947657, 335.0377908989821}, {559.1100908947657, 335.1197908989821}, {564.9850908947657, 332.7857908989821}, { 567.0320908947657, 332.3357908989821}, {569.0800908947657, 331.8857908989821}, {571.1270908947657, 331.4347908989821}, { 569.0800908947657, 330.9847908989821}, {567.0320908947657, 330.5347908989821}, {564.9850908947657, 330.0837908989821}, { 559.1100908947657, 327.7497908989821}, {558.9890908947657, 327.8117908989821}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{648.1934000000001, 329.}, { 700.3364000000001, 329.}}}]}}, { {Thickness[0.0012626262626262627`], {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{664.623, 329.}, {664.623, 207.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{664.623, 208.01949999999997`}, { 668.2270000000001, 210.21049999999997`}, {668.307, 210.08749999999998`}, {665.975, 204.21249999999998`}, {665.523, 202.16549999999995`}, {665.0740000000001, 200.11949999999996`}, { 664.623, 198.07249999999996`}, {664.172, 200.11949999999996`}, { 663.7230000000001, 202.16549999999995`}, {663.2710000000001, 204.21249999999998`}, {660.937, 210.08749999999998`}, {661., 210.21049999999997`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{179.6226, 303.1777}, {176.0196, 300.98670000000004`}, {175.9376, 301.10970000000003`}, {178.2716, 306.98470000000003`}, {178.7216, 309.0317}, {179.1716, 311.0787}, { 179.6226, 313.1257}, {180.0726, 311.0787}, {180.5226, 309.0317}, { 180.9736, 306.98470000000003`}, {183.3076, 301.10970000000003`}, { 183.2456, 300.98670000000004`}}}]}}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{0, 2, 0}, {0, 1, 0}}}, {{{397.12387098913996`, 217.0052483104812}, {314.3427393261901, 217.0052483104812}, { 314.3427393261901, 317.07172140479906`}}}, CurveClosed->{0}]}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{523.7188389458395, 330.76629515507955`}, {564.9684992280038, 332.0140800987946}}]}, {AbsoluteThickness[1], StrokeForm[{GrayLevel[0.], Opacity[0.64]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{316.3147046983335, 330.78655481789576`}, {347.4053462125612, 330.786554817896}}]}}, {AbsoluteThickness[1.5667510701081488`], StrokeForm[{GrayLevel[0.], Opacity[0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{532.8769950916001, 330.93149967466684`}, {534.1247800353149, 218.38630614516103`}}]}, { InsetBox["", {227.29126922539376`, 329.68859319586625`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontWeight->Bold, FontColor->GrayLevel[0.], FontOpacity->1., Background->GrayLevel[1.]], {245.6199527883859, 329.68859319586625`}, { Left, Baseline}, Alignment->{Left, Top}], StyleBox[InsetBox[ StyleBox[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"x", "(", RowBox[{"k", "+", "1"}], ")"}], "=", RowBox[{ RowBox[{"a", " ", RowBox[{"x", "(", "k", ")"}]}], "+", RowBox[{"b", " ", RowBox[{"u", "(", "k", ")"}]}]}]}], TraditionalForm]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {363.0159260870473, 327.97106950193603`}, {Left, Baseline}, {157.43472607382995`, 38.37742006823411}, {{1., 0.}, {0., 1.}}, Alignment->{Left, Top}], FontSize->12], StyleBox[InsetBox[ StyleBox[Cell["\[Kappa]", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {432.17434350199306`, 215.82821214459966`}, { Left, Baseline}, {34.78067012754152, 29.791132184491065`}, {{1., 0.}, { 0., 1.}}, Alignment->{Left, Top}], FontSize->13], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["u", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {336.69170578814266`, 341.4713183435042}, {Left, Baseline}, Alignment->{Left, Top}], FontSize->13], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["y", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {655.7236151982551, 341.7163175500202}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], InsetBox["", {313.69792030807093`, 231.49921696555123`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {405.3413381230317, 391.2206023001969}, {Left, Baseline}, Alignment->{Left, Top}], {AbsoluteThickness[1.023292992280754], StrokeForm[{GrayLevel[0.], Opacity[0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{533.404088212509, 216.40466685739796`}, {487.8931453277352, 216.40466685739796`}}]}, {Thickness[0.0012626262626262627`], EdgeForm[{AbsoluteThickness[ 1.023292992280754], JoinForm[{"Miter", 4.}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 630.3792868152244, 306.8516532817028}, {568.3735766420643, 306.8516532817028}, {568.3735766420643, 359.5584599920413}, { 630.3792868152244, 359.5584599920413}}}]}, {{ {FaceForm[{RGBColor[0.8630000000000001, 0.894, 0.89], Opacity[1.]}], FilledCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 326.6901564341327, 329.49126303719174`}, {326.6901564341327, 321.7592630371917}, {320.42215643413266`, 315.49126303719174`}, { 312.6901564341327, 315.49126303719174`}, {304.9581564341326, 315.49126303719174`}, {298.6901564341327, 321.7592630371917}, { 298.6901564341327, 329.49126303719174`}, {298.6901564341327, 337.22326303719177`}, {304.9581564341326, 343.49126303719174`}, { 312.6901564341327, 343.49126303719174`}, {320.42215643413266`, 343.49126303719174`}, {326.6901564341327, 337.22326303719177`}, { 326.6901564341327, 329.49126303719174`}}}]}, {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 326.6522555803771, 331.3026010219939}, {326.6522555803771, 323.90677493503733`}, {320.6567773195075, 317.9112966741678}, { 313.260951232551, 317.9112966741678}, {305.8651251455944, 317.9112966741678}, {299.8696468847249, 323.90677493503733`}, { 299.8696468847249, 331.3026010219939}, {299.8696468847249, 338.6984271089504}, {305.8651251455944, 344.69390536981996`}, { 313.260951232551, 344.69390536981996`}, {320.6567773195075, 344.69390536981996`}, {326.6522555803771, 338.6984271089504}, { 326.6522555803771, 331.3026010219939}}}, CurveClosed->{1}]}}, FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{294.1132073260418, 345.17383887988825`}, {294.1132073260418, 346.57783887988825`}, {296.96720732604183`, 346.57783887988825`}, { 296.96720732604183`, 349.40983887988824`}, {298.3492073260418, 349.40983887988824`}, {298.3492073260418, 346.57783887988825`}, { 301.20220732604184`, 346.57783887988825`}, {301.20220732604184`, 345.17383887988825`}, {298.3492073260418, 345.17383887988825`}, { 298.3492073260418, 342.3218388798883}, {296.96720732604183`, 342.3218388798883}, {296.96720732604183`, 345.17383887988825`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 293.76715306915196`, 319.42401874778835`}, {300.76315306915205`, 319.42401874778835`}, {300.76315306915205`, 318.19601874778834`}, { 293.76715306915196`, 318.19601874778834`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{314.52783995442275`, 307.3729291397311}, {310.80138736583405`, 305.3991912885968}, { 310.7268169557421, 305.52683240242277`}, {313.4047106947715, 311.25326812828814`}, {313.9751698024199, 313.27001932923014`}, { 314.54556967742775`, 315.28577228596635`}, {315.1170270292818, 317.3024642542679}, {315.44498770680315`, 315.23240367689345`}, { 315.773007616965, 313.1633413437248}, {316.1019665386921, 311.0932215337099}, {318.08387675207626`, 305.0902878420818}, { 318.01469999654097`, 304.97117622848504`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{289.95152908215715`, 330.7187883180905}, {287.7605290821571, 334.3217883180905}, { 287.88352908215717`, 334.4037883180905}, {293.75852908215717`, 332.0697883180905}, {295.8055290821572, 331.6197883180905}, { 297.8515290821571, 331.1697883180905}, {299.89852908215715`, 330.7187883180905}, {297.8515290821571, 330.2687883180905}, { 295.8055290821572, 329.8187883180905}, {293.75852908215717`, 329.3677883180905}, {287.88352908215717`, 327.0337883180905}, { 287.7605290821571, 327.0957883180905}}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{272.46470541213625`, 330.75278567242134`}, {289.8083336524264, 330.75278567242134`}}]}, StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["c", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {595.0107993639633, 328.29773511062376`}, { Left, Baseline}, {23.565450295275596`, 32.72979207677166}, {{1., 0.}, { 0., 1.}}, Alignment->{Left, Top}], FontSize->14], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["r", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {189.97551238539552`, 336.9708138549281}, {Left, Baseline}, Alignment->{Left, Top}], FontSize->13], StyleBox[InsetBox[ StyleBox[Cell[BoxData[ FormBox["x", TraditionalForm]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {532.1721278248807, 345.8084900277363}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{486.75484614911244`, 215.79454052720783`}, {488.91390027341464`, 212.1723070603625}, { 488.7901799817784, 212.09139791726076`}, {482.93604851803735`, 214.4772574699665}, {480.89310776098085`, 214.94534086334556`}, { 478.851166964827, 215.41341541403426`}, {476.80823505046084`, 215.88249876831603`}, {478.8591342288502, 216.3143801873212}, { 480.90903344633676`, 216.74627044901678`}, {482.9599414674165, 217.17915182892466`}, {488.8553506098037, 219.46110976978545`}, { 488.9777975540267, 219.39802454290572`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{350.6444937510316, 330.8618614596187}, { 348.4554937510316, 334.4648614596187}, {348.5764937510316, 334.5468614596187}, {354.4514937510316, 332.2128614596187}, { 356.4984937510316, 331.7628614596187}, {358.5464937510316, 331.3128614596187}, {360.59349375103164`, 330.8618614596187}, { 358.5464937510316, 330.4118614596187}, {356.4984937510316, 329.9618614596187}, {354.4514937510316, 329.5108614596187}, { 348.5764937510316, 327.1768614596187}, {348.4554937510316, 327.2388614596187}}}]}}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{631.9097288954656, 332.07946173464575`}, {672.3849900484279, 333.3272466783608}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{674.3744774083532, 333.3831304436646}, { 672.1854774083533, 336.9861304436646}, {672.3064774083532, 337.0681304436646}, {678.1814774083532, 334.7341304436646}, { 680.2284774083532, 334.2841304436646}, {682.2764774083532, 333.8341304436646}, {684.3234774083533, 333.3831304436646}, { 682.2764774083532, 332.9331304436646}, {680.2284774083532, 332.4831304436646}, {678.1814774083532, 332.0321304436646}, { 672.3064774083532, 329.6981304436646}, {672.1854774083533, 329.7601304436646}}}]}, {EdgeForm[{Opacity[1.], AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 11.700000000000001`}]}], FaceForm[{RGBColor[ 1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 276.02674212864736`, 305.4658228768797}, {220.97931210794843`, 305.4658228768797}, {220.97931210794843`, 354.4033649806663}, { 276.02674212864736`, 354.4033649806663}}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{179.7865467042711, 330.1390230319719}, {209.40542775355004`, 330.1390230319719}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{210.36992122857788`, 329.7005546009679}, { 208.18092122857803`, 333.3035546009679}, {208.3019212285779, 333.3855546009679}, {214.1769212285779, 331.0515546009679}, { 216.22392122857792`, 330.6015546009679}, {218.27192122857792`, 330.1515546009679}, {220.31892122857795`, 329.7005546009679}, { 218.27192122857792`, 329.2505546009679}, {216.22392122857792`, 328.8005546009679}, {214.1769212285779, 328.3495546009679}, { 208.3019212285779, 326.0155546009679}, {208.18092122857803`, 326.0775546009679}}}]}, InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["f", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {246.64305617977533`, 323.8923133730337}, { Left, Baseline}, Alignment->{Left, Top}]}, ImagePadding->{{0., 1.}, {1., 3.1072}}, ImageSize->{446., Automatic}, PlotRange->{{165.012625, 711.261375}, {187.584524, 401.708578}}, PlotRangePadding->Automatic]}]], "Input", CellChangeTimes->{{3.527948988559887*^9, 3.527948997243039*^9}, { 3.527949033931761*^9, 3.527949047967971*^9}, {3.5279491487368107`*^9, 3.527949153515368*^9}, {3.527949215698564*^9, 3.527949564579357*^9}, { 3.527949601771641*^9, 3.527949657136855*^9}, {3.527953788211113*^9, 3.527953792849533*^9}, {3.528068603210266*^9, 3.528068605607337*^9}, 3.528069140992613*^9, 3.5280696174529467`*^9, {3.5282061823349447`*^9, 3.528206183861218*^9}}], Cell[CellGroupData[{ Cell["Objective", "Subsubsection", CellChangeTimes->{{3.5281247346507998`*^9, 3.528124755327036*^9}}], Cell[TextData[{ "Select state-feedback gain ", Cell[BoxData[ FormBox["\[Kappa]", TraditionalForm]]], " and reference gain ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " for the discretized model of a dc motor speed control system (\[Tau] = \ 0.02 s) to obtain:\ni) zero steady-state error due to a unit step\nii) \ damping ratio ~ 0.7\niii) settling time ~ 1 s" }], "Text", CellChangeTimes->{{3.528068664615162*^9, 3.528068722880962*^9}, { 3.528068764367908*^9, 3.52806890537744*^9}, {3.528069248675118*^9, 3.528069268808476*^9}, {3.52812474452809*^9, 3.528124747727936*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Model", "Subsubsection", CellChangeTimes->{{3.5281248281157293`*^9, 3.528124836190053*^9}}], Cell["Motor model and its discretized state-space representation", "Text", CellChangeTimes->{{3.528124841168666*^9, 3.5281248985180187`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"tfmotor", " ", "=", " ", RowBox[{"TransferFunctionModel", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"s", "+", "1"}], ")"}], RowBox[{"(", RowBox[{"s", "+", "10"}], ")"}]}], ")"}]}], ",", "s"}], "]"}]}], ";"}]], "Input"], Cell[BoxData[ RowBox[{"ssmotor", "=", RowBox[{"StateSpaceModel", "@", RowBox[{"ToDiscreteTimeModel", "[", RowBox[{"tfmotor", ",", "0.02", ",", "z", ",", RowBox[{"Method", "\[Rule]", " ", "\"\\""}]}], "]"}]}]}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Compute Pole Locations", "Subsubsection", CellChangeTimes->{{3.528124942714229*^9, 3.52812494841208*^9}, { 3.528124979923476*^9, 3.5281249991526127`*^9}}], Cell["\<\ Translate performance specifications to desired pole locations in z-domain. \ With settling time = 0.9 s,\ \>", "Text", CellChangeTimes->{{3.528069103688066*^9, 3.528069120716528*^9}, { 3.5280692055080643`*^9, 3.528069233190469*^9}, {3.52806927995954*^9, 3.528069280299159*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"specsToPoles", "[", RowBox[{"\[Zeta]_", ",", "ts_"}], "]"}], ":=", RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"wn", "=", RowBox[{"4.6", "/", RowBox[{"(", RowBox[{"\[Zeta]", " ", "ts"}], ")"}]}]}], "}"}], ",", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"E", "^", RowBox[{"(", RowBox[{"0.02", "#"}], ")"}]}], "&"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "\[Zeta]"}], " ", "wn"}], " ", "+", " ", RowBox[{"I", " ", "wn", " ", RowBox[{"Sqrt", "[", RowBox[{"1", "-", RowBox[{"\[Zeta]", "^", "2"}]}], "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{"-", "\[Zeta]"}], " ", "wn"}], " ", "-", " ", RowBox[{"I", " ", "wn", " ", RowBox[{"Sqrt", "[", RowBox[{"1", "-", RowBox[{"\[Zeta]", "^", "2"}]}], "]"}]}]}]}], "}"}]}], "]"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"z1", ",", "z2"}], "}"}], "=", RowBox[{"specsToPoles", "[", RowBox[{"0.7", ",", "0.9"}], "]"}]}]}], "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Design of a 2-DOF DC Servo Motor Controller\ \>", "Section", CellChangeTimes->{3.5280693288228273`*^9, 3.528070396385273*^9, 3.528146246283596*^9}], Cell[CellGroupData[{ Cell["Verify Controllability", "Subsubsection", CellChangeTimes->{{3.528125281920528*^9, 3.528125314485552*^9}, { 3.5281263767566833`*^9, 3.5281263788388166`*^9}}], Cell["Pole placement possible since the system is controllable :", "Text", CellChangeTimes->{{3.528069375463283*^9, 3.528069398444154*^9}}], Cell[BoxData[ RowBox[{"ControllableModelQ", "[", "ssmotor", "]"}]], "Input", CellChangeTimes->{{3.5280694009709187`*^9, 3.528069420250041*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["State-feedback Controller", "Subsubsection", CellChangeTimes->{{3.5281253481098022`*^9, 3.528125393320953*^9}}], Cell["State-feedback gain matrix for pole placement :", "Text", CellChangeTimes->{{3.528069431118875*^9, 3.5280694542492113`*^9}}], Cell[BoxData[ RowBox[{"\[Kappa]", "=", " ", RowBox[{"StateFeedbackGains", "[", RowBox[{"ssmotor", ",", RowBox[{"{", RowBox[{"z1", ",", "z2"}], "}"}]}], "]"}]}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Feedforward Gain", "Subsubsection", CellChangeTimes->{{3.528126330592148*^9, 3.528126347008772*^9}}], Cell[TextData[{ "Compute the reference gain matrix to ensure perfect tracking: ", Cell[BoxData[ FormBox[ RowBox[{"f", "=", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"c", ".", SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["I", "n"], "-", RowBox[{"(", RowBox[{"a", "-", RowBox[{"b", ".", " ", "\[Kappa]"}]}], ")"}]}], ")"}], RowBox[{"-", "1"}]]}], "b"}], ")"}], RowBox[{"-", "1"}]]}], TraditionalForm]]], " " }], "Text", CellChangeTimes->{{3.528069519691341*^9, 3.528069589365501*^9}, { 3.52807044440562*^9, 3.528070446095487*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"sskm", "=", RowBox[{"SystemsModelStateFeedbackConnect", "[", RowBox[{"ssmotor", ",", "\[Kappa]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"acl", ",", "bcl", ",", "ccl"}], "}"}], "=", " ", RowBox[{ RowBox[{"Normal", "[", "sskm", "]"}], "[", RowBox[{"[", RowBox[{"1", ";;", "3"}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"f", "=", RowBox[{"Inverse", "[", RowBox[{"ccl", ".", RowBox[{"Inverse", "[", RowBox[{"(", RowBox[{ RowBox[{"IdentityMatrix", "[", "2", "]"}], "-", "acl"}], ")"}], "]"}], ".", "bcl"}], "]"}]}]}], "Input", CellChangeTimes->{{3.52806972532067*^9, 3.528069813875868*^9}, { 3.5280699196299*^9, 3.52806994267058*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Construct 2 DOF Control System", "Subsubsection", CellChangeTimes->{{3.528069846944724*^9, 3.5280698660827312`*^9}, { 3.528126447375163*^9, 3.528126460362618*^9}}], Cell[BoxData[ RowBox[{"sscl", " ", "=", " ", RowBox[{"SystemsModelSeriesConnect", "[", RowBox[{"sskm", ",", RowBox[{"StateSpaceModel", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"{", "0", "}"}], "}"}], ",", RowBox[{"{", RowBox[{"{", "0", "}"}], "}"}], ",", RowBox[{"{", RowBox[{"{", "0", "}"}], "}"}], ",", "f"}], "}"}], ",", RowBox[{"SamplingPeriod", "\[Rule]", " ", "0.02"}]}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.5280699656595488`*^9, 3.5280699664675407`*^9}}] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Design of a 2-DOF DC Motor Servo Controller\ \>", "Section", CellChangeTimes->{3.528070010415289*^9, 3.528070407584447*^9}], Cell[CellGroupData[{ Cell["Closed-loop response", "Subsubsection", CellChangeTimes->{{3.528070079978671*^9, 3.5280700920676394`*^9}, { 3.5281265339481583`*^9, 3.528126538596943*^9}, {3.528202539722197*^9, 3.528202540174038*^9}}], Cell["Simulate the closed-loop behavior. Tracking is achieved.", "Text", CellChangeTimes->{{3.5281265521358213`*^9, 3.5281266002096786`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"OutputResponse", "[", RowBox[{"sscl", ",", RowBox[{"Table", "[", RowBox[{"1", ",", RowBox[{"{", RowBox[{"k", ",", "0", ",", "120"}], "}"}]}], "]"}]}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{"ListPlot", "[", RowBox[{ RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Thread", "[", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Range", "[", RowBox[{"Length", "[", "#", "]"}], "]"}], "-", "1"}], ")"}], "0.02"}], ",", "#"}], "}"}], "]"}], "&"}], ",", "%"}], "]"}], ",", RowBox[{"Joined", "\[Rule]", "True"}], ",", RowBox[{"InterpolationOrder", "\[Rule]", "0"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", "All"}], ",", RowBox[{"Frame", "\[Rule]", " ", "True"}]}], "]"}]}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Other Design Goals", "Subsubsection", CellChangeTimes->{{3.5281266241073008`*^9, 3.5281266420272427`*^9}}], Cell["Verify overall damping and settling time:", "Text", CellChangeTimes->{{3.528070294249928*^9, 3.528070303274825*^9}, 3.5281266567320538`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"\[Zeta]", ",", "ts"}], "}"}], "=", RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"p2", "=", RowBox[{ RowBox[{"Most", "[", RowBox[{"Eigenvalues", "[", RowBox[{ RowBox[{"Normal", "[", RowBox[{"ToContinuousTimeModel", "@", "sscl"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}]}], "}"}], ",", RowBox[{ RowBox[{"{", RowBox[{"\[Zeta]1", ",", " ", RowBox[{"4.6", "/", RowBox[{"(", RowBox[{"\[Zeta]1", " ", "wn1"}], ")"}]}]}], "}"}], "/.", RowBox[{"Flatten", "@", RowBox[{"Solve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "\[Zeta]1"}], " ", "wn1"}], "\[Equal]", RowBox[{"Re", "[", "p2", "]"}]}], ",", RowBox[{ RowBox[{"wn1", " ", "*", " ", RowBox[{"Sqrt", "[", RowBox[{"1", "-", RowBox[{"\[Zeta]1", "^", "2"}]}], "]"}]}], "\[Equal]", " ", RowBox[{"Im", "[", "p2", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"\[Zeta]1", ",", " ", "wn1"}], "}"}]}], "]"}]}]}]}], "]"}]}]], "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ LQG/LTR Design: Optimal Control of a Remotely Piloted Aircraft \ \>", "Section", CellChangeTimes->{{3.528070955713223*^9, 3.528070962387936*^9}}], Cell[CellGroupData[{ Cell["Model", "Subsubsection", CellChangeTimes->{{3.5281267512780933`*^9, 3.528126762149353*^9}}], Cell["\<\ State-space representation of the longitudinal dynamics, with 2 feedback \ inputs, 2 stochastic inputs and 2 sensor outputs.\ \>", "Text", CellChangeTimes->{{3.528071289128763*^9, 3.5280713694135427`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"a", "=", RowBox[{"(", GridBox[{ { RowBox[{"-", "0.025"}], RowBox[{"-", "40"}], RowBox[{"-", "20"}], RowBox[{"-", "32"}]}, {"0.0001", RowBox[{"-", "2"}], "1", RowBox[{"-", "0.0007"}]}, {"0.01", "12", RowBox[{"-", "2.5"}], "0.0009"}, {"0", "0", "1", "0"} }], ")"}]}], ";", RowBox[{"b", "=", RowBox[{"(", GridBox[{ {"3.25", RowBox[{"-", "0.8"}], "3.25", RowBox[{"-", "0.8"}]}, { RowBox[{"-", "0.2"}], RowBox[{"-", "0.005"}], RowBox[{"-", "0.2"}], RowBox[{"-", "0.005"}]}, { RowBox[{"-", "32"}], "22", RowBox[{"-", "32"}], "22"}, {"0", "0", "0", "0"} }], ")"}]}], ";", RowBox[{"c", "=", RowBox[{"(", GridBox[{ {"0", "1", "0", "0"}, {"0", "0", "0", "1"} }], ")"}]}], ";", RowBox[{"ss", "=", RowBox[{"StateSpaceModel", "[", RowBox[{"{", RowBox[{"a", ",", "b", ",", "c"}], "}"}], "]"}]}], ";"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Objectives", "Subsubsection", CellChangeTimes->{{3.528126813164752*^9, 3.5281268235639267`*^9}}], Cell[TextData[{ "i) Regulate a perturbed flight condition {0,0.1,0,0} to zero within 10 s \ with ", Cell[BoxData[ FormBox[ RowBox[{"|", SubscriptBox["u", "i"], "|", RowBox[{"\[LessEqual]", " ", SuperscriptBox["5", "\[Degree]"]}]}], TraditionalForm]]], "\nii) Attempt to recover LQR robustness, lost due to Kalman filtering, by \ selecting noise covariance matrices" }], "Text", CellChangeTimes->{{3.5280714662778883`*^9, 3.528071710925425*^9}, { 3.528126796491061*^9, 3.528126810302527*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Procedure", "Subsubsection", CellChangeTimes->{{3.528154726993909*^9, 3.528154741678138*^9}, { 3.5281549964908953`*^9, 3.528154999569951*^9}}], Cell["\<\ i) Design pure LQR by selecting weighting matrices {q,r} to achieve desired \ performance. ii) Select noise covariance matrices {w,v} = {\[Rho] v, v} to retrieve LQR \ robustness\ \>", "Text", CellChangeTimes->{{3.528154748961483*^9, 3.5281547599420633`*^9}, { 3.5281549472662582`*^9, 3.528155092616284*^9}, {3.528155126711261*^9, 3.528155158096861*^9}, {3.528155192707241*^9, 3.5281552591193542`*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["System Properties", "Subsubsection", CellChangeTimes->{{3.528126842790489*^9, 3.528126861473302*^9}}], Cell["System is bounded-input-bounded-output unstable,", "Text", CellChangeTimes->{{3.5281268751750307`*^9, 3.528126877668387*^9}, { 3.528127043227104*^9, 3.528127116571508*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Eigenvalues", "[", "a", "]"}], "//", "TableForm"}]], "Input"], Cell["but controllable and observable", "Text", CellChangeTimes->{{3.528127119263236*^9, 3.5281271359595013`*^9}}], Cell[BoxData[ RowBox[{"Through", "@", RowBox[{ RowBox[{"{", RowBox[{"ControllableModelQ", ",", "ObservableModelQ"}], "}"}], "@", "ss"}]}]], "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ LQG/LTR Design: Optimal Control of a Remotely Piloted Aircraft \ \>", "Section", CellChangeTimes->{3.528071822094759*^9}], Cell[CellGroupData[{ Cell["Regulator Design for Deterministic System", "Subsection", CellChangeTimes->{{3.528071869355756*^9, 3.5280718830816298`*^9}, { 3.5281467313336*^9, 3.5281467654540377`*^9}}], Cell[BoxData[ RowBox[{"\t\t\t\t\t\t\t\t", GraphicsBox[{ {EdgeForm[{AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}]}], FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 542.4062506317256, 291.80956007083853`}, {380.4035020081383, 291.80956007083853`}, {380.4035020081383, 363.19087148429287`}, { 542.4062506317256, 363.19087148429287`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{194.6226, 329.}, {204.7006, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{203.6802, 329.}, {201.49020000000002`, 332.603}, {201.6122, 332.685}, {207.4872, 330.351}, {209.5342, 329.901}, {211.58120000000002`, 329.451}, {213.62820000000002`, 329.}, { 211.58120000000002`, 328.55}, {209.5342, 328.1}, {207.4872, 327.649}, { 201.6122, 325.315}, {201.49020000000002`, 325.377}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{274.92769999999996`, 329.}, { 294.96369999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{293.9438, 329.}, {291.7538, 332.603}, { 291.8758, 332.685}, {297.7508, 330.351}, {299.7978, 329.901}, {301.8448, 329.451}, {303.8918, 329.}, {301.8448, 328.55}, {299.7978, 328.1}, { 297.7508, 327.649}, {291.8758, 325.315}, {291.7538, 325.377}}}]}, { {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{317.63669999999996`, 353.749}, { 321.23969999999997`, 355.939}, {321.32169999999996`, 355.817}, { 318.98769999999996`, 349.942}, {318.5377, 347.89500000000004`}, { 318.0877, 345.848}, {317.63669999999996`, 343.80100000000004`}, { 317.1867, 345.848}, {316.7367, 347.89500000000004`}, { 316.28569999999996`, 349.942}, {313.95169999999996`, 355.817}, { 314.0137, 355.939}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{331.2451, 329.}, {345.6441, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{372.0752687617261, 327.7522150562854}, { 369.8852687617261, 331.35521505628543`}, {370.0072687617261, 331.4372150562854}, {375.8822687617261, 329.1032150562854}, { 377.92926876172606`, 328.65321505628543`}, {379.9762687617261, 328.20321505628544`}, {382.02326876172606`, 327.7522150562854}, { 379.9762687617261, 327.30221505628543`}, {377.92926876172606`, 326.85221505628544`}, {375.8822687617261, 326.4012150562854}, { 370.0072687617261, 324.0672150562854}, {369.8852687617261, 324.12921505628543`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{482.8262, 329.}, {500.03119999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{499.01169999999996`, 329.}, { 496.82269999999994`, 332.603}, {496.9437, 332.685}, {502.8187, 330.351}, {504.86569999999995`, 329.901}, {506.91369999999995`, 329.451}, {508.9607, 329.}, {506.91369999999995`, 328.55}, { 504.86569999999995`, 328.1}, {502.8187, 327.649}, {496.9437, 325.315}, {496.82269999999994`, 325.377}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{623.7818802206084, 331.4347908989821}, { 621.5928802206084, 335.0377908989821}, {621.7138802206084, 335.1197908989821}, {627.5888802206084, 332.7857908989821}, { 629.6358802206084, 332.3357908989821}, {631.6838802206084, 331.8857908989821}, {633.7308802206085, 331.4347908989821}, { 631.6838802206084, 330.9847908989821}, {629.6358802206084, 330.5347908989821}, {627.5888802206084, 330.0837908989821}, { 621.7138802206084, 327.7497908989821}, {621.5928802206084, 327.8117908989821}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{648.1934000000001, 329.}, { 700.3364000000001, 329.}}}]}}, { {Thickness[0.0012626262626262627`], {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{664.623, 329.}, {664.623, 207.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{664.623, 208.01949999999997`}, { 668.2270000000001, 210.21049999999997`}, {668.307, 210.08749999999998`}, {665.975, 204.21249999999998`}, {665.523, 202.16549999999995`}, {665.0740000000001, 200.11949999999996`}, { 664.623, 198.07249999999996`}, {664.172, 200.11949999999996`}, { 663.7230000000001, 202.16549999999995`}, {663.2710000000001, 204.21249999999998`}, {660.937, 210.08749999999998`}, {661., 210.21049999999997`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{179.6226, 303.1777}, {176.0196, 300.98670000000004`}, {175.9376, 301.10970000000003`}, {178.2716, 306.98470000000003`}, {178.7216, 309.0317}, {179.1716, 311.0787}, { 179.6226, 313.1257}, {180.0726, 311.0787}, {180.5226, 309.0317}, { 180.9736, 306.98470000000003`}, {183.3076, 301.10970000000003`}, { 183.2456, 300.98670000000004`}}}]}}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{0, 2, 0}, {0, 1, 0}}}, {{{592.3003906520612, 217.0052483104812}, {314.3427393261901, 217.0052483104812}, { 314.3427393261901, 317.07172140479906`}}}, CurveClosed->{0}]}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{542.1317181593226, 330.76629515507955`}, {621.4346621493521, 332.0140800987946}}]}, {AbsoluteThickness[1], StrokeForm[{GrayLevel[0.], Opacity[0.64]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{316.3147046983335, 330.78655481789576`}, {370.72832654963986`, 330.786554817896}}]}}, {AbsoluteThickness[1.5667510701081488`], StrokeForm[{GrayLevel[0.], Opacity[0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{593.0257338556451, 329.7039743937679}, {594.2735187993599, 217.1587808642621}}]}, { InsetBox["", {227.29126922539376`, 329.68859319586625`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontWeight->Bold, FontColor->GrayLevel[0.], FontOpacity->1., Background->GrayLevel[1.]], {245.6199527883859, 329.68859319586625`}, { Left, Baseline}, Alignment->{Left, Top}], StyleBox[InsetBox[ StyleBox[Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"-", "\[Kappa]"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"s", " ", SubscriptBox["I", "n"]}], "-", "a"}], ")"}], RowBox[{"-", "1"}]], "b"}]}], TraditionalForm]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {385.11138114322716`, 326.7435442210371}, { Left, Baseline}, {151.29709966933558`, 38.37742006823411}, {{1., 0.}, { 0., 1.}}, Alignment->{Left, Top}], FontSize->12], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["u", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {345.2843827544348, 341.4713183435042}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], InsetBox["", {313.69792030807093`, 231.49921696555123`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {405.3413381230317, 391.2206023001969}, {Left, Baseline}, Alignment->{Left, Top}], {{ {FaceForm[{RGBColor[0.8630000000000001, 0.894, 0.89], Opacity[1.]}], FilledCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 326.6901564341327, 329.49126303719174`}, {326.6901564341327, 321.7592630371917}, {320.42215643413266`, 315.49126303719174`}, { 312.6901564341327, 315.49126303719174`}, {304.9581564341326, 315.49126303719174`}, {298.6901564341327, 321.7592630371917}, { 298.6901564341327, 329.49126303719174`}, {298.6901564341327, 337.22326303719177`}, {304.9581564341326, 343.49126303719174`}, { 312.6901564341327, 343.49126303719174`}, {320.42215643413266`, 343.49126303719174`}, {326.6901564341327, 337.22326303719177`}, { 326.6901564341327, 329.49126303719174`}}}]}, {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 326.6522555803771, 331.3026010219939}, {326.6522555803771, 323.90677493503733`}, {320.6567773195075, 317.9112966741678}, { 313.260951232551, 317.9112966741678}, {305.8651251455944, 317.9112966741678}, {299.8696468847249, 323.90677493503733`}, { 299.8696468847249, 331.3026010219939}, {299.8696468847249, 338.6984271089504}, {305.8651251455944, 344.69390536981996`}, { 313.260951232551, 344.69390536981996`}, {320.6567773195075, 344.69390536981996`}, {326.6522555803771, 338.6984271089504}, { 326.6522555803771, 331.3026010219939}}}, CurveClosed->{1}]}}, FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{294.1132073260418, 345.17383887988825`}, {294.1132073260418, 346.57783887988825`}, {296.96720732604183`, 346.57783887988825`}, { 296.96720732604183`, 349.40983887988824`}, {298.3492073260418, 349.40983887988824`}, {298.3492073260418, 346.57783887988825`}, { 301.20220732604184`, 346.57783887988825`}, {301.20220732604184`, 345.17383887988825`}, {298.3492073260418, 345.17383887988825`}, { 298.3492073260418, 342.3218388798883}, {296.96720732604183`, 342.3218388798883}, {296.96720732604183`, 345.17383887988825`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 293.76715306915196`, 319.42401874778835`}, {300.76315306915205`, 319.42401874778835`}, {300.76315306915205`, 318.19601874778834`}, { 293.76715306915196`, 318.19601874778834`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{314.52783995442275`, 307.3729291397311}, {310.80138736583405`, 305.3991912885968}, { 310.7268169557421, 305.52683240242277`}, {313.4047106947715, 311.25326812828814`}, {313.9751698024199, 313.27001932923014`}, { 314.54556967742775`, 315.28577228596635`}, {315.1170270292818, 317.3024642542679}, {315.44498770680315`, 315.23240367689345`}, { 315.773007616965, 313.1633413437248}, {316.1019665386921, 311.0932215337099}, {318.08387675207626`, 305.0902878420818}, { 318.01469999654097`, 304.97117622848504`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{289.95152908215715`, 330.7187883180905}, {287.7605290821571, 334.3217883180905}, { 287.88352908215717`, 334.4037883180905}, {293.75852908215717`, 332.0697883180905}, {295.8055290821572, 331.6197883180905}, { 297.8515290821571, 331.1697883180905}, {299.89852908215715`, 330.7187883180905}, {297.8515290821571, 330.2687883180905}, { 295.8055290821572, 329.8187883180905}, {293.75852908215717`, 329.3677883180905}, {287.88352908215717`, 327.0337883180905}, { 287.7605290821571, 327.0957883180905}}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{251.59677563685534`, 330.75278567242134`}, {289.8083336524264, 330.75278567242134`}}]}, StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["r", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {270.9921809247213, 338.1983391358269}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], StyleBox[InsetBox[ StyleBox[Cell[BoxData[ FormBox["y", TraditionalForm]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {609.5062205215099, 340.8983889041407}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{372.73994880721136`, 330.8618614596187}, {370.5509488072114, 334.4648614596187}, { 370.6719488072114, 334.5468614596187}, {376.5469488072114, 332.2128614596187}, {378.5939488072114, 331.7628614596187}, { 380.6419488072114, 331.3128614596187}, {382.68894880721143`, 330.8618614596187}, {380.6419488072114, 330.4118614596187}, { 378.5939488072114, 329.9618614596187}, {376.5469488072114, 329.5108614596187}, {370.6719488072114, 327.1768614596187}, { 370.5509488072114, 327.2388614596187}}}]}}, InsetBox[ StyleBox[Cell[TextData[{ "design parameters : ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"q", ",", "r"}]}], TraditionalForm]], FormatType->"TraditionalForm"], "}" }], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->12, Background->GrayLevel[1.]], {385.3534129213484, 246.5582206764043}, { Left, Baseline}, Alignment->{Left, Top}]}, ImagePadding->{{0., 1.}, {1., 3.1072}}, ImageSize->{446., Automatic}, PlotRange->{{165.012625, 711.261375}, {187.584524, 401.708578}}, PlotRangePadding->Automatic]}]], "Input", CellChangeTimes->{{3.527959067247854*^9, 3.527959073409318*^9}, { 3.528071925509041*^9, 3.528071926887467*^9}, {3.528155411317737*^9, 3.528155496609354*^9}, {3.528155930072933*^9, 3.5281559321548443`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"q", "=", " ", RowBox[{"0.01", RowBox[{"IdentityMatrix", "[", "4", "]"}]}]}], ";", RowBox[{"r", "=", RowBox[{"IdentityMatrix", "[", "2", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"\[Kappa]", "=", " ", RowBox[{"LQRegulatorGains", "[", RowBox[{ RowBox[{"{", RowBox[{"ss", ",", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{"q", ",", "r"}], "}"}]}], "]"}]}]}], "Input"], Cell["\<\ The system has been stabilized:\ \>", "Text", CellChangeTimes->{{3.528071959629813*^9, 3.528071985137908*^9}, { 3.528132639819468*^9, 3.528132645250827*^9}, {3.5281327347497797`*^9, 3.5281327413524523`*^9}, 3.52813277453905*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"sscl", "=", " ", RowBox[{"SystemsModelStateFeedbackConnect", "[", RowBox[{"ss", ",", "\[Kappa]", ",", "All", ",", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"ListPlot", "[", RowBox[{ RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "#", "]"}], ",", RowBox[{"Im", "[", "#", "]"}]}], "}"}], "&"}], ",", RowBox[{"Eigenvalues", "[", RowBox[{ RowBox[{"Normal", "[", "sscl", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "]"}]}], "]"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", "All"}], ",", RowBox[{"AxesOrigin", "\[Rule]", " ", RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}]}], ",", RowBox[{"PlotMarkers", "\[Rule]", " ", RowBox[{"{", RowBox[{"\"\\"", ",", "Medium"}], "}"}]}], ",", RowBox[{"PlotStyle", "\[Rule]", " ", "Red"}]}], "]"}]}], "Input", CellChangeTimes->{{3.528133859004681*^9, 3.5281338597021112`*^9}}], Cell["Meets the design goals:", "Text", CellChangeTimes->{{3.528132720766961*^9, 3.528132776986958*^9}, 3.52813354670152*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"x", "=", RowBox[{"StateResponse", "[", RowBox[{ RowBox[{"{", RowBox[{"sscl", ",", " ", RowBox[{"{", RowBox[{"0", ",", "0.1", ",", "0", ",", "0"}], "}"}]}], "}"}], ",", "0", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"GraphicsGrid", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"Plot", "[", RowBox[{"x", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", "All"}], ",", RowBox[{"FrameLabel", "\[Rule]", " ", RowBox[{"Style", "[", RowBox[{"\"\\"", ",", "Bold", ",", RowBox[{"FontSize", "\[Rule]", " ", "14"}]}], "]"}]}], ",", RowBox[{"Frame", "\[Rule]", " ", "True"}]}], "]"}], ",", RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"\[Kappa]", ".", RowBox[{"Transpose", "[", "x", "]"}]}]}], "*", RowBox[{"180", "/", "Pi"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", "All"}], ",", RowBox[{"FrameLabel", "\[Rule]", " ", RowBox[{"Style", "[", RowBox[{"\"\\"", ",", "Bold", ",", RowBox[{"FontSize", "\[Rule]", " ", "14"}]}], "]"}]}], ",", RowBox[{"Frame", "\[Rule]", " ", "True"}]}], "]"}]}], "}"}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", " ", "500"}]}], "]"}]}], "Input", CellChangeTimes->{{3.5280719988101273`*^9, 3.528072019724771*^9}, { 3.52807206938958*^9, 3.528072082444865*^9}, {3.528072121373631*^9, 3.5280721524930973`*^9}, 3.528132667141308*^9, 3.528132758771245*^9}] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ LQG/LTR Design: Optimal Control of a Remotely Piloted Aircraft \ \>", "Section", CellChangeTimes->{3.5280724552905293`*^9}], Cell[CellGroupData[{ Cell["\<\ Regulator Design for Stochastic System\ \>", "Subsection", CellChangeTimes->{{3.528071869355756*^9, 3.5280718830816298`*^9}, { 3.528072436312901*^9, 3.528072445030752*^9}, {3.528146799577298*^9, 3.5281468053880663`*^9}}], Cell["\<\ Design two LQG regulators for two sets of process-noise covariance matrices\ \>", "Text", CellChangeTimes->{{3.528072503963697*^9, 3.528072579004156*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"v", "=", RowBox[{"IdentityMatrix", "[", "2", "]"}]}], ";", RowBox[{"\[Rho]1", "=", "5"}], ";", RowBox[{"\[Rho]2", "=", "100"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"lqg1", ",", "lqg2"}], "}"}], "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"LQGRegulator", "[", RowBox[{ RowBox[{"{", RowBox[{"ss", ",", "All", ",", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"#", " ", "v"}], ",", "v"}], "}"}], ",", RowBox[{"{", RowBox[{"q", ",", "r"}], "}"}]}], "]"}], "&"}], ",", RowBox[{"{", RowBox[{"\[Rho]1", ",", "\[Rho]2"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.528072617977147*^9, 3.528072623633503*^9}, 3.528072659321999*^9, 3.528157764969501*^9}] }, Closed]], Cell[CellGroupData[{ Cell["Loop-transfer recovery", "Subsection", CellChangeTimes->{{3.528072670701819*^9, 3.528072689590206*^9}}], Cell[BoxData[ RowBox[{ GraphicsBox[{ {EdgeForm[{AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}]}], FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 543.8605438504817, 288.31100081503973`}, {315.5072376703588, 288.31100081503973`}, {315.5072376703588, 359.6923122284941}, { 543.8605438504817, 359.6923122284941}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{194.6226, 329.}, {204.7006, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{203.6802, 329.}, {201.49020000000002`, 332.603}, {201.6122, 332.685}, {207.4872, 330.351}, {209.5342, 329.901}, {211.58120000000002`, 329.451}, {213.62820000000002`, 329.}, { 211.58120000000002`, 328.55}, {209.5342, 328.1}, {207.4872, 327.649}, { 201.6122, 325.315}, {201.49020000000002`, 325.377}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{274.92769999999996`, 329.}, { 294.96369999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{293.9438, 329.}, {291.7538, 332.603}, { 291.8758, 332.685}, {297.7508, 330.351}, {299.7978, 329.901}, {301.8448, 329.451}, {303.8918, 329.}, {301.8448, 328.55}, {299.7978, 328.1}, { 297.7508, 327.649}, {291.8758, 325.315}, {291.7538, 325.377}}}]}, { {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{331.2451, 329.}, {345.6441, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{372.0752687617261, 327.7522150562854}, { 369.8852687617261, 331.35521505628543`}, {370.0072687617261, 331.4372150562854}, {375.8822687617261, 329.1032150562854}, { 377.92926876172606`, 328.65321505628543`}, {379.9762687617261, 328.20321505628544`}, {382.02326876172606`, 327.7522150562854}, { 379.9762687617261, 327.30221505628543`}, {377.92926876172606`, 326.85221505628544`}, {375.8822687617261, 326.4012150562854}, { 370.0072687617261, 324.0672150562854}, {369.8852687617261, 324.12921505628543`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{482.8262, 329.}, {500.03119999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{499.01169999999996`, 329.}, { 496.82269999999994`, 332.603}, {496.9437, 332.685}, {502.8187, 330.351}, {504.86569999999995`, 329.901}, {506.91369999999995`, 329.451}, {508.9607, 329.}, {506.91369999999995`, 328.55}, { 504.86569999999995`, 328.1}, {502.8187, 327.649}, {496.9437, 325.315}, {496.82269999999994`, 325.377}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{614.1726238169356, 326.5628395555966}, { 611.9836238169356, 330.1658395555966}, {612.1046238169356, 330.2478395555966}, {617.9796238169356, 327.9138395555966}, { 620.0266238169356, 327.4638395555966}, {622.0746238169356, 327.01383955559663`}, {624.1216238169357, 326.5628395555966}, { 622.0746238169356, 326.1128395555966}, {620.0266238169356, 325.66283955559663`}, {617.9796238169356, 325.2118395555966}, { 612.1046238169356, 322.8778395555966}, {611.9836238169356, 322.9398395555966}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{648.1934000000001, 329.}, { 700.3364000000001, 329.}}}]}}, { {Thickness[0.0012626262626262627`], {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{664.623, 329.}, {664.623, 207.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{664.623, 208.01949999999997`}, { 668.2270000000001, 210.21049999999997`}, {668.307, 210.08749999999998`}, {665.975, 204.21249999999998`}, {665.523, 202.16549999999995`}, {665.0740000000001, 200.11949999999996`}, { 664.623, 198.07249999999996`}, {664.172, 200.11949999999996`}, { 663.7230000000001, 202.16549999999995`}, {663.2710000000001, 204.21249999999998`}, {660.937, 210.08749999999998`}, {661., 210.21049999999997`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{179.6226, 303.1777}, {176.0196, 300.98670000000004`}, {175.9376, 301.10970000000003`}, {178.2716, 306.98470000000003`}, {178.7216, 309.0317}, {179.1716, 311.0787}, { 179.6226, 313.1257}, {180.0726, 311.0787}, {180.5226, 309.0317}, { 180.9736, 306.98470000000003`}, {183.3076, 301.10970000000003`}, { 183.2456, 300.98670000000004`}}}]}}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{0, 2, 0}, {0, 1, 0}}}, {{{588.3021922953203, 222.90132515231497`}, {242.4574921341378, 221.95213551305866`}, { 242.4574921341378, 317.06482281062813`}}}, CurveClosed->{0}]}, {AbsoluteThickness[1], StrokeForm[{GrayLevel[0.], Opacity[0.64]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{253.2830220686622, 329.22241382925426`}, {303.7392896035099, 329.2224138292545}}]}}, {AbsoluteThickness[1.5667510701081488`], StrokeForm[{GrayLevel[0.], Opacity[0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{587.0766122474863, 327.15282875460673`}, {588.3243971912011, 222.54096787339074`}}]}, { InsetBox["", {227.29126922539376`, 329.68859319586625`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontWeight->Bold, FontColor->GrayLevel[0.], FontOpacity->1., Background->GrayLevel[1.]], {245.6199527883859, 329.68859319586625`}, { Left, Baseline}, Alignment->{Left, Top}], StyleBox[InsetBox[ StyleBox[Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"-", "\[Kappa]"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"s", " ", SubscriptBox["I", "n"]}], "-", "a", "+", RowBox[{"b", " ", "\[Kappa]"}], " ", "+", " ", RowBox[{"l", " ", "c"}]}], ")"}], RowBox[{"-", "1"}]], "l", " ", "c", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"s", " ", SubscriptBox["I", "n"]}], "-", "a"}], ")"}], RowBox[{"-", "1"}]], "b"}]}], TraditionalForm]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {324.87961050460626`, 325.4329870655599}, { Left, Baseline}, {290.0074564109087, 45.742571753627374`}, {{1., 0.}, { 0., 1.}}, Alignment->{Left, Top}], FontSize->12], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["u", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {278.0420289759084, 339.3708363271943}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], InsetBox["", {313.69792030807093`, 231.49921696555123`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {405.3413381230317, 391.2206023001969}, {Left, Baseline}, Alignment->{Left, Top}], {{ {FaceForm[{RGBColor[0.8630000000000001, 0.894, 0.89], Opacity[1.]}], FilledCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 258.7835476589956, 330.489889636826}, {258.7835476589956, 322.75788963682595`}, {252.51554765899556`, 316.489889636826}, { 244.7835476589956, 316.489889636826}, {237.0515476589955, 316.489889636826}, {230.7835476589956, 322.75788963682595`}, { 230.7835476589956, 330.489889636826}, {230.7835476589956, 338.221889636826}, {237.0515476589955, 344.489889636826}, { 244.7835476589956, 344.489889636826}, {252.51554765899556`, 344.489889636826}, {258.7835476589956, 338.221889636826}, { 258.7835476589956, 330.489889636826}}}]}, {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 256.7483936059712, 330.3039744223591}, {256.7483936059712, 322.90814833540253`}, {250.75291534510166`, 316.91267007453297`}, { 243.35708925814515`, 316.91267007453297`}, {235.96126317118853`, 316.91267007453297`}, {229.96578491031903`, 322.90814833540253`}, { 229.96578491031903`, 330.3039744223591}, {229.96578491031903`, 337.6998005093156}, {235.96126317118853`, 343.69527877018515`}, { 243.35708925814515`, 343.69527877018515`}, {250.75291534510166`, 343.69527877018515`}, {256.7483936059712, 337.6998005093156}, { 256.7483936059712, 330.3039744223591}}}, CurveClosed->{1}]}}, FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{219.2162123534641, 349.1683452784252}, {219.2162123534641, 350.5723452784252}, {222.07021235346414`, 350.5723452784252}, { 222.07021235346414`, 353.4043452784252}, {223.4522123534641, 353.4043452784252}, {223.4522123534641, 350.5723452784252}, { 226.30521235346416`, 350.5723452784252}, {226.30521235346416`, 349.1683452784252}, {223.4522123534641, 349.1683452784252}, { 223.4522123534641, 346.3163452784252}, {222.07021235346414`, 346.3163452784252}, {222.07021235346414`, 349.1683452784252}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 220.86741129584303`, 316.4281389488846}, {227.86341129584312`, 316.4281389488846}, {227.86341129584312`, 315.2001389488846}, { 220.86741129584303`, 315.2001389488846}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{242.62672478074813`, 307.3729291397307}, {238.90027219215943`, 305.39919128859634`}, { 238.82570178206748`, 305.5268324024223}, {241.50359552109688`, 311.2532681282877}, {242.07405462874527`, 313.2700193292297}, { 242.64445450375314`, 315.2857722859659}, {243.2159118556072, 317.3024642542674}, {243.54387253312854`, 315.232403676893}, { 243.8718924432904, 313.16334134372437`}, {244.2008513650175, 311.0932215337094}, {246.18276157840165`, 305.09028784208135`}, { 246.11358482286636`, 304.9711762284846}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{221.0462937073857, 330.71878831809005`}, {218.85529370738567`, 334.32178831809006`}, { 218.97829370738572`, 334.40378831809005`}, {224.85329370738572`, 332.06978831809005`}, {226.90029370738574`, 331.61978831809006`}, { 228.94629370738568`, 331.16978831809007`}, {230.9932937073857, 330.71878831809005`}, {228.94629370738568`, 330.26878831809006`}, { 226.90029370738574`, 329.81878831809007`}, {224.85329370738572`, 329.36778831809005`}, {218.97829370738572`, 327.03378831809005`}, { 218.85529370738567`, 327.09578831809006`}}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{182.6915402620839, 330.75278567242066`}, {220.90309827765498`, 330.75278567242066`}}]}, StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["r", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {198.09243915141235`, 340.1955923350955}, {Left, Baseline}, Alignment->{Left, Top}], FontSize->13], StyleBox[InsetBox[ StyleBox[Cell[BoxData[ FormBox["y", TraditionalForm]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {596.795611397175, 334.59469875049354`}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{306.1192120356484, 329.06882178971307`}, {303.9302120356484, 332.6718217897131}, { 304.0512120356484, 332.7538217897131}, {309.9262120356484, 330.41982178971307`}, {311.97321203564843`, 329.9698217897131}, { 314.02121203564843`, 329.5198217897131}, {316.06821203564846`, 329.06882178971307`}, {314.02121203564843`, 328.6188217897131}, { 311.97321203564843`, 328.1688217897131}, {309.9262120356484, 327.71782178971307`}, {304.0512120356484, 325.38382178971307`}, { 303.9302120356484, 325.4458217897131}}}]}}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{545.633115546703, 327.25759257370004`}, {613.0191847679703, 327.2575925736998}}]}, StyleBox[InsetBox[ StyleBox[Cell[TextData[{ StyleBox["noise covariance matrices {\[Rho] ", FontSize->14], Cell[BoxData[ FormBox["v", TraditionalForm]], FontSize->14], StyleBox[", ", FontSize->14], Cell[BoxData[ FormBox["v", TraditionalForm]], FontSize->14], StyleBox["}", FontSize->14] }], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {311.3114218464351, 204.58687734369278`}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->12]}, ImagePadding->{{0., 1.}, {1., 3.1072}}, ImageSize->{548., Automatic}, PlotRange->{{165.012625, 711.261375}, {187.584524, 401.708578}}, PlotRangePadding->Automatic], GraphicsBox[{{ {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{317.63669999999996`, 353.749}, { 321.23969999999997`, 355.939}, {321.32169999999996`, 355.817}, { 318.98769999999996`, 349.942}, {318.5377, 347.89500000000004`}, { 318.0877, 345.848}, {317.63669999999996`, 343.80100000000004`}, { 317.1867, 345.848}, {316.7367, 347.89500000000004`}, { 316.28569999999996`, 349.942}, {313.95169999999996`, 355.817}, { 314.0137, 355.939}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{331.2451, 329.}, {345.6441, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{372.0752687617261, 327.7522150562854}, { 369.8852687617261, 331.35521505628543`}, {370.0072687617261, 331.4372150562854}, {375.8822687617261, 329.1032150562854}, { 377.92926876172606`, 328.65321505628543`}, {379.9762687617261, 328.20321505628544`}, {382.02326876172606`, 327.7522150562854}, { 379.9762687617261, 327.30221505628543`}, {377.92926876172606`, 326.85221505628544`}, {375.8822687617261, 326.4012150562854}, { 370.0072687617261, 324.0672150562854}, {369.8852687617261, 324.12921505628543`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{482.8262, 329.}, {500.03119999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{499.01169999999996`, 329.}, { 496.82269999999994`, 332.603}, {496.9437, 332.685}, {502.8187, 330.351}, {504.86569999999995`, 329.901}, {506.91369999999995`, 329.451}, {508.9607, 329.}, {506.91369999999995`, 328.55}, { 504.86569999999995`, 328.1}, {502.8187, 327.649}, {496.9437, 325.315}, {496.82269999999994`, 325.377}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{648.1934000000001, 329.}, { 700.3364000000001, 329.}}}]}}, {Thickness[0.0012626262626262627`], {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{664.623, 329.}, {664.623, 207.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{664.623, 208.01949999999997`}, { 668.2270000000001, 210.21049999999997`}, {668.307, 210.08749999999998`}, {665.975, 204.21249999999998`}, {665.523, 202.16549999999995`}, {665.0740000000001, 200.11949999999996`}, { 664.623, 198.07249999999996`}, {664.172, 200.11949999999996`}, { 663.7230000000001, 202.16549999999995`}, {663.2710000000001, 204.21249999999998`}, {660.937, 210.08749999999998`}, {661., 210.21049999999997`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{179.6226, 303.1777}, {176.0196, 300.98670000000004`}, {175.9376, 301.10970000000003`}, {178.2716, 306.98470000000003`}, {178.7216, 309.0317}, {179.1716, 311.0787}, { 179.6226, 313.1257}, {180.0726, 311.0787}, {180.5226, 309.0317}, { 180.9736, 306.98470000000003`}, {183.3076, 301.10970000000003`}, { 183.2456, 300.98670000000004`}}}]}}, { InsetBox["", {313.69792030807093`, 231.49921696555123`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {405.3413381230317, 391.2206023001969}, {Left, Baseline}, Alignment->{Left, Top}]}, {AbsoluteThickness[1], Dashing[{Medium, Medium}], Arrowheads[{{0.04897788193684462, 1, { GraphicsBox[{ EdgeForm[None], Dashing[{}], PolygonBox[{{-1.5833333333333333`, 0.4166666666666667}, {-1.5410500000000003`, 0.369283333333333}, {-1.448333333333333, 0.255583333333333}, {-1.3991000000000005`, 0.18721666666666673`}, {-1.3564666666666663`, 0.11826666666666673`}, {-1.3268499999999999`, 0.05408333333333341}, {-1.3166666666666667`, 0.}, {-1.3268499999999999`, -0.048950000000000195`}, \ {-1.3564666666666663`, -0.11228333333333372`}, {-1.3991000000000005`, \ -0.18353333333333333`}, {-1.448333333333333, -0.2562833333333335}, \ {-1.5410500000000003`, -0.38048333333333345`}, {-1.5833333333333333`, \ -0.43333333333333335`}, {0., 0.}, {-1.5833333333333333`, 0.4166666666666667}, {-1.5833333333333333`, 0.4166666666666667}}]}], 1.31}}}], StrokeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], ArrowBox[{{240.71557676630442`, 299.6051866467392}, {638.5271664402176, 299.6051866467392}}]}, InsetBox[ StyleBox[Cell["increase \[Rho]", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontSize->13, Background->GrayLevel[1.]], {329.7778729619573, 341.1675915380438}, { Left, Baseline}, Alignment->{Left, Top}]}, ImagePadding->{{0., 1.}, {1., 3.1072}}, ImageSize->{185., Automatic}, PlotRange->{{165.012625, 711.261375}, {187.584524, 401.708578}}, PlotRangePadding->Automatic], GraphicsBox[{ {EdgeForm[{AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}]}], FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 542.4062506317256, 291.80956007083853`}, {380.4035020081383, 291.80956007083853`}, {380.4035020081383, 363.19087148429287`}, { 542.4062506317256, 363.19087148429287`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{194.6226, 329.}, {204.7006, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{203.6802, 329.}, {201.49020000000002`, 332.603}, {201.6122, 332.685}, {207.4872, 330.351}, {209.5342, 329.901}, {211.58120000000002`, 329.451}, {213.62820000000002`, 329.}, { 211.58120000000002`, 328.55}, {209.5342, 328.1}, {207.4872, 327.649}, { 201.6122, 325.315}, {201.49020000000002`, 325.377}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{274.92769999999996`, 329.}, { 294.96369999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, { 0, 1, 0}, {0, 1, 0}}}, {{{293.9438, 329.}, {291.7538, 332.603}, { 291.8758, 332.685}, {297.7508, 330.351}, {299.7978, 329.901}, {301.8448, 329.451}, {303.8918, 329.}, {301.8448, 328.55}, {299.7978, 328.1}, { 297.7508, 327.649}, {291.8758, 325.315}, {291.7538, 325.377}}}]}, { {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{317.63669999999996`, 353.749}, { 321.23969999999997`, 355.939}, {321.32169999999996`, 355.817}, { 318.98769999999996`, 349.942}, {318.5377, 347.89500000000004`}, { 318.0877, 345.848}, {317.63669999999996`, 343.80100000000004`}, { 317.1867, 345.848}, {316.7367, 347.89500000000004`}, { 316.28569999999996`, 349.942}, {313.95169999999996`, 355.817}, { 314.0137, 355.939}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{331.2451, 329.}, {345.6441, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{372.0752687617261, 327.7522150562854}, { 369.8852687617261, 331.35521505628543`}, {370.0072687617261, 331.4372150562854}, {375.8822687617261, 329.1032150562854}, { 377.92926876172606`, 328.65321505628543`}, {379.9762687617261, 328.20321505628544`}, {382.02326876172606`, 327.7522150562854}, { 379.9762687617261, 327.30221505628543`}, {377.92926876172606`, 326.85221505628544`}, {375.8822687617261, 326.4012150562854}, { 370.0072687617261, 324.0672150562854}, {369.8852687617261, 324.12921505628543`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{482.8262, 329.}, {500.03119999999996`, 329.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{499.01169999999996`, 329.}, { 496.82269999999994`, 332.603}, {496.9437, 332.685}, {502.8187, 330.351}, {504.86569999999995`, 329.901}, {506.91369999999995`, 329.451}, {508.9607, 329.}, {506.91369999999995`, 328.55}, { 504.86569999999995`, 328.1}, {502.8187, 327.649}, {496.9437, 325.315}, {496.82269999999994`, 325.377}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{623.7818802206084, 331.4347908989821}, { 621.5928802206084, 335.0377908989821}, {621.7138802206084, 335.1197908989821}, {627.5888802206084, 332.7857908989821}, { 629.6358802206084, 332.3357908989821}, {631.6838802206084, 331.8857908989821}, {633.7308802206085, 331.4347908989821}, { 631.6838802206084, 330.9847908989821}, {629.6358802206084, 330.5347908989821}, {627.5888802206084, 330.0837908989821}, { 621.7138802206084, 327.7497908989821}, {621.5928802206084, 327.8117908989821}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{648.1934000000001, 329.}, { 700.3364000000001, 329.}}}]}}, { {Thickness[0.0012626262626262627`], {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}}}, {{{664.623, 329.}, {664.623, 207.}}}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{664.623, 208.01949999999997`}, { 668.2270000000001, 210.21049999999997`}, {668.307, 210.08749999999998`}, {665.975, 204.21249999999998`}, {665.523, 202.16549999999995`}, {665.0740000000001, 200.11949999999996`}, { 664.623, 198.07249999999996`}, {664.172, 200.11949999999996`}, { 663.7230000000001, 202.16549999999995`}, {663.2710000000001, 204.21249999999998`}, {660.937, 210.08749999999998`}, {661., 210.21049999999997`}}}]}, {FaceForm[{RGBColor[1., 1., 1.], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{179.6226, 303.1777}, {176.0196, 300.98670000000004`}, {175.9376, 301.10970000000003`}, {178.2716, 306.98470000000003`}, {178.7216, 309.0317}, {179.1716, 311.0787}, { 179.6226, 313.1257}, {180.0726, 311.0787}, {180.5226, 309.0317}, { 180.9736, 306.98470000000003`}, {183.3076, 301.10970000000003`}, { 183.2456, 300.98670000000004`}}}]}}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{0, 2, 0}, {0, 1, 0}}}, {{{592.3003906520612, 217.0052483104812}, {314.3427393261901, 217.0052483104812}, { 314.3427393261901, 317.07172140479906`}}}, CurveClosed->{0}]}, {AbsoluteThickness[1.023292992280754], JoinForm[{"Miter", 4.}], StrokeForm[Opacity[0.62]], EdgeForm[Opacity[1.]], EdgeForm[None], LineBox[{{542.1317181593226, 330.76629515507955`}, {621.4346621493521, 332.0140800987946}}]}, {AbsoluteThickness[1], StrokeForm[{GrayLevel[0.], Opacity[0.64]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{316.3147046983335, 330.78655481789576`}, {370.72832654963986`, 330.786554817896}}]}}, {AbsoluteThickness[1.5667510701081488`], StrokeForm[{GrayLevel[0.], Opacity[0.65]}], EdgeForm[{GrayLevel[0.], Opacity[1.]}], EdgeForm[None], LineBox[{{593.0257338556451, 329.7039743937679}, {594.2735187993599, 217.1587808642621}}]}, { InsetBox["", {227.29126922539376`, 329.68859319586625`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[Cell["", GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], FontWeight->Bold, FontColor->GrayLevel[0.], FontOpacity->1., Background->GrayLevel[1.]], {245.6199527883859, 329.68859319586625`}, { Left, Baseline}, Alignment->{Left, Top}], StyleBox[InsetBox[ StyleBox[Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"-", "\[Kappa]"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"s", " ", SubscriptBox["I", "n"]}], "-", "a"}], ")"}], RowBox[{"-", "1"}]], "b"}]}], TraditionalForm]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {385.11138114322716`, 326.7435442210371}, { Left, Baseline}, {151.29709966933558`, 38.37742006823411}, {{1., 0.}, { 0., 1.}}, Alignment->{Left, Top}], FontSize->12], StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["u", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {345.2843827544348, 341.4713183435042}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], InsetBox["", {313.69792030807093`, 231.49921696555123`}, {Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {405.3413381230317, 391.2206023001969}, {Left, Baseline}, Alignment->{Left, Top}], {{ {FaceForm[{RGBColor[0.8630000000000001, 0.894, 0.89], Opacity[1.]}], FilledCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 326.6901564341327, 329.49126303719174`}, {326.6901564341327, 321.7592630371917}, {320.42215643413266`, 315.49126303719174`}, { 312.6901564341327, 315.49126303719174`}, {304.9581564341326, 315.49126303719174`}, {298.6901564341327, 321.7592630371917}, { 298.6901564341327, 329.49126303719174`}, {298.6901564341327, 337.22326303719177`}, {304.9581564341326, 343.49126303719174`}, { 312.6901564341327, 343.49126303719174`}, {320.42215643413266`, 343.49126303719174`}, {326.6901564341327, 337.22326303719177`}, { 326.6901564341327, 329.49126303719174`}}}]}, {JoinForm[{"Miter", 4.}], JoinedCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}}}, {{{ 326.6522555803771, 331.3026010219939}, {326.6522555803771, 323.90677493503733`}, {320.6567773195075, 317.9112966741678}, { 313.260951232551, 317.9112966741678}, {305.8651251455944, 317.9112966741678}, {299.8696468847249, 323.90677493503733`}, { 299.8696468847249, 331.3026010219939}, {299.8696468847249, 338.6984271089504}, {305.8651251455944, 344.69390536981996`}, { 313.260951232551, 344.69390536981996`}, {320.6567773195075, 344.69390536981996`}, {326.6522555803771, 338.6984271089504}, { 326.6522555803771, 331.3026010219939}}}, CurveClosed->{1}]}}, FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{294.1132073260418, 345.17383887988825`}, {294.1132073260418, 346.57783887988825`}, {296.96720732604183`, 346.57783887988825`}, { 296.96720732604183`, 349.40983887988824`}, {298.3492073260418, 349.40983887988824`}, {298.3492073260418, 346.57783887988825`}, { 301.20220732604184`, 346.57783887988825`}, {301.20220732604184`, 345.17383887988825`}, {298.3492073260418, 345.17383887988825`}, { 298.3492073260418, 342.3218388798883}, {296.96720732604183`, 342.3218388798883}, {296.96720732604183`, 345.17383887988825`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}}}, {{{ 293.76715306915196`, 319.42401874778835`}, {300.76315306915205`, 319.42401874778835`}, {300.76315306915205`, 318.19601874778834`}, { 293.76715306915196`, 318.19601874778834`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{314.52783995442275`, 307.3729291397311}, {310.80138736583405`, 305.3991912885968}, { 310.7268169557421, 305.52683240242277`}, {313.4047106947715, 311.25326812828814`}, {313.9751698024199, 313.27001932923014`}, { 314.54556967742775`, 315.28577228596635`}, {315.1170270292818, 317.3024642542679}, {315.44498770680315`, 315.23240367689345`}, { 315.773007616965, 313.1633413437248}, {316.1019665386921, 311.0932215337099}, {318.08387675207626`, 305.0902878420818}, { 318.01469999654097`, 304.97117622848504`}}}]}, {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{289.95152908215715`, 330.7187883180905}, {287.7605290821571, 334.3217883180905}, { 287.88352908215717`, 334.4037883180905}, {293.75852908215717`, 332.0697883180905}, {295.8055290821572, 331.6197883180905}, { 297.8515290821571, 331.1697883180905}, {299.89852908215715`, 330.7187883180905}, {297.8515290821571, 330.2687883180905}, { 295.8055290821572, 329.8187883180905}, {293.75852908215717`, 329.3677883180905}, {287.88352908215717`, 327.0337883180905}, { 287.7605290821571, 327.0957883180905}}}]}, {Thickness[0.0012626262626262627`], JoinForm[{"Miter", 4.}], LineBox[{{251.59677563685534`, 330.75278567242134`}, {289.8083336524264, 330.75278567242134`}}]}, StyleBox[InsetBox[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox["r", TraditionalForm]]]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[1.]], {270.9921809247213, 338.1983391358269}, { Left, Baseline}, Alignment->{Left, Top}], FontSize->13], StyleBox[InsetBox[ StyleBox[Cell[BoxData[ FormBox["y", TraditionalForm]], GeneratedCell->False, CellAutoOverwrite->False, CellBaseline->Baseline, TextAlignment->Left], Background->GrayLevel[ 1.]], {603.3685941170155, 343.35343946593855`}, {Left, Baseline}, Alignment->{Left, Top}], FontSize->13], {Thickness[0.0012626262626262627`], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {1, 3, 3}, {1, 3, 3}, {0, 1, 0}, {0, 1, 0}}}, {{{372.73994880721136`, 330.8618614596187}, {370.5509488072114, 334.4648614596187}, { 370.6719488072114, 334.5468614596187}, {376.5469488072114, 332.2128614596187}, {378.5939488072114, 331.7628614596187}, { 380.6419488072114, 331.3128614596187}, {382.68894880721143`, 330.8618614596187}, {380.6419488072114, 330.4118614596187}, { 378.5939488072114, 329.9618614596187}, {376.5469488072114, 329.5108614596187}, {370.6719488072114, 327.1768614596187}, { 370.5509488072114, 327.2388614596187}}}]}}}, ImagePadding->{{0., 1.}, {1., 3.1072}}, ImageSize->{446., Automatic}, PlotRange->{{165.012625, 711.261375}, {187.584524, 401.708578}}, PlotRangePadding->Automatic]}]], "Input", CellChangeTimes->{{3.527958026409628*^9, 3.5279580546218777`*^9}, { 3.527958294088612*^9, 3.527958310577566*^9}, {3.5279583568862743`*^9, 3.527958367235661*^9}, {3.5279589451160727`*^9, 3.527959045914156*^9}, { 3.5279590935494547`*^9, 3.527959094462413*^9}, {3.5281556032740917`*^9, 3.528155638604478*^9}, {3.5281556749236403`*^9, 3.528155688619938*^9}}], Cell[TextData[{ "How does the LQG return ratio ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", SuperscriptBox[ RowBox[{"\[Kappa]", "(", RowBox[{ RowBox[{"s", " ", SubscriptBox["I", "n"]}], "-", "a", "+", RowBox[{"b", " ", "\[Kappa]"}], " ", "+", " ", RowBox[{"l", " ", "c"}]}], ")"}], RowBox[{"-", "1"}]]}], "l", " ", "c", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"s", " ", SubscriptBox["I", "n"]}], "-", "a"}], ")"}], RowBox[{"-", "1"}]], "b"}], TraditionalForm]], FormatType->"TraditionalForm"], " compare with the pure LQR return ration ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", SuperscriptBox[ RowBox[{"\[Kappa]", "(", RowBox[{ RowBox[{"s", " ", SubscriptBox["I", "n"]}], "-", "a"}], ")"}], RowBox[{"-", "1"}]]}], " ", "b"}], TraditionalForm]], FormatType->"TraditionalForm"], " for two estimator gains?" }], "Text", CellChangeTimes->{{3.528073158218573*^9, 3.52807335512845*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"sysk", "=", " ", RowBox[{"StateSpaceModel", "[", RowBox[{"{", RowBox[{"a", ",", "b", ",", RowBox[{"-", "\[Kappa]"}]}], "}"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"sysKG1", ",", "sysKG2"}], "}"}], "=", " ", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"SystemsModelSeriesConnect", "[", RowBox[{"#", ",", RowBox[{"SystemsModelExtract", "[", RowBox[{"ss", ",", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}]}], "]"}]}], "]"}], "&"}], ",", RowBox[{"{", RowBox[{"lqg1", ",", "lqg2"}], "}"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.528073737984467*^9, 3.528073738726901*^9}, 3.528073947145917*^9}] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ LQG/LTR Design: Optimal Control of a Remotely Piloted Aircraft\ \>", "Section", CellChangeTimes->{{3.528073379078985*^9, 3.528073383249219*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"<<", "PlotLegends`"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"p1", "=", RowBox[{"SingularValuePlot", "[", RowBox[{"sysk", ",", RowBox[{"{", RowBox[{ SuperscriptBox["10", RowBox[{"-", "2"}]], ",", SuperscriptBox["10", "2"]}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", " ", "Red"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"p2", "=", RowBox[{"SingularValuePlot", "[", RowBox[{"sysKG1", ",", RowBox[{"{", RowBox[{ SuperscriptBox["10", RowBox[{"-", "2"}]], ",", SuperscriptBox["10", "2"]}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", " ", "Blue"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"p3", "=", RowBox[{"SingularValuePlot", "[", RowBox[{"sysKG2", ",", RowBox[{"{", RowBox[{ SuperscriptBox["10", RowBox[{"-", "2"}]], ",", SuperscriptBox["10", "2"]}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", "Cyan"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"ShowLegend", "[", RowBox[{ RowBox[{"Show", "[", RowBox[{ RowBox[{"{", RowBox[{"p1", ",", "p2", ",", "p3"}], "}"}], ",", RowBox[{"Frame", "\[Rule]", " ", "True"}], ",", RowBox[{"PlotLabel", "\[Rule]", " ", RowBox[{"Style", "[", RowBox[{"\"\\"", ",", "Bold", ",", RowBox[{"FontSize", "\[Rule]", " ", "14"}]}], "]"}]}]}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Transpose", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"Red", ",", RowBox[{"Thickness", "[", ".05", "]"}], ",", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "0"}], "}"}]}], "}"}], "]"}]}], "}"}], "]"}], ",", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"Blue", ",", RowBox[{"Thickness", "[", ".05", "]"}], ",", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "0"}], "}"}]}], "}"}], "]"}]}], "}"}], "]"}], ",", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"Cyan", ",", RowBox[{"Thickness", "[", ".05", "]"}], ",", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "0"}], "}"}]}], "}"}], "]"}]}], "}"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Style", "[", RowBox[{"\"\\"", ",", "Bold", ",", RowBox[{"FontSize", "\[Rule]", " ", "14"}]}], "]"}], ",", RowBox[{"Style", "[", RowBox[{"\"\<\[Rho] = 5\>\"", ",", "Bold", ",", RowBox[{"FontSize", "\[Rule]", " ", "14"}]}], "]"}], ",", RowBox[{"Style", "[", RowBox[{"\"\<\[Rho] = 100\>\"", ",", "Bold", ",", RowBox[{"FontSize", "\[Rule]", " ", "14"}]}], "]"}]}], "}"}]}], "}"}], "]"}], ",", RowBox[{"LegendShadow", "\[Rule]", "None"}], ",", RowBox[{"LegendSize", "\[Rule]", RowBox[{"{", RowBox[{".5", ",", ".20"}], "}"}]}], ",", RowBox[{"LegendPosition", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "0.80"}], ",", ".05"}], "}"}]}]}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.5278507368638*^9, 3.527850743788673*^9}, { 3.527850785779695*^9, 3.527850819811924*^9}, {3.527851147432455*^9, 3.527851187309786*^9}, {3.5278512284035873`*^9, 3.5278512611421013`*^9}, { 3.5278512978629093`*^9, 3.527851300300029*^9}, {3.527851335388856*^9, 3.527851349130028*^9}, {3.527851831591915*^9, 3.527851834420164*^9}, { 3.527851950560062*^9, 3.527851991955587*^9}, {3.527852040601034*^9, 3.5278520630383883`*^9}, {3.527852152127487*^9, 3.5278522153437157`*^9}, { 3.527852249969605*^9, 3.527852273997458*^9}, {3.527852821906481*^9, 3.527852871382571*^9}, {3.527852992931477*^9, 3.527852999111709*^9}, { 3.527853114281892*^9, 3.527853260388335*^9}, {3.5278533352916937`*^9, 3.527853529404852*^9}, {3.5278540411300573`*^9, 3.527854192534237*^9}, { 3.527854865012426*^9, 3.527854871580127*^9}, {3.527855184447052*^9, 3.527855191593625*^9}, {3.527855253937353*^9, 3.527855389655808*^9}, { 3.527855547408545*^9, 3.52785572527172*^9}, {3.527855871275577*^9, 3.527855889596451*^9}, {3.527855946085412*^9, 3.5278559492985992`*^9}, { 3.527857626388011*^9, 3.527857654617571*^9}, {3.5278576905066357`*^9, 3.527857708867955*^9}, {3.52785777196716*^9, 3.5278578010567293`*^9}, { 3.527857857324374*^9, 3.527857861807851*^9}, 3.527858096716456*^9, { 3.527858154186503*^9, 3.527858159045965*^9}, {3.527858249759841*^9, 3.527858389932221*^9}, {3.527858429581476*^9, 3.527858458537468*^9}, { 3.527870733513567*^9, 3.5278707486003523`*^9}, {3.527871803514402*^9, 3.527871803744506*^9}, {3.527872300783537*^9, 3.527872327252438*^9}}], Cell["\<\ Design with \[Rho] = 100 offers better loop-transfer recovery, but \ performance goals not met\ \>", "Text", CellChangeTimes->{{3.528073493528625*^9, 3.52807350108867*^9}, { 3.528073693375691*^9, 3.52807369480545*^9}, {3.528073782694652*^9, 3.528073788439788*^9}, {3.528134805765039*^9, 3.5281348065528316`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"x1", "=", RowBox[{ RowBox[{"StateResponse", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"SystemsModelFeedbackConnect", "[", RowBox[{"ss", ",", "lqg1", ",", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{ "0", ",", "0.1", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}]}], "}"}], ",", "0", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}]}], "]"}], "//", "Flatten"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"x2", "=", " ", RowBox[{ RowBox[{"StateResponse", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"SystemsModelFeedbackConnect", "[", RowBox[{"ss", ",", "lqg2", ",", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{ "0", ",", "0.1", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}]}], "}"}], ",", "0", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}]}], "]"}], "//", "Flatten"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"KX1", "=", RowBox[{"-", RowBox[{"\[Kappa]", ".", RowBox[{"Transpose", "[", RowBox[{"{", RowBox[{"x1", "[", RowBox[{"[", RowBox[{"5", ";;", "8"}], "]"}], "]"}], "}"}], "]"}]}]}]}], ",", RowBox[{"KX2", "=", RowBox[{"-", RowBox[{"\[Kappa]", ".", RowBox[{"Transpose", "[", RowBox[{"{", RowBox[{"x2", "[", RowBox[{"[", RowBox[{"5", ";;", "8"}], "]"}], "]"}], "}"}], "]"}]}]}]}], ",", RowBox[{"KX", "=", RowBox[{ RowBox[{"-", RowBox[{"\[Kappa]", ".", RowBox[{"Transpose", "[", "x", "]"}]}]}], "*", RowBox[{"180", "/", "Pi"}]}]}]}], "}"}], ",", RowBox[{"GraphicsGrid", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"KX1", "*", RowBox[{"180", "/", "Pi"}]}], ")"}], "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{ RowBox[{"(", RowBox[{"KX2", "*", RowBox[{"180", "/", "Pi"}]}], ")"}], "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{ RowBox[{"(", "KX", ")"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "2"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", "All"}], ",", RowBox[{"PlotStyle", "\[Rule]", " ", RowBox[{"{", RowBox[{"Blue", ",", "Cyan", ",", "Red"}], "}"}]}], ",", RowBox[{"PlotLabel", "\[Rule]", " ", RowBox[{"Style", "[", RowBox[{ "\"\<\!\(\*SubscriptBox[\(u\), \(1\)]\)\>\"", ",", "Bold", ",", RowBox[{"FontSize", "\[Rule]", " ", "14"}]}], "]"}]}]}], "]"}], ",", RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"KX1", "*", RowBox[{"180", "/", "Pi"}]}], ")"}], "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{ RowBox[{"(", RowBox[{"KX2", "*", RowBox[{"180", "/", "Pi"}]}], ")"}], "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{ RowBox[{"(", "KX", ")"}], "[", RowBox[{"[", "2", "]"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "2"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", "All"}], ",", RowBox[{"PlotStyle", "\[Rule]", " ", RowBox[{"{", RowBox[{"Blue", ",", "Cyan", ",", "Red"}], "}"}]}], ",", RowBox[{"PlotLabel", "\[Rule]", " ", RowBox[{"Style", "[", RowBox[{ "\"\<\!\(\*SubscriptBox[\(u\), \(2\)]\)\>\"", ",", "Bold", ",", RowBox[{"FontSize", "\[Rule]", " ", "14"}]}], "]"}]}]}], "]"}]}], "}"}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", " ", "700"}], ",", RowBox[{"Frame", "\[Rule]", " ", "True"}]}], "]"}]}], "]"}]}], "Input", CellChangeTimes->{{3.528073640860676*^9, 3.52807367118832*^9}}], Cell["\<\ Conclusion : Design with \[Rho] = 5 recovers pure LQR performance, and \ provides larger roll - off at higher frequencies (lower sensitivity to \ measurement noise)\ \>", "Text", CellChangeTimes->{{3.5281915007393427`*^9, 3.528191507383312*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Questions?", "Section", CellChangeTimes->{{3.528191538089341*^9, 3.5281915412446203`*^9}}], Cell["", "Text", CellChangeTimes->{{3.528073865241042*^9, 3.528073875263371*^9}, { 3.5281342219965763`*^9, 3.5281342234347887`*^9}, 3.528191495536333*^9}] }, Open ]] }, Open ]] }, AutoGeneratedPackage->None, ScreenStyleEnvironment->"SlideShow", WindowSize->{1405, 856}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, ShowSelection->True, CellContext->Notebook, FrontEndVersion->"8.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (October 5, \ 2011)", StyleDefinitions->Notebook[{ Cell[ StyleData[StyleDefinitions -> "Default.nb"]], Cell[ CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[ StyleData[All, "Condensed"], MenuSortingValue -> None], Cell[ StyleData[All, "SlideShow"], DockedCells -> { FEPrivate`FrontEndResource["FEExpressions", "SlideshowToolbar"], Cell[ BoxData[ GraphicsBox[ TagBox[ RasterBox[CompressedData[" 1:eJzt3U9oXGe65/HAgEBSIVllV1UQjkEkyGAnYNyy2w43ThzjONyOO3YTbDqI 65tO0rG5gY7s0Ol0B4duQRbugAlO2pmFMY17Ydkh2Wnh3cVcr41hGBjMLGYx i6rFLGYxi4FhHp9H53mf854/daokOXL0/dxqR64657zv+5z36EL9/J4z85vf /eo3/+mpp556W/73r6NPPfXo5w4AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAACwubWdNTlI4QErPvoRaU9aA/KjGGLHsm74o5VVyW9W XcaKLesfZIgW64xiuJ7U7OQQRys8lfkj5Hu73kNrZz2GHddEu7a12rHvvuva 7fUzULsbsP8AAAAAAAAAgDV39OjR559/XtMH//7BRP79vXv3yi47ZmbsW2LZ fX5+/pNPPjl79qzssm3btmgvbaLiK2U52tFy2g05QuGnsm++RX9Y62o0Ovmo OnPRkRYe2TvdbN4Y/HWx2dy6dascfOgd29mobue2be9NTfktrzab8o68H9Vn f/aA+aP5w150W8qOcijdUv70DVUcJH+c0+440WY6iqvZHhaOorD+hcWpQ7aX 41zKFlCOdmTr1joHlE+lOOdzJ0veiXouf17NblBxfNn3Ro26VfTquV27jpx6 61f/dk5e8sPufXNlZYx2FHOvHv7FO2d035+9fGh6+/bqfaUt2UWaqF//3Qd+ Lrvkr1B9f+6Vl6Xdvi/ZzC7k4XbURh/tWLmvDNAqUPjbRnev2boczVqPSlen ejpSf5DC8ygb/NPxN/Qk/vOZf5F2bb7l9+pbAf8a6EQDAAAAAAAAADaUdrt9 9+7da9euRSGFvv/w4UNN3/z7i4uL8r5sL+/Lp/pXcf/+ff3h1q1bzz//vA+S 5M1PPvmkIgc5evTow3JyQNlXjlC2gfRBepKPzLRv8/Pz2rT/6Pbt2/KR9FM+ KovGtALNylBG3j/fbD4Y/HWj2dyyZUtz2B19t3du23Zpaqpil4vN5jNTUzoK cWTrVv/pxMRERRFuuC1lR+mwVlI+8gc5n/SqrFDRcQo3llFcrBy4jNFGUVb/ fHH60mTwXmXB99dI8Wr2POrwveTURPPTOuZPq25p9e87qCOn3vrTP/5+5T/+ PXot/nBbPqoYjlxKv/q3c3+9s5zf952/XHx2drYwJZS/yl6yjewo29Q5BY9K 8Z//JrtYaBW9X/+l00mOMNyOel7qbC/Vk2FqVBoNcO7VwwM1Lcexs6Clk9fu fXOFk6GsdP4g0QbSn8JqyAkqHELNCthLDq6zkYgQAAAAAAAAAJ447Xb7q6++ un//vqYq9r30wYMHNXo7fvy4foVu2y8vL9+6dUuzLflZtvn4448709MTifn5 +fuJ6e3bNSOQY2o+WBHcaD54+PDhiZwtKc0Ho0+f27VLWteIMAry5AeNLJeW lqKmLR+8fPlyWaolA9EKSCsVoczQ+eD1yclGoyGtD7ejdWl/s3mnMhzU1/fJ Ejw9xVE+qN0oHKAub7QtXxwbs3LJR1ET+5Nsrs5xPmw0olBS+vZ9jVHISC2q K6x/VJy+ogyu4nW62SxMQ/ZXZouFPX9masrv8mFS//wklJpEm1WcKW/HzMy5 Ly9ZGCQ///KD9+UlP1jqt/DtN4Up3u59c4s/3LYs7Mznn8mOpy8sfHrjugVD /3zmX/Kl8CGXHLxOSmsh1wsHD0S/ZPR96cBHV7/W10LFK21OjqA7Sm8H2lHn s3a+bEd5x0dsO/fsiYK8uVderj5C9Dp08oTFmlY66flApTtx7oP89vLzr39/ wXorbekEkLPpT64NQfe1CtSsnsyKin8SAAAAAAAAAADYyNrttgZh+/bts+hB 3jx79qwuCVxcXPRRzo6ZGQ0ENQrU1XlRkCeH8tGb0F00Usx/mSzv+HxwS1YS yzRt/WAjiZaibTQifG7XLh8MnTx5UnND+fPZ2VkfrMgPt27d0vgv+si6pAGi tlgRyvy4+WAUIVW//pjUTU5HlA+Ojo7WzwdtMuTzwe+npsqWIubzQR2Cblkz HNSXLbiz1alRPlgxnPy5q16xGL2OjY83s8ljzXAw6rk4n32/NToaHTm/zFC2 8UUrI1eoLRv85Qfvb23LfqMN57X5tzUl1IV+Pt/ZvW/OPnrpxJvRjnJky8g0 lorieAu55CWt9A2PbLmf5YMWVOn70v/RRKOS/kLw+eDuAz8faEefD1bvKGWx EnWmp/08tHxwZGSkb9PWuj5Z0pdORl2/dLKx/gMGfyLeXfyzHurcl5fkrFln ZEv589DJEzaEA68fsyFYBWQa1K8e+SAAAAAAAAAAPInaydMDHz58ePbsWfuq XFy7dm15efny5cvyp//+WUM3DfKWlpbu3r1r4aB+Py9/ys+yo628k3csUqyZ D+qaGk/esXxQQ0N9U1u03W0Vkq6LlO7JzzY6H0DcunVLbx96/vz5/LfcunxS N6jOm2Qv2Vc2kI6NOh82Gj60Gkn4DewLdh8hvTg2lt8y0kgX30Whmzbkj3Bs fNxyty8mJnQg65cPPihfCleRD+ZDuu8mJ6XnhaPQ15XJSX9byKHzwagO95KO tUZHtd3ZsTEpWnU0GXWsov7yOjU+rtWTnvddQphfPGinryKO8QvHLODbkrNz z57FH25/euN6Z3racskdMzMaG8n7mirmg3h558znn+nxXzh4wGeaUT4oh5ID NisXcvp80CfLUT6Y70aeXg4+H2wU/UOCsh2jfLCwaL56WigphZ+HP3v5UM0j qGa63jnKB+UlTWypnMO+RA2XGsufb/zmX6MJYHVoJit8Hx356aflLOsqSGso 5INJutq3es1hn/UJAAAAAAAAAPjRabyljyDc4pbSaKKnKwRthV07faKfZoKy 1+XLl6NVJHpAXX7YmZ7Wr5Hr54NbSu6tJ+/Y/UWjb86j3bUnus5RutFoNKST 0tUtbvWi5oNLS0vy0f37931K0nH3XD1//nzffFCPpqmol88H84HF1uRxflE+ WOebef1afn923y8mJjSWst3lh9boqLyv4WAjDaHWLx98UHKX0Yp8cOe2bVE4 2MrmpzoKqaHfbDa506nOyeHywXb2kYj3kmNGp0l+ODU+7tt9b2rK5tjpGvWX o2nIKMfRj0KSVbSEsHBRpF88WD2u53btsuV71hNt0VIwjbNlzreefnqL+ycB v3jnjO6rK870avVJvaX/C8kqwk9vXPehnuWDv/vbFb0Z6UdXvy67nG1W9M0H bdK2KukQogNquers2HHpmD9NeVq9QydPaAZqv9OifLDiCPmm86WLaltRunw+ qNnluS8v2SWfnwCyvUwAOdH+5sktlw8OWj0AAAAAAAAAwBOn1WppHGbRm8Vt z87O6h1E7Sv65eXlpaUlzQct9asO7B5DPqgf6S1SNbuxZFO2P378eHQD1Vaa D+7du9cPUNvVBZXSYc0HG/0e+mbZhIluIGmhVRQZaJYa5YN1ApH8ujmNkLQV jXWUrhiyNZ562HXNB2Ww+WijVZIPyvtRyjabJqTRKJ7JrtSznO5RUDJUPhjl kpbfbUlXwloc5qNJG50c3z+48M7UlIaDhfXX02pTV6sXLSGUDtgkHHrxoGZ8 f72z7OPgaBdtXbNpHalOJ70r6ZmLn1n9W7mlYfJX2eXA68csSdzibkqsIddH f7viY8qKbrfq5YOFvzQKlR2w5r7a5+rrXas3vX17tNAvygfrzEB/zHzp3jz7 2zqli0o09+phOzV1JkDTPbM1ygcHqh4AAAAAAAAA4InTarX8Iwj1Tp4aFzYa Db3LqIYm/uGD8k5Z6hflfRVb5rfPr1ux9M3fX9QHbdJ56a2mlvZ9+O3bt/XO qNoB2cBGoUfTfHB0dPTmzZu6utAWUv3hD3/QxY/6WMO++WBe/UVtUcrmA7i+ TfjETVcI5vfVCFKfeWfFXNd8MFpkV3gcnw9ezBZqZGTEr2myUTST24ralldc TjdcPhgVoZUGaj4U0+pFCaZFddGIKuqvKaG/GaO+73t+Z2qqMDOtv3hQjnnu y0uW8VXniW2n47KhuVcP9w3IHp2LdOMt7o6+KyHX1a/lJP7yg/f73mX0ScwH O2lIajdZtTqvQT7oSld9l9HCEtlxPr1xvXA2Ro3qVewvUvJBAAAAAAAAANg8 2tlHEDabzeXl5WvXrumis8XFRVtaGD0lcM3zQV2ceCtrfn5ec0DNB5cS9unD hLzTmZ62XEmHIztqP+VNvY+oDx81H2w0Grq60JYQ7piZ0TBRPtJ8ML9isU5J 1zsfzCduZU1ESdA65YP3sj/Pjo35VKgsHyxM2SxyinoSldSOMEQ+2M7eHfS7 ZK/Cystfo/u42lLQ6MSV5Uo+iInWVBYuIZQj33ErE2suHtSGotty1p+0tmxN 7zlZPQOlJwvffqMN2cY+5NLVi33vMvqE5oNaAcsHpaE1zAd96SruMlqWD+rT J/XmooXXUZ0KkA8CAAAAAAAAwGbQTh9BqCvs7Ml9moudPHnSlhb6O3yuRz4o Hfg4Rz7SuyBq6/4j6bO8s3fvXrt5o35RbwsA7V6RBw8elHeOHz9uN6XUfNAe pCg/a5Lob0y6wfPBfD5VJ0lZp3zwWPY5fdFdRuvngxUp2+mikg6dD9bcSxeL RaXWqTLcifNHLlxCONziwY7LjF6bf3ugWE3MvfKyf3Zeda7kwyn/AD4LufR6 3LlnT/VdRp/cfHD3vjndeGu7tbb5YFS6sruMluWDQwfEHfJBAAAAAAAAANh8 WskjCO/evTsxMaGB4LOzsxqCNJOnB+rSQsvUqp8qOHQ+KNtb0uf5fDBJkFa2 0SxP/vRP2pKxyEB0/aPd1FH+1DftaH4sehztrWwmH+n7P1Y+2MzeYTV/t9Wy HWvemDSfD9q9OiPyZs18cGRkxN//80H2LqM188Fj4+NleVDU7ceWD3aSxWL5 UsvGq8wHOyVLCIdbPKhH++udZct36sdqspnPB/uuO6uTD+qVe+LcBxV3GX1C 88F2eh9XvY3n2t5fNCpd2V1GC0s0dN2iCpAPAgAAAAAAAMAm0UofQfjs7Ozi 4qIGhT5K07jNP3xQPqrIB/2qw4HyQQsft6Z0uV/LPX/QttE4z3qr8ZkuFZRR HE4cTckQ/KJCnw9uSR5QKBvovUatG3695OPJB7+bnJSNbzSbZS97ht0a5oPX Kxu9V9JKPh9sjY6W3WW0Zj5YMYoNng/qYxMtVrN7ilaHvPklhP51Z2qq/uLB jst3Bk3H8tlW/Xww//A7C7mkA9Pbty/+cLvsLqPrnQ9q3drlCqvnI7882ewX 75zRLV868aZ/ZmVUQ7vtat/WC0snf1bcZfQx5IMDVQ8AAAAAAAAA8CRqp48g nJ+f1xuNWihmwdzhw4dtkZ3mdxaxRfmFHK3mnUhte7/esOx7aX9MWxVoSwjt K/2vvvrqYTl7xmK0FlIPvry8LMPXr+jtzceZD/Z9lcVzq8kH67+q80GpW9ld RgvvI/qk5IOFpY7yQR+rycFvFFVPXz7k7RQtIbTXqfHx+osHO6tY/5XPtqqr VzMf3JawI+fvMlozH7TrvSJvjQ545uJnb579rfSn4qX/8MDva9WT/ki3869f vHPmT//4u24mTfhnVvoaSrerW5ctfdNlpau4y+i65oN9q3fk1Fs15yQAAAAA AAAAYCNrp48gXFpa0uf0+fUjuiLv5s2bunzP1hVqEqd3IvULjlrJHT5tWV+z /E6k1rrPB8vioXxa106eDadtaUPSjfv371++fHk0WXg14chflxN6BJ8Pyl47 ZmZkR3+3UjnyBswHbb3SRssHNSuRvhXeZTS6T+lmzgejJzO2S5YQ3pma0jWJ 9VOejZYP6m8P6cavf3+h8C6jffPBOi/7NTXojvnEreaO8jp9YcEmvP0ushr2 fUmhfLcrSld2l9F1zQf7vha+/aZ+bA0AAAAAAAAA2MhaySMIdZFdu93262Lk B83Obt68aTmIbK9LDpeXl6e3b2+6R+bpcSxk9Hcita++I8Plg53k3o+6hFAX BurP0nThcwxlGws0fT4oR9PjSCf9eqUNmA/acTZgPqgR7c5t2/J3GZWPNk8+ 2MreTDWfD0ZPuJMfmkkg6DcbdPFgZ0Pmg7px2V1G1yQfLDzgmYufSd/09ebZ 3xa+ol81Vr3f/e3KR/K6+vVC+pKf9SMZhRxwx8yMhYN2dnwN33j/3eqmZbzR vmWlK7vL6HqvH6wewqGTJ8gHAQAAAAAAAOCnoZU+gnB5eTl6Bpn8oA/vO3/+ vF/eopna/cTi4uInibt37+rj//wTDOWdpaWljxOf5Mg2Q+eDtoRQ+jC9fbv0 U9ct2rf3llrKzxpoat+WEn6ZpD7zy25w+qPkgy+OjY0kRktYh9cwHxxJFbZ4 3a0H7JsPSpHl0/eyUZeumPtp54OagVo+eDG5j6g+2FFePvvLt6WX0unc4sFB I56NmQ/q6ArvMto3H5Tj27RslCjMB6UC+RXEkbL1g9pi1IpGhIs/3NZP9TL0 u/sa2hEqmm72Wz9opSu8y+i65oO798317T/5IAAAAAAAAAD8NLSTRxAuLS2d PXs2SsTkh5MnT8pHe/fujb7Gl5+f27Xr8uXLGgvqvT0PHz7svwMXulhP3cqR LQ8ePCg/7Nu3r+xrZ3lnfn5eN25mH3co2x89elSOfPz48cL+2xFky8XFRX26 4mLCf1GvGaI9layixTrFHDoftKVJhXxv1zAfLFxuKeTNQfNBXQ0XLaA7Nj7u j1OWD743NbUB80Fpom8++KLLBy1rljFqtiXjrW5Lfi4bXfXZ9DZmPmgFyd9l tE4+qEFV2eUQBW0+H7SniBbSf7fQKnr+YGGL0kP99J+Ov+H/jURZDfu23qp8 /mBUuvxdRtc3H6xXPcJBAAAAAAAAAPgJ0K+Xm82mfh/uwwVN1uyrcn9bPI0I LUsqXCBj2/i1J/kv+bVp/ea8sHvSrm0TfTNvfTCF4YjvrbUbLQKKRl3YYp1i 1oyf8jGffjPvVz5GLArxq9JOjY+vJh/0yxK96LmBdfLBlUVPubuMfu96W5YP fpjElGX5YGFJH0M++Ex2OeRsST444W7Jq6Wzq8m3daWorbL0c6BUWjb+0z/+ fuU//n3u1cOrzAerc8lB88FOyV1GH527fvmgv99v9RXhd9QD2q+gMtGgrALa GTu+/h5Y+PYbXUJYGL3la2iToW/rfUu3JXeXUSvdOuWDg1YPAAAAAAAAAPDk amfX0EUf+bV1+V0sCrHQreXunmdxSeFSFP26u/D4hd3Lb6Pd25oqO44f4Cpb 7FvJofPBmslOK/uQO03W6iQC+Xyw+taaA+WDrfS+iPm7jD7I9tbywavu/T9O TJTlGmubD25tt/z9PO9VFqGwXM3sEwOjfNZSQp0/l3LZ6DrlgxYPDZ0PSmUG ygdt1lWHXO2iu4xKD+vkg/VzrlbJgsSa+2rf8p3XT63zhU/fy+eD9c9dndJF dxmVmhTmg+/85aI+QHD1+eCg/ygCAAAAAAAAAPDkqlgb0vcjv5wnn9/5DfLL f2ouS6nTveqDDLoKZrjFMo8nH/Sp03eTk4W5RmHf1jsf1B2bubuMFuaDvlB3 pqYqCuVXIGqSuJr1g/uzXfIPEIy29Me/5/JBP7ovypPNVm6lZ37Ltc0HT19Y qBkWGxm4ZkMvHDxQJx/8651l2filE2/WzAc7RXcZlW10Ud7Gzwe185q+Lf5w u/X00/kzuE75oLUe3WVUS1d/CWf9CpAPAgAAAAAAAAAwhMeQD8oGp7MJlz0C L4ot9jeb+90z2h5PPtguustoPh9sZZ+79yB5UmF+FNFyP5+yDZcPdpJEzPft yuRkYaa2M7eZ3T3yvdx9RwuTnfxm+VO8Vvmgpm+f3rg+aDz06NGcyc0/X5t/ u+8MfG7XrnyQVCecamXvMir7Pin5oG7w7OysbiMdi2bpuuaDndxdRmWzwnzw yKm3qkdRpwLkgwAAAAAAAAAADGGV+WCz35O/1DNTU9Ez/vYnd3nV2GIlL9i6 9V76kT5S7fHkg53Ku4z6fFC28cvr7mUjQj8Kv03LPSSusNR+90LRbT91DaA9 6k632blt2/fZbfQBkZpLyqf+I9lydmwsOndRrBl1z9dq9fmgHOTA68fqLwPU HtqJtsVx1dGS7PLu4p91DaA+qs9K3TfkaufuMiobPyn5YNstIZSx6xJCH4iv az7Yzt5lVCqjpYvyQYtuXzrxZn6a5Y/pPyUfBAAAAAAAAABgNYbOB7+YmPiw 0ZB9K16n08c7Rkvn9HVpako2OJIEczeymZpFhI8nH+yU32XU8sFHmUtubeCD ZBfpf34UD9LFg5ayRfngnampvgU8nxRQRAGr7n4+qd7ppJLRp3oerQJyhIu1 6/8gXTxYdgPJNckHpWO2xCwKsKIt313885FTb9mzNYUtjjvz+Weak+bjIXln 7pWXLaLyY6kTcnVydxnVtYRRILUx88FO5RLC9c4HO7m7jGrpohJJfyzB1NuQ tkqexCpnX+aAf7gq+SAAAAAAAAAAAKsxdD5Y53UjXSSo6dulogV6ha8/Juvj ZMfHlg9qiLY/1xOfD2rqIX2rOYovJiZGRkb8Y+/O19vRvzT/EvuT5LTOLt9N TrZGRy2+0Qo0k2WDNRvVWLMweVmTfFDP1+59c/pwwE9vXH92draZW4u6Y2ZG QzR5yQZ2d1A5U5pSaUTYmZ72SynVkVNv2cF1LJZM1Qy5Otm7jBYGUmubD1bz +/bNByuWEObzwb4rWNvu3qT1S2d3GfUprZVIa6vnSP6UOjSTNDyaALoCVF5y QvUsd0rywZrVAwAAAAAAAAAA7fXMB6+nT8prJeSwdcI1v+buseWDnTRSiSI8 nw920tSj8E6k+WBxZGTEh3TtofJBPYKUQiPCO/2avp6Eg9GtQdtJpibvfNGv /vdcONgsWtPXXqN8UKu9c88eTYjk9c5fLh54/djPXj6kr1///oJ9dOjkCR/w adyp4ZcuT5ONbccjp96yVPHTG9e3tltWw0HzwXb2LqOPAql9c2ubD742//YL Bw9Y58tec6+8nF891zehK1xC6PPB3Qd+Xqf153btsjk8UOnsLqPyeuP9d32J bAOLXxe+/UbOXeEEOH1hYSL5NwNRPlizenLWChcnAgAAAAAAAACwOeXzwbKv /YfLB+3hfRZRvTg2dmVysnD7LyYmZsfGfDi1mnzQApGa+WAnjf/8OrsPG41o 6ZmOQvpZlrXJ6HQU+VhqiHzQ355UuvFMclfSwoWE301Onhoft1AyWvrn63+9 qP73svWvut9jNh+szokq6IienZ1dSJ5PV/j66OrXz+3aFY3IxnLg9WN+cV/0 +uUH71s46KtRP+TqpDnm6QsLfdcP+nlSZ+wWYtZ8+flcJx/0SwjltWNmxq5E n3jWeUm5dO2eGLR0dpfRfISqG3Smp898/llZ03J+Xzh4oJFehlE+WPMlpbax 1zk7AAAAAAAAAAD8tLXb7Z3JMr0Xx8bkNeuW3UVbalqnm9V8RUezWEfebI2O ygYfNhryOjU+Lj/rqjcNHezL/GeSR/vZAS1tLBzI/uRperplK43V8j2vPsij NU3btvkh5NMlDbYehSNuFPLyo9A4w8LBfKlrvnxv5U/pibTbSNqSumm7x8bH NdezaM+3G41O6z+bHNzXf2RkJF//wjnjT0rFhKnDRrRzz57TFxY+uvq1veSv Lxw8UNYlP5ZDJ0+c+fwz2/Hcl5dem39bk0GtRj4qPXLqrYVvv9FVaWWTwXdy evt2OayGlb688tGvf39BDqUrHOvXwXb0Q16oeH37jQ1Es0V5xy+pq+i5tiI9 tMx39765AZpO99WQetDSyY5aupdOvJkvkU2AHTMzctZ8l+Sczr162K6mqOyP KuA2rh6C9LbsogAAAAAAAAAAYBPyaZd+CR+lWn5LTWR0yzryR9PmNBGw4+j3 /357fytF21iVfcmv3dOBKHtame959UHywywMy3QU1lx+FNZ6fi+/S50CRr31 w6zZbv50169/oeiklE2YmqKCN5zqQfmx2EBsx4rJHM2HvqGezyJtPthHvhT1 62A7RkMu44NO32h15wt7bvOwZtPWul8FPETpJkqS62hKR+3awKN0WN4ZtP/k gwAAAAAAAAAAGEu7VN90SZ+FV0fZ0aLjNJvNsu3r9616Y/9R39sM1my0/ijW toCaENlx6rRbv+d68Jq716xn/f74eFfzrOq4s+NSQr+v7lh9InT4Nftv21uV op5H79ccctTtalaKQSdzvuf2Zs2mLWZVg5ZOR1pRIl/GOhMgirnr9L/vRAIA AAAAAAAAYBOyL//rb1nHQIfqu2X9gazyIJ1VFGTQ7Ycu4KDtrvkRhm634mgt Z9CTHu1Yv3r1m8hvP3QB8+Ot5psYqNHCjQdqvXDf1ZeubLM6E2A11QMAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAKu0FwAAAAAAAAAAAMCmcRwAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAADx2/wUAAAAAAAAAAADApvFfAQAAAAAAAAAAAGwa/w0AAAAA AAAAAADApvHfAQAAAAAAAAAAAGwa/wMAAAAAAAAAAADApvE/AQAAAAAAAAAA AGwavV6325P/e6SXSP4b/v7oP92VP/S/yebpZ+knulH6f72wU2/lsCuH7NrO tsPK31YaSBuJ9nV9tBbtp244SM96sNLV9OC9MCDbI90k0zHX+soxQs97boN0 k25aEevhyptpi91wvJ4dc2XEvlnby8prTVlprSV3oK71I3SzG85StH2352qT diQdSuhYqGA3FNBmg3U8LcpK8z0/BDuRvfhYPfeBFbNnzdt/er4Nq174M9Q7 PfduivZC7dKJECanVbnrRpbOIl+OcELSHf0ZCX/YxApvhnqFieMuLXfVZDoU Trad1FDEbnp9hl5mLkG78uzvtrGb0t3Qa39+wsfxJZr5BeCnf9phuxgzF3KY Lt1sA+FMuV8dbm64C8AuL2s/nQx2zt1e+aGF6WCfWzHSee1mlZ369HIIZ9ed E1e2MG/So/tpZU3YWQhHzbwVZkaYMtak3zLU0lUhHUJ2kFZZm5/WaqiVTT+b va6m0YUeCmuT1c6573A4J242hTGGazJcfe4y8vMpdCPMt3A8d/343V11s7vb PEjLFSayfeh2DNO1Gy44ezt7gfsDukngBpfW11UoO1GtEVfxzKD9VRT6nvY8 TIPMWe1lf+ravt3sgGx696Kxuesgcx35Trh5YW/ZmXdn019wXXds11DadbeV u47dNWZzPMyDcJbCvq6mmV8YmWKEizh/qWYuxvhysB76wttlHPbwo/e/OsJx w7xbqb772e2W6ar70zrr5lh6lqzmdo6ti9Z4N3Qrc4WEM+0OkDk3XWsq7G8T wW1lTYdj2jXXdU2GmWMXTC9sGtXIShhOZVq9cC1mz4Wfs+5K7/ki+E/DZPVF zn7myuZmSTokNzHS3rj6+usjzJvwQbfru2unLfQmned2TVlNbZalRUwL6y6q tNd2/nzx/QXbDXu5C9Q3aucilKMXupt2I1xYdtLTKe3npdvQz9Xw6yS6nl0D aVf8TAw9dtO8l/KTKXMR2DwOBcxOlNBzX7xeGEg32iYz0/2lE00Vt4Hvo2/d rmRXWn8+wwzI1iW3izu6r4n/1E2xsFlmsHYiXK/CYbMt+pL6muSLFs6hv5ys sfwZcc12XVt2qGgXX1WbvzZGfwSbnPHuXSc6P644vdzu3agPNuqVUtgPmcpE c8APO1N1u6yyMzBzBt2sCNPbLl8/EneWsifUTXU/W7IynXQ/xGffF9xdCP74 fmL03MSIehidkcwF66vtO+9OcXyJ+b18r3qZ39vpR+63np6FtI2VPcJwe67p 7C98a3blBHXD/2MJ150vibXXC73ypzT8LvBDtV/NaV/ChFyZRDYSP7Z0CoVO Z+Zb5uq1SR/qbnPfXzDpDAxFz9a8F9oPs3OlKit9T0dg14GVzVU1DNNORi9z iYSehTMY6uBOauaisZ+jeeROYzfU0bpjVbLuWnlcbdNz1A3thIH7k5UpU/jZ RmGdDg2Ec2FdDdeETbKe71s37O6nra9MOIVhqrkz3Muc0zCpuq7zYbZmy+bm UZi1rgp+Trl51/MFSlvs2jjCnEknRXpGQg+i0fptwzhC0cPsDIW0SdMr+Z/v lZ1Dd+7C3zM/ui6kfQgHctVMP89OLZuH7kLpZT8J89nKbafCih4at5/TWtq5 yfU8FNJdkKGb6bTx10/4W5jF3dBUuFAyU6bnTqF1JDNh7S/hZPiz28s0YjXo ht5kRtwLXQuDcB3phQ267mBp8fJdCHPVamA1sivHXWM9P/Uy0yG9rkKlwjVm pzecrqibYT7aibFaujnjpq9NUF+T0CebPJne2SWRHtlOgRtzdPatdnZSwrHt 6nCfu3lqkyUddNfv54vQC1eTTS33Yc+3YhdB+HXSsyvQdTucrMx14KprV2xu ovhzblXpWufjXwjpmQsF74XW3IjTs5ueVPupG0YYpoCfU+HEuXnsrj07/d2w b3aDcNr9BRnqEGZFdC7C5HEH9XMn1MLezRzEeh8unjDeTA39qcrOfzcT/fwL 5ylt3nYJm4RLyPrYDQ2EOqbTxpro+b/YCNyc6rle+evAxmAnywrY/V8AAAAA AAAAAAAANo3/DQAAAAAAAAAAAGDT+D8AAAAAAAAAAAAANo3/CwAAAAAAAAAA AGDT+H8AAAAAAAAAAAAANo3/D2zci8M= "], {{0, 90}, {2400, 0}}, {0, 255}, ColorFunction -> RGBColor], BoxForm`ImageTag[ "Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable -> False], BaseStyle -> "ImageGraphics", ImageSize -> Magnification[1], ImageSizeRaw -> {2400, 90}, PlotRange -> {{0, 2400}, {0, 90}}]], "", CellMargins -> 0, CellChangeTimes -> {{3.516548605022827*^9, 3.516548633085456*^9}, { 3.516616961864057*^9, 3.516616982571135*^9}, 3.518187891728025*^9, {3.518199033657151*^9, 3.5181990340305967`*^9}}]}, CellMargins -> 0, CellBracketOptions -> { "Color" -> RGBColor[0.739193, 0.750317, 0.747173]}], Cell[ BoxData[ GraphicsBox[ TagBox[ RasterBox[CompressedData[" 1:eJzt3U9oXGe65/HAgEBSIVllV1UQjkEkyGAnYNyy2w43ThzjONyOO3YTbDqI 65tO0rG5gY7s0Ol0B4duQRbugAlO2pmFMY17Ydkh2Wnh3cVcr41hGBjMLGYx i6rFLGYxi4FhHp9H53mf854/daokOXL0/dxqR64657zv+5z36EL9/J4z85vf /eo3/+mpp556W/73r6NPPfXo5w4AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAACwubWdNTlI4QErPvoRaU9aA/KjGGLHsm74o5VVyW9W XcaKLesfZIgW64xiuJ7U7OQQRys8lfkj5Hu73kNrZz2GHddEu7a12rHvvuva 7fUzULsbsP8AAAAAAAAAgDV39OjR559/XtMH//7BRP79vXv3yi47ZmbsW2LZ fX5+/pNPPjl79qzssm3btmgvbaLiK2U52tFy2g05QuGnsm++RX9Y62o0Ovmo OnPRkRYe2TvdbN4Y/HWx2dy6dascfOgd29mobue2be9NTfktrzab8o68H9Vn f/aA+aP5w150W8qOcijdUv70DVUcJH+c0+440WY6iqvZHhaOorD+hcWpQ7aX 41zKFlCOdmTr1joHlE+lOOdzJ0veiXouf17NblBxfNn3Ro26VfTquV27jpx6 61f/dk5e8sPufXNlZYx2FHOvHv7FO2d035+9fGh6+/bqfaUt2UWaqF//3Qd+ Lrvkr1B9f+6Vl6Xdvi/ZzC7k4XbURh/tWLmvDNAqUPjbRnev2boczVqPSlen ejpSf5DC8ygb/NPxN/Qk/vOZf5F2bb7l9+pbAf8a6EQDAAAAAAAAADaUdrt9 9+7da9euRSGFvv/w4UNN3/z7i4uL8r5sL+/Lp/pXcf/+ff3h1q1bzz//vA+S 5M1PPvmkIgc5evTow3JyQNlXjlC2gfRBepKPzLRv8/Pz2rT/6Pbt2/KR9FM+ KovGtALNylBG3j/fbD4Y/HWj2dyyZUtz2B19t3du23Zpaqpil4vN5jNTUzoK cWTrVv/pxMRERRFuuC1lR+mwVlI+8gc5n/SqrFDRcQo3llFcrBy4jNFGUVb/ fHH60mTwXmXB99dI8Wr2POrwveTURPPTOuZPq25p9e87qCOn3vrTP/5+5T/+ PXot/nBbPqoYjlxKv/q3c3+9s5zf952/XHx2drYwJZS/yl6yjewo29Q5BY9K 8Z//JrtYaBW9X/+l00mOMNyOel7qbC/Vk2FqVBoNcO7VwwM1Lcexs6Clk9fu fXOFk6GsdP4g0QbSn8JqyAkqHELNCthLDq6zkYgQAAAAAAAAAJ447Xb7q6++ un//vqYq9r30wYMHNXo7fvy4foVu2y8vL9+6dUuzLflZtvn4448709MTifn5 +fuJ6e3bNSOQY2o+WBHcaD54+PDhiZwtKc0Ho0+f27VLWteIMAry5AeNLJeW lqKmLR+8fPlyWaolA9EKSCsVoczQ+eD1yclGoyGtD7ejdWl/s3mnMhzU1/fJ Ejw9xVE+qN0oHKAub7QtXxwbs3LJR1ET+5Nsrs5xPmw0olBS+vZ9jVHISC2q K6x/VJy+ogyu4nW62SxMQ/ZXZouFPX9masrv8mFS//wklJpEm1WcKW/HzMy5 Ly9ZGCQ///KD9+UlP1jqt/DtN4Up3u59c4s/3LYs7Mznn8mOpy8sfHrjugVD /3zmX/Kl8CGXHLxOSmsh1wsHD0S/ZPR96cBHV7/W10LFK21OjqA7Sm8H2lHn s3a+bEd5x0dsO/fsiYK8uVderj5C9Dp08oTFmlY66flApTtx7oP89vLzr39/ wXorbekEkLPpT64NQfe1CtSsnsyKin8SAAAAAAAAAADYyNrttgZh+/bts+hB 3jx79qwuCVxcXPRRzo6ZGQ0ENQrU1XlRkCeH8tGb0F00Usx/mSzv+HxwS1YS yzRt/WAjiZaibTQifG7XLh8MnTx5UnND+fPZ2VkfrMgPt27d0vgv+si6pAGi tlgRyvy4+WAUIVW//pjUTU5HlA+Ojo7WzwdtMuTzwe+npsqWIubzQR2Cblkz HNSXLbiz1alRPlgxnPy5q16xGL2OjY83s8ljzXAw6rk4n32/NToaHTm/zFC2 8UUrI1eoLRv85Qfvb23LfqMN57X5tzUl1IV+Pt/ZvW/OPnrpxJvRjnJky8g0 lorieAu55CWt9A2PbLmf5YMWVOn70v/RRKOS/kLw+eDuAz8faEefD1bvKGWx EnWmp/08tHxwZGSkb9PWuj5Z0pdORl2/dLKx/gMGfyLeXfyzHurcl5fkrFln ZEv589DJEzaEA68fsyFYBWQa1K8e+SAAAAAAAAAAPInaydMDHz58ePbsWfuq XFy7dm15efny5cvyp//+WUM3DfKWlpbu3r1r4aB+Py9/ys+yo628k3csUqyZ D+qaGk/esXxQQ0N9U1u03W0Vkq6LlO7JzzY6H0DcunVLbx96/vz5/LfcunxS N6jOm2Qv2Vc2kI6NOh82Gj60Gkn4DewLdh8hvTg2lt8y0kgX30Whmzbkj3Bs fNxyty8mJnQg65cPPihfCleRD+ZDuu8mJ6XnhaPQ15XJSX9byKHzwagO95KO tUZHtd3ZsTEpWnU0GXWsov7yOjU+rtWTnvddQphfPGinryKO8QvHLODbkrNz z57FH25/euN6Z3racskdMzMaG8n7mirmg3h558znn+nxXzh4wGeaUT4oh5ID NisXcvp80CfLUT6Y70aeXg4+H2wU/UOCsh2jfLCwaL56WigphZ+HP3v5UM0j qGa63jnKB+UlTWypnMO+RA2XGsufb/zmX6MJYHVoJit8Hx356aflLOsqSGso 5INJutq3es1hn/UJAAAAAAAAAPjRabyljyDc4pbSaKKnKwRthV07faKfZoKy 1+XLl6NVJHpAXX7YmZ7Wr5Hr54NbSu6tJ+/Y/UWjb86j3bUnus5RutFoNKST 0tUtbvWi5oNLS0vy0f37931K0nH3XD1//nzffFCPpqmol88H84HF1uRxflE+ WOebef1afn923y8mJjSWst3lh9boqLyv4WAjDaHWLx98UHKX0Yp8cOe2bVE4 2MrmpzoKqaHfbDa506nOyeHywXb2kYj3kmNGp0l+ODU+7tt9b2rK5tjpGvWX o2nIKMfRj0KSVbSEsHBRpF88WD2u53btsuV71hNt0VIwjbNlzreefnqL+ycB v3jnjO6rK870avVJvaX/C8kqwk9vXPehnuWDv/vbFb0Z6UdXvy67nG1W9M0H bdK2KukQogNquers2HHpmD9NeVq9QydPaAZqv9OifLDiCPmm86WLaltRunw+ qNnluS8v2SWfnwCyvUwAOdH+5sktlw8OWj0AAAAAAAAAwBOn1WppHGbRm8Vt z87O6h1E7Sv65eXlpaUlzQct9asO7B5DPqgf6S1SNbuxZFO2P378eHQD1Vaa D+7du9cPUNvVBZXSYc0HG/0e+mbZhIluIGmhVRQZaJYa5YN1ApH8ujmNkLQV jXWUrhiyNZ562HXNB2Ww+WijVZIPyvtRyjabJqTRKJ7JrtSznO5RUDJUPhjl kpbfbUlXwloc5qNJG50c3z+48M7UlIaDhfXX02pTV6sXLSGUDtgkHHrxoGZ8 f72z7OPgaBdtXbNpHalOJ70r6ZmLn1n9W7mlYfJX2eXA68csSdzibkqsIddH f7viY8qKbrfq5YOFvzQKlR2w5r7a5+rrXas3vX17tNAvygfrzEB/zHzp3jz7 2zqli0o09+phOzV1JkDTPbM1ygcHqh4AAAAAAAAA4InTarX8Iwj1Tp4aFzYa Db3LqIYm/uGD8k5Z6hflfRVb5rfPr1ux9M3fX9QHbdJ56a2mlvZ9+O3bt/XO qNoB2cBGoUfTfHB0dPTmzZu6utAWUv3hD3/QxY/6WMO++WBe/UVtUcrmA7i+ TfjETVcI5vfVCFKfeWfFXNd8MFpkV3gcnw9ezBZqZGTEr2myUTST24ralldc TjdcPhgVoZUGaj4U0+pFCaZFddGIKuqvKaG/GaO+73t+Z2qqMDOtv3hQjnnu y0uW8VXniW2n47KhuVcP9w3IHp2LdOMt7o6+KyHX1a/lJP7yg/f73mX0ScwH O2lIajdZtTqvQT7oSld9l9HCEtlxPr1xvXA2Ro3qVewvUvJBAAAAAAAAANg8 2tlHEDabzeXl5WvXrumis8XFRVtaGD0lcM3zQV2ceCtrfn5ec0DNB5cS9unD hLzTmZ62XEmHIztqP+VNvY+oDx81H2w0Grq60JYQ7piZ0TBRPtJ8ML9isU5J 1zsfzCduZU1ESdA65YP3sj/Pjo35VKgsHyxM2SxyinoSldSOMEQ+2M7eHfS7 ZK/Cystfo/u42lLQ6MSV5Uo+iInWVBYuIZQj33ErE2suHtSGotty1p+0tmxN 7zlZPQOlJwvffqMN2cY+5NLVi33vMvqE5oNaAcsHpaE1zAd96SruMlqWD+rT J/XmooXXUZ0KkA8CAAAAAAAAwGbQTh9BqCvs7Ml9moudPHnSlhb6O3yuRz4o Hfg4Rz7SuyBq6/4j6bO8s3fvXrt5o35RbwsA7V6RBw8elHeOHz9uN6XUfNAe pCg/a5Lob0y6wfPBfD5VJ0lZp3zwWPY5fdFdRuvngxUp2+mikg6dD9bcSxeL RaXWqTLcifNHLlxCONziwY7LjF6bf3ugWE3MvfKyf3Zeda7kwyn/AD4LufR6 3LlnT/VdRp/cfHD3vjndeGu7tbb5YFS6sruMluWDQwfEHfJBAAAAAAAAANh8 WskjCO/evTsxMaGB4LOzsxqCNJOnB+rSQsvUqp8qOHQ+KNtb0uf5fDBJkFa2 0SxP/vRP2pKxyEB0/aPd1FH+1DftaH4sehztrWwmH+n7P1Y+2MzeYTV/t9Wy HWvemDSfD9q9OiPyZs18cGRkxN//80H2LqM188Fj4+NleVDU7ceWD3aSxWL5 UsvGq8wHOyVLCIdbPKhH++udZct36sdqspnPB/uuO6uTD+qVe+LcBxV3GX1C 88F2eh9XvY3n2t5fNCpd2V1GC0s0dN2iCpAPAgAAAAAAAMAm0UofQfjs7Ozi 4qIGhT5K07jNP3xQPqrIB/2qw4HyQQsft6Z0uV/LPX/QttE4z3qr8ZkuFZRR HE4cTckQ/KJCnw9uSR5QKBvovUatG3695OPJB7+bnJSNbzSbZS97ht0a5oPX Kxu9V9JKPh9sjY6W3WW0Zj5YMYoNng/qYxMtVrN7ilaHvPklhP51Z2qq/uLB jst3Bk3H8tlW/Xww//A7C7mkA9Pbty/+cLvsLqPrnQ9q3drlCqvnI7882ewX 75zRLV868aZ/ZmVUQ7vtat/WC0snf1bcZfQx5IMDVQ8AAAAAAAAA8CRqp48g nJ+f1xuNWihmwdzhw4dtkZ3mdxaxRfmFHK3mnUhte7/esOx7aX9MWxVoSwjt K/2vvvrqYTl7xmK0FlIPvry8LMPXr+jtzceZD/Z9lcVzq8kH67+q80GpW9ld RgvvI/qk5IOFpY7yQR+rycFvFFVPXz7k7RQtIbTXqfHx+osHO6tY/5XPtqqr VzMf3JawI+fvMlozH7TrvSJvjQ545uJnb579rfSn4qX/8MDva9WT/ki3869f vHPmT//4u24mTfhnVvoaSrerW5ctfdNlpau4y+i65oN9q3fk1Fs15yQAAAAA AAAAYCNrp48gXFpa0uf0+fUjuiLv5s2bunzP1hVqEqd3IvULjlrJHT5tWV+z /E6k1rrPB8vioXxa106eDadtaUPSjfv371++fHk0WXg14chflxN6BJ8Pyl47 ZmZkR3+3UjnyBswHbb3SRssHNSuRvhXeZTS6T+lmzgejJzO2S5YQ3pma0jWJ 9VOejZYP6m8P6cavf3+h8C6jffPBOi/7NTXojvnEreaO8jp9YcEmvP0ushr2 fUmhfLcrSld2l9F1zQf7vha+/aZ+bA0AAAAAAAAA2MhaySMIdZFdu93262Lk B83Obt68aTmIbK9LDpeXl6e3b2+6R+bpcSxk9Hcita++I8Plg53k3o+6hFAX BurP0nThcwxlGws0fT4oR9PjSCf9eqUNmA/acTZgPqgR7c5t2/J3GZWPNk8+ 2MreTDWfD0ZPuJMfmkkg6DcbdPFgZ0Pmg7px2V1G1yQfLDzgmYufSd/09ebZ 3xa+ol81Vr3f/e3KR/K6+vVC+pKf9SMZhRxwx8yMhYN2dnwN33j/3eqmZbzR vmWlK7vL6HqvH6wewqGTJ8gHAQAAAAAAAOCnoZU+gnB5eTl6Bpn8oA/vO3/+ vF/eopna/cTi4uInibt37+rj//wTDOWdpaWljxOf5Mg2Q+eDtoRQ+jC9fbv0 U9ct2rf3llrKzxpoat+WEn6ZpD7zy25w+qPkgy+OjY0kRktYh9cwHxxJFbZ4 3a0H7JsPSpHl0/eyUZeumPtp54OagVo+eDG5j6g+2FFePvvLt6WX0unc4sFB I56NmQ/q6ArvMto3H5Tj27RslCjMB6UC+RXEkbL1g9pi1IpGhIs/3NZP9TL0 u/sa2hEqmm72Wz9opSu8y+i65oO798317T/5IAAAAAAAAAD8NLSTRxAuLS2d PXs2SsTkh5MnT8pHe/fujb7Gl5+f27Xr8uXLGgvqvT0PHz7svwMXulhP3cqR LQ8ePCg/7Nu3r+xrZ3lnfn5eN25mH3co2x89elSOfPz48cL+2xFky8XFRX26 4mLCf1GvGaI9layixTrFHDoftKVJhXxv1zAfLFxuKeTNQfNBXQ0XLaA7Nj7u j1OWD743NbUB80Fpom8++KLLBy1rljFqtiXjrW5Lfi4bXfXZ9DZmPmgFyd9l tE4+qEFV2eUQBW0+H7SniBbSf7fQKnr+YGGL0kP99J+Ov+H/jURZDfu23qp8 /mBUuvxdRtc3H6xXPcJBAAAAAAAAAPgJ0K+Xm82mfh/uwwVN1uyrcn9bPI0I LUsqXCBj2/i1J/kv+bVp/ea8sHvSrm0TfTNvfTCF4YjvrbUbLQKKRl3YYp1i 1oyf8jGffjPvVz5GLArxq9JOjY+vJh/0yxK96LmBdfLBlUVPubuMfu96W5YP fpjElGX5YGFJH0M++Ex2OeRsST444W7Jq6Wzq8m3daWorbL0c6BUWjb+0z/+ fuU//n3u1cOrzAerc8lB88FOyV1GH527fvmgv99v9RXhd9QD2q+gMtGgrALa GTu+/h5Y+PYbXUJYGL3la2iToW/rfUu3JXeXUSvdOuWDg1YPAAAAAAAAAPDk amfX0EUf+bV1+V0sCrHQreXunmdxSeFSFP26u/D4hd3Lb6Pd25oqO44f4Cpb 7FvJofPBmslOK/uQO03W6iQC+Xyw+taaA+WDrfS+iPm7jD7I9tbywavu/T9O TJTlGmubD25tt/z9PO9VFqGwXM3sEwOjfNZSQp0/l3LZ6DrlgxYPDZ0PSmUG ygdt1lWHXO2iu4xKD+vkg/VzrlbJgsSa+2rf8p3XT63zhU/fy+eD9c9dndJF dxmVmhTmg+/85aI+QHD1+eCg/ygCAAAAAAAAAPDkqlgb0vcjv5wnn9/5DfLL f2ouS6nTveqDDLoKZrjFMo8nH/Sp03eTk4W5RmHf1jsf1B2bubuMFuaDvlB3 pqYqCuVXIGqSuJr1g/uzXfIPEIy29Me/5/JBP7ovypPNVm6lZ37Ltc0HT19Y qBkWGxm4ZkMvHDxQJx/8651l2filE2/WzAc7RXcZlW10Ud7Gzwe185q+Lf5w u/X00/kzuE75oLUe3WVUS1d/CWf9CpAPAgAAAAAAAAAwhMeQD8oGp7MJlz0C L4ot9jeb+90z2h5PPtguustoPh9sZZ+79yB5UmF+FNFyP5+yDZcPdpJEzPft yuRkYaa2M7eZ3T3yvdx9RwuTnfxm+VO8Vvmgpm+f3rg+aDz06NGcyc0/X5t/ u+8MfG7XrnyQVCecamXvMir7Pin5oG7w7OysbiMdi2bpuuaDndxdRmWzwnzw yKm3qkdRpwLkgwAAAAAAAAAADGGV+WCz35O/1DNTU9Ez/vYnd3nV2GIlL9i6 9V76kT5S7fHkg53Ku4z6fFC28cvr7mUjQj8Kv03LPSSusNR+90LRbT91DaA9 6k632blt2/fZbfQBkZpLyqf+I9lydmwsOndRrBl1z9dq9fmgHOTA68fqLwPU HtqJtsVx1dGS7PLu4p91DaA+qs9K3TfkaufuMiobPyn5YNstIZSx6xJCH4iv az7Yzt5lVCqjpYvyQYtuXzrxZn6a5Y/pPyUfBAAAAAAAAABgNYbOB7+YmPiw 0ZB9K16n08c7Rkvn9HVpako2OJIEczeymZpFhI8nH+yU32XU8sFHmUtubeCD ZBfpf34UD9LFg5ayRfngnampvgU8nxRQRAGr7n4+qd7ppJLRp3oerQJyhIu1 6/8gXTxYdgPJNckHpWO2xCwKsKIt313885FTb9mzNYUtjjvz+Weak+bjIXln 7pWXLaLyY6kTcnVydxnVtYRRILUx88FO5RLC9c4HO7m7jGrpohJJfyzB1NuQ tkqexCpnX+aAf7gq+SAAAAAAAAAAAKsxdD5Y53UjXSSo6dulogV6ha8/Juvj ZMfHlg9qiLY/1xOfD2rqIX2rOYovJiZGRkb8Y+/O19vRvzT/EvuT5LTOLt9N TrZGRy2+0Qo0k2WDNRvVWLMweVmTfFDP1+59c/pwwE9vXH92draZW4u6Y2ZG QzR5yQZ2d1A5U5pSaUTYmZ72SynVkVNv2cF1LJZM1Qy5Otm7jBYGUmubD1bz +/bNByuWEObzwb4rWNvu3qT1S2d3GfUprZVIa6vnSP6UOjSTNDyaALoCVF5y QvUsd0rywZrVAwAAAAAAAAAA7fXMB6+nT8prJeSwdcI1v+buseWDnTRSiSI8 nw920tSj8E6k+WBxZGTEh3TtofJBPYKUQiPCO/2avp6Eg9GtQdtJpibvfNGv /vdcONgsWtPXXqN8UKu9c88eTYjk9c5fLh54/djPXj6kr1///oJ9dOjkCR/w adyp4ZcuT5ONbccjp96yVPHTG9e3tltWw0HzwXb2LqOPAql9c2ubD742//YL Bw9Y58tec6+8nF891zehK1xC6PPB3Qd+Xqf153btsjk8UOnsLqPyeuP9d32J bAOLXxe+/UbOXeEEOH1hYSL5NwNRPlizenLWChcnAgAAAAAAAACwOeXzwbKv /YfLB+3hfRZRvTg2dmVysnD7LyYmZsfGfDi1mnzQApGa+WAnjf/8OrsPG41o 6ZmOQvpZlrXJ6HQU+VhqiHzQ355UuvFMclfSwoWE301Onhoft1AyWvrn63+9 qP73svWvut9jNh+szokq6IienZ1dSJ5PV/j66OrXz+3aFY3IxnLg9WN+cV/0 +uUH71s46KtRP+TqpDnm6QsLfdcP+nlSZ+wWYtZ8+flcJx/0SwjltWNmxq5E n3jWeUm5dO2eGLR0dpfRfISqG3Smp898/llZ03J+Xzh4oJFehlE+WPMlpbax 1zk7AAAAAAAAAAD8tLXb7Z3JMr0Xx8bkNeuW3UVbalqnm9V8RUezWEfebI2O ygYfNhryOjU+Lj/rqjcNHezL/GeSR/vZAS1tLBzI/uRperplK43V8j2vPsij NU3btvkh5NMlDbYehSNuFPLyo9A4w8LBfKlrvnxv5U/pibTbSNqSumm7x8bH NdezaM+3G41O6z+bHNzXf2RkJF//wjnjT0rFhKnDRrRzz57TFxY+uvq1veSv Lxw8UNYlP5ZDJ0+c+fwz2/Hcl5dem39bk0GtRj4qPXLqrYVvv9FVaWWTwXdy evt2OayGlb688tGvf39BDqUrHOvXwXb0Q16oeH37jQ1Es0V5xy+pq+i5tiI9 tMx39765AZpO99WQetDSyY5aupdOvJkvkU2AHTMzctZ8l+Sczr162K6mqOyP KuA2rh6C9LbsogAAAAAAAAAAYBPyaZd+CR+lWn5LTWR0yzryR9PmNBGw4+j3 /357fytF21iVfcmv3dOBKHtame959UHywywMy3QU1lx+FNZ6fi+/S50CRr31 w6zZbv50169/oeiklE2YmqKCN5zqQfmx2EBsx4rJHM2HvqGezyJtPthHvhT1 62A7RkMu44NO32h15wt7bvOwZtPWul8FPETpJkqS62hKR+3awKN0WN4ZtP/k gwAAAAAAAAAAGEu7VN90SZ+FV0fZ0aLjNJvNsu3r9616Y/9R39sM1my0/ijW toCaENlx6rRbv+d68Jq716xn/f74eFfzrOq4s+NSQr+v7lh9InT4Nftv21uV op5H79ccctTtalaKQSdzvuf2Zs2mLWZVg5ZOR1pRIl/GOhMgirnr9L/vRAIA AAAAAAAAYBOyL//rb1nHQIfqu2X9gazyIJ1VFGTQ7Ycu4KDtrvkRhm634mgt Z9CTHu1Yv3r1m8hvP3QB8+Ot5psYqNHCjQdqvXDf1ZeubLM6E2A11QMAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAKu0FwAAAAAAAAAAAMCmcRwAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAADx2/wUAAAAAAAAAAADApvFfAQAAAAAAAAAAAGwa/w0AAAAA AAAAAADApvHfAQAAAAAAAAAAAGwa/wMAAAAAAAAAAADApvE/AQAAAAAAAAAA AGwavV6325P/e6SXSP4b/v7oP92VP/S/yebpZ+knulH6f72wU2/lsCuH7NrO tsPK31YaSBuJ9nV9tBbtp244SM96sNLV9OC9MCDbI90k0zHX+soxQs97boN0 k25aEevhyptpi91wvJ4dc2XEvlnby8prTVlprSV3oK71I3SzG85StH2352qT diQdSuhYqGA3FNBmg3U8LcpK8z0/BDuRvfhYPfeBFbNnzdt/er4Nq174M9Q7 PfduivZC7dKJECanVbnrRpbOIl+OcELSHf0ZCX/YxApvhnqFieMuLXfVZDoU Trad1FDEbnp9hl5mLkG78uzvtrGb0t3Qa39+wsfxJZr5BeCnf9phuxgzF3KY Lt1sA+FMuV8dbm64C8AuL2s/nQx2zt1e+aGF6WCfWzHSee1mlZ369HIIZ9ed E1e2MG/So/tpZU3YWQhHzbwVZkaYMtak3zLU0lUhHUJ2kFZZm5/WaqiVTT+b va6m0YUeCmuT1c6573A4J242hTGGazJcfe4y8vMpdCPMt3A8d/343V11s7vb PEjLFSayfeh2DNO1Gy44ezt7gfsDukngBpfW11UoO1GtEVfxzKD9VRT6nvY8 TIPMWe1lf+ravt3sgGx696Kxuesgcx35Trh5YW/ZmXdn019wXXds11DadbeV u47dNWZzPMyDcJbCvq6mmV8YmWKEizh/qWYuxvhysB76wttlHPbwo/e/OsJx w7xbqb772e2W6ar70zrr5lh6lqzmdo6ti9Z4N3Qrc4WEM+0OkDk3XWsq7G8T wW1lTYdj2jXXdU2GmWMXTC9sGtXIShhOZVq9cC1mz4Wfs+5K7/ki+E/DZPVF zn7myuZmSTokNzHS3rj6+usjzJvwQbfru2unLfQmned2TVlNbZalRUwL6y6q tNd2/nzx/QXbDXu5C9Q3aucilKMXupt2I1xYdtLTKe3npdvQz9Xw6yS6nl0D aVf8TAw9dtO8l/KTKXMR2DwOBcxOlNBzX7xeGEg32iYz0/2lE00Vt4Hvo2/d rmRXWn8+wwzI1iW3izu6r4n/1E2xsFlmsHYiXK/CYbMt+pL6muSLFs6hv5ys sfwZcc12XVt2qGgXX1WbvzZGfwSbnPHuXSc6P644vdzu3agPNuqVUtgPmcpE c8APO1N1u6yyMzBzBt2sCNPbLl8/EneWsifUTXU/W7IynXQ/xGffF9xdCP74 fmL03MSIehidkcwF66vtO+9OcXyJ+b18r3qZ39vpR+63np6FtI2VPcJwe67p 7C98a3blBHXD/2MJ150vibXXC73ypzT8LvBDtV/NaV/ChFyZRDYSP7Z0CoVO Z+Zb5uq1SR/qbnPfXzDpDAxFz9a8F9oPs3OlKit9T0dg14GVzVU1DNNORi9z iYSehTMY6uBOauaisZ+jeeROYzfU0bpjVbLuWnlcbdNz1A3thIH7k5UpU/jZ RmGdDg2Ec2FdDdeETbKe71s37O6nra9MOIVhqrkz3Muc0zCpuq7zYbZmy+bm UZi1rgp+Trl51/MFSlvs2jjCnEknRXpGQg+i0fptwzhC0cPsDIW0SdMr+Z/v lZ1Dd+7C3zM/ui6kfQgHctVMP89OLZuH7kLpZT8J89nKbafCih4at5/TWtq5 yfU8FNJdkKGb6bTx10/4W5jF3dBUuFAyU6bnTqF1JDNh7S/hZPiz28s0YjXo ht5kRtwLXQuDcB3phQ267mBp8fJdCHPVamA1sivHXWM9P/Uy0yG9rkKlwjVm pzecrqibYT7aibFaujnjpq9NUF+T0CebPJne2SWRHtlOgRtzdPatdnZSwrHt 6nCfu3lqkyUddNfv54vQC1eTTS33Yc+3YhdB+HXSsyvQdTucrMx14KprV2xu ovhzblXpWufjXwjpmQsF74XW3IjTs5ueVPupG0YYpoCfU+HEuXnsrj07/d2w b3aDcNr9BRnqEGZFdC7C5HEH9XMn1MLezRzEeh8unjDeTA39qcrOfzcT/fwL 5ylt3nYJm4RLyPrYDQ2EOqbTxpro+b/YCNyc6rle+evAxmAnywrY/V8AAAAA AAAAAAAANo3/DQAAAAAAAAAAAGDT+D8AAAAAAAAAAAAANo3/CwAAAAAAAAAA AGDT+H8AAAAAAAAAAAAANo3/D2zci8M= "], {{0, 90}, {2400, 0}}, {0, 255}, ColorFunction -> RGBColor], BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable -> False], BaseStyle -> "ImageGraphics", ImageSize -> Magnification[1], ImageSizeRaw -> {2400, 90}, PlotRange -> {{0, 2400}, {0, 90}}]], "", CellMargins -> 0, CellChangeTimes -> {{3.516548605022827*^9, 3.516548633085456*^9}, { 3.516616961864057*^9, 3.516616982571135*^9}, 3.518187891728025*^9, { 3.518199033657151*^9, 3.5181990340305967`*^9}}], Cell[ BoxData[ RowBox[{ RowBox[{"(*", RowBox[{"Evaluate", " ", "the", " ", "following", " ", "to", " ", "copy", " ", "the", " ", "style", " ", "of", " ", "the", " ", "cell", " ", "above", " ", "into", " ", "\[IndentingNewLine]", "the", " ", "docked", " ", "cell", " ", "style", " ", "of", " ", RowBox[{"the", " ", "'"}], RowBox[{"Notebook", "'"}], " ", "definition", " ", RowBox[{"(", RowBox[{"2", " ", "cells", " ", "above"}], ")"}], " ", "\[IndentingNewLine]", "These", " ", "two", " ", "cell", " ", "can", " ", "be", " ", "removed", " ", "once", " ", "the", " ", "docked", " ", "cell", " ", "is", " ", RowBox[{"created", "."}]}], "\[IndentingNewLine]", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"SelectionMove", "[", RowBox[{ RowBox[{"SelectedNotebook", "[", "]"}], ",", "Previous", ",", "Cell", ",", "2"}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"celldata", "=", RowBox[{"NotebookRead", "[", RowBox[{"SelectedNotebook", "[", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"SelectionMove", "[", RowBox[{ RowBox[{"SelectedNotebook", "[", "]"}], ",", "Previous", ",", "Cell", ",", "1"}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"SetOptions", "[", RowBox[{ RowBox[{"NotebookSelection", "[", RowBox[{"SelectedNotebook", "[", "]"}], "]"}], ",", RowBox[{"DockedCells", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"FEPrivate`FrontEndResource", "[", RowBox[{ "\"FEExpressions\"", ",", "\"SlideshowToolbar\""}], "]"}], ",", "celldata"}], "}"}]}]}], "]"}], ";"}]}]}]], "Input", FontWeight -> "Bold"]}, Open]], Cell[ CellGroupData[{ Cell["Notebook Options Settings", "Section"], Cell[ StyleData["Notebook"], CellBracketOptions -> { "Color" -> RGBColor[0.739193, 0.750317, 0.747173]}]}, Open]], Cell[ CellGroupData[{ Cell["Styles for Title and Section Cells", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Title"], ShowCellBracket -> Automatic, ShowGroupOpener -> False, CellMargins -> {{58, 0}, {30, 0}}, CellBracketOptions -> {"Margins" -> {0, 0}}, CellGroupingRules -> {"TitleGrouping", 0}, PageBreakBelow -> False, CellFrameMargins -> {{20, 20}, {20, 20}}, DefaultNewInlineCellStyle -> "None", InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica", "gridMathematica" -> FormBox[ RowBox[{"grid", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], "webMathematica" -> FormBox[ RowBox[{"web", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, CellChangeTimes -> {3.479211616867702*^9, 3.483202458952606*^9}, TextAlignment -> Left, LineSpacing -> {1, 0}, LanguageCategory -> "NaturalLanguage", CounterIncrements -> "Title", CounterAssignments -> {{"Section", 0}, {"Equation", 0}, { "Figure", 0}, {"Subtitle", 0}, {"Subsubtitle", 0}}, FontFamily -> "Helvetica", FontSize -> 48, FontWeight -> "Plain", FontSlant -> "Plain", FontTracking -> "Plain", FontVariations -> { "Masked" -> False, "Outline" -> False, "Shadow" -> False, "StrikeThrough" -> False, "Underline" -> False}, FontColor -> RGBColor[ 0.8156862745098039, 0.07058823529411765, 0.07058823529411765], Background -> None], Cell[ StyleData["Title", "Presentation", StyleDefinitions -> None], CellMargins -> {{55, 3}, {14, 125}}, LineSpacing -> {1, 5}, FontSize -> 28], Cell[ StyleData[ "Title", "SlideShow", StyleDefinitions -> StyleData["Title", "Presentation"]], CellMargins -> {{55, 3}, {14, 35}}, FontSize -> 55], Cell[ StyleData["Title", "Printout", StyleDefinitions -> None], CellMargins -> {{2, 0}, {0, 10}}, LineSpacing -> {1, 18}, FontSize -> 15]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Subtitle"], ShowCellBracket -> False, CellMargins -> {{58, 0}, {0, 5}}, CellBracketOptions -> {"Margins" -> {0, 0}}, CellGroupingRules -> {"TitleGrouping", 10}, PageBreakBelow -> False, DefaultNewInlineCellStyle -> "None", InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica", "gridMathematica" -> FormBox[ RowBox[{"grid", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], "webMathematica" -> FormBox[ RowBox[{"web", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, TextAlignment -> Left, LanguageCategory -> "NaturalLanguage", CounterIncrements -> "Subtitle", CounterAssignments -> {{"Section", 0}, {"Equation", 0}, { "Figure", 0}, {"Subsubtitle", 0}}, FontFamily -> "Helvetica", FontSize -> 20, FontWeight -> "Plain", FontSlant -> "Plain", FontColor -> RGBColor[ 0.34901960784313724`, 0.5254901960784314, 0.5176470588235295], Background -> None], Cell[ StyleData["Subtitle", "Presentation", StyleDefinitions -> None], CellMargins -> {{58, 0}, {0, 5}}, FontSize -> 20], Cell[ StyleData[ "Subtitle", "SlideShow", StyleDefinitions -> StyleData["Subtitle", "Presentation"]]], Cell[ StyleData["Subtitle", "Printout", StyleDefinitions -> None], CellMargins -> {{2, 0}, {0, 5}}, FontSize -> 14, Background -> GrayLevel[1]]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Subsubtitle", StyleDefinitions -> StyleData["Subtitle"]], FontSize -> Inherited + 3], Cell[ StyleData["Subsubtitle", "Presentation"], FontSize -> Inherited + 3], Cell[ StyleData[ "Subsubtitle", "SlideShow", StyleDefinitions -> StyleData["Subsubtitle", "Presentation"]]], Cell[ StyleData["Subsubtitle", "Printout"], FontSize -> Inherited + 0]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Section"], CellFrame -> {{0, 0}, {0.2, 0}}, ShowGroupOpener -> False, CellMargins -> {{88, 50}, {10, 25}}, FontSize -> 36, FontWeight -> "Plain", FontColor -> RGBColor[ 0.8156862745098039, 0.07058823529411765, 0.07058823529411765]], Cell[ StyleData["Section", "Presentation"], CellFrame -> {{0, 0}, {0.2, 0}}, CellMargins -> {{58, 50}, {10, 35}}], Cell[ StyleData[ "Section", "SlideShow", StyleDefinitions -> StyleData["Section", "Presentation"]], CellMargins -> {{58, 50}, {10, 35}}], Cell[ StyleData["Section", "Printout"], ShowGroupOpener -> False, CellMargins -> {{2, 0}, {7, 22}}, FontSize -> 14]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Subsection"], CellDingbat -> None, ShowGroupOpener -> True, CellMargins -> {{60, Inherited}, {0, 12}}, CellGroupingRules -> {"SectionGrouping", 40}, PageBreakBelow -> False, DefaultNewInlineCellStyle -> "None", InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica", "gridMathematica" -> FormBox[ RowBox[{"grid", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], "webMathematica" -> FormBox[ RowBox[{"web", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, LanguageCategory -> "NaturalLanguage", CounterIncrements -> "Subsection", CounterAssignments -> {{"Subsubsection", 0}}, FontFamily -> "Helvetica", FontSize -> 24, FontWeight -> "Plain", FontSlant -> "Plain", FontColor -> RGBColor[ 0.34901960784313724`, 0.5254901960784314, 0.5176470588235295]], Cell[ StyleData["Subsection", "Presentation"], CellMargins -> {{60, 50}, {6, 15}}, LineSpacing -> {1, 0}, FontFamily -> "Helvetica"], Cell[ StyleData["Subsection", "SlideShow"], CellMargins -> {{60, 50}, {8, 12}}, LineSpacing -> {1, 0}, FontFamily -> "Helvetica"], Cell[ StyleData["Subsection", "Printout"], ShowGroupOpener -> False, CellMargins -> {{2, 0}, {2, 22}}, FontSize -> 12]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Subsubsection"], CellDingbat -> None, ShowGroupOpener -> True, CellMargins -> {{60, Inherited}, {0, 12}}, CellGroupingRules -> {"SectionGrouping", 50}, PageBreakBelow -> False, DefaultNewInlineCellStyle -> "None", InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica", "gridMathematica" -> FormBox[ RowBox[{"grid", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], "webMathematica" -> FormBox[ RowBox[{"web", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, LanguageCategory -> "NaturalLanguage", CounterIncrements -> "Subsubsection", FontFamily -> "Helvetica", FontSize -> 24, FontWeight -> "Plain", FontSlant -> "Plain", FontColor -> RGBColor[ 0.34901960784313724`, 0.5254901960784314, 0.5176470588235295]], Cell[ StyleData["Subsubsection", "Presentation"], CellMargins -> {{60, 50}, {6, 20}}, LineSpacing -> {1, 0}], Cell[ StyleData[ "Subsubsection", "SlideShow", StyleDefinitions -> StyleData["Subsubsection", "Presentation"]]], Cell[ StyleData["Subsubsection", "Printout"], ShowGroupOpener -> False, CellMargins -> {{2, 0}, {2, 22}}, FontSize -> 11]}, Open]]}, Open]], Cell[ CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[ CellGroupData[{ Cell["Standard", "Subsection"], Cell[ CellGroupData[{ Cell[ StyleData["Text"], CellMargins -> {{60, 10}, {7, 7}}, InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica", "gridMathematica" -> FormBox[ RowBox[{"grid", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], "webMathematica" -> FormBox[ RowBox[{"web", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, LineSpacing -> {1, 3}, CounterIncrements -> "Text", FontFamily -> "Helvetica", FontSize -> 17], Cell[ StyleData["Text", "Presentation"], CellMargins -> {{60, 50}, {10, 10}}, FontSize -> 17], Cell[ StyleData[ "Text", "SlideShow", StyleDefinitions -> StyleData["Text", "Presentation"]]], Cell[ StyleData["Text", "Printout"], CellMargins -> {{2, 2}, {6, 6}}, TextJustification -> 0.5, Hyphenation -> True, FontSize -> 10]}, Open]]}, Open]], Cell[ CellGroupData[{ Cell["Display", "Subsection"], Cell[ CellGroupData[{ Cell[ StyleData["Item", StyleDefinitions -> StyleData["Text"]], CellDingbat -> Cell["\[FilledSmallCircle]", FontWeight -> "Bold"], CellMargins -> {{84, 10}, {7, 7}}, ReturnCreatesNewCell -> True, CellGroupingRules -> {"GroupTogetherNestedGrouping", 15000}, CounterIncrements -> "Item"], Cell[ StyleData["Item", "Presentation"], CellMargins -> {{124, 10}, {7, 7}}], Cell[ StyleData[ "Item", "SlideShow", StyleDefinitions -> StyleData["Item", "Presentation"]]], Cell[ StyleData["Item", "Printout"], CellMargins -> {{2, 2}, {0, 6}}]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Subitem", StyleDefinitions -> StyleData["Item"]], CellMargins -> {{108, 10}, {7, 7}}, ReturnCreatesNewCell -> True, CellGroupingRules -> {"GroupTogetherNestedGrouping", 15150}, CounterIncrements -> "Subitem"], Cell[ StyleData["Subitem", "Presentation"], CellMargins -> {{146, 10}, {7, 7}}], Cell[ StyleData[ "Subitem", "SlideShow", StyleDefinitions -> StyleData["Subitem", "Presentation"]]], Cell[ StyleData["Subitem", "Printout"], CellMargins -> {{30, 2}, {0, 6}}]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["ItemNumbered", StyleDefinitions -> StyleData["Text"]], CellDingbat -> Cell[ TextData[{ CounterBox["ItemNumbered"], "."}]], CellMargins -> {{84, 10}, {7, 7}}, ReturnCreatesNewCell -> True, CellGroupingRules -> {"GroupTogetherNestedGrouping", 15000}, CounterIncrements -> "ItemNumbered"], Cell[ StyleData["ItemNumbered", "Presentation"], CellMargins -> {{124, 10}, {7, 7}}], Cell[ StyleData[ "ItemNumbered", "SlideShow", StyleDefinitions -> StyleData["ItemNumbered", "Presentation"]]], Cell[ StyleData["ItemNumbered", "Printout"], CellMargins -> {{2, 2}, {0, 6}}]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData[ "SubitemNumbered", StyleDefinitions -> StyleData["ItemNumbered"]], CellDingbat -> Cell[ TextData[{ CounterBox["SubitemNumbered", CounterFunction :> (Part[ CharacterRange["a", "z"], #]& )], "."}]], CellMargins -> {{108, 10}, {7, 7}}, ReturnCreatesNewCell -> True, CellGroupingRules -> {"GroupTogetherNestedGrouping", 15150}, CounterIncrements -> "SubitemNumbered"], Cell[ StyleData["SubitemNumbered", "Presentation"], CellMargins -> {{146, 10}, {7, 7}}], Cell[ StyleData[ "SubitemNumbered", "SlideShow", StyleDefinitions -> StyleData["SubitemNumbered", "Presentation"]]], Cell[ StyleData["SubitemNumbered", "Printout"], CellMargins -> {{30, 2}, {0, 6}}]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData[ "ItemParagraph", StyleDefinitions -> StyleData["Item"]], CellDingbat -> None, CellMargins -> {{84, 10}, {7, 7}}, ReturnCreatesNewCell -> True, CellGroupingRules -> {"GroupTogetherNestedGrouping", 15100}, CounterIncrements -> "ItemParagraph"], Cell[ StyleData["ItemParagraph", "Presentation"], CellMargins -> {{124, 10}, {7, 7}}], Cell[ StyleData[ "ItemParagraph", "SlideShow", StyleDefinitions -> StyleData["ItemParagraph", "Presentation"]]], Cell[ StyleData["ItemParagraph", "Printout"], CellMargins -> {{14, 2}, {0, 6}}]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData[ "SubitemParagraph", StyleDefinitions -> StyleData["Subitem"]], CellDingbat -> None, ReturnCreatesNewCell -> True, CellGroupingRules -> {"GroupTogetherNestedGrouping", 15200}, CounterIncrements -> "SubitemParagraph"], Cell[ StyleData["SubitemParagraph", "Presentation"]], Cell[ StyleData[ "SubitemParagraph", "SlideShow", StyleDefinitions -> StyleData["SubitemParagraph", "Presentation"]]], Cell[ StyleData["SubitemParagraph", "Printout"]]}, Open]]}, Open]]}, Open]], Cell[ CellGroupData[{ Cell["Styles for Formulas and Programming", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["DisplayFormula"]], Cell[ StyleData["DisplayFormula", "Presentation"], CellMargins -> {{60, Inherited}, {Inherited 1.5, Inherited 1.5}}, FontSize -> 17], Cell[ StyleData[ "DisplayFormula", "SlideShow", StyleDefinitions -> StyleData["DisplayFormula", "Presentation"]]], Cell[ StyleData["DisplayFormula", "Printout"], CellMargins -> {{39, Inherited}, {Inherited, Inherited}}, FontSize -> 10]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData[ "DisplayFormulaNumbered", StyleDefinitions -> StyleData["DisplayFormula"]], CellFrameLabels -> {{None, Cell[ TextData[{"(", CounterBox["DisplayFormulaNumbered"], ")"}]]}, {None, None}}, CounterIncrements -> "DisplayFormulaNumbered"], Cell[ StyleData["DisplayFormulaNumbered", "Presentation"], CellMargins -> {{60, Inherited}, {Inherited 1.5, Inherited 1.5}}, FontSize -> 17], Cell[ StyleData[ "DisplayFormulaNumbered", "SlideShow", StyleDefinitions -> StyleData["DisplayFormulaNumbered", "Presentation"]]], Cell[ StyleData["DisplayFormulaNumbered", "Printout"], CellMargins -> {{39, Inherited}, {Inherited, Inherited}}]}, Open]]}, Open]], Cell[ CellGroupData[{ Cell["Styles for Inline Formatting", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["InlineFormula"]], Cell[ StyleData["InlineFormula", "Presentation"], FontSize -> 17], Cell[ StyleData[ "InlineFormula", "SlideShow", StyleDefinitions -> StyleData["InlineFormula", "Presentation"]]], Cell[ StyleData["InlineFormula", "Printout"]]}, Open]]}, Open]], Cell[ CellGroupData[{ Cell["Styles for Input and Output Cells", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Input"], ShowCellBracket -> True, CellMargins -> {{103, 10}, {5, 7}}, CellBracketOptions -> { "Color" -> RGBColor[0.734936, 0.713848, 0.694041]}, Evaluatable -> True, CellGroupingRules -> "InputGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GroupPageBreakWithin -> False, DefaultFormatType -> DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"}, AutoItalicWords -> {}, LanguageCategory -> "Mathematica", FormatType -> InputForm, ShowStringCharacters -> True, NumberMarks -> True, LinebreakAdjustments -> {0.85, 2, 10, 0, 1}, CounterIncrements -> "Input", FontWeight -> "Bold"], Cell[ StyleData["Input", "Presentation"], CellMargins -> {{110, 50}, {8, 10}}, LineSpacing -> {1, 0}, FontSize -> 17], Cell[ StyleData[ "Input", "SlideShow", StyleDefinitions -> StyleData["Input", "Presentation"]]], Cell[ StyleData["Input", "Printout"], CellMargins -> {{39, 0}, {4, 6}}, LinebreakAdjustments -> {0.85, 2, 10, 1, 1}, FontSize -> 9]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["InputOnly"], ShowCellBracket -> True, CellMargins -> {{103, 10}, {7, 7}}, CellBracketOptions -> { "Color" -> RGBColor[0.734936, 0.713848, 0.694041]}, Evaluatable -> True, CellGroupingRules -> "InputGrouping", CellHorizontalScrolling -> True, DefaultFormatType -> DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"}, AutoItalicWords -> {}, LanguageCategory -> "Mathematica", FormatType -> InputForm, ShowStringCharacters -> True, NumberMarks -> True, LinebreakAdjustments -> {0.85, 2, 10, 0, 1}, CounterIncrements -> "Input", MenuSortingValue -> 1550, FontWeight -> "Bold"], Cell[ StyleData["InputOnly", "Presentation"], CellMargins -> {{110, Inherited}, {8, 10}}, LineSpacing -> {1, 0}, FontSize -> 17], Cell[ StyleData[ "InputOnly", "SlideShow", StyleDefinitions -> StyleData["InputOnly", "Presentation"]]], Cell[ StyleData["InputOnly", "Printout"], CellMargins -> {{39, 0}, {4, 6}}, LinebreakAdjustments -> {0.85, 2, 10, 1, 1}, FontSize -> 9]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Output"], ShowCellBracket -> True, CellMargins -> {{103, 10}, {7, 5}}, CellBracketOptions -> { "Color" -> RGBColor[0.734936, 0.713848, 0.694041]}, CellEditDuplicate -> True, CellGroupingRules -> "OutputGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GroupPageBreakWithin -> False, GeneratedCell -> True, CellAutoOverwrite -> True, DefaultFormatType -> DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, AutoItalicWords -> {}, LanguageCategory -> None, FormatType -> InputForm, CounterIncrements -> "Output"], Cell[ StyleData["Output", "Presentation"], CellMargins -> {{110, 50}, {10, 8}}, LineSpacing -> {1, 0}, FontSize -> 17], Cell[ StyleData[ "Output", "SlideShow", StyleDefinitions -> StyleData["Output", "Presentation"]]], Cell[ StyleData["Output", "Printout"], CellMargins -> {{39, 0}, {6, 4}}, FontSize -> 9]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Code"], CellMargins -> {{103, 10}, {5, 10}}], Cell[ StyleData["Code", "Presentation"], CellMargins -> {{110, 50}, {8, 10}}, LineSpacing -> {1, 0}, FontSize -> 17], Cell[ StyleData[ "Code", "SlideShow", StyleDefinitions -> StyleData["Code", "Presentation"]]], Cell[ StyleData["Code", "Printout"], CellMargins -> {{39, 0}, {4, 6}}, LinebreakAdjustments -> {0.85, 2, 10, 1, 1}, FontSize -> 9]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Print"], CellMargins -> {{103, Inherited}, {Inherited, Inherited}}, FontSize -> 14], Cell[ StyleData["Print", "Presentation"], CellMargins -> {{70, Inherited}, {Inherited 1.5, Inherited 1.5}}, FontSize -> 17, Magnification -> Inherited 1.5], Cell[ StyleData[ "Print", "SlideShow", StyleDefinitions -> StyleData["Print", "Presentation"]]], Cell[ StyleData["Print", "Printout"], CellMargins -> {{39, Inherited}, {Inherited, Inherited}}]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData[ "WolframAlphaShortInput", StyleDefinitions -> StyleData["Input"]], CellMargins -> {{98, 10}, {5, 7}}, EvaluationMode -> "WolframAlphaShort", CellEventActions -> {"ReturnKeyDown" :> FrontEndTokenExecute[ EvaluationNotebook[], "HandleShiftReturn"]}, CellFrameLabels -> {{ Cell[ BoxData[ DynamicBox[ FEPrivate`FrontEndResource["WABitmaps", "Equal"]]], CellBaseline -> Baseline], None}, {None, None}}, FormatType -> TextForm, FontFamily -> "Helvetica"], Cell[ StyleData["WolframAlphaShortInput", "Presentation"], CellMargins -> {{107, 50}, {8, 10}}], Cell[ StyleData[ "WolframAlphaShortInput", "SlideShow", StyleDefinitions -> StyleData["WolframAlphaShortInput", "Presentation"]]], Cell[ StyleData["WolframAlphaShortInput", "Printout"], CellFrameLabelMargins -> 3]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData[ "WolframAlphaLong", StyleDefinitions -> StyleData["Input"]], CellMargins -> {{100, 10}, {5, 7}}, StyleKeyMapping -> { "=" -> "Input", "Backspace" -> "WolframAlphaShort"}, EvaluationMode -> "WolframAlphaLong", CellEventActions -> {"ReturnKeyDown" :> FrontEndTokenExecute[ EvaluationNotebook[], "HandleShiftReturn"]}, CellFrameLabels -> {{ Cell[ BoxData[ DynamicBox[ FEPrivate`FrontEndResource["WABitmaps", "SpikeyEqual"]]]], None}, {None, None}}, DefaultFormatType -> TextForm, FormatType -> TextForm, FontFamily -> "Helvetica"], Cell[ StyleData["WolframAlphaLong", "Presentation"], CellMargins -> {{107, 50}, {8, 10}}], Cell[ StyleData[ "WolframAlphaLong", "SlideShow", StyleDefinitions -> StyleData["WolframAlphaLong", "Presentation"]], CellMargins -> {{107, 50}, {8, 10}}], Cell[ StyleData["WolframAlphaLong", "Printout"], CellFrameLabelMargins -> 3]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Program"], CellMargins -> {{60, 4}, {6, 8}}], Cell[ StyleData["Program", "Presentation"], CellMargins -> {{60, 50}, {8, 10}}, LineSpacing -> {1, 0}, FontSize -> 17], Cell[ StyleData[ "Program", "SlideShow", StyleDefinitions -> StyleData["Program", "Presentation"]]], Cell[ StyleData["Program", "Printout"], FontSize -> 9]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["CellLabel"]], Cell[ StyleData["CellLabel", "Presentation"], FontSize -> 12], Cell[ StyleData[ "CellLabel", "SlideShow", StyleDefinitions -> StyleData["CellLabel", "Presentation"]]], Cell[ StyleData["CellLabel", "Printout"], FontSize -> 8, FontColor -> GrayLevel[0.]]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["ManipulateLabel"]], Cell[ StyleData["ManipulateLabel", "Presentation"], FontSize -> 15], Cell[ StyleData[ "ManipulateLabel", "SlideShow", StyleDefinitions -> StyleData["ManipulateLabel", "Presentation"]]], Cell[ StyleData["ManipulateLabel", "Printout"], FontSize -> 8]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["GraphicsLabel"]], Cell[ StyleData["GraphicsLabel", "Presentation"], FontSize -> 14], Cell[ StyleData[ "GraphicsLabel", "SlideShow", StyleDefinitions -> StyleData["GraphicsLabel", "Presentation"]]], Cell[ StyleData["GraphicsLabel", "Printout"], FontSize -> 8]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Graphics3DLabel"]], Cell[ StyleData["Graphics3DLabel", "Presentation"], FontSize -> 14], Cell[ StyleData[ "Graphics3DLabel", "SlideShow", StyleDefinitions -> StyleData["Graphics3DLabel", "Presentation"]]], Cell[ StyleData["Graphics3DLabel", "Printout"], FontSize -> 8]}, Open]]}, Open]], Cell[ CellGroupData[{ Cell[ "Styles for SlideShow", "Section", CellChangeTimes -> {{3.514665148412793*^9, 3.5146651505550737`*^9}}], Cell[ StyleData["slideshowheader"], ShowCellBracket -> False, CellMargins -> {{0, 0}, {0, -2}}, Evaluatable -> False, CellHorizontalScrolling -> False, PageBreakBelow -> False, CellFrameMargins -> 0, ImageMargins -> {{0, 0}, {0, 0}}, ImageRegion -> {{0, 1}, {0, 1}}, Magnification -> 1, Background -> GrayLevel[1], CellPadding -> 0, CellFramePadding -> 0], Cell[ CellGroupData[{ Cell[ StyleData["hidefromslideshowgraphic"], ShowCellBracket -> False, CellMargins -> {{0, 0}, {0, 0}}, Evaluatable -> False, CellHorizontalScrolling -> False, PageBreakBelow -> False, CellFrameMargins -> 0, ImageMargins -> {{0, 0}, {0, 0}}, ImageRegion -> {{0, 1}, {0, 1}}, Magnification -> 1, Background -> GrayLevel[1], CellPadding -> 0], Cell[ StyleData["hidefromslideshowgraphic", "SlideShow"], ShowCellBracket -> False, CellElementSpacings -> { "CellMinHeight" -> 0, "ClosedCellHeight" -> 0, "ClosedGroupTopMargin" -> 0}, CellOpen -> False, CellHorizontalScrolling -> False]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["slideshowheader2"], ShowCellBracket -> False, CellMargins -> {{0, 0}, {0, 0}}, Evaluatable -> False, CellHorizontalScrolling -> False, PageBreakBelow -> False, ImageMargins -> {{0, 0}, {0, 0}}, ImageRegion -> {{0, 1}, {0, 1}}, Magnification -> 1, Background -> GrayLevel[1]], Cell[ StyleData["ConferenceGraphicCell", "SlideShow"], ShowCellBracket -> False, CellElementSpacings -> { "CellMinHeight" -> 0, "ClosedCellHeight" -> 0, "ClosedGroupTopMargin" -> 0}, CellOpen -> False, CellHorizontalScrolling -> True], Cell[ StyleData["slideshowheader", "Printout"], FontSize -> 8, Magnification -> 0.75]}, Open]], Cell[ StyleData[ "ConferenceGraphicCellSlideShowOnly", StyleDefinitions -> StyleData["ConferenceCellGraphic"]], ShowCellBracket -> False, CellMargins -> 0, CellElementSpacings -> { "CellMinHeight" -> 0, "ClosedCellHeight" -> 0, "ClosedGroupTopMargin" -> 0}, CellOpen -> False], Cell[ CellGroupData[{ Cell[ StyleData["SlideShowNavigationBar"], Editable -> True, Selectable -> False, CellFrame -> 0, ShowGroupOpener -> False, CellMargins -> {{0, 0}, {3, 3}}, CellOpen -> True, CellFrameMargins -> 0, CellFrameColor -> None, Background -> None], Cell[ StyleData["SlideShowNavigationBar", "Printout"], PageBreakAbove -> Automatic]}, Open]]}, Open]]}, Visible -> False, FrontEndVersion -> "8.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (October 5, 2011)", StyleDefinitions -> "Default.nb"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{ "SlideShowHeader"->{ Cell[15150, 277, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[15638, 301, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[47858, 971, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[67372, 1370, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[74432, 1595, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[100827, 2120, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[136926, 2830, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[143976, 3033, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[145099, 3066, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[148050, 3173, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[150448, 3257, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[152401, 3327, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[158961, 3549, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[162306, 3663, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[165069, 3754, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[183828, 4131, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[186928, 4232, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[189363, 4310, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[213984, 4808, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[217684, 4932, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[220873, 5036, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[224568, 5165, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[244202, 5572, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[283126, 6333, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[294514, 6633, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"]}, "3.2"->{ Cell[68684, 1403, 2071, 65, 41, "DisplayFormula", CounterAssignments->{{"Section", 1}, {"NumberedEquation", 2}}, CellTags->"3.2"]} } *) (*CellTagsIndex CellTagsIndex->{ {"SlideShowHeader", 360731, 7974}, {"3.2", 362984, 8025} } *) (*NotebookFileOutline Notebook[{ Cell[1235, 30, 12512, 209, 2, "hidefromslideshowgraphic"], Cell[CellGroupData[{ Cell[13772, 243, 465, 11, 85, "Title"], Cell[14240, 256, 420, 6, 25, "Subtitle"], Cell[14663, 264, 355, 5, 20, "Subsubtitle"], Cell[15021, 271, 92, 1, 35, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[15150, 277, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[15239, 282, 146, 2, 93, "Section"], Cell[15388, 286, 201, 9, 169, "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[15638, 301, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[15727, 306, 205, 3, 93, "Section"], Cell[15935, 311, 663, 9, 37, "Text"], Cell[CellGroupData[{ Cell[16623, 324, 223, 4, 48, "Subsection"], Cell[16849, 330, 213, 4, 59, "Text"], Cell[17065, 336, 9535, 182, 206, "Input"], Cell[26603, 520, 327, 6, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[26967, 531, 231, 4, 40, "Subsection"], Cell[27201, 537, 338, 8, 59, "Text"], Cell[27542, 547, 18810, 367, 206, "Input"], Cell[46355, 916, 1442, 48, 191, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[47858, 971, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[47947, 976, 178, 3, 93, "Section"], Cell[CellGroupData[{ Cell[48150, 983, 268, 4, 48, "Subsection"], Cell[48421, 989, 194, 4, 37, "Text"], Cell[48618, 995, 17469, 324, 206, "Input"], Cell[66090, 1321, 713, 27, 118, "Input"], Cell[66806, 1350, 347, 7, 59, "Text"], Cell[67156, 1359, 155, 4, 35, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[67372, 1370, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[67461, 1375, 255, 4, 93, "Section"], Cell[CellGroupData[{ Cell[67741, 1383, 387, 6, 48, "Subsection"], Cell[68131, 1391, 550, 10, 81, "Text"], Cell[68684, 1403, 2071, 65, 41, "DisplayFormula", CounterAssignments->{{"Section", 1}, {"NumberedEquation", 2}}, CellTags->"3.2"], Cell[70758, 1470, 1649, 58, 308, "Input"], Cell[CellGroupData[{ Cell[72432, 1532, 207, 3, 50, "Subsubsection"], Cell[72642, 1537, 184, 4, 37, "Text"], Cell[72829, 1543, 240, 5, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[73106, 1553, 232, 5, 34, "Subsubsection"], Cell[73341, 1560, 138, 1, 37, "Text"], Cell[73482, 1563, 273, 6, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[73792, 1574, 160, 2, 34, "Subsubsection"], Cell[73955, 1578, 140, 1, 37, "Text"], Cell[74098, 1581, 261, 6, 35, "Input"] }, Closed]] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[74432, 1595, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[74521, 1600, 228, 4, 93, "Section"], Cell[CellGroupData[{ Cell[74774, 1608, 340, 5, 48, "Subsection"], Cell[75117, 1615, 252, 5, 37, "Text"], Cell[75372, 1622, 23527, 435, 197, "Input"], Cell[98902, 2059, 120, 2, 35, "Input"], Cell[99025, 2063, 653, 19, 139, "Input"], Cell[CellGroupData[{ Cell[99703, 2086, 205, 3, 50, "Subsubsection"], Cell[99911, 2091, 168, 3, 37, "Text"], Cell[100082, 2096, 367, 8, 58, "Input"], Cell[100452, 2106, 302, 6, 35, "Input"] }, Closed]] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[100827, 2120, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[100916, 2125, 154, 3, 93, "Section"], Cell[CellGroupData[{ Cell[101095, 2132, 436, 6, 48, "Subsection"], Cell[101534, 2140, 472, 8, 37, "Text"], Cell[102009, 2150, 32278, 608, 303, "Input"], Cell[134290, 2760, 70, 2, 35, "Input"], Cell[134363, 2764, 2502, 59, 346, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[136926, 2830, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[137015, 2835, 123, 2, 93, "Section"], Cell[137141, 2839, 564, 9, 59, "Text"], Cell[137708, 2850, 2190, 50, 320, "Input"], Cell[139901, 2902, 312, 6, 37, "Text"], Cell[140216, 2910, 1046, 34, 184, "DisplayFormula"], Cell[141265, 2946, 2662, 81, 124, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[143976, 3033, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[144065, 3038, 69, 1, 93, "Section"], Cell[144137, 3041, 636, 12, 59, "Text"], Cell[144776, 3055, 274, 5, 59, "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[145099, 3066, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[145188, 3071, 190, 6, 143, "Section"], Cell[145381, 3079, 103, 1, 37, "Text"], Cell[145487, 3082, 713, 25, 102, "Input"], Cell[CellGroupData[{ Cell[146225, 3111, 367, 5, 50, "Subsubsection"], Cell[146595, 3118, 164, 5, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[146796, 3128, 128, 2, 34, "Subsubsection"], Cell[146927, 3132, 137, 1, 37, "Text"], Cell[147067, 3135, 413, 13, 58, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[147517, 3153, 112, 1, 34, "Subsubsection"], Cell[147632, 3156, 156, 3, 37, "Text"], Cell[147791, 3161, 198, 5, 35, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[148050, 3173, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[148139, 3178, 130, 2, 93, "Section"], Cell[CellGroupData[{ Cell[148294, 3184, 113, 1, 50, "Subsubsection"], Cell[148410, 3187, 463, 13, 35, "Input"], Cell[148876, 3202, 240, 4, 37, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[149153, 3211, 156, 2, 34, "Subsubsection"], Cell[149312, 3215, 132, 1, 37, "Text"], Cell[149447, 3218, 413, 13, 58, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[149897, 3236, 107, 1, 34, "Subsubsection"], Cell[150007, 3239, 177, 4, 37, "Text"], Cell[150187, 3245, 200, 5, 35, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[150448, 3257, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[150537, 3262, 131, 2, 93, "Section"], Cell[CellGroupData[{ Cell[150693, 3268, 124, 1, 50, "Subsubsection"], Cell[150820, 3271, 181, 2, 37, "Text"], Cell[151004, 3275, 167, 5, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[151208, 3285, 102, 1, 34, "Subsubsection"], Cell[151313, 3288, 179, 4, 37, "Text"], Cell[151495, 3294, 250, 6, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[151782, 3305, 119, 1, 34, "Subsubsection"], Cell[151904, 3308, 184, 4, 37, "Text"], Cell[152091, 3314, 249, 6, 35, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[152401, 3327, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[152490, 3332, 132, 2, 93, "Section"], Cell[CellGroupData[{ Cell[152647, 3338, 161, 2, 50, "Subsubsection"], Cell[152811, 3342, 173, 4, 37, "Text"], Cell[152987, 3348, 1016, 35, 81, "DisplayFormula"] }, Closed]], Cell[CellGroupData[{ Cell[154040, 3388, 159, 2, 34, "Subsubsection"], Cell[154202, 3392, 1039, 32, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[155278, 3429, 103, 1, 34, "Subsubsection"], Cell[155384, 3432, 182, 2, 37, "Text"], Cell[155569, 3436, 2027, 62, 80, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[157633, 3503, 161, 2, 34, "Subsubsection"], Cell[157797, 3507, 263, 5, 37, "Text"], Cell[158063, 3514, 352, 11, 17, "DisplayFormula"], Cell[158418, 3527, 482, 15, 35, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[158961, 3549, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[159050, 3554, 128, 2, 93, "Section"], Cell[CellGroupData[{ Cell[159203, 3560, 207, 3, 50, "Subsubsection"], Cell[159413, 3565, 548, 18, 65, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[159998, 3588, 190, 2, 34, "Subsubsection"], Cell[160191, 3592, 242, 5, 37, "Text"], Cell[160436, 3599, 147, 4, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[160620, 3608, 112, 1, 34, "Subsubsection"], Cell[160735, 3611, 312, 6, 37, "Text"], Cell[161050, 3619, 1195, 37, 58, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[162306, 3663, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[162395, 3668, 81, 1, 93, "Section"], Cell[162479, 3671, 148, 3, 37, "Text"], Cell[CellGroupData[{ Cell[162652, 3678, 115, 1, 50, "Subsubsection"], Cell[162770, 3681, 645, 17, 80, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[163452, 3703, 117, 1, 34, "Subsubsection"], Cell[163572, 3706, 119, 1, 37, "Text"], Cell[163694, 3709, 395, 8, 35, "Input"], Cell[164092, 3719, 129, 2, 37, "Text"], Cell[164224, 3723, 784, 24, 65, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[165069, 3754, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[165158, 3759, 81, 1, 93, "Section"], Cell[CellGroupData[{ Cell[165264, 3764, 214, 3, 50, "Subsubsection"], Cell[165481, 3769, 175, 4, 37, "Text"], Cell[165659, 3775, 16524, 300, 237, "Input"], Cell[182186, 4077, 243, 5, 37, "Text"], Cell[182432, 4084, 1335, 40, 141, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[183828, 4131, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[183917, 4136, 79, 1, 93, "Section"], Cell[CellGroupData[{ Cell[184021, 4141, 162, 2, 50, "Subsubsection"], Cell[184186, 4145, 138, 2, 37, "Text"], Cell[184327, 4149, 1370, 37, 112, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[185734, 4191, 111, 1, 34, "Subsubsection"], Cell[185848, 4194, 177, 4, 37, "Text"], Cell[186028, 4200, 839, 25, 65, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[186928, 4232, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[187017, 4237, 79, 1, 93, "Section"], Cell[CellGroupData[{ Cell[187121, 4242, 102, 1, 50, "Subsubsection"], Cell[187226, 4245, 287, 5, 37, "Text"], Cell[187516, 4252, 1786, 51, 180, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[189363, 4310, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[189452, 4315, 339, 6, 93, "Section"], Cell[189794, 4323, 20733, 371, 199, "Input"], Cell[CellGroupData[{ Cell[210552, 4698, 102, 1, 50, "Subsubsection"], Cell[210657, 4701, 595, 13, 103, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[211289, 4719, 98, 1, 34, "Subsubsection"], Cell[211390, 4722, 142, 1, 37, "Text"], Cell[211535, 4725, 355, 12, 35, "Input"], Cell[211893, 4739, 260, 6, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[212190, 4750, 163, 2, 34, "Subsubsection"], Cell[212356, 4754, 294, 6, 37, "Text"], Cell[212653, 4762, 1270, 39, 58, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[213984, 4808, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[214073, 4813, 161, 4, 93, "Section"], Cell[CellGroupData[{ Cell[214259, 4821, 166, 2, 50, "Subsubsection"], Cell[214428, 4825, 140, 1, 37, "Text"], Cell[214571, 4828, 145, 2, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[214753, 4835, 118, 1, 34, "Subsubsection"], Cell[214874, 4838, 131, 1, 37, "Text"], Cell[215008, 4841, 189, 5, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[215234, 4851, 107, 1, 34, "Subsubsection"], Cell[215344, 4854, 668, 21, 53, "Text"], Cell[216015, 4877, 805, 23, 80, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[216857, 4905, 172, 2, 34, "Subsubsection"], Cell[217032, 4909, 591, 16, 35, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[217684, 4932, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[217773, 4937, 134, 3, 93, "Section"], Cell[CellGroupData[{ Cell[217932, 4944, 213, 3, 50, "Subsubsection"], Cell[218148, 4949, 142, 1, 37, "Text"], Cell[218293, 4952, 890, 26, 80, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[219220, 4983, 113, 1, 34, "Subsubsection"], Cell[219336, 4986, 151, 2, 37, "Text"], Cell[219490, 4990, 1322, 39, 58, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[220873, 5036, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[220962, 5041, 156, 3, 143, "Section"], Cell[CellGroupData[{ Cell[221143, 5048, 98, 1, 50, "Subsubsection"], Cell[221244, 5051, 216, 4, 37, "Text"], Cell[221463, 5057, 1061, 39, 131, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[222561, 5101, 103, 1, 34, "Subsubsection"], Cell[222667, 5104, 518, 13, 71, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[223222, 5122, 151, 2, 34, "Subsubsection"], Cell[223376, 5126, 418, 8, 59, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[223831, 5139, 108, 1, 34, "Subsubsection"], Cell[223942, 5142, 181, 2, 37, "Text"], Cell[224126, 5146, 96, 2, 35, "Input"], Cell[224225, 5150, 115, 1, 37, "Text"], Cell[224343, 5153, 164, 5, 35, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[224568, 5165, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[224657, 5170, 132, 3, 143, "Section"], Cell[CellGroupData[{ Cell[224814, 5177, 180, 2, 48, "Subsection"], Cell[224997, 5181, 15315, 276, 199, "Input"], Cell[240315, 5459, 504, 16, 58, "Input"], Cell[240822, 5477, 246, 5, 37, "Text"], Cell[241071, 5484, 1066, 28, 80, "Input"], Cell[242140, 5514, 130, 2, 37, "Text"], Cell[242273, 5518, 1868, 47, 124, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[244202, 5572, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[244291, 5577, 134, 3, 143, "Section"], Cell[CellGroupData[{ Cell[244450, 5584, 236, 5, 48, "Subsection"], Cell[244689, 5591, 165, 3, 37, "Text"], Cell[244857, 5596, 189, 5, 35, "Input"], Cell[245049, 5603, 707, 21, 35, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[245793, 5629, 110, 1, 40, "Subsection"], Cell[245906, 5632, 35306, 632, 239, "Input"], Cell[281215, 6266, 1075, 35, 44, "Text"], Cell[282293, 6303, 772, 23, 58, "Input"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[283126, 6333, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[283215, 6338, 155, 3, 143, "Section"], Cell[283373, 6343, 5658, 135, 305, "Input"], Cell[289034, 6480, 329, 6, 37, "Text"], Cell[289366, 6488, 4840, 132, 366, "Input"], Cell[294209, 6622, 256, 5, 59, "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[294514, 6633, 64, 1, 1, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[294603, 6638, 97, 1, 93, "Section"], Cell[294703, 6641, 158, 2, 37, "Text"] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *) (* NotebookSignature 7xDjw7h#45RF#C1TTPnkmhQQ *)