(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1106097, 34505]*) (*NotebookOutlinePosition[ 1107174, 34543]*) (* CellTagsIndexPosition[ 1107069, 34536]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(\(H\^\[Infinity]\ Control\ Design\ on\ Frequency\ Domain\)\(\ \[IndentingNewLine]\) \), FontFamily->"Times New Roman", FontSize->30]], "Title", TextAlignment->Center], Cell["\<\ B\[EAcute]la Pal\[AAcute]ncz Budapest Technical University, Hungary palancz@epito.bme.hu http://geo2.fmt.bme.hu/palancz Budapest, 2006 \ \>", "Author", TextAlignment->Center, FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[CellGroupData[{ Cell["Abstract", "Subsubsection"], Cell[TextData[{ "In this case study we demonstrate, how to employ the ", StyleBox["Mathematica", FontSlant->"Italic"], " Application, Control System Professional Suite, (CSPS) together with the \ Professor Helton's package, OPTDesign to develope a robust controller for a \ nonlinear plant. First, CSPS is used to linearize the nonlinear system in \ state space then to transform it into the frequency domain. In the second \ step, the OPTDesign package is applied to design optimal ", Cell[BoxData[ \(H\^\[Infinity]\)]], " controller on frequency domain considering different constrains \ formulated by Helton and Merino (1998). In addition a new technique is \ suggested to consider the disturbance model as a constrains inequality. In \ the last step, we use CSPS again to transform the Laplace representation of \ the controller dynamics back to state space and to find a minimal state space \ representation of it. This state space model of the controller was connected \ to the nonlinear plant model, and closed loop simulation of the controlled \ plant dynamics has been achieved." }], "Text"], Cell[BoxData[ \(\(Off[Syntax::stresc];\)\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \(\(Off[General::spell];\)\)], "Input", CellLabel->"In[19]:="] }, Open ]], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ Let us consider a water storage vessel with a time dependent inlet flow, \ Qin[t]. 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"Text"] }, Open ]], Cell[CellGroupData[{ Cell["Model equations", "Section"], Cell["The mass balance:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(deq = \(Qin[t] - Qout[t]\)\/A \[Equal] \(h'\)[t]\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[ RowBox[{\(\(Qin[t] - Qout[t]\)\/A\), "\[Equal]", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]], "Output", CellLabel->"Out[1]="] }, Open ]], Cell["\<\ The outlet flow rate can be expressed by the valve \"constant\" and the water \ level:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Qout[t_] = c[t]\ \@h[t]\)], "Input", CellLabel->"In[2]:="], Cell[BoxData[ \(c[t]\ \@h[t]\)], "Output", CellLabel->"Out[2]="] }, Open ]], Cell["\<\ Combining them, we get the mathematical model of the system, which is a \ nonlinear, non-autonomous ordinary differential equation for the \ state-variable h[t].\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(deq\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ RowBox[{\(\(\(-c[t]\)\ \@h[t] + Qin[t]\)\/A\), "\[Equal]", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]], "Output", CellLabel->"Out[3]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Nonlinear model", "Section"], Cell["The general form of the nonlinear model is ", "Text"], Cell[BoxData[{ StyleBox[\(x\& . = f \((x, u)\)\), FontFamily->"Times New Roman"], "\[IndentingNewLine]", StyleBox[\(y = g \((x, u)\)\), FontFamily->"Times New Roman"]}], "Text", TextAlignment->Center], Cell["In our case", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f = deq[\([1]\)]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \(\(\(-c[t]\)\ \@h[t] + Qin[t]\)\/A\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell["and", "Text"], Cell[BoxData[ \(x = {h[t]}; y = {h[t]}; u = {Qin[t], c[t]}; g = y;\)], "Input", CellLabel->"In[5]:="], Cell[TextData[{ "To design the control for the system , it should be linearized. Now, we \ load the CSPS application. However, in order to remove any packages easily, \ first we save the original content of the ", StyleBox["$ContextPath,", FontFamily->"Courier New"] }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"BasicContextPath", "=", StyleBox["$ContextPath", FontFamily->"Courier New"]}], StyleBox[";", FontFamily->"Courier New"]}]], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \(<< ControlSystems`\)], "Input", CellLabel->"In[7]:=", CellTags->"1.1"] }, Open ]], Cell[CellGroupData[{ Cell["Linearization", "Section"], Cell["\<\ The linearization should be carried out at a steady state, namely at (h0, \ Qin0, c0):\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ControlObjectSS = Linearize[f, g, {{h[t], h0}}, {{Qin[t], Qin0}, {c[t], c0}}]\)], "Input",\ CellLabel->"In[8]:="], Cell[BoxData[ \(StateSpace[{{\(-\(c0\/\(2\ A\ \@h0\)\)\)}}, {{1\/A, \(-\(\@h0\/A\)\)}}, \ {{1}}, {{0, 0}}]\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell["\<\ From the state representation, one can get the equation form of the \ linearized system:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ControlObjectSS // EquationForm\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ FormBox[ TagBox[ FormBox[GridBox[{ { RowBox[{\(\[ScriptX]\& . \), "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-\(c0\/\(2\ A\ \@h0\)\)\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/A\), \(-\(\@h0\/A\)\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]}, { RowBox[{"\[ScriptY]", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]} }, ColumnAlignments->{"="}, AllowScriptLevelChange->False], "TraditionalForm"], (EquationForm[ StateSpace[ SlotSequence[ 1]]]&)], TraditionalForm]], "Output", CellLabel->"Out[9]//EquationForm="] }, Open ]], Cell[TextData[{ "Here we should keep in mind, that the variables ", StyleBox["x and u", FontSlant->"Italic"], " represent the deviation from the steady state. Our model in matrix form \ is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm[%]\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ FormBox[ TagBox[ FormBox[ SubsuperscriptBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-\(c0\/\(2\ A\ \@h0\)\)\), \(1\/A\), \ \(-\(\@h0\/A\)\)}, {"1", "0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, RowLines->{True, False}, ColumnLines->{True, False}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "TraditionalForm"], FormBox["\[Bullet]", "TraditionalForm"], FormBox[ StyleBox["\<\"\[ScriptCapitalS]\"\>", Editable->False, ShowStringCharacters->False], "TraditionalForm"], MultilineFunction->None], "TraditionalForm"], (StateSpace[ SlotSequence[ 1]]&)], TraditionalForm]], "Output", CellLabel->"Out[10]//TraditionalForm="] }, Open ]], Cell["or its representation on frequency domain:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ControlObjectTF = TransferFunction[s, ControlObjectSS] // Simplify\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(TransferFunction[ s, {{\(2\ \@h0\)\/\(c0 + 2\ A\ \@h0\ s\), \(-\(\(2\ h0\)\/\(c0 + 2\ A\ \@h0\ s\)\)\)}}]\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell["The numerical values are", "Text"], Cell[BoxData[ \(h0 = 9. ; A = 10. ; c0 = 3. ; Qin0 = 9. ;\)], "Input", CellLabel->"In[12]:="], Cell["\<\ The transfer function between the level h[t] and the inlet flow rate Qin[t], \ as disturbance:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pd[s_] = \(2\ \@h0\)\/\(c0 + 2\ A\ \@h0\ s\)\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ \(6.`\/\(\(\(3.`\)\(\[InvisibleSpace]\)\) + 60.`\ s\)\)], "Output", CellLabel->"Out[13]="] }, Open ]], Cell["\<\ The transfer function between the level h[t] and the control valve position, \ c[t] as actuator:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(p[s_] = \(-\(\(2\ h0\)\/\(c0 + 2\ A\ \@h0\ s\)\)\)\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ \(\(-\(18.`\/\(\(\(3.`\)\(\[InvisibleSpace]\)\) + 60.`\ s\)\)\)\)], "Output", CellLabel->"Out[14]="] }, Open ]], Cell[TextData[{ "Now, we remove ", StyleBox["ControlSystems", FontFamily->"Courier New"], " application and load the ", StyleBox["OPTDesign", FontFamily->"Courier New"], " package, which is available on ", StyleBox["http://www.math.ucsd.edu/~anopt/", "Address"] }], "Text"], Cell[BoxData[ \(\(Remove[t, s, s$];\)\)], "Input", CellLabel->"In[15]:="], Cell[BoxData[ \(\($ContextPath = BasicContextPath;\)\)], "Input", CellLabel->"In[16]:="], Cell[CellGroupData[{ Cell[BoxData[ \(<< OPTDesign`\)], "Input", CellLabel->"In[20]:="], Cell[BoxData[ \(" \n Anopt 4.0 (c) Copyright 1991-2001 \n J.W. 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consider C(s) and P(s) the transfer functions for the compensator and \ plant, respectively, then the closed-loop transfer function is,\ \>", "Text"], Cell[BoxData[ \(T \((s)\)\ = \ \(P \((s)\)\ C \((s)\)\)\/\(1 + P \((s)\)\ C \ \((s)\)\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ Roughly speaking, to design a system one must choose all of the system's \ parts that have not been given in advance. This is done according to some \ criteria or requirements. In this case, the most obvious way is to specify \ C(s). However, now we use another way, namely we specify T(s) instead of \ C(s). If T(s) is specified, the one can calculate C(s) as \ \>", "Text"], Cell[BoxData[ \(C \((s)\) = \(T \((s)\)\)\/\(\((1 - T \((s)\))\)\ P \((s)\)\)\)], "Text",\ TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "Under these circumstanses we refer to T(s) as a variable, as a designable \ transfer function.\nThe designable transfer function cannot take an arbitrary \ form. The set of admissible functions to which the designable T(s) must \ belong is a subset of \[ScriptCapitalR]", Cell[BoxData[ \(\[ScriptCapitalH]\^\[Infinity]\)]], ". These rational functions are stable, proper and real (the coefficients \ are real). A rational function is stable if every pole of the function has \ negative real part and the relative degree is not positve. The relative \ degree of a rational function F(s) = ", Cell[BoxData[ \(\(N \((s)\)\)\/\(D \((s)\)\)\)]], " is," }], "Text"], Cell["d(F(s)):= degree(D(s)) - degree(N(s))", "Text", TextAlignment->Center], Cell["\<\ A function F(s) is called proper if d(F(s)) \[GreaterEqual] 0 and strictly \ proper if d(F(s)) > 0. If F(s) proper and stable, then it is bounded,\ \>", "Text"], Cell["\[VerticalSeparator]F(i \[Omega])\[VerticalSeparator] \[LessEqual] M \ for all \[Omega]\[Element]R", "Text", TextAlignment->Center], Cell[BoxData[ \(\(\(The\ smallest\ such\ M\ denoted\ \((\(\[VerticalSeparator]\)\(F\)\(\ \[VerticalSeparator]\))\)\_\[Infinity]\)\(,\)\)\)], "Text", FontFamily->"Times New Roman"], Cell[BoxData[ \(\((\(\[VerticalSeparator]\)\(F\)\(\[VerticalSeparator]\))\)\_\[Infinity]\ := sup \[VerticalSeparator] F \((i\ \[Omega])\) \[VerticalSeparator] \ for\ \ all\ \ \[Omega] \[Element] R\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "In addition a designable transfer function should ensure internal \ stability, namely there should be no cancellation of \ left-half-plane(unstable) poles and zeros in the product P(s) C(s).\nThe \ usual convention is to denote the set of all rational functions that are \ stable and real by \[ScriptCapitalR]", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalH]\^\[Infinity]\)]], ". Note that functions in \[ScriptCapitalR]", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalH]\^\[Infinity]\)]], " can be described as proper rational functions with real coefficents and \ with no poles in the closed rigth half-plane RHP." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Performance Requirements ", "Section"], Cell["\<\ A frequency domain performance requirement is an inequality that the \ designable transfer function must satisfy, and an interval in \[Omega] for \ which the inequality is required to hold.\ \>", "Text"], Cell[CellGroupData[{ Cell["Disk inequalities", "Subsubsection"], Cell["\<\ All of the frequency domain performance requirements treated here can be \ written in the form of a disk inequality\ \>", "Text"], Cell[TextData[{ "\[VerticalSeparator]K(\[ImaginaryI] \[Omega]) - T(\[ImaginaryI] \[Omega])\ \[VerticalSeparator] \[LessEqual] R(\[ImaginaryI] \[Omega]) \ for ", Cell[BoxData[ \(\[Omega]\_\(\(a\)\(\ \)\)\)]], "\[LessEqual] \[Omega] \[LessEqual] ", Cell[BoxData[ \(\[Omega]\_b\)]] }], "Text", TextAlignment->Center], Cell["\<\ which T must satisfy. 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The gain margin g, phase margin \[Phi], and gain - phase margin m" }], "Text", TextAlignment->Center], Cell[TextData[{ "If a given stable system has m near 0, then it is close to the unstable \ case, which is undesirable. If ", Cell[BoxData[ \(\[Alpha]\_g\)]], " denotes the largest value of 1/m considered to be acceptable, then" }], "Text"], Cell[BoxData[ \(\(\(\[VerticalSeparator]\)\(1\ - T \((\[ImaginaryI]\ \[Omega])\)\)\(\[VerticalSeparator]\)\(\(\ \[LessEqual]\)\(\ \ \ \ \)\(\[Alpha]\_g\ \ for\ \ \ all\ \ \ \ \[Omega]\)\)\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"] }, Open ]], Cell[CellGroupData[{ Cell["Disturbance rejection", "Subsubsection"], Cell["\<\ Systems are effected by disturbances, and one of the purposes of feedback \ control is to minimize the effect of them. 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StyleBox["Fig. 4", FontSlant->"Italic"], ". The control scheme with the disturbance transfer function" }], "Text", TextAlignment->Center], Cell[TextData[{ "where Qin(t) is the disturbance - generally ", Cell[BoxData[ \(\(d\^*\)\)]], "(t). Let us suppose that the disturbance has the transfer function ", Cell[BoxData[ \(p\_d\)], TextAlignment->Center, FontFamily->"Times New Roman"], ", then" }], "Text"], Cell[BoxData[ \(d \((s)\) = \(p\_d\) \((s)\)\ \(d\^*\) \((s)\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["The transfer function from disturbance d(s) to plant is", "Text"], Cell[BoxData[ \(T\_\(d\ \[Rule] \ h\)\ = \ \(1\/\(1 + P\ C\) = 1 - T\)\)], "Text", TextAlignment->Center], Cell["Now, we want to restrict the size of", "Text"], Cell[BoxData[ \(T\_\(\(d\^*\) \[Rule] h\) = \ \(h\/\(d\^*\) = \(\(h\/d\) d\/\(d\^*\) = \((1 - T)\)\ p\_d\)\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["namely", "Text"], Cell[BoxData[ \(\(\(\[VerticalSeparator]\)\(T\_\(\(d\^*\) \[Rule] y\)\)\(\[VerticalSeparator]\)\) = \ \(\(\[VerticalSeparator]\)\(\((1 - T)\)\ p\_d\)\(\[VerticalSeparator]\)\(\(\[LessEqual]\)\(\ \)\(c\ \)\)\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ therefore the corresponding disk inequality constraint for T is\ \>", "Text"], Cell[BoxData[ \(\(\(\[VerticalSeparator]\)\(1 - T\)\(\[VerticalSeparator]\)\(\(\[LessEqual]\)\(c\/\(\(\ \[VerticalSeparator]\)\(p\_d\)\(\[VerticalSeparator]\)\)\)\)\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["where c is a contant value.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Bandwidth", "Subsubsection"], Cell[TextData[{ "Bandwidth of a system is commonly defined as the frequency ", Cell[BoxData[ \(\[Omega]\_b\)]], " at which \[VerticalSeparator]T(\[ImaginaryI] \[Omega]) \ \[VerticalSeparator] falls below a given constant times the low - frequency \ value of the input. 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"Text"], Cell[BoxData[ \(T\ = \ \(P\ C\)\/\(1\ - \ P\ C\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["for large frequency ", "Text"], Cell[BoxData[ \(\(\(\[VerticalSeparator]\)\(\ \)\(T\ \((\[ImaginaryI]\ \[Omega])\)\)\(\ \[VerticalSeparator]\)\(\(\[LessEqual]\)\(\ \ \ \)\(\(\(\[VerticalSeparator]\)\(P\ \((\[ImaginaryI]\ \[Omega])\)\ C \((\ \[ImaginaryI]\ \[Omega])\)\)\(\[VerticalSeparator]\)\)\/\(\(\(1\)\(\ \ \)\(-\)\) \[VerticalSeparator] \ P \((\[ImaginaryI]\ \[Omega])\)\ C \((\[ImaginaryI]\ \ \[Omega])\) \[VerticalSeparator] \)\)\(\ \)\(\[TildeTilde]\)\)\(\ \ \ \ \ \ \)\(\[VerticalSeparator]\)\(P\ \((\[ImaginaryI]\ \[Omega])\)\ C \((\ \[ImaginaryI]\ \[Omega])\)\)\(\[VerticalSeparator]\)\(\ \ \ \ \ \)\(for\ \ large\ \[VerticalBar] \[Omega] \[VerticalBar] \)\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ To obtain an inequality, C(\[ImaginaryI] \[Omega]) must be eliminated. \ Therefore let\ \>", "Text"], Cell[BoxData[ \(C 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Requirements envelope" }], "Text", TextAlignment->Center], Cell["\<\ In our case, the parameters for the performance requirements are chosen as it \ follows,\ \>", "Text"], Cell[BoxData[ \(c = 0.9; \[Alpha]r = 185. ; n = 2; \[Omega]d = 0.275; \[Omega]b = 2. ; \[Omega]r = 4. ; \[Alpha]g = 2.5; \[Alpha]b = 0.9;\)], "Input", CellLabel->"In[21]:="], Cell["Then the disk inequalities can be written for the center,", "Text"], Cell[BoxData[ \(\(k0[\[Omega]_] = Which[0 \[LessEqual] Abs[\[Omega]] < \[Omega]b, 1.0, \[Omega]b \[LessEqual] Abs[\[Omega]], 0];\)\)], "Input", CellLabel->"In[22]:="], Cell["and for the radius", "Text"], Cell[BoxData[ \(\(\(r0[\[Omega]_] = Which[\ 0 \[LessEqual] \ Abs[\[Omega]] < \[Omega]d, c/Abs[pd[\[ImaginaryI]\ \[Omega]]], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[Omega]d \[LessEqual] Abs[\[Omega]] < \[Omega]b, \[Alpha]g, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[Omega]b \[LessEqual] Abs[\[Omega]] < \[Omega]r, \[Alpha]b, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[Omega]r \[LessEqual] Abs[\[Omega]], \ \[Alpha]r\ Abs[ p[\[ImaginaryI]\ \[Omega]]]/\[Omega]\^n];\)\(\ \)\)\)], \ "Input", CellLabel->"In[23]:="], Cell[TextData[{ "Now we remove the ", StyleBox["Controlsystems", FontFamily->"Courier New"], " application and load the ", StyleBox["OPTDesign", FontFamily->"Courier New"], " package," }], "Text"], Cell["\<\ The detailed description of these packages can be found in Helton and Merino \ (1998). 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L p .40211 .48093 L F P 0 g s 1 .821 .589 r .40263 .48411 m .39699 .48382 L .40211 .48093 L F .39729 .48544 m .39699 .48382 L p .40263 .48411 L F P 0 g s 1 .82 .594 r .46321 .84966 m .44851 .86368 L p .39764 .4886 L F P 0 g s 1 .82 .594 r .39699 .48382 m .39786 .48861 L p .39764 .4886 L F P 0 g s 1 .82 .594 r .39786 .48861 m .46321 .84966 L p .39764 .4886 L F P 0 g s .554 .981 1 r .3854 .48844 m .37876 .48841 L p .37917 .4883 L F P 0 g s .1 .1 .1 r .41488 .479 m .42028 .47595 L p .39163 .48684 L F P 0 g s .554 .981 1 r .40888 .48051 m .4037 .48226 L p .40701 .48099 L F P 0 g s .1 .1 .1 r .39833 .4843 m .42028 .47595 L .39347 .48585 L F .42028 .47595 m .39711 .48378 L p .39347 .48585 L F P 0 g s .1 .1 .1 r .39711 .48378 m .39699 .48382 L p .39192 .48673 L F P 0 g s .554 .981 1 r .4037 .48226 m .3855 .4884 L p .40701 .48099 L F P 0 g s .217 .1 .1 r .42028 .47595 m .42502 .47142 L p .39713 .48376 L F P 0 g s .217 .1 .1 r .39699 .48382 m .39711 .48378 L p .39713 .48376 L F P 0 g s .217 .1 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The design problem is the following:\ \>", "Text"], Cell[TextData[{ "Given\n1. a plant, P(s)\n2. a set of performance requirements, \ \[ScriptCapitalP]\n3. a set \[ScriptCapitalI] = {T(s) \[Element] \ \[ScriptCapitalR]", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalH]\^\[Infinity]\)]], "} where T(s) is the closed loop transfer function of an internally stable \ system with plant P(s)." }], "Text"], Cell["\<\ Determine T \[Element] \[ScriptCapitalI] if there exist that satisfy all the \ performance requirements in \[ScriptCapitalP]. Such a T is called feasible.\ \>", "Text"], Cell["\<\ Optimal design means that find a feasible T that yields the best performance. \ If many satisfactory T exist, tighten or loosen the specification, and \ repeat the design.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Generating a performance function", "Subsubsection"], Cell["\<\ As we have seen the performance requirements can be expressed in from of disk \ inequalities, which can be written in the following way, too: \ \>", "Text"], Cell[BoxData[ \(\(1\/\((R \((\[ImaginaryI]\ \[Omega])\))\)\^2\) \((\(\ \[VerticalSeparator]\)\(K \((\[ImaginaryI]\ \[Omega])\)\ - \ T \((\ \[ImaginaryI]\ \[Omega])\)\)\(\[VerticalSeparator]\))\)\^2 \[LessEqual] \ 1\ \ \ for\ \ \ \ \(\[ForAll] \ \[Omega]\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ Now, for a given T calculate the largest value of the left-hand side of this, \ see Fig. 9.,\ \>", "Text"], Cell[BoxData[ \(\[Gamma] \((T)\) = \ \(sup\+\[Omega]\) 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The largest values of different T functions" }], "Text", TextAlignment->Center], Cell["\<\ If a particular T is avaiable, then one can check if the original performance \ requirements are satisfied. In case of \[Gamma](T) \[LessEqual] 1, T is \ satisfactory.\ \>", "Text"], Cell["\<\ For designable T and given center and radius functions, K and R respectively, \ the expression\ \>", "Text"], Cell[BoxData[ \(\(1\/\((R \((\[ImaginaryI]\ \[Omega])\))\)\^2\) \((\(\ \[VerticalSeparator]\)\(K \((\[ImaginaryI]\ \[Omega])\)\ - \ T \((\ \[ImaginaryI]\ \[Omega])\)\)\(\[VerticalSeparator]\))\)\^2\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ is called the circular performance function, or simply performance function. \ The number \[Gamma](T) is the worst-case performance of T, but we will call \ it simply, the performance function of T.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Finding T with the best performance", "Subsubsection"], Cell[BoxData[ \(A\ feasible\ function\ \(T\^*\)\ with\ property\ that\)], "Text", FontFamily->"Times New Roman"], Cell[BoxData[ \(\[Gamma] \((\(T\^*\))\) \[LessEqual] \[Gamma] \((T)\)\ \ \(\[ForAll] \ T\ feasible\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ is called optimal. The performance of the optimal designable transfer \ function is\ \>", "Text"], Cell[BoxData[ \(\(\[Gamma]\^*\)\ = \ \[Gamma] \((T)\)\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"] }, Open ]], Cell[CellGroupData[{ Cell["Optimization", "Subsubsection"], Cell[TextData[{ "The process of finding the optimal ", Cell[BoxData[ \(TraditionalForm\`\(T\^*\)\)]], "and the corresponding optimal performance ", Cell[BoxData[ \(TraditionalForm\`\(\[Gamma]\^*\)\)]], "is called optimization. More precisely, we say that ", Cell[BoxData[ \(TraditionalForm\`\(T\^*\)\)]], "is a solution to" }], "Text"], Cell[BoxData[ \(\(\[Gamma]\^*\)\ = \ inf\+T\ \(sup\+\[Omega]\) \(1\/\((R \((\[ImaginaryI]\ \[Omega])\))\)\^2\ \) \((\(\[VerticalSeparator]\)\(K \((\[ImaginaryI]\ \[Omega])\)\ - \ T \((\ \[ImaginaryI]\ \[Omega])\)\)\(\[VerticalSeparator]\))\)\^2\)], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "There exist A, B \[Element] \[ScriptCapitalR]", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalH]\^\[Infinity]\)]], ", that if T \[Element]\[ScriptCapitalI], then " }], "Text"], Cell[BoxData[ RowBox[{"T", " ", "=", " ", RowBox[{\(A\ + \ B\ \(T\^+\)\ \ \ \ \ \ \ \ for\ \ \ some\ \ \(T\^+\)\), " ", "\[Element]", " ", RowBox[{"\[ScriptCapitalR]", FormBox[\(\[ScriptCapitalH]\^\[Infinity]\), "TraditionalForm"]}]}]}]], "Text", TextAlignment->Center, FontFamily->"Times New Roman"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Design with ", StyleBox["OPTDesign", FontSlant->"Italic"] }], "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(OPTDesign[p, Radius \[Rule] r0, Center \[Rule] k0, Nsmth \[Rule] 100, Ngrid \[Rule] 512, Cdecay \[Rule] 2]\)], "Input", CellLabel->"In[27]:="], Cell[BoxData[ \("\nParameterization for internal stability:"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \(" T = A + B * T1, where T1 is RHP stable and"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ InterpretationBox[\("A[s]="\[InvisibleSpace]1\/\((\(\(2.`\)\(\ \[InvisibleSpace]\)\) + s)\)\^3\), SequenceForm[ "A[s]=", Power[ Plus[ 2.0, s], -3]], Editable->False]], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ InterpretationBox[\("B[s]="\[InvisibleSpace]1\/\((\(\(2.`\)\(\ \[InvisibleSpace]\)\) + s)\)\^3\), SequenceForm[ "B[s]=", Power[ Plus[ 2.0, s], -3]], Editable->False]], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \(" "\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("Processing performance function..."\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("Sampling Radius and Center ..."\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("Smoothing Radius and Center ..."\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("Optimization routine Anopt now number-crunching ..."\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \(" "\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("It : Current Value : Step : Optimality Tests : Error : Sm. \ : Grid"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \(" : gammastar : : flat grAlign : ned : \ :"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("--------------\ ------"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \(" 0 : 1.55678 E+00 : N/A : 9.99848E-01 : 0 E+00 : N/A : \ NON : 512"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \(" 1 : 8.93373 E-02 : 1.85209E+01 : 2.11386E-05 : 0 E+00 : \ 1.91705E-05 : NON : 512"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \(" "\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \(" "\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("Summary"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("-------"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("gammastar = 8.93373 E-02"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("flat = 2.11386 E-05"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("grAlign = 0 E+00"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("ned = 1.91705E-05"\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("Calculating output functions ..."\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("Resetting options ..."\)], "Print", CellLabel->"From In[27]:="], Cell[BoxData[ \("Done! "\)], "Print", CellLabel->"From In[27]:="] }, Open ]], Cell[TextData[{ "The output parameter ", Cell[BoxData[ \(\(\[Gamma]\^*\)\)]], " (gammastar) is less than 1. 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.368 .576 r .55251 .43362 m .52922 .44155 L .52854 .45218 L p F P 0 g s .538 .216 .42 r .50382 .46965 m .4807 .47746 L .47833 .48596 L p F P 0 g s .538 .216 .42 r .45529 .49371 m .47833 .48596 L .4807 .47746 L p F P 0 g s .662 .466 .673 r .55236 .42209 m .57574 .41408 L .57478 .40177 L p F P 0 g s .642 .421 .63 r .57574 .41408 m .55236 .42209 L .55251 .43362 L p F P 0 g s .306 .1 .1 r .3474 .9202 m .32485 .92637 L .30865 .94073 L p F P 0 g s .306 .1 .1 r .28601 .94688 m .30865 .94073 L .32485 .92637 L p F P 0 g s .768 .818 .986 r .48694 .24361 m .46379 .25227 L .471 .26183 L p F P 0 g s .768 .818 .986 r .471 .26183 m .4941 .25323 L .48694 .24361 L p F P 0 g s .473 .103 .301 r .47833 .48596 m .45529 .49371 L .45208 .50099 L p F P 0 g s .473 .103 .301 r .42912 .50869 m .45208 .50099 L .45529 .49371 L p F P 0 g s .02 w .09379 .57965 Mdot .5 Mabswid .702 .567 .77 r .5941 .36722 m .61766 .35901 L .61453 .34513 L p F P 0 g s .691 .537 .742 r .61766 .35901 m .5941 .36722 L .59653 .3807 L p F P 0 g s .764 .79 .964 r .51736 .24457 m .4941 .25323 L .50112 .26386 L p F P 0 g s .764 .79 .964 r .50112 .26386 m .52433 .25527 L .51736 .24457 L p F P 0 g s .748 .714 .902 r .5844 .26208 m .56092 .27067 L .56707 .2837 L p F P 0 g s .748 .714 .902 r .56707 .2837 m .59053 .27518 L .5844 .26208 L p F P 0 g s .677 .504 .71 r .57478 .40177 m .59827 .39368 L .59653 .3807 L p F P 0 g s .662 .466 .673 r .59827 .39368 m .57478 .40177 L .57574 .41408 L p F P 0 g s .02 w .09348 .5784 Mdot .5 Mabswid .42 .1 .159 r .45214 .87725 m .42917 .88358 L .41607 .90145 L p F P 0 g s .42 .1 .159 r .39301 .90775 m .41607 .90145 L .42917 .88358 L p F P 0 g s .759 .763 .943 r .54771 .24661 m .52433 .25527 L .5311 .26685 L p F P 0 g s .759 .763 .943 r .5311 .26685 m .55445 .25826 L .54771 .24661 L p F P 0 g s .02 w .09317 .57712 Mdot .5 Mabswid .728 .644 .84 r .62524 .29423 m .60162 .30269 L .60647 .31682 L p F P 0 g s .728 .644 .84 r .60647 .31682 m .6301 .30841 L .62524 .29423 L p F P 0 g s .72 .62 .818 r .6301 .30841 m .60647 .31682 L .61079 .33102 L p F P 0 g s .72 .62 .818 r .61079 .33102 m .63444 .32265 L .6301 .30841 L p F P 0 g s .712 .594 .795 r .63444 .32265 m .61079 .33102 L .61453 .34513 L p F P 0 g s .02 w .09286 .5758 Mdot .5 Mabswid .735 .667 .861 r .61991 .28024 m .59629 .28876 L .60162 .30269 L p F P 0 g s .735 .667 .861 r .60162 .30269 m .62524 .29423 L .61991 .28024 L p F P 0 g s .691 .537 .742 r .59653 .3807 m .62013 .37252 L .61766 .35901 L p F P 0 g s .677 .504 .71 r .62013 .37252 m .59653 .3807 L .59827 .39368 L p F P 0 g s .754 .738 .922 r .57796 .24961 m .55445 .25826 L .56092 .27067 L p F P 0 g s .754 .738 .922 r .56092 .27067 m .5844 .26208 L .57796 .24961 L p F P 0 g s .02 w .09255 .57445 Mdot .5 Mabswid .712 .594 .795 r .61453 .34513 m .63821 .33681 L .63444 .32265 L p F P 0 g s .702 .567 .77 r .63821 .33681 m .61453 .34513 L .61766 .35901 L p F P 0 g s .742 .691 .882 r .61417 .26659 m .59053 .27518 L .59629 .28876 L p F P 0 g s .742 .691 .882 r .59629 .28876 m .61991 .28024 L .61417 .26659 L p F P 0 g s .617 .368 .576 r .52854 .45218 m .55191 .44425 L .55251 .43362 L p F P 0 g s .584 .301 .509 r .55191 .44425 m .52854 .45218 L .5271 .46179 L p F P 0 g s .584 .301 .509 r .50382 .46965 m .5271 .46179 L .52854 .45218 L p F P 0 g s .642 .421 .63 r .55251 .43362 m .57597 .42562 L .57574 .41408 L p F P 0 g s .617 .368 .576 r .57597 .42562 m .55251 .43362 L .55191 .44425 L p F P 0 g s .02 w .09224 .57305 Mdot .5 Mabswid .538 .216 .42 r .5271 .46179 m .50382 .46965 L .50154 .47815 L p F P 0 g s .538 .216 .42 r .47833 .48596 m .50154 .47815 L .50382 .46965 L p F P 0 g s .662 .466 .673 r .57574 .41408 m .5993 .40601 L .59827 .39368 L p F P 0 g s .642 .421 .63 r .5993 .40601 m .57574 .41408 L .57597 .42562 L p F P 0 g s .306 .1 .1 r .37013 .914 m .3474 .9202 L .33146 .93454 L p F P 0 g s .306 .1 .1 r .30865 .94073 m .33146 .93454 L .3474 .9202 L p F P 0 g s .768 .818 .986 r .51026 .23488 m .48694 .24361 L .4941 .25323 L p F P 0 g s .768 .818 .986 r .4941 .25323 m .51736 .24457 L .51026 .23488 L p F P 0 g s .02 w .09192 .57161 Mdot .13672 .47843 Mdot .13795 .4782 Mdot .13549 .47865 Mdot .13916 .47797 Mdot .13425 .47888 Mdot .14037 .47773 Mdot .133 .4791 Mdot .14157 .47749 Mdot .13173 .47932 Mdot .5 Mabswid .473 .103 .301 r .50154 .47815 m .47833 .48596 L .47521 .49323 L p F P 0 g s .473 .103 .301 r .45208 .50099 m .47521 .49323 L .47833 .48596 L p F P 0 g s .02 w .14277 .47724 Mdot .13045 .47955 Mdot .14396 .47699 Mdot .12916 .47978 Mdot .14515 .47673 Mdot .14634 .47647 Mdot .12786 .48001 Mdot .5 Mabswid .702 .567 .77 r .61766 .35901 m .64138 .35073 L .63821 .33681 L p F P 0 g s .691 .537 .742 r .64138 .35073 m .61766 .35901 L .62013 .37252 L p F P 0 g s .02 w .14752 .4762 Mdot .5 Mabswid .764 .79 .964 r .54079 .23584 m .51736 .24457 L .52433 .25527 L p F P 0 g s .764 .79 .964 r .52433 .25527 m .54771 .24661 L .54079 .23584 L p F P 0 g s .02 w .12654 .48025 Mdot .14871 .47592 Mdot .14989 .47564 Mdot .12521 .48051 Mdot .09161 .57013 Mdot .15107 .47536 Mdot .5 Mabswid .748 .714 .902 r .60806 .25343 m .5844 .26208 L .59053 .27518 L p F P 0 g s .748 .714 .902 r .59053 .27518 m .61417 .26659 L .60806 .25343 L p F P 0 g s .02 w .12386 .48078 Mdot .15225 .47506 Mdot .5 Mabswid .677 .504 .71 r .59827 .39368 m .62194 .38553 L .62013 .37252 L p F P 0 g s .662 .466 .673 r .62194 .38553 m .59827 .39368 L .5993 .40601 L p F P 0 g s .02 w .15343 .47477 Mdot .12249 .48107 Mdot .15461 .47446 Mdot .5 Mabswid .377 .1 .137 r .47521 .49323 m .45208 .50099 L .44815 .50697 L p F P 0 g s .377 .1 .137 r .4251 .51468 m .44815 .50697 L .45208 .50099 L p F P 0 g s .02 w .12111 .48138 Mdot .15579 .47415 Mdot .15697 .47384 Mdot .5 Mabswid .42 .1 .159 r .41607 .90145 m .4393 .8951 L .45214 .87725 L p F P 0 g s .02 w .0913 .5686 Mdot .11972 .48172 Mdot .15816 .47352 Mdot .5 Mabswid .759 .763 .943 r .57127 .23789 m .54771 .24661 L .55445 .25826 L p F P 0 g s .759 .763 .943 r .55445 .25826 m .57796 .24961 L .57127 .23789 L p F P 0 g s .02 w .15935 .47319 Mdot .11831 .4821 Mdot .16054 .47286 Mdot .16173 .47253 Mdot .11689 .48252 Mdot .09099 .56702 Mdot .16293 .47218 Mdot .5 Mabswid .728 .644 .84 r .64904 .2857 m .62524 .29423 L .6301 .30841 L p F P 0 g s .728 .644 .84 r .6301 .30841 m .65391 .29994 L .64904 .2857 L p F P 0 g s .72 .62 .818 r .65391 .29994 m .6301 .30841 L .63444 .32265 L p F P 0 g s .02 w .11545 .48299 Mdot .16413 .47183 Mdot .5 Mabswid 1 .711 .561 r .26353 .95298 m .19603 .83219 L .18341 .83789 L p F P 0 g s 1 .729 .577 r .19603 .83219 m .11415 .67101 L .11046 .67218 L p F P 0 g s 1 .729 .577 r .11046 .67218 m .18341 .83789 L .19603 .83219 L p F P 0 g s .72 .62 .818 r .63444 .32265 m .65826 .31423 L .65391 .29994 L p F P 0 g s .712 .594 .795 r .65826 .31423 m .63444 .32265 L .63821 .33681 L p F P 0 g s .02 w .16534 .47148 Mdot .5 Mabswid .735 .667 .861 r .64371 .27165 m .61991 .28024 L .62524 .29423 L p F P 0 g s .735 .667 .861 r .62524 .29423 m .64904 .2857 L .64371 .27165 L p F P 0 g s .02 w .11401 .48351 Mdot .16655 .47112 Mdot .5 Mabswid .691 .537 .742 r .62013 .37252 m .64391 .36428 L .64138 .35073 L p F P 0 g s .677 .504 .71 r .64391 .36428 m .62013 .37252 L .62194 .38553 L p F P 0 g s .02 w .09068 .56539 Mdot .16776 .47075 Mdot .11257 .4841 Mdot .16898 .47038 Mdot .5 Mabswid .754 .738 .922 r .60165 .24089 m .57796 .24961 L .5844 .26208 L p F P 0 g s .754 .738 .922 r .5844 .26208 m .60806 .25343 L .60165 .24089 L p F P 0 g s .02 w .17021 .47 Mdot .11112 .48475 Mdot .5 Mabswid .146 .1 .1 r .28601 .94688 m .26353 .95298 L .24485 .96354 L p F P 0 g s 1 .711 .561 r .18341 .83789 m .24485 .96354 L .26353 .95298 L p F P 0 g s .712 .594 .795 r .63821 .33681 m .66207 .32843 L .65826 .31423 L p F P 0 g s .702 .567 .77 r .66207 .32843 m .63821 .33681 L .64138 .35073 L p F P 0 g s .02 w .17144 .46962 Mdot .09037 .56371 Mdot .17268 .46923 Mdot .5 Mabswid .742 .691 .882 r .63797 .25794 m .61417 .26659 L .61991 .28024 L p F P 0 g s .742 .691 .882 r .61991 .28024 m .64371 .27165 L .63797 .25794 L p F P 0 g s .617 .368 .576 r .55191 .44425 m .57546 .43627 L .57597 .42562 L p F P 0 g s .584 .301 .509 r .57546 .43627 m .55191 .44425 L .55056 .45387 L p F P 0 g s .584 .301 .509 r .5271 .46179 m .55056 .45387 L .55191 .44425 L p F P 0 g s .02 w .10967 .48548 Mdot .5 Mabswid .642 .421 .63 r .57597 .42562 m .59961 .41757 L .5993 .40601 L p F P 0 g s .617 .368 .576 r .59961 .41757 m .57597 .42562 L .57546 .43627 L p F P 0 g s .02 w .17392 .46883 Mdot .5 Mabswid .538 .216 .42 r .55056 .45387 m .5271 .46179 L .52492 .47028 L p F P 0 g s .538 .216 .42 r .50154 .47815 m .52492 .47028 L .5271 .46179 L p F P 0 g s .02 w .17517 .46843 Mdot .09006 .56197 Mdot .10824 .48629 Mdot .17643 .46802 Mdot .5 Mabswid .662 .466 .673 r .5993 .40601 m .62304 .39788 L .62194 .38553 L p F P 0 g s .642 .421 .63 r .62304 .39788 m .5993 .40601 L .59961 .41757 L p F P 0 g s .306 .1 .1 r .39301 .90775 m .37013 .914 L .35444 .9283 L p F P 0 g s .306 .1 .1 r .33146 .93454 m .35444 .9283 L .37013 .914 L p F P 0 g s .02 w .1777 .4676 Mdot .10681 .48719 Mdot .5 Mabswid .768 .818 .986 r .53375 .22608 m .51026 .23488 L .51736 .24457 L p F P 0 g s .768 .818 .986 r .51736 .24457 m .54079 .23584 L .53375 .22608 L p F P 0 g s .02 w .08977 .56017 Mdot .17897 .46718 Mdot .5 Mabswid .473 .103 .301 r .52492 .47028 m .50154 .47815 L .49852 .48542 L p F P 0 g s .473 .103 .301 r .47521 .49323 m .49852 .48542 L .50154 .47815 L p F P 0 g s .02 w .18025 .46675 Mdot .1054 .48818 Mdot .5 Mabswid .702 .567 .77 r .64138 .35073 m .66528 .3424 L .66207 .32843 L p F P 0 g s .691 .537 .742 r .66528 .3424 m .64138 .35073 L .64391 .36428 L p F P 0 g s .764 .79 .964 r .5644 .22705 m .54079 .23584 L .54771 .24661 L p F P 0 g s .764 .79 .964 r .54771 .24661 m .57127 .23789 L .5644 .22705 L p F P 0 g s .02 w .18154 .46632 Mdot .08948 .55832 Mdot .5 Mabswid .748 .714 .902 r .63189 .24471 m .60806 .25343 L .61417 .26659 L p F P 0 g s .748 .714 .902 r .61417 .26659 m .63797 .25794 L .63189 .24471 L p F P 0 g s .02 w .10401 .48927 Mdot .18284 .46587 Mdot .5 Mabswid .677 .504 .71 r .62194 .38553 m .64578 .37731 L .64391 .36428 L p F P 0 g s .662 .466 .673 r .64578 .37731 m .62194 .38553 L .62304 .39788 L p F P 0 g s 1 .729 .569 r .18341 .83789 m .11046 .67218 L .10651 .67258 L p F P 0 g s .02 w .18415 .46542 Mdot .5 Mabswid .377 .1 .137 r .49852 .48542 m .47521 .49323 L .47138 .4992 L p F P 0 g s .377 .1 .137 r .44815 .50697 m .47138 .4992 L .47521 .49323 L p F P 0 g s .02 w .0892 .5564 Mdot .10265 .49046 Mdot .18546 .46497 Mdot .5 Mabswid .759 .763 .943 r .595 .2291 m .57127 .23789 L .57796 .24961 L p F P 0 g s .759 .763 .943 r .57796 .24961 m .60165 .24089 L .595 .2291 L p F P 0 g s .02 w .18679 .46451 Mdot .10133 .49176 Mdot .08893 .55443 Mdot .18812 .46404 Mdot .5 Mabswid .728 .644 .84 r .67301 .2771 m .64904 .2857 L .65391 .29994 L p F P 0 g s .728 .644 .84 r .65391 .29994 m .67789 .2914 L .67301 .2771 L p F P 0 g s .72 .62 .818 r .67789 .2914 m .65391 .29994 L .65826 .31423 L p F P 0 g s .02 w .18947 .46356 Mdot .10004 .49317 Mdot .08869 .55241 Mdot .5 Mabswid .72 .62 .818 r .65826 .31423 m .68227 .30575 L .67789 .2914 L p F P 0 g s .712 .594 .795 r .68227 .30575 m .65826 .31423 L .66207 .32843 L p F P 0 g s .02 w .19082 .46307 Mdot .5 Mabswid .236 .1 .1 r .47138 .4992 m .44815 .50697 L .44353 .51155 L p F P 0 g s .735 .667 .861 r .66768 .263 m .64371 .27165 L .64904 .2857 L p F P 0 g s .735 .667 .861 r .64904 .2857 m .67301 .2771 L .66768 .263 L p F P 0 g s .02 w .0988 .49468 Mdot .5 Mabswid .691 .537 .742 r .64391 .36428 m .66787 .35598 L .66528 .3424 L p F P 0 g s .677 .504 .71 r .66787 .35598 m .64391 .36428 L .64578 .37731 L p F P 0 g s .02 w .19219 .46258 Mdot .08846 .55032 Mdot .19357 .46208 Mdot .09761 .4963 Mdot .5 Mabswid .754 .738 .922 r .62551 .2321 m .60165 .24089 L .60806 .25343 L p F P 0 g s .754 .738 .922 r .60806 .25343 m .63189 .24471 L .62551 .2321 L p F P 0 g s .02 w .08825 .54819 Mdot .19496 .46158 Mdot .5 Mabswid .146 .1 .1 r .30865 .94073 m .28601 .94688 L .26757 .95739 L p F P 0 g s .146 .1 .1 r .24485 .96354 m .26757 .95739 L .28601 .94688 L p F P 0 g s .712 .594 .795 r .66207 .32843 m .68611 .31999 L .68227 .30575 L p F P 0 g s .702 .567 .77 r .68611 .31999 m .66207 .32843 L .66528 .3424 L p F P 0 g s .02 w .09648 .49803 Mdot .19636 .46106 Mdot .5 Mabswid .742 .691 .882 r .66196 .24923 m .63797 .25794 L .64371 .27165 L p F P 0 g s .742 .691 .882 r .64371 .27165 m .66768 .263 L .66196 .24923 L p F P 0 g s .617 .368 .576 r .57546 .43627 m .59918 .42823 L .59961 .41757 L p F P 0 g s .584 .301 .509 r .59918 .42823 m .57546 .43627 L .5742 .44589 L p F P 0 g s .584 .301 .509 r .55056 .45387 m .5742 .44589 L .57546 .43627 L p F P 0 g s .02 w .08807 .546 Mdot .5 Mabswid .642 .421 .63 r .59961 .41757 m .62342 .40946 L .62304 .39788 L p F P 0 g s .617 .368 .576 r .62342 .40946 m .59961 .41757 L .59918 .42823 L p F P 0 g s .02 w .09541 .49985 Mdot .19777 .46054 Mdot .5 Mabswid .538 .216 .42 r .5742 .44589 m .55056 .45387 L .54847 .46236 L p F P 0 g s .538 .216 .42 r .52492 .47028 m .54847 .46236 L .55056 .45387 L p F P 0 g s 1 .729 .569 r .10651 .67258 m .16973 .84084 L .18341 .83789 L p F P 0 g s 1 .73 .561 r .16973 .84084 m .10651 .67258 L .10237 .6722 L p F P 0 g s .02 w .08792 .54376 Mdot .1992 .46 Mdot .0944 .50178 Mdot .5 Mabswid .662 .466 .673 r .62304 .39788 m .64695 .38969 L .64578 .37731 L p F P 0 g s .642 .421 .63 r .64695 .38969 m .62304 .39788 L .62342 .40946 L p F P 0 g s .306 .1 .1 r .41607 .90145 m .39301 .90775 L .37759 .92201 L p F P 0 g s .306 .1 .1 r .35444 .9283 m .37759 .92201 L .39301 .90775 L p F P 0 g s .02 w .08781 .54148 Mdot .5 Mabswid .768 .818 .986 r .55742 .21722 m .53375 .22608 L .54079 .23584 L p F P 0 g s .768 .818 .986 r .54079 .23584 m .5644 .22705 L .55742 .21722 L p F P 0 g s .02 w .20064 .45946 Mdot .09347 .50379 Mdot .5 Mabswid .473 .103 .301 r .54847 .46236 m .52492 .47028 L .52199 .47754 L p F P 0 g s .473 .103 .301 r .49852 .48542 m .52199 .47754 L .52492 .47028 L p F P 0 g s .02 w .08773 .53916 Mdot .20209 .45892 Mdot .0926 .50589 Mdot .5 Mabswid .702 .567 .77 r .66528 .3424 m .68936 .334 L .68611 .31999 L p F P 0 g s .691 .537 .742 r .68936 .334 m .66528 .3424 L .66787 .35598 L p F P 0 g s .764 .79 .964 r .58818 .21819 m .5644 .22705 L .57127 .23789 L p F P 0 g s .764 .79 .964 r .57127 .23789 m .595 .2291 L .58818 .21819 L p F P 0 g s .02 w .0877 .5368 Mdot .0918 .50806 Mdot .20355 .45836 Mdot .5 Mabswid .748 .714 .902 r .65589 .23593 m .63189 .24471 L .63797 .25794 L p F P 0 g s .748 .714 .902 r .63797 .25794 m .66196 .24923 L .65589 .23593 L p F P 0 g s .02 w .08771 .5344 Mdot .09108 .5103 Mdot .5 Mabswid .677 .504 .71 r .64578 .37731 m .66979 .36904 L .66787 .35598 L p F P 0 g s .662 .466 .673 r .66979 .36904 m .64578 .37731 L .64695 .38969 L p F P 0 g s .02 w .20503 .45779 Mdot .08777 .53199 Mdot .09044 .5126 Mdot .08788 .52955 Mdot .5 Mabswid .377 .1 .137 r .52199 .47754 m .49852 .48542 L .49478 .49137 L p F P 0 g s .377 .1 .137 r .47138 .4992 m .49478 .49137 L .49852 .48542 L p F P 0 g s .02 w .08986 .51495 Mdot .08805 .5271 Mdot .08936 .51733 Mdot .20653 .45722 Mdot .08828 .52464 Mdot .08893 .51975 Mdot .08857 .52219 Mdot .5 Mabswid .759 .763 .943 r .61891 .22025 m .595 .2291 L .60165 .24089 L p F P 0 g s .759 .763 .943 r .60165 .24089 m .62551 .2321 L .61891 .22025 L p F P 0 g s .02 w .20803 .45663 Mdot .20955 .45604 Mdot .5 Mabswid .519 .519 .852 r .69236 .27072 m .67301 .2771 L .67789 .2914 L p F P 0 g s .519 .519 .852 r .67789 .2914 m .69664 .28329 L .69236 .27072 L p F P 0 g s .508 .488 .826 r .69664 .28329 m .67789 .2914 L .68227 .30575 L p F P 0 g s .02 w .21109 .45543 Mdot .5 Mabswid .508 .488 .826 r .68227 .30575 m .70049 .29589 L .69664 .28329 L p F P 0 g s .492 .455 .797 r .70049 .29589 m .68227 .30575 L .68611 .31999 L p F P 0 g s .236 .1 .1 r .49478 .49137 m .47138 .4992 L .46685 .50376 L p F P 0 g s .236 .1 .1 r .44353 .51155 m .46685 .50376 L .47138 .4992 L p F P 0 g s .527 .547 .877 r .68769 .25832 m .66768 .263 L .67301 .2771 L p F P 0 g s .527 .547 .877 r .67301 .2771 m .69236 .27072 L .68769 .25832 L p F P 0 g s 1 .73 .561 r .10237 .6722 m .15512 .84095 L .16973 .84084 L p F P 0 g s 1 .731 .554 r .15512 .84095 m .10237 .6722 L .09806 .67103 L p F P 0 g s .691 .537 .742 r .66787 .35598 m .692 .34762 L .68936 .334 L p F P 0 g s .677 .504 .71 r .692 .34762 m .66787 .35598 L .66979 .36904 L p F P 0 g s .02 w .21264 .45482 Mdot .5 Mabswid .754 .738 .922 r .64955 .22326 m .62551 .2321 L .63189 .24471 L p F P 0 g s .754 .738 .922 r .63189 .24471 m .65589 .23593 L .64955 .22326 L p F P 0 g s .02 w .21421 .45419 Mdot .5 Mabswid .146 .1 .1 r .33146 .93454 m .30865 .94073 L .29046 .9512 L p F P 0 g s .146 .1 .1 r .26757 .95739 m .29046 .9512 L .30865 .94073 L p F P 0 g s .492 .455 .797 r .68611 .31999 m .70388 .30841 L .70049 .29589 L p F P 0 g s .472 .416 .766 r .70388 .30841 m .68611 .31999 L .68936 .334 L p F P 0 g s .02 w .2158 .45355 Mdot .5 Mabswid .532 .572 .901 r .68269 .24622 m .66196 .24923 L .66768 .263 L p F P 0 g s .532 .572 .901 r .66768 .263 m .68769 .25832 L .68269 .24622 L p F P 0 g s .617 .368 .576 r .59918 .42823 m .62308 .42012 L .62342 .40946 L p F P 0 g s .584 .301 .509 r .62308 .42012 m .59918 .42823 L .59801 .43785 L p F P 0 g s .584 .301 .509 r .5742 .44589 m .59801 .43785 L .59918 .42823 L p F P 0 g s .642 .421 .63 r .62342 .40946 m .64741 .40129 L .64695 .38969 L p F P 0 g s .617 .368 .576 r .64741 .40129 m .62342 .40946 L .62308 .42012 L p F P 0 g s .1 .1 .1 r .46685 .50376 m .44353 .51155 L .43826 .51467 L p F P 0 g s .02 w .2174 .45291 Mdot .5 Mabswid .538 .216 .42 r .59801 .43785 m .5742 .44589 L .5722 .45438 L p F P 0 g s .538 .216 .42 r .54847 .46236 m .5722 .45438 L .5742 .44589 L p F P 0 g s 1 .711 .55 r .24485 .96354 m .18341 .83789 L .16973 .84084 L p F P 0 g s .662 .466 .673 r .64695 .38969 m .67104 .38144 L .66979 .36904 L p F P 0 g s .642 .421 .63 r .67104 .38144 m .64695 .38969 L .64741 .40129 L p F P 0 g s .02 w .21901 .45225 Mdot .5 Mabswid .306 .1 .1 r .4393 .8951 m .41607 .90145 L .40091 .91568 L p F P 0 g s .306 .1 .1 r .37759 .92201 m .40091 .91568 L .41607 .90145 L p F P 0 g s .768 .818 .986 r .58127 .2083 m .55742 .21722 L .5644 .22705 L p F P 0 g s .768 .818 .986 r .5644 .22705 m .58818 .21819 L .58127 .2083 L p F P 0 g s .473 .103 .301 r .5722 .45438 m .54847 .46236 L .54564 .46961 L p F P 0 g s .473 .103 .301 r .52199 .47754 m .54564 .46961 L .54847 .46236 L p F P 0 g s .02 w .22065 .45158 Mdot .5 Mabswid .472 .416 .766 r .68936 .334 m .70676 .3207 L .70388 .30841 L p F P 0 g s .446 .372 .729 r .70676 .3207 m .68936 .334 L .692 .34762 L p F P 0 g s .764 .79 .964 r .61215 .20927 m .58818 .21819 L .595 .2291 L p F P 0 g s .764 .79 .964 r .595 .2291 m .61891 .22025 L .61215 .20927 L p F P 0 g s .02 w .2223 .4509 Mdot .5 Mabswid .533 .596 .925 r .6774 .23454 m .65589 .23593 L .66196 .24923 L p F P 0 g s .533 .596 .925 r .66196 .24923 m .68269 .24622 L .6774 .23454 L p F P 0 g s 1 .731 .554 r .09806 .67103 m .13973 .83813 L .15512 .84095 L p F P 0 g s 1 .733 .547 r .13973 .83813 m .09806 .67103 L .09365 .66908 L p F P 0 g s .677 .504 .71 r .66979 .36904 m .69399 .36071 L .692 .34762 L p F P 0 g s .662 .466 .673 r .69399 .36071 m .66979 .36904 L .67104 .38144 L p F P 0 g s .306 .1 .1 r .46271 .88871 m .4393 .8951 L .42442 .90929 L p F P 0 g s .306 .1 .1 r .42442 .90929 m .44809 .90286 L .46271 .88871 L p F P 0 g s .146 .1 .1 r .47196 .89638 m .44809 .90286 L .43151 .91302 L p F P 0 g s .146 .1 .1 r .43151 .91302 m .45566 .90649 L .47196 .89638 L p F P 0 g s 1 .819 .598 r .41502 .52243 m .47196 .89638 L .45566 .90649 L p F P 0 g s 1 .819 .598 r .45566 .90649 m .40905 .52405 L .41502 .52243 L p F P 0 g s .02 w .22397 .4502 Mdot .5 Mabswid .377 .1 .137 r .54564 .46961 m .52199 .47754 L .51835 .48348 L p F P 0 g s .377 .1 .137 r .49478 .49137 m .51835 .48348 L .52199 .47754 L p F P 0 g s .759 .763 .943 r .64299 .21133 m .61891 .22025 L .62551 .2321 L p F P 0 g s .759 .763 .943 r .62551 .2321 m .64955 .22326 L .64299 .21133 L p F P 0 g s .02 w .22565 .44949 Mdot .22736 .44877 Mdot .5 Mabswid .236 .1 .1 r .51835 .48348 m .49478 .49137 L .49035 .49591 L p F P 0 g s .236 .1 .1 r .46685 .50376 m .49035 .49591 L .49478 .49137 L p F P 0 g s .02 w .22908 .44803 Mdot .5 Mabswid .446 .372 .729 r .692 .34762 m .7091 .33264 L .70676 .3207 L p F P 0 g s .413 .32 .687 r .7091 .33264 m .692 .34762 L .69399 .36071 L p F P 0 g s .02 w .23082 .44728 Mdot .5 Mabswid .53 .619 .948 r .67187 .2234 m .64955 .22326 L .65589 .23593 L p F P 0 g s .53 .619 .948 r .65589 .23593 m .6774 .23454 L .67187 .2234 L p F P 0 g s .146 .1 .1 r .35444 .9283 m .33146 .93454 L .31353 .94496 L p F P 0 g s .146 .1 .1 r .29046 .9512 m .31353 .94496 L .33146 .93454 L p F P 0 g s .02 w .23257 .44651 Mdot .5 Mabswid .617 .368 .576 r .62308 .42012 m .64715 .41196 L .64741 .40129 L p F P 0 g s .584 .301 .509 r .64715 .41196 m .62308 .42012 L .622 .42976 L p F P 0 g s .584 .301 .509 r .59801 .43785 m .622 .42976 L .62308 .42012 L p F P 0 g s 1 .733 .547 r .09365 .66908 m .12374 .83234 L .13973 .83813 L p F P 0 g s 1 .735 .541 r .12374 .83234 m .09365 .66908 L .08919 .66638 L p F P 0 g s .642 .421 .63 r .64741 .40129 m .67158 .39305 L .67104 .38144 L p F P 0 g s .617 .368 .576 r .67158 .39305 m .64741 .40129 L .64715 .41196 L p F P 0 g s .1 .1 .1 r .49035 .49591 m .46685 .50376 L .46168 .50684 L p F P 0 g s .1 .1 .1 r .43826 .51467 m .46168 .50684 L .46685 .50376 L p F P 0 g s .538 .216 .42 r .622 .42976 m .59801 .43785 L .5961 .44634 L p F P 0 g s .538 .216 .42 r .5722 .45438 m .5961 .44634 L .59801 .43785 L p F P 0 g s .02 w .23434 .44573 Mdot .5 Mabswid .662 .466 .673 r .67104 .38144 m .6953 .37313 L .69399 .36071 L p F P 0 g s .642 .421 .63 r .6953 .37313 m .67104 .38144 L .67158 .39305 L p F P 0 g s .306 .1 .1 r .40091 .91568 m .42442 .90929 L .4393 .8951 L p F P 0 g s .768 .818 .986 r .60529 .1993 m .58127 .2083 L .58818 .21819 L p F P 0 g s .768 .818 .986 r .58818 .21819 m .61215 .20927 L .60529 .1993 L p F P 0 g s .02 w .23613 .44492 Mdot .5 Mabswid .473 .103 .301 r .5961 .44634 m .5722 .45438 L .56947 .46162 L p F P 0 g s .473 .103 .301 r .54564 .46961 m .56947 .46162 L .5722 .45438 L p F P 0 g s .764 .79 .964 r .63629 .20028 m .61215 .20927 L .61891 .22025 L p F P 0 g s .764 .79 .964 r .61891 .22025 m .64299 .21133 L .63629 .20028 L p F P 0 g s .02 w .23794 .4441 Mdot .5 Mabswid .413 .32 .687 r .69399 .36071 m .71088 .34411 L .7091 .33264 L p F P 0 g s .37 .258 .637 r .71088 .34411 m .69399 .36071 L .6953 .37313 L p F P 0 g s .368 .867 .821 r .46168 .50684 m .43826 .51467 L .4324 .51625 L p F P 0 g s .02 w .23976 .44325 Mdot .5 Mabswid .377 .1 .137 r .56947 .46162 m .54564 .46961 L .5421 .47553 L p F P 0 g s .377 .1 .137 r .51835 .48348 m .5421 .47553 L .54564 .46961 L p F P 0 g s .523 .641 .971 r .66617 .21293 m .64299 .21133 L .64955 .22326 L p F P 0 g s .523 .641 .971 r .64955 .22326 m .67187 .2234 L .66617 .21293 L p F P 0 g s .02 w .2416 .44237 Mdot .5 Mabswid 1 .735 .541 r .08919 .66638 m .10735 .82358 L .12374 .83234 L p F P 0 g s 1 .738 .536 r .10735 .82358 m .08919 .66638 L .08474 .66294 L p F P 0 g s .1 .1 .1 r .26757 .95739 m .24485 .96354 L .22432 .96995 L p F P 0 g s 1 .711 .55 r .16973 .84084 m .22432 .96995 L .24485 .96354 L p F P 0 g s .02 w .24345 .44147 Mdot .5 Mabswid .236 .1 .1 r .5421 .47553 m .51835 .48348 L .51403 .48799 L p F P 0 g s .236 .1 .1 r .49035 .49591 m .51403 .48799 L .51835 .48348 L p F P 0 g s .146 .1 .1 r .44809 .90286 m .42442 .90929 L .40755 .91951 L p F P 0 g s .146 .1 .1 r .40755 .91951 m .43151 .91302 L .44809 .90286 L p F P 0 g s .1 .1 .1 r .45566 .90649 m .43151 .91302 L .41307 .91889 L p F P 0 g s .1 .1 .1 r .41307 .91889 m .4375 .91229 L .45566 .90649 L p F P 0 g s 1 .818 .602 r .40905 .52405 m .45566 .90649 L .4375 .91229 L p F P 0 g s 1 .818 .602 r .4375 .91229 m .40256 .52411 L .40905 .52405 L p F P 0 g s .02 w .24531 .44053 Mdot .5 Mabswid .146 .1 .1 r .37759 .92201 m .35444 .9283 L .33677 .93867 L p F P 0 g s .146 .1 .1 r .31353 .94496 m .33677 .93867 L .35444 .9283 L p F P 0 g s .02 w .24718 .43956 Mdot .5 Mabswid .617 .368 .576 r .64715 .41196 m .67141 .40374 L .67158 .39305 L p F P 0 g s .584 .301 .509 r .67141 .40374 m .64715 .41196 L .64617 .4216 L p F P 0 g s .584 .301 .509 r .622 .42976 m .64617 .4216 L .64715 .41196 L p F P 0 g s .642 .421 .63 r .67158 .39305 m .69593 .38476 L .6953 .37313 L p F P 0 g s .617 .368 .576 r .69593 .38476 m .67158 .39305 L .67141 .40374 L p F P 0 g s .1 .1 .1 r .51403 .48799 m .49035 .49591 L .48528 .49896 L p F P 0 g s .1 .1 .1 r .46168 .50684 m .48528 .49896 L .49035 .49591 L p F P 0 g s .538 .216 .42 r .64617 .4216 m .622 .42976 L .62019 .43824 L p F P 0 g s .538 .216 .42 r .5961 .44634 m .62019 .43824 L .622 .42976 L p F P 0 g s .02 w .24907 .43854 Mdot .5 Mabswid 1 .738 .536 r .08474 .66294 m .09078 .81189 L .10735 .82358 L p F P 0 g s .37 .258 .637 r .6953 .37313 m .71208 .35497 L .71088 .34411 L p F P 0 g s .315 .181 .574 r .71208 .35497 m .6953 .37313 L .69593 .38476 L p F P 0 g s .768 .818 .986 r .6295 .19024 m .60529 .1993 L .61215 .20927 L p F P 0 g s .768 .818 .986 r .61215 .20927 m .63629 .20028 L .6295 .19024 L p F P 0 g s .473 .103 .301 r .62019 .43824 m .5961 .44634 L .59347 .45357 L p F P 0 g s .473 .103 .301 r .56947 .46162 m .59347 .45357 L .5961 .44634 L p F P 0 g s .02 w .25097 .43748 Mdot .5 Mabswid .576 .554 .854 r .71316 .2637 m .69236 .27072 L .69664 .28329 L p F P 0 g s .576 .554 .854 r .69664 .28329 m .71698 .27492 L .71316 .2637 L p F P 0 g s .565 .525 .829 r .71698 .27492 m .69664 .28329 L .70049 .29589 L p F P 0 g s .509 .661 .994 r .66035 .20324 m .63629 .20028 L .64299 .21133 L p F P 0 g s .509 .661 .994 r .64299 .21133 m .66617 .21293 L .66035 .20324 L p F P 0 g s .565 .525 .829 r .70049 .29589 m .72042 .28616 L .71698 .27492 L p F P 0 g s .552 .493 .801 r .72042 .28616 m .70049 .29589 L .70388 .30841 L p F P 0 g s .584 .581 .878 r .70901 .25263 m .68769 .25832 L .69236 .27072 L p F P 0 g s .584 .581 .878 r .69236 .27072 m .71316 .2637 L .70901 .25263 L p F P 0 g s .368 .867 .821 r .48528 .49896 m .46168 .50684 L .45592 .5084 L p F P 0 g s .368 .867 .821 r .4324 .51625 m .45592 .5084 L .46168 .50684 L p F P 0 g s .02 w .25287 .43636 Mdot .5 Mabswid .377 .1 .137 r .59347 .45357 m .56947 .46162 L .56603 .46752 L p F P 0 g s .377 .1 .137 r .5421 .47553 m .56603 .46752 L .56947 .46162 L p F P 0 g s .552 .493 .801 r .70388 .30841 m .72345 .29732 L .72042 .28616 L p F P 0 g s .534 .458 .771 r .72345 .29732 m .70388 .30841 L .70676 .3207 L p F P 0 g s 1 .711 .54 r .22432 .96995 m .16973 .84084 L .15512 .84095 L p F P 0 g s .589 .606 .902 r .70457 .24183 m .68269 .24622 L .68769 .25832 L p F P 0 g s .589 .606 .902 r .68769 .25832 m .70901 .25263 L .70457 .24183 L p F P 0 g s .02 w .25478 .43518 Mdot .5 Mabswid .1 .1 .1 r .29046 .9512 m .26757 .95739 L .24728 .96374 L p F P 0 g s .1 .1 .1 r .22432 .96995 m .24728 .96374 L .26757 .95739 L p F P 0 g s .236 .1 .1 r .56603 .46752 m .5421 .47553 L .53788 .48002 L p F P 0 g s .236 .1 .1 r .51403 .48799 m .53788 .48002 L .5421 .47553 L p F P 0 g s .542 .976 .997 r .45592 .5084 m .4324 .51625 L .426 .51627 L p F P 0 g s .542 .976 .997 r .40256 .52411 m .426 .51627 L .4324 .51625 L p F P 0 g s .02 w .2567 .43393 Mdot .5 Mabswid .534 .458 .771 r .70676 .3207 m .72604 .30827 L .72345 .29732 L p F P 0 g s .512 .417 .737 r .72604 .30827 m .70676 .3207 L .7091 .33264 L p F P 0 g s .592 .63 .925 r .69989 .2314 m .6774 .23454 L .68269 .24622 L p F P 0 g s .592 .63 .925 r .68269 .24622 m .70457 .24183 L .69989 .2314 L p F P 0 g s .146 .1 .1 r .40091 .91568 m .37759 .92201 L .36018 .93233 L p F P 0 g s .146 .1 .1 r .33677 .93867 m .36018 .93233 L .37759 .92201 L p F P 0 g s .02 w .25863 .43261 Mdot .5 Mabswid .617 .368 .576 r .67141 .40374 m .69585 .39545 L .69593 .38476 L p F P 0 g s .584 .301 .509 r .69585 .39545 m .67141 .40374 L .67052 .41338 L p F P 0 g s .584 .301 .509 r .64617 .4216 m .67052 .41338 L .67141 .40374 L p F P 0 g s .315 .181 .574 r .69593 .38476 m .71268 .36512 L .71208 .35497 L p F P 0 g s .241 .1 .493 r .71268 .36512 m .69593 .38476 L .69585 .39545 L p F P 0 g s .1 .1 .1 r .53788 .48002 m .51403 .48799 L .50906 .49102 L p F P 0 g s .1 .1 .1 r .48528 .49896 m .50906 .49102 L .51403 .48799 L p F P 0 g s .538 .216 .42 r .67052 .41338 m .64617 .4216 L .64446 .43007 L p F P 0 g s .538 .216 .42 r .62019 .43824 m .64446 .43007 L .64617 .4216 L p F P 0 g s .651 1 1 r .426 .51627 m .40256 .52411 L .3956 .52258 L p F P 0 g s .146 .1 .1 r .42442 .90929 m .40091 .91568 L .38378 .92594 L p F P 0 g s .146 .1 .1 r .38378 .92594 m .40755 .91951 L .42442 .90929 L p F P 0 g s 1 .816 .606 r .40256 .52411 m .4375 .91229 L .41763 .91351 L p F P 0 g s 1 .816 .606 r .41763 .91351 m .3956 .52258 L .40256 .52411 L p F P 0 g s .02 w .26056 .43119 Mdot .5 Mabswid .488 .68 1 r .65448 .19445 m .6295 .19024 L .63629 .20028 L p F P 0 g s .488 .68 1 r .63629 .20028 m .66035 .20324 L .65448 .19445 L p F P 0 g s .473 .103 .301 r .64446 .43007 m .62019 .43824 L .61766 .44546 L p F P 0 g s .473 .103 .301 r .59347 .45357 m .61766 .44546 L .62019 .43824 L p F P 0 g s .512 .417 .737 r .7091 .33264 m .72816 .31891 L .72604 .30827 L p F P 0 g s .485 .37 .698 r .72816 .31891 m .7091 .33264 L .71088 .34411 L p F P 0 g s .02 w .2625 .42968 Mdot .5 Mabswid .591 .654 .948 r .695 .22147 m .67187 .2234 L .6774 .23454 L p F P 0 g s .591 .654 .948 r .6774 .23454 m .69989 .2314 L .695 .22147 L p F P 0 g s .368 .867 .821 r .50906 .49102 m .48528 .49896 L .47961 .50048 L p F P 0 g s .368 .867 .821 r .45592 .5084 m .47961 .50048 L .48528 .49896 L p F P 0 g s .377 .1 .137 r .61766 .44546 m .59347 .45357 L .59014 .45945 L p F P 0 g s .377 .1 .137 r .56603 .46752 m .59014 .45945 L .59347 .45357 L p F P 0 g s .02 w .26445 .42805 Mdot .26641 .42631 Mdot .5 Mabswid .1 .1 .1 r .31353 .94496 m .29046 .9512 L .27043 .95748 L p F P 0 g s .1 .1 .1 r .24728 .96374 m .27043 .95748 L .29046 .9512 L p F P 0 g s .236 .1 .1 r .59014 .45945 m .56603 .46752 L .56191 .47199 L p F P 0 g s .236 .1 .1 r .53788 .48002 m .56191 .47199 L .56603 .46752 L p F P 0 g s .542 .976 .997 r .47961 .50048 m .45592 .5084 L .44961 .50838 L p F P 0 g s .542 .976 .997 r .426 .51627 m .44961 .50838 L .45592 .5084 L p F P 0 g s .485 .37 .698 r .71088 .34411 m .72979 .3291 L .72816 .31891 L p F P 0 g s .45 .314 .652 r .72979 .3291 m .71088 .34411 L .71208 .35497 L p F P 0 g s .02 w .26839 .42444 Mdot .5 Mabswid .588 .677 .971 r .68998 .21213 m .66617 .21293 L .67187 .2234 L p F P 0 g s .588 .677 .971 r .67187 .2234 m .695 .22147 L .68998 .21213 L p F P 0 g s .146 .1 .1 r .36018 .93233 m .38378 .92594 L .40091 .91568 L p F P 0 g s .241 .1 .493 r .69585 .39545 m .71266 .37444 L .71268 .36512 L p F P 0 g s .144 .1 .388 r .71266 .37444 m .69585 .39545 L .69505 .40509 L p F P 0 g s .584 .301 .509 r .67052 .41338 m .69505 .40509 L .69585 .39545 L p F P 0 g s .1 .1 .1 r .56191 .47199 m .53788 .48002 L .53301 .48302 L p F P 0 g s .1 .1 .1 r .50906 .49102 m .53301 .48302 L .53788 .48002 L p F P 0 g s .02 w .27038 .42242 Mdot .5 Mabswid .538 .216 .42 r .69505 .40509 m .67052 .41338 L .66891 .42185 L p F P 0 g s .538 .216 .42 r .64446 .43007 m .66891 .42185 L .67052 .41338 L p F P 0 g s .651 1 1 r .44961 .50838 m .426 .51627 L .41913 .5147 L p F P 0 g s .651 1 1 r .3956 .52258 m .41913 .5147 L .426 .51627 L p F P 0 g s .473 .103 .301 r .66891 .42185 m .64446 .43007 L .64203 .43728 L p F P 0 g s .473 .103 .301 r .61766 .44546 m .64203 .43728 L .64446 .43007 L p F P 0 g s .02 w .2724 .42027 Mdot .5 Mabswid .368 .867 .821 r .53301 .48302 m .50906 .49102 L .50349 .49251 L p F P 0 g s .368 .867 .821 r .47961 .50048 m .50349 .49251 L .50906 .49102 L p F P 0 g s .71 .981 1 r .41913 .5147 m .3956 .52258 L .38825 .51947 L p F P 0 g s 1 .815 .609 r .3956 .52258 m .41763 .91351 L .39623 .90994 L p F P 0 g s 1 .815 .609 r .39623 .90994 m .38825 .51947 L .3956 .52258 L p F P 0 g s .02 w .27446 .41796 Mdot .5 Mabswid .45 .314 .652 r .71208 .35497 m .73089 .33876 L .72979 .3291 L p F P 0 g s .404 .245 .594 r .73089 .33876 m .71208 .35497 L .71268 .36512 L p F P 0 g s .377 .1 .137 r .64203 .43728 m .61766 .44546 L .61443 .45132 L p F P 0 g s .377 .1 .137 r .59014 .45945 m .61443 .45132 L .61766 .44546 L p F P 0 g s .602 .57 .854 r .73486 .2563 m .71316 .2637 L .71698 .27492 L p F P 0 g s .602 .57 .854 r .71698 .27492 m .73827 .26635 L .73486 .2563 L p F P 0 g s .592 .541 .829 r .73827 .26635 m .71698 .27492 L .72042 .28616 L p F P 0 g s .58 .701 .995 r .68485 .2035 m .66035 .20324 L .66617 .21293 L p F P 0 g s .58 .701 .995 r .66617 .21293 m .68998 .21213 L .68485 .2035 L p F P 0 g s .592 .541 .829 r .72042 .28616 m .74136 .27643 L .73827 .26635 L p F P 0 g s .579 .511 .802 r .74136 .27643 m .72042 .28616 L .72345 .29732 L p F P 0 g s .61 .596 .877 r .73115 .24638 m .70901 .25263 L .71316 .2637 L p F P 0 g s .61 .596 .877 r .71316 .2637 m .73486 .2563 L .73115 .24638 L p F P 0 g s .02 w .27656 .41549 Mdot .5 Mabswid .1 .1 .1 r .33677 .93867 m .31353 .94496 L .29375 .95117 L p F P 0 g s .1 .1 .1 r .27043 .95748 m .29375 .95117 L .31353 .94496 L p F P 0 g s .236 .1 .1 r .61443 .45132 m .59014 .45945 L .58612 .4639 L p F P 0 g s .236 .1 .1 r .56191 .47199 m .58612 .4639 L .59014 .45945 L p F P 0 g s .579 .511 .802 r .72345 .29732 m .74409 .28642 L .74136 .27643 L p F P 0 g s .563 .476 .773 r .74409 .28642 m .72345 .29732 L .72604 .30827 L p F P 0 g s .542 .976 .997 r .50349 .49251 m .47961 .50048 L .47341 .50042 L p F P 0 g s .542 .976 .997 r .44961 .50838 m .47341 .50042 L .47961 .50048 L p F P 0 g s .615 .621 .9 r .7272 .2367 m .70457 .24183 L .70901 .25263 L p F P 0 g s .615 .621 .9 r .70901 .25263 m .73115 .24638 L .7272 .2367 L p F P 0 g s .02 w .27871 .41286 Mdot .5 Mabswid 1 .711 .54 r .15512 .84095 m .20211 .97195 L .22432 .96995 L p F P 0 g s 1 .712 .529 r .20211 .97195 m .15512 .84095 L .13973 .83813 L p F P 0 g s .02 w .28092 .41007 Mdot .5 Mabswid .563 .476 .773 r .72604 .30827 m .74643 .29622 L .74409 .28642 L p F P 0 g s .543 .438 .74 r .74643 .29622 m .72604 .30827 L .72816 .31891 L p F P 0 g s .1 .1 .1 r .58612 .4639 m .56191 .47199 L .55715 .47496 L p F P 0 g s .1 .1 .1 r .53301 .48302 m .55715 .47496 L .56191 .47199 L p F P 0 g s .144 .1 .388 r .69505 .40509 m .71202 .38281 L .71266 .37444 L p F P 0 g s .1 .1 .249 r .71202 .38281 m .69505 .40509 L .69354 .41356 L p F P 0 g s .538 .216 .42 r .66891 .42185 m .69354 .41356 L .69505 .40509 L p F P 0 g s .619 .646 .923 r .72303 .22735 m .69989 .2314 L .70457 .24183 L p F P 0 g s .619 .646 .923 r .70457 .24183 m .7272 .2367 L .72303 .22735 L p F P 0 g s .651 1 1 r .47341 .50042 m .44961 .50838 L .44284 .50676 L p F P 0 g s .651 1 1 r .41913 .5147 m .44284 .50676 L .44961 .50838 L p F P 0 g s .02 w .28322 .40714 Mdot .5 Mabswid .404 .245 .594 r .71268 .36512 m .73147 .34776 L .73089 .33876 L p F P 0 g s .344 .16 .521 r .73147 .34776 m .71268 .36512 L .71266 .37444 L p F P 0 g s .473 .103 .301 r .69354 .41356 m .66891 .42185 L .66658 .42905 L p F P 0 g s .473 .103 .301 r .64203 .43728 m .66658 .42905 L .66891 .42185 L p F P 0 g s .02 w .28561 .40405 Mdot .5 Mabswid .567 .723 1 r .67969 .19566 m .65448 .19445 L .66035 .20324 L p F P 0 g s .567 .723 1 r .66035 .20324 m .68485 .2035 L .67969 .19566 L p F P 0 g s .543 .438 .74 r .72816 .31891 m .74835 .30573 L .74643 .29622 L p F P 0 g s .518 .393 .702 r .74835 .30573 m .72816 .31891 L .72979 .3291 L p F P 0 g s .368 .867 .821 r .55715 .47496 m .53301 .48302 L .52755 .48447 L p F P 0 g s .368 .867 .821 r .50349 .49251 m .52755 .48447 L .53301 .48302 L p F P 0 g s .436 .918 .946 r .24728 .96374 m .22432 .96995 L .20211 .97195 L p F P 0 g s .71 .981 1 r .44284 .50676 m .41913 .5147 L .41187 .51153 L p F P 0 g s .71 .981 1 r .38825 .51947 m .41187 .51153 L .41913 .5147 L p F P 0 g s .377 .1 .137 r .66658 .42905 m .64203 .43728 L .63891 .44313 L p F P 0 g s .377 .1 .137 r .61443 .45132 m .63891 .44313 L .64203 .43728 L p F P 0 g s .62 .67 .946 r .7187 .21845 m .695 .22147 L .69989 .2314 L p F P 0 g s .62 .67 .946 r .69989 .2314 m .72303 .22735 L .7187 .21845 L p F P 0 g s .02 w .28811 .40084 Mdot .29074 .39751 Mdot .5 Mabswid .1 .1 .1 r .36018 .93233 m .33677 .93867 L .31725 .94481 L p F P 0 g s .1 .1 .1 r .29375 .95117 m .31725 .94481 L .33677 .93867 L p F P 0 g s .236 .1 .1 r .63891 .44313 m .61443 .45132 L .61052 .45575 L p F P 0 g s .236 .1 .1 r .58612 .4639 m .61052 .45575 L .61443 .45132 L p F P 0 g s .738 .944 1 r .41187 .51153 m .38825 .51947 L .38061 .51477 L p F P 0 g s 1 .813 .611 r .38825 .51947 m .39623 .90994 L .37351 .90141 L p F P 0 g s 1 .813 .611 r .37351 .90141 m .38061 .51477 L .38825 .51947 L p F P 0 g s .02 w .2935 .39408 Mdot .5 Mabswid .542 .976 .997 r .52755 .48447 m .50349 .49251 L .49738 .49241 L p F P 0 g s .542 .976 .997 r .47341 .50042 m .49738 .49241 L .50349 .49251 L p F P 0 g s .02 w .29642 .39058 Mdot .5 Mabswid .518 .393 .702 r .72979 .3291 m .74984 .31484 L .74835 .30573 L p F P 0 g s .486 .34 .657 r .74984 .31484 m .72979 .3291 L .73089 .33876 L p F P 0 g s .02 w .2995 .38702 Mdot .5 Mabswid .618 .694 .97 r .71424 .21009 m .68998 .21213 L .695 .22147 L p F P 0 g s .618 .694 .97 r .695 .22147 m .7187 .21845 L .71424 .21009 L p F P 0 g s .02 w .30277 .38345 Mdot .5 Mabswid .1 .1 .1 r .61052 .45575 m .58612 .4639 L .58147 .46684 L p F P 0 g s .1 .1 .1 r .55715 .47496 m .58147 .46684 L .58612 .4639 L p F P 0 g s .344 .16 .521 r .71266 .37444 m .73151 .35601 L .73147 .34776 L p F P 0 g s .262 .1 .426 r .73151 .35601 m .71266 .37444 L .71202 .38281 L p F P 0 g s .02 w .30623 .37987 Mdot .5 Mabswid .651 1 1 r .49738 .49241 m .47341 .50042 L .46673 .49876 L p F P 0 g s .651 1 1 r .44284 .50676 m .46673 .49876 L .47341 .50042 L p F P 0 g s .02 w .30988 .37633 Mdot .31374 .37285 Mdot .31781 .36944 Mdot .32209 .36614 Mdot .34124 .3542 Mdot .33616 .35698 Mdot .32658 .36295 Mdot .33127 .3599 Mdot .3465 .35156 Mdot .35194 .34904 Mdot .5 Mabswid .1 .1 .249 r .69354 .41356 m .71076 .39015 L .71202 .38281 L p F P 0 g s .1 .1 .1 r .71076 .39015 m .69354 .41356 L .69133 .42075 L p F P 0 g s .473 .103 .301 r .66658 .42905 m .69133 .42075 L .69354 .41356 L p F P 0 g s .02 w .35754 .34664 Mdot .3633 .34433 Mdot .36921 .34209 Mdot .37526 .33989 Mdot .5 Mabswid .368 .867 .821 r .58147 .46684 m .55715 .47496 L .55179 .47638 L p F P 0 g s .368 .867 .821 r .52755 .48447 m .55179 .47638 L .55715 .47496 L p F P 0 g s .02 w .38146 .33772 Mdot .5 Mabswid .436 .918 .946 r .27043 .95748 m .24728 .96374 L .22532 .96566 L p F P 0 g s .436 .918 .946 r .20211 .97195 m .22532 .96566 L .24728 .96374 L p F P 0 g s .71 .981 1 r .46673 .49876 m .44284 .50676 L .43567 .50354 L p F P 0 g s .71 .981 1 r .41187 .51153 m .43567 .50354 L .44284 .50676 L p F P 0 g s .377 .1 .137 r .69133 .42075 m .66658 .42905 L .66357 .43488 L p F P 0 g s .377 .1 .137 r .63891 .44313 m .66357 .43488 L .66658 .42905 L p F P 0 g s .486 .34 .657 r .73089 .33876 m .75087 .32346 L .74984 .31484 L p F P 0 g s .445 .275 .602 r .75087 .32346 m .73089 .33876 L .73147 .34776 L p F P 0 g s .02 w .38781 .33554 Mdot .5 Mabswid .612 .718 .994 r .70971 .20235 m .68485 .2035 L .68998 .21213 L p F P 0 g s .612 .718 .994 r .68998 .21213 m .71424 .21009 L .70971 .20235 L p F P 0 g s .02 w .3943 .33333 Mdot .5 Mabswid .622 .582 .853 r .75732 .24858 m .73486 .2563 L .73827 .26635 L p F P 0 g s .622 .582 .853 r .73827 .26635 m .76039 .25762 L .75732 .24858 L p F P 0 g s .612 .554 .828 r .76039 .25762 m .73827 .26635 L .74136 .27643 L p F P 0 g s .612 .554 .828 r .74136 .27643 m .76317 .26668 L .76039 .25762 L p F P 0 g s .6 .524 .802 r .76317 .26668 m .74136 .27643 L .74409 .28642 L p F P 0 g s .63 .608 .876 r .754 .23966 m .73115 .24638 L .73486 .2563 L p F P 0 g s .63 .608 .876 r .73486 .2563 m .75732 .24858 L .754 .23966 L p F P 0 g s .1 .1 .1 r .38378 .92594 m .36018 .93233 L .34093 .93841 L p F P 0 g s .1 .1 .1 r .31725 .94481 m .34093 .93841 L .36018 .93233 L p F P 0 g s .02 w .40096 .33106 Mdot .5 Mabswid .236 .1 .1 r .66357 .43488 m .63891 .44313 L .63511 .44753 L p F P 0 g s .236 .1 .1 r .61052 .45575 m .63511 .44753 L .63891 .44313 L p F P 0 g s .738 .944 1 r .43567 .50354 m .41187 .51153 L .4043 .50678 L p F P 0 g s .738 .944 1 r .38061 .51477 m .4043 .50678 L .41187 .51153 L p F P 0 g s .542 .976 .997 r .55179 .47638 m .52755 .48447 L .52154 .48433 L p F P 0 g s .542 .976 .997 r .49738 .49241 m .52154 .48433 L .52755 .48447 L p F P 0 g s .6 .524 .802 r .74409 .28642 m .76563 .27567 L .76317 .26668 L p F P 0 g s .585 .491 .773 r .76563 .27567 m .74409 .28642 L .74643 .29622 L p F P 0 g s .636 .633 .899 r .75047 .23094 m .7272 .2367 L .73115 .24638 L p F P 0 g s .636 .633 .899 r .73115 .24638 m .754 .23966 L .75047 .23094 L p F P 0 g s .02 w .40777 .32871 Mdot .5 Mabswid .751 .905 1 r .4043 .50678 m .38061 .51477 L .37275 .50853 L p F P 0 g s 1 .811 .613 r .38061 .51477 m .37351 .90141 L .34975 .88784 L p F P 0 g s 1 .811 .613 r .34975 .88784 m .37275 .50853 L .38061 .51477 L p F P 0 g s .02 w .41477 .32627 Mdot .5 Mabswid .262 .1 .426 r .71202 .38281 m .731 .36342 L .73151 .35601 L p F P 0 g s .152 .1 .298 r .731 .36342 m .71202 .38281 L .71076 .39015 L p F P 0 g s .585 .491 .773 r .74643 .29622 m .76776 .28447 L .76563 .27567 L p F P 0 g s .567 .454 .741 r .76776 .28447 m .74643 .29622 L .74835 .30573 L p F P 0 g s .1 .1 .1 r .63511 .44753 m .61052 .45575 L .60597 .45866 L p F P 0 g s .1 .1 .1 r .58147 .46684 m .60597 .45866 L .61052 .45575 L p F P 0 g s .64 .658 .922 r .74675 .22253 m .72303 .22735 L .7272 .2367 L p F P 0 g s .64 .658 .922 r .7272 .2367 m .75047 .23094 L .74675 .22253 L p F P 0 g s .651 1 1 r .52154 .48433 m .49738 .49241 L .4908 .4907 L p F P 0 g s .651 1 1 r .46673 .49876 m .4908 .4907 L .49738 .49241 L p F P 0 g s .445 .275 .602 r .73147 .34776 m .75143 .33148 L .75087 .32346 L p F P 0 g s .391 .195 .532 r .75143 .33148 m .73147 .34776 L .73151 .35601 L p F P 0 g s .02 w .42197 .32373 Mdot .5 Mabswid .602 .742 1 r .70516 .19534 m .67969 .19566 L .68485 .2035 L p F P 0 g s .602 .742 1 r .68485 .2035 m .70971 .20235 L .70516 .19534 L p F P 0 g s 1 .712 .529 r .13973 .83813 m .17845 .96932 L .20211 .97195 L p F P 0 g s 1 .713 .52 r .17845 .96932 m .13973 .83813 L .12374 .83234 L p F P 0 g s .02 w .42939 .32106 Mdot .5 Mabswid .567 .454 .741 r .74835 .30573 m .76951 .29301 L .76776 .28447 L p F P 0 g s .544 .411 .704 r .76951 .29301 m .74835 .30573 L .74984 .31484 L p F P 0 g s .368 .867 .821 r .60597 .45866 m .58147 .46684 L .57621 .46822 L p F P 0 g s .368 .867 .821 r .55179 .47638 m .57621 .46822 L .58147 .46684 L p F P 0 g s .436 .918 .946 r .29375 .95117 m .27043 .95748 L .24871 .95931 L p F P 0 g s .436 .918 .946 r .22532 .96566 m .24871 .95931 L .27043 .95748 L p F P 0 g s .641 .682 .944 r .74289 .21452 m .7187 .21845 L .72303 .22735 L p F P 0 g s .641 .682 .944 r .72303 .22735 m .74675 .22253 L .74289 .21452 L p F P 0 g s .71 .981 1 r .4908 .4907 m .46673 .49876 L .45965 .49549 L p F P 0 g s .71 .981 1 r .43567 .50354 m .45965 .49549 L .46673 .49876 L p F P 0 g s .1 .1 .1 r .69133 .42075 m .70889 .39635 L .71076 .39015 L p F P 0 g s .1 .1 .1 r .70889 .39635 m .69133 .42075 L .68842 .42656 L p F P 0 g s .377 .1 .137 r .66357 .43488 m .68842 .42656 L .69133 .42075 L p F P 0 g s .02 w .43704 .31828 Mdot .44496 .31538 Mdot .5 Mabswid .1 .1 .1 r .40755 .91951 m .38378 .92594 L .36479 .93195 L p F P 0 g s .1 .1 .1 r .34093 .93841 m .36479 .93195 L .38378 .92594 L p F P 0 g s .236 .1 .1 r .68842 .42656 m .66357 .43488 L .65988 .43925 L p F P 0 g s .236 .1 .1 r .63511 .44753 m .65988 .43925 L .66357 .43488 L p F P 0 g s .738 .944 1 r .45965 .49549 m .43567 .50354 L .42819 .49873 L p F P 0 g s .738 .944 1 r .4043 .50678 m .42819 .49873 L .43567 .50354 L p F P 0 g s .542 .976 .997 r .57621 .46822 m .55179 .47638 L .54588 .47619 L p F P 0 g s .542 .976 .997 r .52154 .48433 m .54588 .47619 L .55179 .47638 L p F P 0 g s .02 w .45317 .31237 Mdot .5 Mabswid .544 .411 .704 r .74984 .31484 m .77088 .30118 L .76951 .29301 L p F P 0 g s .515 .36 .661 r .77088 .30118 m .74984 .31484 L .75087 .32346 L p F P 0 g s .391 .195 .532 r .73151 .35601 m .75152 .33882 L .75143 .33148 L p F P 0 g s .318 .1 .442 r .75152 .33882 m .73151 .35601 L .731 .36342 L p F P 0 g s .641 .706 .968 r .73893 .20699 m .71424 .21009 L .7187 .21845 L p F P 0 g s .641 .706 .968 r .7187 .21845 m .74289 .21452 L .73893 .20699 L p F P 0 g s .751 .905 1 r .42819 .49873 m .4043 .50678 L .39652 .50049 L p F P 0 g s .751 .905 1 r .37275 .50853 m .39652 .50049 L .4043 .50678 L p F P 0 g s .02 w .4617 .30927 Mdot .5 Mabswid .152 .1 .298 r .71076 .39015 m .72995 .36988 L .731 .36342 L p F P 0 g s .1 .1 .127 r .72995 .36988 m .71076 .39015 L .70889 .39635 L p F P 0 g s .1 .1 .1 r .65988 .43925 m .63511 .44753 L .63067 .45041 L p F P 0 g s .1 .1 .1 r .60597 .45866 m .63067 .45041 L .63511 .44753 L p F P 0 g s .651 1 1 r .54588 .47619 m .52154 .48433 L .51505 .48257 L p F P 0 g s .651 1 1 r .4908 .4907 m .51505 .48257 L .52154 .48433 L p F P 0 g s .756 .868 1 r .39652 .50049 m .37275 .50853 L .36477 .50081 L p F P 0 g s 1 .809 .615 r .37275 .50853 m .34975 .88784 L .32525 .86922 L p F P 0 g s 1 .809 .615 r .32525 .86922 m .36477 .50081 L .37275 .50853 L p F P 0 g s .02 w .47058 .30608 Mdot .5 Mabswid .515 .36 .661 r .75087 .32346 m .77184 .30891 L .77088 .30118 L p F P 0 g s .477 .298 .607 r .77184 .30891 m .75087 .32346 L .75143 .33148 L p F P 0 g s .02 w .47983 .30283 Mdot .5 Mabswid .368 .867 .821 r .63067 .45041 m .60597 .45866 L .60083 .46 L p F P 0 g s .368 .867 .821 r .57621 .46822 m .60083 .46 L .60597 .45866 L p F P 0 g s .637 .731 .992 r .73491 .20003 m .70971 .20235 L .71424 .21009 L p F P 0 g s .637 .731 .992 r .71424 .21009 m .73893 .20699 L .73491 .20003 L p F P 0 g s .436 .918 .946 r .31725 .94481 m .29375 .95117 L .27228 .95292 L p F P 0 g s .436 .918 .946 r .24871 .95931 m .27228 .95292 L .29375 .95117 L p F P 0 g s .71 .981 1 r .51505 .48257 m .4908 .4907 L .48381 .48737 L p F P 0 g s .71 .981 1 r .45965 .49549 m .48381 .48737 L .4908 .4907 L p F P 0 g s .638 .592 .852 r .78047 .24058 m .75732 .24858 L .76039 .25762 L p F P 0 g s .638 .592 .852 r .76039 .25762 m .78323 .24874 L .78047 .24058 L p F P 0 g s .629 .564 .828 r .78323 .24874 m .76039 .25762 L .76317 .26668 L p F P 0 g s .02 w .48948 .29954 Mdot .5 Mabswid .629 .564 .828 r .76317 .26668 m .78574 .25692 L .78323 .24874 L p F P 0 g s .617 .535 .802 r .78574 .25692 m .76317 .26668 L .76563 .27567 L p F P 0 g s .646 .617 .875 r .77748 .23251 m .754 .23966 L .75732 .24858 L p F P 0 g s .646 .617 .875 r .75732 .24858 m .78047 .24058 L .77748 .23251 L p F P 0 g s .1 .1 .1 r .43151 .91302 m .40755 .91951 L .38884 .92545 L p F P 0 g s .1 .1 .1 r .36479 .93195 m .38884 .92545 L .40755 .91951 L p F P 0 g s .617 .535 .802 r .76563 .27567 m .78798 .26502 L .78574 .25692 L p F P 0 g s .603 .502 .773 r .78798 .26502 m .76563 .27567 L .76776 .28447 L p F P 0 g s .318 .1 .442 r .731 .36342 m .75112 .3454 L .75152 .33882 L p F P 0 g s .218 .1 .319 r .75112 .3454 m .731 .36342 L .72995 .36988 L p F P 0 g s .652 .642 .898 r .77431 .22464 m .75047 .23094 L .754 .23966 L p F P 0 g s .652 .642 .898 r .754 .23966 m .77748 .23251 L .77431 .22464 L p F P 0 g s .1 .1 .1 r .68842 .42656 m .70643 .40133 L p .70361 .40414 L F P 0 g s .236 .1 .1 r .65988 .43925 m .68485 .43091 L .68842 .42656 L p F P 0 g s .722 .948 .569 r .70643 .40133 m .68842 .42656 L .68485 .43091 L p F P 0 g s .722 .948 .569 r .70338 .40502 m .70643 .40133 L p .70086 .40896 L F P 0 g s .722 .948 .569 r .68485 .43091 m .70338 .40502 L p .70086 .40896 L F P 0 g s .1 .1 .1 r .70643 .40133 m .70889 .39635 L p .70361 .40414 L F P 0 g s .759 1 .747 r .70338 .40502 m .69978 .40736 L p .70592 .40105 L F P 0 g s .1 .1 .1 r .70643 .40133 m .70338 .40502 L p .70927 .39849 L F P 0 g s .1 .1 .1 r .72624 .37967 m .70643 .40133 L p .70927 .39849 L F P 0 g s .1 .1 .1 r .72836 .37532 m .70889 .39635 L .70643 .40133 L p F P 0 g s .1 .1 .1 r .70643 .40133 m .72624 .37967 L .72836 .37532 L p F P 0 g s .1 .1 .1 r .72361 .38287 m .72624 .37967 L p .71759 .38926 L F P 0 g s .1 .1 .1 r .70338 .40502 m .72361 .38287 L p .71759 .38926 L F P 0 g s .1 .1 .1 r .74888 .35592 m .72836 .37532 L .72624 .37967 L p F P 0 g s .1 .1 .1 r .72624 .37967 m .74706 .35974 L .74888 .35592 L p F P 0 g s .1 .1 .1 r .74706 .35974 m .72624 .37967 L .72361 .38287 L p F P 0 g s .697 1 .741 r .72361 .38287 m .72049 .38486 L p .72968 .37641 L F P 0 g s .1 .1 .1 r .74478 .36253 m .74706 .35974 L p .73774 .36893 L F P 0 g s .1 .1 .1 r .72361 .38287 m .74478 .36253 L p .73774 .36893 L F P 0 g s .1 .1 .1 r .76875 .34124 m .74706 .35974 L .74478 .36253 L p F P 0 g s .639 .982 .733 r .74478 .36253 m .74208 .36423 L p .75188 .35608 L F P 0 g s .1 .1 .1 r .76678 .34367 m .76875 .34124 L p .75879 .35008 L F P 0 g s .1 .1 .1 r .74478 .36253 m .76678 .34367 L p .75879 .35008 L F P 0 g s .1 .1 .1 r .79121 .3239 m .76875 .34124 L .76678 .34367 L p F P 0 g s .1 .1 .1 r .7895 .32604 m .79121 .3239 L p .78062 .33247 L F P 0 g s .1 .1 .1 r .76678 .34367 m .7895 .32604 L p .78062 .33247 L F P 0 g s .1 .1 .1 r .81434 .30752 m .79121 .3239 L .7895 .32604 L p F P 0 g s .1 .1 .1 r .7895 .32604 m .81285 .30942 L .81434 .30752 L p F P 0 g s .544 .935 .717 r .7895 .32604 m .78745 .32731 L p .79844 .31957 L F P 0 g s .544 .935 .717 r .81285 .30942 m .7895 .32604 L p .79844 .31957 L F P 0 g s .588 .957 .725 r .76678 .34367 m .76443 .34514 L p .77483 .33722 L F P 0 g s .588 .957 .725 r .7895 .32604 m .76678 .34367 L p .77483 .33722 L F P 0 g s .639 .982 .733 r .76678 .34367 m .74478 .36253 L p .75188 .35608 L F P 0 g s .697 1 .741 r .74478 .36253 m .72361 .38287 L p .72968 .37641 L F P 0 g s .759 1 .747 r .72361 .38287 m .70338 .40502 L p .70592 .40105 L F P 0 g s .738 .944 1 r .48381 .48737 m .45965 .49549 L .45225 .49062 L p F P 0 g s .738 .944 1 r .42819 .49873 m .45225 .49062 L .45965 .49549 L p F P 0 g s .02 w .49956 .29623 Mdot .5 Mabswid .542 .976 .997 r .60083 .46 m .57621 .46822 L .57041 .46799 L p F P 0 g s .542 .976 .997 r .54588 .47619 m .57041 .46799 L .57621 .46822 L p F P 0 g s .477 .298 .607 r .75143 .33148 m .77239 .31609 L .77184 .30891 L p F P 0 g s .428 .222 .54 r .77239 .31609 m .75143 .33148 L .75152 .33882 L p F P 0 g s .603 .502 .773 r .76776 .28447 m .78991 .27297 L .78798 .26502 L p F P 0 g s .586 .466 .742 r .78991 .27297 m .76776 .28447 L .76951 .29301 L p F P 0 g s .02 w .51009 .29291 Mdot .5 Mabswid .656 .667 .92 r .77098 .21704 m .74675 .22253 L .75047 .23094 L p F P 0 g s .656 .667 .92 r .75047 .23094 m .77431 .22464 L .77098 .21704 L p F P 0 g s .751 .905 1 r .45225 .49062 m .42819 .49873 L .42048 .49238 L p F P 0 g s .751 .905 1 r .39652 .50049 m .42048 .49238 L .42819 .49873 L p F P 0 g s .63 .757 1 r .73088 .19373 m .70516 .19534 L .70971 .20235 L p F P 0 g s .63 .757 1 r .70971 .20235 m .73491 .20003 L .73088 .19373 L p F P 0 g s .1 .1 .1 r .68485 .43091 m .65988 .43925 L .65556 .4421 L p F P 0 g s .1 .1 .1 r .63067 .45041 m .65556 .4421 L .65988 .43925 L p F P 0 g s .1 .1 .127 r .70889 .39635 m .72836 .37532 L .72995 .36988 L p F P 0 g s .651 1 1 r .57041 .46799 m .54588 .47619 L .5395 .47438 L p F P 0 g s .651 1 1 r .51505 .48257 m .5395 .47438 L .54588 .47619 L p F P 0 g s .572 .974 1 r .22532 .96566 m .20211 .97195 L .17845 .96932 L p F P 0 g s .756 .868 1 r .42048 .49238 m .39652 .50049 L .38862 .4927 L p F P 0 g s .756 .868 1 r .36477 .50081 m .38862 .4927 L .39652 .50049 L p F P 0 g s .02 w .5211 .28959 Mdot .5 Mabswid .586 .466 .742 r .76951 .29301 m .79152 .28066 L .78991 .27297 L p F P 0 g s .564 .424 .706 r .79152 .28066 m .76951 .29301 L .77088 .30118 L p F P 0 g s .659 .691 .942 r .76753 .2098 m .74289 .21452 L .74675 .22253 L p F P 0 g s .659 .691 .942 r .74675 .22253 m .77098 .21704 L .76753 .2098 L p F P 0 g s .02 w .53262 .28629 Mdot .5 Mabswid .368 .867 .821 r .65556 .4421 m .63067 .45041 L .62563 .45171 L p F P 0 g s .368 .867 .821 r .60083 .46 m .62563 .45171 L .63067 .45041 L p F P 0 g s .436 .918 .946 r .34093 .93841 m .31725 .94481 L .29603 .94648 L p F P 0 g s .436 .918 .946 r .27228 .95292 m .29603 .94648 L .31725 .94481 L p F P 0 g s .71 .981 1 r .5395 .47438 m .51505 .48257 L .50816 .47919 L p F P 0 g s .71 .981 1 r .48381 .48737 m .50816 .47919 L .51505 .48257 L p F P 0 g s 1 .713 .52 r .12374 .83234 m .15359 .9619 L .17845 .96932 L p F P 0 g s 1 .716 .511 r .15359 .9619 m .12374 .83234 L .10735 .82358 L p F P 0 g s .428 .222 .54 r .75152 .33882 m .77252 .32265 L .77239 .31609 L p F P 0 g s .361 .124 .453 r .77252 .32265 m .75152 .33882 L .75112 .3454 L p F P 0 g s .564 .424 .706 r .77088 .30118 m .79278 .28802 L .79152 .28066 L p F P 0 g s .537 .375 .663 r .79278 .28802 m .77088 .30118 L .77184 .30891 L p F P 0 g s .218 .1 .319 r .72995 .36988 m .75024 .35112 L .75112 .3454 L p F P 0 g s .1 .1 .154 r .75024 .35112 m .72995 .36988 L .72836 .37532 L p F P 0 g s .02 w .54467 .28301 Mdot .5 Mabswid .659 .716 .966 r .76399 .203 m .73893 .20699 L .74289 .21452 L p F P 0 g s .659 .716 .966 r .74289 .21452 m .76753 .2098 L .76399 .203 L p F P 0 g s .1 .1 .1 r .38884 .92545 m .41307 .91889 L .43151 .91302 L p F P 0 g s .738 .944 1 r .50816 .47919 m .48381 .48737 L .4765 .48245 L p F P 0 g s .738 .944 1 r .45225 .49062 m .4765 .48245 L .48381 .48737 L p F P 0 g s .542 .976 .997 r .62563 .45171 m .60083 .46 L .59513 .45972 L p F P 0 g s .542 .976 .997 r .57041 .46799 m .59513 .45972 L .60083 .46 L p F P 0 g s .02 w .55727 .27975 Mdot .5 Mabswid .751 .905 1 r .4765 .48245 m .45225 .49062 L .44462 .4842 L p F P 0 g s .751 .905 1 r .42048 .49238 m .44462 .4842 L .45225 .49062 L p F P 0 g s .537 .375 .663 r .77184 .30891 m .79369 .29497 L .79278 .28802 L p F P 0 g s .502 .316 .611 r .79369 .29497 m .77184 .30891 L .77239 .31609 L p F P 0 g s .1 .1 .1 r .65556 .4421 m .68063 .43373 L .68485 .43091 L p F P 0 g s .86 1 .749 r .70338 .40502 m .68485 .43091 L .68063 .43373 L p F P 0 g s .651 1 1 r .59513 .45972 m .57041 .46799 L .56413 .46613 L p F P 0 g s .651 1 1 r .5395 .47438 m .56413 .46613 L .57041 .46799 L p F P 0 g s .572 .974 1 r .24871 .95931 m .22532 .96566 L .20189 .96293 L p F P 0 g s .572 .974 1 r .17845 .96932 m .20189 .96293 L .22532 .96566 L p F P 0 g s .657 .741 .989 r .76042 .19672 m .73491 .20003 L .73893 .20699 L p F P 0 g s .657 .741 .989 r .73893 .20699 m .76399 .203 L .76042 .19672 L p F P 0 g s .756 .868 1 r .44462 .4842 m .42048 .49238 L .41265 .48453 L p F P 0 g s .756 .868 1 r .38862 .4927 m .41265 .48453 L .42048 .49238 L p F P 0 g s .02 w .57045 .2765 Mdot .5 Mabswid .361 .124 .453 r .75112 .3454 m .77221 .32851 L .77252 .32265 L p F P 0 g s .269 .1 .335 r .77221 .32851 m .75112 .3454 L .75024 .35112 L p F P 0 g s .651 .599 .851 r .80421 .23232 m .78047 .24058 L .78323 .24874 L p F P 0 g s .651 .599 .851 r .78323 .24874 m .80671 .23972 L .80421 .23232 L p F P 0 g s .642 .572 .827 r .80671 .23972 m .78323 .24874 L .78574 .25692 L p F P 0 g s .642 .572 .827 r .78574 .25692 m .80899 .24713 L .80671 .23972 L p F P 0 g s .631 .543 .801 r .80899 .24713 m .78574 .25692 L .78798 .26502 L p F P 0 g s .659 .625 .874 r .80152 .22501 m .77748 .23251 L .78047 .24058 L p F P 0 g s .659 .625 .874 r .78047 .24058 m .80421 .23232 L .80152 .22501 L p F P 0 g s .368 .867 .821 r .68063 .43373 m .65556 .4421 L .65063 .44336 L p F P 0 g s .368 .867 .821 r .62563 .45171 m .65063 .44336 L .65556 .4421 L p F P 0 g s .436 .918 .946 r .36479 .93195 m .34093 .93841 L .31997 .93999 L p F P 0 g s .436 .918 .946 r .29603 .94648 m .31997 .93999 L .34093 .93841 L p F P 0 g s .71 .981 1 r .56413 .46613 m .5395 .47438 L .5327 .47095 L p F P 0 g s .71 .981 1 r .50816 .47919 m .5327 .47095 L .5395 .47438 L p F P 0 g s .631 .543 .801 r .78798 .26502 m .81102 .25447 L .80899 .24713 L p F P 0 g s .617 .512 .773 r .81102 .25447 m .78798 .26502 L .78991 .27297 L p F P 0 g s .665 .65 .896 r .79866 .21788 m .77431 .22464 L .77748 .23251 L p F P 0 g s .665 .65 .896 r .77748 .23251 m .80152 .22501 L .79866 .21788 L p F P 0 g s .02 w .58424 .27323 Mdot .5 Mabswid .1 .1 .154 r .72836 .37532 m .74888 .35592 L .75024 .35112 L p F P 0 g s .502 .316 .611 r .77239 .31609 m .79422 .30142 L .79369 .29497 L p F P 0 g s .456 .243 .546 r .79422 .30142 m .77239 .31609 L .77252 .32265 L p F P 0 g s .617 .512 .773 r .78991 .27297 m .81279 .26165 L .81102 .25447 L p F P 0 g s .601 .476 .742 r .81279 .26165 m .78991 .27297 L .79152 .28066 L p F P 0 g s .652 .768 1 r .75683 .19103 m .73088 .19373 L .73491 .20003 L p F P 0 g s .652 .768 1 r .73491 .20003 m .76042 .19672 L .75683 .19103 L p F P 0 g s .67 .674 .918 r .79566 .21099 m .77098 .21704 L .77431 .22464 L p F P 0 g s .67 .674 .918 r .77431 .22464 m .79866 .21788 L .79566 .21099 L p F P 0 g s .738 .944 1 r .5327 .47095 m .50816 .47919 L .50094 .47421 L p F P 0 g s .738 .944 1 r .4765 .48245 m .50094 .47421 L .50816 .47919 L p F P 0 g s .542 .976 .997 r .65063 .44336 m .62563 .45171 L .62005 .45139 L p F P 0 g s .542 .976 .997 r .59513 .45972 m .62005 .45139 L .62563 .45171 L p F P 0 g s .02 w .59868 .26993 Mdot .5 Mabswid .601 .476 .742 r .79152 .28066 m .81426 .26861 L .81279 .26165 L p F P 0 g s .58 .435 .707 r .81426 .26861 m .79152 .28066 L .79278 .28802 L p F P 0 g s .751 .905 1 r .50094 .47421 m .4765 .48245 L .46895 .47597 L p F P 0 g s .751 .905 1 r .44462 .4842 m .46895 .47597 L .4765 .48245 L p F P 0 g s .673 .699 .941 r .79257 .20443 m .76753 .2098 L .77098 .21704 L p F P 0 g s .673 .699 .941 r .77098 .21704 m .79566 .21099 L .79257 .20443 L p F P 0 g s .269 .1 .335 r .75024 .35112 m .77148 .3336 L .77221 .32851 L p F P 0 g s .141 .1 .175 r .77148 .3336 m .75024 .35112 L .74888 .35592 L p F P 0 g s .651 1 1 r .62005 .45139 m .59513 .45972 L .58895 .45782 L p F P 0 g s .651 1 1 r .56413 .46613 m .58895 .45782 L .59513 .45972 L p F P 0 g s .572 .974 1 r .27228 .95292 m .24871 .95931 L .22552 .95649 L p F P 0 g s .572 .974 1 r .20189 .96293 m .22552 .95649 L .24871 .95931 L p F P 0 g s .756 .868 1 r .46895 .47597 m .44462 .4842 L .43686 .47629 L p F P 0 g s .756 .868 1 r .41265 .48453 m .43686 .47629 L .44462 .4842 L p F P 0 g s .456 .243 .546 r .77252 .32265 m .79438 .30731 L .79422 .30142 L p F P 0 g s .394 .149 .461 r .79438 .30731 m .77252 .32265 L .77221 .32851 L p F P 0 g s .58 .435 .707 r .79278 .28802 m .81544 .27526 L .81426 .26861 L p F P 0 g s .555 .388 .665 r .81544 .27526 m .79278 .28802 L .79369 .29497 L p F P 0 g s .02 w .61379 .26655 Mdot .5 Mabswid .86 1 .749 r .68063 .43373 m .69978 .40736 L .70338 .40502 L p F P 0 g s .674 .724 .964 r .78941 .19826 m .76399 .203 L .76753 .2098 L p F P 0 g s .674 .724 .964 r .76753 .2098 m .79257 .20443 L .78941 .19826 L p F P 0 g s .938 1 .866 r .69978 .40736 m .68063 .43373 L .67582 .43495 L p F P 0 g s .368 .867 .821 r .65063 .44336 m .67582 .43495 L .68063 .43373 L p F P 0 g s .436 .918 .946 r .38884 .92545 m .36479 .93195 L .3441 .93345 L p F P 0 g s .436 .918 .946 r .31997 .93999 m .3441 .93345 L .36479 .93195 L p F P 0 g s .71 .981 1 r .58895 .45782 m .56413 .46613 L .55743 .46264 L p F P 0 g s .71 .981 1 r .5327 .47095 m .55743 .46264 L .56413 .46613 L p F P 0 g s 1 .716 .511 r .10735 .82358 m .12783 .9496 L .15359 .9619 L p F P 0 g s 1 .72 .503 r .12783 .9496 m .10735 .82358 L .09078 .81189 L p F P 0 g s .555 .388 .665 r .79369 .29497 m .81629 .28154 L .81544 .27526 L p F P 0 g s .522 .33 .614 r .81629 .28154 m .79369 .29497 L .79422 .30142 L p F P 0 g s .738 .944 1 r .55743 .46264 m .5327 .47095 L .52557 .46591 L p F P 0 g s .738 .944 1 r .50094 .47421 m .52557 .46591 L .5327 .47095 L p F P 0 g s .02 w .62962 .26306 Mdot .5 Mabswid .542 .976 .997 r .67582 .43495 m .65063 .44336 L .64515 .443 L p F P 0 g s .542 .976 .997 r .62005 .45139 m .64515 .443 L .65063 .44336 L p F P 0 g s .673 .749 .987 r .78621 .19256 m .76042 .19672 L .76399 .203 L p F P 0 g s .673 .749 .987 r .76399 .203 m .78941 .19826 L .78621 .19256 L p F P 0 g s .394 .149 .461 r .77221 .32851 m .79416 .31256 L .79438 .30731 L p F P 0 g s .309 .1 .347 r .79416 .31256 m .77221 .32851 L .77148 .3336 L p F P 0 g s .141 .1 .175 r .74888 .35592 m .77032 .33786 L .77148 .3336 L p F P 0 g s .1 .1 .1 r .77032 .33786 m .74888 .35592 L .74706 .35974 L p F P 0 g s .751 .905 1 r .52557 .46591 m .50094 .47421 L .49348 .46767 L p F P 0 g s .751 .905 1 r .46895 .47597 m .49348 .46767 L .50094 .47421 L p F P 0 g s .651 1 1 r .64515 .443 m .62005 .45139 L .61397 .44944 L p F P 0 g s .651 1 1 r .58895 .45782 m .61397 .44944 L .62005 .45139 L p F P 0 g s .572 .974 1 r .29603 .94648 m .27228 .95292 L .24933 .94999 L p F P 0 g s .572 .974 1 r .22552 .95649 m .24933 .94999 L .27228 .95292 L p F P 0 g s .756 .868 1 r .49348 .46767 m .46895 .47597 L .46127 .46799 L p F P 0 g s .756 .868 1 r .43686 .47629 m .46127 .46799 L .46895 .47597 L p F P 0 g s .522 .33 .614 r .79422 .30142 m .81681 .28736 L .81629 .28154 L p F P 0 g s .479 .259 .55 r .81681 .28736 m .79422 .30142 L .79438 .30731 L p F P 0 g s .02 w .64622 .25941 Mdot .5 Mabswid .669 .776 1 r .78302 .1874 m .75683 .19103 L .76042 .19672 L p F P 0 g s .669 .776 1 r .76042 .19672 m .78621 .19256 L .78302 .1874 L p F P 0 g s .436 .918 .946 r .41307 .91889 m .38884 .92545 L .36842 .92685 L p F P 0 g s .436 .918 .946 r .3441 .93345 m .36842 .92685 L .38884 .92545 L p F P 0 g s .71 .981 1 r .61397 .44944 m .58895 .45782 L .58235 .45427 L p F P 0 g s .71 .981 1 r .55743 .46264 m .58235 .45427 L .58895 .45782 L p F P 0 g s .759 1 .747 r .69978 .40736 m .72049 .38486 L .72361 .38287 L p F P 0 g s .868 1 .888 r .72049 .38486 m .69978 .40736 L .69567 .40831 L p F P 0 g s .938 1 .866 r .67582 .43495 m .69567 .40831 L .69978 .40736 L p F P 0 g s .309 .1 .347 r .77148 .3336 m .79355 .31711 L .79416 .31256 L p F P 0 g s .189 .1 .191 r .79355 .31711 m .77148 .3336 L .77032 .33786 L p F P 0 g s .479 .259 .55 r .79438 .30731 m .817 .29266 L .81681 .28736 L p F P 0 g s .421 .169 .467 r .817 .29266 m .79438 .30731 L .79416 .31256 L p F P 0 g s .653 .97 1 r .20189 .96293 m .17845 .96932 L .15359 .9619 L p F P 0 g s .738 .944 1 r .58235 .45427 m .55743 .46264 L .55039 .45754 L p F P 0 g s .738 .944 1 r .52557 .46591 m .55039 .45754 L .55743 .46264 L p F P 0 g s .969 1 .928 r .69567 .40831 m .67582 .43495 L .67046 .43454 L p F P 0 g s .542 .976 .997 r .64515 .443 m .67046 .43454 L .67582 .43495 L p F P 0 g s .02 w .66365 .25555 Mdot .5 Mabswid .1 .1 .1 r .74706 .35974 m .76875 .34124 L .77032 .33786 L p F P 0 g s .751 .905 1 r .55039 .45754 m .52557 .46591 L .51819 .4593 L p F P 0 g s .751 .905 1 r .49348 .46767 m .51819 .4593 L .52557 .46591 L p F P 0 g s .651 1 1 r .67046 .43454 m .64515 .443 L .63918 .44099 L p F P 0 g s .651 1 1 r .61397 .44944 m .63918 .44099 L .64515 .443 L p F P 0 g s .572 .974 1 r .31997 .93999 m .29603 .94648 L .27333 .94345 L p F P 0 g s .572 .974 1 r .24933 .94999 m .27333 .94345 L .29603 .94648 L p F P 0 g s .756 .868 1 r .51819 .4593 m .49348 .46767 L .48586 .45962 L p F P 0 g s .756 .868 1 r .46127 .46799 m .48586 .45962 L .49348 .46767 L p F P 0 g s .421 .169 .467 r .79416 .31256 m .81684 .29738 L .817 .29266 L p F P 0 g s .341 .1 .356 r .81684 .29738 m .79416 .31256 L .79355 .31711 L p F P 0 g s .189 .1 .191 r .77032 .33786 m .79257 .32091 L .79355 .31711 L p F P 0 g s .1 .1 .1 r .79257 .32091 m .77032 .33786 L .76875 .34124 L p F P 0 g s .02 w .68197 .25141 Mdot .5 Mabswid .436 .918 .946 r .4375 .91229 m .41307 .91889 L .39293 .92021 L p F P 0 g s .436 .918 .946 r .36842 .92685 m .39293 .92021 L .41307 .91889 L p F P 0 g s .71 .981 1 r .63918 .44099 m .61397 .44944 L .60747 .44584 L p F P 0 g s .71 .981 1 r .58235 .45427 m .60747 .44584 L .61397 .44944 L p F P 0 g s 1 .72 .503 r .09078 .81189 m .1015 .93241 L .12783 .9496 L p F P 0 g s .697 1 .741 r .72049 .38486 m .74208 .36423 L .74478 .36253 L p F P 0 g s .824 1 .897 r .74208 .36423 m .72049 .38486 L .71692 .3856 L p F P 0 g s .868 1 .888 r .69567 .40831 m .71692 .3856 L .72049 .38486 L p F P 0 g s .919 1 .961 r .71692 .3856 m .69567 .40831 L .69108 .40783 L p F P 0 g s .969 1 .928 r .67046 .43454 m .69108 .40783 L .69567 .40831 L p F P 0 g s .653 .97 1 r .22552 .95649 m .20189 .96293 L .17725 .95539 L p F P 0 g s .653 .97 1 r .15359 .9619 m .17725 .95539 L .20189 .96293 L p F P 0 g s .738 .944 1 r .60747 .44584 m .58235 .45427 L .57541 .44911 L p F P 0 g s .738 .944 1 r .55039 .45754 m .57541 .44911 L .58235 .45427 L p F P 0 g s .341 .1 .356 r .79355 .31711 m .81634 .30146 L .81684 .29738 L p F P 0 g s .228 .1 .203 r .81634 .30146 m .79355 .31711 L .79257 .32091 L p F P 0 g s .751 .905 1 r .57541 .44911 m .55039 .45754 L .5431 .45087 L p F P 0 g s .751 .905 1 r .51819 .4593 m .5431 .45087 L .55039 .45754 L p F P 0 g s .02 w .70127 .24693 Mdot .5 Mabswid .1 .1 .1 r .76875 .34124 m .79121 .3239 L .79257 .32091 L p F P 0 g s .975 1 .955 r .69108 .40783 m .67046 .43454 L .6646 .43248 L p F P 0 g s .651 1 1 r .63918 .44099 m .6646 .43248 L .67046 .43454 L p F P 0 g s .572 .974 1 r .3441 .93345 m .31997 .93999 L .29752 .93685 L p F P 0 g s .572 .974 1 r .27333 .94345 m .29752 .93685 L .31997 .93999 L p F P 0 g s .756 .868 1 r .5431 .45087 m .51819 .4593 L .51065 .45119 L p F P 0 g s .756 .868 1 r .48586 .45962 m .51065 .45119 L .51819 .4593 L p F P 0 g s .436 .918 .946 r .39293 .92021 m .41763 .91351 L .4375 .91229 L p F P 0 g s .71 .981 1 r .6646 .43248 m .63918 .44099 L .63279 .43733 L p F P 0 g s .71 .981 1 r .60747 .44584 m .63279 .43733 L .63918 .44099 L p F P 0 g s .228 .1 .203 r .79257 .32091 m .81551 .30486 L .81634 .30146 L p F P 0 g s .1 .1 .1 r .81551 .30486 m .79257 .32091 L .79121 .3239 L p F P 0 g s .824 1 .897 r .71692 .3856 m .73897 .36482 L .74208 .36423 L p F P 0 g s .887 1 .977 r .73897 .36482 m .71692 .3856 L .71294 .38509 L p F P 0 g s .919 1 .961 r .69108 .40783 m .71294 .38509 L .71692 .3856 L p F P 0 g s .639 .982 .733 r .74208 .36423 m .76443 .34514 L .76678 .34367 L p F P 0 g s .782 1 .903 r .76443 .34514 m .74208 .36423 L .73897 .36482 L p F P 0 g s .653 .97 1 r .24933 .94999 m .22552 .95649 L .2011 .94883 L p F P 0 g s .653 .97 1 r .17725 .95539 m .2011 .94883 L .22552 .95649 L p F P 0 g s .738 .944 1 r .63279 .43733 m .60747 .44584 L .60062 .44061 L p F P 0 g s .738 .944 1 r .57541 .44911 m .60062 .44061 L .60747 .44584 L p F P 0 g s .02 w .72164 .24206 Mdot .5 Mabswid .935 1 .989 r .71294 .38509 m .69108 .40783 L .68606 .40591 L p F P 0 g s .975 1 .955 r .6646 .43248 m .68606 .40591 L .69108 .40783 L p F P 0 g s .751 .905 1 r .60062 .44061 m .57541 .44911 L .5682 .44238 L p F P 0 g s .751 .905 1 r .5431 .45087 m .5682 .44238 L .57541 .44911 L p F P 0 g s .572 .974 1 r .36842 .92685 m .3441 .93345 L .3219 .9302 L p F P 0 g s .572 .974 1 r .29752 .93685 m .3219 .9302 L .3441 .93345 L p F P 0 g s .756 .868 1 r .5682 .44238 m .5431 .45087 L .53563 .44269 L p F P 0 g s .756 .868 1 r .51065 .45119 m .53563 .44269 L .5431 .45087 L p F P 0 g s .1 .1 .1 r .79121 .3239 m .81434 .30752 L .81551 .30486 L p F P 0 g s .968 1 .961 r .68606 .40591 m .6646 .43248 L .6583 .42876 L p F P 0 g s .71 .981 1 r .63279 .43733 m .6583 .42876 L .6646 .43248 L p F P 0 g s .02 w .74319 .2367 Mdot .5 Mabswid .782 1 .903 r .73897 .36482 m .76172 .34561 L .76443 .34514 L p F P 0 g s .857 1 .99 r .76172 .34561 m .73897 .36482 L .7355 .36428 L p F P 0 g s .887 1 .977 r .71294 .38509 m .7355 .36428 L .73897 .36482 L p F P 0 g s .91 1 1 r .7355 .36428 m .71294 .38509 L .70857 .38329 L p F P 0 g s .935 1 .989 r .68606 .40591 m .70857 .38329 L .71294 .38509 L p F P 0 g s .588 .957 .725 r .76443 .34514 m .78745 .32731 L .7895 .32604 L p F P 0 g s .743 1 .906 r .78745 .32731 m .76443 .34514 L .76172 .34561 L p F P 0 g s .653 .97 1 r .27333 .94345 m .24933 .94999 L .22514 .94222 L p F P 0 g s .653 .97 1 r .2011 .94883 m .22514 .94222 L .24933 .94999 L p F P 0 g s .738 .944 1 r .6583 .42876 m .63279 .43733 L .62603 .43205 L p F P 0 g s .738 .944 1 r .60062 .44061 m .62603 .43205 L .63279 .43733 L p F P 0 g s .751 .905 1 r .62603 .43205 m .60062 .44061 L .5935 .43381 L p F P 0 g s .751 .905 1 r .5682 .44238 m .5935 .43381 L .60062 .44061 L p F P 0 g s .934 1 .993 r .70857 .38329 m .68606 .40591 L .68067 .40255 L p F P 0 g s .968 1 .961 r .6583 .42876 m .68067 .40255 L .68606 .40591 L p F P 0 g s .572 .974 1 r .39293 .92021 m .36842 .92685 L .34648 .9235 L p F P 0 g s .572 .974 1 r .3219 .9302 m .34648 .9235 L .36842 .92685 L p F P 0 g s .756 .868 1 r .5935 .43381 m .5682 .44238 L .56081 .43413 L p F P 0 g s .756 .868 1 r .53563 .44269 m .56081 .43413 L .5682 .44238 L p F P 0 g s .697 .939 1 r .17725 .95539 m .15359 .9619 L .12783 .9496 L p F P 0 g s .02 w .76604 .2308 Mdot .5 Mabswid .857 1 .99 r .7355 .36428 m .75868 .34506 L .76172 .34561 L p F P 0 g s .887 1 1 r .75868 .34506 m .7355 .36428 L .73169 .36261 L p F P 0 g s .91 1 1 r .70857 .38329 m .73169 .36261 L .7355 .36428 L p F P 0 g s .743 1 .906 r .76172 .34561 m .78508 .32768 L .78745 .32731 L p F P 0 g s .828 1 .999 r .78508 .32768 m .76172 .34561 L .75868 .34506 L p F P 0 g s .913 1 1 r .73169 .36261 m .70857 .38329 L .70388 .38023 L p F P 0 g s .934 1 .993 r .68067 .40255 m .70388 .38023 L .70857 .38329 L p F P 0 g s .653 .97 1 r .29752 .93685 m .27333 .94345 L .24938 .93556 L p F P 0 g s .653 .97 1 r .22514 .94222 m .24938 .93556 L .27333 .94345 L p F P 0 g s .956 .982 .957 r .68067 .40255 m .6583 .42876 L .65165 .42342 L p F P 0 g s .738 .944 1 r .62603 .43205 m .65165 .42342 L .6583 .42876 L p F P 0 g s .544 .935 .717 r .78745 .32731 m .81106 .31053 L .81285 .30942 L p F P 0 g s .708 1 .907 r .81106 .31053 m .78745 .32731 L .78508 .32768 L p F P 0 g s .751 .905 1 r .65165 .42342 m .62603 .43205 L .619 .42518 L p F P 0 g s .751 .905 1 r .5935 .43381 m .619 .42518 L .62603 .43205 L p F P 0 g s .572 .974 1 r .41763 .91351 m .39293 .92021 L .37125 .91675 L p F P 0 g s .572 .974 1 r .34648 .9235 m .37125 .91675 L .39293 .92021 L p F P 0 g s .756 .868 1 r .619 .42518 m .5935 .43381 L .5862 .42549 L p F P 0 g s .756 .868 1 r .56081 .43413 m .5862 .42549 L .5935 .43381 L p F P 0 g s .925 .983 .986 r .70388 .38023 m .68067 .40255 L .67497 .39778 L p F P 0 g s .956 .982 .957 r .65165 .42342 m .67497 .39778 L .68067 .40255 L p F P 0 g s .697 .939 1 r .2011 .94883 m .17725 .95539 L .1517 .94296 L p F P 0 g s .697 .939 1 r .12783 .9496 m .1517 .94296 L .17725 .95539 L p F P 0 g s .828 1 .999 r .75868 .34506 m .78241 .32714 L .78508 .32768 L p F P 0 g s .865 1 1 r .78241 .32714 m .75868 .34506 L .75534 .3435 L p F P 0 g s .887 1 1 r .73169 .36261 m .75534 .3435 L .75868 .34506 L p F P 0 g s .893 1 1 r .75534 .3435 m .73169 .36261 L .72759 .3598 L p F P 0 g s .913 1 1 r .70388 .38023 m .72759 .3598 L .73169 .36261 L p F P 0 g s .708 1 .907 r .78508 .32768 m .80898 .31081 L .81106 .31053 L p F P 0 g s .803 1 1 r .80898 .31081 m .78508 .32768 L .78241 .32714 L p F P 0 g s .906 .982 1 r .72759 .3598 m .70388 .38023 L .69892 .37591 L p F P 0 g s .925 .983 .986 r .67497 .39778 m .69892 .37591 L .70388 .38023 L p F P 0 g s .653 .97 1 r .3219 .9302 m .29752 .93685 L .2738 .92884 L p F P 0 g s .653 .97 1 r .24938 .93556 m .2738 .92884 L .29752 .93685 L p F P 0 g s .942 .946 .947 r .67497 .39778 m .65165 .42342 L .64471 .41648 L p F P 0 g s .751 .905 1 r .619 .42518 m .64471 .41648 L .65165 .42342 L p F P 0 g s .572 .974 1 r .37125 .91675 m .39623 .90994 L .41763 .91351 L p F P 0 g s .756 .868 1 r .64471 .41648 m .619 .42518 L .61178 .41679 L p F P 0 g s .756 .868 1 r .5862 .42549 m .61178 .41679 L .619 .42518 L p F P 0 g s .697 .939 1 r .22514 .94222 m .2011 .94883 L .17576 .93627 L p F P 0 g s .697 .939 1 r .1517 .94296 m .17576 .93627 L .2011 .94883 L p F P 0 g s .865 1 1 r .75534 .3435 m .77948 .32568 L .78241 .32714 L p F P 0 g s .875 1 1 r .77948 .32568 m .75534 .3435 L .75175 .34093 L p F P 0 g s .893 1 1 r .72759 .3598 m .75175 .34093 L .75534 .3435 L p F P 0 g s .914 .944 .972 r .69892 .37591 m .67497 .39778 L .66902 .39164 L p F P 0 g s .942 .946 .947 r .64471 .41648 m .66902 .39164 L .67497 .39778 L p F P 0 g s .803 1 1 r .78241 .32714 m .80664 .31028 L .80898 .31081 L p F P 0 g s .845 1 1 r .80664 .31028 m .78241 .32714 L .77948 .32568 L p F P 0 g s .889 .979 1 r .75175 .34093 m .72759 .3598 L .72325 .35589 L p F P 0 g s .906 .982 1 r .69892 .37591 m .72325 .35589 L .72759 .3598 L p F P 0 g s .897 .942 .985 r .72325 .35589 m .69892 .37591 L .69374 .37039 L p F P 0 g s .914 .944 .972 r .66902 .39164 m .69374 .37039 L .69892 .37591 L p F P 0 g s .653 .97 1 r .34648 .9235 m .3219 .9302 L .29843 .92206 L p F P 0 g s .653 .97 1 r .2738 .92884 m .29843 .92206 L .3219 .9302 L p F P 0 g s .875 1 1 r .75175 .34093 m .77631 .32332 L .77948 .32568 L p F P 0 g s .874 .977 1 r .77631 .32332 m .75175 .34093 L .74793 .33737 L p F P 0 g s .889 .979 1 r .72325 .35589 m .74793 .33737 L .75175 .34093 L p F P 0 g s .845 1 1 r .77948 .32568 m .80405 .30892 L .80664 .31028 L p F P 0 g s .859 1 1 r .80405 .30892 m .77948 .32568 L .77631 .32332 L p F P 0 g s .929 .912 .935 r .66902 .39164 m .64471 .41648 L .63757 .40802 L p F P 0 g s .756 .868 1 r .61178 .41679 m .63757 .40802 L .64471 .41648 L p F P 0 g s .881 .939 .996 r .74793 .33737 m .72325 .35589 L .71871 .35091 L p F P 0 g s .897 .942 .985 r .69374 .37039 m .71871 .35091 L .72325 .35589 L p F P 0 g s .697 .939 1 r .24938 .93556 m .22514 .94222 L .20001 .92952 L p F P 0 g s .697 .939 1 r .17576 .93627 m .20001 .92952 L .22514 .94222 L p F P 0 g s .902 .908 .957 r .69374 .37039 m .66902 .39164 L .6629 .38418 L p F P 0 g s .929 .912 .935 r .63757 .40802 m .6629 .38418 L .66902 .39164 L p F P 0 g s .719 .9 1 r .1517 .94296 m .12783 .9496 L .1015 .93241 L p F P 0 g s .886 .905 .968 r .71871 .35091 m .69374 .37039 L .6884 .36372 L p F P 0 g s .902 .908 .957 r .6629 .38418 m .6884 .36372 L .69374 .37039 L p F P 0 g s .859 1 1 r .77631 .32332 m .80126 .30674 L .80405 .30892 L p F P 0 g s .86 .974 1 r .80126 .30674 m .77631 .32332 L .77295 .32007 L p F P 0 g s .874 .977 1 r .74793 .33737 m .77295 .32007 L .77631 .32332 L p F P 0 g s .653 .97 1 r .37125 .91675 m .34648 .9235 L .32326 .91524 L p F P 0 g s .653 .97 1 r .29843 .92206 m .32326 .91524 L .34648 .9235 L p F P 0 g s .867 .936 1 r .77295 .32007 m .74793 .33737 L .74395 .33286 L p F P 0 g s .881 .939 .996 r .71871 .35091 m .74395 .33286 L .74793 .33737 L p F P 0 g s .872 .902 .977 r .74395 .33286 m .71871 .35091 L .71404 .34491 L p F P 0 g s .886 .905 .968 r .6884 .36372 m .71404 .34491 L .71871 .35091 L p F P 0 g s .697 .939 1 r .2738 .92884 m .24938 .93556 L .22446 .92271 L p F P 0 g s .697 .939 1 r .20001 .92952 m .22446 .92271 L .24938 .93556 L p F P 0 g s .86 .974 1 r .77295 .32007 m .79829 .30377 L .80126 .30674 L p F P 0 g s .855 .933 1 r .79829 .30377 m .77295 .32007 L .76943 .31598 L p F P 0 g s .867 .936 1 r .74395 .33286 m .76943 .31598 L .77295 .32007 L p F P 0 g s .719 .9 1 r .17576 .93627 m .1517 .94296 L .12556 .92562 L p F P 0 g s .719 .9 1 r .1015 .93241 m .12556 .92562 L .1517 .94296 L p F P 0 g s .859 .898 .985 r .76943 .31598 m .74395 .33286 L .73983 .32745 L p F P 0 g s .872 .902 .977 r .71404 .34491 m .73983 .32745 L .74395 .33286 L p F P 0 g s .653 .97 1 r .39623 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In our case" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Tdy[s_] = \((1 - Trat[s])\)\ pd[s] // FullSimplify\)], "Input", CellLabel->"In[46]:="], Cell[BoxData[ \(\(0.1`\ \((\(\(0.2802992924437547`\)\(\[InvisibleSpace]\)\) + s)\)\ \((\ \(\(3.175979626438095`\)\(\[InvisibleSpace]\)\) + s\ \ \((\(\(1.6430085285389604`\)\(\[InvisibleSpace]\)\) + s)\))\)\ \ \((\(\(19.938819051393555`\)\(\[InvisibleSpace]\)\) + s\ \ \((\(\(7.30163149189027`\)\(\[InvisibleSpace]\)\) + \ s)\))\)\)\/\(\((\(\(0.049999999999999996`\)\(\[InvisibleSpace]\)\) + s)\)\ \ \((\(\(2.`\)\(\[InvisibleSpace]\)\) + s)\)\^3\ \((\(\(6.268981882908511`\)\(\ \[InvisibleSpace]\)\) + s\ \ \((\(\(3.2249393128729857`\)\(\[InvisibleSpace]\)\) + s)\))\)\)\)], "Output", CellLabel->"Out[46]="] }, Open ]], Cell["\<\ The magnitude of this transfer function can indicate the effectivness of the \ disturbace rejection. 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But first, we should remove the OPTDesign application by \ restarting ", StyleBox["Mathematica ", FontSlant->"Italic"], StyleBox["and continue the compuation from this point!", FontVariations->{"CompatibilityType"->0}] }], "Text"], Cell[BoxData[ \(\(Off[General::spell];\)\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ \(<< ControlSystems`\)], "Input", CellLabel->"In[1]:="], Cell["Let us load the transfer function of the compensator", "Text"], Cell[BoxData[ \(<< foo.mx\)], "Input", CellLabel->"In[2]:=", CellTags->"DumpSave"], Cell["The transfer function of the compensator,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(K = TransferFunction[s, \(-comp1[s]\)]\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \(TransferFunction[ s, {{\(61.739894122701735`\ \((\(\(0.049999999999999996`\)\(\ \[InvisibleSpace]\)\) + s)\)\ \ \((\(\(1.749372561662118`\)\(\[InvisibleSpace]\)\) + 1.1888268755328026`\ s + \ s\^2)\)\)\/\(\((\(\(0.28029929244375457`\)\(\[InvisibleSpace]\)\) + s)\)\ \((\ \(\(3.175979626438095`\)\(\[InvisibleSpace]\)\) + 1.6430085285389604`\ s + \ s\^2)\)\ \((\(\(19.938819051393555`\)\(\[InvisibleSpace]\)\) + \ 7.30163149189027`\ s + s\^2)\)\)}}]\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(SK = StateSpace[K]; EquationForm[SK]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ FormBox[ TagBox[ FormBox[GridBox[{ { RowBox[{\(\[ScriptX]\& . \), "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1.`", "0.`", "0.`", "0.`"}, {"0", "0.`", "1.`", "0.`", "0.`"}, {"0", "0.`", "0.`", "1.`", "0.`"}, {"0", "0.`", "0.`", "0.`", "1.`"}, {\(-17.750032041814467`\), \(-79.00788347020072`\), \ \(-65.79119481511636`\), \(-37.61861776014642`\), \(-9.224939312872985`\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"}, {"0"}, {"0"}, {"0"}, {"1.`"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]}, { RowBox[{"\[ScriptY]", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"5.400303836908934`", "111.67597900945955`", "76.48504013175264`", "61.739894122701735`", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]"}]} }, ColumnAlignments->{"="}, AllowScriptLevelChange->False], "TraditionalForm"], (EquationForm[ StateSpace[ SlotSequence[ 1]]]&)], TraditionalForm]], "Output", CellLabel->"Out[4]//EquationForm="] }, Open ]], Cell["\<\ One may try to reduce the states looking for minimal realization\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(SKM = MinimalRealization[SK]; EquationForm[SKM]\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ \(MinimalRealization::"nsc" \(\(:\)\(\ \)\) "No states deleted."\)], "Message", CellLabel->"From In[5]:="], Cell[BoxData[ FormBox[ TagBox[ FormBox[GridBox[{ { RowBox[{\(\[ScriptX]\& . \), "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-2.8714656782718957`\), "0.22060063126961119`", "0.007869531774128673`", \ \(-0.003512863994978452`\), \(-0.0033320564353568616`\)}, {\(-32.995709243109765`\), \(-4.284071059408608`\), \ \(-0.14957607425899735`\), "0.0665263276438742`", "0.06313611552835874`"}, {"56.52812470587592`", "7.6245094807196265`", \(-1.3642731917838824`\), "0.6696671860858928`", "0.6082873150293869`"}, {"65.68653973391537`", "9.425369085860474`", \(-1.5693682684412944`\), \ \(-0.34618883343310447`\), \(-0.060590610126163805`\)}, {"59.600417094396995`", "7.53593407132337`", \(-1.7696899004704392`\), \ \(-0.08746777247104556`\), \(-0.3589405499754887`\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-0.03244719372671142`\)}, {\(-0.2970418213321456`\)}, {"0.5125006548999308`"}, {"0.6000377331367794`"}, {"0.5366666875614323`"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]}, { RowBox[{"\[ScriptY]", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-145.5060933545219`\), \(-13.625254416514856`\), "9.935980069807071`", \(-26.325981135793434`\), "3.6071755669875856`"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]"}]} }, ColumnAlignments->{"="}, AllowScriptLevelChange->False], "TraditionalForm"], (EquationForm[ StateSpace[ SlotSequence[ 1]]]&)], TraditionalForm]], "Output", CellLabel->"Out[5]//EquationForm="] }, Open ]], Cell["\<\ It means, that the number of the states can not be reduced, which is in \ harmony with Fig. 11. Consequently we consider the original state space \ model, which is in traditional form\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TK = TraditionalForm[SK]\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ FormBox[ TagBox[ FormBox[ SubsuperscriptBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1.`", "0.`", "0.`", "0.`", "0"}, {"0", "0.`", "1.`", "0.`", "0.`", "0"}, {"0", "0.`", "0.`", "1.`", "0.`", "0"}, {"0", "0.`", "0.`", "0.`", "1.`", "0"}, {\(-17.750032041814467`\), \(-79.00788347020072`\), \ \(-65.79119481511636`\), \(-37.61861776014642`\), \(-9.224939312872985`\), "1.`"}, {"5.400303836908934`", "111.67597900945955`", "76.48504013175264`", "61.739894122701735`", "0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}, RowLines->{False, False, False, False, True, False}, ColumnLines->{False, False, False, False, True, False}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "TraditionalForm"], FormBox["\[Bullet]", "TraditionalForm"], FormBox[ StyleBox["\<\"\[ScriptCapitalS]\"\>", Editable->False, ShowStringCharacters->False], "TraditionalForm"], MultilineFunction->None], "TraditionalForm"], (StateSpace[ SlotSequence[ 1], ControlSystems`Common`Private`noD -> True]&)], TraditionalForm]], "Output", CellLabel->"Out[6]//TraditionalForm="] }, Open ]], Cell["\<\ Now, considering the Fig.2, the compensator dynamics can be represented in \ state space as\ \>", "Text"], Cell["x'(t) = AA x(t) + BB (h(t) - h0)", "Text", TextAlignment->Center], Cell["c(t ) - c0 = CC x(t) + DD (h(t) - h0)", "Text", TextAlignment->Center], Cell["These matrices can be deduced from TK,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(AA = TK[\([1, 1]\)]; MatrixForm[AA]\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1.`", "0.`", "0.`", "0.`"}, {"0", "0.`", "1.`", "0.`", "0.`"}, {"0", "0.`", "0.`", "1.`", "0.`"}, {"0", "0.`", "0.`", "0.`", "1.`"}, {\(-17.750032041814467`\), \(-79.00788347020072`\), \ \(-65.79119481511636`\), \(-37.61861776014642`\), \(-9.224939312872985`\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[7]//MatrixForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(BB = TK[\([1, 2]\)]; MatrixForm[BB]\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"}, {"0"}, {"0"}, {"0"}, {"1.`"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[8]//MatrixForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CC = TK[\([1, 3]\)]; MatrixForm[CC]\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"5.400303836908934`", "111.67597900945955`", "76.48504013175264`", "61.739894122701735`", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[9]//MatrixForm="] }, Open ]], Cell["and DD is zero, therefore", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Length[TK[\([1]\)]]\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \(3\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell["\<\ We shall create the system of the differential equations describing the state \ of the compensator dynamics,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(n = Length[Transpose[AA]]\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(5\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell[BoxData[ \(\(X = Table[x\_i[t], {i, 1, n}];\)\)], "Input", CellLabel->"In[12]:="], Cell[CellGroupData[{ Cell[BoxData[ \(deqx = Map[# \[Equal] 0 &, Flatten[Table[ D[X[\([i]\)], t] - AA[\([i]\)] . 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X]\)], "Input", CellLabel->"In[17]:="], Cell[BoxData[ RowBox[{"{", RowBox[{\(\(1\/A\) \((Qin[ t] - \@h[t]\ \((c0 + 5.400303836908934`\ x\_1[t] + 111.67597900945955`\ x\_2[t] + 76.48504013175264`\ x\_3[t] + 61.739894122701735`\ x\_4[t])\))\)\), "\[Equal]", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "}"}]], "Output", CellLabel->"Out[17]="] }, Open ]], Cell["\<\ The disturbance function, in our case, Qin(t) should be defined. Let us \ consider a unit step function at t = 10\ \>", "Text"], Cell[BoxData[ \(\(Qin[t_] = Qin0 + UnitStep[t - 10];\)\)], "Input", CellLabel->"In[18]:="], Cell["\<\ Th numerical values are for the plant model and for the initial (steady \ state) values,\ \>", "Text"], Cell[BoxData[ \(h0 = 9. ; A = 10. ; c0 = 3. ; Qin0 = 9. ;\)], "Input", CellLabel->"In[19]:="], Cell["\<\ The initial conditions for the controller state variables are\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(incx = Map[\((# /. t \[Rule] 0)\) \[Equal] 0 &, X]\)], "Input", CellLabel->"In[20]:="], Cell[BoxData[ \({x\_1[0] \[Equal] 0, x\_2[0] \[Equal] 0, x\_3[0] \[Equal] 0, x\_4[0] \[Equal] 0, x\_5[0] \[Equal] 0}\)], "Output", CellLabel->"Out[20]="] }, Open ]], Cell["\<\ Then the system of differential equations, deqx and deqh can be solved \ simultanuesly,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol = NDSolve[Join[deqx, deqh, {h[0] \[Equal] h0}, incx], Join[X, {h}], {t, 0, 150}, \ MaxSteps \[Rule] 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ool100000040oooo000K0?ooool000001@0000010?ooo`00o`3ooolQ0?ooo`00o`3ooolQ0?ooo`00 \ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-20.2789, 2.97457, 0.606639, \ 0.0013982}}] }, Open ]], Cell[TextData[{ StyleBox["Fig. 21", FontSlant->"Italic"], " The change of the control valve position as actuator during the control \ operation" }], "Text", TextAlignment->Center] }, Open ]], Cell[CellGroupData[{ Cell["Remark", "Section"], Cell["\<\ Employing higher order approximation of the numerically evaluated closed loop \ transfer function, which provides smaller approximation error (see in \ Producing Rational Compensator), we get smaller deviation from the steady \ state, see Figures 22-23.\ \>", "Text"], Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@0005dH0@0008`0000400000P00024100230@00 00000000001F6`00gB<00215CDH0004065d003@4000300003`0006`00000000000D00004000X0@00 l000000000000000000004241020Z@<0EP1905<0B@1?0000A01b0640M`1Y06h0I`000000000U0000 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