(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 1565129, 21290] NotebookOptionsPosition[ 1265109, 16277] NotebookOutlinePosition[ 1557269, 21019] CellTagsIndexPosition[ 1557187, 21014] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Mathematica assistance in proving theorems in nonlinear control\ \>", "Title", CellMargins->0, CellFrameMargins->{{60, 9}, {9, 12}}, CellChangeTimes->{{3.41252692941323*^9, 3.412526931828525*^9}, { 3.412532832874496*^9, 3.412532837988109*^9}, 3.432292838191875*^9}, FontColor->GrayLevel[1], Background->GrayLevel[0]], Cell["\[CapitalUDoubleDot]lle Kotta & Palle Kotta", "Author", CellChangeTimes->{{3.43229284767625*^9, 3.43229285198875*^9}}], Cell["\<\ Institute of Cybernetics at Tallinn University of Technology, Estonia \ \>", "Affiliation", CellChangeTimes->{3.43229285786375*^9}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"], ButtonFrame->"None"], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"], ButtonFrame->"None"] }], "PreviousNext"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Control system", FontColor->RGBColor[ 0.6502021820401312, 0.7862973983367666, 0.08026245517662318]]], "Section", CellChangeTimes->{{3.432292890410625*^9, 3.432292893285625*^9}}], Cell["\<\ Control systems are systems that have at least one variable that can be \ changed (within certain limits) in order to control other system\ \[CloseCurlyQuote]s variables.\ \>", "Text", CellChangeTimes->{ 3.432292898848125*^9, {3.432384090348125*^9, 3.43238410055125*^9}}, FontFamily->"Arial", FontSize->20], Cell[CellGroupData[{ Cell["Example", "Subsection", CellChangeTimes->{ 3.3824524252213373`*^9, 3.3979195443087*^9, {3.432292993035625*^9, 3.4322930003325*^9}}], Cell["\<\ A simple example of control system is TV remote control. We can change input variable (by pushing the channel button on the remote) in \ order to control output variable (channel on TV).\ \>", "Text", CellChangeTimes->{{3.43229306577*^9, 3.432293069191875*^9}}, FontFamily->"Arial", FontSize->20] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Input-output vs state equations", "Section", CellChangeTimes->{{3.432292890410625*^9, 3.432292893285625*^9}, 3.432293100879375*^9}], Cell["Control system can be described in many ways, including:", "Text", CellChangeTimes->{ 3.432292898848125*^9, {3.4322931060825*^9, 3.432293112504375*^9}}, FontSize->20], Cell[CellGroupData[{ Cell[TextData[StyleBox["input-output equations", "Item"]], "Subsection", CellChangeTimes->{{3.43229306577*^9, 3.432293069191875*^9}, { 3.4322931233325*^9, 3.432293142941875*^9}, {3.432295912691875*^9, 3.43229591448875*^9}, {3.432295983879375*^9, 3.432295989441875*^9}}], Cell["\<\ Input-output equations describe the relation between \ system\[CloseCurlyQuote]s inputs and outputs, and possibly their derivatives.\ \ \>", "Text", CellChangeTimes->{{3.432296025098125*^9, 3.4322960380825*^9}, 3.432298714348125*^9, {3.432386483785625*^9, 3.432386484504375*^9}, 3.4323866638325*^9}, FontSize->20], Cell[BoxData[ StyleBox["Example", "Subsubsection"]], "Input", FontSize->20], Cell["\<\ The channel shown on TV (output) is related to the button on the remote \ (input).\ \>", "Text", CellChangeTimes->{{3.432296046504375*^9, 3.432296057348125*^9}, { 3.432296098379375*^9, 3.432296101691875*^9}, 3.43229871567625*^9, { 3.43238459002*^9, 3.432384594191875*^9}}, FontSize->20] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["state equations", "Item"]], "Subsection", CellChangeTimes->{{3.43229306577*^9, 3.432293069191875*^9}, { 3.4322931233325*^9, 3.432293142941875*^9}, {3.432295912691875*^9, 3.43229591448875*^9}, {3.43229597230125*^9, 3.432295982535625*^9}}], Cell["\<\ State 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RowBox[{"\[InvisiblePrefixScriptBase]", "g"}], TraditionalForm]]], ": gravitational constant, ", Cell[BoxData[ FormBox["l", TraditionalForm]]], ": length of pendulum, ", Cell[BoxData[ FormBox["m", TraditionalForm]]], ": mass of pendulum, ", "\[Phi]", ": angle of pendulum, ", Cell[BoxData[ FormBox["u", TraditionalForm]]], ": torque" }], "Text", CellChangeTimes->{{3.432386543316875*^9, 3.43238660598875*^9}, { 3.4323866439575*^9, 3.4323866439575*^9}}, FontSize->20] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Model", "Section", CellChangeTimes->{{3.432299214285625*^9, 3.43229923342625*^9}}], Cell["\<\ In order to control processes more accurately and easily, we need a process \ (system) model.\ \>", "Text", FontSize->20] }, Open ]], Cell[CellGroupData[{ Cell["Modelling", "Section"], Cell["\<\ In modelling a system, we usually measure the inputs and outputs of the \ system and based on these measurements build up the input-output equations.\ \>", "Text", CellChangeTimes->{{3.432299248223125*^9, 3.432299248723125*^9}, { 3.432299303754375*^9, 3.43229930452*^9}}, FontSize->20], Cell["\<\ Input-output equations don\[CloseCurlyQuote]t usually come from first \ principles and may not correspond to the inner logic of the system.\ \>", "Text", CellChangeTimes->{{3.432299308004375*^9, 3.432299309410625*^9}, 3.4324700566294904`*^9}, FontSize->20] }, Open ]], Cell[CellGroupData[{ Cell["Identification", "Section"], Cell["\<\ Building an (input-output) equation based on measurements of system\ \[CloseCurlyQuote]s inputs and outputs, is called identification.\ \>", "Text", CellChangeTimes->{{3.4322992625825*^9, 3.43229926402*^9}, { 3.43238649048875*^9, 3.4323864910825*^9}, 3.432386664941875*^9}, FontSize->20] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Why state equations?", "Section"], Cell["\<\ Most control system related theory uses state equations.To apply control \ theory on systems to be controlled, we should obtain state equations from \ input-output equations.Obtaining state equations from input-output equations \ is called realization.\ \>", "Text", CellChangeTimes->{{3.432299371895*^9, 3.432299372035625*^9}, { 3.43229943277*^9, 3.43229943873875*^9}}, FontSize->20] }, Open ]], Cell[CellGroupData[{ Cell["Realization", "Section"], Cell["\<\ Obtaining state equations from input-output equations may sometimes be \ impossible.\ \>", "Text", CellChangeTimes->{{3.43229939167625*^9, 3.432299391848125*^9}, { 3.432299441598125*^9, 3.43229944255125*^9}, 3.43238696036375*^9}, FontSize->20], Cell["\<\ It is good to know, if an input-output equation has a state equation.If we \ would knew that the model structure does not have state equation, we could \ choose (due to certain freedom) another similar input-output model structure \ to describe the system.\ \>", "Text", CellChangeTimes->{{3.432299392645*^9, 3.432299395848125*^9}, { 3.432299445879375*^9, 3.432299450629375*^9}}, FontSize->20] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Linear control systems", "Section"], Cell["\<\ From linear input-output equations, it is always possible to obtain state \ equations.\ \>", "Text", CellChangeTimes->{{3.43238331898875*^9, 3.432383319145*^9}, { 3.432386492848125*^9, 3.43238649348875*^9}}, FontSize->20], Cell["\<\ Linear control systems are simple but often not suitable for applications.\ \>", "Text", FontSize->20] }, Open ]], Cell[CellGroupData[{ Cell["Nonlinear control system", "Section"], Cell["\<\ Unlike linear systems, it is not always possible to find state equations for \ nonlinear systems.\ \>", "Text", CellChangeTimes->{{3.432384126895*^9, 3.43238412705125*^9}}, FontSize->20] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Simple nonlinear control systems", "Section"], Cell["\<\ In practice, most often two types of nonlinear control systems are used:\ \>", "Text", CellChangeTimes->{{3.432383346941875*^9, 3.43238335217625*^9}, 3.432384701191875*^9, 3.43238680236375*^9}, FontSize->20], Cell[CellGroupData[{ Cell["Bilinear", "Item", CellChangeTimes->{{3.432383355191875*^9, 3.4323833614575*^9}, { 3.432384128754375*^9, 3.432384129145*^9}, {3.432384715785625*^9, 3.4323847168325*^9}, {3.432387213941875*^9, 3.432387215129375*^9}, { 3.432387874785625*^9, 3.43238787555125*^9}}, FontSize->20], Cell["Quadratic", "Item", FontSize->20] }, Open ]], Cell["\<\ These two types of systems are easy enough to operate with and flexible \ enough to describe a system with desired accuracy.\ \>", "Text", CellChangeTimes->{3.432383357816875*^9}, FontSize->20] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Realizability", "Section"], Cell["\<\ Realizability is a property of input-output equation which means it has state \ equations.\ \>", "Text", CellChangeTimes->{{3.43238649598875*^9, 3.43238649661375*^9}}, FontSize->20], Cell["\<\ Modellers aim to describe system with realizable input-output equations in \ order to use tools from control systems theory.\ \>", "Text", CellChangeTimes->{{3.432384129816875*^9, 3.4323841299575*^9}, { 3.432386498723125*^9, 3.432386499316875*^9}}, FontSize->20] }, Open ]], Cell[CellGroupData[{ Cell["Checking realizability", "Section"], Cell[CellGroupData[{ Cell["\<\ In the literature, there exists many methods for checking if the system is \ realizable or not.\ \>", "Item", CellChangeTimes->{{3.432384130473125*^9, 3.432384130629375*^9}}, FontSize->20], Cell["\<\ Checking realizability and building state equation requires huge amount of \ complex symbolic computations.\ \>", "Item", CellChangeTimes->{{3.432384131129375*^9, 3.43238413127*^9}}, FontSize->20] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Problem", "Section"], Cell[CellGroupData[{ Cell["\<\ Modellers should describe system with realizable input-output equations.\ \>", "Item", CellChangeTimes->{{3.4323865017075*^9, 3.43238650230125*^9}}, FontSize->20], Cell["\<\ Modellers don\[CloseCurlyQuote]t know beforehand whether the input-output \ equations are realizable or not.\ \>", "Item", CellChangeTimes->{{3.432384131785625*^9, 3.43238413192625*^9}, { 3.432386504410625*^9, 3.432386505035625*^9}, 3.432470058702548*^9}, FontSize->20], Cell["Checking the realizability takes much effort and time.", "Item", CellChangeTimes->{{3.432384132348125*^9, 3.43238413248875*^9}}, FontSize->20] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Our goal", "Section"], Cell["\<\ Our goal was to obtain the subclasses of bilinear and quadratic input-output \ equations which are realizable, i.e.guaranteed to have state equations.\ \>", "Text", CellChangeTimes->{{3.43238413286375*^9, 3.432384133004375*^9}, { 3.432386507098125*^9, 3.432386507723125*^9}}, FontSize->20], Cell["\<\ Using these subclasses, modeller could simply by direct inspection (without \ any calculations) check from the input-output equation parameters, if the \ system has state equations or not.\ \>", "Text", CellChangeTimes->{{3.432384133441875*^9, 3.432384133598125*^9}, { 3.432386509395*^9, 3.432386510004375*^9}}, FontSize->20] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Tools used to obtain the results", "Section"], Cell["We used theoretical results from the following book:", "Text", CellChangeTimes->{ 3.4323834413325*^9, {3.432384134066875*^9, 3.4323841360825*^9}, { 3.432384807535625*^9, 3.432384818145*^9}, 3.432386808816875*^9}, FontSize->20], Cell["\<\ G.Conte, C.H.Moog, A.M.Perdon \[OpenCurlyDoubleQuote]Algebraic Methods for \ Nonlinear Control Systems.Theory and Applications\[CloseCurlyDoubleQuote], \ 2nd Edition, Springer, 2007.\ \>", "Item", CellChangeTimes->{{3.432384813910625*^9, 3.4323848160825*^9}, 3.43238653048875*^9}, FontSize->20], Cell["\<\ We used Mathematica together with nonlinear control system toolkit NLControl \ developed in our institute.\ \>", "Text", FontSize->20] }, Open ]], Cell[CellGroupData[{ Cell["Mathematica assistance", "Section"], Cell[CellGroupData[{ Cell["\<\ Mathematica assisted finding the realizability conditions for bilinear and \ quadratic input-output equations.\ \>", "Item", CellChangeTimes->{{3.432386511785625*^9, 3.43238651236375*^9}}, FontSize->20], Cell["\<\ Mathematica assisted finding the state equations for some simpler cases.\ \>", "Item", CellChangeTimes->{{3.43238413723875*^9, 3.432384137395*^9}}, FontSize->20], Cell["Mathematica assisted making hypothesis.", "Item", CellChangeTimes->{{3.4323841379575*^9, 3.4323841380825*^9}}, FontSize->20], Cell["Mathematica assisted proving theorems.", "Item", CellChangeTimes->{{3.432384138473125*^9, 3.43238413861375*^9}}, FontSize->20] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Finding results: low order systems", "Section", CellChangeTimes->{3.432386810035625*^9}], Cell["\<\ Checking general realizability conditions requires solving a certain number \ (depending on the system order) of systems of nonlinear equations. For each \ system of equations there may exist many possible solutions. After finding all solutions for one system of equations, we omitted the \ solutions, which contained the other solutions. As a result, we got \ restrictions on system parameters (\"branches\"). For each set of \ restrictions, we repreated the process with the following system of nonlinear \ equations until the last system of nonlinear equations. Finally, after processing all \[OpenCurlyDoubleQuote]branches\ \[CloseCurlyDoubleQuote], we omitted these sets of restrictions which were \ special cases of another sets of restrictions.\ \>", "Text", CellChangeTimes->{{3.43238526436375*^9, 3.43238535861375*^9}, { 3.432470999543419*^9, 3.432471048408351*^9}, {3.4324710860195403`*^9, 3.4324712485753202`*^9}, 3.432471344294473*^9, {3.4324714498021994`*^9, 3.432471495082144*^9}, {3.432471616753485*^9, 3.432471830340776*^9}}, FontSize->20] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Finding results: subclasses", "Section", CellChangeTimes->{3.432385734285625*^9}], Cell[CellGroupData[{ Cell["We found results for systems of order n.", "Item", CellChangeTimes->{{3.432384151723125*^9, 3.432384156191875*^9}, { 3.432385388191875*^9, 3.432385394129375*^9}}, FontSize->20], Cell["We made hypothesis for order n + 1.", "Item", CellChangeTimes->{{3.43238539623875*^9, 3.43238539823875*^9}}, FontSize->20], Cell["We checked the hypothesis for order n + 1.", "Item", CellChangeTimes->{{3.4323854002075*^9, 3.432385402035625*^9}}, FontSize->20], Cell["\<\ After doing above steps for a few times, we made hypothesis for the general \ case. We proved the hypothesis for the general case theoretically.\ \>", "Item", FontSize->20] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Finding results: methods", "Section", CellChangeTimes->{{3.432385737645*^9, 3.432385738910625*^9}}], Cell[CellGroupData[{ Cell["\<\ We used two different methods for finding the necessary and sufficient \ conditions for bilinear input-output equations.\ \>", "Item", CellChangeTimes->{{3.432384157098125*^9, 3.43238416173875*^9}, { 3.432384206645*^9, 3.432384218879375*^9}, {3.432385742441875*^9, 3.432385743910625*^9}, {3.432386514145*^9, 3.432386514723125*^9}}, FontSize->20], Cell["\<\ We used one method for computations on paper, and another one on computer \ algebra system Mathematica.\ \>", "Item", CellChangeTimes->{{3.432385747395*^9, 3.432385749598125*^9}}, FontSize->20], Cell["\<\ Both methods yielded the same results, as expected, but doing complex \ symbolic calculations on paper took more effort, and resulted in many \ intermediate mistakes.\ 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