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The symbolic tools are used to derive a set of nonlinear \ differential equations using Euler-Lagrange equations of motion. The model is \ converted to C++ using ", StyleBox["MathCode C++", FontSlant->"Italic"], ", which produces an efficient implementation of the large expressions used \ in the model. The exported code is used for simulations, which illustrates \ that ", StyleBox["Mathematica", FontSlant->"Italic"], " in combination with ", StyleBox["MathCode C++", FontSlant->"Italic"], " can be used to do accurate and powerful simulations of nonlinear systems. \ Controller synthesis is performed where the resulting controller is exported \ to C++ and run externally. The applications presented are a seesaw/pendulum \ process and aerodynamics of a fighter aircraft." }], "Text", FontFamily->"Times New Roman"] }, Open ]], Cell[CellGroupData[{ Cell["1 Introduction", "Section"], Cell[TextData[{ "Control system design is a discipline where advanced mathematics is \ applied to real world problems ranging from paper and pulp manufacturing, \ aircraft, and CD players to bio-chemical processes, logistics, and financial \ applications. ", StyleBox["Mathematica", FontSlant->"Italic"], " is very powerful for dealing with the mathematics in these control \ problems and the application package Control System Professional (CSP) \ implements many useful features. When it comes to nonlinear systems the \ combination of the symbolic and numeric capabilities of ", StyleBox["Mathematica", FontSlant->"Italic"], " makes modeling and control system design in this environment very \ attractive. The notebook concept for documentation of different trade-offs \ and decisions during the design process also contributes to making ", StyleBox["Mathematica", FontSlant->"Italic"], " suitable for this kind of work." }], "Text"], Cell[TextData[{ "The application package ", StyleBox["MathCode C++", FontSlant->"Italic"], " adds automatic C++ code generation to ", StyleBox["Mathematica,", FontSlant->"Italic"], " which can be used both to enhance performance of simulations of large \ systems and to generate stand-alone C++ code to be used in applications \ separately from ", StyleBox["Mathematica.", FontSlant->"Italic"], " Performance will be the issue in the seesaw/pendulum example below while \ the stand-alone feature is explored in the fighter aircraft example. The \ stand-alone code generation feature of ", StyleBox["MathCode C++", FontSlant->"Italic"], " makes it possible to design and prototype a controller in ", StyleBox["Mathematica ", FontSlant->"Italic"], "and then \"lift out\" the resulting stand-alone code to be used in the \ real control system. This minimizes the need of coding the control law \ manually from the algorithms designed in ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ "In this document we will try to illustrate the use of ", StyleBox["Mathematica", FontSlant->"Italic"], " together with ", StyleBox["MathCode C++", FontSlant->"Italic"], " for modeling, control system design, and code generation. The document is \ organized as follows. In the rest of this section we give some basic facts \ about dynamic systems needed to formulate controller design problems. In \ Section 2 a nonlinear mechanical system is modeled and simulated using \ generated C++ code. In Section 3 controller design for an aircraft is done \ and in Section 4 we give some conclusions. " }], "Text"], Cell[CellGroupData[{ Cell["Preliminaries", "Subsection"], Cell["\<\ Many dynamic systems can be modeled by a set of first order \ differential equations, see e.g. [1, 2]\ \>", "Text"], Cell[BoxData[{ RowBox[{"\t\t", Cell[TextData[Cell[BoxData[ RowBox[{ RowBox[{Cell[""], \(dx[t]\/dt\)}], " ", "=", " ", \(f[x[t], u[t]]\)}]]]]]}], "\[IndentingNewLine]", RowBox[{"\t \t", Cell[TextData[Cell[BoxData[ \(TraditionalForm\`y[t] = \ g[x[t]]\)]]]]}]}], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(f : \[DoubleStruckCapitalR]\^n\), "TraditionalForm"], "\[Cross]", \(\[DoubleStruckCapitalR]\^m\)}], "\[Rule]", \(\[DoubleStruckCapitalR]\^n\)}], TraditionalForm]]], ". Here ", Cell[BoxData[ \(TraditionalForm\`x[t] \[Element] \[DoubleStruckCapitalR]\^n\)]], ", ", Cell[BoxData[ \(TraditionalForm\`u[t] \[Element] \[DoubleStruckCapitalR]\^m\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`y[t] \[Element] \[DoubleStruckCapitalR]\^p\)]], " are called the ", StyleBox["states", FontSlant->"Italic"], ", ", StyleBox["inputs,", FontSlant->"Italic"], " and ", StyleBox["outputs", FontSlant->"Italic"], " of the system, respectively. Equation (1) is called the ", StyleBox["state space form", FontSlant->"Italic"], " of the equations describing the behavior of the system and is \ particularly useful for control system design. We will also use the \ dot-notation ", Cell[BoxData[ \(TraditionalForm\`x\& . \)]], " for denoting differentiation w.r.t. time ", Cell[BoxData[ \(TraditionalForm\`dx\/dt\)]], ". If the system is linear the state space form can be written as" }], "Text"], Cell[BoxData[ RowBox[{"\t\t", RowBox[{Cell[TextData[Cell[BoxData[ \(dx[t]\/dt = \[ScriptCapitalA]\ . x[t] + \[ScriptCapitalB] . u[t]\)]]]], ",", "\[IndentingNewLine]", "\t\t ", Cell[ TextData[Cell[BoxData[ \(TraditionalForm\`y[t] = \[ScriptCapitalC] . x[t]\)]]]]}]}]], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalA] \[Element] \[DoubleStruckCapitalR]\ \^\(n\[Cross]n\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalB] \[Element] \[DoubleStruckCapitalR]\ \^\(n\[Cross]m\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalC] \[Element] \[DoubleStruckCapitalR]\ \^\(p\[Cross]n\)\)]], " are constant real matrices. If ", Cell[BoxData[ \(TraditionalForm\`f[0, 0] = 0\)]], " in (1) the system can be linearized around x=0, u=0. The \ \[ScriptCapitalA], \[ScriptCapitalB], and \[ScriptCapitalC] matrices are then \ computed from ", Cell[BoxData[ \(TraditionalForm\`f\)]], " and ", Cell[BoxData[ \(TraditionalForm\`g\)]], " by taking partial derivatives as follows" }], "Text"], Cell[BoxData[ RowBox[{"\t\t", RowBox[{Cell[TextData[{ "\[ScriptCapitalA] = ", Cell[BoxData[ RowBox[{\(\[PartialD]f[x, u]\/\[PartialD]x\), SubscriptBox["\[VerticalSeparator]", GridBox[{ {\(x = 0\)}, {\(u = 0\)} }]]}]]] }]], ",", "\[IndentingNewLine]", "\t\t", Cell[TextData[{ "\[ScriptCapitalB] = ", Cell[BoxData[ RowBox[{\(\[PartialD]f[x, u]\/\[PartialD]u\), SubscriptBox["\[VerticalSeparator]", GridBox[{ {\(x = 0\)}, {\(u = 0\)} }]]}]]] }]], ",", "\[IndentingNewLine]", "\t\t", Cell[TextData[{ "\[ScriptCapitalC] = ", Cell[BoxData[ \(\(\(\[PartialD]g[ x]\/\[PartialD]x\)\( \[VerticalSeparator] \_\(x = 0\)\)\)\)]] }]]}]}]], "NumberedEquation"], Cell["\<\ The linearized model is usually a good approximation whenever the \ states and input to the system are small. There exists a large number of \ control design methods for linear systems which makes it desirable to work \ with linear models if possible during controller synthesis. The performance \ of the resulting controller can then be studied in simulations where the \ linear model has been exchanged with the nonlinear one.\ \>", "Text"], Cell["\<\ A linear state-feedback controller is a very simple type of \ controller where the control signal is a linear combination of the states \ that are assumed to be measurable. Hence, a linear state-feedback law has the \ form\ \>", "Text"], Cell[BoxData[ RowBox[{ "\t\t", Cell["u[t] = -\[ScriptCapitalL].x[t]"]}]], "NumberedEquation"], Cell[TextData[{ "There exists many methods for computing the matrix \[ScriptCapitalL] but a \ necessary condition is that the chosen ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\)]], " makes the closed loop system asymptotically stable, which essentially \ means that non-zero states converge to zero. More or less advanced methods \ makes it possible to compute ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\)]], " such that different trade offs between performance and robustness can be \ obtained, see e.g. [1, 4, 5]." }], "Text"], Cell[TextData[{ "A slightly more advanced controller is the linear dynamic controller, \ which includes an internal model of the system to be controlled. A so called \ observer is used to estimated the states ", Cell[BoxData[ \(TraditionalForm\`x[t]\)]], " of the system from measured signals ", Cell[BoxData[ \(TraditionalForm\`y[t]\)]], " and a linear feedback law is used to compute the input to the system to \ be controlled. Using the notation ", Cell[BoxData[ \(TraditionalForm\`x\&^[t]\)]], " for estimated states the linear dynamic controller can be written as \ follows" }], "Text"], Cell[BoxData[ RowBox[{"\t\t", RowBox[{Cell[TextData[Cell[BoxData[ \(\(d x\&^[t]\)\/dt\ = \ \[ScriptCapitalA]\_c\ . x\&^[t] + \[ScriptCapitalB]\_c . y[t]\)]]]], ",", "\[IndentingNewLine]", "\t\t ", Cell[TextData[Cell[BoxData[ \(u[ t]\ = \ \(-\[ScriptCapitalL] . \(x\&^\)[ t]\)\)]]]]}]}]], "NumberedEquation"], Cell["\<\ Hence, a dynamic controller requires real time solving of a system \ of differential equations. In the seesaw/pendulum example below we will use a \ linear state-feedback controller of the form (1) and in the fighter aircraft \ example we will use the observer based controller of the form (5).\ \>", \ "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2 The Seesaw/Pendulum Process", "Section"], Cell[TextData[{ "In this section we will illustrate how the symbolic capabilities of ", StyleBox["Mathematica", FontSlant->"Italic"], " can be used to derive a model of a nonlinear system. Using ", StyleBox["MathCode C++", FontSlant->"Italic"], " the large symbolic expressions in the nonlinear model are then converted \ to C++ code and used to simulate the system very efficiently. The system is a \ laboratory process frequently used in control education known as the \ seesaw/pendulum process. One implementation of the seesaw/pendulum process \ has been developed by Quanser Consulting (", ButtonBox["http://www.quanser.com", ButtonData:>{ URL[ "http://www.quanser.com"], None}, ButtonStyle->"Hyperlink"], ")." }], "Text"], Cell[CellGroupData[{ Cell["The System", "Subsection"], Cell[TextData[{ "The process consists of a seesaw, two carts called ", Cell[BoxData[ \(TraditionalForm\`C\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`C\_2\)]], ", two parallel tracks, an inverted pendulum, and a weight. Each cart can \ be driven by a DC motor controlled by an input voltage. Cart ", Cell[BoxData[ \(TraditionalForm\`C\_1\)]], " carries the weight and cart ", Cell[BoxData[ \(TraditionalForm\`C\_2\)]], " carries the inverted pendulum attached by a friction free joint. The \ carts can be moved along the tracks by controlling the input voltages of the \ DC motors." }], "Text"], Cell[TextData[{ "We start by modeling the open loop system, i.e., without any feedback from \ measured signals. The forces ", Cell[BoxData[ \(TraditionalForm\`F\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`F\_2\)]], " acting on each cart are chosen as inputs. We will use the Lagrangian \ methodology to obtain a nonlinear model of the system and then linearize it. \ The linearized model is used for control system design where the Control \ System Professional (CSP) application package is used. The closed loop system \ are then simulated both within ", StyleBox["Mathematica", FontSlant->"Italic"], " and by external code generated using ", StyleBox["MathCode C++", FontSlant->"Italic"], "." }], "Text"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgWoo000>Ool3000KOol4000JOol3000JOol4000KOol3 000KOol4000JOol3000JOol4000HOol4000JOol4000JOol3000KOol3000J Ool4000JOol4000JOol3000JOol4000IOol4000JOol4000JOol3000JOol4 000IOol4000KOol3000JOol3000KOol3000IOol4000KOol3000JOol4000J Ool3000JOol3000KOol3000JOol4000JOol3000IOol3000KOol4000JOol3 000JOol4000iOol000moo`<001]oo`@001Yoo`<001Yoo`@001]oo`<001]o o`@001Yoo`<001Yoo`@001Qoo`@001Yoo`@001Yoo`<001]oo`<001Yoo`@0 01Yoo`@001Yoo`<001Yoo`@001Uoo`@001Yoo`@001Yoo`<001Yoo`@001Uo o`@001]oo`<001Yoo`<001]oo`<001Uoo`@001]oo`<001Yoo`@001Yoo`<0 01Yoo`<001]oo`<001Yoo`@001Yoo`<001Uoo`<001]oo`@001Yoo`<001Yo o`@003Qoo`0047oo0`006goo10006Woo0`006Woo10006goo0`006goo1000 6Woo0`006Woo100067oo10006Woo10006Woo0`006goo0`006Woo10006Woo 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ogooogoo;goo003oOoooOolGOol00`00Oomoo`08Ool014ic000007Nm0goo 00<8@P00E[D0ogooogoo;Woo003oOoooOolROol00bU:000YBP04Ool00c6< 000aS03oOoooOol^Ool00?mooomoob5oo`04Mkd00000CW<4Ool015Je0000 07Nmogooogoo;Goo003oOoooOolQOol014Ha0008@WNm1Goo00<8@P00CW<0 ogooogoo;Goo003oOoooOolQOol00aS6000aS006Ool00c6<000HaP3oOooo Ool]Ool00?mooomooomooomooeYoo`00ogooogooogooogooFWoo003oOooo OoooOoooOomJOol00?mooomooomooomooeYoo`00ogooogooogooogooFWoo 003oOoooOoooOoooOomJOol00?mooomooomooomooeYoo`00ogooogooogoo ogooFWoo003oOoooOoooOoooOomJOol00?mooomooomooomooeYoo`00ogoo ogooogooogooFWoo003oOoooOoooOoooOomJOol00?mooomooomooomooeYo o`00ogooogooogooogooFWoo0000\ \>"], "Graphics", ImageSize->{407.875, 239.563}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell["Figure 1. The Pendulum/Seesaw Process.", "Caption"], Cell["\<\ In Figure 1 we introduce names of variables and constants \ describing the process.\ \>", "Text"], Cell[TextData[{ "Variables:\n\t", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], "\ttranslation of cart 1 from center of track\n\t", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], "\ttranslation of cart 2 from center of track\n\t\[Theta]\tangle of seesaw \ with vertical\n\t\[Alpha]\tangle of inverted pendulum with normal to track\n\t\ ", Cell[BoxData[ \(TraditionalForm\`F\_1\)]], "\tforce applied to cart 1\n\t", Cell[BoxData[ \(TraditionalForm\`F\_2\)]], "\tforce applied to cart 2" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Constants:\n\tJ\tinertia of seesaw with track at height h\n\t", Cell[BoxData[ \(TraditionalForm\`m\_s\)]], "\tmass of seesaw with track\n\tc\theight of center of gravity of the \ seesaw from pivot point\n\th\theight of track from pivot point\n\t", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], "\tmass of cart 1 (weight cart, on the back track)\n\t", Cell[BoxData[ \(TraditionalForm\`m\_2\)]], "\tmass of cart 2\t (pendulum cart, on the front track)\n\t", Cell[BoxData[ \(TraditionalForm\`m\_p\)]], "\tmass of pendulum\n\t", Cell[BoxData[ \(TraditionalForm\`l\_p\)]], "\tcenter of mass of pendulum rod (half of full length)\n\tg\tgravitational \ acceleration" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The physical values of the above constants are stored as rules in ", StyleBox["Mathematica", FontSlant->"Italic"], " to be used later on in simulations." }], "Text"], Cell[BoxData[ \(physicalvalues := {\ \ J \[Rule] 1.6, m\_s \[Rule] 6.6, c \[Rule] 0.06, h \[Rule] 0.115, m\_1 \[Rule] 0.48 + 0.38, m\_2 \[Rule] 0.48, m\_p \[Rule] 0.2, l\_p \[Rule] 0.61\/2 + 0.03, g \[Rule] 9.81}\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Modeling", "Subsection"], Cell["\<\ Following the Lagrangian methodology [3] the system is divided in a \ number of subsystems whose potential energy and kinetic energy are computed \ in terms of the generalized coordinates introduced in Figure 1 above. The \ Lagrangian, which is the difference between the total kinetic and potential \ energy, can then be used to derive the equations of motion for the \ system.\ \>", "Text"], Cell[CellGroupData[{ Cell["Computation of the Lagrangian", "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell["Coordinates of the center of mass of the seesaw", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(x\_s := c\ Sin[\[Theta][t]]\), "\n", \(y\_s := c\ Cos[\[Theta][t]]\)}], "Input", AspectRatioFixed->True], Cell["The potential and kinetic energies of the seesaw", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(V\_s := m\_s\ g\ y\_s\), "\n", \(T\_s := Simplify[1\/2\ J\ \((\[PartialD]\_t \[Theta][t])\)\^2]\)}], "Input", AspectRatioFixed->True], Cell["Coordinates of the center of track", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(x\_c := h\ Sin[\[Theta][t]]\), "\n", \(y\_c := h\ Cos[\[Theta][t]]\)}], "Input", AspectRatioFixed->True], Cell["Coordinates of cart 1", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(x\_m1 := x\_c + x\_1[t]\ Cos[\[Theta][t]]\), "\n", \(y\_m1 := y\_c - x\_1[t]\ Sin[\[Theta][t]]\)}], "Input", AspectRatioFixed->True], Cell["The potential and kinetic energies of cart 1", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(V\_m1 := m\_1\ g\ y\_m1\), "\n", \(T\_m1 := Simplify[1\/2\ m\_1\ \((\((\[PartialD]\_t x\_m1)\)\^2 + \ \((\[PartialD]\_t y\_m1)\)\^2)\)]\)}], "Input", AspectRatioFixed->True], Cell["Coordinates of cart 2", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(x\_m2 := x\_c + x\_2[t]\ Cos[\[Theta][t]]\), "\n", \(y\_m2 := y\_c - x\_2[t]\ Sin[\[Theta][t]]\)}], "Input", AspectRatioFixed->True], Cell["The potential and kinetic energies of cart 2", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(V\_m2 := m\_2\ g\ y\_m2\), "\n", \(T\_m2 := Simplify[1\/2\ m\_2\ \((\((\[PartialD]\_t x\_m2)\)\^2 + \ \((\[PartialD]\_t y\_m2)\)\^2)\)]\)}], "Input", AspectRatioFixed->True], Cell["Coordinates of the center of mass of the pendulum", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(x\_p := x\_m2 + l\_p\ Sin[\[Alpha][t] + \[Theta][t]]\), "\n", \(y\_p := y\_m2 + l\_p\ Cos[\[Alpha][t] + \[Theta][t]]\)}], "Input", AspectRatioFixed->True], Cell["The potential and kinetic energies of the pendulum", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(V\_p := m\_p\ g\ y\_p\), "\n", \(T\_p := Simplify[1\/2\ m\_p\ \((\((\[PartialD]\_t x\_p)\)\^2 + \ \((\[PartialD]\_t y\_p)\)\^2)\)]\)}], "Input", AspectRatioFixed->True], Cell["The total potential energy", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(V\_tot := V\_s + V\_m1 + V\_m2 + V\_p\)], "Input", AspectRatioFixed->True], Cell["The total kinetic energy", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(T\_tot := T\_s + T\_m1 + T\_m2 + T\_p\)], "Input", AspectRatioFixed->True], Cell["The Lagrangian of the system ", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(L := T\_tot - V\_tot\)], "Input", AspectRatioFixed->True], Cell[" The Lagrangian of the system becomes", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[L]\)], "Input"], Cell[BoxData[ RowBox[{\(1\/2\), " ", RowBox[{"(", RowBox[{\(\(-2\)\ c\ g\ Cos[\[Theta][t]]\ m\_s\), "-", \(2\ g\ m\_1\ \((h\ Cos[\[Theta][t]] - Sin[\[Theta][t]]\ x\_1[t])\)\), "-", \(2\ g\ m\_2\ \((h\ Cos[\[Theta][t]] - Sin[\[Theta][t]]\ x\_2[t])\)\), "-", \(2\ g\ m\_p\ \((h\ Cos[\[Theta][t]] + Cos[\[Alpha][t] + \[Theta][t]]\ l\_p - Sin[\[Theta][t]]\ x\_2[t])\)\), "+", RowBox[{"J", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "2"]}], "+", RowBox[{\(m\_1\), " ", RowBox[{"(", RowBox[{ RowBox[{\((h\^2 + x\_1[t]\^2)\), " ", SuperscriptBox[ RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "2"]}], "+", RowBox[{"2", " ", "h", " ", RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], " ", RowBox[{ SubsuperscriptBox["x", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", SuperscriptBox[ RowBox[{ SubsuperscriptBox["x", "1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "2"]}], ")"}]}], "+", RowBox[{\(m\_2\), " ", RowBox[{"(", RowBox[{ RowBox[{\((h\^2 + x\_2[t]\^2)\), " ", SuperscriptBox[ RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "2"]}], "+", RowBox[{"2", " ", "h", " ", RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], " ", RowBox[{ SubsuperscriptBox["x", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", SuperscriptBox[ RowBox[{ SubsuperscriptBox["x", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "2"]}], ")"}]}], "+", RowBox[{\(m\_p\), " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\((h\ Cos[\[Theta][t]] - Sin[\[Theta][t]]\ x\_2[t])\), " ", RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", RowBox[{\(Cos[\[Alpha][t] + \[Theta][t]]\), " ", \(l\_p\), " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["\[Alpha]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "+", RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], ")"}]}], "+", RowBox[{\(Cos[\[Theta][t]]\), " ", RowBox[{ SubsuperscriptBox["x", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]}], ")"}], "^", "2"}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\((h\ Sin[\[Theta][t]] + Cos[\[Theta][t]]\ x\_2[t])\), " ", RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", RowBox[{\(Sin[\[Alpha][t] + \[Theta][t]]\), " ", \(l\_p\), " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["\[Alpha]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "+", RowBox[{ SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], ")"}]}], "+", RowBox[{\(Sin[\[Theta][t]]\), " ", RowBox[{ SubsuperscriptBox["x", "2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]}], ")"}], "^", "2"}]}], ")"}]}]}], ")"}]}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Computation of the Equations of Motion", "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The equations of motion are given by the partial differential equations \ that ", Cell[BoxData[ \(TraditionalForm\`L\)]], " must satisfy" }], "Text"], Cell[TextData[{ "\t\t", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\(d\/\(d\ t\)\) \[PartialD]L\/\[PartialD]q\& . \_i - \ \[PartialD]L\/\[PartialD]q\_i \[Equal] Q\_i\)], StyleBox[","], StyleBox[" "], StyleBox[\(i = 1\)], StyleBox[","], StyleBox["\[Ellipsis]"], StyleBox[" "], StyleBox[","], StyleBox["n"]}], TraditionalForm]]], " ," }], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`L\)]], " is the Lagrangian, ", Cell[BoxData[ \(TraditionalForm\`q\_i, \ i = 1, \[Ellipsis], n\)]], " are the generalized coordinates, and ", Cell[BoxData[ \(TraditionalForm\`Q\_i, \ i = 1, \[Ellipsis], n\)]], " are the generalized force associated with each coordinate. In this case \ the quadruple ", Cell[BoxData[ \(TraditionalForm\`{q\_1, q\_2, q\_3, q\_4}\)]], " corresponds to ", Cell[BoxData[ \(TraditionalForm\`{x\_1, x\_2, \[Theta], \[Alpha]}\)]], " and ", Cell[BoxData[ \(TraditionalForm\`Q\_1 = F\_1, \ Q\_2 = F\_2, Q\_3 = \(Q\_4 = 0\)\)]], ". We derive each equation and simplify it" }], "Text"], Cell[BoxData[{ \(\(eqn\_1 = Simplify[\[PartialD]\_t\((\[PartialD]\_\(\[PartialD]\_t x\_1[t]\)L)\) \ - \[PartialD]\_\(x\_1[t]\)L == F\_1];\)\), "\n", \(\(eqn\_2 = Simplify[\[PartialD]\_t\((\[PartialD]\_\(\[PartialD]\_t x\_2[t]\)L)\) \ - \[PartialD]\_\(x\_2[t]\)L == F\_2];\)\), "\n", \(\(eqn\_3 = Simplify[\[PartialD]\_t\((\[PartialD]\_\(\[PartialD]\_t \ \[Theta][t]\)L)\) - \[PartialD]\_\(\[Theta][t]\)L == 0];\)\), "\n", \(\(eqn\_4 = Simplify[\[PartialD]\_t\((\[PartialD]\_\(\[PartialD]\_t \ \[Alpha][t]\)L)\) - \[PartialD]\_\(\[Alpha][t]\)L == 0];\)\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "These equations are coupled ordinary differential equations (ODEs) of \ second order in the generalized coordinates ", Cell[BoxData[ \(TraditionalForm\`x\_1, \ \[Theta], \ x\_2, \ \[Alpha]\)]], ". To rewrite these in standard state space form (a system of first order \ ODEs) we introduce the following states:" }], "Text"], Cell[TextData[{ "\t\t", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{"(", GridBox[{ {\(x\_1\), "\[Theta]", \(x\_2\), "\[Alpha]", \(x\& . \_1\), \(\[Theta]\& . \), \(x\& . \ \_2\), \(\[Alpha]\& . \)} }], ")"}]}], TraditionalForm]]], "." }], "Text"], Cell["The inputs to the system are", "Text"], Cell[TextData[{ "\t\t", Cell[BoxData[ FormBox[ RowBox[{"u", "=", Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {\(F\_1\), \(F\_2\)} }], ")"}], TraditionalForm]]]]]}], TraditionalForm]]], "." }], "Text"], Cell[TextData[{ "We observe that the second order time derivatives of the positions and \ angles appears linearly which makes it easy to solve for these in terms the \ states. Second order time derivatives will ", StyleBox["always", FontSlant->"Italic"], " appear linearly in the equations derived from the partial differential \ equation (6) that the Lagrangian of the system has to satisfy. This is due to \ the fact that the Lagrangian only consists of at most first order time \ derivatives and the second order derivatives appear according to the chain \ rule when differentiating the term ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]L\/\[PartialD]\(q\& . \)\_i\)]], " in (6) w.r.t. time." }], "Text"], Cell["Solve for second order derivatives", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(sol = Solve[\ {\ eqn\_1, eqn\_2, eqn\_3, eqn\_4\ }, {\ \[PartialD]\_{t, 2}x\_1[t], \[PartialD]\_{t, 2}x\_2[ t], \[PartialD]\_{t, 2}\[Theta][ t], \[PartialD]\_{t, 2}\[Alpha][t]\ }\ ];\)\)], "Input", AspectRatioFixed->True], Cell["\<\ These solutions will become a part of the right hand sides of the \ differential equations used for simulating the system.\ \>", "Text"], Cell["\<\ We introduce conversion rules to get rid of explicit time \ dependence. This makes some expressions look simpler and takes less \ space.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(ssvariables := {\ \ x\_1[t] \[Rule] x\_1, \[Theta][t] \[Rule] \[Theta], x\_2[t] \[Rule] x\_2, \[Alpha][ t] \[Rule] \[Alpha], \[PartialD]\_t x\_1[t] \[Rule] dx\_1, \[PartialD]\_t \[Theta][t] \[Rule] d\[Theta], \[PartialD]\_t x\_2[t] \[Rule] dx\_2, \[PartialD]\_t \[Alpha][t] \[Rule] d\[Alpha]\ };\)\)], "Input", AspectRatioFixed->True], Cell["\<\ We compute the right hand side of the nonlinear state space form \ (1) of the equations\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "\t\t", Cell[BoxData[ \(TraditionalForm\`x\& . [t] = f[x[t]\ , u[t]]\)]], " " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "which becomes very large! Observe that the first four entries in ", Cell[BoxData[ \(TraditionalForm\`f\)]], " correspond to pure integrations. This allow us to write the model as a \ system of first order differential equations as desired." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"f", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", GridBox[{ {\(\[PartialD]\_t\ x\_1[t]\)}, {\(\[PartialD]\_t \[Theta][t]\)}, {\(\[PartialD]\_t x\_2[t]\)}, {\(\[PartialD]\_t \[Alpha][t]\)}, {\(\[PartialD]\_{t, 2}x\_1[t]\)}, {\(\[PartialD]\_{t, 2}\[Theta][t]\)}, {\(\[PartialD]\_{t, 2}x\_2[t]\)}, {\(\[PartialD]\_{t, 2}\[Alpha][t]\)} }], ")"}], "/.", "sol"}], "/.", "ssvariables"}], "//", "Flatten"}]}]], "Input"], Cell[BoxData[ \({dx\_1, d\[Theta], dx\_2, d\[Alpha], \(-\(\(\(-F\_1\) + m\_1\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_1)\)\)\/m\_1\)\) + h\ \((\((Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-F\_2\) + m\_2\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_2)\) + m\_p\ \((\(-g\)\ Sin[\[Theta]] + \((\(-d\[Alpha]\^2\ \)\ Sin[\[Alpha]] - 2\ d\[Alpha]\ d\[Theta]\ Sin[\[Alpha]] - d\[Theta]\^2\ Sin[\[Alpha]])\)\ l\_p - d\[Theta]\^2\ x\_2)\))\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((\(-g\)\ Sin[\[Alpha] + \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\))\)/\((Cos[\[Alpha]]\ \ l\_p\ m\_p\ \((h\ m\_2 + \((h + Cos[\[Alpha]]\ l\_p)\)\ m\_p)\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\))\) - \((\((Cos[\[Alpha]]\^2\ \ l\_p\%2\ m\_p\%2 - l\_p\%2\ m\_p\ \((m\_2 + m\_p)\))\)\ \((\(-\((Cos[\[Alpha]]\ l\_p\ m\_p\ \ \((\(-F\_2\) + m\_2\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_2)\) + m\_p\ \((\(-g\)\ Sin[\[Theta]] + \ \((\(-d\[Alpha]\^2\)\ Sin[\[Alpha]] - 2\ d\[Alpha]\ d\[Theta]\ \ Sin[\[Alpha]] - d\[Theta]\^2\ Sin[\[Alpha]])\)\ l\_p - d\[Theta]\^2\ x\_2)\))\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((\(-g\)\ Sin[\[Alpha] + \ \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\))\)\)\ \ \((\(-l\_p\)\ m\_1\ m\_p\ \((h\ m\_2 + h\ m\_p + Cos[\[Alpha]]\ l\_p\ m\_p)\)\ \((h\ Cos[\ \[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\) + Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-h\^2\)\ \ m\_1\%2 + m\_1\ \((J + h\^2\ m\_p + 2\ h\ Cos[\[Alpha]]\ l\_p\ m\_p + l\_p\%2\ m\_p + m\_1\ \((h\^2 + x\_1\%2)\) + 2\ Sin[\[Alpha]]\ l\_p\ m\_p\ x\_2 + m\_p\ x\_2\%2 + m\_2\ \((h\^2 + x\_2\%2)\))\))\))\) + \ \((Cos[\[Alpha]]\ l\_p\ m\_p\ \((h\ m\_2 + \((h + Cos[\[Alpha]]\ l\_p)\)\ m\_p)\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\))\)\ \((\(-l\_p\)\ \ m\_1\ m\_p\ \((h\ m\_2 + h\ m\_p + Cos[\[Alpha]]\ l\_p\ m\_p)\)\ \((\(-g\)\ \ Sin[\[Alpha] + \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\) + Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-h\)\ m\_1\ \((\ \(-F\_1\) + m\_1\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_1)\))\) + m\_1\ \((2\ d\[Theta]\ Sin[\[Alpha]]\ \ dx\_2\ l\_p\ m\_p - c\ g\ Sin[\[Theta]]\ m\_s + m\_1\ \((\(-g\)\ h\ Sin[\[Theta]] + \ \((\(-g\)\ Cos[\[Theta]] + 2\ d\[Theta]\ dx\_1)\)\ x\_1)\) + 2\ d\[Theta]\ dx\_2\ m\_p\ x\_2 + d\[Alpha]\^2\ Cos[\[Alpha]]\ l\_p\ \ m\_p\ x\_2 + 2\ d\[Alpha]\ d\[Theta]\ Cos[\[Alpha]]\ l\_p\ m\_p\ x\_2 + m\_2\ \((\(-g\)\ h\ Sin[\[Theta]] + \ \((\(-g\)\ Cos[\[Theta]] + 2\ d\[Theta]\ dx\_2)\)\ x\_2)\) - m\_p\ \((\((d\[Alpha]\^2\ h\ Sin[\ \[Alpha]] + 2\ d\[Alpha]\ d\[Theta]\ h\ Sin[\[Alpha]] + g\ Sin[\[Alpha] + \[Theta]])\)\ l\_p \ + g\ \((h\ Sin[\[Theta]] + Cos[\[Theta]]\ \ x\_2)\))\))\))\))\))\))\)/\((\((J\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\ \ m\_p\%2 + J\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_p\%3 - J\ Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_p\%3 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\%2\ m\_2\ m\_p\%2\ \ x\_1\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\%2\ m\_p\%3\ x\_1\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\%2\ m\_p\%3\ x\_1\%2 \ + Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\%2\ m\_p\%2\ x\_2\%2 + 2\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ \ x\_2\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ x\_2\%2 - Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ l\_p\%3\ m\_1\ m\_2\ \ m\_p\%3\ x\_2\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_p\%4\ x\_2\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_p\%4\ x\_2\%2 - Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ l\_p\%3\ m\_1\ \ m\_p\%4\ x\_2\%2)\)\ \((Cos[\[Alpha]]\ l\_p\ m\_p\ \((h\ m\_2 + \((h + Cos[\[Alpha]]\ l\_p)\)\ m\_p)\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\))\))\))\), \(-\(\((Cos[\ \[Alpha]]\ l\_p\ m\_p\ \((\(-F\_2\) + m\_2\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_2)\) + m\_p\ \((\(-g\)\ Sin[\[Theta]] + \((\(-d\[Alpha]\^2\)\ \ Sin[\[Alpha]] - 2\ d\[Alpha]\ d\[Theta]\ Sin[\[Alpha]] - d\[Theta]\^2\ Sin[\[Alpha]])\)\ l\_p - d\[Theta]\^2\ x\_2)\))\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((\(-g\)\ Sin[\[Alpha] + \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\))\)/\((Cos[\[Alpha]]\ \ l\_p\ m\_p\ \((h\ m\_2 + \((h + Cos[\[Alpha]]\ l\_p)\)\ m\_p)\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\))\)\)\) + \((\((Cos[\[Alpha]]\^2\ \ l\_p\%2\ m\_p\%2 - l\_p\%2\ m\_p\ \((m\_2 + m\_p)\))\)\ \((\(-\((Cos[\[Alpha]]\ l\_p\ m\_p\ \ \((\(-F\_2\) + m\_2\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_2)\) + m\_p\ \((\(-g\)\ Sin[\[Theta]] + \((\(-d\ \[Alpha]\^2\)\ Sin[\[Alpha]] - 2\ d\[Alpha]\ d\[Theta]\ Sin[\[Alpha]] - d\[Theta]\^2\ Sin[\[Alpha]])\)\ l\_p \ - d\[Theta]\^2\ x\_2)\))\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((\(-g\)\ Sin[\[Alpha] + \[Theta]] \ + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\))\)\)\ \((\(-l\ \_p\)\ m\_1\ m\_p\ \((h\ m\_2 + h\ m\_p + Cos[\[Alpha]]\ l\_p\ m\_p)\)\ \((h\ \ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\) + Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-h\^2\)\ m\_1\%2 + m\_1\ \((J + h\^2\ m\_p + 2\ h\ Cos[\[Alpha]]\ l\_p\ m\_p + l\_p\%2\ m\_p + m\_1\ \((h\^2 + x\_1\%2)\) + 2\ Sin[\[Alpha]]\ l\_p\ m\_p\ x\_2 + m\_p\ x\_2\%2 + m\_2\ \((h\^2 + x\_2\%2)\))\))\))\) + \ \((Cos[\[Alpha]]\ l\_p\ m\_p\ \((h\ m\_2 + \((h + Cos[\[Alpha]]\ l\_p)\)\ m\_p)\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\))\)\ \((\(-l\_p\)\ m\_1\ \ m\_p\ \((h\ m\_2 + h\ m\_p + Cos[\[Alpha]]\ l\_p\ m\_p)\)\ \((\(-g\)\ Sin[\ \[Alpha] + \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\) + Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-h\)\ m\_1\ \ \((\(-F\_1\) + m\_1\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_1)\))\) + m\_1\ \((2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2\ \ l\_p\ m\_p - c\ g\ Sin[\[Theta]]\ m\_s + m\_1\ \((\(-g\)\ h\ Sin[\[Theta]] + \ \((\(-g\)\ Cos[\[Theta]] + 2\ d\[Theta]\ dx\_1)\)\ x\_1)\) + 2\ d\[Theta]\ dx\_2\ m\_p\ x\_2 + d\[Alpha]\^2\ Cos[\[Alpha]]\ l\_p\ m\_p\ \ x\_2 + 2\ d\[Alpha]\ d\[Theta]\ Cos[\[Alpha]]\ l\_p\ m\_p\ x\_2 + m\_2\ \((\(-g\)\ h\ Sin[\[Theta]] + \ \((\(-g\)\ Cos[\[Theta]] + 2\ d\[Theta]\ dx\_2)\)\ x\_2)\) - m\_p\ \((\((d\[Alpha]\^2\ h\ \ Sin[\[Alpha]] + 2\ d\[Alpha]\ d\[Theta]\ h\ Sin[\[Alpha]] + g\ Sin[\[Alpha] + \[Theta]])\)\ l\_p \ + g\ \((h\ Sin[\[Theta]] + Cos[\[Theta]]\ \ x\_2)\))\))\))\))\))\))\)/\((\((J\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\ \ m\_p\%2 + J\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_p\%3 - J\ Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_p\%3 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\%2\ m\_2\ m\_p\%2\ x\_1\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\%2\ m\_p\%3\ x\_1\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\%2\ m\_p\%3\ x\_1\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\%2\ m\_p\%2\ x\_2\%2 + 2\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ x\_2\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ x\_2\%2 - Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ l\_p\%3\ m\_1\ m\_2\ \ m\_p\%3\ x\_2\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_p\%4\ x\_2\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_p\%4\ x\_2\%2 - Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ l\_p\%3\ m\_1\ m\_p\%4\ \ x\_2\%2)\)\ \((Cos[\[Alpha]]\ l\_p\ m\_p\ \((h\ m\_2 + \((h + Cos[\[Alpha]]\ l\_p)\)\ m\_p)\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\))\))\), \(-Sec[\[Alpha]]\)\ \((\ \(-g\)\ Sin[\[Alpha] + \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\) + \((Sec[\[Alpha]]\ l\_p\ \ \((\(-\((Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-F\_2\) + m\_2\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_2)\) + m\_p\ \((\(-g\)\ Sin[\[Theta]] + \((\(-d\ \[Alpha]\^2\)\ Sin[\[Alpha]] - 2\ d\[Alpha]\ d\[Theta]\ Sin[\[Alpha]] - d\[Theta]\^2\ Sin[\[Alpha]])\)\ l\_p \ - d\[Theta]\^2\ x\_2)\))\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((\(-g\)\ Sin[\[Alpha] + \[Theta]] \ + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\))\)\)\ \((\(-l\ \_p\)\ m\_1\ m\_p\ \((h\ m\_2 + h\ m\_p + Cos[\[Alpha]]\ l\_p\ m\_p)\)\ \((h\ \ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\) + Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-h\^2\)\ m\_1\%2 + m\_1\ \((J + h\^2\ m\_p + 2\ h\ Cos[\[Alpha]]\ l\_p\ m\_p + l\_p\%2\ m\_p + m\_1\ \((h\^2 + x\_1\%2)\) + 2\ Sin[\[Alpha]]\ l\_p\ m\_p\ x\_2 + m\_p\ x\_2\%2 + m\_2\ \((h\^2 + x\_2\%2)\))\))\))\) + \ \((Cos[\[Alpha]]\ l\_p\ m\_p\ \((h\ m\_2 + \((h + Cos[\[Alpha]]\ l\_p)\)\ m\_p)\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\))\)\ \((\(-l\_p\)\ m\_1\ \ m\_p\ \((h\ m\_2 + h\ m\_p + Cos[\[Alpha]]\ l\_p\ m\_p)\)\ \((\(-g\)\ Sin[\ \[Alpha] + \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\) + Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-h\)\ m\_1\ \ \((\(-F\_1\) + m\_1\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_1)\))\) + m\_1\ \((2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2\ \ l\_p\ m\_p - c\ g\ Sin[\[Theta]]\ m\_s + m\_1\ \((\(-g\)\ h\ Sin[\[Theta]] + \ \((\(-g\)\ Cos[\[Theta]] + 2\ d\[Theta]\ dx\_1)\)\ x\_1)\) + 2\ d\[Theta]\ dx\_2\ m\_p\ x\_2 + d\[Alpha]\^2\ Cos[\[Alpha]]\ l\_p\ m\_p\ \ x\_2 + 2\ d\[Alpha]\ d\[Theta]\ Cos[\[Alpha]]\ l\_p\ m\_p\ x\_2 + m\_2\ \((\(-g\)\ h\ Sin[\[Theta]] + \ \((\(-g\)\ Cos[\[Theta]] + 2\ d\[Theta]\ dx\_2)\)\ x\_2)\) - m\_p\ \((\((d\[Alpha]\^2\ h\ \ Sin[\[Alpha]] + 2\ d\[Alpha]\ d\[Theta]\ h\ Sin[\[Alpha]] + g\ Sin[\[Alpha] + \[Theta]])\)\ l\_p \ + g\ \((h\ Sin[\[Theta]] + Cos[\[Theta]]\ \ x\_2)\))\))\))\))\))\))\)/\((J\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\ m\_p\%2 + J\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_p\%3 - J\ Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_p\%3 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\%2\ m\_2\ m\_p\%2\ x\_1\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\%2\ m\_p\%3\ x\_1\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\%2\ m\_p\%3\ x\_1\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\%2\ m\_p\%2\ x\_2\%2 + 2\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ x\_2\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ x\_2\%2 - Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ \ x\_2\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_p\%4\ x\_2\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_p\%4\ x\_2\%2 - Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ l\_p\%3\ m\_1\ m\_p\%4\ \ x\_2\%2)\) + Sec[\[Alpha]]\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\)\ \((\((Cos[\[Alpha]]\ l\_p\ m\_p\ \ \((\(-F\_2\) + m\_2\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_2)\) + m\_p\ \((\(-g\)\ Sin[\[Theta]] + \((\(-d\[Alpha]\^2\ \)\ Sin[\[Alpha]] - 2\ d\[Alpha]\ d\[Theta]\ Sin[\[Alpha]] - d\[Theta]\^2\ Sin[\[Alpha]])\)\ l\_p - d\[Theta]\^2\ x\_2)\))\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((\(-g\)\ Sin[\[Alpha] + \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\))\)/\((Cos[\[Alpha]]\ \ l\_p\ m\_p\ \((h\ m\_2 + \((h + Cos[\[Alpha]]\ l\_p)\)\ m\_p)\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\))\) - \((\((Cos[\[Alpha]]\^2\ \ l\_p\%2\ m\_p\%2 - l\_p\%2\ m\_p\ \((m\_2 + m\_p)\))\)\ \((\(-\((Cos[\[Alpha]]\ l\_p\ m\_p\ \ \((\(-F\_2\) + m\_2\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_2)\) + m\_p\ \((\(-g\)\ Sin[\[Theta]] + \ \((\(-d\[Alpha]\^2\)\ Sin[\[Alpha]] - 2\ d\[Alpha]\ d\[Theta]\ \ Sin[\[Alpha]] - d\[Theta]\^2\ Sin[\[Alpha]])\)\ l\_p - d\[Theta]\^2\ x\_2)\))\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((\(-g\)\ Sin[\[Alpha] + \ \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\))\)\)\ \ \((\(-l\_p\)\ m\_1\ m\_p\ \((h\ m\_2 + h\ m\_p + Cos[\[Alpha]]\ l\_p\ m\_p)\)\ \((h\ Cos[\ \[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\) + Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-h\^2\)\ \ m\_1\%2 + m\_1\ \((J + h\^2\ m\_p + 2\ h\ Cos[\[Alpha]]\ l\_p\ m\_p + l\_p\%2\ m\_p + m\_1\ \((h\^2 + x\_1\%2)\) + 2\ Sin[\[Alpha]]\ l\_p\ m\_p\ x\_2 + m\_p\ x\_2\%2 + m\_2\ \((h\^2 + x\_2\%2)\))\))\))\) + \ \((Cos[\[Alpha]]\ l\_p\ m\_p\ \((h\ m\_2 + \((h + Cos[\[Alpha]]\ l\_p)\)\ m\_p)\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\))\)\ \((\(-l\_p\)\ \ m\_1\ m\_p\ \((h\ m\_2 + h\ m\_p + Cos[\[Alpha]]\ l\_p\ m\_p)\)\ \((\(-g\)\ \ Sin[\[Alpha] + \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\) + Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-h\)\ m\_1\ \((\ \(-F\_1\) + m\_1\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_1)\))\) + m\_1\ \((2\ d\[Theta]\ Sin[\[Alpha]]\ \ dx\_2\ l\_p\ m\_p - c\ g\ Sin[\[Theta]]\ m\_s + m\_1\ \((\(-g\)\ h\ Sin[\[Theta]] + \ \((\(-g\)\ Cos[\[Theta]] + 2\ d\[Theta]\ dx\_1)\)\ x\_1)\) + 2\ d\[Theta]\ dx\_2\ m\_p\ x\_2 + d\[Alpha]\^2\ Cos[\[Alpha]]\ l\_p\ \ m\_p\ x\_2 + 2\ d\[Alpha]\ d\[Theta]\ Cos[\[Alpha]]\ l\_p\ m\_p\ x\_2 + m\_2\ \((\(-g\)\ h\ Sin[\[Theta]] + \ \((\(-g\)\ Cos[\[Theta]] + 2\ d\[Theta]\ dx\_2)\)\ x\_2)\) - m\_p\ \((\((d\[Alpha]\^2\ h\ Sin[\ \[Alpha]] + 2\ d\[Alpha]\ d\[Theta]\ h\ Sin[\[Alpha]] + g\ Sin[\[Alpha] + \[Theta]])\)\ l\_p \ + g\ \((h\ Sin[\[Theta]] + Cos[\[Theta]]\ \ x\_2)\))\))\))\))\))\))\)/\((\((J\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\ \ m\_p\%2 + J\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_p\%3 - J\ Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_p\%3 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\%2\ m\_2\ m\_p\%2\ \ x\_1\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\%2\ m\_p\%3\ x\_1\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\%2\ m\_p\%3\ x\_1\%2 \ + Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\%2\ m\_p\%2\ x\_2\%2 + 2\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ \ x\_2\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ x\_2\%2 - Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ l\_p\%3\ m\_1\ m\_2\ \ m\_p\%3\ x\_2\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_p\%4\ x\_2\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_p\%4\ x\_2\%2 - Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ l\_p\%3\ m\_1\ \ m\_p\%4\ x\_2\%2)\)\ \((Cos[\[Alpha]]\ l\_p\ m\_p\ \((h\ m\_2 + \((h + Cos[\[Alpha]]\ l\_p)\)\ m\_p)\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\))\))\))\), \ \(-\(\((\(-\((Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-F\_2\) + m\_2\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_2)\) + m\_p\ \((\(-g\)\ Sin[\[Theta]] + \ \((\(-d\[Alpha]\^2\)\ Sin[\[Alpha]] - 2\ d\[Alpha]\ d\[Theta]\ \ Sin[\[Alpha]] - d\[Theta]\^2\ Sin[\[Alpha]])\)\ l\_p - d\[Theta]\^2\ x\_2)\))\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((\(-g\)\ Sin[\[Alpha] + \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\))\)\)\ \ \((\(-l\_p\)\ m\_1\ m\_p\ \((h\ m\_2 + h\ m\_p + Cos[\[Alpha]]\ l\_p\ m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\) + Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-h\^2\)\ m\_1\%2 + m\_1\ \((J + h\^2\ m\_p + 2\ h\ Cos[\[Alpha]]\ l\_p\ m\_p + l\_p\%2\ m\_p + m\_1\ \((h\^2 + x\_1\%2)\) + 2\ Sin[\[Alpha]]\ l\_p\ m\_p\ x\_2 + m\_p\ x\_2\%2 + m\_2\ \((h\^2 + x\_2\%2)\))\))\))\) + \((Cos[\[Alpha]]\ \ l\_p\ m\_p\ \((h\ m\_2 + \((h + Cos[\[Alpha]]\ l\_p)\)\ m\_p)\) - l\_p\ m\_p\ \((m\_2 + m\_p)\)\ \((h\ Cos[\[Alpha]] + l\_p + Sin[\[Alpha]]\ x\_2)\))\)\ \((\(-l\_p\)\ m\_1\ m\_p\ \ \((h\ m\_2 + h\ m\_p + Cos[\[Alpha]]\ l\_p\ m\_p)\)\ \((\(-g\)\ Sin[\ \[Alpha] + \[Theta]] + 2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2 + d\[Theta]\^2\ \((h\ Sin[\[Alpha]] - Cos[\[Alpha]]\ x\_2)\))\) + Cos[\[Alpha]]\ l\_p\ m\_p\ \((\(-h\)\ m\_1\ \((\(-F\_1\) \ + m\_1\ \((\(-g\)\ Sin[\[Theta]] - d\[Theta]\^2\ x\_1)\))\) + m\_1\ \((2\ d\[Theta]\ Sin[\[Alpha]]\ dx\_2\ l\_p\ \ m\_p - c\ g\ Sin[\[Theta]]\ m\_s + m\_1\ \((\(-g\)\ h\ Sin[\[Theta]] + \((\(-g\)\ \ Cos[\[Theta]] + 2\ d\[Theta]\ dx\_1)\)\ x\_1)\) + 2\ d\[Theta]\ dx\_2\ m\_p\ x\_2 + d\[Alpha]\^2\ Cos[\[Alpha]]\ l\_p\ m\_p\ x\_2 \ + 2\ d\[Alpha]\ d\[Theta]\ Cos[\[Alpha]]\ l\_p\ m\_p\ x\_2 + m\_2\ \((\(-g\)\ h\ Sin[\[Theta]] + \((\(-g\)\ \ Cos[\[Theta]] + 2\ d\[Theta]\ dx\_2)\)\ x\_2)\) - m\_p\ \((\((d\[Alpha]\^2\ h\ Sin[\[Alpha]] + 2\ d\[Alpha]\ d\[Theta]\ h\ Sin[\ \[Alpha]] + g\ Sin[\[Alpha] + \[Theta]])\)\ l\_p + g\ \((h\ Sin[\[Theta]] + Cos[\[Theta]]\ \ x\_2)\))\))\))\))\))\)/\((J\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\ m\_p\%2 + J\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_p\%3 - J\ Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_p\%3 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\%2\ m\_2\ m\_p\%2\ x\_1\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\%2\ m\_p\%3\ x\_1\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\%2\ m\_p\%3\ x\_1\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\%2\ m\_p\%2\ x\_2\%2 + 2\ Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ x\_2\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ x\_2\%2 - Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ l\_p\%3\ m\_1\ m\_2\ m\_p\%3\ \ x\_2\%2 + Cos[\[Alpha]]\ l\_p\%3\ m\_1\ m\_p\%4\ x\_2\%2 - Cos[\[Alpha]]\^3\ l\_p\%3\ m\_1\ m\_p\%4\ x\_2\%2 - Cos[\[Alpha]]\ Sin[\[Alpha]]\^2\ l\_p\%3\ m\_1\ m\_p\%4\ \ x\_2\%2)\)\)\)}\)], "Output"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["The Linearized Model", "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Since we will design a controller using methods for linear systems \ we need to linearize the nonlinear state space model of the system around the \ origin. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"x", "=", RowBox[{"(", GridBox[{ {"0", "0", "0", "0", "0", "0", "0", "0"} }], ")"}]}], " ", ",", " ", RowBox[{"u", "=", RowBox[{ RowBox[{"(", GridBox[{ {"0", "0"} }], ")"}], " ", "\[Implies]", " ", \(x\& . [ t] \[TildeTilde] \[ScriptCapitalA]\ . x[t] + \ \[ScriptCapitalB] . \ u[t]\)}]}]}], TraditionalForm]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This can easily be obtained using the ", StyleBox["Linearize", FontFamily->"Courier"], " function in CSP" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(ss = Linearize[ f, {x\_1, \[Theta], x\_2, \[Alpha]}, \n\t\t{{x\_1, 0}, {\[Theta], 0}, {x\_2, 0}, {\[Alpha], 0}, {dx\_1, 0}, {d\[Theta], 0}, {dx\_2, 0}, {d\[Alpha], 0}}, \n\t\t{{F\_1, 0}, {F\_2, 0}}] // Simplify;\)\), "\n", \(Map[MatrixForm, ss]\)}], "Input"], Cell[BoxData[ RowBox[{"StateSpace", "[", RowBox[{ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0", "1", "0", "0", "0"}, {"0", "0", "0", "0", "0", "1", "0", "0"}, {"0", "0", "0", "0", "0", "0", "1", "0"}, {"0", "0", "0", "0", "0", "0", "0", "1"}, {\(-\(\(g\ h\ m\_1\)\/J\)\), \(g - \(c\ g\ h\ m\_s\)\/J\), \ \(-\(\(g\ h\ \((m\_2 + m\_p)\)\)\/J\)\), "0", "0", "0", "0", "0"}, {\(\(g\ m\_1\)\/J\), \(\(c\ g\ m\_s\)\/J\), \(\(g\ \((m\_2 + m\_p)\)\)\/J\), "0", "0", "0", "0", "0"}, {\(-\(\(g\ h\ m\_1\)\/J\)\), \(g - \(c\ g\ h\ m\_s\)\/J\), \ \(-\(\(g\ h\ \((m\_2 + m\_p)\)\)\/J\)\), \(-\(\(g\ m\_p\)\/m\_2\)\), "0", "0", "0", "0"}, {\(-\(\(g\ m\_1\)\/J\)\), \(-\(\(c\ g\ m\_s\)\/J\)\), \ \(-\(\(g\ \((m\_2 + m\_p)\)\)\/J\)\), \(\(g\ \((m\_2 + m\_p)\)\)\/\(l\_p\ m\_2\)\), "0", "0", "0", "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)], ",", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", "0"}, {"0", "0"}, {"0", "0"}, {\(h\^2\/J + 1\/m\_1\), \(h\^2\/J\)}, {\(-\(h\/J\)\), \(-\(h\/J\)\)}, {\(h\^2\/J\), \(h\^2\/J + 1\/m\_2\)}, {\(h\/J\), \(h\/J - 1\/\(l\_p\ m\_2\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)], ",", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0", "0", "0"}, {"0", "1", "0", "0", "0", "0", "0", "0"}, {"0", "0", "1", "0", "0", "0", "0", "0"}, {"0", "0", "0", "1", "0", "0", "0", "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)], ",", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", "0"}, {"0", "0"}, {"0", "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]}], "]"}]], "Output"] }, Open ]], Cell["\<\ We extract the matrices from the linearization and substitute with \ the physical values of the parameters to get numerical entities\ \>", "Text"], Cell[BoxData[ \(\({\[ScriptCapitalA], \[ScriptCapitalB], \[ScriptCapitalC], \ \[ScriptCapitalD]} = \(ss /. StateSpace \[Rule] List\) /. physicalvalues;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\[ScriptCapitalA] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0", "1", "0", "0", "0"}, {"0", "0", "0", "0", "0", "1", "0", "0"}, {"0", "0", "0", "0", "0", "0", "1", "0"}, {"0", "0", "0", "0", "0", "0", "0", "1"}, {\(-0.6063806250000001`\), "9.530782875`", \(-0.47946374999999997`\), "0", "0", "0", "0", "0"}, {"5.272875`", "2.4279749999999996`", "4.16925`", "0", "0", "0", "0", "0"}, {\(-0.6063806250000001`\), "9.530782875`", \(-0.47946374999999997`\), \(-4.0875`\), "0", "0", "0", "0"}, {\(-5.272875`\), \(-2.4279749999999996`\), \(-4.16925`\), "41.48507462686568`", "0", "0", "0", "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[ScriptCapitalB] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", "0"}, {"0", "0"}, {"0", "0"}, {"1.1710563226744186`", "0.008265625`"}, {\(-0.07187500000000001`\), \(-0.07187500000000001`\)}, {"0.008265625`", "2.0915989583333334`"}, {"0.07187500000000001`", \(-6.147030472636817`\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[ScriptCapitalC] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0", "0", "0"}, {"0", "1", "0", "0", "0", "0", "0", "0"}, {"0", "0", "1", "0", "0", "0", "0", "0"}, {"0", "0", "0", "1", "0", "0", "0", "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[ScriptCapitalD] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", "0"}, {"0", "0"}, {"0", "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell["\<\ The linear model of the system allow us to use standard methods for \ controller design. \ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Controller Design", "Subsection"], Cell[TextData[{ "The method for computing the feedback law requires the linear model to be \ controllable. This implies that the system can be stabilized and that the \ method will find a numerical instantiation of ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\)]], " such that this is obtained. To check controllability we use the ", StyleBox["Controllable", FontFamily->"Courier"], " command in CSP." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Controllable[ss /. physicalvalues]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "Compute a full state feedback controller using the LQG-method (", StyleBox["LQRegulatorGains", FontFamily->"Courier"], " is a CSP command)." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\[ScriptCapitalL] = LQRegulatorGains[\[IndentingNewLine]ss /. physicalvalues, \[IndentingNewLine]DiagonalMatrix[{1, 5. , 1, 5. , 0, 0, 0, 0}], \[IndentingNewLine]0.1\ IdentityMatrix[ 2]\ ];\)\), "\n", \(\[ScriptCapitalL] // MatrixForm\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"26.792917634061048`", "52.94001757251486`", "21.36763243632607`", "11.826075752239758`", "7.239817088310864`", "18.30749866807136`", "6.540279309740164`", "2.289195364739435`"}, {\(-7.651828094122264`\), \(-26.790811938538504`\), \ \(-10.07752853452522`\), \(-27.725924950007695`\), \(-1.628894247352639`\), \ \(-8.778166989710856`\), \(-5.388408575785637`\), \(-4.549390017098888`\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "The control law to be used is ", Cell[BoxData[ \(TraditionalForm\`u[t] = \(-\[ScriptCapitalL] . x[t]\)\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`x[t]\)]], " are measurements of the states. This gives the following closed loop \ system to simulate ", Cell[BoxData[ \(TraditionalForm\`x\& . [t] = f\ [x[t]\ , \(-\[ScriptCapitalL] . x[t]\)]\)]], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Simulation and Code Generation", "Subsection"], Cell[CellGroupData[{ Cell["Preliminaries", "Subsubsection"], Cell["We store the states as a vector", "Text"], Cell[BoxData[ \(x\_ss := {x\_1, \[Theta], x\_2, \[Alpha], dx\_1, d\[Theta], dx\_2, d\[Alpha]}\)], "Input"], Cell["\<\ and close the feedback loop which gives the following right hand \ side of the state space equations\ \>", "Text"], Cell[BoxData[ \(\(f1 = f /. Thread[{F\_1, F\_2} \[Rule] \(-\[ScriptCapitalL] . x\_ss\)];\)\)], "Input"], Cell["\<\ Change names of variables to only ASCII characters (needed for code \ generation)\ \>", "Text"], Cell[BoxData[ \(\(Cvariables\ = \[IndentingNewLine]{x\_1 \[Rule] \ x1, \[Theta]\ \[Rule] \ theta, x\_2\ \[Rule] \ x2, \[Alpha]\ \[Rule] \ alpha, \[IndentingNewLine]dx\_1\ \[Rule] \ dx1, d\[Theta]\ \[Rule] \ dtheta, dx\_2\ \[Rule] \ dx2, \ d\[Alpha] \[Rule] \ dalpha};\)\)], "Input"], Cell["\<\ The right hand side of the closed loop state space equations \ (intended for code generation)\ \>", "Text"], Cell[BoxData[ \(\(rhs = \(f1 /. physicalvalues\)\ /. \ Cvariables;\)\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "We define the function ", StyleBox["ClosedLoopPendulumEquationsRHS", FontFamily->"Courier"], ", which has the huge right hand side expression from above as its body. \ The type declarations ", StyleBox["Real ", FontFamily->"Courier"], StyleBox["and", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Courier"], " ", StyleBox["\[Rule] Real[8]", FontFamily->"Courier"], " below are the syntax in ", StyleBox["MathCode C++", FontSlant->"Italic"], " for providing type information. Explicit type declarations is needed to \ generate efficient C++ code. " }], "Text"], Cell[BoxData[ \(\(ClosedLoopPendulumEquationsRHS[\n\tReal\ x1_, Real\ theta_, Real\ x2_, Real\ alpha_, \n\tReal\ dx1_, Real\ dtheta_, Real\ dx2_, Real\ dalpha_] \[Rule] Real[8] = rhs;\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Simulation of the Nonlinear Model within ", StyleBox["Mathematica", FontSlant->"Italic"] }], "Subsubsection"], Cell[TextData[{ "We define ", Cell[BoxData[ \(TraditionalForm\`f\_cl[t]\)]], " to be a short notation for the ClosedLoopPendulumEquationsRHS" }], "Text"], Cell[BoxData[ \(f\_cl[t_] := ClosedLoopPendulumEquationsRHS[x1[t], theta[t], x2[t], alpha[t], dx1[t], dtheta[t], dx2[t], dalpha[t]]\)], "Input"], Cell["The state vector", "Text"], Cell[BoxData[ \(\(x[t_] := {x1[t], theta[t], x2[t], alpha[t], dx1[t], dtheta[t], dx2[t], dalpha[t]};\)\)], "Input"], Cell["The differential equations for the closed loop system", "Text"], Cell[BoxData[ \(deq := Thread[\(x'\)[t] == f\_cl[t]]\)], "Input"], Cell["\<\ We need to specify some initial conditions for the simulation. \ Assume that cart 1 is positioned at -1, the seesaw is horizontal, cart 2 is \ positioned at +1 and the pendulum has a small deviation of 5.7\[Degree] from \ the vertical axis. Furthermore, assume that the system is at rest at these \ positions (all time derivatives are zero).\ \>", "Text"], Cell[BoxData[ \(initeq := Thread[x[0] == {\(-1\), 0, 1, 0.1, 0, 0, 0, 0}]\)], "Input"], Cell[TextData[StyleBox["Nonlinear System Simulation", FontWeight->"Bold"]], "Text"], Cell["We simulate the system using NDSolve.", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(dsolStandard = NDSolve[\ deq\ \[Union] \ initeq, {x1, theta, x2, alpha, dx1, dtheta, dx2, dalpha}, {t, 0, 10}\ ];\) // Timing\), "\n", \(\(NDSolveStandardTime = First[%];\)\)}], "Input"], Cell[BoxData[ \({0.9519999999999982`\ Second, Null}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Linear System Simulation", FontWeight->"Bold"]], "Text"], Cell[BoxData[ \(dlineq := Thread[\(x'\)[ t] == \((\[ScriptCapitalA] - \[ScriptCapitalB] . \ \[ScriptCapitalL])\) . x[t]] // Chop\)], "Input"], Cell["We simulate the linear system using NDSolve.", "Text"], Cell[BoxData[ \(\(dsollin = NDSolve[\ dlineq\ \[Union] \ initeq, {x1, theta, x2, alpha, dx1, dtheta, dx2, dalpha}, {t, 0, 8}\ ];\)\)], "Input"], Cell["\<\ Plot the result for the first 4 state variables from the nonlinear \ simulation together with the result for the linear system.\ \>", "Text"], Cell[BoxData[ \(\(Map[\n\t Plot[#, {t, 0, 8}, PlotRange \[Rule] All, DisplayFunction \[Rule] Identity] &, {Part[ Flatten[x[t] /. dsolStandard], Range[4]], \n\t\tPart[ Flatten[x[t] /. dsollin], Range[4]]} // Transpose\ ];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[GraphicsArray[Partition[%, 2]], Display\n\t\tFunction \[Rule] $DisplayFunction];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.0147151 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NcQ_N`008fmk00<006mkKg/0;6mk2G`0ofmk>Vmk000SKg/00`00Kg]_N`3oKg]_Kg/002=_N`03001_ Nfmk0?m_Nfm_N`008fmk00<006mkKg/0ofmkKfmk003oKg^EKg/00?m_NiE_N`00ofmkUFmk003oKg^E Kg/00?m_NiE_N`00ofmkUFmk003oKg^EKg/00?m_NiE_N`00ofmkUFmk003oKg^EKg/00?m_NiE_N`00 \ \>"], ImageRangeCache->{{{104.813, 508.438}, {452.125, 203.125}} -> {-0.598238, \ 1.02785, 0.00521236, 0.00521236}, {{114.75, 297.438}, {446.188, 333.25}} -> \ {-6.79951, 2.18575, 0.0504292, 0.0119734}, {{315.75, 498.438}, {446.188, \ 333.25}} -> {-18.9112, 0.403541, 0.0543925, 0.00309412}, {{114.75, 297.438}, \ {321.938, 209}} -> {-7.36664, -0.290006, 0.0523359, 0.0113353}, {{315.75, \ 498.438}, {321.938, 209}} -> {-20.0202, 0.00871514, 0.0566173, 0.00187926}}] }, Open ]], Cell["\<\ Figure 2. Initial value response for the linear (red) and nonlinear \ (blue) system. \ \>", "Caption"], Cell["\<\ We notice that the linear response differs significantly from the \ nonlinear one. The difference between the linear and nonlinear response will \ be much smaller if the initial values are chosen to be smaller since the \ linear model only is a good approximation for small values of the states and \ inputs.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["External Simulation of the Nonlinear Model", "Subsubsection"], Cell[TextData[{ "To generate simulation code to be run outside ", StyleBox["Mathematica", FontSlant->"Italic"], " we have to provide a differential equation solver that can be compiled or \ linked when the executable program is generated. In this case we have chosen \ to implement a very simple Runge-Kutta solver i ", StyleBox["Mathematica,", FontSlant->"Italic"], " which will be included in the generated code. Another solution could be \ to use a solver from a publicly available software library like the solvers \ at ", ButtonBox["www.netlib.org", ButtonData:>{ URL[ "http://www.netlib.org"], None}, ButtonStyle->"Hyperlink"], "." }], "Text"], Cell[TextData[StyleBox["Code for External Simulation of the Nonlinear Model", FontWeight->"Bold"]], "Text"], Cell["\<\ Here follows an implementation of a state equation solver using the \ Runge-Kutta method\ \>", "Text"], Cell[BoxData[ \(RK[Integer\ n_, Real\ h_, Real\ t0_, Real[_]\ startv_, Integer\ dimen_, Integer\ dimPlusOne_] \[Rule] Real[n, dimPlusOne] \ := \[IndentingNewLine]Module[{\[IndentingNewLine]Real[dimen]\ {x, k1, k2, k3, k4}, \[IndentingNewLine]Integer\ i, \[IndentingNewLine]Real\ t, \ \[IndentingNewLine]Real[n, dimPlusOne]\ res\ \[IndentingNewLine]}, \[IndentingNewLine]t = t0; \[IndentingNewLine]x = startv; \[IndentingNewLine]res\[LeftDoubleBracket]1, 1\[RightDoubleBracket] = t; \[IndentingNewLine]res\[LeftDoubleBracket]1, 2 | dimPlusOne\[RightDoubleBracket] = x; \[IndentingNewLine]For[ i = 1, \ i \[LessEqual] n - 1, \ \(i++\), \[IndentingNewLine]k1 = h\ ClosedLoopPendulumEquationsRHS[ x\[LeftDoubleBracket]1\[RightDoubleBracket], x\[LeftDoubleBracket]2\[RightDoubleBracket], x\[LeftDoubleBracket]3\[RightDoubleBracket], x\[LeftDoubleBracket]4\[RightDoubleBracket], x\[LeftDoubleBracket]5\[RightDoubleBracket], x\[LeftDoubleBracket]6\[RightDoubleBracket], x\[LeftDoubleBracket]7\[RightDoubleBracket], x\[LeftDoubleBracket]8\[RightDoubleBracket]]; \ \[IndentingNewLine]k2 = h\ ClosedLoopPendulumEquationsRHS[ x\[LeftDoubleBracket]1\[RightDoubleBracket] + k1\[LeftDoubleBracket]1\[RightDoubleBracket]/2, x\[LeftDoubleBracket]2\[RightDoubleBracket] + k1\[LeftDoubleBracket]2\[RightDoubleBracket]/2, x\[LeftDoubleBracket]3\[RightDoubleBracket] + k1\[LeftDoubleBracket]3\[RightDoubleBracket]/2, x\[LeftDoubleBracket]4\[RightDoubleBracket] + k1\[LeftDoubleBracket]4\[RightDoubleBracket]/2, x\[LeftDoubleBracket]5\[RightDoubleBracket] + k1\[LeftDoubleBracket]5\[RightDoubleBracket]/2, x\[LeftDoubleBracket]6\[RightDoubleBracket] + k1\[LeftDoubleBracket]6\[RightDoubleBracket]/2, x\[LeftDoubleBracket]7\[RightDoubleBracket] + k1\[LeftDoubleBracket]7\[RightDoubleBracket]/2, x\[LeftDoubleBracket]8\[RightDoubleBracket] + k1\[LeftDoubleBracket]8\[RightDoubleBracket]/ 2]; \[IndentingNewLine]k3 = h\ ClosedLoopPendulumEquationsRHS[ x\[LeftDoubleBracket]1\[RightDoubleBracket] + k2\[LeftDoubleBracket]1\[RightDoubleBracket]/2, x\[LeftDoubleBracket]2\[RightDoubleBracket] + k2\[LeftDoubleBracket]2\[RightDoubleBracket]/2, x\[LeftDoubleBracket]3\[RightDoubleBracket] + k2\[LeftDoubleBracket]3\[RightDoubleBracket]/2, x\[LeftDoubleBracket]4\[RightDoubleBracket] + k2\[LeftDoubleBracket]4\[RightDoubleBracket]/2, x\[LeftDoubleBracket]5\[RightDoubleBracket] + k2\[LeftDoubleBracket]5\[RightDoubleBracket]/2, x\[LeftDoubleBracket]6\[RightDoubleBracket] + k2\[LeftDoubleBracket]6\[RightDoubleBracket]/2, x\[LeftDoubleBracket]7\[RightDoubleBracket] + k2\[LeftDoubleBracket]7\[RightDoubleBracket]/2, x\[LeftDoubleBracket]8\[RightDoubleBracket] + k2\[LeftDoubleBracket]8\[RightDoubleBracket]/ 2]; \[IndentingNewLine]k4 = h\ ClosedLoopPendulumEquationsRHS[\n\t\t\t\t\t\tx\ \[LeftDoubleBracket]1\[RightDoubleBracket] + k3\[LeftDoubleBracket]1\[RightDoubleBracket], x\[LeftDoubleBracket]2\[RightDoubleBracket] + k3\[LeftDoubleBracket]2\[RightDoubleBracket], x\[LeftDoubleBracket]3\[RightDoubleBracket] + k3\[LeftDoubleBracket]3\[RightDoubleBracket], x\[LeftDoubleBracket]4\[RightDoubleBracket] + k3\[LeftDoubleBracket]4\[RightDoubleBracket], x\[LeftDoubleBracket]5\[RightDoubleBracket] + k3\[LeftDoubleBracket]5\[RightDoubleBracket], x\[LeftDoubleBracket]6\[RightDoubleBracket] + k3\[LeftDoubleBracket]6\[RightDoubleBracket], x\[LeftDoubleBracket]7\[RightDoubleBracket] + k3\[LeftDoubleBracket]7\[RightDoubleBracket], x\[LeftDoubleBracket]8\[RightDoubleBracket] + k3\[LeftDoubleBracket]8\[RightDoubleBracket]]; \ \[IndentingNewLine]x = x + 1/6\ \((k1 + 2\ k2 + 2 k3 + k4)\); \[IndentingNewLine]t = t + h; \[IndentingNewLine]res\[LeftDoubleBracket]i + 1, 1\[RightDoubleBracket] = t; \[IndentingNewLine]res\[LeftDoubleBracket]i + 1, 2 | dimPlusOne\[RightDoubleBracket] = x;\[IndentingNewLine]]; \ \[IndentingNewLine]res\[IndentingNewLine]]\)], "Input"], Cell["We use RK in the following ODE solver", "Text"], Cell[BoxData[ \(ndSolve[initvalue_, maxstepsize_, {x_, xmin_, xmax_}] := \[IndentingNewLine]Module[{n, dimen}, \[IndentingNewLine]n = \ IntegerPart[\((xmax - xmin)\)/maxstepsize] + 1; \[IndentingNewLine]dimen = Length[initvalue]; \[IndentingNewLine]res = RK[n, maxstepsize, xmin, initvalue, dimen, dimen + 1]; \[IndentingNewLine]Array[\((Interpolation[ AppendRows[ Transpose[{res\[LeftDoubleBracket]_, 1\[RightDoubleBracket]}], Transpose[{res\[LeftDoubleBracket]_, # + 1\[RightDoubleBracket]}]]])\) &, \ {8}]\ \[IndentingNewLine]]\)], "Input"], Cell["\<\ To get a correct estimate of the time spent by the external code \ for simulating the system we define the following functions that repeats the \ external simulation n times.\ \>", "Text"], Cell[BoxData[ \(RKRepeated[Integer\ n_, Real\ h_, Real\ t0_, Real[_]\ startv_, Integer\ dimen_, Integer\ dimPlusOne_, Integer\ loops_] \[Rule] Real[n, dimPlusOne] := \[IndentingNewLine]Module[{Real[n, dimPlusOne]\ res}, \[IndentingNewLine]Do[ res = RK[n, h, t0, startv, dimen, dimPlusOne], {loops}]; \ \[IndentingNewLine]res\[IndentingNewLine]]\)], "Input"], Cell[BoxData[ \(ndSolveRepeated[\[IndentingNewLine]initvalue_, maxstepsize_, {x_, xmin_, xmax_}, loops_] := \[IndentingNewLine]Module[{n, dimen}, \[IndentingNewLine]n = \ IntegerPart[\((xmax - xmin)\)/maxstepsize] + 1; \[IndentingNewLine]dimen = Length[initvalue]; \[IndentingNewLine]res = RKRepeated[n, maxstepsize, xmin, initvalue, dimen, dimen + 1, \ loops]; \[IndentingNewLine]Array[\((Interpolation[ AppendRows[ Transpose[{res\[LeftDoubleBracket]_, 1\[RightDoubleBracket]}], Transpose[{res\[LeftDoubleBracket]_, # + 1\[RightDoubleBracket]}]]])\) &, \ {8}]\ \[IndentingNewLine]]\)], "Input"], Cell[TextData[StyleBox["Compilation", FontWeight->"Bold"]], "Text"], Cell["\<\ We are now ready to compile the package. The right hand side of the \ state equations are optimized using common subexpression elimination\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CompilePackage[ EvaluateFunctions \[Rule] {ClosedLoopPendulumEquationsRHS}]\)], "Input"], Cell[BoxData[ \("Successful compilation to C++: 3 function(s)"\)], "Print"] }, Open ]], Cell["The corresponding code can be inspected in a text editor", "Text"], Cell[BoxData[ \(\(Run["\"];\)\)], "Input"], Cell[TextData[{ "All necessary files for building binaries are now generated. The ", StyleBox["MakeBinary", FontFamily->"Courier"], " command can build either a standalone application using the code \ specified by an option ", StyleBox["GenerateMainFileAndFunction", FontFamily->"Courier"], " or a ", StyleBox["MathLink", FontSlant->"Italic"], " version which can be easily installed into ", StyleBox["Mathematica", FontSlant->"Italic"], ". We build the ", StyleBox["MathLink", FontSlant->"Italic"], " version" }], "Text"], Cell[BoxData[ \(\(MakeBinary[];\)\)], "Input"], Cell[TextData[{ "To use the generated code for the simulation we only have to install the \ code using the ", StyleBox["InstallCode", FontFamily->"Courier"], " command from ", StyleBox["MathCode C++", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[StyleBox["External Simulation", FontWeight->"Bold"]], "Text"], Cell[TextData[{ StyleBox["Uncompiled code (computations within ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[")", FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(dsolUncompiled = ndSolve[{\(-1\), 0, 1, 0.1, 0, 0, 0, 0}, 0.01, {t, 0. , 10. }];\) // Timing\), "\n", \(\(ndSolveUncompiledTime = First[%];\)\)}], "Input"], Cell[BoxData[ \({85.14199999999998`\ Second, Null}\)], "Output"] }, Open ]], Cell["We plot the result to verify the Runge-Kutta solver", "Text"], 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Initial value response for the nonlinear system computed \ internally by ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Caption"], Cell[TextData[StyleBox["Compiled code (external computations)", FontWeight->"Bold"]], "Text"], Cell["We install the executable code", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(InstallCode[];\)\)], "Input"], Cell[BoxData[ InterpretationBox[\("Global"\[InvisibleSpace]" is installed."\), SequenceForm[ "Global", " is installed."], Editable->False]], "Print"] }, Open ]], Cell["\<\ The external simulation is performed 200 times to get a accurate \ measure of its timing\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(dsolCompiled = ndSolveRepeated[{\(-1\), 0, 1, 0.1, 0, 0, 0, 0}, 0.01, {t, 0. , 10. }, 200];\) // AbsTime\), "\n", \(\(ndSolveCompiledTime = First[%]/200;\)\)}], "Input"], Cell[BoxData[ \({12.`6.7988\ Second, Null}\)], "Output"] }, Open ]], Cell["We plot the result to verify the compiled Runge-Kutta solver", "Text"], Cell[BoxData[ \(\(Map[ Plot[#[t], {t, 0, 8}, PlotRange \[Rule] All, DisplayFunction \[Rule] Identity, PlotStyle \[Rule] Blue] &, 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Initial value response for the nonlinear system computed \ by the external code. \ \>", "Caption"], Cell["The responses in Figure 3 and Figure 4 are identical.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Performance Comparisons", "Subsubsection"], Cell["\<\ The difference in performance between the uncompiled and compiled \ version of the ODE solver is \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ndSolveUncompiledTime/ndSolveCompiledTime\)], "Input"], Cell[BoxData[ \(1419.033333333333`\)], "Output"] }, Open ]], Cell["\<\ A somewhat unfair comparison between the timings of the built-in \ ODE solver and the compiled solver gives\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NDSolveStandardTime/ndSolveCompiledTime\)], "Input"], Cell[BoxData[ \(15.866666666666637`\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Clean-up", "Subsection"], Cell["Uninstall the code and delete the temporary files. ", "Text"], Cell[BoxData[{ \(\(UninstallCode[];\)\), "\n", \(\(CleanMathCodeFiles[Confirm \[Rule] False, CleanAllBut \[Rule] {}];\)\)}], "Input"], Cell["Remove the functions for which we have generated code.", "Text"], Cell[BoxData[ \(Remove[ClosedLoopPendulumEquationsRHS, RK, ndSolve, RKRepeated, ndSolveRepeated]\)], "Input"], Cell[TextData[{ "Unset the symbol ", Cell[BoxData[ \(TraditionalForm\`f\)]], " storing the right hand side of the state space equations." }], "Text"], Cell[BoxData[ \(\(f =. ;\)\)], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["3 A Fighter Aircraft", "Section"], Cell[TextData[{ "In this section a controller for a linear system is designed using LQG \ techniques, see e.g. [1, 4]. The resulting filter is transformed into \ discrete form and C++ code is generated corresponding to the filter \ equations. A extremely simple simulation is used to compare the generated \ code with the uncompiled ", StyleBox["Mathematica", FontSlant->"Italic"], " functions." }], "Text"], Cell[CellGroupData[{ Cell["The System", "Subsection"], Cell["\<\ We will consider a linearization of a nonlinear aircraft dynamics \ for some specific flight conditions. 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A fighter aircraft.", "Caption"] }, Open ]], Cell[CellGroupData[{ Cell["Modeling", "Subsection"], Cell[TextData[{ "Introduce the following notation:\n\n\t\[Bullet] ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\)]], "\tcourse angle\n\t\[Bullet] ", Cell[BoxData[ \(TraditionalForm\`v\_y\)]], "\tvelocity across the aircraft in the ", Cell[BoxData[ \(TraditionalForm\`y\)]], "-direction\n\t\[Bullet] ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\)]], "\troll angle\n\t\[Bullet] ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_a\)]], "\taileron deflection\n\t\[Bullet] ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_r\)]], "\trudder deflection" }], "Text"], Cell[TextData[{ "and choose the state vector ", Cell[BoxData[ \(TraditionalForm\`x = {v\_y, p, r, \[Phi], \[Psi], \[Delta]\_a, \[Delta]\_r}\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`p \[TildeTilde] \[Phi]\& . \)]], " and ", Cell[BoxData[ \(TraditionalForm\`r \[TildeTilde] \[Psi]\& . \)]], "." }], "Text"], Cell["\<\ Notation for a Linear Time 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Some wind disturbances acting on the aircraft.", "Caption"], Cell["\<\ How will the system respond to these disturbances? 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The response in roll angle, \[Phi] (blue) and course \ angle, \[Psi] (red) to the wind disturbances plotted in Figure 6.\ \>", \ "Caption"], Cell[CellGroupData[{ Cell["An LQ State-Feedback Design", "Subsubsection"], Cell[TextData[{ "We compute a state-feedback, i.e., an ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalL]\)]], " matrix, by minimizing a quadratic criterion with weight matrices ", Cell[BoxData[ \(TraditionalForm\`Q\_1\)]], "and ", Cell[BoxData[ \(TraditionalForm\`Q\_2\)]], ". The quadratic criterion is given by " }], "Text"], Cell[BoxData[ RowBox[{"\t\t", RowBox[{\(min\_\[ScriptCapitalL]\), Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox[\(\[Integral]\_0\%\[Infinity]\), ScriptLevel->0], \(\(x\^T\)[t] \(Q\_1\) x[t]\)}], "+", RowBox[{\(\(u\^T\)[t]\), \(Q\_2\), \(u[t]\), StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]t\), ScriptLevel->0]}]}], TraditionalForm]]], "\t" }]]}]}]], "NumberedEquation"], Cell[TextData[{ "We play around with the matrices ", Cell[BoxData[ \(TraditionalForm\`Q\_1\)]], "and ", Cell[BoxData[ \(TraditionalForm\`Q\_2\)]], " until we are satisfied with the result." }], "Text"], Cell[BoxData[{ \(\(Q\_1 = I\_2;\)\), "\n", \(\(Q\_2 = 0.1 I\_2;\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(L = LQOutputRegulatorGains[ StateSpace[\[ScriptCapitalA], \[ScriptCapitalB], \[ScriptCapitalM], \ \[ScriptCapitalD]\_z], Q\_1, Q\_2]\)], "Input"], Cell[BoxData[ \({{\(-0.004259639421133564`\), 0.4428238066232245`, 0.15574675772480318`, 3.146240145581956`, 0.3697636880741098`, 1.6914932364059445`, 0.03690271334248907`}, {\(-0.007895194385691317`\), 0.058786654759773606`, \(-0.8368376928935927`\), 0.3053818297819882`, \(-3.1405851071069972`\), 0.1426694010612083`, 0.2678200700110251`}}\)], "Output"] }, Open ]], Cell["The closed loop eigenvalues are given by", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalA] - \[ScriptCapitalB] . L // Eigenvalues\) // TableForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(-21.89134196834625`\)}, {\(-20.017272176637324`\)}, {\(\(-9.495761366581114`\) + 11.336054671991217`\ \[ImaginaryI]\)}, {\(\(-9.495761366581114`\) - 11.336054671991217`\ \[ImaginaryI]\)}, {\(\(-2.3517063602268893`\) + 3.7387879485075604`\ \[ImaginaryI]\)}, {\(\(-2.3517063602268893`\) - 3.7387879485075604`\ \[ImaginaryI]\)}, {\(-0.17826749657076055`\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {-21.89134196834625, -20.017272176637324, Complex[ -9.4957613665811138, 11.336054671991217], Complex[ -9.4957613665811138, -11.336054671991217], Complex[ -2.3517063602268893, 3.7387879485075604], Complex[ -2.3517063602268893, -3.7387879485075604], \ -.17826749657076055}]]], "Output"] }, Open ]], Cell["\<\ A necessary and sufficient condition for asymptotic stability of \ the closed loop system is that these eigenvalues belong to the left half \ plane.\ \>", "Text"], Cell["We simulate the closed loop output response:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(plot2 = SimulationPlot[ StateSpace[\[ScriptCapitalA] - \[ScriptCapitalB] . 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A comparison between the response in roll angle, \[Phi] \ (blue) and course angle, \[Psi] (red) to the wind disturbances plotted in \ Figure 6 for the controlled (dashed) and uncontrolled system, respectively.\ \ \>", "Caption"] }, Open ]], Cell[CellGroupData[{ Cell["Kalman Filter", "Subsubsection"], Cell["\<\ Now suppose that the states cannot be measured. Here we will design \ a linear dynamic controller known as a Kalman filter which is used to \ estimate the states from measured signals.\ \>", "Text"], Cell["\<\ Assume that the noise have the following covariance matrices:\ \>", \ "Text"], Cell[BoxData[{ \(\(R\_1 = \(10\^\(-2\)\) I\_4;\)\), "\n", \(\(R\_2 = \(10\^\(-6\)\) I\_2;\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(K = LQEstimatorGains[ StateSpace[\[ScriptCapitalA], \[ScriptCapitalN], \[ScriptCapitalC]], R\_1, R\_2]\)], "Input"], Cell[BoxData[ \(RiccatiSolve::"meig" \(\(:\)\(\ \)\) "Matrix with multiple eigenvalues encountered. Result may be \ inaccurate."\)], "Message"], Cell[BoxData[ \({{\(-55.17197088914429`\), \(-1720.1902449092781`\)}, \ {237.79305636806635`, 15.419148377526918`}, {\(-1.666759868958176`\), 34.82319934923566`}, {21.799074265127732`, 0.5027919829223733`}, {0.5027919829223697`, 8.330282043267214`}, {0.`, 0.`}, {0.`, 0.`}}\)], "Output"] }, Open ]], Cell["\<\ We check the eigenvalues of the dynamic controller to verify that \ it is stable.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalA] - K . \[ScriptCapitalC] // Eigenvalues\) // TableForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(-20.`\)}, {\(-20.`\)}, {\(\(-11.406418423478712`\) + 11.40466243355356`\ \[ImaginaryI]\)}, {\(\(-11.406418423478712`\) - 11.40466243355356`\ \[ImaginaryI]\)}, {\(\(-4.4828114977251925`\) + 5.192706136180229`\ \[ImaginaryI]\)}, {\(\(-4.4828114977251925`\) - 5.192706136180229`\ \[ImaginaryI]\)}, {\(-0.20389646598717645`\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {-.2*^2, -.2*^2, Complex[ -11.406418423478712, 11.404662433553559], Complex[ -11.406418423478712, -11.404662433553559], Complex[ -4.4828114977251925, 5.1927061361802291], Complex[ -4.4828114977251925, -5.1927061361802291], \ -.20389646598717645}]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["A Dynamic Controller", "Subsubsection"], Cell["The Kalman filter equations are", "Text"], Cell[BoxData[ \(\(\(\t\t\)\(x\& . \_e = \[ScriptCapitalA]\ x\_e + \[ScriptCapitalB]\ u \ + \ K \((y - \[ScriptCapitalC]\ x\_e)\)\)\)\)], "NumberedEquation"], Cell["and the computed state-feedback control law is of the form", "Text"], Cell[BoxData[ \(\(\(\t\t\)\(u = \(-L\)\ x + \(L\_r\) \(\(r\)\(.\)\)\)\)\)], \ "NumberedEquation"], Cell[TextData[{ "if we add a reference signal ", Cell[BoxData[ \(TraditionalForm\`r\)]], "." }], "Text"], Cell["\<\ Using the separation principle [1, 4] we can use the estimated \ states in (9) instead of the true states and still get an optimal design. In \ that case (9) and (10) gives the following LQG controller\ \>", "Text"], Cell[BoxData[{ \(\t\t\t\tx\& . \_e = \((\[ScriptCapitalA] - \[ScriptCapitalB]\ L - K\ \[ScriptCapitalC])\) x\_e + \[ScriptCapitalB]\ \(L\_r\) r + K\ y\), "\n", \(\t\t\t\tu = \(-L\)\ x\_e + \(L\_r\) r\)}], "NumberedEquation"], Cell[TextData[{ "Here we choose ", Cell[BoxData[ \(TraditionalForm\`L\_r\)]], " such that the static gain from ", Cell[BoxData[ \(TraditionalForm\`r\)]], " to ", Cell[BoxData[ \(TraditionalForm\`z\)]], " becomes an identity matrix:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(L\_r = \((\[ScriptCapitalM] . \((\[ScriptCapitalB] . L - \ \[ScriptCapitalA])\)\^\(-1\) . \[ScriptCapitalB])\)\^\(-1\); MatrixForm[L\_r]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"3.1518926551851045`", "0.3697636880741078`"}, {"0.3710950070919182`", \(-3.140585107106998`\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell["\<\ We represent the controller given by (11) in state-space \ form\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"contKalmanFilter", "=", RowBox[{"StateSpace", "[", RowBox[{\(\[ScriptCapitalA] - \[ScriptCapitalB] . L - K . \[ScriptCapitalC]\), ",", RowBox[{"BlockMatrix", "[", RowBox[{"(", GridBox[{ {\(\[ScriptCapitalB] . L\_r\), "K"} }], ")"}], "]"}], ",", \(-L\), ",", RowBox[{"BlockMatrix", "[", RowBox[{"(", GridBox[{ {\(L\_r\), \(0\_\(2\[Cross]2\)\)} }], ")"}], "]"}]}], "]"}]}], ";"}]], "Input"], Cell["\<\ The continuous Kalman filter above is transformed to discrete time \ for implementation purposes\ \>", "Text"], Cell[BoxData[ \(\(discKalmanFilter = Chop[ToDiscreteTime[contKalmanFilter, Sampled \[Rule] Period[0.1]], 10\^\(-8\)];\)\)], "Input"], Cell[TextData[{ "We define a simple test signal, which corresponds to a unit step in the \ reference signal for the roll angle, ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\)]], " while keeping the reference signal for the course angle ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\)]], " and the sensors for measuring the output signals ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Psi]\)]], " equal to zero." }], "Text"], Cell[BoxData[ \(\(ry = Table[{1, 0, 0, 0}, {10}]\^T;\)\)], "Input"], Cell[CellGroupData[{ 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The controller outputs ", Cell[BoxData[ \(TraditionalForm\`u\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`u\_2\)]], " when sensor signals are kept to zero and there is a unit step in the \ reference signal for the roll angle, ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\)]], ". " }], "Caption"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Simulation and Code Generation", "Subsection"], Cell[TextData[{ "The system matrices of the discrete Kalman filter are declare as a global \ real constants (type information given to ", StyleBox["MathCode C++", FontSlant->"Italic"], ")" }], "Text"], Cell[BoxData[ \(Declare[\[IndentingNewLine]Constant\ Real[_, _]\ Ac = discKalmanFilter\[LeftDoubleBracket]1\[RightDoubleBracket], \ \[IndentingNewLine]Constant\ Real[_]\ Bc = \ discKalmanFilter\[LeftDoubleBracket]2\[RightDoubleBracket], \ \[IndentingNewLine]Constant\ Real[_, _]\ Cc = discKalmanFilter\[LeftDoubleBracket]3\[RightDoubleBracket], \ \[IndentingNewLine]Constant\ Real\ Dc = discKalmanFilter\[LeftDoubleBracket]4\[RightDoubleBracket]\ \[IndentingNewLine]]\)], "Input"], Cell["\<\ Define the system functions (right hand sides of the controller \ equations).\ \>", "Text"], Cell[BoxData[{ \(\(f[Real[7]\ xe_, \ Real[4]\ in_] \[Rule] \ Real[_] := \ Ac . xe + Bc . in;\)\), "\n", \(\(g[Real[7]\ xe_, \ Real[4]\ in_] \[Rule] Real[_]\ := \ Cc . xe\ + \ Dc . \ in;\)\)}], "Input"], Cell["\<\ We perform a simulation of the controller equations using these \ right hand sides. Initial condition:\ \>", "Text"], Cell[BoxData[ \(\(x\_0 = {0, 0, 0, 0, 0, 0, 0};\)\)], "Input"], Cell[TextData[{ "FoldList can be used to compute an iteration ", Cell[BoxData[ \(TraditionalForm\`x\_\(n + 1\) = f[x\_n, u\_n]\)]], ", i.e., a discrete simulation:" }], "Text"], Cell[BoxData[ \(\(xesim\ = \ FoldList[f, x\_0, ry\^T];\)\)], "Input"], Cell["Remove the last value.", "Text"], Cell[BoxData[ \(\(xesim = \ Drop[xesim, \(-1\)];\)\)], "Input"], Cell["Compute the output from the calculated input and state", "Text"], Cell[BoxData[ \(\(usim\ = \ MapThread[g, {xesim, ry\^T}];\)\)], "Input"], Cell[TextData[{ "A plot of output ", Cell[BoxData[ \(TraditionalForm\`u\_2\)]], " with samples instead of time on the x-axis:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(ListPlot[\((usim\^T)\)\[LeftDoubleBracket]2\[RightDoubleBracket], PlotJoined \[Rule] True, PlotStyle \[Rule] Red];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 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The controller output ", Cell[BoxData[ \(TraditionalForm\`u\_2\)]], " when sensor signals are kept to zero and there is a unit step in the \ reference signal for the roll angle, ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\)]], ". " }], "Caption"], Cell["\<\ Create a loop that will be the main loop in the external \ code.\ \>", "Text"], Cell[BoxData[ \(\(DiscreteSimulate[Real[n_, _]\ u_, Real[_]\ x0_] \[Rule] Real[_, _]\ := \[IndentingNewLine]\ Module[\[IndentingNewLine]{Real[_]\ x = x0, \[IndentingNewLine]Real[n, 2]\ y\ = \ 0, \[IndentingNewLine]Integer\ i}, \[IndentingNewLine]For[ i = 1, i \[LessEqual] \ n, \(i++\), \[IndentingNewLine]y\[LeftDoubleBracket] i\[RightDoubleBracket]\ = \ g[x, u\[LeftDoubleBracket] i\[RightDoubleBracket]]; \[IndentingNewLine]x\ = \ f[x, u\[LeftDoubleBracket] i\[RightDoubleBracket]]\[IndentingNewLine]]; \ \[IndentingNewLine]y\[IndentingNewLine]];\)\)], "Input"], Cell[TextData[{ "From construction, the function f and g can be expanded to remove the \ dot-products and to reduce the computational effort. This is especially good \ when the system matrices contain many zeroes. The expansion is achieved using \ the ", StyleBox["EvaluateFunctions", FontFamily->"Courier"], " option." }], "Text"], Cell["\<\ Generate the code and compile it. The functions f and g will be \ expanded and subexpression optimized before being code generated.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(BuildCode[EvaluateFunctions \[Rule] {f, g}]\)], "Input"], Cell[BoxData[ \("Successful compilation to C++: 3 function(s)"\)], "Print"] }, Open ]], Cell["We inspect the resulting C++ code", "Text"], Cell[BoxData[ \(\(Run["\"];\)\)], "Input"], Cell[TextData[{ "Install the compiled code into ", StyleBox["Mathematica", FontSlant->"Italic"], " and test if the same simulation result as in Figure 11 is achieved." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(InstallCode[];\)\)], "Input"], Cell[BoxData[ InterpretationBox[\("Global"\[InvisibleSpace]" is installed."\), SequenceForm[ "Global", " is installed."], Editable->False]], "Print"] }, Open ]], Cell[BoxData[ \(\(usim2\ = \ DiscreteSimulate[ry\^T, \ x\_0];\)\)], "Input"], Cell[TextData[{ "A comparison between controller outputs computed by ", StyleBox["Mathematica", FontSlant->"Italic"], " and by the external code. 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The same plot as in Figure 11 but computed by the \ external code.\ \>", "Caption"], Cell[TextData[{ "We observe that the external simulation gives the same result as the \ simulation computed within ", StyleBox["Mathematica.", FontSlant->"Italic"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Clean-up", "Subsection"], Cell["Uninstall the code and delete the temporary files. ", "Text"], Cell[BoxData[{ \(\(UninstallCode[];\)\), "\n", \(\(CleanMathCodeFiles[Confirm \[Rule] False, CleanAllBut \[Rule] {}];\)\)}], "Input"], Cell["Delete the temporary directory and all files it contains.", "Text"], Cell[BoxData[{ \(\(SetDirectory[ToFileName["\<..\>"]];\)\), "\[IndentingNewLine]", \(\(DeleteDirectory["\", DeleteContents \[Rule] True];\)\)}], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["4 Conclusions", "Section"], Cell[TextData[{ "In this notebook we have demonstrated the use of ", StyleBox["Mathematica", FontSlant->"Italic"], " in modeling of dynamic systems and controller design. The symbolic \ capabilities of ", StyleBox["Mathematica", FontSlant->"Italic"], " are very useful for deriving dynamic equations according to the \ Lagrangian formalism. Furthermore, rewriting these equations in state-space \ form, suitable for controller design, is also easily done using ", StyleBox["Solve.", FontFamily->"Courier"], " A unique solution is always obtained since the second order derivatives \ to solve for always appear linearly in the equations derived from the partial \ differential equation (6) that the Lagrangian of the system has to satisfy." }], "Text"], Cell[TextData[{ "We have also shown how efficient C++ code for simulation can be generated \ using the application package ", StyleBox["MathCode C++", FontSlant->"Italic"], ". The pendulum/seesaw example is only a toy example compared to many \ industrial applications, which motivates the efforts to be able to \ automatically generate C++ code that can be compiled and run ouside ", StyleBox["Mathematica", FontSlant->"Italic"], " for increased performance." }], "Text"], Cell[TextData[{ "In the fighter aircraft example we illustrated how ", StyleBox["Mathematica", FontSlant->"Italic"], " can be used as a prototyping environment for controller design. First the \ model of the system is analyzed and different controllers can be evaluated. \ When a satisfactory solution has been found ", StyleBox["MathCode C++", FontSlant->"Italic"], " can be used to generate stand-alone C++ code that can be used in the real \ application." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "[1]\tT. Glad and L. Ljung. ", StyleBox["Reglerteori - Flervariabla och olinj\[ADoubleDot]ra metoder", FontSlant->"Italic"], ". Studentlitteratur, 1997.\n\n[2]\tL. Ljung and T. Glad. ", StyleBox["Modeling of Dynamic Systems", FontSlant->"Italic"], ". Prentice Hall, 1994. \n\n[3]\tH. Goldstein. ", StyleBox["Classical Mechanics", FontSlant->"Italic"], ". Addison-Wesley, second edition, 1980.\n\n[4]\tJ. M. Maciejowski. ", StyleBox["Multivariable Feedback Design", FontSlant->"Italic"], ". Electronic Systems Engineering Series. Addison-Wesley, 1989.\n\n[5]\tK. \ Zhou, J. C. Doyle, and K. Glover. ", StyleBox["Robust and Optimal Control", FontSlant->"Italic"], ". 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Cell[StyleData["Subsection"], CellDingbat->"\[FilledSquare]", CellMargins->{{38, 30}, {2, 20}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Times", FontSize->14, FontWeight->"Bold"], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{35, 30}, {0, 20}}], Cell[StyleData["Subsection", "Printout"], CellMargins->{{18, 30}, {0, 10}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{55, 30}, {4, 10}}, CellGroupingRules->{"SectionGrouping", 60}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Subsubsection", FontFamily->"Times", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{31, 30}, {0, 12}}], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{18, 30}, {0, 12}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{55, 10}, {6, 6}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, TextJustification->1, Hyphenation->True, LineSpacing->{1, 2}, FontFamily->"Times"], Cell[StyleData["Text", "Presentation"], CellMargins->{{20, 10}, {6, 6}}, TextAlignment->Left, TextJustification->0, LineSpacing->{1.3, 0}, FontSize->14], Cell[StyleData["Text", "Printout"], CellMargins->{{18, 4}, {4, 4}}, LineSpacing->{1, 3}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Commentary"], CellMargins->{{55, 10}, {2, 6}}, TextJustification->1, Hyphenation->True, LineSpacing->{1, 2}, FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 0.4]], Cell[StyleData["Commentary", "Presentation"], CellMargins->{{60, 30}, {2, 6}}, TextJustification->1, LineSpacing->{1.3, 0}, FontSize->12], Cell[StyleData["Commentary", "Printout"], CellMargins->{{18, 30}, {3, 0}}, LineSpacing->{1, 3}, FontFamily->"Times", FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellFrame->{{1, 1}, {0, 1}}, CellMargins->{{55, 10}, {0, 0}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelPositioning->Automatic, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, FontSize->12, FontWeight->"Bold", Background->GrayLevel[0.966674]], Cell[StyleData["Input", "Presentation"], CellMargins->{{60, 10}, {0, 10}}, Background->GrayLevel[0.850004]], Cell[StyleData["Input", "Printout"], CellMargins->{{55, 10}, {0, 10}}, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->10, Background->GrayLevel[0.850004]] }, Closed]], Cell[StyleData["InlineInput"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[StyleData["Output"], CellFrame->{{1, 1}, {1, 0}}, CellMargins->{{55, 10}, {15, 0}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", FormatType->InputForm, FontSize->12, Background->GrayLevel[0.850004]], Cell[StyleData["Output", "Presentation"], CellMargins->{{60, Inherited}, {10, 0}}], Cell[StyleData["Output", "Printout"], CellMargins->{{55, Inherited}, {10, 0}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["InputOnly"], CellFrame->1, CellMargins->{{55, 10}, {15, 0}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelPositioning->Automatic, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, FontSize->12, FontWeight->"Bold", Background->GrayLevel[0.966674]], Cell[StyleData["InputOnly", "Presentation"], 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CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, ImageMargins->{{35, Inherited}, {Inherited, 0}}, StyleMenuListing->None, Background->GrayLevel[0.850004]], Cell[StyleData["Graphics", "Presentation"], CellMargins->{{60, Inherited}, {0, 0}}, ImageMargins->{{10, 10}, {10, 10}}], Cell[StyleData["Graphics", "Printout"], CellMargins->{{55, Inherited}, {0, 0}}, ImageSize->{0.0625, 0.0625}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontSlant->"Oblique", FontColor->RGBColor[0.6, 0, 0.6]], Cell[StyleData["CellLabel", "Presentation"], CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[StyleData["CellLabel", "Printout"], CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Unique Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Author"], CellMargins->{{20, 30}, {45, 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CellMargins->{{60, 65}, {6, 4}}, FontSize->10], Cell[StyleData["Caption", "Printout"], CellMargins->{{55, 55}, {5, 4}}, LineSpacing->{1, 2}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Reference"], CellMargins->{{24, 40}, {6, 6}}, TextJustification->1, Hyphenation->True, LineSpacing->{1, 0}, FontFamily->"Times"], Cell[StyleData["Reference", "Presentation"], CellMargins->{{20, 40}, {Inherited, 6}}, TextJustification->0, LineSpacing->{1, 4}, FontSize->12], Cell[StyleData["Reference", "Printout"], CellMargins->{{18, 4}, {4, 4}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PictureGroup"], CellFrame->{{1, 1}, {0, 0}}, CellMargins->{{55, Inherited}, {0, 0}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, ShowCellLabel->False, ImageMargins->{{35, Inherited}, {Inherited, 0}}, StyleMenuListing->None, Background->GrayLevel[0.850004]], Cell[StyleData["PictureGroup", "Presentation"], CellMargins->{{60, Inherited}, 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FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ 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TextJustification->1, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", AutoSpacing->False, ScriptLevel->1, ScriptBaselineShifts->{0.6, Automatic}, SingleLetterItalics->False, ZeroWidthTimes->True], Cell[StyleData["ChemicalFormula", "Presentation"], CellMargins->{{60, 10}, {Inherited, 6}}], Cell[StyleData["ChemicalFormula", "Printout"], CellMargins->{{18, 4}, {4, 4}}, LineSpacing->{1, 3}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellMargins->{{55, 10}, {Inherited, 0}}, CellHorizontalScrolling->True, LanguageCategory->"Formula", FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{60, 10}, {Inherited, 6}}], Cell[StyleData["Program", "Printout"], CellMargins->{{18, 4}, {4, 4}}, LineSpacing->{1, 3}, FontSize->9.5] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], Cell["\<\ The following styles are useful for numbered equations, figures, \ etc. They automatically give the cell a FrameLabel containing a reference to \ a particular counter, and also increment that counter.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], CellMargins->{{55, 85}, {Inherited, Inherited}}, CellFrameLabels->{{None, Cell[ TextData[ {"(", CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, CounterIncrements->"NumberedEquation", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedEquation", "Presentation"], CellMargins->{{60, 10}, {Inherited, 6}}, LineSpacing->{1, 0}], Cell[StyleData["NumberedEquation", "Printout"], CellMargins->{{18, 4}, {4, 4}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], CellMargins->{{55, 95}, {Inherited, Inherited}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure ", CounterBox[ "NumberedFigure"]}], FontWeight -> 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StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9], Cell[StyleData["PageNumber"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontWeight->"Bold"], Cell[StyleData["Footer"], TextAlignment->Center, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Notation Package Styles", "Section", GeneratedCell->True, CellTags->"NotationPackage"], Cell["\<\ The cells below define certain styles needed by the Notation \ package. 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