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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 113316, 3506]*) (*NotebookOutlinePosition[ 114103, 3535]*) (* CellTagsIndexPosition[ 114032, 3529]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "Fully Symbolic Design of Glucose Control with ", StyleBox["CSPS", FontSlant->"Italic"], " of ", StyleBox["Mathematica", FontSlant->"Italic"] }], "Title", TextAlignment->Center, FontSize->36], Cell[TextData[StyleBox["Beny\[OAcute], B.-Beny\[OAcute], Z.-Pal\[AAcute]ncz, \ B.-Kov\[AAcute]cs, L. and Szil\[AAcute]gyi L.\nBudapest Technical \ University,Hungary\npalancz@epito.bme.hu\nBudapest, 2003", FontSize->12]], "Author", TextAlignment->Center], Cell[CellGroupData[{ Cell[TextData[StyleBox["Abstract", FontVariations->{"CompatibilityType"->0}]], "Section"], Cell[TextData[{ StyleBox["In", FontFamily->"Times New Roman"], StyleBox[" this case study a fully symbolic design and modelling method are \ presented for blood glucose control of diabetic patiens under intensive \ care. The analysis is based on a modified two compartment model proposed by \ BERGMAN et al. (1). The applied feedback control law decoupling even the \ nonlinear model leads to a fully symbolic solution of the closed loop \ equations. The effectivity of the applied symbolic procedures being mostly \ built-in the new version of ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["Control System Professional Suite (CSPS) Application of \ Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"], StyleBox[" - have been demonstrated for controller design in case of a \ glucose control for treament of diabetes mellitus. The results are in good \ agreement with the earlier presented symbolic-numeric analysis by BENYO et \ al. (7).", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["\n", FontFamily->"Times New Roman", FontSize->10, CharacterEncoding->"WindowsEastEurope"], StyleBox["Keywords:", FontFamily->"Times New Roman", FontWeight->"Bold", CharacterEncoding->"WindowsEastEurope"], StyleBox[" symbolic computation, diabetes mellitus, blood glucose control \ and modeling with ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "From the engineering point of view, treatment of diabetes mellitus can be \ represented by outer control loop to replace the partially or totally failing \ blood-glucose-control system of the human body. To maintain the glucose level \ in a diabetic under intensive care is currently an actively researched topic \ in the field of Biomedical Engineering. Many different models and strategies \ have been designed and applied to the problem SANO (6), FISCHER (3), CANDAS \ (2), JUHASZ (4) and BENYO et al. (7). The authors orientated to BENYO (7), \ considering as the best appropiate model.\nIn this paper symbolic computation \ was used to design multivariable modal control based on the space \ representation of a verified nonlinear model. The computations were carried \ out with ", StyleBox["Mathematica", FontSlant->"Italic"], " Version 4.2, and the article was written in", StyleBox[" Mathematica", FontSlant->"Italic"], " and presented as a live worksheet" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Model equations", "Section"], Cell["\<\ To simulate the insulin-glucouse interaction in human body the following \ two-compartment model was employed:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(deq1 = \(p\_1\) X[t] + p\_2\ h[t] \[Equal] \(X'\)[t]\)], "Input"], Cell[BoxData[ RowBox[{\(h[t]\ p\_2 + p\_1\ X[t]\), "==", RowBox[{ SuperscriptBox["X", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]], "Output"] }, Open ]], Cell["and", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(deq2 = \((p\_3 - X[t])\)\ Y[t] + i[t] + p\_4 \[Equal] \(Y'\)[ t]\)], "Input"], Cell[BoxData[ RowBox[{\(i[t] + p\_4 + \((p\_3 - X[t])\)\ Y[t]\), "==", RowBox[{ SuperscriptBox["Y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]], "Output"] }, Open ]], Cell["\<\ The terms h(t) and i(t) as exogenous insulin and glucose take the impacts on \ glucose level into consideration, X(t) and Y(t) stand for the concentration \ of glucose in the plasma and that of the insulin remote from plasma, \ respectively.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Nonlinear model", "Section"], Cell[TextData[{ StyleBox["In order to handle the problem with ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["CSPS Application", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"], StyleBox[" the following general form of a nonlinear model should be \ considered:", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"] }], "Text"], Cell[BoxData[{ StyleBox[\(x\& . = f \((x, u)\)\), FontFamily->"Times New Roman"], "\[IndentingNewLine]", StyleBox[\(y = g \((x, u)\)\), FontFamily->"Times New Roman"]}], "Text", TextAlignment->Center], Cell[TextData[{ StyleBox["where ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["x(t)", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"], StyleBox[" is the state variable, ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["u(t)", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"], StyleBox[" and ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["y(t)", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"], StyleBox[" are the input and output variables, respectively. ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], "In our case ", StyleBox["the casting is the following. There are two state variables", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"] }], "Text"], Cell[BoxData[ \(\({x\_1, x\_2} = {X[t], Y[t]};\)\)], "Input"], Cell["\<\ The two output variables supposed to be the same as the state variables \ provided the dynamical performances of the measurement and actuator devices \ are considerably faster than that of the system itself.\ \>", "Text"], Cell[BoxData[ \(\({y\_1, y\_2} = {X[t], Y[t]};\)\)], "Input"], Cell["The model has two input variables:", "Text"], Cell[BoxData[ \(\({u\_1, u\_2} = {h[t], i[t]};\)\)], "Input"], Cell[TextData[{ "The function", StyleBox[" f ", FontSlant->"Italic"], "and ", StyleBox["g", FontSlant->"Italic"], " can be defined as:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({f\_1, f\_2} = {deq1[\([1]\)], deq2[\([1]\)]}\)], "Input"], Cell[BoxData[ \({h[t]\ p\_2 + p\_1\ X[t], i[t] + p\_4 + \((p\_3 - X[t])\)\ Y[t]}\)], "Output"] }, Open ]], Cell[BoxData[ \(\({g\_1, g\_2} = {y\_1, y\_2};\)\)], "Input"], Cell[TextData[{ "To design the control for the system , it should be linearized. Now, we \ load the ", StyleBox["CSP2 Application", FontSlant->"Italic"], ":" }], "Text"], Cell[BoxData[ \(<< ControlSystems`\)], "Input", CellTags->"1.1"] }, Closed]], Cell[CellGroupData[{ Cell["Linearization", "Section"], Cell[TextData[{ "The linearization should be carried out at a steady state, namely at (", Cell[BoxData[ FormBox[ StyleBox[\(X0, \ \(Y0\_\(\(,\)\(\ \)\)\) h0, \ i0\), FontSlant->"Italic"], TraditionalForm]]], "):" }], "Text"], Cell[BoxData[ \(\(ControlObjectSS = Linearize[{f\_1, f\_2}, {g\_1, g\_2}, {{X[t], X0}, {Y[t], Y0}}, {{h[t], h0}, {i[t], i0}}];\)\)], "Input"], Cell["\<\ From the state representation, one can get the equation form of the \ linearized system:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ControlObjectSS // EquationForm\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ FormBox[GridBox[{ { RowBox[{\(\[ScriptX]\& . \), "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(p\_1\), "0"}, {\(-Y0\), \(p\_3 - X0\)} }, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(p\_2\), "0"}, {"0", "1"} }, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]}, { RowBox[{"\[ScriptY]", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }, ColumnAlignments->{Decimal}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", "0"} }, ColumnAlignments->{Decimal}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]} }, ColumnAlignments->{"="}, AllowScriptLevelChange->False], "TraditionalForm"], (EquationForm[ StateSpace[ SlotSequence[ 1]]]&)], TraditionalForm]], "Output"] }, Open ]], Cell["or its representation on the frequency domain:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ControlObjectTF = TransferFunction[s, ControlObjectSS] // Simplify\)], "Input"], Cell[BoxData[ \(TransferFunction[ s, {{p\_2\/\(s - p\_1\), 0}, {\(-\(\(Y0\ p\_2\)\/\(\((s - p\_1)\)\ \((s + X0 - p\_3)\)\)\)\), 1\/\(s + X0 - p\_3\)}}]\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Controllability test", "Section"], Cell[TextData[{ StyleBox["First, we test the controllability of the linearized system.", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], "The controllability matrix is" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ControllabilityMatrix[ControlObjectSS]\)], "Input"], Cell[BoxData[ \({{p\_2, 0, p\_1\ p\_2, 0}, {0, 1, \(-Y0\)\ p\_2, \(-X0\) + p\_3}}\)], "Output"] }, Open ]], Cell["\<\ In case of controllable system the rank of this matrix must be equal with \ that of the system matrix\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Rank[%] \[Equal] Rank[ControlObjectSS[\([1]\)]]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["\<\ Therefore the system controllable. This can be checked by the following \ function, too:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Controllable[ControlObjectSS]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Steady state values", "Section"], Cell[TextData[{ StyleBox["The linearization should be carried out", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox[" ", FontFamily->"Times New Roman", FontSize->10, CharacterEncoding->"WindowsEastEurope"], StyleBox["a", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], "t", " steady state, ", StyleBox["X[0] = 0", FontSlant->"Italic"], ", ", StyleBox["h[0] = 0", FontSlant->"Italic"], ", ", StyleBox["i[0] = 0", FontSlant->"Italic"], " and ", StyleBox["Y[0] = Y0", FontSlant->"Italic"], ". " }], "Text"], Cell[BoxData[ \(X0 = \(X[0] = 0\); h0 = \(h[0] = 0\); i0 = \(i[0] = 0\); Y0 = Y[0];\)], "Input"], Cell[TextData[{ "We need to compute", StyleBox["Y0", FontSlant->"Italic"], ".The first model equation is identity,the second one is" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq = deq2[\([1]\)] /. t \[Rule] 0\)], "Input"], Cell[BoxData[ \(p\_4 + p\_3\ Y[0]\)], "Output"] }, Open ]], Cell[TextData[{ "It can be easily solved for ", StyleBox["Y[0]", FontSlant->"Italic"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol = Solve[eq \[Equal] 0, Y[0]]\)], "Input"], Cell[BoxData[ \({{Y[0] \[Rule] \(-\(p\_4\/p\_3\)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Y[0] = Y[0] /. sol[\([1]\)]\)], "Input"], Cell[BoxData[ \(\(-\(p\_4\/p\_3\)\)\)], "Output"] }, Open ]], Cell["Now, the linearized model is ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ControlObjectSS // EquationForm\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ FormBox[GridBox[{ { RowBox[{\(\[ScriptX]\& . \), "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(p\_1\), "0"}, {\(p\_4\/p\_3\), \(p\_3\)} }, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(p\_2\), "0"}, {"0", "1"} }, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]}, { RowBox[{"\[ScriptY]", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }, ColumnAlignments->{Decimal}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", "0"} }, ColumnAlignments->{Decimal}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]} }, ColumnAlignments->{"="}, AllowScriptLevelChange->False], "TraditionalForm"], (EquationForm[ StateSpace[ SlotSequence[ 1]]]&)], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Introducing ", StyleBox["A", FontSlant->"Italic"], " and ", StyleBox["B", FontSlant->"Italic"], " for the matrices as usual, the equillibrum is stable, because the model \ parameters ", StyleBox["p1", FontSlant->"Italic"], " and ", StyleBox["p3", FontSlant->"Italic"], " are negative." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(A = ControlObjectSS[\([1]\)]; MatrixForm[A]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(p\_1\), "0"}, {\(p\_4\/p\_3\), \(p\_3\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(B = ControlObjectSS[\([2]\)]; MatrixForm[B]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(p\_2\), "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Control Design", "Section"], Cell[TextData[{ StyleBox["Let us consider the ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["\[Lambda]1", FontFamily->"Times New Roman Greek", FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"], StyleBox[", ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["\[Lambda]2", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"], StyleBox[" as the eigenvalues of the matrix ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["A-KB", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"], StyleBox[", where ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["K", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"], StyleBox[" is the gain matrix. Then ", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"], StyleBox["K", FontFamily->"Times New Roman", FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"], StyleBox[" can be computed as:", FontFamily->"Times New Roman", CharacterEncoding->"WindowsEastEurope"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(K = Inverse[B] . \((A - DiagonalMatrix[{\[Lambda]1, \[Lambda]2}])\) // Simplify; MatrixForm[K]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(\(-\[Lambda]1\) + p\_1\)\/p\_2\), "0"}, {\(p\_4\/p\_3\), \(\(-\[Lambda]2\) + p\_3\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["We can check this result by computing the eigenvalues:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Eigenvalues[A - K . B] // Simplify\)], "Input"], Cell[BoxData[ \({\[Lambda]1, \[Lambda]2}\)], "Output"] }, Open ]], Cell[TextData[{ "This is a feasible control, because both of the model parameters (", Cell[BoxData[ \(TraditionalForm\`p\_1\)]], "and ", Cell[BoxData[ \(TraditionalForm\`p\_3\)]], ") are negative. " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Closed loop system", "Section"], Cell["The state space form of our closed loop model is:", "Text"], Cell[BoxData[ \(\(CLControlObjectSS = StateFeedbackConnect[ControlObjectSS, K];\)\)], "Input"], Cell["The equation form of this closed loop model is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CLControlObjectSS // EquationForm\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ FormBox[GridBox[{ { RowBox[{\(\[ScriptX]\& . \), "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(p\_1 + \((\[Lambda]1\/p\_2 - p\_1\/p\_2)\)\ p\_2\), "0"}, {"0", "\[Lambda]2"} }, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(p\_2\), "0"}, {"0", "1"} }, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]}, { RowBox[{"\[ScriptY]", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }, ColumnAlignments->{Decimal}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", "0"} }, ColumnAlignments->{Decimal}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]} }, ColumnAlignments->{"="}, AllowScriptLevelChange->False], "TraditionalForm"], (EquationForm[ StateSpace[ SlotSequence[ 1]]]&)], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ It is interesting to see, that the closed loop system is decoupled.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Simulation of the linear closed loop model", "Section"], Cell["\<\ As disturbance, let the initial conditions Xs = X - X0 > 0 and Ys - Y0 > 0, \ then we can compute the system responses even in symbolic form:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(OutputResponse[CLControlObjectSS, {0, 0}, t, InitialConditions \[Rule] {Xs, Ys}] // Simplify\)], "Input"], Cell[BoxData[ \({\[ExponentialE]\^\(t\ \[Lambda]1\)\ Xs, \[ExponentialE]\^\(t\ \ \[Lambda]2\)\ Ys}\)], "Output"] }, Open ]], Cell[TextData[{ "It can be seen, that greater absolute value of \[EDoubleDot]1, \ \[EDoubleDot]2, make the system reach the steady state faster. With other \ words, the quality of the control will be improved with the increase of the \ absolute value of the \[EDoubleDot]'s, however in real cases, the dynamical \ performance ability of the actuators can be the bottle-neck.\nSo, we were \ able to get symbolic results for control design of our system and could draw \ certain conclusions concerning the control performance, which demonstrate one \ of the unique features of the ", StyleBox["Control System Professional Application", FontSlant->"Italic"], "! However this is not the end, but just the end of the beginning. Now, we \ continue our study with further symbolic computations!" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Nonlinear closed loop model", "Section"], Cell["\<\ The variables of the linearized model represent the deviations from the \ steady state instead of the total values. Therefore to get the nonlinear \ closed loop model, one should take into considaration the steady values. So, \ the control vector is:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({h[t], i[t]} = {h0, i0} - K . {X[t] - X0, Y[t] - Y0}\)], "Input"], Cell[BoxData[ \({\(-\(\(\((\(-\[Lambda]1\) + p\_1)\)\ X[ t]\)\/p\_2\)\), \(-\(\(p\_4\ X[ t]\)\/p\_3\)\) - \((\(-\[Lambda]2\) + p\_3)\)\ \((p\_4\/p\_3 + Y[t])\)}\)], "Output"] }, Open ]], Cell["Then our model equations are", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(deq1\)], "Input"], Cell[BoxData[ RowBox[{\(p\_1\ X[t] - \((\(-\[Lambda]1\) + p\_1)\)\ X[t]\), "==", RowBox[{ SuperscriptBox["X", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(deq2\)], "Input"], Cell[BoxData[ RowBox[{\(p\_4 - \(p\_4\ X[t]\)\/p\_3 + \((p\_3 - X[t])\)\ Y[ t] - \((\(-\[Lambda]2\) + p\_3)\)\ \((p\_4\/p\_3 + Y[t])\)\), "==", RowBox[{ SuperscriptBox["Y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]], "Output"] }, Open ]], Cell["\<\ The first equation is independent from the second one, therefore it can be \ solved as a single equation:\ \>", "Text"], Cell[BoxData[ \(Clear[X, X0]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(sol1 = DSolve[{deq1, X[0] \[Equal] X0}, X[t], t]\)], "Input"], Cell[BoxData[ \({{X[t] \[Rule] \[ExponentialE]\^\(0.7`\ t\ p\_1\)\ X0}}\)], "Output"] }, Open ]], Cell["Substituting this result into the second equation:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(deq2m = deq2 /. sol1[\([1]\)] // Simplify\)], "Input"], Cell[BoxData[ RowBox[{\(-\(\(\((\[ExponentialE]\^\(t\ \[Lambda]1\)\ XO - \[Lambda]2)\)\ \ \((p\_4 + p\_3\ Y[t])\)\)\/p\_3\)\), "==", RowBox[{ SuperscriptBox["Y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]], "Output"] }, Open ]], Cell["the solution can be achieved in symbolic form:", "Text"], Cell[BoxData[ \(Clear[Y, Y0]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(sol2 = DSolve[{deq2m, Y[0] \[Equal] Y0}, Y[t], t] // FullSimplify\)], "Input"], Cell[BoxData[ \({{Y[ t] \[Rule] \(\(-p\_4\) + \[ExponentialE]\^\(\(XO - \ \[ExponentialE]\^\(t\ \[Lambda]1\)\ XO + t\ \[Lambda]1\ \[Lambda]2\)\/\ \[Lambda]1\)\ \((Y0\ p\_3 + p\_4)\)\)\/p\_3}}\)], "Output"] }, Open ]], Cell["\<\ Then the time functions of the state variables of the closed loop system \ are\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Xc[t_] = X[t] /. sol1[\([1]\)]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(t\ \[Lambda]1\)\ XO\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Yc[t_] = \((Y[t] /. sol2[\([1]\)])\) // Simplify\)], "Input"], Cell[BoxData[ \(\(\(-p\_4\) + \[ExponentialE]\^\(\(XO - \[ExponentialE]\^\(t\ \ \[Lambda]1\)\ XO + t\ \[Lambda]1\ \[Lambda]2\)\/\[Lambda]1\)\ \((Y0\ p\_3 + p\ \_4)\)\)\/p\_3\)], "Output"] }, Open ]], Cell["The control variables:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(h[t_] = \((h[t] /. X[t] \[Rule] Xc[t])\) // Simplify\)], "Input"], Cell[BoxData[ \(\(\[ExponentialE]\^\(t\ \[Lambda]1\)\ XO\ \((\[Lambda]1 - \ p\_1)\)\)\/p\_2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(i[t_] = \((i[t] /. {X[t] \[Rule] Xc[t], Y[t] \[Rule] Yc[t]})\) // Simplify\)], "Input"], Cell[BoxData[ \(\(\(-\[ExponentialE]\^\(t\ \[Lambda]1\)\)\ XO\ p\_4 + \[ExponentialE]\^\ \(\(XO - \[ExponentialE]\^\(t\ \[Lambda]1\)\ XO + t\ \[Lambda]1\ \[Lambda]2\)\ \/\[Lambda]1\)\ \((\[Lambda]2 - p\_3)\)\ \((Y0\ p\_3 + p\_4)\)\)\/p\_3\)], \ "Output"] }, Open ]], Cell["\<\ Now, let us finish our analysis with a numerical example for the nonlinear \ closed loop control simulation.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Numerical simulation of the nonlinear closed loop model", "Section"], Cell[TextData[{ "Let us consider the following numerical values ", StyleBox["JUHASZ et.al. 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The symbolic results shows, \ that it is possible to carry out the design algorithm fully symbolical way! According to the first results, the system is expected - after the necessary \ further verifications - to provide a useful help to control of blood glucose \ level in diabetics under intensive care, and to the optimisation process of \ diabetic administration.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["References", "Section"], Cell["\<\ [1] Bergman, B.N.-Ider, Y.Z.-Bowden, C.R.-Cobelli C. (1979). Quantitive \ estimation of insulin sensitivity., American Journal of Physiology, Vol. 236, \ pp. 667-677. [2] Candas, B.-Radziuk, J. (1991). An adaptive controller of glycemia based \ on a physiological model of insulin-dependent glucose removal. Ann.Int.Conf. \ of the IEEE Eng. In Medicine and Biology Soc., Vol. 13, No. 5, pp. 2285-2286. [3] Fischer, M.E.-Teo, K.L. (1989). Optimal Insulin Infusion Resulting from a \ Mathematical Model of Blood Glucose Dynamics. IEEE Trans. On BME, Vol. 36, \ No. 3, pp. 479-486. [4] Juh\[AAcute]sz, Cs. - Asztalos, B. (1996). AdASDiM: An Adaptive Control \ Approach to Diabetic Management. Innovation et Technologie en Biologie et \ Medicine, Vol. 17, No. 1. [5] Ogunye, A.B. (1996). Process Control and Symbolic Computation: An \ Overview with Maple V. MapleTech, Vol. 3, No. 1, pp. 94-103. [6] Sano, A. (1986). Adaptive and optimal schemes for control of blood \ glucose levels. Biomedical Measurements, Informatics and Control, Vol. 1, No. \ 1, pp. 16-22. [7] Beny\[OAcute], Z., Pal\[AAcute]ncz, B., Juh\[AAcute]sz, Cs., \ V\[AAcute]rady, P. (1998). Design of Glucose Control via Symbolic \ Computation., Proceedings of the 20th Annual International Conference of the \ IEEE Engineering in Medicine and Biology Society, Vol. 20, No. 6, pp. \ 3116-3119. \ \>", "Text"] }, Open ]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1400}, {0, 977}}, WindowSize->{681, 668}, WindowMargins->{{111, Automatic}, {Automatic, 15}}, StyleDefinitions -> "Classroom.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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