(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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To design the optimal controller, the disturbance rejection LQ \ method based on the minimax differential game is applied", FontFamily->"Times New Roman"], ".", StyleBox[" The critical, minimax value of the scaling parameter ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\[Gamma]\_crit\)]], StyleBox[" is determined by using ", FontFamily->"Times New Roman"], "the M", StyleBox["odified Riccati Control Algebraic (MCARE) equation employing ", FontFamily->"Times New Roman"], "reduced Groebner basis solution on rational field", StyleBox[". The numerical results are in good agreement with those of the \ Control Toolbox of MATLAB. It turned out, that in order to get positive \ definite solution stabilizing the closed loop, \[Gamma] should be greater \ than ", FontFamily->"Times New Roman"], Cell[BoxData[ StyleBox[\(\[Gamma]\_crit\), FontFamily->"Times New Roman"]]], ". ", StyleBox["The", FontFamily->"Times New Roman"], " obtained results are compared with the classical LQ technique on the \ original non-linear system, using a standard meal disturbance situation. ", StyleBox["It is also demonstrated that for \[Gamma] >> ", FontFamily->"Times New Roman"], Cell[BoxData[ StyleBox[\(\[Gamma]\_crit\), FontFamily->"Times New Roman"]]], StyleBox[", the gain matrix approaches the traditional LQ optimal control \ design solution. The symbolic and numerical computations were carried out \ with ", FontFamily->"Times New Roman"], StyleBox["Mathematica ", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["5.2, and with the CSPS Application 2, as well as with MATLAB 6.5.\ \n\n", FontFamily->"Times New Roman"], StyleBox["Keywords", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[": disturbance rejection LQ control, LQ control, glucose-insulin \ control, diabetes mellitus, symbolic computations, ", FontFamily->"Times New Roman"], StyleBox["Mathematica.", FontFamily->"Times New Roman", FontSlant->"Italic"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "Diabetes mellitus is one of the most serious diseases which need to be \ artificially regulated. The newest statistics of the World Health \ Organization (WHO) predate an increase of adult diabetes population from 4% \ (in 2000, meaning 171 million people) to 5.4% (366 million worldwide) by the \ year 2030, [1]. This warns that diabetes could be the \ \[OpenCurlyDoubleQuote]disease of the future\[CloseCurlyDoubleQuote].\nIn \ many biomedical systems, external controller provides the necessary input, \ because the human body could not ensure it. The outer control might be \ partially or fully automatized. The self-regulation has several strict \ requirements, but once it has been designed it permits not only to facilitate \ the patient's life suffering from the disease, but also to optimize (if \ necessary) the amount of the used dosage.\nThe blood-glucose control is one \ of the most difficult control problems to be solved in biomedical \ engineering. One of the main reasons is that patients are extremely diverse \ in their dynamics and in addition their characteristics are time-varying. Due \ to the inexistence of an outer control loop, replacing the partially or \ totally deficient blood-glucose-control system of the human body, patients \ are regulating their glucose level manually. Based on the measured glucose \ levels (obtained from extracted blood samples), they decide on their own what \ is the necessary insulin dosage to be injected. Although this process is \ supervised by doctors (diabetologists), mishandled situations often appear. \ Hyper- (deviation over the basal glucose level) and hypoglycemia (deviation \ under the basal glucose level) are both dangerous cases, but on short term \ the latter is more dangerous, leading for example to coma.\nStarting from the \ late Sixties lot of researchers investigate the problem of the \ glucose-insulin interaction and control. The closed-loop glucose regulation \ as it was several times formulated [2], [3], [4], requires three components \ and the current paper focuses on the last component of them:\n\[CenterDot]\t\ Glucose sensor (already realized even for 10 min. frequent readings - MiniMed \ [5], Glucowatch [6]);\n\[CenterDot]\tInsulin pump, for insulin injection \ (MiniMed [7], Disetronic [8]);\n\[CenterDot]\tControl algorithm, which based \ on the glucose measurements, is able to determine the necessary insulin \ dosage.\nTo design an appropriate control, an adequate model is necessary. In \ the last 50 years several models appeared. The mostly used and also the \ simplest one proved to be the minimal model of Bergman, [9], [10], but its \ shortcoming is its big sensitivity to variance in the parameters. \ Henceforward, the plasma insulin concentration must be known as a function of \ time. Therefore, extensions of this minimal model have been proposed [11], \ [12], [13], [14] trying to capture the changes in patient dynamics of the \ glucose-insulin interaction, particularly with respect to insulin \ sensitivity. Other more general, but more complicated models appeared in the \ literature, [15], [16].\nRegarding the control strategies applied, the \ palette is very wide [17], starting form classical control strategies like \ PID control [18], optimal control [19], to the modern control techniques like \ adaptive control [11], neuro-fuzzy algorithms [20], [21], model predictive \ control [2], [22], [23], but also post-modern control strategies, like ", Cell[BoxData[ \(TraditionalForm\`H\_\[Infinity]\)]], " control [3], [4], [24], ", Cell[BoxData[ \(TraditionalForm\`H\_2\)]], " / ", Cell[BoxData[ \(TraditionalForm\`H\_\[Infinity]\)]], " control [25], [26], \[Mu]-synthesis, [27], Linear Parameter Varying (LPV) \ technique [28].\nThe investigations of [2] discourage the use of a low \ complexity control such as PID, if high level of performance is desired. \ However, probably the best way to approach the problem is to consider the \ system model and the applied control technique together, [17], [29].\nThis \ article presents the ", Cell[BoxData[ \(TraditionalForm\`H\_2\)]], "/", Cell[BoxData[ \(TraditionalForm\`H\_\[Infinity]\)]], " control (disturbance rejection LQ method) of the Bergman minimal model \ [10] in symbolic way using Mathematica 5.2 together with its Control System \ Professional Suite (CSPS). The paper is structured firstly of a brief \ description of the model, which then is reduced to a two-state model \ eliminating the unmeasurable slow state variable. Secondly, the LQ and \ disturbance rejection LQ (minimax control, LQR) methods are presented and the \ symbolic computations are performed to determine a general solution of the \ considered model for these control methods. Finally, the obtained results are \ compared on a standard meal disturbance situation for the original non-linear \ system and checked under MATLAB 6.5 as well.\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Model equations", "Section", TextAlignment->Left], Cell["\<\ Several different models of diabetic systems exist in the literature \ including, for example, the very detailed 21st-order metabolic model of \ Sorensen [16]. However, to have a system that on one hand, can be readily \ handled from the point of view of control design, but on the other hand \ represents the biological process properly, we consider the three- state \ minimal patient model of Bergman [10],\ \>", "Text"], Cell[BoxData[ \(\(deq1NL = \(G'\)[t] \[Equal] \(-p1\)\ G[t] - X[t]\ \((G[t] + GB)\) + h[t];\)\)], "Input"], Cell[BoxData[ \(\(deq2NL = \(X'\)[t] \[Equal] \(-p2\)\ X[t] + p3\ Y[t];\)\)], "Input"], Cell[BoxData[ \(\(deq3NL = \(Y'\)[t] \[Equal] \(-p4\)\ \((Y[t] + YB)\) + i[t]/VL;\)\)], "Input"], Cell[TextData[{ "where the three state variables: \n\tG(t) - Plasma glucose deviation, \ [mg/dL]\n\tX(t) - Remote compartment insulin utilization, [1/min]\n\tY(t) - \ Plasma insulin deviation, [mU/dL]\nThe control variable:\n\ti(t) - Exogenous \ insulin infusion rate, [mU/min]\nThe disturbance is,\n\th(t) - Exogenous \ glucose infusion rate, [mg/dL min]\nThe physical parameters:\n\tGB - Basal \ glucose level, [mg/dL]\n\tYB - Basal insulin level, [mU/dL]\n\tVL - Insulin \ distribution volume, [dL]\nThe model parameters: \n\tp1 - [1/min]\n\tp2 - \ [1/min]\n\tp3 - [dL/m", StyleBox["U ", FontFamily->"Times New Roman"], Cell[BoxData[ \(min\^2\)], FontFamily->"Times New Roman"], StyleBox["]\n\tp4 - [1/min]", FontFamily->"Times New Roman"] }], "Text"], Cell["\<\ The values of the model and physical parameters are from Furler et al [30]:\ \>", "Text"], Cell[BoxData[ \(\(NumericalValues = {p1 \[Rule] 28/1000, p2 \[Rule] 25/1000, p3 \[Rule] 13/100000, p4 -> 5/54, GB -> 110, YB \[Rule] 1.5, VL -> 120};\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Model reduction", "Section", TextAlignment->Left], Cell["\<\ Assuming that the unmeasurable variable X(t), remote compartment insulin \ utilization, is a slow variable, namely X'[t] \[TildeTilde] 0, then\ \>", "Text"], Cell[BoxData[ \(\(X[t] = \(p3\/p2\) Y[t];\)\)], "Input"], Cell["\<\ it can be eliminated by substituting it into the first equation. Now our \ reduced model is\ \>", "Text"], Cell[BoxData[ \(\(deq1NML = \(G'\)[ t] \[Equal] \(-p1\)\ G[t] - \ \(p3\/p2\) Y[t]\ \((G[t] + GB)\) + h[t];\)\)], "Input"], Cell[BoxData[ \(\(deq2NML = \(Y'\)[t] \[Equal] \(-p4\)\ \((Y[t] + YB)\) + i[t]/VL;\)\)], "Input"], Cell["\<\ To design optimal control, the first step is the linearization of the \ nonlinear model, which has the following general form, \t\ \>", "Text"], Cell[BoxData[{ StyleBox[\(x\& . = f \((x, u)\)\), FontFamily->"Times New Roman"], "\[IndentingNewLine]", StyleBox[\(y = g \((x, u)\)\), FontFamily->"Times New Roman"]}], "NumberedEquation", TextAlignment->Center], Cell["In our case", "Text"], Cell[BoxData[{ \(\(f = {deq1NML[\([2]\)], deq2NML[\([2]\)]};\)\), "\[IndentingNewLine]", \(\(x = {G[t], Y[t]};\)\), "\[IndentingNewLine]", \(\(y = x;\)\), "\[IndentingNewLine]", \(\(u = {h[t], i[t]};\)\), "\[IndentingNewLine]", \(\(g = y;\)\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Linearized model ", "Section", TextAlignment->Left], Cell[TextData[{ "In order to linearize our model, the ", StyleBox["Control System", FontSlant->"Italic"], " ", StyleBox["Applicaton", FontSlant->"Italic"], " of ", StyleBox["Mathematica", FontSlant->"Italic"], " should be loaded" }], "Text"], Cell[BoxData[ \(<< ControlSystems`\)], "Input"], Cell[TextData[{ "The linearization will be carried out at the equilibrium point, namely at \ (", StyleBox["X0, Y0, h0, i0", FontSlant->"Italic"], "). The linearized model as control object in state space is" }], "Text"], Cell[BoxData[ \(\(ControlObjectSS = Linearize[f, g, \[IndentingNewLine]{{G[t], G0}, {Y[t], Y0}}, {{h[t], h0}, {i[t], i0}}];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(ControlObjectSS // EquationForm\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ FormBox[GridBox[{ { RowBox[{\(\[ScriptX]\& . \), "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-p1\) - \(p3\ Y0\)\/p2\), \(-\(\(\((G0 + GB)\)\ p3\)\/p2\)\)}, {"0", \(-p4\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", \(1\/VL\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]}, { RowBox[{"\[ScriptY]", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]} }, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "TraditionalForm"], (EquationForm[ StateSpace[ SlotSequence[ 1]]]&)], TraditionalForm]], "Output"] }, Open ]], Cell["The steady state values are", "Text"], Cell[BoxData[ \(h0 = 0; i0 = p4\ YB\ VL; G0 = 0; Y0 = 0;\)], "Input"], Cell["Then", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ControlObjectSS // EquationForm\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ FormBox[GridBox[{ { RowBox[{\(\[ScriptX]\& . \), "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-p1\), \(-\(\(GB\ p3\)\/p2\)\)}, {"0", \(-p4\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", \(1\/VL\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]}, { RowBox[{"\[ScriptY]", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]} }, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "TraditionalForm"], (EquationForm[ StateSpace[ SlotSequence[ 1]]]&)], TraditionalForm]], "Output"] }, Open ]], Cell["or in traditional form", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ControlObjectSS // TraditionalForm\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ FormBox[ SubsuperscriptBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-p1\), \(-\(\(GB\ p3\)\/p2\)\), "1", "0"}, {"0", \(-p4\), "0", \(1\/VL\)}, {"1", "0", "0", "0"}, {"0", "1", "0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, RowLines->{False, True, False}, ColumnLines->{False, True, False}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "TraditionalForm"], FormBox["\[Bullet]", "TraditionalForm"], FormBox[ StyleBox["\<\"\[ScriptCapitalS]\"\>", Editable->False, ShowStringCharacters->False], "TraditionalForm"], MultilineFunction->None], "TraditionalForm"], (StateSpace[ SlotSequence[ 1]]&)], TraditionalForm]], "Output"] }, Open ]], Cell["The system matrices can be easily separated", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(A = %[\([1]\)]; MatrixForm[A]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-p1\), \(-\(\(GB\ p3\)\/p2\)\)}, {"0", \(-p4\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(B = %%[\([2]\)]; MatrixForm[B]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", \(1\/VL\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ Let us check the controllability of the linearized system. The \ controllability matrix is\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(MC = ControllabilityMatrix[ControlObjectSS]; MatrixForm[MC]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", \(-p1\), \(-\(\(GB\ p3\)\/\(p2\ VL\)\)\)}, {"0", \(1\/VL\), "0", \(-\(p4\/VL\)\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "If the rank of this matrix is equal with the rank of matrix ", StyleBox["A", FontSlant->"Italic"], ", then the system can be controlled" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixRank[MC] \[Equal] MatrixRank[A]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["or we can use built in function directly", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Controllable[ControlObjectSS]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Classical LQ control", "Section", TextAlignment->Left], Cell["\<\ It is well-known , that the dynamic of a linear time invariant system can be \ described in the following way \ \>", "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`x\& . \)]], "(t) = A x(t) + B u(t)\n\ny(t) = C x(t) + D u(t)" }], "NumberedEquation", TextAlignment->Center], Cell["\<\ where A,B,C and D are constant matrices. \tUsing classical LQ control, the requirement is to minimize the following \ quadratic cost functional \t\ \>", "Text"], Cell[BoxData[ \(J \((u \((t)\))\) = \ \(1\/2\) \(\[Integral]\_0\%\[Infinity]\( y\^T\) \((t)\)\ Q\ y \((t)\)\) + \(u\^T\) \((t)\)\ R\ u \ \((t)\) \[DifferentialD]t\)], "NumberedEquation", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ "LQ", " ", "attempts", " ", "to", " ", "find", " ", "an", " ", "optimal", " ", "control", " ", SuperscriptBox[ StyleBox["u", FontSlant->"Italic"], "*"], \((t)\)}], ",", " ", \(t\ \[Element] \ \([0, \ \[Infinity]]\)\ \ based\ on\ the\ \ CARE\ \((Control\ Algebraic\ Riccati\ Equation)\)\ \ such\ that\)}], "\[IndentingNewLine]"}]}]], "Text", FontFamily->"Times New Roman"], Cell[BoxData[ \(J \((\(u\^*\) \((t)\))\) \[LessEqual] J \((u \((t)\))\)\)], "NumberedEquation", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "\nfor all ", StyleBox["u", FontSlant->"Italic"], "(t) on t \[Element] [0, \[Infinity]] under properly choosen ", StyleBox["R", FontSlant->"Italic"], " and ", StyleBox["Q", FontSlant->"Italic"], " matrices. " }], "Text"], Cell[TextData[{ "The first component of input vector ", StyleBox["u(t)", FontSlant->"Italic"], ", the exogenous glucose (", StyleBox["h(t)", FontSlant->"Italic"], StyleBox[")", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontSlant->"Italic"], "stands for disturbance, therefore its effect on the output should be \ minimized. As a result, the disturbance should be overweighted. Consequently \ ", Cell[BoxData[ \(TraditionalForm\`R\_11\)]], " should be considerable greater than ", Cell[BoxData[ \(TraditionalForm\`R\_22\)]], ". We choose the following matrices, ", StyleBox["R", FontSlant->"Italic"], " is" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(R = \ DiagonalMatrix[{10\^3, \ 10\^\(-3\)}]; Q = \ \(10\^\(-3\)\) IdentityMatrix[2]; \ MatrixForm[R]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1000", "0"}, {"0", \(1\/1000\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "and considering that in the objective function the portion of the injected \ insulin is important and the values of the state variables play not an \ important role, the ", StyleBox["Q", FontSlant->"Italic"], " matrix is choosen as" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[Q]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/1000\), "0"}, {"0", \(1\/1000\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["Then the optimal gain matrix is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(KLQ = LQRegulatorGains[ControlObjectSS /. NumericalValues, Q, R]; MatrixForm[KLQ] // N\)], "Input", CellTags->"10.1"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0.000013958065111811256`", \(-0.00005604828968540799`\)}, {\(-0.4670690807117331`\), "2.621073424234561`"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellTags->"10.1"] }, Open ]], Cell["The first row in K can be neglected", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(KLQ = Chop[%, 0.001]; MatrixForm[KLQ]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {\(-0.4670690807117331`\), "2.621073424234561`"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ In order to check this result, it was compared with the KLQ computed by \ MATLAB:\ \>", "Text"], Cell[BoxData[ RowBox[{"(", GridBox[{ {"0", "0"}, {\(-0.4671\), "2.6211"} }], ")"}]], "Text", TextAlignment->Center, FontFamily->"Times New Roman"] }, Open ]], Cell[CellGroupData[{ Cell["Disturbance rejection LQ control (minimax control)", "Section", TextAlignment->Left], Cell[TextData[{ "The disturbance rejection method, LQR (LQ rejection) represents a \ generalization of the classical LQ method and is based on the minimax \ criteria, where the system dynamics is generally described as before. \ However, now the disturbance ", StyleBox["d(t)", FontSlant->"Italic"], " is separeted from active control input ", Cell[BoxData[ \(TraditionalForm\`u\&_\)]], "(t) and could be considered as unmeasurable, namely" }], "Text"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{\(\(x\& . \) \((t)\)\), "=", RowBox[{\(A\ x \((t)\)\), "+", RowBox[{"B", " ", OverscriptBox[ StyleBox["u", FontSlant->"Italic"], "_"], \((t)\)}], " ", "+", \(L\ d \((t)\)\)}]}], "\[IndentingNewLine]", "\n", \(y \((t)\) = C\ x \((t)\) + D\ u \((t)\)\)}]}]], "NumberedEquation", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ Therefore, in this case the quadratic cost functional will be modified with \ the disturbance explicitly, [31], \ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(J \((u \((t)\))\)\), "=", " ", RowBox[{\(\(1\/2\) \(\[Integral]\_0\%\[Infinity]\( y\^T\) \((t)\)\ \ y \((t)\)\)\), "+", RowBox[{ SuperscriptBox[ OverscriptBox[ StyleBox["u", FontSlant->"Italic"], "_"], "T"], \((t)\), " ", StyleBox[\(u\&_\), FontSlant->"Italic"], \((t)\)}], "-", \(\[Gamma]\^2\ \(d\^T\) \((t)\)\ d \((t)\)\ \ \[DifferentialD]t\)}]}], "\[IndentingNewLine]"}]], "NumberedEquation", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "Now, the disturbace - while it appears with negative sign - attempts to \ maximize the cost, while we want to find a control ", Cell[BoxData[ \(TraditionalForm\`\(\(u\&_\)\(\ \)\((t)\)\(\ \)\(that\)\(\ \ \)\(minimizes\)\(\ \)\(the\)\(\ \)\(maximum\)\(\ \)\(cost\)\(\ \)\)\)]], "achievable by the disturbance (by worst case disturbace). This is a case \ of so called \"worst case\" design and leads to the formulation of a min-max \ differential game, [32], \n" }], "Text"], Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{\(\(\(max\)\(\ \ \)\)\+\(d \((t)\)\)\), " ", "J", RowBox[{"(", RowBox[{ RowBox[{ OverscriptBox[ StyleBox["u", FontSlant->"Italic"], "_"], \((t)\)}], ",", \(d \((t)\)\)}], ")"}]}], "\[Rule]", " ", RowBox[{ UnderscriptBox["min", RowBox[{ OverscriptBox[ StyleBox["u", FontSlant->"Italic"], "_"], \((t)\)}]], " ", \(J \((u \((t)\), d \((t)\))\)\)}]}]}]], "NumberedEquation", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{"where", StyleBox[" ", FontSlant->"Italic"], StyleBox[\(u\&_\), FontSlant->"Italic"], StyleBox[\((t)\), FontSlant->"Italic"], " ", "and", " ", StyleBox["d", FontSlant->"Italic"], StyleBox[\((t)\), FontSlant->"Italic"], " ", "are", " ", "satisfying", " ", "the", " ", "state", " ", \(equation . \ It\), " ", "can", " ", "be", " ", "demonstrated", " ", "that", " ", "the", " ", "unique", " ", "solution", " ", "of", " ", "the", " ", "differential", " ", "game"}], ",", " ", RowBox[{ RowBox[{"{", StyleBox[\(\(u\&_\^*\) \((t)\), \ \ \(d\^*\) \((t)\)\), FontSlant->"Italic"], " ", "}"}], " ", "exists", " ", "and", " ", "satisfies", " ", "the", " ", "saddle", " ", "point", " ", "condition"}], ",", "\[IndentingNewLine]"}]}]], "Text", FontFamily->"Times New Roman"], Cell[BoxData[ RowBox[{ RowBox[{"J", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox[ OverscriptBox[ StyleBox["u", FontSlant->"Italic"], "_"], "*"], \((t)\)}], ",", \(d \((t)\)\)}], ")"}]}], "\[LessEqual]", " ", RowBox[{"J", RowBox[{"(", RowBox[{ RowBox[{ OverscriptBox[ StyleBox["u", FontSlant->"Italic"], "_"], \((t)\)}], ",", \(d \((t)\)\)}], ")"}]}], "\[LessEqual]", RowBox[{"J", RowBox[{"(", RowBox[{ RowBox[{ OverscriptBox[ StyleBox["u", FontSlant->"Italic"], "_"], \((t)\)}], ",", \(\(d\^*\) \((t)\)\)}], ")"}]}]}]], "NumberedEquation", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "\nwhere ", Cell[BoxData[ StyleBox[\(\(u\&_\^*\) \((t)\)\), FontSlant->"Italic"]], FontFamily->"Times New Roman"], " is the optimal control and ", Cell[BoxData[ RowBox[{" ", RowBox[{ StyleBox[\(d\^*\), FontSlant->"Italic"], RowBox[{ StyleBox["(", FontSlant->"Italic"], StyleBox["t", FontSlant->"Italic"], ")"}]}]}]], FontFamily->"Times New Roman"], " is the worst-case disturbance. These functions can be computed as\n" }], "Text"], Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ StyleBox[\(u\&_\), FontSlant->"Italic"], "*"], \((t)\)}], "=", \(\(-B\^T\)\ P\ x \((t)\)\)}], "\[IndentingNewLine]", "\[IndentingNewLine]", \(\(d\^*\) \((t)\)\ = \ \(1\/\[Gamma]\^2\) L\^T\ x \((t)\)\)}]}]], "NumberedEquation", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ where P is the positive definite symmetric solution of the Modified Control \ Algebraic Riccati Equation (MCARE) \ \>", "Text"], Cell[BoxData[ \(P\ A\ + \ A\^T\ P + \(C\^T\) C - P\ \((B\ B\^T - \(1\/\[Gamma]\^2\) L\ L\^T)\)\ P = 0\)], "NumberedEquation", TextAlignment->Center, FontFamily->"Times New Roman"] }, Open ]], Cell[CellGroupData[{ Cell["Solution of MCARE", "Section", TextAlignment->Left], Cell[TextData[{ "Now, one of the system matrices, ", StyleBox["B", FontSlant->"Italic"], " should be modified and a new one, ", StyleBox["L", FontSlant->"Italic"], " will be introduced. " }], "Text"], Cell["For control input (insulin)", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"B", "=", FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", \(1\/VL\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "TraditionalForm"]}], ";"}]], "Input"], Cell["For disturbance (glucose)", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"L", "=", RowBox[{"(", GridBox[{ {"1", "0"}, {"0", "0"} }], ")"}]}], ";"}]], "Input"], Cell["\<\ We are looking for the symmetric solution matrix of the modified Riccati \ equations in the following form, \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(P = Table[If[i > j, p\_\(j, i\), p\_\(i, j\)], {i, 1, 2}, {j, 1, 2}]; MatrixForm[P]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(p\_\(1, 1\)\), \(p\_\(1, 2\)\)}, {\(p\_\(1, 2\)\), \(p\_\(2, 2\)\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "Keeping in mind, that in our case the ", StyleBox["C", FontSlant->"Italic"], " is an identity matrix, the left hand side of the Riccati equation is" }], "Text"], Cell[BoxData[ \(\(RI = P . A + Transpose[A] . P + IdentityMatrix[2] - P . \((B . Transpose[B] - 1\/\[Gamma]\^2\ L . Transpose[L])\) . P;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[RI]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1 - 2\ p1\ p\_\(1, 1\) + p\_\(1, 1\)\%2\/\[Gamma]\^2 - p\_\(1, 2\)\%2\/VL\^2\), \(\(-\(\(GB\ p3\ p\_\(1, 1\)\)\/p2\)\ \) - p1\ p\_\(1, 2\) - p4\ p\_\(1, 2\) + \(p\_\(1, 1\)\ p\_\(1, 2\)\)\/\[Gamma]\^2 - \ \(p\_\(1, 2\)\ p\_\(2, 2\)\)\/VL\^2\)}, {\(\(-\(\(GB\ p3\ p\_\(1, 1\)\)\/p2\)\) - p1\ p\_\(1, 2\) - p4\ p\_\(1, 2\) + \(p\_\(1, 1\)\ p\_\(1, 2\)\)\/\[Gamma]\^2 - \ \(p\_\(1, 2\)\ p\_\(2, 2\)\)\/VL\^2\), \(1 - \(2\ GB\ p3\ p\_\(1, 2\)\)\/p2 + p\_\(1, 2\)\%2\/\[Gamma]\^2 - 2\ p4\ p\_\(2, 2\) - p\_\(2, 2\)\%2\/VL\^2\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "It is easy to see that", StyleBox[" RI", FontSlant->"Italic"], " is a symmetric matrix" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Apply[And, Table[RI[\([i, j]\)] \[Equal] RI[\([j, i]\)], {i, 1, 2}, {j, 1, 2}] // Flatten]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{ "Therefore", ",", \(to\ solve\ the\ modified\ Ricatti\ equation\), ",", " ", RowBox[{ StyleBox[\(R\ I\), FontSlant->"Italic"], "=", " ", RowBox[{"(", GridBox[{ {"0", "0"}, {"0", "0"} }], ")"}]}], ",", \(only\ three\ equations\ should\ be\ solved\ with\ the\ \[Gamma]\ \ as\ parameter\), " ", ",", " ", "namely"}]], "Text", FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[BoxData[ \(eqs = \(Table[RI[\([i, j]\)] \[Equal] 0, {i, 1, 2}, {j, i, 2}] // Flatten\) // FullSimplify\)], "Input"], Cell[BoxData[ \({1 + p\_\(1, 1\)\%2\/\[Gamma]\^2 \[Equal] 2\ p1\ p\_\(1, 1\) + p\_\(1, 2\)\%2\/VL\^2, p\_\(1, 1\)\ \((\(-\(\(GB\ p3\)\/p2\)\) + p\_\(1, 2\)\/\[Gamma]\^2)\) \[Equal] \(p\_\(1, 2\)\ \((\((p1 + \ p4)\)\ VL\^2 + p\_\(2, 2\))\)\)\/VL\^2, 1 + p\_\(1, 2\)\%2\/\[Gamma]\^2 \[Equal] \(2\ GB\ p3\ p\_\(1, 2\)\)\/p2 \ + p\_\(2, 2\)\ \((2\ p4 + p\_\(2, 2\)\/VL\^2)\)}\)], "Output"] }, Open ]], Cell["\<\ The critical solution of this system belongs to the critical value of the \ parameter \[Gamma]. Crossing this critical value with \[Gamma] the solution, \ which was real becomes imaginary and vice versa. On one hand finding this \ critical solution numerically is an ill-conditioned problem, on the other \ hand to solve this polynomial system in fully symbolic form is hard to \ manage. Therefore, we used reduced Groebner basis solution on rational field. \ Substituting the values of the system parameters in rational form, we get\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Eqs = eqs /. NumericalValues\)], "Input"], Cell[BoxData[ \({1 + p\_\(1, 1\)\%2\/\[Gamma]\^2 \[Equal] \(7\ p\_\(1, 1\)\)\/125 + p\_\(1, 2\)\%2\/14400, p\_\(1, 1\)\ \((\(-\(143\/250\)\) + p\_\(1, 2\)\/\[Gamma]\^2)\) \[Equal] \(p\_\(1, 2\)\ \ \((26048\/15 + p\_\(2, 2\))\)\)\/14400, 1 + p\_\(1, 2\)\%2\/\[Gamma]\^2 \[Equal] \(143\ p\_\(1, 2\)\)\/125 + \ \((5\/27 + p\_\(2, 2\)\/14400)\)\ p\_\(2, 2\)}\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{ "To", " ", "get", " ", "the", " ", "solution", " ", "for", " ", StyleBox[\(p\_\(1, 1\)\), FontSlant->"Italic"]}], ",", " ", \(reduced\ Groebner\ basis\ can\ be\ employed\), ","}]], "Text", FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[BoxData[ \(grb21 = GroebnerBasis[ Eqs, {p\_\(1, 1\), p\_\(1, 2\), p\_\(2, 2\)}, {p\_\(1, 2\), p\_\(2, 2\)}] // Expand\)], "Input"], Cell[BoxData[ \({132860250000000000000000\ \[Gamma]\^4 + 2088252940500000000000\ \[Gamma]\^6 - 3869314817519750000\ \[Gamma]\^8 - 14880348000000000000000\ \[Gamma]\^4\ p\_\(1, 1\) - 233884329336000000000\ \[Gamma]\^6\ p\_\(1, 1\) + 433363259562212000\ \[Gamma]\^8\ p\_\(1, 1\) + 265720500000000000000000\ \[Gamma]\^2\ p\_\(1, 1\)\%2 + 4593155625000000000000\ \[Gamma]\^4\ p\_\(1, 1\)\%2 + 4847596591368500000\ \[Gamma]\^6\ p\_\(1, 1\)\%2 - 69048343484786936\ \[Gamma]\^8\ p\_\(1, 1\)\%2 - 14880348000000000000000\ \[Gamma]\^2\ p\_\(1, 1\)\%3 - 233884329336000000000\ \[Gamma]\^4\ p\_\(1, 1\)\%3 + 95265219282212000\ \[Gamma]\^6\ p\_\(1, 1\)\%3 + 3187193644154520\ \[Gamma]\^8\ p\_\(1, 1\)\%3 + 132860250000000000000000\ p\_\(1, 1\)\%4 + 2088252940500000000000\ \[Gamma]\^2\ p\_\(1, 1\)\%4 + 2168150187480250000\ \[Gamma]\^4\ p\_\(1, 1\)\%4 - 56914172217045000\ \[Gamma]\^6\ p\_\(1, 1\)\%4 + 68588956604025\ \[Gamma]\^8\ p\_\(1, 1\)\%4, 132860250000000000000000\ \[Gamma]\^3 + 2088252940500000000000\ \[Gamma]\^5 - 3869314817519750000\ \[Gamma]\^7 - 14880348000000000000000\ \[Gamma]\^3\ p\_\(1, 1\) - 233884329336000000000\ \[Gamma]\^5\ p\_\(1, 1\) + 433363259562212000\ \[Gamma]\^7\ p\_\(1, 1\) + 265720500000000000000000\ \[Gamma]\ p\_\(1, 1\)\%2 + 4593155625000000000000\ \[Gamma]\^3\ p\_\(1, 1\)\%2 + 4847596591368500000\ \[Gamma]\^5\ p\_\(1, 1\)\%2 - 69048343484786936\ \[Gamma]\^7\ p\_\(1, 1\)\%2 - 14880348000000000000000\ \[Gamma]\ p\_\(1, 1\)\%3 - 233884329336000000000\ \[Gamma]\^3\ p\_\(1, 1\)\%3 + 95265219282212000\ \[Gamma]\^5\ p\_\(1, 1\)\%3 + 3187193644154520\ \[Gamma]\^7\ p\_\(1, 1\)\%3 + \ \(132860250000000000000000\ p\_\(1, 1\)\%4\)\/\[Gamma] + 2088252940500000000000\ \[Gamma]\ p\_\(1, 1\)\%4 + 2168150187480250000\ \[Gamma]\^3\ p\_\(1, 1\)\%4 - 56914172217045000\ \[Gamma]\^5\ p\_\(1, 1\)\%4 + 68588956604025\ \[Gamma]\^7\ p\_\(1, 1\)\%4}\)], "Output"] }, Open ]], Cell["\<\ This is a monomial with parameter \[Gamma]. 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FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], StyleBox["solution", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], StyleBox["for", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], StyleBox[\(\[Gamma]\_crit\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}]], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ However, this is just a good approximation for our problem, because\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(p11crit[\[Gamma]crit]\)], "Input"], Cell[BoxData[ \(\(-82.16879589110081`\) - 87.34586796082296`\ \[ImaginaryI]\)], "Output"] }, Open ]], Cell["see Fig.1/b.", "Text"], Cell["\<\ Tuning this approximation a little by trial-error method, we get,\ \>", "Text"], Cell[BoxData[ \(\(\[Gamma]crit = 17.117425694;\)\)], "Input"], Cell["Indeed, now the imaginary part of the solution is zero,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Im[p11crit[\[Gamma]crit]]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[TextData[{ "Employing MATLAB minmax control design we have got the same result, \ namely, the optimal, minimal \[Gamma], ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Gamma]\)\(\ \)\)\_min\)]], "=17.11743 is equal to the critical value ", Cell[BoxData[ \(TraditionalForm\`\[Gamma]\_crit\)]], " computed above, see Fig.1/c. 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New Roman"], Cell[TextData[{ "For this ", StyleBox["N", FontSlant->"Italic"], "\[Gamma] value, the modified Ricatti can be solved numerically" }], "Text"], Cell[BoxData[ \(\(NsolR = Solve[\(eqs /. NumericalValues\) /. \[Gamma] \[Rule] N\[Gamma], {p\_\(1, 1\), p\_\(1, 2\), p\_\(2, 2\)}];\)\)], "Input"], Cell["\<\ Considering the eigenvalues, none of the solutions is positive definite, \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Map[Eigenvalues[#] &, P /. NsolR] // TableForm\) // N\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {\(-9127.665458169891`\), "5.044056555586021`"}, {\(-9121.878877627918`\), "5.044126390473578`"}, {\(-2066.1127562109973`\), "22.97349259835514`"}, {\(-2066.1018333360175`\), "22.974976665013703`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["However, we choose the second solution like MATLAB does", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(solPD = P /. \ NsolR[\([2]\)] // N\)], "Input"], Cell[BoxData[ \({{\(-169.4461095867866`\), 1249.844832875687`}, {1249.844832875687`, \ \(-8947.38864165066`\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[solPD] // N\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-169.4461095867866`\), "1249.844832875687`"}, {"1249.844832875687`", \(-8947.38864165066`\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["MATLAB computed practically the same solution", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"solPDM", "=", RowBox[{"(", GridBox[{ {\(-169.5\), "1250.3"}, {"1250.3", \(-8950.6\)} }], ")"}]}], ";"}]], "Input", TextAlignment->Center, FontFamily->"Courier New"], Cell["indeed, this is not positive definite solution,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Eigenvalues[solPDM] // N\)], "Input"], Cell[BoxData[ \({\(-9125.154532483348`\), 5.054532483347176`}\)], "Output"] }, Open ]], Cell["\<\ although this solution satisfies the modified Ricatti equation\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[ Norm[RI /. Join[NumericalValues, {p\_\(1, 1\) \[Rule] #[\([1, 1]\)], p\_\(1, 2\) \[Rule] #[\([1, 2]\)], p\_\(2, 2\) \[Rule] #[\([2, 2]\)], \[Gamma] \[Rule] N\[Gamma]}]]/Norm[#] &, {solPD}] // First\)], "Input"], Cell[BoxData[ \(7.404002507487596`*^-16\)], "Output"] }, Open ]], Cell["\<\ Considering this solution of the Riccati equation, the corresponding gain \ matrix can be computed as\ \>", "Text"], Cell[BoxData[ \(\(KLQR = Transpose[B] . solPD /. NumericalValues // N;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[KLQR]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0.`", "0.`"}, {"10.415373607297392`", \(-74.5615720137555`\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["The MATLAB result is practically the same", "Text"], Cell[BoxData[ RowBox[{"(", GridBox[{ {"0", "0"}, {"10.4092", \(-74.5880\)} }], ")"}]], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["Let consider the gain of the closed loop", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((A - B . KLQR)\) /. NumericalValues // N\)], "Input"], Cell[BoxData[ \({{\(-0.028`\), \(-0.572`\)}, {\(-0.0867947800608116`\), 0.5287538408553699`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Eigenvalues[%]\)], "Input"], Cell[BoxData[ \({0.6069443390480433`, \(-0.1061904981926734`\)}\)], "Output"] }, Open ]], Cell["\<\ It means that in this case, if we want to control the system only by the \ control input, this control is not stabilizing the system! \ \>", "Text"], Cell["\<\ Let us increasing the value of \[Gamma] up to the value provinding positive \ definite solution. Increasing somewhat the value of \[Gamma] by 0.25 % , \ we get \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N\[Gamma] = \[Gamma]crit\ 1.0025\)], "Input"], Cell[BoxData[ \(17.160219258235`\)], "Output"] }, Open ]], Cell["\<\ Fig.1/d demonstrates, how sensitive the solution. 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Join[NumericalValues, {p\_\(1, 1\) \[Rule] #[\([1, 1]\)], p\_\(1, 2\) \[Rule] #[\([1, 2]\)], p\_\(2, 2\) \[Rule] #[\([2, 2]\)], \[Gamma] \[Rule] N\[Gamma]}]]/Norm[#] &, {solPD}] // First\)], "Input"], Cell[BoxData[ \(5.906846157958238`*^-15\)], "Output"] }, Open ]], Cell["The control matrix is", "Text"], Cell[BoxData[ \(\(KLQR = Transpose[B] . solPD /. NumericalValues // N;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[KLQR]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0.`", "0.`"}, {\(-59.61899756203763`\), "415.64820958649085`"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["The closed loop gain matrix", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((A - B . KLQR)\) /. NumericalValues // N\)], "Input"], Cell[BoxData[ \({{\(-0.028`\), \(-0.572`\)}, {0.49682497968364686`, \ \(-3.556327672480016`\)}}\)], "Output"] }, Open ]], Cell["which is now stabilizing the system", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Eigenvalues[%]\)], "Input"], Cell[BoxData[ \({\(-3.4738564813800217`\), \(-0.11047119109999448`\)}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Comparing classical LQ and disturbance rejection LQ control design\ \>", "Section", TextAlignment->Left], Cell[TextData[{ "The gain matrix ", StyleBox["KLQR", FontSlant->"Italic"], ", provided by the disturbance rejection LQ design depends on the actual \ value of the scaling parameter \[Gamma]. In case \[Gamma] \[Rule] \ \[Infinity], we get the gain matrix designed by LQ method [25], [26], namely" }], "Text"], Cell[BoxData[ \(lim\+\(\[Gamma] \[Rule] \ \[Infinity]\)\ KLQR\ = \ KLQ\)], "NumberedEquation", TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "To demonstrate this we have considered \[Gamma] = 100 ", Cell[BoxData[ \(TraditionalForm\`\[Gamma]\_crit\)]], ", and we repeated the computations for solving Riccati equation," }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N\[Gamma]100 = SetPrecision[100\ N\[Gamma], 20]\)], "Input"], Cell[BoxData[ \(1716.0219258234999415435595437884`20. \)], "Output"] }, Open ]], Cell["Now the modified Ricatti can be solved numerically", "Text"], Cell[BoxData[ \(\(NsolR = Solve[\(eqs /. NumericalValues\) /. \[Gamma] \[Rule] N\[Gamma]100, {p\_\(1, 1\), p\_\(1, 2\), p\_\(2, 2\)}];\)\)], "Input"], Cell[TextData[{ "The solutions providing positive definite ", StyleBox["P", FontSlant->"Italic"], " matrix should be selected" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Map[Eigenvalues[#] &, P /. NsolR] // TableForm\) // N\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {\(-3805.5768950777438`\), \(-46.33645532208641`\)}, {\(\(-1742.2710243940412`\) - 540.1067272427903`\ \[ImaginaryI]\), \(\(\(7.514962009563929`\)\ \(\[InvisibleSpace]\)\) + 2.360691981679045`\ \[ImaginaryI]\)}, {\(\(-1742.2710243940412`\) + 540.1067272427903`\ \[ImaginaryI]\), \(\(\(7.514962009563929`\)\ \(\[InvisibleSpace]\)\) - 2.360691981679045`\ \[ImaginaryI]\)}, {"324.7363480040445`", "3.846896358392604`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["Therefore, the last solution should be selected", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(solPD = P /. Last[\ NsolR] // N\)], "Input"], Cell[BoxData[ \({{13.960628381419776`, \(-56.06337795322945`\)}, {\(-56.06337795322945`\ \), 314.6226159810173`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[solPD] // N\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"13.960628381419776`", \(-56.06337795322945`\)}, {\(-56.06337795322945`\), "314.6226159810173`"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["This solution satisfies the modified Ricatti equation", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[ Norm[RI /. Join[NumericalValues, {p\_\(1, 1\) \[Rule] #[\([1, 1]\)], p\_\(1, 2\) \[Rule] #[\([1, 2]\)], p\_\(2, 2\) \[Rule] #[\([2, 2]\)], \[Gamma] \[Rule] N\[Gamma]100}]]/Norm[#] &, {solPD}] // First\)], "Input"], Cell[BoxData[ \(4.66566169032367`*^-17\)], "Output"] }, Open ]], Cell[TextData[{ "Employing MATLAB minmax control procedure in case of \[Gamma] = 100 ", Cell[BoxData[ \(TraditionalForm\`\[Gamma]\_min\)]], " = 100 17.1174 = 1711.74, the solution of the modified Ricatti equation \ is:" }], "Text"], Cell[BoxData[ RowBox[{"(", GridBox[{ {"13.9606", \(-56.0634\)}, {\(-56.0634\), "314.6227"} }], ")"}]], "Text", TextAlignment->Center, FontFamily->"Times New Roman"], Cell["which is very close to the result above.", "Text"], Cell["\<\ Having the solution of the Riccati equation, the corresponding gain matrix \ can be computed as\ \>", "Text"], Cell[BoxData[ \(\(KLQRinf = Transpose[B] . solPD /. NumericalValues;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[KLQRinf] // N\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0.`", "0.`"}, {\(-0.46719481627691206`\), "2.6218551331751443`"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["which is indeed very close to KLQ, see above.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(KLQ // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {\(-0.4670690807117331`\), "2.621073424234561`"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Nonlinear system performance with LQ control", "Section", TextAlignment->Left], Cell[TextData[{ "The performance of the control is tested by using a standard 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", FontSlant->"Italic"], " Exogenous glucose infusion, H(t)" }], "Text", TextAlignment->Center], Cell["As a result, the disturbance function is", "Text"], Cell[BoxData[ \(\(h[t_] = H[t];\)\)], "Input"], Cell["The steady state insulin injection is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(i0 = i0 /. NumericalValues\)], "Input"], Cell[BoxData[ \(16.666666666666668`\)], "Output"] }, Open ]], Cell["\<\ Considering negative feedback, the insulin injection rate as control variable \ is\ \>", "Text"], Cell[BoxData[ \(i[t_] := i0 - KLQ[\([2]\)] . {G[t], Y[t]} /. NumericalValues\)], "Input"], Cell["\<\ Now, we consider the original 3-states nonlinear model, instead of the \ reduced 2-states linear model which has been employed for control design. The \ cloosed loop equations are\ \>", "Text"], Cell[BoxData[ \(Clear[X]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(deq1NL = \(G'\)[ t] \[Equal] \((\(-p1\)\ G[t] - X[t]\ \((G[t] + GB)\) + h[t])\) /. 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matrix ", StyleBox["B", FontSlant->"Italic"], " in MCARE. In real situation only insuline injection is possible. \ Therefore we can use only the second row of the control matrix." }], "Text"], Cell["Now, the insulin injection rate as control variable is", "Text"], Cell[BoxData[ \(i[t_] := i0 - KLQR[\([2]\)] . {G[t], Y[t]} /. NumericalValues\)], "Input"], Cell[BoxData[ \(Clear[X]\)], "Input"], Cell["The modell equations are", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(deq1NL = \(G'\)[ t] \[Equal] \((\(-p1\)\ G[t] - X[t]\ \((G[t] + GB)\) + h[t])\) /. NumericalValues\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["G", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "\[Equal]", RowBox[{\(-\(\(7\ G[t]\)\/250\)\), "-", \(\((110 + G[t])\)\ X[t]\), "+", RowBox[{ TagBox[\(InterpolatingFunction[{{0.`, 1000.`}}, "<>"]\), False, Editable->False], "[", "t", "]"}]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(deq2NL = \(X'\)[t] \[Equal] \((\(-p2\)\ X[t] + p3\ Y[t])\) /. 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interpretation of a control system, will not \ automatically ensure positive definite solution. It has been demonstrated \ that \"blind\" numerical solution of disturbance rejection LQ problem can be \ misleading, and not always provides positive definite solution. This problem \ can be detected by using symbolic-numeric solution and increasing the value \ of \[Gamma] up to the very value, which already ensures positive eigenvalues \ of the Ricatti matrix. The other advantage of the symbolic-numeric solution, \ whenever it is possible, is its robustness concerning round-off errors. \ However, this problem can be avoided by using another numerical technique for \ minimax control based on robust control on frequency domain proposed by \ Helton [33], and its application for glucose-insulin control see [24]. It was \ also illustrated, that disturbance rejection LQ control could provide better \ control quality than LQ does, see Fig.7. Disturbance rejection LQ control is \ interacting faster as well as employing higher infusion rate (in the \ considered case) than the classical control, see Fig.8.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section", TextAlignment->Left], Cell[TextData[{ "\n", StyleBox["[1] \tS. Wild, G. Roglic, A. Green, R. Sicree and H. King, \ \[OpenCurlyDoubleQuote]Global Prevalence of Diabetes - Estimates for the year \ 2000 and projections for 2030\[CloseCurlyDoubleQuote], Diabetes Care, vol. 27 \ (5), pp. 1047-1053, 2004.\n[2] \tN. Hernjak, F. J. Doyle III, \ \[OpenCurlyDoubleQuote]Glucose control Design Using Nonlinearity Assessment \ Techniques\[CloseCurlyDoubleQuote], AIChE Journal, vol. 51, no. 2, pp. \ 544-554, Febr. 2005.\n[3] \tR. S. Parker, F. J. Doyle III, J. H. Ward, N. 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Campos-Delgado, \:201eBlood glucose \ control for type I diabetes mellitus: A robust tracking ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`H\_\[Infinity]\)]], StyleBox[" problem\[CloseCurlyDoubleQuote], Elsevier Control Engineering \ Practice, vol. 12, pp. 1179-1195, 2004.\n[5] \tMini Med CGMS ", FontFamily->"Times New Roman"], StyleBox["http://www.minimed.com/patientfam/pf_products_cgms_ov_completetic.\ shtml", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], StyleBox[".\n[6] \tGlucowatch, ", FontFamily->"Times New Roman"], StyleBox["http://www.glucowatch.com", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], StyleBox[".\n[7] \tMini Med Paradigm Insulin Pump ", FontFamily->"Times New Roman"], StyleBox["http://www.minimed.com/patientfam/pf_ipt_paradigm_insulin_pump.\ shtml", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], StyleBox[".\n[8] \tDisetronic, ", FontFamily->"Times New Roman"], StyleBox["http://www.disetronic-usa.com/insulin-pumps.htm", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], StyleBox[".\n[9] \tB. N. Bergman, Y. Z. Ider, C. R. Bowden, C. Cobelli, \ \[OpenCurlyDoubleQuote]Quantitive estimation of insulin sensitivity\ \[CloseCurlyDoubleQuote], American Journal of Physiology, vol. 236, pp. \ 667-677, 1979.\n[10] \tR. N. Bergman, L. S. Philips, C. Cobelli, \ \[OpenCurlyDoubleQuote]Physiologic evaluation of factors controlling glucose \ tolerance in man\[CloseCurlyDoubleQuote], Journal of Clinical Investigation, \ vol. 68; pp. 1456-1467, 1981.\n[11]\t J. Lin, J. G. Chase, G. M. Shaw, C. V. \ Doran, C. E. Hann, M. B. Robertson, P. M. Browne, T. Lotz, G. C. Wake, B. \ Broughton, \[OpenCurlyDoubleQuote]Adaptive Bolus-Based Set-Point Regulation \ of Hyperglycemia in Critical \tCare\[CloseCurlyDoubleQuote], Proc. of 26th \ Ann. Int. Conf. of IEEE Eng. in Biomedicine Soc., San Francisco, USA, pp. \ 3463-3466, 2004.\n[12] \tM. Fernandez, D. Acosta, M. Villasana, D. 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Jones, \[OpenCurlyDoubleQuote]Partitioning \ glucose distribution/transport, disposal, and endogenous \tproduction during \ IVGTT\[CloseCurlyDoubleQuote], American Journal of Physiology - Endocrinology \ and Metabolism, vol. 282, pp. E992-E1007.\n[16] \tJ. T. Sorensen, \ \[OpenCurlyDoubleQuote]A Physiologic Model of Glucose Metabolism in Man and \ Its Use to Design and Assess Improved Insulin Therapies for Diabetes\ \[CloseCurlyDoubleQuote], PhD Thesis, Dept. of Chemical Engineering, MIT, \t\t\ Cambridge, MA, USA, 1985.\n[17] \tR. Parker, F. J. Doyle III, N. A. Peppas, \ \[OpenCurlyDoubleQuote]The Intravenous Route to Blood Glucose Control\ \[CloseCurlyDoubleQuote], IEEE Engineering in Medicine and Biology, vol. 20 \ (1), pp. 65-73, 2001.\n[18] \tF. Chee, T. L. Fernando, A. V. 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Bokor, Z. \ Beny\[OAcute], \[OpenCurlyDoubleQuote]LPV Fault Detection of Glucos-Insulin \ System\[CloseCurlyDoubleQuote], MED06 - 14th Mediterranean Conference on \ Control and Automation, Ancona, Italy, electronic publication \tTLA2-4, 2006.\ \n[29] \tA. Makroglou, J. Li, Y. Kuang, \[OpenCurlyDoubleQuote]Mathematical \ models and software tools for the glucose - insulin regulatory system and \ diabetes: an overview\[CloseCurlyDoubleQuote], Elsevier Applied Numerical \ Mathematics, vol. 56 (3-4), pp. \t559-573, 2006.\n[30] \tS. M. Furler, E. W. \ Kraegen, R. H. Smallwood, D. J. Chisolm, \[OpenCurlyDoubleQuote]Blood glucose \ control by intermittent loop closure in the basal mode: Computer simulation \ studies with a diabetic model\[CloseCurlyDoubleQuote], ", FontFamily->"Times New Roman"], StyleBox["Diabetes Care,", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->0}], StyleBox[" vol. 8, \tpp. 553-561, 1985.\n[31]\tK. 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Deutsch, \[OpenCurlyDoubleQuote]A physiological model of \ glucose-insulin interaction in type I diabetes mellitus\ \[CloseCurlyDoubleQuote], Journal of Biomedical Engineering, vol. 14, pp. \ 235-242, 1992.", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->0}], StyleBox["\n", FontSize->8], StyleBox["\n", FontSize->9] }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.1 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 681}}, WindowSize->{1164, 649}, WindowMargins->{{-2, Automatic}, {Automatic, -1}}, PrintingCopies->1, PrintingPageRange->{1, 80}, ShowSelection->True, StyleDefinitions -> "Classic.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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