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However this is not usually the case, \ because mainly the blood plasma glucose measurable in the practise. In this \ study the use of the", StyleBox[" ", FontSlant->"Italic"], "reduced-order estimator (also known as the", StyleBox[" Luenberger observer", FontSlant->"Italic"], ") is considered to reconstruct those state variables that cannot be \ recovered directly from the system outputs. An observer for state variables - \ remote compartment insulin and plasma insulin - is designed in case of \ extended minimal Bergman model, employing ", StyleBox["Polynomial Control System Application ", FontSlant->"Italic"], "of ", StyleBox["Mathematica. ", FontSlant->"Italic"], "The results of nonlinear closed loop simulations with the classical LQ \ control showed that an observer, which about ten times faster than the system \ itself can provide a very good state recovery performance." }], "Abstract"], Cell[CellGroupData[{ Cell["System Equations", "Section", TextAlignment->Left, FontVariations->{"CompatibilityType"->0}], Cell["\<\ Several different models of diabetic systems exist in the literature \ including, for example, the very detailed 21st-order metabolic model of \ Sorensen (1985). 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 (1981),\ \>", "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 (1985):\ \ \>", "Text"], Cell[BoxData[ \(\(numPar = {p1 -> 0.028, p2 -> 0.025, p3 -> 0.00013, GB -> 110, YB -> 1.5, p4 -> 5. /54, VL -> 120};\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Steady State", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}]], "Section", TextAlignment->Left, FontWeight->"Plain"], Cell["\<\ In order to linearize the system, we need its steady-state values. The \ steady-state values of the three state variables are,\ \>", "Text"], Cell[BoxData[ \(\(G0 = \(X0 = \(Y0 = 0\)\);\)\)], "Input"], Cell["\<\ The steady-state value of the control variable can be computed from the third \ differential equation,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[i0/VL \[Equal] p4\ YB, i0] /. numPar\)], "Input"], Cell[BoxData[ \({{i0 \[Rule] 16.666666666666668`}}\)], "Output"] }, Open ]], Cell["which means", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(i0 = i0 /. %[\([1]\)]\)], "Input"], Cell[BoxData[ \(16.666666666666668`\)], "Output"] }, Open ]], Cell["The steady-state value of the disturbance variable is,", "Text"], Cell[BoxData[ \(\(h0 = 0;\)\)], "Input"], Cell["\<\ Now, we save the state of the actual context path, and then load the Control \ System Professional Suite (CSPS) application,\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Linearization", FontVariations->{"CompatibilityType"->0}]], "Section", TextAlignment->Left], Cell[BoxData[ \(<< ControlSystems`\)], "Input"], Cell["\<\ The linearization can be carried out in the vicinity of this steady state, \ using new variables denoting the deviations from the equilibrium,\ \>", "Text"], Cell[BoxData[ \(\({x\_1, x\_2, x\_3\ } = {G[t], X[t], Y[t]};\)\)], "Input"], Cell["\<\ In our model the only measurable output is the G(t) - plasma glucose \ deviation,\ \>", "Text"], Cell[BoxData[ \(\({y\_1} = {G[t]};\)\)], "Input"], Cell[BoxData[ \(\({u\_1, u\_2} = {h[t], i[t]};\)\)], "Input"], Cell[BoxData[ \(\({f\_1, f\_2, f\_3} = {deq1NL[\([2]\)], deq2NL[\([2]\)], deq3NL[\([2]\)]};\)\)], "Input"], Cell[BoxData[ \(\({g\_1} = {y\_1};\)\)], "Input"], Cell["The linearized model, as a control object in CSPS,", "Text"], Cell[BoxData[ \(\(ControlObjectSS = Linearize[{f\_1, f\_2, f\_3}, {g\_1}, \[IndentingNewLine]{{G[t], G0}, {X[t], X0}, {Y[t], Y0}}, {{h[t], h0}, {i[t], i0}}];\)\)], "Input"], Cell["Its state-space form is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ControlObjectSS // EquationForm\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ FormBox[GridBox[{ { RowBox[{\(\[ScriptX]\& . \), "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-p1\), \(-GB\), "0"}, {"0", \(-p2\), "p3"}, {"0", "0", \(-p4\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "0"}, {"0", \(1\/VL\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptU]"}]}, { RowBox[{"\[ScriptY]", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "\[ScriptX]", "+", RowBox[{"(", "\[NoBreak]", GridBox[{ {"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 giving the model matrices,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ss = ControlObjectSS // TraditionalForm\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ FormBox[ SubsuperscriptBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-p1\), \(-GB\), "0", "1", "0"}, {"0", \(-p2\), "p3", "0", "0"}, {"0", "0", \(-p4\), "0", \(1\/VL\)}, {"1", "0", "0", "0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Center}, RowLines->{False, False, True, False}, ColumnLines->{False, False, True, False}, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "TraditionalForm"], FormBox["\[Bullet]", "TraditionalForm"], FormBox[ StyleBox["\<\"\[ScriptCapitalS]\"\>", Editable->False, ShowStringCharacters->False], "TraditionalForm"], MultilineFunction->None], "TraditionalForm"], (StateSpace[ SlotSequence[ 1]]&)], TraditionalForm]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Steady-State Stability", FontVariations->{"CompatibilityType"->0}]], "Section", TextAlignment->Left], Cell["\<\ It is useful to check the stability of the open-loop system. The system \ matrix can be extracted as\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(A = ss[\([1, 1]\)]; MatrixForm[A]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-p1\), \(-GB\), "0"}, {"0", \(-p2\), "p3"}, {"0", "0", \(-p4\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["and its eigenvalues are", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Eigenvalues[A]\)], "Input"], Cell[BoxData[ \({\(-p1\), \(-p2\), \(-p4\)}\)], "Output"] }, Open ]], Cell["or", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(% /. numPar\)], "Input"], Cell[BoxData[ \({\(-0.028`\), \(-0.025`\), \(-0.09259259259259259`\)}\)], "Output"] }, Open ]], Cell["which means the system is stable.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Controllability and Observability", FontVariations->{"CompatibilityType"->0}]], "Section", TextAlignment->Left], Cell["\<\ The controllability and observability can be tested by using the \ corresponding built - in functions,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Controllable[ControlObjectSS]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Observable[ControlObjectSS]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["State Reconstruction", FontWeight->"Bold"]], "Section", TextAlignment->Left, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[{ "In our model only the G(t) - plasma glucose deviation can be measured, the \ other two state variables, the X(t) - remote compartment insulin utilization \ and \tthe Y(t) - plasma insulin deviation, and their corresponding output \ variables, ", Cell[BoxData[ StyleBox[\(y\_2\), FontFamily->"Times New Roman"]]], " and ", Cell[BoxData[ \(y\_3\)]], " should be reconstructed in order to provide feedback signal for the \ controller.\nHere, the use of the", StyleBox[" ", FontSlant->"Italic"], "reduced-order estimator (also known as the", StyleBox[" Luenberger observer", FontSlant->"Italic"], ") is considered, as this can be used to reconstruct those state variables \ that cannot be recovered directly from the system outputs. The reduced-order \ state estimator described in ", ButtonBox["Patel and Munro (1982)", ButtonData:>{"PCS References", "PatelMunro82"}, ButtonStyle->"AddOnsLinkText"], " is based on the Luenberger observable canonical form of the system, " }], "Text"], Cell[TextData[{ "Given an ", Cell[BoxData[ \(TraditionalForm\`n\^"th"\)], "InlineFormula"], "-order linear system in state-space form" }], "Text"], Cell[BoxData[{ FormBox[ RowBox[{ OverscriptBox[ StyleBox["x", FontWeight->"Bold", FontSlant->"Plain"], "."], "=", " ", RowBox[{ RowBox[{ StyleBox["A", FontWeight->"Bold", FontSlant->"Plain"], " ", StyleBox["x", FontWeight->"Bold", FontSlant->"Plain"]}], "+", RowBox[{ StyleBox["B", FontWeight->"Bold", FontSlant->"Plain"], " ", StyleBox["u", FontWeight->"Bold", FontSlant->"Plain"]}]}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ StyleBox["y", FontWeight->"Bold", FontSlant->"Plain"], "=", " ", RowBox[{ RowBox[{ StyleBox["C", FontWeight->"Bold", FontSlant->"Plain"], " ", StyleBox["x", FontWeight->"Bold", FontSlant->"Plain"]}], "+", RowBox[{ StyleBox["D", FontWeight->"Bold", FontSlant->"Plain"], " ", StyleBox["u", FontWeight->"Bold", FontSlant->"Plain"]}]}]}], TraditionalForm]}], "NumberedEquation", FormatType->StandardForm], Cell[TextData[{ "with ", StyleBox["q ", FontSlant->"Italic"], "outputs and ", StyleBox["p", FontSlant->"Italic"], " inputs, then a reduced-order dynamical system of order ", StyleBox["n-q", FontSlant->"Italic"], ", called an estimator, can be constructed with state vector ", StyleBox["z", FontSlant->"Italic"], " such that ", StyleBox["z", FontSlant->"Italic"], " approximates a linear combination of ", StyleBox["n-q", FontSlant->"Italic"], " of the system states given by" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ StyleBox["z", FontWeight->"Bold", FontSlant->"Plain"], StyleBox["=", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["Lx", FontWeight->"Bold", FontSlant->"Plain"]}], TraditionalForm]], "NumberedEquation", FormatType->StandardForm], Cell[TextData[{ "where ", StyleBox["L", FontWeight->"Bold"], " has dimensions (", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["n", FontSlant->"Italic"], "-", StyleBox["q", FontSlant->"Italic"]}], TraditionalForm]]], ") \[Times] ", StyleBox["n.", FontSlant->"Italic"], " An observer of order ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["n", FontSlant->"Italic"], "-", StyleBox["q", FontSlant->"Italic"]}], TraditionalForm]]], " is then defined for the system above by" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox[ StyleBox["z", FontWeight->"Bold", FontSlant->"Plain"], "."], StyleBox["=", FontWeight->"Plain", FontSlant->"Plain"], RowBox[{ StyleBox["Fz", FontWeight->"Bold", FontSlant->"Plain"], StyleBox["+", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["Gu", FontWeight->"Bold", FontSlant->"Plain"], StyleBox["+", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["Hy", FontWeight->"Bold", FontSlant->"Plain"], StyleBox[" ", FontWeight->"Plain", FontSlant->"Plain"]}]}], TraditionalForm]], "NumberedEquation", FormatType->StandardForm], Cell[TextData[{ "where ", StyleBox["F", FontWeight->"Bold"], ", ", StyleBox["G", FontWeight->"Bold"], ", and ", StyleBox["H", FontWeight->"Bold"], " are matrices of constants with dimensions (", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["n", FontSlant->"Italic"], "-", StyleBox["q", FontSlant->"Italic"]}], TraditionalForm]]], ") \[Times] (", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["n", FontSlant->"Italic"], "-", StyleBox["q", FontSlant->"Italic"]}], TraditionalForm]]], "), (", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["n", FontSlant->"Italic"], "-", StyleBox["q", FontSlant->"Italic"]}], TraditionalForm]]], ") \[Times] ", StyleBox["p", FontSlant->"Italic"], ", and (", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["n", FontSlant->"Italic"], "-", StyleBox["q", FontSlant->"Italic"]}], TraditionalForm]]], ") \[Times] ", StyleBox["q", FontSlant->"Italic"], ", respectively. The desired estimate ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["x", FontWeight->"Bold", FontSlant->"Plain"], "^"], TraditionalForm]], CellFrameLabels->{{None, Cell[ TextData[ {"(6.", CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}}], " of the system state vector ", StyleBox["x", FontWeight->"Bold"], " is then recovered from ", StyleBox["z", FontWeight->"Bold"], " and the system outputs ", StyleBox["y", FontWeight->"Bold"], " as" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox[ StyleBox["x", FontWeight->"Bold", FontSlant->"Plain"], "^"], StyleBox["=", FontWeight->"Plain", FontSlant->"Plain"], RowBox[{ RowBox[{ StyleBox[ SubscriptBox[ StyleBox["K", FontWeight->"Bold", FontSlant->"Plain"], StyleBox["1", FontWeight->"Plain"]], FontWeight->"Bold"], StyleBox["y", FontWeight->"Bold", FontSlant->"Plain"]}], "+", RowBox[{ StyleBox[ SubscriptBox[ StyleBox["K", FontWeight->"Bold", FontSlant->"Plain"], StyleBox["2", FontWeight->"Plain"]], FontWeight->"Bold"], StyleBox["z", FontWeight->"Bold", FontSlant->"Plain"]}]}]}], TraditionalForm]], "NumberedEquation",\ FormatType->StandardForm], Cell[TextData[{ "The design of the observer requires the solution of the", StyleBox[" observer equation", FontSlant->"Italic"] }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["LA", FontWeight->"Bold", FontSlant->"Plain"], StyleBox["-", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["FL", FontWeight->"Bold", FontSlant->"Plain"]}], StyleBox["=", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["HC", FontWeight->"Bold", FontSlant->"Plain"]}], TraditionalForm]], "NumberedEquation", FormatType->StandardForm], Cell[TextData[{ "such that the matrix ", StyleBox["L", FontWeight->"Bold"], " has maximum rank, in order to guarantee the recovery of the unmeasurable \ state variables. Thus, not only must", StyleBox[" L", FontWeight->"Bold"], " have maximum rank, but the matrix ", Cell[BoxData[ FormBox[ RowBox[{"[", "\[NegativeThinSpace]", GridBox[{ { StyleBox["C", FontWeight->"Bold", FontSlant->"Plain"]}, { StyleBox["L", FontWeight->"Bold", FontSlant->"Plain"]} }], "]"}], TraditionalForm]], CellFrameLabels->{{None, Cell[ TextData[ {"(6.", CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}}], " must also have maximum rank, since ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["K", FontWeight->"Bold", FontSlant->"Plain"], "1"], TraditionalForm]], CellFrameLabels->{{None, Cell[ TextData[ {"(6.", CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}}], " and ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["K", FontWeight->"Bold", FontSlant->"Plain"], "2"], TraditionalForm]], CellFrameLabels->{{None, Cell[ TextData[ {"(6.", CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}}], " are determined from" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"[", RowBox[{ SubscriptBox[ StyleBox["K", FontWeight->"Bold", FontSlant->"Plain"], "1"], " ", SubscriptBox[ StyleBox["K", FontWeight->"Bold", FontSlant->"Plain"], "2"]}], "]"}], " ", "=", SuperscriptBox[ RowBox[{"[", "\[NegativeThinSpace]", GridBox[{ { StyleBox["C", FontWeight->"Bold", FontSlant->"Plain"]}, { StyleBox["L", FontWeight->"Bold", FontSlant->"Plain"]} }], "]"}], \(-1\)]}], TraditionalForm]], "NumberedEquation", FormatType->StandardForm], Cell[TextData[{ "The reduced-order estimator algorithm developed by Munro (", ButtonBox["Patel and Munro (1982)", ButtonData:>{"PCS References", "PatelMunro82"}, ButtonStyle->"AddOnsLinkText"], ") is based on the Luenberger observable canonical form of the system.\nTo \ determine a reduced-order estimator for a given system, you simply have to \ specify the dynamics of the observer, that is, the ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["n", FontSlant->"Italic"], "-", StyleBox["q", FontSlant->"Italic"]}], TraditionalForm]]], " eigenvalues of the matrix ", StyleBox["F", FontWeight->"Bold"], ". Here, these do not need to be distinct, as is the case with other \ reduced-order observer algorithms, but must be chosen to be faster than the \ fastest eigenvalue of the system matrix ", StyleBox["A", FontWeight->"Bold"], ". An advantage of this algorithm is that the designer can specify exactly \ the speed with which the observer tracks the system states. 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oooo0P000?l20?ooo`030000o`3oool0oooo0480oooo0P000?l20?ooo`030000o`3oool0oooo0480 oooo0P000?l20?ooo`030000o`3oool0oooo04D0oooo00<000000?ooo`3oool0203oool00240oooo 00<000000?ooo`3oool0503oool30000odH0oooo0`000?m60?ooo`<0003oB03oool00`000000oooo 0?ooo`080?ooo`008@3oool00`000000oooo0?ooo`3a0?ooo`030000003oool0oooo00P0oooo000Q 0?ooo`030000003oool0oooo00<0oooo00<000000?ooo`3oool0:`3oool00`000000oooo0?ooo`0/ 0?ooo`030000003oool0oooo02/0oooo00<000000?ooo`3oool0;03oool00`000000oooo0?ooo`0[ 0?ooo`030000003oool0oooo00<0oooo00<000000?ooo`3oool0203oool00240oooom@00000:0?oo o`00\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-169.623, 9.60372, 4.30086, \ 0.0695892}}] }, Open ]], Cell[TextData[StyleBox["Fig.3 Insulin infusion ", FontSize->10]], "Caption", TextAlignment->Center], Cell["\<\ Now we solve this differential equation system consisting of the three states \ variables and the six controller states, simultaneously\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol = NDSolve[{deq1, deq2, deq3, deq4, deq5, X[0] \[Equal] 0, Y\ [0] \[Equal] 0, G[0] \[Equal] 0, z1[0] == 0, z2[0] \[Equal] 0}, {X, Y, G, z1, z2}, {t, 0, 1000}, Method \[Rule] BDF, \ MaxSteps \[Rule] 50000]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"X", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 1000.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"Y", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 1000.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"G", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 1000.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"z1", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 1000.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"z2", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 1000.`}}, "<>"]\), False, Editable->False]}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell["\<\ The dynamics of the remote compartment insulin utilization of the nonlinear \ system\ \>", "Text"], Cell[BoxData[ \(\(P1 = Plot[Evaluate[\((X[t] /. sol)\)], {t, 0, 1000}, PlotStyle \[Rule] {RGBColor[0, 0, 1]}, Frame \[Rule] True, PlotRange \[Rule] All, FrameLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell["\<\ The dynamics of the plasma insulin deviation of the nonlinear system\ \>", "Text"], Cell[BoxData[ \(\(P2 = Plot[Evaluate[\((Y[t] /. sol)\)], {t, 0, 1000}, PlotStyle \[Rule] {RGBColor[0, 0, 1]}, Frame \[Rule] True, PlotRange \[Rule] All, FrameLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell["The recovered states are", "Text"], Cell[BoxData[ \(Xrec[ t_] := \(K1[\([2]\)]\ \((G[t] /. sol)\) + K2[\([2, 1]\)]\ \((z1[t] /. sol)\) + K2[\([2, 2]\)] \((z2[t] /. sol)\) /. numPar\) /. {p\_1 \[Rule] \(-1\), p\_2 \[Rule] \(-1\)}\)], "Input"], Cell[BoxData[ \(Yrec[ t_] := \(K1[\([3]\)]\ \((G[t] /. sol)\) + K2[\([3, 1]\)]\ \((z1[t] /. sol)\) + K2[\([3, 2]\)] \((z2[t] /. sol)\) /. numPar\) /. {p\_1 \[Rule] \(-1\), p\_2 \[Rule] \(-1\)}\)], "Input"], Cell["The recovered X(t) state", "Text"], Cell[BoxData[ \(\(P3 = Plot[Xrec[t], {t, 0, 1000}, PlotStyle \[Rule] {RGBColor[1, 0, 0]}, Frame \[Rule] True, PlotRange \[Rule] All, FrameLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell["The recovered Y(t) state", "Text"], Cell[BoxData[ \(\(P4 = Plot[Yrec[t], {t, 0, 1000}, PlotStyle \[Rule] {RGBColor[1, 0, 0]}, Frame \[Rule] True, PlotRange \[Rule] All, FrameLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell["\<\ We can compare the original (blue) and recovered (red) states\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[{P1, P3}, Frame \[Rule] True, PlotRange \[Rule] All, FrameLabel \[Rule] {"\", "\"}, DisplayFunction \[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 } 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0?ooo`8000001`3oool002X0oooo00<000000?ooo`3oool04@3oool00`3o0000oooo0?ooo`0?0?oo o`030?l0003oool0oooo00?ooo`030?l0003oool0oooo0"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-212.197, -0.695953, \ 4.40801, 0.00867694}}] }, Open ]], Cell[TextData[StyleBox["Fig.5 The original (blue) and recovered (red) states \ of plasma insulin deviation", FontSize->10]], "Caption", TextAlignment->Center] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Classical LQ control", FontWeight->"Bold"]], "Section", TextAlignment->Left, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], 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[{ "We choose the following matrices, ", StyleBox["R", FontSlant->"Italic"], " is" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(R = \ DiagonalMatrix[{10\^4, \ 10\^\(-4\)}]; Q = \ \(10\^\(-3\)\) IdentityMatrix[3]; \ MatrixForm[R]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"10000", "0"}, {"0", \(1\/10000\)} }, 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"}, {"0", \(1\/1000\), "0"}, {"0", "0", \(1\/1000\)} }, 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Exogenous insulin infusion rate in case of \ ideal state feedback", FontSize->10]], "Text", TextAlignment->Center] }, Open ]], Cell[CellGroupData[{ Cell["Simulation with observer", "Section", TextAlignment->Left, FontVariations->{"CompatibilityType"->0}], Cell["\<\ Now in the control law, we should use the recovered states for X(t) and \ Y(t) estimated by the observer, namely \ \>", "Text"], Cell[BoxData[ \(Xrec[t_] := First[\(K1[\([2]\)]\ \((G[t])\) + K2[\([2, 1]\)]\ \((z1[t])\) + K2[\([2, 2]\)] \((z2[t])\) /. numPar\) /. {p\_1 \[Rule] \(-1\), p\_2 \[Rule] \(-1\)}]\)], "Input"], Cell[BoxData[ \(Yrec[t_] := First[\((\(K1[\([3]\)]\ \((G[t])\) + K2[\([3, 1]\)]\ \((z1[t])\) + K2[\([3, 2]\)] \((z2[t])\) /. numPar\) /. {p\_1 \[Rule] \(-1\), p\_2 \[Rule] \(-1\)})\) // Flatten]\)], "Input"], Cell["then", "Text"], Cell[BoxData[ \(i[t_] := i0 - KLQ[\([2]\)] . {G[t], Xrec[t], Yrec[t]}\)], "Input"], Cell["\<\ We solve the state equations with the observer equations, simultaneously\ \>", "Text"], 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Exogenous insulin infusion rate in case of \ observer (red) and in case of ideal state feedback (blue)", FontSize->10]], "Text", TextAlignment->Center], Cell["\<\ There is only a slight difference in the dynamical performances. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Conclusions", "Section", TextAlignment->Left], Cell["\<\ Numerical simulations show, that ten times faster observer - means, that the \ absolute value of the greatest eigenvalue of the observer (F matrix) is ten \ times greater than that of the system - can ensure satisfactory state \ recovery for the unmeasurable state variables, remote compartment insulin \ utilization and plasma insulin deviation. This observer provide good \ performance in case of standard meal disturbance (exogenous glucose infusion) \ for classical LQ as well as for disturbance rejection control. Symbolic form \ of the observer makes easy to change parameter values without designing the \ observer again.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section", TextAlignment->Left], Cell[TextData[{ "Patel, R. V. and N. Munro, ", StyleBox["Multivariable System Theory and Design", FontSlant->"Italic"], ", Oxford: Pergamon Press, 1982.\n\n", StyleBox["R. 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", FontFamily->"Times New Roman"], "\n", StyleBox["J. 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, Cambridge, MA, USA, 1985.\n\n", FontFamily->"Times New Roman"], StyleBox["E. D. Lehmann, T. A. 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