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CellTags->"10.7"], Cell["Load the application. ", "MathCaption", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.1"], Cell[BoxData[ \(<< ControlSystems`\)], "Input", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.1"], Cell["Here is a state\[Hyphen]space realization of the antenna mechanism. ", \ "MathCaption", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.5"], Cell[CellGroupData[{ Cell[BoxData[ \(antenna = StateSpace[{{0, 1}, {0, \(-5\)}}, {{0, 0}, {1, 0.1}}, {{1, 0}}]\)], "Input", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.5"], Cell[BoxData[ FormBox[ TagBox[ FormBox[ SubsuperscriptBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "0", "0"}, {"0", \(-5\), "1", "0.1`"}, {"1", "0", "0", "0"} }, ColumnAlignments->{Decimal}, 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], ControlSystems`Common`Private`noD -> True]&)], TraditionalForm]], "Output", CellMargins->{{54, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[TextData[{ "The variance of the process noise is ", StyleBox["q", "MR"], ", and the variance of the measurement noise is ", StyleBox["r", "MR"], ". We now make the measurement noise far more intense and will try to \ filter it out with the Kalman filter. " }], "MathCaption", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.7"], Cell[BoxData[ \(q = {{1}}; r = {{100}};\)], "Input", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.7"], Cell["This is the estimator gain matrix. ", "MathCaption", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.7"], Cell[CellGroupData[{ Cell[BoxData[ \(L = LQEstimatorGains[antenna, q, r, {1}]\)], "Input", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.7"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0.0019996001599200982`"}, {"1.9992003997761246`*^-6"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Output", CellMargins->{{54, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell["\<\ Notice that because the process noise was made relatively small and the \ measurement noise was relatively large, the relatively small gain in the \ Kalman estimator were achieved. Consequently, the Kalman filter will rely \ more on re\[Hyphen]creating the output signal from the deterministic input \ using the known system dynamics than on actually processing the noisy sensor \ output, which is the optimal strategy for noisy measurements. \ \>", "Text", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.7"], Cell[TextData[{ "This is a new object, ", StyleBox["antenna1", "MR"], ", that incorporates another stochastic input through which the measurement \ noise adds directly to the output. This is done by using the parallel \ connection with no input connected. The expanded system has three inputs and \ one output.", " " }], "MathCaption", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->{"10.7", "i:60"}], Cell[CellGroupData[{ Cell[BoxData[ \(antenna1 = ParallelConnect[antenna, TransferFunction[1], {}, {1, 1}]\)], "Input", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.7"], Cell[BoxData[ FormBox[ TagBox[ FormBox[ SubsuperscriptBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "0", "0", "0"}, {"0", \(-5\), "1", "0.1`", "0"}, {"1", "0", "0", "0", "1"} }, ColumnAlignments->{Decimal}, 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", CellMargins->{{54, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell["\<\ This finds the Kalman estimator. Note that only the first output of the \ estimator gives the filtered output of the system. \ \>", "MathCaption", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.7"], Cell[CellGroupData[{ Cell[BoxData[ \(estimator = KalmanEstimator[%, L, {1}, {1}]\)], "Input", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.7"], Cell[BoxData[ FormBox[ TagBox[ FormBox[ SubsuperscriptBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-0.0019996001599200982`\), "1.`", "0.`", "0.0019996001599200982`"}, {\(-1.9992003997761246`*^-6\), \(-5.`\), "1.`", "1.9992003997761246`*^-6"}, {"1.`", "0.`", "0.`", "0.`"}, {"1.`", "0.`", "0.`", "0.`"}, {"0.`", "1.`", "0.`", "0.`"} }, ColumnAlignments->{Decimal}, 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", CellMargins->{{54, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell["\<\ This picks the subsystem we are currently interested in. It contains all the \ inputs and the first output of the estimator. \ \>", "MathCaption", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.7"], Cell[CellGroupData[{ Cell[BoxData[ \(filter = Subsystem[estimator, All, {1}]\)], "Input", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"10.7"], Cell[BoxData[ FormBox[ TagBox[ FormBox[ SubsuperscriptBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-0.0019996001599200982`\), "1.`", "0.`", "0.0019996001599200982`"}, {\(-1.9992003997761246`*^-6\), \(-5.`\), "1.`", "1.9992003997761246`*^-6"}, {"1.`", "0.`", "0.`", "0.`"} }, ColumnAlignments->{Decimal}, 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", CellMargins->{{54, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[TextData[{ "Then, we connect the expanded system ", StyleBox["antenna1", "MR"], " and the Kalman filter according to the diagram in the figure below. First \ we connect the inputs. In the figure, the inputs and outputs are numbered as \ they appear after this stage. 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