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The block diagram of a PID controller connected to \ the system is shown in the figure\[NonBreakingSpace]below. Only the \ propagation of the reference signal ", StyleBox["r", FontSlant->"Italic"], " to the controlled output ", StyleBox["c", FontSlant->"Italic"], " is taken into account. A typical system would also have a disturbance \ input, not shown on the diagram. The prefilter can be used to further correct \ the dynamics of the system, for example, by eliminating unwanted zeros from \ the closed-loop transfer function. Note that the derivative part of the \ controller includes the term ", Cell[BoxData[ \(TraditionalForm\`1/\((\[Tau] s + 1)\)\)], "InlineFormula", FormatType->StandardForm, CellTags->"S0.0.6"], ", with a presumably small time constant ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\)], "InlineFormula", FormatType->StandardForm, CellTags->"S0.0.6"], ". 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", "MathCaption", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"6.5"], Cell[CellGroupData[{ Cell[BoxData[ \(pid = TransferFunction[s, k\_p + k\_i\/s + \(k\_d\ s\)\/\(1 + \[Tau]\ s\)]\)], "Input", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"6.5"], Cell[BoxData[ FormBox[ TagBox[ FormBox[ SubsuperscriptBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(\[ScriptS]\ k\_d\)\/\(\[ScriptS]\ \[Tau] + 1\) + k\_i\/\[ScriptS] + k\_p\)} }, AllowScriptLevelChange->False], "\[NoBreak]", ")"}], "TraditionalForm"], FormBox[ StyleBox["\<\"\[InvisibleSpace]\"\>", ShowStringCharacters->False], "TraditionalForm"], FormBox[ StyleBox["\<\"\[ScriptCapitalT]\"\>", Editable->False, ShowStringCharacters->False], "TraditionalForm"], MultilineFunction->None], "TraditionalForm"], (TransferFunction[ s, SlotSequence[ 1]]&)], TraditionalForm]], "Output", CellMargins->{{54, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell["\<\ This connects the controller to the plant, closes the feedback loop, and \ simplifies the result. \ \>", "MathCaption", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"6.5"], Cell[CellGroupData[{ Cell[BoxData[ \(\(SeriesConnect[pid, plant] // FeedbackConnect\) // Simplify\)], "Input",\ CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"6.5"], Cell[BoxData[ FormBox[ TagBox[ FormBox[ SubsuperscriptBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "0", "0", "0"}, {"0", \(-\(1\/\[Tau]\)\), \(-1\), "0", "1"}, {"0", "0", "0", "1", "0"}, {\(k\_i\/\[Tau]\), \(k\_i - k\_d\/\[Tau]\^2\), \(-\(\(k\_d + \[Tau]\ k\_p\)\/\ \[Tau]\)\), "0", \(k\_d\/\[Tau] + k\_p\)}, {"0", "0", "1", "0", "0"} }, RowLines->{False, False, False, True, False}, ColumnLines->{False, 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", CellMargins->{{54, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell["This finds the transfer function of the closed-loop system. 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", "MathCaption", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->{"6.5", "i:22"}], Cell[CellGroupData[{ Cell[BoxData[ \(d\_1 = Expand[%\/\[Tau]]\)], "Input", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, CellTags->"6.5"], Cell[BoxData[ \(TraditionalForm\`s\^4 + s\^3\/\[Tau] + \(k\_d\ s\^2\)\/\[Tau] + k\_p\ s\^2 + k\_i\ s + \(k\_p\ s\)\/\[Tau] + k\_i\/\[Tau]\)], "Output", CellMargins->{{54, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[TextData[{ "Suppose now that the closed\[Hyphen]loop system with desired dynamics has \ poles at ", Cell[BoxData[ \(TraditionalForm\`\(-p\_1\)\)], FormatType->StandardForm], ", ", Cell[BoxData[ \(TraditionalForm\`\(-p\_2\)\)], FormatType->StandardForm], ", ", Cell[BoxData[ \(TraditionalForm\`\(-p\_3\)\)], FormatType->StandardForm], ", and ", Cell[BoxData[ \(TraditionalForm\`\(-p\_4\)\)], FormatType->StandardForm], ". This is the denominator of the corresponding transfer function. 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