Descriptor Systems 

Just like time-delay systems, descriptor systems are also fully integrated into Mathematica.

Obtain a descriptor state-space model from a differential-algebraic equation.

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X StateSpaceModel[{Subscript[x, 1]'[t] == Subscript[x, 1][t] + u[t], Subscript[x, 2][t] - 0.4 Subscript[x, 1][t] == 0}, {Subscript[x, 1][ t], Subscript[x, 2][t]}, {u[t]}, Subscript[x, 1][t], t]

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A descriptor state-space model constructed from the state, input, output, transmission, and descriptor matrices.

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X StateSpaceModel[{\!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", "0"}, { RowBox[{"-", "2"}], RowBox[{"-", "1"}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), ({ {1}, {2} }), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ { RowBox[{"-", "1"}], "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"0.5"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), ({ {1, 0}, {0, 0} })}]

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A descriptor state-space model from an improper transfer-function model.

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X StateSpaceModel[ TransferFunctionModel[Unevaluated[{{(s^2 + 4)/(s + 1)}}], s, SamplingPeriod ->None, SystemsModelLabels -> None]]

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Construct a discrete-time descriptor system by specifying a sampling period.

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X StateSpaceModel[{\!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"a"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"b"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"c"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"d"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"e"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\)}, SamplingPeriod -> \[Tau]]

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Tight integration allows workflows for descriptor systems to match those of standard or proper systems.

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X ControllableModelQ[ StateSpaceModel[{{{-1, 0}, {0, -2}}, {{1}, {1}}, {{0, 1}}, {{0}}, {{1, 0}, {0, 0}}}, SamplingPeriod ->None, SystemsModelLabels -> None]]

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If the descriptor matrix is nonsingular, the system can be converted to a standard state-space representation.

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X StateSpaceModel[ StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}, {{e}}}, SamplingPeriod ->None, SystemsModelLabels -> None], DescriptorStateSpace -> False]

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