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Descriptor Systems

Just like time-delay systems, descriptor systems are also fully integrated into the Wolfram Language.

    
Obtain a descriptor state-space model from a differential-algebraic equation:
Wolfram Language code: StateSpaceModel[{Subscript[x, 1]'[t] == Subscript[x, 1][t] + u[t], Subscript[x, 2][t] - 0.4Subscript[x, 1][t] == 0}, {Subscript[x, 1][t], Subscript[x, 2][t]}, {u[t]}, Subscript[x, 1][t], t]
A descriptor state-space model constructed from the state, input, output, transmission and descriptor matrices:
Wolfram Language code: StateSpaceModel[{(⁠| | | | -- | -- | | 1 | 0 | | -2 | -1 |⁠), (⁠| | | - | | 1 | | 2 |⁠), (⁠-1 0⁠), (⁠0.5⁠), (⁠| | | | - | - | | 1 | 0 | | 0 | 0 |⁠)}]
A descriptor state-space model from an improper transfer-function model:
Wolfram Language code: StateSpaceModel[TransferFunctionModel[Unevaluated[{{(s^2 + 4/s + 1)}}], s, SamplingPeriod -> None, SystemsModelLabels -> None]]
Construct a discrete-time descriptor system by specifying a sampling period:
Wolfram Language code: StateSpaceModel[{(⁠a⁠), (⁠b⁠), (⁠c⁠), (⁠d⁠), (⁠e⁠)}, SamplingPeriod -> τ]
Tight integration allows workflows for descriptor systems to match those of standard or proper systems:
Wolfram Language code: ControllableModelQ[StateSpaceModel[{{{-1, 0}, {0, -2}}, {{1}, {1}}, {{0, 1}}, {{0}}, {{1, 0}, {0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]
If the descriptor matrix is nonsingular, the system can be converted to a standard state-space representation:
Wolfram Language code: StateSpaceModel[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}, {{e}}}, SamplingPeriod -> None, SystemsModelLabels -> None], DescriptorStateSpace -> False]