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DualSystemsModel[ssm]

gives the dual of the state-space model ssm.

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DualSystemsModel

DualSystemsModel[ssm]

gives the dual of the state-space model ssm.

Details

  • The state-space model ssm can be a standard or descriptor StateSpaceModel.
  • The original and dual systems for a standard StateSpaceModel[{a,b,c,d}]:
  • The original and dual systems for a descriptor StateSpaceModel[{a,b,c,d,e}]:

Examples

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Basic Examples  (1)

The dual of a system:

Wolfram Language code: DualSystemsModel[StateSpaceModel[{{{0, 1}, {Subscript[a, 1], Subscript[a, 2]}}, {{1}, {-1}}, {{1, 0}, {1, 1}}, {{0}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]

Scope  (3)

A dual model:

Wolfram Language code: DualSystemsModel[StateSpaceModel[{{{-2, 0}, {-3, -4}}, {{1}, {-1}}, {{0.5, 1.5}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]

All variables are considered as complex:

Wolfram Language code: dualssm = DualSystemsModel[StateSpaceModel[{{{Subscript[a, 1], Subscript[a, 2]}, {Subscript[a, 3], Subscript[a, 4]}}, {{Subscript[b, 1]}, {Subscript[b, 2]}}, {{Subscript[c, 1], Subscript[c, 2]}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]

Simplify, assuming all variables are real:

Wolfram Language code: Simplify[dualssm, {Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4], Subscript[b, 1], Subscript[b, 2], Subscript[c, 1], Subscript[c, 2]}∈Reals]

Specify only some variables to be real:

Wolfram Language code: Simplify[dualssm, {Subscript[b, 1], Subscript[b, 2], Subscript[c, 1], Subscript[c, 2]}∈Reals]

The dual of a descriptor system:

Wolfram Language code: DualSystemsModel[StateSpaceModel[{{{1, 0, 0}, {0, 1, 0}, {0, 0, -1/2}}, {{0}, {1}, {1}}, {{-1, 1, 9/16}}, {{0}}, {{0, 1, 0}, {0, 0, 0}, {0, 0, 1}}}, SamplingPeriod -> 1, SystemsModelLabels -> None]]

Properties & Relations  (4)

The controllable and observable realizations have a dual relationship:

Wolfram Language code: m = {{(s + Subscript[z, 11]/s + Subscript[p, 11]), (s + Subscript[z, 12]/s + Subscript[p, 12])}, {(s + Subscript[z, 21]/s + Subscript[p, 21]), (s + Subscript[z, 22]/s + Subscript[p, 22])}};
Wolfram Language code: Subscript[ssm, 1] = StateSpaceModel[TransferFunctionModel[m, s], StateSpaceRealization -> "Controllable"];
Wolfram Language code: Subscript[ssm, 2] = Simplify[DualSystemsModel@StateSpaceModel[TransferFunctionModel[m//Transpose, s], StateSpaceRealization -> "Observable"], Element[{Subscript[p, 11], Subscript[p, 12], Subscript[p, 21], Subscript[p, 22], Subscript[z, 11], Subscript[z, 12], Subscript[z, 21], Subscript[z, 22]}, Reals]];
Wolfram Language code: {Subscript[ssm, 1], Subscript[ssm, 2]}
Wolfram Language code: Normal[Subscript[ssm, 1]] === Expand[Normal[Subscript[ssm, 2]]]

A similar relationship exists between the controllable companion and observable companion forms:

Wolfram Language code: Subscript[ssm, 1] = StateSpaceModel[TransferFunctionModel[m, s], StateSpaceRealization -> "ControllableCompanion"];
Wolfram Language code: Subscript[ssm, 2] = Simplify[DualSystemsModel@StateSpaceModel[TransferFunctionModel[m//Transpose, s], StateSpaceRealization -> "ObservableCompanion"], Element[{Subscript[p, 11], Subscript[p, 12], Subscript[p, 21], Subscript[p, 22], Subscript[z, 11], Subscript[z, 12], Subscript[z, 21], Subscript[z, 22]}, Reals]];
Wolfram Language code: Expand[Normal[Subscript[ssm, 1]]] === Expand[Normal[Subscript[ssm, 2]]]

An uncontrollable but observable model:

Wolfram Language code: ssm = StateSpaceModel[{{{-1, 0, 0}, {0, -3, 1}, {0, 0, -3}}, {{1}, {3}, {0}}, {{1, 1, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]; Through@{ControllableModelQ, ObservableModelQ}@ssm

The dual model is controllable but not observable:

Wolfram Language code: With[{dualssm = DualSystemsModel[ssm]}, Through@{ControllableModelQ, ObservableModelQ}@dualssm]

The controllability matrix of a system is the conjugate transpose of the observability matrix of the dual system and vice versa:

Wolfram Language code: ssm = StateSpaceModel[{{{a[1, 1], a[1, 2]}, {a[2, 1], a[2, 2]}}, {{b[1, 1]}, {b[2, 1]}}, {{c[1, 1], c[1, 2]}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]; dualssm = DualSystemsModel[ssm];
Wolfram Language code: ControllabilityMatrix[ssm] == ObservabilityMatrix[dualssm]
Wolfram Language code: ObservabilityMatrix[ssm] == ControllabilityMatrix[dualssm]

The estimator gains are the conjugate transpose of feedback gains of the dual model and vice versa:

Wolfram Language code: ssm = StateSpaceModel[{{{0, 1, 0}, {0, 0, 1}, {-3, -2, -1}}, {{0}, {0}, {1}}, {{1, 0, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: {StateFeedbackGains[ssm, {-2, -3, -4}], ConjugateTranspose[EstimatorGains[DualSystemsModel[ssm], {-2, -3, -4}]]}
Wolfram Language code: {EstimatorGains[ssm, {-2, -3, -4}], ConjugateTranspose[StateFeedbackGains[DualSystemsModel[ssm], {-2, -3, -4}]]}
Wolfram Research (2010), DualSystemsModel, Wolfram Language function, https://reference.wolfram.com/language/ref/DualSystemsModel.html (updated 2012).

Text

Wolfram Research (2010), DualSystemsModel, Wolfram Language function, https://reference.wolfram.com/language/ref/DualSystemsModel.html (updated 2012).

CMS

Wolfram Language. 2010. "DualSystemsModel." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2012. https://reference.wolfram.com/language/ref/DualSystemsModel.html.

APA

Wolfram Language. (2010). DualSystemsModel. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DualSystemsModel.html

BibTeX

@misc{reference.wolfram_2026_dualsystemsmodel, author="Wolfram Research", title="{DualSystemsModel}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/DualSystemsModel.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_dualsystemsmodel, organization={Wolfram Research}, title={DualSystemsModel}, year={2012}, url={https://reference.wolfram.com/language/ref/DualSystemsModel.html}, note=[Accessed: 12-June-2026]}