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OutputControllabilityMatrix[ssm]

gives the output controllability matrix of the state-space model ssm.

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OutputControllabilityMatrix

OutputControllabilityMatrix[ssm]

gives the output controllability matrix of the state-space model ssm.

Details

  • The state-space model ssm can be given as StateSpaceModel[{a,b,c,d}], where a, b, c, and d represent the state, input, output, and transmission matrices in either a continuous-time or a discrete-time system:
  • continuous-time system
    discrete-time system
  • The output controllability matrix is given by , where is the dimension of the square state matrix .
  • For a descriptor state-space model, OutputControllabilityMatrix returns a matrix where is associated with the slow subsystem and is associated with the fast subsystem.
  • For StateSpaceModel[{a,b,c,d,e}] with singular descriptor matrix e, the output controllability matrix is computed by decoupling the slow and fast subsystems:
  • slow subsystem
    fast subsystem
    output equation
  • The output controllability matrix where has nilpotency index is given by .
  • OutputControllabilityMatrix only takes descriptor systems in which Det[λ e - a]≠0 for some λ.

Examples

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Basic Examples  (1)

The output controllability matrix of a state-space model:

Wolfram Language code: OutputControllabilityMatrix[StateSpaceModel[{{{0, 1, 3}, {-2, -2, -1}, {2, 0, -3}}, {{2}, {3}, {-3}}, {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{0}, {0}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]//MatrixForm

Scope  (2)

The output controllability matrix of a continuous-time system:

Wolfram Language code: OutputControllabilityMatrix[StateSpaceModel[{{{-3, -1, -2}, {-1, 1, -2}, {0, -3, -1}}, {{-2}, {-1}, {0}}, {{1, 1, 1}, {1, -1, -1}}, {{0}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]//MatrixForm

A descriptor system:

Wolfram Language code: OutputControllabilityMatrix[StateSpaceModel[{{{-1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{1}, {0}, {1}}, {{1, 1, 1}}, {{0}}, {{1, 0, 0}, {0, 0, 1}, {0, 0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]

Properties & Relations  (4)

A system is output controllable when the rank of the matrix equals the number of outputs:

Wolfram Language code: ssm = StateSpaceModel[{{{-3, -2, 0}, {1, 3, 2}, {1, 3, 1}}, {{0}, {-2}, {-2}}, {{1, -1, 0}, {1, 0, -1}}, {{0}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: MatrixRank[OutputControllabilityMatrix[ssm]] == Last@SystemsModelDimensions[ssm]
Wolfram Language code: OutputControllableModelQ[ssm]

This system is not output-controllable but is state-controllable:

Wolfram Language code: ssm = StateSpaceModel[{{{1, 1}, {-1, -2}}, {{1}, {-1}}, {{1, -1}, {-1, 1}}, {{0}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: {OutputControllableModelQ[ssm], ControllableModelQ[ssm]}

This system is output-controllable but not state-controllable:

Wolfram Language code: ssm = StateSpaceModel[{{{-1.5, -1.5, 0.5, -0.5}, {-0.5, -3.5, 0.5, -1.5}, {-0.5, -0.5, -2.5, -1.5}, {0.5, -0.5, -0.5, -2.5}}, {{3}, {2}, {2}, {1}}, {{0, 1/3, 0, 1/3}, {0.5, -1/6, 0.5, -1/6}}, {{0}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: {OutputControllableModelQ[ssm], ControllableModelQ[ssm]}

The output controllability matrix does not depend on the sampling period:

Wolfram Language code: OutputControllabilityMatrix[StateSpaceModel[{{{-3, -1, -2}, {-1, 1, -2}, {0, -3, -1}}, {{-2}, {-1}, {0}}, {{1, 1, 1}, {1, -1, -1}}, {{0}, {0}}}, SamplingPeriod -> τ, SystemsModelLabels -> None]]//MatrixForm
Wolfram Research (2010), OutputControllabilityMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/OutputControllabilityMatrix.html (updated 2012).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_outputcontrollabilitymatrix, author="Wolfram Research", title="{OutputControllabilityMatrix}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/OutputControllabilityMatrix.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_outputcontrollabilitymatrix, organization={Wolfram Research}, title={OutputControllabilityMatrix}, year={2012}, url={https://reference.wolfram.com/language/ref/OutputControllabilityMatrix.html}, note=[Accessed: 12-June-2026]}