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ObservabilityMatrix[ssm]

gives the observability matrix of the state-space model ssm.

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ObservabilityMatrix

ObservabilityMatrix[ssm]

gives the observability matrix of the state-space model ssm.

Details

  • For a standard state-space model:
  • continuous-time system
    discrete-time system
  • The observability matrix is given by , where is the dimension of .
  • For a descriptor state-space model:
  • continuous-time system
    discrete-time system
  • The slow and fast subsystems can be decoupled as described in KroneckerModelDecomposition:
  • slow subsystem
    fast subsystem
    output equation
  • ObservabilityMatrix returns a pair of matrices , based on the decoupled slow and fast subsystems. The matrices and are defined as follows, where is the dimension of , and is the nilpotency index of .
  • slow subsystem
    fast subsystem
  • The observability matrices only exist for descriptor systems in which Det[λ e-a]≠0 for some λ.

Examples

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Basic Examples  (1)

The observability matrix of a state-space model:

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

Scope  (2)

The observability matrix of a continuous-time system:

Wolfram Language code: ObservabilityMatrix[StateSpaceModel[{{{0, 1}, {-10, -2}}, {{2}, {6}}, {{1, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]//MatrixForm

A singular system returns two matrices:

Wolfram Language code: om = ObservabilityMatrix[StateSpaceModel[{{{0., -1., 0., 0.}, {-3., 0., 0., 0.}, {10., 0., 0., -1.}, {0., 1., 1., 1.}}, {{0.}, {0.}, {0.}, {-1.}}, {{0., 0., 1., 2.}}, {{0.}}, {{-0.5, 0., 0., 0.}, {0., 0., -1., 0.}, {0., 0., 0., 0.}, {0., 0., 0., 0.}}}, SamplingPeriod -> None, SystemsModelLabels -> None]];
Wolfram Language code: MatrixForm /@ om

Properties & Relations  (3)

A system is observable if and only if its observability matrix has full rank:

Wolfram Language code: ssm = StateSpaceModel[{{{1, -3, -1}, {-1, -2, 2}, {-1, 2, -2}}, {{1}, {-1}, {0}}, {{1, 1, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: q = ObservabilityMatrix[ssm];
Wolfram Language code: {ObservableModelQ[ssm], MatrixRank[q] == Length[q]}

The observability matrix of a discrete-time system does not depend on the sampling period:

Wolfram Language code: ObservabilityMatrix[StateSpaceModel[{{{-3, 1, 0}, {-5, 0, 1}, {-3, 0, 0}}, {{1}, {2}, {3}}, {{1, 0, 0}}, {{0}}}, SamplingPeriod -> τ, SystemsModelLabels -> None]]//MatrixForm

A descriptor system gives one matrix for the slow subsystem and one for the fast subsystem:

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

Complete observability requires both matrices to be full rank:

Wolfram Language code: {ObservableModelQ[ssm], MatrixRank[qf] === Length[qf]}

Observability of the slow subsystem is determined by the first matrix:

Wolfram Language code: {ObservableModelQ[{ssm, "Slow"}], MatrixRank[qs] === Length[qs]}
Wolfram Research (2010), ObservabilityMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/ObservabilityMatrix.html (updated 2012).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_observabilitymatrix, author="Wolfram Research", title="{ObservabilityMatrix}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ObservabilityMatrix.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_observabilitymatrix, organization={Wolfram Research}, title={ObservabilityMatrix}, year={2012}, url={https://reference.wolfram.com/language/ref/ObservabilityMatrix.html}, note=[Accessed: 12-June-2026]}