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:

Scope  (2)

The observability matrix of a continuous-time system:

A singular system returns two matrices:

Properties & Relations  (3)

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

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

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

Complete observability requires both matrices to be full rank:

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

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).

BibTeX

@misc{reference.wolfram_2021_observabilitymatrix, author="Wolfram Research", title="{ObservabilityMatrix}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ObservabilityMatrix.html}", note=[Accessed: 04-August-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_observabilitymatrix, organization={Wolfram Research}, title={ObservabilityMatrix}, year={2012}, url={https://reference.wolfram.com/language/ref/ObservabilityMatrix.html}, note=[Accessed: 04-August-2021 ]}

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