ObservabilityGramian

ObservabilityGramian[ssm]

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

Details and Options

  • 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 observability Gramian:
  • continuous-time system
    discrete-time system
  • For asymptotically stable systems, the Gramian can be computed as the solution of a Lyapunov equation:
  • continuous-time system
    discrete-time system
  • For a StateSpaceModel with a descriptor matrix, ObservabilityGramian returns a pair of matrices {wos,wof}, where wos is associated with the slow subsystem, and wof is associated with the fast subsystem.
  • The observability Gramians 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 Gramian of a system:

Scope  (4)

The observability Gramian of a continuous-time system:

The observability Gramian of a discrete-time system:

The observability Gramian of a symbolic system:

The observability Gramian of a descriptor system:

Applications  (1)

Check if a system is observable:

Properties & Relations  (7)

The observability Gramian is the controllability Gramian of the dual system:

The observability Gramian has the dimension of the state matrix:

If the observability Gramian has full rank, the system is observable:

The observability Gramian of an observable and asymptotically stable system is symmetric and positive definite:

The observability Gramian of a continuous-time (discrete-time) system satisfies a continuous (discrete) Lyapunov equation:

Descriptor systems give two observability Gramians:

The system is completely observable if and only if the sum is positive definite:

The fast and slow subsystem Gramians are computed from the Kronecker decomposition:

Split the decoupled states:

The slow subsystem yields a Gramian for the slow states and a zero matrix:

The fast subsystem yields a Gramian for the fast states and a zero matrix:

Inversing the Kronecker transformation gives the Gramians for the original system:

This gives the same result as using ObservabilityGramian directly:

Possible Issues  (1)

The observability Gramian is not defined for a system that is not asymptotically stable:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_observabilitygramian, author="Wolfram Research", title="{ObservabilityGramian}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ObservabilityGramian.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_observabilitygramian, organization={Wolfram Research}, title={ObservabilityGramian}, year={2012}, url={https://reference.wolfram.com/language/ref/ObservabilityGramian.html}, note=[Accessed: 21-December-2024 ]}