ObservableModelQ

ObservableModelQ[sys]

gives True if the system sys is observable, and False otherwise.

ObservableModelQ[{sys,sub}]

gives True if the subsystem sub is observable.

Details and Options

  • A state-space model is said to be observable at if the trajectory of the model from is distinguishable from that of another state in its neighborhood in finite time.
  • The system sys can be a standard or descriptor StateSpaceModel or AffineStateSpaceModel.
  • The following subsystems sub can be specified:
  • Allwhole system
    "Fast"fast subsystem
    "Slow"slow subsystem
    "Unstable"unstable subsystem
    {λ1,…}subsystem with eigenmodes lambda_(i)
  • The "Fast" and "Slow" subsystems primarily apply to descriptor state-space models as described in KroneckerModelDecomposition.
  • The eigenmodes λi are described in JordanModelDecomposition.
  • ObservableModelQ accepts a Method option with the following settings:
  • Automaticautomatically choose the appropriate test
    "Distribution"use observability distribution's rank
    "Gramian"use observability Gramian's rank or positive definiteness
    "Matrix"use observability matrix's rank
    "PBH"use Popov–Belevitch–Hautus rank test

Examples

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Basic Examples  (2)

An observable system:

An unobservable system, since the second state is not observable:

Scope  (6)

Test the observability of a system with approximate coefficients:

Exact coefficients:

Symbolic coefficients:

Multiple-output system:

Discrete-time system:

A descriptor system:

Observability is equivalent to observability of both slow and fast modes (C-observability):

Test observability of individual eigenmodes:

The system is unobservable because of eigenmode :

This can be seen in the Jordan form, where there is no way to observe the second state:

Test observability of an AffineStateSpaceModel:

If an operating point is given, observability at is tested:

The system is observable at a generic point:

Options  (6)

Method  (6)

By default, the observability matrix is used for exact and symbolic systems:

The system is observable if the ObservabilityMatrix has full rank:

The observability Gramian is used for stable numeric systems:

The system is observable if the ObservabilityGramian has full rank:

For the observability Gramian, this is equivalent to it being positive definite:

The PBH rank test is used for all other numeric systems:

The system is observable because has full rank for all :

The observability codistribution is used for input-linear systems:

For linear systems, the tests based on the observability matrix and codistribution are equivalent:

Observability of the linearized system implies observability of the input-linear system:

The matrix test for input-linear systems uses the "Matrix" method for the linearized system:

Applications  (2)

The positions and velocities of all three masses can be estimated from the measurement of :

An electric circuit with the capacitor voltage and inductor current as states and current as measurement:

In general, the system is observable:

However, if , it is not observable:

Properties & Relations  (6)

A diagonal system is observable, assuming and :

With , there is no way to observe the first state:

With , the second state cannot be observed directly, but indirectly from the first state:

With , the first state cannot be observed directly or indirectly from the second state:

Use JordanModelDecomposition to compute the preceding canonical state-space representation:

Compute the observability of each mode using the "PBH" test:

For descriptor systems, KroneckerModelDecomposition is the generalization of the diagonal form:

Determine the observability of the slow subsystem from its structure:

Compute it using the original system:

Determine the observability of the fast subsystem from its structure:

Compute it using the original system:

If the descriptor matrix of a StateSpaceModel has full rank, there is no fast subsystem:

Hence the complete controllability of the system can be evaluated from the slow subsystem:

For AffineStateSpaceModel, the nonlinearities in the input vectors aid observability:

The system with a linear input vector is not observable:

Unobservable systems have indistinguishable initial states:

The two initial states and produce indistinguishable outputs:

Possible Issues  (1)

The Gramian method is not reliable for systems that are not asymptotically stable:

Wolfram Research (2010), ObservableModelQ, Wolfram Language function, https://reference.wolfram.com/language/ref/ObservableModelQ.html (updated 2014).

Text

Wolfram Research (2010), ObservableModelQ, Wolfram Language function, https://reference.wolfram.com/language/ref/ObservableModelQ.html (updated 2014).

CMS

Wolfram Language. 2010. "ObservableModelQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/ObservableModelQ.html.

APA

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

BibTeX

@misc{reference.wolfram_2024_observablemodelq, author="Wolfram Research", title="{ObservableModelQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ObservableModelQ.html}", note=[Accessed: 24-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_observablemodelq, organization={Wolfram Research}, title={ObservableModelQ}, year={2014}, url={https://reference.wolfram.com/language/ref/ObservableModelQ.html}, note=[Accessed: 24-December-2024 ]}