# 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).

#### 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_2024_observabilitymatrix, author="Wolfram Research", title="{ObservabilityMatrix}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ObservabilityMatrix.html}", note=[Accessed: 12-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_observabilitymatrix, organization={Wolfram Research}, title={ObservabilityMatrix}, year={2012}, url={https://reference.wolfram.com/language/ref/ObservabilityMatrix.html}, note=[Accessed: 12-September-2024 ]}