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

open allclose all

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:

Introduced in 2010
 (8.0)
 |
Updated in 2012
 (9.0)