# ObservableDecomposition

yields the observable subsystem of the system sys.

ObservableDecomposition[sys,{z1,}]

specifies the new coordinates zi.

# Examples

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## Basic Examples(1)

Find the observable subsystem and its transformation:

## Scope(4)

The observable subsystem of an observable system is the complete system:

The observable subsystem of a partially observable continuous-time system:

The observable subsystem of a descriptor system:

The observable subsystem of an affine system:

Specify the new variables:

## Applications(7)

### Linear Systems(4)

Construct the Kalman observable decomposition:

ObservableDecomposition picks out the observable subsystem only:

Kalman observable decomposition puts the observable subsystem first and keeps the rest:

Compute the dimension of the observable subspace:

The observable subspace is the range of p, i.e. the column dimension:

Find the observable subspace for the system below and show what state trajectories you can tell apart from observing the output only:

The system is unobservable, so only a subspace is observable from output:

The range of the transformation p gives the observable subspace:

Simulate trajectories whose initial value projects to a single point on the observable subspace:

From observing the output, all these trajectories look identical:

Determine states that can be estimated using available measurements and design an estimator:

Only the position of mass is measured, and so the system is not completely observable:

The states associated with zero rows in the transformation matrix cannot be observed:

An estimator can be designed to estimate any combination of the first four states:

An estimator that estimates :

Compute the response of for a set of input signals and initial conditions:

The estimated state trajectories:

### Affine Systems(3)

Construct the triangular observability decomposition:

ObservableDecomposition picks out the observable subsystem only:

Triangular observability decomposition puts the observable subsystem first and keeps the rest:

Compute the dimension of the observable subspace:

The dimension can be obtained from the inverse transformation :

Find the subspaces whose outputs are indistinguishable:

The system is unobservable, so only a subspace is observable from output:

The indistinguishable subspace:

Two points on the subspace:

The trajectories of the outputs from the two points are the same:

## Properties & Relations(2)

The transformation matrix p selects the observable subsystem using StateSpaceTransform:

For affine systems, the transformation rules select the observable subsystem:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2023_observabledecomposition, author="Wolfram Research", title="{ObservableDecomposition}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ObservableDecomposition.html}", note=[Accessed: 17-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_observabledecomposition, organization={Wolfram Research}, title={ObservableDecomposition}, year={2014}, url={https://reference.wolfram.com/language/ref/ObservableDecomposition.html}, note=[Accessed: 17-April-2024 ]}