# ControllabilityMatrix

gives the controllability matrix of the state-space model ssm.

# Details

• For a standard state-space model with state equations:
•  continuous-time system discrete-time system
• The controllability matrix is computed as , where is the dimension of .
• For a descriptor state-space model with state equations:
•  continuous-time system discrete-time system
• The slow and fast subsystems can be decoupled as described in KroneckerModelDecomposition:
•  slow subsystem fast subsystem
• ControllabilityMatrix returns a pair of matrices {q1,q2}, based on the decoupled slow and fast subsystems. The matrices q1 and q2 are defined as follows, where is the dimension of , and is the nilpotency index of .
•  slow subsystem fast subsystem
• The controllability matrices only exist for descriptor systems in which Det[λ e-a]0 for some λ.

# Examples

open allclose all

## Basic Examples(2)

The controllability matrix of a state-space model:

The controllability multi-input state-space model:

## Scope(5)

The controllability matrix of a symbolic single-input system:

The controllability matrix of a two-input system has twice as many columns:

The controllability matrix of an uncontrollable single-input system:

The controllability matrix of a diagonal multiple-input system:

The controllability matrix of a third-order system:

A singular descriptor system returns two matrices:

## Properties & Relations(8)

The computation depends only on the state and input matrices:

A system is controllable if and only if its controllability matrix has full rank:

This system is not controllable, but is output-controllable:

This system is controllable, but is not output-controllable:

The controllability matrix of a discrete-time system does not depend on the sampling period:

For descriptor systems, the slow and fast system matrices need to be full rank for controllability:

Controllability of the slow subsystem is determined by the first matrix:

For nonsingular descriptor systems, the fast system matrix is empty:

Each matrix is associated with a subsystem from the Kronecker decomposition:

The controllability matrices match those for the original system:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_controllabilitymatrix, author="Wolfram Research", title="{ControllabilityMatrix}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ControllabilityMatrix.html}", note=[Accessed: 12-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_controllabilitymatrix, organization={Wolfram Research}, title={ControllabilityMatrix}, year={2012}, url={https://reference.wolfram.com/language/ref/ControllabilityMatrix.html}, note=[Accessed: 12-September-2024 ]}