# 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:

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