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 allBasic Examples (2)
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:
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:
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