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ControllabilityMatrix
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ControllabilityMatrix

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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)Summary of the most common use cases

The controllability matrix of a state-space model:

In[1]:=1
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https://wolfram.com/xid/0v5mlez1y02f2-upem7l

The controllability multi-input state-space model:

In[1]:=1
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https://wolfram.com/xid/0v5mlez1y02f2-gf48cp

Scope  (5)Survey of the scope of standard use cases

The controllability matrix of a symbolic single-input system:

In[1]:=1
✖
https://wolfram.com/xid/0v5mlez1y02f2-b2z3gx

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

In[2]:=2
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https://wolfram.com/xid/0v5mlez1y02f2-726ze

The controllability matrix of an uncontrollable single-input system:

In[1]:=1
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https://wolfram.com/xid/0v5mlez1y02f2-etce3u

The controllability matrix of a diagonal multiple-input system:

In[1]:=1
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https://wolfram.com/xid/0v5mlez1y02f2-uoved

The controllability matrix of a third-order system:

In[1]:=1
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https://wolfram.com/xid/0v5mlez1y02f2-jbhvld

A singular descriptor system returns two matrices:

In[1]:=1
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https://wolfram.com/xid/0v5mlez1y02f2-cn4ws
In[2]:=2
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https://wolfram.com/xid/0v5mlez1y02f2-zbfiia
Out[2]=2

Properties & Relations  (8)Properties of the function, and connections to other functions

The computation depends only on the state and input matrices:

In[1]:=1
✖
https://wolfram.com/xid/0v5mlez1y02f2-p3th3c

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

In[1]:=1
✖
https://wolfram.com/xid/0v5mlez1y02f2-07glrp
In[2]:=2
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https://wolfram.com/xid/0v5mlez1y02f2-id9nis
Out[2]=2
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0v5mlez1y02f2-bsrx2
In[2]:=2
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https://wolfram.com/xid/0v5mlez1y02f2-dqmh4k
Out[2]=2
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0v5mlez1y02f2-12umlc
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0v5mlez1y02f2-6ti71r
In[2]:=2
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https://wolfram.com/xid/0v5mlez1y02f2-ft42uu
Out[2]=2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0v5mlez1y02f2-fv6yb1
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0v5mlez1y02f2-9g3yz5

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

In[1]:=1
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https://wolfram.com/xid/0v5mlez1y02f2-dqtgb6
In[2]:=2
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https://wolfram.com/xid/0v5mlez1y02f2-gtdse
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0v5mlez1y02f2-4aserv
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0v5mlez1y02f2-gedfn
Out[4]=4
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0v5mlez1y02f2-go90w
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0v5mlez1y02f2-cd755t
Out[1]=1

The controllability matrices match those for the original system:

In[2]:=2
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https://wolfram.com/xid/0v5mlez1y02f2-kaoaf4
In[3]:=3
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https://wolfram.com/xid/0v5mlez1y02f2-5iqa7
Out[3]=3
Out[3]=3
Wolfram Research (2010), ControllabilityMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/ControllabilityMatrix.html (updated 2012).
✖
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).

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_controllabilitymatrix, author="Wolfram Research", title="{ControllabilityMatrix}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ControllabilityMatrix.html}", note=[Accessed: 29-March-2025 ]}

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@misc{reference.wolfram_2025_controllabilitymatrix, author="Wolfram Research", title="{ControllabilityMatrix}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ControllabilityMatrix.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_controllabilitymatrix, organization={Wolfram Research}, title={ControllabilityMatrix}, year={2012}, url={https://reference.wolfram.com/language/ref/ControllabilityMatrix.html}, note=[Accessed: 29-March-2025 ]}

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@online{reference.wolfram_2025_controllabilitymatrix, organization={Wolfram Research}, title={ControllabilityMatrix}, year={2012}, url={https://reference.wolfram.com/language/ref/ControllabilityMatrix.html}, note=[Accessed: 29-March-2025 ]}