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ControllableModelQ
✖
ControllableModelQ

✖

yields True if the state-space model sys is controllable, and False otherwise.

✖
ControllableModelQ[{sys,sub}]

yields True if the subsystem sub is controllable.

Details and Options

  • ControllableModelQ is also known as a reachable model.
  • A state-space model is said to be controllable if for any initial state and any final state there exists some control input that drives the state from to in finite time.
  • The system sys can be a standard or descriptor StateSpaceModel or AffineStateSpaceModel.
  • The following subsystems sub can be specified: » »
  • Allwhole system
    "Fast"fast subsystem
    "Slow"slow subsystem
    {λ1,…}subsystem with eigenmodes
  • The "Fast" and "Slow" subsystems primarily apply to descriptor state-space models as described in KroneckerModelDecomposition.
  • The eigenmodes λi are described in JordanModelDecomposition.
  • ControllableModelQ accepts a Method option with the following settings:
  • Automaticautomatically choose the appropriate test
    "Distribution"use controllability distribution's rank
    "Gramian"use controllability Gramian's rank or positive definiteness
    "Matrix"use controllability matrix's rank
    "PBH"use Popov–Belevitch–Hautus rank test

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

A controllable system:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-4rseat
Out[1]=1

An uncontrollable system, since there is no way to affect the second state:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-6d1p9u
Out[1]=1

Scope  (6)Survey of the scope of standard use cases

Test the controllability of a system with approximate coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-k6h6p0
Out[1]=1

Exact coefficients:

In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-u4nfst
Out[2]=2

Symbolic coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-cuphvw
Out[3]=3

Multiple-input system:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-k51bv6
Out[1]=1

Discrete-time system:

In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-sr3ack
Out[2]=2

A descriptor system:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-sh5yfz
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-0ax8eu
Out[2]=2

Controllability is equivalent to controllability of both slow and fast modes (C-controllability):

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-j0qby6
Out[3]=3

Test controllability of individual eigenmodes:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-t795wb

The system is uncontrollable because of eigenmode :

In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-hnqko8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-j4uwzz
Out[3]=3

This can be seen in the Jordan form, where there is no way to affect the third state:

In[4]:=4
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-8nmo5d
Out[4]=4

Test controllability of an AffineStateSpaceModel:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-vlx8ex
Out[1]=1

If an operating point is given, controllability from is tested:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-r7kug9
Out[1]=1

The system is controllable from a generic point:

In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-tfrswu
Out[2]=2

Options  (7)Common values & functionality for each option

Method  (7)

By default, the controllability matrix is used for exact and symbolic systems:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-4vg30g
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-onzgng
Out[2]=2

The system is controllable if the ControllabilityMatrix has full rank:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-td87q5
In[4]:=4
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-zq44ny
Out[4]=4

The controllability Gramian is used for stable numeric systems:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-7ctner
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-coldt7
Out[2]=2

The system is controllable if the ControllabilityGramian has full rank:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-qqt69
In[4]:=4
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-ds2c8w
Out[4]=4

For the controllability Gramian, this is equivalent to it being positive definite:

In[5]:=5
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-n2m18a
Out[5]=5

The PBH rank test is used for all other numeric systems:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-9c4vhw
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-jtbvtb
Out[2]=2

The system is controllable because has full rank for all :

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-pkfdbb
In[4]:=4
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-dmeinh
Out[4]=4

The controllability distribution is used for input-linear systems:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-updc86
Out[1]=1

For linear systems, the tests based on the controllability matrix and distribution are equivalent:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-hkj51j
Out[1]=1

Controllability of the linearized system implies controllability of the input-linear system:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-iblsi8
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-9ga5h
In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-s06qyz
Out[3]=3

The matrix test for input linear system uses the "Matrix" method for the linearized system:

In[4]:=4
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-53x0lv
Out[4]=4

Using no control input, the system follows the drift vector field :

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-h2ubuf

A system is weakly controllable if the drift vector field can also be used to move state:

In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-ctp3p8
Out[2]=2

Without including the drift vector field, the system is not controllable:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-emix4e
Out[3]=3

Applications  (5)Sample problems that can be solved with this function

The position and velocities of all three masses can be controlled by the force on :

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-qhpwk5
Out[1]=1

An electric circuit with the capacitor voltage and inductor current as states:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-uex790

In general, the system is controllable:

In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-7jzujl
Out[2]=2

However if , it is not controllable:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-ua7e5t
Out[3]=3

A unicycle with drive and steering as input:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-0ndr10

The system is controllable, but the linearized system is not:

In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-fzpfc4
Out[2]=2

The parking problem:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-xqbiqp

The system is controllable, but the linearized system is not:

In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-c4gw4i
Out[2]=2

A rotating rigid satellite modeled using Euler's equations of motion:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-vqotat

Its controllability for various values of the principal moments and actuator combinations:

In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-1l2zza

A grid to show the results:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-zaulvf

If , the system is controllable with any two actuators:

In[4]:=4
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-cas2r3
Out[4]=4

If , the system is not controllable with actuators 1 and 2 or any one actuator:

In[5]:=5
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-zpo3q8
Out[5]=5

If , the system is controllable only with all 3 actuators:

In[6]:=6
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-dljwdm
Out[6]=6

Properties & Relations  (7)Properties of the function, and connections to other functions

A diagonal system is controllable, assuming and :

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-6nsclp
Out[1]=1

With , there is no way to control the first state:

In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-brtpcp
Out[2]=2

With , the first state cannot be controlled directly, but indirectly from the second state:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-fr7x6d
Out[3]=3

With , the second state cannot be controlled directly or indirectly from the first state:

In[4]:=4
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-bet95
Out[4]=4

Use JordanModelDecomposition to compute the canonical state-space representation above:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-fuaft5
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-m3tfhi
Out[2]=2

Compute the controllability of each mode using the "PBH" test:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-745cjk
Out[3]=3

For descriptor systems, KroneckerModelDecomposition is the generalization of the diagonal form:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-3k6skt
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-ol56y
Out[2]=2

Determine the controllability of the slow subsystem from its structure:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-j5tf2a
Out[3]=3

Compute it using the original system:

In[4]:=4
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-dccvkc
Out[4]=4

Determine the controllability of the fast subsystem:

In[5]:=5
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-v646g9
Out[5]=5

Compute it using the original system:

In[6]:=6
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-n1eh3q
Out[6]=6

If the descriptor matrix of a StateSpaceModel has full rank, there is no fast subsystem:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-wwty4o

Hence the complete controllability of the system can be evaluated from the slow subsystem:

In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-qmci4f
Out[2]=2

Controllability is invariant under a nonsingular StateSpaceTransform:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-b1429y
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-kddl4s
In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-okbxn5
Out[3]=3

Controllability is invariant under state feedback:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-hqzkns
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-gff48o
Out[2]=2

Compute the state feedback using StateFeedbackGains:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-lwj3ql
Out[3]=3

The closed-loop system is also controllable:

In[4]:=4
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-epg8ze
In[5]:=5
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-w6lu4m
Out[5]=5

Controllability does not mean it is output controllable (OutputControllableModelQ):

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-2frou5
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-gw7vd8
Out[2]=2

Output controllability does not imply controllability:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-hh9vk2
In[4]:=4
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-fcw0v9
Out[4]=4

Possible Issues  (2)Common pitfalls and unexpected behavior

The Gramian method is not reliable for systems that are not asymptotically stable:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-3skwej
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-dtf4fu
Out[2]=2

The eigenvalue in the right half of the complex plane leads to instability for continuous-time systems:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-ucbwah
Out[3]=3

For affine systems with nonzero drift, the distribution tests only for accessibility:

In[1]:=1
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-pffk2w
In[2]:=2
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-n9kpwx
Out[2]=2

A random initial condition and random input signal generator:

In[3]:=3
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-oj6d9s

Simulate the system 10 times with random inputs:

In[4]:=4
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-3spfof

The system cannot be moved to the left of the initial point:

In[5]:=5
✖
https://wolfram.com/xid/0cf1p1u1v56jj3n-xi1r4f
Out[5]=5
Wolfram Research (2010), ControllableModelQ, Wolfram Language function, https://reference.wolfram.com/language/ref/ControllableModelQ.html (updated 2014).
✖
Wolfram Research (2010), ControllableModelQ, Wolfram Language function, https://reference.wolfram.com/language/ref/ControllableModelQ.html (updated 2014).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_controllablemodelq, author="Wolfram Research", title="{ControllableModelQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ControllableModelQ.html}", note=[Accessed: 04-June-2025 ]}

✖
@misc{reference.wolfram_2025_controllablemodelq, author="Wolfram Research", title="{ControllableModelQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ControllableModelQ.html}", note=[Accessed: 04-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_controllablemodelq, organization={Wolfram Research}, title={ControllableModelQ}, year={2014}, url={https://reference.wolfram.com/language/ref/ControllableModelQ.html}, note=[Accessed: 04-June-2025 ]}

✖
@online{reference.wolfram_2025_controllablemodelq, organization={Wolfram Research}, title={ControllableModelQ}, year={2014}, url={https://reference.wolfram.com/language/ref/ControllableModelQ.html}, note=[Accessed: 04-June-2025 ]}