The controllability matrix for the constant-coefficient system of ordinary differential equations , with state variables and inputs, where and are positive integers, is the block matrix

The column space of consists of the states reachable from the zero state. The system is controllable, meaning that any initial state can be driven to any final state by a suitable input over a finite time interval, iff has matrix rank (Massachusetts Institute of Technology 2010, pp. 10-1-10-2).

For example, and give

which has matrix rank 2, so the system is controllable.

The same block matrix is also called a reachability matrix in control theory. This usage is distinct from the reachability matrix of a directed graph.

The Wolfram Language computes the controllability matrix of such a state-space model using ControllabilityMatrix[ssm].

See also

Block Matrix, Column Space, Control Theory, Linear Time-Invariant System, Matrix Rank, Reachability Matrix, State-Space Representation

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References

Massachusetts Institute of Technology. "Controllability." Topic 10 in 16.30/31 Feedback Control Systems. 2010. https://ocw.mit.edu/courses/16-30-feedback-control-systems-fall-2010/4bebd71d575bdde1384713e9efb400cc_MIT16_30F10_lec10.pdf.Wolfram Research. "ControllabilityMatrix." https://reference.wolfram.com/language/ref/ControllabilityMatrix.html.

Cite this as:

Weisstein, Eric W. "Controllability Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ControllabilityMatrix.html

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