Controllability Gramian

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In control theory, the controllability Gramian is a Gramian used to determine whether or not a linear system is controllable. For the time-invariant linear system

if all eigenvalues of have negative real part, then the unique solution of the Lyapunov equation

is positive definite if and only if the pair is controllable. is known as the controllability Gramian and can also be expressed as

A related matrix used for determining controllability is

The pair is controllable if and only if the matrix is nonsingular, for any .[1][2] A physical interpretation of the controllability Gramian is that if the input to the system is white gaussian noise, then is the covariance of the state.[3]

Linear time-variant state space models of form


are controllable in an interval if and only if the Gramian matrix is nonsingular, where

See also[edit]


  1. ^ Controllability Gramian Lecture notes to ECE 521 Modern Systems Theory by Professor A. Manitius, ECE Department, George Mason University.
  2. ^ Chen, Chi-Tsong (1999). Linear System Theory and Design Third Edition. New York, New York: Oxford University Press. p. 145. ISBN 0-19-511777-8. 
  3. ^ Franklin, Gene F. (2002). Feedback Control of Dynamic Systems Fourth Edition. Upper Saddle River, New Jersey: Prentice Hall. p. 854. ISBN 0-13-032393-4. 

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