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 . 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.
Linear time-variant state space models of form
are controllable in an interval if and only if the rows of the matrix product where is the state transition matrix are linearly independent. The Gramian is used to prove the linear independency of . To have linear independency Gramian matrix have to be nonsingular, i.e., invertible.
- Controllability Gramian Lecture notes to ECE 521 Modern Systems Theory by Professor A. Manitius, ECE Department, George Mason University.
- 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.
- 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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