Controllability Gramian

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

\dot{x} = A x + B u

if all eigenvalues of $A$ have negative real part, then the unique solution $W_{c}$ of the Lyapunov equation

A W_c + W_c A^T = -BB^T

is positive definite if and only if the pair $(A, B)$ is controllable. $W_{c}$ is known as the controllability Gramian and can also be expressed as

W_c = \int\limits_0^\infty e^{A\tau} B B^T e^{A^T \tau} \; d\tau

A related matrix used for determining controllability is

W_c(t_0, t_1) = \int_{t_0}^{t_1} e^{A(t_0 - \tau)} BB^T e^{A^T(t_0 - \tau)} \; d\tau

The pair $(A,B)$ is controllable if and only if the matrix $W_c(t_0, t_1)$ is nonsingular, for any $t_{1}>t_{0}$ .^[1]^[2] A physical interpretation of the controllability Gramian is that if the input to the system is white gaussian noise, then $W_{c}$ is the covariance of the state.^[3]

Linear time-variant state space models of form

\dot{x}(t) = A(t) x(t) + B(t) u(t)

y(t) = C(t) x(t) + D(t) u(t)

are controllable in an interval $[t_0,t_1]$ if and only if the Gramian matrix $W_c(t_0, t_1)$ is nonsingular, where

W_c(t_0, t_1) = \int\limits_{t_0}^{t_1} \Phi(t_0,\tau)B(\tau)B^T(\tau)\Phi^T(t_0,\tau) \; d\tau

References

↑ 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.

External links

Mathematica function to compute the controllability Gramian

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Controllability Gramian

See also

References

External links