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

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

See also

• Controllability
• Observability Gramian
• Gramian matrix
• Minimum energy control

References

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.

External links

• Mathematica function to compute the controllability Gramian