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

\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) = \int_0^t e^{A\tau} B B^T e^{A^T \tau} d\tau = \int_0^t e^{A(t-\tau)} B B^T e^{A^T(t-\tau)} d\tau

The pair (A,B) is controllable if and only if the matrix W_c(t) is nonsingular, for any 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 rows of the matrix product \Phi(t_0,\tau)B(\tau) where \Phi is the state transition matrix are linearly independent. The Gramian is used to prove the linear independency of \Phi(t_0,\tau)B(\tau). To have linear independency Gramian matrix W_c have to be nonsingular, i.e., invertible.

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

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References[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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