Controllability Gramian

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

the controllability Gramian is given by

W_c (t) = \int\limits_0^t e^{A\tau} B B^T e^{A^T \tau} d\tau = \int\limits_0^t e^{A(t-\tau)} B B^T e^{A^T(t-\tau)} d\tau

If the matrix Wc is nonsingular, i.e. Wc has full rank, for any t > 0, then the pair (A,B) is controllable.

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 [t0,t1] if the rows of the matrix productΦ(t0,τ)B(τ) where Φ is the state transition matrix are linearly independent. The Gramian is used to prove the linear independency of Φ(t0,τ)B(τ). To have linear independency Gramian matrix Wc have to be nonsingular, i.e., invertible.

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

See also

References

External links

  • Controllability Gramian Lecture notes to ECE 521 Modern Systems Theory by Professor A. Manitius, ECE Department, George Mason University.

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