# 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

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

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.

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