# 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

${\displaystyle {\dot {x}}=Ax+Bu}$

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

${\displaystyle AW_{c}+W_{c}A^{T}=-BB^{T}}$

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

${\displaystyle W_{c}=\int \limits _{0}^{\infty }e^{A\tau }BB^{T}e^{A^{T}\tau }\;d\tau }$

A related matrix used for determining controllability is

${\displaystyle W_{c}(t_{0},t_{1})=\int _{t_{0}}^{t_{1}}e^{A(t_{0}-\tau )}BB^{T}e^{A^{T}(t_{0}-\tau )}\;d\tau }$

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

Linear time-variant state space models of form

${\displaystyle {\dot {x}}(t)=A(t)x(t)+B(t)u(t)}$,
${\displaystyle y(t)=C(t)x(t)+D(t)u(t)}$

are controllable in an interval ${\displaystyle [t_{0},t_{1}]}$ if and only if the Gramian matrix ${\displaystyle W_{c}(t_{0},t_{1})}$ is nonsingular, where

${\displaystyle W_{c}(t_{0},t_{1})=\int \limits _{t_{0}}^{t_{1}}\Phi (t_{0},\tau )B(\tau )B^{T}(\tau )\Phi ^{T}(t_{0},\tau )\;d\tau }$