Observability Gramian

The Observability Gramian is a Gramian used in control theory to determine whether or not a linear system is observable.

For a linear system described by

$\dot{x}(t) = A(t) x(t) + B(t) u(t)$

$y(t) = C(t) x(t) + D(t) u(t) \,$

The observability Gramian for a linear time variant system is given by

$W_{o}(t_{0},t_{1})=\int_{t_{0}}^{t_{1}}\Phi^{T}(s,t_{0})C^{T}(s)C(s)\Phi(s,t_{0}) ds$ ,

where $\Phi$ is the state transition matrix.

The system is observable on the interval $t\in[t_{0},t_{1}]$ if and only if $W_{o}(t_{0},t_{1})$ is nonsingular. In the case of a linear time invariant system, this can be simplified to finding the rank of the "observability matrix". If $x(t)$ is a $n$ -dimensional real-valued vector, then the system is observable if and only if