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

${\text{rank}}[C^{T},A^{T}C^{T},...,(A^{T})^{n-1}C^{T}]=n$

See also

• Observability
• Controllability Gramian
• Gramian matrix

External links

• Mathematica function to compute the observability Gramian