LMIs in Control/pages/LMI for the Observability Grammian

LMI for the Observability Grammian

Observability is a system property which says that the state of the system $$x(T)$$ can be reconstructed using the input $$u(t)$$ and output $$y(t)$$ on an interval $$[0,T]$$. This is necessary when knowledge of the full state is not available. If observable, estimators or observers can be created to reconstruct the full state. Observability and controllability are dual concepts. Thus in order to investigate the observability of a system we can study the controllability of the dual system. Although system observability can be determined with multiple methods, one is to compute the rank of the observability grammian.

The System


\begin{align} \dot x(t)&=Ax(t)+Bu(t),\\ y(t)&=Cx(t)+Du(t),\\ x(0)&=x_0, \end{align}$$ where $$x(t)\in \R^n$$, $$u(t)\in \R^m$$, at any $$t\in \R$$.

The Data
The matrices necessary for this LMI are $$A$$ and $$C$$.

The LMI:LMI to Determine the Observability Grammian
$$(A,B)$$ is observable if and only if $$Y>0$$ is the unique solution to
 * $$AY+YA^T+C^TC<0$$,

where $$Y$$ is the observability grammian.

Conclusion:
The above LMI attempts to find the observability grammian $$Y$$of the system $$(A,C)$$. If the problem is feasible and a unique $$Y$$ is found, then the system is also observable. The observability grammian can also be computed as: $$Y=\int_0^{\infty} e^{A^Ts}C^TCe^{As} ds$$. Due to the dual nature of observability and controllability this LMI can be determined by determining the controllability of the dual nature, which results in the above LMI. The Observability and Controllability matricies are written as $$\mathcal{O}$$ and $$\mathcal{C}$$ respectively. They are related as follows:
 * $$\mathcal{O}(C,A)=\mathcal{C}(A^T,C^T)^T$$
 * $$\mathcal{C}(A,B)=\mathcal{O}(B^T,A^T)^T $$

Hence $$(C,A)$$ is observable if and only if $$(A^T,C^T)$$ is controllable. Please refer to the section on controllability grammians.

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/eoskowro/LMI/blob/master/Observability_Gram_LMI.m

Related LMIs
Stabilizability LMI

Hurwitz Stability LMI

Detectability LMI

Controllability Grammian LMI