User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Continuous-time Detectability

Detectability LMI
Detectability is a weaker version of observability. A system is detectable if all unstable modes of the system are observable, whereas observability requires all modes to be observable. This implies that if a system is observable it will also be detectable. The LMI condition to determine detectability of the pair $$(A,C)$$ is shown below.

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 $$C$$. There is no restriction on the stability of $$A$$.

The LMI: Detectability LMI
$$(A,B)$$ is detectable if and only if there exists $$X>0$$ such that
 * $$AX+XA^T-B^TB<0$$.

Conclusion:
If we are able to find $$X>0$$ such that the above LMI holds it means the matrix pair $$(A,C)$$ is detectable. In words, a system pair $$(A,C)$$ is detectable if the unobservable states asymptotically approach the origin. This is a weaker condition than observability since observability requires that all initial states must be able to be uniquely determined in a finite time interval given knowledge of the input $$u(t)$$ and output $$y(t)$$.

Implementation
This implementation requires Yalmip and Sedumi.

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

Related LMIs
Stabilizability LMI

Hurwitz Stability LMI

Controllability Grammian LMI

Observability Grammian LMI