LMIs in Control/pages/Detectability LMI

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{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t),\\x(0)&=x_{0},\end{aligned}}

where $x(t)\in \mathbb {R} ^{n}$ , $u(t)\in \mathbb {R} ^{m}$ , at any $t\in \mathbb {R}$ .

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^{T}B<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 $y(t)$ .