# 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 $y(t)$ .

## Implementation

This implementation requires Yalmip and Sedumi.