# 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 ${\displaystyle (A,C)}$  is shown below.

## The System

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

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

## The Data

The matrices necessary for this LMI are ${\displaystyle A}$  and ${\displaystyle C}$ . There is no restriction on the stability of ${\displaystyle A}$ .

## The LMI: Detectability LMI

${\displaystyle (A,B)}$  is detectable if and only if there exists ${\displaystyle X>0}$  such that

${\displaystyle AX+XA^{T}-B^{T}B<0}$ .

## Conclusion:

If we are able to find ${\displaystyle X>0}$  such that the above LMI holds it means the matrix pair ${\displaystyle (A,C)}$  is detectable. In words, a system pair ${\displaystyle (A,C)}$  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 ${\displaystyle u(t)}$  and output ${\displaystyle y(t)}$ .

## Implementation

This implementation requires Yalmip and Sedumi.