# 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

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

## 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.