# LMIs in Control/pages/Hurwitz detectability

LMIs in Control/pages/Hurwitz detectability

## Hurwitz Detectability

Hurwitz detectability is a dual concept of Hurwitz stabilizability and is defined as the matrix pair $(A,C)$ , is said to be Hurwitz detectable if there exists a real matrix $L$ such that $A+LC$ is Hurwitz stable.

## The System

{\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\\\end{aligned}} where $x(t)\in \mathbb {R} ^{n}$ , $y(t)\in \mathbb {R} ^{m}$ , $u(t)\in \mathbb {R} ^{q}$ , at any $t\in \mathbb {R}$ .

## The Data

• The matrices $A,B,C,D$ are system matrices of appropriate dimensions and are known.

## The Optimization Problem

There exist a symmetric positive definite matrix $P$ and a matrix $W$ satisfying
$A^{T}P+PA+W^{T}C+C^{T}W<0$ There exists a symmetric positive definite matrix $P$ satisfying
$N_{c}^{T}(A^{T}P+PA)N_{c}<0$ with $N_{c}$ being the right orthogonal complement of $C$ .
There exists a symmetric positive definite matrix $P$ such that
$A^{T}P+PA<\gamma C^{T}C$ for some scalar $\gamma >0$ ## The LMI:

Matrix pair $(A,C)$ , is Hurwitz detectable if and only if following LMI holds

• $A^{T}P+PA+W^{T}C+C^{T}W<0.$ • $N_{c}^{T}(A^{T}P+PA)N_{c}<0$ • $A^{T}P+PA-\gamma C^{T}C<0$ ## Conclusion:

Thus by proving the above conditions we prove that the matrix pair $(A,C)$ is Hurwitz Detectable.

## Implementation

Find the MATLAB implementation at this link below
Hurwitz detectability