# 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