# 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 ${\displaystyle (A,C)}$ , is said to be Hurwitz detectable if there exists a real matrix ${\displaystyle L}$  such that ${\displaystyle A+LC}$  is Hurwitz stable.

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

## The Data

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

## The Optimization Problem

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

## The LMI:

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

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

## Conclusion:

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

## Implementation

Find the MATLAB implementation at this link below
Hurwitz detectability