# 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