# LMIs in Control/Stability Analysis/Hurwitz Stability

LMIs in Control/Stability Analysis/Hurwitz Stability

This is a set of LMI conditions for determining Hurwitz Stability of continuous time systems.

## The System

{\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}} ## The Data

• The matrices $A,B,C,D$ are system matrices of appropriate dimensions.
• $x\in \mathbb {R} ^{n}$ , $y\in \mathbb {R} ^{m}$ and $u\in \mathbb {R} ^{r}$ are state vector, output vector and input vector respectively.

## The Optimization Problem

Find a symmetric postive definite matrix $X$ , where $X\in \mathbb {R} ^{n}$ . Thus $X>0$ and $K=ZX^{-1}$ where $Z\in \mathbb {R} ^{r}$ .

## The LMI: The Lyapunov Inequality

Matrix pair $(A,B)$ , is Hurwitz stabilizable if and only if there exist a symmetric positive definite matrix $X$ and a matrix $Z$ satisfying
$AX+XA^{T}+BZ+Z^{T}B^{T}<0$ Proof : Matrix pair $(A,B)$ , is Hurwitz stabilizable if and only if

$rank[sI-AB]=n$ , $\forall s\in \lambda (A)$ and $Re(s)\geq 0$ This is the definition of Hurwitz Stability. Now, using this definition we can prove the above LMI if we find matrix $X>0$ and matrix $Z$ and thus by substituting $Z=KX$ in the above LMI we get,
$(A+BK)X+X(A+BK)^{T}<0$ , which brings us to the Lyapunov Stability Theory.

## Conclusion:

Thus by proving the above conditions we prove that the matrix pair $(A,B)$ is Hurwitz Stabilizable. At the same time, we also prove that the $rank[sI-AB]=n$ i.e. it is full rank and the real part of $s$ is $>0$ .