# 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

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

## The Data

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

## The Optimization Problem

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

## The LMI: The Lyapunov Inequality

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

Proof : Matrix pair ${\displaystyle (A,B)}$, is Hurwitz stabilizable if and only if

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

## Conclusion:

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