LMIs in Control/LMIs for Hurwitz Stability

LMIs in Control/LMIs for 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 $>0$.

Implementation

Please find the MATLAB implementation at this link below
https://github.com/omiksave/LMI

Related LMIs

Links to other closely-related LMIs

• Schur Stability
• Quadratic Hurwitz Stability
• Quadratic Schur Stability
• Quadratic D-Stability

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

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