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{align} \dot x(t)&=Ax(t)+Bu(t),\\ y(t)&=Cx(t)+Du(t)\\ \end{align}$$

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

$$rank[sI - A  B] = n $$, $$\forall s \in \lambda (A)  $$ and $$Re(s) \ge 0$$

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 $$ rank[sI-A B] = n$$ 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

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