LMIs in Control/pages/Generalized Lyapunov Theorem

WIP, Description in progress

The theorem can be viewed as a true essential generalization of the well-known continuous- and discrete-time Lyapunov theorems.

Kronecker Product

The Kronecker Product of a pair of matrices $A\in {ℝ}^{m\times n}$ and $B\in {ℝ}^{p\times q}$ is defined as follows:

$A\otimes B=\left[\begin{array}{cccc}{a}_{11}B& a_{12}B& \cdots & a_{1n}B\\ a_{21}B& a_{22}B& \cdots & a_{2n}B\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}B& a_{m2}B& \cdots & a_{mn}B\end{array}\right]\in {ℝ}^{mp\times nq}$.

Lemma 1: Manipulation Rules of Kronecker Product

Let $A,B,C$ be matrices with appropriate dimensions. Then, the Kronecker product has the following properties:

• $1\otimes A=A$;
• $(A+B)\otimes C=A\otimes C+B\otimes C$
• $(A\otimes B)(C\otimes D)=(AC)\otimes (BD)$
• $(A\otimes B{)}^T=A^T\otimes B^T$
• $(A\otimes B{)}^{-1}=A^{-1}\otimes B^{-1}$
• $\lambda (A\otimes B)={\lambda }_i(A){\lambda }_{j}B$

Theorem

In terms of Kronecker products, the following theorem gives the $𝔻$-stability condition for the general LMI region case: Let $𝔻={𝔻}_{L,M}$ be an LMI region, whose characteristic function is

$F_{𝔻}=L+sM+\bar{s}{M}^T$

Then, a matrix $A\in {ℝ}^{n\times n}$ is $\mathbb{D}_{L,M}$-stable if and only if there exists symmetric positive definite matrix $P$ such that

$R_{𝔻}(A,P)=L\otimes P+M\otimes (AP)+M^T\otimes (AP{)}^T<0$,

where $\otimes$ represents the Kronecker product.

Lemma 2

Given two LMI regions ${𝔻}_1$ and ${𝔻}_2$, a matrix $A$ is both ${𝔻}_1$-stable and ${𝔻}_2$-stable if there exists a positive definite matrix $P$ , such that $R_{{𝔻}_1}(A,P)<0$ and $R_{{𝔻}_2}(A,P)<0$.

${\displaystyle }$ WIP, additional references to be added

External Links

A list of references documenting and validating the LMI.