# 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 ${\displaystyle A\in \mathbb {R} ^{m\times n}}$  and ${\displaystyle B\in \mathbb {R} ^{p\times q}}$  is defined as follows:

${\displaystyle A\otimes B={\begin{bmatrix}a_{11}B&a_{12}B&\cdots &a_{1n}B\\a_{21}B&a_{22}B&\cdots &a_{2n}B\\\vdots &\vdots &\ddots &\vdots \\a_{m1}B&a_{m2}B&\cdots &a_{mn}B\end{bmatrix}}\in \mathbb {R} ^{mp\times nq}}$ .

## Lemma 1: Manipulation Rules of Kronecker Product

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

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

## Theorem

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

${\displaystyle F_{\mathbb {D} }=L+sM+{\overline {s}}M^{T}}$

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

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

where ${\displaystyle \otimes }$  represents the Kronecker product.

## Lemma 2

Given two LMI regions ${\displaystyle \mathbb {D} _{1}}$  and ${\displaystyle \mathbb {D} _{2}}$ , a matrix ${\displaystyle A}$  is both ${\displaystyle \mathbb {D} _{1}}$ -stable and ${\displaystyle \mathbb {D} _{2}}$ -stable if there exists a positive definite matrix ${\displaystyle P}$  , such that ${\displaystyle R_{\mathbb {D} _{1}}(A,P)<0}$  and ${\displaystyle R_{\mathbb {D} _{2}}(A,P)<0}$ .

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