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 ProductEdit

The Kronecker Product of a pair of matrices   and   is defined as follows:

 .

Lemma 1: Manipulation Rules of Kronecker ProductEdit

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

  •  ;
  •  
  •  
  •  
  •  
  •  

TheoremEdit

In terms of Kronecker products, the following theorem gives the  -stability condition for the general LMI region case: Let   be an LMI region, whose characteristic function is

 

Then, a matrix   is $\mathbb{D}_{L,M}$-stable if and only if there exists symmetric positive definite matrix   such that

 ,

where   represents the Kronecker product.

Lemma 2Edit

Given two LMI regions   and  , a matrix   is both  -stable and  -stable if there exists a positive definite matrix   , such that   and  .


WIP, additional references to be added

External LinksEdit

A list of references documenting and validating the LMI.

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