LMIs in Control/pages/Generalized Lyapunov Theorem

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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[edit | edit source]

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


Lemma 1: Manipulation Rules of Kronecker Product[edit | edit source]

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

  • ;

Theorem[edit | edit source]

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 2[edit | edit source]

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 Links[edit | edit source]

A list of references documenting and validating the LMI.

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