WIP, Description in progress
The theorem can be viewed as a true essential generalization of the well-known continuous- and discrete-time Lyapunov theorems.
The Kronecker Product of a pair of matrices
and
is defined as follows:
.
Let
be matrices with appropriate dimensions. Then, the
Kronecker product has the following properties:
;
![{\displaystyle (A+B)\otimes C=A\otimes C+B\otimes C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3898591596f67e49931f125d62f8ef9a2a3ab65c)
![{\displaystyle (A\otimes B)(C\otimes D)=(AC)\otimes (BD)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1654d6ed12a5ed17efc0b424031c62929f86ab)
![{\displaystyle (A\otimes B)^{T}=A^{T}\otimes B^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc4b25a5500a9408d3b2bff42cf9571663be1c8)
![{\displaystyle (A\otimes B)^{-1}=A^{-1}\otimes B^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d76899362e6afa0713ef7e6f4a65bd2e0eee52a)
![{\displaystyle \lambda (A\otimes B)={\lambda _{i}(A)\lambda _{j}{B}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00ffaa3d6e8cdb96ccc5fc9c005a000b951f8be)
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.
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
A list of references documenting and validating the LMI.