Multiple Models Approach in Automation: Takagi-Sugeno Fuzzy Systems

Appendix 1

LMI Regions

A1.1. Definition of an LMI region

A subset D of a complex plane is called an nth-order LMI region if there exist a symmetric matrix α ∈^n.n and a matrix β ∈ ^n.n such that:

[A1.1]

We observe that values for the characteristic function f_D of complex variable z are taken in the set of nth-order Hermitian matrices and that an LMI region is symmetric with the real axis since . This is often verified by regions used for studying the D-stability of a real matrix, since the spectrum of a real matrix is self-adjoint.

THEOREM A1.1.– [CHI 96]: Let A ∈^n.n and D be an LMI region defined by [A1.1]. Matrix A is D-stable if and only if there exists a positive definite symmetric matrix X∈^n.n such that:

[A1.2]

In order to test the D-stability of a real matrix in an intersection of LMI ...