On the D-Stability Criterion of Matrices

Root clustering problems of matrices are considered. Here the conditions are given for eigenvalues of matrices to lie in a prescribed subregion D of the complex plane. The region D (stability region ) is defined by rational functions. A simple necessary and sufficient condition for stability of a single matrix is obtained. For the commuting polynomial family, a necessary and sufficient condition in terms of a common solution to a set of Lyapunov inequalities is derived. A simple sufficient condition for the Hurwitz stability of a commuting quadratic polynomial family is given.


Introduction
Matrix root clustering, also known as D-stability, is an important problem in control theory.The problem of root clustering has attracted a great deal of consideration in the past.Previous work on matrix root clustering has employed different approaches.First, many authors have presented matrix root clustering problem via the Generalized Lyapunov Theorem (GLT) approach (see, e.g., Gutman and Jury, 1981;Gutman, 1990;Yedevalli, 1993;Yedevalli, 1993;Yedevalli, 1985;Juang, Hong, Wang, 1989;Juang, 1991;Jabbari, 1991;Horng, Horng, Chou, 1993).Extensive works on the subject have led to a study by (Gutman and Jury, 1981), and continuously do so (Gutman, 1990;Yedevalli, 1993;Yedevalli, 1993;Yedevalli, 1985;Juang, Hong, Wang, 1989;Juang, 1991;Jabbari, 1991;Horng, Horng, Chou, 1993).The authors considered in (Gutman and Jury, 1981) the root clustering of a matrix called Γ into the unit circle.There are also issues on Ω transformable region which is defined by mapping from two points in Ω in to the left half plane.(Yedevalli, 1985(Yedevalli, , 1993) ) presented some explicit bounds on uncertainty for root clustering of linear state space models in terms of GLT approach and the Kronecker approach.Following this, (Wang, 1994(Wang, , 2000(Wang, , 2003) ) analysed robust root clustering in a specific region in which the GLT is not valid.This clustering region is an intersection of a ring and horizontal strip, located in the left half plane which is the non Ω transformable region providing good ride quality for aircraft.
A further approach was discussed by Chilali and Gahinet in (Chilali, Gahinet, 1996, 1999).In particular, sufficient conditions were derived for a class of convex regions of complex plane via linear matrix inequalities (LMI).Recently Bosche (Bosche, Bachelier, Mehdi, 2005;Rejichi, Bachelier, Chaabane, Mehdi, 2007) addressed the problem of matrix root clustering analysis in EEMI (Extended Ellipsoidal Matrix Inequality) regions which express the set of some non-connected regions.This paper is organized as follows.In section 2, we present a necessary and sufficient condition for D-stability of a matrix.In section 3, we give the necessary and sufficient conditions for D-stability of the commuting family.Finally, our conclusions are presented in section 4. Now, we present a number of notations and results that will be needed in the following sections.
Let R n be the set of real n vectors, R n×n (C n×n ) be the set of n × n real (complex) matrices.For P ∈ R n×n (C n×n ) the symbol P > 0 means that P is symmetric (Hermitian) and positive definite.Let the subregion D of the complex plane C be defined as where f j (z) and g j (z) are polynomials with real coefficients, ḡ is the complex conjugate of g .The inequality Re f j (z)ḡ j (z) < 0 is equivalent from the inequality Rer j (z) < 0, where r j (z) = f j (z) g j (z) .The region D (1) will also be referred to as the stability region.It is a generalization of the known stability regions: By the Lyapunov theorem, the matrix A ∈ R n×n (C n×n ) is Hurwitz stable if and only if there exists P ∈ R n×n (C n×n ), P > 0 such that where A T (A * ) denotes the transpose (conjugate transpose) of A.
Theorem 1.1 (Mori, Mori, Kokame 2001) Let the stability region Ω be defined as where f j (z) are polynomials.Then the matrix A ∈ R n×n is Ω -stable if and only if there exists a matrix P ∈ R n×n , P > 0 such that for all j = 1, 2, . . ., m [ f j (A) In (Narendra, Balakrishnan 1994), the following result on the existence of a common P > 0 for commuting matrices A 1 , A 2 , . . ., A k is given (Narendra, Balakrishnan 1994, Theorem 2).
In this work by using Theorems 1.1 and 1.2 we prove a simple criterion for D-stability of a matrix A. We show that D-stability of a matrix A is equivalent to the Hurwitz stability of the matrices f 1 (A)g −1 1 (A), . . ., f m (A)g −1 m (A) (Theorem 2.1).

Stability of a Single Matrix
In this section we give a criterion for the D-stability of the matrix A ∈ R n×n .
Lemma 2.1 Let f (z) and g(z) be polynomials, A ∈ R n×n .If g(A) is invertible then f (A) and g −1 (A) commute.
Lemma 2.3 If f (z) and g(z) are polynomials, A ∈ R n×n , λ is an eigenvalue of A, g(A) is invertible then g(λ) 0 and f (λ) g(λ) is an eigenvalue of f (A)g −1 (A).Proof.g(λ) is an eigenvalue of g(A).Since g(A) is invertible then g(λ) 0. There exists x ∈ C n×1 , x 0 such that the following can be written: Lemma 2.4 Let f (z) and g(z) be polynomials and, g(A) is invertible.If µ is an eigenvalue of f (A)g −1 (A) then there exists an eigenvalue λ of A such that g(λ) 0 and µ = f (λ) g(λ) .
Example 2.1 (Mori, Mori, Kokame 2001) Let A be given as and the region Ω is as shown in Figure 1.