Latin American Applied Research 38:253-258 (2008) SUFFICIENT CONDITIONS FOR HURWITZ STABILITY OF MATRICES W. ZHANG†,S.Q.SHEN‡ and Z.Z. HAN† †Department of Automation, Shanghai Jiao Tong University, 200240, Shanghai, China [email protected], [email protected] ‡School of Applied Mathematics, University of Electronic Science and Technology of China, 610054, Chengdu, Sichuan, China [email protected] Abstract— New sufficient conditions for the e.g., Wang et al., 1994; Naimark and Zeheb, 1997; Hurwitz stability of a complex matrix are es- Huang, 1998). In the last fifty years, Gerˇsgorin-like tablished based on the concept of α-diagonally criteria made many contributions in linear systems dominance. These criteria depend only on the theory. A lot of designing techniques have been de- entries of a given matrix. Numerical examples veloped by using Gerˇsgorin theorem (see, e.g., Wang are given to illustrate the applications of these et al., 1994; Naimark and Zeheb, 1997; Carotenuto et criteria. al., 2004; Franze et al., 2006). Recently, Huang (1998) presented several Gerˇsgorin-like criteria for Hurwitz Keywords— Hurwitz stability, H-matrices, matrices. In this paper we will follow Huang’s work to α-diagonally dominance investigate the Gerˇsgorin-like criterion. Several new I. INTRODUCTION sufficient conditions for Hurwitz stability will be de- veloped. We would like to emphasize that we directly Hurwitz stability plays a fundamental role in control deal with complex matrices while most of the existing theory since a time-invariant linear system is stable literature are focused on real matrices. if and only if its system matrix is a Hurwitz matrix The paper is organized as follows. The preliminaries (Chen, 1998). Thus, checking the Hurwitz stability is including notations, concepts and some lemmas are important for control systems. Many researchers have presented in Section II. The main results are given in considered the problem and lots of useful criteria for Section III. In Section IV, several numerical examples the Hurwitz stability have been established in the last are given to illustrate the applications of the results. two decades (see, e.g., Wang et al., 1994; Naimark and The conclusions are drawn in Section V. Zeheb, 1997; Huang, 1998; Duan and Patton, 1998; Franze et al., 2006). II. PRELIMINARIES Being similar to the Lyapunov methods for the sta- This section presents preliminaries of the paper that bility of differential equations, there are two kinds of include notations, concepts and lemmas. criteria for the Hurwitz stability of matrices. One is Let Cn×n denote the set of n × n complex matrices. indirect method, i.e., the stability is checked by the N := {1, 2,...,n}.LetI denote the identity matrix n×n eigenvalues. The indirect methods include computing with appropriate dimensions. Let A =[aij ] ∈ C . Jordan canonical form, calculating the invariant fac- Then A is said to be Hurwitz stable if its eigenvalues tors, etc. Generally, it is not easy to complete these are all in the left-half side of the complex plane. A computations due to the computational complexity. matrix is called Hurwitz matrix if it is Hurwitz stable Another is the so-called direct method which deals (see, for example, Chen, 1998). For i ∈ N, we define with the stability based on the entries of a given ma- n n trix directly (see, e.g., Wang et al., 1994; Naimark and Ri(A):= |aij |,Ci(A):= |aji|, Zeheb, 1997; Huang, 1998; Carotenuto et al., 2004; j=1,j=i j=1,j=i Franze et al., 2006). For example, Routh array (Chen, 1998), Hurwitz criterion (Duan and Patton, 1998) and which denote the deleted absolute row and column Lyapunov functions method (Cai and Han, 2006) are sums of A, respectively (see, p. 344 of Horn and John- well-known direct methods. son, 1985a). Without loss of generality, throughout One of the important direct methods is based on this paper we assume that Ri(A) > 0andCi(A) > 0 Gerˇsgorin Theorem (see, e.g., Chap. 6 of Horn and for all i ∈ N.Infact,ifRi(A)=0orCi(A)=0for Johnson, 1985a; Varga, 2004). The advantage is that some i ∈ N, then the investigation of Hurwitz stability the method can directly point out the location of eigen- of A reduces to that of an (n − 1) × (n − 1) matrix, values of a square matrix on the complex plane (see, i.e., 253 W. ZHANG, S. Q. SHEN, Z. Z. HAN are all real and among aii there are p positive numbers ⎡ ⎤ and n−p negative numbers. Then A has p eigenvalues a1,1 ··· a1,i−1 a1,i+1 ··· a1,n ⎢ ⎥ with positive real parts and n − p eigenvalues with ⎢ ··· ··· ··· ··· ··· ··· ⎥ ⎢ ··· ··· ⎥ negative real parts. ⎢ai−1,1 ai−1,i−1 ai−1,i+1 ai−1,n⎥ ∈ Cn×n 1 H ⎢ ⎥ , Let A .ThenB = 2 (A+A ) is a Hermitian ⎢ai+1,1 ··· ai+1,i−1 ai+1,i+1 ··· ai+1,n⎥ H ⎣ ··· ··· ··· ··· ··· ··· ⎦ matrix, where A denotes the conjugate transpose of A.Letλmin(B)andλmax(B) be the minimum and an,1 ··· an,i−1 an,i+1 ··· an,n the maximum eigenvalues of B, respectively. We now which is obtained by deleting the i-th row and i-th can state the following conclusion. Lemma 3 (Horn and Johnson, 1985b). Let A ∈ column from A. n×n ∈ C and λ(A) be an arbitrary eigenvalue of A.Let Let α [0, 1] be a constant. Then we define 1 H B = 2 (A + A ). Then α α 1−α N2 := i ∈ N |aii| >Ri(A) Ci(A) , λmin(B) ≤ Reλ(A) ≤ λmax(B), and α α N1 := N\N2 . where Reλ(A) denotes the real part of λ(A). α If N2 = N,thenA is an H-matrix by Lemma 1. Notice that we have assumed that Ri(A) > 0and α α 1−α Moreover, if A is an H-matrix, then N2 = ∅ by apply- Ci(A) > 0, so Ri(A) and Ci(A) are well-defined ing Ostrowskii Theorem (see Corollary 6.4.11 of Horn for all α ∈ [0, 1]. In the following, we review several and Johnson, 1985a). Hence, throughout this paper useful concepts and conclusions. we assume that N α = ∅ and N α = ∅. Definition 1 (Berman and Plemmons, 1994). Let 1 2 n×n A =[aij ] ∈ C .If|aii| >Ri(A)(Ci(A)) for all III. MAIN RESULTS ∈ i N,thenA is said to be row (column) strictly di- This section presents the main results of the paper. agonally dominant. If there exists a positive diagonal n×n Theorem 1. Let A =[aij ] ∈ C and aii < 0for matrix D such that AD (DA) is row (column) strictly each i ∈ N.Ifthereexistanα ∈ [0, 1] and 0 <xi,yi ≤ diagonally dominant, then A is said to be generalized 1(i ∈ N), such that strictly diagonally dominant (GSDD). A matrix is called H-matrix if it is GSDD (see, e.g., ⎛ ⎞α ⎛ ⎞1−α n n Zhang and Han, 2006; Huang et al., 2006). Clearly, | | 1 ⎝ | | ⎠ ⎝ | | ⎠ aii > α 1−α aij xj aji yj the diagonal entries of an H-matrix are nonzero. There xi yi j=1,j=i j=1,j=i are numbers of equivalent conditions for H-matrix (see, (1) Berman and Plemmons, 1994). α for all i ∈ N ,and Definition 2. (Berman and Plemmons, 1994) Let 1 n×n A =[aij ] ∈ C .ThenM(A)=[mij ]issaidto α 1−α | |≥Ri(A) Ci(A) ∈ α be the comparison matrix of A if mii = |aii|,and aii ,iN2 , (2) xiyi mij = −|aij| for all i = j, 1 ≤ i, j ≤ n. Definition 3. (Berman and Plemmons, 1994) Let then A is a Hurwitz matrix. A =[aij ] be a real square matrix with aii > 0and Proof. Note that, without loss of generality, we aij ≤ 0, i = j, 1 ≤ i, j ≤ n.ThenA is an M-matrix if have assumed that for all i ∈ N A + εI is nonsingular for arbitrary ε ≥ 0. It is known that A is an H-matrix if and only if Ri(A) > 0andCi(A) > 0. (3) the comparison matrix of A is an M-matrix (see, e.g., α Chapter 6 of Berman and Plemmons, 1994). When i ∈ N1 ,wedenote ∈ Cn×n Definition 4.LetA =[aij ] .Ifthereexists ⎛ ⎞α ⎛ ⎞1−α ∈ | | α 1−α an α [0, 1] such that aii >Ri(A) Ci(A) for n n α 1−α ⎜ ⎟ ⎜ ⎟ all i ∈ N,thenA is said to be strictly α-diagonally |aii|xi yi − ⎝ |aij |xj ⎠ ⎝ |aji|yj⎠ j=1 j=1 dominant (α-SDD). j=i j=i The following conclusion states that a strictly α- δi = ⎛ ⎞α ⎛ ⎞1−α . diagonally dominant matrix is also an H-matrix. ⎜ n ⎟ ⎜ n ⎟ Lemma 1.IfA ∈ α-SDD, then A is an H-matrix. ⎝ |aij |xj ⎠ ⎝ |aji|yj⎠ M j=1 j=1 Proof. Let (A)=[mij ] be the comparison ma- j=i j=i trix of A.SinceA ∈ α-SDD, by Ostrowskii Theo- (4) rem (Corollary 6.4.11 of Horn and Johnson, 1985a), From (1) and (3), we have 0 <δi < +∞.Let M(A)+εI is nonsingular for arbitrary ε ≥ 0. There- fore, M(A) is an M-matrix, and it follows that A is an n n |aij |xj |aji|yj H-matrix. 2 j=1,j=i j=1,j=i n×n Lemma 2 (Huang, 1998). Let A =[aij ] ∈ C be φi = δi ,ωi = δi , |aij | |aji| an H-matrix. If the diagonal entries aii, i =1, 2,...,n, j∈N α j∈N α 2 2 254 Latin American Applied Research 38:279-287 (2008) where if |aij | =0(resp. |aji| = 0 ), then we Case 2: |aij | =0, |aji| = 0.