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Latin American applied research

Print version ISSN 0327-0793

Lat. Am. appl. res. vol.38 no.3 Bahía Blanca July 2008

 

Sufficient conditions for Hurwitz stability of matrices

W. Zhang1, S. Q. Shen2 and Z. Z. Han1

1 Department of Automation, Shanghai Jiao Tong University, 200240, Shanghai, China
avee@sjtu.edu.cn, zzhan@sjtu.edu.cn

2 School of Applied Mathematics, University of Electronic Science and Technology of China, 610054, Chengdu, Sichuan, China
shuqianshen@126.com

Abstract — New sufficient conditions for the Hurwitz stability of a complex matrix are established based on the concept of α-diagonally dominance. These criteria depend only on the entries of a given matrix. Numerical examples are given to illustrate the applications of these criteria.

Keywords — Hurwitz Stability. H-Matrices. α-Diagonally Dominance.

I. INTRODUCTION

Hurwitz stability plays a fundamental role in control theory since a time-invariant linear system is stable if and only if its system matrix is a Hurwitz matrix (Chen, 1998). Thus, checking the Hurwitz stability is important for control systems. Many researchers have considered the problem and lots of useful criteria for the Hurwitz stability have been established in the last two decades (see, e.g., Wang et al., 1994; Naimark and Zeheb, 1997; Huang, 1998; Duan and Patton, 1998; Franze et al., 2006).

Being similar to the Lyapunov methods for the stability of differential equations, there are two kinds of criteria for the Hurwitz stability of matrices. One is indirect method, i.e., the stability is checked by the eigenvalues. The indirect methods include computing Jordan canonical form, calculating the invariant factors, etc. Generally, it is not easy to complete these computations due to the computational complexity. Another is the so-called direct method which deals with the stability based on the entries of a given matrix directly (see, e.g., Wang et al., 1994; Naimark and Zeheb, 1997; Huang, 1998; Carotenuto et al., 2004; Franze et al., 2006). For example, Routh array (Chen, 1998), Hurwitz criterion (Duan and Patton, 1998) and Lyapunov functions method (Cai and Han, 2006) are well-known direct methods.

One of the important direct methods is based on Geršgorin Theorem (see, e.g., Chap. 6 of Horn and Johnson, 1985a; Varga, 2004). The advantage is that the method can directly point out the location of eigenvalues of a square matrix on the complex plane (see, e.g., Wang et al., 1994; Naimark and Zeheb, 1997; Huang, 1998). In the last fifty years, Geršgorin-like criteria made many contributions in linear systems theory. A lot of designing techniques have been developed by using Geršgorin theorem (see, e.g., Wang et al., 1994; Naimark and Zeheb, 1997; Carotenuto et al., 2004; Franze et al., 2006). Recently, Huang (1998) presented several Geršgorin-like criteria for Hurwitz matrices. In this paper we will follow Huang's work to investigate the Geršgorin-like criterion. Several new sufficient conditions for Hurwitz stability will be developed. We would like to emphasize that we directly deal with complex matrices while most of the existing literature are focused on real matrices.

The paper is organized as follows. The preliminaries including notations, concepts and some lemmas are presented in Section II. The main results are given in Section III. In Section IV, several numerical examples are given to illustrate the applications of the results. The conclusions are drawn in Section V.

II. PRELIMINARIES

This section presents preliminaries of the paper that include notations, concepts and lemmas.

Let n×n denote the set of n × n complex matrices. N := {1,2,...,n}.Let I denote the identity matrix with appropriate dimensions. Let A =[aij] ∈ n×n. Then A is said to be Hurwitz stable if its eigenvalues are all in the left-half side of the complex plane. A matrix is called Hurwitz matrix if it is Hurwitz stable (see, for example, Chen, 1998). For iN, we define

which denote the deleted absolute row and column sums of A, respectively (see, p. 344 of Horn and Johnson, 1985a). Without loss of generality, throughout this paper we assume that Ri(A) > 0 and Ci(A) > 0 for all iN. In fact, if Ri(A)=0 or Ci(A)=0 for some iN, then the investigation of Hurwitz stability of A reduces to that of an (n − 1) × (n − 1) matrix, i.e.,

which is obtained by deleting the i-th row and i-th column from A.

Let α ∈ [0,1] be a constant. Then we define

and

Notice that we have assumed that Ri(A) > 0and Ci(A) > 0, so Ri(A)α and Ci(A)1−α are well-defined for all α ∈ [0,1]. In the following, we review several useful concepts and conclusions.

Definition 1 (Berman and Plemmons, 1994). Let A =[aij] ∈ n×n.If |aii| > Ri(A)(Ci(A)) for all iN, then A is said to be row (column) strictly diagonally dominant. If there exists a positive diagonal matrix D such that AD (DA) is row (column) strictly diagonally dominant, then A is said to be generalized strictly diagonally dominant (GSDD).

A matrix is called H-matrix if it is GSDD (see, e.g., Zhang and Han, 2006; Huang et al., 2006). Clearly, the diagonal entries of an H-matrix are nonzero. There are numbers of equivalent conditions for H-matrix (see, Berman and Plemmons, 1994).

Definition 2. (Berman and Plemmons, 1994) Let A =[aij] ∈ n×n.Then M(A)=[mij] is said to be the comparison matrix of A if mii = |aii|, and mij = −|aij| for all ij, 1 ≤ i,jn.

Definition 3. (Berman and Plemmons, 1994) Let A =[aij] be a real square matrix with aii > 0 and aij ≤ 0, ij, 1 ≤ i,jn. Then A is an M-matrix if A + εI is nonsingular for arbitrary ε ≥ 0.

It is known that A is an H-matrix if and only if the comparison matrix of A is an M-matrix (see, e.g., Chapter 6 of Berman and Plemmons, 1994).

Definition 4. Let A = [aij] ∈ n×n.If there exists an α ∈ [0,1] such that |aii| >Ri(A)αCi(A)1−α for all iN, then A is said to be strictly α-diagonally dominant (α-SDD).

The following conclusion states that a strictly αdiagonally dominant matrix is also an H-matrix.

Lemma 1. If Aα-SDD, then A is an H-matrix.

Proof. Let (A)=[mij] be the comparison matrix of A. Since A ∈ α-SDD, by Ostrowskii Theorem (Corollary 6.4.11 of Horn and Johnson, 1985a), (A)+ εI is nonsingular for arbitrary ε ≥ 0. Therefore, (A) is an M-matrix, and it follows that A is an H-matrix.

Lemma 2 (Huang, 1998). Let A = [aij] ∈ n×n be an H-matrix. If the diagonal entries aii, i = 1,2,..., n, are all real and among aii there are ppositive numbers and n−pnegative numbers. Then A has peigenvalues with positive real parts and np eigenvalues with negative real parts.

Let An×n.Then B = ½(A+AH) is a Hermitian matrix, where AH denotes the conjugate transpose of A. Let λmin(B)and λmax(B) be the minimum and the maximum eigenvalues of B, respectively. We now can state the following conclusion.

Lemma 3 (Horn and Johnson, 1985b). Let An×n and λ(A) be an arbitrary eigenvalue of A. Let B = ½(A+ AH). Then

λmin(B) ≤ Reλ(A) ≤ λmax(B),

where Reλ(A) denotes the real part of λ(A).

If , then A is an H-matrix by Lemma 1. Moreover, if A is an H-matrix, then by applying Ostrowskii Theorem (see Corollary 6.4.11 of Horn and Johnson, 1985a). Hence, throughout this paper we assume that and .

III. MAIN RESULTS

This section presents the main results of the paper.

Theorem 1. Let A =[aij] ∈ n×n and aii < 0 for each iN. If there exist an α ∈ [0,1] and 0 < xi,yi ≤ 1 (iN), such that

(1)

for all , and

(2)

then A is a Hurwitz matrix.

Proof. Note that, without loss of generality, we have assumed that for all iN

Ri(A) > 0 and Ci(A) > 0. (3)

When , we denote

(4)

From (1) and (3), we have 0 < δi < +∞. Let

where if (resp. ), then we define (resp. ωi = +∞). It is easy to get

Let us define two sets

Thus, for , there exists a positive number ε such that

(5)

Construct two positive diagonal matrices

D = diag(d1, d2, ..., dn), E = diag(e1, e2, ..., en)

where

and

Define G = [gij] = EAD. In the following we will show that for each iN,

(6)

First, let us consider the case of . For each , we have from (4) that

(7)

We now prove that inequality (6) is valid for by the following four cases.

Case 1: . From (1), we can obtain

Case 2: . It implies for all . From (5), we know that ωi > ε, that is,

Equivalently,

(8)

Thus (7) and (8) result in

Case 3: . Using a similar argument to that in Case 2, we can verify that inequality (6) is valid.

Case 4: . It follows from (5) that , that is,

Equivalently,

(9)

From the inequalities (7) and (9), it is true that

From the above discussions, we conclude that the inequality (6) is valid for each .

We now turn to the case of . For each , from the choice of ε and the construction of D and E, we have

0 < di, ei ≤ 1, iN

Now we prove that (6) is valid for by the following four cases.

Case 1: . It is easy to get

Case 2: . Based on (2), we have

Case 3: . By a similar argument to that in Case 2, we can verify that inequality (6) is valid.

Case 4: . It follows from (2) that

Hence, |gii|>Ri(G)αCi(G)1-α for all iN. Thus, G is strictly α-diagonally dominant, and it follows from Lemma 1 that A is an H-matrix. Since aii < 0 for all iN, by Lemma 2, A is a Hurwitz matrix.

Let A =[aij] ∈ n×n . Then, for iN and α ∈ [0,1], we define two real numbers as follows:

Using the above notations, the following corollaries can be obtained directly from Theorem 1.

Corollary 1. Let A=[aij] ∈ n×n and aii < 0 for each iN. If there exists an α ∈ [0,1] such that for all ,

then A is a Hurwitz matrix.

Remark 1. If α = 1, then Corollary 1 is exactly the Theorem 1 of Huang (1998).

Corollary 2. Let A=[aij] ∈ n×n and aii < 0 for each iN. If there exists an α ∈ [0,1] such that for all ,

then A is a Hurwitz matrix.

Corollary 3. Let A = [aij] ∈ n×n and aii < 0 for each iN. If there exists an α ∈ [0,1] such that for all ,

then A is a Hurwitz matrix.

Corollary 4. Let A = [aij] ∈ n×n and aii < 0 for each iN. If there exists an α ∈ [0,1] such that for all ,

then A is a Hurwitz matrix.

Corollary 5. Let A = [aij] ∈ n×n and aii < 0 for Example 2. Consider matrix each iN. If there exists an α ∈ [0,1] such that for ,

then A is a Hurwitz matrix.

Corollary 6. Let A = [aij] ∈ n×n and aii < 0 for each iN. If there exists an α ∈ [0,1] such that for all ,

then A is a Hurwitz matrix.

By applying Lemma 3, we can obtain the following Theorem 2.

Theorem 2. Let A = [aij] ∈ n×n and Re(aii) < 0 for iN. Define B = ½(A + AH). If B satisfies the conditions of Theorem 1 or Corollaries 1-6, then A is a Hurwitz matrix.

IV. NUMERICAL EXAMPLES

This section presents numerical examples to illustrate the applications of the conclusions established in Section III.

Example 1. Consider matrix

Let α = 0.5. Then it is straightforward to check that , . Since

by Corollary 1, A is a Hurwitz matrix.

However, since , ,and

it fails to meet the conditions of Theorem 1 of Huang (1998) in this example.

Example 2. Consider matrix

Let α = 0.5. Then , . Since

by Corollary 2, A is a Hurwitz matrix. However, if we denote then by a direct calculation we can obtain ,

A does not satisfy the conditions of Theorem 1 of Huang (1998) in this case.

Example 3. Consider matrix

The real part of its diagonal entries are negative. Let B = ½(A + AH). Then we have

Let α =0.5. Then , . It is straightforward to check

Note that the diagonal entries of B are all negative real numbers. Therefore, B satisfies the conditions of Corollary 5. It follows from Theorem 2 that A is a Hurwitz matrix.

However, since A is a complex matrix and its diagonal entries are not all real, the criteria in Wang et al. (1994), Huang (1998), Pastravanu and Voicu (2004) are not valid in this case.

V. CONCLUSIONS

Based on the concept of α-diagonally dominance, this paper presents new criteria for Hurwitz stability. More precisely, several Geršgorin-like sufficient conditions for Hurwitz stability of matrices are developed. These conditions are simple since they only depend on the entries of a given matrix. Moreover, we deal with complex rather than real matrix which is considered by most of the existing literature. Further research work may consider, for example, the iterative algorithm and the necessary conditions for Hurwitz matrices.

VI. ACKNOWLEDGMENTS
The authors would like to thank the anonymous reviewers and the subject editor Prof. Jorge A. Solsona for their helpful suggestions and valuable comments. This work was supported by the National Natural Science Foundation of China (No. 60674024).

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Received: April 3, 2007.
Accepted: November 9, 2007.
Recommended by Subject Editor Jorge Solsona.

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