Sufficient Conditions for Hurwitz Stability of Matrices

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 developed. 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 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. It implies for α α α α j∈N2 j∈N2 j∈N2 j∈N2 α define φi =+∞ (resp. ωi =+∞ ). It is easy to get all j ∈ N2 , |aij | = 0. From (5), we know that ωi >ε, φi > 0,ωi > 0. that is, Let us define two sets n | | | | α α δi aji yj >ε aji . Nx = {i ∈ N2 | xi =1} , α j=1,j=i j∈N2 α { ∈ α | } Ny = i N2 yi =1 . Equivalently, α Thus, for i ∈ N1 , there exists a positive number ε n n such that (1 + δi) |aji|yj > |aji|yj + ε |aji|. α j=1,j=i j=1,j=i j∈N2 i i 0 <εy |aij |xj and i j∈N α,j=i 1 ∈ α ∪ α ⎛ ⎞1−α yi,iN1 Ny n ei = α α . yi + ε, i ∈ N2 \ Ny ⎝ ⎠ 1−α × |aji|yj + ε |aji| xi α j=1,j=i j∈N2 Define G =[gij ]=EAD. In the following we will ⎛ ⎞1−α show that for each i ∈ N, ⎜ ⎟ ⎜ ⎟ α 1−α ≥ | ji| j | ji| j |gii| >Ri(G) Ci(G) . (6) ⎝ a y + a (y + ε)⎠ α α α α j∈N1 ∪Ny j∈N2 \Ny α j=i First, let us consider the case of i ∈ N1 .Foreach ⎛ ⎞ α α i ∈ N1 , we have from (4) that ⎝ ⎠ 1−α α × |aij |xj x y ⎛ ⎞α i i n α j∈N1 ,j=i α 1−α ⎝ ⎠ |aii|x y =(1+δi) |aij |xj α 1−α i i = Ri(G) Ci(G) . j=1,j=i ⎛ ⎞1−α n Case 3: |aij | =0 , |aji| =0.Usingasim- ⎝ ⎠ j∈N α j∈N α × |aji|yj . (7) 2 2 j=1,j=i ilar argument to that in Case 2, we can verify that inequality (6) is valid. α We now prove that inequality (6) is valid for i ∈ N1 Case 4: |aij | =0 , |aji| = 0. It follows j∈N α j∈N α by the following four cases. 2 2 Case 1: |aij | = |aji| =0.From(1),we from (5) that φi >ε,thatis, α α j∈N2 j∈N2 can obtain n ⎛ ⎞α δi |aij |xj >ε |aij |. j=1,j=i j∈N α α ⎝ ⎠ 2 |gii| = yi|aii|xi >yi |aij |xj j∈N α,j=i 1 Equivalently, ⎛ ⎞1−α ⎝ ⎠ 1−α n n × |aji|yj xi α (1 + δi) |aij |xj > |aij |xj + ε |aij |. j∈N1 ,j=i j=1,j=i j=1,j=i j∈N α α 1−α 2 = Ri(G) Ci(G) . (9)

255 W. ZHANG, S. Q. SHEN, Z. Z. HAN

α 1−α From the inequalities (7) and (9), it is true that Hence, |gii| >Ri(G) Ci(G) for all i ∈ N.Thus,G is strictly α-diagonally dominant, and it follows from | | | | gii = yi aii xi Lemma 1 that A is an H-matrix. Since aii < 0 for all ⎛ ⎞α n i ∈ N, by Lemma 2, A is a Hurwitz matrix. 2 α ⎝ ⎠ ∈ Cn×n ∈ ∈ = yi (1 + δi) |aij |xj Let A =[aij ] . Then, for i N and α j=1,j=i [0, 1], we deﬁne two real numbers as follows: ⎛ ⎞1−α n α 1−α Ri(A) Ci(A) 1−α α × ⎝ i | ji| j⎠ Qi (A)= , (1 + δ ) a y xi |aii| j=1,j=i ⎛ ⎞α α 1−α |aii| + Ri(A) Ci(A) n Kα(A)= . α ⎝ ⎠ i | | >yi |aij |xj + ε |aij | 2 aii j=1,j=i j∈N α 2 Using the above notations, the following corollaries can ⎛ ⎞1−α n be obtained directly from Theorem 1. n×n ⎝ ⎠ 1−α ij ∈ C ii × |aji|yj + ε |aji| xi Corollary 1. Let A =[a ] and a < 0for α each i ∈ N.Ifthereexistsanα ∈ [0, 1] such that for j=1,j=i j∈N2 ⎛ all i ∈ N α, 1 ≥ α ⎝ | | | | ⎛ ⎞α yi aij xj + aij xj α α j∈N1 ,j=i j∈Nx ⎝ α ⎠ 1−α |aii| > |aij | + Qj (A)|aij | Ci(A) , ⎞α α j∈N α j∈N1 ,=i 2 ⎠ + |aij |(xj + ε) α α then A is a Hurwitz matrix. j∈N2 \Nx ⎛ Remark 1. If α = 1, then Corollary 1 is exactly ⎝ the Theorem 1 of Huang (1998). × |aji|yj + |aji|yj n×n Corollary 2. Let A =[aij ] ∈ C and aii < 0for j∈N α,j=i j∈N α 1 y each i ∈ N.Ifthereexistsanα ∈ [0, 1] such that for ⎞1−α α all i ∈ N1 , ⎠ 1−α | ji| j + a (y + ε) xi ⎛ ⎞1−α α α j∈N \Ny 2 α ⎝ α ⎠ α 1−α |aii| >Ri(A) |aji| + Qj (A)|aji| , = Ri(G) Ci(G) . α α j∈N1 ,=i j∈N2 From the above discussions, we conclude that the α then A is a Hurwitz matrix. inequality (6) is valid for each i ∈ N1 . n×n α α Corollary 3. Let A =[aij ] ∈ C and aii < 0for We now turn to the case of i ∈ N .Foreachi ∈ N , 2 2 each i ∈ N.Ifthereexistsanα ∈ [0, 1] such that for from the choice of ε and the construction of D and E, all i ∈ N α, we have 1 ⎛ ⎞ 0 Qi (A) α + Qj (A)|aij | α Qj (A) α following four cases. j∈N1 ,=i j∈N2 α α Case 1: i ∈ N ∩ N .Itiseasytoget ⎛ ⎞1−α x y ⎝ |aji| ⎠ α 1−α α 1−α × + |aji| , |gii| = |aii| >Ri(A) Ci(A) ≥ Ri(G) Ci(G) . α α Qj (A) α j∈N1 ,=i j∈N2 α α Case 2: i ∈ Nx ,i∈ / Ny . Based on (2), we have then A is a Hurwitz matrix. α 1−α ∈ Cn×n |gii| =(yi + ε)|aii|≥ε|aii| + Ri(A) Ci(A) Corollary 4. Let A =[aij ] and aii < 0for α 1−α α 1−α each i ∈ N.Ifthereexistsanα ∈ [0, 1] such that for >Ri(A) Ci(A) ≥ Ri(G) Ci(G) . α all i ∈ N1 , α α ⎛ ⎞ Case 3: i/∈ Nx ,i ∈ Ny . By a similar argument α | | to that in Case 2, we can verify that inequality (6) is α ⎝ aij ⎠ |aii| >Qi (A) α + |aij | valid. α Qj (A) α α α j∈N1 ,=i j∈N2 Case 4: i/∈ Nx ,i∈ / Ny . It follows from (2) that ⎛ ⎞1−α |aji| |gii| =(yi + ε)|aii|(xi + ε) ⎝ α ⎠ × α + Qj (A)|aji| , 2 α 1−α j∈N α,=i Qj (A) j∈N α ≥ ε |aii| + Ri(A) Ci(A) 1 2 α 1−α ≥ α 1−α >Ri(A) Ci(A) Ri(G) Ci(G) . then A is a Hurwitz matrix.

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n×n Corollary 5. Let A =[aij ] ∈ C and aii < 0for Example 2. Consider matrix each i ∈ N.Ifthereexistsanα ∈ [0, 1] such that for ⎡ ⎤ −31 0−1 all i ∈ N α, 1 ⎢ 1 −4 i −1⎥ A = ⎢ ⎥ . ⎛ ⎞1−α ⎣ 1 −2i −1.70⎦ | | α ⎝ aji ⎠ −2i −10−3 |aii| >Ki (A) α + |aji| j∈N α,=i Kj (A) j∈N α 0.5 0.5 1 2 Let α =0.5. Then N1 = {3},N2 = {1, 2, 4}.Since ⎛ ⎞α | | | |− 0.5 0.5 | | 0.5 | | ⎝ aij α ⎠ a33 R3(A) Q1 (A) a13 + Q2 (A) a23 × + Qj (A)|aij | , α 0.5 0.5 α Kj (A) α | | ≈ j∈N1 ,=i j∈N2 +Q4 (A) a43 0.0881 > 0, by Corollary 2, A is a Hurwitz matrix. However, if then A is a Hurwitz matrix. 1 ∈ Cn×n we denote N0 = {i ∈ N ||aii| = Ri(A)}, then by a Corollary 6. Let A =[aij ] and aii < 0for 1 1 1 ∈ ∈ direct calculation we can obtain N0 = {4},N1 \ N0 = each i N.Ifthereexistsanα [0, 1] such that for 1 α {3},N2 = {1, 2},and all i ∈ N1 ,

⎛ ⎞α R1(A) R2(A) |a33|− |a31|− |a32|−|a34| | | | | | | α ⎝ aij ⎠ a11 a22 |aii| >Ki (A) α + |aij | α Kj (A) α ≈−0.4667 < 0. j∈N1 ,=i j∈N2 ⎛ ⎞ 1−α A does not satisfy the conditions of Theorem 1 of | | ⎝ aji α ⎠ Huang (1998) in this case. × α + Qj (A)|aji| , j∈N α,=i Kj (A) j∈N α Example 3. Consider matrix 1 2 ⎡ ⎤ −4+i 2 i then A is a Hurwitz matrix. A = ⎣ −4 −4+2i 2 ⎦ . By applying Lemma 3, we can obtain the following −3i 2 −3.7 Theorem 2. n×n Theorem 2. Let A =[aij ] ∈ C and Re(aii) < 0 The real part of its diagonal entries are negative. Let 1 H 1 H for i ∈ N. Define B = 2 (A + A ). If B satisfies the B = 2 (A + A ). Then we have conditions of Theorem 1 or Corollaries 1-6, then A is ⎡ ⎤ − − a Hurwitz matrix. 4 12i B = ⎣ −1 −42⎦ . IV. NUMERICAL EXAMPLES −2i 2 −3.7 This section presents numerical examples to illustrate 0.5 0.5 Let α =0.5. Then N1 (B)={3},N2 (B)={1, 2}. the applications of the conclusions established in Sec- It is straightforward to check tion III. 0.5 0.5 Example 1. Consider matrix |b33|−K3(B) (|b13| + |b23|) ⎡ ⎤ × 0.5 | | 0.5 | | 0.5 ≈ −20 −1 Q1 (B) b31 + Q2 (B) b32 0.0955 > 0. ⎣ ⎦ A = 2i −1.9 −1 , Note that the diagonal entries of B are all negative 1 −i −1.9 real numbers. Therefore, B satisfies the conditions of Corollary 5. It follows from Theorem 2 that A is a Let α =0.5. Then it is straightforward to check that Hurwitz matrix. N 0.5 = {3}, N 0.5 = {1, 2}.Since 1 2 However, since A is a complex matrix and its diag- 0.5 0.5 onal entries are not all real, the criteria in Wang et al. R1(A) C1(A) |a33|− |a31| (1994), Huang (1998), Pastravanu and Voicu (2004) |a11| are not valid in this case. 0.5 0.5 0.5 R2(A) C2(A) 0.5 + |a32| C3(A) V. CONCLUSIONS |a22| Based on the concept of α-diagonally dominance, this ≈ 0.0145 > 0, paper presents new criteria for Hurwitz stability. More by Corollary 1, A is a Hurwitz matrix. precisely, several Gerˇsgorin-like sufficient conditions 1 1 for Hurwitz stability of matrices are developed. These However, since N1 = {2, 3},N2 = {1},and conditions are simple since they only depend on the en- R1(A) tries of a given matrix. Moreover, we deal with com- |a22|− |a21|−|a23| = −0.1 < 0, |a11| plex rather than real matrix which is considered by most of the existing literature. Further research work it fails to meet the conditions of Theorem 1 of Huang may consider, for example, the iterative algorithm and (1998) in this example. the necessary conditions for Hurwitz matrices.

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VI. ACKNOWLEDGMENTS Zhang, W. and Z.Z. Han, “Bounds for the spectral The authors would like to thank the anonymous re- radius of block H-matrices,” Electronic J. Linear viewers and the subject editor Prof. Jorge A. Solsona Algebra, 15 269-273 (2006). for their helpful suggestions and valuable comments. This work was supported by the National Natural Sci- ence Foundation of China (No. 60674024). REFERENCES Berman,A.andR.J.Plemmons,Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia (1994). Cai, X.S. and Z.Z. Han, “Universal construction of control Lyapunov functions of linear systems,” Latin American Appl. Research, 36, 15-22 (2006). Carotenuto, L., G. Franze and P. Muraca, “Compu- tational method to analyse the stability of interval matrices,” IEE Proc.-Conrol Theory Appl., 151, 669-674 (2004). Chen, C.T., Linear system theory and design (3rd edi- tion), Oxford University Press (1998). Duan, G.R. and R.J. Patton, “A note on Hurwitz stability of matrices,” Automatica, 34, 509-511 (1998). Franze, G., L. Carotenuto and A. Balestrino, “New inclusion criterion for the stability of interval matrces,” IEE Proc.-Conrol Theory Appl., 153, 478- 482 (2006). Horn, R.A. and C.R. Johnson, Matrix Analysis,Cam- bridge University Press, Cambridge (1985a). Horn, R.A. and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge (1985b). Huang, T.Z., “Stability criteria for matrices,” Auto- matica, 34, 637-639 (1998). Huang, T.Z., W. Zhang and S.Q. Shen, “Regions con- taining eigenvalues of a matrices,” Electronic J. Linear Algebra, 15, 215-224 (2006). Naimark, L. and E. Zeheb, “An extension of the Levy- Desplanque theorem and some stability conditions for matrices with uncertain entries,” IEEE Trans. Circuits Syst. I, 44, 167-170 (1997). Pastravanu, O. and M. Voicu, “Necessary and suﬃ- cient conditions for componentwise stability of interval matrix systems,” IEEE Trans. Automat. Control, 49, 1016-1021 (2004). Varga, R.S., Gerˇsgorin and His Circles, Springer- Verlag, Berlin (2004). Wang, K., A. Mickel and D. Liu, “Necessary and suﬃ- cient conditions for the Hurwitz and Schur stability of interval matrces,” IEEE Trans. Automat. Con- Received: April 3, 2007. trol, 39, 1251-1254 (1994). Accepted: November 9, 2007. Recommended by Subject Editor Jorge Solsona.

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