c 2008 Institute for Scientific ° Computing and Information
INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 4, Number 1, Pages 114–123
THE STABILITY OF DISCRETE CIRCLE-LINKED SYSTEMS YUECHAO MA AND QINGLING ZHANG Abstract. This paper proposes a class of discrete circle-linked systems. A necessary and sufficient condition for the realization of circle-linked systems is given by using circle-linked eigenvectors. The problems of stability for the discrete circle-linked systems and the solution of the Lyapunov equation are discussed. Key Words. Circle-linked eigenvectors, discrete circle-linked systems, Lyapunov equation, stability.
1. Introduction Great progress has been made in the research of control systems. In recent years attention has been on the systems that relate to the real systems. Publications [1-9] describe the problem of symmetric systems, in fact symmetric systems can be found in electric power systems, economics systems and robot systems. [10] studies decentralized control problem of a class of nonlinear circle-linked systems. Circlelinked systems can be found in electric power systems. In this paper we define a class of discrete circle-linked systems and give a sufficient and necessary condition for the realization of discrete circle-linked systems. At last we discuss the existence of the solution for Lyapunov equation for the discrete circle-linked systems. 2. Discrete Circle-linked Systems Consider the following discrete system: (1)
xk+1 = Axk
Definition 1. [9] Suppose A ∈ Rn×n . Let α1 , α2 , · · · , αk be linearly independent vectors in Rn if there exist λ1 , λ2 , · · · , λk ∈ Rn such that Aαk = λk−1 αk−1 , Aαk−1 = λk−2 αk−2 , · · · , Aα2 = λ1 α1 , Aα1 = λk αk . Then α1 , α2 , · · · , αk is called a k-order circle-linked eigenvector group of A. Definition 2. The system (1) is said to be a circle-linked realizable discrete system, if there exists reversible matrix T satisfying ¤1 ¤2 T −1 AT = .. . ¤q Received by the editors January 11, 2007 and, in revised form, March 26, 2007. 2000 Mathematics Subject Classification. 35R35, 49J40, 60G40. This research was supported by the National Science Foundation of China (No. 60574011). 114
THE STABILITY OF DISCRETE CIRCLE-LINKED SYSTEMS
115
where
0 0 λ1 · · · 0 .. . .. .. . . ··· . ¤1 = . , · · · , ¤q = . 0 0 0 · · · λk1 −1 λk1 0 · · · 0 λn with k1 + k2 + · · · + kq = n.
λPq−1 ki +1 i=1 .. . 0 0
··· ··· ··· ···
0 .. . λn−1 0
Definition 3. The system (1) is said to be a fully circle-linked realizable discrete system, if there exists reversible matrix T satisfying 0 λ1 0 · · · 0 0 0 λ2 · · · 0 .. .. . . −1 .. · · · .. T AT = . . 0 0 0 · · · λn−1 λn 0 0 ··· 0 where λi ∈ R, i = 1, 2, · · ·, n. Theorem 1. The system (1) is fully circle-linked realizable discrete system if and only if the n-ordered circle-linked eigenvector group of A constitutes a base of Rn . Proof. This can be proved by definition 1 and definition 3.
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Theorem 2. The system (1) has circle-linked decomposition if and only if there exist the k1 , k2 , · · · , kq order circle-linked eigenvector groups of A which constitute a base of Rn . Proof. This can be obtained by definition 1 and definition 2.
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3. Main results If the system xk+1 = Axk has circle-linked decomposition, then there exists a reversible matrix T, such that ¤1 ¤2 T −1 AT = . .. ¤q Let
¤k1
¤kq
=
0 0 .. .
λ1 0 .. .
= 0 λk 1 0 0 .. . 0 λn
0 0
λt−1 P
0
.. . 0 0
··· ··· ··· ··· ···
0 0 ··· ,
ki +1
i=1
0 λ2 .. .
0 λq−1 P
ki +2
0 0 .. . λk1 −1 0
···
0
···
0
i=1
.. . 0 0
··· ··· ···
... λn−1 0
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Y. MA AND Q. ZHANG
Theorem 3. If a system has circle-linked decomposition, then allp the eigenvalues of the state matrix lie in the circle cantered at origin with radius ki k¤ki k. Proof. Upon computation,
|SI − A|
¯ ¯ SI − ¤k1 ¯ ¯ ¯ = ¯ ¯ ¯ ¯
SI − ¤k2
..
. SI − ¤kq
¯ ¯ = |SI − ¤k1 | · · · ¯SI − ¤kq ¯
|SI − ¤ki |
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
= S ki + (−1)ki |¤ki |
It follows that p the eigenvalues of A all lie in the circle whose center is zero, and the radius is ki k¤ki k (i = 1, 2, · · · , q). ¤ From theorem 3, we have the following result: Theorem 4. The discrete system xk+1 = Axk has circle-linked decomposition and the unique equilibrium xe = 0 is asymptotically stable if and only if k¤ki k < 1 (i = 1, 2, · · · , q). Now we give a sufficient condition to judge stability. Corollary 1. If the discrete system xk+1 = Axk has circle-linked decomposition for every |λi | < 1(i = 1, 2, · · · , n), then the system is asymptotically stable. Example 1. Consider circle-linked discrete system: 0 2 0 0 0 1 0 0 0 0 3 xk+1 = 0 0 0 1 0 xk 0 0 0 0 4 0 0 15 0 0 ° ° ° 0 1 0 ° ° 4 ° ° kDk1 k < 1, kDk2 k = ° ° 01 0 4 ° = 5 < 1 ° 0 0 ° 5
According to theorem 2, the system is asymptotically stable. Example 2. Consider the circle-linked discrete system 0 13 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 2 xk+1 = 0 0 0 0 −1 0 2 0 0 0 0 0 − 13 1 0 0 0 7 0 0
xk
every |λi | < 1(i = 1, 2, · · · , q) as in corollary 1 the system is asymptotically stable. Theorem 5. Let
0 0 .. .
A= 0 λn
λ1 0 .. .
0 λ2 .. .
0 0
0 0
··· ··· ··· ··· ···
0 0 .. . λn−1 0
THE STABILITY OF DISCRETE CIRCLE-LINKED SYSTEMS
117
such that AT U A − U = −W is negative definite, U is symmetric. (1) If |λ1 λ2 · · · λn | = 1, then AT U A − U = −W has no solution. (2) If |λ1 λ2 · · · λn | 6= 1, then AT U A − U = −W has a unique solution. Besides Uii (i = 1, 2, · · · , n) are uniquely determined by the equations:
−1 λ21 0 .. .
0 −1 λ22 .. .
0
0
0 0 0 0 −1 0 .. .. . . 0 0
··· ···
0 0 0 .. .
λ2n 0 0 .. .
λ2n−1
−1
··· ··· ···
U11 U22 U33 .. .
=
Unn
−Wnn
Uij (i 6= j) are uniquely determined by the equations:
−1 0 .. . 0 λ1 λ2 0 .. . 0 0 . .. 0
0 −1 .. . 0 0 λ1 λ3 .. . 0 0 .. . 0
··· ···
0 0 .. .
λ1 λn 0 .. .
0 0 .. .
0 0 0 .. .
−1 0 0 .. .
0 −1 0 .. .
λ1 λn−1 0 .. .
0 0 .. .
0 λ2 λ3 .. .
0
0
0
··· ··· ··· ··· ··· ··· ··· ··· ···
0 λ2 λn .. .
0 0 .. .
0 0 0 .. .
0 0 0 .. .
−1 0 .. .
0 −1 .. .
0
0
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···
0 0 .. .
··· ···
··· 0 ··· 0 ··· 0 ··· .. . ··· 0 ··· 0 ··· .. . ··· 0
···
0 0 .. .
··· ···
0
···
··· 0 ··· 0 ··· −1 · · · .. . ··· 0 ··· 0 ··· .. . ···
0 0 .. . λn−1 λn 0 0 .. . 0 0 .. . −1
−W11 −W22 −W33 .. .
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Y. MA AND Q. ZHANG
U12 U13 .. .
U1,n−1 U1n U23 U24 .. × . U2n U34 .. . U3n .. . Un−1,n
−W12 −W13 .. .
−W1,n−1 −W1n −W23 −W24 = .. . −W 2n −W 34 . . . −W 3n .. . −Wn−1,n
Proof. As AT U A − U = −W holds, direct multiplication of the matrices will yield the above two equation groups. Applying primary row transformations, we will get the results. ¤ Theorem 6. Consider the system
xk+1 = Axk + buk ,
0 0 .. .
where A = 0 λn
λ1 0 .. .
0 λ2 .. .
0 0
0 0
··· ···
0 0 .. .
··· ··· ···
λn−1 0
,
b 6= 0
(1) If kAk = 1, then AU AT − U = bbT has no solution. (2) If kAk 6= 1, then AU AT − U = bbT has a unique solution. Besides Uii (i = 1, 2, · · · , n) are uniquely determined by the equations:
−1 0 .. .
λ21 −1 .. .
0 λ22 .. .
0 λ2n
0 0
0 0
··· ··· ··· ··· ···
0 0 .. . −1 0
0 0 .. .
U11 U22 .. .
λ2 Un−1,n−1 −1 Unn
b21 b22 .. .
= 2 bn−1,n−1 b2n
THE STABILITY OF DISCRETE CIRCLE-LINKED SYSTEMS
Uij (i 6= j) are uniquely determined by the equations: −1 0 ··· 0 0 λ1 λ2 0 0 −1 · · · 0 0 0 λ λ3 1 .. .. .. .. .. .. . . ··· . . . . 0 0 · · · −1 0 0 0 λ1 λn 0 ··· 0 −1 0 0 0 0 · · · 0 0 −1 0 .. .. .. .. .. .. . . ··· . . . . 0 0 · · · 0 0 0 0 0 λ2 λn · · · 0 0 0 0 . .. . . . . .. .. .. .. . .. ··· 0 0 ··· 0 λn−1 λn 0 0 0 0 .. .
0 0 .. .
0 0 .. .
0 0 0 .. .
λ1 λn−1 0 0 .. .
0 0 λ2 λ3 .. .
−1 0 .. .
0 −1 .. .
0 0 .. .
0
0
0
U12 U13 .. .
U1,n−1 U1n U23 U24 .. . × U2−1 U2n U34 .. . U3n .. . Un−1,n
··· ···
0 0 .. .
··· ··· ··· ···
0 0 0 .. .
··· ··· ···
λ2 λn 0 .. .
··· ···
0
b1 b2 b1 b3 .. .
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···
b1 bn−1 b1 bn b2 b3 b2 b4 .. = . b2 bn−1 b2 bn b3 b4 .. . b3 bn .. . bn−1 bn
119
0 0 .. . 0 0 0 .. . 0 0 .. .
−1
The proof is similar to that of theorem 5. Corollary 2. If the system xk+1 = Axk + buk is fully circle-linked and stable, then AU AT − U = bbT has an unique solution.
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Y. MA AND Q. ZHANG
Proof. If xk+1 = Axk + buk is controllable and fully circle-linked, then rank P |SI − A|
P
−1
AP
P −1 b
£ ¤ rank An−1 b, · · · , b = n
=
= S n + (−1)n |A| 0 1 0 0 . .. .. = . 0 0 (−1)n |A| 0 0 0 = . ..
0 1 .. .
0 0 .. .
b=P
= P BP −1 ,
0 0 .. .
1 0
··· ··· ···
0 0
1
A
··· ···
1 AU AT − U = bbT (P BP −1 )U (P BP −1 )T − U
=
P
P B[P
−1
U (P
−1 T
T
T
) ]B P − U
=
B[P −1 U (P −1 )T ]B T − P −1 U (P −1 )T
=
0 0 .. .
P
1
0 ··· .. P . 0 ··· 0 ··· 0 .. .. . . 0
···
1
Let K = P −1 U (P −1 )T = (Kij )
0 BKB T − K = ... 0 According to theorem 6, we get
0 .. T 1) If kAk = 1, BKB − K = . 0
··· ···
··· ···
0 .. . 1
0 .. has no solution. . 1
0 0 .. . 1
0 .. P T . 1
T
THE STABILITY OF DISCRETE CIRCLE-LINKED SYSTEMS
0 .. T 2) If kAk 6= 1, BKB − K = . 0
−1 0 .. . 0 (−1)n |A|
1 0 −1 1 .. .. . . 0 0 0 0
···
0 0 .. .
··· ··· ···
−1 0 .. . 0 (−1)n |A| 0 .. . 0 0 .. . 0
0 −1 .. . 0 0 0 .. . 0 (−1)n |A| .. . 0
1 (−1)n |A|−1 ,
··· ··· ··· ··· ··· ··· ··· ··· ···
i = 1, 2, · · · , n
0 0 .. .
1 0 .. .
−1 0 0 .. .
0 −1 0 .. .
0 0 .. .
0 0 .. .
0 0 0 0 −1 0 .. .. . . 0 0 0 0 .. .. . .
0
(−1)n |A|
··· ···
0 0 .. .
0 0 0 .. .
1 0 0 .. .
0 0 1 .. .
··· ··· ··· ··· ··· ···
··· 0 0 ··· −1 0 · · · .. .. . . ··· 0 0 ···
0 0 .. . 0 0 0 .. . 1 0 .. . 0
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···
0
0 0 .. . 0 0 0 .. . 0 0 .. . 1
0 0 .. .
= 0 1
0 0 .. .
0 0 .. .
0
K11 K22 .. .
1 Kn−1,n−1 Kn,n −1
0 0 .. .
−1 0 .. .
0 0 .. .
−1 0
Solving the equation, we get Kii =
0 .. has a unique solution and . 1
···
··· ···
121
0 1 .. .
0
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···
122
Y. MA AND Q. ZHANG
×
U12 U13 .. .
U1,n−1 U1,n U23 U24 .. . U2,n−1 U2,n U34 .. . U3n .. . Un−1,n
=0
Then Kij = 0, i 6= j. So U = P KP T , where 1 (−1)n |A|−1 ¡ n−1 ¢ P = A b, · · · , b , K =
..
. 1 (−1)n |A|−1
This completes the proof.
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Corollary 3. If the system xk+1 = Axk + buk is controllable, fully circle-linked and stable, then AU AT − U = bbT has a unique solution as follows: U = P KP T where ¡ ¢ P = An−1 b, · · · , b ,
K=
1 (−1)n |A|−1
..
. 1 (−1)n |A|−1
4. Conclusion This paper discusses the problems of the stability for a class of discrete circlelinked systems and the solution of Lyapunov equations. A sufficient and necessary condition for the stability of the discrete circle-linked systems is given. References [1] G.S. Zhai, X.K. Chen, I. Masao, Stability and L2 gain analysis for a class of switched symmetric systems, Proceedings of the IEEE Conference on Decision and Control, 4(2002) 4395-4400. [2] J.N. Xue, K. Yang, Symmetric relations in multistate system, IEEE Transactions on Reliability, 44(1995) 689-693. [3] R. Tanaka, K. Murota, Symmetric failures in symmetric control systems, Linear Algebra and Its Applications, 318(2000) 145-172. [4] I.G. Shaposhnikov, On some systems of generators of symmetric and alternating groups which allow a simple software implementation, Discrete Mathematics and Applications, 14(2004) 103-110. [5] L. Bakule, J. Rodellar, Decentralised control design of uncertain nominally linear symmetric composite systems, IEE Proceedings: Control Theory and Applications, 143(1996) 530-535. [6] J. Rohn, Algorithm for checking stability of symmetric interval matrices, IEEE Transactions on Automatic Control, 41 (1996) 133-136.
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[7] S.L. Osburn, D.S. Bernstein, Robust stability and performance analysis for skew-symmetric uncertainty using the shifted bounded real guaranteed cost bound, Proceedings of the IEEE Conference on Decision and Control, 4(1998) 4392-4393. [8] X.P. Liu, Output regulation of strongly coupled symmetric composite systems, Automatica, 28(1992)1037-104. [9] Z.D. Xu, X.Y. Li, Control design based on state observer for nonlinear delay systems via an LMI approach, International journal of information and systems sciences. This issue. [10] Y.C. Ma, Q.L. Zhang, X.F. Zhang, Decentralized output feedback robust control for a class of uncertain nonlinear circlr-kinked large-scale composite systems, International journal of information and systems sciences, 2006, 2(1):20-30. College of Science, Yanshan University, Qianhuangdao, Hebei Province, China, 066004; E-mail:
[email protected] College of Science, Northeastern University, Shenyang, Liaoning Province, China, 110004. E-mail:
[email protected]
