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A switched server system of order n with all its trajectories converging to (n−1)! limit cycles
Automatica 37 (2001) 303}306
Technical Communique
A switched server system of order n with all its trajectories converging to (n!1)! limit cycles夽 Andrey V. Savkin *, Alexey S. Matveev School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney NSW 2052, Australia Department of Electrical and Electronic Engineering, University of Western Australia, WA 6907, Australia Received 17 May 1999; revised 1 May 2000; received in "nal form 28 July 2000
Abstract The paper analyzes an example of the discrete control of a continuous variable system. We prove that the switched server system with one server and an arbitrary number of bu!ers has a "nite set of periodic trajectories which attracts all other trajectories of the system. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Limit cycles; Dynamic systems; Nonlinear systems; Stability analysis; Manufacturing systems; Logic controllers; Automata theory
1. Introduction Hybrid dynamical systems have attracted considerable attention in recent years. In general, hybrid systems are those that combine continuous and discrete behavior and involve, thereby, both continuous and discrete state variables. One important type of hybrid dynamical systems is the class of discretely controlled continuous-time systems. An interesting example of such a system was introduced in Perkins and Kumar (1989) as a mathematical model for #exible manufacturing systems. Two interesting examples of such systems were inspired by the model from Perkins and Kumar (1989) and introduced in Chase, Serrano and Ramadge (1993). They were called `the switched arrival systema and `the switched server systema. These dynamical systems are of interest on their own right but have also been used to model certain aspects of #exible manufacturing systems. These examples can also be interpreted as models for simple dynamically routed closed queueing networks. It was shown in
夽
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Peter Dorato under the direction of Editor Paul Van den Hof. * Corresponding author. Tel.: #61-8-9380-1621; fax: #61-8-93801065. E-mail addresses: [email protected] (A.V. Savkin), [email protected] (A.S. Matveev).
Chase, Serrano and Ramadge (1993) that the switched arrival system exhibits a chaotic behavior whereas, under certain assumptions, the dynamics of the switched server system is eventually periodic. However, only the case of systems with three bu!ers was considered. The systems with three bu!ers can be reduced to planar systems, which makes their analysis a much easier task. Some other results on three bu!er systems were given in Ushio, Ueda and Hirai (1995), Horn and Ramadge (1997) and Li, Soh and Xu (1997). In Ushio, Ueda and Hirai (1996), the problem of existence and local stability of limit cycles in the switched arrival system with n bu!ers was considered; however, no mathematically rigorous results were presented. In the paper Savkin and Matveev (2000), the switched server system with an arbitrary number of bu!ers and very simple cyclic switching policy was considered. It was proved, that this system has a unique limit cycle and all trajectories of the system converge to this cycle. In the current paper, we give a complete qualitative analysis of the switched server system from Perkins and Kumar (1989) consisting of one server and n bu!ers with a simple and quite natural server switching feedback strategy. This switching strategy is more realistic for manufacturing applications and very similar to switching policies for #exible manufacturing systems considered in Perkins and Kumar (1989). This system has no equilibrium points. Hence, the simplest possible attractors are limit cycles. We prove that the state space of this system can be partitioned into a collection of (n!1)!
0005-1098/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 1 4 4 - 8
304
A.V. Savkin, A.S. Matveev / Automatica 37 (2001) 303}306
unbounded regions such that each of them is invariant and contains one limit cycle. Moreover, all trajectories with initial conditions from a "xed region converge to the corresponding limit cycle. Hence, any trajectory of this system is asymptotically periodic and the switched server system always exhibits a regular stable predictable behavior. This conclusion is very important for applications. Our example shows that it is typical even for very simple hybrid systems to exhibit a qualitative behavior which is quite unusual for either ordinary di!erential equations or discrete event systems.
We assume that (e.g. see Perkins & Kumar, 1989) the following assumption holds:
2. The switched server system
I(x , x ,2, x ) L
Consider the following single-machine #exible manufacturing system (Perkins & Kumar, 1989) or switched server system. This system consists of n bu!ers, with work arriving to the bu!er j at a constant rate p '0 where j"1,2,2, n. H Also, the system contains one machine or server that removes work from any selected bu!er at a constant rate p'0. Furthermore, whenever the server switches from one bu!er to another, a set-up time d'0 is required. A switched server system with n"3 is shown in the following "gure:
x x x x " min j: H "max , ,2, L . p p p p H L H 2 L In other words, I(x , x ,2, x ) is the index j at which L the maximum of x Ip is achieved, and if the maximum is H H achieved at several j, we take the minimum among them. Here, we propose the following simple switching strategy: P1. The server starts with the bu!er j such that j"I(x , x ,2, x). L P2. The server removes work from the current bu!er until it is empty. P3. Whenever, the server has emptied one bu!er at time t, it switches to the bu!er j such that j"I(x (t), x (t),2, x (t)). L This control feedback policy is quite natural and very similar to the `Clear-the-largest-bu!er-levela policy for manufacturing systems introduced in Perkins and Kumar (1989). Now we show that this system can be described by a system of logic-di!erential equations. Indeed, introduce a set of discrete variables
We refer to the contents of bu!ers as `worka; it will be convenient to think of work as a #uid, and a bu!er as a tank. In applications, work can represent a continuous approximation to the discrete #ow of parts in manufacturing systems (as in Perkins & Kumar, 1989), or jobs in a computer system, etc. The example can also be thought of as a simple instance of the switched controller problem (see e.g. Savkin, Petersen, Ska"das & Evans, 1996; Savkin & Evans, 1998). Let x (t) be the amount of work in the bu!er j at time t. H Then x (t) is a continuous variable of this system. The H location of the server is a control variable. This variable is a discrete one. Any trajectory of the switched server system is de"ned by our switching feedback policy and initial condition x (0)"x ∀j"1,2,2, n, H H where x50. H
(1)
Assumption 1. p'p #p #2#p . L It is obvious that if Assumption 1 does not hold then the system is unstable in the sense of the de"nition from Perkins and Kumar (1989): its trajectories are not bounded on [0,R). Switching strategy: Introduce the set K O+(x , x ,2, x )3RL: x 50, x 50,2, x 50,. L L Furthermore, introduce a map I from the set K to the set of indexes +1,2,2, n, as follows:
QO+q , q ,2, q , q ,. L\ L Furthermore, introduce the following vectors:
p p a(q )O $ , p L\ p L
p !p p a(q )O $ , p L\ p L
p p p !p p $ ,2, a(q )O $ . a(q )O L p p L\ L\ p p !p L L
(2)
A.V. Savkin, A.S. Matveev / Automatica 37 (2001) 303}306
305
In this system, x (t) is the amount of work in bu!er j at H time t, the discrete state q corresponds to the case when H the server is removing work from the bu!er j, the discrete state q corresponds to the case when the server is switching from one bu!er to another. Moreover, introduce map IQ from the set K to the set Q as follows:
switching sequence. Then [x( (t),q( (t)] is said to converge to [x(t), q(t)] if the following condition holds: There exists a number N'0 such that
IQ (x , x ,2, x )"q L H
lim x( (tK )"x(t ) GQ>,>H H G >
if I(x , x ,2, x )"j. L
Notation. Let l(t) be a function of time. Then l(t#0)O lim l(t#e). C C Then the above switched server system can be described by the following equations: if q(t)"q then x (t)"a(q ). (3) H H Furthermore, our switching rules P1}P3 can be described as (if q(t)"q j"1,2,2, n and x (t)"0) H H q(q) " : q ∀q3(t, t#d], . (4) then q(t#d#0) " : IQ (x (t), x (t),2, x (t)). L It should be pointed out that, for any solution [x(t), q(t)] of system (3), (4), (1), x(t) is continuous and q(t) is piecewise-constant and left-continuous. Note also that the solution apparently existing, is unique, and can be de"ned on [0,#R). Furthermore, it can be easily seen that for any solution [x(t), q(t)] of system (3), (4), (1), there exists a sequence +t ," such that t "0, t 5t , I I t 't for all k"1,2,2,lim t "#R as kP#R, I> I I q(t )Oq(t #0), and I I q(t)"const ∀t3(t , t ] ∀k"0,1,2,2 . (5) I I>
De5nition 1. The sequence t is called the switching time I sequence of solution [x(t), q(t)]. De5nition 2. A solution [x(t), q(t)] of system (3), (4), (1) is said to be a limit cycle if there exists a time ¹'0 such that x(t#¹)"x(t),
q(t#¹)"q(t) ∀t50.
De5nition 3. Let [x(t), q(t)] be a limit cycle of system (3), (4), (1). Furthermore, let ¹'0 be its minimal period, and let t be its switching time sequence. An integer number I s is said to be the order of this limit cycle if t 4¹(t . Q Q> De5nition 4. Let [x(t), q(t)] be a limit cycle of system (3), (4), (1), and let t be the switching time sequence of I [x(t), q(t)], and s be its order. Furthermore, let [x( (t),q( (t)] be any other solution of this system and let tK be its I
q(t )"q( (tK ) ∀k"0,1,2,3,2 I I>, and ∀j"1,2,2, s.
(6)
Remark. It can be easily seen that De"nition 4 implies the following property: Let [x(t), q(t)] be a limit cycle of system (3), (4), (1), and let [x( (t), q( (t)] be any other trajectory, which converges to [x(t), q(t)]. Then, it follows from condition (6) that lim sup inf ""x( (tK )!x(t)"""0. RH > RK YRH RYRH
(7)
Condition (7) is the standard de"nition of convergence to a limit cycle from the classical qualitative theory of ordinary di!erential equations (see e.g. Petrovski, 1966) and means that the corresponding limit cycle is an attractor. In general, condition (7) does not imply (6).
3. The main result In this section, we present a complete qualitative analysis of the hybrid dynamical system under consideration. We will use the following notation (n!1)!O 1;2;3;2;(n!1). Now we are in a position to present the main result of this paper. Theorem 5. Consider the switched server system (3), (4), (1) where p'0, p '0, p '0,2, p '0 are any para L meters such that condition Assumption 1 holds. Then this system has (n!1)! limit cycles. Furthermore, any trajectory of the system converges to one of them. Proof. Let x"[x , x ,2, x]3K be any initial con L dition of the system. Introduce a permutation b(x)"[b(1), b(2),2, b(n)] of the index set +1,2,2, n, such that x x x x @ 5 @ 52 @L\ 5 @L p p p p @ @ @L\ @L and if x /p "x /p for some j then @H @H @H> @H> b( j)(b( j#1). It follows from the description of the switched server system under consideration that if x (t) x (t) H 5 G p p H G
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A.V. Savkin, A.S. Matveev / Automatica 37 (2001) 303}306
for some jOi and some time t50, then x (t ) x (t ) H H 5 G H p p H G for any t 't such that the server worked with neither H the bu!er j nor the bu!er i during the time interval (t, t ]. H This and the switching rules P1}P3 imply that the discrete variable q(t) of the solution [x(t), q(t)] with initial condition x(0)"x has the following periodic sequence of discrete values: q q q q 2q q q q q q q 2. (8) @ @ @L\ @L @ The total number of all permutations b of the set of n elements is n!. Hence, the set K of initial conditions x can be partitioned into n! sets K@ such that all solu tions with initial conditions from the same set K@ have the periodic sequence of discrete values de"ned by the permutation b. Therefore, the trajectories of the system (3), (4) with initial conditions from K@ are trajectories of the cyclic switched server system considered in Savkin and Matveev (2000) with the bu!er permutation b. Now, Theorem 5.1 of Savkin and Matveev (2000) implies that for any b, there exists a limit cycle of order 2n with initial condition from K@ such that all trajectories with initial condition from K@ converge to this limit cycle. More over, for any permutation b there exist n!1 other permutations which can be obtained by cyclic shifts of the permutation b. Hence, all trajectories with initial conditions from the corresponding sets K@ converge to a common limit cycle. Therefore, the total number of cycles of the system is equal to n!/n"(n!1)!. This completes the proof of the theorem. 䊐
4. Conclusions An example of a hybrid dynamical system called a switched server system was considered. We have
proved that this hybrid system has a "nite number of limit cycles. Moreover, all trajectories of the system converge to one of these limit cycles.
Acknowledgements This work was supported by the Australian Research Council.
References Chase, C., Serrano, J., & Ramadge, P. (1993). Periodicity and chaos from switched #ow systems: Contrasting examples of discretely controlled continuous systems. IEEE Transactions on Automatic Control, 38(1), 70}83. Horn, C., & Ramadge, P. J. (1997). A topological analysis of a family of dynamical systems with nonstandard chaotic and periodic behavior. International Journal of Control, 67(6), 979}1020. Li, Z., Soh, C. B., & Xu, X. (July 1997). Stability of hybrid dynamic systems. Proceedings of the second Asian control conference, Seoul: Korea (pp. 105}108). Perkins, J. R., & Kumar, P. R. (1989). Stable, distributed, real-time scheduling of #exible manufacturing/assembly/disassembly systems. IEEE Transactions on Automatic Control, 34(2), 139}148. Petrovski, I. G. (1966). Ordinary diwerential equations. New York: Pover Publications. Savkin, A. V., & Evans, R. J. (1998). A new approach to robust control of hybrid systems over in"nite time. IEEE Transactions on Automatic Control, 43(9), 1292}1296. Savkin, A. V., & Matveev, A. S. (2000). Cyclic linear di!erential automata: A simple class of hybrid dynamical systems. Automatica, 36(5), 727}734. Savkin, A. V., Petersen, I. R., Ska"das, E., & Evans, R. J. (1996). Hybrid dynamical systems: Robust control synthesis problems. Systems and Control Letters, 29(2), 81}90. Ushio, T., Ueda, H., & Hirai, K. (1995). Controlling chaos in a switched arrival system. Systems and Control Letters, 26, 335}339. Ushio, T., Ueda, H., & Hirai, K. (December 1996). Stabilization of periodic orbits in switched arrival systems with n bu!ers. Proceedings of the 35th IEEE conference on decision and control, Kobe: Japan (pp. 1213}1214).
Technical Communique
A switched server system of order n with all its trajectories converging to (n!1)! limit cycles夽 Andrey V. Savkin *, Alexey S. Matveev School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney NSW 2052, Australia Department of Electrical and Electronic Engineering, University of Western Australia, WA 6907, Australia Received 17 May 1999; revised 1 May 2000; received in "nal form 28 July 2000
Abstract The paper analyzes an example of the discrete control of a continuous variable system. We prove that the switched server system with one server and an arbitrary number of bu!ers has a "nite set of periodic trajectories which attracts all other trajectories of the system. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Limit cycles; Dynamic systems; Nonlinear systems; Stability analysis; Manufacturing systems; Logic controllers; Automata theory
1. Introduction Hybrid dynamical systems have attracted considerable attention in recent years. In general, hybrid systems are those that combine continuous and discrete behavior and involve, thereby, both continuous and discrete state variables. One important type of hybrid dynamical systems is the class of discretely controlled continuous-time systems. An interesting example of such a system was introduced in Perkins and Kumar (1989) as a mathematical model for #exible manufacturing systems. Two interesting examples of such systems were inspired by the model from Perkins and Kumar (1989) and introduced in Chase, Serrano and Ramadge (1993). They were called `the switched arrival systema and `the switched server systema. These dynamical systems are of interest on their own right but have also been used to model certain aspects of #exible manufacturing systems. These examples can also be interpreted as models for simple dynamically routed closed queueing networks. It was shown in
夽
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Peter Dorato under the direction of Editor Paul Van den Hof. * Corresponding author. Tel.: #61-8-9380-1621; fax: #61-8-93801065. E-mail addresses: [email protected] (A.V. Savkin), [email protected] (A.S. Matveev).
Chase, Serrano and Ramadge (1993) that the switched arrival system exhibits a chaotic behavior whereas, under certain assumptions, the dynamics of the switched server system is eventually periodic. However, only the case of systems with three bu!ers was considered. The systems with three bu!ers can be reduced to planar systems, which makes their analysis a much easier task. Some other results on three bu!er systems were given in Ushio, Ueda and Hirai (1995), Horn and Ramadge (1997) and Li, Soh and Xu (1997). In Ushio, Ueda and Hirai (1996), the problem of existence and local stability of limit cycles in the switched arrival system with n bu!ers was considered; however, no mathematically rigorous results were presented. In the paper Savkin and Matveev (2000), the switched server system with an arbitrary number of bu!ers and very simple cyclic switching policy was considered. It was proved, that this system has a unique limit cycle and all trajectories of the system converge to this cycle. In the current paper, we give a complete qualitative analysis of the switched server system from Perkins and Kumar (1989) consisting of one server and n bu!ers with a simple and quite natural server switching feedback strategy. This switching strategy is more realistic for manufacturing applications and very similar to switching policies for #exible manufacturing systems considered in Perkins and Kumar (1989). This system has no equilibrium points. Hence, the simplest possible attractors are limit cycles. We prove that the state space of this system can be partitioned into a collection of (n!1)!
0005-1098/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 1 4 4 - 8
304
A.V. Savkin, A.S. Matveev / Automatica 37 (2001) 303}306
unbounded regions such that each of them is invariant and contains one limit cycle. Moreover, all trajectories with initial conditions from a "xed region converge to the corresponding limit cycle. Hence, any trajectory of this system is asymptotically periodic and the switched server system always exhibits a regular stable predictable behavior. This conclusion is very important for applications. Our example shows that it is typical even for very simple hybrid systems to exhibit a qualitative behavior which is quite unusual for either ordinary di!erential equations or discrete event systems.
We assume that (e.g. see Perkins & Kumar, 1989) the following assumption holds:
2. The switched server system
I(x , x ,2, x ) L
Consider the following single-machine #exible manufacturing system (Perkins & Kumar, 1989) or switched server system. This system consists of n bu!ers, with work arriving to the bu!er j at a constant rate p '0 where j"1,2,2, n. H Also, the system contains one machine or server that removes work from any selected bu!er at a constant rate p'0. Furthermore, whenever the server switches from one bu!er to another, a set-up time d'0 is required. A switched server system with n"3 is shown in the following "gure:
x x x x " min j: H "max , ,2, L . p p p p H L H 2 L In other words, I(x , x ,2, x ) is the index j at which L the maximum of x Ip is achieved, and if the maximum is H H achieved at several j, we take the minimum among them. Here, we propose the following simple switching strategy: P1. The server starts with the bu!er j such that j"I(x , x ,2, x). L P2. The server removes work from the current bu!er until it is empty. P3. Whenever, the server has emptied one bu!er at time t, it switches to the bu!er j such that j"I(x (t), x (t),2, x (t)). L This control feedback policy is quite natural and very similar to the `Clear-the-largest-bu!er-levela policy for manufacturing systems introduced in Perkins and Kumar (1989). Now we show that this system can be described by a system of logic-di!erential equations. Indeed, introduce a set of discrete variables
We refer to the contents of bu!ers as `worka; it will be convenient to think of work as a #uid, and a bu!er as a tank. In applications, work can represent a continuous approximation to the discrete #ow of parts in manufacturing systems (as in Perkins & Kumar, 1989), or jobs in a computer system, etc. The example can also be thought of as a simple instance of the switched controller problem (see e.g. Savkin, Petersen, Ska"das & Evans, 1996; Savkin & Evans, 1998). Let x (t) be the amount of work in the bu!er j at time t. H Then x (t) is a continuous variable of this system. The H location of the server is a control variable. This variable is a discrete one. Any trajectory of the switched server system is de"ned by our switching feedback policy and initial condition x (0)"x ∀j"1,2,2, n, H H where x50. H
(1)
Assumption 1. p'p #p #2#p . L It is obvious that if Assumption 1 does not hold then the system is unstable in the sense of the de"nition from Perkins and Kumar (1989): its trajectories are not bounded on [0,R). Switching strategy: Introduce the set K O+(x , x ,2, x )3RL: x 50, x 50,2, x 50,. L L Furthermore, introduce a map I from the set K to the set of indexes +1,2,2, n, as follows:
QO+q , q ,2, q , q ,. L\ L Furthermore, introduce the following vectors:
p p a(q )O $ , p L\ p L
p !p p a(q )O $ , p L\ p L
p p p !p p $ ,2, a(q )O $ . a(q )O L p p L\ L\ p p !p L L
(2)
A.V. Savkin, A.S. Matveev / Automatica 37 (2001) 303}306
305
In this system, x (t) is the amount of work in bu!er j at H time t, the discrete state q corresponds to the case when H the server is removing work from the bu!er j, the discrete state q corresponds to the case when the server is switching from one bu!er to another. Moreover, introduce map IQ from the set K to the set Q as follows:
switching sequence. Then [x( (t),q( (t)] is said to converge to [x(t), q(t)] if the following condition holds: There exists a number N'0 such that
IQ (x , x ,2, x )"q L H
lim x( (tK )"x(t ) GQ>,>H H G >
if I(x , x ,2, x )"j. L
Notation. Let l(t) be a function of time. Then l(t#0)O lim l(t#e). C C Then the above switched server system can be described by the following equations: if q(t)"q then x (t)"a(q ). (3) H H Furthermore, our switching rules P1}P3 can be described as (if q(t)"q j"1,2,2, n and x (t)"0) H H q(q) " : q ∀q3(t, t#d], . (4) then q(t#d#0) " : IQ (x (t), x (t),2, x (t)). L It should be pointed out that, for any solution [x(t), q(t)] of system (3), (4), (1), x(t) is continuous and q(t) is piecewise-constant and left-continuous. Note also that the solution apparently existing, is unique, and can be de"ned on [0,#R). Furthermore, it can be easily seen that for any solution [x(t), q(t)] of system (3), (4), (1), there exists a sequence +t ," such that t "0, t 5t , I I t 't for all k"1,2,2,lim t "#R as kP#R, I> I I q(t )Oq(t #0), and I I q(t)"const ∀t3(t , t ] ∀k"0,1,2,2 . (5) I I>
De5nition 1. The sequence t is called the switching time I sequence of solution [x(t), q(t)]. De5nition 2. A solution [x(t), q(t)] of system (3), (4), (1) is said to be a limit cycle if there exists a time ¹'0 such that x(t#¹)"x(t),
q(t#¹)"q(t) ∀t50.
De5nition 3. Let [x(t), q(t)] be a limit cycle of system (3), (4), (1). Furthermore, let ¹'0 be its minimal period, and let t be its switching time sequence. An integer number I s is said to be the order of this limit cycle if t 4¹(t . Q Q> De5nition 4. Let [x(t), q(t)] be a limit cycle of system (3), (4), (1), and let t be the switching time sequence of I [x(t), q(t)], and s be its order. Furthermore, let [x( (t),q( (t)] be any other solution of this system and let tK be its I
q(t )"q( (tK ) ∀k"0,1,2,3,2 I I>, and ∀j"1,2,2, s.
(6)
Remark. It can be easily seen that De"nition 4 implies the following property: Let [x(t), q(t)] be a limit cycle of system (3), (4), (1), and let [x( (t), q( (t)] be any other trajectory, which converges to [x(t), q(t)]. Then, it follows from condition (6) that lim sup inf ""x( (tK )!x(t)"""0. RH > RK YRH RYRH
(7)
Condition (7) is the standard de"nition of convergence to a limit cycle from the classical qualitative theory of ordinary di!erential equations (see e.g. Petrovski, 1966) and means that the corresponding limit cycle is an attractor. In general, condition (7) does not imply (6).
3. The main result In this section, we present a complete qualitative analysis of the hybrid dynamical system under consideration. We will use the following notation (n!1)!O 1;2;3;2;(n!1). Now we are in a position to present the main result of this paper. Theorem 5. Consider the switched server system (3), (4), (1) where p'0, p '0, p '0,2, p '0 are any para L meters such that condition Assumption 1 holds. Then this system has (n!1)! limit cycles. Furthermore, any trajectory of the system converges to one of them. Proof. Let x"[x , x ,2, x]3K be any initial con L dition of the system. Introduce a permutation b(x)"[b(1), b(2),2, b(n)] of the index set +1,2,2, n, such that x x x x @ 5 @ 52 @L\ 5 @L p p p p @ @ @L\ @L and if x /p "x /p for some j then @H @H @H> @H> b( j)(b( j#1). It follows from the description of the switched server system under consideration that if x (t) x (t) H 5 G p p H G
306
A.V. Savkin, A.S. Matveev / Automatica 37 (2001) 303}306
for some jOi and some time t50, then x (t ) x (t ) H H 5 G H p p H G for any t 't such that the server worked with neither H the bu!er j nor the bu!er i during the time interval (t, t ]. H This and the switching rules P1}P3 imply that the discrete variable q(t) of the solution [x(t), q(t)] with initial condition x(0)"x has the following periodic sequence of discrete values: q q q q 2q q q q q q q 2. (8) @ @ @L\ @L @ The total number of all permutations b of the set of n elements is n!. Hence, the set K of initial conditions x can be partitioned into n! sets K@ such that all solu tions with initial conditions from the same set K@ have the periodic sequence of discrete values de"ned by the permutation b. Therefore, the trajectories of the system (3), (4) with initial conditions from K@ are trajectories of the cyclic switched server system considered in Savkin and Matveev (2000) with the bu!er permutation b. Now, Theorem 5.1 of Savkin and Matveev (2000) implies that for any b, there exists a limit cycle of order 2n with initial condition from K@ such that all trajectories with initial condition from K@ converge to this limit cycle. More over, for any permutation b there exist n!1 other permutations which can be obtained by cyclic shifts of the permutation b. Hence, all trajectories with initial conditions from the corresponding sets K@ converge to a common limit cycle. Therefore, the total number of cycles of the system is equal to n!/n"(n!1)!. This completes the proof of the theorem. 䊐
4. Conclusions An example of a hybrid dynamical system called a switched server system was considered. We have
proved that this hybrid system has a "nite number of limit cycles. Moreover, all trajectories of the system converge to one of these limit cycles.
Acknowledgements This work was supported by the Australian Research Council.
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