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A hybrid dynamical system of order N with all trajectories converging to (N-1)! limit cycles
A HYBRID DYNAMICAL SYSTEM OF ORDER N WITH ALL TRAJ. ..
14th World Congress ofIFAC
E-2c-05-.5
Copyright © 1999 IFAC 14th Triennial World COngress, Beijing, P.R. China
A HYBRID DYNAMICAL SYSTEM OF ORDER N "WITH ALL TRAJECTORIES CONVERGING TO (N-l)! LIMIT CYCLES
Andrey V. Savkin*
Department of Electrical and Electronic Engineering, University of Western Australia, Nedlands, ,~rA 6907, Australia, email: [email protected], fax: +(618) >I<
93801065
Abstract. The paper analyzes an example of the discrete control of a continuous variable system. We prove that the switched server system ~pith one server and an arbitrary number of buffers has a finite set of periodic trajectories which attracts all other trajectories of the system. Copyright © 1999IFAC
Keywords. Limit Cycles, Stability, Dynamic Systems, Networks, Switching
1. INTRODUCTION Hybrid dynamical systems have attracted considerable attention in recent years. In general, hy~ brid 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 continuoustime systems. Two interesting examples of such systems were inspired by the models for flexible manufacturing systems (Chase et al. 1993). They were called "the switched arrival system't and "the switched server system". These dynan"lical systems are of interest on their own right but have also been used to model certain aspects of flexible manufacturing systems. These examples can also be interpreted as models for simple dynami-
cally routed closed queueing net'\vorks. It is well knovln that the switched arrival system exhibits a chaotic behavior \vhereas, under certain assumptions, the dynamics of the switched server system is eventually periodic (Chase et aL 1993). However, only the case of systems with three buffers was considered. The systems with three buffers can be reduced to planar systems, which makes
lar to switching policies for flexible manufacturing systems. 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 -I)! unbounded regions such that each of them is invariant and contains one limit cycle. Moreover, all trajectories with initial conditions from a fixed region converge to the corresponding limit cycle. Hence any trajectory of this system is asymptotically periodic and the switched server system allvays exhibits a regular stable predictable behavior. This conclusion is very important for appli-
cations. OUf example shows that it is typical even for very simple hybrid systems to exibit a qualitative be-
havior which is quite unusual for either ordinary differential equations or discrete event systems.
2. THE
S'v~ITCHED
SERVER
SYSTEM Consider the following single-machine flexible manufacturing system or switched server system (Perkins and Kumar 1989).
their analysis a much easier task.
This system consists of n buffers, with work ar-
>
In the current paper, ,"ve give a complete quali-
riving to the buffer j at a constant rate Pi
tative analysis of the switched server system con-
where j
sisting of one server and n buffers with a simple and quite natural server switching feedback strategy. This sVt'itching strategy is more realistic for manufacturing applications and very simi-
one machine or server that removes work from any selected buffer at a constant rate p > o. Furthermore 1 whenever the server switches from one buffer to another, a set-up time 0 > 0 is required.
= 1,2, ... , n.
0
Also. the system contains
2155
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
A HYBRID DYNAMICAL SYSTEM OF ORDER N WITH ALL TRAJ. ..
We refer to the contents of buffers as "work"; it will be convinient to to think of ",~ork as a fluid: and a buffer as a tank. In applications, work can represent a continuous approximation to the discrete flow of parts in manufacturing systems (as in (Perkins and Kumar 1989)), or jobs in a computer system) etc.
This control feedback policy is quite natural and very similar to the control policies for manufacturing systems considered in (Perkins and Kllmar
1989). Now we show that this system can be described by a system of logic-differential equations. Indeed, introduce a set of discrete variables
The example can also be thought of as a simple instance of the switched controller problem.
Let
Xj (t)
be the amount of work in the buffer j
at time t. Then Xj(t) is a continuous variable of this system. The location of the server is a control variable. This variable is a discrete one. Any trajectory of the slvitched server system is defined by our switching feedback policy and initial con-
Furthermore, introduce the following vectors
Pt P2
dition
Pn-l
Xj(O)==xj where
Vj==1,2, ... ,n
Pn
(1)
Pt - P P2
xJ ?:: o.
We assume that the following assumption holds
Pn-l
(Perkins and Kumar 1989): Assumption 1
p> Pt
Pn
+ P2 + ... + Pn·
Pt P2 -p
It is obvious that if Assumption 1 does not hold then the system is unstable in the following sense: its traj ectories are not bounded on [0, (0).
, ... , Pn-l
Pn
Switching strategy Introduce the set
Pt
P2
(2) Pn-l
Furthermore, introduce a map I from the set Ko to the set of indeces {1, 2, ... , n} as follows: Z(XI'
X2,
j=re~,n {j: ;: = max [;:.
Pn -p
=
In this system, Xj(t) is the amount of work in buffer j at time t, the discrete state qj corresponds
, ;:]} ·
to the case when the server is removing work from the buffer j, the discrete state qo corresponds to
,
xn)
the case \vhen the server is switching from one In other \vords, I(Xl' X2, . . . , x n ) is the index i at which the maximum of ~ is achieved, and if the maximum is achieved at several j, we take the minimunl among them.
buffer to another.
Moreover, introduce map IQ from the set Ko to the set Q as follows: IQ(Xl' X2, . .• , x n ) ::: qj if I(Xl' X2, .. ,xn ) j.
Here, we propose the follov.ring simple switching
strategy:
P
=
P1: The server starts with the buffer j such that j
== :r(x~) xg, ... , x~).
P2: The server removes work from the current buffer until it is empty.
P3: "lhenever, the server has emptied one buffer at time t it switches to the buffer j such that
Notation Let vet) be a function of time. Then v(t
Then the above switched server system can be described by the following equations:
if q(t)
t
j
+ 0) ~ limE>ol €~O v(t + f).
= I(Xl(t), X2(t), ... , xn(t)).
==
qj then x(t)
==
a(qj).
(3)
Furthermore, our switching rule PI, P2, P3 can
2156
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A HYBRID DYNAMICAL SYSTEM OF ORDER N WITH ALL TRAJ. ..
(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
be described as (if q (t)
= qj
14th World Congress ofIFAC
j == 1, 2, ... , n
and Xj(t) = 0)
q(r) := qo
then
VT E (t~ t
+ 6],
q(t + 8 + 0) := ( IQ(Xl(t), X2(t), .. ", xn(t)).
t.-++oo f~t. t~t.
=
= const
q(t)
(7)
The condition (7) is the standard definition of con~ vergence to a limit cycle from the classical qualitative theory of ordinary differential equations (see e.g. (Petrovski 1966)).
3. THE MAIN RESULT In this section) we present a cOInplete qualitative analysis of the h:rbrid dynamical system under consideration. \Ve will use the following notation
Vt E (tk' tk+l] Yk=O~1)2, ....
Definition 2 A solution [x(t), q(t)] of the system (3), (4), (1) is said to be a limit cycle if there exists a time T > 0 such that
+ T) == x(t),
a
(n - I)! = 1 x 2 x 3 x ~ · · x (n - 1).
(5)
Definition 1 The sequence tk is called the switching time sequence of the solution [x(t)) q(t)}.
x(t
== O.
. (4)
It should be pointed out that, for any solution [x(t), q(t)] of the system (3), (4), (1), x(t) is continuous and q(t) is piecewise-constant and leftcontinuous. Note also that the solution appar~ ently exists, is unique~ and can be defined on [0 , +00). Furthermore: it can be easily seen, that for any solution [x(t), q(t)] of the system (3), (4), (1), there exists a sequence {tk} Ik'=o such that to 0, t 1 ~ to, t k +1 > t k for all k = 1, 2, .. -~ limtk +00 as k --t +00, q(tk) =I q(tk + 0), and
=
sup inf Ilx(t) - x(t)1l
lim
)
q(t + T) = q(t)
Vt
> o.
Definition 3 Let [x(t), q(t)] be a limit cycle of the system (3), (4), (1). Furthermore, let T > 0 be its mimimal period, and let tk be its switching time sequence. An integer number s is said to be the order of this limit cycle if
Now we are in a position to present the main result of this paper. Theorelll 1 Consider the switched server system (3), (4)~ (1) ¥t"here p> O,Pl > O,P2 > 0, .. . ,Pn > oare any parameters such that condition Assumption 1 holds. Then this system has (n - I)! limit cycles. Furthermore, any trajectory of the system converges to one of them.
The proof of this result will be given in the full version of the paper.
REFERENCES Definition 4 Let [x(t), q(t)] be a limit cycle of the system (3), (4)t (1), and let tk be the switching time sequence of [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 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
and
Chase~ C.~
J. Serrano and P. Ramadge (1993)~ 'Periodicity and chaos from switched flow sys~ terns: Contrasting examples of discretely CODtrolled continuous systems). IEEE Transac-
tions on Automatic Control 38(1),70-83.
Perkins, J. and P.R. Kumar (1989). 'Stable, distributed, real-time scheduling offlexible manufacturing/assembly/ disassembly systems'. IEEE Transactions on Automatic Control 34(2), 139-148. Petrovski t I. (1966). Ordinary Differential Equations. Payer Publications Inc .. New York.
. lim X(ii8+N+j) == x(tj)
~--t+oo
\rij
=
1,2, ... ,8.
(6)
Remark It can be easily seen that Definition 4 implies the follo\ving property:
Let [x(t), q(t)] be a limit cycle of the system (3),
2157
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
E-2c-05-.5
Copyright © 1999 IFAC 14th Triennial World COngress, Beijing, P.R. China
A HYBRID DYNAMICAL SYSTEM OF ORDER N "WITH ALL TRAJECTORIES CONVERGING TO (N-l)! LIMIT CYCLES
Andrey V. Savkin*
Department of Electrical and Electronic Engineering, University of Western Australia, Nedlands, ,~rA 6907, Australia, email: [email protected], fax: +(618) >I<
93801065
Abstract. The paper analyzes an example of the discrete control of a continuous variable system. We prove that the switched server system ~pith one server and an arbitrary number of buffers has a finite set of periodic trajectories which attracts all other trajectories of the system. Copyright © 1999IFAC
Keywords. Limit Cycles, Stability, Dynamic Systems, Networks, Switching
1. INTRODUCTION Hybrid dynamical systems have attracted considerable attention in recent years. In general, hy~ brid 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 continuoustime systems. Two interesting examples of such systems were inspired by the models for flexible manufacturing systems (Chase et al. 1993). They were called "the switched arrival system't and "the switched server system". These dynan"lical systems are of interest on their own right but have also been used to model certain aspects of flexible manufacturing systems. These examples can also be interpreted as models for simple dynami-
cally routed closed queueing net'\vorks. It is well knovln that the switched arrival system exhibits a chaotic behavior \vhereas, under certain assumptions, the dynamics of the switched server system is eventually periodic (Chase et aL 1993). However, only the case of systems with three buffers was considered. The systems with three buffers can be reduced to planar systems, which makes
lar to switching policies for flexible manufacturing systems. 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 -I)! unbounded regions such that each of them is invariant and contains one limit cycle. Moreover, all trajectories with initial conditions from a fixed region converge to the corresponding limit cycle. Hence any trajectory of this system is asymptotically periodic and the switched server system allvays exhibits a regular stable predictable behavior. This conclusion is very important for appli-
cations. OUf example shows that it is typical even for very simple hybrid systems to exibit a qualitative be-
havior which is quite unusual for either ordinary differential equations or discrete event systems.
2. THE
S'v~ITCHED
SERVER
SYSTEM Consider the following single-machine flexible manufacturing system or switched server system (Perkins and Kumar 1989).
their analysis a much easier task.
This system consists of n buffers, with work ar-
>
In the current paper, ,"ve give a complete quali-
riving to the buffer j at a constant rate Pi
tative analysis of the switched server system con-
where j
sisting of one server and n buffers with a simple and quite natural server switching feedback strategy. This sVt'itching strategy is more realistic for manufacturing applications and very simi-
one machine or server that removes work from any selected buffer at a constant rate p > o. Furthermore 1 whenever the server switches from one buffer to another, a set-up time 0 > 0 is required.
= 1,2, ... , n.
0
Also. the system contains
2155
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
A HYBRID DYNAMICAL SYSTEM OF ORDER N WITH ALL TRAJ. ..
We refer to the contents of buffers as "work"; it will be convinient to to think of ",~ork as a fluid: and a buffer as a tank. In applications, work can represent a continuous approximation to the discrete flow of parts in manufacturing systems (as in (Perkins and Kumar 1989)), or jobs in a computer system) etc.
This control feedback policy is quite natural and very similar to the control policies for manufacturing systems considered in (Perkins and Kllmar
1989). Now we show that this system can be described by a system of logic-differential equations. Indeed, introduce a set of discrete variables
The example can also be thought of as a simple instance of the switched controller problem.
Let
Xj (t)
be the amount of work in the buffer j
at time t. Then Xj(t) is a continuous variable of this system. The location of the server is a control variable. This variable is a discrete one. Any trajectory of the slvitched server system is defined by our switching feedback policy and initial con-
Furthermore, introduce the following vectors
Pt P2
dition
Pn-l
Xj(O)==xj where
Vj==1,2, ... ,n
Pn
(1)
Pt - P P2
xJ ?:: o.
We assume that the following assumption holds
Pn-l
(Perkins and Kumar 1989): Assumption 1
p> Pt
Pn
+ P2 + ... + Pn·
Pt P2 -p
It is obvious that if Assumption 1 does not hold then the system is unstable in the following sense: its traj ectories are not bounded on [0, (0).
, ... , Pn-l
Pn
Switching strategy Introduce the set
Pt
P2
(2) Pn-l
Furthermore, introduce a map I from the set Ko to the set of indeces {1, 2, ... , n} as follows: Z(XI'
X2,
j=re~,n {j: ;: = max [;:.
Pn -p
=
In this system, Xj(t) is the amount of work in buffer j at time t, the discrete state qj corresponds
, ;:]} ·
to the case when the server is removing work from the buffer j, the discrete state qo corresponds to
,
xn)
the case \vhen the server is switching from one In other \vords, I(Xl' X2, . . . , x n ) is the index i at which the maximum of ~ is achieved, and if the maximum is achieved at several j, we take the minimunl among them.
buffer to another.
Moreover, introduce map IQ from the set Ko to the set Q as follows: IQ(Xl' X2, . .• , x n ) ::: qj if I(Xl' X2, .. ,xn ) j.
Here, we propose the follov.ring simple switching
strategy:
P
=
P1: The server starts with the buffer j such that j
== :r(x~) xg, ... , x~).
P2: The server removes work from the current buffer until it is empty.
P3: "lhenever, the server has emptied one buffer at time t it switches to the buffer j such that
Notation Let vet) be a function of time. Then v(t
Then the above switched server system can be described by the following equations:
if q(t)
t
j
+ 0) ~ limE>ol €~O v(t + f).
= I(Xl(t), X2(t), ... , xn(t)).
==
qj then x(t)
==
a(qj).
(3)
Furthermore, our switching rule PI, P2, P3 can
2156
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A HYBRID DYNAMICAL SYSTEM OF ORDER N WITH ALL TRAJ. ..
(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
be described as (if q (t)
= qj
14th World Congress ofIFAC
j == 1, 2, ... , n
and Xj(t) = 0)
q(r) := qo
then
VT E (t~ t
+ 6],
q(t + 8 + 0) := ( IQ(Xl(t), X2(t), .. ", xn(t)).
t.-++oo f~t. t~t.
=
= const
q(t)
(7)
The condition (7) is the standard definition of con~ vergence to a limit cycle from the classical qualitative theory of ordinary differential equations (see e.g. (Petrovski 1966)).
3. THE MAIN RESULT In this section) we present a cOInplete qualitative analysis of the h:rbrid dynamical system under consideration. \Ve will use the following notation
Vt E (tk' tk+l] Yk=O~1)2, ....
Definition 2 A solution [x(t), q(t)] of the system (3), (4), (1) is said to be a limit cycle if there exists a time T > 0 such that
+ T) == x(t),
a
(n - I)! = 1 x 2 x 3 x ~ · · x (n - 1).
(5)
Definition 1 The sequence tk is called the switching time sequence of the solution [x(t)) q(t)}.
x(t
== O.
. (4)
It should be pointed out that, for any solution [x(t), q(t)] of the system (3), (4), (1), x(t) is continuous and q(t) is piecewise-constant and leftcontinuous. Note also that the solution appar~ ently exists, is unique~ and can be defined on [0 , +00). Furthermore: it can be easily seen, that for any solution [x(t), q(t)] of the system (3), (4), (1), there exists a sequence {tk} Ik'=o such that to 0, t 1 ~ to, t k +1 > t k for all k = 1, 2, .. -~ limtk +00 as k --t +00, q(tk) =I q(tk + 0), and
=
sup inf Ilx(t) - x(t)1l
lim
)
q(t + T) = q(t)
Vt
> o.
Definition 3 Let [x(t), q(t)] be a limit cycle of the system (3), (4), (1). Furthermore, let T > 0 be its mimimal period, and let tk be its switching time sequence. An integer number s is said to be the order of this limit cycle if
Now we are in a position to present the main result of this paper. Theorelll 1 Consider the switched server system (3), (4)~ (1) ¥t"here p> O,Pl > O,P2 > 0, .. . ,Pn > oare any parameters such that condition Assumption 1 holds. Then this system has (n - I)! limit cycles. Furthermore, any trajectory of the system converges to one of them.
The proof of this result will be given in the full version of the paper.
REFERENCES Definition 4 Let [x(t), q(t)] be a limit cycle of the system (3), (4)t (1), and let tk be the switching time sequence of [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 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
and
Chase~ C.~
J. Serrano and P. Ramadge (1993)~ 'Periodicity and chaos from switched flow sys~ terns: Contrasting examples of discretely CODtrolled continuous systems). IEEE Transac-
tions on Automatic Control 38(1),70-83.
Perkins, J. and P.R. Kumar (1989). 'Stable, distributed, real-time scheduling offlexible manufacturing/assembly/ disassembly systems'. IEEE Transactions on Automatic Control 34(2), 139-148. Petrovski t I. (1966). Ordinary Differential Equations. Payer Publications Inc .. New York.
. lim X(ii8+N+j) == x(tj)
~--t+oo
\rij
=
1,2, ... ,8.
(6)
Remark It can be easily seen that Definition 4 implies the follo\ving property:
Let [x(t), q(t)] be a limit cycle of the system (3),
2157
Copyright 1999 IFAC
ISBN: 0 08 043248 4