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Existence and stability of periodic trajectories in switched server systems
EXISTENCE AND STABILITY OF PERIODIC TRAJECTORIES 1...
Copyright © 1999 IFAC 14th Triennial World Congress,
14th World Congress ofIFAC
G-2e-15-1 Beijing~ P.R.
China
EXISTENCE AND STABILITY OF PERIODIC TRAJECTORIES IN SWITCHED SERVER SYSTEMS
Alldrey V. Savkin'" and Alexey S. Matveev*
* Departluent of Electrical and Electronic Engineering University of Western _~us traiia, Nedlands, 'VA 6907~ Australia) enlaiL [email protected]\va.edu.au, fax: +(618) 1
93801065
Abstract. The paper analyzes an example of the discrete cont.rol of a continuous varia.ble system. We prove that the s\vitched server system with one server and an arbitrary number of buffers has a set of periodic trajectories which at.tracts all other trajectories of the system. Copyright © 19991FA(7 Keywords. Lilnit Cycles, Stability, Dynan1.ic Systems, Networks r
1. IN'T'RODUC'T'JON
S\,,~itching
the state space of this system can be partitioned into an infinite set of polyt.apes such that each of them is invariant and contains one periodic trajectory. fvloreover, all trajectories with initial conditions from a fixed polytope converge to the corresponding periodic trajectory. Hence the switched server system alwa.ys exhibits a regular stable predictable behavior. 'rhis conclusion is very important for applications.
Hybrid dyna.mical systems have attracted considerable attention in recent years. In general, hybrid systems are those that combine continuous and discret.e behavjor and invoJve.t t.hereby, both continuous and discrete state variables.
One important type of hybrid dynamical systems is the class of discretely controlled continuoustime systems. Recently, two interesting exarrlpIes of such systems have been extensively studied. They "vere called ~'the s,vitched arrival system~~ and "the 8\vltched server systern" (Chase et al. 1993). These dynamical systems are of interest on their own right but have also been used to lTIodel certain aspects of flexible manufacturing systems. 'I'hese examples can also be interpret.ed a..o;; rnodels for simple dynan1ically routed closed queueing networks. It was sho\vn in (Chase et al. 1993) that the sv.ritched arrival system exhibits a chaotic behavior whereas, under certain assumptions the dynamics of the switched server systelll is eventually periodic (see also (Horn and Rarnadge 1997). I-Iowever, only the case of systems with three buffers was considered. rrhe systems with three buffers can be reduced to planar systems, 1j,vhich makes their analysis a much easier task.
2. DEFINl'TIONS I n this paper '"vc ,"viII consider the following switched server system. 1
'The system consists of n buffers and one server. We refer to the contents of a buffer as "work", it will be convenient to think of work as a fluid, and a buffer as a tank. '"York arrives to the buffer j at a constant rate Pj > 0 where j == 1,2, ... n. Also, the server rerYlOves work from any selected buffer at a constant rate p > o. l
1
Our description of the switched server systenl has been phrased in terlllS of work , buffers, and tanks. However, in applications, work can represent a continuous approximation to the discrete flow of parts ill a luanufacturing system (Perkins and Kumar 1989), or jobs in a computer system, etc.
In the current paper 3 we give a complete qualitative analysis of the svvitched server systenl consisting of one server and an arbitrary nUIllber of buffers with a simple and quite natural cyclic server switching feedback policy. ''-le prove that
'Tv"re assume that the system is closed: P
= PI + P2 + ... + Pn·
(1)
3544
Copyright 1999 IFAC
ISBN: 0 08 043248 4
EXISTENCE AND STABILITY OF PERIODIC TRAJECTORIES I...
14th World Congress of IFAC
1"'\he location of the server is a control variable, and may be selected using a feedback policy. Here) we study the following simple cyclic server switching feed back policy:
1) The server starts with the buffer 1.
scribed by the follovving equations:
if q(t)
==
(3)
qj then x(t) = a(gj).
Furthermore, our s\vitching rule 1),2), 3) can be described as
=
(if q(t) == 9j and x j (t) 0) then q(t + 0) :== qj+l
2) \;vrhenever.. the server has emptied the buffer j, + 1 for i == 1, 2~ ... , n-l.
it s\vitches to the buffer j
Vj=11 .. ~,n-l,
3) Whenever~ the server has emptied the buffer n, it switches to the buffer 1.
(if q(t) == qn and X n (t) =: 0) tllcn q(t + 0) :== qt.
v,,"e show that this systenl can be described by a system of logic-differential equations. Indeed, let
(4)
NOl,r\r
Also: ,~.re should define initial conditions for the system as
(5) (2)
It should be pointed out that, for any solutiDn [x(t), q(t)J of the system (3), (4), (5), x(i) is continuous and q (t) is piecewise-constant and leftcontinuous~ Note also that the solution apparently exist.s, is unique and can be defined on [0, +co). Furthermore, x (t) == [Xl (t), .. ~ , X n (t)] E K for all t 2: 0 and~ moreover, Xj(t) > 0 for any t > 0 and all indices j == 1, ... , n except for at
where Ql, Q2, . . . ,qn are symbols. Let Xj (t) be the amount of "~ork in the buffer j at time t. The discrete state qj corresponds to the case \vhen the server is removing work from the buffer j and the discrete state variable q(t) describes the state of the server at time t. Furthermore! introduce t.he follov\ting vectors:
l
1
nlost one.
Pt - P
It can be casily seen, that for any solution [x(t),q(t)] of the system (3), (4), (5), there exists a sequenc.e {tk} ir'=o such that to == 0, t k+ 1 ;?:: tk, in(k+l) > ink for all k == 0 1 1, . .J 1imtk +00 as k ----t- +
P2
a(ql)
== Pn-l
=
>
Pn
Pt P2 - P
q(t) = qj
Vt E U~o(t'in+j-l1 tin+j] ~j==1,2, ... ,n.
(6)
Definition 1 The sequence tk is called the switching time sequence of the solution [x(t), q(t)].
Pn-l
Pn
=
Observation In a singular case ~~hen x? xg = ... == xj == 0 for SOln€' 1 ::; j < n r we have for the corresponding so] ution that t 1 :::= t 2 :=: •.. == tj == o. IIo",~ever, since (2)&(5)=> XO =F 0, 1,-ve have that tk+l > tk for all k > n - 1. It implies that for all i == 1, 2, . ~. and j =: l~ 2, ... ,11 the foIloVII·ing condition holds
PI P2 Pn-l -
P
Pn
PI
pz
Xj Pn-l
(tni+j) == 0
and
Xli
(tn.i+j)
>0
Vs:l j.
Definition 2 A solution [x(t), q(t)] of the systen'l (3), (4) 1 (5) is said to be an elementa.ry periodic trajectory if the fo]]owjng condition holds:
Pn - p
Notation Let v(t) be a function of tinle. rfhen
v(t
+ 0) ~ limc>ol
.;-+0 v{t
Let tk be t.he switching time sequence of this solution, then x(t n ) x(O).
+ t).
=
Then the above switched server system can be de-
Remark It folloyvs iUIlnediately from the definition of the switching tlrIle sequence) that if
3545
Copyright 1999 IFAC
ISBN: 0 08 043248 4
EXISTENCE AND STABILITY OF PERIODIC TRAJECTORIES I...
14th World Congress of IFAC
[x(t), q{t)J is an elementary periodic trajectory of the system (3)~ (4), (5), then tk+n -= tk + tn~ x(t + tn) x(t) and g(t + tn) == q(t) for all k == 0, 1, 2, ... and all t > O. Therefore this solution [x(t)) q(t)] is period~ v{ith the period T ~ in.
=
I
Defini tion 3 Let [x (t) , q Ct)] be an elenlentary periodic trajectory of the system (3), (4), (5) ~ and let tk be the switching time sequence of [x(t), q(t)]. Furthermore, let (x(t), q(t)] be any other solution of this system and let i k be it.s switching sequence. '-rhen [xCt), q(t)] is said to converge to [x(t.L q(t)] if the followjng condition holds:
sup inf 11£(£) - x(t)1I t2 t •
lim
= O.
The proof of this result VY'ill be given in the full version of the paper.
Chase, C., J. Serrana and P. Ramadge (1993). 'Periodicity and chaos [roIn switched flo~'- systems: Contrasting examples of discretely controlled continuous systems'~ IEEE Transactions on Automatjc Control 38(1), 70-83. IIorn, C. and P.J. Ran'ladge (1997). ~A topological analysis of a falnily of dynanlical systems \~. . i t h nonstandard chaotic and periodic
behavior'. International Journal of Control 979-1020. Perkins, J. and P.R. Kurnar (1989). 'Stable, distributed, real-time sc.heduling of flexible manufacturing/assembly I disassembly systems'. IEEE Transactions on . .- \utomatic Control 34(2): 139-148. 67(6)~
Let [x(i), q(t)] be an elementary periodic trajectory of the system (3), (4), (5), and let [x(tL ~j'(t)J be any other trajectory, which converges to [x(t), q(t)]. Then, it follows from condition (7) t.hat --++co t?:.,t.
J
REFERENCES
Re III ark It can be easily seen that Defini tion 3 implies the following property:
t ..
any trajectory [x(t) q(t)J ,vith the initial condition x{O) E 1<"" converges to [x(t), q(t)J.
Petrovski, 1. (1966)~ Ordinary Differential Equat.ions. Payer Publications Inc .. Ne'i,i\l' York.
(8)
The condition (8) is the standard definition of convergence to a limit cycle from the classical qualitative theory of ordinary differential equations (see
e.g. (Petrovski 1966)).
3. THE l\1AIN RES1JLT
>
Let f
0 be a given constant.
Introduce the
following set I'C",; C !{: (Xl,X2,"
1< ""'!
~
Xl
{
~ 0,
X2
.,xn ) ERn: 2: 0, ... , X n :? 0
and Xl
+ X2 + ... + X n
=:
'l
}-
(9)
Rernark It obviously follo"vs fro ill condition (1) that 1<, is an invariant set of the systern (3), (4), (5): any solution [x(t)~ q(t)] of (3)} (4), (5) with the initial condition X O E [{, satisfies x(t) E K-, for all t 2: o.
Kow \ve are in a position to present the main result of this paper.
Theorem 1 Consider the switched server system (3), (4) ~ (5) \v here p > 0, PI> 0, pz > 0, ... Pn > o are any parameters such that condjtion (1) holds. Let I > 0 be a given constant, and J{-y be the set defined by (9). fIhcn there exists an elementary periodic trajectory [X(t)1 q(t)] with the initial condition x(O) E !{,.,( such that 1
3546
Copyright 1999 IFAC
ISBN: 0 08 043248 4
Copyright © 1999 IFAC 14th Triennial World Congress,
14th World Congress ofIFAC
G-2e-15-1 Beijing~ P.R.
China
EXISTENCE AND STABILITY OF PERIODIC TRAJECTORIES IN SWITCHED SERVER SYSTEMS
Alldrey V. Savkin'" and Alexey S. Matveev*
* Departluent of Electrical and Electronic Engineering University of Western _~us traiia, Nedlands, 'VA 6907~ Australia) enlaiL [email protected]\va.edu.au, fax: +(618) 1
93801065
Abstract. The paper analyzes an example of the discrete cont.rol of a continuous varia.ble system. We prove that the s\vitched server system with one server and an arbitrary number of buffers has a set of periodic trajectories which at.tracts all other trajectories of the system. Copyright © 19991FA(7 Keywords. Lilnit Cycles, Stability, Dynan1.ic Systems, Networks r
1. IN'T'RODUC'T'JON
S\,,~itching
the state space of this system can be partitioned into an infinite set of polyt.apes such that each of them is invariant and contains one periodic trajectory. fvloreover, all trajectories with initial conditions from a fixed polytope converge to the corresponding periodic trajectory. Hence the switched server system alwa.ys exhibits a regular stable predictable behavior. 'rhis conclusion is very important for applications.
Hybrid dyna.mical systems have attracted considerable attention in recent years. In general, hybrid systems are those that combine continuous and discret.e behavjor and invoJve.t t.hereby, both continuous and discrete state variables.
One important type of hybrid dynamical systems is the class of discretely controlled continuoustime systems. Recently, two interesting exarrlpIes of such systems have been extensively studied. They "vere called ~'the s,vitched arrival system~~ and "the 8\vltched server systern" (Chase et al. 1993). These dynamical systems are of interest on their own right but have also been used to lTIodel certain aspects of flexible manufacturing systems. 'I'hese examples can also be interpret.ed a..o;; rnodels for simple dynan1ically routed closed queueing networks. It was sho\vn in (Chase et al. 1993) that the sv.ritched arrival system exhibits a chaotic behavior whereas, under certain assumptions the dynamics of the switched server systelll is eventually periodic (see also (Horn and Rarnadge 1997). I-Iowever, only the case of systems with three buffers was considered. rrhe systems with three buffers can be reduced to planar systems, 1j,vhich makes their analysis a much easier task.
2. DEFINl'TIONS I n this paper '"vc ,"viII consider the following switched server system. 1
'The system consists of n buffers and one server. We refer to the contents of a buffer as "work", it will be convenient to think of work as a fluid, and a buffer as a tank. '"York arrives to the buffer j at a constant rate Pj > 0 where j == 1,2, ... n. Also, the server rerYlOves work from any selected buffer at a constant rate p > o. l
1
Our description of the switched server systenl has been phrased in terlllS of work , buffers, and tanks. However, in applications, work can represent a continuous approximation to the discrete flow of parts ill a luanufacturing system (Perkins and Kumar 1989), or jobs in a computer system, etc.
In the current paper 3 we give a complete qualitative analysis of the svvitched server systenl consisting of one server and an arbitrary nUIllber of buffers with a simple and quite natural cyclic server switching feedback policy. ''-le prove that
'Tv"re assume that the system is closed: P
= PI + P2 + ... + Pn·
(1)
3544
Copyright 1999 IFAC
ISBN: 0 08 043248 4
EXISTENCE AND STABILITY OF PERIODIC TRAJECTORIES I...
14th World Congress of IFAC
1"'\he location of the server is a control variable, and may be selected using a feedback policy. Here) we study the following simple cyclic server switching feed back policy:
1) The server starts with the buffer 1.
scribed by the follovving equations:
if q(t)
==
(3)
qj then x(t) = a(gj).
Furthermore, our s\vitching rule 1),2), 3) can be described as
=
(if q(t) == 9j and x j (t) 0) then q(t + 0) :== qj+l
2) \;vrhenever.. the server has emptied the buffer j, + 1 for i == 1, 2~ ... , n-l.
it s\vitches to the buffer j
Vj=11 .. ~,n-l,
3) Whenever~ the server has emptied the buffer n, it switches to the buffer 1.
(if q(t) == qn and X n (t) =: 0) tllcn q(t + 0) :== qt.
v,,"e show that this systenl can be described by a system of logic-differential equations. Indeed, let
(4)
NOl,r\r
Also: ,~.re should define initial conditions for the system as
(5) (2)
It should be pointed out that, for any solutiDn [x(t), q(t)J of the system (3), (4), (5), x(i) is continuous and q (t) is piecewise-constant and leftcontinuous~ Note also that the solution apparently exist.s, is unique and can be defined on [0, +co). Furthermore, x (t) == [Xl (t), .. ~ , X n (t)] E K for all t 2: 0 and~ moreover, Xj(t) > 0 for any t > 0 and all indices j == 1, ... , n except for at
where Ql, Q2, . . . ,qn are symbols. Let Xj (t) be the amount of "~ork in the buffer j at time t. The discrete state qj corresponds to the case \vhen the server is removing work from the buffer j and the discrete state variable q(t) describes the state of the server at time t. Furthermore! introduce t.he follov\ting vectors:
l
1
nlost one.
Pt - P
It can be casily seen, that for any solution [x(t),q(t)] of the system (3), (4), (5), there exists a sequenc.e {tk} ir'=o such that to == 0, t k+ 1 ;?:: tk, in(k+l) > ink for all k == 0 1 1, . .J 1imtk +00 as k ----t- +
P2
a(ql)
== Pn-l
=
>
Pn
Pt P2 - P
q(t) = qj
Vt E U~o(t'in+j-l1 tin+j] ~j==1,2, ... ,n.
(6)
Definition 1 The sequence tk is called the switching time sequence of the solution [x(t), q(t)].
Pn-l
Pn
=
Observation In a singular case ~~hen x? xg = ... == xj == 0 for SOln€' 1 ::; j < n r we have for the corresponding so] ution that t 1 :::= t 2 :=: •.. == tj == o. IIo",~ever, since (2)&(5)=> XO =F 0, 1,-ve have that tk+l > tk for all k > n - 1. It implies that for all i == 1, 2, . ~. and j =: l~ 2, ... ,11 the foIloVII·ing condition holds
PI P2 Pn-l -
P
Pn
PI
pz
Xj Pn-l
(tni+j) == 0
and
Xli
(tn.i+j)
>0
Vs:l j.
Definition 2 A solution [x(t), q(t)] of the systen'l (3), (4) 1 (5) is said to be an elementa.ry periodic trajectory if the fo]]owjng condition holds:
Pn - p
Notation Let v(t) be a function of tinle. rfhen
v(t
+ 0) ~ limc>ol
.;-+0 v{t
Let tk be t.he switching time sequence of this solution, then x(t n ) x(O).
+ t).
=
Then the above switched server system can be de-
Remark It folloyvs iUIlnediately from the definition of the switching tlrIle sequence) that if
3545
Copyright 1999 IFAC
ISBN: 0 08 043248 4
EXISTENCE AND STABILITY OF PERIODIC TRAJECTORIES I...
14th World Congress of IFAC
[x(t), q{t)J is an elementary periodic trajectory of the system (3)~ (4), (5), then tk+n -= tk + tn~ x(t + tn) x(t) and g(t + tn) == q(t) for all k == 0, 1, 2, ... and all t > O. Therefore this solution [x(t)) q(t)] is period~ v{ith the period T ~ in.
=
I
Defini tion 3 Let [x (t) , q Ct)] be an elenlentary periodic trajectory of the system (3), (4), (5) ~ and let tk be the switching time sequence of [x(t), q(t)]. Furthermore, let (x(t), q(t)] be any other solution of this system and let i k be it.s switching sequence. '-rhen [xCt), q(t)] is said to converge to [x(t.L q(t)] if the followjng condition holds:
sup inf 11£(£) - x(t)1I t2 t •
lim
= O.
The proof of this result VY'ill be given in the full version of the paper.
Chase, C., J. Serrana and P. Ramadge (1993). 'Periodicity and chaos [roIn switched flo~'- systems: Contrasting examples of discretely controlled continuous systems'~ IEEE Transactions on Automatjc Control 38(1), 70-83. IIorn, C. and P.J. Ran'ladge (1997). ~A topological analysis of a falnily of dynanlical systems \~. . i t h nonstandard chaotic and periodic
behavior'. International Journal of Control 979-1020. Perkins, J. and P.R. Kurnar (1989). 'Stable, distributed, real-time sc.heduling of flexible manufacturing/assembly I disassembly systems'. IEEE Transactions on . .- \utomatic Control 34(2): 139-148. 67(6)~
Let [x(i), q(t)] be an elementary periodic trajectory of the system (3), (4), (5), and let [x(tL ~j'(t)J be any other trajectory, which converges to [x(t), q(t)]. Then, it follows from condition (7) t.hat --++co t?:.,t.
J
REFERENCES
Re III ark It can be easily seen that Defini tion 3 implies the following property:
t ..
any trajectory [x(t) q(t)J ,vith the initial condition x{O) E 1<"" converges to [x(t), q(t)J.
Petrovski, 1. (1966)~ Ordinary Differential Equat.ions. Payer Publications Inc .. Ne'i,i\l' York.
(8)
The condition (8) is the standard definition of convergence to a limit cycle from the classical qualitative theory of ordinary differential equations (see
e.g. (Petrovski 1966)).
3. THE l\1AIN RES1JLT
>
Let f
0 be a given constant.
Introduce the
following set I'C",; C !{: (Xl,X2,"
1< ""'!
~
Xl
{
~ 0,
X2
.,xn ) ERn: 2: 0, ... , X n :? 0
and Xl
+ X2 + ... + X n
=:
'l
}-
(9)
Rernark It obviously follo"vs fro ill condition (1) that 1<, is an invariant set of the systern (3), (4), (5): any solution [x(t)~ q(t)] of (3)} (4), (5) with the initial condition X O E [{, satisfies x(t) E K-, for all t 2: o.
Kow \ve are in a position to present the main result of this paper.
Theorem 1 Consider the switched server system (3), (4) ~ (5) \v here p > 0, PI> 0, pz > 0, ... Pn > o are any parameters such that condjtion (1) holds. Let I > 0 be a given constant, and J{-y be the set defined by (9). fIhcn there exists an elementary periodic trajectory [X(t)1 q(t)] with the initial condition x(O) E !{,.,( such that 1
3546
Copyright 1999 IFAC
ISBN: 0 08 043248 4