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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38. NO. 1, JANUARY 1993
Periodicity and Chaos from Switched Flow
Systems: Contrasting Examples of
Discretely Controlled
Continuous Systems
Christopher Chase, Member, IEEE, Joseph Serrano, Member, IEEE, and
Peter J. Ramadge, Senior Member, IEEE
Abstract-We analyze two examples of the discrete control of a
On the other hand, it is well known that simple nonlincontinuous variable system. These examples exhibit what may be ear systems can exhibit “chaotic” behavior, e.g., characterregarded as the two extremes of complexity of the closed-loop
behavior: one is eventually periodic, the other is chaotic. Our istics such as sensitivity to initial conditions, dense unstaexamples are derived from sampled deterministic flow models. ble periodic orbits, etc. Intuitively, the state trajectory of
These are of interest in their own right but have also been used such a system appears “random” and “unpredictable”
a models for certain aspects of manufacturing systems. In each even though it is generated by a deterministic device.
case, we give a precise characterization of the closed-loop behav- Interestingly, some of these systems can be analyzed rather
ior.
I. INTRODUCTION
E analyze two examples of the discrete control of a
continuous variable system. The behavior of such a
hybrid system can be very complex, and it is not clear at
what level a useful model can be formulated. Is it possible, for example, to reduce the continuous components to
a ‘higher-level’ automaton model? Or will the continuous
dynamics, by the introduction of chaotic behavior, make
the adoption of a statistical model more appropriate? Our
examples illustrate that both alternatives can arise.
Our main concern is the “complexity” of the behavior
of the closed-loop mixed variable system. This is discussed
in a precise way in terms of periodicity, chaos, and statistical stability. Periodicity is analyzed through the notion of
the algebraic reduction of the closed-loop system to a
finite automaton. Intuitively, this means that the relevant
dynamics of the closed-loop system are determined by a
finite state automaton, and perforce must be eventually
periodic. Such systems exhibit a simple regular behavior
that can be found by simulation or on-line observation, or,
in some cases, by an off-line algorithm. In a very precise
sense, the finite automaton is an “aggregated” or
“higher-level” model for the closed-loop system.
Manuscript received April 18, 1991; revised May 13, 1992. Paper
recommended by Associate Editor, E. H. Abed. This work was supported
in part by NASA under Grant NAG 2-558 and in part by the National
Science Foundation under Grants ECS-8715217 and ECS-9022634.
C. Chase is with the The Applied Physics Laboratory, The Johns
Hopkins University, Laurel, MD 20723-6009.
J. Serrano is with the IBM Corporation, San Jose, CA 95193.
P. Ramadge is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544.
IEEE Log Number 9204916.
nicely if one thinks of the initial state as a random
variable on an underlying probability space and looks at
the evolution of the induced measure on the state space
(typically represented as a density). In the case of principal interest, the induced measures converge in an appropriate sense to a unique invariant measure, and the system is “statistically stable.” In the example considered
here, this statistical viewpoint leads to useful results on
the relevant dynamics of the closed-loop system.
The specific systems analyzed are examples of real-time
event-driven feedback scheduling. We were inspired to
consider these examples by a manufacturing model in a
recent paper by Perkins and Kumar [17]. Our examples
can also be interpreted as sampled fluid models for simple
dynamically routed closed queueing networks. To our
knowledge these are the first examples of fluid models for
queueing systems that exhibit periodic and chaotic behavior. However, the implications of this behavior for the
stochastic system remain to be explored.
Several authors have recently considered aspects of
mixed variable or hybrid control systems. For example,
[22], [23], [ll],and [71 deal with the issue of quantization;
[12], [21], and [16] concern modeling frameworks for hybrid systems; and [181 considers the dynamic behavior of a
class of hybrid systems. We briefly comment on some of
this work below.
In [181, Ramadge presents results applicable to the
supervision of a class of continuous variable processes.
The main result in [181 concerns the eventual periodicity
of symbolic observations of the state of a “multimodal”
system. However, the result says nothing about the initial
transient in the observation sequence or the possible
number of periodic orbits. In any case, it is by no means
clear how to verify the assumption of [18] that the state
0018-9286/93$03.00 0 1993 IEEE
CHASE et al.: PERIODICITY AND CHAOS FROM SWITCHED FLOW SYSTEMS
trajectory has no limit points on the switching boundary.
One of our examples can be regarded as a detailed case
study of the verification of the conditions required in [18].
However, in the context of this example we are able to
strengthen the conclusions to obtain information about
the transient behavior of the observations as well as the
steady-state periodic orbits.
In our analysis, we employ results from the ergodic
theory of deterministic systems (see e.g., [13]); particularly
results on iterated maps of the unit interval [141, [131, [2l.
Similar methods have been used to analyze the statistical
properties of nonlinear quantized systems. For example,
Gray [ l l ] has used ergodic theory to evaluate the marginal
distribution of the binary quantization noise for the single-loop sigma-delta modulator, and Delchamps [7] has
used tools from the ergodic theory of deterministic systems to study the chaotic behavior that can result when a
linear system is controlled using quantized state feedback.
In related work Ushio and Hsu [22], [23] have used mixed
mappings to study the chaotic effects of roundoff errors in
digital control systems.
Control systems that exhibit chaotic behavior are not
new to the control community. Indeed, in addition to the
quantization work mentioned above, research has been
reported in nonlinear and adaptive control settings on the
introduction of chaotic behavior by the application of
continuous nonlinear feedback control. See for example,
[l], [241, [191, and [lo]. In addition, several authors have
considered the problem of synthesizing a controller for a
chaotic system. For example, [9] proposes suboptimal
stochastic control methods to reduce the effects of chaotic
behavior; and [151 proposes a method to effectively remove chaos by stabilizing a chaotic system about an
unstable periodic orbit embedded in a chaotic attractor.
The latter method is further elaborated in [3].
The remainder of the paper is organized into three
parts: in Section I1 we introduce the two examples; Sections I11 and IV present the analysis of these systems; and
Sections V and VI contain the technical detail and proofs
of the main results.
71
PI + Pz
I
+
P3 = 1
I
/ I
i1
(a)
I
Y
!
p3
p2
PI + P2
+
P3 = 1
Fig. 1. (a) The switched arrival system. (b) The switched server system.
tion to the discrete flow of parts in a manufacturing
system, or jobs in a computer system, etc. Since each
buffer acts as an integrator, the example can also be
thought of as a simple instance of the discrete control of
three 0.d.e.s.
The control scheme investigated is a threshold policy of
the following form. Assign a threshold to each buffer, and
instantaneously move the server to any buffer in which
the level of work falls below the assigned threshold. Note
that the location of the server is selected based on a
quantized observation of the buffer state, and the move11. SWITCHED FLOW
SYSTEMS
ment of the server is triggered by a “discrete event.” In
A. The SwitchedAm‘ual System
the simplest version of this scheme, we take all the
Consider a system consisting of N buffers, and one thresholds equal to zero, and switch the arrival server
server. We refer to the contents of a buffer as “work;” it each time a buffer empties.
Let w,(t) denote the amount of work in buffer i at time
will be convenient to think of work as a fluid, and a buffer
as a tank. Work is removed from buffer i at a fixed rate t 2 0, and let w ( t ) = (w,(t),...,w,,,(t)). At t = 0 we aspi > 0. To compensate, the server delivers material to any sume that w,(O) 2 0 with Cfl,w,(O) = 1. We call w ( t ) the
selected buffer at unit rate. We assume that the system is state of the buffers at time t. Let l ( t ) denote the server
pi = 1. The location of the server is a location at time t. We assume that I(.) is right continuous.
closed, so that I;“,,
control variable, and may be selected using a feedback If at time t the server is in location j , then the server will
policy. Moving the server alters the topology of the flow, remain at j until a buffer empties. This event will occur
.
t Is 5 t 7 the
and hence permits control over the buffer levels. Fig. l(a) after a time T = min, { w , ( t ) / p , ) For
shows the set-up with N = 3 and the server in location 1. buffer state is determined by the set of linear equations
Our description of the example has been phrased in
terms of work, fluid, buffers, and tanks. However, in
applications, work can represent a continuous approxima-
+,
+
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 1, JANUARY 1993
72
At time t + r one or more buffers are empty. If exactly
o n e buffer is empty then its index is k =
argmini, (wi(t)/pi). In this case, the arrival server is
instantaneously moved to fill buffer k , and the above
process repeats. If more than one buffer is empty we
assume the system stops, so that w ( s ) = w(t + r ) for all
s>t+r.
Let {t,) be the sequence of times when buffers empty.
We refer to these as the clearing times of the system. Let
r, = t, - t n p l , i 2 1. {r,} is the sequence of interevent
times, i.e., the times between the emptying of buffers. The
case of particular interest is when C;=,r, = 00, i.e., there
are an infinite number of events and limn t , = W. We
verify in due course that this situation is in fact “typical.”
In this case, the buffer state trajectory is a well-defined
function w:[0, 00) -+ RN,completely determined by the
system parameters p,,..., p N , and the initial condition
w(0). Since the rate of work being processed is equal to
the rate of work arriving, the total work in the buffer
system is constant. The buffer state thus evolves in the
region of RN defined by the intersection of the hyperplane Ciwi = 1 with the regions wi > 0, i = l;.., N .
The system evolves in continuous time. However, by
sampling the trajectories at the clearing times we obtain
an equivalent discrete-time model. This event-driven sampling can be thought of as a form of time aggregation.’
Let x k ( n ) = w,(t,), and x ( n ) = ( x , ( n ) ; . . , x N ( n ) ) .At each
clearing time, the index of the empty buffer determines
the new location of the server, and once this is known the
value of the state until the next clearing time is determined by (1). Thus the sampled sequence { x ( n ) : n > 01,
completely specifies the buffer trajectory. For simplicity
we assume that 0 is a clearing time, i.e., that the initial
condition has one buffer empty. This ensures that the
initial condition is the first element in the sampled sequence.
The sequence { x ( n ) } lies in the set X = {x: X i x i = 1,
x i 2 0, and for some j, xi = 01. If GI:X + X denotes the
map that describes the transformation of X that results
by placing the server in location j until a buffer empties,
then
~
1.0)
3c
01)
Fig. 2. (a) Transitions under G, and (b) transitions under H .
tions that give rise to such trajectories is easily shown to
be of zero Bore1 measure.
To illustrate what is happening under G consider the
system with N = 3. In this case the state space X can be
visualized geometrically as the equilateral triangle in R
with “edges” X I ,X,, X,, where X , = {x: C:x, = 1, xI > 0
for j # i , x , = 0); and “vertexes” U , = (l,O,O), U , =
(0,1,0), and v g = (0,0, 1). The vertexes represent states
where two buffers have emptied simultaneously; these are
the fixed points of G. Clearly, X is a one dimensional
manifold.
Let x E XI, i.e., x = (0, x,, x,), where x 2 > 0, x, > 0.
Since x, = 0, the server starts filling buffer 1 at rate 1,
while buffers 2 and 3 empty at rates p2 > 0 and p, > 0. If
( x 2 / p 2 )< ( x 3 / p 3 ) ,then buffer 2 empties first and G ( x )
E X , . If ( x 2 / p 2 ) > ( x 3 / p 3 ) , then buffer 3 empties first
and G ( x ) E X,. When ( x 2 / p 2 )= ( x 3 / p 3 ) , both buffers
empty simultaneously, and G ( x ) = u l . This last event
occurs when x = P I , where P, is the point (0, ( p 2 / p 2 +
p,), ( p 3 / p 2 pJ). The point P, partitions X , into Q,,
= { x E X , l x , < ( p3/p2 + p3)) and Q I 2 = {x E X , h , >
( p 3 / p 2 + p,)}. G maps Q,, linearly onto X,, and Q12
linearly onto X,. Similarly in X , and X , there are points
P2 and P,, respectively, such that G(P,) = U , and G(P,)
= U , ; and the points P, and P, partition X , and X , into
regions on which G is linear. The state transitions under
G are illustrated in Fig. 2(a).
+
where p is the vector of the pi, and l j is the vector of
zeros except for a 1 in the j t h position. Note that if x E X
with xk = 0 and k # j , then G j ( x )= x. So Gj only modifies x E X when xi is the only zero element of x. The
transition function of the sampled system, G: X + X , is
then given by G(x) = Gq(.,(x)where q(x) = argmin, { x i } .
If q(x) is not uniquely defined, any of the indexes mini- B. The Switched Server System
mizing xi can be selected. This corresponds to the unOur second system consists of N buffers, with work
likely event that two or more buffers empty at exactly the
same time. The state so reached is a fixed point of the arriving to buffer i at a constant rate of pi > 0, and a
transition function, i.e., G ( x ) = x . The set of initial condi- server that removes work from any selected buffer at unit
rate. As in the previous example the location of the server
is a control variable that can be chosen using a feedback
’ This is analogous to forming a Poincari map.
13
CHASE et al.: PERIODICITY AND CHAOS FROM SWITCHED FLOW SYSTEMS
policy. Again, we assume that the system is closed, so that serve at state x. Clearly, the S(x)th component of H ( x )
C ipi = 1. Fig. l(b) shows the set-up when N = 3 and the will be zero.
server is in location
We have not specified an exact form for the switching
An interesting supervision policy can be formulation as function S. This will not be necessary. However, as stated
follows. The server remains in its current location until informally in our description of the system we do assume
the associated buffer empties. Then the server instanta- that S satisfies the following condition:
neously switches to a new buffer determined by a deterSO) For every state x E X , x s ( x )> 0. This ensures that
ministic function of the current buffer state.
when the switching function S is applied to a buffer state
To determine the system equations we proceed as fol- x, the buffer selected for service is nonempty. In due
lows. First, for t 2 0, let w,(t) denote the amount of work course, we will need to impose some additional regularity
in buffer i , and let w ( t ) = (w,(t>;.., wN(t)). Let l(t>re- conditions, these will be brought in when needed.
To illustrate what is happening under H consider the
present the server location at time t. We take 1(.) to be
left continuous. If the server is in location j at time t , system with N = 3, and the switching function S(x) =
then the server remains there until the event “buffer j argmax {xi}.In this case, the state space is the one dimenempty” occurs. This takes a time T = w j ( t ) / ( l - p,). For sional manifold X discussed in the previous subsection.
t _< s _< t T , the buffer state changes linearly:
The map S partitions X into three regions Si= S-’(i),
i = 1,2,3 each of which is a connected component of X .
Three boundary points b,, b,, b, separate these regions.
Let x E S , , with x E X,. So x = (x,,0, x,) with x, > x,.
Then buffer 3 is selected to be cleared and the next state
When the buffer empties at time t + T the server instan- is H , ( x ) EX,. Clearly, H maps S , into X,. The state
taneously switches to the buffer determined by a given transition is illustrated in Fig. 2(b).
feedback rule, S : R N + { l , . . .N, } , and then the process
C. Relationship Between Examples
repeats.
As with the switched arrival system, we let {tn}and { T ~ } Intuitively, the switched arrival and switched server
denote, respectively, the sequences of clearing times and systems are “inverses” of each other. To state this preinterevent times. Once more, the interesting case is when cisely we need the following definition. Let X , c X be the
C ~ =T m.
~ The regularity conditions we later place on the set {x: x, = 0 and x, > 0, j # i}, i.e., X , is the subset of
switching function will guarantee that this holds, and the state set where the ith, and only the ith, buffer is
hence, we restrict attention to this case. This ensures that empty. Note that, by definition, for x E X , , G ( x ) = G,(x),
= x . In addition, for x E U , X , , H , ( x ) E X , .
the buffer state trajectory is a well-defined function and H,(x)
w: [O, m) -+ R N that is completely determined by the sys- We can now state the following results.
. in
Lemma 2.1: For i = 1,2,3:
tem parameters { p,} and the initial condition ~ ( 0 )As
the previous example, the total work in the buffer system
1 ) x E X\X, implies G, 0 H,(x)
= x.
is constant, and the buffer state evolves according to (3) in
2)
x
E X , implies Hi0 G , ( x ) = x.
the region of R N defined by the intersection of the
For all switching finctions S:
hypersurface C,w, = 1 with the regions w, 2 0, i =
3) if x E X , then G H ( x ) = x,
l;.., N.
4) if x E H ( X ) , then H 0 G ( x ) = x.
By sampling the system trajectories at the clearing
Proo$ These results following easily from (2) and (4),
times we obtain an equivalent discrete-time model. Let
and
the
definition of X,.
x , ( n ) = w k ( t n ) and x ( n > = (x,(n),-.., x,(n)). Reasoning
as in the previous case, this sequence completely deter111. ANALYSIS OF THE SWITCHED ARRIVAL SYSTEM
mines the buffer state for all t 2 0. For convenience
We
restrict our analysis here to the case N = 3. In
assume that time 0 is a clearing time. The sequence { x ( n ) }
lies in the region X of R N defined in the previous contrast to the benign nature of the open-loop system, the
subsection. Let H,: X + X, be the map describing the behavior of the sampled controlled switched arrival systransformation of X due to clearing buffer j . Then H, is tem is very complex. Technically, the system exhibits characteristics of chaos 18, p. 501: it has sensitive dependence
linear with
on initial conditions, is topologically transitive, and its
periodic orbits are dense in the state space, It is clear that
there are three fixed points and it is easy to show that
each of these is unstable. Similarly, it is possible to show
\ 0,
otherwise.
that each periodic orbit is unstable. Thus, although the
The transition function H : X + X of the sampled system state trajectory remains bounded, it is highly unlikely that
is defined by H ( x ) = Hs(x)(x)7where S: X + X is the it will settle into a periodic pattern. A sample trajectory is
switching function that determines the next buffer to shown in Fig. 3.
An alternative to direct analysis of the state trajectory
is to model the initial condition as a random variable and
This is a closed version of the model of [17].
+
0
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 1, JANUARY 1993
74
Fig. 4. g on the unit interval ( p l = 0.6, p 2 = 0.1, p 3 = 0.3). The
vertical dotted lines indicate the points p , , 1/3, p 2 , 2 / 3 , and p3.
Specifically, for z
E
[O, 11
(b)
Fig. 3. A trajectory of the sampled switched arrival system: work in
buffers 1 and 2 as a function of the clearing index n. ( p1 = .6, p 2 = .l,
p3
=
.3)
examine the corresponding sequence of induced measures
on the state space. Assuming certain regularity conditions
these measures have density functions, and we can study
the evolution of these functions with time. From the
behavior of the density trajectories we can draw quite
strong conclusions about the underlying transformation
G, and the statistical properties of the buffer trajectories.
where p1 = ( 1 / 3 ) ( p 3 / p 2 + p 3 ) , p 2 = (1/3) + ( 1 / 3 ) *
(Pl/Pl + P3), and P3 - (2/3) + (1/3) ( p 2 / p 1 + ~ 2 ) .
These are the preimages under g of (2/3), 0, and (1/3),
A . Statistical Analysis of the Switched Am’val System
respectively. The graph of g is shown for particular values
The theory of the statistical analysis of deterministic of the parameters in Fig. 4.
Our underlying measure space is (Z,&’, m ) with I =
functions on the unit interval is fairly well developed. For
[0,1]
and Lebesgue measure m . Let L , denote the family
convenience in appealing to these results we first recast
c L , denote the
our system as a map on the unit interval. Let c$l: X , + of integrable functions on [0,1], and
(0,1/3), &: X , --, (1/3,2/3), and &: X 3 (2/3, l), family of density functions. The transformation g is nonsingular with respect to m , i.e., for each A E &’,m ( A ) = 0
where
implies m ( g - ’ ) ( A ) )= 0. Hence, g induces a Markov
operator Pg:L , --, L , , called the Frobenius-Perron operator of g [13, p. 371. If the initial condition z(0) for the
system ([O, 11, g ) is a random variable on the probability
space ( I ,&’,m ) with density function fo, then the next
state z(1) = g(z(0)) is a random variable with a density
function f, = Pg(fo).So Pg describes the evolution of the
state density function.
The fixed points of Pg in D are called stationary densiWe map (0,1,0) to 0, (O,O, 1) to 1/3, and (1,0,0) to 2/3.
This defines a one-to-one and onto map 4: X + [0, 1). ties. These are of particular interest, since they represent
Geometrically, this parametrization of X can be thought possible statistically stationary regimes of operation.
of as “cutting” the triangle X at the point (0,1,0) and The map g is statistically stable if there exists a stationary density f* such that for any density f E D,
“unfolding it” onto the unit interval.
We then bring in the induced transition function limn IIP,”(f>- f* (1, = 0. In this case, regardless of the
g : [0, 11 + [0,1] defined on [0,1) by g = 4 0 G 4-’, and initial density the state will asymptotically be distributed
at the point 1 by g(1) = 0. Since the changes of coordi- with density f*. In this sense the system has a unique
nates
&, 43 are affine, g is also piecewise linear. “steady state.”
+
~~
0
+,,
75
CHASE et al.: PERIODICITY AND CHAOS FROM SWITCHED FLOW SYSTEMS
A measure po on LZ’ is invariant under g, or equivalently g is measure preserving with respect to po, if for
.
fo E S induces
~
every A E&, p o ( g - ’ ( A ) )= p O ( A )Every
an absolutely continuous (w.r.t. m ) probability measure
po on d , defined by p O ( A )= JAf0 dm, and it is a standard result that if fo is a stationary density, then po is
invariant under g [13, theorem 4.1.11. So a stationary
density gives rise to an invariant measure.
A set A ~dis invariant under g if g - ’ ( A ) = A . If for
every invariant set in A E & either p O ( A )= 0 or p o ( A )
= 1, then the map g is said to be ergodic with respect to
the measure po.As discussed in more detail later, if g is
both ergodic and measure preserving with respect to po,
then we can appeal to the Birkhoff ergodic theorem to
equate (almost surely) sample path averages and expectations with respect to po.
We are now ready to state our main result for the
sampled switched arrival system.
Theorem 3.1: The map g representing the sampled
switched arrival system on [0,1] is statistically stable and
the unique stationary density is the piecewise constant
function
‘ -3
PA1 - P I )
2 P1 P2 + P1 P 3 + P2 P3
3
-
-
P2(1 - P2)
2 P1 P2 + P1 P3 + P2 P3
3
[
P3(1 - P2)
2 P1 P2 + P1 P3 + P2 P3
,
if z
E
[0,1/3);
,
if z
E
[1/3,2/3);
,
if z
E
[2/3,1].
P1
[
[
( 2 -
p3
P2 + P3
P2 + P3
( 2 -
p1 -
9 P 2
P2
$3
+ P3
)
)
I
,From the expression for Pg we can make some simple
observations. First, let gcdenote the set of density functions that are constant on each of the subintervals [0,1/2),
[1/3,2/3) and [2/3,1], i.e., if f €Sc,
then f has the form
f ( z ) = a11[0,1,3)(2)+ a21f1/3,2/3)(z)
+ (Y31[2/3,11(2). Then
p3
P1 + P 3
PI + P2
Z E
f)P*
(2
+ P3
[O,f)
-)
1 2
3’3
-
P1
-
.E[-
p1 + p 2
f[i- i ) P 3
.
~
0
P1 + P3
tP1
P1 + P2
~
P1 + P2
P3 _
-)
-
P2
0
_
(2 -
P1 + P2
PI + P3
P2 + P3
- P2 + P3
1-
_ P1 _
P2
_ P3
PI + P2
-
f 3P2 + P3
p2
gp=
P1 _
_
0
+ -f( P1
zpl
---f(p2
PI + P3
r
[4
2
2 E
respect to f * allows us to appeal to the Birkhoff ergodic
theorem to equate (almost surely) sample path averages
with expectations. Specifically, for any bounded measurable function, h , on I , and for almost all initial conditions
z(0) E I
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 1, JANUARY 1993
76
z(0) E I
the state space is the one dimensional manifold X defined
in Section 11. Rather than use the transformation +: X +
[O,l) defined in Section I11 to map X onto the unit
interval, here we analyze the system directly on the manifold X .
To state our main result for this system we need to
introduce some additional terminology and assumptions.
We use the topology on X induced by the Euclidean
topology of R3. For B cX, denotes the closure of B
and dB denotes the set of boundary points of B. We let p
denote the one-dimensional Hausdorff measure on X .
The p-measure of a connected component in X corresponds to its path length.
The parameter space for the system is 0 =
{( p , , p 2 , p3)ICp, = 1, pI > 0, i = 1,2,3}. On 0 we use the
topology induced by the Euclidean norm in R3.
The switching function S: X + {1,2,3) partitions X
into switching sets, {S,};, with SI = S-’(i). We assume:
i.e., time averages of h ( z ( n ) ) are equal a.s. to the expected value of h with respect to f * .
For example, let T: X -+ R + with T ( X ) = min,,, x , / p , ,
where j is any index for which x, = 0. For x E X , d x ) is
the service time at the buffer state x, i.e., the time until
the next buffer empties when the buffer state is x. For a
buffer trajectory { x ( n ) } , d x ( n ) ) is the ( n + 1)th inwe can
terevent time. Using the coordinate change
obtain an equivalent measurable function t = d4-I)
which gives the service time as a function of the state
z E I. The average interevent time for initial state z(0) is
7 = limn
(l/n)C:=,?(g’(z(O))>, and by the Birkhoff ergodic theorem this a.s. equals ld?(z)f*(z) dz. Evaluation
of this integral yields 7 = 1/(4d), where d = p1 p2 +
Sl) Each switching set Sf has a finite number of conp1 p3 + p2 p3. Since 7 > 0, the interevent times, {T,},sum
nected components.
S2) There exists a > 0 such that for all x E X , x s ( x )2
to infinity almost surely. This justifies our previous statement that this is the “typical” case (see Section 11-A).
a,.
When the system is stationary, e.g., the initial state is a
Let dS = U, dS,. We refer to points in d S as the
random variable with density f * , the interevent times
have fixed first-order statistics. For example, if p1 2 p2 2 switching points or bounday points. Assumption Sl) ensures that d s is a finite set. Indeed, Sl) and the topology
le density f,“ of the service times is given by
of X imply that each switching point is a boundary point
of exactly two switching sets. So the number of switching
points is equal to the total number of connected components of the switching sets.
Assumption S2) is a strengthening of SO). It guarantees
that a minimum amount of work is present in the buffer
selected for service. This is equivalent to the Clear-AFraction property introduced in [17]. This in turn guarantees the earlier assumption (Section 11-B) that the interevent times sum to infinity. To see this let the state of
the buffers be x E X , and let i = S(x). Then the time
required to empty buffer i is
where d = p , p2 + p1 p3 + pz p3. In addition, if we only
have access to the location of the server then knowledge
a
a
Xi
of the history of the server location gives us no more
7 = -22
1
- Pi
1
- Pi
1
- maxi pi
information about the future behavior of the server than
does knowledge of the present server location alone. In
this case, the server location can be modeled as a finite which gives a lower bound for every interevent time.
state, aperiodic stationary Markov chain in which each
Let r odenote the equivalence relation on X induced
state represents a particular buffer being served.
by S , i.e., the equivalences classes of no are the switching
In summary, our analysis illustrates that the discrete sets Sf, and let r be the equivalence relation on X with
control of a deterministic continuous variable system can x = y(mod n-> iff for all n 2 0, S ( N ” ( x ) ) = S(H”(y)), i.e.,
lead to a closed-loop system which naturally admits a iff x and y produce the same control sequence. It is easy
statistical model. Indeed, the complicated nature of the to see that n- is a congruence on ( X , H ) , i.e., that x 3
closed-loop dynamics may make the effective computa- y(mod n-) implies that H ( x ) H(y)(mod n->. If S ( x ) is
tional use of a deterministic model difficult, if not impos- thought of as an “observation” of the state x E X , then nsible. The crucial feature of the system that gives rise to is the obsemabiliiy congruence of the system ( X , H , S ) .
this phenomenon is the expansive nature of the dynamics.
Let
denote the set of equivalence classes of n-, a y
let r also denote the canonical pLoje_ction-.rr: X -+ X .
IV. ANALYSIS OF THE SWITCHED
SERVERSYSTEM
Then there exists a unique map H : X + X such that
We now consider the sampled switched server system n-H = prr. Since 7r is finer than r o ,and x y(mod r 0 )
with attention restricted to the case N = 3. In this case iff S ( x ) = S ( y ) , it is clear that there exists a map
+
+,
--)3c
x
s:x
~
77
CHASE et al.: PERIODICITY AND CHAOS FROM SWITCHED FLOW SYSTEMS
{l,.-.,N) such that S = %. The above observations are
summarized in the commutative diagram:
-The
_ _situation of interest is when the quotient system
( X , H , S ) has a finite number of states, i.e.,
7~ has finite
index. In this case the control structure of the switched
server system will be reduced to a finite automation, i.e.,
the switching sequence and hence the control policy will
be completely determined by a deterministic finite state
system. As a result, the control policy will be eventually
periodic. Our aim is determined conditions under which
this holds.
and for all
Finally, for each switching function
-S n, H " + ' ( X ) c H " ( X ) . so x 2 H ( X ) 2 H 2 ( X )2 *.' .
The limit set is the forward attractor A = n ; = , H " ( X ) .
This is the set of limit points of all possible trajectories. A
will play an important role in our analysis.
We are now ready to state our main result for the
sampled switched server system.
Theorem 4.1: For each fixed p E 0 there is a set r, c X
of measure zero, such that for all switching functions S
having switching points outside of r,, the following hold:
1) The observability congruence 7r has finite index.
2) A contains at most 2IdSl possible periodic cycles.
3) All buffer trajectories converge uniformly exponentially to periodic orbits.
4) Items 11-31 continue to hold for sufficiently small
variations of the switching function S (meaning small
changes in the switching points).
easy, in this example, to formulate an algorithm by which
this may be $oLe,_and this algorithm indicates that the
automaton ( X , H , S ) is structurally stable with respect to
variations of the switching function S , and the parameters
p. The details will not be presented here. However, since
the algorithm makes use of the (expansive) inverse map
G, it would appear to have certain undesirable characteristics. So even though the steady-state behavior is tame,
computing it in advance may be difficult.
v . PROOF OF THEOREM3.1
We will appeal to several results from the literature.
From [14] we have the following result.
Theorem 5.1: Suppose that g : [O, 11 + [O, 11 is piecewise
twice differentiable, and that Ig'(x)l > 1 for all x where
the derivative is defined. Then there exist stationary densities fl,..-,f, E D with essentially disjoint supports such
that every fixed point f E L , of Pg can be written as
f=
for suitable ai E R. Moreover, for each i =
l;-.,n, the support of fi is: a finite union of closed
intervals, invariant under g, and contains at least one
point of discontinuity of g or g' in its interior.
The following theorem is a combination of [13, theorem
5.3.21 and [13, theorem 6.4.11.
Theorem 5.2 (Spectral Decomposition): If g: J -+ J is a
piecewise-linear map on an interval J c R, with Ig'(x)l > 1
for x E J , then there exist: an integer r, r densities f , €9
with disjoint supports, r linear functionals Ai E L,, and
an operator Q: L , + L,, such that for all f E L,, P g ( f )
may be written as
where IIP~(Q(f))IIl+ 0 as n + CO, and there is a permuIn addition, there exists a set r c X of measure zero such tation of a of 1;--, r such that for each i, Pg<f,>= facl,.
A key observation is that if r = 1 in Theorem 5.2, then
that for each switching function having switching points
outside r there is an open dense set in the parameter the corresponding operator g is statistically stable, [13,
space for which items 1)-3) hold.
theorem 5.6.11.
We are now ready to prove Theorem 3.1.
The proof of the theorem is contained in Section VI.
Proofi Since Ig'(x)l > 1 and g is piecewise linear, by
Roughly, the theorem says that for almost all switching
functions S the control policy of the sampled switched Theorem 5.2, Pg has a spectral representation in terms of
server system is determined by a finite automaton and is densities fl;..,fr
where, for each i, Pg(f,>
= facl, for a
thus eventually periodic. This in turn implies that the fixed permutation a of {l,..., r}. For each i, let n, satisfy
buffer trajectory is asymptotically periodic. Moreover, the a"c(i) = i. Then since Pg" = Pgn, we have Pgn,(f,)=
system retains this qualitative behavior for sufficiently P:i(fi> = fanrc,, = f,. Note that gni is piecewise linear with
small changes in the system or controller parameters. So Idg"i/dxl > 1. Thus by Theorem 5.1 applied to g"1, the
these characteristics are structurally stable.
support of f, is a finite union of closed intervals.
Suppose that for any interval J , lim, g k ( J > = [O, 11.
Notice that the quotient automaton gives all important
information regarding the controller dynamics. It displays Since the support of f, is invariant under g" for some n
both the transient as well as the steady-state controller (Theorem 5.1) and by the above argument contains an
behavior. Given the steady-state controller behavior it is a interval, the support of f, is all of the unit interval. But
simple matter to actually compute the asymptotic periodic since the support of the f, are disjoint, this means that
there can be only one term in the spectral representation,
orbit for the buffer state.
i.e.,
r = 1. It then follows from the remark after Theorem
There remains
the
problem
of
computing
the
finite
_ - _
automaton ( X , H , SI without simulating the system. It is 5.2 that g is statistically stable.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 1, JANUARY 1993
78
So to complete the proof, we only need to show that if J
is any interval in [O, 11, then lim, g k ( J ) = [O, 11. For this,
let J be any interval, I its length, and A = inf Ig'(x)l > 1.
Let int ( J ) denote the interior of the interval J . If p , E
int(J), then g ( J ) is an interval of length at least Al.
Further, if p2 E int (&.
forIj)
=)
0,1,2,...,
,
k - 1, then
g k ( J ) is an interval at least of length Akl. But for all
k , gk(.J) c [O, 13, and so we conclude that for some k,,
p2 E int ( g k o ( J ) ) .
Since gko(.O is an interval containing p 2 , g k o + l ( J ) 3
[O,a]U[b,ll, for some a E [O,p,) and some b E [p3,1].
But this implies [O, g2'(a)]U[g2'(b), 11 c g ( k o + 1 ) +(2Ji ) for
all i > 0.
For some i o , g2'o(a) > p2 or g2'o(b) < p2. In the first
case, g[O, g2'o(a)l 3 [0, 11, while in the second,
g[g2'o(b), 11 3 [O, 11. So [O, 13 c g(ko+1)+2io+1
( J ) . Hence, if
J is any interval, lim, gk(.O = [o, 11.
Let p* be the measure induced by f *. It is standard
result that if f* is a stationary density, then p* is
invariant [13, theorem 4.1.11. Moreover, since the stationary density f* is unique and f* > 0, it follows from [13,
theorem 4.2.21 that g is ergodic with respect to p*.
VI. PRELIMINARY LEMMASAND PROOF OF THEOREM
4.1
We begin with some simple but useful observations.
First, the maps Hi, G;, and G are continuous on X.The
continuity of G follows from the continuity of the G;, and
the fact that the vertexes of X are fixed. Since X is
compact, for any continuous function F : X -+ X and any
B c X , F(B)= F ( B ) . Next, it is easy to see that S2)
implies that, for i = 1,2,3,
S2')
3, n Xi= 0.
On the other hand, since X is compact, S2') implies that
Thus S2) and
there is a finite distance between and
S2') are equivalent.
To prove Theorem 4.1 we need a variety of preliminary
results. Some of these are of interest in their own right.
We begin in the following subsection with some technical
lemmas.
x.
Lemma 6.3: For all switching functions S satisfying S2),
if B c X and y E H ( B ) , then there exists a unique j
such that y E XIand G(y) E
B.
Lemma 6.4: If b E HM+l(X)
for M 2 1 then Gk(b)
E
for k = l , - . - , M .
qn
B. Structural Properties of the Switched Server System
By noting that H, is contractive on S,, it is not difficult
to show the following properties of A.
Proposition 6.5: For all switching functions S satisfying
S1) and S2):
1) p ( A ) = 0, and
2) G(A) c A.
In addition, if A n dS
switching points, then
=
0, i.e., A does not contain any
3) H ( A ) = G(A) = A, and
4) the observability congruence n- has finite index.
Proofi
1) First note that if B c X,is connected with endpoints
b, and b,, then p ( B ) = IJb, - b21I2, and if B C X is
connected, then [email protected]) = p ( B ) .
We claim that if B G X has a finite number of connected components, then p ( H ( B ) ) I
y p ( B ) . To see this
first suppose that B c S, n X I is connected, and has endpoints b, and b,. (We can assume i # j , since S, n X,=
0).
Because of the continuity of H,, H ( B ) = H , ( B ) is a
Let z , and 2, be the endpoints of
connected set in X,.
H ( B ) . Then by Lemma 6.2
p ( H ( B ) ) = 1121 - 2,112 I
yllb, - ~
~ I =I zy p ( B ) .
Next, suppose that B c S, n XI has connected components B,,..., B,. Then
n
-
p ( H ( B ) ) = p ( H ( U k B k ) ) IC p ( H ( B , ) )
k=1
n
p(B,)
I
y
=
yp(B).
k=l
If B,, B, c X are connected, then B, n B, has a finite
number of connected components. This follows from the
A. Properties of the Transition Functions
fact that there are only two distinct paths connecting any
two
points in X . By extension, the intersection of sets
All the results listed in this section are proven in the
with a finite number of connected components is a set
Appendix.
Let R = H ( X ) . The range has the following properties. with a finite number of connected components.
Finally, suppose that B C X has a finite number of
Lemma 6.1:
c U iXi
and X \ contains a nonempty
connected components B,;.., B,. Then B, n S, n X I is a
open set.
The maps Hi have contractive properties which will be set with a finite number of connected components. So
exploited for many of the results that follow. The follow- applying the above result we have:
ing lemma establishes these properties.
p ( H 0 ) = ~ ( H W,,,,]B, n S, n xl))
Lemma 6.2:
H,(B, n s, n XI))
= P( U
1) For i = 1,2,3, IIHi(x) - Hi(y)lll I
IIx - yII1.
I
2) There exists 0 < y < 1 such that for each i , j , i # j ,
p ( H , ( B kn S, n X,))
k,t,i
and all x , y E
7
IIH;(x) - Hi(Y>llZ I
yllx -yllz.
Finally, we note the following two technical lemmas.
P(B,
IY
k,i,i
Iy p ( B ) .
n S, n x,)
79
CHASE et al.: PERIODICITY AND CHAOS FROM SWITCHED FLOW SYSTEMS
If B C X has a finite number of connected components, then H ( B ) = U ,H,(B n Si)has a finite number of
connected components, because H, is continuous and the
B n Si have a finite number of connected components. So
by induction, H k ( X ) ,k 2 1, has a finite number of connected components. Hence, we can apply the above result
repeatedly to obtain p ( H k ( X ) )s - y p ( H k - ' ( X ) ) I
-ykp(X>,k 2 1.
Since A c H k ( X ) for all k 2 1 and y < 1, it follows
that p ( A ) = 0.
2) Let w E A = n H k ( X ) .Let { i ( k ) } : be a sequence
with f ( k ) E H k ( X ) and lim,+= i ( k ) = w. Then there
exists a sequence {x(k)}: c X such that P(k) = H k ( x ( k ) ) .
Hence, lim,+= H k ( x ( k ) )= w. Since G is continuous on
X , lim, - = G ( H k ( x ( k ) ) )= G(w) = z. But G is the inverse
of H , so
The sequence { H k - l ( x ( k ) ) } T = n - lc H "( X ) , forn > 1.
Thus, z E H " ( X ) for n > 1, and since H 2 ( X ) cH ( X ) ,
z E A. Hence, G ( A ) c A.
3) If A does not intersect dS then there exists M 2 1
such that H M ( X >does not intersect dS. To see this
suppose otherwise. Then there exists {si}: c dS such that
s, E H i ( X ) . Since X is compact and dS is closed, dS
contains a limit point b of {si):. But (sJT c H k ( X ) since
H k + ' ( X ) cH k ( X ) .So b E H k ( X )for k L 1. Thus b E
A; a contradiction.
The switching points are at least some finite distance
r > 0 from H ' ( X ) since dS and H M ( X ) are disjoint
closed subsets of the compact space X . For x E H M ( X > ,
let B,(x) be the open ball centered at x of radius r with
respect to X , i.e., B,(x) = { y E XI IIx - yIIl < r ) . Since
B , ( x ) is connected, B , ( x ) is interior to one of the switching regions. Therefore, for all y E B,(x), H ( y ) = H S ( J y ) .
Since Hs(,, is continuous at x , this implies that H is
continuous at x , and hence on all of H M ( X ) . Since
H k +' ( X I c H k ( X ) for k 2 1, H is a continuous map of
H M ( X )into H M ( X ) .Similarly, H" is a continuous map
of H M ( X )into H M ( X ) .Therefore,
On the other hand, by the same reasoning we obtain
m
m
n=l
n=l
m
c
nw(x)=~.
n=l
So A = n : , , H " ( X ) c R . Then using Lemma 2.1 and
the fact that G(A) c A (Lemma 6 3 , A = H ( G ( A ) ) C
H ( A ) c A. So H ( A ) = A. Finally, appealing to Lemma
2.1 again, A = G ( H ( A ) ) = G(A).
4) By Lemma 6.2, Hiis nonexpansive in the 1-norm on
Si,i = 1,2,3. From the proof of part l), H M ( X ) is a
compact H invariant subset of X . Hence, by [18, prop.
2.11 there exists an open H-invariant set Z with
H M (X ) c Z , a finite set Q, maps p: Q + Q and 3:Q +
{1,2,3} and a map p : Z + Q such that
H
Z-Z-{1,2,3}
S
.I I f /
p
Q-Q.
Since Q is finite, there can only be a finite number of
distinct server location sequences after M switches. This
implies that S can result in only a finite number of
distinct server location sequences. Since each equivalence
class in 7~ corresponds to a unique server location sequence, 7~ must have finite index.
When 7~ has finite index the buffer location sequence is
eventually periodic. As shown in the following proposition, this implies that the buffer trajectory is asymptotically periodic.
Proposition 6.6: If the server location sequence
{S(x(i))}: for some buffer trajectory, {x(i)):, is eventually
periodic, then the buffer trajectory converges exponentially to a unique periodic orbit that depends only on the
periodic part of the server location sequence, and the
parameters pl, p2, and p3.
Proofi It will be sufficient to show that the buffer
trajectory converges exponentially after some time N .
Without loss of generality we will assume N = 0 and that
the location sequence is periodic beginning at time 0. Let
M be the period of the location sFquence. Let b, =
S(x(i)). The range of Hb, is
Let H, = FI~"=,'I;I,,,,
for
k = O;.., M - 1. Note in the expression for H,, Hb,+,
operates on the range of H b , + k -for
l i = l;.., M - 1 and
Hb, operates on the range of
n u s Ei, maps
X b k + , - ,into itself.
Assumption S2) on the switching function guarantees
that the system will not attempt to clear the same buffer
twice in a row. Thus, b, # b,+ for i 2 0 and b, # b k + M for k 2 0. Then we can apply Lemma 6.2 so see that Hb,
contracts by y on X b , _ , It
. follows that H, contracts by
q.
c ;H(H"(X))
n=M
So H ( A ) c A .
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 1, JANUARY 1993
80
c.
-
Now x(m) E
Since B, n dS # 0 and B, is connected, there is a b E dSl,-, with b E B,. Then H I m _ $ b )
x ( ( n + l)M + k ) = H M ( x ( n M + k ) )
E A and HI -,<b)E B m - l . Now for k = l;..,m - 1,
H ( B , ) = H ( B , ) c B,-'. Hence H m - l ( g m - , < bE) )R
= tj,(x(nM + k ) )
and Hm-'(h1:-,<b)) E Bo. Thus Bo n R # 0;contradictfor n > 0 and each k . Applying the contraction mapping ing our original assumption.
theorem we obtain
If m = 1, then as above, there is a b in dSl, flB , . So
Hk,(b) is in R and Bo which is again a contradiction.
lim x(nM + k ) = lim I j , " ( x ( k ) ) =Yk
Therefore, A c fi.
W
n-30
n+m
Lemma 6.7 allows one to obtain an approximation to A
where the limit conv-rges as o ( y M " ) .Furthermore, since
by simulating only the trajectories of the finite set A . The
y , depends only on H k , each Yk depends only on pl, p 2 , p3
range of the trajectory of a periodic h e d point of H is a
W
and the corresponding location sequence cycle.
finite set in A. We call such a set a periodic cycle in A. By
We can obtain further information about A from the
the previous lemma, each periodic cycle must be a limit of
forward orbits of the switching points. Let
the trajectories of the set A. Then under certain conditions the number of possible periodic cycles can be
A = ( y l y = H,(b), b E dS and b E
bounded by the size of A.
Proposition 6.8: If A dS = 0, then the number of
As pointed out earlier, each b E dS is a limit point of
distinct periodic cycles in A is bounded above by the
exactly 2 of the SI.Therefore, IAI = 21dSI. Let
number of points in A .
Proofi If A does not contain any switching points
R = UHk(A)
7~ is finite by Lemma 6.5. Therefore, every location
then
k=O
sequence is eventually periodic and by Lemma 6.6 every
R is the set of points swept out by A under H . We note buffer trajectory converges to a periodic cycle. Specifithe following.
cally, the trajectory of each point in A converges to a
Lemma 6.7: A c fi
periodic cycle. Let U c X be the union of the periodic
Proofi Suppose otherwise. Then there exists w E A cycles corresponding to the trajectories of points in A. Let
and r > 0 such that B , ( w ) n R = 0. Let x(0) = w and M = IAI. Then U contains at most M periodic cycles.
x ( k + 1) = G ( x ( k ) ) for k 2 0. Note that { x ( k ) ] c A by
Suppose there is a periodic cycle V E A distinct from
Proposition 6.5, and so, { x ( k ) } c H ( K ) . Then applying the periodic cycles in U. Then there is a distance r > 0
Lemma 6.3 with B = X , for each x ( k ) there exists j, such separating the sets U and V. Let Z = U x E u B r , Z ( ~ ) .
that x ( k ) E X I , and x ( k 1) E
and by Lemma 2.1 Because A is finite and for each d E A , H Y d ) converges
part 3, H I J x ( k + 1) = H l j G ( x ( k ) ) ) = x ( k ) . Since the X , to a periodic cycle in U, there exists an N such that
are open we can take r small enough such that B,(x(O) c H " ( d ) E Z for d E A and n > N. Thus there are only a
XI,. For k 2 1 define
finite number of points of R = n z = l H k ( A )outside Z.
Clearly, H ( V ) = V. So no point in V can be in R. But
then points of V are not limit points of R since at most a
Let y E Hl -k - l ( B
, ) , i.e., y = Hj,_,<z)
for some z E B,. Thus finite number of points in R are_within distance r/2 of
the points in V. Therefore, I/ CZ R which contradicts the
y , x ( k - 1) E X j k - , and z , x ( k ) E x/,. Then by Lemma
conclusion of Lemma 6.7. Therefore, A can contain at
6.2
most M periodic cycles.
W
y M < 1 on X b , - ]into itself. Now
q).
n
51
+
C. Proof of Theorem 4.1
I yIIz - x(kIll2
< yy-,r.
Finally we bring the preliminary results to bear on
The last inequality follows from the definition of B, and Theorem 4.1
Fix p . We known that the map g corresponding to this
implies y E B,-'. Hence, for k 2 1 Hjk-I(B,)c B k - l .
Since y-' > 1, there exists n such that B, = x/,. Note p is statistically stable with an invariant measure that has
that by assumption SO), for i = 1,2,3, ui E Si. So the support on all of I . By the Birkhoff ergodic theorem, for
are in different switching sets. Therefore, every point x outside a set A , c I of measure zero,
endpoints of
) ] ~is dense in I. Note that 4-l is continuous on
x/, must contain a switching point, i.e., boundary of two ([O,g "1)( xand
+-'([O, 1)) = X . If 0 c [O, 1) then
different switching sets. So B, dS # 0 for some k 2 1.
Let m be the smallest such k .
Suppose m > 1. Then for 1 I k < m, B, dS = 0
and since B, is connected, it must be interior to a single
we must since 4-l is continuous and [0,1) is a subset of a compact
switching set. Since x ( k ) E B, and x ( k ) E
have B, E i d s j k -,). Otherwise, B, would contain a set. If 0 c I is dense in I , then 0 \ (1) is dense in [0, 1).
boundary point of si,-,.
so Bk C si,-,, for 1 I
k < m.
Then by (51, 4-'(0) is dense in X .
n
CHASE et al.: PERIODICITY AND CHAOS FROM SWITCHED FLOW SYSTEMS
Note that for x E I and y = +-'(x), +-'({g"(x)};) =
{G"(y))!. Let x E [0,1) \AP and y = +-'(x). By the
above, since {g"(x));"is dense in I, (G"(y)}Yis dense in X .
We note that + - I maps sets of measure zero in [O, 11 to
sets of measure zero in X . Thus, r, = + - ' ( A P )is a set of
measure zero in X . Then for y E r,, (G"(y))y is dense
in X .
Suppose_wechoose an S with switching points in X \ r,.
Since X \ R contains a nonempty open set (Lemma 6.1),
and { G " ( x ) )is dense in X , there is an M such that for
for some
each switching point b, G k ( b ) is outside
k < M . So by Lemma 6.4, H M ( X ) cannot contain any
switching points. Since, A c H M ( X ) ,this implies that A
does not contain any switching points. Then (1) follows
from Proposition 6.5 and (2) follows from Proposition 6.8.
To prove (3) note that (1) implies that the location
sequence must be periodic after a finite number of
switches. By Proposition 6.6, the buffer trajectory converges exponentially to a periodic orbit. Furthermore,
since there are a finite number distinct periodic cycles, the
orbits must converge uniformly exponentially. This proves
the first part of the theorem.
Following the reasoning above, there is an M > 0 such
that for each switchingpoint 6, there exists k < M such
that G k ( b )is outside R. The maps G and H,, i = 1,2,3,
are continuous. Thus for any switching point b, G k ( b )and
H,(b) vary continuously with continuous changes in b.
There are only a finite number of switching points. So
there is a 6 > 0 such that variations in the switching
points less than 6 will guarantee that G k ( b ) remains
outside the range for every switching point b. Thus, for all
variations in the switching points less than 6, H M ( X )will
not contain any switching point and the result follows
from the above argument.
To prove the second part of the theorem, we first make
some observations about the transition functions. Fix x
and consider G ( x ) and H,(x),
i = 1,2,3, as functions of
p E (0, 113. Let GJ p ) = G ( x ) and HEX(
p ) = H , ( x ) . G,( p )
is a rational function of p l , p 2 , and p3 (see the equations
for G in Section 11-A and Section 111). Since each p, is
nonzero, GI( p ) is continuous. Thus G,k( p ) is a continuous function of p for any k. Likewise, each H,,(p) is a
rational function of p l , p z , and p3 and is continuous as a
function of p (see Section 11-B).
Now let r = U rpE where Q is the set of all p with
rational component values. Since Q is countable, r is a
set of measure zero. Consider a switching function S with
switching points in X\ r. Fix p E Q. Then we known
there exists M such that H ' ( X ) will not contain any
switching points. That means for each switching point b,
there is a k < M such that G k ( b )is outside R.
The set is defined completely by the set A because A
marks the endpoints of the connected components of R.
For y E A , y = H,(b) for some b E dS,. As noted above,
H,(b) varies continuously as a function of p. Since there
are only a finite number of switching points, there is a
6( p ) > 0 such that if the parameter is varied by no more
81
than 6( p ) then the changes in G k ( b )and Hi(b) for each
switching point b will be small enough to guarantee that
G k ( b )remain outside R. Hence, for any variation in p
less than 6( p), A will not contain any switching points of
S and items 1, 2, 3, and 4 of the theorem hold.
Let B,(p) denote the open ball with radius 6 in the
space of parameters centered at p. If we select S with
switching points in X\ r then, by the above argument,
items 1)-3) hold for every parameter vector in I/=
U E QB,(P,( p ) , where 6( p ) is defined above. Clearly, V
is an open-dense set in the space of parameters.
VII. CONCLUSIONS
We have analyzed two examples of a continuous variable system supervised by a discrete controller. The first
example, the switched arrival system, exhibited chaotic
behavior and this behavior could be analyzed in terms of
the action of the dynamics on density functions on the
state space. This led to an interesting statistical description of the system. In contrast, the second example, the
switched server system, was shown to be generically periodic. In this case, the control function is governed by a
finite state automaton. For this example, it is possible to
formulate an off-line algorithm to compute the finite state
automaton. However, the complexity of the algorithm for
higher dimensional systems seems prohibitive. In this regard, there may be some interesting connections to recent
work on complexity theory and chaos reported in [4].
The two examples we have analyzed are of interest in
their own right. They are continuous models for simple
real-time event-driven scheduling. However, the examples
are also intended to be simplifications of what one might
expect in more complex control situations where continuous systems, governed by differential equations, say, are
supervised by discrete control. The examples have illustrated the broad range of closed-loop behaviors possible
in such systems and we have illustrated tools and methods
that may prove of value in analysis and design. Of course,
the examples analyzed here have a very simple piecewise
linear structure and this greatly facilitated our analysis. In
more complex situations it may not be possible, for example, to obtain an explicit expression for the stationary
density, and the closed-loop system may fall between the
extremes of behavior illustrated here.
There are several technical difficulties involved in extending our results to higher dimensional systems. The
available results on the statistical stability of higher dimensional systems are inapplicable, and a complete analysis of discontinuous piecewise contractions in higher dimensions is an open problem. Some issues that arise in
proving statistical stability have been examined in a general setting in [20] following the method of [14]. In addition, the state transition function of the sampled N buffer
switched server system is an example of a Markov map [2],
and the statistical stability of such maps is currently being
investigated. Work on contractive systems has specifically
concerned the N buffer system [6], and discrete-time
~
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 1, JANUARY 1993
82
systems on the unit interval where a controller selects
among a finite number of contractive transition maps [5].
The method of analysis used here for the switched server
system does not immediately extend to the higher dimensional case. Aside from the results of [6], this remains an
open problem.
APPENDIX
LEMMAS
PROOFS OF TECHNICAL
G ( y ) = lim G ( H ( x ( n , ) ) ) = lim x(n,>
But { ~ ( n , ) ) E
:
First note that H,(Sl)= Hi(%) c H,(X\x,)
c X,. Thus
i= I
i= 1
The second part is immediate because X\
contains u l , u 2 , and U,.
E
is open and
A.2) Proof of Lemma 6.2
1) The induced 1-norm of Hiis given by the 1-norm for
the matrix representation of Hi in the standard
coordinates. The 1-norm for a matrix is the max of
the 1-norm of its column vectors. From (4) one can
quickly compute that llHjlll = 1. Hence, Hi is nonexpansive in the 1-norm and the result follows.
2) Take i = 2, j = 1. U ,and u2 are the endpoints of
H2(u,) = u3 since the second coordinate of U , is
zero (see Section 111-A). H2(U , ) = p 2 . For x,y E
x # y , there is an a # 0 such that
F,
F,
IIH2(x) - H2(y)l12 IlH,(&, - u2>)1I2
IIX - yll2
Ila(u, - U21112
2
= Y21
110, - p2112 < 2 so y I 2< 1. For i, j , i f j , an analogous expression holds with y,, < 1. So let y =
1, J
ylJ*
J # I
A.3) Proof of Lemma 6.3
From Lemma 6.1, H ( B )c c U I X I .Since the X , are
disjoint, there is a unique j with y E X J .
If y E H ( B ) , then for some x E B, y = H ( x ) = Hk(x)
with k = S(x). By Lemma 6.1, y is interior to H , ( X )
=
Since the X , are disjoint, k = j . Now using Lemma
2.1, x = G ( H ( x ) )= G(y). Since x E SJ n B , we have
K.
G ( Y )E
k+m
k+m
A.1) Proof of Lemma 6.1
i= 1
there exists a convergent subsequence {x(nk)) c {x(n))
such that {x(nk)} c Si for some i E {1,2,3). Then
H(x(n,)))y c X l with the limit point y. But y is in Xi
which is open. Therefore, i = j and {x(n,>)y c Sj flB.
Since G is continuous and the inverse of H , and
{H(x(n,)))converges to y , we have
qn E.
If y E H ( B ) \ H ( B ) then there exists {x(n))y such that
{ H ( x ( n ) ) ) y converges to y. Since X = U I S l is compact,
sj n B. SO
G ( Y )E
=
x
qn E.
A.4) Proof of Lemma 6.4
For M = 1, b E H ( R ) and by Lemma 6.3 with B = R ,
G ( b ) E R. Assume the result holds for M = n , and con,
by Lemma 6.3
sider M = n 1. If b E H n + 2 ( X ) then
with B = H ” + ’ ( X ) ,G ( b ) E H ” + ’ ( X ) cE. Using the result for M = n , Gk”(b) E E for k = 2,.--,n. Hence, the
result holds for M = n + 1.
+
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Christopher J. Chase (S’89-M’91) was born in
Los Angeles, CA, in 1962. He received the B.Sc.
degrees in electrical engineering and mathematics from Brigham Young University, Provo, UT,
in 1987. He received the M.Sc. and Ph.D. degrees in electrical engineering from Princeton
University, Princeton, NJ, in 1989 and 1992,
respectively.
Between 1983 and 1988, he was a member of
the Computer Architecture and Sensor System
Modeling Departments of The Aerospace Corporation, El Segundo, CA. He is currently with the Computer Science
and Technology Group at The Johns Hopkins University Applied Physics
Laboratory. His research interests include chaotic dynamical yystems,
discrete event systems, networks, image processing, and pattern recognition.
Joseph Serrano (M’84-S’88-M’90) received the
B.S.E.E degree from The Johns Hopkins University, Baltimore, MD, in 1984, and the Ph.D.
degree in electrical engineering from Princeton
University, Princeton, NJ, in 1990.
He has been a Design Engineer for the Harris
Corporation and is currently in the Servo System Development Group at the IBM Corporation San Jose.
Peter J. Ramadge (S’79-M82-SM’92) received
the Bachelor of Science and Bachelor of Engineering (electrical) Hons. Class I degrees from
the University of Newcastle, Australia, in 1976
and 1978, respectively; the Master of Engineering degree from the University of Newcastle, in
1980; and the Ph.D degree in electrical engineering from the University of Toronto, Canada,
in 1983.
He joined the Faculty of Princeton University,
Princeton. NJ. in SeDtember 1984 where he is
currently as Associate Professor of Electrical Engineering. His current
research interests are in the theoretical aspects of computer science and
control theory, with an emphasis on applications of computers in signal
processing, learning, and control.
Dr. Ramadge is a recipient of the University Medal from the University of Newcastle, Australia. He is a member of SIAM.
,
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