70 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 p@) = 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. + REFERENCES J. Baillieul, R. W. Brockett, and R. B. Washburn, “Chaotic motion in nonlinear feedback systems,” IEEE Trans. Circuits Syst., vol. 27, no. 11, pp. 990-997, Nov. 1980. A. Boyarsky and M. Scarowsky, “On a class of transformations which have unique absolutely continuous invariant measures,” Trans. Amer. Math. Soc., vol. 255, pp. 243-262, Nov. 1979. E. Bradley, “Control algorithms for chaotic systems,” A. I. Memo no. 1278, Mass. Inst, of Tech., Mar. 1991. S. Buss, C. H. Papadimitriou, and J. Tsitsiklis, “On the predictability of coupled automata: an allegory about chaos,” in Proc. 31st Ann. Symp. Foundations Comput. Sci., Oct. 1990, pp. 788-793. 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Ramadge, “On the periodicity of symbolic observations of piecewise smooth discrete-time systems,” IEEE Trans. Automat. Contr., vol. 35, no. 7, pp. 807-813, July 1990. ~ 83 CHASE et al.: PERIODICITY AND CHAOS FROM SWITCHED FLOW SYSTEMS F. R. Rubio, J. Aracil, and E. F. Camacho, “Chaotic motion in an adaptive control system,” Int. J . Contr., vol 42, no. 2, pp. 353-360, 1985. L. J. Serrano, “The effects of time sampling and quantization in the discrete Lontrol of continuous systems,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, Oct. 1990. J. A. Stiver and P. J. Antsaklis, “A novel discrete event system approach to modeling and analysis of hybrid control systems,” Control Systems Tech Rep. #71, Dept. of Elect. Eng., Univ. of Notre Dame, Notre Dame, IN, June 1991. T. Ushio and C. S. Hsu, “Simple example of a digital control system with chaotic rounding errors,” Int. J . Contr., vol. 45, no. 1, pp. 17-31, 1987. -, “Chaotic rounding error in digital control systems,” IEEE Trans. Circuits Syst., vol. 34, no. 2, pp. 133-139, Feb. 1987. M. Varghese, A. Fuchs, and Mukundan, R., “Characterization of chaos in the zero dynamics of kinematically redundant robots,” in Proc. 1991 Amer. Contr. Conf., Boston, MA, June 1991, pp. 225-230. 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. , I