Chapter I Introduction to discrete event systems Learning objectives : Introduce fundamental concepts of system theory Understand features of event-driven dynamic systems Textbook : C. Cassandras and S. Lafortune, Introduction to Discrete Event Systems, Springer, 2007 ftp://[email protected] or fttp://public.sjtu.edu.cn (user: xie, passwd: public) 1 Plan • System basics • Discrete-event system by an example of a queueing system • Discrete event systems 2 System basics 3 3 The concept of system •System: A combination of components that act together to perform a function not possible with any of the individual parts (IEEE) •Salient features : Interacting components Function the system is supposed to perform 4 The Input-Output Modeling process • Define a set of measurable variables • Select a subset of variables that can be changed over time (Input variables) • Select another set of variables directly measurable (Output variables, responses, stimulus) • Derive the Input-Output relation Output Input y(t) = g(u, t) u(t) SYSTEM MODEL 5 The Input-Output Modeling process Example 1 : An electric circuit with two resistances r and R r u(t) R y(t) y(t)/u(t)= R/(r+R) Example 2 : An electric circuit with a resistance R and a capacitor C R u(t) = vR(t) + y(t) vR(t) = iR i=C.dy(t)/dt C u(t) y(t) Y(s)/U(s) = 1/(1+CRs) 6 Static and dynamic systems Static systems : • Output y(t) independent of the past values of the input u(t), for t < t. • The IO relation is a function : y(t) = g(u(t)) Dynamic systems : • Output y(t) depends on past values of the input u(t), for t < t. • Memory of the input history is needed to determine y(t) • The IO relation is a differential equation. 7 The concept of state Definition : • The state of a system at time t0 is the information required at t0 such that the output y(t), for all t ≥ t0 is uniquely determined from this information and from u(t), t ≥ t0. The state us generally a vector of state variables x(t). 8 System dynamics State equation : • The set of equations required to specify the state x(t) for all t≥ t0, given x(t0) and the function u(t), t≥ t0. x  t   f  x  t  , u  t  , t  , x  t   x0 State space : The state space of a system is a set of all possible values that the state may take. Output equation : y t   g  x t  , u t  , t  9 System dynamics : sample path x(t) x0 t 10 Discrete system • The system is observed at regular intervals at time t = nD for all constant elementary period D. xn 1  fn  xn , un  , x0  x0 y n  g n  xn , u n  x0 xn t 11 A queueing system 12 • State of the system : x(t) = number of customers in the system • Random customer arrivals • Random service times • FIFO service Customer arrivals Server Queue 13 Customer departures System dynamic The state of the system remains unchanged except at the following instants (events) • arrival times t of customers where x(t+0) = x(t-1) +1 • departure times t of customers where x(t+0) = x(t-1) -1 Sample path x(t) 14 Discrete event systems 15 The concept of event • An event occurs instantaneously and causes transitions from one discrete state to another • An event can be  a specific action taken (press a button)  a spontaneous occurrence dictated by nature (failures)  sudden fulfillment of some conditions (buffer full). • Notation : e = event, E = set of event. • Queueing system: E = {a, d} with a = arrival, d = departure 16 Time-driven and event-driven systems Time-driven systems Continuous time systems Discrete systems (driven by regular clock ticks) State transitions are synchronized by the clock Event-driven systems State changes at various time instants (may not known in advance) with some event e announcing that it is occurring State transitions as a result of combining asynchronous and concurrent event processes. 17 Characteristics of discrete event systems Definition. A Discrete Event Systems (DES) is a discrete-state, event-driven system, that is, its state evolution depends entirely on the occurrence of asyncrhonuous discrete events over time. Essential defining elements: E : a discrete-event set X : a discrete state space 18 Two Points of Views Untimed models (logical behavior) Input : event sequence {e1, e2, ...} without information about the occurrence times. Sample path: sequence of states resulting from {s1, s2, ...} Timed models (quantitative behavior) Input : timed event sequence {(e1, t1), (e2, t2), ...}. Sample path : the entire sample path over time. Also called a realization. e1 e2 e3 e4 e5 e1 e2 e3 e4 e5 s1 s2 t1 19 s3 t2 s4 t3 s5 t4 s6 t5 A manufacturing system 2 1 part departures part arrivals A two-machine transfer line with an intermediate buffer of capacity 3. Essential defining elements: E = {a, c1, d2} X = {(x1, x2) : x1 ≥ 0, x2  {0, 1, 2, 3, B}} 20 System classifications • • • • • • • Static vs dynamic systems Time-varying vs time-invariant systems Linear vs nonlinear systems continuous-state vs discrete state systems time-drived vs event-driven systems deterministic vs stochastic systems discrete-time vs continuous-time systems 21 Goals of system theory • • • • • Modeling and analysis Design and synthesis Control Performance evaluation Optimization 22