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# Learning objectives : Introduce fundamental concepts of system theory Understand features of event-driven dynamic systems Textbook : C. Cassandras and.

## Presentation on theme: "Learning objectives : Introduce fundamental concepts of system theory Understand features of event-driven dynamic systems Textbook : C. Cassandras and."— Presentation transcript:

1 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 or fttp://public.sjtu.edu.cn (user: xie, passwd: public) Chapter I Introduction to discrete event systems 1

2 Plan System basics Discrete-event system by an example of a queueing system Discrete event systems 2

3 3 3 System basics

4 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

5 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 5

6 The Input-Output Modeling process y(t)/u(t)= R/(r+R) Example 1 : An electric circuit with two resistances r and R u(t) = v R (t) + y(t) v R (t) = iR i=C.dy(t)/dt Y(s)/U(s) = 1/(1+CRs) Example 2 : An electric circuit with a resistance R and a capacitor C 6

7 Static and dynamic systems Static systems : Output y(t) independent of the past values of the input u( ), for < 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( ), for < t. Memory of the input history is needed to determine y(t) The IO relation is a differential equation. 7

8 The concept of state Definition : The state of a system at time t 0 is the information required at t 0 such that the output y(t), for all t t 0 is uniquely determined from this information and from u(t), t t 0. The state us generally a vector of state variables x(t). 8

9 System dynamics State equation : The set of equations required to specify the state x(t) for all t t 0, given x(t 0 ) and the function u(t), t t 0. State space : The state space of a system is a set of all possible values that the state may take. Output equation : 9

10 System dynamics : sample path 10

11 Discrete system The system is observed at regular intervals at time t = n for all constant elementary period. 11

12 12 A queueing system

13 State of the system : x(t) = number of customers in the system Random customer arrivals Random service times FIFO service 13

14 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) Sample path

15 15 Discrete event systems

16 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

17 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

18 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

19 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. 19 e1e1 e2e2 e3e3 e4e4 e5e5 t1t1 t2t2 t3t3 t4t4 t5t5 s1s1 s2s2 s3s3 s4s4 s5s5 e1e1 e2e2 e3e3 e4e4 e5e5 s6s6

20 A manufacturing system Essential defining elements: E = {a, c 1, d 2 } X = {(x1, x2) : x1 0, x2 {0, 1, 2, 3, B}} part arrivals part departures A two-machine transfer line with an intermediate buffer of capacity 3.

21 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

22 Goals of system theory Modeling and analysis Design and synthesis Control Performance evaluation Optimization 22

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