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

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)

Chapter IIntroduction to discrete event systems

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Plan

• System basics• Discrete-event system by an example of a queueing

system• Discrete event systems

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33

System basics

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

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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

SYSTEM

Input Output

u (t) y (t) = g ( u , t)

MODEL

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The Input-Output Modeling process

r

R u(t) y(t) y(t)/u(t)= R/(r+R)

Example 1 : An electric circuit with two resistances r and R

R

C u(t) y(t)

u(t) = vR(t) + y(t)vR(t) = iRi=C.dy(t)/dt

Y(s)/U(s) = 1/(1+CRs)

Example 2 : An electric circuit with a resistance R and a capacitor C

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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.

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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).

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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.

State space : The state space of a system is a set of all possible values that the state may take.

Output equation :

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0, , ,t t t t t x f x u x x

, ,t t t ty g x u

System dynamics : sample path

10

x0

x(t)

t

Discrete system

• The system is observed at regular intervals at time t = nD for all constant elementary period D.

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1 0 0, ,n n n n x f x u x x

,n n n ny g x u

x0

t

xn

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A queueing system

• State of the system :x(t) = number of customers in the system

• Random customer arrivals• Random service times• FIFO service

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Customer arrivals

Queue Server

Customer departures

System dynamic The state of the system remains unchanged except at the

following instants (events)

• arrival times t of customers wherex(t+0) = x(t-1) +1

• departure times t of customers wherex(t+0) = x(t-1) -1

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x(t)

Sample path

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Discrete event systems

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 = departure16

Time-driven and event-driven systems

Time-driven systems Continuous time systemsDiscrete systems (driven by regular clock ticks)

State transitions are synchronized by the clock

Event-driven systemsState 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.

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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 setX : a discrete state space

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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.

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e1 e2 e3 e4 e5

t1 t2 t3 t4 t5

s1 s2 s3 s4 s5

e1 e2 e3 e4 e5

s6

A manufacturing system

Essential defining elements: E = {a, c1, d2}X = {(x1, x2) : x1 ≥ 0, x2 {0, 1, 2, 3, B}}

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1 2

part arrivals part departures

A two-machine transfer line with an intermediate buffer of capacity 3.

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

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Goals of system theory

• Modeling and analysis• Design and synthesis• Control• Performance evaluation• Optimization

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