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://xie@public.sjtu.edu.cn or
fttp://public.sjtu.edu.cn (user: xie, passwd: public)
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Plan
• System basics
• Discrete-event system by an example of a queueing
system
• Discrete event systems
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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
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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
Output
Input
y(t) = g(u, t)
u(t)
SYSTEM
MODEL
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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)
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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.
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 
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System dynamics : sample path
x(t)
x0
t
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Discrete system
• The system is observed at regular intervals at time t =
nD for all constant elementary period D.
x n 1  f n  x n , u n  , x 0  x 0
yn  gn xn ,un 
x0
xn
t
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A queueing system
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• State of the system :
x(t) = number of customers in the system
• Random customer arrivals
• Random service times
• FIFO service
Customer
arrivals
Server
Queue
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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)
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Discrete event systems
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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
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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.
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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 set
X : 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.
e1
e2
e3
e4
e5
e1
e2
e3
e4
e5
s1
s2
t1
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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}}
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System classifications
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•
•
•
•
•
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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
•
•
•
•
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Modeling and analysis
Design and synthesis
Control
Performance evaluation
Optimization
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