Learning objectives :Introduce fundamental concepts of system theoryUnderstand features of event-driven dynamic systems
Textbook :C. Cassandras and S. Lafortune, Introduction to Discrete
Event Systems, Springer, 2007ftp://[email protected] orfttp://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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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 componentsFunction 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
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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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