learning objectives : introduce fundamental concepts of system theory
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/1.jpg)
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
1
![Page 2: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/2.jpg)
Plan
• System basics• Discrete-event system by an example of a queueing
system• Discrete event systems
2
![Page 3: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/3.jpg)
33
System basics
![Page 4: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/4.jpg)
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
4
![Page 5: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/5.jpg)
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
5
![Page 6: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/6.jpg)
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
6
![Page 7: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/7.jpg)
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
![Page 8: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/8.jpg)
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
![Page 9: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/9.jpg)
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 :
9
0, , ,t t t t t x f x u x x
, ,t t t ty g x u
![Page 10: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/10.jpg)
System dynamics : sample path
10
x0
x(t)
t
![Page 11: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/11.jpg)
Discrete system
• The system is observed at regular intervals at time t = nD for all constant elementary period D.
11
1 0 0, ,n n n n x f x u x x
,n n n ny g x u
x0
t
xn
![Page 12: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/12.jpg)
12
A queueing system
![Page 13: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/13.jpg)
• State of the system :x(t) = number of customers in the system
• Random customer arrivals• Random service times• FIFO service
13
Customer arrivals
Queue Server
Customer departures
![Page 14: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/14.jpg)
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
14
x(t)
Sample path
![Page 15: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/15.jpg)
15
Discrete event systems
![Page 16: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/16.jpg)
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
![Page 17: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/17.jpg)
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.
17
![Page 18: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/18.jpg)
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
18
![Page 19: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/19.jpg)
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
e1 e2 e3 e4 e5
t1 t2 t3 t4 t5
s1 s2 s3 s4 s5
e1 e2 e3 e4 e5
s6
![Page 20: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/20.jpg)
A manufacturing system
Essential defining elements: E = {a, c1, d2}X = {(x1, x2) : x1 ≥ 0, x2 {0, 1, 2, 3, B}}
20
1 2
part arrivals part departures
A two-machine transfer line with an intermediate buffer of capacity 3.
![Page 21: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/21.jpg)
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
![Page 22: Learning objectives : Introduce fundamental concepts of system theory](https://reader036.vdocuments.site/reader036/viewer/2022062315/56816333550346895dd3ba10/html5/thumbnails/22.jpg)
Goals of system theory
• Modeling and analysis• Design and synthesis• Control• Performance evaluation• Optimization
22