#### Transcript ECE 877-J - Wichita State University

```ECE 877-J
Discrete Event Systems
224 McKinley Hall
Class Objectives
• Theory
Concepts
Definitions
Terminology
• Applications
• New Ideas
Education
• Sharing
• Dialog
• Customized to meet the needs of
1) our program
You
2) our industrial sponsors
System
Set of objects that interact with each other
to perform a given task
System Classification
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•
•
•
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Linear or nonlinear
Continuous-time or discrete-time
Time-invariant or time-varying
Deterministic or stochastic
Centralized or decentralized
Large-scale or reduced-order
Signals
• Time functions that are used to operate a
system
• Examples:
Current
Voltage
Force
Torque
Signal Classification
• Continuous or discrete
• Deterministic or random (stochastic)
• Periodic or non-periodic
Alternate Classification
of Systems
Signal-driven vs. Event-driven
• Signal-driven: Continuous-Variable Dynamic
Systems (CVDS)
• Event-driven: Discrete Event Dynamic Systems,
a.k.a. Discrete Event Systems (DES)
DES
• State space is a discrete set
• State transition mechanism is event-driven
Queueing System
• Customer
• Server
• Queue
An Example
Computer System
•
•
•
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Arrival from outside
Departure from CPU to outside
Departure from CPU to disk
Return from disk to CPU
An Example
System Engineering
• Modeling
• Analysis
• Design
Modeling
• Signal-driven: Differential equations,
Transfer function (linear, nonlinear, timeinvariant, time varying, coupled, highorder, …)
• Event-driven: ??????????
Languages and Automata
Language
• Events  Alphabet
• String (of events) is a sequence of events
• Language: Given a set of events, we
define a language over such set in terms
of its strings
Language
Mathematical Definition
A language defined over an event set E is
a set of finite-length strings formed from
events in E
Example
E = {a,b,g}
L1 = {a,abb}
L2 = {ε,a,abb}
where ε denotes an empty string, i.e. a
string that consists of no events.
Operations
on Languages
Concatenation
Let La and Lb be two languages.
The concatenation of La and Lb is the
language LaLb. A string is in LaLb if it can
be written as the concatenation of a string
in La with a string in Lb.
Terminology
Consider a string that consists of three
events as follows:
s = tuv
t is called a prefix of s
u is called a substring of s
v is called a suffix of s
Kleene-Colsure
For a set of events E, we define the
Kleene-closure as the set of all finite
strings of elements of E, including the
empty string ε. It is denoted by E*.
Example:
E = {a,b,c}
E* = {ε,a,b,c,aa,ab,ac,ba,bb,bc,ca,cb,cc,aaa,…}
Note that E* is countably infinite
Prefix-Closure
The prefix-closure of a given language A is
a language that consists of all the prefixes
of all the strings in the given language.
The prefix-closure of A is denoted by Ā.
Examples:
A1 = {g}
Ā1 = {ε,g}
A2 = {ε,a,abb}
Ā2 = {ε,a,ab,abb}
Automaton
A device capable of representing a
language according to well-defined rules.
We define a set of states and a set of
events (alphabet). The occurrence of an
event results in transition from one state to
another.
Automaton
Mathematical Definition
An automaton is defined in terms of six items as follows:
G = (X,E,f,Γ,x0,Xm)
X: set of states
E: set of events
f: transition function
Γ: X  2E, active event function. Γ(x) is the set of all
events e for which f(x,e) is defined.
2E is the power set of E, i.e., the set of all
subsets of E.
x0: initial state
Xm: set of marked states
An Example
Example
Terminology
Event set: E = {a,b,g}
State set: X = {x,y,z}
Initial state: x (identified by an arrow)
Marked states: x, z (identified by double
circles)
Transition function: f
Example
Transition Function
f: X x E  X
f(y,a) = x means the following
If the automaton is in state y, then upon
the occurrence of event a, the automaton
will make an instantaneous transition to
state x.
Example
State Transition
f(x,a) = x
f(x,g) = z
f(y,a) = x
f(y,b) = y
f(z,b) = z
f(z,a) = f(z,g) = y
Languages
Generated vs. Marked
For the automaton G = (X,E,f,Γ,x0,Xm), we define the
following:
L(G) is the Language generated by G
all the strings, s, in E*, such that f(x0,s) is defined.
Lm(G) is the Language marked by G
all the strings, s, in L(G), such that f(x0,s) belongs to the
marked set Xm.
Control
Modeling
Analysis
Design
Analysis
Control
Supervisory Control
The transition function of the automaton
G = (X,E,f,Γ,x0,Xm)
is controlled by the
supervisor S in the sense that, at least
some of the events of G can be
dynamically enabled or disabled by S.
Supervisory Control
Mathematical Definition
A supervisor S is a function from the
language generated by the automaton G
to the power set of E.
Therefore, we write
S: L(G)  2E
Controllability
E consists of two types of events,
controllable and uncontrollable.
Ec: Set of controllable events that can be
disabled by the supervisor
Euc: Set of uncontrollable events that cannot
be prevented from happening by the
supervisor
Observability
Furthermore, E consists of two types of
events, observable and unobservable.
Eo: Set of observable events that can be
seen by the supervisor
Euo: Set of unobservable events that
cannot be seen by the supervisor
Decentralized Control
• Interconnected
• Hierarchical
• Cooperative
• Competitive
Clock Structure
Clock Structure
Terminology
vk = tk – tk-1
The kth event is activated at tk-1.
It has a lifetime vk
The event is active during vk
The clock ticks down during the lifetime.
At tk, the clock reaches zero (the lifetime expires).
At tk, the event occurs, causing a state transition.
Clock Structure
Further Definitions
Consider a time t within the event lifetime
tk-1 ≤ t ≤ tk
t divides the lifetime into two parts
yk = tk - t
zk = t – tk-1
yk is called the clock (residual lifetime) of the event
zk is called the age of the event
Stochastic Process
A stochastic (or random) process X(ω,t) is
a collection of random variables indexed
by t.
The random variables are defined over a
common probability space, and the
variable t ranges over some given set.
Classification of
Stochastic processes
• Stationary processes: stochastic behavior
is always the same at any point in time.
Strict-sense stationary or Wide-sense stationary.
• Independent processes: the random
variables are all mutually independent.
Markov Chain
• The future is conditionally independent of
the past history, given the present state.
• The entire past history is summarized in
the present state.
Controlled Markov Chains
Markov Decision Problem
• Cost
• Decision
Dynamic Programming
Control of Queueing Systems