ECE 877-J - Wichita State University

Download Report

Transcript ECE 877-J - Wichita State University

ECE 877-J
Discrete Event Systems
224 McKinley Hall
Class Objectives
• Theory
• Applications
• New Ideas
• Sharing
• Dialog
• Customized to meet the needs of
1) our program
2) our industrial sponsors
Set of objects that interact with each other
to perform a given task
System Classification
Linear or nonlinear
Continuous-time or discrete-time
Time-invariant or time-varying
Deterministic or stochastic
Centralized or decentralized
Large-scale or reduced-order
• Time functions that are used to operate a
• Examples:
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)
• State space is a discrete set
• State transition mechanism is event-driven
Queueing System
• Customer
• Server
• Queue
An Example
Computer System
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
• Signal-driven: Differential equations,
Transfer function (linear, nonlinear, timeinvariant, time varying, coupled, highorder, …)
• Event-driven: ??????????
Languages and Automata
• 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
Mathematical Definition
A language defined over an event set E is
a set of finite-length strings formed from
events in E
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.
on Languages
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.
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
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*.
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
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.
A1 = {g}
Ā1 = {ε,g}
A2 = {ε,a,abb}
Ā2 = {ε,a,ab,abb}
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
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
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
Transition function: f
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.
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
Generated vs. Marked
For the automaton G = (X,E,f,Γ,x0,Xm), we define the
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.
Supervisory Control
Control Paradigm
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
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
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
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
• Admission Problem
• Routing Problem
• Scheduling Problem
More Information
Control Systems Group