ECE 877-JDiscrete Event Systems

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System

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

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 Classificationof 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

• 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, time-invariant, time varying, coupled, high-order, …)

• 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

LanguageMathematical 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.

AutomatonMathematical Definition

An automaton is defined in terms of six items as follows:

G = (X,E,f,Γ,x0,Xm)X: set of statesE: set of eventsf: transition functionΓ: 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 stateXm: set of marked states

An Example

ExampleTerminology

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

ExampleTransition 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.

ExampleState Transition

f(x,a) = xf(x,g) = zf(y,a) = xf(y,b) = yf(z,b) = zf(z,a) = f(z,g) = y

LanguagesGenerated 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

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 ControlMathematical 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 StructureTerminology

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 StructureFurther 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

www.engineering.wichita.edu/esawan/news.htm