ECE 877-J Discrete Event Systems 224 McKinley Hall

Class Objectives Theory Concepts Definitions Terminology Applications New Ideas

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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 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 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

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} L 1 = {a,abb} L 2 = {ε,a,abb} where ε denotes an empty string, i.e. a string that consists of no events.

Operations on Languages Concatenation Let L a and L b be two languages. The concatenation of L a and L b is the language L a L b. A string is in L a L b if it can be written as the concatenation of a string in L a with a string in L b.

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: A 1 = {g} Ā 1 = {ε,g} A 2 = {ε,a,abb} Ā 2 = {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,x 0,X m ) X: set of states E: set of events f: transition function Γ: X  2 E, active event function. Γ(x) is the set of all events e for which f(x,e) is defined. 2 E is the power set of E, i.e., the set of all subsets of E. x 0: initial state X m : 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,x 0,X m ), we define the following: L(G) is the Language generated by G all the strings, s, in E *, such that f(x 0,s) is defined. L m (G) is the Language marked by G all the strings, s, in L(G), such that f(x 0,s) belongs to the marked set X m.

Control Modeling Analysis Design Analysis Control

Supervisory Control

Control Paradigm The transition function of the automaton G = (X,E,f,Γ,x 0,X m ) 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)  2 E

Controllability E consists of two types of events, controllable and uncontrollable. E c : Set of controllable events that can be disabled by the supervisor E uc : 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. E o : Set of observable events that can be seen by the supervisor E uo : Set of unobservable events that cannot be seen by the supervisor

Decentralized Control Interconnected Hierarchical Cooperative Competitive

Clock Structure

Clock Structure Terminology v k = t k – t k-1 The k th event is activated at t k-1. It has a lifetime v k The event is active during v k The clock ticks down during the lifetime. At t k, the clock reaches zero (the lifetime expires). At t k, the event occurs, causing a state transition.

Clock Structure Further Definitions Consider a time t within the event lifetime t k-1 ≤ t ≤ t k t divides the lifetime into two parts y k = t k - t z k = t – t k-1 y k is called the clock (residual lifetime) of the event z k 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