#### 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 • • • • • • 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, 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 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 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 • Admission Problem • Routing Problem • Scheduling Problem More Information Control Systems Group www.engineering.wichita.edu/esawan/news.htm