This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 5: Lyapunov Functions and Storage Functions 1 This lecture gives an introduction into system analysis using Lyapunov functions and their generalizations. 5.1 Recognizing Lyapunov functions There exists a number of slightly different ways of defining what constitutes a Lyapunov function for a given system. Depending on the strength of the assumptions, a variety of conclusions about a systems behavior can be drawn. 5.1.1 Abstract Lyapunov and storage functions In general, Lyapunov functions are realvalued functions of systems state which are mono tonically nonincreasing on every signal from the systems behavior set. More gener ally, stotage functions are realvalued functions of systems state for which explicit upper bounds of increments are available. Let B = { z } be a behavior set of a system (i.e. elements of B are are vector sig nals, which represent all possible outputs for autonomous systems, and all possible in put/output pairs for systems with an input). Remember that by a state of a system we mean a function x : B [0 , ) X such that two signals z 1 , z 2 B define same state of B at time t whenever x ( z 1 ( ) , t ) = x ( z 2 ( ) , t ) (see Lecture 1 notes for details and examples). Here X is a set which can be called the state space of B . Note that, given the behavior set B , state space X is not uniquelly defined. 1 Version of September 19, 2003 2 Definition A realvalued function V : X R defined on state space X of a system with behavior set B and state x : B [0 , ) X is called a Lyapunov function if t V ( t ) = V ( x ( t )) = V ( x ( z ( ) , t )) is a nonincreasing function of time for every z B . According to this definition, Lyapunov functions provide limited but very explicit information about system behavior. For example, if X = R n and V ( x ( t )) =  x ( t )  2 is a Lyapunov function then we now that system state x ( t ) remains bounded for all times, though we may have no idea of what the exact value of x ( t ) is....
View
Full Document
 Spring '09
 AlexandreMegretski
 Dynamics, Continuous function, Lyapunov, Lyapunov function

Click to edit the document details