lec5_6243_2003

lec5_6243_2003 - Massachusetts Institute of Technology...

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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 real-valued functions of systems state which are mono- tonically non-increasing on every signal from the systems behavior set. More gener- ally, stotage functions are real-valued 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 real-valued 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 non-increasing 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....
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lec5_6243_2003 - Massachusetts Institute of Technology...

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