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 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 system's behavior can be drawn 5.1.1 Abstract Lyapunov and storage functions In general, Lyapunov functions are real-valued functions of system' s 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=2l 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 21,∈ B define same state of B at time t whenever x(1), t)=z(22(), 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 ehavior se t B, state space X is not uniquely defined I Version of September 19. 2003Massachusetts 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 system’s behavior can be drawn. 5.1.1 Abstract Lyapunov and storage functions In general, Lyapunov functions are real-valued functions of system’s state which are mono￾tonically non-increasing on every signal from the system’s behavior set. More gener￾ally, stotage functions are real-valued functions of system’s 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 z1, z2 ≤ B define same state of B at time t whenever x(z1(·), t) = x(z2(·), 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. 1Version of September 19, 2003