Free pdf to word converter has been designed to allow users to edit pdf. Lyapunov function in the classical theory of ordinary di. Some of these will be described in later sections of this paper. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. Lyapunov theory with inputs and outputs notesgen notesgen.
Finally, we look at alternate feedback control laws and closed loop dynamics. We then analyze and apply lyapunov s direct method to prove these stability properties, and develop a nonlinear 3axis attitude pointing control law using lyapunov theory. Cute pdf writer is a customizable tool that exports windowsgenerated files. Mathematical theory of lyapunov exponents this article has been downloaded from iopscience. Lyapunov theory with inputs and outputs 1419 if v is the value function, this method recovers the optimal control law weve used lyapunov methods to generate a suboptimal control law. Furthermore, we assume that there exists a continuously di. Noltllt numerical solution of the lyapunov equation by.
A free powerpoint ppt presentation displayed as a flash slide show on id. In using lypunov s direct method for stability analysis, it is sometimes possible to manipulate into the form 3. Lyapunov stability theory was come out of lyapunov, a russian mathematician in 1892, and came from his doctoral dissertation. Linear quadratic lyapunov theory stanford university.
An important means for stability analysis of a mas is to construct a lyapunov. Lyapunovs second or direct method provides tools for studying asymp totic stability properties of an equilibrium point of a dynamical system or systems of dif ferential equations. Specifically, fixed point theory and order theory are used to prove the existence of a least fixed point as the unique nash equilibrium ne of the game and the optimal choice of all players. Tutorial on lyapunovs stability utsa college of engineering. Lyapunov function partial differential equations for. Communications in applied analysis 12 lyapunov theory. Standard methods for the numerical solution of the lyapunov equation 3, 7 make use of the real schur. Moreover lyapunov did not himself make application in this field, his own interest being in the stability of rotating fluid masses with astronomical application. Lyapunov stability theory an overview sciencedirect topics. Lyapunov s theory for characterizing and studying the stability of equilibrium points is presented for timeinvariant and timevarying systems modeled by ordinary differential equations. The aim of this course and these notes is to present an exposition of the basics of stability theory, stable group theory, and geometric stability theory.
We will demonstrate that c is a necessary and sufficient condition for a and b. The theory of lyapunov function is nice and easy to learn, but nding a good lyapunov function can often be a big scienti c problem. Lyapunov stability theory research papers academia. Lyapunov, considerably anticipated the time for its implementation in science and technology. If in addition to being lyapunov stable, all motions n which start out at t 0inside a. Fundamentals of lyapunov theory the objective of this chapter is to present lyapunov stability theorem and illustrate its use in the analysis and the design of nonlinear systems. Department of mathematics indian institute of science bangalore 560 012, india december 9, 2003 abstract in this article, di. This discoverywent against the intuition drawn from the classical theory of furstenberg.
A function is the gradient of a scalar function if and only if the matrix. Lecture notes on nonlinear vibrations cornell university. Lyapunov time in modern physics, primarily exploring very unbalanced conditions and chaotic systems, there is a technical term lyapunov time. D r satisfying 3 and 4 is called a lya punov function. Lyapunov exponents for continuoustime dynamical systems. First of all, the lyapunov stability theory is understood through the picture. Lyapunov functions are also basis for many other methods in analysis of dynamical system, like frequency criteria and the method of comparing with other systems. Sources and studies in the history of mathematics and. Nonlinear dynamical systems and control presents and develops an extensive treatment of stability analysis and control design of nonlinear dynamical systems, with an emphasis on lyapunov based methods. Sources and studies in the history of mathematics and physical sciences managing editor j. Lecture 12 basic lyapunov theory stanford university. Rninto rnand there is at least one equilibrium point x, that is f x 0. Lyapunov s stability theory 100 years on 277 central limit theorem and other deep investigations in areas of mechanics and mathematical analysis.
Mathematical methods in control and system engineering april, 2015 1 26. Let v x,t be a nonnegative function with derivative v. When engineers analyze and design nonlinear dynamical systems in elec trical circuits, mechanical systems, control systems, and other engineering disciplines, they need to be able to use a wide range of nonlinear analysis tools. Kalman brings lyapunov theory to the eld of automatic control kalman and bertram \control system analysis and design via the second method of lyapunov. If the curved panel is stable at this mach number, then the free stream mach number is increased. Still, the pseudohelmholtz free energy function serves as the lyapunov function. View lyapunov stability theory research papers on academia. Introduction to dynamic systems network mathematics. Pdf a new lyapunov function for sirs epidemic models. Lyapunov theory is used to make conclusions about trajectories of a system x. The krasovskiis method is used to construct a lyapunov function and obtain the conditions to examine the stability of the ne. Q moreover, if a is hurwitz, then p is the unique solution idea of the proof. Ee363 winter 200809 lecture 14 lyapunov theory with inputs and outputs systems with inputs and outputs reacha. At this time, many of the later uses of lyapunov s work could hardly have been foreseen.
Theorem a matrix a is hurwitz if and only if for any q qt 0 there is p pt 0 that satis. I will assume knowledge of my autumn 2002 model theory lecture notes 1. Belur,department of electrical engineering,iit bombay. It designates a period, when a certain process physical, mechanical, quantum, or even biological moves beyond the bounds of precise or probabalistic predictability and enters a chaotic mode. Free excelto pdf converter allows you to easily convert your excel files to. Dynamical system theory lies at the heart of mathematical sciences and engineering. Lyapunov stability theory, we assume that the ode in consideration has a smooth at least c1 righthand side and the origin is an equilibrium. In that setting, the existence of a smooth lyapunov function is both necessary and su. His main preoccupations were the stability of equilibria and the motion of mechanical systems, and the study of particles under the influence of gravity. Lecture notes stability theory math 414 spring 2003. A lyapunov function is a scalar function of a systems state that decreases with time. The numerical simulations show that, even if these additional assumptions do not hold, the global stability of the diseasefree equilibrium and. In control theory, the discrete lyapunov equation is of the form. Lyapunov equations we assume a 2 rn n, p pt 2 rn n.
Stability analysis of nonlinear systems using lyapunov theory nptel. Direct method to prove these stability properties, and develop a nonlinear 3axis attitude pointing control law using lyapunov theory. Lyapunov exponents for continuoustime dynamical systems t. Lyapunov functions and feedback in nonlinear control. It is p ossible to ha v e stabilit y in ly apuno without ha ving asymptotic stabilit y, in whic h case w e refer to the equilibrium p oin t as mar ginal ly stable. This and related equations are named after the russian mathematician aleksandr lyapunov.
Until now, the theory of lyapunov stability is still the main theoretical basis of almost all systemcontroller design chen, 1984. When every point has this property, and when additionally the origin has the familiar local stability property known as lyapunov stability, it is said in the literature to be gac. Aleksandr lyapunov in 1876 lyapunov contributed to several fields, including differential equations, potential theory, dynamical systems and probability theory. Nonmonotonic lyapunov functions for stability of nonlinear. The mathematical theory of stability of motion, founded by a.