Lyapunov stability theorem pdf

D r be a continuously differentiable function, such that v0 0 and vx 0 in d. Then the origin is a globally asymptotically stable equilibrium of x0 fx. And were going to throw in our equations of motion, and attitude, and rotation and everything. Lyapunov s 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. R2 1 we have two methods to get insight about the structure of the phase portrait. Lyapunov direct method is the most effective method for studying nonlinear and timevarying systems and is a basic method for stability analysis and control law desgin. Lyapunov stability game the adversary picks a region in the state space of radius. Lyapunov was a pioneer in successful endeavoring to develop the global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium. When the jacobian matrix of a dynamical system at equilibrium consists of the eigenvalues with negative real parts this equilibrium is asymptotically stable.

So a lyapunov function is always a scalar function subject to this dynamical system. Lyapunovs theory for characterizing and studying the stability of equilibrium points is presented for timeinvariant and timevarying systems modeled by ordinary differential equations. Lecture linear quadratic lyapunov theory the lyapunov equation lyapunov stability conditions the lyapunov operator and integral evaluating quadratic integrals analysis of are discretetime results linearization theorem 1. Lyapunov stability of equilibrium may be discussed by the lyapunov theory in which the stability can be proven without defining the initial state of system. The lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. It follows from the theorem that the existence of a lower semicontinuous lyapunov pair v,w with v everywhere. The conditions in the theorem are summarized in table 4.

Stability in the sense of lyapunov theequilibriumpoint. In this example we demonstrate the necessary steps required in applying the lyapunov. Stability is a classical issue in dynamical system theory. The proposed smallgain criterion for the stability analysis of the network the spectral radius of the gain operator is less than one can b e check ed in a computationally e. In control theory, the discrete lyapunov equation is of the form. Lyapunov functions and feedback in nonlinear control. Pdf a lyapunovbased smallgain theorem for infinite. On 12 october 1892 by the modern calendar alexandr mikhailovich lyapunov defended his doctoral thesis the general problem of the stability of motion at moscow university. Here we introduce a theorem which can be viewed as a dual to lyapunovs result. Lyapunovs second theorem is a standard tool for stability analysis of ordinary differential equations. Lyapunov stability lyapunovs stability theorem theorem 4. Assuming that each subsystem is exponentially inputtostate stable. In the theory of ordinary differential equations odes, lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ode. Until now, the theory of lyapunov stability is still the main theoretical basis of almost all systemcontroller design chen, 1984.

Global lyapunov stability and lasalles invariance principle. The first method usually requires the analytical solution of the differential equation. This theory still dominates modern notions of stability, and provides the foundation upon which alternative notions of stability continue to be built. One of the most widely adopted stability concepts is lyapunov stability, which plays important roles in sys tem and control theory and in the analysis of engineering systems. Lyapunov stability an overview sciencedirect topics. Pdf a lyapunovbased smallgain theorem for infinite networks.

This theory still dominates modern notions of stability, and provides the foundation upon which alternative notions of. This does not imply lyapunov stability at the origin, however, so it cannot characterize global asymptotic controllability. Anand pillay september 29, 2003 1 introduction and preliminaries 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. A lyapunov function for an autonomous dynamical system. I will assume knowledge of my autumn 2002 model theory lecture notes 1. Stability is one of the properties of solutions of any differential systems. The following theorem is a direct application of the standard discretetime lyapunov stability theorem for general dynamical systems to the dynamical system 4. Pdf on application of lyapunov and yoshizawas theorems. Yao 3 for autonomous systems, when wx in the above theorem is only positive semidefinite, asymptotic stability may still be obtained by applying the following simplified version of lasalles theorem. Lyapunov was a pioneer in successful endeavoring to develop the global approach to the analysis of the stability of nonlinear dynamical systems by. A problem, which attracted a great deal of early interest was the problem of stability of the solar system, generalized under the title the nbody stability problem. Many famous mathematicians, physicists, and astronomers worked on axiomatizing the concepts of stability. Subsequent applications of these ideas in the control. Lyapunov stability theory was come out of lyapunov, a russian mathematician in 1892, and came from his doctoral dissertation.

There are however some major di erences with that work. A wikipedia page gives a general idea about the lyapunov stability. One of the most widely adopted stability concepts is lyapunov stability, which plays important roles in system and control theory and in the analysis of engineering systems. There are two lyapunov methods for stability analysis. An interesting aspect of lyapunov theory for lti systems is that the existence. Lyapunov stability theory an overview sciencedirect topics. Some of these will be described in later sections of this paper. 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. Failure of a lyapunov function candidate to satisfy the conditions for stability or asymptotic stability does not mean that the equilibrium point is not stable or asymptotically stable. It does not, however, give a prescription for determining the lyapunov function. Lyapunov functions and lyapunovs stability theorem duration.

Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. Lyapunov s second theorem is a standard tool for stability analysis of ordinary differential equations. Lyapunov stability is named after aleksandr mikhailovich lyapunov, a russian mathematician who defended the thesis the general problem of stability of motion at kharkov university in 1892. Structured and simultaneous lyapunov functions for system. It only means that such stability property cannot be established by using this lyapunov function. R be a continuously di erentiable function such that v0 0, x 0. This and related equations are named after the russian mathematician.

First of all, the lyapunov stability theory is understood through the picture. Lyapunov stability theory provides a means of stabilizing unstable nonlinear systems using feedback control. Lyapunov function theorem for local asymptotic stability with respect to closed sets. A matrix a is a stability matrix or hurwitz matrix, that is, re. Lecture 12 basic lyapunov theory stanford university. This paper presents a smallgain theorem for networks composed of a countably infinite number of finitedimensional subsystems. At this time, many of the later uses of lyapunovs work could hardly have been foreseen. A brief history of lyapunovs life and tragic death is given, and followed by a section highlighting the important ideas in his thesis of 1892. The idea is that if one can select a suitable lyapunov function and force it to decrease along the trajectories of the system, the resulting system will converge to its equilibrium. One the major stability technique for non linear dynamic system is called lyapunov. In the classical lyapunov stability theory, we assume that the ode in consideration has a smooth at. Lyapunovs stability theory100 years on 277 central limit theorem and other deep investigations in areas of mechanics and mathematical analysis. A detailed post on the lyapunov stability criteria will be uploaded soon.

This and related equations are named after the russian mathematician aleksandr lyapunov. Suppose that for a given system there exists a lyapunov function which also satisfies the stronger third condition 0 dv v dt stability theory math 414 spring 2003. Pdf on application of lyapunov and yoshizawas theorems on. In other words, a system has to be in a stable state before it can be asymptotically stable which means. The study of the stability of dynamical systems has a very rich history.