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. 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. It only means that such stability property cannot be established by using this lyapunov function. Lyapunov was a pioneer in successful endeavoring to develop the global approach to the analysis of the stability of nonlinear dynamical systems by. Lyapunov stability lyapunovs stability theorem theorem 4. One the major stability technique for non linear dynamic system is called lyapunov. Lyapunov stability theory provides a means of stabilizing unstable nonlinear systems using feedback control. And were going to throw in our equations of motion, and attitude, and rotation and everything. This does not imply lyapunov stability at the origin, however, so it cannot characterize global asymptotic controllability. Lyapunov functions and lyapunovs stability theorem duration. In the classical lyapunov stability theory, we assume that the ode in consideration has a smooth at. We take a close look at lyapunov stability for lti systems and discuss how to relate chapter 4s linearization theorem to lyapunov stability through lyapunovs indirect method.
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. This paper presents a smallgain theorem for networks composed of a countably infinite number of finitedimensional subsystems. This and related equations are named after the russian mathematician aleksandr lyapunov. A function v satisfying the hypotheses of this theorem is called a lyapunov function for the. And were talking, if it is continuous this function and there exists in neighborhood, such that for any states, that we are arguing local stability here with lyapunov. Until now, the theory of lyapunov stability is still the main theoretical basis of almost all systemcontroller design chen, 1984. Lyapunov s second theorem is a standard tool for stability analysis of ordinary differential equations. 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. Lyapunovs theory for characterizing and studying the stability of equilibrium points is presented for timeinvariant and timevarying systems modeled by ordinary differential equations. It does not, however, give a prescription for determining the lyapunov function. The study of the stability of dynamical systems has a very rich history.
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. It follows from the theorem that the existence of a lower semicontinuous lyapunov pair v,w with v everywhere. Stability in the sense of lyapunov theequilibriumpoint. 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. Lyapunov stability theory was come out of lyapunov, a russian mathematician in 1892, and came from his doctoral dissertation. 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. 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. Pdf on application of lyapunov and yoshizawas theorems on. Then the origin is a globally asymptotically stable equilibrium of x0 fx.
There are two lyapunov methods for stability analysis. Pdf a lyapunovbased smallgain theorem for infinite networks. D r be a continuously differentiable function, such that v0 0 and vx 0 in d. Pdf on application of lyapunov and yoshizawas theorems. Lyapunovs second theorem is a standard tool for stability analysis of ordinary differential equations. Stability is one of the properties of solutions of any differential systems. A wikipedia page gives a general idea about the lyapunov stability. A lyapunov function for an autonomous dynamical system. The conditions in the theorem are summarized in table 4. Lecture 12 basic lyapunov theory stanford university. I will assume knowledge of my autumn 2002 model theory lecture notes 1. An interesting aspect of lyapunov theory for lti systems is that the existence.
A matrix a is a stability matrix or hurwitz matrix, that is, re. First, we can study stability of an equilibrium using linearization of 1 around this equilibrium. So a lyapunov function is always a scalar function subject to this dynamical system. 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 functions and feedback in nonlinear control. Lyapunov stability theory an overview sciencedirect topics. The first method usually requires the analytical solution of the differential equation. 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. Some of these will be described in later sections of this paper. 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.
In other words, a system has to be in a stable state before it can be asymptotically stable which means. Named after the russian mathematician aleksandr mikhailovich lyapunov, lyapunov functions also called the lyapunovs second method for stability are important to stability theory of dynamical systems and control. Global lyapunov stability and lasalles invariance principle. R be a continuously di erentiable function such that v0 0, x 0. 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. Here we introduce a theorem which can be viewed as a dual to lyapunov s result. Lyapunov stability game the adversary picks a region in the state space of radius. Structured and simultaneous lyapunov functions for system. Lyapunov stability the stability of solutions to odes was.
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. Lyapunovs stability theory100 years on 277 central limit theorem and other deep investigations in areas of mechanics and mathematical analysis. There are however some major di erences with that work. A smooth converse lyapunov theorem for robust stability. In control theory, the discrete lyapunov equation is of the form. A detailed post on the lyapunov stability criteria will be uploaded soon. The following theorem is a direct application of the standard discretetime lyapunov stability theorem for general dynamical systems to the dynamical system 4. A dynamical system in a state of equilibrium is said to be stable. Lyapunov function theorem for local asymptotic stability with respect to closed sets. R2 1 we have two methods to get insight about the structure of the phase portrait. The lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. Lyapunov stability an overview sciencedirect topics.
Here we introduce a theorem which can be viewed as a dual to lyapunovs result. Many famous mathematicians, physicists, and astronomers worked on axiomatizing the concepts of stability. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. This and related equations are named after the russian mathematician. Subsequent applications of these ideas in the control. Assuming that each subsystem is exponentially inputtostate stable. 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. When the jacobian matrix of a dynamical system at equilibrium consists of the eigenvalues with negative real parts this equilibrium is asymptotically stable. Stability is a classical issue in dynamical system theory. This theory still dominates modern notions of stability, and provides the foundation upon which alternative notions of. At this time, many of the later uses of lyapunovs work could hardly have been foreseen. In this example we demonstrate the necessary steps required in applying the lyapunov. Various types of stability are there for solution of differential equation of the dynamic systems. This theory still dominates modern notions of stability, and provides the foundation upon which alternative notions of stability continue to be built.
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