Lyapunovs theory for characterizing and studying the stability of equilibrium points is presented for timeinvariant and timevarying systems modeled by ordinary differential equations. The possible function definiteness is introduced which forms the building block of lyapunov s direct method. Lyapunov stability analysis stability theory systems. 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. Lyapunovs second or direct method provides tools for studying asymptotic stability properties of an equilibrium point of a dynamical system or systems of 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. One the major stability technique for non linear dynamic system is called lyapunov. We present four techniques for examining interior stability.
A small private ceremony in the mountains, just the two of us. Lyapunov asymptotic stability overview of lyapunov. 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. Yang et al lyapunov stability and strong passivity analysis for nonlinear descriptor systems 1005 iii globally asymptotically stable if it is stable and for any consistent initial condition, it holds that remark 2. Lyapunov stability game the adversary picks a region in the state space of radius. Lyapunov functions and feedback in nonlinear control. Hi, like jordan mentioned, wikipedia has a very good description about lyapunov stability. Lyapunovs stability analysis technique is very common and dominant. This subject was proposed to lyapunov by chebyshev as a topic for his masters thesis which he submitted in 1884 with the title on the stability of ellipsoidal forms of rotating fluids. Lyapunov stability theory research papers academia. Lyapunov stability the stability of solutions to odes was. The application of dynamical systems has crossed interdisciplinary boundaries from chemistry to.
Indirect relation between sensitivity analysis and lyapunov stability perturbations affecting the nominal structure of a system. 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. Lyapunovbased approach introduces advanced tools for stability analysis of nonlinear systems. Lyapunov stability stability theory applied mathematics. Lyapunov was a pioneer in successful endeavoring to develop the global approach to the analysis of the stability of nonlinear dynamical systems by. This theory still dominates modern notions of stability, and provides the foundation upon which alternative notions of stability continue to be built. Lyapunov function that takes an optimized workspace. R n is an equilibrium point of the system if fxe 0 xe is an equilibrium point xt xe is a trajectory suppose xe is an equilibrium point system is globally asymptotically stable g. Sensitivity analysis and lyapunov stability sciencedirect.
Dynamical system theory lies at the heart of mathematical sciences and engineering. Lyapunov stability analysis with solved examples electrical. November 15, 2009 1 1 lyapunov theory of stability introduction. I will assume knowledge of my autumn 2002 model theory lecture notes 1. Lyapunov s direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. I am struggling to see how differentiation of the lyapunov function is carried out. In particular, for vx xtpxquadratic form, where pis a real symmetric matrix, vx is positive semide nite if and. But then there is an extra step and that extra step is youre talking about v dot be negative semi definite, thats what guarantees stability.
There are a number of stability concepts used in the study of dynamical systems. Lyapunov stability free download as powerpoint presentation. But perhaps the most important stability concept is that of stability in the sense of lyapunov or simply 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. Apr 25, 2015 4 lyapunov stability a state of an autonomous system is called an equilibrium state, if starting at that state the system will not move from it in the absence of the forcing input. Lyapunov stability tools for sliding modes request pdf. A detailed post on the lyapunov stability criteria will be uploaded soon. Pdf ecological and lyapunov stability researchgate. 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. The lyapunov first and second methods are investigated and the stability analysis of fractional differential systems is highlighted.
Lyapunov functions and stability problems gunnar s oderbacka, workshop ghana, 29. Lyapunov based approach introduces advanced tools for stability analysis of nonlinear systems. In the classical lyapunov stability theory, we assume that the ode in consideration has a smooth at. In particular, it does not require that trajectories starting close to the origin tend to the origin asymptotically. Ecological and lyapunov stability article pdf available in philosophy of science 754 october 2008 with 215 reads how we measure reads. A wikipedia page gives a general idea about the lyapunov stability. How to use a lyapunov function to test stability of a non. If the lyapunov function satis es the conditions of asymptotic stability over a domain. Stability in the sense of lyapunov stability is one of the most important properties characterizing a systems qualitative behavior. Lyapunov s second or direct method provides tools for studying asymptotic stability properties of an equilibrium point of a dynamical system or systems of differential equations.
Lecture 12 basic lyapunov theory stanford university. A major theme in lyapunov s research was the stability of a rotating fluid mass with possible astronomical application. 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. 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 lyapunovbased methods. Lyapunov stability analysis free download as powerpoint presentation. The geometrical implication of stability is indicated in fig. The knowledge of lyapunov stability permits one frequently to deduce some property of solutions of differential equations if a parameter other than an initial condition is varied. His surname is sometimes romanized as ljapunov, liapunov, liapounoff or. Lecture notes stability theory math 414 spring 2003. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. The system in 1 is autonomous time invariant if f fx. View lyapunov stability theory research papers on academia. It only means that such stability property cannot be established by using this lyapunov function. 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.
So a lyapunov function is always a scalar function subject to this dynamical system. The main deficiency, which severely limits its utilization, in reality, is the complication linked with the development of the lyapunov function which is needed by the technique. And were going to throw in our equations of motion, and attitude, and rotation and everything. Stability notions from lyapunov stability 1982 to fixedtime stability 2012 are observed. May 25 1857 november 3, 1918 was a russian mathematician, mechanician and physicist. Scribd is the worlds largest social reading and publishing site. Stability is a classical issue in dynamical system theory. Various types of stability are there for solution of differential equation of the dynamic systems.
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