Lecture 12 basic lyapunov theory stanford university. 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 theory enables to investigate the system stability without the necessity. In this paper, a fourlegged robot with intelligent controllers is designed and simulated. Introduction lyapunov stability examplesconclusions observations the lyapunov stability theorem can be applied without solving the ode system the theorem provides a su cient condition for stability the theorem does not provide a systematic method for constructing the lyapunov function v of a system.
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. Lyapunov functions and feedback in nonlinear control. He was inspired by the following theorem from mechanics lagrange in 1800. Lyapunov theory lyapunov theory is used to make conclusions about trajectories of a system x. Control system engineeringii 3 10 modulei 10 hours state variable analysis and design. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. Hafstein reykjavik university, menntavegur 1, 101 reykjavik, iceland. Assume next that is a spherical region dfx jkx nxk r 0g d. Application of hydraulic system in wave energy converter. Lyapunov functions are the centerpiece of the lyapunovstability theory for. 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. Numerical methods for lyapunov equations methods for lyapunov equations this chapter is about numerical methods for a particular type of equation expressed as a matrix equality.
Principled reward shaping for reinforcement learning via. We revisit the canonical continuoustime and discretetime matrix algebraic and ma. The idea of lyapunov stability can be extended to infinitedimensional manifolds, where it is known as structural stability, which concerns the behavior of different but nearby solutions to differential equations. It explores the state of the art of chaos in plasma physics, plasma harmonics, and overtone coupling.
Matlab implementation skuli gudmundsson sigurdur f. The potential of the proposed methods are demonstrated by simulation examples. I will assume knowledge of my autumn 2002 model theory lecture notes 1. Lyapunov stability of the solution of an equation given on is lyapunov stability of the point relative to the family of mappings. The lyapunov equation is the most common problem in the class of problems called matrix equations. This and related equations are named after the russian mathematician. 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 control theory, the discrete lyapunov equation is of the form. We then use these results to give an algorithm for constructing such a common lyapunov function for the systems x. Linear quadratic lyapunov theory stanford university. 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. Quadruped robots have increasingly been used in complex terrains where barriers and gaps exist. November 15, 2009 1 1 lyapunov theory of stability introduction. It only means that such stability property cannot be.
Introduction stability of nonlinear dynamic systems plays an important role in. Inputtostate stability iss applies lyapunov notions to. These tools will be used in the next section to analyze the stability properties of a robot controller. Lyapunov theory for zeno stability georgia institute of. Pdf lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and. Lyapunov function and relate it to these various stability notions.
Pdf computational methods for lyapunov functions researchgate. Perhaps you can argue this must practically be the case, since a reasonable optimization functional will also force the system to be stable again, you can think of the lqr case but. 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. 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. 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. View lyapunov stability theory research papers on academia. Pdf on feb 9, 1993, aristide halanay and others published applications of lyapunov. For a feedback after a linearization, it is possible to write dependence. Lyapunov function based on approximation theory and the abilities of artificial neural networks. Theorem4 suppose that lis a lyapunov functional on oand that p. To distinguish between the first and second cases, the lyapunov function that satisfies.
Theorem a matrix a is hurwitz if and only if for any q qt 0 there is p pt 0 that satis. Authors in 1 have proposed a lyapunov theorybased adaptive filtering laf algorithm. Suppose that the independent random variables have finite means, variances and absolute moments, and. Lyapunovs second or direct method provides tools for studying asymptotic stability properties of an equilibrium point of a. The possible function definiteness is introduced which forms the building block of lyapunovs direct method. It has been shown in 11 that, unlike many adaptive filtering schemes using gradient search in the parameter space, laf algorithm uses a lyapunov functionvk, which is positive definite, with a unique global minimum in the state space. Furthermore, we assume that there exists a continuously di. Theorem3 supposse that lis a lyapunov functional on oand xt is an orbit lying in a closed bounded set k. Introduction, concepts of state, sate variables and state model, state models for linear continuoustime systems, state variables and linear discretetime. Chaos theory available for download and read online in other formats. Lyapunov theory for zeno stability andrew lamperski and aaron d. Lyapunovs theorem in probability theory is a theorem that establishes very general sufficient conditions for the convergence of the distributions of sums of independent random variables to the normal distribution. 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.
Rninto rnand there is at least one equilibrium point x, that is f x 0. Q moreover, if a is hurwitz, then p is the unique solution idea of the proof. Linear quadratic lyapunov theory lyapunov equations we assume a 2 rn n, p pt 2 rn n. We will demonstrate that c is a necessary and sufficient condition for a and b. Arts lab, scuola superiore santanna, pisa, italy department of information engineering, univ. By using the monotonicity theory combined with the lyapunov stability, some sufficient conditions are developed to ensure the closedloop system with the given eventtriggering scheme is inputto. Generation of lyapunov functions by neural networks.
Lyapunov stability theory research papers academia. Fundamentals of lyapunov theory autonomous systems lyapunov s pioneering work on stability is based on finding a energylike function with special features. Pdf applications of lyapunov methods in stability researchgate. This is due to the fact that we cannot write a simple formula relating the. Buy lyapunov matrix equation in system stability and control dover civil and mechanical engineering on free shipping on qualified orders. In control theory, a controllyapunov function is a lyapunov function for a system with control inputs. Lyapunovs theory for characterizing and studying the stability of equilibrium points is presented for timeinvariant and timevarying systems modeled by ordinary differential equations. Index termsapproximation theory, lyapunov function, nonlinear system, neural network. Statebased lyapunov functions overview of lyapunov.
Common lyapunov functions for specific classes of switched linear systems. The lyapunov and lasalle theorems university of michigan. Overview of lyapunov stability theory lyapunovs direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. If a is stable, lyapunov equation has a unique solution p, for any q qt. Lyapunov stability theory, we assume that the ode in consideration has a smooth at least c1 righthand side and the origin is an equilibrium.
Lyapunov time in modern physics, primarily exploring very unbalanced conditions and chaotic systems, there is a technical term lyapunov time. A lyapunovbased approach for the control of biomimetic robotic systems with periodic forcing inputs domenico campolo. The ordinary lyapunov function is used to test whether a dynamical system is stable more restrictively, asymptotically stable. The precise statement of lyapunovs theorem is as follows. 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. This behavior commonly arises in mechanical systems undergoing impacts and optimal. Lyapunovs direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. Dynamical system theory lies at the heart of mathematical sciences and engineering. Communications in applied analysis 12 lyapunov theory. This discoverywent against the intuition drawn from the classical theory of furstenberg. Ames abstractzeno behavior is a dynamic phenomenon unique to hybrid systems in which an in. As it so often goes with easy ideas, it turns out that lyapunov exponents are not natural for study of dynamics, and we would have passed them.