We also discuss the case where the switched systems are indexed by an arbitrary compact set. Lyapunov functions for nonlinear discretetime systems. Lyapunov theory is used to make conclusions about trajectories of a system x. One the major stability technique for non linear dynamic system is called lyapunov. Lyapunov stability theory an overview sciencedirect topics.
Computation of lyapunov functions and stability of interconnected systems. Example of stability problem we consider the system x0 y x3. This behavior commonly arises in mechanical systems undergoing impacts and optimal. These tools will be used in the next section to analyze the stability properties of a robot controller. Lyapunov functions and feedback in nonlinear control. Hence, if given q qt o, the lyapunov equation 8 has a symmetric positivede. On merging constraint and optimal controllyapunov functions. Lyapunov function from a set of demonstrations, 3 proposing an optimal control problem based on the learned lyapunov function to ensure stability of nonlinear autonomous ds, and 4 extending the classical clf control scheme and present it in the context of learning robot discrete motions from demonstrations. T wo control lyapunov functions v 1 and v 2 for 1 ha ve the control sharing pr operty if there exists a locallybounded control law u. The most useful and general approach for studying the stability of nonlinear control systems is the theory introduced in alexandr mikhailovich lyapunov. Kellett and teel, on the robustness of klstability for difference inclusions. Many physical laws are described in terms of such generally homogeneous functions.
The possible function definiteness is introduced which forms the building block of lyapunovs direct method. This article is devoted to a brief description of the basic stability theory, criteria, and methodologies of lyapunov, as well as a few related important stability concepts, for nonlinear dynamical systems. 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. Sami fadali, antonio visioli, in digital control engineering second edition, 20. 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 theory. To successfully merge two clfs, the controlsharing condition is crucial. Lyapunovs theory for characterizing and studying the stability of equilibrium points is presented for timeinvariant and timevarying systems modeled by ordinary differential equations. A particular focus is on the problem of the existence of liapunov functions converse liapunov theorems and their regularity, whose interest is especially motivated by applications. Lecture notes stability theory math 414 spring 2003. There are several approaches in the literature addressing this problem. 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. The most important type is that concerning the stability of solutions near to a point of equilibrium.
Lyapunov functions, in many cases it is almost impossible to get ones hands onto one such lyapunov function. Learning control lyapunov function to ensure stability of. In stability theory these functions as the widest class of the first approximation equations are used. Computation of lyapunov functions and stability of. Overview of lyapunov stability theory lyapunovs direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. Stability conditions for nonlinear control processes using. Lecture 12 basic lyapunov theory stanford university. Stability analysis and stabilization of nonlinear systems are two of the most important and extensively studied problems in control theory.
Lyapunov theory for zeno stability andrew lamperski and aaron d. Control lyapunovrazumikhin functions and robust stabilization of time delay. A detailed post on the lyapunov stability criteria will be uploaded soon. View lyapunov stability theory research papers on academia. Institute of control sciences, russian academy of sciences. Generation of lyapunov functions by neural networks. 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. That example motivates the development of stability theory in which the c condition on v is relaxed. We present a survey of the results that we shall need in the sequel, with no proofs. The idea is that if one can select a suitable lyapunov function and force it to decrease along the trajectories of the system, the. Lyapunovs direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. Lyapunov theory lyapunov theory is used to make conclusions about trajectories of a system x. 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. In simple terms, if the solutions that start out near an equilibrium point stay near forever.
Lyapunov matrix equation in system stability and control pdf. The ordinary lyapunov function is used to test whether a dynamical system is stable more restrictively, asymptotically stable. The ob jectiv this c hapter is to formalize the notion of in ternal stabilit y for general nonlinear statespace mo dels. An sosbased control lyapunov function design for polynomial fuzzy control of nonlinear systems. 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. Describing functions optimization theory variational optimization, dynamic. Techniques of nonlinear control systems analysis and design phase plane analysis. 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. Stability of nonlinear dynamic systems plays an important role in systems theory and engineering. Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. 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. We describe how to use the dual lyapunov function in dual space and the hamiltonjacobi inequality to formulate sufficient conditions for the stability of the zero solution in primary and dual space.
Following posts gives a very basic example to hel user use the lyapunov function in. This may be discussed by the theory of aleksandr lyapunov. Detecting new e ective families of lyapunov functions can be seen as a serious advance. We investigate the problem of lyapunov stability for nonautonomous odes for which the righthand side is only lebesguemeasurable with respect to time t, x. Common and multiple lyapunov functions in stability. In the classical lyapunov stability theory, we assume that the ode in consideration has a smooth at. Lyapunov functions are used for stability analysis and control lyapunov functions clf are used in the design of stabilizing feedback controllers. Pdf liapunov functions and stability in control theory. This book presents a modern and selfcontained treatment of the liapunov method for stability analysis, in the framework of mathematical nonlinear control theory. Theory and applications of stochastic systems lecture 2 b60.
In control theory, a controllyapunov function is a lyapunov function for a system with control inputs. However, it is a remarkable fact that the converse of theorem 4. We introduce multiple lyapunov functions as a tool for analyzing lyapunov stability and use iterated function systems ifs theory as a tool for lagrange stability. 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. For switched systems with generally homogeneous functions as nonlinear righthand sides, a method to construct the common lyapunov function clf from the lyapunov functions for partial systems is proposed. Lyapunov design for eventtriggered exponential stabilization. Is there any standard procedure to find out the lyapunov fucnion. Lecture notes in control and information sciences 267. We then analyze and apply lyapunovs direct method to prove these stability properties, and develop a nonlinear 3axis attitude pointing control law using lyapunov theory. In particular, for vx xtpxquadratic form, where pis a real symmetric matrix, vx is positive semide nite if and.
Lyapunov stability game the adversary picks a region in the state space of radius. Lyapunov stability theory provides a means of stabilizing unstable nonlinear systems using feedback control. Dynamical system theory lies at the heart of mathematical sciences and engineering. Stability is a classical issue in dynamical system theory. Stability analysis of ts fuzzy systems has been pursued mainly based on lyapunov stability theory but with. Overview of definite function overview of lyapunov. Up to 2nd order or maxi 3rd order system graphical method differential geometry feedback linearization lyapunov theory intelligent techniques.
Neural networks, fuzzy logic, genetic algorithm etc. Liapunov functions and stability in control theory. Stability analysis of nonlinear systems using lyapunov theory. A wikipedia page gives a general idea about the lyapunov stability. T wo control lyapunov functions v 1 and v 2 for 1 ha ve the controlsharing pr operty if there exists a locallybounded control law u.
Lyapunov stability theory research papers academia. The application of dynamical systems has crossed interdisciplinary boundaries from chemistry to. I will assume knowledge of my autumn 2002 model theory lecture notes 1. Multiple lyapunov functions and other analysis tools for. Ames abstractzeno behavior is a dynamic phenomenon unique to hybrid systems in which an in. Computation of lyapunov functions and stability of interconnected systems huijuan li to cite this version.
Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. First, we cover stability definitions of nonlinear dynamical systems, covering the difference between local and global stability. The case of v being continuous and locally lipschitzian has been treated. Pdf controlsharing and merging control lyapunov functions. Q moreover, if a is hurwitz, then p is the unique solution idea of the proof.
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