Sontag 5 gave a constructive proof providing a formula for the. A lyapunov approach to output feedback control using second. This serves as a basis for a general procedure of searching for lyapunov functions or storage functions. Universal stabilization using control lyapunov functions, adaptive derivative. Suchafunctionisa smooth controllyapunovfunctionono 1,thiscanbeseenbychoosing ud1.
Lyapunov functions article about lyapunov functions by the. Numerical computation of control lyapunov functions in the sense. Thus, for the given system, there is a lyapunov function, and its derivative is negative everywhere except at the origin. Thisv 2isa smooth controllyapunovfunctionono 2,asverifiedbyud1.
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. On piecewise quadratic controllyapunov functions for. This smooth function approximates the patchy function when working with clarkes generalized gradient. A piecewise smooth controllyapunov function framework for. Universal stabilization using control lyapunov functions, adaptive. A lyapunov boundedness theorem suppose there is a function v that satis. Smooth patchy control lyapunov functions sciencedirect. Concepts of generalized derivatives and nonsmooth lyapunov functions are considered. Hybrid control lyapunov functions for the stabilization of.
Suppose that the gains k 1 and k 2 of the sf controller 2. In this talk, we start with a simple and fundamental epidemic model to illustrate the technique of lyapunov functional in global stability analysis of equilibria for differential systems. Common polynomial lyapunov functions for linear switched systems. The set of all realvalued functions of system state which do not increase along system trajectories is convex, i. In chapter3and4, the discontinuities in the system are due to the fact that sensing occurs within a network.
A generalization to a class of piecewisepolynomial lyapunov functions is given. In other words, we shall prove the following theorems. Lecture 12 basic lyapunov theory stanford university. Discontinuous lyapunov functions for nonasymptotic stability. Q moreover, if a is hurwitz, then p is the unique solution idea of the proof. Control lyapunov functions for adaptive nonlinear stabilization miroslav krsti6, petar v.
One the major stability technique for non linear dynamic system is called lyapunov. Proceedings of 1994 ieee international conference on neural networks icnn94, 47504754. A smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control. Kokotovi6 department of electrical and computer enoineerino, university of california, santa barbara, ca 93106, usa received 6 august 1994. The control lyapunov function is used to test whether a system is feedback stabilizable, that is whether for any state x there exists a control.
Starting with a locally lipschitz patchy lyapunov function for a given switched system, we provide the construction of a continuously differentiable smooth lyapunov function, obtained via a convolutionbased approach. A wikipedia page gives a general idea about the lyapunov stability. Switched systems, stability, polynomial lyapunov function. Smooth patchy control lyapunov functions archive ouverte hal. International audiencea smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy certain further increase or decrease conditions. A lyapunovlike characterization of asymptotic controllability. L yapunov functions and feedbac k in nonlinear control.
Then, design partial lyapunov functions and intermediate virtual control for each subsystem until back to the entire system. A smooth patchy control lyapunov function for a nonlinear control system consists of an ordered family of smooth local control lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy a certain arrangement property. An algorithm for constructing lyapunov functions 74 8. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Control lyapunov functions for adaptive nonlinear stabilization. Control lyapunov barrier functions epub bayreuth universitat. The main objective of control is to modify the behavior of a dynamical system, typically with the purpose of regulating certain variables or of tracking desired signals. Smooth approximation of patchy lyapunov functions for. Abstract a smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose. More formally, suppose we are given an autonomous dynamical system. Starting with a locally lipschitz patchy lyapunov function for a given switched system, we provide the construction of a continuously di erentiable smooth lyapunov function, obtained via a convolutionbased approach. Counterexample guided synthesis of control lyapunov functions for switched systems.
A barrier lyapunov function may be symmetric or asymmetric, as illustrated in fig. Rninto rnand there is at least one equilibrium point x, that is f x 0. A detailed post on the lyapunov stability criteria will be uploaded soon. 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. This paper studies switching stabilization problems for general switched nonlinear systems. Named after the russian mathematician aleksandr mikhailovich lyapunov, lyapunov functions also called the lyapunov s second method for stability are important to stability theory of dynamical systems and control theory. Application to stabilization of switched nonlinear systems is illustrated through an numerical example. The advantage of this method is that we do not need to know the actual solution xt. R suppose there is a continuous stabilizing state feedback control u. Following posts gives a very basic example to hel user use the lyapunov function in matlab. For nonlinear systems, it turns out to be essential to consider nonsmooth lyapunov functions, even if the underlying control dynamics are themselves smooth.
To extend these concepts to control system, in section 1. In section 3 we introduce the concept of hybrid control lyapunov function, and we show that its existence guarantees the existence of a standard control lyapunov function for the closed. Sympathetic nervous system definition of sympathetic. In control theory and engineering, feedback stabilization methods for nonlinear dynamical systems are of both theoretical and practical importance, and control lyapunov functions clfs constitute. Kellett and teel, on the robustness of klstability for difference inclusions. In addition, this method allows to study the stability of equilibrium points of nonrough systems, for example, in the case when the equilibrium point is a center. On almost lyapunov functions university of illinois at. N2 we study asymptotic stability properties of nonlinear systems in the presence of almost lyapunov functions which decrease along solutions in a given region not everywhere but rather on the complement of a set of small volume. Recently, 17 proposed control lyapunov barrier functions clbfs for the stabi lization of nonlinear dynamical systems with state constraints. We continue this line of work in the current paper. A smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy a decrease condition when the domains overlap. Part of the communications and control engineering book series cce abstract. Smooth patchy control lyapunov functions citeseerx. Start studying introduction to the autonomic nervous system.
Introduction to the autonomic nervous system questions and. Lyapunov functions and feedback in nonlinear control universite. Lyapunov functions are crucial in the present book aims, given the strict relation between lyapunov functions and invariant sets. This result answers a question raised by dayawansa and martin. Counterexample guided synthesis of control lyapunov. Citeseerx smooth patchy control lyapunov functions. Examples of lyapunov functions generated by linear programming 78 9. Mathematical modeling and modal switching control of a novel tiltrotor uav system 32 is globally asymptotically stable with respect to the lyapunov functions 18 and 26.
857 78 180 789 526 367 1346 928 181 1306 631 1251 1362 552 979 465 493 11 516 577 840 1257 1286 1307 772 1048 695 366 33 1197 376 40