Control lyapunov functions for adaptive nonlinear stabilization. The main result of the paper shows that the existence of a vector control lyapunov function is a necessary and sufficient. Smooth control lyapunov functions clfs are instrumental in many feedback control designs and can be traced back to artstein who introduced this lyapunov characterization of asymptotic controllability in 4. 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. Request pdf liapunov functions and stability in control theory this book.
Examples of lyapunov functions generated by linear programming 78 9. Advances in the direct computation of lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth nonlinear systems. Kellett and teel, on the robustness of klstability for difference inclusions. Multilayer minimum projection method with singular point. Nonsmooth stochastic lyapunov functions with weak extension of viscosity solutions. Recent results on vector lyapunov functions are utilized. Control lyapunov functions the consideration of nonsmooth lyapunov functions for proving stability of feedback discontinuous systems is an important extension to. Smooth patchy control lyapunov functions citeseerx. Lecture 12 basic lyapunov theory stanford university. This paper studies the use of vector lyapunov functions for the design of globally stabilizing feedback laws for nonlinear systems. R2 1 we have two methods to get insight about the structure of the phase portrait.
Abstract a smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose. Nonsmooth stochastic lyapunov functions with weak extension. This smooth function approximates the patchy function when working with clarkes generalized gradient. Rafal goebels research works loyola university chicago. More formally, suppose we are given an autonomous dynamical system. For nonlinear systems, it turns out to be essential to consider nonsmooth lyapunov functions, even if the underlying control dynamics are themselves smooth. This cited by count includes citations to the following articles in scholar.
Below, we restrict ourselves to the autonomous systems \\ \\mathbfx. Mar 01, 2009 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 certain further increase or decrease conditions. We prove that such a control lyapunov function exists for any asymptotically controllable nonlinear system. Teel, fellow, ieee, and rafal goebel abstracthybrid systems are dynamical systems where the. Smooth approximation of patchy lyapunov functions for switched systems matteo della rossa, aneel tanwani, luca zaccarian to cite this version. Further, patchy control lyapunov function 16 and nonpathological lyapunov functions 17, 18 are also related topics. The main idea behind a patchy control lyapunov function is to cover. Citeseerx smooth patchy control lyapunov functions. Proceedings of the 45th ieee conference on decision and control. Definition of the lyapunov function a lyapunov function is a scalar function defined on the phase space, which can be used to prove the stability of an equilibrium point. A piecewise smooth controllyapunov function framework for switching.
Mar 01, 20 read multilayer minimum projection method with singular point assignment for nonsmooth control lyapunov function design, asian journal of control on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. This paper studies switching stabilization problems for general switched nonlinear systems. Liapunov functions and stability in control theory request pdf. 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. Comparison of the minimum projection method with these lyapunov functions also remains. 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. Smooth patchy control lyapunov functions ieee conference. 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, such that the system can be brought to the zero state by applying the control u. Numerical construction of nonsmooth control lyapunov functions.
A smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose open. The method of lyapunov functions plays a central role in the study of the controllability and stabilizability of control systems. The lyapunov function method is applied to study the stability of various differential equations and systems. Smooth patchy control lyapunov functions, automatica 10. 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.
Smooth approximation of patchy lyapunov functions for. However, as argued earlier, system dynamics like artsteins circles or. 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. The lyapunov function concept can be extended to control systems, but in that case it turns out that nonsmooth functions are essential. A piecewise smooth controllyapunov function framework for. Lyapunov functions and stability problems gunnar s oderbacka, workshop ghana, 29. Lyapunov functions are discussed and results stronger than the existing methods in the literature are obtained. It is a global control lyapunov function if it is radially. Proceedings of the 45th ieee conference on decision and control, 327276. 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 certain further increase or decrease conditions. 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. Automatica smooth patchy control lyapunov functions gipsalab.
An algorithm for constructing lyapunov functions 74 8. Local transverse feedback linearization with partial. Smooth approximation of patchy lyapunov functions for switched systems. Converse lyapunov theorems and robust asymptotic stability. 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. Every asymptotically controllable to a compact set nonlinear system admits a smooth. Semiconcave controllyapunov functions and stabilizing. Lyapunov functions for nonlinear discretetime systems. A smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose open domai. In a sense, the latter was already known, from results by sontag and. 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. Numerical construction of nonsmooth control lyapunov functions 3 to a mixed integer linear programming formulation with the ability to construct continuous piecewise a ne control lyapunov functions for dynamical systems which admit a smooth control lyapunov function.
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. For instance, one of the useful characteristic of smooth clfs is the existence of universal formulas for stabilization. Thisv 2isa smooth controllyapunovfunctionono 2,asverifiedbyud1. Existence is equivalent to robustness chaohong cai, student member, ieee, andrew r. Lyapunov functions and feedback in nonlinear control. Multilayer minimum projection method with singular. Suchafunctionisa smooth controllyapunovfunctionono 1,thiscanbeseenbychoosing ud1. Here we present a feedback motionplanning algorithm which uses rigorously computed stability regions to build a sparse tree of lqrstabilized trajectories. A piecewise smooth controllyapunov function psclf approach is proposed and a constructive way to.
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