Energy functions in neural networks with continuous local. The technique presented here has been extended to treat performance analysis and optimal control problems in 3 and smooth nonlinear systems in 4. Citeseerx smooth patchy control lyapunov functions. Named after the russian mathematician aleksandr mikhailovich lyapunov, lyapunov functions also. In this paper we consider a nonlinear system with distributed parameters. The ordinary lyapunov function is used to test whether a dynamical system is stable more restrictively, asymptotically stable. In control theory, people are good at putting forward new control laws, many of which are derived from the use of a lyapunov function.
Control lyapunov functions the consideration of nonsmooth lyapunov functions for proving stability of feedback discontinuous systems is an important extension to. The intuitive picture is that of a scalar output function, often thought. Smooth approximation of patchy lyapunov functions for switched systems. 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. Abstract 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.
Learning control lyapunov function to ensure stability of. The lyapunov function method is applied to study the stability of various differential equations and systems. 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. Control lyapunov functions and partial differential equations.
Applications of the framework to switched linear systems with quadratic and piecewise quadratic control lyapunov functions are discussed and results stronger than the existing methods in the literature are obtained. The following lemma formalizes the result for general forms of barrier functions and is used in the control design and analysis for strict feedback system to ensure that output or state constraints are not violated download. Kellett and teel, on the robustness of klstability for difference inclusions. A patchy approximation of explicit model predictive control. Lyapunov s realization was that stability can be proven without requiring knowledge of the true physical energy, provided a lyapunov function can be found to satisfy the above constraints. Control lyapunov functions and eduardo control lyapunov function is a very powerful tool for stabilization of nonlinear control system in. In this context, it can be shown that the optimal control law is a. Barrier lyapunov functions for the control of outputconstrained nonlinear systems article pdf available in automatica 454. Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be squareintegrable. Converse lyapunov theorems and robust asymptotic stability. The control design principles are based on receding horizon optimisation and generally use linear prediction models for the system dynamics. It does not mean that the transient behavior of the solutions is of no. Pdf barrier lyapunov functions for the control of output.
Already, the types of results reported herein have been used in 27 to develop the notion of a patchy, smooth control lyapunov function clf for systems that do not admit a standard smooth clf. 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. Lyapunov function was proposed as the total energy function plus a crossterm bilinear in position and velocity, resulting in a local strict lyapunov function. Barrier lyapunov functionbased sliding mode control for. The rst group of systems starts with general invariant sets for general dynamical systems in section2. 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. 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. 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. 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.
Let us mention that this tool has been strongly developed by eduardo. In control theory, a controllyapunov function is a lyapunov function for a system with control inputs. Suchafunctionisa smooth controllyapunovfunctionono 1,thiscanbeseenbychoosing ud1. 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. Computation of piecewise quadratic lyapunov functions for. Robust hybrid control systems university of california. Smooth lyapunov functions have the potential to be just as relevant for hybrid systems. Construction of a smooth lyapunov function for the robust and exact secondorder differentiator tonametlsanchez,jaimea. Lyapunov functions for nonlinear discretetime systems. The explicit solution of multiparametric optimisation problems mpop has been used to construct an offline solution to relatively small and mediumsized constrained control problems.
In a sense, the latter was already known, from results by. Lyapunov functions and feedback in nonlinear control. Grizzle, paulo tabuada abstractsafety critical systems involve the tight coupling between potentially con. We use the method of lyapunov functions for the stability analysis. Control of lyapunov exponents in nonlinear lattice and stimulation of the epileptic brain dynamics panos m. The lyapunov function concept can be extended to control systems, but in that case it turns out that nonsmooth functions are essential. Control barrier function based quadratic programs for safety critical systems aaron d.
We propose a decentralized errorbounded sliding mode control mechanism that ensures the prescribed tracking performance of a robot manipulator. Every asymptotically controllable to a compact set nonlinear system admits a smooth. More formally, suppose we are given an autonomous dynamical system. Local transverse feedback linearization with partial. For nonlinear systems, it turns out to be essential to consider nonsmooth lyapunov functions, even if the underlying control dynamics are themselves smooth. Smooth patchy control lyapunov functions sciencedirect. 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. The use of piecewise quadratic lyapunov functions appears to be a powerful extension of quadratic stability that also covers polytopic lyapunov functions. Lyapunov functions and storage functions 1 this lecture gives an introduction into system analysis using lyapunov functions and their generalizations.
Generation of lyapunov functions by neural networks. On piecewise quadratic controllyapunov functions for. Research article construction of a smooth lyapunov function. November 15, 2009 1 1 lyapunov theory of stability introduction. Review on computational methods for lyapunov functions 3 general settings and systems, before we then describe more general types of lyapunov functions. But theres a large literature on lyapunov functions in control theory, and roozbehani and his colleagues are hopeful that much of it will prove applicable to software verification. On the existence of nonsmooth controllyapunov functions in the sense of generalized gradients article in esaim control optimisation and calculus of variations 66. Yatsenko ise department and mcknight brain institute university of florida 303 weil hall, gainesville, fl 326116595 usa abstract. Why a smooth control lyapunov function is associated with fullstate feedback law while a non smooth one non smooth clf defines discontinuous or. Below, we restrict ourselves to the autonomous systems \\ \\mathbfx. This multiplicative version of the central limit theorem is sometimes called gibrats law.
Inverse optimal control with linearlysolvable mdps data which is what the above irl methods do. The bounds on the control are known a priori and can be adjusted by. These generalized control lyapunov functions play an important role in. His main preoccupations were the stability of equilibria and the motion of mechanical systems, and. When modeling robot discrete motions with ds, ensuring stability of the learned ds is a key requirement to provide a useful policy. 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. Based on the rise control methodology, the proposed controller is able to utilize the bene.
On the existence of nonsmooth controllyapunov functions. Finding lyapunov functions1 this lecture gives an introduction into basic methods for. Strict lyapunov functions for control of robot manipulators. 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. That is, whether the system starting in a state in some domain d will. 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. Generation of lyapunov functions by neural networks navid noroozi, paknoosh karimaghaee, fatemeh safaei, and hamed javadi abstractlyapunov function is generally obtained based on. Mar 19, 20 of course, most dynamical systems are more complex than pendulums, and finding lyapunov functions that characterize them can be difficult. For nonlinear systems, it turns out to be essential to consider nonsmooth lyapunov functions, even if the. In this paper we broach the problem of finding strict global lyapunov functions for a class of global regulators. I am new to control theory, so could anyone give me an illustrative example about how to derive a control law or other similar applications by finding out a suitable lyapunov function. Previous works have shown that lyapunov or energy functions could be derived for networks of binary elements, thus allowing a rather complete char acterization. One reason to use data instead of solving the forward problem directly is that an appropriate cost function which captures the control objectives may be hard to design. It is a global control lyapunov function if it is radially.
In the classical lyapunov stability theory, we assume that the ode in consideration has a smooth at least c1 righthand side and the origin is an equilibrium. Abstract 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. In particular, in his following seminal works the lyapunov approach is a key step. 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. Lecture 12 basic lyapunov theory stanford university. Aleksandr lyapunov in 1876 lyapunov contributed to several fields, including differential equations, potential theory, dynamical systems and probability theory. Proceedings of the 45th ieee conference on decision and control, 327276.
Neural networks with continuous local transition func tions have been recently used for a variety of applications, especially in learning tasks and combinatorial optimization. This smooth function approximates the patchy function when working with clarkes generalized gradient. Jan 29, 2016 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. Semiconcave controllyapunov functions and stabilizing. Application to stabilization of switched nonlinear systems is illustrated through an numerical example. Let the function \v\left \mathbfx \right\ have the form. Smooth approximation of patchy lyapunov functions for. A tracking errortransformed sliding surface was constructed and the barrier lyapunov function blf was used to ensure the transient and steadystate time performance of the positioning function of a robot manipulator as well as satisfy the ordinary. The method of lyapunov functions plays a central role in the study of the controllability and stabilizability of control systems. Thisv 2isa smooth controllyapunovfunctionono 2,asverifiedbyud1. The definition for discretetime systems is almost identical to that for continuoustime systems. For instance, one of the useful characteristic of smooth. Smooth patchy control lyapunov functions, automatica 10. Later, koditschek 1988a presented in a formal framework the construction of a class of strict lyapunov functions for dissipative mechanical.
Barrier lyapunov functions for the control of output. Control of lyapunov exponents in nonlinear lattice and. Feedback stabilization and lyapunov functions siam. Control barrier function based quadratic programs for. In this paper we propose an imitation learning approach that exploits the power of control lyapunov function clf control scheme to ensure global asymptotic stability of nonlinear ds. 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. 11th ifac symposium on nonlinear control systems nolcos. A barrier lyapunov function may be symmetric or asymmetric, as illustrated in fig. Such a converse controllyapunov function theorem justi. Smooth approximation of patchy lyapunov functions for switched systems matteo della rossa, aneel tanwani, luca zaccarian to cite this version. 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. 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.
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