Smooth approximation of patchy lyapunov functions for. Lecture 14 lyapunov theory with inputs and outputs systems with inputs and outputs reachability bounding bounds on rms gain boundedreal lemma feedback synthesis via control lyapunov functions 141. 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. The ordinary lyapunov function is used to test whether a dynamical system is stable more restrictively, asymptotically stable. Then construct the observer as, where is the observer state vector, is the output, is the input, and is the estimated state vector. Systems with inputs we now consider systems with inputs, i. 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. Thisv 2isa smooth controllyapunovfunctionono 2,asverifiedbyud1. 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. The use of piecewise quadratic lyapunov functions appears to be a powerful extension of quadratic stability that also covers polytopic lyapunov functions.
Lyapunov functions are a fundamental tool used in control theory, and other disciplines, for studying the stability of systems evolving through time. Ieee trans actions on automatic control, 434, 555a559. 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. Branicky, member, ieee abstract in this paper, we introduce some analysis tools for switched and hybrid systems. We prove that such a control lyapunov function exists for any.
Sevilla lyapunov krasovskii functionals for the study of stability and stabilisation of timedelay systems with application to networked control systems2 15. Citeseerx smooth patchy control lyapunov functions. Professor michael maliso 1 motivation the construction of strict lyapunov functions is important for proving stability and robustness properties for nonlinear control systems. A general power system control technique based on lyapunov s function francesco torelli, federico milano. If the system is stable, then there exists a lyapunov function if a lyapunov function cannot be found, nothing is known about the stability of the system for general nonlinear systems, these functions can be hard to find recent computational tools in lmis and polynomial functions can provide numerical computations of lyapunov functions. A similar approach was used in the rst authors work on bernoulli di eomorphism on surfaces k1 and related later work on smooth m. 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. A is n n matrix lemma let ux be a snd quadratic form. Stratified semiconcave controllyapunov functions and the. We propose a decentralized errorbounded sliding mode control mechanism that ensures the prescribed tracking performance of a robot manipulator. Intuitively, a lyapunov function is a scalar function of the system state on which system trajectories descend monotonically. In the opposite case, we will show that pa 1 and pa 2 of proposition 5. 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. Jun 01, 2009 read robust and nonlinear control literature survey no.
A piecewise smooth controllyapunov function framework for switching. A barrier lyapunov function may be symmetric or asymmetric, as illustrated in fig. 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. Feedback stabilization and lyapunov functions siam. In probability theory, the central limit theorem clt establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution informally a bell curve even if the original variables themselves are not normally distributed. Contents 1 introduction to the lyapunov krasovskii functionals 2 objectives 3 ncs model 4 general procedure 5 example of application 6 summary univ. The method is a generalization of the idea that if there is some measure of energy in a system, then. Goebels 10 research works with 441 citations and 227 reads, including. Classical converse theorems in lyapunovas second method. Multiple lyapunov functions and other analysis tools for.
Converse lyapunov theorems and robust asymptotic stability. It gives a very precise definition of a lyapunov candidate function and says a lot about lyapunov candidate functions, but it doesnt bother to define a lyapunov function, except in a sketchy way near the start. On linear copositive lyapunov functions and the stability of switched positive linear systems o. A continuous lyapunov function implies robustness to strictly causal perturbations. For instance, one of the useful characteristic of smooth clfs is the existence of universal formulas for stabilization. The control design principles are based on receding horizon optimisation and generally use linear prediction models for the system dynamics. Finding common quadratic lyapunov functions for switched. 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.
Proceedings of the 48h ieee conference on decision and control cdc held jointly with 2009 28th chinese control conference, 42284233. The advantage of this method is that we do not need to know the actual solution xt. Lyapunovs direct method also called the second method of lyapunov allows us to determine the stability of a system without explicitly integrating the di. Stability, robustness, delays, and state constraints lecture series lecturer. Existence of lipschitz and semiconcave controllyapunov. Smooth patchy control lyapunov functions, automatica 10. Lecture 12 basic lyapunov theory stanford university. Lyapunov function is equivalent to stabilizability of the ori gin with relaxed controls. 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. Hence, the function \v\left \mathbfx \right\ is a lyapunov function and the zero solution of the system is stable in the sense of lyapunov. Kellett and teel, on the robustness of klstability for difference inclusions. In control theory, a control lyapunov function is a lyapunov function for a system with control inputs. Thus, lyapunov functions allow to determine the stability or instability of a system. Shorten abstract we consider the problem of common linear copositive function existence for positive switched linear systems.
Q moreover, if a is hurwitz, then p is the unique solution idea of the proof. We prove that such a control lyapunov function exists for any asymptotically controllable nonlinear system. We also establish that the feedback in question possesses a robustness. A general power system control technique based on lyapunovs. A lyapunov function is a function which is decreasing along solutions of the ode. Lyapunov functions 26 are a natural way of analysing the basin of attraction. A proper smooth patchy control lyapunov function for 1 is a smooth patchy control lyapunov function for 1. A piecewise smooth controllyapunov function framework for. A smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose open. The explicit solution of multiparametric optimisation problems mpop has been used to construct an offline solution to relatively small and mediumsized constrained control problems. Lyapunov stability of the solution of an equation given on is lyapunov stability of the point relative to the family of mappings. The theorem is a key concept in probability theory because it implies that.
Abstract a smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose. However, while it is easy to see that for every asymptotically stable system there exists many, even smooth, such lyapunov functions, in many cases it is almost impossible to get ones hands onto one such lyapunov function. Goebel, results on existence of smooth lyapunov functions for hybrid systems with nonopen basin of attraction, proceedings of the 2007 american control conference, to appear. Computation of piecewise quadratic lyapunov functions for hybrid systems. 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. Computation of piecewise quadratic lyapunov functions for. Proceedings of the 45th ieee conference on decision and control, 327276. One can think of these functions as energy description of a system to understand what the reasoning behind it is. Lyapunov function from a classkl estimate involving two positive semidefinite functions. Automatica smooth patchy control lyapunov functions gipsalab. By lyapunov control, we present a proposal to drive quasiparticles into a topological mode in quantum systems described by a quadratic hamiltonian. 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. In particular, for vx xtpxquadratic form, where pis a real symmetric matrix, vx is positive semide nite if and. Optimal robust stabilization of control systems siam.
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. 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. Recall that d is called a lyapunov function for system 1. The technique presented here has been extended to treat performance analysis and optimal control problems in 3 and smooth nonlinear systems in 4. Hybrid control lyapunov functions for the stabilization of. Lyapunov stability of the solution of a differential equation given on is lyapunov stability of the point relative to the family of mappings, where is the cauchy operator of this equation. Lyapunov functions for nonlinear discretetime systems. In this context, it can be shown that the optimal control. Semiconcave controllyapunov functions and stabilizing. The method of lyapunov functions is commonly used to establish global stability results for biological models.
Journal the theory and application of control lyapunov functions were developed by z. Robust stabilization of nonlinear systems with pointwise norm. The bad thing about this page is that it doesnt really say. The method of lyapunov functions plays a central role in the study of the controllability and stabilizability of control systems. Rninto rnand there is at least one equilibrium point x, that is f x 0. In control theory, a controllyapunov function is a lyapunov function for a system with control inputs. It is a global control lyapunov function if it is radially. Asymptotic stability and smooth lyapunov functions core. Barrier lyapunov functions for the control of output. Suchafunctionisa smooth controllyapunovfunctionono 1,thiscanbeseenbychoosing ud1. However a small modification of the proof of this result gives the existence of a strict proper pclf.
Multilayer minimum projection method with singular point. Sergio grammatico, anantharaman subbaraman, andrew r. A converse result shows that suitable lyapunov functions of this type exist under mild assumptions. Lyapunov function exists which holds vx lyapunov stability givenit just says that system is bounded by some valueof energy, it does not. Every asymptotically controllable to a compact set nonlinear system admits a smooth. We prove that such a control lyapunov function exists for any asymptotically controllable. Lyapunov functions and feedback in nonlinear control. A journal of ifac the international federation of automatic control, issn 00051098, vol. A smooth patchy control lyapunov function with finite number of patches for 1 with the attractor a consists of a set q and a collection of functions vq and sets q, q for each q q, such that i q z is a finite set. In the theory of ordinary differential equations odes, lyapunov functions are scalar functions that may be used to prove the stability of. Barrier lyapunov functionbased sliding mode control for. 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.
Local existence of a solution, lyapunov functions, asymptotic behaviour of the solution, uniform lp. Homogeneous lyapunov function for homogeneous continuous. The condition of asymptotic stability is not satisfied for this, the derivative \\large\fracdvdt ormalsize\ must be negative. The explicit construction of lyapunov functions for a given system, however, is a di. Duartemermoud department of electrical engineering and advanced mining technology center amtc, university of chile, avda. 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. Smooth patchy control lyapunov functions citeseerx.
This paper studies switching stabilization problems for general switched nonlinear systems. If all eigenvalues of the matrix a have negative real parts, then the di erential equation. Lyapunov functions, stability, stabilizability, discontinuous control. Lyapunov functions and cone families 5 everywhere, expanding foliation whose leaves include local expanding manifolds as open sets. 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 a decrease condition when the domains overlap. The lyapunov s direct method consists in building a. 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. An almost strict smooth patchy control lyapunov function for in the sense ofdefinition 5. Smooth lyapunov functions have the potential to be just as relevant for hybrid systems. A patchy approximation of explicit model predictive control.
Given a locally defined, nondifferentiable but lipschitz lyapunov function, we employ it in order to construct a discontinuous feedback law which stabilizes the underlying system to any given tolerance. This smooth function approximates the patchy function when working with clarkes generalized gradient. Lyapunov functions and lp estimates for a class of reaction. The primary qualitative technique we use is lyapunov control design. Local transverse feedback linearization with partial. 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. 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. For nonlinear systems, it turns out to be essential to consider. 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. Smooth patchy control lyapunov functions sciencedirect. First, choose an appropriate and such that the lyapunov equation yields a nonsingular solution.
Stability of continuous systems stability of linear systems. In a sense, the latter was already known, from results by sontag and. Preparation of topological modes by lyapunov control. Lyapunov functions are discussed and results stronger than the existing methods in the literature are obtained.
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