This serves as a basis for a general procedure of searching for lyapunov functions or storage functions. An example is provided that illustrates the soundness of the method. Continuoustime nonlinear system x fx x, 0 with state x tr. The ordinary lyapunov function is used to test whether a dynamical system is stable more restrictively, asymptotically stable. 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. Kellett and teel, on the robustness of klstability for difference inclusions. Smooth patchy control lyapunov functions citeseerx. Contents 1 introduction to the lyapunovkrasovskii functionals 2 objectives 3 ncs model 4 general procedure 5 example of application 6 summary univ. We propose a decentralized errorbounded sliding mode control mechanism that ensures the prescribed tracking performance of a robot manipulator. It is a global control lyapunov function if it is radially. But if one extends in a suitable way the concept of lyapunov function to nonsmooth functions, then the existence of a lyapunov function becomes a necessary and su. 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. Sevilla lyapunov krasovskii functionals for the study of stability and stabilisation of timedelay systems with application to networked control systems2 15. Asymptotic stability and smooth lyapunov functions. A be a discretetime stable lti system, and p pt 0 be a solution of the discretetime lyapunov equation atpa. 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. Lyapunov approach for analysis and design of second order. Lyapunovkrasovskii functionals for the study of stability. Since wellknown concepts of generalized solutions of differential equations with discontinuous righthand side can be described in terms of. Lyapunov functions for nonlinear discretetime systems. Lyapunov functions and stability problems gunnar s oderbacka, workshop ghana, 29. When modeling robot discrete motions with ds, ensuring stability of the learned ds is a key requirement to provide a useful policy.
Smooth approximation of patchy lyapunov functions for. This might be somehow unexpected, for the di erential. We propose an approach for constructing lyapunov function in quadratic form of a differential system. Lecture 12 basic lyapunov theory stanford university.
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. In section 2, 3, 4 open problems are indicated where there is a gap in the theory. First, positive polynomial system is obtained via the local property of the lyapunov function as well as its derivative. On common quadratic lyapunov functions for stable discrete. In section 4, we construct lyapunov functionals for the corresponding reactiondi. So actually a lot has been happening with lyapunov functions, and a good source of this are the underactuated robotics notes.
Lyapunov stability is named after aleksandr mikhailovich lyapunov, a russian mathematician who defended the thesis the general problem of stability of motion at kharkov university in 1892. Abstract a smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Lyapunovs direct method also called the second method of lyapunov allows us to determine the stability of a system without explicitly integrating the di. Inputtostate stability iss applies lyapunov notions to systems with inputs. In 606162 63, by using the method of smooth lyapunov functions slflf, there have.
The following lemma relates the discretetime lyapunov equation 1 to the continuoustime lyapunov equation 6, and introduces a notation that we shall use to state our results. Two well known approaches to nonlinear control involve the use of control lyapunov functions clfs and receding horizon control rhc, also known as model predictive control mpc. Switched systems, stability, polynomial lyapunov function. Robustness of control barrier functions for safety. 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. The nonholonomic integrator is an example of this phenomenon. However, as argued earlier, system dynamics like artsteins circles or. Then, the positive polynomial system is converted into an equation system by adding some variables. Pdf a lyapunov approach to output feedback control using.
Q moreover, if a is hurwitz, then p is the unique solution idea of the proof. Branicky, member, ieee abstract in this paper, we introduce some analysis tools for switched and hybrid systems. Smooth patchy control lyapunov functions archive ouverte hal. Multiple lyapunov functions and other analysis tools for. Billard 2014, learning control lyapunov function to ensure stability of dynamical systembased robot reaching motions. Hybrid control lyapunov functions for the stabilization of. 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. Lyapunov stability analysis for feedback control design lyapunov theorems lyapunov analysis allows one to analyze the stability of continuoustime dynamical systems and discretetime dynamical systems of the form. It is also shown how to moderate the requirement that, for each individual robot, there exists a control lyapunov function. 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. Sevilla lyapunovkrasovskii functionals for the study of stability and stabilisation of timedelay systems. The main result of the paper shows that the existence of a vector control lyapunov function is a necessary and sufficient. Asymptotic stability and smooth lyapunov functions core.
Common polynomial lyapunov functions for linear switched systems. Numerical construction of nonsmooth control lyapunov functions. 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. This paper studies the use of vector lyapunov functions for the design of globally stabilizing feedback laws for nonlinear systems. R n is an equilibrium point of the system if fxe 0 xe is an equilibrium point xt xe is a trajectory suppose xe is an equilibrium point system is globally asymptotically stable g. The natural extension of barrier functions to a system with control inputs is a control barrier function cbf, rst proposed by wieland and allg ower 2007. Lyapunov stability game the adversary picks a region in the state space of radius. Smooth patchy control lyapunov functions sciencedirect. This smooth function approximates the patchy function when working with clarkes generalized gradient. Common polynomial lyapunov functions for linear switched. 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. The main result is a suite of theorems about formation maintenance, task completion time, and formation velocity. Finally, numerical technique is applied to solve the equation system. Lyapunov stability analysis for feedback control design.
The method is a generalization of the idea that if there is some measure of energy in a system, then. Lyapunov function can be upper bounded by a negativede. The uni cation of control lyapunov functions clfs with cbfs appeared at the. First, we can study stability of an equilibrium using linearization of 1 around this equilibrium. 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. We present a formula for a stabilizing feedback law under the assumption that a piecewise smooth control. A control barrier function enables controller synthesis for safety requirements speci ed by forward invariance of a set using a lyapunovlike condition.
Constructing the lyapunov function through solving. 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. Stability of continuous systems stability of linear systems. The advantage of this method is that we do not need to know the actual solution xt. Lyapunov like functions and lie brackets franco rampazzo monica motta 11th meeting on nonlinear hyperbolic pdes and applications on the occasion of the 60th birthday of alberto bressan trieste, june 2016 franco rampazzomonica motta lyapunov like functions and lie brackets. Thus, lyapunov functions allow to determine the stability or instability of a system. Numerical construction of nonsmooth control lyapunov. We prove that such a control lyapunov function exists for any asymptotically controllable nonlinear system. R2 1 we have two methods to get insight about the structure of the phase portrait. 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. Lyapunov functions and feedback in nonlinear control universite.
More formally, suppose we are given an autonomous dynamical system. A generalization to a class of piecewisepolynomial lyapunov functions is given. The use of piecewise quadratic lyapunov functions appears to be a powerful extension of quadratic stability that also covers polytopic lyapunov functions. In the study ofweakas opposed to strong asymptotic stability, the existence of a smooth lyapunov function is rather exceptional. The design of control or observation algorithms based on second order sliding modes has been performed until now using either geometric or homogeneous approaches, but not lyapunov methods. The idea of lyapunov stability can be extended to infinitedimensional manifolds, where it is known as structural stability, which concerns the behavior of different but nearby solutions to differential equations. 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. 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. Barrier lyapunov functionbased sliding mode control for. The online eulerlagrange computation of receding horizon control is naturally viewed in terms of optimal control, whereas.
Citeseerx smooth patchy control lyapunov functions. A lyapunov approach to output feedback control using second. For nonlinear systems, it turns out to be essential to consider nonsmooth lyapunov functions, even if the underlying control dynamics are themselves smooth. Robustness of control barrier functions for safety critical. Lyapunov was a pioneer in successful endeavoring to develop the global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local. 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. Furthermore, contraction metrics and finslerlyapunov functions as a related method to study.
Suchafunctionisa smooth controllyapunovfunctionono 1,thiscanbeseenbychoosing ud1. 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 control lyapunov function approach to multiagent. Linear quadratic regulator lqr control for the inverted pendulum. Control lyapunov functions for adaptive nonlinear stabilization. In particular, for vx xtpxquadratic form, where pis a real symmetric matrix, vx is positive semide nite if and. To extend these concepts to control system, in section 1. Thisv 2isa smooth controllyapunovfunctionono 2,asverifiedbyud1. A relatively interesting paper presented in ieee transactions on automatic control, it presents an argument for extending control lyapunov functinos, a standard result for smooth equations, to discretetime hybrid systems. Lyapunovs second or direct method provides tools for studying asymp. This can be thought of as being analogous to how sontag extended lyapunov functions to control lypaunov functions in order to enable controller synthesis for stabilization tasks. Control lyapunov for hybrid system university of michigan. I am struggling to see how differentiation of the lyapunov function is carried out. Lyapunov functions are also basis for many other methods in analysis of dynamical system, like frequency criteria and the method of comparing with other systems.
Recent results on vector lyapunov functions are utilized. The technique presented here has been extended to treat performance analysis and optimal control problems in 3 and smooth nonlinear systems in 4. Lyapunovlike functions and lie brackets franco rampazzo monica motta 11th meeting on nonlinear hyperbolic pdes and. This result answers a question raised by dayawansa and martin. Smooth lyapunov functions for hybrid systems part ii. Furthermore, contraction metrics and finsler lyapunov functions as a related method to study. Contents 1 introduction to the lyapunov krasovskii functionals 2 objectives 3 ncs model 4 general procedure 5 example of application 6 summary univ.
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