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. For nonlinear systems, it turns out to be essential to consider nonsmooth lyapunov functions, even if the underlying control dynamics are themselves smooth. Suchafunctionisa smooth controllyapunovfunctionono 1,thiscanbeseenbychoosing ud1. Smooth patchy control lyapunov functions sciencedirect.
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. 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. Lyapunovlike functions and lie brackets franco rampazzo monica motta 11th meeting on nonlinear hyperbolic pdes and. Teel, smooth patchy control lyapunov functions, in. Hybrid control lyapunov functions for the stabilization of. In particular, in his following seminal works the lyapunov approach is. In particular, for vx xtpxquadratic form, where pis a real symmetric matrix, vx is positive semide nite if and. Furthermore, contraction metrics and finslerlyapunov functions as a related method to study.
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. The structure of the level surfaces of a lyapunov function. Switched systems, stability, polynomial lyapunov function. Thisv 2isa smooth controllyapunovfunctionono 2,asverifiedbyud1. This paper studies switching stabilization problems for general switched nonlinear systems. The technique presented here has been extended to treat performance analysis and optimal control problems in 3 and smooth nonlinear systems in 4. Smooth patchy control lyapunov functions archive ouverte hal. Sevilla lyapunovkrasovskii functionals for the study of stability and stabilisation of timedelay systems. 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. Lyapunov functions and feedback in nonlinear control universite. The use of piecewise quadratic lyapunov functions appears to be a powerful extension of quadratic stability that also covers polytopic lyapunov functions.
Lyapunovkrasovskii functionals for the study of stability. This might be somehow unexpected, for the di erential. Converse lyapunov theorems and robust asymptotic stability. 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. 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. On piecewise quadratic controllyapunov functions for. We present a formula for a stabilizing feedback law under the assumption that a piecewise smooth control.
A piecewise smooth controllyapunov function framework for. Contents 1 introduction to the lyapunovkrasovskii functionals 2 objectives 3 ncs model 4 general procedure 5 example of application 6 summary univ. Control lyapunov functions and eduardo control lyapunov function is a very powerful tool for stabilization of nonlinear control system in. This result answers a question raised by dayawansa and martin.
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. Suchafunctionisasmooth controllyapunovfunctionono 1,thiscanbeseenbychoosing ud1. What is an intuitive explanation of the lyapunov function. In fact a smooth controllyapunov function is associated with fullstate feedback to meet the conditions of stability and robustness also for more details please look at link and attached files in. Lyapunov stability game the adversary picks a region in the state space of radius. In extending the technique of lyapunov functions to control systems, a number of new issues arise. Robustness of control barrier functions for safety. 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. We prove that such a control lyapunov function exists for any asymptotically controllable nonlinear system. Lecture 14 lyapunov theory with inputs and outputs systems with inputs and outputs reachability bounding bounds on rms gain boundedreal lemma feedback synthesis via controllyapunov functions 141.
Numerical construction of nonsmooth control lyapunov. Billard 2014, learning control lyapunov function to ensure stability of dynamical systembased robot reaching motions. See on the construction of lyapunov functions using the sum of squares decomposition. Then construct the observer as, where is the observer state vector, is the output, is the input, and is the estimated state vector. Basically the sos approximation constraint the polynomial optimization of finding a lyapunov function to be a semidefinite program, which is generally considered solvable for low. Later, in the classical works of massera, barbashin and krasovskii, and kurzweil, this su. Smooth approximation of patchy lyapunov functions for. Department of mathematics, university of michigan, ann arbor, michigan 48104 received february 24, 1966 let f be a realvalued cl function which is defined on euclidian space r. A lyapunov function is a nonnegative function of a systems state such that as the state changes, the value of the function at the systems current state decreases or simply does not increase.
This smooth function approximates the patchy function when working with clarkes generalized gradient. This paper develops several important extensions to the notion of a control barrier. A control barrier function enables controller synthesis for safety requirements speci ed by forward invariance of a set using a lyapunovlike condition. An almost strict smooth patchy control lyapunov function for in the sense ofdefinition 5. For nonlinear systems the sum of square polynomial lyapunov function can be constructed algorithmically. So actually a lot has been happening with lyapunov functions, and a good source of this are the underactuated robotics notes. Computation of piecewise quadratic lyapunov functions for. The structure of the level surfaces of a lyapunov function f. Let us mention that this tool has been strongly developed by eduardo.
In other words, we shall prove the following theorems. Asymptotic stability and smooth lyapunov functions core. Abstract a smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions, whose. Systems with inputs we now consider systems with inputs, i. Lyapunov functions for nonlinear discretetime systems. Lyapunov functions and cone families 5 everywhere, expanding foliation whose leaves include local expanding manifolds as open sets. Lyapunov functions and feedback in nonlinear control.
This serves as a basis for a general procedure of searching for lyapunov functions or storage functions. The nonholonomic integrator is an example of this phenomenon. Linear quadratic regulator lqr control for the inverted pendulum. A similar approach was used in the rst authors work on bernoulli di eomorphism on surfaces k1 and related later work on smooth m. For instance, one of the useful characteristic of smooth.
To extend these concepts to control system, in section 1. In the opposite case, we will show that pa 1 and pa 2 of proposition 5. Common polynomial lyapunov functions for linear switched. A smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control. First, choose an appropriate and such that the lyapunov equation yields a nonsingular solution. Relaxed characterizations of smooth patchy control lyapunov functions, proceedings of the ifac symposium on nonlinear control systems, 2007. Smooth patchy control lyapunov functions citeseerx. Kellett and teel, on the robustness of klstability for difference inclusions. A generalization to a class of piecewisepolynomial lyapunov functions is given. 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. Control lyapunov functions and partial differential equations.
However, as argued earlier, system dynamics like artsteins circles or. Lyapunov function can be upper bounded by a negativede. 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. Thisv 2isa smoothcontrollyapunovfunctionono 2,asverifiedbyud1. 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.
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