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20
Hybrid systems: Generalized solutions and robust stability
 In IFAC Symposium on Nonliear Control Systems
, 2004
"... Abstract: Robust asymptotic stability for hybrid systems is considered. For this purpose, a generalized solution concept is developed. The first step is to characterize a hybrid time domain that permits an efficient description of the convergence of a sequence of solutions. Graph convergence is used ..."
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Cited by 46 (13 self)
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Abstract: Robust asymptotic stability for hybrid systems is considered. For this purpose, a generalized solution concept is developed. The first step is to characterize a hybrid time domain that permits an efficient description of the convergence of a sequence of solutions. Graph convergence is used. Then a generalized solution definition is given that leads to continuity with respect to initial conditions and perturbations of the system data. This property enables new results on necessary conditions for asymptotic stability in hybrid systems.
A Globalization Procedure for Locally Stabilizing Controllers
, 2000
"... For a nonlinear system with a singular point that is locally asymptotically nullcontrollable we present a class of feedbacks that globally asymptotically stabilizes the system on the domain of asymptotic nullcontrollability. The design procedure is twofold. In a neighborhood of the singular point ..."
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Cited by 36 (0 self)
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For a nonlinear system with a singular point that is locally asymptotically nullcontrollable we present a class of feedbacks that globally asymptotically stabilizes the system on the domain of asymptotic nullcontrollability. The design procedure is twofold. In a neighborhood of the singular point we use linearization arguments to construct a sampled (or discrete) feedback that yields a feedback invariant neighborhood of the singular point and locally exponentially stabilizes without the need for vanishing sampling rate as the trajectory approaches the equilibrium. On the remainder of the domain of controllability we construct a piecewise constant patchy feedback that guarantees that all Carathéodory solutions of the closed loop system reach the previously constructed neighborhood.
Homogeneous State Feedback Stabilization of Homogeneous Systems
, 2000
"... : We show that for any asymptotically controllable homogeneous system in euclidian space (not necessarily Lipschitz at the origin) there exists a homogeneous control Lyapunov function and a homogeneous, possibly discontinuous state feedbacklaw stabilizing the corresponding sampled closed loop system ..."
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Cited by 29 (1 self)
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: We show that for any asymptotically controllable homogeneous system in euclidian space (not necessarily Lipschitz at the origin) there exists a homogeneous control Lyapunov function and a homogeneous, possibly discontinuous state feedbacklaw stabilizing the corresponding sampled closed loop system. We also show the relation between the degree of homogeneity and the bounds on the sampling rates which ensure asymptotic stability. 1Introduction In this paper we consider the problem of asymptotic state feedback stabilization of homogeneous control systems in R n . This problem has been considered by anumber of authors during the last years, see e.g. [14, 15, 16, 17, 20, 21, 22, 24], to mention just a few examples. Homogeneous systems appear naturally as local approximations to nonlinear systems, cf. e.g. [13]. In order to make use of this approximation property in the design of locally stabilizing feedbacks for nonlinear systems the main idea lies in the construction of homogeneous ...
Clocks and Insensitivity to Small Measurement Errors
, 1999
"... This paper deals with the problem of stabilizing a system in the presence of small measurement errors. It is known that, for general stabilizable systems, there may be no possible memoryless state feedback which is robust with respect to such errors. In contrast, a precise result is given here, sho ..."
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Cited by 28 (1 self)
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This paper deals with the problem of stabilizing a system in the presence of small measurement errors. It is known that, for general stabilizable systems, there may be no possible memoryless state feedback which is robust with respect to such errors. In contrast, a precise result is given here, showing that, if a (continuoustime, finitedimensional) system is stabilizable in any way whatsoever (even by means of a dynamic, time varying, discontinuous, feedback) then it can also be semiglobally and practically stabilized in a way which is insensitive to small measurement errors, by means of a hybrid strategy based on the idea of sampling at a "slow enough" rate.
On the existence of nonsmooth controlLyapunov functions in the sense of generalized gradients
 ESAIM Control Optim. Calc. Var
"... Abstract. Let x ̇ = f(x, u) be a general control system; the existence of a smooth controlLyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we re ..."
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Cited by 11 (2 self)
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Abstract. Let x ̇ = f(x, u) be a general control system; the existence of a smooth controlLyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a controlLyapunov function in the case of a nonsmooth function; it is based on Clarke’s generalized gradient. Finally, with an inedite proof we prove that the existence of this type of controlLyapunov function is equivalent to the existence of a classical controlLyapunov function. This property leads to a generalization of a result on the systems with integrator. Abstract. Soit x ̇ = f(x, u) un système commande ́ ; l’existence d’une fonction Lyapunov lisse associée a ̀ ce système ne garantit généralement pas l’existence d’un retour d’état stabilisant continu. Cependant, nous montrons qu’elle conduit toujours a ̀ la construction d’un retour d’état stabilisant au sens de Krasovskii (ou de Filippov). En outre, nous rappelons une définition de fonction Lyapunov dans le cas d’une fonction seulement Lipschitzienne; celleci est caractérisée par une condition sur les gradients généralisés de Clarke. Et Nous démontrons par une preuve inédite que l’existence d’une telle fonction est équivalente a ̀ celle d’une fonction Lyapunov lisse classique. Cette dernière propriéte ́ nous permet de généraliser un résultat sur le problème intégrateur au cas nonlisse. 1.
Weak converse Lyapunov theorems and control Lyapunov functions
 SIAM J. Control Optim
, 2004
"... Abstract. Given a weakly uniformly globally asymptotically stable closed (not necessarilycompact) set A for a differential inclusion that is defined on R n, is locallyLipschitz on R n \A, and satisfies other basic conditions, we construct a weak Lyapunov function that is locally Lipschitz on R n. Us ..."
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Cited by 10 (2 self)
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Abstract. Given a weakly uniformly globally asymptotically stable closed (not necessarilycompact) set A for a differential inclusion that is defined on R n, is locallyLipschitz on R n \A, and satisfies other basic conditions, we construct a weak Lyapunov function that is locally Lipschitz on R n. Using this result, we show that uniform global asymptotic controllabilityto a closed (not necessarilycompact) set for a locallyLipschitz nonlinear control system implies the existence of a locallyLipschitz controlLyapunov function, and from this controlLyapunov function we construct a feedback that is robust to measurement noise.
State Constrained Feedback Stabilization
 SIAM J. Control Optim
, 2003
"... Abstract A standard finite dimensional nonlinear control system is considered, along with a state constraint set S and a target set Σ. It is proven that open loop Sconstrained controllability to Σ implies closed loop Sconstrained controllability to the closed δneighborhood of Σ, for any specified ..."
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Cited by 8 (3 self)
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Abstract A standard finite dimensional nonlinear control system is considered, along with a state constraint set S and a target set Σ. It is proven that open loop Sconstrained controllability to Σ implies closed loop Sconstrained controllability to the closed δneighborhood of Σ, for any specified δ > 0. When the target set Σ satisfies a small time Sconstrained controllability condition, conclusions on closed loop Sconstrained stabilizability ensue. The (necessarily discontinuous) feedback laws in question are implemented in the sampleandhold sense and possess a robustness property with respect to state measurement errors. The feedback constructions involve the quadratic infimal convolution of a control Lyapunov function with respect to a certain modification of the original dynamics. The modified dynamics in effect provide for constraint removal, while the convolution operation provides a useful semiconcavity property.
Quasioptimal robust stabilization of control systems
, 2005
"... In this paper, we investigate the problem of semiglobal minimal time robust stabilization of analytic control systems with controls entering linearly, by means of a hybrid state feedback law. It is shown that, in the absence of minimal time singular trajectories, the solutions of the closedloop s ..."
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Cited by 8 (5 self)
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In this paper, we investigate the problem of semiglobal minimal time robust stabilization of analytic control systems with controls entering linearly, by means of a hybrid state feedback law. It is shown that, in the absence of minimal time singular trajectories, the solutions of the closedloop system converge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small measurement noise, external disturbances and actuator noise.
Discontinuous Feedback and Nonlinear Systems
 in Proc. of 8th IFAC Symposium on Nonlinear Control Systems
, 2010
"... Abstract: This tutorial paper is devoted to the controllability and stability of control systems that are nonlinear, and for which, for whatever reason, linearization fails. We begin by motivating the need for two seemingly exotic tools: nonsmooth controlLyapunov functions, and discontinuous feedba ..."
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Cited by 5 (0 self)
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Abstract: This tutorial paper is devoted to the controllability and stability of control systems that are nonlinear, and for which, for whatever reason, linearization fails. We begin by motivating the need for two seemingly exotic tools: nonsmooth controlLyapunov functions, and discontinuous feedbacks. Then, after a (very) short course on nonsmooth analysis, we build a theory around these tools. We proceed to apply it in various contexts, focusing principally on the design of discontinuous stabilizing feedbacks.
Nonsmooth Optimal Regulation and Discontinuous Stabilization
, 1997
"... For affine control systems, we study the relationship between an optimal regulation problem on the infinite horizon and stabilizability. We are interested in the case the value function of the optimal regulation problem is not smooth and feedback laws involved in stabilizability may be discontinuous ..."
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Cited by 5 (3 self)
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For affine control systems, we study the relationship between an optimal regulation problem on the infinite horizon and stabilizability. We are interested in the case the value function of the optimal regulation problem is not smooth and feedback laws involved in stabilizability may be discontinuous.