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- control-lyapunov function
- all continuous
- general sense
- gac systems
- exist ? ?
- pi ?
- trajectories through sampling

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Remarks on Input to State Stabilization

1 Michael Malisoff Department of Mathematics 304 Lockett Hall Louisiana State University and A. & M. College Baton Rouge LA 708034918 USA malisoff@lsu.edu

Ludovic Rifford Institut Girard Desargues Universit´eLyon1 BˆatimentBraconnier 21 Avenue Claude Bernard 69622 Villeurbanne Cedex France rifford@desargues.univlyon1.fr

2 Eduardo Sontag Department of Mathematics RutgersNew Brunswick Hill CenterBusch Campus 110 Frelinghuysen Road Piscataway NJ 088548019 USA sontag@control.rutgers.edu

Abstract— Weannounce a new construction of a stabilizingother applications where Brockett’s necessary condition is feedback law for nonlinear globally asymptotically controllable not satisﬁed, and which therefore cannot be stabilized by (GAC) systems. Given a control afﬁne GAC system, our feed continuous state feedbacks (cf. [2], [17]). back renders the closed loop system input to state stable with Our results alsostrengthen[3], which constructed feed respect to actuator errors and small observation noise. We backs for GAC systems which render the systems globally also announce a variant of our result for fully nonlinear GAC systems.asymptotically stable. Our main tool will be the recent constructions of semiconcave control Lyapunov functions (CLF’s) for GAC systems from [11], [12], [13]. Our con I. INTRODUCTION struction also applies in the more general situation where The theory of input to state stable (ISS) systems formsmeasurement noise may occur. In particular, our feedbackK the basis for much current research in mathematical controlhas the additional feature that theperturbedsystem theory (cf. [6], [7], [17]). The ISS property was introduced ˙x=f(x) +G(x)K(x+e) +G(x)u(3) in [15]. In the past decade, there has been a great deal of research done to ﬁnd ISS stabilizing control laws (cf. [5], is ISS (with respect to the actuator erroru) when the [6], [7], [9]). In this note, we study the ISS stabilizability ofn observation errore: [0,∞)→Rin the controller is control afﬁne systems of the form sufﬁciently small(cf. the deﬁnitions below). In this context, the precise value ofe(t)is unknown to the controller, but x˙ =f(x) +G(x)u(1) information about upper bounds on the magnitude ofe(t) n wherefandGare locally Lipschitz vector ﬁelds onRbe used to design the feedback. The following theorem, can m f(0) = 0, and the controluis valued inR(but see alsois shown in [8]: §V for extensions for fully nonlinear systems). We assumeTheorem 1:If (1) is GAC, then there exists a feedbackK throughout this note that the system (1) is globally asympfor which (3) is ISS for Euler solutions. totically controllable (GAC), and we construct a feedback Theorem 1 characterizes the uniform limits of sampling n m K:R→Rfor whichsolutions of (3) (cf.§II for the deﬁnitions of Euler and sampling solutions). From a computational standpoint, it is x˙ =f(x) +G(x)K(x) +G(x)u(2) also desirable to know how frequently to sample in order to achieve ISS for sampling solutions. This information is is ISS (cf.§II for the relevant deﬁnitions). As pointed out in [2], [17], acontinuousprovided by the following theorem:(timeinvariant) stabilizing feedback Theorem 2:If (1) is GAC, then there exists a feedbackK Kfails to exist in general. This fact forces us to considerdiscontinuouswhich (3) is ISS for sampling solutions.feedbacks for K, so our solutions will be interpreted in the more generalThis note is organized as follows. In§II, we review sense of sampling and Euler solutions for dynamics whichthe relevant background on CLF’s, GAC and ISS systems, are discontinuous in the state (cf. [3], [17]). By an Eulernonsmooth analysis, and discontinuous feedbacks. In§III, solution, we mean a uniform limit of sampling solutions,we sketch the proofs of the above theorems (cf. [8] for their taken as the frequency of sampling becomes inﬁnite (cf.detailed proofs). This is followed in§IV by a comparison §of our feedback construction with the known feedback conII for precise deﬁnitions). Our construction extends [15], o o [16], which show how to makeCstabilizable systems ISSstructions forCstabilizable systems, and an application of to actuator errors. In particular, our feedback applies to theour results to the nonholonomic integrator. We close in§V nonholonomic integrator (cf. [2], [12], and§IV below) andby announcing an extension of our results for fully nonlinear systems. 1 This author was supported in part by Louisiana Board of Regents Support Fund Contract LEQSF(20022004)ENHTR26 as part of the project “Interdisciplinary Education, Outreach, and Research in ControlAND MAIN LEMMASII. DEFINITIONS Theory at LSU”. This author last revised this paper on August 27, 2003. We letK∞denote the set of all continuous functions 2 This author was supported in part by US Air Force Grant F4962001 10063, and by NSF Grant CCR0206789.ρ: [0,∞)→[0,∞)for which (i)ρ(0) = 0and (ii)ρis