Publications about 'control-Lyapunov functions'
Articles in journal or book chapters
  1. M. Malisoff and E.D. Sontag. Asymptotic controllability and input-to-state stabilization: the effect of actuator errors. In Optimal control, stabilization and nonsmooth analysis, volume 301 of Lecture Notes in Control and Inform. Sci., pages 155-171. Springer, Berlin, 2004. [PDF] Keyword(s): input to state stability, control-Lyapunov functions, nonlinear control, feedback stabilization, ISS.
    We discuss several issues related to the stabilizability of nonlinear systems. First, for continuously stabilizable systems, we review constructions of feedbacks that render the system input-to-state stable with respect to actuator errors. Then, we discuss a recent paper which provides a new feedback design that makes globally asymptotically controllable systems input-to-state stable to actuator errors and small observation noise. We illustrate our constructions using the nonholonomic integrator, and discuss a related feedback design for systems with disturbances.

  2. M. Malisoff, L. Rifford, and E.D. Sontag. Global Asymptotic Controllability Implies Input-to-State Stabilization. SIAM J. Control Optim., 42(6):2221-2238, 2004. [PDF] [doi:] Keyword(s): input to state stability, control-Lyapunov functions, nonlinear control, feedback stabilization.
    The main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable (GAC) control affine systems that render the closed loop systems input to state stable with respect to actuator errors. Extensions for fully nonlinear GAC systems with actuator errors are also discussed. Our controllers have the property that they tolerate small observation noise as well.

  3. M. Malisoff and E.D. Sontag. Universal formulas for feedback stabilization with respect to Minkowski balls. Systems Control Lett., 40(4):247-260, 2000. [PDF] Keyword(s): nonlinear control, feedback stabilization, saturation, control-Lyapunov functions, bounded inputs.
    This note provides explicit algebraic stabilizing formulas for clf's when controls are restricted to certain Minkowski balls in Euclidean space. Feedbacks of this kind are known to exist by a theorem of Artstein, but the proof of Artstein's theorem is nonconstructive. The formulas are obtained from a general feedback stabilization technique and are used to construct approximation solutions to some stabilization problems.

  4. E.D. Sontag. Control-Lyapunov functions. In Open problems in mathematical systems and control theory, Comm. Control Engrg. Ser., pages 211-216. Springer, London, 1999. Keyword(s): control-Lyapunov functions.

  5. E.D. Sontag. Stability and stabilization: discontinuities and the effect of disturbances. In Nonlinear analysis, differential equations and control (Montreal, QC, 1998), volume 528 of NATO Sci. Ser. C Math. Phys. Sci., pages 551-598. Kluwer Acad. Publ., Dordrecht, 1999. [PDF] Keyword(s): feedback stabilization, nonlinear control, input to state stability.
    In this expository paper, we deal with several questions related to stability and stabilization of nonlinear finite-dimensional continuous-time systems. We review the basic problem of feedback stabilization, placing an emphasis upon relatively new areas of research which concern stability with respect to "noise" (such as errors introduced by actuators or sensors). The table of contents is as follows: Review of Stability and Asymptotic Controllability, The Problem of Stabilization, Obstructions to Continuous Stabilization, Control-Lyapunov Functions and Artstein's Theorem, Discontinuous Feedback, Nonsmooth CLF's, Insensitivity to Small Measurement and Actuator Errors, Effect of Large Disturbances: Input-to-State Stability, Comments on Notions Related to ISS.

  6. F. Albertini and E.D. Sontag. Continuous control-Lyapunov functions for asymptotically controllable time-varying systems. Internat. J. Control, 72(18):1630-1641, 1999. [PDF] Keyword(s): control-Lyapunov functions.
    This paper shows that, for time varying systems, global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control-Lyapunov function with respect to the set.

  7. E.D. Sontag and H.J. Sussmann. General classes of control-Lyapunov functions. In Stability theory (Ascona, 1995), volume 121 of Internat. Ser. Numer. Math., pages 87-96. Birkhäuser, Basel, 1996. [PDF] Keyword(s): control-Lyapunov functions.
    Shorter and more expository version of "Nonsmooth control-Lyapunov functions"

  8. Y. Lin and E.D. Sontag. Control-Lyapunov universal formulas for restricted inputs. Control Theory Adv. Tech., 10(4, part 5):1981-2004, 1995. [PDF] Keyword(s): control-Lyapunov functions, saturation, bounded inputs.
    We deal with the question of obtaining explicit feedback control laws that stabilize a nonlinear system, under the assumption that a "control Lyapunov function" is known. In previous work, the case of unbounded controls was considered. Here we obtain results for bounded and/or positive controls. We also provide some simple preliminary remarks regarding a set stability version of the problem and a version for systems subject to disturbances.

  9. Y. Lin and E.D. Sontag. A universal formula for stabilization with bounded controls. Systems Control Lett., 16(6):393-397, 1991. [PDF] [doi:] Keyword(s): stabilization, nonlinear systems, saturation, bounded inputs, control-Lyapunov functions, real-analytic functions.
    We provide a formula for a stabilizing feedback law using a bounded control, under the assumption that an appropriate control-Lyapunov function is known. Such a feedback, smooth away from the origin and continuous everywhere, is known to exist via Artstein's Theorem. As in the unbounded-control case treated in a previous note, we provide an explicit and ``universal'' formula given by an algebraic function of Lie derivatives. In particular, we extend to the bounded case the result that the feedback can be chosen analytic if the Lyapunov function and the vector fields defining the system are analytic.

  10. E.D. Sontag. A ``universal'' construction of Artstein's theorem on nonlinear stabilization. Systems Control Lett., 13(2):117-123, 1989. [PDF] Keyword(s): control-Lyapunov functions, stabilization, real-analytic functions.
    This note presents an explicit proof of the theorem - due to Artstein - which states that the existence of a smooth control-Lyapunov function implies smooth stabilizability. Moreover, the result is extended to the real-analytic and rational cases as well. The proof uses a "universal" formula given by an algebraic function of Lie derivatives; this formula originates in the solution of a simple Riccati equation.

  11. E.D. Sontag. A Lyapunov-like characterization of asymptotic controllability. SIAM J. Control Optim., 21(3):462-471, 1983. [PDF] Keyword(s): control-Lyapunov functions.
    It is shown that a control system in Rn is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls.

  12. E.D. Sontag. A characterization of asymptotic controllability. In A. Bednarek and L. Cesari, editors, Dynamical Systems II, pages 645-648. Academic Press, NY, 1982. [PDF] Keyword(s): control-Lyapunov functions.
    This paper was a conference version of the SIAM paper that introduced the idea of control-Lyapunov functions for arbitrary nonlinear systems. (The journal paper was submitted in 1981 but only published in 1983.)

Conference articles
  1. D. Liberzon, E.D. Sontag, and Y. Wang. On integral-input-to-state stabilization. In Proc. American Control Conf., San Diego, June 1999, pages 1598-1602, 1999. [PDF] Keyword(s): input to state stability, integral input to state stability, iISS, ISS, control-Lyapunov functions.
    This paper continues the investigation of the recently introduced integral version of input-to-state stability (iISS). We study the problem of designing control laws that achieve iISS disturbance attenuation. The main contribution is an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The results are compared and contrasted with the ones available for the ISS case.

  2. E.D. Sontag. Recent results on discontinuous stabilization and control-Lyapunov functions. In Proc. Workshop on Control of Nonlinear and Uncertain Systems, London, Feb. 1998, 1998. Keyword(s): control-Lyapunov functions.

  3. F. Albertini and E.D. Sontag. Control-Lyapunov functions for time-varying set stabilization. In Proc. European Control Conf., Brussels, July 1997, 1997. Note: (Paper WE-E A5, CD-ROM file ECC515.pdf, 6 pages). Keyword(s): control-Lyapunov functions.

  4. F.H. Clarke, Y.S. Ledyaev, E.D. Sontag, and A.I. Subbotin. Asymptotic controllability and feedback stabilization. In Proc. Conf. on Information Sciences and Systems (CISS 96)Princeton, NJ, pages 1232-1237, 1996. Keyword(s): control-Lyapunov functions, feedback stabilization.

  5. E.D. Sontag and H.J. Sussmann. Nonsmooth control-Lyapunov functions. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pages 2799-2805, 1995. [PDF] Keyword(s): control-Lyapunov functions.
    It is shown that the existence of a continuous control-Lyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finite-dimensional control systems. The CLF condition is expressed in terms of a concept of generalized derivative (upper contingent derivative). This result generalizes to the non-smooth case the theorem of Artstein relating closed-loop feedback stabilization to smooth CLF's. It relies on viability theory as well as optimal control techniques. A "non-strict" version of the results, analogous to the LaSalle Invariance Principle, is also provided.

  6. Y. Lin and E.D. Sontag. On control-Lyapunov functions under input constraints. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 1994, IEEE Publications, 1994, pages 640-645, 1994. Keyword(s): control-Lyapunov functions.



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