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Publications about 'nonlinear control'
Articles in journal or book chapters
  1. E.D. Sontag. Stability and feedback stabilization. In Robert Meyers, editor, Mathematics of Complexity and Dynamical Systems, pages 1639-1652. Springer-Verlag, Berlin, 2011. [PDF] Keyword(s): stability, nonlinear control, feedback stabilization.
    Abstract:
    The problem of stabilization of equilibria is one of the central issues in control. In addition to its intrinsic interest, it represents a first step towards the solution of more complicated problems, such as the stabilization of periodic orbits or general invariant sets, or the attainment of other control objectives, such as tracking, disturbance rejection, or output feedback, all of which may be interpreted as requiring the stabilization of some quantity (typically, some sort of ``error'' signal). A very special case, when there are no inputs, is that of stability. This short and informal article provides an introduction to the subject.


  2. E.D. Sontag. Stability and Feedback Stabilization. In Robert Meyers, editor, Encyclopedia of Complexity and Systems Science. Springer-Verlag, Berlin, 2007. Keyword(s): stability, nonlinear control, feedback stabilization.
    Abstract:
    The problem of stabilization of equilibria is one of the central issues in control. In addition to its intrinsic interest, it represents a first step towards the solution of more complicated problems, such as the stabilization of periodic orbits or general invariant sets, or the attainment of other control objectives, such as tracking, disturbance rejection, or output feedback, all of which may be interpreted as requiring the stabilization of some quantity (typically, some sort of ``error'' signal). A very special case, when there are no inputs, is that of stability. This short and informal article provides an introduction to the subject.


  3. M. Malisoff, M. Krichman, and E.D. Sontag. Global stabilization for systems evolving on manifolds. Journal of Dynamical and Control Systems, 12:161-184, 2006. [PDF] Keyword(s): nonlinear stability, nonlinear control, feedback stabilization.
    Abstract:
    This paper shows that any globally asymptotically controllable system on any smooth manifold can be globally stabilized by a state feedback. Since discontinuous feedbacks are allowed, solutions are understood in the ``sample and hold'' sense introduced by Clarke-Ledyaev-Sontag-Subbotin (CLSS). This work generalizes the CLSS Theorem, which is the special case of our result for systems on Euclidean space. We apply our result to the input-to-state stabilization of systems on manifolds relative to actuator errors, under small observation noise.


  4. 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.
    Abstract:
    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.


  5. 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:http://dx.doi.org/10.1137/S0363012903422333] Keyword(s): input to state stability, control-Lyapunov functions, nonlinear control, feedback stabilization.
    Abstract:
    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.


  6. D. Liberzon, A. S. Morse, and E.D. Sontag. Output-input stability and minimum-phase nonlinear systems. IEEE Trans. Automat. Control, 47(3):422-436, 2002. [PDF] Keyword(s): input to state stability, nonlinear control, minimum phase, adaptive control.
    Abstract:
    This paper introduces and studies a new definition of the minimum-phase property for general smooth nonlinear control systems. The definition does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics. In the spirit of the ``input-to-state stability'' philosophy, it requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The class of minimum-phase systems thus defined includes all affine systems in global normal form whose internal dynamics are input-to-state stable and also all left-invertible linear systems whose transmission zeros have negative real parts. As an application, we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control.


  7. D. Liberzon, E.D. Sontag, and Y. Wang. Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation. Systems Control Lett., 46(2):111-127, 2002. Note: Errata here: http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/iiss-clf-errata.pdf. [PDF] Keyword(s): input to state stability, nonlinear control, feedback stabilization.
    Abstract:
    We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closed-loop system integral-input-to-state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The same method applies to the problem of input-to-state stabilization. Converse results and techniques for generating iISS-CLFs are also discussed.


  8. E.D. Sontag. The ISS philosophy as a unifying framework for stability-like behavior. In Nonlinear control in the year 2000, Vol. 2 (Paris), volume 259 of Lecture Notes in Control and Inform. Sci., pages 443-467. Springer, London, 2001. [PDF] Keyword(s): input to state stability.
    Abstract:
    (This is an expository paper prepared for a plenary talk given at the Second Nonlinear Control Network Workshop, Paris, June 9, 2000.) The input to state stability (ISS) paradigm is motivated as a generalization of classical linear systems concepts under coordinate changes. A summary is provided of the main theoretical results concerning ISS and related notions of input/output stability and detectability. A bibliography is also included, listing extensions, applications, and other current work.


  9. 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.
    Abstract:
    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.


  10. 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.
    Abstract:
    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.


  11. Y.S. Ledyaev and E.D. Sontag. A Lyapunov characterization of robust stabilization. Nonlinear Anal., 37(7, Ser. A: Theory Methods):813-840, 1999. [PDF] Keyword(s): nonlinear control, feedback stabilization.
    Abstract:
    One of the fundamental facts in control theory (Artstein's theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in this paper establishes that the existence of smooth Lyapunov functions implies the existence of (in general, discontinuous) feedback stabilizers which are insensitive to small errors in state measurements. Conversely, it is shown that the existence of such stabilizers in turn implies the existence of smooth control Lyapunov functions. Moreover, it is established that, for general nonlinear control systems under persistently acting disturbances, the existence of smooth Lyapunov functions is equivalent to the existence of (possibly) discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances.


  12. E.D. Sontag. Clocks and insensitivity to small measurement errors. ESAIM Control Optim. Calc. Var., 4:537-557, 1999. [PDF] Keyword(s): nonlinear control, feedback stabilization, hybrid systems, discontinuous feedback, measurement noise.
    Abstract:
    This paper provides a precise result which shows that insensitivity to small measurement errors in closed-loop stabilization can be attained provided that the feedback controller ignores observations during small time intervals.


  13. F. Albertini and E.D. Sontag. Further results on controllability properties of discrete-time nonlinear systems. Dynam. Control, 4(3):235-253, 1994. [PDF] [doi:http://dx.doi.org/10.1007/BF01985073] Keyword(s): discrete-time, nonlinear control.
    Abstract:
    Controllability questions for discrete-time nonlinear systems are addressed in this paper. In particular, we continue the search for conditions under which the group-like notion of transitivity implies the stronger and semigroup-like property of forward accessibility. We show that this implication holds, pointwise, for states which have a weak Poisson stability property, and globally, if there exists a global "attractor" for the system.


  14. E.D. Sontag. Feedback stabilization using two-hidden-layer nets. IEEE Trans. Neural Networks, 3:981-990, 1992. [PDF] Keyword(s): neural networks, feedback stabilization.
    Abstract:
    This paper compares the representational capabilities of one hidden layer and two hidden layer nets consisting of feedforward interconnections of linear threshold units. It is remarked that for certain problems two hidden layers are required, contrary to what might be in principle expected from the known approximation theorems. The differences are not based on numerical accuracy or number of units needed, nor on capabilities for feature extraction, but rather on a much more basic classification into "direct" and "inverse" problems. The former correspond to the approximation of continuous functions, while the latter are concerned with approximating one-sided inverses of continuous functions - and are often encountered in the context of inverse kinematics determination or in control questions. A general result is given showing that nonlinear control systems can be stabilized using two hidden layers, but not in general using just one.


  15. Y. Wang and E.D. Sontag. Generating series and nonlinear systems: analytic aspects, local realizability, and i/o representations. Forum Math., 4(3):299-322, 1992. [PDF] Keyword(s): identifiability, observability, realization theory, input/output system representations.
    Abstract:
    This paper studies fundamental analytic properties of generating series for nonlinear control systems, and of the operators they define. It then applies the results obtained to the extension of facts, which relate realizability and algebraic input/output equations, to local realizability and analytic equations.


  16. E.D. Sontag. Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control, 34(4):435-443, 1989. [PDF] Keyword(s): input to state stability, input to state stability.
    Abstract:
    This paper shows that coprime right factorizations exist for the input to state mapping of a continuous time nonlinear system provided that the smooth feedback stabilization problem be solvable for this system. In particular, it follows that feedback linearizable systems admit such factorizations. In order to establish the result a Lyapunov-theoretic definition is proposed for bounded input bounded output stability. The main technical fact proved relates the notion of stabilizability studied in the state space nonlinear control literature to a notion of stability under bounded control perturbations analogous to those studied in operator theoretic approaches to systems; it states that smooth stabilization implies smooth input-to-state stabilization. (Note: This is the original ISS paper, but the ISS results have been much improved in later papers. The material on coprime factorizations is still of interest, but the 89 CDC paper has some improvements and should be read too.)


  17. E.D. Sontag. Finite-dimensional open-loop control generators for nonlinear systems. Internat. J. Control, 47(2):537-556, 1988. [PDF]
    Abstract:
    This paper concerns itself with the existence of open-loop control generators for nonlinear (continuous-time) systems. The main result is that, under relatively mild assumptions on the original system, and for each fixed compact subset of the state space, there always exists one such generator. This is a new system with the property that the controls it produces are sufficiently rich to preserve complete controllability along nonsingular trajectories. General results are also given on the continuity and differentiability of the input to state mapping for various p-norms on controls, as well as a comparison of various nonlinear controllability notions.


  18. E.D. Sontag. Controllability and linearized regulation. IEEE Trans. Automat. Control, 32(10):877-888, 1987. [PDF]
    Abstract:
    A nonlinear controllable plant, under mild technical conditions, admits a precompensator with the following property: along control trajectories joining pairs of states, the composite system (precompensator plus plant) is, up to first order, isomorphic to a parallel connection of integrators.


  19. E.D. Sontag. An algebraic approach to bounded controllability of linear systems. Internat. J. Control, 39(1):181-188, 1984. [PDF] Keyword(s): saturation.
    Abstract:
    In this note we present an algebraic approach to the proof that a linear system with matrices (A,B) is null-controllable using bounded inputs iff it is null-controllable (with unbounded inputs) and all eigenvalues of A have nonpositive real parts (continuous time) or magnitude not greater than one (discrete time). We also give the analogous results for the asymptotic case. Finally, we give an interpretation of these results in the context of local nonlinear controllability.


  20. E.D. Sontag. Nonlinear regulation: the piecewise linear approach. IEEE Trans. Automat. Control, 26(2):346-358, 1981. [PDF] Keyword(s): hybrid systems.
    Abstract:
    Development of an approach to nonlinear control based on mixtures of linear systems and finite automata. File obtained by scanning.


Conference articles
  1. M. Malisoff, L. Rifford, and E.D. Sontag. Remarks on input to state stabilization. In Proc. IEEE Conf. Decision and Control, Maui, Dec. 2003, IEEE Publications, 2003, pages 1053-1058, 2003. [PDF] Keyword(s): nonlinear control, feedback stabilization.


  2. M. Arcak, D. Angeli, and E.D. Sontag. Stabilization of cascades using integral input-to-state stability. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 2001, IEEE Publications, 2001, pages 3814-3819, 2001. Keyword(s): nonlinear control, feedback stabilization, input to state stability.


  3. B.P. Ingalls, D. Angeli, E.D. Sontag, and Y. Wang. Asymptotic characterizations of IOSS. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 2001, IEEE Publications, 2001, pages 881-886, 2001. Keyword(s): nonlinear control, feedback stabilization, input to state stability.


  4. E.D. Sontag. From linear to nonlinear: some complexity comparisons. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pages 2916-2920, 1995. [PDF] Keyword(s): theory of computing and complexity, computational complexity, controllability, observability.
    Abstract:
    This paper deals with the computational complexity, and in some cases undecidability, of several problems in nonlinear control. The objective is to compare the theoretical difficulty of solving such problems to the corresponding problems for linear systems. In particular, the problem of null-controllability for systems with saturations (of a "neural network" type) is mentioned, as well as problems regarding piecewise linear (hybrid) systems. A comparison of accessibility, which can be checked fairly simply by Lie-algebraic methods, and controllability, which is at least NP-hard for bilinear systems, is carried out. Finally, some remarks are given on analog computation in this context.


  5. E.D. Sontag. Spaces of observables in nonlinear control. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Basel, pages 1532-1545, 1995. Birkhäuser. [PDF] Keyword(s): observability, dynamical systems.
    Abstract:
    Invited talk at the 1994 ICM. Paper deals with the notion of observables for nonlinear systems, and their role in realization theory, minimality, and several control and path planning questions.



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