Publications about 'Lie algebras'
Articles in journal or book chapters
  1. A. C. B. de Oliveira, M. Siami, and E. D. Sontag. Regularising numerical extremals along singular arcs: a Lie-theoretic approach. In M.A. Belabbas, editor, Geometry and Topology in Control System Design. American Institute of Mathematical Sciences, 2024. Note: To appear.Keyword(s): optimal control, nonlinear control, Lie algebras, robotics.
    Numerical ``direct'' approaches to time-optimal control often fail to find solutions that are singular in the sense of the Pontryagin Maximum Principle. These approaches behave better when searching for saturated (bang-bang) solutions. In previous work by one of the authors, singular solutions were theoretically shown to exist for the time-optimal problem for two-link manipulators under hard torque constraints. The theoretical results gave explicit formulas, based on Lie theory, for singular segments of trajectories, but the global structure of solutions remains unknown. In this work, we show how to effectively combine these theoretically found formulas with the use of general-purpose optimal control softwares. By using the explicit formula given by theory in the intervals where the numerical solution enters a singular arcs, we not only obtain an algebraic expression for the control in that interval, but we are also able to remove artifacts present in the numerical solution. In this way, the best features of numerical algorithms and theory complement each other and provide a better picture of the global optimal structure. We showcase the technique on a 2 degrees of freedom robotic arm example, and also propose a way of extending the analyzed method to robotic arms with higher degrees of freedom through partial feedback linearization, assuming the desired task can be mostly performed by a few of the degrees of freedom of the robot and imposing some prespecified trajectory on the remaining joints.

  2. M. Chyba, N. E. Leonard, and E.D. Sontag. Singular trajectories in multi-input time-optimal problems: Application to controlled mechanical systems. Journal of Dynamical and Control Systems, 9(1):103-129, 2003. [PDF] [doi:] Keyword(s): optimal control.
    This paper addresses the time-optimal control problem for a class of control systems which includes controlled mechanical systems with possible dissipation terms. The Lie algebras associated with such mechanical systems enjoy certain special properties. These properties are explored and are used in conjunction with the Pontryagin maximum principle to determine the structure of singular extremals and, in particular, time-optimal trajectories. The theory is illustrated with an application to a time-optimal problem for a class of underwater vehicles.

  3. F. Albertini and E.D. Sontag. Discrete-time transitivity and accessibility: analytic systems. SIAM J. Control Optim., 31(6):1599-1622, 1993. [PDF] [doi:] Keyword(s): controllability, discrete-time systems, accessibility, real-analytic functions.
    A basic open question for discrete-time nonlinear systems is that of determining when, in analogy with the classical continuous-time "positive form of Chow's Lemma", accessibility follows from transitivity of a natural group action. This paper studies the problem, and establishes the desired implication for analytic systems in several cases: (i) compact state space, (ii) under a Poisson stability condition, and (iii) in a generic sense. In addition, the paper studies accessibility properties of the "control sets" recently introduced in the context of dynamical systems studies. Finally, various examples and counterexamples are provided relating the various Lie algebras introduced in past work.

  4. B. Jakubczyk and E.D. Sontag. Controllability of nonlinear discrete-time systems: a Lie-algebraic approach. SIAM J. Control Optim., 28(1):1-33, 1990. [PDF] [doi:] Keyword(s): discrete-time.
    This paper presents a geometric study of controllability for discrete-time nonlinear systems. Various accessibility properties are characterized in terms of Lie algebras of vector fields. Some of the results obtained are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena appear.

  5. E.D. Sontag. Remarks on the preservation of various controllability properties under sampling. In Mathematical tools and models for control, systems analysis and signal processing, Vol. 3 (Toulouse/Paris, 1981/1982), Travaux Rech. Coop. Programme 567, pages 623-637. CNRS, Paris, 1983. [PDF] Keyword(s): controllability, sampling, nonlinear systems, real-analytic functions.
    This note studies the preservation of controllability (and other properties) under sampling of a nonlinear system. More detailed results are obtained in the cases of analytic systems and of systems with finite dimensional Lie algebras.



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Last modified: Thu Jun 27 23:19:31 2024
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