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Jacobi curves are far going generalizations of the spaces of “Jacobi fields” along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. Differential geometry of these curves provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. In the present paper we mainly discuss two principal invariants: the generalized Ricci curvature...
Smooth closed-form curves on the Lie group of rigid body motions are constructed via the De Casteljau algorithm. Due to the lack of a bi-invariant metric on SE(3), the resulting curve depends on the choice of the metric tensor. The two most common cases are analyzed.
A design methodology is presented for tracking control of a class of second-order nonholonomic systems. The method consists of three steps. In the first step we transform the system into an extended chained-form system. This extended chained-form system is in cascade form and we apply a linear feedback to the first subsystem. In the second step, the second subsystem is exponentially stabilized by...
The static output feedback stabilization problem for linear and nonlinear (affine) systems is discussed. A novel necessary and sufficient condition for linear systems is proposed. For nonlinear systems a sufficient condition is established and a (partial) converse is also discussed. The nonlinear formulation is used to derive a simple characterization of stabilizing static output feedback control...
The stability of non-linear fractional differential equations is studied. A sufficient stability condition on the non-linearity is given for the input-output stability, thanks to many different reformulations of the system using diffusive representations of dissipative pseudo-differential operators. The problem of asymptotic internal stability is analyzed by a more involved functional analytic method...
Linear control systems on Lie groups were introduced by Markus [3] and also studied by Ayala and Tirao in [1]. For this class of control systems we establish controllability results in the compact case and also in the semi-simple non-compact case. Wealso show that the forward orbit from the identity is not in general a semigroup.
This paper is concerned with stability robustness of nonlinear multivariable systems under input-output feedback linearization. A procedure is presented that allows plant uncertainty to be propagated through the control design, yielding an uncertainty description of the closed-loop in polytopic form. As feedback linearization aims for a linear closed-loop system, plant uncertainty in the nonlinear...
It is well known that external stability of nonlinear input systems can be investigated by means of a suitable extension of the Liapunov functions method. We prove that a complete characterization by means of continuous Liapunov functions is actually possible, provided that the definition of external stability is appropriately strengthened.
In this paper a new algorithm is used to regulate flux and torque in induction machine. The aim is to show the possibility of attenuating the harmonics as early as the phase of the design of the control law while preserving good regulation performances. The control law results from an optimisation algorithm associated to a cost function. By adding some harmonic weighting factor to the regulation term...
The problem of deriving conditions for a stabilising linear compensator in a uncertain nonlinear control system is addressed, for some types of memoryless nonlinearities like the saturation or the dead-zone. The approach is to incorporate to QFT conditions given by the application of harmonic balance and multiplier techniques, providing the designer with a very transparent tool for synthesising stabilising...
We consider nonlinear filtering problems, nonlinear robust control problems and the partial differential equations that characterize their solutions. These include the Zakai equation, and in the robust control case two coupled Dynamic Programming equations. We then characterize equivalence between two such problems when we can compute the solution of one from the solution of the other using change...
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...
Optimal control problems naturally lead, via the Maximum Principle, to implicit Hamiltonian systems. It is shown that symmetries of an optimal control problem lead to symmetries of the corresponding implicit Hamiltonian system. Using the reduction theory described in [3,2] one can reduce the system to a lower dimensional implicit Hamiltonian system. It is shown that for symmetries coming from the...
The present paper is devoted to the study of absolute stability of delay systems with nonlinearities subject to sector conditions, and with uncertain delays. We construct Lyapunov-Krasovskii functionals candidates for various such systems, whose decreasingness along the trajectories is expressed in terms of Linear Matrix Inequalities (LMIs). We then show that feasibility of the latter implies some...
L’objectif de cet article est de fournir le cadre géométrique pour faire une analyse de la singularité de l’application exponentielle le long d’une direction anormale en géométrie sous-Riemannienne. Il utilise les calculs de [9], [12], et conduit dans le cas Martinet à une stratification de la singularité en secteurs Lagrangiens.
This paper is concerned with the development of basic concepts and constructs for a nonequilibrium theory of nonlinear control. Motivated by an example of nonstabilizability of rigid spacecraft about an equilibrium (reference attitude) but stabilizability about a revolute motion, we review recent work on the structure of those compact attractors which are Lyapunov stable. These results are illustrated...
We derive a method for the computation of robust domains of attraction based on a recent generalization of Zubov’s theorem on representing robust domains of attraction for perturbed systems via the viscosity solution of a suitable partial differential equation. While a direct discretization of the equation leads to numerical difficulties due to a singularity at the stable equilibrium, a suitable regularization...
We clarify in which sense Ryan’s generalization of Brockett’s condition to discontinuous stabilizability applies when solutions of the implemented system are intended in Filippov’s sense. Moreover, by means of an example, we see how the interpretation of solutions of systems with discontinuous righthand side may influence a stabilizability result for a system which does not admit a continuous stabilizing...
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