Implicit Euler numerical simulations of sliding mode systems

In this report it is shown that the implicit Euler time-discretization of some classes of switching systems with sliding modes, yields a very good stabilization of the trajectory and of its derivative on the sliding surface. Therefore the spurious oscillations which are pointed out elsewhere when an explicit method is used, are avoided. Moreover the method (an {\em event-capturing}, or {\em time-stepping} algorithm) allows for accumulation of events (Zeno phenomena) and for multiple switching surfaces (i.e., a sliding surface of codimension $\geq 2$). The details of the implementation are given, and numerical examples illustrate the developments. This method may be an alternative method for chattering suppression, keeping the intrinsic discontinuous nature of the dynamics on the sliding surfaces. Links with discrete-time sliding mode controllers are studied

Dissipative Systems Analysis and Control, Theory and Applications: Addendum/Erratum

This is an up-dated addendum/erratum to the second edition of the book Dissipative Systems Analysis and Control, Theory and Applications, Springer-Verlag London, 2nd Edition, 2007

Dissipative Systems Analysis and Control, Theory and Applications: Addendum/Erratum

This is an up-dated addendum/erratum to the second edition of the book Dissipative Systems Analysis and Control, Theory and Applications, Springer-Verlag London, 2nd Edition, 2007

Analysis of explicit and implicit discrete-time equivalent-control based sliding mode controllers

Different time-discretization methods for equivalent-control based sliding mode control (ECB-SMC) are presented. A new discrete-time sliding mode control scheme is proposed for linear time-invariant (LTI) systems. It is error-free in the discretization of the equivalent part of the control input. Results from simulations using the various discretized SMC schemes are shown, with and without perturbations. They illustrate the different behaviours that can be observed. Stability results for the proposed scheme are derived

On the control of complementary-slackness juggling mechanical systems

International audienceThis paper studies the feedback control of a class of complementary-slackness hybrid mechanical systems. Roughly, the systems we study are composed of an uncontrollable part (the "object") and a controlled one (the "robot"), linked by a unilateral constraint and an impact rule. A systematic and general control design method for this class of systems is proposed. The approach is a nontrivial extension of the one degree-of-freedom (DOF) juggler control design. In addition to the robot control, it is also useful to study some intermediate controllability properties of the object's impact Poincaré mapping, which generally takes the form of a non-linear discrete-time system. The force input mainly consists of a family of dead-beat feedback control laws, introduced via a recur-sive procedure, and exploiting the underlying discrete-time structure of the system. The main goal of this paper is to highlight the role of various physical and control properties characteristic of the system on its stabilizability properties and to propose solutions in certain cases

On the Control of a One Degree-of-Freedom Juggling Robot

International audienceThis paper is devoted to the feedback control of a one degree-of-freedom (dof) juggling robot, considered as a subclass of mechanical systems subject to a unilateral constraint. The proposed approach takes into account the whole dynamics of the system, and focuses on the design of a force input. It consists of a family of hybrid feedback control laws, that allow to stabilize the object around some desired (periodic or not) trajectory. The closed-loop behavior in presence of various disturbances is studied. Despite good robustness properties, the importance of good knowledge of the system parameters, like the restitution coefficient, is highlighted. Besides its theoretical interest concerning the control of a class of mechanical systems subject to unilateral constraints, this study has potential applications in non-prehensile manipulation, extending pushing robotic tasks to striking-and-pushing tasks

Comments on "Control of a Planar Underactuated Biped on a Complete Walking Cycle"

International audienceThe above paper [1] possesses several approximations and flaws, which we try to explain. Roughly, the topic concerns the problem of trajectory tracking for a class of mechanical Lagrangian systems subject to unilateral constraints on the generalized position (q) 0, (q) 2 IR m. Such multibody mechanical systems also involve a com-plementarity relation between the constraint and a Lagrange multiplier 0 (q) ? 0 (1), and generalized velocity jumps (impacts). The complementarity relations and the velocity jump law, form a specific contact model. A contact model is necessary for the chosen model to be meaningful from a mechanical point of view. When dealing with systems of rigid bodies, the complementarity conditions are the simplest way to deal with the contact dynamics: they state that adhesion or magnetic forces are excluded from the model. Such nonsmooth mechanical systems form a special class of complementarity systems, but other formalisms exist [7]. It is worth noting that the complementarity conditions are not included in the model presented in [1], which is therefore incomplete. Specifically, the authors deal with a particular biped robots model that fits within a class of impulsive ODEs, or measure differential equations. We will come back on this later in this note. The tracking problem is examined when the system undergoes an infinity of cycles, each cycle being composed of three phases of motion: single-support phase, double-support phase, and the impact when the feet hit the ground. Apart from possible underactuation, the problem is quite similar to what is tackled in [2]–[5], that concerns fully actuated Lagrangian systems undergoing cycles which consist of free motion phases, constrained motion phases, and transition phases with impacts. The effects of the impacts and of the complementarity relations do not change from one problem to the other one. This is why it is worth understanding the simplest case before tackling more sophisticated control problems (underactuated systems, flexible joint manipulators, to cite a few). It is worth noting that the infinity of cycles (and consequently Manuscript 1 The symbol ? means that (q) and have to be orthogonal one to each other. Since they are both non-negative, this is equivalent to the componentwise relation 0 (q), 0 , (q) = 0 for all 1 i m

Comparison between explicit and implicit discrete-time implementations of sliding-mode controllers

International audienceDifferent time-discretization methods for sliding mode control (SMC) are presented. A new discrete-time sliding mode control scheme is proposed for linear time-invariant (LTI) systems. It is error-free in the discretization of the equivalent part of the control input. Results from simulations using the various discretized SMC schemes are shown, with and without perturbations. They illustrate the different behaviours that can be observed

Addendum-Erratum to Nonsmooth Modeling and Simulation for Switched Circuits

This second version of the Addendum/Erratum corrects some typos present in the monograph