Tracking Control of Complementarity Lagrangian Systems

In this paper we study the tracking control of Lagrangian systems subject to frictionless unilateral constraints. The stability analysis incorporates the hybrid and nonsmooth dynamical feature of the overall system. The difference between tracking control for unconstrained systems and unilaterally constrained ones, is explained in terms of closed-loop desired trajectories and control signals. This work provides details on the conditions of existence of controllers which guarantee stability. It is shown that the design of a suitable transition phase desired trajectory, is a crucial step. Some simulation results provide information on the convergence of such controller. Finally the extension towards the case of multiple impacts, is considered

Passivity-based tracking control of multiconstraint complementarity Lagrangian systems

In this study one considers the tracking control problem of a class of nonsmooth fully actuated Lagrangian systems subject to frictionless unilateral constraints. A passivity-based switching controller that guarantees some stability properties of the closed-loop system is designed. A particular attention is paid to transition (impacting) and detachment phases of motion. This paper extends previous works on the topic as it considers multiconstraint n-degree-of-freedom systems

A Parametric Multi-Convex Splitting Technique with Application to Real-Time NMPC

A novel splitting scheme to solve parametric multiconvex programs is presented. It consists of a fixed number of proximal alternating minimisations and a dual update per time step, which makes it attractive in a real-time Nonlinear Model Predictive Control (NMPC) framework and for distributed computing environments. Assuming that the parametric program is semi-algebraic and that its KKT points are strongly regular, a contraction estimate is derived and it is proven that the sub-optimality error remains stable if two key parameters are tuned properly. Efficacy of the method is demonstrated by solving a bilinear NMPC problem to control a DC motor.Comment: To appear in Proceedings of the 53rd IEEE Conference on Decision and Control 201

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

Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation

In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem. Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. All extensions preserve the computational efficiency of the classic algorithms, and most of the extensions are illustrated with numerical examples, which are part of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure

A Pseudospectral Approach to High Index DAE Optimal Control Problems

Historically, solving optimal control problems with high index differential algebraic equations (DAEs) has been considered extremely hard. Computational experience with Runge-Kutta (RK) methods confirms the difficulties. High index DAE problems occur quite naturally in many practical engineering applications. Over the last two decades, a vast number of real-world problems have been solved routinely using pseudospectral (PS) optimal control techniques. In view of this, we solve a "provably hard," index-three problem using the PS method implemented in DIDO, a state-of-the-art MATLAB optimal control toolbox. In contrast to RK-type solution techniques, no laborious index-reduction process was used to generate the PS solution. The PS solution is independently verified and validated using standard industry practices. It turns out that proper PS methods can indeed be used to "directly" solve high index DAE optimal control problems. In view of this, it is proposed that a new theory of difficulty for DAEs be put forth.Comment: 14 pages, 9 figure