Trajectory Tracking Control of Multiconstraint Complementarity Lagrangian Systems

Abstract

International audienceIn this study, one considers the tracking control problem of a class of nonsmooth fully actuated Lagrangian systems subject to frictionless unilateral constraints. The task under consideration contains both free-motion and constraint-motion phases. A switching controller that guarantees an approximate tracking is designed. Particular attention is paid to transition (impacting and detachment) between different phases of motion. The exogenous signals that assure the stabilization on (take-off from) some constaints are explicitly defined. This paper extends previous works on the topic as it considers more than one constraint for n -degree-of-freedom systems. Numerical examples illustrate the main results

Follow the Bouncing Ball: Global Results on Tracking and State Estimation With Impacts

“…Some of them address the problem via the larger class of complementarity Lagrangian systems. These systems are a specific class of hybrid systems where the state is subject to a jump or re-initialization rule whenever a unilateral constraint is reached (see [13] for a survey and [19] which also improves the results in [1], [4]). Several additional recent techniques addressing tracking control with impacts both from a theoretical and an experimental viewpoint are provided in the works [21], [22], [15], [16], [28] and references therein.…”

Lyapunov-Based Sufficient Conditions for Exponential Stability in Hybrid Systems

“…Pick h large enough so that 1 > γ := exp − (19) and the fact that ϕ is continuous and bounded away from zero on [0, N ], which is where the variable τ is constrained to evolve for the hybrid system (12).…”

“…For (4), considering (6), (13), (19), for all ξ ∈ Since γ < 1, there exists λ > 0 satisfying (4)-(5).…”

“…This technical note is motivated by analysis problems that arise in tracking and estimation problems for mechanical systems with impacts (see, e.g., [19], [17] and references therein). For such problems, in [7], [8], [6], it has been shown how to construct a radially unbounded Lyapunov function that does not increase during flows, does not increase at impacts, and whose zero level set corresponds to zero position tracking error.…”

Global Tracking and State Estimation with Nonsmooth Impacts for a Mass Confined to an N-Dimensional Half-Space

“…Some of them address the problem via the larger class of complementarity Lagrangian systems. These systems are a specific class of hybrid systems where the state is subject to a jump or re-initialization rule whenever a unilateral constraint is reached (see Heemels and Brogliato (2003) for a survey and Morarescu and Brogliato (2010) for a recent work which gives an updated overview of the results in this field). The parallel problem of tracking trajectories while restricting the control action at the impact times is addressed in Sanfelice et al (2007b) and references therein.…”