Sensitivity analysis of hybrid systems with state jumps with application to trajectory tracking

Abstract

Abstract-This paper addresses the sensitivity analysis for hybrid systems with discontinuous (jumping) state trajectories. We consider state-triggered jumps in the state evolution, potentially accompanied by mode switching in the control vector field as well. For a given trajectory with state jumps, we show how to construct an approximation of a nearby perturbed trajectory corresponding to a small variation of the initial condition and input. A major complication in the construction of such an approximation is that, in general, the jump times corresponding to a nearby perturbed trajectory are not equal to those of the nominal one. The main contribution of this work is the development of a notion of error to clarify in which sense the approximate trajectory is, at each instant of time, a firstorder approximation of the perturbed trajectory. This notion of error naturally finds application in the (local) tracking problem of a time-varying reference trajectory of a hybrid system. To illustrate the possible use of this new error definition in the context of trajectory tracking, we outline how the standard linear trajectory tracking control for nonlinear systems -based on linear quadratic regulator (LQR) theory to compute the optimal feedback gain-could be generalized for hybrid systems.

Trajectory tracking of mechanical systems with unilateral constraints: Experimental results of a recently introduced hybrid PD feedback controller

“…The following section details the hybrid PD plus acceleration feedforward controller presented in [14] for the hybrid system with continuous dynamics (7) with guard map (5) and reset map (6).…”

“…Adopting the notation in [14], we denote with τ ∈ [t 0 , t 1 ] a jump time, f − and f + the corresponding values of the vector field before and after the state reset. In terms of the state x = (q,q), the unilateral constraint (5) is…”

“…In [14], while addressing the same tracking problem studied in [11] and [12], the authors suggest the use of a new notion of tracking error, moving away from the concept of mirror This is the final version of the accepted paper submitted for inclusion in the Proceedings of the IEEE 54th Conference on Decision and Control, Osaka, Japan, Dec. 2015. Gian Paolo Incremona and Antonella Ferrara are with the Dipartimento di Ingegneria Industriale e dell'Informazione, University of Pavia, via trajectory.…”

“…The new notion of error requires the (local) extension of the reference trajectory in each neighborhood of the nominal jump times. These extensions are referred to as extented anteand post-event trajectories [14]. Further details about this control strategy and the synthesis of optimal gains can be found in [15].…”

“…Starting from the hybrid control law introduced in [14], this paper presents the first simulation and experimental results obtained by employing this hybrid controller for tracking a reference trajectory of a 1-DOF robot arm bouncing against an aluminum rod. Moreover, estimates of the coefficient of restitution obtained via experimental tests are also reported.…”