Smooth, time‐invariant regulation of nonholonomic systems via energy pumping‐and‐damping

Abstract

In this article we propose an energy pumping-and-damping technique to regulate nonholonomic systems described by kinematic models. The controller design follows the widely popular interconnection and damping assignment passivity-based methodology, with the free matrices partially structured. Two asymptotic regulation objectives are considered: drive to zero the state or drive the systems total energy to a desired constant value. In both cases, the control laws are smooth, time-invariant, state-feedbacks. For the nonholonomic integrator we give an almost global solution for both problems, with the objectives ensured for all system initial conditions starting outside a set that has zero Lebesgue measure and is nowhere dense. For the general case of higher order nonholonomic systems in chained form, a local convergence result is given. Simulation results comparing the performance of the proposed controller with other existing designs are also provided.

Bounded inputs total energy shaping for a class of underactuated mechanical systems

“…Furthermore, we explain the applicability of the proposed method for fully actuated systems. In (15), except the upper bound of the gravity (c V ), all the other parameters are based on the solution of the PDEs in ( 6) and the free design gains of the controller, K v and k i s defined in (9). Hence, derivation of a suitable solution of the PDEs and appropriate selection of the gains gives the possibility to design the controller considering the limitation of the control input.…”

“…Notice that the reason of difference between the actual and the computed upper bound is that in (15) we consider the worst-case scenario in obtaining the upper bounds. Note that in this case, it is possible to confine ∇ q V d using (9) as by considering V dn = g[1 − cos(q 2 )] and…”

“…The essence of passivity‐based control is energy shaping, 7 which provides a natural procedure to shape the potential energy in the closed‐loop Euler–Lagrange (EL) systems 5,8 . Energy‐shaping control treats the plant and its controller as energy‐transformation devices 9 . In the trajectory tracking of the mechanical systems, the energy‐shaping control based on the classical procedure of PBC does not provide an EL representation of the closed‐loop system.…”

“…5,8 Energy-shaping control treats the plant and its controller as energy-transformation devices. 9 In the trajectory tracking of the mechanical systems, the energy-shaping control based on the classical procedure of PBC does not provide an EL representation of the closed-loop system. In other words, the storage function does not have the expression of total energy.…”

Comments on recent claims about trajectories of control systems valid for particular initial conditions

“…To overcome this obstacle in [25], a smooth, time‐invariant controller based on the ideas of energy pumping‐and‐damping was proposed. As shown in the paper, the regulation objectives are ensured for all system ICs starting outside a set, which is the image of a smooth function [25, Equation (16)]. From the classical Sard's theorem, we know that this set has zero Lebesgue measure.…”