Asymptotic Tracking of Second-Order Nonsmooth Feedback Stabilizable Unknown Systems With Prescribed Transient Response

Abstract

This paper considers the asymptotic tracking control problem for a class of nonlinear systems subject to predefined constraints for the system response, such as maximum overshoot or minimum convergence rate. In particular, by employing discontinuous control protocols and nonsmooth analysis, we extend previous results on funnel control to guarantee at the same time asymptotic trajectory tracking. We consider 2ndorder systems that are affine in the control input and contain completely unknown nonlinear and nonsmooth vector fields, with no boundedness or approximation/parametric factorization assumptions. Simulation results verify the theoretical findings.

Funnel control of nonlinear systems

“…Recently (and unaware of the latter results) it was observed in [37] that asymptotic funnel control is possible for a class of nonlinear single-input single-output systems, albeit more restrictive than the class N m,r of the present paper. Note also that asymptotic tracking via funnel control for systems with relative degree two has been shown by [59,60]. However, the radius of the funnel in these works is bounded away from zero and the property of exact asymptotic tracking is achieved at the expense of a discontinuous control scheme.…”

KDF: Kinodynamic Motion Planning via Geometric Sampling-based Algorithms and Funnel Control

“…Assumption 1 intuitively states that the terms f i (•), g i (•) are sufficiently smooth in the state qi and bounded in time t. The smoothness in qi is satisfied by standard terms that appear in the dynamics of robotic systems (inertia, Coriolis, gravity); friction terms might pose an exception, since they are usually modeled by discontinuous functions of the state [43]. Although smooth friction approximations can be employed [44], the proposed control design can be adapted to account for discontinuous dynamics (as, e.g., in [45]), we consider smooth terms for ease of exposition. Moreover, the incorporation of time dependence in f i (•), g i (•) reflects a time-varying and bounded external disturbance (e.g., wind or adversarial perturbations).…”

Non-Parametric Neuro-Adaptive Control Subject to Task Specifications

“…The control design is inspired by adaptive control methodologies [1], where the time-varying gains d(t), B(t), used to estimate the unknown bounds d, B of (4), adapt to the unknown dynamics in order to ensure closed-loop stability. The discontinuity imposed by ŝ was also employed in the recent work [42], which, however, uses a reciprocal control term, possibly producing unnecessarily large control inputs, as explained in Section 1. Note that (5) does not use any information on the system dynamics f (•) or the constants B, d. The tracking of y d (t) is guaranteed by the following theorem, whose proof is given in the supplementary material file.…”