Asymptotic Stability of Uncertain Lagrangian Systems with Prescribed Transient Response

Abstract

This paper considers the asymptotic tracking problem for 2nd-order nonlinear Lagrangian systems subject to predefined constraints for the system response, such as maximum overshoot or minimum convergence rate. In particular, by employing discontinuous adaptive control protocols and nonsmooth analysis, we extend previous results on funnel control to guarantee at the same time asymptotic trajectory tracking from all the initial conditions that are compliant with the given funnel. The considered system contains parametric and structural uncertainties, with no boundedness or approximation/parametric factorization assumptions. The response of the closed loop system is solely determined by the predefined funnel and is independent from the control gain selection. Finally, simulation results verify the theoretical findings.

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

“…Regarding the latter, note that, if the initial value of the funnel is a design parameter, it can be always set larger than the initial value of the error to be confined in the funnel, rendering thus the results global. This work extends our preliminary results [28], which considered the special instance of time-invariant 2nd-order Lagrangian systems, to systems with more general dynamic terms F , G and relaxation of the positive definitiveness of G.…”

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.…”

Exact output tracking in prescribed finite time via funnel control

“…This controller achieves asymptotic tracking for a class of nonlinear SISO systems of arbitrary relative degree, where the reference signal is generated by a reference model. To the price of a discontinuous control, asymptotic tracking for nonlinear MIMO systems is achieved in [23,24]. In [17] a funnel control scheme is proposed, which achieves asymptotic tracking for a class of nonlinear relative degree one MIMO systems.…”

“…Note that the estimation in (20) is too rough to show (23). Recalling the definition (6c) of γ k (•) = α k (•)e k (•) and exemplary its derivative (14) we see that by…”