Asymptotic Tracking via Funnel Control

Abstract

Funnel control is a powerful and simple method to solve the output tracking problem without the need of a good system model, without identification and without knowlegde how the reference signal is produced, but transient behavior as well as arbitrary good accuracy can be guaranteed. Until recently, it was believed that the price to pay for these very nice properties is that only practical tracking and not asymptotic tracking can be achieved. Surprisingly, this is not true! We will prove that funnel control -without any further assumptions -can achieve asymptotic tracking.

Funnel control of nonlinear systems

“…Exact asymptotic tracking with unbounded ϕ was achieved in [50] for a class of nonlinear relative degree one systems: in [27] a predecessor for linear relative degree one systems was developed utilizing the internal model principle. 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].…”

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

“…These functions can encode maximum overshoot or convergence rate properties. Note that, compared to the majority of the related works on funnel control (e.g., [13], [14], [24], [31]), we do not require arbitrarily small final values lim t→∞ ρ pi (t), which would achieve convergence of y i (t) − y i,d (t) arbitrarily close to zero, since one of the objectives is actual asymptotic stability. In this section, the problem statement is as follows:…”

“…The latter, however, might yield undesired large inputs due to the small funnel values, and can be problematic in real-time systems. Such a scheme was developed in the works [24], [25] for first-order systems, where the funnel converges to zero. This, however, can create numerical ill-conditioning in the practical computation of the control input, since it involves the product of "large" and "small" quantities (the funnel reciprocal and and the error signal) [25].…”

Utility of Edge-wise Funnel Coupling for Asymptotically Solving Distributed Consensus Optimization

“…Remark 5: Let us stress again that asymptotic tracking via funnel control was first achieved in [17], albeit this result seems to have not received much attention. As mentioned in the introduction, we utilize the alternative method developed in [18]. Indeed, the coupling gain 1/ ψ(t)−|x p j −x p i | grows unbounded when asymptotic consensus is achieved, because ψ(t) → 0 as t → ∞ and this implies that ψ(t) − |x p j − x p i | also tends to zero.…”

“…In that work a control structure of the form u(t) = ν e(t) /ψ(t) θ(e(t)), with bounded θ, was utilized. Recently, unaware of this result, it was observed in [18] that if the feedback is chosen to be of the form u(t) = F e(t)/ψ(t) , then asymptotic tracking is possible. In the present paper, we exploit the technique from [18] to achieve asymptotic consensus (without additional dynamics like the PI consensus algorithms or embedding a common internal model).…”