Funnel control in the presence of infinite-dimensional internal dynamics

Abstract

We consider output trajectory tracking for a class of uncertain nonlinear systems whose internal dynamics may be modelled by infinite-dimensional systems which are bounded-input, bounded-output stable. We describe under which conditions these systems belong to an abstract class for which funnel control is known to be feasible. As an illustrative example, we show that for a system whose internal dynamics are modelled by a transport equation, which is not exponentially stable, we obtain prescribed performance of the tracking error.

Adaptive output error feedback for a class of nonlinear infinite-dimensional systems

“…This shows that funnel control becomes more and more attractive for systems driven by infinitedimensional internal dynamics. In that way, this topic has been considered in Berger et al (2020b). The authors proved that some class of infinite-dimensional linear systems fits the required assumptions for funnel control to be feasible.…”

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An adaptive funnel control method is considered for the regulation of the output for a class of nonlinear infinitedimensional systems on real Hilbert spaces. After a decomposition of the state space and some change of variables related to the Byrnes-Isidori form, it is shown that the funnel controller presented in (Berger et al, 2020b) achieves the control objective under that some assumptions on the system dynamics are considered, like well-posedness and BIBO stability assumptions of the nonlinear system. The theory is applied to the regulation of the temperature in a chemical plug-flow tubular reactor whose reaction kinetics are modeled by the Arrhenius nonlinearity.

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Funnel control of nonlinear systems

“…On the one hand, there are systems which have a well-defined relative degree and exhibit infinite-dimensional internal dynamics, see e.g. [11]. Such systems are susceptible to funnel control with the control laws presented in the present paper; for instance, a linearized model of a moving water tank, where sloshing effects appear, is discussed in [10].…”

“…In particular, the class N m,r encompasses systems with arbitrary state space dimension, including systems with infinite-dimensional internal dynamics, see e.g. [11]: we will elaborate further on this in Sect. 4.…”

Funnel control for boundary control systems

“…The funnel control for infinite-dimensional systems has so far only attracted attention in special configurations [10,24,22]. The recent article [10] deals with a linearized model of a moving water tank by showing that this system belongs to the class being treated in [9], see also [11]. In [24], a class of infinite-dimensional systems has been considered where feasibility of the funnel controller can be proved in a similar way as for finite-dimensional systems.…”