Funnel control — A survey

Abstract

The methodology of funnel control was introduced in the early 2000s, and it has developed since then in many respects achieving a level of mathematical maturity balanced by practical applications. Its fundamental tenet is the attainment of prescribed transient and asymptotic behaviour for continuous-time controlled dynamical processes encompassing linear and nonlinear systems described by functional differential equations, differential–algebraic systems, and partial differential equations. Considered are classes of systems specified only by structural properties – such as relative degree and stable internal dynamics. Prespecified are: a funnel shaped through the choice of a function (absolutely continuous), freely selected by the designer, and a class of (sufficiently smooth) reference signals. The aim is to design a single ‘simple’ feedback strategy (using only input and output information) – called the funnel controller – which, applied to any system of the given class and for any reference signal of the given class, achieves the funnel control objective: that is, the closed-loop system is well-posed in the sense that all signals (both internal and external) are bounded and globally defined, and – most importantly – the error between the system’s output and the reference signal evolves within the prespecified funnel.

The survey is organized as follows. In the Introduction, the genesis of funnel control is outlined via the most simple class of systems: the linear prototype of scalar, single-input, single-output systems. Generalizing the prototype, there follows an exposition of diverse system classes (described by linear, nonlinear, functional, partial differential equations, and differential–algebraic equations) for which funnel control is feasible. The structure and properties of funnel control – in its various guises attuned to available output information – are described and analysed. Up to this point, the treatment is predicated on an implicit assumption that system inputs are unconstrained. Ramifications of input constraints and their incorporation in the funnel methodology are then discussed. Finally, practical applications and implementations of funnel control are highlighted.