Non‐parametric linear time‐invariant extensions of non‐invertible and backlash plant

Abstract

SUMMARYA rigorous validation for the use of a set of linear time‐invariant models as a surrogate in the design of controllers for uncertain nonlinear systems, which are invertible as one‐to‐one operators, such as used in the nonlinear quantitative feedback theory (NLQFT) design methodology has been given by Baños and Bailey. This paper presents a similar validation but weakens the requirement on the invertibility of the nonlinear plant by application of Kakutani's fixed‐point theorem and an incremental gain constraint on the plant within its operational envelope. The set of linear time‐invariant models to be used for design is shown to be an extension (termed here the linear time‐invariant extension—LTIE) of the nonlinear plant restricted to the desired output operating space. A new non‐parametric approach to the modelling of the LTIE is proposed which is based on Fourier transforms of the plant I/O data and which accordingly may be based solely on experimental testing without the need for an explicit parametric plant model. This new approach thus extends the application of robust linear controller design methods (including those of NLQFT) to nonlinear plants with set‐valued (multi‐valued) inverses such as those containing backlash and also to plants for which explicit parametric models are difficult to obtain. The method is illustrated by application of the non‐parametric approach to an NLQFT tracking controller design for a mechanical backlashed servomechanism problem. Copyright © 2013 John Wiley & Sons, Ltd.