Global Reconstruction of Nonlinear Systems From Families of Linear Systems

Abstract

This note concerns a fundamental issue in the modelling and realisation of nonlinear systems; namely, whether it is possible to uniquely reconstruct a nonlinear system from a suitable collection of transfer functions and, if so, under what conditions. It is established that a family of frozen-parameter linearisations may be associated with a class of nonlinear systems to provide an alternative realisation of such systems. Nevertheless, knowledge of only the inputoutput dynamics (transfer functions) of the frozen-parameter linearisations is insufficient to permit unique reconstruction of a nonlinear system. The difficulty with the transfer function family arises from the degree of freedom available in the choice of state-space realisation of each linearisation. Under mild structural conditions, it is shown that knowledge of a family of augmented transfer functions is sufficient to permit a large class of nonlinear systems to be uniquely reconstructed. Essentially, the augmented family embodies the information necessary to select state-space realisations for the linearisations which are compatible with one another and with the underlying nonlinear system. The results are constructive, with a state-space realisation of the nonlinear system associated with a transfer function family being obtained as the solution to a number of linear equations. IntroductionThis note concerns a fundamental issue in the modelling and realisation of nonlinear systems; namely, whether it is possible to uniquely reconstruct a nonlinear system from a suitable collection of transfer functions and, if so, under what conditions. Families of linear systems play an important role in many areas of nonlinear systems theory and practice. The construction of nonlinear systems related to a family of linear systems is, for example, the subject of the pseudo-linearisation (e.g. Reboulet & Champetier 1984) and extended linearisation (e.g. Rugh 1986) approaches and plays a central role in the choice of realisation of gain-scheduled controllers (e.g. Lawrence & Rugh 1995, Leith & Leithead 1996,1998a. Families of linear systems also play an important role in system identification practice (e.g. Skeppstedt et al. 1992, McLoone & Irwin 2000.A key issue in many application domains is that the linear systems are specified only to within a linear state transformation; that is, the choice of state realisation is available as a degree of freedom. This is usually the situation, for example, in divide and conquer identification (because only input-output data is measurable) and many forms of gain-scheduling design (because the linear methods used to carry out point designs are generally insensitive to the choice of state-space realisation). The objective of this note is to investigate the conditions, if any, under which unique, global reconstruction of a nonlinear system is possible. In order to focus on structural factors and to improve the clarity of the development, attention is restricted here to situations where the linearisation family is wellposed...

Application of Varying Parameters Modelling with Gaussian Processes

“…It is important to note that a local linear input-output model only specifies parameters up to a co-ordinate transformation (Leith and Leithead (2002)).…”