NONLINEARITY QUANTIFICATION AND ITS APPLICATION TO NONLINEAR SYSTEM IDENTIFICATION

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Abstract
In a series of previous works (Nikolaou, 1993) we introduced an inner product and a corresponding 2-norm for discrete-time nonlinear dynamic systems. Unlike induced norms of nonlinear systems, which are difficult to compute (albeit extremely useful), the 2-norm mentioned above is straightforward to compute, through Monte Carlo calculations with either experimental or simulated data. Loosely speaking, the 2-norm captures the average effect of a class of inputs on the output of a dynamic system. In this presentation we will give a brief introduction to this 2-norm, based on our previous results, and will discuss our latest work and applications on this subject. In particular, we will address the following points: (a) How is the nonlinearity of a dynamic system quantified by the 2-norm? (b) How adequate is a linear model for the representation of a nonlinear system? (c) What nonlinear model can be used for the representation of a nonlinear system for which a linear model is inadequate? An important result of this theory is that appropriate orthogonal bases for the representation of a nonlinear dynamic system can be constructed, that allow the successive refinement of a moving average nonlinear model through inclusion of additional basis terms, without requirement for readjustment of the entire model. Parallel (neural) implementation issues for the proposed algorithms are discussed. Nonlinear models based on Volterra-Legendre series are discussed in detail; and (d) How does feedback alter the nonlinearity characteristics of a dynamic system? Examples on four chemical processes are presented to elucidate the computational and conceptual merits of the proposed methodologies.