Structure, nonlinearity, and system theory

Abstract

SummaryThe suggestion of writing, for some problems, nonlinear state equations not as dx/dt = F(x,u,t), but as dx/dt = [A(t,x)]x + [B(t,x)]u(t), which is more ‘constructive’ as re system perception and possible structural generalizations, is considered, supported by arguments related to the classification of switched circuits as linear and nonlinear. The point of the distinction is mainly that when solving dx/dt = F(x,u,t), one immediately dwells into the analytical problems related to pure mathematics, whereas for dx/dt = [A(t,x)]x + [B(t,x)]u(t), considering first a constant matrix [A], one introduces the system's physical structure and considering then [A(x)] sees the nonlinearity of the system as a dependence of the structure on the processes in it or on system's input. (This might be named structural response). The thinking in terms of structure better observes the engineering and physical degrees of freedom, which are relevant regarding applications. Some electronic systems and physical systems (e.g., hydrodynamic) are considered in these terms. The logical side is always the focus, and the pedagogical (even philosophical) side is not ignored. Copyright © 2013 John Wiley & Sons, Ltd.