Análise de sistemas não-lineares e síntese de operadores inversos por séries de volterra diagonais

Abstract

This work proposes innovative strategies for the analysis of nonlinear systems and the synthesis of inverse operators using the Volterra diagonal series. By expressing the output explicitly from the input, the Volterra series enable nonlinear analysis in the frequency domain. However, the multidimensional nature of the model confers several difficulties to its systematic use. This work takes a new look at the so-called Volterra series in diagonal coordinates, in which Volterra operators are expressed as a set of linear and one-dimensional filters that process nonlinear polynomial terms of the input. The proposition of the rational form for these filters leads to exact and compact Volterra models, which exhibit a direct connection with modern nonlinear formalisms, notably the Wiener and Hammerstein block structured models, and the non-linear, autoregressive polynomial models with exogenous input (NARX). In particular, it is proposed a strategy to obtain diagonal Volterra models from the polynomial NARX. The strategy is called derivative method, because it depends only on the established results of the differential calculus. This is important because a NARX model can fit relatively well to experimental data to describe a wide variety of practical systems. A subsequent study through the Volterra series comes as an additional natural step of analysis. This result also opens up possibilities for non-linear synthesis. A problem that has received increasing attention in systems engineering is that of the synthesis of inverse nonlinear operators, through which it tries to reverse distortions generated by the underlying system, preserving the integrity of the information of interest. In this case we propose a strategy of synthesis of Volterra inverse diagonal operators for particular classes of nonlinear polynomial models. It is a numerical approach where the synthesis is driven by an optimization problem that is inspired by the classic inverse p-order operator. Keywords: Non-linear systems, Volterra series diagonals, systems identification, nonlinear analysis, dynamic inversion