Giovanni L. Sicuranza

In this paper, we first introduce a novel sub-class of recursive linear-in-the-parameters nonlinear filters, called recursive functional link polynomial filters, which are derived by using the constructive rule of Volterra filters. These filters are universal approximators, according to the Stone-Weierstrass theorem, and offer a remedy to the main drawback of their finite memory counterparts, that is the curse of dimensionality. Since recursive nonlinear filters become, in general, unstable for large input signals, we then consider a simple stabilization procedure by slightly modifying the input-output relationship of recursive functional link polynomial filters. The resulting filters are always stable and, even though no more universal approximators, still offer good modeling performance for nonlinear systems.

We consider the identification of nonlinear filters using periodic sequences. Perfect periodic sequences have already been proposed for this purpose. A periodic sequence is called perfect for a nonlinear filter if it causes the basis functions to be orthogonal and the autocorrelation matrix to be diagonal. In this paper, we introduce for the same purpose the quasi-perfect periodic sequences. We define a periodic sequence as quasi-perfect for a nonlinear filter if the resulting auto-correlation matrix is highly sparse. The sequence is obtained by means of a simple combinatorial rule and is formed by samples having few discrete levels. These characteristics allow an efficient implementation of the least-squares method for the approximation of certain linear-in-the-parameters nonlinear filters. A real-world experiment shows the good performance obtained.

In this paper, a novel class of recursive nonlinear filters is presented as an extension of finite-memory even mirror Fourier nonlinear filters. These filters are characterized by two relevant properties: i) they are able to arbitrarily well approximate any discrete-time, time-invariant, causal, infinite-memory, continuous, nonlinear system and ii) they are always stable according to the bounded-input-bounded-output criterion. Even though recursive models can represent many systems with fewer coefficients than their finite-memory counterparts, it is still possible to further reduce their computational complexity. In fact, while in general simplified structures lead to a loss of performance, it is pointed out in the paper that in various common real-world situations, they are able of giving remarkable complexity reductions without negatively affecting the modeling capabilities.

In this study, the unconstrained linear combination of the outputs of even mirror Fourier non‐linear filters is considered. These filters are new members of the class of causal, shift‐invariant, finite‐memory and linear‐in‐the parameters non‐linear filters. Their name derives from the even symmetry of their trigonometric basis functions. Even mirror Fourier non‐linear filters are universal approximators for causal, time invariant, finite‐memory and continuous non‐linear systems. Moreover, their basis functions are mutually orthogonal for white uniform input signals in the interval [−1, +1]. The authors show in this study how to exploit these characteristics, in the framework of the unconstrained linear combination of non‐linear filters, for modelling unknown non‐linear systems. In particular, they show that the filters whose outputs are combined can be adapted avoiding the choice of the step sizes, by using a simple algorithm presented in this study. The analysis of the proposed structures is accompanied by a set of simulation results that confirm the good performance obtained in different situations.