Identification of FIR Wiener systems with unknown, non-invertible, polynomial non-linearities

Abstract

Wiener systems consist of a linear dynamic system whose output is measured through a static non-linearity. In this paper we study the identification of single-input single-output Wiener systems with finite impulse response dynamics and polynomial output non-linearities. Using multi-index notation, we solve a least squares problem to estimate products of the coefficients of the non-linearity and the impulse response of the linear system. We then consider four methods for extracting the coefficients of the non-linearity and impulse response: direct algebraic solution, singular value decomposition, multi-dimensional singular value decomposition and prediction error optimization.

Direct identification of the linear block in Wiener system

“…It has computational complexity O.N / and can be evaluated with direct, well-elaborated linear least squares procedures. In contrast to the methods proposed, for example, in [12,21], the nonlinear optimization stage is eliminated. Obviously, computing the series of estimates O j ; j D 1; 2; : : : ; N , (15) requires N repeats.…”

“…This, in general, forces to restrict the class of admissible input signals to, for example, Gaussian ( [5,6]) or sine wave ( [7,8]), or to apply sophisticated and time-consuming nonlinear optimization techniques [9]. Furthermore, most of algorithms assume finite and known order of the difference equation that describes the linear dynamic subsystem (e.g., [10][11][12]) and restrict the class of admitted static characteristics to polynomials or strictly monotonous functions [13].…”

Spatio-Temporal Modeling of Nonlinear Distributed Parameter Systems

“…The assumption of invertible nonlinearities is common in most existing Wiener identification algorithms (Hagenblad, 1999) because it is particularly convenient for control system design, whereas others (Wigren, 1994; Lacy & Bernstein, 2003) allow noninvertible nonlinearities. Typically, the polynomial representation is chosen because it is simple to implement and analyze.…”

Identification of Wiener systems with non-invertible nonlinearity

“…Identification of Wiener systems based on available input and output data has been a hot topic for a long time, see for examples in [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [17], [18], [19], [20], [21], and [22]. Most of these existing schemes assume the random inputs as shown in [5], [6], [7], [11], [16] and an invertible nonlinear static function such as in [6], [12], [13], and [15].…”