MINLIP for the identification of monotone Wiener systems

Abstract

This paper studies the MINLIP estimator for the identification of Wiener systems consisting of a sequence of a linear FIR dynamical model, and a monotonically increasing (or decreasing) static function. Given T observations, this algorithm boils down to solving a convex quadratic program with O(T ) variables and inequality constraints, implementing an inference technique which is based entirely on model complexity control 1 . The resulting estimates of the linear submodel are found to be almost consistent when no noise is present in the data, under a condition of smoothness of the true nonlinearity and local Persistency of Excitation (local PE) of the data. This result is novel as it does not rely on classical tools as a 'linearization' using a Taylor decomposition, nor exploits stochastic properties of the data. It is indicated how to extend the method to cope with noisy data, and empirical evidence contrasts performance of the estimator against other recently proposed techniques.

Identifiability and convergence analysis of the MINLIP estimator

“…In order to make the arguments as clean as possible, only the noiseless case is considered. For extensions of MINLIP for noisy data, see Pelckmans (2011). Two types of consistency results are derived: (1) Exclusion with probability one (in Theorem 1) and (2) Almost Sure Convergence (in Theorem 2).…”

“…This estimator was initially studied in Pelckmans (2010Pelckmans ( , 2011, based on earlier work by Bai and Reyland (2008); Zhang, Iouditski, and Ljung (2006). The motivation underlying this estimator does not rely on a loss function as tradition dictates (e.g.…”

“…, N. This particular class of systems occur often in case the output of a linear system is seen after an unknown (1) quantization, (2) saturation, or (3) transformation step. For more practical information, see Pelckmans (2011). The focus to FIR linear subsystem is not so restrictive in practice as one can always has d be high enough to approximate the impulse response of any stable linear system arbitrary well, typically one takes d = O(10 3 ).…”

Convex Optimization for Blind Identification of Monotone Wiener Systems**This work was supported by the Swedish Research Council under contract 621-2007-6364.

“…We argued before Pelckmans (2011) that this assumption is often satisfied, e.g. when studying saturation effects, quantisation effects, transformations and others.…”

Gradient-Based Iterative Identification for Wiener Nonlinear Dynamic Systems with Moving Average Noises

“…Differing from the work in [8,9], this paper focuses on the identification problem for Wiener nonlinear systems with moving average noises which are called Wiener output error moving average (OEMA) systems. In most existing works, the nonlinear part of Wiener systems is assumed a linear combination or a piecewise-linear function [10], or has a invertible and monotone function representation over the operating range [11,12]. Wang and Ding derived a least squares-based and a gradient-based iterative identification algorithms for Wiener nonlinear systems by separating one bilinear cost function into two linear cost functions [13].…”