Abstract

This article introduces the spline approximation concept, in the context of system identification, aiming to obtain useful autoregressive models of reduced order. Models with a small number of poles are extremely useful in real time control applications, since the corresponding regulators are easier to design and implement. The main goal here is to compare the identification models complexity when using two types of experimental data: raw (affected by noises mainly produced by sensors) and smoothed. The smoothing of raw data is performed through a least squares optimal stochastic cubic spline model. The consecutive data points necessary to build each polynomial of spline model are adaptively selected, depending on the raw data behavior. In order to estimate the best identification model (of ARMAX class), two optimization strategies are considered: a two-step one (which provides first an optimal useful model and then an optimal noise model) and a global one (which builds the optimal useful and noise models at once). The criteria to optimize rely on the signal-to-noise ratio, estimated both for identification and validation data. Since the optimization criteria usually are irregular in nature, a metaheuristic (namely the advanced hill climbing algorithm) is employed to search for the model optimal structure. The case study described in the end of the article is concerned with a real plant with nonlinear behavior, which provides noisy acquired data. The simulation results prove that, when using smoothed data, the optimal useful models have significantly less poles than when using raw data, which justifies building cubic spline approximation models prior to autoregressive identification.

Highlights

  • Over the last few decades, the spline functions theory has been applied in many fields of science and engineering, such as: signal processing [1], computer graphics [2], system modeling [3] and identification [4], statistics [5], industrial design [6], geodetics [7], etc

  • The superior tanks are filled by means of a vertical pipe, which tanks are filled by means of a vertical pipe, which is branched in two small horizontal pipes in the is branched in two small horizontal pipes in the upper side

  • This article aimed to emphasize the advantage of employing the stochastic spline approximation concept in System Identification (SI), which consists of obtaining a useful model with reduced number of poles

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Summary

Introduction

Over the last few decades, the spline functions theory has been applied in many fields of science and engineering, such as: signal processing [1], computer graphics [2], system modeling [3] and identification [4], statistics [5], industrial design [6], geodetics [7], etc. Substantial effort has been made towards developing the mathematical bases of spline functions [8]. The spline function is a piecewise polynomial continuous curve passing through some pre-defined points, referred to as joint/control points or knots. Such a knot consists of a time instant and a data value. Two major approaches are commonly reported into the scientific literature, for deriving a spline function [10]: (a) by enforcing the polynomials to pass through successive knots in a table, subjected to some continuity and, possibly, derivative constraints at their knots; (b) by simultaneously determining the polynomials and their knots, while optimizing some cost function, subject to similar constraints, as previously described. The spline function is a data interpolator and can be determined

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