Nonlinear-Adaptive Mathematical System Identification

Timothy Sands

doi:10.3390/computation5040047

Abstract

By reversing paradigms that normally utilize mathematical models as the basis for nonlinear adaptive controllers, this article describes using the controller to serve as a novel computational approach for mathematical system identification. System identification usually begins with the dynamics, and then seeks to parameterize the mathematical model in an optimization relationship that produces estimates of the parameters that minimize a designated cost function. The proposed methodology uses a DC motor with a minimum-phase mathematical model controlled by a self-tuning regulator without model pole cancelation. The normal system identification process is briefly articulated by parameterizing the system for least squares estimation that includes an allowance for exponential forgetting to deal with time-varying plants. Next, towards the proposed approach, the Diophantine equation is derived for an indirect self-tuner where feedforward and feedback controls are both parameterized in terms of the motor’s math model. As the controller seeks to nullify tracking errors, the assumed plant parameters are adapted and quickly converge on the correct parameters of the motor’s math model. Next, a more challenging non-minimum phase system is investigated, and the earlier implemented technique is modified utilizing a direct self-tuner with an increased pole excess. The nominal method experiences control chattering (an undesirable characteristic that could potentially damage the motor during testing), while the increased pole excess eliminates the control chattering, yet maintains effective mathematical system identification. This novel approach permits algorithms normally used for control to instead be used effectively for mathematical system identification.

Highlights

The feedback is parameterized in a Diophantine equation [2], referred to in the literature as a Bezout identity [3], or, alternatively, as an Aryabhatta equation [4]
This description describes a method of system identification where the identified system parameters are the time-varying terms in a nonlinear adaptive controller
A DC motor with a plant given in the Laplace domain is represented in Equation (1), which was discretized as Equation (2). This is indicated by the discrete time-shift operator variable q, where t indicates time (shifted in discrete steps (i.e., 1, 2, . . . , n)) used on the output y side of the equation, and where the pole excess d0 is written in terms of the time-shift operator, requiring a separate index m used on the control u side of the equation

Summary

Introduction

As described in [1], utilize an assumed mathematical structure of the plant to be controlled. The feedback is parameterized in a Diophantine equation [2], referred to in the literature as a Bezout identity [3], or, alternatively, as an Aryabhatta equation [4]. Without saying it, this description describes a method of system identification where the identified system parameters are the time-varying terms in a nonlinear adaptive controller. Since the plant’s mathematical model derives from first principles, the plant’s mathematical equation can instead be considered the output of such algorithms, assuming the algorithm demonstrates parameter convergence

Methods

Discussion

Conclusion