System identification of MISO fractional systems: Parameter and differentiation order estimation

Abstract

This paper deals with continuous-time system identification of multiple-input single-output (MISO) fractional differentiation models. When differentiation orders are assumed to be known, coefficients are estimated using the simplified refined instrumental variable method for continuous-time fractional models extended to the MISO case. For unknown differentiation orders, a two-stage optimization algorithm is proposed with the developed instrumental variable for coefficient estimation and a gradient-based algorithm for differentiation order estimation. A new definition of structured-commensurability (or S-commensurability) is introduced to better cope with differentiation order estimation. Three variants of the algorithm are then proposed: (i) first, all differentiation orders are set as integer multiples of a global S-commensurate order, (ii) then, the differentiation orders are set as integer multiples of a local S-commensurate orders (one S-commensurate order for each subsystem), (iii) finally, all differentiation orders are estimated by releasing the S-commensurability constraint. The first variant has the smallest number of parameters and is used as a good initial hit for the second variant which in turn is used as a good initial hit for the third variant. Such a progressive increase of the number of parameters allows better performance of the optimization algorithm evaluated by Monte Carlo simulation analysis.