Maximum Likelihood Estimation of Stochastic Fractional Singular Models

Komeil Nosrati,Masoud Shafiee,Derek Abbott

doi:10.1109/access.2021.3112636

Abstract

Using the non-causal nature of a fractional-order singular (FOS) model, this paper deals with the modification of an estimation algorithm developed by Nosrati and Shafiee, and demonstrates how the derived estimation procedure can be adjusted by additional information related to the future dynamics. The procedure adopts the maximum likelihood (ML) method leading to a 3-block fractional singular Kalman filter (FSKF). In addition to some conditions on existence and uniqueness of solutions for discrete-time linear stochastic FOS models, the estimability analyses are given and an optimal filter is presented. Finally, the performance of the derived filter is verified and validated via numerical simulation on a three machine infinite bus system.

Highlights

F RACTIONAL or non-integer calculus steers us to a more general form called fractional-order singular (FOS) models, which are utilized to model various physical systems and scientific processes [2]– [4], and at the same time, share characteristics of both non-integer theories and singular systems
There have been some notable studies in stability [5], normalization and stabilization [6], estimator and observer design problems [1], [7]– [8], issues have been reported for deterministic FOS models such as control designs for nonlinear and rectangular FOS systems, admissibility conditions and stabilization problems for convex intervals 1 < α < 2 of fractional order or based on the complex domain that has broader descriptions and more complex behaviors than linear square FOS systems with fractional order 0 < α < 1
Owing to the conception of fractional calculus and singular theory, the corresponding linear FOS model of the three machines infinite bus system is described as follows, Dαδ1 =ω1 Dαδ2 = ω2 Dαδ3 = ω3

Summary

Introduction

F RACTIONAL or non-integer calculus steers us to a more general form called fractional-order singular (FOS) models, which are utilized to model various physical systems and scientific processes [2]– [4], and at the same time, share characteristics of both non-integer theories and singular systems. In the state estimator problem, the Kalman filter (KF) is well established for optimal state observers [9]. The extension of this algorithm to the discrete time non-integer order model was elaborated in [10], which has been called the fractional KF (FKF). That study transformed a singular model to a normal form to apply the classic KF algorithms to estimate the states of the system [12]. Some normal approaches such as the least-squares (LS), ML and deterministic approaches [13]– [15] were applied to solve the estimation problem without the need for transformation, yielding a singular KF (SKF)

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Conclusion