Maximum Likelihood Estimation of Stochastic Fractional Singular Models

Abstract

Using the non-causal nature of a fractional-order singular (FOS) model, this paper deals with the modification of an estimation algorithm developed in [1] 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.

Extended fractional singular kalman filter

“…This solution is unique if and only if A T k R k A k is an invertible matrix. Necessity: Assume that the solution of (11), or equivalently (14), is unique, that is the matrix…”

“…Fractional-order singular (FOS) models, which consist of both fractional and singular features, can be also considered for modeling of more complicated dynamical processes [7][8][9]. Some primary research were directed towards solvability [10], stability [11], and estimability [1,[12][13][14] of linear FOS models in the both time invariant (TI) and time varying (TV) forms; however, there have been few reports connected to the nonlinear cases.…”

“…The filtering problem of the both TI and TV linear FOS systems has been considered in literature based on direct [1,13,14] and indirect methods [26]. Using the deterministic filtering and ML techniques, the authors in [1] and [14] derived a kalman-type algorithm to estimate the states of the TI cases, and introduced the fractional singular KF (FSKF) in a direct methodology.…”

“…The filtering problem of the both TI and TV linear FOS systems has been considered in literature based on direct [1,13,14] and indirect methods [26]. Using the deterministic filtering and ML techniques, the authors in [1] and [14] derived a kalman-type algorithm to estimate the states of the TI cases, and introduced the fractional singular KF (FSKF) in a direct methodology. In the following and according to a same formulation, the designed FSKF algorithm was extended to a TV FOS system based on the given measurement including the correlated noises [13].…”

“…As seen, not only the EFSKF covers the three other nonlinear filtering ESKF, EFKF and EKF but also it covers FSKF, SKF, FKF and KF algorithms introduced in [1,13], [17][18][19], [15] and [14], respectively, when the system (8) collapses to a linear FOS system. In a hierarchical structure, Figure 1 illustrates these connections in visual perspective.…”