Asymptotic properties of the OBE parameter estimation algorithm under diffuse initialization

The parameters estimation problem of autoregressive with exogenous inputs models is considered. Our approach is based on the study of the optimal bounding ellipsoid (OBE) recurrent algorithm properties. It is assumed that the shape matrix of the initial ellipsoid in the OBE algorithm is proportional to a large positive parameter. Asymptotic expansions of the relations that describe the OBE algorithm are obtained. The limit recursive algorithm (diffuse) not depending on a large parameter which may lead the OBE algorithm to a divergence is proposed and explored. The robust properties of the proposed algorithm are illustrated by a numerical example. By simulation, it is shown that it may provide better the tracking ability compared to the diffuse least-squares (RLS) algorithm and the diffuse RLS algorithm with the sliding window.


Introduction
Problems related to parameters estimation of autoregressive with exogenous inputs (ARX) models arise in various applications including in particular identification systems, signals processing and control systems. Traditional approaches to their solution depend on the used assumptions regarding the nature of the disturbances (noises). Within the framework of statistical interpretation, the recursive least-squares (RLS) algorithm is commonly used to provide the high convergence rate to optimal values of the parameters model and robustness when processing data in a changing environment [1]. However in many cases, the use of statistics is not entirely justified and may lead to erroneous decisions. In such situations, the approach based on optimal bounding ellipsoid (OBE) recurrent algorithms can be a good alternative [2][3][4][5]. Within this approach, it is assumed that noises are unknown but bounded by specified bounds. As opposed to the RLS algorithm that looks for point estimates of unknown parameters, the OBE algorithms allow us to obtain them in such a way that they are only compatible with the model assumptions. More exactly, the important advantage of the OBE algorithms compared to the RLS algorithm is that they update the estimate when the new data can improve its.
The OBE algorithms produce a sequence of ellipsoids (analogs to covariance matrices) in some time instants characterized by their centers that are used as estimates of unknown parameters and sizes. The difference between all known the OBE algorithms is determined primarily by the choice of the optimality criterion by which these ellipsoids are determined. In this paper we will interested in studying of the ODE algorithm properties developed in [5] in the absence of a priori information about unknown parameters of the ARX models. The obtained results are an extension of the approach to the construction of diffuse RLS algorithms proposed in [6] on bounding ellipsoid estimation. Taking into account that the ODE algorithm is nonlinear in contrast to the RLS algorithm and their structures differ significantly, the corresponding generalization is not obvious. In this work, it is assumed that the shape matrix of the initial ellipsoid in the OBE algorithm is proportional to a large positive parameter. Asymptotic expansions of the relations that describe the OBE algorithm are obtained. The limit recursive algorithm (diffuse) not depending on a large parameter which may lead the OBE algorithm to a divergence is proposed and explored.

Problem statement
Consider the ARX model where t y , 1 R u t ∈ are the measurable output and input, respectively, , ,..., ,..., 2 , 1 , = are unknown parameters and the noise t ξ satisfies the restriction with a specified constant 2 γ . Introducing the notations The OBE parameter estimation algorithm is described by the relations [5] To evaluate α with help of this algorithm you need to set the initial condition for the matrix t P . Taking into account that there is no a priori information aboutα , the standard approach is to use soft initialization assuming is selected by a user and characterizing the degree of the initial uncertainty ofα . The limiting cases ∞ → µ we will be call the diffuse initialization.
It is required to study the OBE algorithm described by the relations (4) -(12) as . ∞ → µ More exactly, we will consider properties of t and develop a limit algorithm based on them which we call the diffuse version of the OBE (DOBE algorithm).

Main results
The proof of Theorem 1 is based on the following auxiliary statement. Lemma 1. Let t Ω be an n n × matrix defined by the expression Then t P and t K possess the uniform asymptotic expansions with respect to t on any bounded set N} 1,2,..., (19) Consequence 1. Neglecting the terms of the order of smallness in (17) gives where Neglecting the terms of the order of smallness ) / 1 ).
Consequence 3. It follows from the expansion (16) that the term in t P which is proportional to a large parameter µ (the diffuse component) vanishing when (27) Consider the system for the state transition matrix of the homogeneous part of (20) (28) The following auxiliary statement is used in the proof of Theorem 2.

Lemma 2. The representations
, 0 , Consequence 5. In the absence of perturbations, the estimation error of the DOBE turns to zero in a finite time as for the diffuse RLS algorithm [6]. So if , be any sequence of numbers satisfying the condition 1 0 Then the estimate t α from (20), (21) does not depend on t λ for Summing up, let us write the limit relations for quantities t , ), where t λ is defined by expressions (9), (10) up to the point of replacing 2

Simulation
To illustrate the capability of the proposed DOBE algorithm we use the ARX model of the form [5] ξ is the uncorrelated uniformly distributed random process with zero means and known covariance 1 ) ( = T t t E ξ ξ . Suppose that each parameter undergoes a ten-percent step change in magnitude at every 200 sampling points. Figure 1shows the trajectories of actual parameters, their estimates by the DOBE algorithm, the diffuse RLS (DRLS) algorithm and the finite impulse response (DFIRS) algorithm. It is seen that the DOBE algorithm may provide better the tracking ability compared to the DRLS and the DFIRLS algorithms. The update sample number of the DOBE was 1585. The used DRLS algorithm is described by the relations [6] ) (

Conclusion
The parameters estimation problem of the ARX models with help of the OBE algorithm in the absence of a priori information about their parameters was considered. It is assumed that the shape matrix of the initial ellipsoid in the OBE algorithm is proportional to a large positive parameter. Asymptotic expansions of the algorithm are obtained. The limit recursive algorithm not depending on a large parameter which may lead the OBE algorithm to a divergence is developed and explored. By simulation, it is shown that the proposed DOBE algorithm may provide better the tracking ability compared to the diffuse RLS algorithm and the diffuse RLS algorithm with the sliding window.