Data Filtering Based Recursive Least Squares Algorithm for Two-Input Single-Output Systems with Moving Average Noises

This paper studies identification problems of two-input single-output controlled autoregressivemoving average systems by using an estimated noise transfer function to filter the input-output data.Through data filtering, we obtain two simple identificationmodels, one containing the parameters of the system model and the other containing the parameters of the noise model. Furthermore, we deduce a data filtering based recursive least squares method for estimating the parameters of these two identification models, respectively, by replacing the unmeasurable variables in the information vectors with their estimates. The proposed algorithm has high computational efficiency because the dimensions of its covariance matrices become small.The simulation results indicate that the proposed algorithm is effective.


Introduction
Studies on identification methods have been active in recent years [1][2][3].The recursive least squares algorithm is a popular and important identification method for many different systems [4][5][6].Recently, Wang and Ding presented an inputoutput data filtering based recursive least squares parameter estimation for CARARMA systems [7]; Wang et al. proposed a data filtering based recursive least squares algorithm for Hammerstein systems using the key-term separation principle [8]; and Ding and Duan presented a two-stage parameter estimation algorithm for Box-Jenkins systems [9].Hu proposed an iterative and recursive least squares estimation algorithm for moving average systems [10].
The filtering technique has received much attention in the field of system identification [7,11,12] and signal processing [13,14].For example, Xie et al. studied recursive least squares parameter estimation methods for nonuniformly sampled systems based on data filtering [11]; Wang et al. discussed filtering based recursive least squares algorithm for Hammerstein nonlinear FIR-MA systems [12]; Wang proposed a filtering and auxiliary model-based recursive least squares identification algorithm for output error moving average systems [15]; Shi and Fang developed a recursive algorithm for parameter estimation by modifying the Kalman filterbased algorithm after designing a missing output estimator [16]; and Wang et al. derived a hierarchical generalized stochastic gradient algorithm and a filtering based hierarchical stochastic gradient algorithm to estimate the parameter vectors and parameter matrix of the multivariable colored noise systems by using the hierarchical identification principle [17].
For several decades, multiple-input single-output systems [18] or multiple-input multiple-output systems [19,20] have attracted researchers' attention, but most of the work focused on the single-input single-output systems [21].For example, Li proposed parameter estimation for Hammerstein controlled autoregressive moving average systems based on the Newton iteration [22].Yao and Ding derived a two-stage least squares based iterative identification algorithm for controlled autoregressive moving average (CARMA) systems; the basic idea is to decompose a CARMA system into two subsystems and to identify each subsystem, respectively [23].This paper considers the identification problems of two-input singleoutput controlled autoregressive moving average systems by using input-output data filtering and derives a data filtering The two-input single-output system with moving average noise.
based recursive least squares method.The proposed algorithm has high computational efficiency because the dimensions of its covariance matrices become small.Although this paper focuses on two-input single-output systems, the proposed method can be extended to multiple-input singleoutput systems.
The rest of the paper is organized as follows.Section 2 proposes a data filtering based recursive least squares algorithm for a two-input single-output system with moving average noise.Section 3 introduces the recursive extended least squares algorithm for comparison.In Section 4, we give an example to prove the effectiveness of the proposed algorithm.Finally, concluding remarks are given in Section 5.
Define the parameter vector  and the information vector () as The goal of this paper is to apply the data filtering technique and to develop a new recursive least squares for estimating the system parameters.
If we use the rational fraction 1/() (a liner filter) to filter the input-output data, we can get a simple "equation error model" which is easy to identify, then the recursive least squares algorithm can be applied.Because 1/() is unknown, we use its estimate 1/ D(, ) to filter the inputoutput data [7].The identification method based on this approach will be referred to as the data filtering based recursive least squares (F-RLS) method.
For the model in (1), define the filtered inputs  1 () and  2 (), the filtered output   (), and the filtered information vector   () as Dividing both sides of (1) by () gives It can be written as This filtered model is an equation error model and can be rewritten in a vector form Define the inner variable: For two identification models ( 7) and ( 8), we can obtain the following recursive least squares algorithm for computing the estimates θ () and θ () of   and   : Note that the filtered input  1 (), the filtered input  2 (), and the filtered output   () are all unknown because of the unknown polynomial () and the unmeasurable noise term V() in the information vector   () and () are unknown.So it is impossible to implement the algorithm in ( 9)-( 14).The solution we adopted here is to replace the unknown variables with their estimates according to the auxiliary model identification idea [24][25][26].

The RELS Algorithm
To show the advantages of the algorithm we proposed, we give the recursive extended least squares (RELS) algorithm for comparison.

Example
Consider the following example: The inputs { 1 ()}, { 2 ()} are taken as two uncorrelated persistent excitation signal sequences with zero mean and unit variance, {V()} as a white noise sequence with zero mean and variance  2 = 0.50 2 and  2 = 0.10 2 , and the corresponding noise-to-signal ratio are  ns = 59.70% and  ns = 11.94%,respectively.Applying the RELS and the F-RLS algorithms to estimate the parameters of the system, the parameter estimates and their errors are shown in Tables 1  and 2, and the estimation errors  := ‖ θ − ‖/‖‖ versus  are shown in Figure 2 with  2 = 0.10 2 .
From Tables 1 and 2 and Figure 2, we can draw the following conclusions.
(i) The parameter estimation errors become (generally) smaller and smaller with the data length  increasing.This shows that the proposed algorithm is effective.(ii) The F-RLS algorithm is more accurate than the RELS algorithm.This means that the proposed F-RLS algorithm has better identification performance compared with the RELS algorithm.
(iii) The parameter estimates given by the F-RLS algorithm converge fast to their true values compared with the RELS algorithm.
(iv) The F-RLS algorithm has a higher computational efficiency than the RELS algorithm because the dimensions of its covariance matrices are smaller than those of the covariance in the RELS algorithm.

Conclusions
The data filtering based recursive least squares algorithm for the two-input single-output system with moving average noise is proposed by means of the data filtering technique.