# Coupled Least Squares Identification Algorithms for Multivariate Output-Error Systems
## 1 Introduction  
Multivariable systems are popular in industrial processes [1; 2; 3] and a number of successful methods have been developed to solve the identification and control problems of multivariable systems [4; 5; 6; 7]. For example, [PERSON] and [PERSON] used a candidate-pool approach to identify the feedback and feedforward transfer function matrices and presented a frequency-domain technique for identifying multivariable feedback and feedforward systems [8]; [PERSON] and [PERSON] proposed a recursive algorithm to estimate the parameters of the dynamic linear part and the static nonlinear part of multivariable Hammerstein systems [9].  
The idea of the auxiliary model is to use the measurable information to construct a dynamical model and to replace the unknown variables in the information vector with the output of the auxiliary model [10; 11]. There are two typical identification methods for multivariate output-error systems: stochastic gradient (SG) algorithms [12; 13] and the recursive least squares (RLS) algorithms [14; 15]. The SG algorithm requires lower computational cost, but the RLS algorithm has a faster convergence rate than the SG algorithm [16]. The RLS algorithm has been applied to the identification of various systems [17; 18]. For example, on the basis of the work in [19], [PERSON] et al. proposed an auxiliary model based recursive least squares algorithm for autoregressive output-error autoregressive systems [20]; and [PERSON] and [PERSON] presented an auxiliary model based recursive least squares algorithm for a class of linear-in-parameter output-error moving average systems [21].  
Although the RLS algorithm can be applied to identify the parameter of the multivariate output-error systems, it requires computing the matrix inversion (see Remark 1 in the following section), resulting in a large computational burden [22]. This motivates us to study a new coupled least squares algorithm without involving matrix inversion. The coupling identification concept is useful for simplifying the parameter estimation of the coupled parameter multivariable systems [23]. It is based on the coupled relationship of the parameter estimates between the subsystems of a multivariable system [24; 25; 26]. The purpose of the coupling identification is to reduce the redundantestimation of the subsystem parameter vectors and to avoid computing the matrix inversion of the RLS algorithm. Recently, a coupled least squares algorithm has been proposed for multiple linear regression systems [22].  
This paper focuses on the parameter estimation of multivariate output-error systems, and the main contributions of this paper are the following:  
* for multivariate output-error systems, this paper derives two coupled least squares parameter estimation algorithms by using the auxiliary model identification idea and the coupling identification concept;
* the proposed algorithms can generate more accurate parameter estimates, and avoid computing the matrix inversion in the multivariable RLS algorithm, for the purpose of reducing computational load.  
The rest of this paper is organized as follows: Section 2 gives some definitions and the identification model of multivariate output-error systems. Section 3 presents two new coupled auxiliary model identification algorithms. Section 4 gives two simulation examples to validate the effectiveness of the proposed methods. Finally, some concluding remarks are offered in Section 5.