# Coupled Least Squares Identification Algorithms for Multivariate Output-Error Systems
## The Multivariate Auxiliary Model Coupled Identification Algorithm
### The Coupled Subsystem Auxiliary Model Based Recursive Least Squares Algorithm  
The coupling identification is usually used to reduce the redundant estimates of the system parameter vectors, based on the coupled relationship of the parameter estimates between subsystems [22]. Let \(\boldsymbol{\phi}_{i}^{\mathsf{ T}}(t)\in\mathbb{R}^{1\times n_{0}}\) be the \(i\)th row of the information matrix \(\boldsymbol{\Phi}(t)\), i.e.,  
\[\boldsymbol{\Phi}(t):= \begin{bmatrix}\boldsymbol{\phi}_{1}^{ \mathsf{ T}}(t)\\ \boldsymbol{\phi}_{2}^{\mathsf{ T}}(t)\\ \vdots\\ \boldsymbol{\phi}_{m}^{\mathsf{ T}}(t)\end{bmatrix} \in\mathbb{R}^{m\times n_{0}}. \tag{12}\]  
From (5), we obtain \(m\) identification models (subsystems)  
\[y_{i}(t)=\boldsymbol{\phi}_{i}^{\mathsf{ T}}(t)\boldsymbol{\vartheta}+v_{i}(t),\quad i=1,2,\cdots,m. \tag{13}\]  
From here, all subsystems contain a common parameter vector \(\boldsymbol{\vartheta}\in\mathbb{R}^{n_{0}}\). In general, one of the subsystems can be used to identify the parameter vector \(\boldsymbol{\vartheta}\); however, in order to improve the parameter estimation precision, we should make full use of the information in all subsystems for identifying \(\boldsymbol{\vartheta}\). Based on the RLS algorithm in (6)-(11), and applying the auxiliary model idea, we replace the unknown variables \(\boldsymbol{\phi}_{i}(t)\) in the identification algorithm with their estimates \(\hat{\boldsymbol{\phi}}_{i}(t)\), and obtain \(m\) RLS algorithms from (13), namely, the subsystem recursive least squares (S-RLS) algorithm,  
\[\hat{\boldsymbol{\vartheta}}_{i}(t) =\hat{\boldsymbol{\vartheta}}_{i}(t-1)+\boldsymbol{L}_{i}(t)[y_{ i}(t)-\hat{\boldsymbol{\phi}}_{i}^{\mathsf{ T}}(t)\hat{\boldsymbol{\vartheta}}_{i}(t-1)],\quad\hat{\boldsymbol{\vartheta}}_{i}(0)= \boldsymbol{1}_{n_{0}}/p_{0}, \tag{14}\] \[\boldsymbol{L}_{i}(t) =\boldsymbol{P}_{i}(t)\hat{\boldsymbol{\phi}}_{i}(t)=\boldsymbol {P}_{i}(t-1)\hat{\boldsymbol{\phi}}_{i}(t)[1+\hat{\boldsymbol{\vartheta}}_{i }^{\mathsf{ T}}(t)\boldsymbol{P}_{i}(t-1)\hat{\boldsymbol{\phi}}_{i}(t)]^{-1},\] (15) \[\boldsymbol{P}_{i}(t) =[\boldsymbol{I}_{n_{0}}-\boldsymbol{L}_{i}(t)\hat{\boldsymbol{ \vartheta}}_{i}^{\mathsf{ T}}(t)]\boldsymbol{P}_{i}(t-1),\quad\boldsymbol{P}_{i}(0)=p_{0} \boldsymbol{I}_{n_{0}},\quad i=1,2,\cdots,m. \tag{16}\]  
From here, we can see that there is no coupled relationship between the subsystem parameter estimation vector \(\hat{\boldsymbol{\vartheta}}_{i}(t)\). **Remark 2**. : _For \(i=1,2,\cdots,m\), we can obtain \(m\) estimation vectors \(\hat{\boldsymbol{\vartheta}}_{i}(t)\) from (14)-(16), and they are all the estimates of the common parameter vector \(\boldsymbol{\vartheta}\) in all subsystems, resulting in a large amount of redundant parameter estimates. One way is to use their average as the estimate of \(\boldsymbol{\vartheta}\), that is_  
\[\hat{\boldsymbol{\vartheta}}(t):=\frac{\hat{\boldsymbol{\vartheta}}_{1}(t)+ \hat{\boldsymbol{\vartheta}}_{2}(t)+\cdots+\hat{\boldsymbol{\vartheta}}_{m} (t)}{m}\in\mathbb{R}^{n_{0}}. \tag{17}\]  
If we regard the parameter estimate \(\hat{\boldsymbol{\vartheta}}(t)\) in (17) as the output parameter vector, then each S-RLS identification algorithm is still independent.