# 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
According to the coupling identification concept,we use \(\hat{\boldsymbol{\vartheta}}(t-1)\) to replace \(\hat{\boldsymbol{\vartheta}}_{i}(t-1)\) in the S-RLS algorithm, and get the coupled subsystem AM-RLS (C-S-AM-RLS) algorithm:  
\[\hat{\boldsymbol{\vartheta}}_{i}(t) =\hat{\boldsymbol{\vartheta}}(t-1)+\boldsymbol{L}_{i}(t)[y_{i}(t )-\hat{\boldsymbol{\vartheta}}_{i}^{\top}(t)\hat{\boldsymbol{\vartheta}}(t-1 )], \tag{18}\] \[\boldsymbol{L}_{i}(t) =\boldsymbol{P}_{i}(t)\hat{\boldsymbol{\vartheta}}_{i}(t)= \boldsymbol{P}_{i}(t-1)\hat{\boldsymbol{\vartheta}}_{i}(t)[1+\hat{ \boldsymbol{\vartheta}}_{i}^{\top}(t)\boldsymbol{P}_{i}(t-1)\hat{\boldsymbol {\phi}}_{i}(t)]^{-1},\] (19) \[\boldsymbol{P}_{i}(t) =[\boldsymbol{I}_{n_{0}}-\boldsymbol{L}_{i}(t)\hat{\boldsymbol{ \vartheta}}_{i}^{\top}(t)]\boldsymbol{P}_{i}(t-1),\quad i=1,2,\cdots,m,\] (20) \[\hat{\boldsymbol{\vartheta}}(t) =\frac{\hat{\boldsymbol{\vartheta}}_{1}(t)+\hat{\boldsymbol{ \vartheta}}_{2}(t)+\cdots+\hat{\boldsymbol{\vartheta}}_{m}(t)}{m},\] (21) \[\hat{\boldsymbol{\Phi}}_{x}(t) =[-\mathbf{x}_{\text{a}}(t-1),-\mathbf{x}_{\text{a}}(t-2), \cdots,-\mathbf{x}_{\text{a}}(t-n_{a})],\] (22) \[\hat{\boldsymbol{\vartheta}}_{i}(t) =\hat{\boldsymbol{\vartheta}}_{i}(t-1)+\boldsymbol{L}_{i}(t)[y_{ i}(t)-\hat{\boldsymbol{\phi}}_{i}^{\mathsf{ T}}(t)\hat{\boldsymbol{\vartheta}}_{i}(t-1)],\quad\hat{\boldsymbol{\vartheta}}_{i}(0)= \boldsymbol{1}_{n_{0}}/p_{0}, \tag{14}\]0 (23) \[\hat{\boldsymbol{\vartheta}}_{i}(t) =\hat{\boldsymbol{\vartheta}}_{i}(t-1)+\boldsymbol{L}_{i}(t)[y_{ i}(t)-\hat{\boldsymbol{\phi}}_{i}^{\mathsf{ T}}(t)\hat{\boldsymbol{\vartheta}}_{i}(t-1)],\quad\hat{\boldsymbol{\vartheta}}_{i}(0)= \boldsymbol{1}_{n_{0}}/p_{0}, \tag{14}\]1 (24) \[\hat{\boldsymbol{\vartheta}}_{i}(t) =\hat{\boldsymbol{\vartheta}}_{i}(t-1)+\boldsymbol{L}_{i}(t)[y_{ i}(t)-\hat{\boldsymbol{\phi}}_{i}^{\mathsf{ T}}(t)\hat{\boldsymbol{\vartheta}}_{i}(t-1)],\quad\hat{\boldsymbol{\vartheta}}_{i}(0)= \boldsymbol{1}_{n_{0}}/p_{0}, \tag{14}\]2  
The steps of computing the parameter estimation vector \(\hat{\boldsymbol{\vartheta}}(t)\) by the C-S-AM-RLS algorithm in (18)-(25) are listed in the following:  
1. Set the initial values: \(t=1\), \(\hat{\boldsymbol{\vartheta}}(0)=\boldsymbol{1}_{n_{0}}/p_{0}\), \(\boldsymbol{P}_{i}(0)=p_{0}\boldsymbol{I}_{n_{0}}\), \(\mathbf{x}_{\text{a}}(t-j)=\boldsymbol{1}_{m}/p_{0}\), \(j=1\), \(2\), \(\cdots\), \(n_{a}\), \(p_{0}=10^{6}\). Set the data length \(L\). 2. Collect the observation data \(\{\boldsymbol{\Phi}_{s}(t),\boldsymbol{y}(t)\}\) and form the information matrix \(\hat{\boldsymbol{\Phi}}_{x}(t)\) by (22). 3. Form \(\boldsymbol{\Phi}(t)\) by (23) and read \(\hat{\boldsymbol{\varphi}}_{i}(t)\) from \(\boldsymbol{\Phi}(t)\) in (24). 4. For each \(i\), \(i=1,2,\cdots,m\), compute \(\boldsymbol{L}_{i}(t)\) by (19), and \(\boldsymbol{P}_{i}(t)\) by (20), and update the parameter estimation vector \(\hat{\boldsymbol{\vartheta}}_{i}(t)\) by (18). 5. Compute \(\hat{\boldsymbol{\vartheta}}(t)\) by (21) and \(\mathbf{x}_{\text{a}}(t)\) by (25). 6. If \(t=L\), stop the recursive computation and obtain the parameter estimates; otherwise, increase \(t\) by \(1\) and go to Step 2. **Remark 3**. : _The C-S-AM-RLS algorithm in (18)-(25) uses the estimate \(\hat{\boldsymbol{\vartheta}}(t-1)\) on the right-hand side of (18) instead of \(\hat{\boldsymbol{\vartheta}}_{i}(t-1)\) on the right-hand side of (14) for \(i=1,2,\cdots,m\). Thus, the C-S-AM-RLS algorithm is different from the S-RLS algorithm._