## 4 The Model Equivalence-Based Recursive Least Squares Algorithm  
For the EEAR system in Equation (1), the information vector \(\mathbf{\varphi}(t)\) of the recursive generalized least squares algorithm contains the unknown noise terms \(w(t-i)\). The solution is replacing the unknown noise terms \(w(t-i)\) with their estimates. However, the existence of the unknown noise terms in the information vector \(\mathbf{\varphi}(t)\) affects the accuracy of the parameter estimates to some extent.  
The method proposed in this paper is transforming the original system with colored noise into an equation-error system using the model equivalent transformation, so that the information vector in the identification model is composed of the available input \(u(t-i)\) and output \(y(t-i)\). Since there are no noise terms to be estimated in the information vector, the identification accuracy can be improved.  
Consider the CARAR system in Figure 1, which is rewritten as follows,  
\[A(z)y(t)=B(z)u(t)+\frac{1}{C(z)}v(t) \tag{14}\]  
Multiplying both sides of it by \(C(z)\) makes  
\[A(z)C(z)y(t)=B(z)C(z)u(t)+v(t) \tag{15}\]  
For simplicity, let \(n_{p}:=n_{a}+n_{c}\) and \(n_{q}:=n_{b}+n_{c}\); define the polynomials:  
\[P(z) := C(z)A(z)=1+p_{1}z^{-1}+p_{2}z^{-2}+\cdots+p_{n_{p}}z^{-n_{p}} \tag{16}\] \[Q(z) := C(z)B(z)=q_{1}z^{-1}+q_{2}z^{-2}+\cdots+q_{n_{q}}z^{-n_{q}} \tag{17}\]  
Inserting Equations (16) and (17) into Equation (15) yields  
\[P(z)y(t)=Q(z)u(t)+v(t) \tag{18}\]  
It is clear that Equation (14) reduces to an equation-error model, whose parameters can be estimated by the recursive least squares algorithm. Define the parameter vector \(\boldsymbol{\vartheta}\) and the information vector \(\boldsymbol{\phi}(t)\) as  
\[\boldsymbol{\vartheta} := [p_{1},p_{2},\cdots,p_{n_{p}},q_{1},q_{2},\cdots,q_{n_{q}}]^{ \mathrm{ T}}\in\mathbb{R}^{n_{p}+n_{q}},\] \[\boldsymbol{\phi}(t) := [-y(t-1),-y(t-2),\cdots,-y(t-n_{p}),u(t-1),u(t-2),\cdots,u(t-n_{ q})]^{\mathrm{ T}}\in\mathbb{R}^{n_{p}+n_{q}}\]  
In this case, Equation (18) can be equivalently written as  
\[y(t)=\boldsymbol{\phi}^{\mathrm{ T}}(t)\boldsymbol{\vartheta}+v(t) \tag{19}\]  
That is the identification model of Equation (18). Let \(\hat{\boldsymbol{\vartheta}}(t)\) be the estimate of \(\boldsymbol{\vartheta}\) at time \(t\). We obtain the recursive least squares algorithm for identifying \(\boldsymbol{\vartheta}\) in Equation (19):  
\[\hat{\boldsymbol{\vartheta}}(t) = \hat{\boldsymbol{\vartheta}}(t-1)+\boldsymbol{L}_{2}(t)[y(t)- \boldsymbol{\phi}^{\mathrm{ T}}(t)\hat{\boldsymbol{\vartheta}}(t-1)] \tag{20}\] \[\boldsymbol{L}_{2}(t) = \boldsymbol{P}_{2}(t-1)\boldsymbol{\phi}(t)[1+\boldsymbol{\phi}^ {\mathrm{ T}}(t)\boldsymbol{P}_{2}(t-1)\boldsymbol{\phi}(t)]^{-1}\] (21) \[\boldsymbol{P}_{2}(t) = [\boldsymbol{I}-\boldsymbol{L}_{2}(t)\boldsymbol{\phi}^{\mathrm{ T}}(t)]\boldsymbol{P}_{2}(t-1),\ \boldsymbol{P}_{2}(0)=p_{0}\boldsymbol{I}\] (22) \[\hat{v}(t) = y(t)-\boldsymbol{\phi}^{\mathrm{ T}}(t)\hat{\boldsymbol{\vartheta}}(t)\] (23) \[\boldsymbol{\phi}(t) = [-y(t-1),-y(t-2),\cdots,-y(t-n_{p}),u(t-1),u(t-2),\cdots,u(t-n_{ q})]^{\mathrm{ T}}\] (24) \[\hat{\boldsymbol{\vartheta}}(t) = [\hat{p}_{1}(t),\hat{p}_{2}(t),\cdots,\hat{p}_{n_{p}}(t),\hat{q}_ {1}(t),\hat{q}_{2}(t),\cdots,\hat{q}_{n_{q}}(t)]^{\mathrm{ T}} \tag{25}\]  
From Equations (20)-(25), we can compute the parameter estimate \(\hat{\boldsymbol{\vartheta}}(t)\), _i.e._, the estimates of the parameters \(p_{i}\) and \(q_{i}\). The following derives the model equivalence-based recursive least squares algorithm.