## 2 The Identification Model for an EEAR System  
Let us define some notation. \"\(A=:X\)\" or \"\(X:=A\)\" represents \"\(A\) is defined as \(X\)\"; the superscript T denotes the matrix/vector transpose; the norm of a matrix \(\mathbf{X}\) is defined by \(\|\mathbf{X}\|^{2}:=\mathrm{tr}[\mathbf{X}\mathbf{X}^{\mathrm{T}}]\); \(\mathbf{I}\) stands for an identity matrix of appropriate size; \(\hat{\mathbf{X}}(t)\) represents the estimate of \(\mathbf{X}\) at time \(t\).  
Consider the following equation-error autoregressive system, _i.e._, the controlled autoregressive autoregressive (CARAR) system, in Figure 1,  
\[A(z)y(t)=B(z)u(t)+\frac{1}{C(z)}v(t) \tag{1}\]where \(u(t)\) and \(y(t)\) are the measured input and output of the system, respectively, \(v(t)\) represents stochastic white noise with zero mean and variance \(\sigma^{2}\) and \(A(z)\), \(B(z)\) and \(C(z)\) denote the polynomials in the unit backward shift operator \(z^{-1}\) [_i.e._, \(z^{-1}y(t)=y(t-1)\)]:  
\[A(z) := 1+a_{1}z^{-1}+a_{2}z^{-2}+\cdots+a_{n_{a}}z^{-n_{a}},\] \[B(z) := b_{1}z^{-1}+b_{2}z^{-2}+\cdots+b_{n_{b}}z^{-n_{b}},\] \[C(z) := 1+c_{1}z^{-1}+c_{2}z^{-2}+\cdots+c_{n_{c}}z^{-n_{c}}\]  
Suppose that \(u(t)=0\), \(y(t)=0\), \(v(t)=0\) for \(t\leqslant 0\), the orders \(n_{a}\), \(n_{b}\) and \(n_{c}\) are known, and \(n:=n_{a}+n_{b}+n_{c}\).  
Define the intermediate variable (the correlated stochastic noise):  
\[w(t) := \frac{1}{C(z)}v(t) \tag{2}\]  
Inserting Equation (2) into Equation (1) yields  
\[A(z)y(t)=B(z)u(t)+w(t) \tag{3}\]  
Define the parameter vector \(\mathbf{\theta}_{\mathrm{s}}\) and the information vector \(\mathbf{\varphi}_{\mathrm{s}}(t)\) of the system model and the parameter vector \(\mathbf{\theta}_{\mathrm{n}}\) and the information vector \(\mathbf{\varphi}_{\mathrm{n}}(t)\) of the noise model as  
\[\mathbf{\theta} := \begin{bmatrix}\mathbf{\theta}_{\mathrm{s}}\\ \mathbf{\theta}_{\mathrm{n}}\end{bmatrix}\in\mathbb{R}^{n},\] \[\mathbf{\theta}_{\mathrm{s}} := [a_{1},a_{2},\cdots,a_{n_{a}},b_{1},b_{2},\cdots,b_{n_{b}}]^{ \mathrm{ T}}\in\mathbb{R}^{n_{a}+n_{b}},\] \[\mathbf{\theta}_{\mathrm{n}} := [c_{1},c_{2},\cdots,c_{n_{c}}]^{\mathrm{ T}}\in\mathbb{R}^{n_{c}},\] \[\mathbf{\varphi}(t) := \begin{bmatrix}\mathbf{\varphi}_{\mathrm{s}}(t)\\ \mathbf{\varphi}_{\mathrm{n}}(t)\end{bmatrix}\in\mathbb{R}^{n},\] \[\mathbf{\varphi}_{\mathrm{s}}(t) := [-y(t-1),-y(t-2),\cdots,-y(t-n_{a}),u(t-1),u(t-2),\cdots,u(t-n_{b} )]^{\mathrm{ T}}\in\mathbb{R}^{n_{a}+n_{b}},\] \[\mathbf{\varphi}_{\mathrm{n}}(t) := [-w(t-1),-w(t-2),\cdots,-w(t-n_{c})]^{\mathrm{ T}}\in\mathbb{R}^{n_{c}}\]  
where subscripts \(\mathrm{s}\) and \(\mathrm{n}\) denote the first letters of the words \"system\" and \"noise\", respectively. \(\mathbf{\varphi}_{\mathrm{s}}(t)\) is the known information vector, which consists of measured input-output data \(u(t-i)\) and \(y(t-i)\); \(\mathbf{\varphi}_{\mathrm{n}}(t)\) is the unknown information vector, which consists of noise terms \(w(t-i)\). By means of the above definitions, Equations (2) and (3) can be expressed as:  
Figure 1: A system described by the equation-error autoregressive (EEAR) model.  
\[w(t) = [1-C(z)]w(t)+v(t)\] \[= \mathbf{\varphi}_{\mathrm{n}}^{\mathrm{r}}(t)\mathbf{\theta}_{\mathrm{n}}+v(t)\] \[y(t) = [1-A(z)]y(t)+B(z)u(t)+w(t)\] \[= \mathbf{\varphi}_{\mathrm{s}}^{\mathrm{r}}(t)\mathbf{\theta}_{\mathrm{s}}+w (t)\] \[= \mathbf{\varphi}^{\mathrm{r}}(t)\mathbf{\theta}+v(t) \tag{4}\]  
This is the identification model for the EEAR system in Equation (1). The objective of this paper is to propose new identification algorithms for estimating the parameters of EEAR systems.