## 3 The Dual-Rate Identification Model of the Hammerstein System  
In this section, we transform the Hammerstein system in Equations (1) and (2) into a dual-rate linear-in-parameter identification model, which is suitable for the dual-rate sampled data, by using the polynomial transformation technique [20] and the key variable separation technique [12].  
Referring to [20], assume that the roots of \(A(z)\) are \(z_{i},\ i=1,2,\cdots,n\). Then, we have:  
\[A(z)=(1-z_{1}z^{-1})(1-z_{2}z^{-1})\cdots(1-z_{n}z^{-1}).\]  
Define:  
\[\epsilon(z) := (1+z_{1}z^{-1}+z_{1}^{2}z^{-2}+\cdots+z_{1}^{q-1}z^{-q+1})(1+z_{2 }z^{-1}+z_{2}^{2}z^{-2}+\cdots+z_{2}^{q-1}z^{-q+1})\cdots\] \[(1+z_{n}z^{-1}+z_{n}^{2}z^{-2}+\cdots+z_{n}^{q-1}z^{-q+1})\] \[= 1+\epsilon_{1}z^{-1}+\epsilon_{2}z^{-2}+\cdots+\epsilon_{n(q-1) }z^{-n(q-1)}.\]  
Multiplying both sides of Equation (2) by \(\epsilon(z)\) gives:  
\[\alpha(z)y(k)=\beta(z)x(k).\]  
Using the identity:  
\[1-z_{i}^{q}z^{-q}\equiv(1-z_{i}z^{-1})(1+z_{i}z^{-1}+z_{i}^{2}z^{-2}+\cdots+z_ {i}^{q-1}z^{-(q-1)}),\]one obtains:  
\[\alpha(z) := \epsilon(z)A(z)\] \[= (1+z_{1}z^{-1}+z_{1}^{2}z^{-2}+\cdots+z_{1}^{q-1}z^{-(q-1)})(1-z_{1} z^{-1})\] \[(1+z_{2}z^{-1}+z_{2}^{2}z^{-2}+\cdots+z_{2}^{q-1}z^{-(q-1)})(1-z_{2 }z^{-1})\] \[\cdots\] \[(1+z_{n}z^{-1}+z_{n}^{2}z^{-2}+\cdots+z_{n}^{q-1}z^{-(q-1)})(1-z_{ n}z^{-1})\] \[= (1-z_{1}^{q}z^{-q})(1-z_{2}^{q}z^{-q})\cdots(1-z_{n}^{q}z^{-q})\] \[=: 1+\alpha_{1}z^{-q}+\alpha_{2}z^{-2q}+\cdots+\alpha_{n}z^{-nq},\] \[\beta(z) := \epsilon(z)B(z)\] \[= (1+\epsilon_{1}z^{-1}+\epsilon_{2}z^{-2}+\cdots+\epsilon_{n(q-1) }z^{-n(q-1)})(b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots+b_{n}z^{-n})\] \[=: \beta_{0}+\beta_{1}z^{-1}+\beta_{2}z^{-2}+\cdots+\beta_{nq}z^{- nq}.\]  
Consider a disturbance in a practical system; introducing a zero-mean random white noise \(v(k)\), we have:  
\[\alpha(z)y(k)=\beta(z)x(k)+v(k). \tag{8}\]  
Substituting the polynomials \(\alpha(z)\) and \(\beta(z)\) into Equation (8) gives:  
\[y(k) = [\beta_{0}+\beta_{1}z^{-1}+\beta_{2}z^{-2}+\cdots+\beta_{nq}z^{- nq}]x(k) \tag{9}\] \[-[\alpha_{1}z^{-q}+\alpha_{2}z^{-2q}+\cdots+\alpha_{n}z^{-nq}]y(k )+v(k),\] \[= \beta_{0}x(k)+\sum_{i=1}^{nq}\beta_{i}x(k-i)-\sum_{i=1}^{n}\alpha _{i}y(k-iq)+v(k). \tag{10}\]  
In order to get a unique solution, we assume that \(\beta_{0}=b_{0}=1\)[21]. The first term \(x(k)\) (its coefficient is one) on the right-hand side is a separated key variable, and \(x(k-i)\left(i=1,2,\cdots,n_{\beta}\right)\) are not separated variables. Substituting (3) into the separated key variable \(x(k)\) in Equation (10) gives:  
\[y(k)=m_{2}u(k)+(m_{1}-m_{2})h(k)u(k)+\sum_{i=1}^{nq}\beta_{i}x(k-i)-\sum_{i=1} ^{n}\alpha_{i}y(k-iq)+v(k).\]  
Define the information vector \(\mathbf{\psi}(k)\) and the parameter vector \(\mathbf{\Theta}\) as:  
\[\mathbf{\psi}(k) := [u(k),h(k)u(k),x(k-1),x(k-2),\cdots,x(k-nq),-y(k-q),-y(k-2q),\cdots,\] \[-y(t-nq)]^{\mbox{\tiny T}}\in\mathbb{R}^{nq+n+2},\] \[\mathbf{\Theta} := [m_{2},m_{1}-m_{2},\beta_{1},\beta_{2},\cdots,\beta_{nq},\alpha_{ 1},\alpha_{2},\cdots,\alpha_{n}]^{\mbox{\tiny T}}\in\mathbb{R}^{nq+n+2}.\]  
Then, we have:  
\[y(k)=\mathbf{\psi}^{\mbox{\tiny T}}(k)\mathbf{\Theta}+v(k).\]  
Replacing \(k\) with \(kq\) gives the following dual-rate linear-in-parameter identification model,  
\[y(kq) = \mathbf{\psi}^{\mbox{\tiny T}}(kq)\mathbf{\Theta}+v(kq), \tag{11}\]with:  
\[\boldsymbol{\psi}(kq) := [u(kq),h(kq)u(kq),x(kq-1),x(kq-2),\cdots,x(kq-nq),-y(kq-q),-y(kq-2q ),\cdots, \tag{12}\] \[-y(kq-nq)]^{\mathrm{ T}}.\]  
The output of the obtained model is linear about all parameters. Note that \(y(kq-iq)\) is available by sampling the output signal in every \(qT\) interval, and \(y(kq)\) in Equation (11) is a linear function about \(\boldsymbol{\Theta}\). Thus, it is easy to estimate the parameter vector \(\boldsymbol{\Theta}\) by using the standard least squares method.