## 4 The Dual-Rate/Single-Rate Parameter Estimation  
In this section, the parameter estimates (in the following \(\hat{\boldsymbol{\Theta}}(kq)\)) of the dual-rate model are computed by a recursive least squares algorithm; then by comparing the coefficients of \(z^{-i}\) on both sides of a coefficient polynomial equation containing the parameters of the dual-rate/single-rate models, the parameter estimates (in the following \(\hat{\boldsymbol{\theta}}(kq)\)) of the single-rate model are derived.  
Suppose that the data length \(N\gg nq+n+2\). Define a quadratic cost function,  
\[J(\boldsymbol{\Theta}):=\sum_{k=1}^{N}[y(kq)-\boldsymbol{\psi}^{\mathrm{ T}}(kq)\boldsymbol{\Theta}]^{2}.\]  
The information vector \(\boldsymbol{\psi}(kq)\) in \(J(\boldsymbol{\Theta})\) contains the unknown internal variables \(\{x(kq-i),\)\(i=1,2,\cdots,nq\}\). Here, the solution is to replace \(x(kq-i)\) with its estimate \(\hat{x}(kq-i)\); then the estimate of \(\boldsymbol{\psi}(kq)\) can be written as:  
\[\hat{\psi}_{(kq)} := [u(kq),h(kq)u(kq),\hat{x}(kq-1),\hat{x}(kq-2),\cdots,\hat{x}(kq- nq),-y(kq-q),-y(kq-2q),\cdots, \tag{13}\] \[-y(kq-nq)]^{\mathrm{ T}}.\]  
the estimate \(\hat{x}(kq-i)\) can be computed by replacing \(m_{1}\) and \(m_{2}\) in Equation (3) with their estimates \(\hat{m}_{1}(kq)\) and \(\hat{m}_{2}(kq)\),  
\[\hat{x}(kq-i)=\hat{m}_{2}(kq)u(kq-i)+[\hat{m}_{1}(kq)-\hat{m}_{2}(kq)]h(kq-i) u(kq-i).\]  
Minimizing the cost functions \(J(\boldsymbol{\Theta})\), replacing \(x(kq-i)\) in \(\boldsymbol{\psi}(kq)\) with its estimate \(\hat{x}(kq-i)\) and by using the least squares method, we obtain the following key variable separation-based recursive least squares algorithm for estimating \(\boldsymbol{\Theta}\) of the dual-rate Hammerstein system,  
\[\hat{\boldsymbol{\Theta}}(kq) = \hat{\boldsymbol{\Theta}}(kq-q)+\boldsymbol{L}(kq)[y(kq)-\hat{ \boldsymbol{\psi}}^{\mathrm{ T}}(kq)\hat{\boldsymbol{\Theta}}(kq-q)], \tag{14}\] \[\boldsymbol{L}(kq) = \frac{\boldsymbol{P}(kq-q)\hat{\boldsymbol{\psi}}(kq)}{1+\hat{ \boldsymbol{\psi}}^{\mathrm{ T}}(kq)\boldsymbol{P}(kq-q)\hat{ \boldsymbol{\psi}}(kq)},\] (15) \[\boldsymbol{P}(kq) = [\boldsymbol{I}-\boldsymbol{L}(kq)\hat{\boldsymbol{\psi}}^{ \mathrm{ T}}(kq)]\boldsymbol{P}(kq-q),\;\boldsymbol{P}(0)=p_{0}\boldsymbol{I},\] (16) \[\hat{\boldsymbol{\psi}}(kq) = [u(kq),h(kq)u(kq),\hat{x}(kq-1),\hat{x}(kq-2),\cdots,\hat{x}(kq- nq),\] (17) \[-y(kq-q),-y(kq-2q),\cdots,-y(kq-nq)]^{\mathrm{ T}},\] \[\hat{x}(kq+i) = \hat{m}_{2}(kq)u(kq+i)+[\hat{m}_{1}(kq)-\hat{m}_{2}(kq)]h(kq+i)u( kq+i),\] (18) \[\hat{\boldsymbol{\Theta}}(kq) = [\hat{m}_{2}(kq),\hat{m}_{1}(kq)-\hat{m}_{2}(kq),\hat{\boldsymbol {\Theta}}_{1}(kq),\hat{\boldsymbol{\Theta}}_{2}(kq),\cdots,\hat{\boldsymbol {\Theta}}_{nq}(kq),\]\[\hat{\mathbf{\alpha}}(kq,z) = \hat{\beta}(kq,z)\hat{A}(kq,z).\]  
By comparing the coefficients of \(z^{-i}\) on both sides of this equation, we can establish a series of linear equations about \(\hat{a}_{i}(kq)\) and \(\hat{b}_{i}(kq)\) and build a matrix equation as:  
\[\mathbf{T}(kq)\hat{\mathbf{\theta}}(kq)=\mathbf{\rho}(kq), \tag{20}\]  
where:  
\[\hat{\mathbf{\theta}}(kq) = [\hat{a}_{1}(kq),\hat{a}_{2}(kq),\cdots,\hat{a}_{n}(kq),\hat{b}_ {1}(kq),\hat{b}_{2}(kq),\cdots,\hat{b}_{n}(kq)]^{\intercal}\in\mathbb{R}^{2n},\] \[\mathbf{\rho}(kq) = [\hat{\beta}_{1}(kq)-\lambda_{1}(kq),\hat{\beta}_{2}(kq)-\lambda _{2}(kq),\cdots,\hat{\beta}_{nq}(kq)-\lambda_{nq}(kq),0,\cdots,0]^{\intercal} \in\mathbb{R}^{nq+n}, \tag{21}\] \[\mathbf{T}(kq) = [-\mathbf{T}_{\beta}(kq)\;\mathbf{T}_{\alpha}(kq)]\in\mathbb{R}^{(nq+n) \times 2n},\] (22) \[\mathbf{T}_{\alpha}(kq) = \left[\begin{array}{cccc}1&0&\cdots&0\\ \lambda_{1}(kq)&1&&&\vdots\\ \lambda_{2}(kq)&\lambda_{1}(kq)&\ddots&0\\ \vdots&\lambda_{2}(kq)&\ddots&1\\ \vdots&&\ddots&\lambda_{1}(kq)\\ \lambda_{nq}(kq)&&&\lambda_{2}(kq)\\ 0&\lambda_{nq}(kq)&&&\vdots\\ \vdots&&\ddots&\vdots\\ 0&\cdots&&&\lambda_{nq}(kq)\\ \end{array}\right]\in\mathbb{R}^{(nq+n)\times n}, \tag{23}\]