## 2 The Description of the Dual-Rate Hammerstein System  
Let us introduce some notation. The symbol \(\mathbf{I}\) stands for an identity matrix of appropriate size; the superscript T denotes the matrix/vector transpose. \(\hat{x}(t)\) stands for the estimate of \(x\) at time t.  
A Hammerstein system with two sampling periods in input-output signals is depicted in Figure 1, where \(P_{c}\) is a continuous-time Hammerstein process. Under the dual-rate framework, the discrete control signal \(u(kT_{1})\) is generated by a computer; the input \(u(t)\) is produced by a zero-order hold \(H_{T_{1}}\) with period \(T_{1}=T\) and takes the piecewise constant values \(u(kT)\) within the updating intervals; the output signal \(y(t)\) is sampled by a sampler \(S_{T_{2}}\) with period \(T_{2}:=qT\), yielding a discrete-time signal \(y(kqT)\). That means that the multiple of the output sampling period \((T_{2}=qT)\) to the input sampling period \((T_{1}=T)\) is \(q\). Adopting a simple expression, \(u(kT)\) is written as \(u(k)\), and \(y(kqT)\) is written as \(y(kq)\). Assume that the corresponding discrete-time singe-rate Hammerstein system consists of a two-segment piecewise-linear nonlinearity with slopes \(m_{1}\) and \(m_{2}\) and a linear model with polynomials \(A(z)\) and \(B(z)\) as depicted in Figure 2,  
\[x(k) = \left\{ \begin{array}{ll}m_{1}u(k),\;u(k)\geq 0,\\ m_{2}u(k),\;u(k)<0,\end{array} \right. \tag{1}\] \[A(z)y(k) = B(z)x(k), \tag{2}\]  
where \(x(k)\) is the internal variable; \(A(z)\) and \(B(z)\) are defined by:  
\[A(z) := 1+a_{1}z^{-1}+a_{2}z^{-2}+\cdots+a_{n}z^{-n},\]  
Figure 1: The dual-rate Hammerstein system. Introduce a switching function \(h(k)\) as:  
\[h(k)=h[u(k)]=0.5\{1+\mathrm{sgn}[u(k)]\},\]  
where:  
\[\mathrm{sgn}(u)=\left\{ \begin{array}{rl}1,&u>0,\\ 0,&u=0,\\ -1,&u<0.\end{array} \right.\]  
Then, Equation (1) can be rewritten as:  
\[x(k)=m_{2}u(k)+(m_{1}-m_{2})h(k)u(k). \tag{3}\]  
From Equation (2), we have:  
\[y(k)=b_{0}x(k)+b_{1}x(k-1)+\cdots+b_{n}x(k-n)-a_{1}y(k-1)-a_{2}y(k-2)-\cdots-a _{n}y(k-n). \tag{4}\]  
Inserting Equation (3) into \(x(k-i)\) in Equation (4) gives:  
\[y(k)=\sum_{i=0}^{n}b_{i}[m_{2}u(k-i)+(m_{1}-m_{2})h(k-i)u(k-i)]-\sum_{i=1}^{n}a _{i}y(k-i).\]  
Define:  
\[\mathbf{U}(k) := \left[ \begin{array}{rl}u(k)&h(k)u(k)\\ u(k-1)&h(k-1)u(k-1)\\ \vdots\\ u(k-n)&h(k-n)u(k-n)\end{array} \right]\in\mathbb{R}^{(n+1)\times 2},\] \[\mathbf{\phi}(k) := [-y(k-1),-y(k-2),\cdots,-y(k-n)]^{\mathrm{ T}}\in\mathbb{R}^{n},\] \[\mathbf{a} := [a_{1},a_{2},\cdots,a_{n}]^{\mathrm{ T}}\in\mathbb{R}^{n},\] \[\bar{\mathbf{b}} := [b_{0},b_{1},b_{2},\cdots,b_{n}]^{\mathrm{ T}}\in\mathbb{R}^{n+1},\] \[\mathbf{M} := [m_{2},m_{1}-m_{2}]^{\mathrm{ T}}\in\mathbb{R}^{2}.\]  
Then, we have:  
\[y(k)=\bar{\mathbf{b}}^{\mathrm{ T}}\mathbf{U}(k)\mathbf{M}+\mathbf{\phi}^{ \mathrm{ T}}(k)\mathbf{a},\]  
Figure 2: The discrete-time Hammerstein system. Replacing \(k\) with \(kq\) gives:  
\[y(kq)=\bar{\mathbf{b}}^{\intercal}\mathbf{U}(kq)\mathbf{M}+\mathbf{\phi}^{\intercal}(kq)\mathbf{a}. \tag{5}\]  
The identification problems caused by dual-rate Hammerstein systems with piecewise-linear nonlinearity exist in two aspects. * For the dual-rate Hammerstein system in this paper, all of the input data \(\{u(k):k=0,1,2,\cdots\}\) are measurable; only a part of the output data \(\{y(kq):k=0,1,2,\cdots\}\) can be measured, where \(q\geq 2\) is a positive integer. Intersample outputs (missing outputs) \(\{y(kq-i):k=0,1,2,\cdots,\)\(0<i<q\}\) are unknown. Because there exist unavailable outputs \(y(kq-i)\) in the information vector \(\mathbf{\phi}(kq)\), with \(kq-i\) not equaling the integer multiples of \(q\), the single-rate model in Equation (5) cannot be used as the identification model for the dual-rate system.