## 5 The Parameter Estimation of the Original System  
According to the acquired estimates \(\hat{p}_{i}(t)\) and \(\hat{q}_{i}(t)\) of the parameters \(p_{i}\) and \(q_{i}\), we can compute the parameter estimates \(\hat{a}_{i}(t)\), \(\hat{b}_{i}(t)\) and \(\hat{c}_{i}(t)\) of the original system. The key idea is using the coefficient equivalent principle, and the details are as follows. Assume that the estimates of \(A(z)\), \(B(z)\) and \(C(z)\) are  
\[\hat{A}(t,z) := 1+\hat{a}_{1}(t)z^{-1}+\hat{a}_{2}(t)z^{-2}+\cdots+\hat{a}_{n_{a} }(t)z^{-n_{a}},\] \[\hat{B}(t,z) := \hat{b}_{1}(t)z^{-1}+\hat{b}_{2}(t)z^{-2}+\cdots+\hat{b}_{n_{b}}( t)z^{-n_{b}},\] \[\hat{C}(t,z) := 1+\hat{c}_{1}(t)z^{-1}+\hat{c}_{2}(t)z^{-2}+\cdots+\hat{c}_{n_{c }}(t)z^{-n_{c}}\]  
According to Equations (16) and (17), we can approximately suppose  
\[\hat{P}(t,z) = \hat{C}(t,z)\hat{A}(t,z)=1+\hat{p}_{1}(t)z^{-1}+\hat{p}_{2}(t)z^{ -2}+\cdots+\hat{p}_{n_{p}}(t)z^{-n_{p}} \tag{26}\] \[\hat{Q}(t,z) = \hat{C}(t,z)\hat{B}(t,z)=\hat{q}_{1}(t)z^{-1}+\hat{q}_{2}(t)z^{-2 }+\cdots+\hat{q}_{n_{q}}(t)z^{-n_{q}} \tag{27}\]  
Based on the above assumptions, we let  
\[\hat{B}(t,z)\hat{P}(t,z)=\hat{A}(t,z)\hat{Q}(t,z) \tag{28}\]  
Using \(\hat{B}(t,z)\), \(\hat{P}(t,z)\), \(\hat{A}(t,z)\) and \(\hat{Q}(t,z)\) gives  
\[[\hat{b}_{1}(t)z^{-1} + \cdots+\hat{b}_{n_{b}}(t)z^{-n_{b}}][1+\hat{p}_{1}(t)z^{-1}+\cdots +\hat{p}_{n_{p}}(t)z^{-n_{p}}]\] \[= [1+\hat{a}_{1}(t)z^{-1}+\cdots+\hat{a}_{n_{a}}(t)z^{-n_{a}}][ \hat{q}_{1}(t)z^{-1}+\cdots+\hat{q}_{n_{q}}(t)z^{-n_{q}}]\]  
Expanding the above equation and comparing the coefficients of the same power of \(z^{-1}\) on both sides, we can set up \((n_{b}+n_{p})\) equations:  
\[z^{-1}: \hat{b}_{1}(t)=\hat{q}_{1}(t),\] \[z^{-2}: \hat{b}_{1}(t)\hat{p}_{1}(t)+\hat{b}_{2}(t)=\hat{q}_{1}(t)\hat{a} _{1}(t)+\hat{q}_{2}(t),\] \[\hat{B}(t,z) := \hat{b}_{1}(t)z^{-1}+\hat{b}_{2}(t)z^{-2}+\cdots+\hat{b}_{n_{b}}( t)z^{-n_{b}},\]0 \[\hat{B}(t,z) := \hat{b}_{1}(t)z^{-1}+\hat{b}_{2}(t)z^{-2}+\cdots+\hat{b}_{n_{b}}( t)z^{-n_{b}},\]1 \[\hat{B}(t,z) := \hat{b}_{1}(t)z^{-1}+\hat{b}_{2}(t)z^{-2}+\cdots+\hat{b}_{n_{b}}( t)z^{-n_{b}},\]2 \[\hat{B}(t,z) := \hat{b}_{1}(t)z^{-1}+\hat{b}_{2}(t)z^{-2}+\cdots+\hat{b}_{n_{b}}( t)z^{-n_{b}},\]3  
which can be written in a matrix form,  
\[\hat{B}(t,z) := \hat{b}_{1}(t)z^{-1}+\hat{b}_{2}(t)z^{-2}+\cdots+\hat{b}_{n_{b}}( t)z^{-n_{b}},\]4  
where  
\[\hat{B}(t,z) := \hat{b}_{1}(t)z^{-1}+\hat{b}_{2}(t)z^{-2}+\cdots+\hat{b}_{n_{b}}( t)z^{-n_{b}},\]5 \[\hat{B}(t,z) := \hat{b}_{1}(t)z^{-1}+\hat{b}_{2}(t)z^{-2}+\cdots+\hat{b}_{n_{b}}( t)z^{-n_{b}},\]6_Algorithms_**2015**, \(8\)  
\[\mathbf{S}_{p}(t):=\left[ \begin{array}{cccccc}1&0&\cdots&\cdots&0\\ \hat{p}_{1}(t)&1&\ddots&&\vdots\\ \hat{p}_{2}(t)&\hat{p}_{1}(t)&1&\ddots&\vdots\\ \vdots&\hat{p}_{2}(t)&\hat{p}_{1}(t)&\ddots&0\\ \hat{p}_{n_{p}-1}(t)&&\ddots&\ddots&1\\ \hat{p}_{n_{p}}(t)&\hat{p}_{n_{p}-1}(t)&&\ddots&\hat{p}_{1}(t)\\ 0&\hat{p}_{n_{p}}(t)&\ddots&&\hat{p}_{2}(t)\\ \vdots&\ddots&\ddots&\hat{p}_{n_{p}-1}(t)&\vdots\\ \vdots&&\ddots&\hat{p}_{n_{p}}(t)&\hat{p}_{n_{p}-1}(t)\\ 0&\cdots&\cdots&0&\hat{p}_{n_{p}}(t)\\ \end{array} \right]\in\mathbb{R}^{(n_{b}+n_{p})\times n_{b}},\quad\mathbf{B}(t):= \left[ \begin{array}{c}\hat{q}_{1}(t)\\ \hat{q}_{2}(t)\\ \vdots\\ \hat{q}_{n_{q}}(t)\\ 0\\ \vdots\\ 0\end{array} \right]\in\mathbb{R}^{n_{b}+n_{p}},\]  
\[\mathbf{S}_{q}(t):=\left[ \begin{array}{cccccc}0&0&\cdots&\cdots&0\\ \hat{q}_{1}(t)&0&\ddots&&\vdots\\ \hat{q}_{2}(t)&\hat{q}_{1}(t)&0&\ddots&\vdots\\ \vdots&\hat{q}_{2}(t)&\hat{q}_{1}(t)&\ddots&0\\ \hat{q}_{n_{q}-1}(t)&&\ddots&\ddots&0\\ \hat{q}_{n_{q}}(t)&\hat{q}_{n_{q}-1}(t)&&\ddots&\hat{q}_{1}(t)\\ 0&\hat{q}_{n_{q}}(t)&\ddots&&\hat{q}_{2}(t)\\ \vdots&\ddots&\ddots&\hat{q}_{n_{q}-1}(t)&\vdots\\ \vdots&&\ddots&\hat{q}_{n_{q}}(t)&\hat{q}_{n_{q}-1}(t)\\ 0&\cdots&\cdots&0&\hat{q}_{n_{q}}(t)\\ \end{array}