## 2 Problem Description and Identification Model
\begin{array}{ccccc}0&0&\cdots&0&0\\ \mathbf{f}\mathbf{h}&0&\cdots&0&0\\ \mathbf{f}\mathbf{G}\mathbf{h}&\mathbf{f}\mathbf{h}&\ddots&\vdots&\vdots\\ \vdots&\vdots&\ddots&0&0\\ \mathbf{f}\mathbf{G}^{n-2}\mathbf{h}&\mathbf{f}\mathbf{G}^{n-3}\mathbf{h}&\cdots&\mathbf{f}\mathbf{h}&0\\ \end{array} \right]\in\mathbb{R}^{n\times n},\] \[\mathbf{\phi}_{\text{y}}(k):= [y(k-n),y(k-n+1),\cdots,y(k-1)]^{\intercal}\in\mathbb{R}^{n},\]\[\boldsymbol{\phi}_{\mathrm{u}}(k) :=[u(k-n),u(k-n+1),\cdots,u(k-1)]^{\mathrm{ T}}\in\mathbb{R}^{n},\] \[\boldsymbol{\phi}_{\mathrm{e}}(k) :=[e(k-n),e(k-n+1),\cdots,e(k-1)]^{\mathrm{ T}}\in\mathbb{R}^{n}.\]  
According to Equations (5)-(8), we have:  
\[\boldsymbol{\phi}_{\mathrm{y}}(k+n) =\boldsymbol{T}\boldsymbol{x}(k)+\boldsymbol{M}\boldsymbol{\phi} _{\mathrm{u}}(k+n)+\boldsymbol{\phi}_{\mathrm{e}}(k+n)\] \[=\boldsymbol{x}(k)+\boldsymbol{M}\boldsymbol{\phi}_{\mathrm{u}}(k +n)+\boldsymbol{\phi}_{\mathrm{e}}(k+n). \tag{10}\]  
Equation (10) can be rewritten as:  
\[\boldsymbol{x}(k)=\boldsymbol{\phi}_{\mathrm{y}}(k+n)-\boldsymbol{M} \boldsymbol{\phi}_{\mathrm{u}}(k+n)-\boldsymbol{\phi}_{\mathrm{e}}(k+n). \tag{11}\]  
Define the parameter vectors \(\boldsymbol{\phi}\), \(\boldsymbol{\phi}_{\mathrm{g}}\) and \(\boldsymbol{\phi}_{\mathrm{h}}\) and the information vectors \(\boldsymbol{\phi}_{\mathrm{s}}(k)\) and \(\boldsymbol{\phi}_{\mathrm{n}}(k)\) as:  
\[\boldsymbol{\phi} :=[\boldsymbol{\phi}_{\mathrm{g}}^{\mathrm{ T}},\boldsymbol{\phi}_{\mathrm{h}}^{\mathrm{ T}}]^{\mathrm{ T}}\in\mathbb{R}^{2n},\] \[\boldsymbol{\phi}_{\mathrm{g}} :=[\boldsymbol{f}\boldsymbol{G}^{n}]^{\mathrm{ T}}\in\mathbb{R}^{n},\] \[\boldsymbol{\phi}_{\mathrm{h}} :=[-\boldsymbol{f}\boldsymbol{G}^{n}\boldsymbol{M}+[\boldsymbol{f }\boldsymbol{G}^{n-1}\boldsymbol{h},\boldsymbol{f}\boldsymbol{G}^{n-2} \boldsymbol{h},\cdots,\boldsymbol{f}\boldsymbol{h}]]^{\mathrm{ T}}\in\mathbb{R}^{n},\] \[\boldsymbol{\phi}_{\mathrm{s}}(k) := \begin{bmatrix}\boldsymbol{\phi}_{\mathrm{y}}(k)\\ \boldsymbol{\phi}_{\mathrm{u}}(k)\end{bmatrix} \in\mathbb{R}^{2n},\quad \boldsymbol{\phi}_{\mathrm{n}}(k):= \begin{bmatrix}-\boldsymbol{\phi}_{ \mathrm{e}}(k)\\ \boldsymbol{0}\end{bmatrix} \in\mathbb{R}^{2n}.\]  
Inserting (11) into (9) yields:  
\[y(k+n) =\boldsymbol{f}\boldsymbol{G}^{n}[\boldsymbol{\phi}_{\mathrm{y}}( k+n)-\boldsymbol{M}\boldsymbol{\phi}_{\mathrm{u}}(k+n)-\boldsymbol{\phi}_{ \mathrm{e}}(k+n)]\] \[\quad+[\boldsymbol{f}\boldsymbol{G}^{n-1}\boldsymbol{h}, \boldsymbol{f}\boldsymbol{G}^{n-2}\boldsymbol{h},\cdots,\boldsymbol{f} \boldsymbol{h}]\boldsymbol{\phi}_{\mathrm{u}}(k+n)+e(k+n)\] \[=\boldsymbol{\phi}_{\mathrm{y}}^{\mathrm{ T}}(k+n)\boldsymbol{\phi}_{\mathrm{g}}+ \boldsymbol{\phi}_{\mathrm{u}}^{\mathrm{ T}}(k+n)\boldsymbol{\phi}_{\mathrm{h}}- \boldsymbol{\phi}_{\mathrm{e}}^{\mathrm{ T}}(k+n)\boldsymbol{\phi}_{\mathrm{g}}+e(k+n)\] \[=\boldsymbol{\phi}_{\mathrm{s}}^{\mathrm{ T}}(k+n)\boldsymbol{\phi}+ \boldsymbol{\phi}_{\mathrm{n}}^{\mathrm{ T}}(k+n)\boldsymbol{\phi}+e(k+n). \tag{12}\]  
Define the intermediate variable:  
\[w(k):=\boldsymbol{\phi}_{\mathrm{n}}^{\mathrm{ T}}(k)\boldsymbol{\phi}+e(k). \tag{13}\]  
Replacing \(k+n\) in Equation (12) with \(k\) gives:  
\[y(k) =\boldsymbol{\phi}_{\mathrm{s}}^{\mathrm{ T}}(k)\boldsymbol{\phi}+ \boldsymbol{\phi}_{\mathrm{n}}^{\mathrm{ T}}(k)\boldsymbol{\phi}+e(k) \tag{14}\] \[=\boldsymbol{\phi}_{\mathrm{s}}^{\mathrm{ T}}(k)\boldsymbol{\phi}+w(k). \tag{15}\]  
Equation (14) or (15) is the system identification model of the state-space system in Equation (1), where the information vector \(\boldsymbol{\phi}_{\mathrm{s}}(k)\) is composed of the observed data.