# Identification Approach for Nonlinear MIMO Dynamics of Closed-Loop Active Magnetic Bearing System
## 4 System Identification of Rotor/RAMB Dynamics
To reduce the uncertainties of the Hammerstein-Wiener model but keep the nonlinearities of the dynamic model, we can measure the characteristics of the gap sensors prior to the process of parameter estimation. The parameters of the inverse equations of static nonlinear elements shown in Equation (25) are known. The remaining parameters to be estimated are the part of the Hammerstein model, i.e., Equations (23) and (24). Equations (23) and (24) are organized into one multivariable regression equation system:  
Figure 18: Schematic diagram of MIMO Hammerstein–Wiener model for rotor/RAMB dynamics [32]. \[\mathbf{V}(t)=\sum_{k=1}^{n}\boldsymbol{\varphi}_{d_{k}}(t)\boldsymbol{\theta}_{d_{k }}+\sum_{l=1}^{n_{h}}\boldsymbol{\varphi}_{ac_{l}}(t)\boldsymbol{\theta}_{ac_{l }}+v(t) \tag{29}\]  
where \(v(t)\) are the colored noises. Normally, colored-noise terms can be regarded as the white noises filtered with the transfer functions. In this study, the colored-noise is assumed to be the ARMA-type. Therefore, Equation (29) can be represented as the auxiliary model:  
\[\mathbf{V}(t)=\sum_{k=1}^{n}\boldsymbol{\varphi}_{d_{k}}(t)\boldsymbol{\theta} _{d_{k}}+\sum_{l=1}^{n_{h}}\boldsymbol{\varphi}_{ac_{l}}(t)\boldsymbol{\theta} _{ac_{l}}+\sum_{l=1}^{m}\boldsymbol{\varphi}_{e_{l}}(t)\boldsymbol{\theta}_{e _{l}}+\sum_{l=1}^{m}\boldsymbol{\varphi}_{f_{l}}(t)\boldsymbol{\theta}_{f_{l}}+ \omega(t) \tag{30}\]  
where  
\[\mathbf{V}(t)= \begin{bmatrix}v_{1}(t)\\ v_{2}(t)\\ \vdots\\ v_{j}(t)\end{bmatrix} , \tag{31}\]  
\[\boldsymbol{\varphi}_{d_{k}}(t)=- \begin{bmatrix}v_{1}(t-k)&&0\\ v_{2}(t-k)&&\\ &\ddots&\\ 0&&v_{j}(t-k)\end{bmatrix} ,\,k=1,2,..,n \tag{32}\]  
\[\boldsymbol{\theta}_{d_{k}}= \begin{bmatrix}d_{1k}\;d_{2k}\;\cdots d_{jk} \end{bmatrix} ^{T},\,k=1,2,..,n \tag{33}\]  
\[\boldsymbol{\varphi}_{ac_{l}}(t)= \begin{bmatrix}h_{1}(\mathbf{U}(t-j))^{T}&0 \\ h_{2}(\mathbf{U}(t-j))^{T}&&\\ &\ddots&\\ 0&&h_{i}(\mathbf{U}(t-j))^{T}\end{bmatrix} ,\,l=1,2,..,n_{h} \tag{34}\]  
\[\boldsymbol{\theta}_{ac_{l}}= \begin{bmatrix}\mathrm{G}_{l}^{T}(;1)^{T}\; \mathrm{G}_{l}^{T}(;2)^{T}\;\cdots\;\mathrm{G}_{l}^{T}(;n)^{T}\end{bmatrix} ^{T},\,l=1,2,..,n\times n_{h} \tag{35}\]  
\[\boldsymbol{\varphi}_{e_{l}}(t)= \begin{bmatrix}\omega_{1}(t-l)&&0\\ \omega_{2}(t-l)&&\\ &\ddots&\\ 0&&\omega_{j}(t-l)\end{bmatrix} ,\,l=1,2,..,m \tag{36}\]  
\[\boldsymbol{\varphi}_{e_{l}}= \begin{bmatrix}e_{1l}\;e_{2l}\;\cdots\;e_{jl} \end{bmatrix} ^{T},\,l=1,2,..,m \tag{37}\]  
\[\boldsymbol{\varphi}_{f_{l}}(t)=- \begin{bmatrix}v_{1}(t-l)&&0\\ v_{2}(t-l)&&\\ &\ddots&\\ 0&&v_{j}(t-l)\end{bmatrix} ,\,l=1,2,..,m \tag{38}\]  
\[\boldsymbol{\theta}_{f_{l}}= \begin{bmatrix}f_{1l}f_{2l}\;\cdots\;f_{jl}\end{bmatrix} ^{T},\,l=1,2,..,m \tag{39}\]  
\[\boldsymbol{\varphi}(t)= \begin{bmatrix}\omega_{1}(t)\\ \omega_{2}(t)\\ \vdots\\ \omega_{j}(t)\end{bmatrix} , \tag{40}\]  
where the vectors \(\mathrm{G}_{l}^{T}(;n)^{T}\) shown in Equation (35) are extracted from \(\mathrm{G}_{n\times n_{h}}=\mathrm{C}_{n}a_{n_{h}}\). Equation (29) is rewritten in the multivariable linear regression form as:\[\mathbf{V}(t) =\boldsymbol{\varphi}(t)\mathbf{\theta}+\mathbf{\omega}(t) \tag{41}\] \[=\left[ \begin{array}{cc}\boldsymbol{\varphi}_{L}(t)&\boldsymbol{ \varphi}_{N}(t)\end{array} \right]\left[ \begin{array}{c}\mathbf{\theta}_{L} \\ \mathbf{\theta}_{N}\end{array} \right]+\mathbf{\omega}(t)\]  
where \(\boldsymbol{\varphi}_{L}(t)=\left[\boldsymbol{\varphi}_{d_{1}}(t)\cdots \ \boldsymbol{\varphi}_{d_{n}}(t)\right]\boldsymbol{\varphi}_{n}(t)\), \(\boldsymbol{\varphi}_{N}(t)=\left[\boldsymbol{\varphi}_{\Omega}(t)\ \cdots\ \boldsymbol{\varphi}_{\Omega}(t)\ \boldsymbol{\varphi}_{f_{1}}(t)\ \cdots\ \boldsymbol{\varphi}_{f_{m}}(t)\ \right]\), \(\boldsymbol{\theta}_{L}=\left[\boldsymbol{\up}_{d_{1}}^{T}\cdots\ \boldsymbol{\up}_{d_{n}}^{T}\ \boldsymbol{\up}_{d_{n}}^{T}\ \cdots\ \boldsymbol{\up}_{d_{n}}^{T} \right]^{T}\), and \(\boldsymbol{\theta}_{N}=\left[\boldsymbol{\up}_{c_{1}}^{T}\cdots\ \boldsymbol{\up}_{c_{n}}^{T}\ \boldsymbol{\up}_{f_{1}}^{T}\ \cdots\ \boldsymbol{\up}_{f_{m}}^{T}\right]^{T}\).