# Identification Approach for Nonlinear MIMO Dynamics of Closed-Loop Active Magnetic Bearing System
## 4 System Identification of Rotor/RAMB Dynamics  
The 5-DOF (degree of freedom) rotor/active magnetic bearing (rotor/AMB) unit at TMP can be divided into a 4-DOF rotor/radial active magnetic bearing (rotor/RAMB) module and a 1-DOF rotor/axial active magnetic bearing (rotor/AAMB) module. Except the controller for active magnetic bearings (AMBs), the hardware within a closed-loop AMB system comprises power amplifiers, rotor/AMB module, and gap sensors. Compared with the dynamic responses of the mechanical part, i.e., rotor/RAMB module, the dynamic responses of the power amplifier and the gap sensors are much fast. That is, it is reasonable to describe the input/output characteristics of the power amplifier and the gap sensors with static models in the rotor/AMB system. It is noted that the scheme of the Hammerstein-Wiener model [31] is close to the composition of the hardware within a closed-loop AMB system. Hence, the model of the rotor/RAMB unit is described by a MIMO Hammerstein-Wiener model representation in this study. The schematic diagram of the MIMO Hammerstein-Wiener model is depicted in Figure 18. The MIMO Hammerstein-Wiener model comprises a MIMO nonlinear input element, a MIMO linear element, and several SISO nonlinear output elements. The nonlinearities and the dynamic properties of the rotor/RAMB module can be well described by the combination of the Hammerstein model and the Wiener model. As to the Hammerstein model, the nonlinear static element, H, is cascaded ahead of the linear dynamic element, L. In contrast, the linear dynamic element, L, is cascaded ahead of the nonlinear static elements, \(w_{k},\ k=1,2,..,j\), in the Wiener model. The nonlinear element of the Hammerstein model, H, is to represent the actuator nonlinearities or any other potential nonlinear effects embedded on the system input side. In comparison, the nonlinear elements of the Wiener model, \(w_{k},\ k=1,2,..,j\), are referred to as the output nonlinearities, which are the nonlinear characteristics of gap sensors in this study. The symbols, U and \(y_{k},\ k=1,2,..,j\), denote the inputs and the outputs of the Hammerstein-Wiener model. The symbols, \(\omega_{k},\ k=1,2,..,j\) and \(v_{k},\ k=1,2,..,j\), are the white noises and colored noises, respectively. The relations between the elements of the Hammerstein-Wiener model are summarized as:  
\[\mathbf{X}=\mathbf{H}(\mathbf{U})=\mathbf{U}+\sum_{l=2}^{n_{h}}a_{l}h_{l}( \mathbf{U}),\ \text{where}\ \mathbf{U}=[u_{1},u_{2},\cdots,u_{i}]^{T} \tag{23}\]  
Figure 17: Constructed input signals applied to plant. \[\mathbf{V} =\mathbf{L}\cdot\mathbf{X}+\mathbf{N}\cdot\varpi \tag{24}\] \[=\left(\sum\limits_{k=1}^{n}\mathbf{C}_{k}z^{-n}\right)\mathbf{X} \Bigg{/}\left(1+\sum\limits_{l=1}^{n}\mathbf{D}_{l}z^{-n}\right)+\left(1+\sum \limits_{k=1}^{m}\mathbf{E}_{k}z^{-m}\right)\varpi\Bigg{/}\left(1+\sum \limits_{l=1}^{m}\mathbf{F}_{l}z^{-m}\right)\] \[\mathrm{v}_{k}(t)=\mathrm{w}_{k}^{-1}(y_{k})=\sum\limits_{l=1}^{ n_{m}}b_{kl}y_{k}^{l}(t),k=1,2,\cdots,j \tag{25}\]  
where \(i\) is the number of inputs, \(j\) is the number of outputs, \(n_{h}\) is the degree of the nonlinear input functions, and \(n_{w}\) is the degree of inverse nonlinear output functions. \(n\) is the order of the linear dynamic model, and \(m\) is the order of the colored-noise model. \(a_{l}\) and \(b_{kl}\) are the scale parameters, while \(\mathrm{C}_{k}\) denotes the parameter matrices. The diagonal parameter matrices, \(\mathrm{D}_{l}\), \(\mathrm{E}_{k}\), and \(\mathrm{F}_{l}\), are represented as:  
\[\mathrm{D}_{l} = \begin{bmatrix}d_{1l}&&&0\\ &d_{2l}&&\\ &&\ddots&\\ 0&&&d_{jl}\end{bmatrix} ,\;\textit{where}\;l=1,2,..,n \tag{26}\] \[\mathrm{E}_{k} = \begin{bmatrix}e_{1k}&&&0\\ &e_{2k}&&\\ &\ddots&\\ 0&&&e_{jk}\end{bmatrix} ,\;\textit{where}\;k=1,2,..,m\] (27) \[\mathrm{F}_{l} = \begin{bmatrix}f_{1l}&&&0\\ &f_{2l}&&\\ &&\ddots&\\ 0&&&f_{jl}\end{bmatrix} ,\;\textit{where}\;l=1,2,..,m \tag{28}\]  
The Hammerstein-Wiener model is a gray-box model, which processes the merits of the white-box model and the black-box model.