# Adaptive Control of Active Magnetic Bearing against Milling Dynamics
## 2 Control Strategy for ECAMA against Milling Dynamics
### Fuzzy Model-Reference Adaptive Control (FMRAC)  
The block diagram of the fuzzy model-reference adaptive control (FMRAC) strategy is shown in Figure 6. The fuzzy milling dynamic models are constructed by experiments for the purpose of estimating the cutting force, \(F_{c}\)[3] online. Once the estimated cutting force, \(\widetilde{F}_{c}\), is obtained according to the present axial cut depth, \(a_{C}\), feedrate, \(f\), and spindle speed, \(\Omega\), the compensatory magnetic force, \(F_{m}\), can be determined to counter-balance \(F_{c}\), _i.e._, \(F_{m}=-\widetilde{F}_{c}\). Similarly, the reference model of spindle dynamics is also built by experiments. On the other hand, the control inputs provided by power amplifiers to the ECAMA are nothing but electric currents. In other words, the control commands to the ECAMA are coil currents, instead of magnetic force requirements. Since the nonlinear magnetic force is a function of coil current and air gap, the characteristics of the ECAMA can also be explicitly revealed by experiments [28]. By employing the fuzzy milling dynamic model and the magnetic force model, the corresponding coil current for required compensatory magnetic force can be determined by the controller. Although all the employed dynamic models are constructed by experiments, these models are just approximate copies and impossible to completely represent the actual dynamics of milling operation and magnetic force. As a result, a certain degree of estimation error might exist between the actual compensatory force, \(F_{m}\), and the actual cutting force, \(F_{c}\), due to the inaccuracy of modeling. In order to enhance the performance of spindle position regulation, a fuzzy compensator is proposed to modulate the control command via adjusting the feedback gain of controller. The rules for fuzzy compensator are based on the error, \(e_{2}\), between the compensatory force determined by controller, \(\widetilde{F}_{m}\), and the compensatory force evaluated by the reference model of spindle dynamics, \(\hat{F}_{m}\). Assume there are \(\ell\) terms of rules in the rule base and the rule \(L^{i}\) can be expressed as follows:  
\[ \begin{array}{ll}L^{i}\,:&\quad\text{if }\,x(k)\,\text{is }D_{1}^{i},x(k-1)\,\text{is }D_{2}^{i},\cdots,x(k-n+1)\,\text{is }D_{n}^{i}\\ &\quad\text{then }x(k+1)=a_{1}^{i}x(k)+a_{2}^{i}x(k-1)+\cdots+a_{n}^{i}x(k-n+1) \end{array}  \tag{1}\]  
where \(i=1,2,\cdots,\ell\), \(D_{j}^{i}\) is the fuzzy set and \(a_{j}^{i}\) is constant. Rule \(L^{i}\) can be further rewritten as:  
\[X(k+1)=M_{i}X(k) \tag{2}\]  
where:  
\[X(k+1)=\left[ \begin{array}{cccc}x(k+1)&x(k)&\cdots&x(k-n+2)\end{array} \right] ^{T} \tag{3a}\] \[X(k)=\left[ \begin{array}{cccc}x(k)&x(k-1)&\cdots&x(k-n+1)\end{array} \right] ^{T} \tag{3b}\]  
Figure 5: Block diagram of milling process regulated by the ECAMA. \[M_{i}=\left[ \begin{array}{cccccc}a_{1}^{i}&a_{2}^{i}&\cdots&a_{n-1}^{i}&a_{n}^{ i}\\ 1&0&\cdots&0&0\\ 0&1&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&1&0\end{array} \right] \tag{3c}\]  
The defuzzification of Equation (2) can be stated as follows:  
\[X(k+1)=\frac{\sum\limits_{i=1}^{\ell}w_{i}\,M_{i}\,X(k)}{\sum\limits_{i=1}^{ \ell}w_{i}} \tag{4}\]  
where \(w_{i}\) is the fitness function for rule \(L^{i}\) and is defined as follows:  
\[w_{i}=\min_{j}\,(D_{j}^{i}(x)),\ \omega_{i}\geq 0 \tag{5}\]  
The membership functions for fuzzification are shown in Figure 7, where the horizontal and vertical axes are corresponding to the normalized error, \(E_{2}\) and its derivative, \(E_{2}\) respectively. The  
Figure 6: Block diagram of the fuzzy model reference adaptive control system. obtained fuzzy variables, \(e_{2}\) and \(\dot{e}_{2}\) are firstly normalized within the interval, [-1,1], and it ends up with seven triangular membership functions for all individual variables. The complete rule base for the membership functions is listed in Table 1. The fuzzification plane in Figure 7 can be further divided into several sectors.