## 3 Discrete Time Sliding Mode Control with Type-2 Fuzzy Compensation  
Generally the overall stiffness of machine tools is computed using the static loading test. The stiffness calculated by the static test, however, in general cases shows hysterics characteristics. This is due to the fact that the contact area of the bearing changes as the load direction changes [45; 46]. The stiffness has been modeled considering it to be a nonlinear in nature and is demonstrated effectively in [14]. The hysteric behavior can be handled effectively using the Bouc-Wen model. The behavior of the structure can be modeled using Bouc-Wen method which separates Equation (15) into two parts (elastic and inelastic) as [47]:  
\[f_{\rho;i}=\epsilon_{\rho i}k_{\rho i}x_{\rho i}+(1-\epsilon_{\rho i})k_{\rho i }\phi_{\rho i} \tag{15}\]  
representing the positive numbers by \(\rho\). The function illustrating nonlinearities (\(\phi_{\rho i}\)) is:  
\[\phi_{\rho i}=-\frac{1}{\eta_{\rho i}}\left[\lambda_{\rho i}\left|\dot{x}_{ \rho i}\right|\left|a_{\rho i}\right|^{\eta_{i}-1}a_{\rho i}v_{\rho i}-\gamma_ {\rho i}\left|\dot{x}_{\rho i}\right|\left|a_{\rho i}\right|^{\eta_{i}-1}v_{ \rho i}a_{\rho i}sign(\dot{x}_{\rho i}a_{\rho i})\right]-\kappa\dot{x}_{\rho i}\right] \tag{16}\]  
In the above equation, the positive numbers are \(\kappa,\lambda_{\rho i},\gamma_{\rho i},\alpha_{\rho i},n\) and \(\eta\). Moreover, \(\eta_{\rho i}\) and \(v_{\rho i}\) represent the stiffness degradation control factor and strength degradation control factor, respectively. The term, \(E_{\rho i}=(1-\alpha_{\rho i})\int_{0}^{t}\frac{\dot{x}_{\rho i}a_{\rho i}}{ \Delta_{\rho i}a_{\rho i}}dt\) is defined as dissipated hysteretic energy in normal conditions. Moreover, \(\Delta_{\rho i}=(\lambda_{\rho i}+\gamma_{\rho i})^{-\frac{1}{\eta\rho i}}\). The stiffness \(\mathbf{f}_{kn}\) will be considered as nonlinear. The continuous time model of the milling process, which is a closed loop system from Equation (14), is illustrated as:  
\[M_{m}\ddot{\mathbf{x}}_{c}(t)+C_{m}\dot{\mathbf{x}}_{c}(t)+\mathbf{f}_{kn}( \mathbf{x})=\mathbf{F}_{m}(\mathbf{t})+\mathbf{u}_{c}(t)-\mathbf{d}_{c} \tag{17}\]  
It is very important to discretize the milling process model for digitalization and to make it appropriate for the design of computer based control. For this, the following steps are implemented: \(V_{\mathbf{1}}\left(t\right)=\mathbf{x}_{c}\) and \(V_{\mathbf{2}}\left(t\right)=\dot{\mathbf{x}}_{c}\). The model represented by Equation (17) is illustrated as a state space model by:  
\[\dot{Z}(t)= AZ+B\mathbf{u}+F_{kn}+f_{n} \tag{18}\]  
Figure 2: AVD on Spindle Top. \[\text{Again},\;\;V(t)=\left[ \begin{array}{c}V_{1}(t)\\ V_{2}(t)\end{array} \right],\;\;A_{D}=\left[ \begin{array}{cc}0&0\\ 0&-M_{m}^{-1}C_{m}\end{array} \right],\;\;B_{D}=\left[ \begin{array}{c}0\\ M_{m}^{-1}\end{array} \right],\;\;F_{kn}\;\;=\;\;M_{m}^{-1}\mathbf{f}_{kn},\;\;f_{ n}\;=\;\;M_{m}^{-1}\mathbf{f}_{kn},\;\;f_{n}\;=\;\;\]  
In the linear system \(\dot{V}\!=\!A_{D}V\!+\!\mathbf{u}_{c}B_{D}\), \(F_{kn}\) and \(f_{n}\) are considered to be the uncertainties. When there is no external excitations, the tool will be at the stable position. So it is justifiable to consider that \(F_{kn}\) is bounded, \(||F_{kn}||d_{st}\). The cutting forces and friction forces are bounded, \(||f_{n}(t)||d_{n}\). The continuous time model is discretized by assuming that the control force and the external forces are constant during the sampling period \(T\).