## 4 Validation and Results  
The total control scheme that is essential for having a brief overview of the control process is shown in Figure 3. Initially, to simulate the cutting forces, the cutting conditions are essential. This cutting force has influence on the tool vibration. In the first instance, the cutting conditions of the milling process illustrated in [52] are extracted for tool vibration simulation so as to validate the effectiveness of the developed control mechanism. In Table 1, the tool and cutting parameters are displayed. For the validation of the significant performances of DSMC-T2 fuzzy, the results of DSMC-T2 fuzzy are compared with DSMC-T1 fuzzy and discrete time PID (D-PID). The ideal continuous time PID controller can be expressed as [53]:  
\[\mathbf{u}(\mathbf{t})=-K_{p}\mathbf{e}(t)-K_{i}\int_{0}^{t}\mathbf{e}(t)d \tau-K_{d}\dot{\mathbf{e}}(t) \tag{61}\]  
Figure 3: Brief overview of control process. where, \(\mathbf{u}(\mathbf{t}),K_{p},K_{i},K_{d}\) and \(\mathbf{e}(t)\) control output, proportional gain, integral gain, derivative gain and error, respectively. The discrete time PID (D-PID) controller in the z-domain is given by [54]:  
\[\mathbf{u}(z)=-K_{p}\mathbf{e}(z)-\frac{K_{i}T_{\mathrm{s}}z}{z-1}\mathbf{e}(z )-\frac{K_{d}}{T_{s}}(\frac{z-1}{z})\mathbf{e}(z) \tag{62}\]  
where \(T_{\mathrm{s}}\) is the sampling time. For the purpose of numerical analysis and at par with the best results, the gains for the \(x\) and \(y\) components are selected as follows:  
\[ \begin{array}{l}K_{px}=200,K_{dx}=84,K_{ix}=1500\\ K_{py}=190,K_{dy}=76,K_{iy}=1400\end{array}  \tag{63}\]  
Matlab/Simulink is utilized as the software environment. Simulation results are presented to validate that the tool chatter can be mitigated significantly by implementing the combined technique of AVD and DSMC-T2 fuzzy. The proposed control strategy result is then compared with DSMC, D-PID, DSMC-T1 fuzzy to prove the capabilities of DSMC-T2 fuzzy in vibration mitigation. For the simulation process, a total duration of 0.1 s to 0.6 s is considered. The weight of the AVD is considered to be 5% of the main device and this assumption is implemented in the simulation process. For the comparison of the results, two subsystem blocks of the milling model are developed using Simulink, where one block includes control systems and the other block has no control. The inputs to the process model are cutting forces and damper forces. The value 650 rad/s is set as the frequency of the simulation procedure. The arrangements of numerical integrators and filters are implemented to convert the acceleration signals to required velocities and positions. The four sets of tests are conducted with DSMC, with D-PID, with DSMC -T1 fuzzy and DSMC-T2 fuzzy for the generation and comparison of results. The toolbox named IT2-FLSs designed by [PERSON] et al. [55] is utilized for processing type-2 fuzzy operations. The membership functions selected for position error and velocity error are three and two, respectively. The process of normalization is carried out in the form \([-2,2]\). The methodology implemented for the defuzzification of the type-2 fuzzy logic system is Karnik-Mendel technique [26]. In this paper, for the control of chatter considering the type-2 fuzzy logic concept, six fuzzy rules for \(\hat{\mathit{h}}\) as well as four fuzzy rules for \(\hat{\mathit{h}}\) are sufficient for effective control. Considering the type-1 fuzzy logic concept, for the control of chatter, nine fuzzy rules for \(\hat{\mathit{j}}\) and six fuzzy rules for \(\hat{\mathit{h}}\) are sufficient. The membership functions are designed using Gaussian function. IF-THEN rules are applied for both the types of fuzzy system.The chosen learning rates are \(\Omega_{1}=\Omega_{2}=0.9\).Theorem 1 is utilized for selecting \(\sigma\) which is 0.17. The vibration minimization obtained by implementing the controllers DSMC, D-PID, DSMC-T1 fuzzy and DSMC-T2 fuzzy are compared for validating the effectiveness and the results are shown in the plots given by Figures 4-11.