**6. Simulations and results**

The motor model transfer function as described in Eq. (5) along with the constraints from Eqs. (6)–(10) is simulated using MATLAB Simulink.

To model the uncertainty of plant parameters, the specified motor parameters can be used as variables. The PID controller of 2-DOF is characterized to have tunable gains. The analysis point is where disturbance torques are calculated and disturbance sensitiveness is determined. The performance goals as examined in Section 5 are selected and H∞ minimization of the scalar function relating to the described performance goals is evolved in this work. This results in optimal tuning of controller gains and is tabulated as shown in **Table 2**.

The outcomes are presented in the following figures.

The **Figure 7** represents response of TE (tracking error) of the tuned function of closed loop transfer function. This plot represents the tracking error that is


### **Table 2.**

*The resulting tuned parameters of 2-DOF PID from simulation for optimization.*

**Figure 7.** *The performance goal 1: Desired v/s achieved.*

*PID Gain Tuning for Robust Control of PMDC Motor for External Disturbance Rejection with… DOI: http://dx.doi.org/10.5772/intechopen.102546*

**Figure 8.** *The performance goal 2: Desired v/s achieved.*

**Figure 9.** *The performance goal 3: Desired v/s achieved.*

accomplished over all ranges of frequencies below the specified margin. Correspondingly, the max-LG and achieved min-loop v/s ideal values are plotted in **Figures 8** and **9**.

**Figure 8** shows the achieved transient duration of proposed model is much less that desired transient duration. **Figures 10** and **11** demonstrates tracking performance of the proposed model for any arbitrary input. In all cases, the set point of tracking can be accomplished well within the prescribed 2 seconds limitations in system response. From these plots, we infer that simulation of the proposed model guarantees motor internal parameters to remain within tolerable limits of variations for any applied arbitrary external disturbances. Further, the proposed model is tested for simulation with 30 different randomly chosen external disturbances.

**Figure 10.** *The performance of tracking with 2-DOF PID controller.*

**Figure 11.** *The performance of speed control to the pulse command (tracking better than the 2 sec).*

It is significant to note that achieved performances are far superior to the focused objectives. Additionally, H<sup>∞</sup> advancement of targeted goals outperforms over expected performance. Out of several performance goals specified for PMDC motors, our work presents simulation results for only three objectives. However, other goals remain to be explored.

**Figure 12** represents the disturbance rejection achieved for two different values of proportional and derivative coefficients. The response for randomly chosen load disturbances (30 arbitrary signals) with implicit effect on internal parameters is presented in **Figure 13**. It can be observed that the response for the proposed robust model for controller is well within the accepted bounds. It is worth to observe that the disturbances attenuate well below the prescribed limit of 2 sec.

*PID Gain Tuning for Robust Control of PMDC Motor for External Disturbance Rejection with… DOI: http://dx.doi.org/10.5772/intechopen.102546*

**Figure 12.** *Disturbance rejection with the controller of 2-DOF PID.*

**Figure 13.** *The response to 30 random disturbances with implicit parameter variations.*
