# Design of PIDD(r) Controller for Robust Performance of Process Plants
## 3 Simulation Study
### Second-Order Model with Inertia and Time Delay  
The current section focuses on the model object, which is based on [33] and represents a second-order system with inertia and time delay. The block diagram of this system is comparable to the first-order plant discussed in Section 3.1. The transfer function of the model, which has a time delay of 0.2 s, is as follows:  
\[G_{p}(s)=\frac{0.37}{1.4299\mathrm{s}^{2}+2.46s+1}e^{-0.2s}. \tag{11}\]  
Following the methodology outlined in the preceding section, the controller parameters, transient response characteristics, step responses, and set-point tracking of various controllers have been illustrated in Table 3 and Figures 9-11. Based on the simulation results, it is clear that both PIDD\({}^{2}\) and the proposed PIDD\({}^{\alpha}\) controllers outperform PID and FOPID controllers in terms of overshoot values. Specifically, PIDD\({}^{2}\) achieved an impressive overshoot value of 0%, while the proposed PIDD\({}^{\alpha}\) achieved an overshoot value of 0.0864%.  
In comparison, PID and FOPID controllers only managed to achieve overshoot values of 3.7692% and 2.9108%, respectively. Furthermore, the utilisation of fractional parameters in the integral control of the FOPID controller resulted in a significant reduction in the response time of the plant, from 6.11 s to 3.17 s.  
However, this came at the cost of a slightly increased \(t_{r}\) of 0.3869 s at the start of a step change and an overshoot of 2.9108%. While the PID controller had the lowest \(t_{r}\) of 0.3821 s, it had the highest overshoot value of 3.7692% among all the other controllers. On the other hand, the PIDD\({}^{2}\) controller managed to achieve an impressive overshoot value of 0%, but at the expense of a higher \(t_{r}\) of 0.4606 s. Overall, the proposed controller demonstrated the best overall performance when compared to the other controllers, achieving a lower \(t_{r}\) of 0.4037 s and \(t_{s}\) of 6.1118 s compared to the PIDD\({}^{2}\) controller, with only a slight increase in overshoot of 0.0864%.  
During the tracking performance, the set-point changes to 2, 3, and 1.5 at 5 s, 12 s, and 20 s, respectively, encompassing both positive and negative changes occurring in the controller as illustrated in Figure 10. The intervals of the set-point changes are precisely planned to ensure that the step responses can attain their steady-state values, which are tabulated in Table 3.  
Here, each controller structure can control the output for achieving each set-point change following the step response. The proposed PIDD\({}^{\kappa}\) controller outperforms its competitors, delivering the best overall performance. Further, to simul  
Figure 11: Disturbance rejection of controller structure for second-order system.  
Figure 10: Variable set-point tracking of controller structure for second-order system.  
external disturbances occur, a disturbance magnitude of \(-0.2\) is introduced to the controller structure at 15 s for disturbance rejection. Figure 11 demonstrates that each controller structure can reject the disturbance by controlling the output to attain the set-point following the step response. The proposed PIDD\({}^{\alpha}\) controller stands out from its competitors, delivering the best overall performance.