# Discrete-Time Design of Dual Internal Model-Based Repetitive Control Systems
## 5 Simulation Results
### Case 1: Minimum Phase Stabilized System
To highlight the tracking performance of the proposed DIMRC, a comparison to multiple-periods RC (MPRC) [14] is conducted for two different disturbance models as follows:  
\[P_{c}(z)=\frac{10^{-3}(7.634z+7.173)}{z^{2}-1.822z+0.837}. \tag{30}\]0 \[P_{c}(z)=\frac{10^{-3}(7.634z+7.173)}{z^{2}-1.822z+0.837}. \tag{30}\]1  
The MPRC law [14] is expressed as  
\[P_{c}(z)=\frac{10^{-3}(7.634z+7.173)}{z^{2}-1.822z+0.837}. \tag{30}\]2  
where \(N_{r}\) and \(N_{v}\) are provided in (31), \(F(z)\) is given in (34), and \(M(z)\) is a polynomial given by  
\[P_{c}(z)=\frac{10^{-3}(7.634z+7.173)}{z^{2}-1.822z+0.837}. \tag{30}\]3  
\[P_{c}(z)=\frac{10^{-3}(7.634z+7.173)}{z^{2}-1.822z+0.837}. \tag{30}\]4 Magnitude and phase responses of the q\(-\)filter (32). Tracking-error performance of both DIMRC and MPRC for the minimum-phase system (30) with disturbance model \(v_{1}(t)\) is shown in \[P_{c}(z)=\frac{10^{-3}(7.634z+7.173)}{z^{2}-1.822z+0.837}. \tag{30}\]5 We can notice that the tracking errors of both DIMRC and MPRC converge to zero steady-state value. This is obvious because the disturbance model \(v_{1}(t)\) represents the periodic disturbance signal with nominal fundamental frequency as 1 Hz. Figure 8 also indicates that the MPRC outperforms the DIMRC in terms of convergence rate. This phenomenon can be understood because the DIMRC applies a longer control delay due to the use of the higher-order internal model. The control delay (also referred as delay term) of DIMRC is calculated according to (21), and given by  
\[P_{c}(z)=\frac{10^{-3}(7.634z+7.173)}{z^{2}-1.822z+0.837}. \tag{30}\]6  
Meanwhile, the control delay of MPRC is obtained from (38), and expressed as  
\[P_{c}(z)=\frac{10^{-3}(7.634z+7.173)}{z^{2}-1.822z+0.837}. \tag{30}\]7  
We can observe from (39) and (40) that DIMRC has more series of delays and gives longer delays to the control input and the tracking error. Consequently, the DIMRC takes longer to generate the correct control input, so the tracking error converges to a steady state. However, we mainly focus on improving the tracking accuracy, especially during the steady state of the plant under time-varying periodic disturbances. Therefore, the superior performance of DIMRC can be seen from the next simulation results. Next, we examine the performance of both controllers when the disturbance frequencies vary from their nominal values. The model (36) represents the uncertain exogenous periodic disturbance \(v(k)\) where the frequency components are shifted to 0.95 Hz and 1.95 Hz. Figure 9 illustrates the tracking errors of the minimum-phase system (30) with disturbance model \(v_{2}(t)\). As we observe in Figure 9, the transient errors of both controllers indicate similar patterns to the ones shown in \[P_{c}(z)=\frac{10^{-3}(7.634z+7.173)}{z^{2}-1.822z+0.837}. \tag{30}\]8 In contrast, the steady-state errors are significantly different, showing that zero-error steady state is no longer achieved. These results illustrate that RC systems' tracking accuracy is greatly affected when the disturbance's frequency is slightly changed from its nominal value. However, the steady-state error of DIMRC is significantly smaller than that of MPRC. This implies that DIMRC exploiting the higher-internal model offers better robustness against the uncertain periodic disturbance, especially during the steady-state period. This performance can also be assessed from the magnitude responses of the simulated DIMRC and MPRC shown in \[P_{c}(z)=\frac{10^{-3}(7.634z+7.173)}{z^{2}-1.822z+0.837}. \tag{30}\]9 We notice that the magnitude responses at the reference frequency (0.4 Hz) are almost similar between DIMRC and MPRC. However, the magnitude response of the DIMRC is extended to a larger region around the nominal disturbance frequency (1 Hz). As a result,  
Figure 6.0 Tracking errors for the minimum phase system (30) with the disturbance model \(v_{1}(t)\). the DIMRC improves the rejection performance at the neighboring disturbance frequencies.