# Discrete-Time Design of Dual Internal Model-Based Repetitive Control Systems
## 5 Simulation Results  
### Case 1: Minimum Phase Stabilized System  
To validate the effectiveness of the proposed controller, the following discrete-time model of Quanser SRV02 servo plant [32] is used in the simulation:  
\[P(z)=\frac{\theta(z)}{V(z)}=\frac{10^{-4}(7.6337z+7.173)}{z^{2}-1.83z+0.8289}, \tag{29}\]  
where \(P(z)\) is an open-loop plant model, \(\theta(z)\) is an angle position (rad), and \(V(z)\) is an input voltage (Volt). The open-loop plant (29) is associated with the sampling time \(T_{s}=0.005\) s. The conventional controller used to stabilize the open-loop plant (29) is a proportional controller with the gain 10, i.e., \(C(z)=10\). The stabilized plant model \(P_{c}(z)\) (25) is obtained as follows:  
\[P_{c}(z)=\frac{10^{-3}(7.634z+7.173)}{z^{2}-1.822z+0.837}. \tag{30}\]  
The stabilized plant model (30) has a stable zero and two stable poles located at \(z_{1}=-0.94\), \(p_{1}=0.911+i0.083\), and \(p_{2}=0.911-i0.083\). Thus, the model (30) is classified as the minimum-phase system. The control objective is to track a periodic triangle reference \(r(k)\) with the fixed fundamental frequency \(f_{r}=0.4\) Hz and to suppress the uncertain periodic disturbance with known nominal frequency as \(\bar{f}_{v}=1\) Hz. We can notice that \(f_{r}\) and \(\bar{f}_{v}\) are uncorrelated, which is inline with Assumption 3. The reference \(r(k)\) is illustrated in Figure 6. The integer numbers \(N_{r}\) and \(N_{v}\) can be calculated as follows:  
\[N_{r}=\frac{1}{0.4\times 0.005}=500,\ N_{v}=\frac{1}{1\times 0.005}=200. \tag{31}\]  
Suppose that the q-filters \(q_{r}(z)\) and \(q_{v}(z)\) are equally chosen as  
\[q_{r}(z)=q_{v}(z)=0.25z^{-1}+0.5+0.25z. \tag{32}\]  
The chosen filter above gives the tracking/rejection performance with a bandwidth of 36.35 Hz. Note that the q-filter here determines the bandwidth, which selects the reference frequencies to be passed and the disturbance frequencies to be attenuated. The coefficients of the q-filter are calculated according to (6) and (7) to give a unity gain at the targeted bandwidth and a zero phase to all frequency components. The filter behaviors can be  
Figure 6: Repetitive reference \(r(k)\). seen in Figure 7. Increasing the filter's degree does not guarantee that the bandwidth is enlarged. The coefficients must be carefully tuned to give the preferred bandwidth. Here, the chosen q-filter already accommodates the targeted reference and disturbance frequencies, as specified before. Let us pick the order of HOMRC \(m\) as 2. Then, the following weights are chosen to satisfy conditions (12) and (13) :  
\[m=2\rightarrow\ w_{1}=2,\ w_{2}=-1. \tag{33}\]  
Based on (31) and (33), the polynomial \(\mathcal{S}_{d}(z)\) in (21) can now be synthesized. Next, we need to obtain \(F(z)\) to stabilize the closed-loop DIMRC system. A straightforward design for \(F(z)\) is  
\[F(z)=k_{r}P_{\epsilon}^{-1}(z)=k_{r}\frac{z^{2}-1.822z+0.837}{10^{-3}(7.634z+7. 173)}, \tag{34}\]  
where \(k_{r}\) is the learning gain and \(P_{\epsilon}^{-1}(z)\) is the inverse model of (30). It can be seen that \(F(z)\) is non-causal by a factor of 1 (\(d_{F}=1\)). Hence, the proposed DIMRC can be realized by following the controller structure shown in Figure 5. The learning gain \(k_{r}\) satisfying condition (28) is chosen as 0.95. As we have \(\mathcal{S}_{d}(z)\) and \(F(z)\), the proposed DIMRC (17) can now be constructed.