# Discrete-Time Design of Dual Internal Model-Based Repetitive Control Systems
## 5 Simulation Results
### Case 2: Non-Minimum Phase Stabilized System  
In this case, the open-loop plant model (29) is stabilized using the conventional controller as follows:  
\[C(z)=\frac{z+1.25}{z+0.75}. \tag{41}\]  
The resulting close-loop plant model is  
\[P_{c}(z)=10^{-3}\frac{0.763z^{2}+0.145z+0.896}{2^{3}-1.079z^{2}-0.541z+0.623}. \tag{42}\]  
The closed-loop plant model (42) has three poles given by \(p_{1}=0.988\), \(p_{2}=0.841\), \(p_{3}=-0.750\), and two zeros located at \(z_{1}=-1.25\), and \(z_{2}=-0.939\). It is obvious that the model (42) has stable poles and one unstable zero (\(z_{1}=-1.25\)) confirming that (42) is a stable non-minimum phase system. Note that the stabilizing controller \(F(z)\) designed according to (34) is not applicable to the non-minimum phase system. The design method (34) results in unstable stabilizing controller \(F(z)\). In this case, we employ a zero-phase  
Figure 10: Magnitude responses of the simulated DIMRC and MPRC.  
Figure 9: Tracking errors for the minimum phase system (\(30\)) with the disturbance model \(v_{2}(t)\).  
tracking error controller (ZPETC) design technique developed in [26]. The ZPETC-based stabilizing controller has the following transfer function:  
\[F(z)=k_{r}\frac{\mathcal{D}_{c}(z)\mathcal{N}_{c}^{u}(z^{-1})}{k_{p}\mathcal{N}_ {c}^{z}(z)}. \tag{43}\]  
Here, \(k_{r}\) is the learning gain similar to the ones in (34), \(k_{p}\) is the stabilized-plant gain, \(\mathcal{D}_{c}(z)\) denotes the denominator of (42), \(\mathcal{N}_{c}^{s}(z)\) and \(\mathcal{N}_{c}^{u}(z^{-1})\) represent stable and unstable factors of \(\mathcal{N}_{c}(z)\), where \(\mathcal{N}_{c}(z)\) is the numerator of (42). The term \(\mathcal{N}_{c}^{u}(z^{-1})\) is obtained from \(\mathcal{N}_{c}^{u}(z)\) with the operator \(z\) replaced by the backward shift operator \(z^{-1}\). Using the Equation (43) and choosing the learning gain as \(k_{r}=0.2\), we obtain \(F(z)\) expressed as  
\[F(z)=0.2\frac{1.252z^{4}-0.351z^{3}-1.756z^{2}+0.239z+0.623}{10^{-3}(0.763z^{2}+ 0.716z)} \tag{44}\]  
We notice that the \(F(z)\) above is non-causal by a factor of 2. This gives \(d_{F}\) as 2, and the DIMRC structure can be adjusted according to Figure 5.  
The tracking-error performance of the non-minimum phase system (42) for two different disturbance models \(v_{1}(t)\) and \(v_{2}(t)\) are, respectively, depicted in Figures 11 and 12. From Figures 11 and 12, we notice that the DIMRC system converges more slowly and exhibits larger peaks of the tracking error during transience compared to the MPRC system. However, when the disturbance model is uncertain, the DIMRC system offers better steady-state error. These results indicate that, in term of the transient responses, the MPRC outperforms the DIMRC for both minimum and non-minimum phase systems when there is no variation in the disturbance model. Nevertheless, when the disturbance model slightly changes from its nominal value, the DIMRC significantly outperforms the MPRC, especially during the steady-state period.  
In order to highlight the controllers' performance during the steady state, the detailed comparison in terms of the steady-state time (\(t_{ss}\)) and root-mean-square of the steady-state error (rms-\(e_{ss}\)) are summarized in Table 1. The \(t_{ss}\) is defined as the time needed to reach 2.5% error (\(\pm\) 0.025). Meanwhile, the rms-\(e_{ss}\) is calculated by  
\[\text{rms-}e_{ss}:=\sqrt{\frac{1}{(n_{t_{s}}-n_{ss})}\sum_{k=n_{ss}}^{n_{t_{s}}} e^{2}(k)}, \tag{45}\]  
where \(n_{t_{s}}=t_{s}/T_{s}\), \(n_{ss}=t_{ss}/T_{s}\), and \(t_{ss}\) is the simulation period. It can be seen from Table 1 that the proposed method has a smaller mean square error for both minimum and non-minimum phase systems under uncertain periodic disturbance \(v_{2}(t)\). However, the proposed controller offers longer steady-state time compared to the MPRC. All these results indicate that the proposed controller provides better robustness against uncertain periodic disturbances, but at the expense of a slower transient response.