## 2 MRDP Design of HO-ARCs for IPDT Model
### Closed-Loop Responses Based on the IPDT Model of the Sopdtuiz Process  
The setpoint and disturbance step responses corresponding to \(PD_{n}^{m}R\) controllers (7) and (8) implemented according to Figure 1, with odd \(m\in[1,7]\), are shown in Figure 3. The controller parameters are based on the IPDT model of process (12) using different values of tuning parameter \(T_{e}\) (11) and setpoint prefilter (A7) with \(p\in[0,2]\). For simplicity, these responses for \(m>1\) do not cover the full set of possible prefilter options. To remain simple, the introductory analysis also avoids the control constraints, which were chosen to be sufficiently high (\(u\in[-10,10]\)). As a preliminary, the amplitude of the measurement noise was set to zero. In addition to the output and input of the process, the reconstructed disturbance signal \(d_{ircc}\), which is given by the negative value of the controller feedback from \(F_{R}\), is also recorded. For the setpoint step responses with the unit step setpoint \(w=1\) and the external disturbance \(d_{i}=0\), the controller steady-state value corresponding to process (12) is \(u(\infty)=-1/S(0)=-0.5\). Since the internal feedback of the Sopdtuiz process is not taken into account by the IPDT model, it forms the lumped disturbance together with the external disturbances (as with ADRC). As the steady-state value of the reconstructed disturbance \(d_{ircc}\) shows, it assumes a non-zero value during the setpoint change, although the actual external disturbance is zero. The values of the actual and the reconstructed disturbance are only equal (\(d_{ircc}=d_{i}=1\)) near to steady state at \(w=0\). For a large \(T_{e}=2T_{dp}\) (top, Figure 3), it can be seen that all processes are sufficiently smooth despite the extremely simple IPDT approximation of the process. The speed of the \(PD_{m}^{m}R\) response increases and becomes slightly more oscillatory with increasing controller order \(m\). For \(T_{e}=T_{dp}\) (center, Figure 3), the control signal of the \(PD_{7}^{m}R\) controller shows some spikes thanks to the \"aggressive\" setting. This shows that the tuning parameter \(T_{em}\) of the individual \(PD_{n}^{m}R\) controller cannot be arbitrarily small with regard to the stability of closed loops (previously analyzed for an IPDT system and \(m\in[0,2]\) in [28]). For the even more aggressive setting with \(T_{e}=0.5T_{dp}\) (bottom, Figure 3), the responses corresponding to \(m=5\) and \(m=7\) are already unstable. The closed-loop results for all setpoint responses with the simplest prefilter numerator \(p=0\) and \(N_{p}=1\) are significantly slower compared to the filtered PID according to [8]. As the value of \(p\in[0,2]\) increases, the speed of the setpoint step responses of the \(PD_{m}^{m}R\) controllers increases. It must be mentioned that not all possible prefilters for \(m>1\) are shown for reasons of clarity of the images. It should also be noted that all process outputs for setpoint step responses of \(PD_{m}^{m}R\) controllers are strictly monotonic, while [8] provides slight overshoots. This overshoot also contributes to the fact that [8] provides faster setpoint closed-loop responses. The \(PD_{n}^{m}R\) controllers provide more efficient disturbance rejection at \(T_{e}\leq T_{dp}\) than [8]. For \(T_{e}=2T_{dp}\), the comparison is not clear and requires a more detailed analysis. Due to higher values of \(n\) and \(T_{e}\), one can intuitively expect, even without a more detailed quantitative evaluation of the obtained waveforms, that controllers corresponding to higher values of \(m\) can filter the measurement noise more intensively. This is, of course, in contrast to the widely held belief that the use of controllers with a derivative component is not suitable for noisy systems [15; 29]. The following evaluation will confirm this intuitive expectation and show that \(m=1\) can lead to over-damped responses when the measurement signals need to be filtered more intensively.