## 3 Speed-Effort and Speed-Wobbling Characteristics  
The design of HO-AR controllers offers several possibilities for improving the setpoint responses, depending on the number of poles \(p\) that are canceled by the prefilter numerator \(N_{p}(s)\) (see Appendix A.1). For \(m=7\), for example, the pre-filter denominator degree \(deg(D_{p})\) formally allows up to \(p=8\) dominant poles to be canceled by the pre-filter numerator \(N_{p}\) (A7). However, a consistent evaluation of all possible prefilters and controllers withodd \(m\in[1,7]\) would take a lot of time and space. For the sake of brevity, we leave that to a separate paper and limit ourselves here to disturbance of measurement noise on controller tuning on disturbance responses. Note that the uncertainties in the models used for the controller design have a similar impact as external disturbances and good disturbance rejection is the first requirement for the robustness of the proposed controller design. Therefore, the following simulation analysis is based on the evaluation of disturbance responses as a function of the basic tuning parameters, i.e., the derivative degree \(m\), the equivalent delay \(T_{e}\), and the order \(n\) of the low-pass filter \(Q_{n}(s)\). The speed-effort (SE) and speed-wobling (SW) characteristics introduced in [25; 30] relate the speed of the closed-loop responses (expressed here in terms of \(IAE_{d}\)) to excessive control effort and excessive process output wobbling. They are used for detailed analysis of the tuning of higher-order \(PD_{n}^{m}R\) tuning and for comparison with the filtered PID controller (4). The SE and SW characteristics of the disturbance responses are evaluated in the \(\xi,\eta\) planes, which are labeled with the coordinates  
\[ \begin{array}{ll}\text{SE:}&\xi=TV_{1}(u_{d}),\eta=IAE_{d};\\ \text{SW:}&\xi=TV_{1}(y_{d}),\eta=IAE_{d}.\end{array}  \tag{14}\]  
The integral of the absolute error (\(IAE_{d}\)) is defined as  
\[IAE_{d}=\int_{0}^{\infty}\lvert e_{d}(t)\rvert dt\ ;\ e_{d}=w_{d}-y_{d}\, \tag{15}\]  
where \(w_{d}=0\) is the desired reference setpoint, \(y_{d}\) is the process output, and \(e_{d}\) is the control error. \(TV_{1}(y_{d})\) and \(TV_{1}(u_{d})\) represent the deviations of the process output and input from the ideal disturbance step responses of the integral first-order plants with a one-pulse (1P) shape. Such responses consist of two monotonic intervals separated by an extreme point \(y_{m}\
otin(y_{0},y_{\infty})\), or \(u_{m}\
otin(u_{0},u_{\infty})\)  
\[ \begin{array}{ll}TV_{1}(y_{d})=\sum_{i}\lvert y_{i+1}-y_{i}\rvert-\lvert 2y_{m}-y_{ \infty}-y_{0}\rvert;\\ TV_{1}(u_{d})=\sum_{i}\lvert u_{i+1}-u_{i}\rvert-\lvert 2u_{m}-u_{\infty}-u_{0}\rvert \end{array}  \tag{16}\]  
The evaluation of the responses was based on performance measurements obtained with the simulation time \(t_{sim}=100\). The length of the simulation time was chosen with respect to the slowest responses corresponding to \(m=1\) and \(T_{e}=2\). Stable (non-distorted) disturbance step responses are obtained by using the following parameters \(T_{e}\) e:  
\[ \begin{array}{ll}m=1,T_{e1}\in[0.1,2.0];\ \Delta T_{e1}=0.1\\ m=3,T_{e3}=T_{e1}+0.4;\\ m=5,T_{e5}=T_{e1}+0.9;\\ m=7,T_{e7}=T_{e1}+1.5.\end{array}  \tag{17}\]  
with three different filter degrees. In the following figures, \(m=1\) corresponds to the black, \(m=3\) to the blue, \(m=5\) to the green, and \(m=7\) to the red curves in combination with the filter orders \(n=m\) (full), \(n=m+1\) (dashed), and \(n=m+2\) (dotted). The special range of values in (17) is explained in more detail in the following section. For each controller order \(m,20\) different control loop responses with gradually increasing \(T_{e}\) are measured. From the loop responses of the \(PD_{1}^{1}R\) controller (equivalent to the PID controller) in the control loop where the particular values of \(T_{e}\) are given by the vector \(T_{e1}\) (17) (see Figure 4), it can be seen that the responses are accelerated by decreasing the parameter \(T_{e}\).