## 3 Speed-Effort and Speed-Wobbling Characteristics
However, the oscillations of the process inputs and outputs increase. It is therefore not easy to find the optimum value of \(T_{em}\) for individual values of \(m\). To illustrate the effects of noise in the controller design, we will first evaluate the SE and SW characteristics without measurement noise (Figure 5). It can be seen that as the filter degree \(n\) increases, the excessive increments of the process input and output signals increase. The reason for this is that the numerical integration used in computer simulations is never perfect. By choosing a sufficiently fine step and low permissible simulation errors, we try to reduce the numerical errors below the limit that would lead to a loss of significance of the observed transients. Increasing the derivative degree \(m\) increases the excessive control effort and the closed loop speed (\(IAE_{d}\) decreases). However, increasing \(m\) also reduces the excessive wobbling in process output. The performance metrics of the filtered PID controller (4) and (13), marked with a \"+\" sign, show that the excessive control effort at the controller output is lower than when using \(PD_{n}^{m}R\) controllers. This confirms the first visual impression from the responses in Figure 3.  
Next, to illustrate the control problem in the POCD context with non-zero measurement noise, the SW and SE characteristics in Figure 6 are calculated at \"smaller\" (\(|\delta|\leq 0.05\)) and \"larger\" (\(|\delta|\leq 0.5\)) amplitudes of measurement noise. This noise was generated in Matlab/Simulink using the \"Uniform Random Number\" block with a sampling period \(T_{s}=0.001\). Figure 4: Disturbance responses of \(PD_{1}^{1}R\) controller for the inverse-response process (2) approximated by the IPDT model and tuned according to (11) for \(T_{e}\in[0.1,2.0],\Delta T_{e}=0.1\): details of the process output and input, \(T_{s}=0.001\), no noise. Figure 5: Speed–wobbling (**left**) and speed–effort (**right**) characteristics of \(PD_{n}^{m}R\) controllers for the process (2) approximated by the IPDT model and tuned according to (11) for \(T_{e}\) (17), no noise, black curves correspond to \(m=1\), blue curves to \(m=3\), green curves to \(m=5\), and red curves to \(m=7\), combined with \(n=m\) (full), \(n=m+1\) (dashed) and \(n=m+2\) (dotted); \(T_{s}=0.001\). The performance measures corresponding to filtered PID controller (4), (13) are denoted by ‘+’. The amplitudes of the applied noise were chosen with respect to the output amplitudes of the analyzed disturbance responses in Figure 4 and correspond to about 4% and 40% of the output amplitude. Although the adjectives \"smaller\" and \"larger\" have limited validity from the point of view of generality, the SE and SW characteristics obtained by considering non-zero amplitudes of the noise already point to generally valid conclusions: The excessive increments of the heuristically optimized PID controller are lower than in the case of equally fast \(PD_{m}^{m}R\) control, but higher than in the case of \(PD_{n}^{m}R\) controllers with \(n>m\). The given conclusion would probably be valid until the level of external noise is reduced to the level of internal noise by numerical integration. For both noise levels considered, the detailed quantitative evaluation of the responses obtained clearly shows the advantages of the new controller design method. The analysis shows that, compared to the PID controller (4), the measurement noise can be significantly reduced by using \(PD_{n}^{m}R\) controllers with \(n>m\), which enables a clear horizontal shift of the performance points to the left (by several decades of the logarithmic scale). Alternatively, by keeping a constant excessive amount, the performance point can be shifted downwards to lower \(IAE_{d}\) values. According to the SE and SW characteristics in Figure 5 (without measurement noise), the use of HO low-pass filters with \(n>m\) leads to an increase in the excessive deviations of the process output from the 1P shape.