## 3 Analytical Tuning of Series PIDA Controller
### Evaluation of the Dynamics of Transient Responses  
Since the MRDP-PIDA controller is now used to control a stable system, the overshoot of the output temperature (beyond the set point, as seen in Figure 9) is lower than it would be in the case of the actual integrating process. At higher \(T_{e}\) values, it may even be absent altogether. Instead of absolute values of the shape deviations \(TV_{0}(y)\) and \(TV_{1}(y)\) (38), the relative shape deviations will be used defined as  
\[ \begin{array}{l}tv_{0}(y)=TV_{0}(y)/|y_{\infty}-y_{0}|;\;TV_{0}(y)=TV-|y_{ \infty}-y_{0}|;\\tv_{1}(y)=TV_{1}(y)/|2y_{m}-y_{\infty}-y_{0}|;\\tv_{1}(u)=TV_{1}(u)/|2u_{m}-u_{\infty}-u_{0}|\end{array}  \tag{42}\]  
These numbers indicate how much the total sum of the absolute values of the individual increments is greater than the total useful change of the given signal. Ideally, the relative shape deviations should take on zero values. The larger the given numbers, the greater was the range of useless redundant changes of the given signals. Shape deviations of individual responses are reduced by the fact that they take place to a considerable extent with the (same) limit value of the control signal and their dynamics  
Figure 8: Temperature control corresponding to the series PIDA controllers from previous example with the simplest prefilter (26) and (27), a setpoint step from \(w=30\) °C to \(w=40\) °C at \(t=100\) followed by a setpoint step to \(w=30\) °C at \(t=200\), a setpoint step \(w=35\) °C at \(t=500\) and a disturbance step produced by a fan control \(u_{f}=10\) applied at \(t=600\); \(T_{d}=1\) s; \(T_{e}=0.6\) s; \(n=4\); \(P_{n}(s)=(1+T_{f}s)^{4}\); \(T_{f}=T_{e}/4\); \(K_{sp}=K_{s}=0.1\). are about the same. The lowest shape deviations of the MRDP-PIDA controller (see Table 2) are the result of a longer actuating of the (smooth) limit value of the control signal. The slow control signal decrease after the setpoint is reached by output is due to thermal conduction neglected by the IPDT model (the actual process is second-order with an additional zero). During the setpoint decrease from \(w=40\)\({}^{\circ}\)C to \(w=30\)\({}^{\circ}\)C (see Figure 10 and Table 3), when the control signal is mostly \(u=0\), the higher order of the actual process associated with multiple heat transfer modes is most evident. When testing the MRDP \begin{table}
\begin{tabular}{c c c c c} \hline - & \(\mathbf{IAE_{1}}\) & \(\mathbf{tv_{0}(y_{1})}\) & \(\mathbf{\Delta\eta(y_{1})}\) & \(\mathbf{tv_{1}(u_{1})}\) \\ \hline MRDP-PIDA & 76.2942 & 0.5854 & 3.2934 & 0.5257 \\ \hline APR-PIDA & 75.4978 & 0.6154 & 0.5000 & 0.5317 \\ \hline PPM-PIDA & 73.8782 & 0.6633 & 0.2994 & 1.4753 \\ \hline \end{tabular}
\end{table} 
Table 2: Performance measures \(IAE_{1}\), \(tv_{0}(y_{1})\), \(\Delta\eta(y_{1})\) and \(tv_{1}(u_{1})\) corresponding to the first setpoint step with \(y_{1}=y(t_{1})\), \(u_{1}=u(t_{1})\), \(t_{1}\in[100,200)\) s, \(\Delta\eta(y_{1})\)—percentage overshoot over the setpoint value. Figure 9: Detail of temperature control corresponding to the proposed series PIDA controllers with the simplest prefilter (26) and (27) and a setpoint step from \(w=30\)\({}^{\circ}\)C to \(w=40\)\({}^{\circ}\)C at \(t=100\)\(T_{d}=1\) s; \(T_{e}=0.6\) s; \(n=4\); \(P_{n}(s)=(1+T_{f}s)^{4}\); \(T_{f}=T_{e}/4\); \(K_{sp}=K_{s}=0.1\). PIDA controller, the process was significantly cooler than with the APR-PIDA and PPM-PIDA controllers, which gives the lowest \(IAE\) value. Since the identified parameters of the IPDT model actually correspond to heat dissipation by radiation, the controlled system is significantly slower than the approximate model and there are no major undershoots in process temperature below the set point--the \"fast\" controller has enough time to stop the fall.