## 3 Analytical Tuning of Series PIDA Controller
### Illustrative Examples of PIDA Control
\begin{array}{l}\tau_{i}=T_{i}/T_{d}=0.2320;\ \kappa=K_{p}K_{s}T_{d}=0.0837;\\ \tau_{D}=T_{D}/T_{d}=2.815;\ \tau_{A}=\sqrt{T_{A}^{2}/T_{d}^{2}}=0.7741\end{array}  \tag{34}\]  
The actual experiments resulted in slightly modified parameters to obtain better transients. Optimal settings are finally found from the performance portrait corresponding to the parameter ranges  
\[ \begin{array}{l}\kappa\in[0.155,0.2];\ \Delta\kappa=0.005;\\ \tau_{D}\in[2.6,3.05];\ \Delta\tau_{D}=0.005;\\ \tau_{A}\in[0.75,0.87];\ \Delta\tau_{A}=0.015;\\ \tau_{i}\in[0.33,0.45];\ \Delta\tau_{i}=0.015;\end{array}  \tag{35}\]  
The differences in \(\tau_{i}\) and \(\kappa\) values are mainly due to the simplification in computing the initial parameter estimates (34). To simplify the calculations, only one performance portrait based on unit disturbance steps with control signal constraints \(U_{min}=-1.1;U_{max}=0.1\) was created for process gain \(K_{s}=1\) and delay \(T_{dp}=1\). The controller was implemented using the second-order filter (\(n=2\)) (16) with a relatively small time constant \(T_{f}=0.1\). When the half-rule-equivalence [41] was chosen, the resulting circuit delay was  
\[T_{d}=T_{dp}+T_{f}=1.1 \tag{36}\]  
Optimal transients satisfactorily corresponded to the values  
\[TV_{1}(y_{d})\leq\epsilon_{yd};\ TV_{1}(u_{d})\leq\epsilon_{ud};\ \epsilon_{yd}= \epsilon_{ud}=0.001 \tag{37}\]  
whereby  
\[ \begin{array}{l}TV_{1}(y)=TV(y)-|2y_{m}-y_{\infty}-y_{0}|;\ y_{m}\
otin(y_{0 },y_{\infty});\ TV(y)=\sum_{i}|y_{i+1}-y_{i}|\\ TV_{1}(u)=TV(u)-|2u_{m}-u_{\infty}-u_{0}|;\ u_{m}\
otin(u_{0},u_{\infty});\ TV(u)=\sum_{i}|u_{i+1}-u_{i}| \end{array}  \tag{38}\]  
with \(y_{m}\) and \(u_{m}\) representing extreme points separating two monotonic intervals of one-pulse (1P) disturbance step responses at the process input and output (see, e.g., [13; 44]). The parameters of the optimal controller have been identified as  
\[K_{p}=0.1545;\ T_{D}=2.9150;\ K_{d}=0.4504;\ T_{A}=0.9075;\ K_{d}=0.1272;\ T_{i}=0. 3960 \tag{39}\]  
For the loop dead time (36) it yields the following dimensionless parameters:  
\[ \begin{array}{l}\kappa=K_{p}K_{s}T_{d}=0.17;\\ \tau_{D}=T_{D}/T_{d}=2.650;\\ \tau_{A}=T_{A}/T_{d}=0.825;\\ \tau_{i}=T_{i}/T_{d}=0.36.\end{array}  \tag{40}\]  
When comparing the calculated dimensionless setting with the analytically tuned controllers in Figure 7, the average-residence-time-equivalence [41] yielding \(T_{f}=T_{e}/n\) was used to calculate the time constant of the fourth-order filter \(Q_{4}(s)\) with \(T_{e}=0.8\), chosen to be comparable to the process time delay \(T_{dp}=1\). Figure 7 shows the responses of the constrained PIDA controller tuned with the MRDP method (24), the approximate tuning (33) in which the imaginary part of the numerator zeros of the MRDP controller is neglected, the PIDA controller with the PPM tuning (40) and the constrained PIDA controller modified by the conditioning technique of [14,47]. The corresponding performance measures expressing the speed and shapes of the transients are shown in Table 1.  
The design based on the parallel PIDA controller tuned with the MRDP method and modified with AW based on the conditioning technique is found to be equivalent to the series PIDA controller tuned with the MRDP method. However, the approximate design (APR) with a reduced \(T_{i}\) value (33) shows smoother responses with a minimum number of monotonic segments at the process output. In addition, the shape deviations at the process input have been partially decreased, while the \(IAE\) values partially increased. The PPM significantly reduces both the \(IAE\) values and the maximum deviations of the process output signals due to the applied disturbances, while maintaining the minimum