## 3 Analytical Tuning of Series PIDA Controller
### Illustrative Examples of PIDA Control
The replacement of the constants appearing in the role \(T_{i}\) can only be realized by neglecting the imaginary part of the pole (31), when we use the approximation  
\[T_{A}^{2}s^{2}+T_{D}s+1\approx 0.0538s^{2}T_{d}^{2}+0.4640T_{d}s+1=(0.2320T_{d}s +1)^{2} \tag{32}\]  
Figure 4: IPDT process \(\dot{y}(t)=u(t-1)+d_{i}\) and series PIDA controller with different prefilter tuning corresponding to a setpoint step to \(w=1\) at \(t=0\) followed by a disturbance step \(d_{i}=-0.2\) at \(t=20\); \(T_{e}=0.8\); \(n=4\); \(P_{n}(s)=(1+T_{f}s)^{4}\); \(T_{f}=T_{e}/4\); \(K_{sp}=K_{s}=1\). In such a case, we recalculate parameters of the controller transfer function (24) as follows:  
\[ \begin{array}{l}T_{i}=0.2320T_{d};\ K_{p}=\frac{0.932273\ 0.232}{2.583K_{s}T_{d}}= \frac{0.0837}{K_{s}T_{d}};\\ T_{\rm D}=2.583 Td+0.232 Td=2.815T_{d};\ K_{d}=K_{p}T_{D}=0.2356/K_{s};\\ T_{A}^{2}=2.583 Td\ 0.232T_{d}=0.599T_{d}^{2};\ K_{a}=T_{A}^{2}K_{p}=0.0502T_{d}/K_{s} \end{array}  \tag{33}\]  
Transient responses in Figure 6 indicate the removal of output overshoots and control signal undershoots, thereby achieving the same number of monotonic sections of changes at the input and output as with transient responses without control signal saturation. However, setpoint step responses are somewhat slower than with unmodified controller parameters. Figure 5: IPDT process \(\dot{y}(t)=u(t-1)+d_{i}\) and constrained series PIDA controller with different prefilter tuning corresponding to a setpoint step to \(w=1\) at \(t=0\) followed by a disturbance step \(d_{i}=-0.2\) at \(t=20\); \(T_{e}=0.8\); \(n=4\); \(P_{n}(s)=(1+T_{f}s)^{4}\); \(T_{f}=T_{e}/4\); \(K_{sp}=K_{s}=1\); \(U_{max}=0.22;U_{min}=-0.02\). Calculation of the Series PIDA Controller for the IPDT System Using the Performance Portrait Method (PPM)  
Since the imaginary part of the poles of the polynomial \(T_{A}^{2}s^{2}+T_{D}s+1\) should be neglected to get a polynomial with real time constants applicable as \(T_{i}\), the analytical retuning of the PIDA controller according to [13; 43] provides limited performance. Although we can calculate the optimal controller settings using analytical methods just approximately, they are still useful, by offering simplifications and time savings, when we search for optimal solutions using numerical methods. The core of the performance portrait method (PPM) is to search all relevant solutions and find the best one based on performance measures. An important advantage of numerical optimization of controller parameters based on a set (database) of system responses [44; 45; 46] is that it can be easily reused in case of changed specifications for the target system performance. In the first phase of the method, the transient responses are determined by simulation or experimentation on a real process with different relevant sets of control parameters. Then, the closed-loop step responses are evaluated using performance measures that approximate actual real-world requirements. The obtained data are stored in a set (database) of relevant solutions, also referred to as a performance portrait (PP). After creating a sufficiently large database, it is then relatively easy to find a controller that comes as close as possible to the expected desired performance. Of course, such a controller should be included in the final database of solutions. An important performance specification is that the closed-loop responses should be as fast as possible, i.e., that the minimum \(IAE\) value (1) for the setpoint and disturbance responses or their weighted combination is guaranteed. Furthermore, the selected con troller should guarantee smooth process responses with a minimum number of monotonic intervals at the process input and output. The estimation of the optimal parameters of the series PIDA controller can start with the modified parameter set (33). To simplify the notation, we introduce here dimensionless parameters  
\[