## 3 Analytical Tuning of Series PIDA Controller
### Illustrative Examples of PIDA Control  
Figure 4 shows example of setpoint and disturbance step responses for an IPDT process (\(T_{dp}=1\)) and a PIDA controller with different prefilter tuning completed by a fourth-order filter with \(T_{e}=0.8\) tuned by the average residence time equivalence \(T_{e}=4T_{f}\). Control signal (process input) \(u(t)\) serves in the first phase of transients to stabilize the output around the new setpoint signal value \(w=1\). Because this signal also enters the disturbance observer, a \"phantom\" disturbance signal is reconstructed at its output with a delay given by the time constant \(T_{i}\), despite the fact that no external disturbance actually occurs. By using a pre-filter with canceling of several poles of the closed circuit, the setpoint responses are accelerated, but the sensitivity of the circuit is also increased, which is also manifested in the partial deformation of the control signal transients at the fastest setting. From the point of view of the output, however, the responses remain monotonic after the setpoint change. After the disturbance step, the output monotonically returns to the required setpoint value with gradual reconstruction and compensation of the external disturbance value. The correct values of the external constant disturbance are reconstructed only after reaching a sufficiently close neighborhood of steady states. When wishing to compensate for piecewise constant input disturbances \(d_{i}\in[-0.2,0]\), the control signal constraints must allow the compensating signal \(-d_{i}\) to be set and at the same time leave a sufficient reserve for imposing the required dynamics of the circuit, i.e., to fulfill for all considered \(d_{i}\) with a sufficient margin the conditions \(U_{min}<-d_{i}<U_{max}\). Figure 5 shows an example of setpoint and disturbance step responses for an IPDT process (\(T_{dp}=1\)) and a PIDA controller with different prefilter tuning completed by a fourth-order filter with \(T_{e}=0.8\) tuned by the average residence time equivalence \(T_{e}=4T_{f}\) with the control signal constraints chosen as (19). In the case of setpoint step responses, the limitations of the process input are manifested only when using pre-filters with canceling of several poles of the closed loop. The consequence of the limitation here is an overshoot of the output over the required setpoint value. In the case of disturbance step responses, when a significant part of the permissible range of process input is used up to compensate for an external disturbance, a substantial extension of transient length occurs. From the point of view of the control action transition from a fully saturated level to a steady state, the critical parameter of the controller is the size of the time constant \(T_{i}\)[5, 7]. From the shape of the transient of \(u(t)\) with undershoot, which manifests itself during the output \(y(t)\) with overshoot, it can be concluded that the value \(T_{i}=2.583T_{d}\) (24) is too large. **Remark 4** (Opposing requirements for choosing \(T_{i}\)). : _If we want to preserve the good properties of the stabilizing PDA controller even after the introduction of positive feedback with the time constant \(T_{i}\), the value of \(T_{i}\) must be as large as possible and, in addition, the coefficients \(K_{p},K_{d}\) and \(K_{a}\) must be increased (to achieve a quintuple real dominant pole). However, to achieve high-quality transients with a limited control signal, the value of \(T_{i}\) should be as small as possible._  
Calculated controller transfer function (24) and (25) with complex polynomial zeros \(T_{A}^{2}s^{2}+T_{D}s+1=0.0484s^{2}Td^{2}+0.4168 Tds+1\)  
\[s_{1,2}=\left(-4.305785124\pm 1.456492874j\right)/Td, \tag{31}\]  
however, does not give a direct possibility to choose some smaller time constant from the calculated parameters for \(T_{i}\), as was the case in the design of the PID controller in [13, 43].