## 4 Linear and Constrained Controller Design  
_Stability and Absolute Stability of the Closed Loop with MRDP Controller_  
The selection of the controller's low-pass filters is usually neglected when designing and tuning PID controllers. By default, such a filter is only present in the derivative term, which is usually realized by the following relation: \(sT_{D}/(sT_{D}/N+1)\), where \(N\approx 5\)-\(100\)[31]. As mentioned above, the generalization of such an approach to the design of PIDA or higher-order controllers can be difficult. Designing a separate filter for each of the derivative components might be inefficient due to the complex computations involved. Another issue regarding the effectiveness of separate filters arises from noise attenuation, as filters with a higher-order denominator prove to be more effective. An important consideration for \(m>0\) and \(n\geq m\) is also the choice of \(T_{e}\) (or \(T_{f}=T_{e}/n\)), which guarantees the stability of the closed loop. The control signal constraints and anti-windup protection can have a significant impact on the closed-loop performance when the control signals are limited [25]. Therefore, consideration of the constraints deserves due attention when designing the controller. This can be done using the circle criterion of absolute stability [9; 32; 33]. Here, the \(PD_{n}^{m}R\) controller from Figure 1 must be converted into the standard non-linear form, which expresses deviations from a required steady state and is composed of the saturation non-linearity and the associated linear loop blocks [17; 18]  
\[L_{s}(s)=PD_{n}^{m}(s)S(s)-\frac{1}{1+T_{i}s} \tag{18}\]  
The use of HO-ARCs on IPDT models in [17; 18] has shown that their MRDP design for odd \(m\) directly satisfies the conditions for absolute stability. However, for \(m>1\), this is only true for suitably chosen \(T_{e}\). Figure 7 shows the Nyquist curves \(L_{s}(s)\) corresponding to \(S(s)\) (9) drawn for the controller parameters corresponding to the IPDT model of the SOPDTUZ process (12) for the set of \(T_{em}\) values  
\[ \begin{array}{l}m=1,T_{c1}=0.1;\\ m=3,T_{c3}=0.3;\\ m=5,T_{e5}=0.9;\\ m=7,T_{e7}=1.4.\end{array}  \tag{19}\]  
These curves do not cross the critical line representing the critical circle (CC) limit case with the center at minus infinity, so they guarantee the absolute stability of the considered circuit. A further reduction in some \(T_{em}\) values in order to speed up the responses would lead to the intersection of the corresponding Nyquist curve with the vertical line with the real value \(-1\) and thus to a deterioration of the control performance. Figure 6: \(PD_{n}^{m}R\) controllers for the process (2) approximated by the IPDT model and tuned according to (11) for \(T_{e}\) (17), speed–wobbling (**left**) and speed–effort (**right**) characteristics in the not distorted (smooth transients) region, and a measurement noise with \(|\delta|\leq 0.05\) (**top**) and \(|\delta|\leq 0.5\) (**bottom**); black curves correspond to \(m=1\), blue to \(m=3\), green to \(m=5\), and red 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 Nyquist curves in Figure 8 have the calculated parameters of the \(PD_{n}^{m}R\) controllers when considering the original SOPDTUZ process (12). They fulfil the absolute stability conditions for a slightly increased set of values \(T_{em}\)  
\[ \begin{array}{l}m=1,\,T_{e1 min}=0.1;\\ m=3,\,T_{e3 min}=0.5;\\ m=5,\,T_{e5 min}=1.0;\\ m=7,\,T_{e7 min}=1.6.\end{array}  \tag{20}\]  
which correspond to the minimum values considered in (17). In Figure 9, this set of \(T_{e}\) values is illustrated by responses in the time domain. With the exception of \(T_{e}=0.1\) for \(m=1\), the significant difference between the actual SOPDTUZ process and the IPDT model used to design the \(PD_{n}^{m}R\) controller is compensated by slowing down the transients (increasing the equivalent delay \(T_{em}\)).