## 4 Linear and Constrained Controller Design
The choice of the minimum value \(T_{e}=0.1\) for \(m=1\) was motivated by the analysis of the SE and SW characteristics in Figure 6, which correspond to a circuit with non-zero measurement noise. Although the \(PD_{1}^{1}R\) controller guarantees absolute stability conditions also for lower values of \(T_{e}\), it is not necessary to reduce them, since the SE and SW curves corresponding to equally low \(IAE_{d}\) values and \(m>1\) already give lower values of excess increments. It turns out that for \(m>1\) the circle criterion, which is a generalization of the Nyquist criterion for circuits with a sector non-linearity and a linear part \(L_{s}(s)\), can also be useful in the design of HO-AR controllers. According to the Nyquist curves in Figure 8, the circuit fulfils the stability conditions when the Nyquist curve passes the real axis on the right side of the critical point \((-1,0j)\). The conditions of absolute stability are fulfilled if the Nyquist curve lies on the right-hand side of the vertical line with the real value \(-1\). Figure 7: Nyquist curves of the linear part \(L_{s}(s)\) (drawn with the help of the average residence time equivalence \(T_{f}=T_{e}/n\) for the IPDT process (12), \(T_{dp}=1.526\) and \(PD_{n}^{m}R\) controllers tuned with \(T_{e}=T_{emmin}\) values (20). A non-linearity \(u=F(\tilde{e})\) is said to satisfy the sector conditions \([\alpha,\beta]\) if it satisfies the conditions \(\alpha\leq F(\tilde{e})/\tilde{e}\leq\beta\). While the position of the Nyquist curve is evaluated according to the critical point \((-1,0j)\), the circle criterion replaces this with a critical circle (CC), which is defined by the vertices \(-1/\alpha,-1/\beta\) (lying on the real axis and related to the sector \([\alpha,\beta]\)). As already mentioned, the control loop is stable if the linear part of \(L_{s}(s)\) crosses the real axis on the right-hand side of the critical point. To guarantee absolute stability, the same requirement applies to the critical circle and the Nyquist curve cannot even pass through CC. Absolute stability guarantees that there is a \(c>0\) and \(\gamma>0\), so that each transient of the system fulfils the requirement of a monotonic decrease in the equivalent deviation \(\tilde{e}\)  
\[|\tilde{e}(t)|\leq ce^{-\gamma t}|\tilde{e}(0)|,\forall t>0. \tag{21}\]  
Such a monotonic decrease can also be used for tuning the loop performance, e.g., to determine the minimum values of the tuning parameter \(T_{em}\) for unconstrained control. The considered saturation non-linearity fulfills the sector conditions \([0,1]\). Such a sector corresponds to the critical circle defined by the points \(-1/\alpha=-\infty\) and \(-1/\beta=-1\) of the real axis, with a radius \(R=|\alpha-\beta|/2\to\infty\). Such a limit-critical boundary is a vertical line that intersects the point \(-1\) of the real axis (Figure 8). For \(m=3\) and \(m=7\), the nominal Nyquist curves (solid lines) lie on the right-hand side of CC. For \(m=5\), the Nyquist curve crosses it. Therefore, for \(m=5\), the only permissible sector is \([\epsilon,1]\), where \(\tan(\epsilon)=1/\tilde{e}_{0}\) limits the maximum permissible value of the deviation \(\tilde{e}_{0}\). The limit of absolute stability CC should be replaced by a smaller circle with the vertices \(-1/\epsilon\) and \(-1\). Figure 8: Nyquist curves of the linear part \(L_{s}(s)\) (drawn with the help of the average residence time equivalence \(T_{f}=T_{e}/n\) for the SOPDTUZ process (12) and \(PD_{n}^{m}R\) controllers tuned with \(T_{e}=T_{emmin}\) values (20) (full) and values decreased for \(m>1\) by 0.1 (dashed). The conclusions regarding the absolute stability are relatively accurate if the smoothness of the process output and process input curves is taken into account. This is illustrated in Figure 9 by the responses obtained for \(m=7\) and \(T_{e}=1.6\). For \(T_{e}=1.5\), the process input is already oscillating. The same can be observed for \(m=5,T_{e}=1\).