## 5 Inspection of the Constrained Loops by Circle Criterion--The Saturation Nonlinearity  
Traditional sensitivity functions \(M_{s}\) and \(M_{p}\) cannot be used to analyze the occurrence of overshooting when using PIDA control with limited input--the transfer functions of the linear loop \(L(s)\) corresponding to both possible parameterizations of the series controllers are (ideally) the same. Although we partially violate this equality by approximate factorization based on the neglect of imaginary parts, the difference in individual parameterizations can be seen rather by means of procedures common in the analysis of absolute stability of nonlinear systems with sector nonlinearity and linear part (see Figure 13 above) by circle criterion.  
The circle criterion can be considered as a generalization of the Nyquist criterion for circuits with sector nonlinearity and a linear part described by the transfer function \(L_{s}(s)\). A nonlinearity \(u=F(\vec{e})\) satisfies sector conditions \([\alpha,\beta]\), if it for all \(\vec{e}\) holds \(\alpha\leq F(\vec{e})/\vec{e}\leq\beta\). As with the Nyquist criterion, the position of the Nyquist curve is used, except that the critical point \(-1\) of the real axis is replaced by a critical circle defined by vertices \(-1/\alpha\), \(-1/\beta\) on the real axis. If the transfer function of the linear part \(L_{s}(s)\) does not have an unstable pole and the Nyquist curve must pass the critical point on the left hand to achieve closed-circuit stability with an omega increase from \(0\) to \(\infty\), the same requirement applies to the critical circle in absolute stability. This means that the Nyquist curve cannot even penetrate the  
Figure 13: Standard nonlinear loop with saturation for analysis of the absolute stability by circle criterion (above) and Nyquist curve of the linear part \(L_{s}(s)\) (below) drawn with the help of the average residence time equivalence.  
critical circle. Fulfillment of the conditions of absolute stability means that we can find \(c>0\) and \(\delta>0\) such that any closed-loop solution of the system satisfies the following relation  
\[|\vec{\varepsilon}(t)|\leq ce^{-\delta t}|\vec{\varepsilon}(0)|,\forall t>0 \tag{43}\]  
of a monotonic decrease of the equivalent disturbance \(\vec{\varepsilon}\).  
In our case, the considered saturation nonlinearity satisfies sector conditions \([0,1]\). Such a sector corresponds to the critical circle defined by the points \(-\infty\) and \(-1\) of the real axis, with a radius \(R\to\infty\). Its boundary is therefore projected as a vertical line passing through the point \(-1\) of the real axis (see Figure 13). By transforming the circuit with PIDA controller from Figure 3 into the standard form in Figure 13, expressing deviations from a required steady-state, the combination of PDA controller (3) with \(Q_{n}(s)\) (16), the positive feedback \(1/(1+T_{i}s)\) and process (2) gives  
\[L_{s}(s)=R(s)Q_{n}(s)S(s)-\frac{1}{1+T_{i}s} \tag{44}\]  
With respect to the critical point \((-1,0j)\), apparently all Nyquist curves from Figure 13 correspond to stable linear parts \(L_{s}(s)\). However, the critical disk is only crossed by curves corresponding to the MRDP-PIDA controller. This means that there will be a problem with meeting the condition (43) of a monotonic decrease of the deviation from the steady state, which is well demonstrated by previous responses.