## 3 Illustrative Example  
Consider the non-interacting control system of Figure 3 with the plant uncertainty, \(P\), is given by  
\[P(s)\;=\;\frac{5\big{(}\frac{s}{a}\;+1\big{)}}{s\big{(}\frac{s}{b}\;+1\big{)}},\;a\;\in\;\;[0.5,\;0.7],\;b\;\in\;[0.04,\;0.1]. \tag{9}\]  
This system is much studied and a challenging one to address the problem statement [13; 16]. The nominal values are \(a_{0}=0.5\), \(b_{0}=0.1\) and the control input saturation limits are \(\pm 1\). As indicated in [11] many high-performance systems are forced to be conditionally stable. The well-known difficulty is that an unstable behavior will appear when the system is driven by large signals, which drive it into saturation [13]. Figure 4: Saturation nonlinearity representation using the describing function \(N(A)\). The controller \(G(s)\) considered here is same as the one in [13],  
\[G(s)=\frac{6(s\ +\ 5)}{s}.\]  
The Nyquist plot of the open-loop transmission function \(L=PG\) (for 15 randomly selected plants) is shown in Figure 5 along with the \(-1/N(A)\) locus. As seen, the intersection of \(L\) with the \(-1/N(A)\) locus at some frequency illustrates that the closed-loop system may experience a limit cycle if the control signal saturates. This is confirmed in the step response (amplitude of 10) of the closed-loop system (nominal case) as shown in Figure 6a and a diverging limit cycle appears with the saturated input in Figure 6b. Figure 7 shows the bounds for the design of an anti-windup (AW) controller (\(E\)) for \(\mathbf{x}=10\) at each design frequency \(\omega=[0.5,\,1,\,10]\). The bounds on \(E\) are generated using the proposed inequality (Equation (8)). The AW controller (\(E\)) is shaped such that is satisfies the bounds at each chosen design frequency. Additionally, the AW controller follows the low-order structure, i.e., gain with integrator and one real pole. The designed AW controller \(E\) is given as follows. \[E(s)=\frac{-1}{s\Big{(}\frac{s}{0.4}\ +\ 1\Big{)}}. \tag{10}\]  
The Nyquist plot of the closed-loop system with the designed AW controller (Equation (10)) is shown in Figure 8 and none of them intersect the \(-1/N(A)\) locus. So, the closed loop system with the designed AW controller does not develop the limit cycle even if the plant input saturates. Figure 5: Limit cycle condition. Here, the Nyquist plot of the open-loop system \(PG\) (for 15 randomly generated plants from Equation (9)) and the line \(-1/N(A)\) are denoted by ‘solid green line’ and ‘solid blue line with \({}^{*}\), respectively. Figure 9 compares the step response of the closed-loop system (for 15 randomly generated plants) with an AW controller for different methods. Especially, the saturated control signals of the different methods are shown in Figure 9. It is observed from the figures that the response time of the closed-loop system with the proposed design is faster (almost five times) than the [PERSON] et al. method [16] and similar to the [PERSON] et al. method [13]. On the other hand, the overshoot is less than [13]. Therefore, it can be concluded that the benefits of the proposed method lie between existing methods showing higher stability and faster response time. It is finally remarked that in a practical problem the designer  
Figure 6: Closed-loop system without anti-windup controller. (**a**) Nominal closed-loop system for a step reference \(r(t)=10\times 1(t)\), (**b**) saturated control input. of the proposed control method must consider the balancing between conservativeness and heuristics in terms of performance enhancement of the uncertain linear system with input saturation. Figure 8: Limit cycle condition checking. Here, the Nyquist plot of the closed-loop system \(\frac{(PG-E)}{(1+E)}\) for 15 randomly generated plants and the line \(-1/N(A)\) are denoted by the ‘dashed red line’ and the ‘solid blue line with \(\star\)’, respectively. The Nyquist plot of the open-loop system \(PG\) is shown by the ‘solid green line’.