## 2 Proposed Design Method  
The problem is to design the feedback controller \(G\), the prefilter \(F\), and an AW controller \(E\) in a 3-DOF non-interfering structure as shown in Figure 3, such that is satisfies the robust stability and robust performance in the presence of plant uncertainty and actuator saturation. A two-step design procedure is commonly used to address the above problem statement. The first step involves synthesizing \(G\) and \(F\) using the standard QFT method, ignoring the actuator saturation. Next, an AW compensator is designed using the QFT to counteract saturation when it occurs and maintain the stability/performance.  
We propose a method to design the AW controller (second step) with the assumption that the \(G\) and \(F\) are designed using the standard QFT (first step). The proposed method begins with the design of the anti-windup compensator \(E\) for the non-interfering control structure. The open-loop transmission function is defined as \(L(s)=G(s)P(s)\). Define, \(L_{n}(s)\), the transfer function between V(s) and \(-\)U(s) when the loop is opened at the plant input, as the loop transmission around the saturation element as shown in Figure 3.  
\[L_{n}(s)=-U(s)/V(s)=(G(s)P(s)-E(s))/(I+E(s)) \tag{1}\]  
and it can be written further as,  
\[L_{n}(s)=(L(s)-E(s))(1+E(s)) \tag{2}\]  
as in [13], we will also make use of describing function (DF) analysis to avoid the limit cycle. The DF of the saturation nonlinearity is shown in Figure 4 and its value is given as  
\[N(A)\;\in\;[0,1], \tag{3}\]  
where the amplitude of the input sine signal to the saturation is \(A\) and the first harmonic output is denoted as \(N(A)\). So, the saturation nonlinearity can be taken as an uncertain gain in the open-loop transmission function [13].  
The condition for the limit cycle avoidance says that if the open-loop frequency function around the nonlinearity intersects \(-1/N(A)\), then the limit cycle may develop if saturation occurs [13]. This is given as follows  
\[L_{n}(s)\;\
eq\;-1/N(A), \tag{4}\]  
where \(N(A)\) is denoted as the gain uncertainty defined by Equation (3). Replacing the \(L_{n}\) term in Equation (4) with Equation (2) yields the following equation.  
\[\frac{L(s)\;-\;E(s)}{1\;+\;E(s)}\
eq\frac{-1}{N(A)}. \tag{5}\]  
The Laplace operator '\(s\)' and \(A\) associated with the \(N(A)\) are dropped hereafter unless its required. Equation (5) can be manipulated (simplified) as follows  
\[N(PG\;-\;E)\;\
eq\;-1\;-\;E. \tag{6}\]  
This can be written further as,  
\[(1\;+\;E)\;+N(PG-\;E)\;\
eq\;0\]or equivalently  
\[(1\;+\;NPG)\;+E(1\;-\;N)\;\
eq\;0.\]  
This is equivalent to the sensitivity function specifications in terms of absolute value,  
\[\left|\frac{1}{(1\;+\;NPG)\;+\;E(1\;-\;N)}\right|<\;\infty. \tag{7}\]  
Following the same line of argument in [13], the right-hand side of Equation (7) is replaced by a smaller value (x dB), that is given by  
\[\left|\frac{1}{(1\;+\;NPG)\;+\;E(1\;-\;N)}\right|<\;x. \tag{8}\]  
The design of the anti-windup controller \(E\) such that is satisfies the above equation and the design procedure is complete. This inequality looks similar to the one in [13] but it is derived from the non-interfering control structure (similar to Figure 3) directly as opposed to the 3-DOF structure (similar to Figure 2) followed in [13] which follows \(G\) and \(F\) design first, next the anti-windup design \(H\), and one more design element \(C\). This simplicity brings the inherent benefit of the proposed design method.