## 1 Introduction  
Unwanted oscillatory behavior is the most common reason for instability in many applications, especially electrical circuits. A stable oscillatory motion (limit cycle) has been observed in many practical systems, for instance, in electrical circuits and systems such as buck converters [1], Chua's circuits [2], power systems [3], bang-bang clocks and data recovery circuits [4], and Micro-electro mechanical systems (MEMS) oscillators [5]. The intense research on the limit cycle behaviors focuses on establishing their existence, investigating their stability and bifurcations analysis [2], and so on. For single-input single-output (SISO) systems, the existence problem was investigated by the well-known describing function (DF) method from the classical control techniques [6]. The DF method leads to results which are not rigorous in nature but the empirical evidence over many decades has proven the power of the DF method. The actuator input saturation is the main reason for the limit cycle development. All actuator devices are subject to amplitude saturation [7]. For example, current, voltage, and every conceivable physical input is ultimately limited. The reliable operation and acceptable performance of most systems must be accessed in the light of actuator saturation. Anti-windup (AW) techniques have been proposed to account a priori for the presence of saturation nonlinearity. AW designs based on model-based linear matrix inequality (LMI) methods [8], Internal model control (IMC) method [9], and sector methods are becoming popular in recent years. As mentioned in [9], most of the abovementioned works assume that if the linear design is robust, then the constrained system with AW compensation will continue to be robust. However, this view was proven wrong in [10]. Input saturation is a common behavior, and significant nonlinearity exists in all practical control systems because of the physical limits of actuators. The saturated actuator can break the feedback loop and hence the closed-loop system becomes unstable or the limit cycle occurs. This finally degrades the performance (anti-windup) when the controller is implemented. AW design with uncertainty is a challenging problem at least in a non-conservative way. It is known that integral quadratic constraint (IQC)-based design characterizing the saturation and uncertainty is a less conservative and promising one [11]. But, in general, this approach is non-convex and thus intractable. By converting the AW problem into a graphical-style design, for the SISO systems, the non-convexity associated with an LMI-based approach is effectively and largely avoided. One such a graphical design method for the uncertain system is the quantitative feedback theory (QFT). QFT design naturally results in relatively simple design structure and low-order AW compensator which is useful from a practical point of view. To name an earlier QFT AW design approach, the describing function-based approach to analyze the limit cycle for an uncertain system is proposed in [12]. The saturation nonlinearity is replaced by an uncertain gain as part of the open-loop uncertainty and the QFT design is performed in a two degrees of freedom (2-DOF) structure as shown in Figure 1. In this figure, the uncertain system, the feedback controller, and the prefilter are denoted by \(P\), \(G\), and \(F\), respectively. Moreover, the saturation input and output are represented by \(U\) and \(V\). The 2-DOF structure does not make it possible to avoid limit cycling in a conditionally stable system. This work is extended for the 3-DOF structure in [13], similar to the one shown in Figure 2, suggested by [PERSON] [14]. In this figure, \(C\) and \(H\) represent the feedback controller and the AW controller, respectively. In addition, harmonic balance and the multiplier technique are applied to analyze the limit cycle within the QFT framework in [15].