## 1 Introduction
The simulation results confirm that the combined approach yields quasi-optimal system performance, underscoring the effectiveness of embedding LQR within fuzzy modeling structures. The investigation performed in [26] presents a comparative evaluation of the Linear Quadratic Regulator (LQR) and the Proportional-Derivative (PD) controller, applied to the attitude control of a quadrotor, a system characterized by nonlinear dynamics. MATLAB 2014 based simulations are employed to assess key performance indicators, including settling time and overshoot percentage. The results indicate that the LQR, despite being a linear control strategy, is capable of delivering satisfactory performance in controlling the nonlinear system, thus validating its applicability in such contexts. The work presented in [27] introduces an enhancement to the Linear Quadratic Regulator (LQR) through the inclusion of an integral feedback component aimed at improving steady-state accuracy. This modification achieves a notable reduction in steady-state error--from over 40 cm to less than 1 cm--demonstrating the effectiveness of integral action as an extension to the LQR for applications demanding high precision. A comparable methodology is explored in [28], where an optimal control scheme based on the LQR is designed for the position and attitude regulation of a quadcopter UAV. In this case, integral compensation is selectively implemented for vertical position control. The simulation outcomes confirm that the proposed control structure successfully eliminates steady-state error in altitude regulation. The methodology applied in [29] presents a comparative evaluation of a state feedback controller (SFC), designed using a linearized model and a Linear Quadratic Regulator (LQR), against a conventional PID controller. The findings indicate that, with appropriate parameter tuning, the SFC markedly improves the dynamic performance of the system. Specifically, it achieves a reduction in steady-state error and delivers an enhanced transient response, thereby offering a more robust control solution compared to the traditional PID approach. Extremum Seeking Control (ESC), initially introduced in [30], constitutes an adaptive control strategy that enables a system to converge toward the optimum of a measured performance index without requiring a priori knowledge of the plant model. The method operates by iteratively minimizing a cost function through the adjustment of specific optimization parameters. A significant resurgence of interest in ESC was driven by the contribution in [31], where a formal proof of convergence was established for the standard ESC scheme utilizing periodic perturbations in the context of nonlinear systems. The application of Extremum Seeking Control (ESC) for PID tuning is explored in [32], where ESC is utilized to minimize a cost function and, thereby, identify the optimal parameters of a PID controller. The comparative analysis reveals that ESC achieves superior performance--specifically, a reduced settling time--relative to traditional tuning techniques such as Ziegler-Nichols (ZN), Internal Model Control (IMC), and Iterative Feedback Tuning (IFT). A related contribution is presented in [33], which proposes a novel PI controller design method grounded in ESC principles. This approach is formulated to address real-time optimization tasks, thereby illustrating the adaptability of ESC for dynamic control environments. As discussed in [34], enhancing process performance typically involves the application of real-time optimization (RTO) or model-based control techniques. Although these strategies can be effective, they often entail considerable computational overhead, and model-based controllers are notably susceptible to disturbances not accounted for in the model. In contrast, Extremum Seeking Control (ESC) offers a model-free alternative that circumvents these issues but is constrained by slow convergence rates.