## 1 Introduction  
Data-Driven Control (DDC) algorithms have emerged as a powerful alternative to traditional Model-Based Control (MBC) techniques, particularly for systems characterized by nonlinearities, modeling challenges, or inaccessible models. Unlike MBC, which relies on mathematical models, DDC techniques derive their flexibility from their reliance solely on data acquired directly from the control system. By leveraging this data, DDC algorithms dynamically assess the behavior of the controlled process to achieve the desired objectives. The foundational work in this field includes the comprehensive survey by [PERSON] [1], with an updated version presented in [2]. These surveys categorize DDC algorithms based on how data is processed (online or offline), the structure of the algorithm (fixed or unknown), and whether gradient calculations are employed during the control process. Gradient-based methods include Iterative Feedback Tuning (IFT) [3], Iterative Learning Control (ILC) [4], Noniterative Data-Driven Model Reference Control [5], Virtual Reference Feedback Tuning(VRFT) [6], and Model-Free Adaptive Control (MFAC) [7]. Non-gradient methods, on the other hand, encompass Model-Free Control (MFC) [8], Correlation-Based Tuning (CbT) [9], and Unfalsification with Analytic Update [10]. Model-Free methods, which rely exclusively on real-time data from the system, are primarily represented by two approaches: MFAC, introduced by [PERSON] [7], and MFC, developed by [PERSON] [8]. While both methods share some structural similarities, they differ fundamentally in their underlying principles. MFAC employs dynamic linearization and operates as a gradient-based method, whereas MFC utilizes an ultralocal model and is non-gradient-based. This paper seeks to clarify the distinctions between these two approaches and provide a detailed analysis of their application to nonlinear systems. The adaptive variant introduced by [PERSON] in [7] dynamically linearizes the plant using process data, with its complexity varying based on the volume of data utilized. This method has been successfully applied in diverse domains, including unmanned surface vessels in [11], absorption refrigeration systems [12], flying robots [13], parafoil systems [14], multivehicle systems with constraints [15] and quadrotor formations [16]. Beyond its standard implementation, MFAC has been integrated with other control techniques such as Iterative Feedback Tuning (IFT) [17], Predictive Control (PC) [18], Sliding Mode Control (SMC) [19; 20], Iterative Learning Concept (ILC) [21], Fault Tolerant Control (FTC) [22], Virtual Reference Feedback Tuning (VFFT) [23] or Error Minimized Regularized Online Sequential Extreme Learning Machine [24]. The method is well-established, with [7] providing stability analyses for all MFAC variants. On the other hand, MFC has been applied in areas such as lateral vehicle control [25], ramp metering [26], variable-speed wind turbines [27], quadrotors used for acrobatic flight [28], machine tools [29], and non-minimum phase systems and switched systems [30]. Also, the MFC approach can be used together with other control techniques to be improved, such as in the case of SMC [31], VRFT [32] or IFT [33], to enhance its performance. While MFC is relatively newer compared to MFAC, ongoing research, including [34; 35], is addressing its theoretical foundations, particularly in stability analysis. This paper conducts a comparative analysis of gradient-based (MFAC) and non-gradient-based (MFC) Model-Free methods. MFAC employs dynamic linearization techniques, such as the Compact Form Dynamic Linearization (CFDL) and Partial Form Dynamic Linearization (PFDL), which utilize pseudo partial derivatives (PPD) and pseudo gradients (PG), respectively. To optimize performance, two algorithms are introduced to determine the initial values of PPD and PG, addressing the influence of these parameters on system dynamics, as highlighted in [36; 37].