However, due to the nonlinear, time-varying uncertainties; strong disturbances often present in practical systems; and the difficulty in tuning PID controller parameters with unsatisfactory performance, the application of PID control is limited in complex and high-performance systems [23]. To overcome the limitations of traditional PID control, such as poor tracking performance and slow convergence, scholars have actively combined the PID with intelligent algorithms to achieve adaptive changes in PID gains [24]. However, there are still suboptimal phenomena, such as an excessively high sampling frequency. MFAC was proposed by Professor [PERSON] in his doctoral thesis [25]. The basic idea of MFAC is to dynamically linearize a nonlinear system by introducing the concept of pseudo-partial derivatives and establishing an equivalent dynamically linearized data model. Based on this virtual data model, the controller and parameter update laws are designed. This algorithm can achieve both parameter adaptation and structure adaptation and has been applied in various industrial fields after years of development [26; 27]. However, traditional MFAC represents the system's input-outputdata and uncertainties (primarily including parameter estimation errors and external disturbances) using a dynamic linearized data model. Furthermore, parameter estimation errors and uncertainties are estimated through parameter estimation algorithms. In the case of significant system disturbances, this approach may lead to issues such as over-linearization [28; 29]. Additionally, research results indicate [30] that in the absence of external disturbances, conventional MFAC schemes can converge the system output error to zero. However, when measurement disturbances are present, applying conventional MFAC schemes will cause the system's output error to converge to a non-zero constant. Therefore, the presence of measurement disturbances significantly deteriorates the control performance of the MFAC scheme. In [31], the advantages of MFAC and iPID were combined to propose an MFAC PID scheme based on compact-form dynamic linearization (CFDL). This approach extends the applicability of iPID to nonlinear and non-affine systems, addressing the issues of poor robustness in the PID and weak disturbance rejection capability in MFAC. However, in the dynamic linearization process in [31], only the output at the next time instant in relation to the input at the previous time instant was considered, resulting in a relatively simplistic design. Based on the analysis presented above, this paper proposes an iFFDL-MFA-PID scheme by combining the advantages of MFAC and iPID. Firstly, we introduce the gradient estimation algorithm and time difference algorithm based on the general formula for nonlinear systems to effectively estimate time-varying parameters and uncertain nonlinear terms. Moreover, by incorporating additional tracking error information using the iPID algorithm, the iFFDL-MFAC system compensates for the decrease in system information accuracy caused by excessive linearization. Finally, we redefine the learning update law and parameter update law of the system to improve the controller structure. Simulation results demonstrate that the proposed control strategy, iFFDL-MFA-PID, more effectively suppresses the harmful effects of measurement disturbances on system stability, thereby achieving more accurate tracking of the system output. The main contributions of this paper are as follows:  
1. In the dynamic linearization process, this study fully considers the relationship between the system's input and output, resulting in a more flexible controller design process. Compared with the CFDL data model previously used, the FFDL method adopted in this study comprehensively considers the relationship between the output change at the next moment and the input and output within a fixed-length sliding time window. The FFDL data model is insensitive to the system's structure and parameter time-varying characteristics. 2.