# An Improved Model-Free Adaptive Nonlinear Control and Its Automatic Application
## 1 Introduction  
For many practical industrial production processes, especially complex ones, there exists an increasing amount of data as well as significant nonlinearities and coupling relationships. It becomes difficult to find an accurate mathematical model (with the order and parameters being hard to obtain). This results in model-based control methods that are cumbersome to design and do not easily achieve the desired control performance. Model-free control is an effective way to settle such problems. Typical model-free methods include PID [1] and PID combined with other algorithms [2]. There are some other model-free control methods. A model-free RBF neural network control was proposed in reference [3]. In [4], a model-free \(H_{\infty}\) control method was proposed for the security control problem of systems with unknown dynamic model physical information under DoS attack. Reference [5] studied the data-driven repetitive learning controller. In addition, there is a classical virtual reference feedback rectification method [6]. Through the research and promotion of many experts and scholars, the above methods achieved different degrees of development and were successfully applied to many practical systems. Although the commonly used PID control method does not require the creation of a model, it is optimal for linear systems rather than nonlinear, time-varying, and stronglycoupled systems. As a result, its response speed is slow and control performance is unsatisfactory [7; 8]. To address these issues, two alternative control methods are proposed in [9; 10]. The first method is a supervised control approach based on the deep belief network (DBN) algorithm, which is specifically designed for systems with nonlinear disturbances. This method enhances the control performance by learning and adapting to changes in the system structure. Furthermore, [9] describes an extended state observer (ESO). The observer ensures fast convergence with sufficient speculative accuracy, thus guaranteeing stability in the presence of nonlinear air gaps within the train system. In [10], the control method integrates a radial basis function (RBF) network to tackle uncertainty problems. By incorporating the RBF network into the control scheme, the system can quickly recover from disturbances and maintain high-quality control. Another effective solution is the MFAC method. This method has the advantages of being both parameter-adaptive and structure-adaptive, as described in [11; 12]. This nonlinear adaptive control method offers flexibility and adaptability in various scenarios. MFAC was proposed by professor [PERSON] [13]. The general idea is to build an equivalent model in the working point of the discrete system. After years of development, many effective algorithms have been proposed [14; 15; 16]. And, they have been applied to practical industrial production [17; 18; 19]. For complex industrial processes with strong nonlinearity and strong coupling, research into MFAC is very important. However, the tracking speed and robustness of MFAC need to be further improved. To address the slow tracking speed of MFAC, [PERSON]. [20] proposes an adaptive control law for the output error rate based on the CFDL data model. The effectiveness of the improved control strategy is verified and satisfactory results are obtained through the dynamic model of the absorption refrigerator. However, the CFDL data model is not suitable for multi-order systems. In [21], a model-free adaptive denoising control method with an improved tracking differentiator is proposed. The simulation results show that the method can quickly track the given signal. However, if the given signal is time-varying, the tracking effect will be affected. When there is external disturbance in the system, the output error of the system will converge to a constant [22]. Therefore, the existence of disturbance will significantly affect the robustness of the MFAC scheme. Aiming at the problem of poor robustness of MFAC, [PERSON].