# An Improved Model-Free Adaptive Nonlinear Control and Its Automatic Application
## 1 Introduction
[23; 24] introduced dead-zone links and a decay factor into the MFAC control law, which makes the control algorithm stop updating to realize disturbance suppression under certain conditions. By introducing a low-pass filter, [PERSON]. [25] first filters out the high-frequency measurement noise, and then inputs the filtered data into the MFAC controller. However, the most common types of white noise in the industry are complex, and it is difficult to obtain accurate noise information. Based on the viewpoints discussed above, this paper presents an improved MFAC method with OER based on the dynamic linearization approach. Firstly, the nonlinear model is converted into an FFDL data model, which relates the output to both input and output variables. This conversion helps to enhance the tracking speed of MFAC. Additionally, the OER is introduced into the new performance index function to further improve the tracking performance by considering the output error. Secondly, in order to mitigate the impact of the output error, the two components are combined and weighted appropriately. By effectively fusing these components, the control scheme can better handle deviations caused by the output error. Moreover, recognizing the practicality of MIMO systems, this paper extends the proposed scheme to accommodate such systems. Towards the end of the paper, the validity of the proposed method is demonstrated through numerical simulations involving several typical objects. Furthermore, the method is successfully applied to drum water level control in a CFB boiler. The simulation results establish that the MFAC-OER method exhibits superior control performance compared to both MFAC and PID methods under different operational conditions. The main contributions of this work are highlighted as follows:  
1. Model-free adaptive control: At each sampling point, the equivalent dynamic linearization data model of the nonlinear system is established, and the time-varyingparameter vector is estimated online through the I/O data of the control system. The FFDL method used in this paper takes into account the relationship between the output variation at the next moment and the input and output during a certain period. Although the dimension of the data model has increased, the design and selection of parameter estimation algorithms are more flexible. It is more adaptable to the time-varying structure and parameters of the system. 2. Model-free adaptive control with output error rate: In this work, a new control law based on the MFAC method is proposed. In order to improve the tracking and convergence speed, the output error rate is introduced into the control input performance index function. Thus, the improved FFDL-MFAC-OER control scheme is obtained and validated by the circulating fluidized bed boiler drum water level control system. 3. Extension of MIMO scheme: Considering that most of the practical systems are MIMO systems and in order to improve the applicability of the method in this paper, the MFAC-OER control scheme is further extended to MIMO systems. And, it is verified by numerical simulation. The rest of the structure of this paper is as follows. Section 2 introduces the basic theory of FFDL-MFAC. Section 3 presents the improved MFAC-OER based on the output error rate, including the complete convergence and stability analysis and the promotion of the MIMO scheme. In Section 4, the effectiveness and superiority of the proposed method are verified by numerical analysis and field experiments of the CFB boiler drum water level system.