Fuzzy Adaptive Tracking Control of Uncertain Strict-Feedback Nonlinear Systems with Disturbances Based on Generalized Fuzzy Hyperbolic Model

In this paper, a fuzzy adaptive tracking control for uncertain strict-feedback nonlinear systems with unknown bounded disturbances is proposed. The generalized fuzzy hyperbolic model (GFHM) with better approximation per-formance is used to approximate the unknown nonlinear function in the system. The dynamic surface control (DSC) is used to design the controller, which not only avoids the “explosion of complexity” problem in the process of repeated derivation, but also makes the control system simpler in structure and lower in computational cost because only one adaptive law is designed in the controller design process. Through the Lyapunov stability analysis, all signals in the closed loop system designed in this paper are semi-globally uniformly ultimately bounded (SGUUB). Finally, the effectiveness of the method is verified by a simulation example.


Introduction
As an effective tool to solve the uncertainty of nonlinear systems, fuzzy logic systems are widely used in adaptive control design [1] [2] because of their good approximation capabilities. T-S (Takagi-Sugeno) fuzzy logic controller is widely used as a nonlinear function approximator [3] because it has less learning parameters to adjust online. The adaptive T-S fuzzy adaptive control method is proposed for the pure feedback nonlinear system in [4] and the uncertain MIMO

System Description
This paper considers the following SISO strictly feedback uncertain nonlinear systems with unknown disturbances: is the state variable of the system, u is the control input and y is the output variable of the system.
where the constant 0 s η > .
Definition 1 [6]: A given system has n input variables  Lemma 1 [6]: If there are input variables x and output variables ( ) y t for the same system as in Definition 1, and define the generalized fuzzy hyperbolic rule base and generalized input variables according to Definition 1, and defines membership functions x P and x N as Equation (2), then the following model can be obtained: , . We call model (4) the generalized fuzzy hyperbolic model.
Remark 1 [21]: As an extension of a fuzzy hyperbolic regular basis function, we can obtain the following equivalent functions of GFHM:

Design of Adaptive Fuzzy Tracking Controller
In this section, a fuzzy adaptive DSC control method based on GFHM is designed by using the preparatory knowledge in the previous section for system (1).
In addition, let the normal number W be According to backstepping, the design process of the controller includes n steps.
First of all, in the DSC design, there are the following transformations: ( ) ( ) where 2, , j n = . Then, the transformed error system is as follows where r and q are the positive constants of the design.
Remark 2: Compared with [9], this method only needs the information of r y and r y , while the common backstepping design needs the information of ( ) In addition, the DSC using the first-order filter (Equation (11)) avoids the repeated derivative problem of nonlinear function ( ) j j g x in the design of controller Equation (13), Equation (14) and Equation (16).
Remark 3: In this method, n adaptive laws need not be designed. Only one adaptive parameter needs to be adjusted, which greatly reduces the computational burden.

Stability Analysis
The stability of the designed control method is proved in this section.
Theorem 1: If there are virtual control variables such as Equation (13) and Equation (14), such as the actual control variables of Equation (16), and the adaptive law of Equation (17), then the nonlinear system (1) is semi globally uniformly ultimately bounded, and the tracking error is kept in a small range.
Proof: Define Lyapunov functional as Notice the following equation From Equation (26), we can conclude that

Simulation Example
References [8] [22] considers the tracking control problem of a single joint manipulator driven by a brush DC motor, and the simulation results are verified by the control method designed in this paper. The nonlinear dynamic equation of the system is as follows: . U is the motor input voltage. Other parameters are set as 3 x I = , so Equation (28) The initial state value is set as − is the given reference signal.
The trajectory curve of output signal y tracking reference signal r y are shown in Figure 1. The tracking error e is shown in Figure 2. The control input u is shown in Figure 3. It can be seen from the simulation results that the tracking error e and the control input u are semi-globally uniformly ultimately bounded, and the tracking error e converges rapidly to a compact set near the origin.
Compared with [8], the simulation results show that the proposed method can obtain faster adaptive and higher tracking accuracy.

Conclusion
In this paper, the problem of fuzzy adaptive tracking control for a class of uncertain SISO strict feedback nonlinear systems with disturbance is studied. In this control method, GFHM nonlinear function approximator is introduced to improve the approximation performance, and DSC technology is used to obtain better tracking performance. It not only avoids the problem of calculation ex-pansion, but also obtains higher tracking accuracy. In addition, only one adaptive law is designed in the controller design, which greatly reduces the calculation cost. The SGUUB of the system is proved by Lyapunov stability theory. The effectiveness of the control method is further proved by simulation examples.