RBF NN-Based Backstepping Adaptive Control for a Class of Nonlinear Systems

There are many control methods for nonlinear systems, but some of them can not control nonlinear mismatched systems very well. Backstepping control has obvious advantages in controlling nonlinear mismatched systems. Thus we proposed a new radial-basis-function (RBF) neural network-based backstepping adaptive controller combining RBF neural network (RBF NN) and backstepping control for a class of nonlinear mismatched systems. We adopted RBF NN to approximate the system uncertainty. And we analyzed the controller stability using Lyapunov stability theory. Finally we chose sine signal as simulation input signal, simulation results show that the proposed control strategy has better adaptive ability and robustness than PID control, validating the effectivess of the proposed control strategy.


Introduction
Nonlinear systems have been studied by many scholars, and the control methods used are as follows: iterative learing control [1], feedback linearization [2], fuzzy control [3], sliding mode control [4], etc. However, when the system is a nonlinear mismatched system, the disadvantages of the above methods will be revealed.
Backstepping control can be used to control nonlinear mismatched systems [5]. RBF neural network (RBF NN) has universal approximation capability and can approximate arbitrary nonlinear functions with arbitrary precision [6]. The control algorithm combining RBF NN and backstepping control has been successfully applied by scholars [7][8][9][10][11].
In this paper, we use RBF NN-based backstepping control for a class of nonlinear mismatched systems. RBF NN is used to approximate system uncertainty. Simulation results verify the feasibility of the proposed control algorithm.

Problem description
Consider the following nonlinear mismatched system with strict feedback [12]: where  is the construction error of RBF NN.
In this paper, our main task is to design an adaptive controller so as to make system output 1 x asymptotically stable tracking the expected output 1d x , namely 11 lim 0 d t xx   .

Controller design and stability analysis
We use a RBF NN-based backstepping adaptive control strategy for a class of nonlinear system, Lyapunov stability theory is used to analyze the system stability. The detailed design process is as follows.
Step 1 Define the tracking error between the actual system output 1 x and the expected system output 1d x as : Its derivative is If 2 x is considered the actual input of system (4), then there exist an ideal control * 2 x [13]: such that x is not available, we design a virtual control 2d x as follows.
where 1 W is the estimated value of * 1 W . We choose the following Lyapunov function 1 V : is a design parameter.
The derivative of 1 V is as follows : such that where ˆi W is the estimated value of * i W .
We choose the following Lyapunov function i V : We choose the following adaptive law : where 12 [] n      , 0 n  is a design parameter.
The derivative of n V is as follows : Since   ZZ , and   , substitute Eq.(23) into (24), we have According to the actual system, the inequality ˆ0   is clearly true. In addition, according to Schwarz inequality, the following inequality is true :

Simulation analysis
We adopt the following system to validate the effectiveness of the proposed control strategy.
The control goal is to make the system output 1 x trace the desired output 1 0.5sin ( . The following curves are comparision curves between PID control and the proposed strategy in this paper, namely RBF NN-based backstepping adaptive control(RBFNNB). Figure 1 and figure 2 are curves of tracking sine signal and error using PID control, respectively. Figure 3 and figure 4 are curves of tracking sine signal and error using RBFNNB, respectively. From figure 1-4, it is obvious for us that RBFNNB has better dynamic and steady performance than PID control.

Conclusion
For a class of nonlinear mismatched systems, we proposed a new RBF NN-based backstepping adaptive control. We designed the adaptive controller combining RBF NN and backstepping control, and analyzed its stability. In simulation analysis, sine signal is chosen as input signal, system has larger tracking error and worse adaptive tracking ability when using PID control, but system has smaller tracking error and better adaptive tracking ability when using RBFNNB, validating the superiority of the proposed control strategy. However, the