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The problem of fuzzy-based direct adaptive tracking control is considered for a class of pure-feedback stochastic nonlinear systems. During the controller design, fuzzy logic systems are used to approximate the packaged unknown nonlinearities, and then a novel direct adaptive controller is constructed via backstepping technique. It is shown that the proposed controller guarantees that all the signals in the closed-loop system are bounded in probability and the tracking error eventually converges to a small neighborhood around the origin in the sense of mean quartic value. The main advantages lie in that the proposed controller structure is simpler and only one adaptive parameter needs to be updated online. Simulation results are used to illustrate the effectiveness of the proposed approach.

During the past decades, many control methods have been developed to control design of nonlinear systems, such as adaptive control [

Pure-feedback nonlinear systems, which have no affine appearance of the state variables that can be used as virtual control and the actual control input, stand for a more representative form than strict-feedback systems. Many practical systems are in nonaffine structure, such as biochemical process [

Motivated by the above observations, we will develop a novel fuzzy-based direct adaptive tracking control scheme for a class of pure-feedback stochastic nonlinear systems. The presented controller guarantees that all the signals in the closed-loop system remain bounded in probability and the tracking error converges to a small neighborhood around the origin in the sense of mean quartic value. The main contributions of this paper lie in that the structure of the proposed controller is simpler and only one adaptive parameter needs to be updated online. As a result, the computational burden is significantly alleviated and the control scheme may be more implemented in practice.

The remainder of this paper is organized as follows. The problem formulation and preliminaries are given in Section

To introduce some useful conceptions and lemmas, consider the following stochastic system:

For any given

The term

The trajectory

Suppose that there exist a

In this paper, we consider a class of pure-feedback stochastic nonlinear systems described by

The control objective is to design a fuzzy-based adaptive tracking control law

For the system (

The signs of

The reference signal

In this note, fuzzy logic system will be used to approximate a continuous function

: IF

Then

where

Let

The following lemma will be used in this note.

For

In this section, a fuzzy-based adaptive tracking control scheme is proposed for the system (

For simplicity, in the following, the time variable

Since

Apparently, Assumption

Based on the coordinate transformation

Furthermore, it can be verified by substituting (

From the definition of

Substituting (

Further, fuzzy logic system

Then, constructing the virtual control signal

The adaptive law

According to the definition of

Furthermore, (

Now, the actual control signal

Consider the pure-feedback stochastic nonlinear system (

all the signals in the closed-loop system are bounded in probability;

there exists a finite time

where the time

(i) For the stability analysis of the closed-loop system, choose the Lyapunov function as

(ii) From (

In this section, to illustrate the effectiveness of the proposed control scheme, we consider the following second-order pure-feedback stochastic nonlinear system:

The simulation results are shown in Figures

System output

The state variable

The adaptive parameter

The true control input

In this paper, a novel fuzzy-based adaptive control scheme has been presented for pure-feedback stochastic nonlinear systems. The proposed controller guarantees that all the signals in the closed-loop systems are bounded in probability and tracking error eventually converges to a small neighborhood around the origin in the sense of mean quartic value. The main advantages of this control scheme are that the controller is simpler than the existing ones and only one adaptive parameter needs to be estimated online for an

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is partially supported by the Natural Science Foundation of China (61304002, 61304003, and 11371071), the Program for New Century Excellent Tallents in University (NECT-13-0696), the Program for Liaoning Innovative Research Team in University under Grant (LT2013023), the Program for Liaoning Excellent Talents in University under Grant (LR2013053), and the Education Department of Liaoning Province under the general project research under Grant (no. L2013424).