This paper is concerned with an integral terminal sliding mode tracking control for a class of uncertain nonaffine nonlinear systems. Firstly, the nonaffine nonlinear systems is approximated to facilitate the desired control design via a novel dynamic modeling technique. Next, for the unmeasured disturbance of nonlinear systems, integral terminal sliding mode disturbance observer is presented. The developed disturbance observer can guarantee the disturbance approximation error to converge to zero in the finite time. Subsequently, based on approximated nonlinear model and the designed disturbance observer, the integral terminal sliding mode tracking control is presented for nonaffine nonlinear systems with uncertainty. Different from traditional terminal sliding-mode control, this paper accomplishes finite convergence time for nonaffine nonlinear systems and avoids the singular problem in the controller design. Furthermore, the control system is forced to start on the terminal sliding hyperplane, so that the reaching time of the sliding modes is eliminated. Finally, two numerical simulation results are given to illustrate the effectiveness of the proposed method.

In recent years, there has been significant progress in the area of designing controllers for nonlinear systems [

Sliding-mode control (SMC) is a well-known efficient control scheme which has been widely applied for both linear and nonlinear systems [

The uncertainty is inherent in practical systems. Designing controller capability of handing uncertainty is of practical interest and is academically challenging. Neural networks (NNs) have been proposed recently as an adaptive controller for nonlinear systems. By the use of their universal approximation capability, the adaptive controller based on neural networks can be designed without significant prior knowledge of the system dynamics [

Based on the above works, this paper is to develop an integral terminal sliding mode control approach for a class of nonaffine nonlinear systems with uncertainty, parameters perturbation, and external disturbances. The organization of this paper is as follows. Following the introduction, the problem formulation is described briefly, some assumptions which will play a basic role in our analysis are introduced in Section

Consider a class of uncertain nonaffine nonlinear systems that can be expressed in the following form:

After combining the uncertainty and disturbance together, the nonlinear system (

To achieve the proposed control objective, the following assumptions are required.

There exist known positive constants

Consider

In many actual process control systems and flight control systems,

In this paper, the control objective is to design the disturbance-observer-based integral terminal sliding mode tracking control and make the system output follow a given desired output of the nonlinear system in the presence of uncertainty and external disturbance. For the desired tracking signal

The problem of controlling the plants characterized by models that are nonaffine in the control input vector is a difficult one. Especially for the tracking control, the liberalization may result in the design of sufficiently accurate controllers in the case of stabilization around the operating point, in the case of tracking of desired trajectories the problem becomes much more difficult, because the linearized model is time-varying. Hence, there is a clear need for the development of systematic control design techniques for nonlinear models that are nonaffine in

For the nonaffine nonlinear model (

Equation (

In Assumption

The traditional model simplification method does not global. It can be seen that (

By (

In this section, the design process of the sign integral terminal sliding mode disturbance observer will be given. Firstly, the following auxiliary sign integral terminal sliding mode vector

If

Next, to keep the system on the sign integral terminal sliding surface

Considering the uncertain nonaffine nonlinear system (

Based on (

Comparing with the existing results [

It is worth noting that the known upper boundary of the dynamic error is required in the design of disturbance observer. However, upper boundary

In this section, we develop the tracking control scheme for the case where all states are available using fractional integral terminal sliding mode control approach. Before the discussion, the tracking error is defined as

If the surface

From solving the error dynamic equation (

Next, to keep the system on the integral terminal sliding surface

According to Assumption

In this subsection, we assume that

The above design procedure of the terminal sliding mode control can be summarized in the following theorem, which contains the results for disturbance-observer-based terminal sliding mode tracking control of uncertain nonaffine systems with external disturbance.

Considering the uncertain nonaffine system (

Choose the Lyapunov function candidate:

The characteristics of the proposed fractional integral terminal sliding mode control including

Aside from the characteristics in Remark

In order to reduce chattering which is caused by discontinuous sign function,

In Section

It is clear that

Based on the sign integral terminal sliding mode disturbance observer, the fractional integral terminal sliding mode tracking control is designed as

The above design procedure and analysis can be summarized in the following theorem, which contains the results for the simplified model (

Considering the uncertain nonaffine system (

Considering the time-varying simplified model (

From (

To verify the validity of the proposed nonaffine nonlinear approximation in Section

The initial conditions are chosen as

State response by the proposed approximation.

State response using the approach developed in [

From Figures

To verify the validity of the fractional integral terminal sliding mode control, the differential equations governing the near space vehicle (NSV) dynamics with coordinated turn are given by

In this paper, we will focus on the model of rate dynamics, that is, (

The initial state conditions are arbitrarily chosen as

The desired command is considered as

To estimate the uncertainty, we apply integral terminal sliding mode disturbance observer in Section

States

Control input for near space vehicle system with coordinated turn.

From these simulation results of two cases, we can obtain that the proposed method is valid. And the developed sign integral terminal sliding mode disturbance observer can modify the control performance of the fractional integral terminal sliding mode control.

In this paper, the disturbance-observer-based terminal sliding mode tracking control has been proposed for a class of uncertain nonaffine nonlinear systems. To design tracking controller, an on-line approximation has been proposed for a class of nonaffine nonlinear systems. To improve the ability of the disturbance attenuation and system performance robustness, the sign integral terminal sliding mode disturbance observer has been developed to approximate the system disturbance in the finite time. Based on the output of the disturbance observer, the disturbance-observer-based fractional integral terminal sliding mode tracking control has been presented for the uncertain nonlinear system with the time-varying external disturbance. By innovating the fractional error integration, finite-time convergence of tracking errors and integral errors is achieved without singular problem. Furthermore, the finite convergence time is easily calculated in contrast to the traditional high-order sliding mode control. The stability of the closed-loop system has been proved using rigorous Lyapunov analysis. Finally, simulation results have been used to illustrate the effectiveness of the proposed robust terminal sliding mode tracking control scheme. In addition, based on the proposed approach, how to relax Assumption

This work was supported by Natural Science Foundation of Shandong Province under Grant ZR2012F Q030. The authors wish to thank the reviewers for their constructive comments and suggestions which have helped to improve the presentation of the paper.