Journal of Control Science and Engineering

Volume 2015, Article ID 218198, 7 pages

http://dx.doi.org/10.1155/2015/218198

## Nonlinear Integral Sliding Mode Control for a Second Order Nonlinear System

Xi’an High Technology Research Institute, Xi’an 710025, China

Received 23 November 2014; Accepted 7 January 2015

Academic Editor: Onur Toker

Copyright © 2015 Xie Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A nonlinear integral sliding-mode control (NISMC) scheme is proposed for second order nonlinear systems. The new control scheme is characterized by a nonlinear integral sliding manifold which inherits the desired properties of the integral sliding manifold, such as robustness to system external disturbance. In particular, compared with four kinds of sliding mode control (SMC), the proposed control scheme is able to provide better transient performances. Furthermore, the proposed scheme ensures the zero steady-state error in the presence of a constant disturbance or an asymptotically constant disturbance is proved by Lyapunov stability theory and LaSalle invariance principle. Finally, both the theoretical analysis and simulation examples demonstrate the validity of the proposed scheme.

#### 1. Introduction

Control strategies design for nonlinear systems has attracted considerable research interest in the recent past [1–4]. Sliding mode control (SMC) as an effective robust control scheme has been successfully applied to a wide variety of systems [5, 6]. In spite of claimed robustness, the SMC suffers certain drawbacks, mainly the chattering phenomenon [7, 8]. This phenomenon is extremely harmful to the actuators of physical systems [9]. To prevent chattering in the SMC, Slotine proposed quasi-SMC which introduced the bounded layer in SMC [10]. In [11], another scheme is based on the observer design which suppresses the high frequency oscillations of the control input. Though the chattering phenomenon could be attenuated by above-mentioned approaches, the approaches would bring in large steady-state error in the presence of the nonlinear system with disturbances. In [12], Chern and Wu first proposed integral sliding mode control (ISMC) which significantly enhanced the robustness against external disturbance of the nonlinear system.

The disturbance and uncertainty can be effectively restrained by ISMC; nevertheless, the control performance (overshoot and respond time) would become very poor if the initial errors of the system are very large. Much worse, duo to the effect of “integral windup” phenomenon, the ISMC system maybe unstable in the presence of control signal control input constraints [13–15]. To alleviate the above shortcomings, Cho et al. proposed a small gain ISMC, but the small gain got the response speed of the system slow [16]. In [17], Lee proposed global integral sliding mode control, this approach achieved that all the states locate in the sliding mode manifold at the beginning and keeps this stage to eliminate the reaching stage.

In this work, aiming at improving control performance for second order systems, a nonlinear ISMC scheme was proposed. Firstly, a nonlinear integral sliding manifold with saturated function was designed, which could eliminate the effect of system initial error. Secondly, based on the proposed sliding manifold, the sliding mode control law design was carried out in two parts: (1) case of control input without constraints and (2) case of control input constraints. In addition, by the Lyapunov stability theory and LaSalle invariance principle, we have proved that the proposed scheme ensured the zero steady-state error in the presence of a constant disturbance or an asymptotically constant disturbance. Furthermore, through theory analysis, the research proved that the proposed control scheme could be equivalent to PD controller with nonlinear integral control scheme. Finally, a numerical example has been provided to demonstrate the effectiveness of the obtained results.

#### 2. Problem Statement

Consider a second order nonlinear system described by the state equation: where , are the state variables of the system and is the state vector, is a sufficiently smooth vector field, denotes the input control signal, and denotes external disturbances. For simplicity, it is assumed.

*Assumption 1. * is a bounded function with uncertainty, and consists of nominal parts which are known a priori and uncertain parts which are bounded and unknown [18]. Furthermore, satisfy

*Assumption 2. * denotes external disturbances of the system, which satisfy

Define the tracking error as
where denotes the reference signal. The role of the controller is to ensure that system output accurately track the reference signal.

#### 3. Nonlinear Integral Sliding Mode Manifold Design

When the system is perturbed or uncertain, the finite time stabilization is not ensured. Hence, in this section a reaching law based discontinuous control law is developed which rejects the uncertainties of the system and ensures that the control objectives are fulfilled.

For system (1), if using the traditional sliding mode control scheme, the sliding manifold can be described by the following equation: where is a strictly positive constant.

To reduce the steady-state error, an integral term of tracking error is introduced into (6), which makes up the traditional integral sliding manifold: where is a strictly positive constant.

To enhance the robustness of closed loop system, a sliding manifold design incorporates global integral sliding manifold which results in the elimination of reaching phase [19]. The global integral sliding manifold can be described as follows:

A nonlinear integral sliding manifold is proposed in this research. With the integral manifold is given as where is a new nonlinear saturation function which enhances small errors and will be saturated with large errors in shaping the tracking errors. In order to research the properties of , a potential energy function is introduced as where is the design parameter of . The time derivative of (8) along can be written as

There is the following lemma for (9) and (10).

Lemma 3 (see [13]). * and satisfy the following: *(1)* for , for ;*(2)* is a second order continuous differentiable function. is a strictly monotone increasing function for , and is a saturated function for .*

*The above properties of and can be obtained by some simple mathematical operations.*

*Figure 1 shows the curves of the potential function , , and . It is obvious from Figure 1 that the proposed integral sliding manifold enhances small errors and will be saturated with large errors in shaping the tracking errors. Namely, when , ; when , . Furthermore, the desired control performance would be obtained by choosing a proper design parameter .*