Abstract

This paper proposes an integral sliding mode control (ISMC) method of a class of uncertain chaotic systems with saturation inputs. Firstly, fuzzy logic system (FLS) is used to estimate the unknown nonlinear function. Then, a disturbance observer is constructed to estimate a compound disturbance, which contains the external disturbance, the error of saturation input and control output, and the fuzzy estimation error. Subsequently, a proposed integral sliding mode controller can ensure that all signals of the closed-loop system are ultimately bounded, and based on the dynamic system of the integral sliding mode variable itself, the ultimate bound of the tracking error can be estimated. Simulation results show that the proposed ISMC method is more effective than the traditional ISMC method.

1. Introduction

Due to the unpredictability and the sensitivity to initial conditions, the chaotic system makes itself useful in many fields such as information processing, secure communications, and mechanical systems [1, 2]. However, some actual systems, such as flutter of aircraft wings and vibration of horizontal platforms, may cause bad results if the chaotic behavior occurs. Therefore, it is necessary to suppress such harmful chaotic behavior. At present, many methods have been proposed to stabilize or synchronize chaotic systems, such as adaptive control method [3, 4], backstepping control [5, 6], impulsive control [7], intermittent control [8], and sliding mode control [9–14].

As we all know, the advantage of using sliding mode control method to control chaotic systems and other nonlinear systems is its good robustness. For example, Wang et al. [11] proposed a nonsingular terminal SMC method to ensure that all states of the chaotic system reach the designed sliding surface in finite time. By using adaptive terminal SMC, Yang and Ou [12] studied the synchronization problem of chaotic gyros. Similarly, by employing adaptive SMC, Chen et al. [13] investigated the synchronization for multiple uncertain coupled chaotic systems. It can be seen that the abovementioned sliding mode control method only considers whether the arrival condition of the sliding surface can be satisfied and does not consider the estimation problem for system uncertainties. Meanwhile, the hysteresis of control switching will cause chattering phenomenon. In order to eliminate this disadvantage, Haghighi and Mobayen [15] proposed a high-order terminal SMC technique for a class of fourth-order systems. However, the higher-order sliding mode controller includes larger amount of higher-order derivatives of sliding mode variable, which may lead to increased noise in closed-loop systems.

Recently, in order to estimate system uncertainties or external disturbances, Zhou et al. [16–20] explored the construction principle of disturbance observer and combined different control methods to achieve the stability of closed-loop systems. Based on the disturbance observer and fuzzy terminal SMC, Vahidi-Moghaddam et al. [17] investigated the finite-time asymptotic stability of uncertain MIMO systems. Xu in [18] proposed composite terminal SMC learning control schemes for quadrotor dynamics via the disturbance observer. However, control results of [17–20] only ensure that all signals of the closed-loop system approach a small region, but the ultimate bound of this area has not been estimated.

Inspired by the abovementioned works, this paper investigates the problem of integral sliding mode control for a class of uncertain chaotic systems with saturation inputs, and the proposed integral SMC method can not only guarantee all signals of the closed-loop system are bounded but also accurately estimate system uncertainties. The main contributions of this paper are summarized as follows:(1)Compared with the traditional integral sliding mode variable, the integrated sliding mode variable proposed in this paper can estimate the error bound by its own dynamics when the boundedness of the integrated sliding mode variable is guaranteed.(2)The established disturbance observer and fuzzy parameter law can accurately estimate system uncertainties.(3)Compared with the traditional ISMC method, the proposed method in this paper can eliminate the chattering phenomenon.

The rest of this work is arranged as follows. Some assumptions and lemmas and the problem statement are presented in Section 2. Section 3 gives sliding mode control design and stability analysis. In Section 4, some comparison results are presented to show the validity of the proposed method. At last, the conclusion is included in Section 5.

2. Preliminaries

In general, the chaotic system can be described aswhere is the state vector and is the nonlinear vector function. Many classic chaotic systems such as Lorenz chaotic system, Chen chaotic system, and L chaotic system can be abbreviated as the form of system (1), and the controlled chaotic system (1) is expressed aswhere is the external disturbance vector, is the control output vector, and is the control input vector subject to saturation type nonlinearity, and is described aswhere is the unknown bound of symmetric input saturation.

Define the tracking error vector , and is the reference signal. According to (2), we obtain the following error dynamic system:

In order to achieve effective tracking of the reference signal by the state , the following assumptions about , , and are made.

Assumption 1. The nonlinear function is unknown but bounded, and the state is measurable.

Assumption 2. The reference signal is , and and are available.

Assumption 3. The time-varying external disturbance and its derivative are unknown but bounded.

Define , where , .

Remark 1. For the error system (4), we can use method [21] to track the reference signal . Firstly, let , where is known. Define , and assume that , where is unknown. Then, construct the disturbance observer as , where is the intermediate variable and satisfieswhere is the positive constant, and design the following control law aswhere , are the positive constants. This method can guarantee that and are uniformly bounded. However, this method does not cover two situations: one is that the function is unknown; the other is whether can be accurately estimated. In order to consider the abovementioned two situations, this paper will use the integral sliding mode control method combined with the disturbance observer and fuzzy logic technology to design a control method to realize the fast stability of tracking error . In addition, the controller and error state will effectively eliminate chattering phenomenon.

Define , where is the positive design constant. By using FLSs, can approximate aswhere are the ideal weight vectors, is the approximation error, and is the upper bound of . , where , are the Gaussian functions. Obviously, there exists a positive constant such that .

Define , where is the estimation of . So, we obtain

Denote , , . So, in (4) can be expressed as

Let . The error system (4) can be rewritten as

Because the compounded disturbance is unknown, it cannot appear directly in the controller design. But from a practical system point of view, is clearly bounded. Combining with the structural ideas of disturbance observer [18, 21, 22], we also make the following assumption.

Assumption 4. The derivative of satisfies , where is an unknown positive constant.

The following lemmas are introduced for the subsequent discussions.

Lemma 1 (see [23]). Let , thenwhere is the real number.

Lemma 2 (see [24]). For any , , and , the following inequality holds:where and is the unique solution of .

Remark 2. When , we get a famous inequality: , where . In this paper, since both the sliding mode and the controller contain functions, the inequalities (12) of Lemma 2 play an important role.

The aim of this paper is (1) to propose an effective control method so that the error state can be remained within a small neighborhood of zero; (2) to construct the disturbance observer so that can be accurately estimated; (3) to design a new integral sliding variable such that the chatter phenomenon of controller can be eliminated.

3. Sliding Mode Control Design and Stability Analysis

3.1. Construction of Integral Sliding Mode and Disturbance Observer

In order to design a sliding mode robust controller to make the tracking error stable, the integral sliding variable needs to be designed. In [25], an integral terminal sliding variable is designed aswhere , , , and is a positive parameter. By using the traditional sliding mode control method, the occurrence of chatter phenomenon is mainly due to the hysteresis of control switching. To avoid the chatter phenomenon, we modify the sliding surface (13) as follows:where , , is the designed positive parameter and is a small positive constant, and and are the same as in (13).

Remark 3. In general, the aim of using the integral sliding variable in (13) is to improve its control performance and eliminate the reaching phase under the nominal control law. Different from the former, based on the dynamic system of integral sliding variable in (14), this paper is to explore the ultimate boundary of the tracking error .

According to (10) and (14), the dynamic system of is obtained as follows:

In order to show the performance of RBFNNs, defining as the estimation of , we havewhere , is the positive design constant, and is the estimation of . Furthermore, the derivative of is obtained as

Now, the disturbance observer is constructed aswhere is an auxiliary variable vector and and are positive design parameters.

The derivative of can be written aswhere , and then, we obtain

3.2. Main Results

Now, the controller is designed asand the parameter adaptive laws are proposed aswhere , , and are the design parameters, , and is a small positive constant. We give the main result as follows.

Theorem 1. For the error system (10) under Assumptions 1–4 and the disturbance observer (18), the controller (21) and parameter adaptive laws (22) guarantee that all the closed-loop system signals in (23) are uniformly ultimately bounded.

Proof. Consider the Lyapunov function aswhereSubstituting (21) into (15), the derivative of can be obtained asFrom (20), the derivative of yieldsFrom (17), the derivative of yieldsFrom (22), the derivative of yieldsUsing (25), (26), (27) and (28), we haveSince the following inequalities hold,where .
By using Lemma 1, one hasSince , the following inequality holds by using Lemma 2:Substituting (32) into (31) yieldsSubstituting (30) and (33) into (29) results inwhere . By choosing parameters and satisfyand let . It can be obtained asFrom (36), we knowObviously, it can conclude that all signals in (23) are ultimately uniformly bounded. The proof is completed.