Adaptive Sliding Mode Control of a Novel Class of Fractional Chaotic Systems

Abstract

Recently, control and synchronization of fractional chaotic systems have increasingly attracted much attention in the fractional control community. In this paper we introduce a novel class of fractional chaotic systems in the pseudo state space and propose an adaptive sliding mode control scheme to stabilize the chaotic systems in the presence of uncertainties and external disturbances whose bounds are unknown. To verify the effectiveness of the proposed adaptive sliding mode control technique, numerical simulations of control design of fractional Lorenz's system and Chen's system are presented.

1. Introduction

Fractional calculus is an old and yet novel topic whose infancy dates back to the end of the 17th century, the time when Newton and Leibniz established the foundations of classical calculus. For three centuries, fractional calculus developed mainly as a pure theoretical mathematical field without applications. However, in the last two decades it has attracted the interest of researchers in several areas including mathematics, physics, chemistry, material, engineering, finance, and even social science. The stability of fractional differential equations (FDEs) and fractional control have both gained rapid development very recently [1–3].

One of the most important areas of application is the chaos theory. In recent years, fractional chaotic systems have intensively attracted a great deal of attention due to the ease of their electronics implementations and the rapid development of the stability of FDEs. More and more fractional dynamics described in the pseudo state space exhibiting chaos have been found, such as the fractional Chua circuit [4], the fractional Van der Pol oscillator [5–7], the fractional Lorenz system [8, 9], the fractional Chen system [10–12], the fractional Lü system [13], the fractional Liu system [14], the fractional Rössler system [15, 16], the fractional Arneodo system [17], the fractional Newton-Leipnik system [18–20], the fractional Lotka-Volterra system [21, 22], the fractional finance system [20, 23], and the fractional Rucklidge system [24]. Most of the above papers have used numerical methods to present chaotic behaviors.

In particular, control and synchronization of fractional chaotic systems have increasingly attracted much attention in the fractional control community. Moreover, several control and synchronization methods have been proposed based on the stability of fractional differential equations in the pseudo state space [22]. The linear state feedback control algorithm based upon the stability criterion of linear FDEs has been used in [10, 25–33], the nonlinear feedback control scheme in [34–38], the fractional PID control method in [39, 40], and active control technique in [41]. However, it is important to note that the above four control methods have ignored modeling inaccuracies and external noises which can be hardly avoided in real world application. Robust control and adaptive control are two major and complementary approaches to deal with model uncertainty: sliding mode control design provides a systematic approach to the problem of maintaining stability and consistent performance in the face of modeling imprecision, while adaptive control is a suitable approach to maintain consistent performance of a system in the presence of uncertainty or unknown variation in plant parameters [42]. The fractional sliding mode control methodology has been designed in [43–47], and the motivation of choosing the particular structures for sliding surfaces has been described in [48]. The adaptive sliding mode control, which is good at maintaining robustness and handling inevitable parameter variation and parameter uncertainty, has been used in [49, 50].

Nevertheless, as stated recently in [51, 52], there arise several questionable or wrong ideas in the field of fractional systems analysis and control using the representation state space description, particularly if Caputo’s definition is used. To correct these wrong ideas, the authors [53–55] have proposed alternative solutions and equivalent frequency distributed model of FDEs. Particularly, Lorenzo and Hartley [56–58] have derived a simple process of initialization of FDEs. Motivated by the continuous frequency distributed model proposed by Trigeassou et al. and the initialization method addressed by Lorenzo and Hartley, this paper utilizes a new Lyapunov approach [59] to analyze the stability of FDEs and impose physically coherent initial conditions to FDEs using the initialization function [60].

The rest of the paper is organized as follows. Section 2 presents some basic definitions about fractional calculus. Section 3 introduces a novel class of fractional chaotic systems. Section 4 proposes the sliding mode control design and adaptive sliding mode control design of fractional chaotic systems. Numerical simulations are presented to show the effectiveness of the proposed method in Section 5. Finally, the paper is concluded in Section 6.

2. Basic Definitions and Preliminaries

Definition 1. The most important function used in fractional calculus is Euler’s Gamma function, which is defined as Γ(𝑛)=^∞₀𝑡^𝑛−1𝑒^−𝑡𝑑𝑡. (1)

Definition 2. Another important function is a two-parameter function of the Mittag-Leffler type defined as 𝐸_𝛼,𝛽(𝑧)=^∞_𝑘=0𝑧^𝑘Γ𝛼𝑘+𝛽,𝛼>0,𝛽>0. (2)

Fractional calculus is a generalization of integration and differentiation to noninteger-order fundamental operator _𝑎𝐷^𝛼_𝑡, where 𝑎 and 𝑡 are the bounds of the operation and 𝑎∈𝑅. The three most frequently used definitions for the general fractional calculus are the Grünwald-Letnikov definition, the Riemann-Liouville definition, and the Caputo definition [23, 61–63].

Definition 3. The Grünwald-Letnikov derivative definition of order 𝛼 is described as _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=lim_ℎ→01ℎ^𝛼∞_𝑗=0(−1)^𝑗𝛼𝑗𝑓𝑡−𝑗ℎ. (3)

Definition 4. The Riemann-Liouville derivative definition of order 𝛼 is described as _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑛𝑡_𝑎𝑓(𝜏)𝑑𝜏(𝑡−𝜏)^{𝛼−𝑛+1},𝑛−1<𝛼<𝑛. (4)

Definition 5. The Caputo definition of fractional derivative can be written as _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎𝑓^(𝑛)(𝜏)𝑑𝜏(𝑡−𝜏)^{𝛼−𝑛+1},𝑛−1<𝛼<𝑛. (5)

In the rest of the paper, we will use the Caputo approach to describe the fractional chaotic systems and use the Grünwald-Letnikov approach to propose numerical simulations. To simplify the notation, we denote the fractional derivative of order 𝛼 as 𝐷^𝛼 instead of ₀𝐷^𝛼_𝑡 in this paper.

3. System Description

In [64], the authors have introduced a class of integer-order chaotic systems covering about half of the recently published integer-order chaotic models. In [43], Yin et al. have designed a novel class of fractional chaotic model covering several fractional chaotic systems. Inspired by the above two contributions, we introduce and control another novel class of fractional chaotic systems (6) which are described in the pseudo state space and cover the fractional Lorenz system [8, 9], the fractional Chen system [10–12], the fractional Lü system [13], the fractional Liu system [14], the fractional Lotka-Volterra system [22], and the fractional Rucklidge system [24], presented in Table 1: 𝐷^𝑞1𝐷𝑥=𝑓𝑥,𝑦,𝑧−𝛼𝑥,_𝑞2𝐷𝑦=𝑥𝑔𝑥,𝑦,𝑧−𝛽𝑦,_𝑞3𝑧=𝑥ℎ𝑥,𝑦,𝑧−𝛾𝑧, (6) where 𝑥, 𝑦, and 𝑧 are the pseudo state variables, 𝛼, 𝛽, and 𝛾 are nonnegative known constants, and 𝑞₁,𝑞₂, and 𝑞₃∈(0,1]. Each of the three functions 𝑓(⋅), 𝑔(⋅), and ℎ(⋅) is assumed to be continuous and satisfies the Lipschitz condition to guarantee the existence and uniqueness of solutions of initial value problems.

Table 1 

List of published fractional chaotic systems covered by (7).

4. Control Design

4.1. Sliding Mode Control Design

Sliding mode control methodology is a simple approach to robust control and good at dealing with dynamic uncertainty. The control design procedure consists of two steps: first constructing a sliding surface which presents the desired dynamics and second selecting a switching control law so as to verify sliding condition. The control input 𝑢(𝑡) is added to the first state equation in order to control chaos. Then, the controlled plant can be described as 𝐷^𝑞1𝐷𝑥=𝑓𝑥,𝑦,𝑧−𝛼𝑥+𝑢,_𝑞2𝐷𝑦=𝑥𝑔𝑥,𝑦,𝑧−𝛽𝑦,_𝑞3𝑧=𝑥ℎ𝑥,𝑦,𝑧−𝛾𝑧. (7)

The main purpose of this paper is to design a suitable controller 𝑢(𝑡) to guarantee the stability of the chaotic system (6).

4.1.1. Sliding Surface Design

In order to achieve the stability of system (6), a sliding surface 𝑆(𝑡) is constructed as 𝑠(𝑡)=𝐷^𝑞1−1𝑥(𝑡)+𝐷⁻¹𝜙(𝑡), (8) where 𝜙(𝑡)=𝑦𝑔(𝑥,𝑦,𝑧)+𝑧ℎ(𝑥,𝑦,𝑧)+𝛼𝑥.

By differentiating (8), one derives 𝑠(̇𝑡)=𝐷^𝑞1𝑥(𝑡)+𝜙(𝑡). (9)

4.1.2. Sliding Mode Dynamics Analysis

In this part we will show that given any initial conditions, the problem of stabilization of system (6) is equivalent to that of remaining on the surface 𝑆(𝑡) for all 𝑡>0.

When the system operates in the sliding surface, it satisfies 𝑠=0 and 𝑠̇=0.

This yields the following sliding mode dynamics: 𝐷^𝑞1𝐷𝑥=−𝑦𝑔𝑥,𝑦,𝑧−𝑧ℎ𝑥,𝑦,𝑧−𝛼𝑥,_𝑞2𝐷𝑦=𝑥𝑔𝑥,𝑦,𝑧−𝛽𝑦,_𝑞3𝑧=𝑥ℎ𝑥,𝑦,𝑧−𝛾𝑧. (10)

In the following we will prove that the above sliding mode dynamics (10) is globally asymptotically stable, so that it is able to be considered as our desired dynamics. To this end, the FDEs are firstly converted into an exactly equivalent infinite dimensional ODEs, namely, continuous frequency distributed model of the fractional integrator [53–55]. Then, the indirect Lyapunov approach is applied to derive the stability [59] of the fractional dynamics (10).

Theorem 6. The fractional sliding mode dynamics as in (10) is globally asymptotically stable.

Proof. In terms of the continuous frequency distributed model of the fractional integrator, the fractional system (10) is exactly equivalent to the following infinite dimensional ODEs: 𝜕𝑧₁(𝜔,𝑡)𝜕𝑡=−𝜔𝑧₁()𝑥(𝑡)=𝜔,𝑡−𝑦𝑔𝑥,𝑦,𝑧−𝑧ℎ𝑥,𝑦,𝑧−𝛼𝑥,_∞0𝜇₁(𝜔)𝑧₁(𝜔,𝑡)𝑑𝜔,𝜕𝑧₂(𝜔,𝑡)𝜕𝑡=−𝜔𝑧₂()𝑦(𝑡)=𝜔,𝑡+𝑥𝑔𝑥,𝑦,𝑧−𝛽𝑦,_∞0𝜇₂(𝜔)𝑧₂(𝜔,𝑡)𝑑𝜔,𝜕𝑧₃(𝜔,𝑡)𝜕𝑡=−𝜔𝑧₃()𝑧(𝑡)=𝜔,𝑡+𝑥ℎ𝑥,𝑦,𝑧−𝛾𝑧,_∞0𝜇₃(𝜔)𝑧₃(𝜔,𝑡)𝑑𝜔, (11) with 𝜇_𝑖(𝜔)=((sin(𝑞_𝑖𝜋))/𝜋)𝜔^−𝑞𝑖, 𝑖=1,2,3.
In the above continuous frequency distributed model, 𝑧_𝑖(𝜔,𝑡) are the true state variables, while 𝑥(𝑡),𝑦(𝑡), and 𝑧(𝑡) are the pseudo state variables.
Let us define two types of Lyapunov functions:(i)𝑣_𝑖(𝑤,𝑡), that is, the monochromatic Lyapunov functions corresponding to the elementary frequency;(ii)𝑉_𝑖(𝑡), that is, the Lyapunov functions summing all the monochromatic 𝑣_𝑖(𝑤,𝑡) with the weighting functions 𝜇_𝑖(𝜔).
Exactly, 𝑣_𝑖()=1𝑤,𝑡2𝑧²_𝑖,𝑉_𝑖(𝑡)=^∞₀𝜇_𝑖(𝜔)𝑣_𝑖()1𝜔,𝑡𝑑𝜔=2^∞₀𝜇_𝑖(𝜔)𝑧²_𝑖(𝜔,𝑡)𝑑𝜔. (12)
Then 𝜕𝑣_𝑖(𝜔,𝑡)=𝜕𝑡𝜕𝑣_𝑖(𝜔,𝑡)𝜕𝑧_𝑖𝜕𝑧_𝑖𝜕𝑡=𝑧_𝑖𝜕𝑧_𝑖,𝜕𝑡𝑑𝑉₁=𝑑𝑡^∞₀𝜇₁(𝜔)𝜕𝑣₁(𝜔,𝑡)𝜕𝑡𝑑𝜔. (13)(14)
Substituting the first equation of (11) into (14) yields 𝑑𝑉₁=𝑑𝑡^∞₀𝜇₁(𝜔)𝑧₁−𝜔𝑧₁−𝑦𝑔−𝑧ℎ−𝛼𝑥𝑑𝜔=−^∞₀𝜇₁(𝜔)𝜔𝑧²₁+𝑑𝜔_∞0𝜇₁(𝜔)𝑧₁−𝑦𝑔−𝑧ℎ−𝛼𝑥𝑑𝜔=−^∞₀𝜇₁(𝜔)𝜔𝑧²₁+𝑑𝜔−𝑦𝑔−𝑧ℎ−𝛼𝑥^∞₀𝜇₁(𝜔)𝑧₁𝑑𝜔. (15) Substituting the second equation of (11) into the integral term of (15) yields 𝑑𝑉₁𝑑𝑡=−^∞₀𝜇₁(𝜔)𝜔𝑧²₁𝑑𝜔+𝑥−𝑦𝑔−𝑧ℎ−𝛼𝑥. (16) Similarly, 𝑑𝑉₂𝑑𝑡=−^∞₀𝜇₂(𝜔)𝜔𝑧²₂𝑑𝜔+𝑦𝑥𝑔−𝛽𝑦,𝑑𝑉₃𝑑𝑡=−^∞₀𝜇₃(𝜔)𝜔𝑧²₃𝑑𝜔+𝑧𝑥ℎ−𝛾𝑧. (17) Finally, let us define 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡). (18)
Then, 𝑑𝑉𝑑𝑡=−³_𝑖=1^∞₀𝜇_𝑖(𝜔)𝜔𝑧²_𝑖𝑑𝜔−𝛼𝑥²+𝛽𝑦²+𝛾𝑧². (19)
Owing to the Lemma of the appendix, 𝑑𝑉/𝑑𝑡<0, which implies the stability of the sliding mode dynamics (10).

4.1.3. Control Design via Sliding Mode Methodology

A Lyapunov candidate is selected as 𝑉=1𝑥,𝑦,𝑧2𝑠². (20)

Taking time derivative of (29) yields 𝑑𝑉𝑠̇𝐷𝑑𝑡=𝑠=𝑠^𝑞𝑓(𝑡).𝑥+𝜙=𝑠𝑥,𝑦,𝑧−𝛼𝑥+𝑢+𝑦𝑔𝑥,𝑦,𝑧+𝑧ℎ𝑥,𝑦,𝑧+𝛼𝑥 (21)

Then the control law is constructed as 𝑢(𝑡)=−𝑓𝑥,𝑦,𝑧−𝑦𝑔𝑥,𝑦,𝑧−𝑧ℎ𝑥,𝑦,𝑧−𝑘₁sgn(𝑠)−𝑘₂𝑠, (22) where sgn(𝑠)=1,𝑠>00,𝑠=0−1,𝑠<0 (23) and 𝑘 is a known strictly positive constant to be chosen later.

Substituting the control law (22) into (21) yields 𝑑𝑉𝑑𝑡=𝑠−𝑘₁sgn(𝑠)−𝑘₂𝑠=−𝑘₁|𝑠|−𝑘₂𝑠=−𝑘₁√𝑉−𝑘₂𝑉 (24) which implies that the Lyapunov function 𝑉 tends to zero in a finite time, and the same holds for the sliding surface (8). Furthermore, the finite time vanishing of the sliding surface guarantees that solutions 𝑥(𝑡), 𝑦(𝑡), and 𝑧(𝑡) of the fractional system (10) will tend globally and asymptotically to zero [65]. This proves that the fractional system (6) can be stabilized via the proposed sliding mode control law (22).

4.2. Adaptive Sliding Mode Control Design

In this subsection, the system uncertainty Δ𝑓(𝑥,𝑦,𝑧) and external disturbance 𝑑(𝑡) will be considered and added to (6). It is assumed that Δ𝑓(𝑥,𝑦,𝑧) and 𝑑(𝑡) are both bounded; that is, |Δ𝑓(𝑥,𝑦,𝑧)|<𝜃₁, |𝑑(𝑡)|<𝜃₂, where both 𝜃₁ and 𝜃₂ are unknown nonnegative constants. 𝜃₁ and 𝜃₂ represent, respectively, the estimates of 𝜃₁ and 𝜃₂. To estimate the two unknown parameters, the adaptive control technique will be employed in the following.

The controlled plant is 𝐷^𝑞1𝐷𝑥=𝑓𝑥,𝑦,𝑧−𝛼𝑥+Δ𝑓+𝑑+𝑢,_𝑞2𝐷𝑦=𝑥𝑔𝑥,𝑦,𝑧−𝛽𝑦,_𝑞3𝑧=𝑥ℎ𝑥,𝑦,𝑧−𝛾𝑧. (25)

In order to propose the control design, a Lyapunov candidate is chosen as 𝑉=1𝑥,𝑦,𝑧2𝑠²+1𝜇₁𝜃₁−𝜃₁²+1𝜇₂𝜃₂−𝜃₂². (26)

By taking its derivative with respect to time, one has 𝑑𝑉̇1𝑑𝑡=𝑠𝑠+𝜇₁𝜃₁−𝜃₁̇𝜃₁+1𝜇₂𝜃₂−𝜃₂̇𝜃₂=𝑠𝐷^𝑞+1𝑥+𝜙𝜇₁𝜃₁−𝜃₁̇𝜃₁+1𝜇₂𝜃₂−𝜃₂̇𝜃₂+1=𝑠𝑓−𝛼𝑥+Δ𝑓+𝑑+𝑢+𝑦𝑔+𝑧+𝛼𝑥𝜇₁𝜃₁−𝜃₁̇𝜃₁+1𝜇₂𝜃₂−𝜃₂̇𝜃₂. (27)

If we chose the control law and adaptive law as 𝑢(𝑡)=−𝑓−𝑦𝑔−𝑧ℎ−𝜃₁+𝜃₂+−𝑘₁sgn(𝑠)−𝑘₂̇𝜃𝑠,₁=𝜇₁|𝑠|,̇𝜃₂=𝜇₂|𝑠|. (28)(29)

Then substituting the control law (28) and adaptive law (29) into (27) follows that 𝑑𝑉𝜃𝑑𝑡=𝑠Δ𝑓+𝑑−₁+𝜃₂+−𝑘₁(𝑠)+𝜃sgn₁−𝜃₁|𝑠|+𝜃₂−𝜃₂|𝑠=𝜃|Δ𝑓+𝑑𝑠−₁+𝜃₂+𝑘₁|𝑠|+𝜃₁−𝜃₁|𝑠|+𝜃₂−𝜃₂|𝑠|−𝑘₂𝑠≤Δ𝑓+|𝑑||𝑠|−𝜃₁+𝜃₂+𝑘₁|𝑠|+𝜃₁−𝜃₁|𝑠|+𝜃₂−𝜃₂|𝑠|−𝑘₂𝑠≤𝜃₁+𝜃₂|𝑠|−𝜃₁+𝜃₂+𝑘₁|𝑠|+𝜃₁−𝜃₁|𝑠|+𝜃₂−𝜃₂|𝑠|=−𝑘|𝑠|−𝑘₂𝑠=−𝑘₁√𝑉−𝑘₂𝑉. (30)

By now it is proved that the fractional system (25) with uncertainty and external disturbance can be stabilized via the proposed sliding mode control law (28) and adaptive law (29).

5. Numerical Simulations

In this section, we present two examples, namely, fractional Lorenz’s system and fractional Chen’s system, to evaluate the performance of the sliding mode control and adaptive sliding mode control technique. Numerical simulations are implemented using the MATLAB software.

5.1. Control of the Fractional Lorenz’s System

The fractional Lorenz’s system is described as 𝐷^𝑞1,𝐷𝑥=−𝑎𝑥−𝑦_𝑞2𝐷𝑦=𝑟𝑥−𝑦−𝑥𝑧,_𝑞3𝑧=−𝑏𝑧+𝑥𝑦, (31) where 𝑎=10,𝑟=28, and 𝑏=8/3.

The equilibrium points of the system with the above parameters are 𝐸₁=(0,0,0), 𝐸₂=(6√2,6√2,5/3), and 𝐸₃=(−6√2,−6√2,5/3).

Owning to the initialization method described in [60], the initial conditions for fractional differential equations with order between 0 and 1 is the constant function of time. So the initial conditions for the fractional Lorenz’s system can be chosen as 𝑥(𝑡)=𝑥0⁺𝑦(𝑡)0=1,=𝑦⁺𝑧(𝑡)0=1,=𝑧⁺=1, (32) for −∞≤𝑡≤0.

The fractional Lorenz’s system exhibits chaos with fractional orders 𝑞₁=𝑞₂=𝑞₃=0.993 and the above initial conditions, as depicted in Figure 1.

(a)

(b)

(c)

(d)

Figure 1 

The chaotic trajectories of the fractional Lorenz’s system with simulation time 𝑇sim=100s and time step ℎ=0.005: (a) the 𝑥−𝑦−𝑧 space; (b) the 𝑥−𝑡 space; (c) the 𝑦−𝑡 space; (d) the 𝑧−𝑡 space.

The numerical algorithm is based on Grünwald-Letnikov’s definition: 𝑥𝑡_𝑘=−𝑎𝑥𝑡_𝑘−1−𝑦𝑡_𝑘−1ℎ^𝑞1−^𝑘_𝑗=2𝑐^(𝑞1)_𝑗𝑥𝑡_𝑘−𝑗,𝑦𝑡_𝑘=𝑟𝑥𝑡_𝑘−𝑦𝑡_𝑘−1−𝑥𝑡_𝑘𝑧𝑡_𝑘−1ℎ^𝑞2−^𝑘_𝑗=2𝑐^(𝑞2)_𝑗𝑦𝑡_𝑘−𝑗,𝑧𝑡_𝑘=−𝑏𝑧𝑡_𝑘−1+𝑥𝑡_𝑘𝑦𝑡_𝑘ℎ^𝑞3−^𝑘_𝑗=2𝑐^(𝑞3)_𝑗𝑧𝑡_𝑘−𝑗, (33) where 𝑇_sim is the simulation time, 𝑘=1,2,…,𝑁, for 𝑁=[𝑇_sim/ℎ].

5.1.1. Sliding Mode Control Design of Lorenz’s System

In this part, we will firstly consider a simple case: the system uncertainty and external disturbance will be ignored. By introducing the control input to the first state equation of fractional Lorenz’s system, the controlled system is derived as 𝐷^𝑞1(𝑡),𝐷𝑥=−𝑎𝑥−𝑦+𝑢_𝑞2𝐷𝑦=𝑟𝑥−𝑦−𝑥𝑧,_𝑞3𝑧=−𝑏𝑧+𝑥𝑦. (34)

Owning to (8), the sliding surface is 𝑠(𝑡)=𝐷^𝑞1𝑥(𝑡)+𝐷⁻¹𝜙(𝑡), (35) where 𝜙(𝑡)=𝑟𝑦(𝑡)+𝑎𝑥(𝑡). (36)

In terms of (22), the control law is 𝑢(𝑡)=−(𝑎+𝑟)𝑦−𝑘₁sgn(𝑠)−𝑘₂𝑠. (37)

The numerical simulations with 𝑘₁=0.05 and 𝑘₂=0.1 are illustrated in Figure 2. It is clear that the control law (37) is efficient for controlling the fractional Lorenz’s system.

(a)

(b)

(c)

(d)

Figure 2 

Sliding mode control of the fractional Loren’s system with simulation time 𝑇sim=5s and time step ℎ=0.0005: (a) the 𝑥−𝑡 space; (b) the 𝑦−𝑡 space; (c) the 𝑧−𝑡 space; (d) the 𝑢−𝑡 space.

5.1.2. Adaptive Sliding Mode Control Design of Uncertain Lorenz’s System

In this part, we will consider a little intricacy case: the system uncertainty and external disturbance in the fractional Loren’s system will be considered. By introducing the control input to the first state equation, the controlled system is derived as 𝐷^𝑞1(𝑡),𝐷𝑥=−𝑎𝑥−𝑦+Δ𝑓+𝑑+𝑢_𝑞2𝐷𝑦=𝑟𝑥−𝑦−𝑥𝑧,_𝑞3𝑧=−𝑏𝑧+𝑥𝑦. (38)

By (28) and (29), the control law and adaptive law are selected as 𝑢(𝑡)=−(𝑎+𝑟)𝑦−𝜃₁+𝜃₂+𝑘₁sgn(𝑠)−𝑘₂̇𝜃𝑠,₁=𝜇₁|𝑠|,̇𝜃₂=𝜇₂|𝑠|. (39)(40)

The numerical simulations with the above control law (39) and adaptive law (40) are illustrated in Figure 3, with the gain of control law 𝑘₁=0.02 and 𝑘₂=0.02, the coefficients of adaptive law 𝜇₁=0.03 and 𝜇₂=0.02, and the initial conditions of the adaptive parameters 𝜃₁(0)=0.03, 𝜃₂(0)=0.02, Δ𝑓=0.1−0.1sin(𝜋𝑥), and 𝑑(𝑡)=0.1cos(𝑡). We can see from Figure 3 that the control law (39) and adaptive laws (40) are capable at controlling the fractional Loren’s system in the presence of uncertainty and external disturbance.

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(b)

(c)

(d)

(e)

(f)

Figure 3 

Adaptive sliding mode control of the fractional Loren’s system with dynamics uncertainty and external disturbance with simulation time 𝑇sim=5s and time step ℎ=0.0005: (a) the 𝑥−𝑡 space; (b) the 𝑦−𝑡 space; (c) the 𝑧−𝑡 space; (d) the 𝑢−𝑡 space; (e) online estimate of 𝜃1; (f) online estimate of 𝜃2.

5.2. Control of the Fractional Chen’s System

The fractional Chen’s system is described as 𝐷^𝑞1,𝐷𝑥=𝑎𝑦−𝑥_𝑞2()𝐷𝑦=𝑐−𝑎𝑥−𝑥𝑧+𝑐𝑦,_𝑞3𝑧=𝑥𝑦−𝑏𝑧, (41) where 𝑎=35, 𝑏=3, and 𝑐=28.

The equilibrium points of the system with the above parameters are 𝐸₁=(0,0,0), 𝐸₂=(3√7,3√7,21), and 𝐸₃=(−3√7,−3√7,21).

The initial condition for the fractional Chen’s system is chosen as 𝑥(𝑡)=𝑥0⁺𝑦(𝑡)0=1,=𝑦⁺𝑧(𝑡)0=1,=𝑧⁺=1, (42) for −∞≤𝑡≤0.

The fractional Chen’s system exhibits chaos with fractional orders 𝑞₁=𝑞₂=𝑞₃=0.9 and the above initial conditions, as depicted in Figure 4. The numerical algorithm is based on Grünwald-Letnikov’s definition: 𝑥𝑡_𝑘=𝑎𝑦𝑡_𝑘−1−𝑥𝑡_𝑘−1ℎ^𝑞1−^𝑘_𝑗=2𝑐^(𝑞1)_𝑗𝑥𝑡_𝑘−𝑗,𝑦𝑡_𝑘=(𝑐−𝑎)𝑥𝑡_𝑘−𝑥𝑡_𝑘𝑧𝑡_𝑘−1+𝑐𝑦𝑡_𝑘−1ℎ^𝑞2−^𝑘_𝑗=2𝑐^(𝑞2)_𝑗𝑦𝑡_𝑘−𝑗,𝑧𝑡_𝑘=𝑥𝑡_𝑘𝑦𝑡_𝑘−𝑏𝑧𝑡_𝑘−1ℎ^𝑞3−^𝑘_𝑗=2𝑐^(𝑞3)_𝑗𝑧𝑡_𝑘−𝑗. (43)

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Figure 4 

The chaotic trajectories of the fractional Chen’s system with simulation time 𝑇sim=100s and time step ℎ=0.005: (a) the 𝑥−𝑦−𝑧 space; (b) the 𝑥−𝑡 space; (c) the 𝑦−𝑡 space; (d) the 𝑧−𝑡 space.

5.2.1. Sliding Mode Control Design of Chen’s System

The system uncertainties and external disturbances will not be considered, and a control input is introduced to the second equation. The dynamics of the controlled system is described as 𝐷^𝑞1,𝐷𝑥=𝑎𝑦−𝑥_𝑞2()𝐷𝑦=𝑐−𝑎𝑥−𝑥𝑧+𝑐𝑦+𝑢,_𝑞3z=𝑥𝑦−𝑏𝑧. (44)

Using (8), the sliding surface is constructed as 𝑠(𝑡)=𝐷^𝑞𝑦(𝑡)+𝐷⁻¹𝜙(𝑡), (45) where 𝜙(𝑡)=𝑎𝑥+𝑐𝑦+𝑥𝑧.

By (22), the control law is determined as 𝑢(𝑡)=−𝑐𝑥−2𝑐𝑦−𝑘₁sgn(𝑠)−𝑘₂𝑠. (46)

The numerical simulations with 𝑘₁=0.1 and 𝑘₂=0.1 are illustrated in Figure 5. It is clear that the proposed control law (46) is feasible and efficient for controlling the fractional Chen’s system.

(a)

(b)

(c)

(d)

Figure 5 

Sliding mode control of the fractional Chen’s system with simulation time 𝑇sim=5s and time step ℎ=0.0005: (a) the 𝑥−𝑡 space; (b) the 𝑦−𝑡 space; (c) the 𝑧−𝑡 space; (d) the 𝑢−𝑡 space.

5.2.2. Adaptive Sliding Mode Control Design of Uncertain Chen’s System

The system uncertainties and external disturbances are added to the second equation and the control input is introduced to the second equation. Then the controlled system can be described as 𝐷^𝑞1,𝐷𝑥=𝑎𝑦−𝑥_𝑞2()𝐷𝑦=𝑐−𝑎𝑥−𝑥𝑧+𝑐𝑦+Δ𝑓+𝑑+𝑢,_𝑞3𝑧=𝑥𝑦−𝑏𝑧. (47)

By (28) and (29), the control law and adaptive law are selected as 𝑢(𝑡)=−𝑐𝑥−2𝑐𝑦−𝜃₁+𝜃₂+𝑘₁sgn(𝑠)−𝑘₂̇𝜃𝑠,₁=𝜇₁|𝑠|,̇𝜃₂=𝜇₂|𝑠|. (48)(49)

The numerical simulations with the above control law (48) and adaptive law (49) are illustrated in Figure 6, with the coefficients of control law 𝑘₁=0.175 and 𝑘₂=0.1, the coefficients of adaptive law 𝜇₁=0.2 and 𝜇₂=0.2, and the initial conditions of the adaptive parameters 𝜃₁(0)=0.2, 𝜃₂(0)=0.2, Δ𝑓=0.1−0.1sin(𝜋𝑥), and 𝑑(𝑡)=0.1−0.1sin(𝜋𝑥). It is clear that the proposed control law (48) and adaptive laws (49) are good at controlling the chaotic system in the presence of uncertainty and external disturbance.

(a)

(b)

(c)

(d)

Figure 6 

Adaptive sliding mode control of the fractional Chen’s system with dynamics uncertainty and external disturbance with simulation time 𝑇sim=5s and time step ℎ=0.0005: (a) the 𝑥−𝑡 space; (b) the 𝑦−𝑡 space; (c) the 𝑧−𝑡 space; (d) the 𝑢−𝑡 space; (e) online estimate of 𝜃1; (f) online estimate of 𝜃2.

6. Conclusion

In this paper, a novel class of fractional chaotic system in the pseudo state space is introduced and stabilized. To guarantee the stability of sliding mode dynamics, a novel fractional integral type sliding surface is constructed, with which an adaptive sliding mode control law is designed to stabilize the proposed fractional chaotic system with uncertainties and external disturbance whose bounds are unknown. Numerical simulations of fractional Lorenz’s system and Chen’s system have been presented to demonstrate the effectiveness of the proposed control technique.

Appendix

Lemma A.1. Consider 𝑊=𝑊₁+𝑊₂, (A.1) where 𝑊₁=^𝑚_𝑖=1𝑊_1𝑖,𝑊₂=^𝑚_𝑖=1𝑎_𝑖𝑊_2𝑖 (A.2) with 𝑊_1𝑖=∫^∞₀𝜇_𝑖(𝜔)𝜔𝑧²_𝑖(𝜔,𝑡)𝑑𝜔 and 𝑊_2𝑖=𝑥²_𝑖, 𝑖=1,2,…,𝑚.
The frequency discretizations of 𝑊_1𝑖 give 𝑊_1𝑖=^𝐽_𝑖=1𝜔_𝑖𝑗𝜇_𝑖𝜔_𝑖𝑗𝑧²_𝑖𝑗𝜔_𝑖𝑗,𝑡×Δ𝜔_𝑖𝑗=^𝐽_𝑖=1𝜔_𝑖𝑗𝑐_𝑖𝑗𝑧²_𝑖𝜔_𝑖𝑗,𝑡. (A.3)
It is clear that 𝑊_1𝑖, 𝑖=1,2,…,𝑚, are all positive definite quadratic forms and can be expressed in the matrix form as 𝑊_1𝑖=𝑍^𝑇_𝑖𝑀_𝑖𝑍_𝑖 (A.4) with 𝑀_𝑖=𝜔_𝑖1𝑐_𝑖1𝜔_𝑖2𝑐_𝑖2⋱𝜔_𝑖𝐽𝑐_𝑖𝐽. (A.5)
It is clear that 𝑊_2𝑖, 𝑖=1,2,…,𝑚, are also positive.
Because 𝑥_𝑖=𝐶^𝑇_𝑖𝑍_𝑖, 𝑊_2𝑖 can be rewritten as 𝑊_2𝑖=𝑍^𝑇_𝑖𝐶_𝑖𝐶^𝑇_𝑖𝑍_𝑖. (A.6) Let us define 𝑍=𝑍₁𝑍₂⋯𝑍_𝑚^𝑇,𝑀_𝐶=𝑀₁+𝑎₁𝐶₁𝐶^𝑇₁𝑀₂+𝑎₂𝐶₂𝐶^𝑇₂⋱𝑀_𝑚+𝑎_𝑚𝐶_𝑚𝐶^𝑇_𝑚. (A.7)
Then 𝑊=𝑍^𝑇𝑀_𝐶𝑍.
Since 𝑊_1𝑖 and 𝑊_2𝑖, 𝑖=1,2,…,𝑚, are all positive, then nonnegative values of 𝑎_𝑖, 𝑖=1,2,…,𝑚, implies 𝑊≥0, and positive values of 𝑎_𝑖 implies 𝑊>0.

Consequently, we have the following lemma.

The quadratic form 𝑊 is positive semi-definite if 𝑎_𝑖≥0 and is positive definite if 𝑎_𝑖>0, 𝑖=1,2,…,𝑚.