Sliding Mode Control in Finite Time Stabilization for Synchronization of Chaotic Systems

Abstract

An adaptive sliding mode control for chaotic systems synchronization is considered. The design of robust finite time convergent controller is based on geometric homogeneity and integral sliding mode manifold. The knowledge of the upper bound of the system uncertainties is not prior required. The chaos synchronization is presented to system stability based on the Lyapunov stability theory. The simulation results show the effectiveness of the proposed method.

1. Introduction

Nowadays, chaos has been seen to have a lot of useful applications in many engineering systems such as secure communications, optics, power converters, chemical and biological systems, and neural networks [1–5]. Chaotic systems are dynamical systems and their response exhibits a lot of specific characteristics, including an excessive sensitivity to the initial conditions, fractal properties of the motion in phase space, and broad spectrums of Fourier transform. The main feature of chaotic systems is that a very small change in initial conditions leads to very large differences in the system states.

Several other control methods have been successfully applied to chaotic motion control. For example, adaptive control [6, 7] presents chaos control of chaotic dynamical systems by using backstepping design method and so forth. Sliding mode control (SMC) is a popular robust control approach for its robustness against parameter variations and external disturbances under matching conditions of nonlinear systems operating under uncertainty conditions, as the controllers can be designed to compensate for the uncertainties or disturbances [8–15]. References [8–11] are concerned with sliding mode control of continuous-time switched stochastic systems. In practice, classic SMC suffers from high frequency chattering, as the infinite switching frequency required by ideal sliding mode is not achievable. Keeping the main advantages of the standard sliding mode control, the chattering effect is reduced and finite time convergence is provided. Interesting high-order sliding modes (HOSM) are proposed in [14–18] with the robustness of the system during the entire response.

In order to reduce the chattering, the approach in [19] is modified and applied Synchronization in chaotic dynamic systems, so that a continuous feedback is produced combining the robustness of HOSM and finite-time stabilization by continuous control. The aim of the modified method is to deal with unknown but bounded system uncertainties. The upper bounds of uncertainties are not required to be known in advance. System stability is proven by using the Lyapunov theory.

2. Problem Formulation

A class of uncertain chaotic systems with uncertainties is described as 𝑥̇_𝑖=𝑥_𝑖+1,𝑥̇_𝑛=𝑓𝑋,𝑝,𝑡, (1) where 1≤𝑖≤𝑛−1, 𝑋=[𝑥₁,𝑥₂,…,𝑥_𝑟]^𝑇∶=[𝑥₁,𝑥̇₁,…,𝑥^(𝑛−1)]^𝑇, 𝑥∈𝑅^𝑛 are the state variable and the control input, respectively. 𝑓(⋅)∈𝑅^𝑛 denotes an uncertain nonlinear function. 𝑝 is a vector of uncertain parameters whose values belong to some closed and bounded set. In fact, several nonlinear chaotic systems can be transformed into the controllable canonical form (1) with some state transformation, for example, Rössler systems, Lur’e-like system, and Duffing-Holmes system.

To control the system effectively we propose to add a control input 𝑢. By adding this input, the equation of the controlled system can be expressed by 𝑦̇_𝑖=𝑦_𝑖+1,𝑦̇_𝑛=𝑓(𝑌,𝑡)+𝑢, (2) where 𝑦∈𝑅^𝑛 is the state vector. The synchronization problem considered in this paper is to design a sliding mode controller 𝑢 based on finite time stabilization, which synchronizes the states of the master system (1) and the slave system (2) in spite of the unknown nonlinear parameter vector. In other words, the aim of synchronization is to make the following: lim_𝑡→∞‖𝑌−𝑋‖=0. (3) Let us define the tracking error as 𝐸(𝑡)(𝑡)(𝑡)=𝑦=𝑌−𝑋₁−𝑥₁,𝑦₂−𝑥₂,…,𝑦_𝑛−𝑥_𝑛^𝑇=𝑒(𝑡),𝑒̇(𝑡),…,𝑒^(𝑛−1)(𝑡)^𝑇=𝑒₁(𝑡),𝑒₂(2),…,𝑒_𝑛(𝑡,𝑥)𝑋=₁,𝑥₂,…,𝑥_𝑟^𝑇∶=𝑥₁,𝑥̇₁,…,𝑥^{(𝑛−1)𝑇}. (4) Subtract (1) from (2) and get 𝑒_𝑖̇=𝑒_𝑖+1𝑒_𝑛̇=𝑓(𝑌,𝑡)−𝑓𝑋,𝑝,𝑡+𝑢. (5)

As a case study one is the chaotic Lur’e-like system considered as drive system. The Lur’e-like system, as master, is considered as follows: 𝑥̇₁=𝑥₂,𝑥̇₂=𝑥₃,𝑥̇₃=𝑎₁𝑥₁+𝑎₂𝑥₂+𝑎₃𝑥₃+12ℎ𝑥₁, (6) where ℎ𝑥₁=𝑘𝑥₁if𝑥₁<1𝑘sign𝑥₁otherwise (7) and 𝑋=[𝑥₁,𝑥₂,𝑥₃]^𝑇 is the state vector. The parameters are selected exactly [4]. 𝑎₁=−6.8, 𝑎₂=−3.9, and 𝑎₃=−1. When 𝑘=1.5 and 𝑘=1.8, [𝑥₁(0),𝑥₂(0),𝑥₃(0)]^𝑇=[1,1,1]^𝑇. Chaotic responses of system (6) are shown in Figure 1 without any control input. System (6) shows a chaotic behavior.

Figure 1 

Chaotic behavior of system 𝑥.

The systems stated above could be only considered as theoretical models. In practice, mostly it is not possible to express the exact model or parameters of the system. It means that the problem of parameter uncertainty is unavoidable, and therefore it must be considered in real systems. If system (6) is considered as the master system, the slave system of (6) with uncertain parameters and perturbation can be rewritten in the form 𝑦̇₁=𝑦₂,𝑦̇₂=𝑦₃,𝑦̇₃=𝑎₁+Δ𝑎₁𝑦₁+𝑎₂+Δ𝑎₂𝑦₂+𝑎₃+Δ𝑎₃𝑦₃+12ℎ𝑦₁−ℎ𝑥₁+𝑢+𝑑(𝑡), (8) where Δ𝑎_𝑖 (𝑖=1,2,3) is an uncertain parameter of the chaos system (8), and 𝑑(𝑡) is the disturbance of system (8). Subtract (6) from (8); the real error dynamics would be obtained as 𝑒₁̇=𝑒₂,𝑒₂̇=𝑒₃,𝑒₃̇=𝑎₁𝑒₁+𝑎₂𝑒₂+𝑎₃𝑒₃+12ℎ𝑦₁−ℎ𝑥₁+𝑢+Δ𝑎₁𝑦₁+Δ𝑎₂𝑦₂+Δ𝑎₃𝑦₃+𝑑(𝑡). (9)

This paper proposes a new adaptive sliding mode controller for chaotic systems to not only preserve the advantages of variable structure control but also release the limitation of knowing the bounds of uncertainties and guarantees the occurrence of sliding motion and the synchronization of the master-slave chaotic systems. An adaptive sliding mode control based on finite time stabilization is established in Section 3.

3. SMC Based on Finite Time Stabilization

In practical terms, the resolution of the finite time stabilization is a delicate task which has generally been studied for homogeneous systems of negative degree with respect to a flow of a complete vector field. Indeed, for this kind of systems, finite time stability is equivalent to asymptotic stability [13–15]. A constructive feedback control law for finite time stabilization of all-dimension chain of integrators without uncertainty has been proposed in [19]. Before designing our robust finite time controller, we introduce the algorithm given in [19] and show its problem in terms of robustness.

3.1. Finite Time Stabilization of an Integrator Chain System

Consider that the nominal system (9), which is represented by SISO independent integrator chains, is defined as follows: 𝑧̇₁=𝑧₂⋮𝑧̇_𝑛−1=𝑧_𝑛,𝑧̇_𝑛=𝜔_nom. (10)

Lemma 1 (see [19]). Let 𝑘₁,…,𝑘_𝑛>0 be such that the polynomial 𝜆^𝑛+𝑘_𝑛𝜆^𝑛−1+⋯+𝑘₂𝜆+𝑘₁ is Hurwitz. Consider the system (10) There exists 𝜀∈(0,1) such that, for every 𝛼∈(1−𝜀,1), the origin is a globally finite time stable equilibrium for the system under the feedback 𝜔_nom(𝑧)=−𝑘₁sgn𝑧₁𝑧₁^𝛼1−⋯−𝑘_𝑛sgn𝑧_𝑛𝑧_𝑛^𝛼𝑛, (11) where 𝛼₁,…,𝛼_𝑛 satisfy 𝛼_𝑖−1=𝛼_𝑖𝛼_𝑖+1/(2𝛼_𝑖+1−𝛼_𝑖), 𝑖=2,…,𝑛 with 𝛼_𝑛+1=1 and 𝛼_𝑛=𝛼.

In order to design adaptive sliding mode controller for uncertain chaotic systems with unknown bounded uncertainties, there exist two major phases: first, an integral sliding manifold should be selected such that the sliding motion on the manifold has the desired properties. Second, an adaptive continuous control law should be determined such that the existence of the sliding mode can be guaranteed without knowing the upper bounds of uncertainties from Lemma 1.

3.2. Design of Adaptive Sliding Mode Control Based on Finite Time Stabilization

Consider system (9) which can be trivially rewritten as 𝑒₁̇=𝑒₂,𝑒₂̇=𝑒₃,𝑒₃̇=𝑎₁𝑒₁+𝑎₂𝑒₂+𝑎₃𝑒₃+12ℎ𝑦₁−ℎ𝑥₁+𝑢+Δ𝑎₁𝑦₁+Δ𝑎₂𝑦^𝛽₂+Δ𝑎₃𝑦₃+𝑑(𝑡). (12) It yields 𝑒₁̇=𝑒₂,𝑒₂̇=𝑒₃,𝑒₃̇=𝑎₁𝑒₁+𝑎₂𝑒₂+𝑎₃𝑒₃+12ℎ𝑦₁−ℎ𝑥₁+𝑢+𝛽(⋅), (13) where |𝛽(⋅)|≤𝑀 and 𝑀 is the upper bounds of uncertainties with unknown bound.

Define the integral sliding manifold for system (9): 𝑠=𝑒₃−𝑒₃(0)−^𝑡₀𝜔_nom𝑑𝜈. (14) It is obvious that 𝑠̇(𝑡)=𝑒₃̇−𝜔_nom, (15) where the construction of control law 𝜔_nom is given in Lemma 1. In order to stabilize in finite time system (13) with uncertainties, we define the following control law: 𝑢=𝑢₀+𝑢₁, (16) where 𝑢₀=−[𝑎₁𝑒₁+𝑎₂𝑒₂+𝑎₃𝑒₃+12(ℎ(𝑦₁)−ℎ(𝑥₁))]+𝜔_nom being the ideal control and stabilizes in finite time (13) at the origin when there are no uncertainties. The control law 𝑢₁ is designed in order to ensure that the sliding motion on the sliding manifold is guaranteed for 𝑡>0 in spite of uncertainties with unknown bound and is given by 𝑢₁=−𝜂+𝐺sign(𝑠), (17) where 𝜂>0. The adaptive law is 𝐺̇=𝑞‖𝑠‖,𝐺(0)=𝐺₀, (18) where 𝑞>0 and 𝐺₀ is the given bounded initial value.

Theorem 2. Consider the error system (13) with parametric uncertainties and disturbances; then, the control law (16) and (17) and the adaptation law (18) can ensure the establishment error state trajectory converges to the sliding manifold (14) 𝑠=0 in finite time.

Proof. Choose the following Lyapunov function: 1𝑉=2𝑠²+𝑞⁻¹𝜃², (19) where 𝜃∈𝑅 denotes the adaptation error that we will define later.
By taking the time derivative of 𝑉, and we get 𝑉̇=𝑠𝑠̇+𝑞⁻¹𝜃𝜃̇=𝑠𝑒₃̇−𝜔_nom+𝑞⁻¹𝜃𝜃̇=𝑠𝑎₁𝑒₁+𝑎₂𝑒₂+𝑎₃𝑒₃+12ℎ𝑦₁−ℎ𝑥₁+𝑢+𝛽(⋅)−𝜔_nom+𝑞⁻¹𝜃𝜃̇. (20) Using (16)–(18), we get 𝑉̇=𝑠𝑢₁+𝛽(⋅)+𝑞⁻¹𝜃𝜃̇=𝑠−𝜂+𝐺sign(𝑠)+𝛽(⋅)+𝑞⁻¹𝜃𝜃̇, (21) where the adaptation error defined as 𝜃=𝑀−𝐺, we get ̇(⋅)‖𝑠‖‖𝑠‖−𝐺‖𝑠‖‖𝑠‖‖𝑠‖‖𝑠‖.𝑉=−𝜂𝑠−𝐺𝑠+𝛽𝑠+𝜃≤−𝜂+𝑀+𝜃≤−𝜂 (22) Equation (22) implies that the manifold {𝑥∈𝑅^𝑛∶𝑠=0} can be reached in spite of uncertainties. Substituting (16) into (13), we get the equivalent closed-loop control, in the sliding manifold by differentiating (14) with respect to time in sliding mode 𝑢_eq=−𝑎₁𝑒₁+𝑎₂𝑒₂+𝑎₃𝑒₃+12ℎ𝑦₁−ℎ𝑥₁+𝛽(⋅). (23) Therefore, a higher order sliding mode with respect to the system trajectories converges to zero in finite time using Lemma 1.

4. Simulation

In this section, the presented control algorithm is demonstrated. In these numerical simulations, the fourth-order Runge-Kutta method is used to solve Lur’e-like system with time step size 0.001 in Matlab/Simulink. The parameters are selected as follows.

A perturbation 𝛽(⋅)=0.5sin(2𝑦₁)+10sin(𝑡) is considered, where 𝑛=3, 𝛼₃=3/4, 𝛼₂=3/5, 𝛼₁=1/2, 𝑘₁=3, 𝑘₂=2.5, 𝑘₃=1, and 𝜂=1.5. The simulation results are illustrated in Figure 2. From the figure, we can see that the synchronization errors 𝑒₁, 𝑒₂, and 𝑒₃ will converge to zero in the finite time. Figures 3 and 4 show the control input and the corresponding sliding manifold 𝑆(𝑡). In particular, it is worthy of note that no information of upper bounds of uncertainties is used in our control design. Estimate value of adaptive gain 𝐺 is described in Figure 5. The adaptation gain parameter and initial value are set as 𝑞=2 and 𝐺(0)=0. Figure 5 shows that the adaptation parameter tends to a constant value.

Figure 2 

Time responses of error states.

Figure 3 

Time response of control input 𝑢(𝑡).

Figure 4 

Time response of the corresponding sliding manifold 𝑆(𝑡).

Figure 5 

Time response of parameter estimation value 𝐺.

5. Conclusions

This work proposes an adaptive SMC controller for nonlinear systems with parametric uncertainties. This method can be viewed as the finite time stabilization based on geometric homogeneity and integral sliding mode control. The knowledge of the upper bound of the system uncertainties is not prior required. Simulation results demonstrate that the proposed control method is effective.