Terminal Sliding Mode Control with Adaptive Feedback Control in a Class of Chaotic Systems

Abstract

This paper analyzes semifinite time stability for a general chaotic system. By cooperating methods terminal sliding mode (TSM) with adaptive feedback control (AFC), a controller based on the two methods is derived to achieve semifinite time stability. The theoretical analysis employs the theories of linear matrix inequalities and Lyapunov functional method. Finally, numerical simulation is given to illustrate the derived theoretical results.

1. Introduction

Chaos phenomenon can be found in many physics and engineering systems in practice. However, to improve the system’s performance, it is often desirable to avoid chaos, and various methods are proposed. Due to different emphases, controllers have different merits and drawbacks. For example, TSM establishes terminal sliding mode surface to couple system variables and control them to reach equilibrium points. Its control is effective, but it can only control system states coupled in the sliding mode surface; readers are referred to [1–7] for more detailed information. As for impulse control [8–16], they add impulse effects to continuous differential equation and, by constructing comparison system, establish relationships between parameters of system and impulse. Their controllers are effective, but design processes of their controllers are too much complex. For adaptive feedback control as in [17–21], similar to TSM, they have unified the format with different parameters. AFC has a wide range of applications in various fields, but its dynamic is not as good as the first two.

For the system’s structure constructed in this paper, we design controllers from methods TSM and AFC, both of which have unified formats. If TSM is used only, we should design several TSM surfaces. If AFC is used only, its controller is very simple and flexible, but it can only achieve asymptotical stability. Combining their merits and drawbacks, a cooperative controller is proposed in the paper. TSM method finite-timely controls system states, which are coupled in TSM surface, as in [4–7]. Simple AFC is introduced as a supplementary control into the remaining states of the system, controlling system states which are outside TSM surface and making them asymptotically stable [18, 20], and, finally, the overall system tends to be semifinite-time stable [5].

This design scheme can control main elements of system finite-time stability, firstly, then use AFC method to ensure that other dimensions are asymptotically stable, and finally realize the overall system’s semifinite time stability. Compared with TSM only, this design can greatly reduce the control input and simplify the design process of controller; compared with AFC only, it has obvious advantages in time sequence.

The rest of the paper is organized as follows. In Section 2, a general chaotic system model and some preliminaries are presented. In Section 3, we will show theoretical analysis, establish several sufficient conditions for SFTS, and formulate controller. In Section 4, numerical simulation is presented to verify the validity of theoretical results. Finally, the conclusions are drawn in Section 5.

2. Problem Statement and Preliminaries

We use the following differential equation to describe general dynamic chaos system: 𝑦̇=𝑓𝑦. (1)

In this paper, we are committed to solve the stability analysis of chaotic systems which can be organized into the following form: 𝑥̇₁=𝐴₁₁𝑥₁+𝐴₁₂𝑥₂𝑥̇₂=𝐴₂₁𝑥₁+𝐴₂₂𝑥₂+𝑓₂𝑥₁,𝑥₂,𝑥₃𝑥̇₃=𝑓₃𝑥₁,𝑥₂,𝑥₃, (2) where 𝑥₁=(𝑦₁,…,𝑦_𝑟)^𝑇𝑥₂=(𝑦_𝑟+1,…,𝑦_𝑠)^𝑇𝑥₃=(𝑦_𝑠+1,…,𝑦_𝑡)^𝑇,𝑓₂𝑓₃, are nonlinear part.

Definition 1 (finite-time stability [22, 23]). Consider that 𝑥(𝑡)∈𝑅^𝑛 are system states. If there exists constant 𝑇>0, such that lim_𝑡→𝑇‖𝑥‖=0 and if 𝑡≥𝑇,  ‖𝑥(𝑡)‖=0, then the system realizes finite-time stability.

Definition 2 (semifinite-time stability [5]). Take dynamic system (2), for example; after adding controllers, if the states 𝑥₁ and 𝑥₂ are finite-time stable and the states 𝑥₃ are asymptotically stable, then we call the system semifinite-time stable.

Lemma 3 (see [23]). Assume that a continuous, positive-definite function 𝑉(𝑡) satisfies the following differential inequality: 𝑉̇(𝑡)≤−𝑐𝑉^𝜏(𝑡),∀𝑡≥𝑡₀,𝑉𝑡₀≥0, (3) where 𝑐>0,  0<𝜏≤1 are two constants. Then, for any given 𝑡₀, 𝑉(𝑡) satisfies the following results. 𝑉(𝑡)≡0,  ∀𝑡≥𝑡₁ with 𝑡₁ being given by 𝑡₁=𝑡₀+𝑉^1−𝜏𝑡₀𝑐().1−𝜏 (4)

Assumption 4. Assume that there exist positive constants 𝑙_𝑘(𝑘=1,…,𝑚),𝑓₂,𝑓₃ satisfying 𝑓(𝑥)≤^𝑚_𝑘=1𝑙_𝑘‖𝑥‖^𝑘,‖𝑥‖=max𝑦_𝑖,(𝑖=𝑠+1,…,𝑡). (5)

3. Main Results

In this section, controller is designed from TSM and AFC separately with detailed theoretical analysis. TSM portion is used to derive 𝑥₁𝑥₂ finite-time stability, and AFC is for asymptotical stability of 𝑥₃. Finally, system (2) tends to semifinite time stability.

3.1. Terminal Sliding Mode Portion Design

Take the system's first two parts of the states and add controlling part: 𝑥̇₁=𝐴₁₁𝑥₁+𝐴₁₂𝑥₂𝑥̇₂=𝐴₂₁𝑥₁+𝐴₂₂𝑥₂+𝑓₂𝑥₁,𝑥₂,𝑥₃+𝑢₁. (6)

Generally speaking, the process of terminal sliding mode control can be divided into two stages: the first stage is to establish the nonsingular terminal sliding model surface; the second is to design TSM controller, which can make the system variables reach and maintain the TSM surface within finite time.

So in this paper, a nonsingular TSM surface is introduced as follows: 𝑠=𝐶₁𝑥₁+𝐶₂𝑥₂+𝐶₃𝑥₁^𝑞/𝑝, (7) where parameters 𝐶₁,𝐶₂,𝐶₃,0<𝑞<𝑝 are odd matrices determined by the designer.

Theorem 5. In the system (6), we add the following controller: 𝑢₁1=−𝐶₂𝐶₁+𝐶₃𝑞𝑝𝑥₁^{(𝑞−𝑝)/𝑝}𝐴₁₁𝑥₁+𝐴₁₂𝑥₂−𝐴₂₁𝑥₁+𝐴₂₂𝑥₂+𝑓₂𝑥₁,𝑥₂,𝑥₃−𝜓𝑠𝐶₂𝜓=𝜇+𝜂‖𝑠‖^𝜉−1, (8) where 𝜇>0,  𝜂>0,  and 0<𝜉<1. Then the system states slide on the switch surface 𝑠=0.

Proof. Introduce the following Lyapunov function: 𝑉₁=12𝑠^𝑇𝑠. (9) Differentiating the function along the trajectories of 𝑠, we will obtain 𝑉̇₁=𝑠^𝑇𝑠̇=𝑠^𝑇𝐶₁+𝐶₃𝑞𝑝𝑥₁^{(𝑞−𝑝)/𝑝}𝐴₁₁𝑥₁+𝐴₁₂𝑥₂+𝑠^𝑇𝐶₂𝐴₂₁𝑥₁+𝐴₂₂𝑥₂+𝑓₂𝑥₁,𝑥₂,𝑥₃+𝑠^𝑇𝑢₁=−𝑠^𝑇𝜓𝑠=−𝜇𝑠^𝑇𝑠−𝜂‖𝑠‖^𝜉+1≤−𝜂‖𝑠‖^𝜉+1≤−2^(𝜉+1)/2𝜂𝑉₁^(𝜉+1)/2. (10)
From Lemma 3 and Definition 1, TSM surface could reach the equilibrium point within finite time.
When the system states slide on the switch surface 𝑠=0, they satisfy the following equations: ̇𝑠=0,𝑠=0. (11)
Then 𝑥₂=−𝐶₂⁻¹𝐶₁𝑥₁+𝐶₃𝑥₁^𝑞/𝑝. (12)
Thus, we can get the following sliding mode dynamics: 𝑥̇₁=𝐴₁₁−𝐴₁₂𝐶₁𝐶₂⁻¹𝑥₁−𝐴₁₂𝐶₃𝐶₂⁻¹𝑥₁^𝑞/𝑝. (13)

Theorem 6. For differential equation (13), if there exist a positive symmetric matrix 𝑃 and matrices 𝐶₁𝐶₂, introduced in (7), satisfying the following conditions: 𝑃𝐴₁₁−𝐴₁₂𝐶₁𝐶₂⁻¹+𝐴₁₁−𝐴₁₂𝐶₁𝐶₂^−1𝑇𝑃<0𝑃𝐴₁₂𝐶₃𝐶₂⁻¹=diag𝑏_𝑖,𝑏_𝑖𝑏>0,𝑖=1,…,𝑟,𝑏=min_𝑖, (14) then the system (6) is finite-time stable.

Proof. Introduce the following Lyapunov function: 𝑉₂=𝑥^𝑇₁𝑃𝑥₁, (15) where 𝑃 is a symmetric positive definite matrix satisfying the conditions in (14).
It can be proved easily that 𝑥₁≥𝑉𝑥₁𝜆,𝜆=𝜆_max(𝑃), (16) where 𝜆_max(𝑃) is the maximum eigenvalue of 𝑃.
Calculate its derivative along the solution of system (13) and we can obtain 𝑉̇₂=2𝑥^𝑇₁𝑃𝑥̇₁=2𝑥^𝑇₁𝑃𝐴₁₁−𝐴₁₂𝐶₁𝐶₂⁻¹𝑥₁−𝐴₁₂𝐶₃𝐶₂⁻¹𝑥₁^𝑞/𝑝=𝑥^𝑇₁𝑃𝐴₁₁−𝐴₁₂𝐶₁𝐶₂⁻¹+𝐴₁₁−𝐴₁₂𝐶₁𝐶₂^−1𝑇𝑃𝑥₁−2𝑥^𝑇₁𝑃𝐴₁₂𝐶₃𝐶₂⁻¹𝑥₁^𝑞/𝑝≤−2^𝑟_𝑖=1𝑏_𝑖𝑥²_1𝑖^{(𝑞+𝑝)/2𝑝}≤−2𝑏𝑥₁^{(𝑝+𝑞)/𝑝}≤−2𝑏𝜆^{(𝑞+𝑝)/2𝑝}𝑉𝑥₁^{(𝑞+𝑝)/2𝑝}. (17)
Then differential equation (13) is exponentially finite-time stable; state 𝑥₁ tends to zero within finite time. By (12), 𝑥₂ also tends to zero; then, the desired objective, finite-time stability of system (6), is achieved.

3.2. Adaptive Feedback Control Portion Design

After the finite-time convergence of system (6), system (4) transforms into 𝑥̇₃=𝑓𝑥₃. (18) Show each state of 𝑥₃ as the following form: 𝑥̇=𝑓(𝑥). (19)

Theorem 7. AFC controller 𝑢₂  is designed as follows and added to the system (19) on the right side: 𝑢₂=^𝑚_𝑘=1𝛽_𝑘𝑥^2𝑘−1, (20) where 𝛽_𝑘(𝑘=1,…,𝑚) are adaptive parameters whose adaptive laws are 𝛽̇_𝑘=−^𝑛_𝑗=1𝛾_𝑘𝑗𝑥^{2(𝑗+𝑘)−2}. (21) Then, the controlled system changes into the following form: 𝑥̇=𝑓(𝑥)+𝑢₂ (22) and it is asymptotically stable.

Proof. We introduce the following Lyapunov function: 𝑉₃=12^𝑛_𝑗=11𝑗𝑥^2𝑗+12^𝑚_𝑘=11𝛾_𝑘𝛽_𝑘+𝐿², (23) where 𝐿 is a constant bigger than the maximum of 𝑚2+𝑙_2𝑘−1𝑘=1,2,…,𝑚+12,𝑚2+𝑛𝑙²_2𝑘4𝑚𝑘=1,2,…,2, (24) where [𝑚/2] denotes the largest integer, no more than 𝑚/2.
The derivative of 𝑉₃ along trajectories of (23) is 𝑉̇₃=^𝑛_𝑗=1𝑥^2𝑗−1𝑥̇+^𝑚_𝑘=11𝛾_𝑘𝛽_𝑘+𝐿𝛽̇_𝑘≤^𝑛_𝑗=1|𝑥|^{𝑚2𝑗−1}_𝑘=1𝑙_𝑘‖𝑥‖^𝑘−𝐿^𝑚𝑛_𝑘=1𝑗=1|𝑥|^{2(𝑗+𝑘)−2}=^𝑛_𝑗=1^[(𝑚+1)/2]_𝑘=1|𝑥|^2𝑗−1𝑙_2𝑘−1‖𝑥‖^2𝑘−1+^𝑛_𝑗=1^[𝑚/2]_𝑘=1|𝑥|^2𝑗−1𝑙_2𝑘‖𝑥‖^2𝑘−𝐿^𝑛𝑚_𝑗=1𝑘=1|𝑥|^{2(𝑗+𝑘)−2}≤^𝑛_𝑗=1^[(𝑚+1)/2]_𝑘=1𝑙_2𝑘−1|𝑥|^{2(𝑗+𝑘)−2}+^𝑛_𝑗=1^[𝑚/2]_𝑘=1|𝑥|^4𝑗−2+^𝑛_𝑗=1^[𝑚/2]_𝑘=1𝑙²_2𝑘4|𝑥|^4𝑘−𝐿^𝑛𝑚_𝑗=1𝑘=1|𝑥|^{2(𝑗+𝑘)−2}≤^𝑛_𝑗=1^[(𝑚+1)/2]_𝑘=1𝑙_2𝑘−1|𝑥|^{2(𝑗+𝑘)−2}+𝑚2^𝑛𝑚_𝑗=1𝑘=1|𝑥|^{2(𝑗+𝑘)−2}+𝑛𝑙²_2𝑘4^𝑛_𝑗=1^[𝑚/2]_𝑘=1|𝑥|^{2(𝑗+𝑘)−2}−𝐿^𝑛𝑚_𝑗=1𝑘=1|𝑥|^{2(𝑗+𝑘)−2}≤−^[(𝑚+1)/2]_𝑘=1𝑚𝐿−2−𝑙_2𝑘−1^𝑚_𝑗=1|𝑥|^{2(𝑗+𝑘)−2}−^[𝑚/2]_𝑘=1𝑚𝐿−2−𝑛𝑙²_2𝑘4^𝑚_𝑗=1|𝑥|^{2(𝑗+𝑘)−2}≤0. (25)
So if and only if 𝑥=0,  𝑉̇₃=0, then asymptotical stability of (22) is proved.

Remark 8. Add related terms of 𝑖 to prove course; introduce Lyapunov function: 1𝑉=2^𝑡𝑛_{𝑖=𝑠+1𝑗=1}1𝑗𝑦_𝑖^2𝑗+12𝛾^𝑡_𝑘𝑖^𝑚_{𝑖=𝑠+1𝑘=1}𝛽_𝑘𝑖+𝐿² (26) by the same method and almost the same process; we can prove the globally asymptotical stability of system state (18) with AFC controller. So AFC acts as a single-state control and also can be designed globally.

4. Simulation Results

After the theoretical analyses are investigated, we discuss the following numerical example to illustrate the derived theoretical results. From [13], we can obtain the system model of Chua’s Oscillator: 𝑥̇₁=𝛼−𝑥₁+𝑥₂−𝑓𝑥₁,𝑥̇₂=𝑥₁−𝑥₂+𝑥₃,𝑥̇₃=−𝛽𝑥₂−𝛾𝑥₃, (27) where 𝑓(𝑥₁)=𝑏𝑥₁+((𝑎−𝑏)/2)(|𝑥₁+1|−|𝑥₁−1|).

The Chua oscillator is illustrated in Figures 1 and 2. For simulation, the system parameters are used as 𝛼=8.72,  𝛽=14.29,  𝛾=0.01,  𝑎=−8/7,  𝑏=−5/7, with initial conditions (𝑥₁(𝑡₀),𝑥₂(𝑡₀),𝑥₃(𝑡₀))^𝑇=(−2.1213,−0.0662,2.8811)^𝑇.

Figure 1 

The time series of uncontrolled Chua oscillator.

Figure 2 

Chaotic attractor of uncontrolled Chua oscillator.

With constraint condition (14), we get 𝐶₁=2,  𝐶₂=𝐶₃=1, introduced from (7); we take 𝜇=𝜂=10,  𝜉=1/2, in (9); via (8), obtain 𝑢₁, added to system (6). Its effectiveness is shown in Figure 3. After the convergence of 𝑥₁,  𝑥₂, there is only 𝑥̇₃=−0.01𝑥₃ in (18). So we let 𝑚=1,  𝑛=2, and  𝛾_𝑘𝑗=1; in (20) and (21), get 𝑢₂, added to system (22). Its effectiveness is shown in Figure 4.

Figure 3 

Time series of Chua oscillator with TSM only.

Figure 4 

Time series of Chua oscillator with both TSM and AFC.

According to simulation results, it is easy to find that, the TSM method can effectively control the first two of system states and realize their finite-time stability. Comparing Figures 3 and 4, we can find that AFC controller effectively controls the remaining one of system state and makes it asymptotically stable. Finally, system states globally tend to semifinite time stable and we complete the expected objectives.

5. Conclusions

A controller, cooperating TSM with AFC, is proposed to control a class of chaotic system as described above in this paper. Two methods are complementary in the procedure and finally achieve good effectiveness. Complex TSM method controls main elements of chaotic system to finite-time stability; then, simple AFC method controls dimension elements of chaotic system to asymptotic stability and finally the overall system goes to semifinite time stability. This design scheme can not only guarantee the system’s convergence but also reduce the system’s control-input spending and also further improve their applications in chaos control.

In this paper, effective performance of the simulation results proves the feasibility of this design scheme. The proposed method can be applied in many famous chaotic systems such as Lorenz, liu chaotic system, and Chua’s circuit.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are greatly indebted to the anonymous referees for their constructive comments. The work described in this paper was partially supported by the National Natural Science Foundation of China (no. 10971240), the Natural Science Foundation Project of CQ CSTC (nos. cstc2012jjA40052, cstc2013jcyjA0973, and cstc2013jcyjA80013), Applying Basic Research Program of Chongqing Education Committee (nos. KJ120615, KJ120630, KJ130611, and KJ1400505), the Foundation Project of Chongqing Normal University (no. 13XLZ01), and the Program of Chongqing Innovation Team Project in University under Grant no. KJTD201308.