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

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.


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][2][3][4][5][6][7] for more detailed information.As for impulse control [8][9][10][11][12][13][14][15][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][18][19][20][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][5][6][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 semifinitetime 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.

Main Results
In this section, controller is designed from TSM and AFC separately with detailed theoretical analysis.TSM portion is used to derive  1  2 finite-time stability, and AFC is for asymptotical stability of  3 .Finally, system (2) tends to semifinite time stability.

Terminal Sliding Mode Portion
Design.Take the system's first two parts of the states and add controlling part: 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: where parameters  1 ,  2 ,  3 , 0 <  <  are odd matrices determined by the designer.
Proof.Introduce the following Lyapunov function: Differentiating the function along the trajectories of , we will obtain 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  = 0, they satisfy the following equations: Then Thus, we can get the following sliding mode dynamics: Theorem 6.For differential equation (13), if there exist a positive symmetric matrix  and matrices  1  2 , introduced in (7), satisfying the following conditions: then the system (6) is finite-time stable.
Proof.Introduce the following Lyapunov function: where  is a symmetric positive definite matrix satisfying the conditions in (14).
It can be proved easily that where  max () is the maximum eigenvalue of .
Calculate its derivative along the solution of system (13) and we can obtain Then differential equation ( 13) is exponentially finite-time stable; state  1 tends to zero within finite time.By (12),  2 also tends to zero; then, the desired objective, finite-time stability of system (6), is achieved.

Adaptive Feedback Control Portion Design.
After the finite-time convergence of system (6), system (4) transforms into Show each state of  3 as the following form: Theorem 7. AFC controller  2 is designed as follows and added to the system (19) on the right side: where   ( = 1, . . ., ) are adaptive parameters whose adaptive laws are Then, the controlled system changes into the following form: and it is asymptotically stable.
Proof.We introduce the following Lyapunov function: where  is a constant bigger than the maximum of where [/2] denotes the largest integer, no more than /2.The derivative of  3 along trajectories of ( 23) is So if and only if  = 0, V 3 = 0, then asymptotical stability of ( 22) is proved.Remark 8. Add related terms of  to prove course; introduce Lyapunov function: 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.
With constraint condition (14), we get  1 = 2,  2 =  3 = 1, introduced from (7); we take  =  = 10,  = 1/2, in (9); via (8), obtain  1 , added to system (6).Its effectiveness is shown in Figure 3.After the convergence of  1 ,  2 , there is only ẋ 3 = −0.01 3 in (18).So we let  = 1,  = 2, and   = 1; in ( 20) and ( 21), get  2 , added to system (22).Its effectiveness is shown in Figure 4.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.

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.

2 Mathematical
Problems in Engineering