Stabilizations of Strange Attractors by Feedback Linearization

This study deals with the control of the Lorenz, Chen and Lu chaotic system. Feedback linearization was successfully implemented on these chaotic systems. Feedback linearization was successful to transform the three attractor systems to a general system that simplify the linear control analysis. Hence, a linear controller is designed for the feedback linearized general system. Furthermore, some numerical simulations were carried out for the closed loop systems. These simulations show that the developed controller design method is effective in stabilizing and regulating the response.


INTRODUCTION
Recently, the control of Lorenz attractor is an active research area.Edward Lorenz demonstrated that the thermal convection loop can be characterized by a three ordinary nonlinear differential equations.Then he showed that the system is chaotic for a certain conditions.Just after Lorenz have found the first chaotic attractor (Lorenz, 1963).Many researchers have been encouraged to look for new chaotic attractors.Recently, some new chaotic systems were found, such as Chen attractor (Chen and Ueta, 1999) and Lu attractor (Lu and Chen, 2002).In the recent years, a variety of nonlinear control techniques have been implemented for a Lorenz, Chen and Lu attractors, such as back-stepping control (Ge et al., 2000;Li, 2006), sliding mode control (Nazzal and Natsheh, 2007;Chiang et al., 2007), feedback linearization control (Alvarez-Gallegos, 1994;Pishkenari et al., 2007).In this study, Feedback linearization is employed to control the three attractors.In order to simplify the control analysis for the three systems, the method of feedback linearization is used to transform the three attractor systems to a general system.Then, a linear control law is proposed for the feedback linearized system.
Dynamical descriptions of the attractors: Three dynamical systems are considered here; The Lorenz attractor, Chen attractor and Lu attractor.In the next subsections, descriptions of the three chaotic systems, their chaotic attractors and their equilibrium points will be presented.The parameters of the chaotic Lorenz attractor is typically set to a = 10, b = 8/3, c = 28.The Lorenz attractor has three equilibrium points located at (0, 0, 0); ((± I(I − 1)), (± I(I − 1), I − 1).The Lorenz attractor is depicted in Fig. 1a and b.
The parameters of the chaotic Lu attractor is typically set to a = 36, b = 3, c = 20.The Lu attractor has three equilibrium points located at (0, 0, 0); Ә ∓ bc , ∓ bc , cә.The Lu attractor is depicted in Fig. 1e and f.

METHODOLOGY
Control strategy: In this section, the controlled dynamical systems for the Lorenz, Chen and Lu attractors are presented.The method of feedback linearization is applied on the attractors.Then, a steady state analysis is presented to facilitate the reference tracking problem for any of the three states (x, y, z).
where, u is the nonlinear control term.

Feedback linearization:
The method of feedback linearization transforms the nonlinear system into a linear system under certain situations.Hence, most of linear control theory can be applied to the feedback linearized system.All the systems (4, 5 and 6) can be transformed easily to the system 7 by selecting the appropriate u as depicted in Table 1: where, v now is a linear control term, thus, the second state y can be controlled easily and also it can be used to derive the other two states, by considering the term (ay) in the first equation and term (xy) in the second equation as an imaginary control terms.
The system 7 can be transformed to: where, v 1 = f (x) and v 2 = f (x, y) we notice that the second state equation in system 7 is linear and totally uncoupled from the other two states, moreover, the other two states are highly dependent on the second state (y).We may select v = -ky in the second state equation.Where, k is the state feedback gain.Then the closed loop system becomes: In order to have a stable closed loop system, the term -(k + 1) must be negative, then k>-1.

Steady state analysis:
In the steady state system 7 becomes: The Eq. ( 11) is well known as reference scaling in the state feedback control design.These relationships will facilitate the set point tracking for any of the three states by a single controller and without the need for designing a new controller for tracking each state of interest as done in Pishkenari et al. (2007).

RESULTS AND DISCUSSION
Each of the chaotic systems studied here are to be controlled using the method of the feedback linearization.The closed loop systems of the Lorenz, Chen and Lu attractor and the feedback linearization controllers were simulated using MATLAB SIMULINK.The state feedback gain was selected to be k = 5, which makes the closed loop system stable and also gives good performance characteristics.The Where the second state y has the fastest settling time even though it has the slowest pole, but this is because the other two states x; z are driven by the second state y, which totally agrees with Eq. ( 8).

CONCLUSION
The method of feedback linearization is implemented on three chaotic attractors.The chaotic oscillations are completely suppressed and the feedback linearized system behaves like a first order system.A linear controller is designed and the closed loop systems show an efficient regulation and reference tracking.
y ss = v, x ss = y ss and consequently Z ss = (x ss y ss ) /b.If v = -ky + r where r is the reference.

Fig. 2 :
Fig. 2: Closed loop response of the three states of Lorenz attractor and the control action

Fig. 4 :
Fig. 4: Closed loop response of the three states of Lu attractor and the control action

Table 1 :
Nonlinear control laws of the attractors