The 2nd
International Conference on Intelligent Control and
Information Processing
ANew Closed-Loop Controller for Fractional Order Chaotic Systems Mao Wang, Wenhu You, Jianni Zhao, Guanghui Sun, Liqun Shen
controlling the chaotic systems, many methods are presented. However, because of the short of basic theory about fractional order chaotic system, control theories in fractional order field develop slowly, and develop faster till the end of 20th century. The TID controller [6] and the CRONE controller [7,8] based on the theory about modem robust control and the fractional lead-lag compensator [7,8] are some of the well-known fractional order controllers. Besides, there are also many methods. The generalized Takagi-Sugeno (T-S) fuzzy model is developed by extending the conventional T-S fuzzy model [9]. For financial system, a sliding mode technique is proposed in [10]. The method is simple, robust and theoretically rigorous. Feedback control method is studied in [11], which can be applied to achieve chaos control in the fractional order modified coupled dynamos system. A simple fractional-order controller is studied in [12], which is a suitable choice in the control of uncertain chaotic systems. The controller has strong ability to eliminate chaotic oscillations or reduce them to regular oscillations in the chaotic systems. The controller has sample structure and is designed very easily. However, it does not apply to systems which have fractional order and nonlinear fractional order chaotic systems. So based on the previous work, we propose a new closed-loop controller. In the paper, we analyze the stability about the fractional order chaotic systems, and present the new closed-loop controller which has simple structure and is designed very easily. Besides, the controller applies to non-integer order and nonlinear fractional order chaotic systems. The controller can stabilize the fractional order chaotic system in a short time. The rest of this letter is organized as follows. In section 2, the closed-loop controller is designed and relevant theorems are given with necessary proof. A fractional order chaotic system is briefly introduced in section 3. The new closed-loop controller is also designed in this section. Numerical simulation results are given in section 4 to illustrate the effectiveness of the proposed controller. One important parameter is also studied in this section which can tune the adjusting time. Finally, some concluding remarks are given in section 5.
Abstract The paper presents a new closed-loop controller to stabilize a kind of fractional order chaotic system to an -
equilibrium point. The proposed controller has simple structure and only two tuning parameters, and is designed easily. The proposed controller can stabilize the fractional order chaotic system in a very short time. Besides, the response time can be changed by tuning the time scaling factor a . To determine the control parameters, one needs only a little knowledge about the plant and therefore, the proposed controller is a suitable choice in the control of the fractional order chaotic systems. Numerical simulations based on fractional order chaotic system show the effectiveness of the method.
I.
INTRODUCTION
LTHOUGH a fractional derivative is a mathematical topic with more than 300 years old history, its application to physics and engineering is just in resent years. It is found that many systems can be described by the fractional differential equations, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, and electromagnetic waves. Such systems are characterized by transfer function of non-integer order, arising from the presence of one or more fractional order elements in the system. Now, fractional order systems have attracted the attention of many researchers. Meanwhile, it has been shown that many fractional order differential systems can demonstrate chaotic behavior, such as the fractional order Lorenz system [1], fractional order Chua circuit [2], fractional order Rossler system [3], fractional order Chen system [4], fractional order unified system [5], and so on. Chaos control, as a new direction of chaos research, has received great attention in the past two decades in order to avoid troubles arising from unusual behaviors of a chaotic system. Since 1900 when OGY was successfully applied to
A
Manuscripe received May 28, 2011. This work was supported by National Natural Science Foundation of China (60904050) and China Postdoctoral Science Foundation Funded Project (20090450997). M. Wang is a professor at Harbin Institute of Technology, and the tutor of Ph.D. His research interests include intelligent systems, inertial technology and digital control. (e-mail:
[email protected]) W. You is an assistant professor at Harbin Institute of Technology. His research interests include optimum control and chaos control. (e-mail: yourmate_
[email protected])
II.
J. Zhao is a M. S candidate in Harbin Institute of Technology. Her research interests
include
fractional
order
system,
chaotic
system.
18745162350, e-mail:
[email protected]) the
control
of
fractional
order
chaotic
system.
(e-mail:
[email protected]) L. Shen is a tutor at Harbin Institute of Technology. His research interests include
chaos
control
and
synchronization,
relevant
control
theory.
(e-mail:
[email protected])
978-l-4577-08l6-9/ll/Y26.00 ©20ll IEEE
IN THE PRESENCE OF A
In general, in order to stabilize an unstable linear system, a controller is designed in feedback structure to drag the unstable poles of the open-loop systems into the stable region for the closed-loop system. However, in the paper, we propose a controller to stabilize the closed-loop system
G. Sun is a tutor at Harbin Institute of Technology. His research interests include
CLOSED-LOOP SYSTEM ORDER
FRACTIONAL-ORDER CONTROLLER
(phone:
22
July 25-28,
2011
fractional derivative and a is a nonzero constant,
without applying any changes on the places of the poles. The controller can stabilize an unstable system through somehow changing the order of the closed-loop system to a specific fractional order to make all eigenvalues (A) of matrix A satisfy 1 arg(A) I> q7r / 2 based on the stability condition. The block diagram of the closed-loop controller is described in Fig.l.
and 0
q1r /2 (5) Where A is the eigenvalue of system matrix A . The other parameter a which can be used to speed up or slow down the closed-loop response time is named as time scaling factor, and also can be confirmed based on the lemma as follows. Lemma 1.[8] Let xJ/) and X2(/) be the responses of and
Rnxn
d'x (6) -=f(x)+u dt' If the equilibrium point is xe ' the Jacobian matrix of this
U E Rn is control input. Consider the controller as follows [12]: C(s) = asq-s
system Dqx, =Ax, ' xJO)=xo
, x"x2 E
Where a -lIq is time scaling factor and therefore, the closed-loop response can be tuned by a suitable choice of a . From the knowledge above, we can obtain that a fractional order chaotic system can be controlled by a suitable choice of q , and can adjust speediness of the transient response of the closed-loop system by tuning the time scaling factor based on lemma 1. We extend it to nonlinear any order chaotic systems to obtain the result as follows. Theorem 1. Consider the nonlinear fractional system given by the form as follows:
Assume that the control objective is to stabilize the closed-loop system. To achieve the goal, the control parameter (q ) should be selected properly. Consider the linear and time-invariant system given by the form as follows: (1) X =Ax+u Rnxn
q < 1
the neighborhood of equilibrium points further, we define: (10) x(/) =xJ/)+ &(/) Thus, dq {xJ/)+ &(/)} It also means:
dlq
=� f(xe(t)+ &(/» a
(11)
dq {&(t)} 1 (12) -'----"-=-f(xe(t)+ &(t» a dlq Make the Taylor expansion of nonlinear function, we can obtain: af (13) f(xe(I)+ &(/» � f(xe( I»+ ax .
Dqx2=a-'Ax2 ,
Consider f(xJ/»,SO:
x2(0)=Xo respectively, where Dq is the operator of Caputo 23
IV.
1 aj (14) -'---'-'''-=-- Ix-x. set) dtq a ax Based on the stability condition of fractional order chaotic system, system (6) is asymptotically stable if and only if all eigenvalues (A) of the Jacobian matrix J 8f / ax at dq (s(t»
_
A.
> atr /2 .
y, (0)
conditions
are
chosen
as
x, (0)
=
1
0 and z, (0) 0, the simulation is showed in Fig.2. From Fig.2, we can see that amplitude of the closed-loop system response reaches below 5% of the initial conditions in 1,5 s. The response time is much shorter than the response time of other controllers based on the previous work. The ultimate states differ according to the initial conditions that we choose. For example, when initial conditions are chosen as x2(0) 10'Y2(0) 0, andz2(0) 0, and corresponding simulation is showed in Fig.3. In this case, amplitude of the closed-loop system response reaches below 5% of the initial conditions in 1.5 s. However, we can see that the fractional order chaotic system stabilizes to different equilibrium points (in Fig.2 state variable x � -6.92810 , in Fig.3 state variable x � 6.92810) .
So it can be
said that nonlinear fractional order system (6) is asymptotically stable under controller (7). D Theorem 1 prescribes a closed-loop controller based on the stability about fractional order chaotic system. The choice of time scaling factor is based on lemma 1 and can be seen as the extension of lemma 1 in fractional order chaotic systems. The following example illustrates the point III.
The simulation of a=1
Initial
=
the equilibrium points satisfy l arg(A) 1
SIMULATION RESULTS
=
=
=
=
SYSTEM DESCRIPTION AND THE DESIGN OF THE CLOSED-LOOP CONTROLLER
We describe the fractional order chaotic system as follows:
=
dq,x
-=a(y- x)
dtq,
dq,x
(15)
-- =(c-a)x-xz+cy
dtq,
dq,x
--=xy-bz
dtq3
When a=40 , b =3 , c=28 , the equilibrium pointsxe are (0, 0, 0) and (±6.928l0, ±6.92820, 16) The eigenvalues of system (15) are: A, Ix�e,={ -32, 20, -3}
(16) ,1,2 Ix�e"e,={-20,2305, 2,6152 ± 13,5268i} Based on the location of equilibrium points and stability condition about fractional order chaotic system, the system can ' t be stable for the first equilibrium point However, for the second and third equilibrium points, the eigenvalues of Jacobian matrix contain conjugate roots which have real part Based on the stability condition, the system will be stable if and only if the system order satisfies the stability condition i.e, q must satisfy the equation below: q, = q2 = q3 2
B.
- I arg(A2,) 1 7f
,13,5268 2,6152
=
=
(17) i
The simulation of different time scalingfactors
When the time scaling factor a 1 , the state response is showed in Fig.2 (the initial conditions are chosen as � (0) l'YI (0) 0, and Zl (0) 0), from which we can see that amplitude of the closed-loop system responses reach below 5% of the initial conditions in 1.5 s. To obtain a response with demanded speed, we should tune the time scaling factor a properly. Now, we set the adjustment time as 1 s. Based on lemma 1, the time scaling factor should satisfy all08 =1.5/1=1.5 , i.e. time scale factor a should be chosen as 1.3832. So the controller is described as follows:
2
