Prediction - Based Feedback Control of Fractional Order System

A. Naceri(1), N. Mansouri(2), A. Charef(2)

(1) IIT: Istituto Italiano di Tecnologia, Robotics, Brain and Cognitive Sciences Dept. Via Morego 30, 16163 Genova, Italy

(2)Laboratoire d’Automatique et Robotique, Département d’électronique, Faculté des Sciences de l’Ingénieur, Université Mentouri, Route de Ain-El-Bey, 25000. Constantine, Algerie

Emails: abdeldjallil.naceri@iit.it, {nor_mansouri, abdelfatah_charef}@yahoo.fr.

Abstract — In this paper, we introduce a control algorithm based on the delayed feedback control method to stabilize a fractional order system on an unstable fixed point. The control input is defined as the difference between the future, and current state. The gain matrix is obtained using the pole placement method. To illustrate the effectiveness of the proposed algorithm, we give simulation results of the control of fractional chaotic Chua’s system.

I. INTRODUCTION Fractional calculus is a 300-year-old topic. Although it has a long mathematical history, the applications of fractional calculus to physics and engineering are only a recent focus of interest [1,2]. Many systems are known to display fractional order dynamics, such as viscoelastic systems [3-5], dielectric polarization, electrode-electrolyte polarization and electromagnetic waves. More recently, many authors begin to investigate the chaotic dynamics of fractional order dynamical systems [7,8]. The study of chaotic systems is motivated by the fact that many interesting natural and man-made phenomena, such as earthquakes, laser systems, and epileptic seizures are of chaotic nature. These phenomena have previously been thought to be stochastic, and therefore unpredictable. However, research has revealed that it is indeed possible to predict time series generated by chaotic systems [9]. On the other hand, deterministic modelling of chaotic systems is limited due to the complexity of the system dynamics [10]. A number of chaos control algorithms have been proposed in the literature and many aspects of controlling chaos have been investigated [11,15]. In particular, Ott, Grebogi and Yorke [11] have shown that chaotic systems may be controlled by making only small changes in an accessible control parameter. This method known as the OGY method is a feedback control method which uses the discrete system model based on the Poincaré map linearization to stabilize an unstable periodic orbit. The delayed feedback method proposed by Pyragas [12] is an alternative method to control chaos. This method allows the control of chaotic systems by finding and stabilizing a T periodic orbit by using a control input which is based on the difference between a T-delayed state and the current state. Ushio and Yamamoto [13] proposed a novel control method, called prediction–based feedback control, for discrete–time chaotic systems. In this method, a stabilized periodic orbit is estimated. The difference between the

estimated periodic orbit and the current state is fed back into the control input. In this paper, we propose to control fractional chaotic systems using a control algorithm based on Pyragas’ delayed feedback control method. The control input is defined as the difference between the future, and current state. The gain matrix is obtained using the pole placement method. Similarly to the OGY method, the control is done by making only small perturbations to the system by exploiting the unstable periodic orbits that are embedded in the chaotic attractor. To illustrate the effectiveness of the proposed algorithm, we give some simulation results obtained on a fractional order Chua system. The paper is organized as follows. The rational approximation is described in section 2. The proposed algorithm of control is developed in section 3 and 4. In section 5, the method is applied in a simulation to the control of a fractional order Chua chaotic system. A conclusion is given in section 5.

II. RATIONAL APPROXIMATION Charef’s [16] method of approximation is used. The transfer function of the fractal order integrator is represented in the frequency domain by the following irrational function

αI s1G = (1)

Where ωjs = is the complex and α is a positive fractional number such that 10 <α< . In given frequency band of practical interest [ ]HL ω,ω .

This fractional order operator can be modelled by a fractional power pole (FPP) whose transfer function is given as follows

( ) α

⎟⎠⎞

⎜⎝⎛ +

=

ωs1

KsG I (2)

If we suppose that [ ]HL ,ωω∈ω we have cω>>ω , therefore we can write

( ) ( )sGs1

sK

s

KsG Imm

cI

c

I ==ω

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ω

=α

α (3)

908978-1-4244-1666-0/08/$25.00 '2008 IEEE

Where Lc 1.0 ω=ω (4)

The method of approximation consists of approximating

the 20m dB/dec slope on a Bode plot of the FPP by a number of zig–zag lines with alternate slopes of 20m dB/dec and zero dB/dec corresponding to alternate poles and zeros on the negative real axis of the s-plane such that

N1N1100 pzzpzp <<<<<< −K (5)

Hence, the approximation is given by:

( )∏

∏

=

−

=

α

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

≅

⎟⎟⎠

⎞⎜⎜⎝

⎛ω

+

=N

0i i

1N

0i iI

c

I

ps1

zs1

Ks1

KsG (6)

Using the graphical method described in [16], the poles and zeros of the approximation are found to be in a geometric progression form. This graphical method of approximation began with specified approximation error y in dB and approximation frequency band maxω . The frequency band maxω can be chosen to be 100 times the high frequency of the frequency band of practical interest Hω , the parameters a, b, p0, z0 and N can be easily determined as follows [19]

( )

( ) 1ablogp

logintegerN

apz and 10p

10b and 10a

0

max

0020

y

c0

10y

110y

+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ω

=

=ω=

==

⎥⎦

⎤⎢⎣

⎡α

⎥⎦

⎤⎢⎣

⎡α

⎥⎦

⎤⎢⎣

⎡α−

Hence, the fractional order integrator can be approximated by a rational function in a given frequency band of practical interest by

( )( ) N,0,1,ifor ,abzz

N,0,1,ifor ,abppi

0i

i0i

K

K

==

==

III. PREDICTION-BASED FEEDBACK CONTROL ALGORITHM

We consider the following continuous fractional order system described by

( ) ( )( )txfdt

txd =α

α

(7)

Where nx ℜ∈ is the state vector, α is the system’s order and we assumed that f is differentiable. The purpose of predictive control is to assure that the system asymptotically converges towards the fixed point or unstable periodic orbits by adding a small signal ( )tu to

the initial system (7). The controlled system will be defined as:

( ) ( )( ) ( )tutxfdt

txd +=α

α

(8)

Fig. 1. Block diagram of Predictive Feedback Control

The control input l)t(u ℜ∈ is chosen to be a function of the error between the future, and current state. In the form

( ) ( ) ( )( )txtxKtu p −= (9)

Where K is gain vector, and ( ),txp ( )tx the future and current state respectively.

Using a one step ahead prediction, the expression of the predictive control becomes:

( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ −=α

α

txdt

txdKtu (10)

Thus, the system (8) becomes:

( ) ( )( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=α

α

α

α

txdt

txdKtxfdt

txd (11)

For simplicity, we deal with stabilisation of unstable fixed points. Let x be a fixed point of (7). The linearized controlled system around x is

( ) ( ) ( ) ( )( )( )( ) ( )

( ) ( )txKBAtxIABKA

txtxABKtxAdt

txd

δ−=

δ−+=

δ−δ+δ=δα

α

(12)

With: ( ) ( ) xtxtx −=δ , ( ) ( )txAdt

txd δ=δα

α

, ( ) 1AIKK −−=

where nnA ×ℜ∈ is the jacobian matrix. All the fixed points are asymptotically stabilized by the prediction–based feedback control if K satifies the following inequality:

( ) απ≥−+2

arctgIABKAseigenvalue (13)

( )tx Fractional order chaotic

system

Prediction

K

( )tx p

- +

909

Fig. 2. Convergence region of the system.

Figure 2 Shows that the poles’ argument of the system

( )KBApI −− must be greater than απ2

. This is the main

difference between the ordinary order system and fractional order system. This problem is called pole placement problem in linear control theory. Suppose the pair (A,B) is controllable. Then, there exists a gain matrix K such that (12) is exponentially stable if ( ) 0AIdet ≠− .

IV. THE POLE PLACEMENT PROBLEM

This matrix gain K assures the controllability of the system. To determine it, we use the poles placement method. K is defined as follows

( ) 1AIKK −−=

With:

1111n1nnn

n21

T]a,,a,a[]k,,k,k[K

−−− −β−β−β=

=KK

KK (14)

Where:{ }n21 a,,a,a KK : are the coefficients of the characteristic polynomial of A, i.e.

( ) ( ) n1n

1n

n

1ii apappApIdet +++=λ−=− −

=∏ KK (15)

And { }n21 ,,, βββ KK : are the coefficients of the characteristic polynomial of ( )KBA − , i.e.:

( ) ( ) n1n

1n

n

1ii ppp)KBA(pIdet β++β+=μ−=−− −

=∏ KK (16)

The matrix T is given by:

T=CW (17)

Where C is the controllability matrix, defined as:

]BABAABB[C 1n2 −= K

And W is a matrix, where the upper elements are the coefficients of the characteristic polynomial of A.

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=−−

−−−

−−−

00000100001a0001aa

001aaa01aaaa1aaaaa

W

1

12

14n3n

14n3n2n

123n2n1n

L

L

L

MMNNNMM

L

L

L

The stabilization is guaranteed in the neighbourhood of the unstable fixed point. The control law will be in the following switching form:

( )( ) ( ) ( ) ( )

⎪⎩

⎪⎨

⎧ε<−−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −= α

α

otherwise0

tx1txiftxdt

txdKtu (18)

Where ε is a sufficiently small number.

V. NUMERICAL RESULTS

We consider the fractional order Chua system. The particular form to be considered here is different from the usual Chua system in that piecewise–linear nonlinearity is replaced by an appropriate cubic nonlinearity which yields very similar behaviour.

y7

100dt

zd

zyxdt

yd7

x2xydt

xd 3

−=

+−=

⎥⎦

⎤⎢⎣

⎡ −+γ=

α

α

α

α

α

α

(19)

For 1=α , we obtain the original integer order Chua system.

The system has three equilibrium points

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

±

±=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

7071.007071.0

x ,000

x 2,31.

Fig. 3. Bifurcation of fractional Chua’s system.

Stabilisation’s region

απ2 σ

ωj

γ

x

910

Fig. 4. Phase plane projections of the fractional order Chua system with

25.11=γ and 9.0=α .

The controlled system is given as follows

y7

100dt

zd

)t(uzyxdt

yd7

x2xydt

xd 3

−=

++−=

⎥⎦

⎤⎢⎣

⎡ −+γ=

α

α

α

α

α

α

(20)

With ( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛−=

α

α

tydt

tydKtu

Fig. 5. State variables of the uncontrolled fractional order Chua system with 25.11=γ , 9.0=α .

With an error of approximately dB2y = from rad/s 10 to10 22−=ω we obtain the integer order

approximation below with a fractional operator of 9.0=α , [9]:

010.0s789.7s150.36s

914.4s270.38s766.1s1

23

2

9.0 +++++= (21)

The linearization of the system around ix , gives the pair (A,B) of matrix:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−

γ−

γ

=010

B ,0

71000

111

07

x61

A

2i

.

For ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

000

x1 , we obtain:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−

γγ

=010

B ,0

71000

111

07

A

By choosing eigenvalues of the characteristic polynomial ( )KBApI −− as follows:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−=μ

5j0.001 5j-0.001

0001.0

The argument of each element of the vector described above is greater than απ

2.

Using the pole placement techniques, we obtain

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

0021.19662.22

6052.0K

Thus, the feedback gain K is determined as follows

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

2534.07487.02301.2

K

Figure 6 shows that the control is activated for s 15t ≥ and when the trajectory of the system is situated in the neighbourhood of a fixed point, the control is switched on. It therefore takes nonzero values, and the controlled system is closed to the fixed point.

x

z

x

y

x

y

z

t ’sec’

911

Fig. 6. State variables of the controlled fractional order Chua system ( 0.01. and 9.0 =ε=α )

VI. CONCLUSION

In this paper, we have studied the chaotic dynamics of the fractional order Chua system. We designed a linear feedback controller to stabilize the fractional order chaotic Chua system. The proposed method requires mathematical models of chaotic fractional order systems but it is not necessary to calculate target unstable periodic orbits beforehand. Future works regarding this topic includes the stabilisation of fractional order Chua system on all its fixed points with only one value of gain matrix K and synchronization of this system.

References [1] I. Podlubny, Fractional Differential Equations, Academic Press,

NewYork, 1999.

[2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, New Jersey, 2001.

[3] R.L. Bagley and R.A. Calico, “Fractional Order State Equations for the Control of Viscoelastic Structures”, J. Guid. Cont. and Dyn., Vol. 14, N o 2, pp. 304–311, 1991

[4] R.C. Koeller, “Application of Fractional Calculus to the Theory of Viscoelasticity” , J. ppl. Mech., Vol 51, pp 299-307, 1984.

[5] R.C. Koeller, “Polynomial Operators, Stieltjes Convolution, and Fractional Calculus in hereditary Mechanics,” Acta Mechanica, Vol 58, pp 251-264, 1986.

[6] H.H. Sun, A. Abdelwahad, and B. Onaral, “Linear Approximation of Transfer Function with Pole of Fractional Order ” IEEE Trans. Auto. Cont., Vol 29, N o 5, pp. 441-444, 1984.

[7] T.T. Hartley, C.F. Lorenzo, and H. K. Qammer, “Chaos in a Fractional Order Chua System”, IEEE Trans. Cir. & Sys. I, Vol 42, N o 8, pp. 485- 490, 1995.

[8] N. Tufillaro, T. Abbott, J. Reilly, An experimental approach to nonlinear dynamics and chaos, Reading MA: Addison-Wesley; 1992.

[9] G. Chen, T. Ueta, Chaos in circuits and systems, World Scientific Publisher; 2002.

[10] L.P. Shil’nikov, “Chua’s circuit: rigorous results and future problems” Int. J. Bifurcation and Chaos, Vol.4, No. 3, 1994, pp. 489-519.

[11] E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett., Vol. 64, No. 11, pp.1196–1199, 1990.

[12] [K. Pyragas, “Continues control of chaos by self controlling feedback,” Physics Letters A, 170, pp.421-428, 1992.

[13] T. Ushio, S. Yamamoto, “Prediction-based control of chaos,” Phys. Lett. A, 246, pp. 30-35, 1999

[14] A. Boukabou and N. Mansouri, “Controlling chaos in higher-order dynamical systems,” Int. J Bifurcation and chaos, Vol. 14, No. 11, pp. 4019-4025, 2004.

[15] [A. Boukabou and N. Mansouri, “Fuzzy predictive controller for unknown discrete chaotic systems”, Int. J Bifurcation and chaos, Vol.17, No. 6, pp. 2141-2148, 2007.

[16] A. Charef, “Analogue realization of fractional order integrator, differentiator and fractional μλDPI controllers”, IEE proceedings of control theory and applications, Vol. 153, Issue 6, Novembre 2006.

t ’sec’

x

y

z

u(t)

912