EISEVIER

16 November 1998

PHYSICS LETTERS A

Physics Letters A 248 (1998) 359-368

Observer-based delayed-feedback control for discrete-time chaotic systems

Keiji Konishi I, Hideki Kokame Department of Electrical and Electronic Systems, Osaka Prefecture University, I-I Gakuen-cho, Sakai, Osaka 599-8531. Japan

Received 27 May 1998; revised manuscript received 17 August 1998; accepted for publication 17 August 1998 Communicated by C.R. Doering

Abstract

This Letter proposes an observer-based delayed-feedback control method that overcomes an inherent weak point of the well-known delayed-feedback control. This method employs a state observer that estimates difference between a system state and a desired unstable fixed point without using location of the point. The estimated difference is used for state feedback control. We give a systematic procedure to design the observer and the controller. To check the theoretical results, some numerical examples are presented. @ 1998 Elsevier Science B.V.

PACS: 05.45.+b; 07.05.D~ Keywords: Chaos; Delayed-feedback control; Stability; Observer

1. Introduction

Stabilizing unstable periodic orbits (UPOs) within chaotic attractors has gathered much attention in nonlinear science [ 1,2]. Pyragas proposed a delayed-feedback control (DFC) method that does not require a reference signal corresponding to a desired UP0 [ 31. The original DFC method is meant for continuous-time chaotic systems; Bielawski, Derozier, and Glorieux modified it for discrete-time systems [4]. The DFC methods both for continuous-time and discrete-time systems are practical tools for stabilizing real chaotic systems. As a consequence, the DFC methods have been successfully applied to several physical systems [ 21, and used for various purposes [S-8]. At the same time, several researchers have investigated the stability of the continuous- time [ 9- 131 and discrete-time DFC systems [ 4,14- 181.

The stability of the discrete-time DFC method for low-dimensional chaotic systems was investigated in [ 4,14- 16]. Ushio showed the following instability condition: the discrete-time DFC method never stabilizes a desired unstable fixed point (UFP) if the Jacobi matrix at the point has an odd number of real eigenvalues greater than 1 [ 171. This is an inherent limitation of the DFC method. Konishi, Ishii, and Kokame [ 181 investigated this limitation in detail, and showed that the extended delayed-feedback control method [ 151 also has the same

’ Corresponding author. E-mail: konishi@ecs.ees.oskafu-u.ac.jp.

0375-9601/98/$ - see front matter @ 1998 Elsevier Science B.V. All rights reserved. PIISO375-9601(98)00673-2

360 K. Konishi. H. K&me/Physics Letters A 248 (1998) 359-368

limitation. The reason the DFC and EDFC methods have such a limitation is due to the fact that these methods

do not use the location of the UFP In order to use the well-known linear state feedback control theory, we

need the difference between a system state and a desired UFP in the control laws (see the control law (2.6)

in [ 193). However, these methods use the only difference between the current and past system states. In other

words, a main advantage of these methods, that is not to use the location of the desired UFP, contributes to the limitation unfortunately. Furthermore, for high-dimensional chaotic systems, no one has given a procedure for

designing the appropriate delayed-feedback controller, since it has been difficult to guarantee the stability of

the DFC system. Even if the dynamical equation of the chaotic system is known exactly, it has been difficult

to design the control system. Very recently, Schuster and Stemmler showed that an oscillating feedback overcomes the inherent limitation

of the DFC method [ 201. They modified the alternating consecutive difference control method [ 141, and

proposed a time-continuous version of it. Nakajima and Ueda [ 211 showed that a half-period delayed feedback can overcome the limitation for symmetrical time-continuous chaotic systems. However, these methods cannot provide a simple systematic procedure for designing the control system.

The present Letter proposes an observer-based delayed-feedback control technique that overcomes the inherent

limitation of the DFC method. Our technique employs a state observer that estimates the difference between the system state and the desired UFP without using the location of it. The estimated difference is used for the

state feedback control. We show a necessary and sufficient condition for the observer-based DFC system to be stable. This condition allows us to design the controller and the observer by the well-known pole placement

technique. Furthermore, we give a simple systematic procedure to design the deadbeat controller and observer.

This Letter is organized as follows. Section 2 describes the observer-based delayed-feedback control technique. In Section 3 we give a systematic procedure to design the controller and the observer. In Section 4 we present

some numerical examples to show the procedure for stabilizing low- and high-dimensional chaotic systems.

Finally, conclusions are presented in Section 5.

2. Control system

Let us consider a chaotic map

u(n + 1) = F(u(n)),

where n is the discrete time, v(n) E R”’ is the system state at time n, and F : R”’ + R”’ is the chaotic map.

The unstable fixed point (UFP) of this map is given by

vf = F(vf).

We consider a chaotic system

(I)

u(n + 1) = F(v(n)) + Bu(n), (2)

where B E R” *” is the input matrix and u(n) E R’ is the control signal. The DFC method uses the following

control law:

u(n) = -Kw(n), w(n) = v(n) - v(n - I),

where K E kxnl is the feedback gain and w(n) E R”’ is the difference between the current state v(n) and the

past state v(n - 1). Fig. la illustrates the DFC system. Ushio showed that there does not exist the feedback gain K such that the system state p(n) converges on the UFP if the Jacobi matrix

K. Konishi, H. Kokmne/Physics Letters A 248 (1998) 359-368 361

Controller

(a) DFC System

Observer

Controller

(b) Observer-based DFC System

Fig. I. Block diagrams of the DFC system and the observer-based DFC system.

has an odd number of real eigenvalues greater than 1 [ 171. This is an inherent limitation of the DFC method.

Konishi, Ishii, and Kokame [ 181 investigated this limitation in detail, and showed that the extended delayed- feedback control (EDFC) method [ 151 also has the same limitation. The reason is due to the fact that the

DFC and EDFC methods do not use the location of the UFP in their control laws. If we could use v(n) - vf in the control law, the limitation could be fully overcome and it would be very easy to design the feedback gain by the well-known pole placement technique [ 191. In this Letter we employ a state observer in modern

control theory to estimate v(n) - vf.

Linearize chaotic system (2) at the UFP, and then we have

(3)

where

362 K. Konishi, H. Kokame/Physics Letters A 248 (1998) 359-368

x1(n) f u(n) - Uf, x2(n) e u(n - 1) - Uf.

We introduce the following state observer [22]:

where HI, HZ E R”‘” are the observer gains we have to determine. Furthermore, we use a state feedback

controller

% (n> u(n) = [ -K1 -K2] 22(n) 9

[ I where KI, K2 E @“’ are the feedback gains. Fig. lb sketches the block diagram of the observer-based DFC system. The controller and the observer replaces the controller of the DFC system shown in Fig. la.

It is undesirable for the control signal u(n) to be added to the chaotic system when the orbit v(n) is far from the UFP, since such a signal may make the control system fall into a divergence regime. To this end, we

employ a watcher [6]. The watcher turns on the switch q5 only if

Ilw(n)II < V? Ilw(n - 1)II < y,

where the threshold v is a small positive value, otherwise it turns off the switch 4.

(6)

3. Design of control system

In this section we shall discuss the stability of the observed-based DFC system, and provide a procedure for

designing the controller and the observer.

Let us introduce new variables

el (n) n = x1(n) - nl(n), e?(n) i x2(n) -i&(n).

Using these variables we derive a necessary and sufficient condition for the control system to be stable. Define

x(n) f [XI (ny x2(n)yT

and

e(n) f [ el (nlT e2(nlTlT,

and then we have

x(n + 1) = (‘4 - B&x(n) + BKe(n),

e(n + 1) = (ii - E?C>e(n),

where

(7)

(8)

8= [KI K2], I?= [H; H;]‘. (9)

K. Konishi. H. Kokame/Physics Letters A 248 (1998) 359-368 363

Eqs. (7) and (8) are written as

[:(‘;+‘:1] = [A--- A._~] [;;;;I.

The behavior of system (10) depends on the characteristic equation

(10)

det [ z lznl - i + ii@ . det[ zZznl - A + Z!Zc] = 0.

The roots of Eq. ( 11) are the roots of the following equations:

(11)

det [ z Z2,,, - d+Et] =o, (12)

det[ zZzm - d + iit?] = 0. (13)

Therefore, the closed-loop system consisting of linearized chaotic system (3)) state observer (4), and state controller (5) is stable if and only if two polynomials ( 12) and ( 13) are stable

The poles of controller and the poles of observer can be determined independently, since they do not influence each other. Hence, we can design the controller and the observer independently. However, it should be noted that a temporal behavior of system ( 10) depends not only on the controller but also the observer.

In this Letter, we design a deadbeat state controller and a deadbeat state observer such that the state of closed loop system ( 10) converges on the origin (i.e., the desired UFF’) within 2m steps. For the deadbeat

behavior, we should design the gains K = [ KI Kz. ] and ii = [ HT Hz ]’ such that all roots of Eqs. ( 12) and ( 13) are zero. Now we shall give a procedure for designing the deadbeat gains.

First of all, we design the deadbeat state controller. Assume that the pair (A, B) is controllable. Substituting Eqs. (9) into Eq. ( 12)) we have

zln, - A + BK, BK2 -Z,, z I”, 1 = 0. (14)

When Kz = 0, the above equation is given as

z”‘det[zZ,-AfBKi] =O. (15)

If the gain KI is chosen such that all roots of Eq. (15) are zero, then all roots of Eq. (14) are zero. The pole placement technique allows us to design the gain K,.

Secondly, we design the deadbeat observer. The pair (AT, CT) is controllable if and only if

rank Z

G”, 0 AT -I, AT(AT -I,,) ... (AT)2n’-2(AT -I,,) =2m

0 . . . 0 1 which can be reduced to

rank [ AT - I,, AT(AT -I,,) ... (AT)2”‘-2(AT -I,)] = m.

From condition (16), we can derive a simple sufficient condition

(16)

det[A-I,,] # 0. (17)

Condition ( 17) implies that the matrix A has no eigenvalues equal to unity. It is well known that if the pair

(AT, CT) is controllable, there exist the deadbeat gain fi such that all roots of Eq. ( 13) are zero. Hence, if the matrix A has no eigenvalues equal to unity, we can obtain the deadbeat state observer by the pole placement technique.

364 K. Konishi, H. Kokmne/Physics Letters A 248 (1998) 359-368

K2 =0 Exit

Determine (H,, H,) suck tkat

all 1he rnOlS of (13) we zero

Fig. 2. Flow chart for designing the deadbeat state controller and the deadbeat state observer.

Fig. 2 shows a flow chart of the systematic procedure. First of all, we have to consider the following assumptions: the chaotic map F and the UFF’ uf are unknown; the matrices A and B are obtained in advance. Secondly, if the pair (A, B) is controllable, then we go to the next step. Otherwise, we cannot design the deadbeat controller easily by the well-known pole placement technique. Thirdly, if the matrix A has no eigenvalues equal to unity, then we go to the next step. Otherwise, we cannot design the deadbeat observer

easily. Fourthly, we set Kz = 0. Fifthly, the gain K1 is designed by the pole placement technique such that all roots of Eq. ( 15) are zero. Finally, we design (HI, Hz) by the same technique such that all roots of Eq. ( 13) are zero.

4. Controlling chaotic systems

4.1. One-dimensional chaotic systems

Let us consider a one-dimensional chaotic system

u(n + I) = f(u(n)> + h(n).

We have

d=[;;], I+)], C=

where

a.f (u) a=av . FU,

[I -11, K= [k, kz I> h= [hl h&

(18)

The pair (a, b) is controllable and a satisfies the condition ( 16) if and only if a # 1 and b + 0. Eqs. ( 12) and ( 13) are written as

K. Konishi, H. Kokame/Physics Letters A 248 (1998) 359-368 365

J Start

$ _~~~

-~~~

0 1000 2000 Time n

Fig. 3. Numerical control of the one-dimensional chaotic system ( 19). The threshold Y is set as v = 0.05. (a) System state u(n). (b) Control signal u(n). (c) The slowly varied parameter c(n)

z*+(bk, -a)z+bk2=0, z2 + (h, - h2 - a)z + ah;! - hl = 0,

respectively. To obtain the deadbeat behavior, we set

k,=;, k2 = 0, a2

h, = - a- 1’

h2=a. U-l

Consider a concrete numerical example. Let us define a nonlinear function

f(u,p,c)=p(u+l) +c v<

=pu+c - < 6

=p(u- +c 0.5 0,

p c the parameters. UFP this

C

vf = l-p

depends on the parameters. The value a described by Eq. ( 18) is equal to the parameter p. Let us assume

that the parameter p is known, but the parameter c is unknown. Location of the UFP is also unknown, for it

depends on the unknown c. We consider a one-dimensional chaotic system

o(n+ 1) = f(v(n), 1.95,O.l) +0.%(n), (19)

which behaves chaotically without control (i.e., u(n) = 0). Judging from the instability conditions [ 17,181, the DFC and EDFC methods never stabilize this system. On the contrary, our scheme can control this system easily. To obtain deadbeat behavior, we take

k, = 3.9, k2 =0, hl = 4.003, h2 = 2.053

from the systematic procedure (see Fig. 2). Fig. 3a shows a numerical result. For n E [ 0,500), the chaotic system without control runs freely. At n = 500,

the control system starts to work. For n E [500,612), the watcher does not turn on the switch 4, since the orbits u(n) , u (n - 1) , and u (n - 2) do not satisfy the watcher’s condition (6). At n = 6 12, the watcher turns on the switch 4 and the controlled orbit u(n) converges on the desired UFP within 2 steps.

366 K. Konishi, H. Kokame/Physics Letters A 248 (1998) 359-368

The system stability and the design procedure discussed in Section 3 are not influenced by the location of UFP, since they use only the matrices A and B. In this example, the parameter c influences the only UFP, so that the above discussions are valid for any c. To check this feature, we vary the parameter c sufficiently slowly as follows:

for n > 1000. Figs. 3a-3c are the system state u(n), the control signal u(n), and the slowly varied parameter c(n), respectively. It can be seen that the controlled orbit v(n) tracks the moved UFP completely.

4.2. High-dimensional chaotic systems

Let us consider a three-dimensional numerical example

1)

[ I[ f(Ul(n), 1.9,cl(n>) +O.‘b2(n) -0.%3(n)

uz(n+l) = 0.34(n) +f(s(n),1.8,~2(n)) (20) u3(n+ I) O.lul(n) - 0.2u3(n)

This system behaves chaotically without control (i.e., ut (n) = uz(n) = 0) and perturbation (i.e., cl (n) = c2( n) = 0). The UFP of of this system is the origin (of = 0). We design the control system in accordance with the flow chart (see Fig. 2). First, assume that we know the linearized system at the UFP

A=

Secondly, we check the controllability of the pair (A, B). Thirdly, we confirm that the matrix A has no eigenvalues equal to unity. Fourthly, the gain K2 is set as K2 = 0. Fifthly, the gain Kl is designed numerically by using a powerful software package (e.g., Matlab’s Control System Toolbox [23])

Finally, the observer gain (HI, Hz) is designed numerically by the same package

[

4.3112 -0.3057 -1.0881

1 I 2.4114 -0.7057 -0.5881

H, = -0.2293 4.3140 0.2206 , H2 = -0.5293 2.5143 0.2206 0.2176 -0.0588 -0.0824 0.1176 -0.0588 0.1179 1

Since the matrix A satisfies the instability condition in [ 17,181, the DFC and EDFC methods never stabilize chaotic system (20). Fig. 4a show a numerical result. For n E [0,500), the chaotic system without control runs freely. For n E [500,540), the orbit v(n) wanders on the chaotic attractor; however, it never visits the neighborhood of the desired UFP and the watcher does not turn on the switch 4. At n = 540, condition (6) is satisfied, and then the watcher turns on the switch 4. The controlled orbit y(n) converges on the desired UFP within 6 steps (see Fig. 5). It can be seen that the deadbeat controller and the deadbeat observer work completely.

The parameters cl (n) and c2 (n) do not influence the matrices A and B; hence, the above control system is stable for any cl(n) and cz(n). We vary the parameters slowly as

cl(n) = +O.lsin@rn;OE), cz(n) = -O.lsin(Zpn~~@)),

K. Konishi. H. Kokame/Physics Letters A 248 (1998) 359-368 361

I

(b)

-0.05 1

J Watcher starts

O:.\pi

545 Time n

Time n

Fig. 4. Numerical control of the three-dimensional chaotic system (20). The threshold v is set as v = 0. I. (a) System state U, (n) (b) Control signal ut (n) (c) The slowly varied parameters ct (n) and c2 (n).

Fig. 5. Controlled orbit of the three-dimensional chaotic system (20). The watcher turns on the switch 4 at n = 540, and then the orbit u(n) = 10, (n) u?(n) rrs(n)lr converges on the UFP (uf = 0) within 6 steps.

for n 2 1000. Figs. 4a-4c show the system state ui (n), the control signal ur (n), and slowly varied parameters c;(n) (i = 1,2), respectively. It can be seen that the controlled orbit ut (n) tracks the moved UFP. We confirm that other states v2 (n) and v3 (n) also track it completely on numerical simulation.

5.. Conclusions

In this paper, we have proposed an observer-based DFC system that overcomes the the DFC method. The deadbeat state controller and observer were used in our control

inherent limitation of system. We provided

a simple systematic procedure for designing the control system. Two numerical examples confirmed that the proposed control system works well.

The well-known linear state feedback control [ 191 requires three pieces of information: the location of UFP (i.e., vf), the Jacobi matrix A around the UFF’, and the input matrix B. These are used in the linear state feedback control law. On the other hand, the DFC method does not require them; nevertheless, one cannot guarantee the stability of the DFC system and it is very difficult to design the feedback gain K theoretically. Furthermore, the DFC method never stabilizes a kind of UFPs due to its inherent limitation. The observer-based DFC method does not require the UFP; however, it uses the matrices A and B in its control law. This method has four advantages: it overcomes the inherent limitation of the DFC method; one can guarantee the theoretical stability; a design of the control system can be given systematically; it can track a moved fixed point of slowly varied chaotic systems. However, the requirement of A and B is not practical for real chaotic systems. Hence, in the future, we have to employ an adaptive observer that does not use A and B in its adaptive law.

The whole results obtained are meant for stabilizing discrete-time systems. It would be a future subject to modify the method for continuous-time chaotic systems.

References

] I] E. Ott, C. Gmbogi, J.A. Yorke, Phys. Rev. Lett. 64 ( 1990) 1196. ]2] G. Chen, Control and synchronization of chaotic systems (a bibliography), ECE Dept. Univ. of Houston, TX - available from ftp:

ftp.egr.uh.edu/pub/TeX/chaos.tex (login name: anonymous; password: your email addtess). 131 K. Pyragas, Phys. Lett. A 170 (1992) 421. [4] S. Bielawski, D. Derozier, P. Glorieux, Phys. Rev. A 47 (1993) 2492.

368 K. Konishi, H. Kokame/Physics Letters A 248 (1998) 359-368

IS] P Parmanada, M. Hildebrand, M. Eiswirth, Phys. Rev. E 56 ( 1997) 239. 161 K. Konishi, M. Ishii, H. Kokame, Phys. Rev. E 54 (1996) 3455. [7] M.E. Brandt, G. Chen, Biol. Cybem. 74 (1996) 1. j 81 B. Crespi, E. Omerti, Phys. Lett. A 216 (1996) 269. 191 K. Pyragas, Phys. Lett. A 206 ( 1995) 323.

[IO] M.E. Bleich. J.E.S. Socolar, Phys. Lett. A 210 (1996) 87. [ I1 1 W. Just et al., Phys. Rev. Lett. 78 ( 1997) 203. f 12 J H. Nakajima, Phys. L&t. A 232 ( 1997) 207. [ 131 H. Nakajima, Y. Ueda, Physica D 111 (1998) 143. [ 141 PV. Bayly, L.N. Virgin, Phys. Rev. E 50 (1994) 604. [ 151 J.E.S. Socolar, D.W. Sukow, D.J. Gauthier, Phys. Rev. E 50 (1994) 3245. 1161 M. Ishii, K. Konishi, H. Kokame, Phys. Lett. A 235 (1997) 603. [ 171 T. Ushio, IEEE CAS-I 43 (1996) 815. [ 181 K. Konishi, M. Ishii, H. Kokame, IEEE CAS-I, in press. 1191 F.J. Romeiras et al., Physica D 58 (1992) 165. 1201 H.G. Schuster, M.B. Stemmler, Phys. Rev. E 56 (1997) 6410. [21] H. Nakajima, Y. Ueda, Phys. Rev. E 58 ( 1998) 1757. 1221 R. Isermann, Digital Control Systems (Springer, Berlin, 1989). 1231 A. Grace, A.J. Laub, J.N. Little, C.M. Thompson, Control System Toolbox, MathWorks. Inc., 1992