Chaos, Solitons and Fractals 13 (2002) 723±730
www.elsevier.com/locate/chaos
A linear feedback synchronization theorem for a class of
chaotic systems
Feng Liu a,*, Yong Ren a, Xiuming Shan a, Zulian Qiu b
a
b
Department of Electronics Engineering, Tsinghua University, Beijing 100084, People's Republic of China
Department of Automatic Control, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
Accepted 8 January 2001
Abstract
A new synchronization theorem for a class of chaotic systems is presented in this paper. We take the last state variable of drive
system as the driving scalar signal. Its linear feedback gain is a function of a free parameter. It is also proved that the global synchronization can be attained through simple linear output error feedback. This approach is illustrated by Chua's circuit and a 4D
hyperchaotic system. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction
Synchronization in chaotic systems has been a hot spot of research since 1990 [1]. Many possible applications, especially in communication, have been discussed [2]. Recently, several approaches have been
proposed to synchronize chaotic systems. Nijmeijer [3] and Molgul [4] regarded the problem of chaos
synchronization as a special case of observer design, so it could be solved by the nonlinear control theory.
The methods they proposed require that the Lipschitz constant of nonlinear function must be smaller than
the minimal real part of linear part eigenvalues, therefore the feedback gain would be too big to realize.
Based upon the nonlinear H1 theory, Suykens et al. [5] proposed the master±slave H1 synchronization for
chaotic Lur'e systems, which is very complicated because its solution could only be achieved by solving a
nonconvex nonlinear optimization problem which takes into account both channel noise and parameter
mismatch.
Wang et al. [6] presented a global synchronization theorem for a subclass of chaotic Lur'e systems, and
its feedback gain is a function of a free parameter. In this paper, a new global synchronization theorem is
presented for a class of chaotic systems, which di€ers from Wang's work and its feedback gain is also a
function of a free parameter. Based on the nonlinear observer design theory and taking the last state
variable of drive system as the driving scalar signal, we give a linear output error feedback control law, and
its asymptotic stability of synchronization is proved. The presented controller is simple and easy to realize.
The paper is organized as follows. In Section 2, the design procedure of a linear output feedback synchronization controller is given for a class of chaotic systems. Section 3 gives main synchronization theorems and their proofs. In Section 4, the synchronization scheme is applied to two chaotic systems and
simulation results are given.
*
Corresponding author.
E-mail address: [email protected] (F. Liu).
0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 9 6 0 - 0 7 7 9 ( 0 1 ) 0 0 0 1 1 - X
724
F. Liu et al. / Chaos, Solitons and Fractals 13 (2002) 723±730
2. Feedback controller design procedure
We consider the drive system described by the following state equation:
x_ ˆ Ax ‡ F …x†; y ˆ Cx;
…1†
T
where x ˆ …x1 ; x2 ; . . . ; xn † 2 Rn is the state variable, y 2 R is the output, A is a constant matrix, and
3
2
a11 a12 a1n
2
3
f1 …x1 ; x2 ; . . . ; xn †
a2n 7
6 a21 a22 6
6 f2 …x2 ; . . . ; xn † 7
.. 7
7
6
6
7
a32 . 7 2 Rnn ; F …x† ˆ 6
A ˆ 60
7 2 Rn ;
..
7
6.
4
5
.
.
.
.
.
4.
.. 5
. . ..
..
.
fn …xn †
0
0
an;n 1 ann
C ˆ …0 0 1† 2 Rn :
…2†
…3†
In our synchronization scheme, we use a linear output error feedback approach to synchronize the drive
and response system. We take xn as the driving scalar signal. The state equation of the response system is
^_ ˆ A^
x
x ‡ F …^
x† ‡ K…y
y^†; y^ ˆ C^
x;
…4†
^ 2 Rn is the state variable, y^ 2 R is the output of the response system and K ˆ …k1 ; k2 ; . . . ; kn †T 2 Rn is
where x
the feedback gain.
For the systems (1) and (4), we make some assumptions as follows
A1: For matrix A, ai;i 1 6ˆ 0…i ˆ 2; . . . ; n†, and aij ˆ 0…j < i 1†;
A2: Denote xi for the vector …0 0xi xn †T 2 Rn , and require that every element fi …xi † ˆ fi …xi ; . . . ; xn †
…i ˆ 1; . . . ; n† in F …x† satis®es Lipschitz condition.
Assumption A1 guarantees the observability of …C; A†, and A2 is a general condition in nonlinear systems. If fi …xi † is not global Lipschitz, it should be at least local Lipschitz almost everywhere.
The design procedure of the feedback gain K is stated as follows:
(1) Construct a constant upper-triangular matrix T ˆ …tij † 2 Rnn , where
t11 ˆ 1; tlk ˆ
k 1
X
tj;k 1 alj ; k ˆ 2; 3; . . . ; n; l ˆ 1; 2; . . . ; k
…5†
jˆl 1
and denote t0i ˆ ai0 ˆ 0…i ˆ 1; 2; . . . ; n†.
(2) Calculate column vector L 2 Rn , where L satis®es the following equation
T
where
1
AT ˆ A0 ‡ LC;
2
0
61
6
6
A0 ˆ 6 0
6.
4 ..
0
…6†
0
0
0
0
1
..
.
0
..
.
..
.
1
0
3
0
07
.. 7
7
. 7 2 Rnn :
.. 7
.5
0
…7†
(3) For a given positive constant h > 0, calculate the feedback gain
K ˆ T …S 1 …h†C T ‡ L†…y
y^†=tnn ;
…8†
where
i‡j
s…h† ˆ …sij …h†† 2 R
nn
; and sij …h† ˆ
n j
… 1† C2n
i
h
2n i j‡1
j
; 1 6 i; j 6 n; Cni ˆ
…n
n!
:
i†!i!
…9†
F. Liu et al. / Chaos, Solitons and Fractals 13 (2002) 723±730
725
Remark. (1) Since t11 ˆ 1 6ˆ 0; tii ˆ ti l; i l ai; i l 6ˆ 0…i ˆ 2; 3; . . . ; n†; T is a nonsingular matrix, so T has
inverse matrix T 1 . T is an upper-triangular matrix, so is T 1 .
(2) Obviously we have
1 1
1
1 1
1
S…h† ˆ h diag n ; n 1 ; . . . ;
S…1† diag n ; n 1 ; . . . ;
:
h h
h
h h
h
And if S…h† is nonsingular, then its inverse matrix is
S 1 …h† ˆ h 1 diag…hn ; hn 1 ; . . . ; h† S 1 …1† diag…hn ; hn 1 ; . . . ; h†:
3. Main theorems and proofs
Now, we give a lemma as follows, which is useful for our proofs of main theorems.
Lemma 1. For matrix A0 and C, there exists a symmetric positive definite matrix S…h† ˆ …sij …h† 2 Rnn which
satisfies the following equation
hS ‡ AT0 S ‡ SA0
C T C ˆ 0:
…10†
Moreover, sij …h† satisfies Eq. (9).
Proof. We look for S1 …h† ˆ limt!1 St …h†, the stationary solution of the following di€erential equation
S_ t …h† ˆ hSt …h† ‡ AT0 St …h† ‡ St …h†A0
C T C; where h > 0:
…11†
We have
AT0 t
St …h† ˆ exp…ht† exp AT0 t S0 …h† exp
Z t
exp… h…t s†† exp AT0 …t s† C T C exp… A0 …t
‡
0
s†† ds:
It is clear that if S0 …h† is symmetric positive de®nite, then St …h† will also be. Moreover, for any a > 0 and
t > a,
Z t
St …h† P
exp… h…t s†† exp AT0 …t s† C T C exp … A0 …t s†† ds;
Z
St …h† P
a
0
a
exp…hu† exp AT0 u C T C exp…A0 u† du;
St …h† P exp… ha†
Z
0
a
exp AT0 u C T C exp…A0 u† du:
Due to the observability of the linear pair …C; A0 †, we have St …h† P dI for some d > 0. Hence, S1 …h† P dI
and S1 …h† is positive de®nite.
An explicit computation shows that for any h > 0; St …h† has a limit S1 …h†. For the sake of simplicity, we
denote S…h† for S1 …h†, and satis®es Eq. (9). It is obvious that S…h† is symmetric.
This ends the proof of Lemma 1.
We de®ne kxkS…h† ˆ …xT S…h†x†1=2 , and consider the feedback gain K of the system (4) when A ˆ A0 .
Theorem 1. For systems (1) and (4), where A ˆ A0 and A2 is satisfied, there exists h0 > 0, when h > h0 and
K ˆ S 1 …h†C T , the drive system (1) and the response system (4) are globally asymptotically synchronized in the
^…t†k ˆ 0.
sense that limt!1 kx…t† x
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F. Liu et al. / Chaos, Solitons and Fractals 13 (2002) 723±730
Proof. Let us consider the error e ˆ …e1 ; e2 ; . . . ; en †T ˆ x
have the error system
e_ ˆ …A0
S 1 …h†C T C†e ‡ …F …x†
^ 2 Rn , and substitute K ˆ S 1 …h†C T into (4), we
x
F …^
x††:
…12†
Construct the Lyapunov function V ˆ eT S…h†e, and V P 0 because S…h† is positive de®nite.
Then its di€erentiation will be
V_ ˆ eT AT0 S…h† ‡ S…h†A0 2C T C e ‡ 2eT S…h†…F …x† F …^
x†† 6 eT … hS…h††e eT C T Ce ‡ 2eT S…h†
F …^
x†† 6 … heT S…h††e ‡ 2eT S…h†…F …x†
…F …x†
F …^
x††:
Using the above norm de®nition and Schwartz inequality [7], we have
V_ 6
2
hkekS…h† ‡ 2kekS…h† F …x†
F …^
x†kS…h† :
^i k.
xi †k 6 li kxi x
Set li as the Lyapunov constant of fi …xi †, thus we have kfi …xi † fi …^
And let lf ˆ maxfli ; i ˆ 1; . . . ; ng and sm ˆ maxfsij …1†; i; j ˆ 1; . . . ; ng, we then have
kF …x†
F …^
x†k2S…h† ˆ
ˆ
n X
fi …xi †
fi …^
xi † Sij …h† fi …xj †
n X
fi …xi †
fi …^
xi †
i;jˆ1
i;jˆ1
6 Sm h
n
X
i;jˆ1
6 Sm h
n
X
i;jˆ1
S …1† ij
fi …xj †
h2n i j‡1
fi …xi † fi …^
xi † fj …xj †
n i‡1
h
hn
li kxi
hn
fj …^
xj †
^i k lj kxj
x
i‡1
hn
fj …^
xj †
fj …^
xj †
!
j‡1
^j k
x
!
j‡1
6 Sm hl2f
n
X
i;jˆ1
^i k kxj
kxi x
n
n i‡1
h
h
Let
1 1
1
ni ˆ diag n ; n 1 ; . . . ;
…xi
h h
h
^i †
x
and
1 1
1
n ˆ diag n ; n 1 ; . . . ;
…x
h h
h
then
x x
i ^i n i‡1 6 kni k 6 knk;
h
h P 1 and k k denotes L2 norm in Euclidean space, i ˆ 1; 2; . . . ; n.
Therefore we have
kF …x†
F …^
x†k2S…h† 6 Sm hn2 l2f knk2 :
Since
2
knk 6
1
2
knkS…1†
kmin …S…1††
2
and knkS…1† 6
1
kx
h
then
kF …x†
2
F …^
x†kS…h† 6
1
Sm n2 l2f kx
kmin …S…1††
^k2S…h† :
x
^k2S…h† ;
x
^i †;
x
^j k
x
j‡1
!
:
F. Liu et al. / Chaos, Solitons and Fractals 13 (2002) 723±730
727
Hence
V_ 6
6
6
2
hkekS…h† ‡ 2kekS…h† kF …x† F …^
x†kS…h†
1=2
1
2
2
hkekS…h† ‡ 2
Sm1=2 nlf kekS…h†
kmin …S…1††
!
1
h 2 1=2
Sm1=2 nlf kek2S…h† :
kmin …S…1††
Then
d 2 kekS…h† 6
dt
So when
h
(
h P h0 ˆ max 2
we have
d
dt
kek2S…h† 6 0
2
1=2
kmin …S…1††
2
Sm1=2 nlf
kekS…h† :
)
1
1=2
kmin …S…1††
and
!
1
Sm1=2 nlf ; 1
;
lim kekS…h† ˆ 0:
t!1
Therefore we conclude that limt!1 kx…t†
This ends the proof of Theorem 1.
^…t†k ˆ 0.
x
A direct computation leads to the following lemma, which is useful for the proof of the next theorem.
Lemma 2. For matrix A in (2) T in (5), there exists Eq. (6), where L 2 Rn is a column vector and A0 satisfies
(7).
Then for the general form of A in (2), we have the following theorem.
Theorem 2. For systems (1) and (4), where A1 and A2 are satisfied, there exists h0 > 0, when h > h0 and
K ˆ T …S 1 …h†C T ‡ L†=tm , the drive system (1) and the response system (4) are globally asymptotically syn^…t†k ˆ 0.
chronized in the sense that limt!1 kx…t† x
^ ˆ T ^z, where z ˆ …z1 ; z2 ; . . . ; zn †T 2 Rn and
Proof. We consider the coordinate transformation x ˆ T z and x
T
n
^z ˆ …^z1 ; ^z2 ; . . . ; ^zn † 2 R , then from (1) and (4) we obtain the following equations:
z_ ˆ T
1
AT z ‡ T
1
F …T z†; y ˆ CT z ˆ tnn Cz;
^z_ ˆ T
1
AT ^z ‡ T
1
F …T ^z† ‡ T
1
…13†
u; y^ ˆ CT ^z ˆ tnn C^z:
^z, and we have the error system
T
e_ ˆ T 1 AT e ‡ T 1 F …T z† F …T ^z†
…14†
Let e ˆ z
1
K…y
y^†:
…15†
Using Lemma 2 and considering K ˆ T …S 1 …h†C T ‡ L†=tnn , we have
e_ ˆ …A0 S 1 …h†C T C†e ‡ T 1 F …T z† F …T ^z† :
Rnn , it is clear
Denoting T 1 ˆ …rij † 2P
P that rnn ˆ 1=tnn .
Obviously the term rkˆi rik fk … njˆk tkj zj †, the ith element of T
be denoted as fi0 …zi †. And then
1
F …T z†, is the function of zi ; . . . ; zn , can
728
F. Liu et al. / Chaos, Solitons and Fractals 13 (2002) 723±730
kfi0 …zi †
fi0 …^zi †k
!
X
n
n
X
^i k
ˆ
r fi …xi † fi …^
j rik j li kxi x
xi † 6
kˆi ik
kˆi
!
!
n
n
X
X
ˆ
j rik j li kT …zi ^zi †k 6
j rik j li kT kkzi ^zi k
kˆi
kˆi
So fi0 …zi † satis®es Lipschitz condition. Then from the proof of Theorem 1, there exists h0 > 0 that
^…t†k ˆ 0 when h > h0 .
limt!1 kx…t† x
This ends the proof of Theorem 2.
4. Simulation
4.1. Chua's circuit
The state equation of Chua's circuit [6] is given by
8
< x_ 1 ˆ a…x2 x1 f …x1 ††;
x_ ˆ x1 x2 ‡ x3 ;
: 2
x_ 3 ˆ bx2 cx3 ;
…16†
and
y ˆ …0 0 1†x
…17†
where
1
f …x1 † ˆ bx1 ‡ …a
2
b†…jx1 ‡ 1j
jx1
1j†:
The system parameters are selected to be a ˆ 10:0; b ˆ 15:0; c ˆ 0:0385; a ˆ 1:27; b ˆ 0:68.
T
The initial states of the drive and the response systems are x…0† ˆ … 1:0; 0:1; 1:5† and
T
T
x^…0† ˆ …1:5; 0:3; 1:0† . We have T and t33 ˆ 15 by (5), and L ˆ … 150:0; 15:43; 11:04† by (6).
Then from (8), the feedback gain is K ˆ T …S 1 …h†C T ‡ L†=t33 .
We ®nd that the synchronization occurs when h P 8:0. When h ˆ 8:0, we have K ˆ … 1:47;
T
2:27; 12:96† , and the error variables varying with time are showed by Fig. 1. It is found that the maximal
T
absolute error is less than 10 3 when t > 17 s. When h ˆ 9:0, we have K ˆ … 3:93; 3:47; 15:96† , and error
variables varying with time are showed by Fig. 2. The maximal absolute error is less than 10 4 when t > 9 s.
The synchronization becomes quicker when h increases.
Fig. 1. Synchronization error in two Chua's circuits performed with h ˆ 8:0.
F. Liu et al. / Chaos, Solitons and Fractals 13 (2002) 723±730
729
Fig. 2. Synchronization error in two Chua's circuits performed with h ˆ 9:0.
Fig. 3. Synchronization error in two 4D hyperchaos performed with h ˆ 2:0.
Fig. 4. Synchronization error in two 4D hyperchaos performed with h ˆ 2:5.
4.2. 4D hyperchaotic system
The state equation of 4D hyperchaotic system [8] is given by
8
x_ 1 ˆ ax1 x2 ;
>
>
<
x_ 2 ˆ x1 x2 x23 ;
x_ ˆ b1 x2 b2 x3 b3 x4 ;
>
>
: 3
x_ 4 ˆ x3 cx4 ;
…18†
730
F. Liu et al. / Chaos, Solitons and Fractals 13 (2002) 723±730
and
y ˆ …0 0 0 1†x;
…19†
where the system parameters are a ˆ 0:56; b1 ˆ 1:0; b2 ˆ 0:3725; b3 ˆ 6:0; c ˆ 0:8.
T
The initial states of the drive and the response systems are x…0† ˆ … 1:0; 1:0; 0:1; 0:1† and
T
T
^…0† ˆ …1:0; 1:0; 0:2; 0:2† . We have T and t44 ˆ 1:0 by (5), and L ˆ … 5:2; 2:712; 6:088; 0:36† by (6).
x
Then from (8), the feedback gain is K ˆ T …S 1 …h†C T ‡ L†=t44 .
We ®nd that the synchronization occurs when h P 2:0 With h ˆ 2:0; K ˆ … 10:04887;
39:00442; 14:2336; 8:36†T , and the error variables varying with time are shown in Fig. 3. It is found
that the maximal absolute error is less than 10 4 when t > 22 s. With h ˆ 2:5; K ˆ … 39:0362;
T
75:69162; 26:8536; 10:36† , and the error variables varying with time are shown in Fig. 4, and the maximal
absolute error is less than 10 4 when t > 14 s. The synchronization also becomes quicker with h increasing.
5. Conclusion
A linear output error feedback approach is proposed to globally synchronize a class of chaotic systems
with the last state variable as the driving signal. The feedback gain is chosen as a function of a free parameter. It is proved that global asymptotic synchronization can be attained when the free parameter is
large enough. Taking Chua's circuit and a 4D hyperchaotic system for example, we show that synchronization can be attained by using the last state variable as the driving signal. These simulation results
indicate that the proposed scheme might be useful for developing practical synchronized chaotic systems.
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