ARTICLE IN PRESS

Chaos, Solitons and Fractals xxx (2007) xxx–xxx www.elsevier.com/locate/chaos

An approach of partial control design for system control and synchronization Wuhua Hu, Jiang Wang *, Xiumin Li School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, PR China Accepted 29 May 2007

Abstract In this paper, a general approach of partial control design for system control and synchronization is proposed. It turns control problems into simpler ones by reducing their control variables. This is realized by utilizing the dynamical relations between variables, which are described by the dynamical relation matrix and the dependence–influence matrix. By adopting partial control theory, the presented approach provides a simple and general way to stabilize systems to their partial or whole equilibriums, or to synchronize systems with their partial or whole states. Further, based on this approach, the controllers can be simplified. Two examples of synchronizing chaotic systems are given to illustrate its effectiveness.  2007 Elsevier Ltd. All rights reserved.

1. Introduction Partial control means stabilizing part or the whole of a system’s states to their equilibriums [1]. And it is developed along with the partial stability theory that was firstly formulated by Lyapunov in 1950s [2]. It has been commonly used in the study of electromagnetics, spacecraft stabilization (especially stabilization by rotors), drift of the gyroscope axis, adaptive stabilization, etc. [2–5]. In this paper, however, we are not going to discuss partial control itself. Instead, we will focus on its way of designing. Insofar, there are yet few papers concerning partial control design. Although some papers have given some constructive designing ways to stabilize the system wholly, including those to control and synchronize the chaotic systems, they are uneasy to realize in a sense [1,2,6–9]. And the advantages of partial control, such as its usage in simplifying the controllers, are not fully recognized. One best example is that: in the field of chaos control and synchronization, because of lacking knowledge about partial control, many authors are still designing some inefficient controllers, or omitting the procedure of configuring controls [10–16], which is actually a key step before any control design [17]. Also, because there is no general way for such design, even though some authors know partial control theory they may find it difficult to use. Considering these, we are to present an approach to alleviate such an embarrassment. Based on this approach, not only can the partial control problem become much clear, but also more effective control can be realized. The following contents of this paper are arranged as follows. First, the preliminaries on partial stability/control and synchronization *

Corresponding author. E-mail address: [email protected] (J. Wang).

0960-0779/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.05.017

Please cite this article in press as: Hu W et al., An approach of partial control design for system ..., Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.017

ARTICLE IN PRESS 2

W. Hu et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx

are given in Section 2. Then, in Section 3 the new approach for partial control design is presented and explained in detail. Next, in Section 4, as examples, this approach is applied to design partial control to synchronize two different sets of chaotic systems. Finally, conclusions and discussions are made in Section 5.

2. Preliminaries Consider the non-autonomous control system  y_ ¼ Y ðt; y; z; uðt; y; zÞÞ; yðt0 Þ ¼ y 0 ; t P t0 ; z_ ¼ Zðt; y; z; uðt; y; zÞÞ; zðt0 Þ ¼ z0 ;

ð1Þ

where y 2 D  Rm with 0 2 D, Y : ½0; 1Þ  D  Rn  Rp ! Rm , Z : ½0; 1Þ  D  Rn  Rp ! Rn , with m 2 Zþ ; n; p 2 f0g [ Zþ , and u(t, y, z) is the control satisfying u(t, 0, z) = 0. Let y ¼ ðy T1 ; y T2 ÞT ; Y ¼ ðY T1 ; Y T2 ÞT , x = (x i)(m+n)·1 = (yT, zT)T, and X = (X i)(m+n)·1 = (YT, ZT)T, with i = 1, 2, . . . , m + n, y 1 ; Y 1 2 Rm1 and y 2 ; Y 2 2 Rm2 . Suppose (1) has a partial equilibrium of y_ ¼ y ¼ 0 (in contrast to the whole equilibrium of x_ ¼ x ¼ 0Þ [1], and we are to stabilize the system to such an equilibrium. In addition, three assumptions are made that: (a) no more requirements are considered other than realizing stabilization; (b) the control u(t, x) is continuous and unrestricted as long as system (1) has solutions for t 2 [0, 1); (c) Xi(t, x, u) are assumed to be polynomials in xi (i = 1, 2, . . . , m + n). And these assumptions are also made for all the systems referred later in this paper. Definition 1. The dynamical system (1) is uniformly y-stable if "e > 0 and 8z0 2 Rn , $d = d(e,z0) > 0, s.t. ky0k < d implies ky(t)k < e for all t P t0 and for all t0 2 [0, 1). Definition 2. The dynamical system (1) is globally uniformly asymptotically y-stable if it is uniformly y-stable and limt!1y(t) = 0 for all y 2 D; z 2 Rn and for all t0 2 [0, 1). Definition 3. Partial control refers to such control u(t, x) in (1) that makes asymptotically y1-stable and consequently asymptotically y-stable. Definition 4. [18] Converging input bounded state (CIBS) condition is defined as: for each allowed control u, limt!1 u(t, x) = 0 and for each initial state x0, the solution of (1) with x(0) = x0 exists for all t P 0 and is bounded. Along with system (1), let us consider the ‘‘truncated’’ system for y1 = 0,  y_ 2 ¼ Y 2 ðt; 0; y 2 ; z; uðt; xÞÞ; y 2 ðt0 Þ ¼ y 20 ; t P t0 z_ ¼ Zðt; 0; y 2 ; z; uðt; xÞÞ;

zðt0 Þ ¼ z0

ð2Þ

Lemma 1. If system (1) is uniformly asymptotically y1-stable and the solution of system (2) is uniformly asymptotically y2stable, then system (1) is uniformly asymptotically y-stable. Remark 1. (a) Both the definitions and the theorem given above extend those given in [1,2,4,5]: Definitions 1 and 2 and Lemma 1 there are all now the particular case of y = x here; and Definition 3 there is now the particular case of y1 = y here. (b) If m2 = n = 0, then the cases above all reduce to the ordinary stability and control. (c) The theories, including all of the below, hold true for all autonomous systems. Actually, Lemma 1 only stands in the sense of local stability. And below, a stronger lemma that holds in the global sense is given (which is a direct result of the theorem in [18] and Lemma 1). Lemma 2. If system (1) is globally uniformly asymptotically y1-stable and satisfies the CIBS property with respect to (w.r.t.) state y2, with y1 as the input, then system (1) is globally uniformly asymptotically y-stable. Next, consider two dynamical systems y_ ¼ Y ðt; yÞ; z_ ¼ Zðt; zÞ;

ð3Þ

where f : ½0; 1Þ  Rm ! Rm , g : ½0; 1Þ  Rn ! Rn , m P n 2 Z+. Subtract Eq. (2) from Eq. (1), and derive the errorcombined system Please cite this article in press as: Hu W et al., An approach of partial control design for system ..., Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.017

ARTICLE IN PRESS W. Hu et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx



3

y_ ¼ Y ðt; yÞ ; e_ ¼ Zðt; zÞ  Y  ðt; yÞ ¼ W ðt; e; yÞ

ð4Þ

where e ¼ z  y  2 D  Rn , with 0 2 D. And without loss of generality, suppose y* = (yi )n·1,Y* = (Yi) i = 1, 2, . . . , n.

n·1,

with

Definition 5. If system (4) is asymptotically e-stable, then it is said that partial complete synchronization occurs in system (3) w.r.t. y* and z. Remark 2. (a) Definition 5 allows for the synchronization of systems with different orders. If m = n, then it reduces to the ordinary complete synchronization, which is similar to those discussed in [19,20]. (b) System (4) can be seen as a particular case of system (1). 3. An approach of partial control design Unless having a particular declaration, it is set that: I = {1, 2, . . . , m} and i, j = 1, 2, . . . , m. Definition 6. Consider system (1) with control u  0, and rewrite it as (

y_ ¼ Y ð1Þ ðt; y; zÞ þ Y ð2Þ ðt; y; zÞ þ Y ð3Þ ðt; zÞ ; z_ ¼ Zðt; y; zÞ

ð5Þ

where Y(1) = [Y(1i)]m·1, Y(2) = [Y(2i)]m·1, Y(3) = [Y(3i)]m·1, and Y(1i)(t, y, z) consists of all terms of Yi(t, y, z) satisfying y_ i ¼ Y ð1iÞ ðt; y; zÞ and is yi-stable, and Y(2i)(t, y, z) consists of all terms, but Y(1i)(t, y, z), of Yi (t, y, z) that are functions of t, y and z, and the rest terms of Yi(t, y, z) comprise Y(3i)(t, z), which is a function only of t and z. Consider system (1) with u(t, x)  0, and we are to analyze such system that some partial control u(t, x) can be designed to stabilize it to y = 0. First, let us rewrite (1) as (5). Then define the so called dynamical relation matrix as DR , ½ðDRij Þmm with

ðDRiðmþ1Þ Þm1 

08  < powerðy ; T ðy ; Y ð2iÞ ðt; xÞÞÞ  k j  j @ DRij , max  :copowerðy; T k ðy j ; Y ð2iÞ ðt; xÞÞÞ

k¼1;2;...;N

91 = A; ;

and DRiðmþ1Þ ,  dðY ð3iÞ ðt; zÞÞ; where 

0; 1;

dðsÞ , ¼

if s ¼ 0 ; if s ¼ 6 0

and N is the number of terms in Y(2i)(t, x)(i 2 I) that contain yj(j 2 I), and Tk (yj,Y(2i)(t, x)) denotes such kth term, and power(yj,Tk (yj,Y(2i)(t, x))) means taking the power of yj in Tk(yj,Y(2i)(t, x)), and copower(y,Tk(yj,Y(2i) (t, x))) means taking the total powers of m elements of y P that are contained in Tk(yj,Y(2i)(t, x)). m Pm In the following definitions, let M ¼ j¼1 k¼1 DRjk . Definition 7. Dynamical dependence factor of yi(i 2 I) is defined as fd ðy i Þ ,

m X

DRij =M:

j¼1

Definition 8. Dynamical influence factor, or dynamical dominance factor, of yi(i 2 I) is defined as fi ðy i Þ ,

m X

DRji =M:

j¼1

Please cite this article in press as: Hu W et al., An approach of partial control design for system ..., Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.017

ARTICLE IN PRESS 4

W. Hu et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx

Based on these definitions, the dynamical dependence–influence matrix can be obtained, namely fdi ðyÞ , ½fd ðyÞ

f i ðyÞ;

where fd(y) = [fd(y i)]m·1, fi(y) = [fi(yi)] m·1. Further, another factor that is useful to characterize the ‘‘dynamical competence’’ of a variable is defined below. Definition 9. Dynamical competence factor of yi(i 2 I) is defined as fc ðy i Þ , f i ðy i Þ=fd ðy i Þ: Next, we are to present the way of selecting variables preferentially to control from elements of y. Assume the current uncontrolled system is not asymptotically y-stable, then carry on the steps as follows. Step 1. Rewrite the current system as (5) and derive DR. Step 2. The variable yi(i 2 I) with DRi(m+1) = 1 is selected to be controlled. Step 3. Let the selected variables be zero and then reduce the control system. Check the stability of the reduced system w.r.t. the non-selected elements of y: if they are all uniformly asymptotically stable, then go to Step 5; otherwise, set to zero the rows and columns of DR that are corresponding to the selected variables and those that are uniformly asymptotically stable in the reduced system. Then recalculate fdi(y) and go to next step. Step 4. Select the variable preferentially to control, obeying one of these rules: Rule 1 – select the variable with maximum fi and then with maximum fd; Rule 2 – select the variable with maximum fi and then with minimum fd. Then go to Step 3. Step 5. The variables preferentially (and relatively necessarily) to be stabilized have been determined, and the selecting procedure ends. Theorem 1. Stabilizing system (5) to be uniformly asymptotically y-stable is equivalent to stabilizing the selected variables (obtained by carrying steps 1  5) to be uniformly asymptotically stable. Proof . The necessity (‘‘ ) ’’) is trivial. And the sufficiency (‘‘ ( ’’) is a direct result of Lemma 1.

h

However, the above theorem only stands in the local sense. To extend it to have a global sense, we should replace Step 3 with the following step and keep all other steps the same. Step 3 0 :If the non-selected elements of y satisfy the property of CIBS (see Definition 4), with the selected variables as the inputs, then go to Step 5; otherwise, set to zero the rows and columns of DR that are corresponding to the selected variables and those that are globally uniformly asymptotically stable in the reduced system. Then recalculate fdi(y) and go to next step. Then, base on steps 1, 2, 3 0 , 4 and 5, the following stronger theorem can be obtained. Theorem 2. Stabilizing system (5) to be globally uniformly asymptotically y-stable is equivalent to stabilizing the selected variables (obtained by carrying steps 1, 2, 3 0 , 4 and 5) to be globally uniformly asymptotically stable. Proof . The necessity (‘‘ ) ’’) is trivial. And the sufficiency (‘‘ ( ’’) is a direct result of Lemma 2.

h

Remark 3. (a) CIBS property can usually be determined by judging a stronger property of it, say the input-to-state stability [21,22]. (b) Though in this approach the variables preferentially to control are selected in order, this does not mean they will have to be stabilized orderly. (c) Rule 1 in Step 4 aims to minimize (relatively) the necessary controls; and Rule 2 aims to minimize (relatively) the complexity of the controls, by reducing their terms. (d) If in Step 4 we do not select the variables by fi and fd, and simply enumerate all possible selections instead, then all feasible sets of control variables guaranteeing the goal can be determined. Further, based on this approach, the way of control design can be extended – it then can proceed hierarchically: With some control components designed first to stabilize part of the variables waiting to be stabilized; then on the foundation of their stabilization, the other control components can be designed to stabilize the rest variables. In this way, the controller would be simpler. Additionally, such a designing method can combine with the traditional ones to improve the control effects. Please cite this article in press as: Hu W et al., An approach of partial control design for system ..., Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.017

ARTICLE IN PRESS W. Hu et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx

5

4. Examples Since stabilization problems and synchronization ones are mutually correlated, here we only give the latter as examples. 4.1. Synchronizing two chaotic Lorenz systems This example aims to show how the presented approach works to select the variables preferentially to control. Consider two identical chaotic Lorenz systems [23] 8 > < x_ 1 ¼ rðx2  x1 Þ x_ 2 ¼ rx1  x2  x1 x3 ; ð6Þ > : x_ 3 ¼ x1 x2  bx3 and 8 > < y_ 1 ¼ rðy 2  y 1 Þ y_ 2 ¼ ry 1  y 2  y 1 y 3 ; > : y_ 3 ¼ y 1 y 2  by 3

ð7Þ

where r = 10, r = 28, b = 8/3. The task is to design some partial control to synchronize them. Now subtract (7) from (6) and obtain the form of (4), namely 8 > < e_ 1 ¼ re1 þ re2 e_ 2 ¼ e2 þ ðr  x3 Þe1  x1 e3  e1 e3 : > : e_ 3 ¼ be3 þ x2 e1 þ x1 e2 þ e1 e2 Together with (6), they construct a system like (5). Then, 0 1 0 1 re1 re2 B C B C Eð1Þ ðe; xÞ ¼ @ e2 A; Eð2Þ ðe; xÞ ¼ @ ðr  x3 Þe1  x1 e3  e1 e3 A; be3 x2 e1 þ x1 e2 þ e1 e2 Hence,

1 0 1 0 0 B C DR ¼ @ 1 0 1 0 A; 1 1 0 0

0 1 0 B C Eð3Þ ðxÞ ¼ @ 0 A: 0

0

0

f di

1 0:2 0:4 B C ¼ @ 0:4 0:4 A: 0:4 0:2

ð8Þ

ð9Þ

According to selecting Rule 1, e2 is selected as the variable first to control. Since it is true that the states e1, e3 of (8) both satisfy CIBS condition w.r.t. the input e2, e2 is the only selected variable. Hence, according to Theorem 2, the partial control synchronizing (7) with (6) equivalently turns into the one synchronizing y2 with x2. Besides, adopting the same approach but not using the selecting Rule 1 or 2, and instead enumerating all available sets of control variables in Step 4, we can find that e1 is an alternative preferential control variable to synchronize the systems. However, e3 is not. And in this case, additional e1 or e2 needs to be controlled. These results are consistent with those in [24,25]. 4.2. Synchronizing two chaotic systems with different orders This example considers a problem of partial complete synchronization, and it mainly aims to show how the proposed approach can be used to simplify the controllers. Consider the chaotic Lorenz system (7) and the hyper-chaotic Chen system [26] 8 x_ 1 ¼ hðx2  x1 Þ þ x4 > > > < x_ ¼ px  x x þ lx 2 1 3 2 1 ; ð10Þ > _ x ¼ x x  kx 3 1 2 3 > > : x_ 4 ¼ x2 x3 þ qx4 Please cite this article in press as: Hu W et al., An approach of partial control design for system ..., Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.017

ARTICLE IN PRESS 6

W. Hu et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx

where h = 35, k = 3, l = 12, p = 7, q = 0.5. For simplicity, we suppose the controls are additive to (7), i.e., system (7) under control looks like y_ ¼ Y ðyÞ þ uðx; yÞ. The goal is to synchronize (y1, y2, y3)T with (x1, x2, x3)T. Subtract (7) with control from (10) and derive 8 > < e_ 1 ¼ re1 þ re2 þ ðr  hÞðx2  x1 Þ  x4 þ u1 ð11Þ e_ 2 ¼ e2 þ ðr  x3 Þe1  x1 e3  e1 e3 þ ðr  pÞx1  ð1 þ lÞx2 þ u2 : > : e_ 3 ¼ be3 þ x2 e1 þ x1 e2 þ e1 e2 þ ðk  bÞx3 þ u3 (11) together with 0 0 B DR ¼ @ 1 1

(10) compose the error-combined control system. Then, 1 1 0 1 C 0 1 1 A: 1 0 1

And hence fdi is the same as (9). Since DR4(1,2,3) = 1, according to Step 2 in Section 3, e1,2,3 are all necessarily selected to be controlled, i.e., it demands u1,2,3 5 0. First, suppose using Lyapunov method to design u. Take the goal Lyapunov function as V(e) = (eTe)/2, and let V_ ðeÞ ¼ re21  e22  be23 . Then the controller can be designed as 8 > < u1 ¼ re2 þ ðh  rÞðx2  x1 Þ þ x4 u2 ¼ ðx3  rÞe1 þ ðp  rÞx1 þ ð1 þ lÞx2 : ð12Þ > : u3 ¼ x2 e1 þ ðb  kÞx3 Second, if we design u totally based on the approach provided in Section 3, making use of the information of fd and fi and adopting the Rule 2 to select the variables preferentially to control, then we should stabilize e1,2,3 in the order of e1 ! e2 ! e3. Namely, we can first design u1 to stabilize e1; then let e1 = 0, design u2 to stabilize e2; last let e1,2 = 0, design u3 to stabilize e3 . (Notice that the boundedness of e should guarantee the CIBS properties here.) As a result, with the ‘‘virtual control goal’’ of e_ 1 = re1, e_ 2 = e2 and e_ 3 = be3, u can be designed as 8 > < u1 ¼ re2 þ ðh  rÞðx2  x1 Þ þ x4 ð13Þ u2 ¼ x1 e3 þ ðp  rÞx1 þ ð1 þ lÞx2 : > : u3 ¼ ðb  kÞx3 And their complexities seem comparative to (12).

e1

10 0 -10

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

e2

50 0 -50

e3

50 0 -50

|||e|2

40 the 1st controller the 2nd controller the 3rd controller

20 0

0

1

2

3

4

5

6

7

8

t Fig. 1. Time evolution of the errors, with the initial states of x = (3, 4, 2, 2)T and y = (12, 15, 30)T.

Please cite this article in press as: Hu W et al., An approach of partial control design for system ..., Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.017

ARTICLE IN PRESS W. Hu et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx

7

Third, however, if in the procedure above, after e1 is independently stabilized by u1, we then use the Lyapunov method to design u2,3. Take V ðer Þ ¼ ðeTr er Þ=2 and V_ ðer Þ ¼ e22  be23 , then the controller can be designed as 8 > < u1 ¼ re2 þ ðh  rÞðx2  x1 Þ þ x4 : ð14Þ u2 ¼ ðp  rÞx1 þ ð1 þ lÞx2 > : u3 ¼ ðb  kÞx3 The controller (14) now becomes much simpler, as compared to (12) or (13). And that (14) can still guarantee the synchronization should be out of expectation in the traditional sense. Fig. 1 shows the effects of these three controllers. And it indicates that the simpler is not always the worse.

5. Conclusions and discussions By establishing the dynamical relation matrix and the dynamical dependence–influence matrix, the inner hierarchical dynamical significance of variables is revealed in a great sense. And based on this, an approach is proposed for partial control design. It not only can seek out the dynamically dominant variables that are preferential to stabilize, but also can simplify the controllers by designing their controls hierarchically. Its effectiveness is justified by the given examples. Also, in this paper, although we simply focus on the theoretical side of this approach, it should not prevent its possible application in real cases to find out the dynamically dominant variables. For example, in the complex chemical industries, though there are large numbers of variables to control, it is possible to find the dynamically dominant variables with the help of this approach so that the industrial control can be met by merely controlling a small subset of variables [17,27–29] (the concept of partial control there is a bit different). Other possible use of this approach can be in the area of complex networks which has attracted great attention recently [30–33]. Since complex networks usually contain enormous variables, they are really hard to analyze, not to mention dominate. But, the approach proposed here can possibly help to seek the ‘‘essential’’ or ‘‘hub’’ variables among them, so that the way to treat the networks may be found [34]. However, it should be seen that there is more work to do with this approach, still. For instance, the assumption (b) made in Section 2 might be too strong, and further study should be done when the conditions are weakened, such as the cases that the bounded control is added. Also, the cases that the system is perturbed should be taken into account in the future work, since robustness should be concerned when simplifying the controllers [35].

Acknowledgements The authors gratefully acknowledge the support of the NSFC (No. 50537030).

References [1] Vorotnikov VI. Partial stability and control. Boston: Birkhauser; 1998. [2] Vorotnikov VI. Partial stability and control: the state-of-the-art and development prospects. Automat Rem Contr 2005;66(4):511–61. [3] Vorotnikov VI. Partial stability, stabilization and control: some recent results. In: 15th IFAC Triennial World Congress, Barcelona, Spain; 2002. [4] Chellaboina VS, Haddad WM. A unification between partial stability and stability theory for time-varing systems. IEEE Contr Syst Mag 2002;22(6):66–75. [5] Chellaboina VS, Haddad WM. Teaching time-varying stability theory using autonomous partial stability theory. In: Proc. IEEE Conf. Dec. Contr., Orlando; 2001. p. 3230–5. [6] Agiza HN. Chaos synchronization of two coupled dynamos systems with unknown system parameters. Int J Modern Phys C 2004;15(6):873–83. [7] Brown R, Rulkov NF. Designing a coupling that guarantees synchronization between identical chaotic systems. Phys Rev Lett 1997;78(22):4189–92. [8] Jiang G-P, Zheng WX. A simple method of chaos control for a class of chaotic discrete-time systems. Chaos, Solitons & Fractals 2005;23:843–9. [9] Jiang G-P, Chen G, Tang WK-S. A new criterion for chaos synchronization using linear state feedback control. Int J Bifurcat Chaos 2003;13(8):2343–51. [10] Huang D. Stabilizing near-nonhyperbolic chaotic systems with applications. Phys Rev Lett 2004;93(21):1–4. 214101. [11] Huang D. Simple adaptive-feedback controller for identical chaos synchronization. Phys Rev E 2005;71(3):1–4. 037203.

Please cite this article in press as: Hu W et al., An approach of partial control design for system ..., Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.017

ARTICLE IN PRESS 8

W. Hu et al. / Chaos, Solitons and Fractals xxx (2007) xxx–xxx

[12] Liu F, Ren Y, et al. A linear feedback synchronization theorem for a class of chaotic systems. Chaos, Solitons & Fractals 2002;13:723–30. [13] Wang XF. On the equivalence of chaos control systems. Chaos, Solitons & Fractals 2003;15:265–9. [14] Ge Z-M, Chen Y-S. Synchronization of unidirectional coupled chaotic systems via partial stability. Chaos, Solitons & Fractals 2004;21:101–11. [15] Ge Z-M, Chen Y-S. Adaptive synchronization of unidirectional and mutual coupled chaotic systems. Chaos, Solitons & Fractals 2005;26:881–8. [16] Yassen MT. Controlling chaos and synchronization for new chaotic system using linear feedback control. Chaos, Solitons & Fractals 2005;26:913–20. [17] Kothare MV, Shinnar R, et al. On defining the partial control problem: concepts and examples. AIChE J 2000;46(12):2456–74. [18] Sontag ED. Remarks on stabilization and input-to-state stability. In: Proceedings of the 28th IEEE conference on decision and control, Tampa, FL, vol. 1–3; 1989. p. 1376–8. [19] Vorotnikov VI. Theory of stability with respect to a part of the variables and the problem of coordinate synchronization for dynamical systems. Doklady Phys 2000;45(12):685–9. [20] Vorotnikov VI. On the coordinate synchronization problem for dynamical systems. Diff Equat 2004;40(1):14–22. [21] Sontag ED, Wang Y. n characterizations of the input-to-state stability property. Syst Contr Lett 1995;24(5):351–9. [22] Sontag ED. On the input-to-state stability property. Eur J Contr 1995;1(1):24–36. [23] Hong Y, Qin H, Chen G. Adaptive synchronization of chaotic systems via state or output feedback control. Int J Bifurcat Chaos 2001;11(4):1149–58. [24] Balmforth N, Tresser C, et al. Master-slave synchronization and the Lorenz equations. Chaos 1997;7(3):392–4. [25] Gao T-G, Chen Z-Q, Yuan Z-Z. Control of chaotic systems based on partial variables feedback. Acta Physica Sinica 2004;53(10):3274–9. [26] Lu J, Cao J. Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters. Chaos 2005;15(4):1–10. 043901. [27] Tyre´us BD. Dominant variables for partial control. 1. A thermodynamic method for their identification. Ind Eng Chem Res 1999;38(4):1432–43. [28] Tyre´us BD. Dominant variables for partial control. 2. Application to the Tennessee Eastman challenge process. Ind Eng Chem Res 1999;38(4):1444–55. [29] Arbel A, Rinard IH, Shinnar R. Dynamics and control of fluidized catalytic crackers. 3. Designing the control system: choice of manipulated and measured variables for partial control. Ind Eng Chem Res 1996;35(7):2215–33. [30] Lu¨ J, Yu X, Chen G. Chaos synchronization of general complex dynamical networks. Physica A 2004;334:281–302. [31] Baraba´si A-L, Albert R. Emergence of scaling in random networks. Science 1999;286:509–12. [32] Strogatz SH. Exploring complex dynamics. Nature 2001;410:268–76. [33] Albert R, Baraba´si A-L. Statistical mechanics of complex networks. Rev Mod Phys 2002;74(1):47–97. [34] Albert R, Jeong H, Baraba´si A-L. Attak and tolerance in complex networks. Nature 2000;406(27):378–81. ˚ stro¨m KJ, Wittenmark B. Adaptive control. 2nd ed. Addison-Wesley Publ. Co; 1995. [35] A

Please cite this article in press as: Hu W et al., An approach of partial control design for system ..., Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.017