Partial synchronization of different chaotic oscillators

Chaos, Solitons and Fractals 33 (2007) 572–581 www.elsevier.com/locate/chaos

Partial synchronization of different chaotic oscillators using robust PID feedback Ricardo Aguilar-Lo´pez

a,*

, Rafael Martinez-Guerra

b

a

Departamento de Energıá, Universidad Autońoma Metropolitana – Azcapotzalco, San Pablo 180, Reynosa-Tamaulipas, Azcapotzalco, 02200 Me´xico, D.F., Mexico b Departamento de Control Automa´tico, CINVESTAV IPN, Apartado Postal 14-740, Me´xico, D.F. C.P. 07360, Mexico Accepted 20 December 2005

Abstract This work deals with the partial synchronization problem of two different chaotic oscillators considering model uncertainties in the slave system via control approach. The slave system is forced to follow the master signal via a linearizing controller based on model uncertainty reconstructor which leads to proportional–integral–derivative (PID) control structure. This reconstructor is related with a proportional–derivative (PD) reduced-order observer, it would be considered as a sub-slave system for the original slave of the synchronization procedure. The asymptotic performance of the synchronization methodology is proven via the dynamic of the synchronization error. Numerical experiment illustrates the closed-loop behavior of the proposed methodology. 2006 Elsevier Ltd. All rights reserved.

1. Introduction As is well known, the study of the synchronization problem for chaotic oscillators has been very important from the non-linear science point of view, in particular, the applications to biology, medicine, cryptography, secure data transmission and so on. In general, the synchronization research has been focused on two areas: the first one, related with the employment of state observers, where the main applications lie on the synchronization of non-linear oscillators with the same model structure and order, but different initial conditions and/or parameters [1–5]; on the other hand, the use of control laws allows to achieve the synchronization between non-linear oscillators, with different structure and order, where the variable states of the slave system are forced to follow the trajectories of the master system. This approach can be seen as a tracking problem [6–10]; some authors design the controller from the dynamic of the synchronization error, because this approach allows transforming the tracking problem to a regulation problem with the origin (zero) as the corresponding set point [11]. Besides, several control approaches have used neural-network, fuzzy, adaptive, and sliding and other techniques [8]. Other traditional control methods [12] consider introducing an additive feedback controller, to force to the system to *

Corresponding author. Tel.: +52 55 5318 9000; fax: +52 55 5394 7378. E-mail addresses: [email protected] (R. Aguilar-Lo´pez), [email protected] (R. Martinez-Guerra).

0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.12.042

R. Aguilar-Lo´pez, R. Martinez-Guerra / Chaos, Solitons and Fractals 33 (2007) 572–581

573

reach the desired reference (set point), i.e. kx(t) xsp(t)k ! 0 as t ! 1. The above mentioned methodologies are based on the cancellation of the non-linear terms of the chaotic systems in order to impose a desired behavior. Under the above philosophy, the non-linear geometric-differential control techniques have been successfully employed [13,14]. They correspond to systems that can be fully or partially linearized by a change of coordinates and/or state feedback following differential-geometric concepts [6]. Such class of non-linear systems can be linearized by a state feedback control, which cancels all the non-linearities assuming perfect knowledge of the mathematical model, producing global asymptotic stability [7]. A drawback of exact linearization techniques and the other model based controllers is that they rely on exact cancellation of non-linearities. In practice, exact knowledge of system dynamics is not possible. A more realistic situation is to know some nominal functions of the corresponding non-linearities, which are employed in the control design. However, the use of nominal model non-linearities can lead to performance degradation and even closed-loop instability. In fact, when the systems possess strong non-linearities, the standard linearizing, generic model, and active controllers, cannot cancel completely such non-linearities and instabilities can be induced. The worst case is when the knowledge of the non-linearities is very poor or null, such that, conventional linearizing techniques are inadequate. To avoid the above problems, the geometric approach for the design of non-linear controllers based on uncertainty observers has been employed; and these kinds of techniques show satisfactory capabilities for a wide range of systems [15,16]. The use of proportional observers coupled with linearizing controllers have been very successful, but the proportional observers have several problems, for example, they are very sensitive for noisy measurements, robustness issues are not completely saved. For these reasons a more sophisticated observers have been designed in order to generate more adequate open-loop and closed-loop performances. PI observers, sliding-mode, numeric, etc. have been developed [17,18]. In particular, in this paper is proposed an uncertainty proportional–derivative reduced-order observer which is able to accelerate the estimation rate and then it is coupled to a linearizing controller to produce a robust proportional–integral–derivative (PID) structure. This methodology is applied to the synchronization of one signal of the slave system, the non-linear oscillators considered here have different order and structure and the examples proposed are de Duffing– Chen systems and the Van der Pol–Lorenz systems.

2. Problem description A generic representation of master system can be described by Eqs. (1a) and (1b) as follows: X_ m ¼ fm ðX m ; S m Þ ¼ Im ðX m Þ þ ‘m ðX m ; S m Þ y m ¼ hm ðX m Þ ¼ CX m þ -m

ð1aÞ ð1bÞ

Here, X m 2 Rn is the vector of states; S m 2 R is the message signal; Im ðÞ : Rnþ1 ! Rn is a non-linear vector field; ‘m ðÞ : Rnþ1 ! Rn is a linear vector of its arguments; -m 2 Rm is an additive bounded measurement noise and y 2 Rm is the vector of transmitted states. Now, considering the following representation for the slave system: X_ s ¼ fs ðX s ; U Þ ¼ Is ðX s Þ þ ‘s ðX s ÞU

ð2aÞ

y s ¼ hs ðX s Þ ¼ CX s þ -s

ð2bÞ

where the vectors and the maps considered are defined in a similar way as mentioned above and U 2 R is the corresponding control command. Defining the vector @ ¼ X m X s (synchronization error), the corresponding dynamic equation considering Eqs. (1) and (2) is as follows: @_ ¼ DIð@Þ þ ‘m ðX m ; S m Þ ‘s ðX s ÞU

ð3Þ

Now, let us consider the following hypothesis: H1. For the realized control input vector U(X(t)), ðkUðX ðtÞÞk 6 UÞ , the nominal closed-loop non-linear system ((1a) and (1b)) is quadratically stable; therefore there exists a Lyapunov function V P 0 that satisfies oV oV b1 ; b2 > 0 ½DIð@Þ þ ‘m ðX m ; S m Þ ‘s ðX s ÞU 6 b1 k@k2 ; o@ 6 b2 k@k; o@ Such that the system is a minimum phase.


574

H2. - is a vector function representing an external (maybe unknown), bounded perturbation k-k2D ¼ } < 1;

0 < D ¼ DT

The normalized matrix D is introduced to ensure the possibility to deal with components of different physical structure and is assumed as given a priori; } represents the power of the corresponding perturbation. H3. The linear vector field ‘()() is bounded, i.e. for any @ 2 Rn , k‘()k 6 ‘+ < 1.

3. Synchronization methodology design Let us assume that there exists a coordinate transformation ðx; tÞ ¼ Wð@Þ such that the system (3) can be locally transformed into canonical form given by x_ i ¼ xiþ1 ; i ¼ 1; 2; . . . ; r 1 x_ r ¼ f ðx; tÞ þ Bðx; tÞu t_ ¼ #ðx; tÞ

ð4Þ

y ¼ x1 Here, the following definitions were used: • • • • • •

Bðx; tÞ ¼ BðxÞ þ DB, where f(x, t) and DB are considered model uncertainties related to the non-linear system, BðxÞ is the nominal value of the control input coefficient, u is the system input, x are the system states, y is the slave measured signal, r is the relative degree of the system. Now, it is possible to define a convenient change of variables ð40 Þ

gðx; uÞ ¼ f ðx; tÞ þ DBu By substitution of Eq. (4 0 ) into Eq. (4), a new system is obtained x_ i ¼ xiþ1 ;

i ¼ 1; 2; . . . ; r 1

x_ r ¼ gðx; uÞ þ BðxÞu t_ ¼ #ðx; tÞ

ð5Þ

y ¼ x1 In order to control this system, let us define the following nominal input–output linearizing feedback control. u ¼ B1 ðxÞ½sg xr gðx; uÞ

with BðxÞ 6¼ 0

ð6Þ

The controller defined by Eq. (6) guarantees exponential stability of non-linear systems with no uncertainties and perfect measurements [14], i.e., DB = 0 and g(x, u) known, under the assumption that the named inner dynamics t_ ¼ #ðx; tÞ i.e. the state equations which are not controlled present a stable dynamic behavior (minimum phase system), such that in accordance with the control theory point of view the system is stabilizable. However, since the uncertainty term, g(x, u), is unknown and, moreover, is the function of the states, x and the control input, u this ideal control law is not causal and therefore is not realizable. Nonetheless, there is another way to develop an input–output linearizing controller that is robust against uncertainties. The procedure shown below defines a method to estimate the uncertainty term, g(x, u). This approach is based on observer theory, where the uncertainty is only a function of the estimation error. Let us define the following dynamic sub-system under the assumption that system (5) has a relative degree r = 1. x_ r ¼ g þ Bu g_ ¼ Uðx; uÞ

ð7Þ ð8Þ

The uncertain term, g, is considered as a new state and U(x, u) is a non-linear unknown function that describes g’s dynamics. We can note that the uncertain term g is observable since g ¼ s_ Bu

ð9Þ


575

From the system given by Eqs. (7) and (8), it can be seen that a standard observer structure design, i.e. a system copy plus output feedback, is not possible to construct since the term U is unknown and considering that variable state xr is the system output, let us propose the following uncertainty observer. g^_ ¼ s1 ðg ^gÞ þ s2 ðg_ ^g_ Þ

ð10Þ

This uncertainty estimator a reduced-order observer, which infers the uncertain term g is obtained from the corresponding state equation. Note that the observer contains proportional and derivative actions; the aim of the derivative term is to improve the speed of the estimation algorithm, because it enhances the anticipatory and stabilizing effects of derivative actions. Substituting the estimate of the uncertain term in the ideal controller defined by Eq. (6), the following non-ideal controller is obtained: u ¼ B1 ½sg xr ^g

ð11Þ

From the corresponding state equation g ¼ x_ r Bðxr Þu; substituting it into Eq. (11) and taking the time derivative of the resulting equation the following expression is obtained: g_ g_ ¼ €xr sg x_ r þ ^

ð12Þ

Introducing this result in Eq. (10) yields: g^_ ¼ s2€xr þ ðs1 s2 sg Þ_xr sg s1 xr

ð13Þ

Since this controller uses an estimated value of the uncertainty, it cannot cancel the system non-linearities completely. Practical stability is achieved as long as the uncertainty estimation error is bounded [16]. Thus, the system trajectories remain inside a neighborhood close to the defined trajectory of reference. The final expression for the input–output linearizing controller with uncertainty estimation can be obtained integrating the estimator (Eq. (13)) and substituting it into the non-ideal controller (Eq. (11)) to obtain Z t u ¼ B1 ðxÞ ðsg ðs1 s2 sg ÞÞxr þ sg s1 ð14Þ xr ðhÞdh s2 x_ r 0

Note that this controller (Eq. (14)) exhibits PID structure and is equivalent to the linearizing controller based on proportional–derivative uncertainty observer (Eqs. (10) and (11)). 3.1. Stability issues of the proposed observer In order to show the stability properties of the closed-loop system, firstly, a convergence analysis of the uncertainty observer has to be done. ^_ ¼ s1 ðg ^ Proposition 1. Let us define g and g^ be the uncertainty term and its estimate. The dynamic system g gÞ þ s2 ðg_ ^g_ Þ is an asymptotic-type reduced-order observer for the systems ((7) and (8)). Proof 1. Let us define the uncertainty estimation error as e ¼ g ^g ¼ Dg

ð15Þ

Now, the dynamic scalar equation of the estimation error is given by Eq. (16), according to Eqs. (8) and (10) as follows. e_ ¼

s1 Uðx; uÞ eþ 1 þ s2 1 þ s2

Integrating this last expression, it renders Z t s1 s1 Uðx; uÞ t þ exp ðt sÞ ds e ¼ e0 exp 1 þ s2 1 þ s 1 þ s2 2 0 Now, consider the following assumption: A1. U(x, u) is bounded, kUk 6 W, with 0 < W < 1. Considering the norms of both sides of Eq. (17) Z t s1 s1 kUðx; uÞk kek 6 ke0 k exp t þ exp ðt sÞ ds 1 þ s2 1 þ s 1 þ s2 2 0

ð16Þ

ð17Þ


576

Applying the assumption A1, the following expression is obtained: s1 W W þ t ke0 k kek 6 exp s1 s1 1 þ s2

ð18Þ

In the limit, when t ! 1: kek 6

W s1

ð19Þ

It is important to analyze the structure of Eq. (18) in order to obtain some characteristics of the proposed observer. As desired, in the limit when t ! 1, the estimation error remains around a closed-ball with radius proportional to W/ s1, which can be made as small as desired by taking s1 large enough. In order to improve the speed to the uncertainty observer convergence to the steady-state estimation error, two actions were taken. The first one is to consider the parameter s1 large enough, which is necessary to obtain a small steady-state error, as it was mentioned above. The second one is faced with the influence of the parameter s2 related to the derivative action of the uncertainty observer. If s2 ! 1, the exponential term of the right side of Eq. (18) can be accelerated enough and consequently, the convergence of the uncertainty observer will exhibit better performance. Note that if the measurements of the system are corrupted by additive noise, i.e. s = xr + m, and this noise is considered bounded, kmk 6 X, a methodology similar to the one used to analyze the estimation error e can be applied in order to prove that the steady-state estimation error becomes kek 6 WþX . This confirms robustness against noisy s1 measurements. Now, it is possible to implement a non-ideal controller using the estimated uncertainty. In order to prove closed-loop stability, it is necessary to analyze the closed-loop equation of the state equation when the non-ideal control law is introduced. 3.2. Stability issues of the proposed controller Consider the state equation of the measured signal of the master system x_ m ¼ fm ðxm Þ þ ‘m ðxm Þ

ð20Þ

With the following assumption: x_ m < M < 1 and the closed-loop of the state equation of the slave signal x_ s ¼ Dg sg ðxs xm Þ

ð21Þ

Defining the synchronization error as @ ¼ xs xm The corresponding dynamic is as follows: @_ ¼ sg @ þ hðxm ; xs ; uÞ

ð22Þ

where h ¼ fm ðxm Þ þ ‘m ðxm Þ Dg Now, for the above considerations the following assumptions are satisfied. There exist sg 2 R and N 2 Rþ such that A1. khðxm ; xs ; uÞk 6R N and limt!t0 sNg ¼ 0. t A2. limt!t0 k expð 0 sg drÞk ¼ 0, where t0 is large enough. Eq. (22) can be solved to obtain the error dynamics in the time domain (Eq. (23)). Z Z t @ ¼ @0 exp sg dt þ expðsg ðt rÞÞhðxm ; xs ; uÞdr ð23Þ 0

Taking norms of both sides of Eq. (23) yields to an inequality (expression (24)), which is limited after assumptions A1 and A2.


Z lim sup t!t0 0 6 lim sup knk 6 kn0 k lim sup exp sg dt þ t!t0

t!t0

Rt R lim sup N 0 exp sg dr t!t0 R 0 6 lim sup knðtÞk 6 lim sup exp sg dt t!t0

R R t exp sg dt hðxm ; xs ; uÞdr 0 R lim sup exp sg dt

577

ð24Þ

t!t0

ð25Þ

t!t0

Expression (25) is an example of the 1/1 case of uniform L’hoˆpital’s rule, which can be applied to solve the undefined quotient (Eq. (25)). Now, taking the limit when t ! t0 it is possible to obtain a bound for the estimation error (Eq. (26)). R N exp sg dt N R ð26Þ sup 0 6 lim sup k@ðtÞk 6 lim sup exp sg dt ksg k ¼ lim ksg k t!t0 t!t0 t!t0 and k@k ! 0

ð27Þ

4. Results and discussion In this section, the results from the numerical experiments carried off to show the goodness of the proposed synchronization methodology are presented. Both of the two cases are considered as application examples; the first one is related with the synchronization of the Van der Pol (master system)–Lorenz (slave system). The mathematical model of both non-linear oscillators is • Van der Pol model: x_ 1;m ¼ x2;m x_ 2;m ¼ lð1 x21 Þx2 x20 x1 ax31 kx51 þ f0 cos xt sm ¼ x2;m

ð28Þ

where l ¼ 0:4; x0 ¼ 0:46; a ¼ 1; k ¼ 0:1; x ¼ 0:86; f 0 ¼ 4:5 with the corresponding initial conditions x1;m ð0Þ ¼ 0 x2;m ð0Þ ¼ 0 • Lorenz model: x_ 1;s ¼ rðx2;s x1;s Þ þ u1 x_ 2;s ¼ qx1;s x2;s x1;s x3;s x_ 3;s ¼ x1;s x2;s bx3;s

ð29Þ

ss ¼ x1;s where r ¼ 10; q ¼ 28; b ¼

8 3

and the corresponding initial conditions x1;s ð0Þ ¼ 1 x2;s ð0Þ ¼ 0 x3;s ð0Þ ¼ 5 The corresponding uncertain term g1 = rx2,s such that x2,m is an unmeasured variable, the control gain was chosen as sg = 10, the observer gains were considered as s1 = 100 and s2 = 0.8. Fig. 1 is related with the synchronization proce-


578

20

15

10

X 2,m , X 2,s

5

0

-5

-10

-15

-20

0

5

10

15

20

25 Time

30

35

40

45

50

Fig. 1. Synchronization of Van der Pol and Lorenz oscillators. Solid line – master signal (x2,m) and dotted line – slave signal (x2,s).

dure, it was turned on at 25 min, where the slave signal tracks the master signal with satisfactory agreement, in accordance with the theory developed. On the other hand, the systems Duffing–Chen whose mathematical models are the following: • Duffing model: z_ 1;m ¼ z2;m z_ 2;m ¼ dx2;m p20 x1;m cx31 þ K 1 cosðx1 t þ h1 Þ þ K 2 cosðx2 t þ h2 Þ

ð30Þ

sm ¼ x2;m where d ¼ 0:2; p ¼ 0:5; c ¼ 1; K 1 ¼ 11; K 2 ¼ 0; h1 ¼ h2 ¼ 0; x1 ¼ 1; x2 ¼ 4 with initial conditions z1;m ð0Þ ¼ 0 z2;m ð0Þ ¼ 0 • Chen model: z_ 1;s ¼ aðz2;s z1;s Þ z_ 2;s ¼ ðc aÞz1;s cz2;s z1;s z3;s þ u2 z_ 3;s ¼ z1;s z2;s bz3;s ss ¼ z2;s with parameters a ¼ 35; b ¼ 3; c ¼ 28 and initial conditions z1;s ¼ 1 z2;s ¼ 1 z3;s ¼ 1

ð31Þ


579

Here the uncertainty considered is the non-linear term g2 = z1,sz3,s, same as in the above case, the terms z1,s and z3,s are unmeasured variables, for this case the observer gains were considered as s1 = 500 and s2 = 0.9 and the controller gain was chosen as sg = 10. The synchronization procedure was started after 10 min; can be observed in Fig. 2 that the slave signal tracks the master signal as expected without any problem. Figs. 3 and 4 are related with the respective control input efforts; note that an acceptable effort is required for the synchronization task. 40

30

20

Z 2,m , Z 2,s

10

0

-10

-20

-30

-40

0

5

10

15

20

25 Time

30

35

40

45

50

Fig. 2. Synchronization of Duffing and Chen oscillators. Solid line – master signal (z2,m) and dotted line – slave signal (z2,s).

250

200

150

100

Control input (u1 )

50

0

-50

-100

-150

-200

-250

0

5

10

15

20

25 Time

30

35

40

45

50

Fig. 3. Control input effort (u1) for the synchronization of Van der Pol and Lorenz oscillators.

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R. Aguilar-Lo´pez, R. Martinez-Guerra / Chaos, Solitons and Fractals 33 (2007) 572–581 800

600

Cont rol input (u2 )

400

200

0

-200

-400

0

5

10

15

20

25 Time

30

35

40

45

50

Fig. 4. Control input effort (u2) for the synchronization of Duffing and Chen oscillators.

5. Conclusions A methodology to synchronize different non-linear oscillators is proposed. The methodology synchronizes one slave signal to one master signal and it is related with an uncertainty proportional–derivative reduced-order observer coupled to linearizing controller, this procedure generates a robust PID feedback structure. Note that the employed observer can be interpreted as a sub-slave of the slave system, such that it provides valuable information (about uncertain terms) to the slave system. The proposed methodology is applied to two examples, the Van der Pol–Lorenz oscillators and the Duffing– Chen oscillators, where the synchronization shows an adequate performance, as can be seen in the numerical simulations.

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[13] Isidori A. Nonlinear control systems. Berlin (Germany): Springer-Verlag; 1989. [14] Nijmeijer H, van der Shaft A. Non-linear dynamical control systems. Springer-Verlag: New York; 1990. [15] Aguilar R, Martıńez-Guerra R, Poznyak A, Maya-Yescas R. Temperature control in catalytic cracking reactors via robust PID controller. J Process Control 2002;12(6):695–705. [16] Aguilar-Lo´pez R, Alvarez-Ramı´rez J. Sliding-mode control scheme for a class of continuous chemical reactors. IEE Proc Control Theory Appl 2002;149(4):263–8. [17] Martıńez-Guerra Rafael, Aguilar Ricardo, Poznyak Alexander. A new robust sliding-mode observer design for monitoring in chemical reactors. J Dyn Syst, Measure Control, ASME J 2004;126(3):473–8. [18] Aguilar Ricardo, Martıńez Rafael, Poznyak Alexander. PI observers for uncertainty estimation in continuous chemical reactors. In: Proceedings of the IEEE conference on control applications/ISIC 2001, Ciudad de Me´xico, Mexico, September 2001. p. 1037– 41.