Adaptive set-point tracking of the Lorenz chaotic system using non-linear feedback
F. Haghighatdar, M. Ataei
*Department of Electronic Engineering, University of Isfahan, Hezar-Jerib St., Postal code: 8174673441, Isfahan, Iran Accepted 17 September 2007
Communicated by Prof. L. Marek-Crnjac
Abstract
In this paper, an adaptive control method for set-point tracking of the Lorenz chaotic system by using non-linear feedback is proposed. The design procedure of the proposed controller is accomplished in two steps. At the first step, using Lyapunov’s direct method, a non-linear state feedback is selected so that without any need to apply identification techniques, in despite of the uncertain parameters existence in the system state equations, the asymptotic stability of the general Lorenz system is guaranteed in a stochastic point of the manifold containing general system equilibrium points.
At the second step, a linear state feedback with adaptive gain is added to the prior controller to eliminate the tracking error. In order to guarantee the system asymptotic stability at desired set-point, the indirect Lyapunov’s method is used.
Finally, to show the effectiveness of the proposed methodology, the simulation results of different experiments including system parameters changes and set-point variation are provided.
Ó2007 Elsevier Ltd. All rights reserved.
1. Introduction
It has been elapsed more than four decades from first studies on the strange changes in the atmosphere, performed by Lorenz[1], which is considered as the first study in the field of chaotic phenomena. Posterior researches indicate that the chaotic phenomena can be detected in various fields of sciences such as, secure communications[2–4], power elec- tronics[5], chemical reactions[6]and biological systems[7]. However, chaotic behavior is usually undesirable in practice and restricts the operating range of many electronic and mechanical devices. Therefore, chaos control has been inten- sively considered by researchers after the pioneering work of Ott et al.[8], in the last two decades. In the field of chaos analysis and control, the Lorenz system is considered as a paradigm, since it captures many of the features of chaotic dynamics, and various methods have been developed for controlling it[9–17]. In the most of publications regarding Lorenz chaos control, knowing the exact value of the model parameters is an essential assumption for derivation of a controller successfully[9–14]. But in practical situations, some or all of the system parameters are unknown or uncer- tain. Moreover, these parameters can change slowly or abruptly. Therefore, the derivation of an adaptive controller for
0960-0779/$ - see front matter Ó2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.chaos.2007.09.087
* Corresponding author. Tel.: +98 311 7933040; fax: +98 311 7933071.
E-mail addresses:[email protected](F. Haghighatdar),[email protected](M. Ataei).
Chaos, Solitons and Fractals 40 (2009) 1938–1945
www.elsevier.com/locate/chaos
controlling the Lorenz chaotic system and any other chaotic system in the presence of system unknown parameters is an important issue. Hence, some researchers and authors have been concentrated to develop adaptive methods for control- ling the Lorenz chaotic system[15–17].
In this paper, the goal is to develop an adaptive method for set-point tracking of the Lorenz chaotic system. This is accomplished by using adaptive non-linear feedback. Already, in[9]non-linear feedback in the form of non-adaptive controller, has been used to control the chaotic behavior of the Lorenz system with fully known parameters. But in this paper, this technique is employed to develop an adaptive approach for controlling the Lorenz chaotic system so that, designed controller have a desired performance in the presence of unknown parameters and also parameter variations in the system model.
The remainder parts of this paper are organized as follow. In the second section the model of under consideration system and the tracking objective are introduced. In the third section, designing of the adaptive controller will be described step by step and the issues related to stability guarantee of the controlled system are presented. The simula- tion results and conclusions are provided at fourth and fifth sections, respectively.
2. The mathematical model of the Lorenz system and the tracking objective
The Lorenz system is known to be a simplified model of several physical systems. At the origin, it was derived from a model of earth’s atmospheric convection flow. This convection flow has a chaotic behavior and resulted from the heated air movement. Furthermore, it has been reported that the Lorenz equations may describe different systems such as laser devices[18]and disc dynamos[19].
After introducing the Chen system in 1999 by Chen and Ueta[20], Lu¨ system was introduced in 2002 by Lu¨ and Chen[21], which both of them have similar but not topological equivalent chaotic attractors with Lorenz attractor.
Finally, in 2002, Lu¨ et al. by considering some restrictions on the coefficients of the linear part of the above mentioned systems proposed a unified form for these systems. This combined chaotic system is called Unified chaotic system and is described by two following equations[22]:
_
x¼aðyxÞ;
_
y¼bxxzþcy;
_
z¼xydz;
8>
<
>: ð1Þ
where:
a¼25aþ10;
b¼2835a;
c¼29a1;
d¼aþ83 : 8>
>>
<
>>
>:
ð2Þ
The system described by relations(1) and (2)has chaotic behavior for anya2[0, 1]. Whena2[0, 0.8), Unified sys- tem, is called the general Lorenz system, fora= 0.8, it becomes the general Lu¨ system and in the case ofa2(0.8, 1], it is called general Chen system.
In this paper, general form of Lorenz equations is chosen as the Lorenz system. In[9,14], two systems related to thermal convection phenomenon in fluids which their dynamic behavior is described by conventional Lorenz equations are investigated. Parameterbwhich is appeared in Eq.(1)is analogous to the Rayleigh number in the conventional form of the Lorenz equations. This parameter that is proportional to applied heat to the bottom half of the fluid, is allowed to vary. So in what follows,bwill be designed to be the system control input. Moreover, in controller design procedure, no restriction is imposed to system parameters. So, the proposed controller is not only able to control conventional form of the Lorenz equations which is a particular form of the general Lorenz equations but also is able to control any system with state equations in the form of Unified system. There is only one limitation that in the physical system described by Eqs.(1) and (2), the parameterbcan be selected as system input.
After describing the mathematical model of the under consideration system, in the following, the control design objective is explained. For this purpose, to determine the system equilibrium points, let time derivation of system states equal to zero, that by considering the first and third equations of relation(1), following two equalities will be obtained:
x¼y; z¼x2
d : ð3Þ
These two equalities describe the general system equilibrium manifold inR3. If the exact value of system parameters be known, the exact situation of three system’s unstable equilibrium points on this manifold will be determined by con- sidering second equation in(1)and two equalities in(3).
In this paper, the goal of adaptive set-point tracking of the Lorenz uncertain system is to stabilize the system, asymp- totically in any desired point of the above mentioned equilibrium manifold, without any knowledge about system parameters. In other words, it is desired that for a given reference of the systemy-state asyref, the controller be able to stabilize the system asymptotically in the related equilibrium pointx¼yref;y¼yref;z¼y2refd
.
3. Design of adaptive controller by using non-linear feedback
As it was mentioned in the previous section, in general Lorenz system described by(1) and (2),bis considered as the system control input. By adding the control signalutob, the closed loop Lorenz system can be rewritten as follows:
_
x¼aðyxÞ;
_
y¼beffxxzþcy;
_
z¼xydz;
8>
<
>: ð4Þ
wherebeff=b+uis called effective Rayleigh number. In order to control the closed loop system described by(4)effec- tively, the proposed strategy is to use a feedback controller consisting of a non-linear feedback term and a linear one.
The non-linear feedback term is designed so that in addition to eliminating the non-linearity in the second equation of (4), it also stabilizes the system asymptotically in a stochastic point on general equilibrium manifold given by(3)simul- taneously. The linear feedback term guarantees the system asymptotic stability in the desired set-point of the above mentioned equilibrium manifold, by eliminating tracking error. Moreover, the proposed controller is adaptive such that it is able to stabilize the system asymptotically in the desired set-point when the undesirable changes occur in the system parameters.
In the following, the details of the controller design procedure will be presented in two steps by two proposed lemmas.
Lemma 1. Consider the general Lorenz system whose chaotic behavior is described by(4). Using the following control law:
u¼zþg ð5Þ
in which parameter g is updated based on the following relation:
_
g¼x2xy ð6Þ
this system, will asymptotically be stabilized in a stochastic point on the general equilibrium manifold given by(3).
Proof. In the control law given by(5), the first expression is used to eliminate the non-linear term in the second state equation of the controlled system. If parameterg, adopts the nominal valueg1=(b+c), the first and the second states of the controlled system will be identical to each other. But, here the exact value ofg1is not definite because the system parameters are unknown.
Now, the auxiliary variableswand~gare defined by the two following relations in terms of the controlled system variables (x,y,z,g):
w¼yx; ð7Þ
~
g¼g1g; ð8Þ
Then the system with following state equations is considered:
_
w¼fðx;y;z;gÞ;
_~
g¼g_¼hðx;y;z;gÞ:
ð9Þ We choose the Lyapunov function for the system described by(9)as follows:
Vðw;~gÞ ¼1
2½w2þ~g2: ð10Þ
The time derivation ofVregarding the previous relations is
V_ðw;~gÞ ¼ ðcaÞw2: ð11Þ
Considering the relation(2), the condition (ca) = (4a11) < 0 fora2[0, 1] is always satisfied. So, it is clear that in the vicinity ofw= 0,Vðw;~gÞis positive definite andV_ðw;gÞ~ is negative definite. Therefore, in the system described by (9), the auxiliary state variablew, asymptotically converges to zero. Consequently, by using control law given in(5) and (6), in the controlled system described by(4), both ofxandystates of the system, asymptotically converge to a sto- chastic value such asY. Furthermore, by solving the third differential equation in(4), in which bothxandyvariables are substituted by the stochastic valueY, the following relation will be resulted forz-state of the controlled system:
zðtÞ ¼edtzð0Þ þY2
d ð1edtÞ; ð12Þ
which showsz(t) converges toYd2whent! 1. So, using the control law given in(5) and (6), the general Lorenz con- trolled system whose dynamic behavior is described by(4), will be asymptotically stabilized in a stochastic point such as
x¼Y;y¼Y;z¼Yd2
on the manifold containing general system equilibrium points. h
Lemma 2. Consider the general Lorenz system whose chaotic behavior is described by(4).Using the following control law:
u¼zþgþkðyrefyÞ ð13Þ
in which parameter g and gain k are updated by(6)and the following relation, respectively:
k_¼csignðyrefÞjyrefyj; c>0; ð14Þ
this system, will asymptotically be stabilized in a desired set-pointE¼x¼yref;y¼yref;z¼y2refd
on the general equilib- rium manifold given by(3).
Proof. In this stage of the proof, we can consider the gaink, which is related to the linear feedback term of the control law, as a constant value. Then, by using Lyapunov’s indirect method, we will determine the admissible range for gaink, in which asymptotic stability of the controlled system will be guaranteed. Hence, the controlled system should be lin- earized around the equilibrium pointEand it is necessary that, parametergadopts the nominal valueg1=(b+c).
Furthermore, it should be noticed that by activating the controller and elimination of non-linear term in the second state equation of the controlled system,xandystates of the controlled system will be independent of the third con- trolled system state,z. Therefore, by considering relation(6), parametergis only a function ofx andystates of the system, and according to chain rule, it satisfies the following differential equation:
og ox
dx dtþog
oy dy dt¼dg
dt ð15Þ
in whichdgdt is given in(6).
Therefore, the Jacobian matrixJand the system characteristic equation around the desired equilibrium point will, respectively, be derived as
J¼
a a 0
bþgþyrefogoxjE ckyrefþyrefogoyjE 0
yref yref d
2 4
3
5; ð16Þ
ðkþdÞ k2þ acþkyrefyrefog oy
E
kþa cþkyrefyrefog oy
E
bgyrefog ox
E
¼0: ð17Þ
And according to Routh–Hurwitz criterion, the controlled system in the desired set-point will be asymptotically stabi- lized if the following two conditions be satisfied:
kyref > caþyrefog oy
E
;
kyref > bþcþgþyrefog oy
E
þyrefog ox
E
: 8>
>>
<
>>
>:
ð18Þ
So, considering two inequalities in(18), ifyref> 0(yref< 0), in order to guarantee the asymptotic stability of the con- trolled system it is necessary that gainkbe greater (smaller) than an uncertain value. Therefore, updating of gaink, related to the linear feedback term, by using(14)which leads to an increasing (decreasing) gain for the case of positive (negative)yrefup to the stabilization of the controlled system, is a suitable adjustment, provided that partial differentials ofgwhich appear in the right hand sides of two inequalities in(18)have a bounded value at the desired equilibrium point. This condition is satisfied because a particular response for the differential equation(15)which satisfies the de- sired conditions can be derived as
gðx;yÞ ¼ ðbþcÞ x2y2ref
2a ; ð19Þ
which have a bounded partial differentials at the desired set-point. h
Thus, by considering the combination of two above proposed lemmas, the procedure of designing the adaptive con- troller by using non-linear state feedback is specified. In the proposed control law, which is given in(13), the adaptive gainkand adaptive parameterg, are updated by(14) and (6), respectively.
4. Simulation results
In this section, to show the effectiveness of the proposed methodology, the simulation results of different experiments are provided. In these experiments, the ability of set-point tracking, by using the proposed controller, in different con- ditions including system parameters changes and set-point variation, separately and also simultaneously, are investi- gated. In all of the experiments, parameter c in control law, given by (14) is selected as c= 100 and the initial values of the system states are considered asx(0) = 3,y(0) = 2,z(0) = 3.
4.1. First experiment
In this experiment, the general Lorenz system described by(1) and (2)is considered witha= 0. By selecting the ref- erence value fory-state of the system asyref= 8.485, the desired set-point isE= (8.485, 8.485, 27). The time response of the states of controlled system, by applying the control law to the system att= 15 (s), is shown inFig. 1. As it can be seen in this figure, after applying the control law, system states converge to the desired equilibrium states, rapidly. The control signal that is applied to the system is also shown inFig. 2.
4.2. Second experiment
In this case, the experimental conditions are the same as first experiment except that in this experiment the reference value of they-state of the system changes fromyref= 8.485 toyref= 12 att= 20 (s). Consequently, desired set-point changes fromE1= (8.485, 8.485, 27) toE2= (12, 12, 54). By applying the control law to the system att= 15 (s), the time response of the controlled system states is as shown inFig. 3, which indicates the desirable performance of the proposed adaptive controller. The control signal applied to the system in this experiment is also shown inFig. 4.
4.3. Third experiment
In this section, the most complete experiment, including parameter changes and set-point variation simultaneously, is accomplished. In this experiment the value of parameterachanges from 0 to 0.7 and the value of theyrefchanges
0 5 10 15 20 25
-30 -20 -10 0 10 20 30 40 50
Time (sec)
x(t),y(t),z(t)
x y z
Fig. 1. The convergence of the system states to the desired set-pointE= (8.485, 8.485, 27) after applying the control law to the general Lorenz system witha= 0, att= 15 (s).
0 5 10 15 20 25 -30
-20 -10 0 10 20 30
Time (sec)
u(t)
Fig. 2. The control signal applied to the system in the first experiment.
0 5 10 15 20 25
-30 -20 -10 0 10 20 30 40 50 60
Time (sec)
x(t),y(t),z(t)
x y z
Fig. 3. The convergence of the system states to the new desired set-pointE= (12, 12, 54), with changing the reference value of the systemy-state att= 20 (s).
0 5 10 15 20 25
-30 -20 -10 0 10 20 30
Time (sec)
u(t)
Fig. 4. The control signal applied to the system in the second experiment.
from 8.485 to 12 at t= 20 (s). Consequently at this time, desired set-point changes from E1= (8.485, 8.485, 27) to (12, 12, 49.655). Again, by applying the control law to the system att= 15 (s), the time response of the system states and the control signal applied to the system can be seen inFigs. 5 and 6, respectively.
5. Conclusion
In this paper the problem of set-point tracking in the Lorenz chaotic system, by using adaptive non-linear state feed- back, in the presence of unknown parameters in the system model has been investigated. The proposed feedback signal can be considered as the addition of a linear feedback term and a non-linear one. The objective of using non-linear feedback term was the elimination of the non-linearity in the second state equation of the system and to identify the first and the second system equilibrium states, and finally to stabilize the system in a stochastic point of the manifold containing general system equilibrium points. At this stage, the asymptotic stability of the controlled system has been guaranteed by using the Lyapunov’s direct method. Then, to eliminate the tracking error, a linear feedback term with adaptive gain has been added to the prior controller. At this stage, the linear feedback gain has been determined based on the stability guarantee condition, by using Lyapunov’s indirect method. Finally, to show the effectiveness of the pro- posed controller, simulation results have been provided.
0 5 10 15 20 25
-30 -20 -10 0 10 20 30 40 50
Time (sec)
x(t),y(t),z(t)
x y z
Fig. 5. The convergence of the system states to the new desired set-pointE= (12, 12, 49.655) with changing the value of the system y-state and parameteraatt= 20 (s).
0 5 10 15 20 25
-30 -20 -10 0 10 20 30
Time (sec)
u(t)
Fig. 6. The control signal applied to the system in the third experiment.
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