Robust chaotic control of Lorenz system by backstepping design 2008

Robust chaotic control of Lorenz system bybackstepping designChao-Chung Peng,Chieh-Li Chen*Department of Aeronautics and Astronautics,National Cheng Kung University,Tainan,TaiwanAccepted 7September 2006Communicated by Prof.Ji-Huan HeAbstractThis work presents a robust chaotic control strategy for the Lorenz chaos via backstepping design.Backstepping technique is a systematic tool of control law design to provide Lyapunov stability.The concept of extended system is used such that a continuous sliding mode control (SMC)effort is generated using backstepping scheme.In the pro-posed control algorithm,an adaptation law is applied to estimate the system parameter and the SMC offers the robust-ness to model uncertainties and external disturbances so that the asymptotical convergence of tracking error can be achieved.Regarding the SMC,an equivalent control algorithm is chosen based on the selection of Lyapunov stability criterion during backstepping approach.The converging rate of error state is relative to the corresponding dynamics of sliding surface.Numerical simulations demonstrate its advantages to a regulation problem and an orbit tracking prob-lem of the Lorenz chaos.Ó2006Elsevier Ltd.All rights reserved.1.IntroductionThe chaotic behavior is a very interesting nonlinear phenomenon which has been intensively studied during the last two decades.The effect of chaotic system is usually undesirable in practice due to its sensitivity to initial conditions,unpredictable behavior and thereby restricts the operation of physical plants.Because of the difficulty of accurate pre-diction of a chaotic system behavior,chaos may cause system instability or degradation in performance,and it should be eliminated in many cases.Regarding the field of chaotic analysis and control,the Lorenz system is often taken as a paradigm,since it captures many of the features of chaotic dynamics.For example,the Lorenz system describes some of the unpredictable behavior which associates with the weather.It also formulates an incompressible fluid between two parallel horizontal bound-aries,with the lower boundary at a higher temperature than the upper boundary.Many approaches and techniques have been proposed for the control of chaos such as OGY method [1],bang–bang control [2],optimal control [3],intelligent control base on neural network [4],feedback linearization [5],differential0960-0779/$-see front matter Ó2006Elsevier Ltd.All rights reserved.doi:10.1016/j.chaos.2006.09.057*Corresponding author.Tel.:+88662389121/4;fax:+88662389940.E-mail address:chiehli@.tw (C.-L.Chen).Available online at Chaos,Solitons and Fractals 37(2008)598–608geometric method [6],adaptive control [7–10],H 1control method [11],and many others [12,13].Sliding mode control (SMC)provides an effective alternative to deal with uncertain chaotic systems.However,it is assumed that the control can be applied with infinite fast switching.In practice,it is impossible due to finite time delays and therefore causes limitation in practical system.This non-ideal switching will result in an undesirable phenomenon so-called chattering which deteriorates system performance and also cause wear and tear in mechanical devices.In order to overcome this chattering phenomenon,the concept of extended systems is used,and it has been successfully applied in controlling chaos [14,15]and physical plant [16].The backstepping approach is one of the most popular nonlinear techniques of control design.It is capable of generating a globally asymptotically stabilizing control laws to suppress and synchronize chaotic system [17–20].Publications regarding Lorenz chaos control,assumed that precise model parameters are available for the feedback linearization and the system is not subjected to external disturbance.However,the ideal condition can not always be provided.In this work,a control strategy was developed for chaos suppression as well as model uncertainties and dis-turbance.The proposed strategy belongs to an input-output control scheme which inherits advantages of adaptive con-trol and SMC via the backstepping design procedure.The adaptation law was design for parameter estimation whereas SMC mainly focuses on the disturbance elimination.The extended model corresponding to the chaotic system is for-mulated to obtain a continuous control input.Simulation results verify that the proposed controller can suppress the Lorenz chaos in spite of model uncertainties and external disturbances.2.System description for uncertain nonlinear systemsConsider the following class of nonlinear systems whose trajectories are contained in a chaotic attractor:_x¼f ðx Þþg ðx ÞÁu ;ð1Þwhere x (t )2R n is a state vector,u 2R is a scalar input,and f (x )and g (x )are smooth vector fields.Generally,the non-linear mode is only an approximate description of the actual plant due to the presence of various uncertainties.Some n -th order nonlinear chaotic dynamical systems with the relative degree n may be directly described or transformed by states transformation as follows:_xi ¼x i þ1;16i 6n À1_x n ¼b 0ðX ;t ÞþD b ðX ;t Þþd ðt Þþu ðt Þ&;x ¼½x 1x 2ÁÁÁx n 2R n ;ð2Þwhere x ðt Þ¼½x 1ðt Þx 2ðt ÞÁÁÁx n ðt Þ ¼½x ðt Þ_xðt ÞÁÁÁx ðn À1Þðt Þ 2R n is the state vector,D b (x ,t )is continuous uncertainty of chaotic systems,u (t )2R is the control input,and d (t )is denoted as the continuous,smooth external disturbance.In general,the uncertain term D b (x ,t )and disturbance term d (t )are assumed bounded,i.e.j D b ðx ;t Þj 6a and j d ðt Þj 6b ;ð3Þwhere a ,b are positive scalar.The control task is to force the system to track an n -dimensional desired vector x d (t ),where x d ðt Þ¼½x d 1ðt Þ;x d 2ðt Þ;ÁÁÁ;x dn ðt Þ ¼½x d ðt Þ;_x d ðt Þ;ÁÁÁ;x ðn À1Þd,which belong to a class of continuous functions on [t 0,1].Define the tracking error ase ðt Þ¼x d ðt ÞÀx ðt Þ¼½x d ðt ÞÀx ðt Þ_xd ðt ÞÀ_x ðt ÞÁÁÁx ðn À1Þd ðt ÞÀx ðn À1Þðt Þ ¼½e ðt Þ_eðt ÞÁÁÁe ðn À1Þðt Þ ¼½e 1ðt Þe 2ðt ÞÁÁÁe n ðt Þ ð4ÞThe control goal considered in this paper is that for any given set point or target orbit,an integral type backsteppingsliding mode control law is designed,such that the resulting state response of tracking error vector satisfieslim t !1k e ðt Þk ¼lim t !1k x d ðt ÞÀx ðt Þk !0:ð5Þwhere k Æk is the Euclidean norm of a vector.3.Backstepping design for the Lorenz chaotic systemIn 1963,Lorenz proposed a simple model which describes the unpredictable behavior of the weather.The dynamic of Lorenz’s equations can be written asC.-C.Peng,C.-L.Chen /Chaos,Solitons and Fractals 37(2008)598–608599_x1¼Àr x1þr x2_x2¼rx1Àx2Àx1x3_x3¼x1x2Àbx3ð6Þwhere r and r are relative to the Prandtl number and Rayleigh number,respectively.The parameter b belongs to a geo-metric factor.Depending on these parameter values,the Lorenz system performs complex dynamics as shown in Fig.1a and b.With the initial condition was placed at(10,10,10)and the values are taken as r=10,r=28and b=8/3,the dynamic behavior(6)acts a two-lobed pattern so called the butterfly effect.3.1.Lorenz system standardization with control input and disturbanceConsider the following chaotic system with external disturbance d._x1¼Àr x1þr x2_x2¼rx1Àx2Àx1x3þuþd_x3¼x1x2Àbx3ð7Þwhere u2R1is a scalar control input and d is denoted as the bounded disturbance,that is j d j6b.One of the control purposes is to stabilize the system to a specific state,and the other task is to force the system response to the desired orbit by u in the direction of x2even when the system is experiencing parameter uncertainty and disturbance.Let the control input u=u1+u2and u1=x1(t)x3(t)such that one can obtain the following decoupled system _x1¼Àr x1þr x2_x2¼rx1Àx2þu2þdð8Þand_x3¼x1x2Àbx3ð9ÞEq.(9)shows that x3(t)represents the internal dynamic of the system,which will converge when x1(t)and x2(t)converge to a specific point.In order to derive the integral type backstepping sliding mode control algorithm,system(8)is reformulated into an extended form as_x1¼Àr x1þr x2_x2¼ x3_ x 3¼Àr r x1þr r x2À x3þ_u2À_dð10Þ600 C.-C.Peng,C.-L.Chen/Chaos,Solitons and Fractals37(2008)598–608Definex c ¼x 1cx 2c x 3c 264375¼T À1x 1x 2 x 3264375ð11Þwhere x c is the new state vector and the transformation matrix is defined by T ¼^r 00^r100^r 1264375ð12ÞThen,dynamic Eq.(10)becomes_x 1c _x 2c _x 3c 264375¼0r =^r 00ð^r Àr Þ10r ð^r þr ÞÀ^r ð1þ^r ÞÀð^r þ1Þ264375x 1c x 2c x 3c 264375þ001264375ð_u 2þ_d Þð13ÞNote that r ¼^rþ~r ,where ^r is the nominal value and ~r denotes the parameter uncertainty.If the exact value of r was known,then (13)can be simplified as_x 1c _x 2c _x 3c 264375¼0100010r ðr À1ÞÀðr þ1Þ264375x 1c x 2c x 3c 264375þ001264375ð_u 2þ_d Þð14ÞEq.(14)is of controllable canonical form.However,the exact parameter value can not always be known previously.Therefore,(13)is considered in the following design procedure.Transform (13)in the form of error equations,one can obtain_e1c ¼e 2c þ~r s ð15:a Þ_e2c ¼e 3c þ~r ðx 2dc Àe 2c Þð15:b Þ_e3c ¼½r ð^r þr ÞÀ^r ð1þ^r Þ e 2c Àð^r þ1Þe 3c þg À_u 2À_d ð15:c Þwhere s ¼ðe 2c Àx 2dc Þ=^r,g ¼½^r ð1þ^r ÞÀr ð^r þr Þ x 2dc þð^r þ1Þx 3dc þ_x 3dc and ~r =^r j j (1,x dc =[x 1dc x 2dc x 3dc ]T is the transformed desired vector,i.e.,x dc =T À1x d3.2.Backstepping sliding mode controller design with adaptation lawIn this section,the backstepping design technique is applied to obtain control law of error system (15).The design procedure is divided into three steps shown as follows.Step.1From (15),treat the system state e 2c as an independent input and lete 2c ¼/1ðe 1c Þ¼Àk 1e 1c ;k 1>0:ð16Þwhere /1(e 1c )is defined as a desired virtual stabilizing algorithm for subsystem (15.a).Select a Lyapunov function V 1¼e 21c =2,and then_V 1¼e 1c _e 1c ¼e 1c ½/1ðe 1c Þþ~r s ¼Àk 1e 21c þ~rs e 1c ð17ÞConsider regulation problem (x 2dc ¼_x 1dc ¼0),(17)can be represented as _V 1¼Àk 1ð1þ~r =^r Þe 21c 60with j ~r=^r j <1.It infers that the asymptotical convergence of e 1c can be achieved without the contribution of adaptation law even inthe presence of uncertainty ~r.However,it is not the case for tracking problem.It can be seen that (17)can not guar-antee the asymptotical stability due to the existence of uncertain term ~rs e 1c .In order to overcome this problem,an adaptation algorithm is introduced as follows.Let_^r¼ls e 1c ð18Þwhere l is an adaptation gain which relative to the convergent rate of parameter estimation.Then,choose a Lyapunov function asV 1¼12e 21c þ12l~r 2;ð19ÞC.-C.Peng,C.-L.Chen /Chaos,Solitons and Fractals 37(2008)598–608601and the derivative of(19)is_V 1¼e1c_e1cþ~r_~r=l¼e1c½/1ðe1cÞþ~r s þ~rðÀls e1cÞ=l¼Àk1e21cð20ÞFrom(19)and(20),it implies that V1(t)<V1(0),and thereby the e1c and~r are bounded.Also,the uniform continuity of _V1can be checked by€V 1¼À2k1e1c_e1c¼À2k1e1c½/1ðe1cÞþ~r s ¼2k21e21cÀ2k1e1c~r sTherefore,€V1is bounded and according to the Barbalat’s lemma[21],(which states that if the differentiable function f(t) has afinite limit as t!1,and€fðtÞis bounded,then_fðtÞ!0as t!1),_V1is uniform continuous and it indicates that_V 1!0as t!1.In other words,limt!1e1cðtÞ!0and it implies that limt!1~rðtÞ!0for tracking task.Step.2:In practice,e2c may be different from/1(e1c)for all time.Therefore,define a new error variable z1= e2cÀ/1(e1c).The dynamic of the subsystem(15.a)can be represented as_e1c¼/1ðe1cÞþz1þ~r s:ð21ÞOne canfind that_V1<0if z l in(21)equals to zero.Further,consider the dynamic of z l_z1¼_e2cÀ_/1ðe1cÞ¼e3cþ~rðx2dcÀe2cÞþk1_e1c¼e3cþ~rðx2dcÀe2cÞþk1½/1ðe1cÞþz1þ~r s¼e3cþk1½/1ðe1cÞþz1 þ~r½ðx2dcÀe2cÞþk1s ¼e3cþk1½/1ðe1cÞþz1 þ~r½ðx2dcÀe2cÞþk1ðe2cÀx2dcÞ=^r¼e3cþk1½/1ðe1cÞþz1 ;for k1¼^rð22ÞIn a similar manner,take the state e3c as an independent input of the form/2(e1c,z1c)as the following, e3c¼/2ðe1c;z1Þ¼Àe1cÀðk1þk2Þz1Àk1/1ðe1cÞ;k22R;R>0:ð23ÞSelect a Lyapunov function of the subsystem(e1c,z1)in the form ofV2¼V1þ12z21:ð24ÞBased on the assumption(23),_V2¼e1c_e1cþ~r_~r=lþz1_z1¼e1c½/1ðe1cÞþz1þ~r s À~r s e1cþz1½e3cþk1ð/1ðe1cÞþz1Þ¼e1c½/1ðe1cÞþz1þ~r s À~r s e1cþz1½/2ðe1c;z1Þþk1ð/1ðe1cÞþz1Þ¼e1c½/1ðe1cÞþz1þ~r s À~r s e1cþz1½Àe1cÀðk1þk2Þz1Àk1/1ðe1cÞþk1ð/1ðe1cÞþz1Þ ¼Àk1e21c Àk2z2160ð25ÞAccording to Barbalat’s lemma,the subsystem(e1c,z1)is asymptotically stable.602 C.-C.Peng,C.-L.Chen/Chaos,Solitons and Fractals37(2008)598–608C.-C.Peng,C.-L.Chen/Chaos,Solitons and Fractals37(2008)598–608603 Step.3Following steps1–2,one canfind that if(23)came into effect,then the desired behavior of the subsystem(e1c, z1)can be achieved.In other words,the asymptotically stability of the error system can be guaranteed.Thus, according to the requirement of Lyapunov stability(25),a sliding manifold is selected as follows s¼e3cÀ/2ðe1c;z1Þ¼e3cþe1cþðk1þk2Þz1þk1/1ðe1cÞ¼e3cþe1cþðk1þk2Þðe2cÀ/1ðe1cÞÞþk1/1ðe1cÞ¼e3cþe1cþðk1þk2Þe2cÀk2/1ðe1cÞ¼e3cþðk1þk2Þe2cþð1þk1k2Þe1cð26ÞEq.(26)shows that once the sliding mode occurs(i.e.,s=0),the settling time of the error responses can be determine by assigning the corresponding eigen-values in the sliding surface.In the absence of uncertainty and disturbance_d,the corresponding equivalent control force_u2eq can be obtained by _s¼0,that is_u2eq¼^rð^rÀ1Þe2cÀð^rþ1Þe3cþð1þk1k2Þe2cþðk1þk2Þe3cþ^g:ð27Þwhere^g¼^rð1À^rÞx2dcþð^rþ1Þx3dcþ_x3dc.Regarding the enhancement of the system robustness such that system states stay on the sliding surface even in the pres-ence of model uncertainties and external disturbances,a switching control action is integrated as follows_u2¼_u2eqþw sgnðsÞð28ÞChoose a Lyapunov function V s=s2/2.Then,the reaching condition can be guaranteed for w>j D bðe c;tÞjþj_dðtÞj, where j D bðe c;tÞj¼j^rð~rþ~rÞþ~r rþ~rðk1þk2Þðx dc2Àe2cÞþ~rð1þk1k2Þsþ~g j is bounded(i.e.,j D b(e c,t)j6a).Hence,_V s ¼s_s¼s½Àw sgnðsÞÀD bðe c;tÞÀ_dðtÞ 6Às½w sgnðsÞþj D bðe c;tÞjþj_dðtÞj 6Àj s j wÀj D bðe c;tÞjÀj_dðtÞjÂÃ60;ð29ÞEq.(29)confirms the existence of sliding mode dynamics.Therefore,the system is globally asymptotically stable.From (27)and(28),an integral type control effort for the Lorenz chaotic system can be represented asu¼u1þu2¼x1x3þZ t_u2eqþw sgnðsÞÂÃdt¼x1x3þZ t^rð^rÀ1Þe2cÀð^rþ1Þe3cþð1þk1k2Þe2cþðk1þk2Þe3cþ^gþw sgnðsÞ½ dt:ð30Þ604 C.-C.Peng,C.-L.Chen/Chaos,Solitons and Fractals37(2008)598–608C.-C.Peng,C.-L.Chen/Chaos,Solitons and Fractals37(2008)598–6086054.Numerical simulationsIn this section,numerical studies of the Lorenz system subjected to disturbance(7)are carried out with initial con-ditions at(x1(0),x2(0),x3(0))=(10,10,10)as shown in Fig.2(a)–(b).4.1.Regulation problemConsider a regulation problem for the specific point,x d=0(i.e.the equilibrium point(0,0,0)).The disturbance term is selected as d(t)=0.5sin(0.5x t)with x=2p.The control gains k1¼^r with^rð0Þ¼9and k2=15are applied for all cases.The nominal value of r is performed by^r¼26and w=150is chosen to satisfy the inequality condition(26). Using the proposed control algorithm,the simulation results of regulation problem by SMC with l=0are shown in Figs.3and4,where the control input is active at t=10.Fig.3shows that irregular state responses converge to the set point asymptotically even in the presence of model uncertainties and disturbances.Fig.4(a)and(b)are the cor-responding responses of control input and sliding surface,respectively.4.2.Tracking problemHowever,regarding the tracking problem,asymptotically stability can not be guaranteed if there exists uncertainty in r(as mentioned in(17)).Therefore,the adaptation law must acts in the tracking task.Let the desired vector to be x d¼Tx dc¼T½kðtÞ_kðtÞ€kðtÞ T;where k¼asinðx tÞwith a=0.4.The control gains are selected as the same values used in regulation problem.The proposed SMC acts at t=5and the adaptation law acts at t=15with l=200.Fig.5shows the trajectory of the controlled Lorenz system has been forced to the assigned orbit.Further,from Fig.6(a)–(b),it can be seen that states x1(t)and x2(t)track the desired value while the control acts.The corresponding phase portrait of x1(t)Àx2(t)is illustrated in Fig.7.It indicates that system trajectory approach to the desired orbit and steer to it when adaptation scheme is applied.Examining the tracking performance,from Fig.8(a),one canfind that the asymptotical convergence can not be reached(before t=15).However,the tracking errors converge to zero owing to the effort of adaptation law(after t=15).In other words,the exact parameter value(r=10)has been estimated accurately as shown in Fig.9.From(9),the internal dynamics belongs to a stable system,but it is subjected to a perturbation by x1(t)x2(t).However,the states x1(t)and x2(t)have been controlled to track a bounded orbit such that the time response of x3(t)is also bounded(which can be proven by bound-input bound-output)as illustrated in Fig.10.The oscillated boundary of the internal state is dominated by the perturbation term x1(t)x2(t),which means that the larger the j x1(t)x2(t)j,the larger the boundary j x3(t)j.As illustrated in Fig.11(a),the proposed synthesized robust controller does not induce chattering behavior.Fig.11(b)shows that the sliding mode dynamics is suppressed to zero as the SMC is activated.606 C.-C.Peng,C.-L.Chen/Chaos,Solitons and Fractals37(2008)598–608C.-C.Peng,C.-L.Chen/Chaos,Solitons and Fractals37(2008)598–6086075.ConclusionsIn this paper,a robust control scheme for perturbed Lorenz chaos is developed.Based on the Lyapunov stability theory,the controller design was integrated with an adaptation algorithm and the SMC via utilizing backstepping design scheme.The SMC offers the robustness to model uncertainties and external disturbances.The adaptation law estimates the system parameter such that the asymptotical convergence of tracking error can be guaranteed.Moreover, under the proposed control algorithm,the convergent rate of the error responses can be determined by selecting the corresponding eigenvalues in 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