International audienceIn this paper two nonlinear observers for a chaotic system are compared. Moreover, the left invertible problem and the observability singularity are discussed. Thus, after a presentation of both observers, a comparison of the two proposed methods and a discussion are done on the basis of simulations results. The last part highlights the fact that the finite time observer is more sensible to the singularity observations, but less sensible to parameter uncertainties and noise in the output of the system

Barbot, Jean-Pierre

Fridman, Leonid

Ghanes, Malek

J. Bejarano, Francisco

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HAL Id: inria-00345027https://hal.inria.fr/inria-00345027Submitted on 9 Dec 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Second order sliding mode and adaptive observer forsynchronization of a chaotic system: a comparativestudyFrancisco J. Bejarano, Malek Ghanes, Jean-Pierre Barbot, Leonid FridmanTo cite this version:Francisco J. Bejarano, Malek Ghanes, Jean-Pierre Barbot, Leonid Fridman. Second order slidingmode and adaptive observer for synchronization of a chaotic system: a comparative study. IFACWorld Congress, Jul 2008, Séoul, North Korea. ￿inria-00345027￿Seond order sliding mode and adaptiveobserver for synhronization of a haotisystem: a omparative studyFraniso J. Bejarano ∗ Malek Ghanes ∗ Jean-Pierre Barbot ∗Leonid Fridman ∗∗∗ Equipe Commande des Systèmes (ECS), ENSEA 6 Avenue duPoneau, 95014 Cergy-Pontoise, Frane, (e-mail: {Franiso.Bejarano,Malek.Ghanes, barbot}ensea.fr)∗∗National Autonomous University of Mexio, Engineering Faulty,Division of Eletrial Engineering, UNAM 04510, Méxio, D.F.,(e-mail: lfridmanservidor.unam.mx)Abstrat:1. INTRODUCTION2. PROBLEM STATEMENTConsider the following haoti systemẋ1 (t) = a(x2 (t) − x1 (t)) + x2 (t)x3 (t) + m1 (t)ẋ2 (t) = b (x1 (t) + x2 (t)) − x1 (t)x3 (t)ẋ3 (t) = −cx3 (t) − ex4 (t) + x1 (t)x2 (t) + m1 (t)ẋ4 (t) = fx3 (t) − dx4 (t) + x1 (t)x3 (t) + m2 (t)y1 (t) = x1 (t)y2 (t) = x2 (t)(1)where xi ∈ R (i = 1, 2, 3, 4) are the states of the system;m1 and m2 represent the information to be transmitted,whih for the observation problem are onsidered as theunknown inputs; y1 and y2 are the outputs of the system,these are the signals to be sent by a publi hannel.Assumption. 2.1. The signals m1 and m2 are assumedto be dierentiable, bounded and with the derivativebounded.The goal is to reonstrut the states x3 and x4 whih allowsto reonstrut the messages m1 and m2. For ahievingsuh a goal two approahes are tested, namely, an ob-server based on the super-twisting algorithm that basiallyallows to obtain information from the derivatives of theoutput and onsequently to reonstrut the states and themessages (unknown inputs), and an adaptive observer. Aomparison between these two methods will be disussed.It should be notie that a singularity appears in the pointx1 = 0, that is, from the outputs x1 and x2 is not possibleto know the states x3 and x4 and onsequently neither themessages m1 and m2.3. SUPER-TWISTING OBSERVERFirst in order to generate a new output, namely the vari-able x3, we use the super-twisting algorithm for designingthe observer for x̂3 in the following way:ẋa,1 = b (x1 + x2) + v1v1 = x̂1,3 + λ1 |s1|1/2 sign sx̂1,3 = α1 sign ss1= x2 − xa,1(2)In this way, the derivative of s1takes the formṡ1 = −x1x3 − v1 (3)Choosing the gains λ1 ≥ (α1+M1)(1+θ)1−θ √ 2α1+M1 and α1 >M1 ≥∣∣ ddt (x1x3)∣∣ (0 < θ < 1), we get, aording to(Levant, 93), (Levant, 98), and (Davila, 05), the seondorder sliding motion, that is, s (t) = 0, ṡ (t) = 0 after somenite time T1. Notie that for s = 0, v1 = x̂3; therefore,from (3), we getx̂1,3 (t) ≡ −x1 (t) x3 (t) (4)From (4) we an reonstrut x3 (t) provided x1 (t) 6= 0.Thus, the observer for x3 is designed in the formx̂3 (t) =− x̂1,3 (t)x1 (t)if |x1 (t)| ≥ εx̂3 (t − τ ) if |x1 (t)| < ε (5)where τ and ε are enough small onstants 1 . Thus, we getthe identityx̂3 (t) ≡ x3 (t) for |x1 (t)| ≥ ε.Reonstrution of x4 Now, dening ȳ3 (t) = x3 (t) asa new output, we an rewrite the dynami equations in(1) as a linear system with output injetion and unknowninputs, that is,1 The onstant τ is hosen enough small but bigger than thesampling time used during the realization of the observer. Theonstant ε should be hosen suiently big to avoid the singularity,but also should be notie that any estimation in this zone an notbe onsidered as an aeptable estimation.ẋ1ẋ2ẋ3ẋ4= -a a 0 0b b 0 00 0 -c -e0 0 f -d ︸ ︷︷ ︸Ax1x2x3x4︸ ︷︷ ︸x+ y2ȳ3-y1ȳ3y1y2y1ȳ3︸ ︷︷ ︸φ(y1,y2,ȳ3)+ 1 00 01 00 1︸ ︷︷ ︸D[m1m2]︸ ︷︷ ︸w[y1y2ȳ3]︸ ︷︷ ︸ȳ=[1 0 0 00 1 0 00 0 1 0]︸ ︷︷ ︸xC (6)Now, let z be dened by the solution of the followingdierential equationz = Az + φ (y1, y2, ȳ3)Thus, dening ez = x− z we obtain the dynami equationfor ezez (t) = Aez (t) + Dw (t)yz = Cez(7)Hene, the system (7) is basially a linear system withunknown inputs, just the sort of systems onsidered in(Bejarano, 07). Then, for the reonstrution of x4 we followin essene, but with a little modiation, the algorithmproposed in (Bejarano, 07). That is, the basi idea pro-posed in (Bejarano, 07) is to obtain an algebrai expressionof ez in terms of the output yz and its derivatives. Thus,we get the equalities 2yz = Cezddt(CD)⊥yz = (CD)⊥CAezthus we get 3ez (t) =[C(CD)⊥CA]+ [yz (t)(CD)⊥yz (t)] (8)Sine we already know the rst 3 states, we solve (8) forez,4, the fourth state of ez. Thus,ez,4 =1e[ėz,1 − ėz,3 + a (ez,1 − ez,2)−cez,3 + ez,1ez,2 − ez,2ez,3](9)3.1 Realization of the observer using super-twistingTo redue the fast dynami and have a smaller gains inthe super-twisting algorithm, we design the following like-linear estimator x̃ whose dynamis is governed by thefollowing dierential equationx̃ =[−c −ef −d]︸ ︷︷ ︸Āx̃ +[y1y2y1ȳ3]+ L (ȳ3 − ỹ3)ỹ3 = [ 1 0 ]︸ ︷︷ ︸C̄x̃(10)The matrix L is hosen so that the eigenvalues of thematrix (Ā − LC̄) have negative real part. In this way wehave that the dynami equations for ē = x̃ − [x3x4] are2 Y ⊥ is a full row rank matrix suh that Y ⊥Y = 0.3 The matrix X+ is dened to be the pseudo-inverse of X. Thematrix onsidered in (8) belongs to sort of matries of full olumnrank. Hene, in suh a ase, X+ = (XT X)−1 XT .ē (t) =(Ā − LC)ē (t) + w (t)where w is dened in (6). Hene we get some upper boundsfor the norm of ē and for the norm of ē that is‖ē (t)‖ ≤ γ exp (−λt) ‖ē (0)‖ + µ ‖w (t)‖γ, λ, µ are positive onstantsThat is, with the estimator (10), ē is onstrained to stayedin a zone depending on the amplitude of m1 and m2; andthis zone an be made smaller by moving the eigenvaluesof (Ā − LC) more to the left in the left half-plane.Then, following the algebrai expression obtained in (9)we design the observer for x4 using the super-twistingalgorithm as followsẋa,2 =1e[a (x2 − x1) + cx̂3 − x1x2 + x2x̂3] + x̃4 + v2v2 = v2,1 + λ2 |s2| sign s2, v̇2,1 = α sign s2s2 =1e(x1 − x3) − xa,2x̂4 (t) ={x̃4 + v2,1 if |x1 (t)| ≥ εx̂4 (t − τ ) if |x1 (t)| < εwhere x̃4 is the seond omponent of the vetor x̃ denedin (10). Then, the time derivative of s2 isṡ2 = x4 − x̃4 + v2Thus, for λ2 ≥ (α2+M2)(1+θ)1−θ √ 2α2+M2 and α2 > M2 ≥∣∣ ddt (x4 − x̃4)∣∣ (0 < θ < 1), after some nite time T2, weget s2 = 0 and ṡ2 = 0; therefore,x̂4 ≡ x4 for |x1 (t)| ≥ ε.3.2 Messages reonstrutionThe reonstrution of m1 is made in the following wayẋa,3 = a (x2 − x1) + x2x̂3 + v3v3 = v3,1 + λ3 |s3| sign s3v̇3,1 = α3 sign s3m̂1 ={v3,1 if |x1 (t)| ≥ εm̂1 (t − τ) if |x1 (t)| < εs3 = x1 − x3aThus, for α3 and λ3 satisfying λ3 ≥ (α3+M3)(1+θ)1−θ √ 2α3+M3and α3 > M3 ≥ |ṁ1| (0 < θ < 1), after some nite time,we get the equalities s3 = 0, ṡ3 = 0. Thus, we get theequalitym̂1 ≡ m1, for |x1 (t)| ≥ εThe reonstrution of m2 is made in a similar way, that is,ẋa,4 = b (x1 + x2) + fx3 − dx4 + v4v4 = v4,1 + λ4 |s4| sign s4v̇4,1 = α4 sign s4m̂2 ={v4,1 if |x1 (t)| ≥ εm̂2 (t − τ) if |x1 (t)| < εs4 = x2 + x4 − xa,4Thus, taking into aount (1) and the derivative of xa,4,with the hoosing of α4 and λ4 satisfying the inequalitiesλ4 ≥ (α4+M4)(1+θ)1−θ√2α4+M4and α4 > M4 ≥ |ṁ2| (0 <θ < 1) (see, (Levant, 93), (Levant, 98)), we get, after somenite time,m̂2 ≡ m2 for |x1 (t)| ≥ εRemark. 3.1. At a glane, it seems that, during the esti-mation of the state x3, it is suient with to use x1 (t) 6= 0instead of |x1 (t)| ≥ ε, but the justiation is given inthe following lines. It is known that during the realiza-tion of the super-twisting algorithm s1 and ṡ1 are notexatly zero, hene, instead of having (4) we really havethe equality x̂1,3 (t) = −x1 (t)x3 (t) + ∆ (t), where ∆represents the estimation error and whih does not tendsto zero. Then, after dividing over x1, it yields the equalityx̄3 := − x̂1,3x1 = x3 −∆x1, whih means that, when x1 isvery lose to zero, the error between x̄3 and x3 is equal toO (1/x1). Therefore, in a small neighborhood of x1 = 0,the error between x̄3 and x3 is extremely big. This justifythe struture of x̂3.4. ADAPTIVE OBSERVERThe model of a haoti system (1) an be rewritten in thefollowing interonneted ompat form{Ẋ1 = A1(y2)X1 + g1(y, X1, X2) + φ1 (t)y1 = C1X1(11){Ẋ2 = A2(y1)X2 + g2(y, X2, X1) + φ2 (t)y2 = C2X2(12)where X1 = (x1, x3, x4, x5)T is the state of the rstsubsystem with x5 := m2, X2 = (x2, x3, x6)T is the stateof the seond subsystem with x6 := m1. y = [x1, x2]T arethe output of the whole system, andA1(y2) =0 y2 0 00 0 −c 00 0 0 10 0 0 0 , A2(y1) = ( 0 −y1 00 0 10 0 0)g1(y, X2, X1) =a(y1 − y2) + x6−ex4 + y1y2 + x6fx3 − dx4 + y1x30g2(y, X2, X1) =(b(y1 + y2)−cx3 − ex4 + y1y20)φi (t) =[00ṁi] , i = 1, 2C1 = ( 1 0 0 0 ) , C2 = ( 1 0 0 ) .Remark. 4.1. The hoie of the variables of eah subsys-tem has been onsidered in order to separate in one sub-system the message m1 and in the other one the messagem2. It is lear that other hoie an be onsidered inorder to represent these subsystems provided the neessaryonditions to design an adaptive are satised.Next, let us introdue an adaptive observer in order toestimate the system's state and the unknown inputs simul-taneously. It is based on interonnetion between severalsubsystems whih satises some required properties, suhthat the property of inputs persisteny ((Hammouri, 90),(Ghanes, 06)).At rst, let us introdue the following assumptions in orderto establish the results onerning the adaptive observerdesign (see more details in (Ghanes, 06)).Assumption. 4.1.1. The signals X1 and X2 are assumed bounded andto be regularly persistent ((Hammouri, 90), (Ghanes,06)) in order to guarantee the observability property ofsubsystems (11) and (12), respetively.2. A1(y2) and A2(y1) are uniformly bounded.3. g1(y, X1, X2) is globally Lipshitz with respet to X2and uniformly with respet to (y, X1).4. g2(y, X2, X1) is globally Lipshitz with respet X1 anduniformly with respet to (y, X2).5. The unknown funtions ṁi(t) (i = 1, 2) are assumed tobe bounded.Then, an adaptive observer for interonneted subsystems(11) and (12) estimating the state and unknown parame-ters is given byŻ1 = A1(y2)Z1 + g1(y, Z1, Z2) + S−11 CT1 (y1 − ŷ1)Ṡ1 = −θ1S1 − AT1 (y2)S1 − S1A1(y2) + CT1 C1ŷ1 = C1Z1 (13)Ż2 = A2(y1)Z2 + g2(y, Z2, Z1) + S−12 CT2 (y2 − ŷ2)Ṡ2 = −θ2S2 − AT2 (y1)S2 − S2A2(y1) + CT2 C2ŷ2 = C2Z2 (14)where Z1 = (x̂1, x̂3, x̂4, x̂5)T ; Z2 = (x̂2, x̂3, x̂6)T Si =STi > 0, i = 1, 2. Note that S−11 CT1 and S−12 CT2 are thegains of the observers (13) and (14), respetively.Remark. 4.2. It is worth notiing that ‖S1‖ and ‖S2‖ arebounded for θ1 and θ2 large enough due to the persistenyof input onsidered in assumption 4.1.Now, in order to guarantee the onvergene of the pro-posed observer, suient onditions are established in thefollowing result. Denote the estimation errors:ǫ1 = X1 − Z1 and ǫ2 = X2 − Z2whose dynamis are given by{ǫ̇1 = [A1(y2) − S−11 CT1 C1]ǫ1+g1(y, X1, X2) − g1(y, Z1, Z2) + φ1 (t)(15){ǫ̇2 = [A2(y1) − S−12 CT2 C2]ǫ2+g2(y, X2, X1) − g2(y, Z2, Z1) + φ2 (t) . (16)The values of θ1 and θ2 are hosen to satisfy the inequali-tiesδ1 = (θ1 − Γη) > 0, δ2 = (θ2 − Γη) > 0 (17)where Γ = µ̃1 + µ̃2, with µ̃i = µi√λmin(S1)√λmin(S2) ,i = 1, 2; and µ1 = k1k2, µ2 = k3k4, η ∈]0, 1[.The parameters k1, k2, k3, k4 are positive onstants andλmin(S1), λmin(S2) are the minimal eigenvalues of S1 andS2 respetively.Lemma 4.1. Consider the system (13)-(14) and that as-sumption 4.1 holds. Then, the system (13)-(14) is a prati-al exponential observer for system (11)-(12) for θ1 and θ2satisfying the inequalities (17). Furthermore, the observeronverges arbitrarily fast with a onvergene rate xed bya parameter δ, δ = min(δ1, δ2).Sketh of proof 4.1. Consider the following Lyapunov fun-tion andidate:Vo = V1 + V2where V1 = ǫT1 S1ǫ1 and V2 = ǫT2 S2ǫ2.From assumption 4.1, the following inequalities hold‖S1‖ ≤ k1;‖{g1(y, X1, X2) − g1(y, Z1, Z2)}‖ ≤ k2 ‖ǫ2‖;‖S2‖ ≤ k3;‖{g2(y, X2, X1) − g2(y, Z2, Z1)}‖ ≤ k4 ‖ǫ1‖.‖φ1‖ ≤ k5.‖φ2‖ ≤ k6.Computing the time derivative of Vo, , by using the aboveinequalities it follows thatV̇o ≤−θ1ǫT1 S1ǫ1 + 2µ1 ‖ǫ1‖ ‖ǫ2‖ + ‖φ1‖ (18)− θ2ǫT2 S2ǫ2 + 2µ2 ‖ǫ2‖ ‖ǫ1‖ + ‖φ2‖Now, onsider that the following inequalities are satisedλmin(Si) ‖ǫi‖2 ≤ ‖ǫi‖2Si ≤ λ max(Si) ‖ǫi‖2, i = 1, 2.By writing (18) in terms of funtions V1 and V2, it followsthaṫVo ≤ −θ1V1 − θ2V2 + 2(µ̃1 + µ̃2)√V1√V2 + k5 + k6where the parameters µ̃1, µ̃2, are dened just before lemma4.1.Next, by using the following inequality √V1√V2 ≤ υ2 V1 +12υ V2, ∀υ ∈]0, 1[, one getV̇o ≤ −(θ1 − Γ)V1 − (θ2 −Γυ)V2 + k5 + k6.where Γ is dened just before lemma 4.1.By taking δ and r suh that δ = min(δ1, δ2) and r = k5+k6one hasV̇o ≤ −δVo + r. (19)Finally, by hoosing θ1 and θ2 suh that the inequalities(17) are satised and suiently large, the inequality (19)shows that arbitrarily bounded perturbation will not resultin large error estimation deviations. This ends the proof.5. NUMERICAL EXAMPLE AND DISCUSSIONSFor the system (1) we use the parameters a = 42.5, b = 24,c = 13, d = 20, e = 50, f = 40. The parameters used forthe super-twisting observer are α1 = 7×107, λ1 = 7×103,α2 = 300, λ2 = 100, α3 = 1000, λ3 = 200, α4 = 600,λ4 = 300. For the adaptive observer the parameters usedare θ1 = θ2 = 400.Figures 1 and 2 show the trajetories of the states x3 andx4 as well as the ones of x̂3 and x̂4 for both the super-twisting and the adaptive observers. We an see in thegures that the trajetories of the super-twisting observeronverge muh faster than the ones of the adaptive ob-server.Figures 3 and 4 show the messages m1 and m2 togetherwith their estimations m̂1 and m̂2, respetively. In thisgures we note that the singularity in the point x1 = 0aets more the estimation of the messages made withthe super-twisting than the one made with the adaptiveobserver. This is lear due to the explanation of Remark3.1 and the fat that, in a ball of radio ε and enter inx1 = 0, it is not done a estimation neither of the statesnor of the messages.In order to test both observers with respet to parametersunertainties, we introdue a variation of 1% in the nomi-nal parameters. Figures 5 and 6 show how the parameters0 2 4 6 8 10−400−2000200400600800Time [s]0 0.02 0.04 0.06 0.08−101Fig. 1. x3 (solid line) and its estimation x̂3 using the super-twisting (dot line) and adaptive (dash line) observers0 2 4 6 8 10−400−2000200400600800Time [s]0 0.0500.511.5Fig. 2. x4 (solid line) and its estimation x̂4 using the super-twisting (dot line) and adaptive (dash line) observersunertainties aets the behavior of the observers. In thease of the adaptive observer the parameter unertaintydestroy ompletely the estimation of the messages. Never-theless, no matter what observe is used, in this ase, theestimation of the messages an be onsider unaeptable.Remark. 5.1. For the super-twisting observer we do somesimulations with x̃ = 0, that is without using a linearestimator. In the simulations, not shown here, we obtaineda bigger error in the estimation of x4 whih had a big eetin the error of the estimation of m2. It was due to thefat that using x̃ = 0, the variable to be estimated withthe super-twisting algorithm beame muh faster and forhaving an aeptable estimation the sampling step mustbe redue onsiderably.REFERENCESH. Hammouri, J. De Leon, `Observer synthesis for state-ane systems'Pro 29th IEEE Conferene on Deisionand Control, Honolulu, Hawaii. 1990.M. Ghanes, J. De Leon and A. Glumineau, `Observabil-ity Study and Observer-Based Interonneted Form forSensorless Indution Motor', Pro 45th IEEE Confer-ene on Deision and Control, San Diego, CA, Deem-ber, USA, 2006, 13-15 Deember, USA.0 2 4 6 8 10−200−150−100−50050100150200250Time [s]0 0.1 0.20204060Fig. 3. Message m1 (solid line) and its estimation m̂1 usingthe super-twisting (dot line) and adaptive (dash line)observers0 2 4 6 8 10−150−100−50050100150Time [s]0 0.2 0.402040Fig. 4. Message m2 (solid line) and its estimation m̂2 usingthe super-twisting (dot line) and adaptive (dash line)observers0 2 4 6 8 10−200−10001002000 2 4 6 8 10−2000020004000Time [s]Fig. 5. Message m1 (solid line) and its estimation m̂1 (dotline) for the system with 1% of unertainty in theparameters. Above with the super-twisting observer,below with the adaptive observer0 2 4 6 8 10−10001002000 2 4 6 8 10−2−1012x 104Time [s]Fig. 6. Message m2 (solid line) and its estimation m̂2 (dotline) for the system with 1% of unertainty in theparameters. Above with the super-twisting observer,below with the adaptive observerA. Levant, 1996, `Sliding order and sliding auray insliding mode ontrol', International Journal of Control,vol. 58, no. 6, pp. 1247-1263.A. Levant, 1998, `Robust exat dierentiation via slidingmode tehnique', International Journal of Control, vol.34, no. 3, pp. 379-384.J. Davila, L. Fridman and A. Levant, 2005, `Seond-OrderSliding-Mode Observer for Mehanial Systems', IEEETransations on Automati Control, vol. 50, no. 11, pp.1785-1789.F.J. Bejarano, L. Fridman and A. Poznyak, 2007, `Exatstate estimation for linear systems with unknown in-puts based on hierarhial super-twisting algorithm',International Journal of Robust and Nonlinear Control,Published on line [doi:10.1002/rn.1190℄.G.Y. Qi, S.Z. Du, G.R. Chen et al., 2005, `On a four-dimensional haoti system', Chaos, Solutions andFratals, vol. 23, no. 5, pp. 1671-1682.

Second order sliding mode and adaptive observer for synchronization of a chaotic system: a comparative study

HAL-Polytechnique

Second order sliding mode and adaptive observer forsynchronization of a chaotic system: a comparativestudyFrancisco J. Bejarano, Malek Ghanes, Jean-Pierre Barbot, Leonid FridmanTo cite this version:Francisco J. Bejarano, Malek Ghanes, Jean-Pierre Barbot, Leonid Fridman. Second ordersliding mode and adaptive observer for synchronization of a chaotic system: a comparativestudy. IFAC World Congress, Jul 2008, Se´oul, North Korea. 2008. <inria-00345027>HAL Id: inria-00345027https://hal.inria.fr/inria-00345027Submitted on 9 Dec 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Seond order sliding mode and adaptiveobserver for synhronization of a haotisystem: a omparative studyFraniso J. Bejarano∗Malek Ghanes∗Jean-Pierre Barbot∗Leonid Fridman∗∗∗Equipe Commande des Systèmes (ECS), ENSEA 6 Avenue duPoneau, 95014 Cergy-Pontoise, Frane, (e-mail: {Franiso.Bejarano,Malek.Ghanes, barbot}ensea.fr)∗∗National Autonomous University of Mexio, Engineering Faulty,Division of Eletrial Engineering, UNAM 04510, Méxio, D.F.,(e-mail: lfridmanservidor.unam.mx)Abstrat:1. INTRODUCTION2. PROBLEM STATEMENTConsider the following haoti systemx˙1 (t) = a(x2 (t)− x1 (t)) + x2 (t)x3 (t) +m1 (t)x˙2 (t) = b (x1 (t) + x2 (t))− x1 (t)x3 (t)x˙3 (t) = −cx3 (t)− ex4 (t) + x1 (t)x2 (t) +m1 (t)x˙4 (t) = fx3 (t)− dx4 (t) + x1 (t)x3 (t) +m2 (t)y1 (t) = x1 (t)y2 (t) = x2 (t)(1)where xi ∈ R (i = 1, 2, 3, 4) are the states of the system;m1 and m2 represent the information to be transmitted,whih for the observation problem are onsidered as theunknown inputs; y1 and y2 are the outputs of the system,these are the signals to be sent by a publi hannel.Assumption. 2.1. The signals m1 and m2 are assumedto be dierentiable, bounded and with the derivativebounded.The goal is to reonstrut the states x3 and x4 whih allowsto reonstrut the messages m1 and m2. For ahievingsuh a goal two approahes are tested, namely, an ob-server based on the super-twisting algorithm that basiallyallows to obtain information from the derivatives of theoutput and onsequently to reonstrut the states and themessages (unknown inputs), and an adaptive observer. Aomparison between these two methods will be disussed.It should be notie that a singularity appears in the pointx1 = 0, that is, from the outputs x1 and x2 is not possibleto know the states x3 and x4 and onsequently neither themessages m1 and m2.3. SUPER-TWISTING OBSERVERFirst in order to generate a new output, namely the vari-able x3, we use the super-twisting algorithm for designingthe observer for xˆ3 in the following way:x˙a,1 = b (x1 + x2) + v1v1 = xˆ1,3 + λ1 |s1|1/2 sign sxˆ1,3 = α1 sign ss1= x2 − xa,1(2)In this way, the derivative of s1takes the forms˙1 = −x1x3 − v1 (3)Choosing the gains λ1 ≥ (α1+M1)(1+θ)1−θ√2α1+M1and α1 >M1 ≥∣∣ ddt (x1x3)∣∣(0 < θ < 1), we get, aording to(Levant, 93), (Levant, 98), and (Davila, 05), the seondorder sliding motion, that is, s (t) = 0, s˙ (t) = 0 after somenite time T1. Notie that for s = 0, v1 = xˆ3; therefore,from (3), we getxˆ1,3 (t) ≡ −x1 (t) x3 (t) (4)From (4) we an reonstrut x3 (t) provided x1 (t) 6= 0.Thus, the observer for x3 is designed in the formxˆ3 (t) = −xˆ1,3 (t)x1 (t)if |x1 (t)| ≥ εxˆ3 (t− τ ) if |x1 (t)| < ε(5)where τ and ε are enough small onstants 1 . Thus, we getthe identityxˆ3 (t) ≡ x3 (t) for |x1 (t)| ≥ ε.Reonstrution of x4 Now, dening y¯3 (t) = x3 (t) asa new output, we an rewrite the dynami equations in(1) as a linear system with output injetion and unknowninputs, that is,1The onstant τ is hosen enough small but bigger than thesampling time used during the realization of the observer. Theonstant ε should be hosen suiently big to avoid the singularity,but also should be notie that any estimation in this zone an notbe onsidered as an aeptable estimation.x˙1x˙2x˙3x˙4=-a a 0 0b b 0 00 0 -c -e0 0 f -d︸ ︷︷ ︸Ax1x2x3x4︸ ︷︷ ︸x+y2y¯3-y1y¯3y1y2y1y¯3︸ ︷︷ ︸φ(y1,y2,y¯3)+1 00 01 00 1︸ ︷︷ ︸D[m1m2]︸ ︷︷ ︸w[y1y2y¯3]︸ ︷︷ ︸y¯=[1 0 0 00 1 0 00 0 1 0]︸ ︷︷ ︸xC(6)Now, let z be dened by the solution of the followingdierential equationz = Az + φ (y1, y2, y¯3)Thus, dening ez = x− z we obtain the dynami equationfor ezez (t) = Aez (t) +Dw (t)yz = Cez(7)Hene, the system (7) is basially a linear system withunknown inputs, just the sort of systems onsidered in(Bejarano, 07). Then, for the reonstrution of x4 we followin essene, but with a little modiation, the algorithmproposed in (Bejarano, 07). That is, the basi idea pro-posed in (Bejarano, 07) is to obtain an algebrai expressionof ez in terms of the output yz and its derivatives. Thus,we get the equalities2yz =Cezddt(CD)⊥yz = (CD)⊥CAezthus we get3ez (t) =[C(CD)⊥CA]+ [yz (t)(CD)⊥yz (t)](8)Sine we already know the rst 3 states, we solve (8) forez,4, the fourth state of ez. Thus,ez,4 =1e[e˙z,1 − e˙z,3 + a (ez,1 − ez,2)−cez,3 + ez,1ez,2 − ez,2ez,3](9)3.1 Realization of the observer using super-twistingTo redue the fast dynami and have a smaller gains inthe super-twisting algorithm, we design the following like-linear estimator x˜ whose dynamis is governed by thefollowing dierential equationx˜ =[−c −ef −d]︸ ︷︷ ︸A¯x˜+[y1y2y1y¯3]+ L (y¯3 − y˜3)y˜3 = [ 1 0 ]︸ ︷︷ ︸C¯x˜(10)The matrix L is hosen so that the eigenvalues of thematrix(A¯− LC¯) have negative real part. In this way wehave that the dynami equations for e¯ = x˜−[x3x4]are2 Y ⊥ is a full row rank matrix suh that Y ⊥Y = 0.3The matrix X+ is dened to be the pseudo-inverse of X. Thematrix onsidered in (8) belongs to sort of matries of full olumnrank. Hene, in suh a ase, X+ =(XTX)−1XT .e¯ (t) =(A¯− LC) e¯ (t) + w (t)where w is dened in (6). Hene we get some upper boundsfor the norm of e¯ and for the norm ofe¯ that is‖e¯ (t)‖ ≤ γ exp (−λt) ‖e¯ (0)‖+ µ ‖w (t)‖γ, λ, µ are positive onstantsThat is, with the estimator (10), e¯ is onstrained to stayedin a zone depending on the amplitude of m1 and m2; andthis zone an be made smaller by moving the eigenvaluesof(A¯− LC) more to the left in the left half-plane.Then, following the algebrai expression obtained in (9)we design the observer for x4 using the super-twistingalgorithm as followsx˙a,2 =1e[a (x2 − x1) + cxˆ3 − x1x2 + x2xˆ3] + x˜4 + v2v2 = v2,1 + λ2 |s2| sign s2, v˙2,1 = α sign s2s2 =1e(x1 − x3)− xa,2xˆ4 (t) ={x˜4 + v2,1 if |x1 (t)| ≥ εxˆ4 (t− τ ) if |x1 (t)| < εwhere x˜4 is the seond omponent of the vetor x˜ denedin (10). Then, the time derivative of s2 iss˙2 = x4 − x˜4 + v2Thus, for λ2 ≥ (α2+M2)(1+θ)1−θ√2α2+M2and α2 > M2 ≥∣∣ ddt (x4 − x˜4)∣∣(0 < θ < 1), after some nite time T2, weget s2 = 0 and s˙2 = 0; therefore,xˆ4 ≡ x4 for |x1 (t)| ≥ ε.3.2 Messages reonstrutionThe reonstrution of m1 is made in the following wayx˙a,3 = a (x2 − x1) + x2xˆ3 + v3v3 = v3,1 + λ3 |s3| sign s3v˙3,1 = α3 sign s3mˆ1 ={v3,1 if |x1 (t)| ≥ εmˆ1 (t− τ) if |x1 (t)| < εs3 = x1 − x3aThus, for α3 and λ3 satisfying λ3 ≥ (α3+M3)(1+θ)1−θ√2α3+M3and α3 > M3 ≥ |m˙1| (0 < θ < 1), after some nite time,we get the equalities s3 = 0, s˙3 = 0. Thus, we get theequalitymˆ1 ≡ m1, for |x1 (t)| ≥ εThe reonstrution of m2 is made in a similar way, that is,x˙a,4 = b (x1 + x2) + fx3 − dx4 + v4v4 = v4,1 + λ4 |s4| sign s4v˙4,1 = α4 sign s4mˆ2 ={v4,1 if |x1 (t)| ≥ εmˆ2 (t− τ) if |x1 (t)| < εs4 = x2 + x4 − xa,4Thus, taking into aount (1) and the derivative of xa,4,with the hoosing of α4 and λ4 satisfying the inequalitiesλ4 ≥ (α4+M4)(1+θ)1−θ√2α4+M4and α4 > M4 ≥ |m˙2| (0 <θ < 1) (see, (Levant, 93), (Levant, 98)), we get, after somenite time,mˆ2 ≡ m2 for |x1 (t)| ≥ εRemark. 3.1. At a glane, it seems that, during the esti-mation of the state x3, it is suient with to use x1 (t) 6= 0instead of |x1 (t)| ≥ ε, but the justiation is given inthe following lines. It is known that during the realiza-tion of the super-twisting algorithm s1 and s˙1 are notexatly zero, hene, instead of having (4) we really havethe equality xˆ1,3 (t) = −x1 (t)x3 (t) + ∆ (t), where ∆represents the estimation error and whih does not tendsto zero. Then, after dividing over x1, it yields the equalityx¯3 := − xˆ1,3x1 = x3 − ∆x1 , whih means that, when x1 isvery lose to zero, the error between x¯3 and x3 is equal toO (1/x1). Therefore, in a small neighborhood of x1 = 0,the error between x¯3 and x3 is extremely big. This justifythe struture of xˆ3.4. ADAPTIVE OBSERVERThe model of a haoti system (1) an be rewritten in thefollowing interonneted ompat form{X˙1 = A1(y2)X1 + g1(y,X1, X2) + φ1 (t)y1 = C1X1(11){X˙2 = A2(y1)X2 + g2(y,X2, X1) + φ2 (t)y2 = C2X2(12)where X1 = (x1, x3, x4, x5)Tis the state of the rstsubsystem with x5 := m2, X2 = (x2, x3, x6)Tis the stateof the seond subsystem with x6 := m1. y = [x1, x2]Tarethe output of the whole system, andA1(y2) =0 y2 0 00 0 −c 00 0 0 10 0 0 0 , A2(y1) =(0 −y1 00 0 10 0 0)g1(y,X2, X1) =a(y1 − y2) + x6−ex4 + y1y2 + x6fx3 − dx4 + y1x30g2(y,X2, X1) =(b(y1 + y2)−cx3 − ex4 + y1y20)φi (t) =[00m˙i], i = 1, 2C1 = ( 1 0 0 0 ) , C2 = ( 1 0 0 ) .Remark. 4.1. The hoie of the variables of eah subsys-tem has been onsidered in order to separate in one sub-system the message m1 and in the other one the messagem2. It is lear that other hoie an be onsidered inorder to represent these subsystems provided the neessaryonditions to design an adaptive are satised.Next, let us introdue an adaptive observer in order toestimate the system's state and the unknown inputs simul-taneously. It is based on interonnetion between severalsubsystems whih satises some required properties, suhthat the property of inputs persisteny ((Hammouri, 90),(Ghanes, 06)).At rst, let us introdue the following assumptions in orderto establish the results onerning the adaptive observerdesign (see more details in (Ghanes, 06)).Assumption. 4.1.1. The signals X1 and X2 are assumed bounded andto be regularly persistent ((Hammouri, 90), (Ghanes,06)) in order to guarantee the observability property ofsubsystems (11) and (12), respetively.2. A1(y2) and A2(y1) are uniformly bounded.3. g1(y,X1, X2) is globally Lipshitz with respet to X2and uniformly with respet to (y,X1).4. g2(y,X2, X1) is globally Lipshitz with respet X1 anduniformly with respet to (y,X2).5. The unknown funtions m˙i(t) (i = 1, 2) are assumed tobe bounded.Then, an adaptive observer for interonneted subsystems(11) and (12) estimating the state and unknown parame-ters is given byZ˙1 = A1(y2)Z1 + g1(y, Z1, Z2) + S−11 CT1 (y1 − yˆ1)S˙1 = −θ1S1 −AT1 (y2)S1 − S1A1(y2) + CT1 C1yˆ1 = C1Z1(13)Z˙2 = A2(y1)Z2 + g2(y, Z2, Z1) + S−12 CT2 (y2 − yˆ2)S˙2 = −θ2S2 −AT2 (y1)S2 − S2A2(y1) + CT2 C2yˆ2 = C2Z2(14)where Z1 = (xˆ1, xˆ3, xˆ4, xˆ5)T; Z2 = (xˆ2, xˆ3, xˆ6)T Si =STi > 0, i = 1, 2. Note that S−11 CT1 and S−12 CT2 are thegains of the observers (13) and (14), respetively.Remark. 4.2. It is worth notiing that ‖S1‖ and ‖S2‖ arebounded for θ1 and θ2 large enough due to the persistenyof input onsidered in assumption 4.1.Now, in order to guarantee the onvergene of the pro-posed observer, suient onditions are established in thefollowing result. Denote the estimation errors:ǫ1 = X1 − Z1 and ǫ2 = X2 − Z2whose dynamis are given by{ǫ˙1 = [A1(y2)− S−11 CT1 C1]ǫ1+g1(y,X1, X2)− g1(y, Z1, Z2) + φ1 (t) (15){ǫ˙2 = [A2(y1)− S−12 CT2 C2]ǫ2+g2(y,X2, X1)− g2(y, Z2, Z1) + φ2 (t) . (16)The values of θ1 and θ2 are hosen to satisfy the inequali-tiesδ1 = (θ1 − Γη) > 0, δ2 = (θ2 − Γη) > 0 (17)where Γ = µ˜1 + µ˜2, with µ˜i =µi√λmin(S1)√λmin(S2),i = 1, 2; and µ1 = k1k2, µ2 = k3k4, η ∈]0, 1[.The parameters k1, k2, k3, k4 are positive onstants andλmin(S1), λmin(S2) are the minimal eigenvalues of S1 andS2 respetively.Lemma 4.1. Consider the system (13)-(14) and that as-sumption 4.1 holds. Then, the system (13)-(14) is a prati-al exponential observer for system (11)-(12) for θ1 and θ2satisfying the inequalities (17). Furthermore, the observeronverges arbitrarily fast with a onvergene rate xed bya parameter δ, δ = min(δ1, δ2).Sketh of proof 4.1. Consider the following Lyapunov fun-tion andidate:Vo = V1 + V2where V1 = ǫT1 S1ǫ1 and V2 = ǫT2 S2ǫ2.From assumption 4.1, the following inequalities hold‖S1‖ ≤ k1;‖{g1(y,X1, X2)− g1(y, Z1, Z2)}‖ ≤ k2 ‖ǫ2‖;‖S2‖ ≤ k3;‖{g2(y,X2, X1)− g2(y, Z2, Z1)}‖ ≤ k4 ‖ǫ1‖.‖φ1‖ ≤ k5.‖φ2‖ ≤ k6.Computing the time derivative of Vo, , by using the aboveinequalities it follows thatV˙o ≤−θ1ǫT1 S1ǫ1 + 2µ1 ‖ǫ1‖ ‖ǫ2‖+ ‖φ1‖ (18)− θ2ǫT2 S2ǫ2 + 2µ2 ‖ǫ2‖ ‖ǫ1‖+ ‖φ2‖Now, onsider that the following inequalities are satisedλmin(Si) ‖ǫi‖2 ≤ ‖ǫi‖2Si ≤ λmax(Si) ‖ǫi‖2, i = 1, 2.By writing (18) in terms of funtions V1 and V2, it followsthatV˙o ≤ −θ1V1 − θ2V2 + 2(µ˜1 + µ˜2)√V1√V2 + k5 + k6where the parameters µ˜1, µ˜2, are dened just before lemma4.1.Next, by using the following inequality√V1√V2 ≤ υ2V1 +12υV2, ∀υ ∈]0, 1[, one getV˙o ≤ −(θ1 − Γ)V1 − (θ2 − Γυ)V2 + k5 + k6.where Γ is dened just before lemma 4.1.By taking δ and r suh that δ = min(δ1, δ2) and r = k5+k6one hasV˙o ≤ −δVo + r. (19)Finally, by hoosing θ1 and θ2 suh that the inequalities(17) are satised and suiently large, the inequality (19)shows that arbitrarily bounded perturbation will not resultin large error estimation deviations. This ends the proof.5. NUMERICAL EXAMPLE AND DISCUSSIONSFor the system (1) we use the parameters a = 42.5, b = 24,c = 13, d = 20, e = 50, f = 40. The parameters used forthe super-twisting observer are α1 = 7×107, λ1 = 7×103,α2 = 300, λ2 = 100, α3 = 1000, λ3 = 200, α4 = 600,λ4 = 300. For the adaptive observer the parameters usedare θ1 = θ2 = 400.Figures 1 and 2 show the trajetories of the states x3 andx4 as well as the ones of xˆ3 and xˆ4 for both the super-twisting and the adaptive observers. We an see in thegures that the trajetories of the super-twisting observeronverge muh faster than the ones of the adaptive ob-server.Figures 3 and 4 show the messages m1 and m2 togetherwith their estimations mˆ1 and mˆ2, respetively. In thisgures we note that the singularity in the point x1 = 0aets more the estimation of the messages made withthe super-twisting than the one made with the adaptiveobserver. This is lear due to the explanation of Remark3.1 and the fat that, in a ball of radio ε and enter inx1 = 0, it is not done a estimation neither of the statesnor of the messages.In order to test both observers with respet to parametersunertainties, we introdue a variation of 1% in the nomi-nal parameters. Figures 5 and 6 show how the parameters0 2 4 6 8 10−400−2000200400600800Time [s]0 0.02 0.04 0.06 0.08−101Fig. 1. x3 (solid line) and its estimation xˆ3 using the super-twisting (dot line) and adaptive (dash line) observers0 2 4 6 8 10−400−2000200400600800Time [s]0 0.0500.511.5Fig. 2. x4 (solid line) and its estimation xˆ4 using the super-twisting (dot line) and adaptive (dash line) observersunertainties aets the behavior of the observers. In thease of the adaptive observer the parameter unertaintydestroy ompletely the estimation of the messages. Never-theless, no matter what observe is used, in this ase, theestimation of the messages an be onsider unaeptable.Remark. 5.1. For the super-twisting observer we do somesimulations with x˜ = 0, that is without using a linearestimator. In the simulations, not shown here, we obtaineda bigger error in the estimation of x4 whih had a big eetin the error of the estimation of m2. It was due to thefat that using x˜ = 0, the variable to be estimated withthe super-twisting algorithm beame muh faster and forhaving an aeptable estimation the sampling step mustbe redue onsiderably.REFERENCESH. Hammouri, J. De Leon, `Observer synthesis for state-ane systems'Pro 29th IEEE Conferene on Deisionand Control, Honolulu, Hawaii. 1990.M. Ghanes, J. De Leon and A. Glumineau, `Observabil-ity Study and Observer-Based Interonneted Form forSensorless Indution Motor', Pro 45th IEEE Confer-ene on Deision and Control, San Diego, CA, Deem-ber, USA, 2006, 13-15 Deember, USA.0 2 4 6 8 10−200−150−100−50050100150200250Time [s]0 0.1 0.20204060Fig. 3. Messagem1 (solid line) and its estimation mˆ1 usingthe super-twisting (dot line) and adaptive (dash line)observers0 2 4 6 8 10−150−100−50050100150Time [s]0 0.2 0.402040Fig. 4. Messagem2 (solid line) and its estimation mˆ2 usingthe super-twisting (dot line) and adaptive (dash line)observers0 2 4 6 8 10−200−10001002000 2 4 6 8 10−2000020004000Time [s]Fig. 5. Message m1 (solid line) and its estimation mˆ1 (dotline) for the system with 1% of unertainty in theparameters. Above with the super-twisting observer,below with the adaptive observer0 2 4 6 8 10−10001002000 2 4 6 8 10−2−1012x 104Time [s]Fig. 6. Message m2 (solid line) and its estimation mˆ2 (dotline) for the system with 1% of unertainty in theparameters. Above with the super-twisting observer,below with the adaptive observerA. Levant, 1996, `Sliding order and sliding auray insliding mode ontrol', International Journal of Control,vol. 58, no. 6, pp. 1247-1263.A. Levant, 1998, `Robust exat dierentiation via slidingmode tehnique', International Journal of Control, vol.34, no. 3, pp. 379-384.J. Davila, L. Fridman and A. Levant, 2005, `Seond-OrderSliding-Mode Observer for Mehanial Systems', IEEETransations on Automati Control, vol. 50, no. 11, pp.1785-1789.F.J. Bejarano, L. Fridman and A. Poznyak, 2007, `Exatstate estimation for linear systems with unknown in-puts based on hierarhial super-twisting algorithm',International Journal of Robust and Nonlinear Control,Published on line [doi:10.1002/rn.1190℄.G.Y. Qi, S.Z. Du, G.R. Chen et al., 2005, `On a four-dimensional haoti system', Chaos, Solutions andFratals, vol. 23, no. 5, pp. 1671-1682.

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Second order sliding mode and adaptive observer for synchronization of a chaotic system: a comparative study

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