Chaotic Systems Part 8 potx

Chaotic Systems 164 ,1 ,1 , , [ ] (.) (.) [ 1] (.) [ ] F j G j TT T jjjj jdj G j d xk d F G uk d G uk ⎡ ⎤ Θ ⎢ ⎥ Θ ⎢ ⎥ ⎡⎤ += +− ⎣⎦ ⎢ ⎥ ⎢ ⎥ Θ ⎢ ⎥ ⎣ ⎦ …  (10) Now define, ][][ dkxk jj += η (11) ,1 , [ ] (.) (.) [ 1] (.) [ ] T TT T jjjj jdj kF Gukd Guk ⎡ ⎤ Φ= +− ⎣ ⎦ … (12) ()( ) ( ) ,1 , T TT T FG G jjj jd ⎡ ⎤ Θ= Θ Θ Θ ⎢ ⎥ ⎣ ⎦ … (13) Using the above definitions Eq. (11) can be written as: j T jj kk ΘΦ= ][][ η (14) ˆ [] j kΘ denotes the estimate of j Θ and it is defined as: ()( )( ) ,1 , ˆˆˆ ˆ [] [] [] [] T TT T FG G jj j jd kk k k ⎡ ⎤ Θ=Θ Θ Θ ⎢ ⎥ ⎣ ⎦ … (15) The error vector can be written as: ]1[ ˆ ][][][ −ΘΦ−= kkkk j T jjj ηε (16) To obtain the estimated parameters, ˆ [] j kΘ , the least squares technique is used. Consider the following objective functions: 2 1 )][ ˆ ][][( ∑ = ΘΦ−= k n j T jjk knnJ η (17) By differentiating Eq. (17) with respect to ˆ Θ and set it to zero we have: [][] ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ΦΦ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Φ Φ ΦΦ=Θ − ][ ]1[ ][]1[ ][ ]1[ ][]1[][ ˆ 1 k k k kk j j jj T j T j jjj η η  (18) Let: [] 1 ][ ]1[ ][]1[][ − ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Φ Φ ΦΦ= k kkP T j T j jjj  (19) Adaptive Control of Chaos 165 Substituting Eq. (19) into Eq. (18) yields: () ][][][]1[ ˆ ]1[][ ][][][][][ ][][][][ ˆ 1 1 1 1 kkkPkkPkP kknnkP nnkPk jjjjjj jj k n jjj k n jjjj η ηη η Φ+−Θ−= Φ+Φ= Φ=Θ − − = = ∑ ∑ (20) Using Eq. (19) one can write: ][][][]1[ 11 kkkPkP T jjjj ΦΦ−=− −− (21) Substituting Eq. (21) into Eq. (20) results in: ][][][]1[ ˆ ][ ˆ kkkPkk jjjjj ε Φ+−Θ=Θ (22) After some matrix manipulations, Eqs. (21) and (22) can be rewritten as: [1][][][1] [] [ 1] 1[][1][] T jjjj jj T jj j Pk k kPk Pk Pk kP k k − ΦΦ − =−− +Φ − Φ (23) [1][][] ˆˆ [] [ 1] 1[][1][] jjj jj T jj j Pk k k kk kP k k ε − Φ Θ=Θ−+ +Φ − Φ (24) The above recursive equations can be solved using a positive definite initial matrix for [0] j P and an arbitrary initial vector for ˆ [0] j Θ . The identification method given by Eqs. (23) and (24) implies that: ∞→ΘΦ→−ΘΦ kaskkk j T jj T j ][]1[ ˆ ][ (25) To show the property (25), note Eq. (19) implies that []Pk is a positive definite matrix. Define: jjj kk Θ−Θ= ][ ˆ ][ δ (26) Consider the following Lyapunov function: ][][][][ 1 kkPkkV jj T jj δδ − = (27) Using Eq. (24), we have: ]1[]1[][][ 1 −−= − kkPkPk jjjj δδ (28) [] j VkΔ can be obtained as follows: Chaotic Systems 166 ][]1[][1 ][ ][]1[][1 ])1[ ˆ ][][( )]1[ ˆ ](1[])1[ ˆ ][ ˆ ( ]1[]1[]1[]1[]1[][ ]1[]1[]1[][][][]1[][ 22 1 11 11 kkPk k kkPk kkk kkPkk kkPkkkPk kkPkkkPkkVkV jj T j j jj T j j T jj jjj T j T j j j T j j j T j j j T j j j T jjj Φ−Φ+ −= Φ−Φ+ −ΘΦ− −= Θ−−Θ−−Θ−Θ= −−−−−−= −−−−=−− − −− − − εη δδδδ δδδδ (29) Equation (29) shows that [ ] j Vkis a decreasing sequence, and consequently: ∞<−= Φ−Φ+ ∑ = ][]0[ ][]1[][1 ][ 1 2 nVV kkPk k jj n k jj T j j ε (30) Hence, 0][limor0 ][]1[][1 ][ lim 2 == Φ−Φ+ ∞→∞→ k kkPk k j k jj T j j k ε ε (31) and consequently: ∞→ΘΦ=→−ΘΦ kaskkkk j T jjj T j ,][][]1[ ˆ ][ η (32) 2.3 Indirect adaptive control of chaos Assume that ( ) [0], [1], , [ 1] F FF F jj j j xxx xd=−is the fixed point of Eq. (9) when [] 0uk ≡ , i.e. ( ) [0] [ ] [0],0, ,0 F FTF F jj j j xxdFx = =Θ or kxdkx F j F j =+ (33) The main goal is stabilizing the fixed point F j x . Let, () () () [] () () () ,2 ,2 ,, 1 [] [],[], ,[ 2] [ ], [ ], , [ 3] 2 [] [] [][] [ ] TF jj j TF jj d TGF F jd jd j j j j j kFxkukukd Gxkuk ukd ukd G xk xkd xkdjxkdj ρ λ = =− + − Θ −+−Θ+−− −Θ++− +−−+− ∑  (34) () () () [] () () () ,2 ,2 ,, 1 ˆ ˆ [] [],[], ,[ 2] [ 1] [ ], [ ], , [ 3] [ 1] 2 [] [ 1] [ ] [ ] [ ] [ ] TF jj j TF jj d TG F F jd jd j j j j j k F xk uk uk d k G xkuk ukd k ukd G xk k xkd xkd j xkd j ρ λ = =− + − Θ − −+−Θ−+−− −Θ−++− +−−+− ∑  (35) Adaptive Control of Chaos 167 where i λ ’s are chosen such that all roots of the following polynomial lie inside the unit circle. 1 1 0 dd d zz λλ − + ++ = (36) Assume that ( ) ,1 ,1 [ ], [ ], , [ 2] [ 1] 0 TG jj Gxkuk ukd k + −Θ −≠and consider the conrol law as given below: () ,1 ,1 ˆ [] [1] [ ], [ ], , [ 2] [ 1] j j TG jj k uk d Gxkuk ukd k ρ +−= + −Θ − (37) To show that the above controller can stabilize the fixed point of Eq. (9), note that from Eq. (25) we have: ,, 1 ,, 1 ˆˆ lim (.) [ 1] (.) [ 1] [ ] (.) (.) [ ] d TF T G j j ji ji j k d TF T G j j ji ji j Fk G kukdi FGukdi = →∞ = Θ−+ Θ − +− = Θ+ Θ + − ∑ ∑ (38) One can write the above equation in the following form: ( ) ( ) ,, , 1 ˆˆ [ ] (.) [ 1] (.) [ 1] [ ] lim [ ] 0 d TF F T G G jjjjjijiji i j k kF k G k ukdi k ε ε = →∞ = Θ−−Θ+ Θ −−Θ +− = ∑ (39) From Eqs. (34), (35) and (39) one can obtain: ( ) ,1 ,1 ,1 ˆ ˆ [] [] [] (.) [ 1] [ 1] TG G jjjjj j kkkG k ukd ρρε − =+ Θ−−Θ++− (40) Using Eqs. (9), (34), (35) and (37) the controlled system can be written as: ( ) () () 1 ,1 ,1 ,1 ,1 [][][] [][] [ ], [ ], , [ 2] ˆ [] ˆ [ ], [ ], , [ 2] [ 1] d FF jjijj i TG jj TG jj x kd k xkd xkdi xkdi Gxkuk ukd k Gxkuk ukd k ρλ ρ = + =− + + − + − − + − +− Θ + +− Θ − ∑ (41) Using Eq. (40) in Eq. (41), we have: ( ) () () 1 ,1 ,1 ,1 ,1 ˆ [ ] [] [] [ ] [ ] [ ] [ ], [ ], , [ 2] ˆ [] ˆ [ ], [ ], , [ 2] [ 1] d FF jjjijj i TG jj TG jj x kd k k xkd xkdi xkdi Gxkuk ukd k Gxkuk ukd k ρε λ ρ = + =− + + + − + − − + − +− Θ + +− Θ − ∑ (42) After some manipulations we get: ][)][][(][][ 1 kidkxidkxdkxdkx j d i F jji F jj ελ +−+−−+−+=+ ∑ = (43) or, ][])[][(])[][( 1 kidkxidkxdkxdkx j d i F jji F jj ελ =−+−−+++−+ ∑ = (44) Chaotic Systems 168 Define, ],1[]1[][,],1[]1[][],[][][ 21 −+−−+=+−+=−= dkxdkxkykxkxkykxkxky F jjd F jj F jj … (45) Eq. (44) can be re-written as: ][ 1 0 0 ][ ][ ][ 0 0 0 ]1[ ]1[ ]1[ 2 1 21 )1()1(2 1 k ky ky ky I ky ky ky j dd dd d ε λλλ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ −−− = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + + + −×−     (46) Let: ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ −−− = −×− 1 0 0 , 0 0 0 21 )1()1(   H I G d dd λλλ (47) Equation (45) can be written as: ][][]1[ kHkGYkY j ε + = + (48) Taking z-transform from both sides of Eq. (48) we get: )()()0()()( 11 zHGzIzYGzIzY j ε −− −+−= (49) Taking inverse yields: )]()[(]0[)( 11 zHGzIzYGkY j k ε −− −+= (50) Note that lim [0] 0 k GY = as k →∞(because all eigen-values of G lie inside the unit circle), besides lim [ ] 0 j k ε = as k →∞ and G is a stable matrix therefore: 0)]()[(lim 11 =− −− →∞ zHGzIz j k ε (51) Consequently we have: ]1[]1[,],[][0][lim −+→−+→⇒= ∞→ dkxdkxkxkxkY F jj F jj k … (52) From Eq. (52) the stability of the proposed controller is established. Remark 1 In practice, control law (37) works well but theoretically there is the remote possibility of division by zero in calculating [.]u . This can be easily avoided. For example [.]u can be calculated as follows: Adaptive Control of Chaos 169 ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = ≠ =−+ 0][ ][ ˆ 0][, ][ ][ ˆ ]1[ k m k k k k dku j j j j j j μ ρ μ μ ρ ε (53) where ( ) ,1 ,1 ˆ [ ] [ ], [ ], , [ 2] [ 1] TG jj j kGxkuk ukd k μ = +− Θ − (54) and 0m ε > is a small positive real number. Remark 2 For the time varying systems, least squares algorithm with variable forgetting factor can be used. The corresponding updating rule is given below (Fortescue et al. 1981): ][]1[][][ ][][]1[ ]1[ ˆ ][ ˆ ][]1[][][ ]1[][][]1[ ]1[ ][ 1 ][ kkPkk kkkP kk kkPkk kPkkkP kP k kP jj T jj jjj jj jj T jj T jjj jj Φ−Φ+ Φ− +−Θ=Θ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Φ−Φ+ −ΦΦ− −−= ν ε ν ν (55) where, ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ + −= min 2 2 , ][1 ][ 1max][ λ ε ε k k kv j j j (56) and min 01 λ << usually set to 0.95. 2.4. Simulation results In this section through simulation, the performance of the proposed adaptive controller is evaluated. Example 1: Consider the logistic map given below: ( ) [ 1] []1 [] [] x kxkxkuk μ += − + (57) For 3.567 μ ≥ and [ ] 0uk = the behavior of the system is chaotic. In this example stabilization of the 2-cycle fixed point of the following logistic map is considered. In this case the governing equation is: ]1[][][][2][][2][ ][][2][)(][]2[ 2222 43332322 ++−+−+ −++−=+ kukukukxkukxku kxkxkxkxkx μμμμ μμμμμ (58) and the system fixed points for 3.6 μ = are [1] 0.8696, [2] 0.4081 FF xx = = and for 3.9 μ = are [1] 0.8974, [2] 0.3590 FF xx== . Equation (58) can be written in the following form: Chaotic Systems 170 0 50 100 150 200 250 300 0 0.5 1 x[k] 0 50 100 150 200 250 300 -0.1 0 0.1 u[k] 0 50 100 150 200 250 300 -2 -1 0 1 k ε [k] Fig. 1. Closed-loop response of the logistic map (59), for stabilizing the 2-cycle fixed point, when the parameter μ changes from 3.6 μ = to 3.9 μ = at 150k = . 0 100 200 300 0 10 20 a 1 0 100 200 300 -100 -50 0 50 a 4 0 100 200 300 -100 0 100 200 b 2 0 100 200 300 -100 -50 0 50 a 2 0 100 200 300 -20 -10 0 10 a 5 0 100 200 300 -10 0 10 b 3 0 100 200 300 -50 0 50 k a 3 0 100 200 300 -50 0 50 k b 1 0 100 200 300 -5 0 5 10 k c Fig. 2. Parameter estimates for the logistic map (59), when the parameter μ changes from 3.6 μ = to 3.9 μ = at 150k = . Adaptive Control of Chaos 171 [][] ]1[][][][1][][][][][]2[ 3 2 1 2 5 4 3 2 1 2432 ++ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ =+ kcuku b b b kxkx a a a a a kukxkxkxkxkx (59) It is assumed that the system parameter μ changes from 3.6 μ = to 3.9 μ = at 150k = . The results are shown in Figs. (1) and (2). As can be seen the 2-cycle fixed point of the system is stabilized and the tracking error tends to zero, although the parameter estimates are not converged to their actual values. Example 2: For the second example, the Henon map is considered, ][][]1[ ][][][1]1[ 212 12 2 11 kukbxkx kukxkaxkx +=+ ++−=+ (60) where for 1.4a = ,0.3b = and 12 [] [] 0uk uk = = the behavior of the system is chaotic. The 1- cycle fixed point is regarded for stabilization. Equation (60) can be written in the following form: ( ) ( ) ()() 1 112 1112 11 2 212 2212 22 [ 1] [], [] [], [] [] [ 1] [], [] [], [] [] TfTg TfTg x kfxkxk gxkxkuk x kfxkxk gxkxkuk += Θ+ Θ += Θ+ Θ (61) where, ( ) ( ) () () 2 11 2 2 1 21 2 11 2 21 2 [], [] 1 [] [] , [], [] 1 [], [] 1, [], [] 1 T fxkxk xk xk fxkxk Gxkxk Gxkxk ⎡⎤ = = ⎣⎦ = = (62) and 11,11,21,3 11 ,, ,1,2 T ffff ff gg ii i θθθ θ θ ⎡⎤ Θ= Θ= Θ= = ⎣ ⎦ (63) Again the 1-cycle fixed point is obtained, using numerical methods, 12 ( 0.6314, 0.1894) FF xx== . Figures (3) and (4) show the results of applying the proposed adaptive controller to the Henon map. It is observed that the 1-cycle fixed point of the system is stabilized. It must be noted that if in the system model the exact functionality is not know, a more general form with additional parameters can be considered. For example system (61) can be modeled as follows: ( ) ( ) 12 12 22 12 1 122 [], [] [], [] 1[][] [][][][] ii T fxkxk gxkxk x kxkxkxkxkxk = ⎡ ⎤ = ⎣ ⎦ (64) Chaotic Systems 172 0 50 100 -2 0 2 x 1 [k] 0 50 100 -0.1 0 0.1 u 1 [k] 0 50 100 -1 0 1 k ε 1 [k] 0 50 100 -0.5 0 0.5 x 2 [k] 0 50 100 -0.1 0 0.1 u 2 [k] 0 50 100 -1 -0.5 0 0.5 k ε 2 [k] Fig. 3. Closed-loop response of the Henon map (61), for stabilizing the 1-cycle fixed point. 0 50 100 0 0.5 1 1.5 θ f 1,1 0 50 100 0 0.5 1 1.5 θ f 1,2 0 50 100 -2 -1 0 1 θ f 1,3 k 0 50 100 0 0.1 0.2 θ f 2,1 0 50 100 0 0.2 0.4 θ f 2,2 0 50 100 -0.2 0 0.2 θ f 2,3 k Fig. 4. Parameter estimates of the Henon map (61). Adaptive Control of Chaos 173 and ,1 ,2 ,3 ,4 ,5 ,6 ,1 ,2 ,3 ,4 ,5 ,6 T f ffffff i iiiiii T g gggggg i iiiiii θθθθθθ θθθθθθ ⎡ ⎤ Θ= ⎣ ⎦ ⎡ ⎤ Θ= ⎣ ⎦ (65) As it is illustrated in Fig. (5), the 1-cycle fixed point is stabilized successfully. It shows that in cases where the system dynamics is not known completely, the proposed method can be applied successfully using the over-parameterized model. 0 50 100 -2 0 2 x 1 [k] 0 50 100 -0.02 0 0.02 u 1 [k] 0 50 100 -1 0 1 k ε 1 [k] 0 50 100 -0.5 0 0.5 x 2 [k] 0 50 100 -0.02 0 0.02 u 2 [k] 0 50 100 -0.1 0 0.1 k ε 2 [k] Fig. 5. Closed-loop response of the Henon map (61), for stabilizing the 1-cycle fixed point, using over-parameterized model. 3. Controlling a class of continuous-time chaotic systems In this section a direct adaptive control scheme for controlling chaos in a class of continuous-time dynamical system is presented. The method is based on the proposed adaptive technique by Salarieh and Alasty (2008) in which the unstable periodic orbits of a stochastic chaotic system with unknown parameters are stabilized via adaptive control. The method is simplified and applied to a non-stochastic chaotic system. 3.1 Problem statement It is assumed that the dynamics of the under study chaotic system is given by: () () () x fx Fx Gxu θ = ++  (66) [...]... −3 (87 ) Applying the control and adaptation laws given by Eqs (85 ) and (86 ) to the system (79) results in the following tracking error bound: E ≤ 2ε f λmax (P ) λmin (Q ) (88 ) in which λ (.) is the eigen-value and is the Euclidian norm ■ Proof: To achieve the control and adaptation laws given by Eq (85 ) and (86 ) consider the following Lyapunov function: V = E T PE + 1 Θ − Θo 2 2 (89 ) 180 Chaotic Systems. .. Sharhrokhi, M (2007) Indirect adaptive control of discrete chaotic systems, Chaos Solitons and Fractals, Vol 34, No 4, 1 188 -1201 Salarieh, H & Alasty, A (20 08) Stabilizing unstable fixed points of chaotic maps via minimum entropy control, Chaos Solitons and Fractals, Vol 37, No 3, 763-769 Salarieh, H & Alasty, A (20 08) Delayed feedback control of chaotic spinning disk via minimum entropy approach, Nonlinear... Applications, Vol 10, No 5, 286 4- 287 2 Sastry, S & Bodson, M (1994) Adaptive control: Stability, Convergence and Robustness, Prentice Hall, ISBN 0-13-004326-5 Schmelcher, P & Diakonos, F.K (1997) Detecting unstable periodic orbits of chaotic dynamical systems Physics Review Letters, Vol 78, No 25, 4733-4736 Tian, Y.-C & Gao, F (19 98) Adaptive control of chaotic continuous-time systems with delay, Physica... Sets and Systems, Vol 139, 81 -93 Hua, C & Guan, X (2004) Adaptive control for chaotic systems, Chaos Solitons and Fractals Vol 22, 55-60 Kiss, I.Z & Gaspar, V., and Hudson, J.L (2000) Experiments on Synchronization and Control of Chaos on Coupled Electrochemical Oscillators, J Phys Chem B, Vol 104, 7554-7560 Konishi, K., Hirai, M & Kokame, H (19 98) Sliding mode control for a class of chaotic systems, ... control scheme for chaotic systems with unknown dynamics is presented 4.1 Problem statement It is assumed that the dynamics of the chaotic system is given by: x (n ) = f (X ) + g (X )u (a) 20 8 6 10 4 x3 15 x3 (79) 5 2 0 10 0 10 x 0 2 -10 -10 -5 0 5 10 x x 1 2 0 -10 (a) -5 -10 0 x 5 10 1 (b) Fig 9 (a) Chaotic attractor of the Rossler system, (b) the UPO with the period of T = 5 .88 2 and the initial... Salarieh, H & Alasty, A (20 08) Chaos control in AFM systems using nonlinear delayed feedback via sliding mode control, Nonlinear Analysis: Hybrid Systems, Vol 2, No 3, 993-1001 Adaptive Control of Chaos 183 Astrom, K.J & Wittenmark, B (1994) Adaptive Control, 2nd edition, Prentice Hall, 1994 Bonakdar, M., Samadi, M., Salarieh, H & Alasty, A (20 08) Stabilizing periodic orbits of chaotic systems using fuzzy... Solitons and Fractals, Vol 36, No 3, 682 -693 Chen, L., Chen, G & Lee, Y.-W (1999) Fuzzy modeling and adaptive control of uncertain chaotic systems, Information Sciences, Vol 121, 27-37 Feng, G & Chen, G (2005) Adaptive control of discrete time chaotic systems: a fuzzy control approach, Chaos Solitons Fractals, Vol 23, 459-467 Fortescue, T.R., Kershenbaum, L.S & Ydstie B.E (1 981 ) Implementation of self-tuning... 5 .88 2 and the initial conditions of: x1 (0) = 0 , x2 (0) = 6. 089 and x3 (0) = 1.301 (Salarieh and Alasty, 20 08) 15 10 10 5 x2 x1 5 0 0 -5 -5 -10 -10 0 5 10 15 0 5 time 10 15 time 8 10 4 0 x3 x3 6 2 -10 10 0 -2 0 5 10 time 15 5 0 x2 -5 -10 0 x1 10 Fig 10 The trajectory of the Rossler system after applying the adaptive control law (69) 1 78 Chaotic Systems 5 50 200 0 150 -50 100 0 -5 -15 3 u u u 1 2 -10 -100... feedback Physics Letters A, Vol 170, No 6, 421-4 28 Pyragas, K., Pyragas, V., Kiss, I.Z & Hudson, J.L (2004) Adaptive control of unknown unstable steady states of dynamical systems, Physical Review E, Vol 70, 026215 1-12  184 Chaotic Systems Pyragas, K (2006), Delayed feedback control of chaos, Philosophical Transactions of The Royal Society A Vol 364( 184 6), 2309-2334 Ramesh, M & Narayanan, S (2001)... in simulation study Fig 12 2π periodic solution of the Duffing system (Layeghi et al 20 08) The variation ranges of x1 and x2 are partitioned into 3 fuzzy sets with Gaussian membership functions, μ ( x ) = exp ⎡ − ⎢ ⎣ − ( xσ x ) 2⎤ ⎥ ⎦ , whose centers are at {−2, −1,0,1, 2} The range 182 Chaotic Systems of t is partitioned to 4 fuzzy sets with Gaussian membership functions and centers at {0, 2, 4,6} . adaptation laws given by Eq. (85 ) and (86 ) consider the following Lyapunov function: 2 1 2 T o VEPE=+Θ−Θ (89 ) Chaotic Systems 180 Differentiating both sides of Eq. (89 ) yields: () T TT o VEPEEPE = ++Θ−ΘΘ . -10 -5 0 5 10 -10 0 10 0 2 4 6 8 x 1 x 2 x 3 (a) (b) Fig. 9. (a) Chaotic attractor of the Rossler system, (b) the UPO with the period of 5 .88 2 T = and the initial conditions of: 1 (0) 0x = , 2 (0) 6. 089 x = and 3 (0). − ⎣ ⎦      (87 ) Applying the control and adaptation laws given by Eqs. (85 ) and (86 ) to the system (79) results in the following tracking error bound: () max min 2() f P E Q ε λ λ ≤ (88 ) in