Observer-based synchronization for a class of unknown chaos systems with adaptive fuzzy-neural network

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(1)IEICE TRANS. FUNDAMENTALS,. VOL.E91-A,. NO.7 JULY 2008 1797. PAPER. Observer-Based Systems. Synchronization. with. Adaptive. for a Class. Fuzzy-Neural Bing-Fei. WU•õa),. Member,. SUMMARY This investigationappliesthe adaptivefuzzy-neuralobserver (AFNO)to synchronizea class of unknown chaoticsystems via scalar transmittingsignal only. The proposedmethod can be used in synchronizationif nonlinearchaoticsystemscan be transformedinto the canonicalformof Lur'e systemtype by the differentialgeometricmethod. In this approach,the adaptivefuzzy-neuralnetwork (FNN)in AFNO is adoptedon line to model the nonlinearterm in the transmitter.Additionally,themaster'sunknownstatescan be reconstructedfromonetransmitted state using observerdesignin the slave end. Synchronization is achieved when all statesare observed.Theutilizedschemecan adaptivelyestimate the transmitterstatesonline,evenif the transmitteris changedinto another chaos system.On the otherhand,the robustnessof AFNOcan be guaranteedwith respectto the modelingerror,andexternalboundeddisturbance. SimulationresultsconfirmthattheAFNOdesignis validforthe application of chaossynchronization. keywords: chaos,fuzzy-neuralnetwork(FNN),adaptivefuzzy-neuralobserver(AFNO),synchronization, robust 1.. Introduction. The synchronization of chaotic systems has been extensively studied and given its potential application to security communications. Synchronization means that the master and slave have identical states as time goes to infinity. Pecora and Carroll first considered the synchronization of chaotic systems [18], in which the drive-response concept is introduced to achieve synchronization by a scalar transmitted signal. Perfectly identical parameters cannot be achieved in real applications. Therefore, the nonlinear robust control [22], [23] concept is employed to chaos, synchronization with previous known states within the margin of synchronization error. An adaptive recurrent neural controller can be utilized to synchronize with respect to unknown systems [19],[20]. However, all statesshould be measurable with thisalgorithm. In contrast,the nonlinear observer is designed to synchronize chaotic systems [3],[8],[16],Morgul and Solak [16] presented global synchronization is possible for a system with Brunowsky canonical form. Grassi and Mascolo [8]provided a systematicmethod for synchroManuscript. received. Manuscript. revised. †The trol. authors. are. Engineering,. November. 15,. February. with. National. the. 23,. 2007.. 2008.. Department. Chiao. Tung. of. Electrical. Univeristy,. and. Con-. Hsinchu,. 300,. Taiwan. ††The. author. Chienkuo †††The. author. Mechanical siung. is with. Technology. 804,. a) E-mail: DOI:. the. Department. University,. is with. the. Engineering,. of. Department National. Electronic. Changhua,. 500,. Engineering. of Mechanical Sun. Yat-sen. ,. Taiwan. and. Electro-. University,. Kaoh-. Taiwan. bwu@cc.nctu.edu.tw 10.1093/ietfec/e91-a.7.1797. Copyright (C). 2008 The Institute. of Electronics,. of Unknown. Chaos. Network Li-Shan. MA•õ,•õ•õ,. and. Jau-Woei. PERNG•õ•õ•õ,. Nonmembers. nizing using a scale transmitted signal. Message-free synchronization has been developed to permit communication with masking message in chaotic signals [14]. Messages can be extracted with message-free synchronization. Moreover, Boutayeb [3] proposed a scheme which is provided to synchronize and extract message simultaneously. Nevertheless, these systems do not consider the robustness of the state observer with respect to parameters mismatch [3], [8], [16]. Adaptive sliding observer design [2], [7] can handle parameters mismatch. Furthermore, a robust observer [13] is designed for synchronization using the Takagi-Sugeno fuzzy model and the LMI approach. Millerioux and Daafouz recently introduced the input-independent global chaos synchronization [15]. In this method, the added message does not affect the synchronization if the observer gain is appropriately designed. Other studies consider nonlinear observer designs for chaos synchronization [1], [17]. However, by the methods of previous descriptions, the chaotic systems should be known previously before synchronization design. Recently, the system identification approaches [5], [6], [9] have been introduced for a scale signal identification and chaos synchronization respectively. In [6], the system identification concepts are applied to approximate the chaos signal. The proposed identification scheme assumes a Lur'e type system as a reference model. This allows us to separate the identification process into two parts, adjusting alternatively the parameters of the linear and the nonlinear part. For modeling the linear system, the autoregressive moving average (ARMA) approach is utilized. On the other hand, the genetic algorithm is applied to optimize the break points parameters of nonlinear static functions to approximate nonlinear mapping. However, this approach is based on off-line identification, and it is not an on-line tuning scheme. Furthermore, the order in linear part identification should be by trial and error. The identification results just imitate the transmission signal and the other states in the master end cannot be achieved to synchronize simultaneously. In addition, the simulation results of this approach seem not very well. According to [5], the recursive identification is applied for chaos synchronization when the slave has exactly identical structure to the master system, but its parameters are unknown. It is shown that the unknown slave system parameters can be found by the concepts of adaptive synchronization. In other words, when the unknown slave system parameters are found, the synchronization is achieved. However, the structures in the master and slave ends should be known previously and exactly the same, although the paInformation. and Communication. Engineers.

(2) IEICE TRANS. FUNDAMENTALS,. VOL.E91-A,. NO.7 JULY 2008. 1798. rameters in the slave end can be estimated by recursive identification. The discussion of robustness is not included too. More recently, an alternative indirect Takagi-Sugeno fuzzy model based adaptive fuzzy observer design has been applied to chaos synchronization under assumptions that states are unmeasurable and parameters are unknown [9]. The adaptive law is designed to estimate the unknown parameters in the T-S fuzzy model of the slave end. When the unknown parameters are estimated correctly, the synchronization is achieved. However, the form of the T-S fuzzy model should be known first, and then the adaptive fuzzy observer is designed by the T-S fuzzy model. In addition, the discussion of robustness is not included. This investigation achieves synchronization with respect to a class of unknown master chaotic systems by introducing the concepts of AFNO [11], Brunowsky canonical form [16] and Lur'e systems [21]. The proposed system includes a chaotic master with canonical form and the slave with AFNO. The AFNO combines a FNN and a linear observer. In this design, the slave should synchronize with the master by a scale transmitted signal. This approach employs adaptive FNN to model the nonlinear term of the master end. The output of the adaptive FNN, robust input and a transmitted state are sent to the linear observer to estimate the states of the slave. The master and slave achieve synchronization when all states are estimated at the slave. Additionally, the adaptive laws are needed to update the weights of the FNN, when the reconstructed and transmitted states differ from each other. The benefits of provided AFNO for synchronization can be stated as follows. AFNO is first applied to chaotic synchronization with only one transmitted signal. Since AFNO is on line learning at the slave, the synchronization can be achieved respect to a switched unknown chaotic system with the Lur' e type. Additionally, the adaptability for parameters change or even system switched in the mater and the robustness for modeling error and external bounded disturbance are also given. AFNO also has FNN's inherent. 2.. Overall Structure of Adaptive Synchronization Fuzzy-Neural Observer Design. 2.1. previous due to soft computing with FNN for fitting it in the slave end. The paper is organized as follows. Section 2 describes the overall structure of adaptive synchronization with the AFNO design. Section 3 then introduces the AFNO design. Next, Sect. 4 includes the simulation results, including two examples to demonstrate the effectiveness of this application. Conclusions are finally are made in Sect. 5.. of Overall. Structure. Assume that the master and slave are all Lur' e type. Figure 1 illustrates the overall structure of adaptive synchronization with AFNO, which is synthesized with an FNN and a linear observer. In this design, only a scalar transmitted signal xM1. issenttotheslavefrom themaster.By theobservedstate xS, fs(xS)canbe computed to approximatefM(xM)with FNN. The adaptivelawsupdatetheweightsinFNN when theerror existsbetween xM1 and xS1.The linearobserverinputsare us=fs(xS), thetransmissionsignalxM1, and the robust inputur.The synchronization isachievedwhen xM=xS. 2.2 Dynamics oftheMasterand SlaveEnds Master End: xM=AMxM+BM(fM(xM)+d) yM1=xM1=CMxM,. (1). Slave End: [11],[12]. xS=ASxS+BS(fs(xS)-Ur)+Koeo yS1=xS1=CSxS,. (2). where. CM=CS=[1. 0 •c0. 0];. d. denotes. an. bounded. external. disturbance; xM=[xM. properties of fault-tolerance, parallelism learning, linguistic information and logic control. By comparing with [5], [6], [9], our presentation provides the on-line, robust and adaptive synchronization for a class of chaos systems. The form of nonlinear functions in the master end cannot be known in. Introduction. with. and. xS=[xS. observer. server. x(n-1)M]T=[xM1 xM2 •c xMn]T•¸Rn,. xS •c gain. As-KoCs. x(n-1)S]T=[xS1. KTo=[k1. strictly pair;. Hurwitz,. caused. FNN. bounded. with. Synchronization. d;. xS2 •c kn]. where ur. by. fM(xM) functions. is. AS). designed is. is [4],. xSn]T•¸Rn;. designed. (CS,. is. fM(xM). fS(xS).. continuous. The. k2 •c. eo=xM1-xS1;. robustness tive. xM •c. to. to. enhance. approximated unknown. satisfy. represents. by (uncertain). obthe adapbut. [25].. Error:. synchronization. error. can. be. defined. as:. esyn=xM-xS,. (3). where esyn=[esyn. The states. are. master estimated. esyn •c. e(n-1)syn]T=[esyn1. and. slave at. the. achieve slave.. esyn2 •c. synchronization. esynn]T•¸Rn.. when. all.

(3) WU et al.: OBSERVER-BASED. SYNCHRONIZATION. FOR A CLASS OF UNKNOWN. CHAOS SYSTEMS 1799. Maiter End Fig. 1. 3.. In. Adaptive. this. tion, to by. Fuzzy-Neural. section, the. AFNO. designed. achieve. and. 3.1. Fuzzy-Neural. The. FNN. fM(xM). as. f(x).. THEN. The rules. function the. the. FNN. to. vector. [10],. be. assumpstates. synthesized. in. which. fuzzy. consists. of is. from. to an rule. fuzzy. input. output. ith. is. Layer IV. IFas. a. is given by. employs. an. Layer III. Fig.2 The fuzzy-neural approximator [11], [12].. such. adopted. engine. a mapping. Layer I Layer II. 2 is utilized. functions. inference. IF-THEN. function. Fig.. engine,. x2•cxn]T•¸Rn Fuzzy. nonlinear. nonlinear. inference. to generate. x=[x1. the. the. [24],. The. y(x)•¸R.. then. depicted. model. approximator.. variable. an master's. [12]. model. FNN. a fuzzy. rules. can. [11],. The to. and. IF-THEN. guistic. Under. estimate. Design. observer.. designed. approximator. Observer. introduced.. Network. fS(xS).. The overall structure of synchronization with AFNO.. can. AFNO. a linear. is. with. an. is. AFNO. synchronization.. an FNN. as. Network. Skive end. lin-. linguistic thus. written. (5). as:. R(i): where. if x1 is Ai1 and •c. Ai1, Ai2, •c,. Ain and. functions ƒÊAij(xj) uct. from. fuzzy. and. fuzzy-neural. is Ain, then sets. membership. By. singleton. approximator. y is Bi,. with. respectively.. center-average, the. xn. Bi are. and ƒÊBi(yi),. inference,. y(x). and. using. fuzzifier, can. be. By. adjusting. laws,. the. prodtive. output. written. proximated. as. neural. the. parameter. uncertain. by. f(x). from. which. the. is defined. in. function. generated. approximator,. termined. vector ƒÆf. nonlinear in. (6).. By. the. estimated. functions. outputs. of the. fuzzy-neural. as. (4). with. f(x). can. using. the. f(x). adapbe. ap-. fuzzy-. can. be de-. approximator,. follows:. (4) f(x|ƒÆf)=ƒÆTfƒ¡(x), where ƒÆf In where,. μAij(xj) denotes. fuzzy. variable. rules,. and. [y1y2…yh]T Γ=[τ1τ2…. yi. xj; is. the. denotes. the h. is. point an. τh]T represents. membership the at. total which. adjustable a fuzzy. function number. value. of. IF-THEN. μBi(yi)=1. parameter basic. of. vector, where. the. four. θf=. vector,. of. (6). and τi. is. an. adjustable. summary,. FNN. layers. can. The as. be. (6). (6). parameter can. overall shown. interpreted. nodes. in Layer. Layer. II. denote. the. structure in. Fig.. by. I represent values. vector.. describe. 2.. Fig. input. of. the. input-output. of the The. 2 in. linguistic. relation. is divided. physical. the. membership. FNN. into. meanings. following. vectors. function. The. of input. Nodes of. in total.

(4) IEICE. TRANS.. FUNDAMENTALS,. VOL .E91-A,. 1800. linguistic variables. Each node in Layer III excuses a fuzzy rule. The output of Layer IV is the output signal modeling the nonlinear function. The connection parameters between layer III and layer IV are adjusted by using adaptive laws. The number of fuzzy rules can be dependent on complex level of nonlinear systems. In general, the more complex the systems are, the more numerous rules are demand. Of course, the computing load is heavy with more numerous rules. On the other hands, when the rules are less, the computing load is slight. This is a trade off problem.. NO.7. JULY. 2008. (13) where ble. φ(xS)=L-1(S)Γ(xS); transfer. function. strictly-positive the. designed. L-1(S). to. real. transform. (SPR). parameter.. a proper. H(s)L(s). transfer. The. denotes. into. function,. function. H(s). and. sta-. a proper γ1 denotes. is represented. as. follows:. 3.2. Adaptive Fuzzy-Neural Network Observer. H(s)=CS(sI-(As-KoCs))-1Bs.. Assumption 1 [11], [12]: The master state vector xMand the slave state vector xS belong to compact sets SM and Ss respectively, where SM={xM∈Rn:‖xM‖_??_εx. M<∞},. and ƒÃxS. The region. are. optimal. with. specified. designed. constant. as. (8). parameters.. parameter. vector ƒÆ*f. radius ƒÃƒÆf.. shown. in. falls. The. in. some. convex. convex. region. can. optimal. 1.. In our. The. for. The design procedure, stability proof and adaptive laws (13) can be referred in [11], [12] Simulation. Results. (9). vector ƒÆ*f. optimal θ*f the. of projection. for fM(xM).. This section verifies the feasibility of AFNO for synchronization using two examples.. (9).. parameter. applications,. operator. error. (15). 4.. can. be. described. 4.1. as:. (10). Refnark. is the. modeling. be. RƒÆf={ƒÆf•¸Rn:•aƒÆf•a_??_ƒÃƒÆf}.. The. in (15). minimal. (7). Ss={xS∈Rn:‖xS‖_??_εxS<∞},. and ƒÃxM. Prf(γ1eoφ(xS)) achieving. (14). is possible. adaptive. in an. laws. ideal. will be. situation.. applied. to tune. Example 1. In this example, AFNO is applied to synchronize a master Chua's circuit under modeling error, different initial conditions and external bounded disturbances. The results will demonstrate the adaptability and robustness of AFNO. The master Chua's circuit is reformed as a canonical form [26].. θf to approach θ*f. The. adaptive. spect. to. under. assumption. fuzzy-neural. a class. of. nonlinear. nonlinear. systems. 1. AFNO. can. be. observer (1)can. designed. with. be. re-. (16). designed. [11],. [12]:. where xS=ASxS+Bs(θTfΓ(xS)-ur)+Koeo. (11). yS1=xS1=CSxS, where θTfΓ(xS) nonlinear. is calculated. functions. denotes. the. robust. ternal. disturbance. FNN.. Based. input and. on. by. fM(xM)in. [11],. FNN. to. to compensate the. ur can. the. effect. designed. and due. modeling. be. the. systems,. approximated. [12],. approximate. dynamical. fM(xM)=14/1805xM1-168/9025xM2+1/38xM3-2/45•~(28/361xM1+7/95xM2+xM3)3. ur. to ex-. error. by. The adaptive laws tune FNN to approach fs(xS). The observer is designed to place poles of As-KoCs in -30 i.e. linear. as follows:. observer Other. ur=-1/γ. λmin(Q)eo,. (12). is. 3•~3. functions where. Q=QT>0,. γ should large. be. ur to. and proper. attenuate. better. attenuation. small. γ is chosen.. Riccati-like derivation The. designed. the. The. effect. performance. satisfied. Lyapunove. adaptive. laws. be. in stability. are. In general,. gamma. will, cause. disturbance.. Indeed,. the. when. the. make. the. obtained. Q=QT>0. function in FNN. constant.. small. of. will. Additionally,. equation with. γ is a positive. will and. [12]. as follows:. adaptive. law. gain. vector. parameters. of. identity. matrix,. for xSi, i=1,. is KTo=[90 AFNO and. 2700. 27000].. are ƒÁ=10, ƒÁ1=0.01,. L-1=1/S+2.. 2, 3 in FNN. The. are given. Q. membership. as follows:. μA1i(xSi)=1/(1+exp(5×(xSi+0.75))), μA2j(xSi)=exp(-(xSi+0.5)2), μA3j(xSi)=exp(-(xS+0.25)2), μA4j(xSi)=exp(-(xSi)2), μA5j(xSi)=exp(-(xSi-0.25)2), μA6j(xSi)=exp(-(xSi-0.5)2), μA7j(xSi)=1/(1+exp(-5×(xSi-0.75))).. (17).

(5) WU. et al.:OBSERVER-BASED. SYNCHRONIZATION. FOR. A CLASS. OF UNKNOWN. CHAOS. SYSTEMS 1801. Table. 1. Three. cases. of the initial. conditions.. Note: In the simulations, the disturbances in the master end are set as Case 1 in Table 2 in three cases. Note: In the simulations, the disturbances in the master end are set as Case 1 in Table 2 in three cases.. Table 2. Note: Table Note:. Three cases of the disturbances.. In the simulations, 1 in three cases. In the simulations,. ble 1 in three. the. the initial. The first. different. initial conditions.. In. this. why. FNN. are. turned. Table. 1.. listed. in Table. 2.. Figures initial. in is. the. first. other,. are chosen. as. the. for. 1 and. xS1. a. 1 in in. that. xS2. adaptive. as Casel. as Casel. Fig. 5 different. The third states initial conditions.. and xS2 in Chua's. circuit. and AFNO. under. and AFNO. under. in. in Ta-. the three. type. AFNO. AFNO. the. synchronizes. form. state. Chua's. 3 different. circuit.. synchronize xM2 initial with. conditions Chua's. In. differ-. Figs.. in. AFNO. in. the. Fig. 6 The first states xM1 and xS1 in Chua's circuit and AFNO under different disturbances.. master that. initial Figures. con4. and. xM3,. differ circuit. 3are. illustrates. and. are are. of. AFNO.. Figure three. slave. disturbances. each. to. weights. fM(xM).. results. in. circuit. ini-. chosen. and. disturbance 2.. with in. to. in Chua's. ac-. The. The. master. simulation. of. Table. are. error.. distinct. for. 343.. xM3 and xS3. under. estimated,. are FNN. the. states. and AFNO. be. laws of the. and xS3. Although. should. adaptive. conditions. Case. circuit. modeling. synchronizes xM1. spectively. each. in. summarize. initial. state. 5 illustrate. chosen. in process. Furthermore,. 3-5. Table. set. ditions. rules. conditions. conditions. distinct. end. by. initial. in. listed. states. fuzzy. parameters. listed. 5, the. conditions. to demonstrating. Different. ent. are. and xS1 in Chua's. three. the. adjustable. be ƒÆf(0)=0 of. states xM1. example,. for. tially. conditions. The second states xM2 initial conditions.. cases.. Fig. 3. counting. initial. Fig. 4 different. refrom. quickly,. well, and adaptively. Moreover, the synchronization error approaches zero as time goes to infinity. The robustness of AFNO can be also specified from Figs. 6-8 with various intensity disturbances in the master end. In Figs. 6-8, the initial conditions of three states are selected as Case 1 in.

(6) IEICE. TRANS.. FUNDAMENTALS,. VOL.E91-A,. NO.7. JULY. 2008. 1802. Table 1 and the different disturbances are chosen as Table 2. Figure 6 demonstrates that the first state xS1 in the slave synchronizes xM1 in the master end immediately and well under three different disturbances. Figures 7 and 8 reveal that xS2 and xS3 synchronize xM2 and xM3, individually. Even if the different disturbances are added in the master Chua's circuit, AFNO synchronizes with the master robustly. 4.2. Example. The original Rossler system can be presented as [16]: z1=z2+az1 z3=b-cz3+z2z3, where. z=[z1. The second disturbances.. z3]T.. xM=Ｔ-1z,. Example 2 demonstrates the adaptability of the utilized method by switched master between Chua's circuit and Rossler system as shown in Fig. 9. When the master is switched to another system, the slave follows to synchronize another chaotic system soon and well. The similar different initial conditions and disturbances listed in Tables 1 and 2 are considered in simulations for demonstrating the robustness of AFNO.. different. z2. Let. 2. Fig. 7. (18). z2=-z1-z3. states. xM2 and xS2 in Chua's. Fig. 9. circuit. and AFNO. The structure. under. (19). where The. Rossler. system. is reformed. as the canonical. form. with. fM(xM)=-cxM1+(ac-1)xM2+(a-c)xM3+ax2M1 -(a2+1)xM1xM2+axM1xM3+2ax2M2-xM2xM3+b. Fig. 8. The third. different. disturbances.. of synchronization. states. with the switched. xM3 and xS3 in Chua's. masters.. circuit. and AFNO. ,. under.

(7) WU et al.: OBSERVER-BASED. SYNCHRONIZATION. FOR A CLASS OF UNKNOWN. CHAOS SYSTEMS 1803. (a). Fig. 12 under. The third states. different. initial. in Chua's. conditions. circuit,. Rossler. and switched. system. and AFNO. masters.. (b) Fig. 10 The first states in Chua's circuit, Rossler system and AFNO under different initial conditions and switched masters: (a) actual figure size. Fig.13 The first statesin Chua'scircuit,RosslersystemandAFNO underdifferentdisturbancesand switchedmasters.. (b) enlarged figure size of local region.. Fig. 11 The secondstatesin Chua's circuit,Rosslersystemand AFNO under differentinitialconditionsandswitchedmasters.. where a=0.2, b=0.2, and c=6.3. Notably, fM(xM) is revised from [16]. The parameters of AFNO at the slave resemble those in Example 1. The initial condition of Rossler system is set [0 0 0]T. Figures. 10-12. indicate. the simulation. results. with re-. spect to each state for diverse initial conditions in AFNO and switched masters. The distinct initial conditions for each state in AFNO are shown in Table 1 and a kind of disturbance in the master end is set as Case 1 in Table 2. Figure 10 illustrates that the first state xS1 in AFNO with three different initial conditions synchronizes xM1 in the master end, even if the switched masters exist at the third second (Chua's circuit to Rossler system) and the sixth second (Rossler system to Chua's circuit). Figures 11 and 12 exhibit that xS2 and xS3 synchronize xM2 and xM3, respectively. Although the initial conditions differ from each other and the switched masters exist, AFNO synchronizes with the switched masters fast, well, and adaptively. On the other hand, simulation results in Figs. 13-15 verify the robustness of AFNO for the different disturbances and the switched systems in the master end. In Figs. 13-15, the initial conditions of three states are chosen as Case 1 in Table 1 and the different disturbances are selected as Table 2. Figure 13 displays that the first state xS1 synchronizes. xM1 immediately. and well under. three dif-. ferent disturbances, even thought the switched masters exist at the third second (Chua's circuit to Rossler system) and the sixth second (Rossler system to Chua's circuit). Fig-.

(8) IEICE. TRANS.. FUNDAMENTALS,. VOL.E91-A,. NO.7. JULY. 2008. 1804. on. line.. Furthermore,. observed. from. sign.. When. tical,. we. to. the. for. chaos. chaos. though. the. and. was. reached.. the. unknown. synchronization. By. into. could. deal. disturbance. Simulation soft. results. AFNO. respect. to. this. altered. bounded. robust with. iden-. master's. AFNO. external. de-. were. was. hand,. were. observer. ends. advantage. and. end. by. master. other. robustness. adaptive. master. slave. the. the. error, its. that. and. estimate. On. modeling. showed. the signal. master. could. system.. demonstrate. in. synchronization. even. chaos. with. the. AFNO. adaptively,. another. states. transmitted. in. the. the. states. scale. states said. scheme,. the. a. was. suitable. a class. unknown. systems.. Acknowledgments. Fig. 14 The second states in Chua's circuit, Rossler under different disturbances and switched masters.. system. and AFNO The. work. Grant. was. supported. by. National. Science. Council. under. no.NSC95-2752-E-009-012-PAE.. References. [1]. J. Amirazodi,. E.E.. performance. in. Yaz,. communication,•h. [2]. 81,. 2002.. A.. Azemi. to. chaotic. M.. systems,•h. 399,. 2004.. B.S.. Chen,. [5]. H.. Fuzzy. Ciruits. ures 14 and 15 reveal that xS2 and xS3 synchronize xM2 xM3, separately. In spite of the different disturbances. and. [6] O.. and the. switched systems are considered in the master end, AFNO synchronizes with the master robustly. It is noted that Figs. 10-15 display the simulation results indicating AFNO synchronizes with Chua's circuit at 0-3sec. The Rossler system also runs dynamically from the initial condition. AFNO synchronizes with Rossler at 3-6sec, while Chua's circuit runs simultaneously. From these simulation results, AFNO can synchronize with a class of unknown chaotic systems adaptively and robustly.. [7]. Syst.. De lar. periodic. M.. Feki, •gSynchronization. G.. [9]. I, vol.44,. C.H.. Hyun,. M.. systems no.10,. 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(9) WU et al.: OBSERVER-BASED. SYNCHRONIZATION. FOR A CLASS OF UNKNOWN. CHAOS SYSTEMS 1805. [15]. G.. Millerioux. and. J.. input-independent switched. systems,•h. pp. 1270-1279, [16]. [17]. and. systems,•h. Phys.. IEEE. approach. synchronization. Trans.. Circuits. E. Solak, •gObserver Rev.. Nijmerijer. and. nization,•h. observer-based. chaos. of Syst.. I,. Bing-Fei Wu was born in Taipei, Taiwan in 1959. He received the B.S. and M.S. degrees in control engineering from National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 1981 and 1983, respectively, and the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 1992. Since 1992, he has been with the De-. for. discrete-time vol.50,. no.10,. 2003.. O. Morgul. H.. Daafouz, •gAn. global. IEEE. based. E, vol.54,. I.M.Y.. Trans.. no.5,. synchronization. Mareels, •gAn Circuits. of chaotic. pp. 4803-4811,. 1996.. observer. Syst.. looks. I, vol.44,. at synchro-. no.10,. pp. 882-890,. 1997. [18]. 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Liu, •gAdaptive dynamical. vol.32,. hybrid. systems,•h. pp. 583-597,. 2002.. inIEEE. gent TransportationSystemsfor manyyears and leads a team to develop the first Taiwansmartcar, TAIWANiTS-1,with autonomousdrivingand active safety system. His currentresearchinterestsinclude vision-based vehicledrivingsafety,intelligentvehiclecontrol,multimediasignal analysis, embeddedsystemsandchip design.Prof.Wufoundedand servedas the Chairof the IEEESystems,Man andCyberneticsSocietyTaipeiChapter in Taiwan,2003. He is also the Chair of the TechnicalCommitteeof IntelligentTransportationSystemsof the IEEE Systems,Man and CyberneticsSocietysince2006. He hasbeenthe Directorof the ResearchGroup of ControlTechnologyof ConsumerElectronicsin the AutomaticControl Sectionof NationalScienceCouncil(NSC),Taiwan,from 1999to 2000. As an activeindustryconsultant,he is also involvedin the chipdesignand applicationsof the flash memorycontrollerand 3C consumerelectronics in multimediasystems.The researchhas been honoredby the Ministryof Educationas the BestIndustry-Academics CooperationResearchAwardin 2003. He receivedthe DistinguishedEngineeringProfessorAwardfrom ChineseInstituteof Engineersin 2002; the OutstandingInformationTechnologyElite Awardfrom TaiwanGovernmentin 2003; the GoldenLinux Awardin 2004; the First Prize Awardof TI China-TaiwanDSP Design Contestin 2006; the OutstandingResearchAwardin 2004from NCTU; the ResearchAwardsfrom NSC in the yearsof 1992, 1994,1996-2000; the GoldenAcer DragonThesisAwardsponsoredby the Acer Foundation in 1998and 2003,respectively;the FirstPrize Awardof the WeWin (Win by Entrepreneurshipand Workwith Innovation& Networking)Competitionhostedby IndustrialBankof Taiwanin 2003; and the SilverAwardof TechnologyInnovationCompetitionsponsoredby the AdvantechFoundationin 2003. Li-Shan. Ma. was born in Changhua, Tai-. wan in 1968. He received the B.S. and M.S. degrees in electrical engineering from Chung Yuan Christian University, Chungli, Taiwan, in 1995 and 1997, respectively. Since 1999, he has been with the Department of Electronic Engineering, Chienkuo Technology University, Changhua, Taiwan, where he is currently a lecturer. Now, he is pursuing a Ph.D. degree in the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu, Taiwan. from 2001.. control,. and intelligent. His research. interests. include. fuzzy. systems,. nonlinear. control.. Jau-Woei Perng was born in Hsinchu, Taiwan in 1973. He received the B.S. and M.S. degrees in electrical engineering from the Yuan Ze University, Chungli, Taiwan, in 1995 and 1997, respectively and the Ph.D. degree in electrical and control engineering from the National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 2003. From 2004 to 2008, he was a Research Assistant Professor with the Department of Electrical and Control Engineering at NCTU. He is currently an Assistant Professor with the Department. of Mechanical. and Electro-Mechanical. Engineering,. National. Sun Yat-sen University. His research interests include robust control, nonlinear control, fuzzy logic control, neural networks, systems engineering, and intelligent. vehicle. control..

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