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Tap chi KHOA HOC & CONG NGHE Do Trung Hai 132(02): 31 -35 APPLICATION OF THE TAKAGI - SUGENO NEURO -FUZZY SYSTEM FOR TRACKING CONTROL OF DYNAMIC NONLINEAR SYSTEM Do Trung Hai College of Technology - TNU SUMMARY This study presents a technique of online identification and tracking control of dynamic nonlinear systems using the neuro-fiizzy system The model of identification and control used in this paper is the Takagi-Sugeno fuzzy rules in which variations of membership fiinction parameters using neural networks follow the backpropagation algorithm and control signal alternations aim at making the output track the reference input The theoretical analysis is applied to identify and control the dynamic model of a I-DOF manipulator Precise tracking results are also presented in simulation session Keywords: Neuro-fuzzy system, Takagi- Sugeno fuzzy controller Identification and control of nonlinear system Notation AJ ^' W y Y y' Meaning Linguistic Variable Membership Function Weight Learning Coefficient Model output value Real output value Set value reviation Takagi - Sugeno T-S Neuro-Fuzzy Controller NFC rNTRODUCTION Identification and control of nonlinear systems have been considered and researched by many scientists for recent decades Therefore, this paper demonstrates an application of the modem control theories using fuzzy logic and neural networks to identify and control nonlinear systems Fuzzy logic and neural networks are used to supplement each other Neural networks aim at providing systems' information from the learning process while ftjzzy logic uses experts' knowledge According to [1], the combinations of fiizzy logic and neural network are shown in the following categories' neuro-fuzzy systems; fuzzy neural networks and fiizzy neural hybrid systems Email- dolrunghai@tnut.edu In this research, the neuro-fuzzy system is applied to identify and control dynamic nonlinear systems In this system, neural networks are used as tools in fuzzy models and there are automatic tuning methods of neural networks Roles of the fuzzy system stay unchanged and neural networks are used to make use of abilities to calculate and process data STRUCTURE OF THE T-S NEURO-FUZZY SYSTEM IDENTIFICATION PROBLEM To identify systems, the neuro-fuzzy system and T-S fuzzy laws are used in this paper T-S fiizzy model was first proposed by Takagi and Sugeno in 1985 Basically, the difference between T-S fuzzy model and Mamdani fuzzy model is the type of consequent clauses of fuzzy rules The consequent clauses in T-S fuzzy are always real value functions replacing fuzzy sets The jth fuzzy rule in T-S fuzzy model is written as: If X| w A,^ and Xj is A^ and and x„ is A^ (1) \i)e3^y=f^=a|^+a|x^-\-c^X2+ +c^„x„ where x, is input variable and y is output variable A,-" is linguistic variable of antecedent clauses with membership ftinction \i ,(x,) a| is coefficients with j=l m, i=I n Do Trung Hai Tap chi KHOA HOC & CONG NGHE If fj is a constant that does not depend on xj, X;, ,Xn, then T-S fiizzy model will be called the zero-order model If fj is defined by (1), T-S fuzzy model will be called the first-order model Without loss of generality, there are two inputs, one output and two rules R| and R2 as follows: /^:IfXi is A) and Xj isA^ ii:eny=f^=ci^+ci^x^+d!^x^ (2) ftilfx, is ^'andXjis J^_1iKny=(^=4+'^x,+4^ 0) where Xi, X2 are input variables Therefore, the typical structure able to carry out fuzzy rules (2), (3) is the 5-layer NFC system as shown in Figure I Layer 5: Calculate output value (defuzzifi cation) -f,- ^^2 system (6) In this section, the backpropagation algorithm is applied Consider the general output with target function that is defined as: £ = j|/'(')-7(of C^) where f*(t) is set value, f{t) is output value Learning rules will change the weights in nodes or parameters of each node (such as; center or width of membership functions) to minimize the error target function by using repetition (for example, adjusting weights) The weight in step (t+1) is calculated through that in step t as shovra in Equation (8) dE M}-w(r) + r?(-—) aw (8) 3w 3(a) 3w Layer I Layer Layer Layer Layer Figure Neural network structure by T-S fuzzy where Layer 1: Receive input signals Layer 2: Implement fuzzification with membership functions Normally, Gauss function is usually chosen: where T\ is learning coefficient and a is an affect function In the NFC system as shown in Figure 1, parameter updating laws of layers are implemented from the output layer to the input layer [4] Parameter updating procedure of online identification problem is shown in Figure CONTROL PROBLEM where m/ and cr/ are position of the center of the peak and width of the membership function, respectively Layer 3: Implement operations of antecedent clauses in which product operator is used After identifying an object, basing on its acquired parameters, a controller is going to be designed It is assumed that the control object has a standard form [2] and without loss of generality, a system including state variables is written as follows: J x j = F(x) + G(x.).u Layer 4: Implement operations of consequent clauses Calculating f, and fj depends on (2), (3) 32 (10) wherex=(x„x,); ^(x), G(2t)are identified functions through the neuro-fuzzy system Do Trung Hai Tap chi KHOA HOC & CONG NGHE 132(02): 31-35 "i r Define values (center and width) of membership functions in NFCs as Wo^ Expected output value (reference value) of the system is denoted by y* Therefore, the error between the expected output and the real , (II output is: Take derivatives of the error target function with respect to W (denoted by AW) e^y' —y aw e-l-a,e+a,e = (III) y* -y + a^t+aJQ = Q ' (y*-i-a,e+a2e-F(x)) G(x) ' Figure 2, Parameter updating procedure The problem is that the control signal u needs to be determined in order to make the output signal of system track the reference signal System (10) can be rewritten as: [y-x, =x, Jy = x , - F ( x ) + G(x).u (11) From (11), u is determined by: A I -DOF manipulator with kinematic equations in (19) is used to test effectiveness of the proposed control system [3] y=9 = X J (12) In (12), R^) and G(x) are estimated by identification process, y needs to be defined From the identified model, output value of the model is y , it is supposed that the output of the system is the same as that of the model (13) (18) From (18), it can be seen that if the object's parameters change then identification process of ftinctionsF(x),G(x)will response and control signal u will be adjusted to be equivalent to changes of the object The general structure of online identification structure and control system is shown in Figure SIMULATION X| U-—!—(y-F(x)) G(x)^ (17) Put y into (14), we have: Set new values to the network and go back to step (I) • (16) From (16), if the roots of the characteristic equation obtain negative real part then e converges to over a finite interval of time Coefficients ai, 02 in (16) can be determined by the pole placement method From (15), (16), we have: Calculating a new value W„„ = W„„+T,.AW ri is the learning coefficient (IV) (15) The error e should become zero in a finite interval of time It is assumed that e is roots of the below linear differential equation (19) (20) = F(x) + 0(x)u wherex=(x,,Xj);x, = is angular position; X j - B is angular speed; m is mass of manipulator; I is length of manipulator; u is 33 D6 Trung Hai Tap chi KHOA HOC & CONG NGHE 132(02): 31-35 torque of the system; d is friction coefficient; g is the acceleration of gravity Modeled dynamic nonlinear system Contrail -er with the rule in (18) fKiH?i- I Flguie Tlie error e ber>\-i expected outpnr signal Iser \alus) 'and die output signal of the object; Figure Structure of identification and control system These parameters are defined as follows: m = 0,9kg; = 0,85m; g = 9,8ni/s^ d = 0,8kg mVs Choose a^ = 11; 0:3 = ^ Modeling system by Matlab-Simulink, the results are shown in Figures 4-6 [radl Figure Changes by time of center and width of membership functions and aj of consequent clauses in the online identification and control process Figures 4-5 show that the output converges to the input within 6s and the error is very tiny In Figure 6, adaptive changes of parameters (including the center and width of the membership funcfions and the coefficients in consequent clauses) are depicted CONCLUSION The paper proposes a technique of using Takagi-Sugeno neuro-fuzzy system to identify and control the dynamic nonlinear system The S-layer structure system and a proper number of neurons will assure high Figure Expected output signal (set value) y , output signal of object y and the error e between speed calculations and adaptive parameter updating processes to minimize the error Control structure in Figure and the control D6 Trung Hai Tap chi KHOA HQC & CONG NGHE law in (18) will make the system stable and converge to the reference trajectory without depending on initial conditions The theoretical analysis has been simulated for the nonlinear system defined in (19) It can be seen from simulation results (Figures 4-6) that we can identify online and control the system simultaneously In addition, these theoretical and the simulation results might be applied to real technical objects 132(02): 31 -35 REFERENCES Chin-Teng Lin and C.S George Lee, (1996), Neuro-fuzzy Systems, Prentice-Hall International Slotine, J.E and Le, W,(1991), Applied nonlinear control, Prentice-Hall, Engleewood Cliffs, NJ X.M.Ren, A.B.Rad, P.T.Chan, Wai Lun Lo, (2003), Identification and control of continuoustime nonlinear systems via dynamic neural networks, IEEE Transactions on industrial electronics, Vol.50 No.3, pp 478-486 Thuan.NT, Hai.TD, (2007), Application of fuzzy and neural network to identify kinetics of strong non-linear system Journal of Science and Technology, Technical Universities, No 21-26 (in Vietnamese) TOM TAT iTNG DUNG HE MOf - NCflRON TAKAGI - SUGENO TRONG ©lEU KHIEN BAM HE © O N G HOC PHI TUYEN D6 Trung H5i* Truong Dai hoc Ky thugl Cong nghiep - DH Thai Nguy Bki bdo de xuSt phucmg ph^p ky thu^t diing he mo - noron de nhan dang true tuyen va dieu khien b4m he dong hipc phi tuySn M6 hinh nhSn dang va difiu khiln su- dung luSt md Takagi - Sugeno v6i su bien doi thong so ciia ham lien thuoc dung mang noron theo thuat toan hoc Ian truyen ngugc va su thay d6i tin hi^u dieu khien vui mong muon dau cua doi tugng bdm theo dau vao cho truoc Viec phan tich ly thuyet dugc irng dung cho vifc nhan dang vk dieu khien m6 hinh dgng hoc tay miy khop noi Ket qua mo phong vdi chinh xac cao da dugc chi Tir khoa: he md noron, luat md Takagi - Sugeno, ddieuf khien bdm he ddng hgc phi tuyen Ngay nhdn bdi:09/12/2014; Ngayphdn bien.24/12/2014; Ngdy duyet dang: 05/3/2015_ Phan biin khoa hoc: TS Nguyin Duy Cuang - Trudng Dgi hoc Ky thuat Cdng nghiep - DHTN ' Email: dolrunghai@lnut edu.vn ... MOf - NCflRON TAKAGI - SUGENO TRONG ©lEU KHIEN BAM HE © O N G HOC PHI TUYEN D6 Trung H5i* Truong Dai hoc Ky thugl Cong nghiep - DH Thai Nguy Bki bdo de xuSt phucmg ph^p ky thu^t diing he mo - noron... error target function with respect to W (denoted by AW) e^y'' —y aw e-l-a,e+a,e = (III) y* -y + a^t+aJQ = Q '' (y*-i-a,e+a2e-F(x)) G(x) '' Figure 2, Parameter updating procedure The problem is that... CONCLUSION The paper proposes a technique of using Takagi- Sugeno neuro-fuzzy system to identify and control the dynamic nonlinear system The S-layer structure system and a proper number of neurons
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