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(IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 A New Approach to Model Reference Adaptive Control using Fuzzy Logic Controller for Nonlinear Systems R.Prakash R.Anita Department of Electrical and Electrnics Engineering, Department of Electrical and Electrnics Engineering, Muthayammal Engineering College, Institute of Road and Transport Technology, Rasipuram, Tamilnadu, India. Erode, Tamilnadu, India. Email: prakashragu@yahoo.co.in Email: anita_irtt@yahoo.co.in Abstract— The aim of this paper is to design a fuzzy logic Adaptive Network-Based Fuzzy Inference System (ANFIS) controller- based model reference adaptive intelligent for speed and position estimation of permanent-magnet controller. It consists of fuzzy logic controller along with a synchronous generator presented in [17].An adaptive fuzzy conventional Model Reference Adaptive Control (MRAC). The output feedback control approach is proposed for Single- idea is to control the plant by conventional model reference Input-Single-Output (SISO) nonlinear systems without the adaptive controller with a suitable single reference model, and measurements of the states. It is discussed in [18]. Gadoue et at the same time control the plant by fuzzy logic controller. In al. presented a fuzzy logic adaptation mechanisms and it is the conventional MRAC scheme, the controller is designed to used in model reference adaptive speed-estimation schemes realize plant output converges to reference model output based that are based on rotor flux[19].An adaptive fuzzy-based on the plant which is linear. This scheme is for controlling dynamic feedback tracking controller will be developed for linear plant effectively with unknown parameters. However, a large class of strict-feedback nonlinear systems involving using MRAC to control the nonlinear system at real time is plant uncertainties and external disturbances and it is difficult. In this paper, it is proposed to incorporate a fuzzy discussed in [20].Chang-Chun Hua et al. [21] presented an logic controller (FLC) in MRAC to overcome the problem. The adaptive fuzzy-logic system and it is investigated for a class control input is given by the sum of the output of conventional of uncertain nonlinear time-delay systems via dynamic MRAC and the output of fuzzy logic controller. The rules for the fuzzy logic controller are obtained from the conventional PI output-feedback approach. A development of Adaptive controller. The proposed fuzzy logic controller-based Model Fuzzy Neural Network Control (AFNNC), including direct Reference Adaptive controller can significantly improve the and indirect frameworks for an n-link robot manipulator, to system’s behavior and force the system to follow the reference achieve high-precision position tracking is discussed in [22]. model and minimize the error between the model and plant An-Min Zou et al. [23] proposed a controller for the robust output. backstepping control of a class of nonlinear pure-feedback systems using fuzzy logic. A set of fuzzy controllers is Keywords-Model Reference Adaptive Controller (MRAC), synthesized to stabilize the nonlinear multiple time-delay Fuzzy Logic Controller (FLC), Proportional-Integral (PI) large-scale system is presented in [24] controller In this paper a proposal of designing a fuzzy logic I. INTRODUCTION controller- based model reference adaptive intelligent controller is designed from a fuzzy logic controller in Model Reference Adaptive Control (MRAC) is one of parallel with a MRAC. From the designed PI controller, the main schemes used in adaptive system. Recently MRAC fuzzy rules are generated and it is used to design a fuzzy has received considerable attention, and many new logic controller. The fuzzy controller is connected in parallel approaches have been applied to practical processes [1], [2]. with an MRAC and its output is added and then given to the In the MRAC scheme, the controller is designed to realize plant input. The fuzzy logic controller is used to compensate plant output converges to reference model output based on the nonlinearity of the plant and it is not taken into the assumption that plant can be linearized. Therefore this consideration in the conventional MRAC. The role of scheme is effective for controlling linear plants with MRAC is to perform the model matching for the uncertain unknown parameters. However, it may not assure for linearized system to a given reference model. Finally to controlling nonlinear plants with unknown structure. It is confirm the effectiveness of proposed method, it is well known that fuzzy technique has been widely used in compared with the simulation results of the conventional many physical and engineering systems, especially for MRAC. systems with incomplete plant information [3]-[8]. In addition to fuzzy logic, it has been widely applied to II. STATEMENT OF THE PROBLEM controller designs for nonlinear systems [9]-[13].A learning To Consider a Single Input and Single Output (SISO), approach of combining MRAC with the use of fuzzy Linear Time Invariant (LTI) plant with strictly proper systems as reference models and controllers for control transfer function dynamical systems can be found in [14]. A hybrid approach by combing fuzzy controller and neural networks for y P (s) Z p (s) (1) G ( s) K learning-based control is proposed in [15]. A problem of u p (s) P R P (s) Fuzzy-Approximation-Based adaptive control for a class of where up is the plant input and yp is the plant output .Also, nonlinear time-delay systems with unknown nonlinearities the reference model is given by and strict-feedback structure is discussed in [16]. An 86 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 (2) ~ G m (s) ym (s) Km Z m (s) and the tracking error e is Strictly Positive Real (SPR), r (s) Rm (s) where r and ym are the model’s input and output. To define [1] and the adaptation rule for the controller gain θ is given the output error as e1 sgn( K p / K m ) (11) e y p ym (3) where e1= yp-ym and is a positive gain. Now the objective is to design the control input u such as The adaptive laws and control schemes developed are that the output error e goes to zero asymptotically for based on a plant model that is free from disturbances, noise arbitrary initial condition, where the reference signal r(t) is and unmodelled dynamics. These schemes are to be piecewise continuous and uniformly bounded. implemented on actual plants that most likely to deviate from the plant models on which their design is based. An actual plant may be infinite in dimensions, nonlinear and its III. STRUCTURE OF AN MRAC DESIGN measured input and output may be corrupted by noise and A. Relative Degree n =1 external disturbances. It is shown by using conventional As in Ref [1] the following input and output filters are MRAC that adaptive scheme is designed for a disturbance- used, free plant model and may go unstable in the presence of small disturbances. 1 F1 gu p (4) 2 F2 gy p IV. PI CONTROLLER-BASED MODEL REFERENCE ADAPTIVE CONTROLLER where F is an (n 1) * (n 1) stable matrix such as that The disturbance and nonlinear component are added to det ( SI F ) is a Hurwitz polynomial whose roots include the plant input of the conventional model reference adaptive the zeros of the reference model and that (F,g) is a controller, in this case the tracking error has not come to controllable pair. It is defined as the “regressor” vector zero and the plant output is not tracked with the reference T T [1 ,2 , y p , r ]T (5) model plant output. The large amplitude of oscillations will In the standard adaptive control scheme, the control u is come with the entire period of the plant output and the structured as tracking error has not come to zero .The disturbance is considered as a random noise signal. To improve the system u T (6) performance, the PI controller-based model reference [1 , 2 , 3 , C 0 ]T adaptive controller is proposed. In this scheme, the where is a vector of adjustable controller is designed by using parallel combination of parameters, and is considered as an estimate of a vector of conventional MRAC system and PI controller. unknown system parameters θ* . The dynamic of tracking error is The transfer function of PI Controller is generally ~ written in the “Parallel form” given (12) by or the “ideal e Gm ( s) p* T (7) * k p form’’ given by (13) P ~ * where k m and ( t ) represents GPI (S ) U pi ( S ) KP Ki (12) parameter error. Now in this case, since the transfer function E (S ) S ~ between the parameter error and the tracking error e is K P (1 1 ) (13) Ti Strictly Positive Real (SPR) [1], the adaptation rule for the controller gain θ is given by where Upi(s) is the control signal, acting on the error signal E(s),Kp is the proportional gain, Ki is the integral gain and Ti e1 sgn p * (8) is the integral time constant. where is a positive gain. The block diagram of the PI controller-based model reference adaptive controller is shown in Fig. 1. B. Relative Degree n =2 In the standard adaptive control scheme, the control u is structured as T u T T T e1 sgn( K p / K m ) (9) T where [1 , 2 , 3 , C 0 ] is a vector of adjustable parameters, and is considered as an estimate of a vector of * unknown system parameters . The dynamic of tracking error is ~ e Gm (s)(s p0 ) p* T (10) k P * p * ~ where and ( t ) k m Fig. 1 PI controller-based MRAC represents the parameter error. Gm (s)(s p0 ) is strictly proper and Strictly Positive Real (SPR). Now in this case, In the PI controller-based model reference adaptive since the transfer function between the parameter error controller, the value for the PI controller gains Kp and Ki are calculated by using the Ziegler–Nichols tuning method. 87 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 The control input U of the plant is given by the following U mr T equation, (17) [1, 2 , 3 , C0 ]T U U mr U pi (14) [ 1 , 2 , y p , r ] T U mr T Stability of the system and adaptability are then achieved where Umr is the output of the adaptive controller and Upi by an adaptive control law Umr tracking the system state x is the output of the PI controller. The input of the PI to a suitable reference model such as that the error e = yp- controller is the error, in which the error is the difference ym =0 asymptotically. The Fuzzy Logic Controller (FLC) between the plant output yp(t) and the reference model provides an adaptive control for better system performance output ym(t). In this case also, the disturbance (random and solution for controlling nonlinear processes. noise signal) and nonlinear component is added to the input The plant output is compared with the model reference of the plant .The PI controller- based model reference output. After comparison, the error and the change in error adaptive controller effectively reduces the amplitude of are calculated and are given as input to the fuzzy controller. oscillations of the plant output. In this case the tracking error has not come to zero. The PI controller-based model The error (e) and error change (ce) are defined as reference adaptive controller improves the performance e(k ) ym (k ) y p (k ) compared with the conventional MRAC. ce ( k ) e( k ) e( k 1) where ym(k) is the response of the reference model at kth V. FUZZY LOGIC CONTROLLER-BASED MODEL sampling interval, yp(k ) is the response of the plant output REFERENCE ADAPTIVE CONTROLLER at kth sampling interval, e(k) is the error signal at kth To make the system adaptable to more quickly and sampling interval, ce(k) is the error change signal at kth efficiently than conventional MRAC system and PI sampling interval. controller-based MRAC system, a new idea is proposed and FLC consists of three stages: fuzzification, rule implemented. The new idea which is proposed in this paper execution, and defuzzification. In the first stage, the crisp is the fuzzy logic controller- based model reference adaptive variables e(kT) and ce(kT) are converted into fuzzy controller. In this scheme, the controller is designed by variables e and ce using the triangular membership using parallel combination of conventional MRAC system functions. Each fuzzy variable is a member of the subsets and fuzzy logic controller. The error and the change in error with a degree of membership varying between ‘0’ (non- are given input to the fuzzy logic controller. The rules and member) and ‘1’ (full member).In the second stage of the membership function of fuzzy logic controller are formed FLC, the fuzzy variables e and ce are processed by an from the input and output waveforms of PI controller of inference engine that executes a set of control rules designed PI controller based MRAC scheme. The block containing in a rule base. In this paper the control rules are diagram of fuzzy logic controller-based model reference formulated using the knowledge of the PI controller of adaptive controller is shown in Fig. 2. designed PI controller-based MRAC system behavior and the experience of Control Engineers. The reverse of fuzzification is called defuzzification. The FLC produces the required output in a linguistic variable (fuzzy number). According to real-world requirements, the linguistic variables have to be transformed to crisp output. As the centroid method is considered to be the best well-known defuzzification method, it is utilized in the proposed method. A. Construction of Fuzzy Rules: Consider an example of a PI controller input (error), change in error and PI controller output waveforms are given by Fig. 3. By using the Fig.3, Fuzzy rules and membership for Fig. 2 Fuzzy logic controller-based MRAC system error (e) and change in error (ce) and output (Ufc ) are The state model of linear time invariant system is given created by the following form The developed fuzzy rules are X (t ) AX (t ) BU(t ) (15) 1. If error is ‘A’ and change in error is ‘A’ then the output is Y (t ) CX (t ) DU (t ) ‘D’ This scheme is restricted to a case of Single Input Single 2. If error is ‘B’ and change in error is ‘B’ then the output is Output (SISO) control, noting that the extension to Multiple ‘F’ Input Multiple Output (MIMO) is possible. To keep the 3. If error is ‘C’ and change in error is ‘D’ then the output is plant output yp converges to the reference model output ym, ‘H’ it is synthesized to control input U by the following 4. If error is ‘D’ and change in error is ‘F’ then the output is equation, ‘J’ U U mr U fc (16) 5. If error is ‘E’ and change in error is ‘C’ then the output is A where Umr is the output of the adaptive controller and Ufc is the output of the fuzzy logic controller 88 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 6. If error is ‘F’ and change in error is ‘I’ then the output is In this proposed fuzzy logic controller- based MRAC ‘K’ method, tracking error became zero within 6 seconds and no 7. If error is ‘G’ and change in error is ‘C’ then the output is oscillation has occurred. The plant output has tracked with B the reference model output. This method is better than 8. If error is ‘H’ and change in error is ‘H’ then the output is conventional MRAC system and PI controller -based ‘I’ MRAC system 9. If error is ‘I’ and change in error is ‘C’ then the output is VI. RESULTS AND DISCUSSION ‘C’ 10. If error is ‘J’ and change in error is ‘E’ then the output is In this section, the results of computer simulations for E conventional MRAC, PI controller-based MRAC and fuzzy logic controller-based MRAC system are reported. The 11. If error is ‘K’ and change in error is ‘G’ then the output results show the effectiveness of the proposed fuzzy logic is ‘G’ controller-based MRAC scheme and reveal its performance superiority to the conventional MRAC technique. Example 1: In this example, the nonlinearity of backlash which is followed by linear system is shown in Fig. 5 Fig. 5 Nonlinear System The disturbance (random noise signal) is also added to the input of the plant As an example, the system taken for the simulation is the Lateral Dynamic Model of a Boeing 747 airplane. The transfer function for the Lateral Dynamic Model of a Boeing 747 airplane System is given by 0.5s 3 0.2608s 2 0.1223s 0.05832 G(s) Fig. 3 PI controller input (error), change in error and 4 s 0.6358s 3 0.9389s 2 0.5116 0.003674 PI controller output (Upi) and the reference model are given by, 1 The FLC has two inputs: error e(kT) and change in error G m s s 3 ce(kT) and one output Ufc(kT). The membership functions The simulation was carried out with MATLAB and the for fuzzy variable error (e), change in error (ce) and output input is chosen as r(t)= 55sin0.7t.The initial value of the (Ufc) are shown in Fig.4. conventional MRAC scheme controller parameters are chosen as (0) = [0.5, 0, 0, 0]T . The conventional model reference adaptive controller is designed by using the equations (6) and (8). The simulations are done for the conventional MRAC, PI controller- based MRAC and fuzzy logic controller-based MRAC system with random noise disturbance and nonlinear component are added to the plant. In the PI controller-based model reference adaptive controller, the value of the PI controller gains Kp and Ki are equal to 10 and 75 respectively. In the fuzzy logic controller- based model reference adaptive controller, each universe of discourse is divided into six fuzzy sets: NH (Negative High), NL (Negative Large), ZE (Zero), PS (Positive Small), PM (Positive Medium) and PH (Positive High). The fuzzy variables e and ce are processed by an inference engine that executes a set of control rules which are contained in a (6x6) rule base as shown in Fig.6. The control rules are formulated using the knowledge of the PI Fig. 4 (a) Membership functions of the fuzzy variables error (e), (b) change controller of designed PI controller based MRAC scheme in error (ce), and output (Ufc) behavior and the experience of Control Engineers. 89 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 Fig. 6 Fuzzy rules table 8(b) The membership functions for fuzzy variable error (e), change in error (ce) and output (Ufc) are shown in Fig. 7 8(c) Fig. 7 Membership functions for fuzzy variable error (e), change in error (ce) and output (Ufc) The results for the conventional MRAC, PI controller- 8(d) based MRAC and fuzzy logic controller -based MRAC system are given in Fig. 8 8( e ) 8(a) 90 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 8(f) Fig. 8 Simulation results:8(a).Plant output yp(t) (solid lines) and the Fig. 9 Fuzzy rules table Reference model output ym (t) (dotted lines) of the conventional MRAC system for the input r(t)= 55sin0.7t. 8(b).Plant output yp(t) (solid lines) and the Reference model output ym (t )(dotted lines) of the PI controller-based MRAC scheme for the input r(t)= 55sin0.7t. 8(c). Plant output yp(t) (solid lines) and the Reference model output ym (t )(dotted lines) of the fuzzy logic controller-based MRAC scheme for the input r(t)= 55sin0.7t. 8(d).Tracking error e for the conventional MRAC.8 (e).Tracking error e for the PI controller-based MRAC scheme and 8(f) Tracking error e for the fuzzy logic controller -based MRAC scheme. Example 2: In this example, the nonlinearity of Dead zone is followed by linear system.The disturbance (random noise signal) is also added to the input of the plant. A second order system with the transfer function is given below 1 G(S ) S 2 3S 10 is used to study and the reference model is chosen as 5 G M (S ) S 2 10S 25 The initial value of conventional MRAC scheme controller parameters are chosen as (0) = [3, 18,-8, 3]T. The conventional model reference adaptive controller is Fig. 10 Fuzzy memberships used for simulation designed by using the equations (9) and (11). The simulation was carried out with MATLAB and the input is chosen as r(t)= 20+5sin4.9t. In the PI controller based model reference The results for the conventional MRAC, PI controller- adaptive controller, the value for the PI controller gains Kp based MRAC and fuzzy logic controller- based MRAC and Ki are equal to 8 and 85 respectively. system are given in Fig .11. In the fuzzy controller based model reference adaptive controller, seven linguistic variables are used for the input variable error and change in error. They are Extremely Negative (EN), High Negative (HN), Medium Negative (MN), Small Negative (SN), zero (ZE), Medium Positive (MP) and High Positive (HP). The seven linguistic variables are used for the output variable as Very Low(VL),Low(L),Nearly Low(NL), Medium(M),Medium High(MH),High(H) and Extremely positive(EP). The control rules are formulated using the knowledge of the PI controller of designed PI controller-based MRAC 11 (a) scheme behavior and the experience of Control Engineers. The fuzzy variables e and ce are processed by an inference engine that executes a set of control rules which are containing in a (7x7) rule base as shown in Fig. 9. The membership functions for fuzzy inputs error (e), change in error (ce) and fuzzy output (Ufc) are shown in Fig. 10. 91 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 11(b) 11(f) Fig. 11 Simulation results:11(a) Plant output yp(t) (solid lines) and the Reference model output ym (t) (dotted lines) of the conventional MRAC system for the input r(t)= 20+5sin4.9t. 11(b) Plant output yp(t) (solid lines) and the Reference model output ym (t )(dotted lines) of the PI controller- based MRAC scheme for the input r(t)= 20+5sin4.9t. 11(c) Plant output yp(t) (solid lines) and the Reference model output ym (t )(dotted lines) of the fuzzy logic controller-based MRAC scheme for the input r(t)= 20+5sin4.9t. 11(d) Tracking error e for the conventional MRAC. 11(e) Tracking error e for the PI controller-based MRAC scheme. 11(f) Tracking error e for the fuzzy logic controller- based MRAC scheme. The nonlinear component and the disturbance (random noise signal) are added to the plant input of conventional MRAC. The plant output is not tracked with the reference 11(c) model output and large amplitude of oscillations occur at the entire plant output signal as shown in Fig. 8(a) and 11(a) and also tracking error has not come to zero as shown in Fig. 8(d) and 11(d). But when the disturbance (random noise signal) and non linear component are added to the input of the plant of PI controller-based model reference adaptive controller and it improves the performance comparing to the conventional MRAC and also reduces the amplitude of oscillations of the plant output as shown in Fig. 8(b) and 11(b).In this case also plant output does not track the reference model output and the tracking error has not come to zero as shown in Fig. 8(e) and 11(e).When the disturbance (random noise signal) and nonlinear component are added to the input of the plant of the proposed fuzzy logic controller-based MRAC scheme, the plant output has 11(d) tracked with the reference model output as shown in Fig. 8(c) and 11(c).The tracking error becomes zero within 6 seconds with less control effort as shown in Fig. 8(f) and 11(f) and no oscillations has occurred. From the plots, one can see clearly that the transient performance, in terms of the tracking error and control signal, has been significantly improved by the proposed MRAC using fuzzy logic controller. The proposed fuzzy logic controller-based MRAC schemes show better control results compared to those by the conventional MRAC and PI controller -based MRAC system. On the contrary, the proposed method has much less error than conventional method in spite of nonlinearities and disturbance. VII. CONCLUSION 11(e) In this section, the response of the conventional model reference adaptive controller is compared with the PI controller-based MRAC system and proposal model reference adaptive controller using fuzzy logic controller. The controller is checked with the two different plants. The proposed fuzzy logic controller -based MRAC controller shows very good tracking results when compared to the 92 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 conventional MRAC and the PI controller- based MRAC [20] Yeong-Chan Chang, “Intelligent Robust Tracking Control for a system. Simulations and analyses have shown that the Class of Uncertain Strict-Feedback Systems,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics vol.31, transient performance can be substantially improved by no.1,.pp. 142 – 155, Feb. 2009 proposed MRAC scheme and also the proposed controller [21] Chang-Chun Hua, Qing-Guo Wang and Xin-Ping Guan“Adaptive shows very good tracking results when compared to Fuzzy Output-Feedback Controller Design for Nonlinear Time- conventional MRAC. Thus the proposed intelligent parallel Delay Systems With Unknown Control Direction,” IEEE controller is found to be extremely effective, efficient and Transactions on Systems, Man, and Cybernetics, Part B: useful Cybernetics, vol.39, no.2,pp. 363 - 374, April 2009 [22] Rong-Jong Wai and Zhi-Wei Yang, “Adaptive Fuzzy Neural REFERENCES Network Control Design via a T–S Fuzzy Model for a Robot Manipulator Including Actuator Dynamics,”IEEE Transactions on [1] K.J. Astrom and B. Wittenmark Adaptive control (2nd Ed.) Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 38, no. Addison-Wesley,1995. 5,pp. 1326 – 1346, Oct. 2008 [2] Petros A loannou, Jing sun. “Robust Adaptive control”, upper [23] An-Min Zou; Zeng-Guang Hou and Min Tan, “Adaptive Control of saddle River, NJ: Prentice-Hall 1996. a Class of Nonlinear Pure-Feedback Systems Using Fuzzy [3] J.Dong,Y.Wang and G.-H. Yang,“Control synthesis of continuous Backstepping Approach,” IEEE Trans. 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College of Technology, affiliated to Bharathiyar [6] R.-J. Wai, M.-A. Kuo, and J.-D. Lee, “Cascade direct adaptive University, Coimbatore, Tamilnadu, India in 2000 and fuzzy control design for a nonlinear two-axis inverted-pendulum completed his M.Tech degree from the College of servomechanism,” IEEE Trans. Syst., Man, Cybern., Part B, vol. Engineering, Thiruvanandapuram, Kerala, India, in 38, no. 2, pp. 439–454, Apr. 2008. 2003. He is currently working for his doctoral degree at [7] T.-H. S. Li, S.-J. Chang, and W.Tong, 2004, “Fuzzy target tracking Anna University, Chennai, India. He has been a member control of autonomous mobile robots by using infrared sensors,” of the faculty Centre for Advanced Research, Muthayammal Engineering IEEE Trans. Fuzzy Systems, vol. 12, no. 4, pp. 491-501,Aug. 2004. College, Rasipuram, Tamilnadu, India since 2008. His research interests [8] K. Tanaka and M. 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