Improvement of Takagi-Sugeno Fuzzy Model for the Estimation of Nonlinear Functions

Two new and efficient approaches are presented to improve the local and global estimation of the Takagi-Sugeno (T-S) fuzzy model. The main aim is to obtain high function approximation accuracy and fast convergence. The main problem is that the T-S identification method can not be applied when the membership functions are overlapped by pairs. The approaches developed here can be considered as generalized versions of T-S method with optimized performance. The first uses the minimum norm approach to search for an exact optimum solution at the expense of increasing complexity and computational cost. The second is a simple and less computational method, based on weighting of parameters. Illustrative examples are chosen to evaluate the potential, simplicity and remarkable performance of the proposed methods and the high accuracy obtained in comparison with the original T-S model

New methods for the estimation of Takagi-Sugeno model based extended Kalman filter and its applications to optimal control for nonlinear systems

This paper describes new approaches to improve the local and global approximation (matching) and modeling capability of Takagi–Sugeno (T-S) fuzzy model. The main aim is obtaining high function approximation accuracy and fast convergence. The main problem encountered is that T-S identification method cannot be applied when the membership functions are overlapped by pairs. This restricts the application of the T-S method because this type of membership function has been widely used during the last 2 decades in the stability, controller design of fuzzy systems and is popular in industrial control applications. The approach developed here can be considered as a generalized version of T-S identification method with optimized performance in approximating nonlinear functions. We propose a noniterative method through weighting of parameters approach and an iterative algorithm by applying the extended Kalman filter, based on the same idea of parameters’ weighting. We show that the Kalman filter is an effective tool in the identification of T-S fuzzy model. A fuzzy controller based linear quadratic regulator is proposed in order to show the effectiveness of the estimation method developed here in control applications. An illustrative example of an inverted pendulum is chosen to evaluate the robustness and remarkable performance of the proposed method locally and globally in comparison with the original T-S model. Simulation results indicate the potential, simplicity, and generality of the algorithm. An illustrative example is chosen to evaluate the robustness. In this paper, we prove that these algorithms converge very fast, thereby making them very practical to use

LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi–Sugeno's form

International audienc

Active fault-tolerant control of nonlinear systems with wind turbine application

The thesis concerns the theoretical development of Active Fault-Tolerant Control (AFTC) methods for nonlinear system via T-S multiple-modelling approach. The thesis adopted the estimation and compensation approach to AFTC within a tracking control framework. In this framework, the thesis considers several approaches to robust T-S fuzzy control and T-S fuzzy estimation: T-S fuzzy proportional multiple integral observer (PMIO); T-S fuzzy proportional-proportional integral observer (PPIO); T-S fuzzy virtual sensor (VS) based AFTC; T-S fuzzy Dynamic Output Feedback Control TSDOFC; T-S observer-based feedback control; Sliding Mode Control (SMC). The theoretical concepts have been applied to an offshore wind turbine (OWT) application study. The key developments that present in this thesis are:• The development of three active Fault Tolerant Tracking Control (FTTC) strategies for nonlinear systems described via T-S fuzzy inference modelling. The proposals combine the use of Linear Reference Model Fuzzy Control (LRMFC) with either the estimation and compensation concept or the control reconfiguration concept.• The development of T-S fuzzy observer-based state estimate fuzzy control strategy for nonlinear systems. The developed strategy has the capability to tolerate simultaneous actuator and sensor faults within tracking and regulating control framework. Additionally, a proposal to recover the Separation Principle has also been developed via the use of TSDOFC within the FTTC framework.• The proposals of two FTTC strategies based on the estimation and compensation concept for sustainable OWTs control. The proposals have introduced a significant attribute to the literature of sustainable OWTs control via (1) Obviating the need for Fault Detection and Diagnosis (FDD) unit, (2) Providing useful information to evaluate fault severity via the fault estimation signals.• The development of FTTC architecture for OWTs that combines the use of TSDOFC and a form of cascaded observers (cascaded analytical redundancy). This architecture is proposed in order to ensure the robustness of both the TSDOFC and the EWS estimator against the generator and rotor speed sensor faults.• A sliding mode baseline controller has been proposed within three FTTC strategies for sustainable OWTs control. The proposals utilise the inherent robustness of the SMC to tolerate some matched faults without the need for analytical redundancy. Following this, the combination of SMC and estimation and compensation framework proposed to ensure the close-loop system robustness to various faults.• Within the framework of the developed T-S fuzzy based FTTC strategies, a new perspective to reduce the T-S fuzzy control design conservatism problem has been proposed via the use of different control techniques that demand less design constraints. Moreover, within the SMC based FTTC, an investigation is given to demonstrate the SMC robustness against a wider than usual set of faults is enhanced via designing the sliding surface with minimum dimension of the feedback signals

Improved Stabilization Conditions For Takagi-sugeno Fuzzy Systems Via Fuzzy Integral Lyapunov Functions

This paper presents new results concerning the design of state feedback controllers for continuous-time Takagi-Sugeno (T-S) fuzzy systems. The conditions, based on a line-integral fuzzy Lyapunov function, are specially suitable for T-S fuzzy systems where no information about the time-derivatives of the membership functions is available. The controller is designed through linear matrix inequalities in a two step procedure: at the first step, a stabilizing fuzzy controller is obtained for a relaxed frozen (i.e. time-invariant) T-S fuzzy system. This control gain is then used as an input data at the second step, that provides a stabilizing control law guaranteed by the line-integral Lyapunov function. An extension to cope with H ∞ guaranteed cost control of T-S fuzzy systems is also provided. Numerical examples illustrate the advantages of the proposed method when compared to other techniques available in the literature. © 2011 AACC American Automatic Control Council.49704975Boeing,Bosch - Invented for Life,Corning,Eaton Corporation,GE Global ResearchTakagi, T., Sugeno, M., Fuzzy identification of systems and its applications to modeling and control (1985) IEEE Trans. Syst., Man, Cybern., SMC-15 (1), pp. 116-132. , JanuaryBoyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., (1994) Linear Matrix Inequalities in System and Control Theory, , Philadelphia, PA: SIAM Studies in Applied MathematicsGahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M., (1995) LMI Control Toolbox User's Guide, , Natick, MA: The Math WorksSturm, J.F., Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones (1999) Optim. Method Softw., 11 (1-4), pp. 625-653. , http://sedumi.mcmaster.ca/Wang, H.O., Tanaka, K., Griffin, M.F., Parallel distributed compensation of nonlinear systems by Takagi-Sugeno fuzzy model Proc. 4th IEEE Int. Conf. Fuzzy Syst. - 2nd Int. Fuzzy Eng. Symp., Yokohama, Japan, March 1995, pp. 531-538Tanaka, K., Wang, H., (2001) Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, , New York, NY: John Wiley & SonsTanaka, K., Hori, T., Wang, H.O., A multiple Lyapunov function approach to stabilization of fuzzy control systems (2003) IEEE Trans. Fuzzy Syst., 11 (4), pp. 582-589. , AugustKim, E., Lee, H., New approaches to relaxed quadratic stability condition of fuzzy control systems (2000) IEEE Trans. Fuzzy Syst., 8 (5), pp. 523-534. , OctoberGuerra, T.M., Vermeiren, L., LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form (2004) Automatica, 40 (5), pp. 823-829. , MayDing, B., Sun, H., Yang, P., Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno's form (2006) Automatica, 42 (3), pp. 503-508. , DOI 10.1016/j.automatica.2005.11.005, PII S0005109805004085Kruszewski, A., Wang, R., Guerra, T.M., Nonquadratic stabilization conditions for a class of uncertain nonlinear discrete time TS fuzzy models: A new approach (2008) IEEE Transactions on Automatic Control, 53 (2), pp. 606-611. , DOI 10.1109/TAC.2007.914278Ding, B., Stabilization of Takagi-Sugeno model via non-parallel distributed compensation law (2010) IEEE Trans. Fuzzy Syst., 18 (1), pp. 188-194. , FebruaryTognetti, E.S., Oliveira, R.C.L.F., Peres, P.L.D., LMI relaxations for nonquadratic stabilization of discrete-time Takagi-Sugeno systems based on polynomial fuzzy Lyapunov functions Proc. 17th Medit. Conf. Control Aut. (MED2009), Thessaloniki, Greece, June 2009, pp. 7-12Tanaka, K., Hori, T., Wang, H.O., A fuzzy Lyapunov approach to fuzzy control system design Proc. 2001 Amer. Control Conf., Arlington, VA, USA, June 2001, pp. 4790-4795Rhee, B.-J., Won, S., A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design (2006) Fuzzy Sets & Syst., 157 (9), pp. 1211-1228. , MayTanaka, K., Ohtake, H., Wang, H.O., A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions (2007) IEEE Transactions on Fuzzy Systems, 15 (3), pp. 333-341. , DOI 10.1109/TFUZZ.2006.880005Mozelli, L.A., Palhares, R.M., Avellar, G.S.C., A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems (2009) Inform. Sci., 179 (8), pp. 1149-1162. , MarchMozelli, L.A., Palhares, R.M., Souza, F.O., Mendes, E.M.A.M., Reducing conservativeness in recent stability conditions of TS fuzzy systems (2009) Automatica, 45 (6), pp. 1580-1583. , JuneTognetti, E.S., Oliveira, R.C.L.F., Peres, P.L.D., Selective stabilization of Takagi-Sugeno fuzzy systems Proc. 2010 IEEE Int. Conf. Fuzzy Syst., Barcelona, Spain, July 2010, pp. 2772-2779Guerra, T.M., Bernal, M., A way to escape from the quadratic framework Proc. 2009 IEEE Int. Conf. Fuzzy Syst., Jeju Island, Korea, August 2009, pp. 784-789Peaucelle, D., Arzelier, D., An efficient numerical solution for H2static output feedback synthesis Proc. 2001 Eur. Control Conf., Porto, Portugal, September 2001Arzelier, D., Peaucelle, D., An iterative method for mixed H2/H∞synthesis via static output-feedback Proc. 41st IEEE Conf. Decision Control, Las Vegas, NV, USA, December 2002, pp. 3464-3469Arzelier, D., Peaucelle, D., Salhi, S., Robust static output feedback stabilization for polytopic uncertain systems: Improving the guaranteed performance bound Proc. 4th IFAC Symp. Robust Control Design, Milan, Italy, June 2003Mehdi, D., Boukas, E.K., Bachelier, O., Static output feedback design for uncertain linear discrete time systems (2004) IMA J. Math. Control Inform., 21 (1), pp. 1-13. , MarchArzelier, D., Gryazina, E.N., Peaucelle, D., Polyak, B.T., Mixed LMI/Randomized methods for static output feedback control design Proc. 2010 Amer. Control Conf., Baltimore, MD, USA, June-July 2010, pp. 4683-4688Tuan, H.D., Apkarian, P., Narikiyo, T., Yamamoto, Y., Parameterized linear matrix inequality techniques in fuzzy control system design (2001) IEEE Transactions on Fuzzy Systems, 9 (2), pp. 324-332. , DOI 10.1109/91.919253, PII S1063670601028259Liu, X., Zhang, Q., New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI (2003) Automatica, 39 (5), pp. 1571-1582. , OctoberSala, A., Arino, C., Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem (2007) Fuzzy Sets and Systems, 158 (24), pp. 2671-2686. , DOI 10.1016/j.fss.2007.06.016, PII S0165011407003284Oliveira, R.C.L.F., Peres, P.L.D., Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations (2007) IEEE Trans. Autom. Control, 52 (7), pp. 1334-1340. , JulyMontagner, V.F., Oliveira, R.C.L.F., Peres, P.L.D., Convergent LMI relaxations for quadratic stabilizability and H ∞ control for Takagi-Sugeno fuzzy systems (2009) IEEE Trans. Fuzzy Syst., 17 (4), pp. 863-873. , AugustLöfberg, J., YALMIP: A toolbox for modeling and optimization in MATLAB Proc. 2004 IEEE Int. Symp. on Comput. Aided Control Syst. Des., Taipei, Taiwan, September 2004, pp. 284-289. , http://control.ee.ethz.ch/~joloef/yalmip.phpDelmotte, F., Guerra, T.M., Ksantini, M., Continuous Takagi-Sugeno's models: Reduction of the number of LMI conditions in various fuzzy control design technics (2007) IEEE Transactions on Fuzzy Systems, 15 (3), pp. 426-438. , DOI 10.1109/TFUZZ.2006.88982

Domain of attraction estimation for nonlinear systems with fuzzy polynomial models

[EN] Many approaches in fuzzy systems literature express LMI conditions for a Takagi-Sugeno model and finish the problem once those conditions are feasible. However, studying the obtained region of attraction and its relationship with the original nonlinear problem is forgotten. This paper proposes to obtain a predefined-shape zone, as large as possible, belonging to the local domain of attraction of the origin of a nonlinear system. In order to do this, local fuzzy polynomial models are used whose analysis can be carried out by convex optimization (sum of squares). Moreover membership information is used in order to do iterations with the fuzzy modeling region, maximizing the size of the proven domain of attraction, which reduces conservatism over existing results.[ES] La mayor parte de referencias de la literatura en control borroso plantean condiciones LMI para un modelo Takagi-Sugeno y dan por terminado el problema una vez se obtienen resultados factibles. No obstante, dejan sin estudiar la región de atracción obtenida. Este tra-baajo propone probar que una zona, de forma prefijada, lo más grande posible, peertenece al dominio de atracción del origen de un sistema no lineal. Para ello, se usan modelos borrosos polinomiales cuyo análisis puede ser llevado a cabo mediante optimización convexa (su-mas de cuadrados). Asimismo, se utiliza información de la forma de las funciones de pertenencia para realizar iteraciones con la región de modelado borroso, maximizando la región de atracción probada, lo cual reduce el conservadurismo sobre otras propuestas.Este trabajo ha sido realizado parcialmente gracias al apoyo del Gobierno de España (DPI2008-06731-C02-01). El primer autor en particular agradece al Ministerio de Ciencia e Innovación (MICINN) por la beca FPI BES-2009-013882.Pitarch, J.; Sala, A.; Ariño, C.; Bedate, F. (2012). Estimación del dominio de atracción de sistemas no lineales mediante modelos borrosos polinomiales. 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Selective ℋ2 And ℋ∞ Stabilization Of Takagi-sugeno Fuzzy Systems

This paper presents new results concerning the stability analysis and design of state-feedback controllers for continuous-time Takagi-Sugeno (T-S) fuzzy systems via fuzzy Lyapunov functions. The membership functions of the T-S fuzzy systems are modeled in a space that is defined by the Cartesian product of simplexes called a multisimplex. If the time derivatives of the membership functions are bounded, the bounds are used to construct a polytope that models the space of the time derivatives of the membership functions. Linear matrix inequality (LMI) relaxations that are based on polynomial matrices are provided for stability analysis and controller design. Extensions for the design of control laws that minimize upper bounds to ℋ2 and ℋ∞ norms are also given. The main novelty of this method is that it allows one to synthesize control gains, which depends only on some premise variables that are selected by the designer. Numerical experiments illustrate the flexibility and advantages of the proposed method. © 2011 IEEE.195890900Takagi, T., Sugeno, M., Fuzzy identification of systems and its applications to modeling and control (1985) IEEE Trans. Syst., Man, Cybern., SMC-15 (1), pp. 116-132. , JanBoyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., (1994) Linear Matrix Inequalities in System and Control Theory, , Philadelphia, PA: SIAMGahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M., (1995) LMI Control Toolbox User's Guide, , Natick, MA: The Math WorksSturm, J.F., Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones (1999) Optim.Method Softw., 11 (1), pp. 625-653. , http://sedumi.mcmaster.ca/, [Online]Wang, H.O., Tanaka, K., Griffin, M.F., Parallel distributed compensation of nonlinear systems by Takagi-Sugeno fuzzy model (1995) Proc. IEEE 4th Int. Conf. Fuzzy Syst., 2nd Int. Fuzzy Eng. Symp., pp. 531-538. , Yokohama, Japan, MarWang, H.O., Tanaka, K., Griffin, M.F., An approach to fuzzy control of nonlinear systems: Stability and design issues (1996) IEEE Transactions on Fuzzy Systems, 4 (1), pp. 14-23. , PII S106367069600639XTanaka, K., Ikeda, T., Wang, H.O., Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs (1998) IEEE Transactions on Fuzzy Systems, 6 (2), pp. 250-265. , PII S1063670698008054Tanaka, K., Wang, H., (2001) Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, , New York, NY: WileyTeixeira, M.C.M., Assunção, E., Avellar, R.G., On relaxed LMIbased designs for fuzzy regulators and fuzzy observers (2003) IEEE Trans. Fuzzy Syst., 11 (5), pp. 613-623. , OctFang, C.-H., Liu, Y.-S., Kau, S.-W., Hong, L., Lee, C.-H., A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems (2006) IEEE Transactions on Fuzzy Systems, 14 (3), pp. 386-397. , DOI 10.1109/TFUZZ.2006.876331Delmotte, F., Guerra, T.M., Ksantini, M., Continuous Takagi-Sugeno's models: Reduction of the number of LMI conditions in various fuzzy control design technics (2007) IEEE Transactions on Fuzzy Systems, 15 (3), pp. 426-438. , DOI 10.1109/TFUZZ.2006.889829Tuan, H.D., Apkarian, P., Narikiyo, T., Yamamoto, Y., Parameterized linear matrix inequality techniques in fuzzy control system design (2001) IEEE Transactions on Fuzzy Systems, 9 (2), pp. 324-332. , DOI 10.1109/91.919253, PII S1063670601028259Montagner, V.F., Oliveira, R.C.L.F., Peres, P.L.D., Convergent LMI relaxations for quadratic stabilizability and H∞ control for Takagi-Sugeno fuzzy systems (2009) IEEE Trans. Fuzzy Syst., 17 (4), pp. 863-873. , AugSala, A., Arino, C., Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem (2007) Fuzzy Sets and Systems, 158 (24), pp. 2671-2686. , DOI 10.1016/j.fss.2007.06.016, PII S0165011407003284Johansson Mikael, Rantzer Anders, Arzen Karl-Erik, Piecewise quadratic stability of fuzzy systems (1999) IEEE Transactions on Fuzzy Systems, 7 (6), pp. 713-722. , DOI 10.1109/91.811241Feng, G., Chen, C.-L., Sun, D., Zhu, Y., H∞ controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions and bilinear matrix inequalities (2005) IEEE Transactions on Fuzzy Systems, 13 (1), pp. 94-103. , DOI 10.1109/TFUZZ.2004.839662Tanaka, K., Hori, T., Wang, H.O., A multiple Lyapunov function approach to stabilization of fuzzy control systems (2003) IEEE Trans. Fuzzy Syst., 11 (4), pp. 582-589. , AugDe Oliveira, M.C., Bernussou, J., Geromel, J.C., A new discrete-time robust stability condition (1999) Systems and Control Letters, 37 (4), pp. 261-265. , PII S0167691199000353Daafouz, J., Bernussou, J., Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties (2001) Systems and Control Letters, 43 (5), pp. 355-359. , DOI 10.1016/S0167-6911(01)00118-9, PII S0167691101001189Ebihara, Y., Hagiwara, T., New dilated LMI characterizations for continuous-timemultiobjective controller synthesis (2004) Automatica, 40 (11), pp. 2003-2009. , NovKim, E., Lee, H., New approaches to relaxed quadratic stability condition of fuzzy control systems (2000) IEEE Trans. Fuzzy Syst., 8 (5), pp. 523-534. , OctGuerra, T.M., Vermeiren, L., LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form (2004) Automatica, 40 (5), pp. 823-829. , MayDing, B., Sun, H., Yang, P., Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno's form (2006) Automatica, 42 (3), pp. 503-508. , DOI 10.1016/j.automatica.2005.11.005, PII S0005109805004085Kruszewski, A., Guerra, T.M., New approaches for the stabilization of discrete takagi-sugeno fuzzy models (2005) Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05, 2005, pp. 3255-3260. , DOI 10.1109/CDC.2005.1582663, 1582663, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05Kruszewski, A., Wang, R., Guerra, T.M., Nonquadratic stabilization conditions for a class of uncertain nonlinear discrete time TS fuzzy models: A new approach (2008) IEEE Transactions on Automatic Control, 53 (2), pp. 606-611. , DOI 10.1109/TAC.2007.914278Ding, B., Huang, B., Reformulation of LMI-based stabilisation conditions for non-linear systems in Takagi-Sugeno's form (2008) International Journal of Systems Science, 39 (5), pp. 487-496. , DOI 10.1080/00207720701832671, PII 791538772Ding, B., Stabilization of Takagi-Sugeno model via nonparallel distributed compensation law (2010) IEEE Trans. Fuzzy Syst., 18 (1), pp. 188-194. , FebTanaka, K., Hori, T., Wang, H.O., A fuzzy Lyapunov approach to fuzzy control system design (2001) Proceedings of the American Control Conference, 6, pp. 4790-4795Rhee, B.-J., Won, S., A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design (2006) Fuzzy Sets Syst., 157 (9), pp. 1211-1228. , MayTanaka, K., Ohtake, H., Wang, H.O., A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions (2007) IEEE Transactions on Fuzzy Systems, 15 (3), pp. 333-341. , DOI 10.1109/TFUZZ.2006.880005Mozelli, L.A., Palhares, R.M., Avellar, G.S.C., A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems (2009) Inf. 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H∞ And H2 Nonquadratic Stabilisation Of Discrete-time Takagi-sugeno Systems Based On Multi-instant Fuzzy Lyapunov Functions

The problem of state feedback control design for discrete-time Takagi-Sugeno (TS) (T-S) fuzzy systems is investigated in this paper. A Lyapunov function, which is quadratic in the state and presents a multi-polynomial dependence on the fuzzy weighting functions at the current and past instants of time, is proposed.This function contains, as particular cases, other previous Lyapunov functions already used in the literature, being able to provide less conservative conditions of control design for TS fuzzy systems. The structure of the proposed Lyapunov function also motivates the design of a new stabilising compensator for Takagi-Sugeno fuzzy systems. The main novelty of the proposed state feedback control law is that the gain is composed of matrices with multi-polynomial dependence on the fuzzy weighting functions at a set of past instants of time, including the current one. The conditions for the existence of a stabilising state feedback control law that minimises an upper bound to the H∞ or H2 norms are given in terms of linear matrix inequalities. 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