In this work, the Kalman Filter (KF) and Takagi–Sugeno fuzzy modeling technique are combined to extend the classical Kalman linear state estimation to the nonlinear field. The framework for such extension is given, and in this sense the discrete-time fuzzy Kalman filter (DFKF) is obtained. It will be shown that the fuzzy version gives some advantages when is compared with the Extended Kalman Filter (EKF), which is the most typical extension of the KF to the nonlinear field. The proposed approach provides a significantly smaller processing time than the processing time of the EKF while the mean square error is also reduced. Finally, some examples, such as the Lorenz chaotic attractor and under actuated mechatronic system (pendubot), are used to compare the DFKF and EKF.
Kalman filtering Fuzzy systems Chaotic systems
Authors want to express their deepest thank to CONACyT and Instituto Politécnico Nacional for the scholarships granted and research projects.
Compliance with ethical standards
Conflict of interest
The authors declare that they do not have conflict of interests.
Abdallah ZS, Gaber MM, Srinivasan B, Krishnaswamy S (2016) AnyNovel: detection of novel concepts in evolving data streams an application for activity recognition. Evolv Syst 7(2):73–93CrossRefGoogle Scholar
Maciel L, Ballini R, Gomide F (2016) Evolving granular analytics for interval time series forecasting, Granular. Computing 1:213–224Google Scholar
Marques Silva A, Caminhas W, Lemos A, Gomide F (2014) A fast learning algorithm for evolving neo-fuzzy neuron. Appl Soft Comput 14:194–209CrossRefGoogle Scholar
Meda-Campaña JA, Castillo-Toledo B, Chen G (2009) Synchronization of chaotic systems from a fuzzy regulation approach. Fuzzy Sets Syst 160:2860–2875MathSciNetCrossRefMATHGoogle Scholar
Meda-Campaña JA, Rodriguez-Valdez J, Hernandez-Cortes T, Tapia-Herreraand R, Nosov V (2015) Analysis of the Fuzzy controllability property and stabilization for a class of T-S Fuzzy Models. IEEE Trans Fuzzy Syst 23(2):291–301CrossRefGoogle Scholar
Moallem P, Mousavi BS, Naghibzadeh S. Sh (2015) Fuzzy inference system optimized by genetic algorithm for robust face and pose detection. Int J Artif Intell 13(2):73–88Google Scholar
Pratama M, Lu J, Lughofer E, Zhang G, Anavatti S (2016a) Scaffolding type-2 classifier for incremental learning under concept drifts. Neurocomputing 191:304–329CrossRefGoogle Scholar
Pratama M, Lu J, Anavatti S, Lughofer E, Lim C-P (2016b) An incremental meta-cognitive-based scaffolding fuzzy neural network. Neurocomputing 171:89–105CrossRefGoogle Scholar
Precup R-E, Tomescu ML, Radac M-B, Petriu EM, Preitl S, Dragos C-A (2012) Iterative performance improvement of fuzzy control systems for three tank systems. Expert Syst Appl 39:8288–8299CrossRefGoogle Scholar
Ruan D, Huang C (2000) Fuzzy sets and fuzzy information Granulation theory, selected papers by Lotfi A. Beijin Normal University Press, ZadehGoogle Scholar
Rubio JJ (2015) Adaptive least square control in discrete time of robotic arms. Soft Comput 19(12):3665–3676CrossRefGoogle Scholar
Rubio JJ (2016) Least square neural network model of the crude oil blending process. Neural Netw 78:88–96CrossRefGoogle Scholar
Rubio JJ, Yu W (2007) Nonlinear system identification with recurrent neural networks and dead-zone Kalman filter algorithm. Neurocomputing 70(13):2460–2466CrossRefGoogle Scholar
Song W, Liang J (2013) Difference equations of Lorenz System. Int J Pure Appl Math 83(1):101–110Google Scholar
Tanaka K, Wang HO (2001) Fuzzy control systems design and analysis: a linear matrix inequality approach. Wiley, New YorkCrossRefGoogle Scholar
Terejanu GA (2008) Extended Kalman Filter Tutorial, Department of Computer Science and Engineering, University at Buffalo, Buffalo, NYGoogle Scholar