Online distributed fuzzy modeling of nonlinear PDE systems: Computation based on adaptive algorithms

With the emergence of novel model-based controllers for partial differential equation (PDE) systems, identifying the mathematical model of PDE systems has become a promising and complicated research topic. This paper suggests a new method to identify an adaptive Takagi–Sugeno (TS) fuzzy PDE model for nonlinear multi-input multi-output (MIMO) first-order PDE systems. The proposed approach is performed online based on the measured input and output data of the nonlinear PDE systems. Furthermore, the identification process will be obtained for the cases that the noise is either white or colored. For the case of white noise, a nonlinear recursive least square (NRLS) approach is applied to identify the nonlinear system. On the other hand, when the colored noise is exerted to the nonlinear PDE system, the fuzzy PDE model of the nonlinear PDE system and also nonlinear colored noise are identified based on the nonlinear extended matrix methods (NEMM). Moreover, the problem of identification for both colored and white noise cases is investigated when premise variables of membership functions are known or unknown. Finally, in order to illustrate the effectiveness and merits of the proposed methods, the identification method is applied to a practical nonisothermal Plug-Flow reactor (PFR) and a hyperbolic PDE system with Lotka–Volterra type applications. As it is expected, the evolutions of the error between the state variables for the obtained TS fuzzy PDE model and the output data converge to the zero in the steady-state conditions. Thus one concludes, the proposed identification algorithm can accurately adjust both consequents and antecedents parameters of TS fuzzy PDE model.