Granular fuzzy PID controller

In this paper an uncertain dynamical system is investigated in which the coefficients are as a class of fuzzy sets and the fuzzy derivative is considered as the granular derivative. Furthermore, the notions of granular second order derivative of a fuzzy function, fuzzy overshoot, fuzzy rise-time, and fuzzy peak-time are introduced. As a result, designing a class of PID controllers called granular Fuzzy PID (gr-FPID) controller is presented based on fuzzy mathematics. The gr-FPID consists of granular integral, granular derivative with fuzzy coefficients. Moreover, the Particle Swarm Optimization (PSO) algorithm is used to tune gr-FPID fuzzy coefficients. It is demonstrated that the gr-FPID controller can effectively control the temperature in a continuous stirred tank reactor in which the parameters are uncertain. •Presenting granular second order derivative of a fuzzy function.•Defining fuzzy overshoot, fuzzy rise time, and fuzzy peak time.•Presenting an approach for the linearization of a nonlinear model around a fuzzy operating point.•A new kind of fuzzy PID controller is designed (gr-FPID).•Explaining the process of PSO algorithm for tuning the fuzzy parameters of gr-FPID.