Gain estimation of nonlinear dynamic systems modeled by an FBFN and the maximum output scaling factor of a self-tuning PI fuzzy controller

• The stability condition is shown for a general case of PI fuzzy controllers with an unrestricted number of input and output membership functions.
• A novel method is shown to extract the local Lipschitz norm from the Fuzzy Basis Function Network (FBFN model).
• The authors combined the above results into a stability condition for a fuzzy control system based on FBFN models.
• The obtained stability condition is applied to the design of a stable self-organizing four-layer fuzzy controller that can control nonlinear systems and accommodate time-varying parameters.

This paper proposes new techniques to calculate the dynamic gains of nonlinear systems represented by fuzzy basis function network (FBFN) models. The dynamic gain of an FBFN can be approximated by finding the maximum of norm values of the locally linearized systems or by solving a non-smooth optimal control problem. From the proposed gain calculation techniques, a novel adaptive multilevel fuzzy controller (AMLFC) with a maximum output scaling factor is presented. To guarantee the system stability, a stability condition is derived, which only requires that the output scaling factor of the AMLFC be bounded. Therefore, this paper provides a systematic and simple design practice for controlling nonlinear systems by using an AMLFC. The AMLFC is simulated in a tower crane control system. Simulation results show that AMLFC is not only robust but also provides improved transient performances compared with the robust adaptive fuzzy controller.