Input/output stability theory for direct neuro-fuzzy controllers

In an input/output (I/O) setting, we undertake a detailed theoretical investigation of the stability of a given direct static multiple-input single-output neuro-fuzzy controller operating under feedback control, dependent only on the functional gain of the plant to be controlled. It is shown that various stability regions in weight space are convex, and necessary and sufficient conditions are given for these stability regions to be open and bounded. The convexity results coupled with the stability test give a practical method for constructing the stability regions. We show that an adaptive neuro-fuzzy controller is stable under feedback if we constrain the weights of the controller to lie within any compact set within the stability region. Combining a projection operator with any standard training law can thus give a stable adaptive controller.