Stable multi-input multi-output adaptive fuzzy/neural control

In this letter, stable direct and indirect adaptive controllers are presented that use Takagi-Sugeno (T-S) fuzzy systems (1985), conventional fuzzy systems, or a class of neural networks to provide asymptotic tracking of a reference signal vector for a class of continuous time multi-input multi-output (MIMO) square nonlinear plants with poorly understood dynamics. The direct adaptive scheme allows for the inclusion of a priori knowledge about the control input in terms of exact mathematical equations or linguistics, while the indirect adaptive controller permits the explicit use of equations to represent portions of the plant dynamics. We prove that with or without such knowledge the adaptive schemes can "learn" how to control the plant, provide for bounded internal signals, and achieve asymptotically stable tracking of the reference inputs. We do not impose any initialization conditions on the controllers and guarantee convergence of the tracking error to zero.