Research reports on designing adaptive controllers for nonlinear systems appear to be abundant. Numerous approaches are utilized; various systems are analyzed; and the behavior of these controllers, and their effect on the systems, under different assumptions, is observed.

In this paper, multiple input, multiple output (MIMO) nonaffine nonlinear systems are observed. Only the outputs in these systems are available for measurement. The research showcases the use of the hierarchical fuzzy-neural network (HFNN) to reduce the computation time compared to the conventional fuzzy-neural networks. The authors build systematic mathematical support for this adaptive controller, with clearly stated assumptions. These formalisms are translated into easy-to-follow algorithms, which are illustrated with simulations at the end of the paper.

Three illustrative examples are presented: “balancing double-inverted pendulums connected by a torsional spring,” “a general robot system with n inputs and n outputs,” and “a two-degree-of-freedom double pendulum.” The dimensionality of the hierarchical controller is lower than that of the conventional fuzzy-neural controller, and thus the time delay contributed to computing the control signal is significantly lower. The controller ensures that the signals “are bounded and that the outputs of the ... system track [the desired output trajectories] asymptotically.”