Multiple DNN identifier for uncertain nonlinear systems based on Takagi-Sugeno inference

In nature, most systems show nonlinear complex behaviors. Among other characteristics, plants present a high degree of oscillation over time. Adaptive algorithms used to approximate such difficult behaviors show some important deficiencies. Many adaptive non-parametric methods cannot reconstruct the trajectories of such complex dynamics. Differential neural networks (DNNs) are no exception. When just one DNN is applied to achieve an approximation, the identification error may significantly differ from zero. A natural trick to overcome this difficulty is to increase the number of neurons or to increase the number of layers. Another possible suggestion is to define a set of neural networks working together (usually in parallel). The members of such a set each work on well-defined trajectories contained in specific subspaces in which the uncertain system may evolve. Nevertheless, a decision system is required to define the contribution of each DNN in the final identification scheme. One of the most successful methodologies for constructing this selector is based on a Takagi-Sugeno (TS) inference system. This paper discusses how to combine the identification properties offered by a continuous neural network and the characteristic decision capabilities of fuzzy methods. The selection of which neural network is activated depends on the decision achieved by a TS fuzzy system. The convergence of this algorithm is proved using a quadratic Lyapunov function. A complete description of the learning laws used for the set of DNN identifiers is also obtained. The Chen circuit and the Rabinovich-Fabrikant system are used to demonstrate the superior performance achieved by this mixed DNN and fuzzy system, usually called a neuro-fuzzy system.