Differential Neuro-Fuzzy Controller for Uncertain Nonlinear Systems

In general, output-based controller design remains an important research area in control theory. Most of the existing solutions use a state estimation algorithm to reconstruct a plausible approximation of the real state. Then, one can apply a nonlinear controller, based on fuzzy logic, for example, to enforce the system trajectories to a desirable stable equilibrium point. Nevertheless, the aforementioned method may not be suitable for uncertain systems affected by external noises. State observers based on the system´s structure cannot be applied in those cases. However, some sort of adaptive estimation may be developed. This paper deals with a fuzzy controller that was designed using the state observer solution when the dynamic model of a plant contains uncertainties or it is partially unknown. Differential neural network (DNN) approach is applied in this uninformative situation. A new learning law, containing an adaptive adjustment rate, is suggested to enforce the stability condition for the observer´s free parameters. On the other hand, nominal weights are adjusted during the preliminary training process using the least mean square method. Lyapunov theory is used to obtain the upper bounds for the weight´s dynamics. The proposed method seems to be a more advanced option to control uncertain systems when the state available information is reduced. Even when several options exist to control this class of nonlinear systems such as PID, the method introduced here uses the knowledge on the system behavior and enforces the reconstruction of the immeasurable states. This last issue is an extra advantage because it serves as a general software sensor. The well-known two-link manipulator is used to show the effectiveness of the proposed algorithm. A couple of cases are used here: the full actuated and the under-actuated systems. In both situations, the controller achieves a better performance than the well-known PID controllers and a fuzzy controller using the estima- ed states produced by a high-order sliding-mode observer. A practical example showing how the fuzzy controller based on the estimated states produced by the differential neural network observer is also presented. The system used to test the controller is the anaerobic digestion. In this case, the benefits of this output-based controller are also demonstrated.