Multi-Step-Ahead Prediction with Gaussian Processes and TS-Fuzzy Models

Fuzzy multiple modeling is a method that combines analytical and black-box modeling in an optimal way. On the other hand, modeling with Gaussian processes is a probabilistic and non-parametric method which allows the prediction of the uncertainty of the model. This is of advantage for one-step ahead or even multi-step ahead predictions with noisy time series and disturbed closed loop control systems. The multi-step ahead prediction assumes the previous outputs and control values to be known as well as the future control values. A "naive" multi-step ahead prediction is a consecutive one-step ahead prediction whereas the outputs in each consecutive step are considered as inputs for the next step of prediction. Usually for closed loop control systems the nominal output trajectory is known in advance. However, because of the uncertainties and disturbances in the control loop the resulting control trajectory is only known up to the present time step but not for the future steps. To obtain the future control inputs for the multi-step ahead prediction the system is modeled by a multiple TS fuzzy model which was trained in advance to generate a nominal control trajectory in closed loop for a given nominal output trajectory. Simulations of nonlinear systems with built-in uncertainties illustrate the good performance of the multi-step ahead prediction with the combination of TS fuzzy models and Gaussian process models