Ensembles of Fuzzy Linear Model Trees for the Identification of Multioutput Systems

We address the task of discrete-time modeling of nonlinear dynamic systems with multiple outputs using measured data. In the area of control engineering, this task is typically converted into a set of classical regression problems, one for each output, which can then be solved with any nonlinear regression approach. Fuzzy models, in the Takagi-Sugeno form, are popular in this context. We use Lolimot, which is a tree learning method, to build fuzzy linear model trees. In this paper, we propose, implement, and empirically evaluate three extensions of fuzzy linear model trees. First, we consider and evaluate multioutput models. Second, we propose to use ensembles of such models. Third, we investigate the use of a search heuristic based on simulation error (as opposed to one-step-ahead prediction error), specific to the context of modeling dynamic systems. Finally, we perform an empirical evaluation and compare these approaches on six multioutput case studies, using both measured and simulated data, with noise: The case studies include modeling of the inverse dynamics of a robot arm, as well as five additional process-industry systems. Ensembles improve the performance of both single- and multioutput trees, while the heuristic specific to modeling dynamic systems only improves performance very slightly. Multioutput model trees exhibit comparable or worse predictive performance to a set of single-output models, while providing a more compact model. Overall, we can recommend the use of bagging of single-output Lolimot models, learned by using the simulation error as a search heuristic.