Nonlinear modeling and control approach to magnetic levitation ball system using functional weight RBF network-based state-dependent ARX model

A hybrid model, which adopts a radial basis function (RBF) neural networks with functional weights (FWRBF) to approximate the coefficients of the state-dependent AutoRegressive model with eXogenous input variables (SD-ARX), is built for modeling a magnetic levitation ball system and is referred to as the functional weight RBF nets-based ARX (FWRBF-ARX) model. This model structure, which may be identified by using the historical input/output data, inherits both the advantages of the FWRBF networks in function approximation and of the state-dependent ARX models in description of nonlinear dynamics. Due to the structured characteristics of the FWRBF-ARX model, an offline structured nonlinear parameter optimization method (SNPOM) is applied to identify the model structure and parameters. Using the input and output observation data of the real system, a FWRBF-ARX model with small residual, small standard variation and small long-term predictive residual can be identified. Based on the local linearity of the built FWRBF-ARX model at certain working point, a locally linearized model predictive controller (MPC) is designed to achieve stable levitating and output-tracking control of the steel ball in the electromagnetic field. From the real-time control results, it is seen that the FWRBF-ARX model-based MPC may control the steel ball to track step signals very well, and may obtain much better control performance within wide step range compared to conventional PID control, the ARX model and RBF-ARX model-based MPCs.