Piecewise polynomial system approximation for nonlinear control

For nonlinear systems which are linear-in-the-control, the differential geometry based control schemes that have recently been developed require that an accurate smooth model of the nonlinear system be available. When the nonlinear dynamics are unknown, or known but in an inappropriate form for differential geometry calculations, difficulties arise. The approximation and control schemes studied here attempt to resolve these difficulties by casting the original problem into a simpler form. Given data from the nonlinear system, its state-space is partitioned into cells, and a polynomial-in-state model is determined for each cell. After identification, the cell models constitute an overall piecewise polynomial system approximation. Feedback linearization techniques are then performed on the approximated cell models. The overall controller is formed by joining the individual cell controllers to form a global nonlinear controller. Both least squares regression and polynomial spline interpolation are examined for the procedures mentioned above. The calculations needed to perform the identification and the feedback linearization control design are shown in a generic form. In addition, the feedback linearization controller derived from spline based identification is shown to converge to a feedback linearization controller designed with perfect knowledge of the system dynamics as the spline mesh size approaches zero. Using least squares smoothing, the problems of noisy data and data regularization, which are common in spline interpolation schemes, are addressed. The results of simulating these control schemes on various example nonlinear systems show that the resulting controllers' performance approaches that of a standard input-output feedback linearization controller designed with perfect knowledge of the system dynamics.