Multivariate B-Spline Scheduling for Linear Parameter Varying Model

Abstract

Linear Parameter-Varying (LPV) models provide means to approximate complex, nonlinear, and time-varying system dynamics using a set of Linear Time-Invariant (LTI) models, interpolated by a scheduling function to ensure smooth transitions across the system’s operating envelope. This study demonstrates that multivariate simplex B-splines can serve as such function, evaluated for State-Space quasi-LPV (SS-qLPV) models by providing a global approximation using local basis functions. The Inverted Pendulum on a Cart Model (IPCM) is used as a demonstrator in an open-loop setting, with an affine LPV representation based on cart velocity and pendulum angle as scheduling parameters. Several scheduling function estimation methods: piecewise constant Zero-Order Hold (ZOH), polynomial uni and multi-variate Ordinary Least Squares (OLS), and multivariate simplex B-splines are evaluated. Results indicate that, at the same polynomial order, B-splines show higher approximation capabilities compared to polynomial methods, as shown by the root mean squared error (RMSE) of the residuals. However, under broader simulation conditions, LPV-ZOH can be computationally less expensive and can achieve lower RMSE, although piecewise constant methods have discontinuities at the switching points, which can have an impact to closed-loop performance. The study highlights trade-offs in scheduling
function selection and suggests future research in optimizing simplices for improved performance. Applying B-spline scheduling functions with gain scheduled controllers in closed-loop control is the next direction for increasing control performance in complex, high-dimensional systems.