A Kernel-Based PCA Approach to Model Reduction of Linear Parameter-Varying Systems

Abstract

This brief presents a model reduction method for linear parameter-varying (LPV) systems using kernel-based principal component analysis (PCA). For state-space LPV models that are affine or rational in the scheduling variables and in which the variation of these variables is confined in a polytope, controller synthesis can be elegantly realized by solving the synthesis problem only at the vertices of the polytope. To exploit the computational simplicity of this approach, it is highly desirable to obtain LPV models of systems of interest in an affine or a rational form. In this respect, kernel PCA allows one to extract principal components of a given data set of scheduling variables in a high-dimensional feature space, reducing complicated coefficient dependencies that otherwise might not be easily reducible in a linear subspace; this gives kernel PCA an advantage over its linear PCA counterpart. We show that high-dimensional scheduling variables can be mapped into a set of low-dimensional variables through a nonlinear kernel PCA-based mapping. Since the kernel PCA mapping is nonlinear, finding the inverse mapping in order to represent the original scheduling variables requires solving a nonlinear optimization problem; consequently, the reduced LPV model is no longer affine in the reduced scheduling variables. To address this, we formulate an optimization problem to obtain a reduced model that is either affine or rational in the reduced scheduling variables. We apply the proposed model reduction method on a robotic manipulator system and use the reduced LPV model to design a gain-scheduled controller that satisfies an induced $\mathcal {L}_{2}$ gain performance. Numerical simulations are used to demonstrate the performance of the resulting LPV controller on the nonlinear manipulator model. The achieved performance of the LPV controller with the kernel PCA-based reduced model is also compared with that of its linear PCA-based counterpart.