Frequency Response Function identification for multivariable motion control:Optimal experiment design with element-wise constraints

Frequency Response Functions (FRFs) are essential in mechatronic systems and its application ranges from system design and validation to controller design and diagnostics. The aim of this paper is to optimally design experiments for FRF identification of multivariable motion systems subject to element-wise power constraints. A multivariable excitation design framework is established that explicitly addresses the frequency-wise directionality of the system to be identified. The design problem involves solving a rank-constrained optimization problem, which is non-convex and NP-hard in most cases. Two algorithms to solving this problem approximately are presented that rely on a convex (semi-definite) relaxation of the original problem. Additionally, exact solutions for several special cases are presented. The two algorithms are shown to overcome the limitations of traditional excitation design. This is confirmed by experimental results from a 7 × 8 wafer stage setup, which show a significant improvement of the FRF quality using the proposed techniques over traditional design approaches.