Robust Controller Design Using Fictitious Parameter Vector and Parametrically Dependent Lyapunov Matrices: An Application to Visual Feedback Control

This note tackles visual-feedback controller design for a cart-driven inverted pendulum using non-calibrated images taken through a fisheye lens. The images have distortion, which can be modeled as an uncertainty in the output matrix for the measurement signals, due to the use of hemispherical lenses. This problem can be reduced to robust controller design for LTI parametrically dependent systems which belong to a subclass of Linear Parameter-Varying (LPV) systems. This research topic, i.e. robust controller design, is slightly obsolete; however, it is still a challenging problem since the well-established change-of-variables procedure for parametrically dependent plant systems produces not constant controllers but parametrically dependent controllers. On this issue, we first introduce a fictitious parameter vector in the change-of-variables procedure to obtain constant state-space matrices of the to-be-designed controllers in exchange for consequently introduced bilinear terms of decision matrices, and then apply overbounding approach based on Elimination lemma with an auxiliary decision matrix introduced for the bilinear terms to derive our proposed method. The method is still formulated in terms of bilinear matrix inequality, but has no structural constraints for the original decision matrices in the initial feasible solution search. We apply the proposed method to the stabilization problem of the cart-driven inverted pendulum. We successfully design a robust visual-feedback stabilizing controller, and lab-level experiments confirm that our controller stabilizes the system without any calibrations for the distorted images.