Synthesis of Optimal Finite Frequency Controllers for Flexible Robotic Manipulator Control

Abstract

In this paper, we explore the relationship between the hybrid passivity and finite gain systems framework and the generalized Kalman-Yakubovich-Popov (GKYP) Lemma. In particular, we investigate how to optimally design finite frequency (FF) controllers which possess strictly positive real (SPR) properties over a low frequency range, and bounded real (BR) properties over a high frequency range. Such FF SPR/BR controllers will be used to control systems which have experienced a passivity violation. We first review the hybrid systems framework and how linear time-invariant hybrid systems relate to FF positive real (PR), FF SPR, and FF BR systems and the GKYP Lemma. A convex optimization problem is posed where constraints are imposed via linear matrix inequalities yielding optimal FF SPR/BR controllers. The FF SPR/BR controllers are optimal in that they approximate the traditional H2 control solution. FF SPR/BR controllers are used to control both single- and two-link flexible manipulators. Experimental results successfully demonstrate closed-loop stability via the hybrid systems framework, and implementation of the proposed controller synthesis scheme.