Synthesis of Optimal Finite-Frequency Controllers Able to Accommodate Passivity Violations

In this paper, we explore the relationship between the hybrid passivity/finite-gain systems framework and the generalized Kalman-Yakubovich-Popov (GKYP) lemma. In particular, we investigate how to optimally design finite-frequency (FF) controllers that 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 that have experienced a passivity violation. We first review the hybrid passive/finite-gain systems framework and how linear time-invariant hybrid passive/finite-gain systems relate to systems with low-frequency FF positive real (PR) or SPR properties, and high-frequency FF BR properties as characterized by the GKYP lemma. Optimal design of FF SPR/BR controllers is considered next. A convex optimization problem constrained by a set of linear matrix inequalities is posed where constraints are imposed using various forms of the GKYP lemma, yielding optimal FF SPR/BR controllers. The FF SPR/BR controllers are optimal in that they approximate the traditional H₂ control solution. Finally, FF SPR/BR controllers are used within a gain-scheduling architecture to control a two-link flexible manipulator. Experimental results successfully demonstrate closed-loop stability and good closed-loop performance.