Multivariable PPF Control of an Active Structure

Abstract

This paper reports experimental implementation of extended positive position feedback (PPF) controller on an active structure consisting of a cantilevered beam with bonded collocated piezoelectric actuators and sensors. Stability conditions for PPF control are rederived to allow for a feedthrough term in the model of the structure which is needed to ensure little perturbation in the in-bandwidth zeros of the model. The set of stabilizing PPF controllers is shown to be a convex set characterized by a set of linear matrix inequalities. A multivariable PPF controller is designed and successfully implemented on the structure. I. INTRODUCTIONA major difficulty in control of flexible structures is due to the fact that they are distributed parameter systems. Consequently, these structures have a very large number of vibration modes and their transfer functions contain many poles close to the jω axis. These systems are generally difficult to control.Very often a small number of in-bandwidth modes of the structure are required to be controlled, and it is possible that some in-bandwidth modes are not targeted to be controlled at all. The presence of uncontrolled modes can lead to the problem of spillover [1]. That is, the control energy is channeled to the residual modes of the system and this process may destabilize the closed-loop system. In particular, the spillover effect is of major concern at higher frequencies where obtaining a precise model of the structure is rather difficult.One approach to overcome the spillover effect is based on using collocated sensors and actuators. Positive position feedback as proposed by Caughey and co-authors [5], [4] is a control design technique for flexible structures with collocated sensors and actuators, which is insensitive to spillover effect. The PPF does not guarantee unconditional stability of the closed-loop system, however, it does guarantee stability in presence of uncontrolled in-bandwidth modes, and it has the additional property that it rolls off quickly at higher frequencies.One of the shortcomings of the PPF as proposed in [5] and [4] is that the effect of out-of-bandwidth modes on the dynamics of the controlled modes is ignored. This effect can be captured by adding a feed-through term to the truncated model of the structure. In this paper we derive stability conditions for PPF controllers when the underlying structure model contains a feed-through term. We also design a multivariable PPF controller for a test structure consisting of a cantilevered beam and several bonded piezoelectric actuators and sensors. Although a drastic simplification of a

Negative Imaginary Systems Theory in the Robust Control of Highly Resonant Flexible Structures

“…In the SISO case, a positive-position feedback controller (e.g., see [5], [6], [12], [38]- [44]) is a controller of the form…”