Dynamics and Modal Control of Piezoelectric Tube Actuators  Jan R. van Hulzen * Georg Schitter ** Paul M.J. Van den Hof * Jan van Eijk *** * Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands, (e-mail:j.r.vanhulzen@tudelft.nl/jr.v.hulzen@nlda.nl). ** Automation and Control Institute, Vienna University of Technology, Gusshausstrasse 27-29, 1040 Vienna, Austria *** Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands Abstract: In piezoelectric positioning systems the achievable bandwidth is often limited by weakly damped resonant modes. System performance may be improved by avoiding the excitation of these modes. If sufficient mechanical damping is present this can be done by shifting resonant modes towards anti-resonant modes through manipulation of the mechanical boundary conditions. In a second approach the anti-resonances may be shifted towards the resonances by the application of modal actuation. Using this method the excitation of the second and higher order modes is avoided by changing the distribution of actuation forces. This paper investigates and compares the application of modal control techniques in systems based on piezoelectric tube actuators such as atomic force microscopes using finite element analysis and experimental verification. Keywords: Modal control, Mode analysis, Mechanical properties, Finite element analysis, Model based control, Microscopes 1. INTRODUCTION Piezoelectric actuators are used in applications where high precision positioning is required over a relatively short range. A typical example of such an application is the atomic force microscope (AFM) described by Binning (1986). A common problem in the design of such systems is the excitation of resonant modes which are close to the controller bandwidth. Possible solutions to this problem were investigated by Balas (1978) and later by Meirovitch (1983) and were based on avoiding excitation of these modes by adapting the control system architecture or by increasing the number of sensors and actuators to enable individual control of resonant modes. This last approach, described in Meirovitch (1982) is a form of distributed control and is commonly referred to as modal actuation. The technique was successfully applied in piezoelectric actuated systems by Lee and Moon (1990) who demon- strated a modal piezoelectric actuator based on electrodes which were shaped by etching. Alternative forms of modal actuators based on porous electrodes or electrodes with a honeycomb motive were described in Preumont (2003) and Preumont (2005). Modal filters based on arrays of individual piezoelectric transducers have been reported by Collins (1994) and by Meirovitch (1985).  This work is part of a project on Model-based subnano-positioning control systems for high-end professional equipment and microsys- tems manipulation sponsored by the Delft Center for Mechatronics and Microsystems. A key aspect of modal actuation is that the response to external excitation is changed without altering the natural modes or the natural frequencies. This is achieved by changing the distribution of forces applied to the system. The design of a modal actuator can be based directly on the mode shapes. Implementation of modal actuation is difficult if the mode shapes are very complex, if the flexible structure can only be actuated partially or in cases where the dynamics are uncertain due to changing load conditions or by the introduction of flexible loads. In an alternative approach proposed by Adriaens (2000), the natural modes are shifted towards the anti-resonances by manipulation of the mechanical boundary conditions. Sufficiently damped natural modes which are close to an anti-resonance will appear less pronounced reducing their impact on the design of the feedback control loop. This paper investigates the application of both forms of modal actuation to positioning systems where the piezoelectric tube actuator is the main flexible component. The objective is to determine which modes are important to the realization of the position control objective and to show that these modes can be controlled through modal actuation. To enable in-situ adaptation to changing load conditions the modal actuator is based on a sectioned electrode which allows tuning of the distribution of the electric field by varying the voltages supplied to each section. 5th IFAC Symposium on Mechatronic Systems Marriott Boston Cambridge Cambridge, MA, USA, Sept 13-15, 2010 978-3-902661-76-0/10/$20.00 © 2010 IFAC 317 10.3182/20100913-3-US-2015.00086