High Speed Inverse Model Implementation for Real-Time Control of Distributed Parameter Systems Operating in Nonlinear and Hysteretic Regimes

Ferroelectric (e.g., PZT), ferromagnetic, and ferroelastic (e.g., shape memory alloy) materials exhibit varying degrees of hysteresis and constitutive nonlinearities at all drive levels due to their inherent domain structure. At low drive levels, these nonlinear effects can be mitigated through feedback mechanisms or certain amplifier architectures (e.g., charge or current control for PZT) so that linear models and control designs provide sufficient accuracy. However, at the moderate to high drive levels where actuators and sensors utilizing these compounds often prove advantageous, hysteresis and constitutive nonlinearities must be incorporated into models and control designs to achieve high accuracy, high speed control specifications. In this paper, we employ a homogenized energy framework to characterize hysteresis in this combined class of ferroic compounds. We then use this framework to construct highly efficient inverse models that can be used to approximately linearize actuator dynamics for subsequent linear control design. For applications such as high speed tracking (e.g., kHz rates) or broadband vibration attenuation, the efficiency of the inverse construction proves crucial for realtime control implementation. We demonstrate aspects of the inverse model implementation in the context of rod models used to characterize PZT and magnetic actuators presently employed in applications ranging from nanopositioning to high speed milling of automotive components. I. FERROELECTRIC AND FERROMAGNETIC TRANSDUCERS Ferroelectric and ferromagnetic materials are employed in a wide variety of applications as both actuators and sensors, including fluid pumps, nanopositioning stages, sonar, vibration control, ultrasonic sources, and high-speed milling. They are attractive because the transducers are solid-state and often very compact. However, the coupling of nonmechanical field to mechanical deformation, which makes these materials effective transducers, also introduces hysteresis and timedependent behavior that must be accommodated by controllers. Classically, this has been attained by operating the transducer in only a small portion of its theoretical operating range and/or operating it at a low frequency. This allows controllers to accommodate nonlinear dynamics but reduces Supported by the US Department of Education GAANN Fellowship and the Air Force Office of Scientific Research under grant AFOSR-FA955004-1-0203. Thomas R. Braun is with the Center for Research in Scientific Computing, Department of Mathematics, North Carolina State University, Raleigh NC, 27606 USA (email: [email protected]) Ralph C. Smith is with the Center for Research in Scientific Computing, Department of Mathematics, North Carolina State University, Raleigh NC, 27606 USA (email: [email protected]) the performance of the transducer. More sophisticated approaches incorporate these dynamics through electromechanical or magnetomechanical models. In theory, this permits high-speed, high-accuracy control, but in practice introduces significant computational overhead which limits effectiveness. If the reference trajectory is known a-priori, a nonlinear control may be utilized which allows the computation to be performed a-priori as well (see [6]). Alternatively, one can employ a model-based inverse compensator to form a composite system which is approximately linear and timeinvariant. We focus on this latter approach and construct highly efficient implementation algorithms. II. DEVICE MODEL Several different models exist for ferroelectric and ferromagnetic materials. These include domain wall models, Preisach models, and homogenized energy models. Information and numerous references for these models may be found in [7]. We employ the homogenized energy model due to its solid basis in the underlying physics of the material, its ability to characterize effects such as minor loops and creep in a natural manner, and its applicability to both ferroelectric and ferromagnetic materials. Since we are typically interested in controlling the displacement and not the polarization or magnetization, the dynamics of the actuator, including its interface in with various plants, must also be incorporated. This is illustrated for the case when the actuator is a rod. A. Homogenized Energy Model Development The homogenized energy model for ferroelectric materials is