Adaptive Robust Repetitive Control of Piezoelectric Actuators

Positioning stages using piezoelectric stack actuators (PEA) possess very high theoretical bandwidth and resolution. However, their tracking performance deteriorates as nonlinear dynamics due to inherent hysteresis starts to dominate when the total length of travel increases. When tracking periodic trajectories, which is common in most industrial applications, the uncertain nonlinearity from hysteresis also becomes periodic. Having identified and separated the fast and slow dynamics of the total stage response, we adopt a simple first-order model assuming the inertial dynamics of the stage is negligible in the low tracking frequency range. By approximating the hysteresis mapping function with simple functions, the overall system model is linearly parameterized for subsequent adaptive robust controller design. Exploiting the periodicity of the uncertain hysteresis nonlinearity, it is further parameterized as finite series of harmonic functions, which eliminates the need for an exact inversion model for hysteresis while achieving high tracking accuracy. Experimental results from tracking control of sinusoidal and typical triangular trajectories show tracking error close to the sensor noise level and demonstrate the effectiveness of the approach. INTRODUCTION Positioning stages for high-precision positioning and tracking applications, especially atomic force microscopy that requires resolution on the nanometer level, almost unanimously use piezoelectric actuators, because they are capable of producAddress all correspondence to this author. 1 ing sub-nanometer displacements due to the inverse piezoelectric coupling effect. And they are also capable of generating large forces, which deliver very high bandwidth when fitted with lowinertia platform and high-stiffness flexures in typical positioning mechanisms [1]. When driven at small strain levels, the dynamics of piezoelectric actuators can be described by the classical equation of linear piezoelectricity [2]. However, as the demand for range and driving frequency increases for today’s applications, we are faced with major nonlinearity inherent in piezoelectric materials, in particular hysteresis, which typically results from ferroelectric phase transitions in most actuators made of lead zirconate titanate ceramics (PZT). It leads to severe positioning errors if not properly modeled and compensated. To compensate the hysteresis effect and achieve higher positioning accuracy, various schemes have been proposed, most of which employ both model-based feedforward and robust feedback control. For quasi-static or low frequency applications, the classic Preisach model is a popular choice for feedforward compensation, which approximates the hysteresis with a set of simple relay operators [3,4]. Despite their success, the large set of operator parameters necessary for higher accuracy makes it difficult to identify and implement online. The classic Preisach model is also invalid for faster operations unless certain dynamic extensions are made, but the inversion of existing dynamic Preisach models are known to be difficult [5]. In addition, due to the loading history dependence of the hysteresis, the actuator often starts from an unknown initial state that differs from the one under which the parameters were measured offline. Thus a different Copyright c © 2005 by ASME approach with less parameters and easy to adapt on-line is needed for better tracking. The difficulty for perfect tracking is much reduced when the desired trajectory is repetitive. When a piezoelectric actuator is driven by a periodic input, it converges to a steady-state hysteresis loop after a certain number of cycles, a phenomenon commonly known as “accommodation” [5]. Therefore for periodic trajectories, the required input is highly periodic after the first few periods. The input can thus be obtained by methods such as iterative learning control (ILC), which updates the input signal using the error signal from the previous period and avoids inverting complex nonlinear models. This has been shown to achieve exact tracking for a piezoelectric positioner in [6], but the convergence of their method is guaranteed only for trajectories that satisfy the classic Preisach model, which are slow or pseudo-static trajectories. Though easy to implement, such a method require large computer memory and are sensitive to noise because the physical dependence of the unknown nonlinearity over the same period is completely overlooked. The known dynamics of the system is also hard to be incorporated. The tracking error in [6] starts from 100% of the total travel and slowly converges to zero after about 50 periods. This may be too long for real world applications. A simple remedy to this problem for repetitive control is proposed by Xu and Yao in [7]. By recognising the physical dependence of the values of periodic uncertainties over the same period and using certain known basis functions to capture such dependence, only the amplitudes of the basis functions are needed for parameterization, which can be easily adapted online. It also overcomes the sensitivity to noise because the basis functions naturally smooths out the effect of random noises. Incorporated with the adaptive robust control (ARC) scheme, it guarantees good transient performance, fast convergence, and perfect tracking when enough number of basis functions are used to parameterize the uncertain nonlinearity. In this paper, we use a simple first order model that describes the dominant relaxation dynamics of the hysteretic response for a piezoelectric stage with negligible inertial dynamics in our desired range of operation. The adaptive robust repetitive control scheme mentioned above is applied to the model, which adapts the unknown parameters using a discontinuous projection based method, and the uncompensated nonlinearities are attenuated by certain robust control laws. No exact model of the hysteresis is needed, and the steady-state tracking error is reduced to almost the sensor noise level for sinusoidal trajectories up to 100 Hz and pseudo-triangular (with smoothed turnaround points) trajectories up to 50 Hz. Transient error is less than 3 percent of the total length and convergence happens within 2 periods, demonstrating the effectiveness of the method. 2 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Control Input voltage (V) O ut pu t v ol ta ge (V ) Pseudo−static Loop Step response Relaxation