Automatica 49 (2013) 2006–2016 Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica An indirect adaptive servocompensator for signals of unknown frequencies with application to nanopositioning ✩ Alex Esbrook 1 , Xiaobo Tan, Hassan K. Khalil Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA article info Article history: Received 28 January 2012 Received in revised form 12 September 2012 Accepted 9 March 2013 Available online 13 May 2013 Keywords: Adaptive systems Servo-compensation Frequency estimation Microsystems: nano- and micro-technologies Hysteresis abstract We propose an adaptive servocompensator utilizing frequency estimation and slow adaptation for systems subject to inputs of unknown frequencies. We show that the proposed controller can achieve zero tracking error for a class of periodic references and disturbances, including scenarios specifically relevant to piezo-actuated nanopositioning systems. In particular, for the case of a sinusoidal reference input, we establish the exponential stability of the closed-loop system in the presence of harmonic disturbances, under certain conditions on the amplitudes of the reference and disturbances. We also prove exponential stability in the case of sinusoidal reference and disturbance with two distinct frequencies. Additionally, we show that the proposed method, in conjunction with approximate hysteresis inversion, can attenuate the effect of hysteresis nonlinearity preceding linear dynamics and ensure the boundedness of the closed- loop system. Experiments conducted on a commercially available nanopositioner confirm our theoretical analysis and demonstrate the effectiveness of the proposed method as compared to Iterative Learning Control, a competitive technique in nanopositioning for tracking periodic references. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction The tracking problem for both linear and nonlinear systems has been a commonly explored topic in the control literature. Among the variety of techniques employed for solving such problems are servocompensators, also known as internal model controllers, which were developed for linear systems in the 1970’s (Davison, 1972; Francis & Wonham, 1975). The most appealing feature of servocompensators is that, in the presence of plant uncertainty, they can completely cancel disturbances whose internal models are contained in the controller as long as the system remains stable. Isidori and Byrnes extended the internal model technique to nonlinear systems in Isidori and Byrnes (1990), and many authors have coupled servocompensators with adaptive controllers to address unknown internal models (Elliott & Goodwin, 1984; Nikiforov, 1998; Serrani, Isidori, & Marconi, 2001). The applications ✩ This work was supported by the National Science Foundation (CMMI 0824830). The material in this paper was partially presented at the 2012 American Control Conference (ACC2012), June 27–29, 2012, and the 50th IEEE Conference on Decision and Control (CDC) and the European Control Conference (ECC), December 12–15, 2011, Orlando, Florida, USA. This paper was recommended for publication in revised form by Associate Editor Andrea Serrani under the direction of Editor Miroslav Krstic. E-mail addresses: esbrooka@msu.edu (A. Esbrook), xbtan@egr.msu.edu (X. Tan), khalil@egr.msu.edu (H.K. Khalil). 1 Tel.: +1 2484444583; fax: +1 517 353 1980. of internal model controllers are also diverse. For example, Isidori, Marconi, and Serrani applied an adaptive servocompensator to an altitude tracking problem in helicopters (Isidori, Marconi, & Serrani, 2003), and Singh and Schy utilized a servocompensator to control an elastic robotic arm (Singh & Schy, 1986). Another popular topic in the literature over the past two decades has been the control of smart materials and other systems with hysteresis (Cavallo, Natale, Pirozzi, & Visone, 2003; Iyer, Tan, & Krishnaprasad, 2005; Tan & Baras, 2004; Tan & Iyer, 2009). Piezoelectric-actuated systems in particular have generated a great deal of interest due to their use in nanopositioner applications, such as Scanning Probe Microscopy (SPM) (Devasia, Eleftheriou, & Moheimani, 2007). From a theoretical perspective, an intriguing element in the control of piezo-actuated systems is dealing with the strong coupling between uncertain vibrational dynamics and the hysteresis nonlinearity (Devasia et al., 2007). In such devices, hysteresis is caused by the existence of multiple stable equilibria of the polarization state for any applied electric field (Smith, 2005), and, along with vibration and creep, it is a major obstacle impeding high-accuracy, high-speed tracking (Croft, Shed, & Devasia, 2001). Due to high performance demands in SPM applications, there are many ongoing efforts to apply advanced control techniques to nanopositioning systems. H ∞ control (Salapaka, Sebastian, Cleveland, & Salapaka, 2002) and 2-degree-of-freedom control (Lee & Salapaka, 2009) have been shown to provide robustness to plant uncertainty and facilitate tracking in the presence of hysteresis. In the work of Zhong and Yao (2008), the hysteresis effect was 0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.03.016