Enhanced robust nanopositioning control for an X-Y piezoelectric stage with sensor delays: An infinite dimensional ${\mathcal{H}}^{\infty }$ optimization approach

Abstract

In nanopositioning systems with capacitive displacement sensors, the analog-to-digital (A/D) conversion time is not negligible, particularly for the sub-nano resolution cases with 16 (or more) digital bits. The sensor induced delay will complicate modeling and control of such systems, due to the infinite dimensionality of the delay block. In this paper, we analyze the piezoelectric nanopositioner with a Hammerstein-like model incorporating uncertainties and a measurement delay. An infinite dimensional ${\mathcal{H}}^{\infty }$ control approach is proposed for robust and nanopositioning control of the piezoelectric stage. To eliminate the fragility problem of the ${\mathcal{H}}^{\infty }$ controller on its implementation, an equivalent structure including a Finite Impulse Response (FIR) filter is derived, and the matrix-function-based rational approximation (MFRA) for the FIR filter is presented. Real time experiments with the proposed control method are conducted where the positioning performance, robustness, and hysteresis compensation capability are comprehensively evaluated. Comparative studies demonstrate significant improvements over conventional methods such as Proportional–Integral–Derivative (PID) control and the finite dimensional robust control.

Introduction

Piezoelectric stages are widely applied in nano servo applications, such as atomic force microscopy [1] and microinjection system [2]. The nano scale motion of piezoelectric stages can be attributed to the compliant structure driven by piezoelectric actuators [3], [4], [5], while the output displacements are captured by nano resolution sensors. To achieve high-precision servo performances, control methods of such systems have attracted significant research efforts in past decades, where complex model dynamics such as uncertainty and hysteresis were extensively studied [6], [7], [8], [9].

In particular, robust control theory has been well developed to achieve desired performance specifications with robustness against various uncertainties [10], [11], [12], [13], [14], [15], [16]. Note that more and more piezoelectric stages are employing capacitive sensors to measure output displacement, thanks to their sub-nano resolution and capability to measure absolute position zero. However, to achieve sub-nano resolution digital displacement signals for digital servo purpose, ultra-high resolution analog-to-digital (A/D) conversion process, as well as signal filters, is required, which inevitably results in a significant delay in the feedback loop [1], [17]. For systems with delays, the above robust control methods cannot be applied directly due to the infinite dimensionality of the delay.

The presence of delays deteriorates stability margin and makes control design much more complicated [18]. To deal with the infinite dimensionality with engineering applicable structures, various studies on robust control for time-delay systems have been reported, for example, the operator-theoretic method [19], [20] and the J-Spectral factorization method [21]. As an alternative, the Padé approximation based method [17] and the Smith predictor based method [22] were well developed for nanopositioning systems with delays to facilitate the application of finite dimensional control methods [23]. However, these two control schemes do not handle the infinite dimensional delay block directly. The robustness of the both methods has attracted long-lasting discussions [18].

In this paper, we investigate the robust nanopositioning control problem for piezoelectric stages subject to a sensor delay and uncertainties. In contrast to most existing results on finite dimensional strategies, an infinite dimensional ${\mathcal{H}}^{\infty }$ control approach handling the delay block directly is considered. The main contributions of this work are highlighted as follows:

• The Hammerstein-like model of the piezoelectric nano stage is transformed into a linear time-delay system with one lumped uncertainty to facilitate formulating control issues as an infinite dimensional ${\mathcal{H}}^{\infty }$ optimization problem.

• An infinite dimensional ${\mathcal{H}}^{\infty }$ control method is presented with a Finite Impulse Response (FIR) filter based controller structure, where the extension of the infinite dimensional ${\mathcal{H}}^{\infty }$ control theory to non-fragile digital implementation is realized.

• Comparative experiments on the piezoelectric stage are conducted, demonstrating significant improvements on positioning accuracy, transient performance, as well as hysteresis compensation capability.

This paper is organized as follows. In Section 2, the dynamic model of the investigated piezoelectric stage is derived and the positioning control problem for the nanopositioning system with capacitive sensors feedback is briefly discussed. The infinite dimensional ${\mathcal{H}}^{\infty }$ control design is proposed in Section 3. Real time experiments and comparative studies are conducted in Section 4, followed by the conclusion in Section 5.