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.

在具有电容式位移传感器的纳米定位系统中，模数（A / D）转换时间不可忽略，特别是对于具有16个（或更多）数字位的亚纳米分辨率情况。由于延迟块的无限维数，传感器引起的延迟将使此类系统的建模和控制复杂化。在本文中，我们使用具有不确定性和测量延迟的类似于Hammerstein的模型来分析压电纳米定位器。无限维${\mathcal{H}}^{\infty }$提出了一种用于压电平台的鲁棒和纳米定位控制的控制方法。为了消除脆弱性的问题${\mathcal{H}}^{\infty }$控制器的实现，推导了包括有限冲激响应（FIR）滤波器的等效结构，并给出了基于FIR滤波器的基于矩阵函数的有理逼近（MFRA）。利用提出的控制方法进行了实时实验，对定位性能，鲁棒性和磁滞补偿能力进行了综合评估。比较研究表明，与常规方法（例如比例-积分-微分（PID）控制和有限维鲁棒控制）相比，已有显着改进。