Two-degrees-of-freedom PI²D controller for precise nanopositioning in the presence of hardware-induced constant time delay

The fast and accurate tracking of periodic and arbitrary reference trajectories is the principal goal in many nanopositioning applications. Flexure-based piezoelectric stack driven nanopositioners are widely employed in applications
where accurate mechanical displacements at these nanometer scales are required. The performance of these nanopositioners is limited by the presence of lightly damped resonances in their dynamic response and actuator nonlinearities. Closed-loop control techniques incorporating both damping and tracking are typically used to address these limitations. However, most tracking schemes employed use a first-order integrator where a triangular trajectory commonly used in nanopositioning applications necessitates a double integral for zero-error tracking. The phase margin of the damped system combined with the hardware-induced delay deem the implementation of a double-integrator unstable. To overcome this limitation, this paper presents the design, analysis and application of a new control scheme based on the structure of the traditional Two-Degrees-of-Freedom PID controller (2DOF-PID). The proposed controller
replaces the integral action of the traditional 2DOF-PID with a double integral action (2DOF-PI2D). Despite its simplicity, the proposed controller delivers superior tracking performance compared to traditional combined damping and
tracking control schemes based on well-reported designs such as positive position feedback (PPF), Integral resonant control (IRC), and Positive Velocity and Position Feedback (PVPF). The stability of the control system is analyzed in
the presence of a time delay in the system. Experimental results validating the efficacy of the proposed chattering-free control of a piezo-driven nanopositioning system are included.