Non-Minimum-Phase Resonant Controller for Active Damping Control: Application to Piezo-Actuated Nanopositioning System

Nanopositioning systems frequently encounter limitations in control bandwidth due to their lightly damped resonance behavior. This paper presents a novel Non-Minimum-Phase Resonant Controller (NRC) aimed at active damping control within dual closed-loop architectures, specifically applied to piezo-actuated nanopositioning systems. The control strategy is structured around formulated objectives for shaping sensitivity functions to meet predetermined system performance criteria. Leveraging non-minimum-phase characteristics, the proposed NRC accomplishes complete damping and the bifurcation of double resonant poles at the primary resonance peak through a constant-gain design accompanied by tunable phase variation. The NRC demonstrates robustness against frequency variations of the resonance arising from load changes and is also capable of damping higher-order flexural modes simultaneously. Furthermore, by establishing high gains at low frequencies within the inner closed-loop and integrating it with a conventional PI tracking controller, the NRC achieves substantial dual closed-loop bandwidths that can exceed the first resonance frequency. Moreover, the NRC significantly diminishes the effect of low-frequency reference signals on real feedback errors while effectively rejecting disturbances proximate to the resonance frequency. All contributions are thoroughly formulated and exemplified mathematically, with the controller’s performance confirmed through an experimental setup utilizing an industrial nanopositioning system. The experimental results indicate dual closed-loop bandwidths of 895 Hz and 845 Hz, characterized by $\pm$3 dB and $\pm$1 dB bounds, respectively, that surpass the resonance frequency of 739 Hz.

💡 Research Summary

The paper addresses the bandwidth limitation inherent in piezo‑actuated nanopositioning stages, which typically exhibit a lightly damped first resonance around 739 Hz. Conventional proportional‑integral (PI) controllers can only achieve a closed‑loop bandwidth of less than 2 % of this resonance because the high‑Q peak introduces large phase lag and low gain margin. Existing remedies—such as notch filters, positive position feedback (PPF), integral resonant control (IRC), and integral force feedback (IFF)—either require highly accurate plant models, add hardware complexity, or lack sufficient design freedom to place resonant poles arbitrarily.

To overcome these drawbacks, the authors propose a Non‑Minimum‑Phase Resonant Controller (NRC) for active damping within a dual‑loop architecture. The dual loop consists of an inner damping controller (C_d(s)) and an outer tracking controller (C_t(s)) (a standard PI). The key idea of the NRC is to exploit the phase‑inverting property of a non‑minimum‑phase (NMP) system while keeping the magnitude of the controller constant. By enforcing (|G(s)C_d(s)| = 1) and (\angle G(s)C_d(s) = \pm\pi) for all frequencies below the resonance, the controller achieves two critical effects: (1) complete attenuation of the first resonant pole pair, and (2) bifurcation (splitting) of the double pole in the frequency domain, which dramatically increases the effective damping ratio.

The design proceeds by constructing (C_d(s)) as a ratio of a second‑order NMP polynomial to its minimum‑phase counterpart, multiplied by a fixed gain (K_c = 1/|G(0)|). A tunable phase offset (\theta) is introduced to adjust the exact pole locations, providing the required phase shift without altering the gain. This constant‑gain, variable‑phase structure satisfies four performance objectives defined for the dual‑loop system:

1. Maximize dual‑loop bandwidth (O1) – ensure (|T_{xr}(j\omega)|\approx1) up to a target frequency (\omega_c) by meeting the magnitude‑phase condition above.
2. Enhance low‑frequency disturbance rejection (O2) – make the outer PI gain large at low frequencies so that the process sensitivity (|PS_{xd}|) is much smaller than the plant magnitude.
3. Strengthen active damping at resonance (O3) – increase the loop gain (|L_D(j\omega_n)|) at the resonance frequency, ideally with a phase of (\pm\pi), to push the sensitivity (|S_{yn}|) well below 1.
4. Attenuate high‑frequency sensor noise (O4) – reduce loop gain beyond the resonance so that (|S_{yn}|) approaches 1, preventing noise amplification.

Robustness to load variations is demonstrated analytically and via simulation: because the controller’s magnitude remains constant, a shift of the resonance frequency by ±10 % does not degrade the damping performance. The approach also extends to higher‑order flexural modes; separate NMP filters can be cascaded to damp multiple resonances simultaneously, achieving multimode active damping without additional sensors.

Experimental validation uses an industrial piezo‑driven nanopositioner (Physik Instrumente) equipped with dual‑plate capacitive sensors. The NRC is implemented on a digital signal processor, while the outer loop employs a conventional PI tuned for high low‑frequency gain. Results show that, without NRC, the closed‑loop bandwidth is limited to about 15 Hz. With NRC, the first resonance is fully suppressed, and the dual‑loop bandwidth reaches 895 Hz (±3 dB) and 845 Hz (±1 dB), both exceeding the natural resonance frequency. Low‑frequency reference signals contribute 30 dB less to the real positioning error, and disturbances near the resonance are attenuated by more than 20 dB compared with the PI‑only case.

In summary, the NRC provides a simple yet powerful solution for active damping in lightly damped nanopositioning systems. By leveraging non‑minimum‑phase dynamics, it achieves constant‑gain control with tunable phase, ensuring complete resonance cancellation, robustness to parameter variations, and simultaneous damping of higher‑order modes. The dual‑loop configuration further allows high tracking performance without sacrificing stability. Future work suggested includes extending the method to multi‑degree‑of‑freedom platforms and incorporating adaptive phase‑tuning to cope with rapid, unpredictable changes in system dynamics.