Non-Minimum-Phase Resonant Damping Controller

• Non-Minimum-Phase Resonant Damping Controller is a feedback architecture using a first-order NMP compensator to actively damp lightly damped resonances in nanopositioning systems.
• It enhances closed-loop bandwidth and suppresses resonance-induced sensitivity peaks by splitting resonant poles through a dual-loop configuration.
• Experimental evaluations in both SISO and MIMO nanopositioners demonstrate robust stabilization, significant sensitivity attenuation, and improved tracking performance.

A Non-Minimum-Phase Resonant Damping Controller (NRC) is a feedback control architecture specifically engineered to achieve active damping of lightly damped resonance modes in precision mechatronic systems. The NRC utilizes a non-minimum-phase (NMP) first-order compensator, implemented as an inner loop in a dual closed-loop configuration, to realize both substantial bandwidth enhancements and robust attenuation of resonance-induced sensitivity peaks. Unlike traditional minimum-phase approaches, the NRC's right-half-plane (RHP) zero introduces a substantial phase lead, enabling the complete splitting and damping of the plant's resonant poles, even as the loop gain is increased beyond typical stability limits. The NRC has been rigorously formulated, tuned, and experimentally validated in both single-input single-output (SISO) and multi-input multi-output (MIMO) piezo-actuated nanopositioner systems (Natu et al., 2024, Natu et al., 17 Jan 2026).

1. Motivation and Control Objectives

Lightly damped resonances fundamentally constrain the achievable bandwidth of nanopositioning and other high-precision electromechanical systems. In both SISO and MIMO nanopositioners, such resonances (e.g., with $\zeta_n\sim0.005$–$0.01$) can lead to high sensitivity peaks, limiting the effectiveness of reference-tracking and disturbance rejection. The NRC is introduced to directly address these limitations via frequency-domain shaping criteria. The chief shaping objectives are:

• Reference tracking up to the closed-loop bandwidth: $|T_{xr}(j\omega)|\approx1$ for $\omega\leq\omega_c$.
• Low-frequency disturbance rejection: $|PS_{yd}(j\omega)|\ll1$ for $\omega\leq\omega_{C_t}$.
• Active damping at resonance: $|S_{yn}(j\omega_n)|\ll1$, achieved by ensuring $|G(j\omega_n)[C_t(j\omega_n)+C_d(j\omega_n)]|\gg1$.
• High-frequency noise attenuation: $|C_t(j\omega)|,|C_d(j\omega)|\ll1$, so $|S_{yn}(j\omega)|\to1$ for $\omega\gg\omega_n$.

These criteria are formally encoded in the dual-loop sensitivity function maps, directly connecting controller structure to frequency-domain performance (Natu et al., 2024).

2. Controller Structure and Mathematical Formulation

The core NRC element is a first-order NMP compensator:

$C_d(s) = k\,\frac{s-\omega_a}{s+\omega_a}$

This structure incorporates:

• A right-half-plane zero at $s=+\omega_a$ (defining the "non-minimum-phase" property).
• A left-half-plane pole at $s=-\omega_a$.
• Constant magnitude $|C_d(j\omega)|=k$ over all frequencies.
• Tunable phase contribution $\angle C_d(j\omega)$, which transitions from $+180^\circ$/$-180^\circ$ to $0^\circ$ as $\omega$ crosses $\omega_a$.

In a standard lightly damped plant,

$$G(s) = \frac{g\,\omega_n^2}{s^2+2\zeta_n\omega_n s+\omega_n^2}$$

the NRC enables closed-form pole-splitting: the resonant conjugate poles of $G(s)$ bifurcate onto the real axis and become fully damped as the normalized NRC corner $n=\omega_a/\omega_n$ increases. The complete pole set with the NRC is

• $p_1 = 0$ (integrator),
• $$p_{2,3} = \frac{-(2\zeta_n + n)\omega_n \pm \sqrt{(2\zeta_n+n)^2\omega_n^2 - 4(2\zeta_n n+2)\omega_n^2}}{2}$$,

yielding "complete (real) damping" when $n \geq 2(\sqrt{2}+\zeta_n)$ (Natu et al., 2024).

3. Tuning Methodology and Implementation Strategy

The NRC admits comprehensive closed-form tuning:

• Normalized corner frequency:

$n = \omega_a/\omega_n$.

• Gain:

$k = \gamma\,g^{-1}$, with $0 < \gamma \leq 1$.

Maximal damping is achieved for $\gamma=1$ and $|C_d(j\omega_n)G(j\omega_n)|\approx1$; this sets the loop DC gain for optimal pole splitting. In practice, the NRC corner $\omega_a$ is set as a multiple of the resonance frequency, typically $n\approx3$ for MIMO systems and $n=8$ for SISO nanopositioners; e.g., $\omega_a\approx8 \cdot 739$ Hz $=5.9$ kHz, $k\approx1.91$ (Natu et al., 2024Natu et al., 17 Jan 2026).

4. Robustness and Higher-Order Mode Damping

The NRC exhibits strong robustness to resonance frequency shifts arising from system loading ($\hat\omega_n=\eta\omega_n,\,\eta<1$), since the pole-splitting criterion $n\ge2\eta(\sqrt{2}+\zeta_n)$ remains satisfied. Additionally, in multi-mode plants where a secondary mode at $\omega_2=\alpha\omega_n$ is present, the mean closed-loop magnitude at $\omega_2$ is $|G_{2d}(j\omega_2)|=1/k=(1+\beta)/\gamma$, allowing attenuation of higher-order modes as $k$ is increased. In MIMO architectures, the NRC can be extended with a band-pass compensator (BPC) targeting specific cross-coupling resonances to enhance directional disturbance rejection without impacting primary axis tracking (Natu et al., 17 Jan 2026).

Parameter	SISO Example (Natu et al., 2024)	MIMO Example (Natu et al., 17 Jan 2026)
Primary resonance	$\omega_n=739$ Hz	$\omega_{n1}=165$ Hz
NRC corner ($\omega_a$)	$5.9$ kHz ($n=8$)	$495$ Hz ($n\approx3$)
NRC gain ($k$ or $k_j$)	$1.91$	Chosen so $|C_{\rm NRC}G|=1$
Band-pass controller	N/A	Centered at $231$ Hz

5. Dual-Loop Architectures: Inner and Outer Loop Synthesis

In both SISO and decentralized MIMO nanopositioners, the NRC is embedded as the inner damping loop. The output of the system is governed by the collective action of inner NRC-based damping and an outer motion/tracking controller, typically PI or PI with additional notch/LPF elements. The standard configuration is:

• Inner loop: NRC active damping

$G_d(s) = \frac{G(s)}{1+G(s)C_d(s)}$

• Outer loop: PI tracking,

$C_t(s) = k_p\left(1+\frac{\omega_i}{s}\right)$

For MIMO systems, $C_{d_j}(s)$ and $C_{t_j}(s)$ are implemented for each decoupled axis, and the band-pass compensator $C_{\rm BPC,j}(s)$ is added in parallel with the NRC for cross-coupling suppression. Key closed-loop maps are defined as $$\mathbf{PS}(s)=(I+\mathbf{G}(s)\mathbf{D}(s))^{-1}\mathbf{G}(s)$$ and $\mathbf{T}(s)=\mathbf{PS}(s)\mathbf{C}_t(s)$ (Natu et al., 2024, Natu et al., 17 Jan 2026).

6. Experimental Evaluation and Performance Metrics

Experimental results confirm the effectiveness of NRC-based architectures:

• SISO nanopositioner (Natu et al., 2024):
  • Plant: PI P-621.1CD, $\omega_n=739$ Hz, $\zeta_n\approx0.01$.
  • NRC: $n=8,\,\omega_a=5.9$ kHz, $k=1.91$.
  • Closed-loop bandwidths: $895$ Hz ($\pm3$ dB), $845$ Hz ($\pm1$ dB).
  • Outer-loop design: $k_p\approx298.4,\;\omega_i=28$ Hz, notches at $1000$ Hz and $2600$ Hz, LPF corner at $5$ kHz.
  • Peak sensitivity attenuation $>20$ dB at resonance. Sine tracking up to $800$ Hz yields $e_\mathrm{rms}<10$ nm.
• MIMO nanopositioner (Natu et al., 17 Jan 2026):
  • Plant: P-562.2CD, resonances $f_{n1}\approx165$ Hz, $f_{n2}\approx231$ Hz, $\zeta_1\sim0.005$, $\zeta_2\sim0.02$.
  • NRC only: Bandwidth $225$ Hz (x), $228$ Hz (y).
  • NRC $+$ band-pass: Bandwidth $214$ Hz (x), $213$ Hz (y).
  • Cross-coupling attenuation at $231$ Hz improved by $11.5$ dB with band-pass path.
  • Disturbance rejection at $231$ Hz: RMS motion reduced by $60\,\%$.
  • Tracking error: $0.1167\,\mu$m (x), $0.1127\,\mu$m (x, full), $0.0964\,\mu$m (y), $0.1124\,\mu$m (y, full).

These results demonstrate robust stabilization, reference tracking to well above first resonance, and cross-mode disturbance suppression without sacrificing overall tracking fidelity.

7. Significance, Extensions, and Limitations

The NRC fundamentally enables bandwidths beyond the primary resonance frequency by leveraging non-minimum-phase dynamics to provide phase resources while decoupling gain shaping, thus overcoming traditional Bode gain-phase limitations in lightly damped systems. The parallel band-pass path in the MIMO extension demonstrates selective modal damping and cross-coupling attenuation unattainable with classic loop-shaping or minimum-phase compensators. All critical performance claims—pole splitting, modal attenuation, robustness to resonance drift, and experimental bandwidth results—are substantiated in full-scale nanopositioner deployments. A plausible implication is that NRCs could generalize to broader precision motion systems with complex multimodal or time-varying resonances, provided collocated sensing and actuation are available.

Further considerations include computational requirements at high bandwidth, interaction with amplifier delays, and extension to non-collocated or strongly non-minimum-phase plants—topics which remain open for future research (Natu et al., 2024, Natu et al., 17 Jan 2026).