Adaptive Input Shaper Design for Unknown Second-Order Systems with Real-Time Parameter Estimation

TL;DR

Addresses vibration suppression for unknown second-order systems via an online adaptive input shaper that estimates $\hat{\omega_n}$ and $\hat{\zeta}$ in real time and computes a closed-form shaper $G(s)$. The method explicitly incorporates an estimation window $\tau$ and fraction $\mathcal{K}$, providing analytical solutions for the undamped case and numerical solutions for the damped case; it supports arbitrary estimation times and delivers robust performance across a wide frequency range. Results show vibration-free tracking with minimal overshoot across slow to fast dynamics and step-wise references, with an open-source implementation available. The work improves upon fixed-estimation-time approaches by decoupling shaper design from a fixed window and lays groundwork for extending online design to broader black-box systems.

Abstract

We propose a feedforward input-shaping framework with online parameter estimation for unknown second-order systems. The proposed approach eliminates the need for prior knowledge of system parameters when designing input shaping for precise switching times by incorporating online estimation for a black-box system. The adaptive input shaping scheme accounts for the system's periodic switching behavior and enables reference shaping even when initial switching instants are missed. The proposed framework is evaluated in simulation and is intended for vibration suppression in motion control applications such as gantry cranes and 3D printer headers.

Figures (5)

• Figure 1: Feedforward control scheme with parameter estimation and optimal input shaping for vibration suppression.
• Figure 2: Reproduced results from stein_aruco_2024 with different estimation times: (a) $\tau < T$, (b) $\tau > T$.
• Figure 3: Performance of the proposed method, (a-d) varying $\zeta$ with $\tau = 2\mathrm{s}$ and $\omega_\mathrm{n} = \pi \ \mathrm{rad/s}$, (e-h) varying $\omega_\mathrm{n}$ with $\tau = 2\mathrm{s}$ and $\zeta = 0.707$, (i-l) varying $\tau$ with $\omega_\mathrm{n} = 3\pi \ \mathrm{rad/s}$ and $\zeta = 0.707$.
• Figure 4: Feedforward control for step-wise reference tracking: (Top) $\omega_\mathrm{n} = 3\pi$ rad/s; (Bottom) $\omega_\mathrm{n} = 30\pi$ rad/s.
• Figure 5: Dependency of input shaper parameters $A$ and $T$ on $\zeta$ and $\omega_\mathrm{n}$ under parameter sweep.