Polynomial Chaos-based Input Shaper Design under Time-Varying Uncertainty

TL;DR

Problem: robust vibration mitigation under time-varying stiffness uncertainty. Approach: intrusive time-dependent PCE with time-delay input shapers and global sensitivity analysis to design robust controllers, expressed via $F(t,\zeta) \approx \sum_{i=0}^P a_i(t) \Psi_i(\zeta)$. Key contributions include time-adaptive PCE basis with Galerkin projection across time intervals, continuity initialization, and MC-comparison demonstrating accuracy and efficiency. Findings show that $\mathbb{E}[V_{res}]$ and $\mathrm{Var}(V_{res})$ from PCE match MC while offering substantial speed-up, enabling efficient, uncertainty-aware vibration suppression for systems with evolving parameters. This framework supports extensions to nonlinear/higher-fidelity models and motivates future experimental validation.

Abstract

The work presented here investigates the application of polynomial chaos expansion toward input shaper design in order to maintain robustness in dynamical systems subject to uncertainty. Furthermore, this work intends to specifically address time-varying uncertainty by employing intrusive polynomial chaos expansion. The methodology presented is validated through numerical simulation of intrusive polynomial chaos expansion formulation applied to spring mass system experiencing time-varying uncertainty in the spring stiffness. The system also evaluates non-robust and robust input shapers through the framework in order to identify designs that minimize residual energy. Results indicate that vibration mitigation is achieved at a similar accuracy, yet at higher efficiency compared to a Monte Carlo framework.

Figures (7)

• Figure 1: Spring-mass system with input $u$.
• Figure 2: Functionality of a GSA TDF over time.
• Figure 3: Expected residual energy and its variance for different MC sample sizes.
• Figure 4: Convergence of PCE for expected value and variance for the position $x$.
• Figure 5: Convergence of PCE for expected value and variance for the velocity $\dot{x}$.