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

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.

💡 Research Summary

The paper addresses the challenge of designing input shapers for dynamical systems whose parameters vary over time, a situation that traditional uncertainty quantification (UQ) methods—typically assuming static probability distributions—cannot handle efficiently. To this end, the authors adopt an intrusive Polynomial Chaos Expansion (PCE) framework that embeds stochastic expansions directly into the governing equations, allowing the stochastic response to be computed with Galerkin projection rather than through repeated sampling.

The test case is a simple undamped spring‑mass system with unit mass. The spring stiffness k, and consequently the natural frequency ωₙ = √(k/m), is modeled as a random variable that changes its distribution at a prescribed switching time t₁. Specifically, ωₙ follows a uniform distribution U(0.75π, 1.25π) on the interval