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

Technical Summary: Polynomial Chaos-based Input Shaper Design under Time-Varying Uncertainty

Problem Statement
Uncertainty quantification (UQ) is critical for designing robust engineering systems, particularly when parameters evolve dynamically over time. Classical probabilistic approaches often rely on fixed distributions and time-invariant assumptions, which limits their effectiveness in complex scenarios such as Unmanned Aerial Vehicle (UAV) payload control and spacecraft attitude tracking. While Polynomial Chaos Expansion (PCE) offers an efficient spectral framework for uncertainty propagation, standard PCE methods can degrade during long-time integrations as the system's probability density functions evolve, causing the fixed orthogonal polynomial basis to become suboptimal. Furthermore, existing robust input shaping techniques often struggle to account for time-varying uncertainties efficiently without resorting to computationally expensive Monte Carlo (MC) simulations. This work addresses the challenge of designing precision motion controllers (specifically input shapers) for dynamical systems subject to time-varying uncertainties.

Methodology
The study proposes an intrusive Polynomial Chaos Expansion (PCE) framework integrated with Time-Delay Filters (TDF) to design robust input shapers. The methodology is validated using an undamped spring-mass system where the spring stiffness (and consequently the natural frequency, $\omega_n$) is subject to interval uncertainty that changes over time.

1. System Modeling: The system is modeled in two time intervals: $[0, t_1]$ and $[t_1, t_2]$. In the first interval, the natural frequency $\omega_n$ follows a uniform distribution. At $t_1$, the system transitions to a second interval where the natural frequency $\omega_m$ follows a different uniform distribution. Crucially, the state of the system at $t_1$ (position and velocity) serves as the initial condition for the second interval, meaning the solution in the second interval depends on both the initial uncertainty ($\zeta_1$) and the new uncertainty ($\zeta_2$).
2. Intrusive PCE Formulation: The authors employ an intrusive approach using Galerkin projections. The system equations are expanded into a series of orthogonal polynomials (Legendre polynomials for uniform distributions). The coefficients of these polynomials are determined by solving the resulting system of differential equations.
   • To handle the time-varying nature, the PCE coefficients at the transition time $t_1$ are mapped from the single-variable expansion (interval 1) to a multi-variable expansion (interval 2) using orthogonality conditions. This ensures continuity of the state and its derivative across the transition.
3. Input Shaper Design: The framework designs Time-Delay Filters (TDFs) to minimize residual energy. Three designs are compared:
   • Non-robust TDF: Places one pair of zeros at the nominal pole location.
   • Robust TDF: Places two sets of zeros at the nominal location.
   • Global Sensitivity Analysis (GSA) TDF: Optimizes the impulse magnitudes ($A_i$) and time delays ($T_i$) to minimize a specific objective function $\Xi$. The objective function is defined as either the expected residual energy ($E[V_{res}]$) or the variance of the residual energy ($Var(V_{res})$).

Key Contributions

• Time-Varying Uncertainty Handling: The paper introduces a specific methodology for applying intrusive PCE to systems where uncertainty parameters change over time. It details the mathematical procedure for transferring PCE coefficients between time intervals with different uncertainty distributions, ensuring the continuity of the solution.
• Efficient Robust Control Design: The study demonstrates that intrusive PCE can be used to design input shapers that account for time-varying uncertainties more efficiently than Monte Carlo simulations.
• Global Sensitivity Analysis Integration: The work integrates Global Sensitivity Analysis into the TDF design process, allowing for the optimization of shaper parameters to specifically target either the mean performance or the robustness (variance) of the system's residual energy.

Results
Numerical simulations were conducted on a spring-mass system with a mass of 1 kg and time-varying stiffness.

• Convergence and Accuracy: The PCE results for expected values and variances of position and velocity converged to the results obtained from Monte Carlo simulations with 10,000 samples. A polynomial order of $P=30$ was found sufficient to match MC accuracy, with no significant difference observed between orders 10 and 30 for the expected value.
• Computational Efficiency: The PCE framework was approximately 12 times faster than the Monte Carlo simulation with 10,000 samples, highlighting its advantage for computationally intensive uncertainty propagation.
• Shaper Performance:
  • All TDF designs significantly reduced the expected residual energy and its variance compared to a non-robust design.
  • The GSA TDF designs outperformed the standard robust TDF. Specifically, the GSA TDF optimized for variance ($\Xi_2$) reduced the expected residual energy by a factor of 2 and the variance by a factor of 5 compared to the standard robust TDF.
  • The GSA TDF optimized for variance resulted in a nearly constant residual energy across different uncertainty pairs, indicating high insensitivity to system variations.
  • While the GSA TDF generally provided superior performance, the authors noted that for specific combinations where the second interval frequency is close to its mean, the standard robust TDF could occasionally yield slightly lower residual energy, though the difference was marginal compared to the significant improvements in borderline cases.

Significance and Claims
The paper claims that the proposed intrusive PCE framework provides a computationally inexpensive and accurate alternative to Monte Carlo methods for designing input shapers in the presence of time-varying uncertainties. By embedding the uncertainty expansion directly into the system equations, the method offers greater analytical insight into the system response and objective function.

The authors state that this approach is particularly advantageous for vibration suppression and residual energy reduction in nonlinear systems where Monte Carlo simulations are computationally prohibitive. They conclude that while the current study focuses on linear spring-mass systems, the methodology lays the groundwork for future extensions to nonlinear and higher-fidelity models. The paper also notes that future work will explore non-intrusive PCE methods for broader versatility and will aim to validate these techniques experimentally, specifically for aerial payload transport applications.