Iterative Learning Control of Fast, Nonlinear, Oscillatory Dynamics (Preprint)

TL;DR

This work addresses the challenge of rapid, nonlinear instabilities in aerospace-relevant systems by developing a slow, iterative parameter-tuning controller that shapes fast dynamics. It combines Time-Lagged Phase Portraits, Earth Mover's Distance, and Gaussian Process Regression within an Iterative Learning Control framework to stabilize and reproduce a desired oscillatory trajectory in the Lorenz system. The results show parameter sensitivity, successful multi-parameter control, and robustness to missing or uncontrolled parameters, suggesting a viable path toward low-speed control of fast dynamics in complex, chaotic systems. The approach offers a flexible, data-driven framework that can adapt to non-repetitive dynamics and may generalize to fluid, plasma, and aeroelastic instabilities in real-world settings.

Abstract

The sudden onset of deleterious and oscillatory dynamics (often called instabilities) is a known challenge in many fluid, plasma, and aerospace systems. These dynamics are difficult to address because they are nonlinear, chaotic, and are often too fast for active control schemes. In this work, we develop an alternative active controls system using an iterative, trajectory-optimization and parameter-tuning approach based on Iterative Learning Control (ILC), Time-Lagged Phase Portraits (TLPP) and Gaussian Process Regression (GPR). The novelty of this approach is that it can control a system's dynamics despite the controller being much slower than the dynamics. We demonstrate this controller on the Lorenz system of equations where it iteratively adjusts (tunes) the system's input parameters to successfully reproduce a desired oscillatory trajectory or state. Additionally, we investigate the system's dynamical sensitivity to its control parameters, identify continuous and bounded regions of desired dynamical trajectories, and demonstrate that the controller is robust to missing information and uncontrollable parameters as long as certain requirements are met. The controller presented in this work provides a framework for low-speed control for a variety of fast, nonlinear systems that may aid in instability suppression and mitigation.

Figures (11)

• Figure 1: The Lorenz system solved at $\sigma=10$, $\rho=28$, and $\beta=8/3$. Subfigures a, b, and c show the time evolution of $x(t)$, $y(t)$, and $z(t)$, respectively, and d and e show two and three-dimensional phase space representations, respectively.
• Figure 2: We change the Lorenz system's parameter, $\rho$, at time $t=0$, and the dynamics change and converge on a solution after a finite amount of time, $\tau_{slow}$. The dynamics of the Lorenz system occur at $\tau_{fast}$ which is over an order of magnitude smaller than $\tau_{slow}$ in this example.
• Figure 3: Block diagram of the a) control loop and b) controller. The system naturally evolves at a "fast" speed, and the controller, which cannot operate that fast, instead operates ("closes the loop") at a "slow" speed by controlling (tuning) the system's parameters.
• Figure 4: The 2D TLPP of the Lorenz system: a) unbinned, and b) binned. Only the first 5000 points are shown in a) for clarity of the trajectories.
• Figure 5: a) Shows that EMD, applied to TLPPs across a range of values for $\rho$, provides an objective function that can be minimized. b-e) Shows the TLPPs at four illustrative values of $\rho$.