Controlling Dynamical Systems into Unseen Target States Using Machine Learning

TL;DR

This work proves the applicability of machine learning-based control mechanisms to previously inaccessible target dynamics to extrapolated chaotic target states over transition times, and demonstrates the method's effectiveness on a nonlinear power system model.

Abstract

We present a novel, model-free, and data-driven methodology for controlling complex dynamical systems into previously unseen target states, including those with significantly different and complex dynamics. Leveraging a parameter-aware realization of next-generation reservoir computing (NGRC), our approach accurately predicts system behavior in unobserved parameter regimes, enabling control over transitions to arbitrary target states utilizing a new prediction evaluation and selection scheme. Crucially, this includes states with dynamics that differ fundamentally from known regimes, such as shifts from periodic to intermittent or chaotic behavior. The method's parameter awareness facilitates non-stationary control with which control scenarios are generated and evaluated on the basis of predefined control objective. In addition to proving the method for transient-free control to extrapolated chaotic target states over transition times, we demonstrate the method's effectiveness on a nonlinear power system model. Our method successfully navigates transitions even in scenarios where system collapse is observed frequently, while ensuring fast transitions and avoiding prolonged transient behavior. By extending the applicability of machine learning-based control mechanisms to previously inaccessible target dynamics, the methodology opens the door to new control applications while maintaining exceptional efficiency.

Figures (8)

• Figure 1: $\textbf{Control method facilitates smooth transitions to unseen target states.}$ The Lorenz system is controlled from a periodic initial state at $\rho_{\text{start}} = 100.5$ to a weakly chaotic state at $\rho_{\text{target}} = 99.5$, using linearly decreasing parameterizations over various transition durations, shown in D. The control method enables a smooth, transient-free transition to the target state (D, blue). In contrast, simulations with the same parameter transitions (D, red) show that shorter transition durations often cause transients. For each transition time step, 100 differently initialized system were tested. A illustrates one of those initial periodic trajectory at $\rho_{\text{start}}$ (A, red). B and C show the controlled dynamics to the weakly chaotic target state with an instantaneous parameter change (B, blue) and its running correlation dimension (C, green).
• Figure 2: $\textbf{Statistical climate of controlled target states.}$ Evaluation of the statistical climate of the $100$ controlled Lorenz systems at $\rho_{target} = 99.5$ given each transition period shown in Fig. \ref{['fig:turbulence_props']}. In A, the correlation dimension is measured and in the B the largest Lyapunov exponent.
• Figure 3: $\textbf{Control method applied to power system control.}$ Fig. A illustrates the voltage behavior of the power system model for several reactive power demands $Q_1$. The NGRC is trained on seven training data samples (shown in light green), to accurately interpolate and extrapolate the system behavior in unseen parameter regions of $Q_1$ (shown in Fig. D). We test the methods control capabilities by applying an instantaneous parameter switch from an unseen initial regime to an unseen target regime, both of which are periodic (Fig. A cyan). From simulating this switch, three different behaviors occur (Fig. B) with their probability for 250 different initial condition shown in Fig. C. The control method applied to the initial system dynamic, ensures fast transitions to the target dynamic while completely avoid prolonged chaotic transients and system collapses (shown in Fig. E and Fig. F).
• Figure 4: $\textbf{Accuracy of the controlled and extrapolated target dynamics.}$ To quantify the accuracy the controlled dynamics in the extrapolated target state, we calculate the mean absolute error of the frequency spectra (Spectral Error) of the simulated and controlled dynamics at the target parametrization. A shows the spectral error of the 250 controlled trajectories as box plots for different control parameters. B shows the mean frequency spectrum of 250 controlled trajectories for the control parameter associated with the smallest spectral error and the target spectrum from simulation.
• Figure 5: $\textbf{Control with additive noise.}$ The upper plots show the system behavior of the controlled dynamics given different control parameters and noise levels. For each configuration we show the probability with which the 250 controlled dynamics exhibiting the objective fast transition to the target dynamic (A) and a system collapse (B). C shows the spectral error between the fast transitioned controlled target dynamics and the target dynamic without noise. D shows the corresponding error for frequencies below $2 Hz$, where the frequencies of the target dynamics are. The results highlight that on the one hand the control method can accurately control the initial dynamics to the target dynamics, even in high noise regimes with a sufficiently high and not higher control parameter $K$. On the other hand choosing an high $K$ leads to additional noise (mainly high frequency) by the control method itself.