Suppressing unknown disturbances to dynamical systems using machine learning

TL;DR

A model-free method to identify and suppress an unknown disturbance to an unknown system based only on previous observations of the system under the influence of a known forcing function is presented.

Abstract

Identifying and suppressing unknown disturbances to dynamical systems is a problem with applications in many different fields. Here we present a model-free method to identify and suppress an unknown disturbance to an unknown system based only on previous observations of the system under the influence of a known forcing function. We find that, under very mild restrictions on the training function, our method is able to robustly identify and suppress a large class of unknown disturbances. We illustrate our scheme with the identification of both deterministic and stochastic unknown disturbances to an analog electric chaotic circuit and with numerical examples where a chaotic disturbance to various chaotic dynamical systems is identified and suppressed.

Figures (10)

• Figure 1: Schematic illustration of our method. In the training phase (top row), a nonlinear system is forced with a training function f(t). Observations of the forced system are used to train a reservoir to approximate the training function. The reservoir subsequently identifies unknown disturbance functions (bottom row).
• Figure 2: Identifying unknown disturbances. (Top) Unknown and reconstructed disturbance functions $[g_x(t),g_y(t)]$ (black curve) and $[u_x(t),u_y(t)]$ (red curve) along with the training forcing functions $[f_x(t),f_y(t)]$ (thick blue curves and symbols). For each case the reservoir was trained with (a) $[f_x(t), f_y(t)] = [\cos(0.05t), \sin(0.05t)]$, (b) $[f_x(t), f_y(t)] = [\cos(0.05t), \cos(0.05t-0.05)]$, and (c) $[f_x(t), f_y(t)] = [\text{sign}(\cos(0.05t)), \text{sign}(\sin(0.05t))]$. (Bottom) Time series for the unknown (solid black) and recovered (dashed red) disturbance functions in the $y$ component, $g_y(t)$ and $u_y(t)$.
• Figure 3: Identifying unknown disturbances: Rössler and Lorenz 96 systems. (Top) Unknown and reconstructed disturbance functions $[g_x(t),g_y(t)]$ (black curve) and $[u_x(t),u_y(t)]$ (red curve) along with the training forcing functions $[f_x(t),f_y(t)]$ (blue curves). For each case the reservoir was trained with $[f_x(t), f_y(t)] = [\cos(0.05t), \sin(0.05t)]$. (Bottom) Time series for the unknown (solid black) and recovered (dashed red) disturbance functions in the second component, $g_y(t)$ and $u_y(t)$ (or $g_{x_2}(t)$ and $u_{x_2}(t)$).
• Figure 4: Experimental results. (Top) Unknown and reconstructed disturbance functions $[g_x(t),g_y(t)]$ (black curve) and $[u_x(t),u_y(t)]$ (red curve) along with the training forcing functions $[f_x(t),f_y(t)]$ (thick blue curves and symbols). For each case the reservoir was trained with (a) a 5 hz square wave out of phase by $\pi/2$ and (b) $[f_x(t), f_y(t)] = [(\cos(10\pi t)),(\sin(10\pi t))]$. (Bottom) Time series for the unknown (solid black) and recovered (dashed red) disturbance functions in the $y$ component, $g_y(t)$ and $u_y(t)$.
• Figure 5: Suppressing unknown disturbances. (a)--(c) For control gains $\alpha=0$, $10$, and $100$, the disturbed attractor obtained from delayed control (black curves) compared to the undisturbed reference attractor (red curves). (d) As a function of the control gain $\alpha$, the average distance $d(\alpha)$ between the disturbed and original attractors as simple and delayed control (blue circles and red crosses) is applied to the disturbed attractor.