Reservoir computing for system identification and predictive control with limited data

TL;DR

This work assesses the ability of RNN variants to both learn the dynamics of benchmark control systems and serve as surrogate models for MPC and finds that echo state networks have a variety of benefits over competing architectures, namely reductions in computational complexity, longer valid prediction times, and reductions in cost of the MPC objective function.

Abstract

Model predictive control (MPC) is an industry standard control technique that iteratively solves an open-loop optimization problem to guide a system towards a desired state or trajectory. Consequently, an accurate forward model of system dynamics is critical for the efficacy of MPC and much recent work has been aimed at the use of neural networks to act as data-driven surrogate models to enable MPC. Perhaps the most common network architecture applied to this task is the recurrent neural network (RNN) due to its natural interpretation as a dynamical system. In this work, we assess the ability of RNN variants to both learn the dynamics of benchmark control systems and serve as surrogate models for MPC. We find that echo state networks (ESNs) have a variety of benefits over competing architectures, namely reductions in computational complexity, longer valid prediction times, and reductions in cost of the MPC objective function.

Figures (12)

• Figure 1: Summary diagram of echo state network (ESN) based model predictive control (MPC). An ESN trained to model plant dynamics acts as a surrogate model capable of rapidly forecasting the system under given control inputs. Online optimizations compute a sequence of control actions that minimize the deviation of the plant from a reference trajectory.
• Figure 2: Surrogate model forecasting results for the spring mass system. Panel A shows the time-series of the validation data (top) and the corresponding control inputs (middle). The time-series of $\mathbf{x}$ has been scaled to improve readability across examples. The bottom of Panel A shows the $\ell_2$ deviation of each surrogate model's forecast from the ground truth as a function of time. All forecasts are reinitialized every 50 time-steps, denoted by the dashed black line in the top plot. Panel B reports the mean $\ell_2$ forecast error obtained across 32 trained surrogate models of each type for varying levels of additive Gaussian white noise. Panel C reports analogous results for the case of zero noise, but a varying number of training samples.
• Figure 3: Surrogate model forecasting results for the stirred tank system. All panels are as labelled in Fig. \ref{['fig:spring_mass_forecasting']}. Panel A shows the time-series of the validation data (top) and the corresponding control inputs (middle). The time-series of $\mathbf{x}$ has been scaled to improve readability across examples. The bottom of Panel A shows the $\ell_2$ deviation of each surrogate model's forecast from the ground truth as a function of time. All forecasts are reinitialized every 50 time-steps, denoted by the dashed black line in the top plot. Panel B reports the mean $\ell_2$ forecast error obtained across 32 trained surrogate models of each type for varying levels of additive Gaussian white noise. Panel C reports analogous results for the case of zero noise, but a varying number of training samples.
• Figure 4: Surrogate model forecasting results for the two tank reservoir system. All panels are as labelled in Fig. \ref{['fig:spring_mass_forecasting']}. Panel A shows the time-series of the validation data (top) and the corresponding control inputs (middle). The time-series of $\mathbf{x}$ has been scaled to improve readability across examples. The bottom of Panel A shows the $\ell_2$ deviation of each surrogate model's forecast from the ground truth as a function of time. All forecasts are reinitialized every 50 time-steps, denoted by the dashed black line in the top plot. Panel B reports the mean $\ell_2$ forecast error obtained across 32 trained surrogate models of each type for varying levels of additive Gaussian white noise. Panel C reports analogous results for the case of zero noise, but a varying number of training samples.
• Figure 5: Surrogate model forecasting results for the Lorenz system with control. All panels are as labelled in Fig. \ref{['fig:spring_mass_forecasting']}. Panel A shows the time-series of the validation data (top) and the corresponding control inputs (middle). The time-series of $\mathbf{x}$ has been scaled to improve readability across examples. The bottom of Panel A shows the $\ell_2$ deviation of each surrogate model's forecast from the ground truth as a function of time. All forecasts are reinitialized every 50 time-steps, denoted by the dashed black line in the top plot. Panel B reports the mean $\ell_2$ forecast error obtained across 32 trained surrogate models of each type for varying levels of additive Gaussian white noise. Panel C reports analogous results for the case of zero noise, but a varying number of training samples.