Controlling chaos using edge computing hardware

Abstract

Machine learning provides a data-driven approach for creating a digital twin of a system - a digital model used to predict the system behavior. Having an accurate digital twin can drive many applications, such as controlling autonomous systems. Often, the size, weight, and power consumption of the digital twin or related controller must be minimized, ideally realized on embedded computing hardware that can operate without a cloud-computing connection. Here, we show that a nonlinear controller based on next-generation reservoir computing can tackle a difficult control problem: controlling a chaotic system to an arbitrary time-dependent state. The model is accurate, yet it is small enough to be evaluated on a field-programmable gate array typically found in embedded devices. Furthermore, the model only requires 25.0 $\pm 7.0$ nJ per evaluation, well below other algorithms, even without systematic power optimization. Our work represents the first step in deploying efficient machine learning algorithms to the computing "edge."

Fig. 1. NG-RC based nonlinear controller realized on edge computing hardware.

(top) Learning phase. A field programmable gate array (FPGA) applies perturbations (red) to a chaotic circuit and the perturbed dynamics of the circuit (blue) are measured by a sensor (analog-to-digital converters). The temporal evolution of the perturbations and responses are transferred to a personal computer (left, purple) to learn the parameters of the NG-RC controller $\mathbf{W}$. These parameters are programmed onto the FPGA as well as the firmware for the controller. (bottom) Control phase. The NG-RC controller implemented on the FPGA measures the dynamics of the chaotic circuit with a sensor (analog-to-digital converters) in real time and receives a desired trajectory $V_{1,des}$ for the $V_1$ variable, and computes a suitable control signal (red) that drives the circuit to the desired trajectory.

Fig. 2. Controlling a dynamical system using the NG-RC for one-step-ahead prediction.

Controlled attractors for a the first task controlling the system to the origin, b controlling back and forth between the two USSs (solid black), and c controlling to a random waveform. The unperturbed attractor (gray) before the control is switched on, the moment the control is switched on (purple ×), the transient for the system to reach the desired trajectory (dashed purple), and the controlled system (solid purple). d Temporal evolution of $V_1$ (blue), and $V_2$ (red) during the second task. When the control begins, the chaotic system follows the desired trajectory (solid black), which is under the blue curve. e The control perturbation before and after the control is switched on (orange). The control gain is set to K = 0.75 for all cases.

Fig. 3. Performance of the one-step ahead NG-RC controller.

a RMSE of the control and b RMS of the control current when controlling the system to the phase-space origin (red circles), back-and-forth between two USSs (green triangles), to a random waveform (blue squares), and stabilized at either nonzero USS (purple diamonds) as a function of the control gain $K$.

Fig. 4. Double scroll electronic circuit.

$R_n=3.15$ kΩ, $C_1=C_2=10$ nF, $L=55$ mH, $R_L=355$ Ω, $R_s=100$ Ω, $R_m=R_L+R_s=355$ Ω, $R_d=7.86$ kΩ, the dotted box encloses the nonlinear coupling $g$.

Fig. 5. The information criteria of features calculated by SysIdentPy.

More negative means less information is contained in the features. The red circles are the features used in the model, and the black xs are excluded.