Abstract
Controlling nonlinear dynamics is a long-standing problem in engineering. Harnessing known physical information to accelerate or constrain stochastic learning pursues a new paradigm of scientific machine learning. By linearizing nonlinear systems, traditional control methods cannot learn nonlinear features from chaotic data for use in control. Here, we introduce Physics-Informed Deep Operator Control (PIDOC), and by encoding the control signal and initial position into the losses of a physics-informed neural network (PINN), the nonlinear system is forced to exhibit the desired trajectory given the control signal. PIDOC receives signals as physics commands and learns from the chaotic data output from the nonlinear van der Pol system, where the output of the PINN is the control. Applied to a benchmark problem, PIDOC successfully implements control with a higher stochasticity for higher-order terms. PIDOC has also been proven to be capable of converging to different desired trajectories based on case studies. Initial positions slightly affect the control accuracy at the beginning stage yet do not change the overall control quality. For highly nonlinear systems, PIDOC is not able to execute control with a high accuracy compared with the benchmark problem. The depth and width of the neural network structure do not greatly change the convergence of PIDOC based on case studies of van der Pol systems with low and high nonlinearities. Surprisingly, enlarging the control signal does not help to improve the control quality. The proposed framework can potentially be applied to many nonlinear systems for nonlinear controls.
Highlights
Controlling chaos and nonlinear dynamics is a long-standing issue in various engineering disciplines, including aerospace systems design [1]; chemical operations [2]; robotics [3]; biological sciences [4]; mechatronics [5]; and, in particular, microelectronics [6], especially for circuits systems involving semiconductors that elicit nonlinearity for signal controls [7,8]
Acknowledging the limitations of physics-informed neural network (PINN) and other neural network (NN)-based controls, a question emerges: can control signals be incorporated in PINNs for chaotic nonlinear dynamical systems? To investigate this, we focus on a system proposed by Balthasar van der Pol in 1920 when he was an engineer working for the Philips Company while studying oscillating circuits [38,39,40]
Inspired by [47] and seeking to compare the application of PINNs [22] to the van der Pol system, we propose Physics-Informed Deep Operator Control (PIDOC), a PINN-based control method that incorporates into the loss function of a PINN the generation losses of NN, the desired control signal, and the initial position of the system
Summary
Controlling chaos and nonlinear dynamics is a long-standing issue in various engineering disciplines, including aerospace systems design [1]; chemical operations [2]; robotics [3]; biological sciences [4]; mechatronics [5]; and, in particular, microelectronics [6], especially for circuits systems involving semiconductors that elicit nonlinearity for signal controls [7,8]. Proportional–integral–derivative controllers (PID) are used for controlling nonlinear systems [14,15], where closed-loop feedback errors are tuned using linearized versions of nonlinear, chaotic equations seeking to come as close to the targeted trajectory as possible [16] Reducing PID to PI (proportional–integral) or P (proportional) might either increase the speed or system stability, but neither can learn the features of nonlinear systematic data, which allows more advanced self-adjust control behavior Another common approach is to model the chaotic behaviors as periodic ones and implement trigonometric commands in hopes of producing predictable periodic system behavior [18]. Model predictive controllers incorporate statistics seeking good results for fuzzy systems [19]
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