Physics-informed Koopman model predictive control of open canal systems

The physical model of open canal systems is described by the Saint-Venant (S-V) equations, which are partial differential equations without explicit solutions. Consequently, the control problem of open canal systems is not trivial. In this paper, a model predictive control (MPC) method based on the framework of the Koopman operator and the physics-informed neural networks is proposed. A continuous-time Koopman model is obtained by mapping the system states, including water levels and discharges, from the original state space to a raised-dimensional observation space. An autoencoder architecture is developed to approximate the mapping to the raised-dimensional space. Specifically, we established a numerical connection between the Koopman model and the S-V equations, and introduced a physics-informed loss function. A two-stage training strategy is implemented to obtain the optimal approximation of the physics-informed Koopman model. Subsequently, a continuous-time stable MPC method for the physics-informed Koopman model of open canal systems is proposed via control parameterization. The proposed method was validated on a one-reach canal system and a cascaded system. The simulation results demonstrate that the physics-informed Koopman model accurately predicts the future dynamics of open canal systems, and the MPC controller effectively tracks the desired water levels.