PDE‐constrained model predictive control of open‐channel systems

Abstract

Abstract A PDE‐constrained model predictive control (MPC) algorithm for open‐channel systems based on the Saint‐Vevant(S‐V) equations is investigated in this paper. The S‐V equations, which precisely model the dynamics of open‐channel systems, are quasi‐linear hyperbolic partial differential equations (PDEs) without analytical solutions. Directly applying the S‐V equations to an MPC controller design becomes sophisticated. In this work, the calculus of variation is used to obtain the adjoint equations and the adjoint analysis method is utilized to deduce the gradients of the MPC optimization problem. Particularly, the physical constraints involving both the state and control variables are also considered. A gradient‐based optimization algorithm in combination with the numerical computation of Preissmann implicit scheme is proposed to solve the constrained MPC optimization problem. The control performances of the developed PDE‐constrained MPC algorithm with respect to the controlled water levels and gate openings are compared with those of the MPC controller designed for the linearized model. All the simulation tests are carried out on an aqueduct reach in Yehe Irrigation District in Hebei Province, China. The results show that the proposed PDE‐constrained MPC algorithm is a promising method in dealing with the constraints in terms of hyperbolic PDEs, control variables and state variables simultaneously.

Keywords

• constraint handling
• partial differential equations
• predictive control