Robust feasibility for control of water flow in a reservoir-canal system

A robust control problem for distant downstream control of a reservoir-canal system modeled by Saint-Venant equations is investigated. The problem is to regulate the release of water at the upstream end such that the measured water level (or stage) at the downstream end does not deviate outside of prescribed bounds under the effect of downstream perturbations. Under the assumption of small perturbations, the Saint-Venant model is linearized around a steady state flow. The resulting linear model is discretized to obtain a linear state-space model using a method of characteristics based numerical scheme. For the state space model, the control is the upstream discharge deviation, the disturbance is the downstream discharge deviation and the output is the downstream stage deviation; the deviations are defined with respect to the steady state. The sets of admissible control, disturbance and output trajectories are modeled by polytopes. It is shown that the control problem can be formulated as a robust feasibility problem. Using linear programming duality, conditions for existence of a robustly feasible solution are derived. These conditions, being affine in the control variables, are checked using linear programming. The proposed method is applied to control a typical reservoir- canal system.