ℒ1 controller design for a high-order 5-pool irrigation canal system

The aim of the work is to present an application of methods for solving the L₁ design problem, based on the scaled-Q approach, to a high-order, nonminimum phase system. We start by describing the system which is an open-channel hydraulic system (e.g. an irrigation canal). From the linearization and discretization of the set of two partial-derivative equations, a state-space model of the system is generated. This model is a high-order MIMO system (five external perturbations w, five control inputs u, five controlled outputs z\´, five measured outputs y, 65 states x) and is nonminimum phase. A controller is then designed by minimizing the L₁ norm of the impulse response of the transfer matrix between the perturbation w and the output z=[z^\´/z^"], where the five additional variables z" are defined as z"=D_uu. Considering this additional transfer (w to z") in the minimization problem leads to a better posed problem and provides much better robustness margins. Time-domain template constraints are added in order to force integrators into the controller. The numerical resolution of the problem proved to be efficient, despite the characteristics of the system. The obtained results are compared in the time-domain to classical PID and LQG controllers, both on linear and non-linear simulated plants. The results proved to be very good in terms of performance and robustness, in particular for the rejection of the worst-case perturbation