Robust control of constrained max-plus-linear systems

SUMMARY Max-plus-linear (MPL) systems are a class of nonlinear systems that can be described by models that are " linear " in the max-plus algebra. We provide here solutions to three types of finite-horizon min-max control problems for uncertain MPL systems, depending on the nature of the control input over which we optimize: open-loop input sequences, disturbances feedback policies, and state feedback policies. We assume that the uncertainty lies in a bounded polytope, and that the closed-loop input and state sequence should satisfy a given set of linear inequality constraints for all admissible disturbance realizations. Despite the fact that the controlled system is nonlinear, we provide sufficient conditions that allow to preserve convexity of the optimal value function and its domain. As a consequence, the min-max control problems can be either recast as a linear program or solved via N parametric linear programs, where N is the prediction horizon. In some particular cases of the uncertainty description (e.g. interval matrices), by employing results from dynamic programming, we show that a min-max control problem can be recast as a deterministic optimal control problem.