Time-Optimal Model Predictive Control for Linear Time-Variant Systems based on Configuration-Constrained Backward Reachability

This paper presents a robust formulation of Time-Optimal Model Predictive Control for Linear Time-Variant systems that leverages backward reachability analysis for time-optimality while achieving constraint handling and disturbance rejection. The formulation minimizes the time required to reach a terminal set while maintaining recursive feasibility and robustness. Unlike most Time-Optimal Model Predictive Control approaches that depend on time-scaling or iterative horizon reduction, the proposed framework computes a backward reachable tube offline. Each reachable set contains the largest set of states that can reach the terminal set within a fixed number of steps. A controller that constrains the state to remain within the backward reachable tube achieves time-optimality while ensuring real-time applicability and feasibility under disturbances, also with a short prediction horizon. The central innovation lies in the use of configuration-constrained polytopes to construct backward reachable tubes offline with fixed complexity, which permits propagation over long time horizons without growing computational cost. Notably, our method introduces a containment check that dynamically identifies the set in the tube that contains the current system state and lies closer to the terminal set, thereby providing an updated time index for the controller. This effectively replaces the originally planned initial constraint with one that reflects the actual progress. As a result, the controller exploits favorable disturbance realizations, accelerates convergence to the terminal set, and significantly reduces conservatism without compromising robustness. The approach is validated in a vertical farming application, where the objective is to drive crop growth toward desired biomass and ripeness levels as quickly as possible.