Model Predictive Control for Constrained Linear Positive Systems on Graphs
For control theorists and engineers working with networked systems (e.g., routing, logistics), this work offers a method to obtain explicit MPC guarantees for constrained positive systems, which are often intractable.
The paper develops a model predictive control (MPC) framework for constrained linear positive systems on graphs, providing explicit suboptimal admissible controllers and computable performance bounds. It shows that leveraging system structure allows for MPC guarantees typically unavailable, including a minimum stabilizing horizon without terminal conditions.
Positive systems describing networks with inherently non-negative states and inputs arise naturally in routing, logistics, and compartmental modelling. We consider problems modelled as positive linear systems in incidence form with linear cost. The addition of capacity constraints on states (storage) and inputs (flows between nodes) significantly increases the problem complexity. Leveraging the analytic structure of the unconstrained problem, an explicit suboptimal admissible controller is constructed. This yields graph-computable performance bounds and a minimum stabilising horizon length for a model predictive controller without terminal conditions. A convex program enables efficient computation of the optimal bound and horizon. These results highlight how system structure enables explicit MPC guarantees that are typically not available.