Globally Certified Invariant-Ellipsoid Control from Data

Abstract

This letter develops a data-based method for designing state feedback for a discrete-time linear system under bounded disturbances. For each admissible feedback gain and scalar design parameter, a Lyapunov equation determines an invariant ellipsoid: a region that the state cannot leave under the permitted disturbances. We optimise the gain and parameter to minimise a trace-based measure of the resulting output enclosure. At each parameter value, value iteration gives a lower cost bound, while separate controller evaluation gives an achievable upper bound. These bounds also let us assess whole parameter intervals without evaluating every point. We prove that the search stops after finitely many evaluations with a stabilising state-feedback gain whose objective is within any prescribed absolute tolerance of the infimum over the chosen ellipsoid family. Neither an initially stabilising gain nor attainment of the infimum is assumed. The guarantee assumes exact arithmetic and a sufficiently informative batch of exact measurements, including disturbances during data collection; the resulting feedback uses only the state. A position--velocity example illustrates the bounds, controller checks, and computational cost.