A Hybrid Systems Model of Feedback Optimization for Linear Systems

Feedback optimization algorithms compute inputs to a system in real time, which helps mitigate the effects of unknown disturbances. However, existing work models both system dynamics and computations in either discrete or continuous time, which does not faithfully model some applications. In this work, we model linear system dynamics in continuous time, and we model the computations of inputs in discrete time. Therefore, we present a novel hybrid systems framework for modeling feedback optimization of linear time-invariant systems that are subject to unknown, constant disturbances. For this setup, we first establish the well-posedness of the hybrid model and establish completeness of solutions while ruling out Zeno behavior. Then, our main result derives a convergence rate and an error bound for the full hybrid computation-in-theloop system and shows that it converges exponentially towards a ball of known radius about a desired fixed point. Simulation results show that this approach successfully mitigates the effects of disturbances, with the magnitude of steady-state error being 81% less than the magnitude of the disturbances in the system.