A Hybrid Systems Model of Feedback Optimization for Linear Systems: Convergence and Robustness

Feedback optimization algorithms compute inputs to a system using real-time output measurements, which helps mitigate the effects of disturbances. However, existing work often models both system dynamics and computations in either discrete or continuous time, which may not accurately 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 model of feedback optimization. We first establish the well-posedness of this hybrid model and establish completeness of solutions while ruling out Zeno behavior. Then we show the state of the system converges exponentially fast to a ball of known radius about a desired goal state. Next we analytically show that this system is robust to perturbations in (i) the values of measured outputs, (ii) the matrices that model the linear time-invariant system, and (iii) the times at which inputs are applied to the system. Simulation results confirm that this approach successfully mitigates the effects of disturbances.