Online optimization of switched LTI systems using continuous-time and hybrid accelerated gradient flows

Abstract

This paper studies the design of feedback controllers to steer a switching linear dynamical system to the solution trajectory of a time-varying convex optimization problem. We propose two types of controllers: (i) a continuous controller inspired by the online gradient descent method, and (ii) a hybrid controller that can be interpreted as an online version of Nesterov’s accelerated gradient method with restarts of the state variables. By design, the controllers continuously steer the system toward a time-varying optimal equilibrium point without requiring knowledge of exogenous disturbances affecting the system. For cost functions that are smooth and satisfy the Polyak–Łojasiewicz inequality, we demonstrate that the online gradient-flow controller ensures uniform global exponential stability when the time scales of the system and controller are sufficiently separated and the switching signal of the system varies slowly on average. For cost functions that are strongly convex, we show that the hybrid accelerated controller can outperform the continuous gradient descent method. When the cost function is not strongly convex, we show that the hybrid accelerated method guarantees global practical asymptotic stability.