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.

本文研究了反馈控制器的设计，以引导切换线性动力系统解决时变凸优化问题的轨迹。我们提出了两种类型的控制器：（i）受在线梯度下降法启发的连续控制器，以及（ii）混合控制器，可以解释为 Nesterov 加速梯度方法的在线版本，并重新启动状态变量。通过设计，控制器不断将系统引导至随时间变化的最佳平衡点，而无需了解外生干扰影响系统。对于平滑且满足 Polyak-Łojasiewicz 不等式的成本函数，我们证明了当系统和控制器的时间尺度充分分离且系统的切换信号缓慢变化时，在线梯度流控制器可确保均匀的全局指数稳定性平均。对于强凸的成本函数，我们表明混合加速控制器可以优于连续梯度下降法。当成本函数不是强凸时，我们表明混合加速方法保证了全局实际渐近稳定性。