We study predictive control in a setting where the dynamics are time-varying and linear, and the costs are time-varying and well-conditioned. At each time step, the controller receives the exact predictions of costs, dynamics, and disturbances for the future $k$ time steps. We show that when the prediction window $k$ is sufficiently large, predictive control is input-to-state stable and achieves a dynamic regret of $O(\lambda^k T)$, where $\lambda < 1$ is a positive constant. This is the first dynamic regret bound on the predictive control of linear time-varying systems. Under more assumptions on the terminal costs, we also show that predictive control obtains the first competitive bound for the control of linear time-varying systems: $1 + O(\lambda^k)$. Our results are derived using a novel proof framework based on a perturbation bound that characterizes how a small change to the system parameters impacts the optimal trajectory.

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基于微扰的线性时变系统预测控制的遗憾分析

我们在动态随时间变化和线性且成本随时间变化且条件良好的环境中研究预测控制。在每个时间步长，控制器都会收到对未来 $k$ 时间步长的成本、动态和干扰的准确预测。我们表明，当预测窗口 $k$ 足够大时，预测控制是输入到状态稳定的，并实现 $O(\lambda^k T)$ 的动态遗憾，其中 $\lambda < 1 $ 是一个正不变。这是线性时变系统预测控制的第一个动态后悔界。在对终端成本的更多假设下，我们还表明预测控制获得了线性时变系统控制的第一个竞争界限：$1 + O(\lambda^k)$。