通过随机控制输入，实现对不稳定连续系统动态的精准识别。
On the Effect of Instability on Learning Continuous-Time Linear Control Systems
- 用随机控制输入估计不稳定的开环矩阵。
- 估计误差随轨迹长度增加而减小，随维度上升而增大。
- 理论工具可推广至非平稳鞅分析，适合控制与系统识别研究者。
我们研究基于单条有限长度状态轨迹的随机连续时间动态系统辨识问题。提出一种通过合理随机化控制输入来估计可能不稳定的开环矩阵的方法，并建立理论性能保证：估计误差随轨迹长度、激励度和信噪比增加而下降，随系统维度增加而上升。数值实验展示了动态学习速率。为进行理论分析，发展了新的技术工具，包括针对高度非平稳鞅的非渐近随机界、广义迭代对数律等，具有独立研究价值。
原文摘要 · Abstract (English)
We study the problem of system identification for stochastic continuous-time dynamics, based on a single finite-length state trajectory. We present a method for estimating the possibly unstable open-loop matrix by employing properly randomized control inputs. Then, we establish theoretical performance guarantees showing that the estimation error decays with trajectory length, a measure of excitability, and the signal-to-noise ratio, while it grows with dimension. Numerical illustrations that showcase the rates of learning the dynamics, will be provided as well. To perform the theoretical analysis, we develop new technical tools that are of independent interest. That includes non-asymptotic stochastic bounds for highly non-stationary martingales and generalized laws of iterated logarithms, among others.
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