This paper studies the uncertainty set estimation of system parameters of linear dynamical systems with bounded disturbances, which is motivated by robust (adaptive) constrained control. Departing from the confidence bounds of least square estimation from the machine-learning literature, this paper focuses on a method commonly used in (robust constrained) control literature: set membership estimation (SME). SME tends to enjoy better empirical performance than LSE's confidence bounds when the system disturbances are bounded. However, the theoretical guarantees of SME are not fully addressed even for i.i.d. bounded disturbances. In the literature, SME's convergence has been proved for general convex supports of the disturbances, but SME's convergence rate assumes a special type of disturbance support: $l_\infty$ ball. The main contribution of this paper is relaxing the assumption on the disturbance support and establishing the convergence rates of SME for general convex supports, which closes the gap on the applicability of the convergence and convergence rates results. Numerical experiments on SME and LSE's confidence bounds are also provided for different disturbance supports.

翻译：本文研究具有有界扰动的线性动力系统参数的不确定性集合估计问题，其动机源于鲁棒（自适应）约束控制。区别于机器学习文献中最小二乘估计的置信界方法，本文聚焦于（鲁棒约束）控制文献中常用的一种方法：集合成员估计（SME）。当系统扰动有界时，SME往往比LSE的置信界具有更好的实证性能。然而，即使对于独立同分布的有界扰动，SME的理论保证仍未得到充分解决。现有文献已证明SME在扰动支撑集为一般凸集时的收敛性，但其收敛速率分析仅针对特殊类型的扰动支撑集：$l_\infty$球。本文的主要贡献在于放宽了对扰动支撑集的假设，为一般凸支撑集建立了SME的收敛速率，从而弥补了收敛性结果与收敛速率结果在适用性上的差距。本文还针对不同扰动支撑集提供了SME与LSE置信界的数值实验对比。