不确定系统设计的随机化方法：样本复杂度与序贯算法
Randomized Methods for Design of Uncertain Systems: Sample Complexity and Sequential Algorithms
AI总结：
本文研究不确定系统反馈设计的随机化方法，推导了样本复杂度，并引入了一类序贯概率验证算法，以降低所需样本数并保证概率精度。
摘要：
本文研究了不确定系统反馈设计的随机化方法。第一个贡献是推导了各种约束控制问题的样本复杂度。特别地，我们展示了二项分布及相关尾不等式所起的关键作用，并计算了样本复杂度。这一贡献通过减少随机化算法中所需样本数，显著改进了现有结果。然后，这些结果被应用于最坏情况性能分析和鲁棒优化设计。本文的第二个贡献是引入了一类通用的序贯算法，称为序贯概率验证（SPV）。在这些序贯算法中，每次迭代时，候选解被概率验证，并在必要时进行修正，以满足所需规格。我们推导的结果提供了样本复杂度，保证了使用SPV算法获得的解满足预先指定的概率精度和置信度。通过一个涉及鲁棒系统辨识的数值示例，展示了这些算法的性能，并与其他现有方法进行了比较。
英文摘要：
In this paper, we study randomized methods for feedback design of uncertain systems. The first contribution is to derive the sample complexity of various constrained control problems. In particular, we show the key role played by the binomial distribution and related tail inequalities, and compute the sample complexity. This contribution significantly improves the existing results by reducing the number of required samples in the randomized algorithm. These results are then applied to the analysis of worst-case performance and design with robust optimization. The second contribution of the paper is to introduce a general class of sequential algorithms, denoted as Sequential Probabilistic Validation (SPV). In these sequential algorithms, at each iteration, a candidate solution is probabilistically validated, and corrected if necessary, to meet the required specifications. The results we derive provide the sample complexity which guarantees that the solutions obtained with SPV algorithms meet some pre-specified probabilistic accuracy and confidence. The performance of these algorithms is illustrated and compared with other existing methods using a numerical example dealing with robust system identification.
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