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Polynomial optimization based approaches to system design, analysis and identification.

机译：基于多项式优化的系统设计，分析和识别方法。

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摘要

In recent developments of system and control theory, a large effort has been devoted to finding equivalent convex formulation of the problems of interest. A successful example is the wide application of linear matrix inequalities (LMIs) in formulating system design and analysis problems. From a theoretic point of view, such problems can be considered solved, as convex optimization can be solved reliably and efficient using interior-point methods or other methods available in the literature and/or commercial software. On the other hand, however, many challenging problems in system and control theory have been proven to be NP-Complete or NP-hard. Therefore, unless proven P=NP, the best way to tackle these problems is to find approximate solutions using limited computational resources. Recent developments in polynomial optimization, which include moment-based approach and its dual sum-of-square method, shed some light on solving some of those challenging problems, as it provides systematic approaches to build asymptotically convergent convex relaxations to a general polynomial optimization problem. In this dissertation, we use this as the main optimization tool to address various important yet difficult problems in system and control theory. The problems addressed are categorized into four topics: 1) chance-constraint optimization, 2) distributional robustness, 3) hybrid system identification, and 4) generalized fixed order interpolation. The first two topics is closely related to the probabilistic framework developed in recent years. In the first topic, we design special polynomial functions and use them to develop deterministic approaches to address the probabilistic constraints. Comparing the scenario approach in the literature of probabilistic control, which give soft bounds on probability, our approaches provide hard bounds. The second topic is connected to system analysis with uncertainty under probabilistic framework, in a distributional-free manner. Instead of assuming some fixed distribution on the uncertainty, it aims at finding the worst-case expected performance of the system, assuming the distribution of uncertainty is unknown but obey some loose conditions. The last two topics addressed concern hybrid system identification and generalized interpolation. We first show that these problems can be equivalently reformulated as polynomial optimization problems. While the recent developed polynomial optimization tools can construct convex relaxations to these problems, the required computational cost is prohibitively large. It is not surprising as a polynomial problem is NP-hard in general. In this dissertation, we exploit the very specific structure of these problems and provide numerically efficient algorithms to solve these problems.
机译：在系统和控制理论的最新发展中，已经付出了巨大的努力来寻找感兴趣问题的等效凸表达。一个成功的例子是线性矩阵不等式（LMI）在制定系统设计和分析问题中的广泛应用。从理论的角度来看，可以考虑解决这些问题，因为可以使用内点方法或文献和/或商业软件中可用的其他方法可靠而有效地解决凸优化问题。但是，另一方面，已证明系统和控制理论中的许多挑战性问题都是NP-Complete或NP-hard。因此，除非证明P = NP，否则解决这些问题的最佳方法是使用有限的计算资源来找到近似解。多项式优化的最新发展，包括基于矩的方法及其对偶平方和方法，为解决一些具有挑战性的问题提供了一些启示，因为它提供了系统的方法来为一般的多项式优化问题建立渐近收敛的凸松弛。 。本文将其作为主要的优化工具，解决系统和控制理论中各种重要而又困难的问题。解决的问题分为四个主题：1）机会约束优化，2）分布鲁棒性，3）混合系统识别和4）广义固定阶插值。前两个主题与近年来开发的概率框架密切相关。在第一个主题中，我们设计特殊的多项式函数，并使用它们来开发确定性方法来解决概率约束。比较概率控制文献中的情景方法，它给出了概率的软边界，而我们的方法则提供了硬边界。第二个主题是在概率框架下以无分布方式与不确定性的系统分析相关。假设不确定性的分布是未知的，但要遵循一些宽松的条件，而不是假设不确定性的分布是固定的，它的目的是找到系统的最坏情况的预期性能。最后讨论的两个主题涉及混合系统识别和广义插值。我们首先表明，这些问题可以等效地重新定义为多项式优化问题。尽管最近开发的多项式优化工具可以构造出针对这些问题的凸松弛，但所需的计算成本却过高。毫不奇怪，因为多项式问题通常是NP难的。本文研究了这些问题的特殊结构，并提供了数值有效的算法来解决这些问题。

著录项

  • 作者

    Feng, Chao.;

  • 作者单位

    The Pennsylvania State University.;

  • 授予单位 The Pennsylvania State University.;
  • 学科 Engineering Electronics and Electrical.
  • 学位 Ph.D.
  • 年度 2013
  • 页码 168 p.
  • 总页数 168
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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