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首页> 外文期刊>Automatic Control, IEEE Transactions on >A Kinship Function Approach to Robust and Probabilistic Optimization Under Polynomial Uncertainty
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A Kinship Function Approach to Robust and Probabilistic Optimization Under Polynomial Uncertainty

机译：多项式不确定性下鲁棒概率优化的亲属函数方法

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摘要

In this paper, we study a class of robust design problems with polynomial dependence on the uncertainty. One of the main motivations for considering these problems comes from robust controller design, where one often encounters systems that depend polynomially on the uncertain parameters. This paper can be seen as integrated in the emerging area of probabilistic robustness, where a probabilistic relaxation of the original robust problem is adopted, thus requiring the satisfaction of the constraints not for all possible values of the uncertainty, but for most of them. Different from the randomized approach for tackling probabilistic relaxations, which is only guaranteed to provide soft bounds on the probability of satisfaction, we present a deterministic approach based on the novel concept of kinship function introduced in this paper. This allows the development of an original framework, which leads to easily computable deterministic convex relaxations of the probabilistic problem. In particular, optimal polynomial kinship functions are introduced, which can be computed a priori and once for all and provide the “best convex bound” on the probability of constraint violation. More importantly, it is proven that the solution of the relaxed problem converges to that of the original robust optimization problem as the degree of the optimal polynomial kinship function increases. Furthermore, by relying on quadrature formulas for computation of integrals of polynomials, it is shown that the computational complexity of the proposed approach is polynomial in the number of uncertain parameters. Finally, unlike other deterministic approaches to robust polynomial optimization, the number of variables in the ensuing optimization problem is not increased by the proposed approximation. An important feature of this approach is that a significant amount of the computational burden is shifted to a one-time offline computation whose results can be stored and provided to -n-nthe end-user.
机译：在本文中，我们研究了一类具有多项式依赖不确定性的鲁棒设计问题。考虑这些问题的主要动机之一是鲁棒的控制器设计，在该设计中，经常会遇到多项式取决于不确定参数的系统。可以将本文视为集成在概率鲁棒性的新兴领域中，在该领域中采用了对原始鲁棒性问题的概率松弛，因此要求对所有不确定性值（而不是所有不确定性）都满足约束条件。与解决随机松弛的随机方法不同，后者只能保证为满足概率提供软边界，我们提出一种基于本文介绍的亲属功能新概念的确定性方法。这允许开发原始框架，从而导致容易计算概率问题的确定性凸松弛。特别地，引入了最佳多项式亲属函数，可以对它进行先验和一次计算，并提供约束违反概率的“最佳凸边界”。更重要的是，事实证明，随着最佳多项式亲缘关系函数的程度增加，松弛问题的解收敛于原始鲁棒优化问题的解。此外，通过使用正交公式来计算多项式的积分，结果表明，该方法的计算复杂度是不确定参数数量的多项式。最后，不同于其他确定性方法来进行鲁棒多项式优化，所提出的近似方法不会增加随后优化问题中的变量数量。这种方法的一个重要特征是大量的计算负担转移到了一次脱机计算中，其结果可以存储并提供给最终用户。

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