原始鲁棒性与半正定锥
Primal robustness and semidefinite cones
AI总结：
本文通过凸优化标准方法重新整理并简化了线性时不变系统鲁棒稳定性和性能的核心工具，将鲁棒性分析直接作为凸优化问题，利用Gramian集的闭包形成半正定锥，从而直接包含各种约束条件。
摘要：
本文通过凸优化标准方法重新整理并简化了线性时不变系统鲁棒稳定性和性能的核心工具。特别地，通过Gramian集的闭包形成半正定锥，将鲁棒性分析直接作为凸优化（半正定规划或SDP）问题。这允许直接包含各种约束，如结构不确定性，并通过原始变量直接构造最坏情况扰动和扰动。已知结果如KYP引理和各种缩放小增益测试可通过标准SDP对偶性直接获得。对于熟悉鲁棒性和SDP的读者，该框架在回顾时似乎显而易见，但这也是其魅力所在，应增强教学，并希望启发新的研究。有一个关键引理证明Gramian的闭包，虽然显而易见，但当前证明显得过于繁琐，本文的最终目标是寻求鲁棒控制和凸优化专家帮助找到更简单的替代方案。
英文摘要：
This paper reformulates and streamlines the core tools of robust stability and performance for LTI systems using now-standard methods in convex optimization. In particular, robustness analysis can be formulated directly as a primal convex (semidefinite program or SDP) optimization problem using sets of gramians whose closure is a semidefinite cone. This allows various constraints such as structured uncertainty to be included directly, and worst-case disturbances and perturbations constructed directly from the primal variables. Well known results such as the KYP lemma and various scaled small gain tests can also be obtained directly through standard SDP duality. To readers familiar with robustness and SDPs, the framework should appear obvious, if only in retrospect. But this is also part of its appeal and should enhance pedagogy, and we hope suggest new research. There is a key lemma proving closure of a grammian that is also obvious but our current proof appears unnecessarily cumbersome, and a final aim of this paper is to enlist the help of experts in robust control and convex optimization in finding simpler alternatives.
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