系统与控制中的线性二次问题：基于协方差表示与线性-锥对偶的有限时域情形
Linear-Quadratic Problems in Systems and Controls via Covariance Representations and Linear-Conic Duality: Finite-Horizon Case
AI总结：
本文提出统一处理系统与控制中线性二次问题的方法，通过联合状态/输入协方差矩阵表示将其转化为无限维半定规划，利用线性-锥对偶揭示LMIs与Riccati方程自然涌现于对偶问题中。
摘要：
系统与控制中出现的线性二次（LQ）问题包括确定性及随机形式的线性二次调节器（LQR）经典最优控制问题，以及$H^\infty$分析（有界实引理）、正实引理和一般积分二次约束（IQCs）测试。我们使用一种将线性二次问题转化为带正定性约束的矩阵值线性-线性问题的方法，对所有这些问题进行了统一处理。这是通过一种系统表示实现的，其中联合状态/输入协方差（确定性情况下的外积）矩阵是基本对象。LQ问题随后变为无限维半定规划，使用的关键工具是线性-锥对偶。线性矩阵不等式（LMIs）作为对偶问题的锥约束自然出现。Riccati方程刻画了这些特殊LMIs的极值，因此提供了对偶问题的解。这些问题中所有最优信号的状态反馈结构，源于原问题与对偶问题之间的对齐（互补松弛）条件。从这种方法中获得的新见解是，首先是LMIs，其次是Riccati方程，它们自然地出现在对偶问题而非原问题中。此外，虽然传统LQ问题建立在信号的$L^2$空间中，但它们等效的协方差表示问题最自然地建立在矩阵值信号的$L^1$空间中。
英文摘要：
Linear-Quadratic (LQ) problems that arise in systems and controls include the classical optimal control problems of the Linear Quadratic Regulator (LQR) in both its deterministic and stochastic forms, as well as $H^\infty$-analysis (the Bounded Real Lemma), the Positive Real Lemma, and general Integral Quadratic Constraints (IQCs) tests. We present a unified treatment of all of these problems using an approach which converts linear-quadratic problems to matrix-valued linear-linear problems with a positivity constraint. This is done through a system representation where the joint state/input covariance (the outer product in the deterministic case) matrix is the fundamental object. LQ problems then become infinite-dimensional semidefinite programs, and the key tool used is that of linear-conic duality. Linear Matrix Inequalities (LMIs) emerge naturally as conal constraints on dual problems. Riccati equations characterize extrema of these special LMIs, and therefore provide solutions to the dual problems. The state-feedback structure of all optimal signals in these problems emerge out of alignment (complementary slackness) conditions between primal and dual problems. Perhaps the new insight gained from this approach is that first LMIs, and then second, Riccati equations arise naturally in dual, rather than primal problems. Furthermore, while traditional LQ problems are set up in $L^2$ spaces of signals, their equivalent covariance-representation problems are most naturally set up in $L^1$ spaces of matrix-valued signals.
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