Infinite-dimensional convex optimization in optimal and robust control theory
Infinite-dimensional convex optimization in optimal and robust control theory
复制标题
最优鲁棒控制理论中的无限维凸优化
DOI：
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发表时间：
1997
影响因子：
6.8
通讯作者：
M. Dahleh
中科院分区：
文献类型：
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作者：
P. Young;M. Dahleh
许多工程问题可以等价于求解半定规划（SP），即，涉及线性矩阵不等式（LMI）的凸优化问题。强大的计算工具可用于在有限维的情况下，这样的问题。然而，在最优和鲁棒控制理论中出现的问题往往是无穷维的，因此没有足够的计算工具。用有限计算工具解决这类问题的关键是要有一个没有对偶间隙的问题的原始-对偶公式。在本文中，我们研究了无限维SP和提升技术重铸SP参数化线性规划（LP）。这使得丰富的理论工具可用于无限维LP扩展到无限维SP。特别是，我们开发了一些新的充分条件缺乏的对偶间隙无限维SP，并给出了这些情况下的原始和对偶问题的精确表征。原始问题和对偶问题都形成为无限维SP问题，每个问题的有限截断分别给出无限维问题的精确解的上界和下界。因此，这些结果可以形成无限维问题的实际计算方案的基础上，只需要有限维的计算工具。为了说明这些工具的力量，我们将结果应用到两个以前未解决的优化问题，即最小化的l/sub 1/范数的闭环系统的频率响应幅度的界限在有限数量的点和/或界限的H/sub 2/范数。
Many engineering problems can be shown to be equivalent to solving semidefinite programs (SPs), i.e., convex optimization problems involving linear matrix inequalities (LMIs). Powerful computation tools are available for such problems in the finite-dimensional case. However, the problems arising in optimal and robust control theory are often infinite dimensional, and so adequate computation tools are not available. The key to tackling such problems with finite computation tools is to have a primal-dual formulation of the problem without duality gap. In this paper we study infinite-dimensional SPs and present a lifting technique to recast SPs as parameterized linear programs (LPs). This enables the wealth of theoretical tools available for infinite-dimensional LPs to be extended to infinite-dimensional SPs. In particular, we develop some new sufficient conditions for the lack of a duality gap for infinite-dimensional SPs and give an exact characterization of the primal and dual problems for these cases. Both primal and dual problems are formed as infinite-dimensional SP problems, with finite truncations to each giving upper and lower bounds, respectively, on the exact solution to the infinite-dimensional problem. Thus, these results can form the basis of practical computation schemes for infinite-dimensional problems, which require only finite-dimensional computation tools. To illustrate the power of these tools we apply the results to two previously unsolved optimization problems, namely minimizing the l/sub 1/ norm of a closed-loop system subject to bounds on the frequency response magnitude at a finite number of points and/or bounds on the H/sub 2/ norm.