Fundamental Research on Effective Methods for Analysis and Design of Control Systems Based on Duality Theory

基于对偶理论的控制系统分析与设计有效方法的基础研究

基本信息

批准号：
11650447
负责人：
OHTA Yoshito
金额：
$ 2.11万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11650447/
关键词：
Control System Design L1 control Linear Programming Duality Frequency Domain Constraint Time Domain Constraint Duality Gap Finite Dimensional Approximation 多目的制御数理計画法 H無限大制御ハンケル作用素テプリッツ作用素

项目摘要

It is often convenient to formulate the control system design and analysis problems on an infinite dimensional space even if the system is finite dimensional. When we compute a solution, we have to restrict the variables on a finite dimensional space. Hence how to approximate the original infinite dimensional problem by a finite dimensional one is of great importance from the engineering point of view. In this research, we study (i) the computational algorithm of Hankel singular values and vectors, and (ii) the multi object control system design using abstract linear programming.As for the Hankel singular values, the orthogonal complement of shift invariant subspace is exploited. If the system consists of finite dimensional part and infinite dimensional inner part, then the finite part and its dual system are represented on the orthogonal complement of the shift invariant subspace corresponding to the inner function. Then the Hamiltonian formula for the singular values is derived based on this representation.As for the multi objective control design, the abstract linear programming gives a powerful approach for the optimal control with time and/or frequency domain constraints. Unlike the finite dimensional case, infinite dimensional linear programming may exhibit duality gap when finite dimensional approximation is not appropriate. In this research, we showed ways to approximate dual problems without introducing duality gap. This way, we can compute the design problem within an arbitrary error using finite dimensional convex optimization problems.

即使系统是有限维的，在无限维空间上制定控制系统设计和分析问题通常也很方便。当我们计算解决方案时，我们必须将变量限制在有限维空间上。因此，如何用有限维问题来逼近原来的无限维问题，从工程的角度来看是非常重要的。在本研究中，我们研究（i）Hankel奇异值和向量的计算算法，以及（ii）使用抽象线性规划的多目标控制系统设计。对于Hankel奇异值，利用了平移不变子空间的正交补。如果系统由有限维部分和无限维内部部分组成，则有限部分及其对偶系统表示在与内部函数对应的平移不变子空间的正交补上。然后基于该表示导出奇异值的哈密顿公式。对于多目标控制设计，抽象线性规划为具有时域和/或频域约束的最优控制提供了有力的方法。与有限维情况不同，当有限维近似不合适时，无限维线性规划可能会表现出对偶间隙。在这项研究中，我们展示了在不引入对偶间隙的情况下近似对偶问题的方法。这样，我们可以使用有限维凸优化问题在任意误差内计算设计问题。