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

即使控制系统是有限维的，在无限维空间上制定控制系统的设计和分析问题往往是方便的。当我们计算一个解时，我们必须在有限维空间中限制变量。因此，如何用有限维问题逼近原来的无限维问题，从工程角度来说具有重要意义。在本研究中，我们研究了(i) Hankel奇异值和向量的计算算法，（ii）多目标控制系统的抽象线性规划设计。对于Hankel奇异值，利用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分组成，则有限维部分及其对偶系统表示在内函数对应的移不变子空间的正交补上。然后在此基础上推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间和/或频域约束的最优控制提供了一种强有力的方法。与有限维情况不同，无限维线性规划在有限维逼近不合适的情况下可能出现对偶间隙。在这项研究中，我们展示了在不引入对偶间隙的情况下逼近对偶问题的方法。这样，我们就可以用有限维凸优化问题来计算任意误差范围内的设计问题。

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.