ABSTRACT

In this paper, the assignability of linear time-invariant (LTI) systems with respect to arbitrary control structures is addressed. It is well established that the closed-loop spectrum of an LTI system with an arbitrary control structure is confined to the set containing the fixed-modes of the system with respect to that control structure. However, the assignment of the closed-loop spectrum is not merely limited by the existence of fixed-modes in practical scenarios. The pole assignment may require excessive control effort or even become infeasible due to the presence of small perturbations in the system dynamics. To offer more insights in such more realistic scenarios, a continuous measure known as fixed-mode radius is developed. However, its evaluation is confronted with a highly non-convex optimization problem combined with the need for a combinatorial search. This study utilises properties of positive-definite cones and duality theory to formulate the assignability assessment as an optimization problem with linear matrix inequality (LMI) constraints. Based on the suggested formulation, three alternative methods are proposed to evaluate the distance to unassignability. The first two methods offer alternative non-iterative and convex programs. The third method proposes an iterative convex optimization while updating the binary variables based on the dual variables. All the proposed methods rely on convex optimization, do not involve gridding over the complex plane and circumvent the combinatorial nature of the problem by using properties of positive definite cones. Simulation results confirm the effectiveness of the proposed methods in the assessment of fixed-mode radius with respect to arbitrary control structures.

摘要

在本文中，讨论了线性时不变（LTI）系统相对于任意控制结构的可分配性。众所周知，具有任意控制结构的 LTI 系统的闭环谱仅限于包含系统相对于该控制结构的固定模式的集合。然而，闭环频谱的分配不仅仅受限于实际场景中固定模式的存在。由于系统动力学中存在小扰动，极点分配可能需要过多的控制工作，甚至变得不可行。为了在这种更现实的场景中提供更多见解，开发了一种称为固定模式半径的连续测量。然而，它的评估面临一个高度非凸的优化问题，并需要组合搜索。本研究利用正定锥的特性和对偶理论将可分配性评估表述为具有线性矩阵不等式 (LMI) 约束的优化问题。基于建议的公式，提出了三种替代方法来评估不可分配性的距离。前两种方法提供了替代的非迭代和凸程序。第三种方法提出了一种迭代凸优化，同时基于对偶变量更新二进制变量。所有提出的方法都依赖于凸优化，不涉及复平面上的网格化，并通过使用正定锥的特性来规避问题的组合性质。