Robust Stability of Linear Dynamical Systems: Algorithms, Theory and Applications

动力系统在我们的现代世界中无处不在，从汽车、飞机到医疗设备，嵌入式控制系统无处不在。为这些系统设计基于反馈的控制器是一个越来越重要的范例。稳定性是动力系统最重要的特性，因此设计控制器使系统即使在存在不确定反馈的情况下也是稳定的是必要的。本研究项目旨在扩展这一重要领域的理论并开发新的计算算法。该项目将为广泛的科学家和工程师社区带来稳定性分析和控制器合成的算法工具，最有效的是通过提供免费软件。由首席研究员开发的开源软件工具箱HIFOO就是为此目的而设计的，它使用固定顺序的控制器来稳定给定的系统，该控制器可以在控制器变量上局部优化适当的目标，例如稳定半径。HIFOO已经成功地应用于各种各样的应用，包括异构多智能体系统和网络的同步，用于稳定光学有效载荷绕轴的角运动的机动框架的设计，飞机飞行系统的控制器，以及微创手术。考虑不确定反馈线性依赖于输出的线性动态系统。这个范例导致一个常微分方程，它的系统矩阵是一个线性分数映射。相关的稳定性半径测量了在保证系统稳定性的同时可以容忍的扰动的大小，即，对于给定量范数有界的所有扰动，系统矩阵的特征值位于复平面的左半部分。该项目的一个关键目标是开发一种高效、可扩展、准确的算法来计算稳定半径，该算法适用于系统矩阵大且稀疏的情况。在不丧失一般性的情况下，所考虑的扰动可以假定为秩为1的矩阵，因此所开发的算法依赖于有效地迭代秩为1的扰动，利用现有的特征解算器有效地利用秩为1的结构。该算法的一个重要方面是，通过利用哈密顿矩阵的虚特征值与相关传递矩阵的奇异值之间的关系定理，使用一种特别设计的新颖方法，在计算这些特征值的误差方面具有鲁棒性，从而确保稳定半径的准确计算。该项目中正在开发的新算法将允许为比以前可能的大得多的系统设计低阶控制器，包括偏微分方程离散系统的控制。

Dynamical systems are ubiquitous in our modern world, with embedded control systems in everything imaginable, from cars and airplanes to medical devices. Designing controllers for these systems based on feedback is a paradigm of increasingly great importance. Stability is the most important property of a dynamical system, so it is essential that controllers be designed so that systems are stable even in the presence of uncertain feedback. This research project aims to extend the theory and develop new computational algorithms in this important area. The project will bring the tools of algorithms for stability analysis and controller synthesis to a wide community of scientists and engineers, most effectively through the provision of freely available software. The open-source software toolbox HIFOO, developed by the principal investigator, was designed for this purpose, stabilizing a given system with a fixed-order controller that locally optimizes appropriate objectives, such as the stability radius, over the controller variables. HIFOO has been used successfully in a wide variety of applications, including synchronization of heterogeneous multi-agent systems and networks, design of motorized gimbals that stabilize an angular motion of an optical payload around an axis, controllers for aircraft flight systems, and minimally invasive surgery. We consider linear dynamical systems with uncertain feedback depending linearly on the output. This paradigm leads to an ordinary differential equation whose system matrix is a linear fractional map. The associated stability radius measures the size of perturbations that can be tolerated while still guaranteeing system stability, i.e., so that the eigenvalues of the system matrix are in the left half of the complex plane for all perturbations that are norm-bounded by a given quantity. A key goal of the project is to develop an efficient, scalable, accurate algorithm for computing the stability radius that is applicable to the case where the system matrix is large and sparse. Without loss of generality, the perturbations under consideration can be assumed to be matrices with rank one, so the algorithm under development depends on efficiently iterating with rank-one perturbations, exploiting the rank-one structure efficiently with existing eigensolvers. An important aspect of the algorithm will be to ensure that the stability radius is computed accurately by making use of a theorem about the relationship between imaginary eigenvalues of a Hamiltonian matrix and singular values of the associated transfer matrix, using a novel approach that is specifically designed to be robust with respect to errors in the computation of these eigenvalues. The new algorithm under development in this project will allow the design of low-order controllers for much larger systems than was previously possible, including control of discretized systems of partial differential equations.