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，任何可以想象到的东西都有嵌入式控制系统。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。稳定性是动态系统最重要的性质，因此设计控制器使得系统即使在存在不确定反馈的情况下也是稳定的是至关重要的。这一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。该项目将把稳定性分析和控制器综合的算法工具带给广泛的科学家和工程师社区，最有效的方法是提供免费的软件。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，它用固定阶控制器稳定给定的系统，该控制器局部优化适当的目标，如稳定半径，超过控制器变量。HIFOO已经成功地应用于各种应用，包括异类多代理系统和网络的同步、稳定光学有效载荷绕轴的角运动的电动万向节的设计、飞机飞行系统的控制器以及微创外科手术。研究了不确定反馈线性依赖于输出的线性动力系统。这个范例导致了一个常微分方程式，它的系统矩阵是一个线性分式映射。相关的稳定性半径衡量了在保证系统稳定性的同时可以容忍的扰动的大小，即对于范数有界于给定量的所有扰动，系统矩阵的特征值都在复平面的左半部。该项目的一个关键目标是开发一种高效、可扩展、准确的稳定半径计算算法，适用于系统矩阵较大且稀疏的情况。在不损失一般性的情况下，所考虑的扰动可以被假设为具有一阶的矩阵，因此正在开发的算法依赖于利用一阶扰动进行高效迭代，并利用现有的特征解析器有效地利用一阶结构。该算法的一个重要方面将是通过利用关于哈密顿矩阵的虚特征值和相关转移矩阵的奇异值之间的关系的定理，使用一种专门设计为对这些特征值的计算中的误差具有鲁棒性的新方法，来确保准确地计算稳定半径。这个项目中正在开发的新算法将允许为比以前可能的更大的系统设计低阶控制器，包括控制离散化的偏微分方程组。

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.