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，从汽车和飞机到医疗设备，嵌入式控制系统无处不在。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。该研究项目旨在扩展该理论并在这一重要领域开发新的计算算法。 该项目将为广大科学家和工程师提供稳定性分析和控制器综合算法工具，最有效的方式是提供免费软件。由主要研究者开发的开源软件工具箱HIFOO就是为此目的而设计的，它用一个固定阶控制器稳定给定的系统，该控制器在控制器变量上局部优化适当的目标，如稳定半径。HIFOO已成功地用于各种各样的应用中，包括异构多智能体系统和网络的同步，稳定光学有效载荷围绕轴的角运动的机动万向节的设计，用于飞机飞行系统的控制器，以及微创手术。我们考虑具有线性依赖于输出的不确定反馈的线性动力系统。这种范式导致一个常微分方程，其系统矩阵是一个线性分式映射。相关联的稳定性半径测量在仍然保证系统稳定性的同时可以容忍的扰动的大小，即，使得系统矩阵的特征值对于所有由给定量范数有界的扰动都在复平面的左半部分。该项目的一个关键目标是开发一种有效的，可扩展的，精确的算法，用于计算稳定半径，适用于系统矩阵大而稀疏的情况。在不失一般性的情况下，所考虑的扰动可以假设为秩为1的矩阵，因此正在开发的算法依赖于有效地迭代秩为1的扰动，有效地利用秩为1的结构与现有的特征值求解器。 该算法的一个重要方面将是确保通过利用关于哈密顿矩阵的虚特征值与相关联的传递矩阵的奇异值之间的关系的定理来精确地计算稳定性半径，使用专门设计为相对于这些特征值的计算中的误差是鲁棒的新方法。该项目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.