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

New York University, New York NY

Investigators

Abstract

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.