Efficient Numerical Analysis of Stability of High-Order Systems with a Time Delay

Abstract

Time delays are a common perturbation in systems with many states, such as networked, distributed, or decentralized systems. Current methods analyzing the stability of large systems with time delay typically produce very conservative results. While more exact methods exist, these become inefficient for large systems. This paper provides a methodology for analyzing the stability of time-delayed systems that is derived from exact methods but is efficient for high-order systems. The computational and memory cost of this new technique is compared to the costs of existing techniques, and its efficiency is shown using a distributed system with over four hundred states I n = n × n identity matrix

An approach to compute and design the delay margin of a large‐scale matrix delay equation

“…Due to the quadratic relationship in the first step, the existing approaches do not scale well for large‐size problems, and as n grows even to a moderate size, these approaches become intractable. This issue has been highlighted in several studies . In the work of Armanious and Lind, the authors note that the computation time to solve the stability problem, or, equivalently the DM‐problem, for n = 400 is approximately 3.3 months.…”

“…This issue has been highlighted in several studies . In the work of Armanious and Lind, the authors note that the computation time to solve the stability problem, or, equivalently the DM‐problem, for n = 400 is approximately 3.3 months. Approach (ii) can be computationally more efficient.…”

Car-following dynamics with multiple delays: network design strategies for reduced traffic jams

“…An eigenvalue-based approach for stability and stabilization of time-delay systems, which is important in the theoretical understanding and practical computation of delay margins, is provided in [123]. Meanwhile, [124] proposed efficient numerical methods for analyzing the stability of high-order systems with time delays.…”

“…Here is an example of the significance of parDMF. It is estimated that computation time of DM in a problem with size S = 300, i.e., with 300 states, should be around three months, for a class of linear systems with delays [124]. However, with parDMF, one can compute the DM of a system with size S = 800 in only 12 minutes.…”