Abstract

Time delays are an inherent feature of real-world dynamical systems, arising whenever information, matter, or energy requires non-negligible time to propagate. Even small delays can destabilize systems that are otherwise well behaved, yet most standard control designs--such as PID or LQR--do not explicitly account for them and may fail when delays are present. This thesis investigates the stability of controlled dynamical systems subject to arbitrary time delays and develops new tools to determine when such systems remain stable. Our approach builds on the notion of intrinsic stability, a strong form of global stability introduced by Webb and Bunimovich, which guaranties robustness to all time delays. Because intrinsic stability is typically more restrictive than classical stability, it is often difficult to verify or achieve through standard control methods. To address this challenge, we derive new sufficient criteria for intrinsic stabilizability based on eigenvalue localization techniques, most notably Gershgorin regions. These criteria allow one to quickly determine whether a controlled system can be made intrinsically stable. Furthermore, we introduce a class of matrices and a constant-time algorithm that constructs intrinsically stable control schemes whenever they exist. The methods developed here provide practical and computationally efficient tools for designing delay-robust controllers. Examples from electrical and mechanical systems illustrate the applicability and effectiveness of our results. Finally, we extend the intrinsic stability framework to controlled switched systems, a broad class of systems in which the dynamics change over time. This extension demonstrates that our techniques apply not only to fixed control laws but also to systems with mode dependent behavior, where questions of delay-robust stability are even more challenging.

Degree

College and Department

Mathematics; Computational, Mathematical, and Physical Sciences

Rights

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