Closed-form Solutions to Robust and Optimal Control Problems of Second-Order Systems

This thesis presents a unified, structure-preserving framework for the optimal and robust control of second-order dynamical systems. Traditional numerical approaches to solving continuous-time algebraic Riccati equations (CAREs), computing coprime factorizations, and synthesizing robust controllers often lose physical insight and become computationally expensive for large-size systems. To overcome these challenges, we develop closed-form solutions and expressions for various control problems that offer high accuracy, scalability, and insight. We begin by proposing novel closed-form stabilizing and destabilizing solutions to the CARE for second-order systems, explicitly in terms of the physically insightful and structurally significant second-order system matrices. They outperform standard solvers in both speed and accuracy, especially for large-size systems. We show that engineering problems, such as the optimal control of vibrating structures, can be addressed more simply and efficiently using our solution. Moreover, this foundation enables closed-form solutions to linear time-varying (LTV) optimal control problems via decomposition into linear time-invariant (LTI) sub-problems. Next, by deriving the closed-form solution to the filtering Riccati equation, we obtain closed-form coprime factorizations for second-order systems. These are especially valuable when frequent recomputation is needed, as in reconfigurable system scenarios. Demonstrations include a coprime factor-based Youla-Kucera controller for satellite networks and mechanical systems, showing robust and high-performance behavior. Finally, we present closed-form and simplified robust control tools for undamped second-order systems. These include explicit expressions for robustness quantification and robust controller design in the presence of parametric uncertainties. Collectively, these contributions provide scalable, real-time implementable, and physically interpretable tools for the control of second-order dynamical systems.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Aerospace Engineering

Department dc:contributor.department

Aerospace and Ocean Engineering

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2025

Author dc:creator

• Rustagi, Vishvendra

Chair dc:contributor.committeechair

• Sultan, Cornel

Committee members dc:contributor.committeemember

• Woolsey, Craig A.
• Boker, Almuatazbellah M.
• Ross, Shane David

• Second-order systems
• Riccati equations
• Coprime factorizations
• Robust stabilization
• Large-size systems

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Dc Identifier Other

vt_gsexam:44310

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/135000