Dynamical systems with random influences mixing logic and physics: a framework enabling control engineers to design for resilient autonomy

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

The goal of this project is to develop the mathematical tools necessary to describe and analyze dynamical systems that mix logic and physics while including random influences. These types of systems are sometimes called stochastic hybrid systems; where stochastic refers to the presence of random influences and hybrid refers to the mix of logic-based decision making and physical laws that determine the system evolution. These mathematical tools would be extremely useful, as they would provide a framework enabling control engineers to better design autonomous dynamical systems that are resilient in the face of uncertainty. The results of this proposal have the potential to impact researchers in a wide variety of application domains, where randomness interacts with worst-case effects and rigorous analysis tools are needed to certify desirable behavior in an engineered system. One possible application domain is the control of gene expression and other particular biological phenomena. Another application area is multi-agent resource allocation algorithms where agents invoke randomness in their allocation strategies to guarantee robustness to malicious agents in the network. Indeed, stochastic hybrid systems can be used to model complicated biological systems, financial systems, resource allocation systems, and traffic management systems, to name just a few. All of the results developed through this proposal will be published in the leading conference venues and journals, to provide avenues for the transition of the work to applications. Beyond contributions to the scientific literature, the proposed work will add to the existing graduate curriculum on stochastic and hybrid dynamical systems. Graduate students will be trained through direct financial support, but a broader group of students will be educated through teaching materials and a course that will be developed around the produced breakthroughs. These students include graduate students in all areas of engineering. To reach students broadly, the PI envisions a graduate textbook on stochastic hybrid systems emerging from this work, to complement the recent book on non-stochastic hybrid systems. The work will spawn collaboration with international researchers, and will enhance student exchange programs with several international universities. The research developments will be applied to areas that are important to the broad population. Moreover, the PI will use this opportunity to consider how to expose freshman engineering students to a broad range of dynamical systems principles. This effort will follow up the previous experience teaching a freshman seminar, to engineering and non-engineering students, on the application of optimization principles to real-world problems. The aim here aligns with the societal goal of keeping younger students engaged in topics related to science, technology, engineering, and mathematics. The specific technical objectives of the project start with establishing a mathematical framework and solution concept that is tractable yet general enough to be widely applicable. Tools from the field of variational analysis are crucial to this development. The objectives continue with the task of developing a wide range of analysis tools that can be used to certify appropriate behavior of an engineered stochastic hybrid system. The techniques will especially focus on a Lyapunov analysis for stability properties like asymptotic stability in probability and recurrence; other ideas that have been fruitful for non-stochastic hybrid systems also will be considered. Next, the aim is to establish strong sequential compactness results for the set of solutions to the models considered and, from these results, establish an invariance principle and converse Lyapunov theorems. These results would parallel recent, very useful results for non-stochastic hybrid systems. The final task is to begin to show explicitly how the developed framework and analysis tools can be used to engineer advanced, resilient, autonomous control systems.