Geometry and Algebra of Nonlinear Control Systems

Arizona State University, Scottsdale AZ

Investigators

Abstract

0072369 Kawski This project builds on the formalism of chronological algebras to study the geometric foundations of nonlinear control systems. Geometrically, the chronological product encodes the interaction of the components of controlled nonlinear dynamics. Algebraically it is the fundamental structure that underlies effective series expansion and formal solution algorithms. Areas of special focus are interconnections of systems, and systems that exhibit full nonlinear dependence on the control. In either case, specific objectives are to develop normal forms and to find effective series (or product) representations (using tools from algebraic combinatorics). Possible applications include algorithms for path planning, feedback stabilization, optimal control and even numerical integration. A complementary second thread develops interactive visualization tools for research (simulation and experimentation) and for communication. This research is motivated and driven by the desire to understand the fundamental, common principles that govern the interactions of dynamical systems on any scale, from molecular levels to astronomical dimensions. The diversity of possible outcomes is a consequence of nonlinear effects that permeate our environment: When combining two subsystems, the result is generally different from just the sum of the parts. This research focuses on further developing the algebra which captures such nonlinear interaction, where the effect of a+b is more than just the sum of the effects of a and b. The chronological algebra provides the formal language to model such interactions where even the order in which pieces are put together matters, where a*b is generally different from b*a. The control perspective is distinguished from the basic study of dynamical systems as it aims beyond just descriptive understanding: The objective of control theory is to exploit the subtle nature of the interactions in order to shape complex systems -- often via only minute interference with how the parts interact. While this mathematical research applies to virtually any kind of dynamical system, including even social, medical and financial environments, this project will focus on mechanical systems (like large satellites with several moving parts) in efforts to demonstrate the general principles. An important component of this project is the development of interactive visualization tools. These are used for research, and for communicating the geometry of the nonlinear interactions, to demonstrate how the profound understanding of the fundamental mathematical structures leads to effective means of controlling complex systems. This visualization effort also provides a rich environment for undergraduate students to connect with advanced theoretical research. It may prove to be an effective means to expand the pipeline bringing new talents into mathematical research.