Collaborative Research: Nonlinear Balancing: Reduced Models and Control

University Of California-San Diego, La Jolla CA

Investigators

Abstract

Fast and accurate computer simulation of complex engineering systems is required for real-time control and engineering design. This grant will support research that will advance balanced truncation model reduction for nonlinear systems, a mathematical framework to produce reliable, accurate, and computationally efficient simulators. Despite the theoretical foundations having been laid in the 1990s, computational implementations that scale to the high dimensionality needed for today’s complex engineering systems are lacking to date. This research will overcome this barrier by developing and employing modern high-performance algorithms that exploit the mathematical structure of the equations that have to be solved. The resulting simulators will, for instance, advance the control and operation of satellites through accurate real-time estimation of atmospheric satellite drag; advance the design of aircraft through low-resource computational models that allow for a large number of design iterations; and optimize our cities’ water networks through efficiently simulating water flows and water quality so that pump stations can be scheduled optimally. This will result in greater benefits to society, improvements of civil infrastructure, and contribute to the industrial competitiveness of the United States. This grant will also support science, technology, engineering and mathematics (STEM) workforce training through a workshop at Virginia Tech that targets early-career researchers, as well as through undergraduate research opportunities. This research seeks to develop a new class of reduced-order models and controllers for complex high-dimensional polynomial nonlinear systems via the concept of nonlinear balanced truncation. To date, this framework has not been applied to model reduction for high-dimensional nonlinear systems since solving the Hamilton-Jacobi-Bellman (HJB) equations, which are at the core of the balancing approach, remained infeasible for large-scale systems. Very recent developments in tensor calculus, nonlinear state transformations, and polynomial feedback laws now make the solution to this problem feasible. This project will develop a scalable tensor-based approach to solve the HJB equations to obtain polynomial expansions of the energy functions required for balanced truncation, as well as high-performance algorithms and numerical analysis to analyze the conditioning of the tensorized problems. Moreover, efficient algorithms for parametric nonlinear balancing will be designed by exploiting the structure in parameter space. Additionally, reduced-order nonlinear controllers will be designed using a simultaneous reduction and control framework, which is far superior to the existing reduce-then-control framework. The project will also develop a theory for the robustness of these controllers, and their stabilizing properties when applied to the high-dimensional systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.