Implicit Learning-Based Optimal Control of Uncertain Nonlinear Systems

University Of Florida, Gainesville FL

Investigators

Abstract

A Summary Project Summary: This project focuses on the synthesis of new implicit learning-based methods that can optimally achieve some control objective for an uncertain nonlinear system. The main research goals include the development and experimental verification of implicit learning and adaptive methods that enable the mismatch between the desired and actual response of an uncertain nonlinear system to converge while optimizing a trade-off between performance and control energy. Efforts will investigate if different learning and adaptive methods have properties that yield more optimal solutions or lead to improved stability margins. Progress on this research topic has been stymied by the challenge of solving a Hamilton-Jacobi equation, and the lack of mathematical tools to asymptotically compensate for generic disturbances with a continuous controller. With the emergence of new implicit learning methods and general Lyapunov analysis techniques, the community is now well positioned to focus increasing attention on simultaneously achieving optimality and stability for uncertain nonlinear systems. The learning capacity of the developed controllers will enable analytical optimal control solutions for a broader class of engineering systems than is currently possible. Optimizing the performance of a control system along with the required control effort will yield improved efficiency that can lead to timely economic and environmental cost savings. Intellectual Merit: Few mathematical tools exist to synthesize controllers for nonlinear systems with model uncertainty and unmodeled disturbances. Of the few tools that exist, either the developed controller requires discontinuous feedback or exhibits degraded steady-state performance in the sense of residual errors. Recent developments have produced a new class of continuous controllers that can implicitly learn such disturbances through a nonlinear differential equation. This advancement opens new possibilities to refocus the nonlinear systems community on the dual stability and optimality problem for general systems. Efforts in this project seek to explore how such implicit learning controllers (and potential permutations) can be used to yield analytical solutions to different optimal control problems. The ability to integrate the proposed class of implicit learning controllers (and such controllers integrated with other adaptive and learning techniques) with optimal control methods is an unexplored concept. New closed-loop error system development, stability analysis, and optimal analysis methods will be required to determine the interplay of optimality, learning capacity, and robustness. Outcomes from these aims may provide an inroad to new ways to augment controllers to incorporate optimality into the design process. Broad Impact: The theoretical discoveries are expected to have a transformative impact on optimal control methods for uncertain nonlinear systems. One approach to solve current optimal control problems is to use numerical methods that only provide local optimal results (at best), typically do not have a proof of stability or optimality, and are typically open-loop. Also, numerical methods are black box approaches, so the designer is shielded from any intuition regarding the effect of the system parameters on the optimality. These issues motivate the need for analytical methods. Yet, the challenge to develop analytical solutions is that they often do not optimize the real engineering problem because of the narrow class of systems that can be analytically examined. The expected outcomes of this project are new mathematical tools to develop analytical stability and optimality solutions for broad classes of nonlinear systems. Further broad impact will be realized by integrating the research outcomes into educational and outreach efforts. Efforts will seek to disseminate the research outcomes to engineers in industry, researchers, and students ranging from grade school through graduate school with an emphasis on under-represented groups. Outcomes of the research will be disseminated to these groups through outlets including: peer-reviewed publications, conference workshops, curriculum development, the development of a new certificate program for industrial control engineers, undergraduate honor?s thesis research, existing University of Florida programs for highschool and under-represented students, and a robotics summer camp for grade school children. A-