Implicit Learning-Based Optimal Control of Uncertain Nonlinear Systems

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A SummaryProject Summary: This project focuses on the synthesis of new implicit learning-based methods thatcan optimally achieve some control objective for an uncertain nonlinear system. The main research goalsinclude the development and experimental verification of implicit learning and adaptive methods that enablethe mismatch between the desired and actual response of an uncertain nonlinear system to convergewhile optimizing a trade-off between performance and control energy. Efforts will investigate if differentlearning and adaptive methods have properties that yield more optimal solutions or lead to improved stabilitymargins. Progress on this research topic has been stymied by the challenge of solving a Hamilton-Jacobiequation, and the lack of mathematical tools to asymptotically compensate for generic disturbances with acontinuous controller. With the emergence of new implicit learning methods and general Lyapunov analysistechniques, the community is now well positioned to focus increasing attention on simultaneously achievingoptimality and stability for uncertain nonlinear systems. The learning capacity of the developed controllerswill enable analytical optimal control solutions for a broader class of engineering systems than is currentlypossible. Optimizing the performance of a control system along with the required control effort will yieldimproved efficiency that can lead to timely economic and environmental cost savings.Intellectual Merit: Few mathematical tools exist to synthesize controllers for nonlinear systems with modeluncertainty and unmodeled disturbances. Of the few tools that exist, either the developed controller requiresdiscontinuous 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 suchdisturbances through a nonlinear differential equation. This advancement opens new possibilities to refocusthe nonlinear systems community on the dual stability and optimality problem for general systems. Efforts inthis project seek to explore how such implicit learning controllers (and potential permutations) can be usedto yield analytical solutions to different optimal control problems. The ability to integrate the proposed classof 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, stabilityanalysis, and optimal analysis methods will be required to determine the interplay of optimality, learningcapacity, and robustness. Outcomes from these aims may provide an inroad to new ways to augmentcontrollers to incorporate optimality into the design process.Broad Impact: The theoretical discoveries are expected to have a transformative impact on optimal controlmethods for uncertain nonlinear systems. One approach to solve current optimal control problems is touse numerical methods that only provide local optimal results (at best), typically do not have a proof ofstability or optimality, and are typically open-loop. Also, numerical methods are black box approaches, sothe 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 solutionsis that they often do not optimize the real engineering problem because of the narrow class of systemsthat can be analytically examined. The expected outcomes of this project are new mathematical tools todevelop analytical stability and optimality solutions for broad classes of nonlinear systems. Further broadimpact will be realized by integrating the research outcomes into educational and outreach efforts. Effortswill seek to disseminate the research outcomes to engineers in industry, researchers, and students rangingfrom grade school through graduate school with an emphasis on under-represented groups. Outcomes ofthe 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 industrialcontrol engineers, undergraduate honor?s thesis research, existing University of Florida programs for highschooland under-represented students, and a robotics summer camp for grade school children.A-