Implicit Learning-Based Optimal Control of Uncertain Nonlinear Systems

项目摘要：本项目的重点是综合新的基于内隐学习的方法，这些方法可以最优地实现不确定非线性系统的某些控制目标。主要研究目标包括内隐学习和自适应方法的开发和实验验证，这些方法使不确定非线性系统的期望和实际响应之间的不匹配收敛，同时优化性能和控制能量之间的权衡。我们将研究不同的学习和自适应方法是否具有产生更优解或提高稳定性裕度的特性。这一研究课题的进展一直受到解决哈密顿-雅可比方程的挑战，以及缺乏用连续控制器渐近补偿一般干扰的数学工具的阻碍。随着新的内隐学习方法和一般李雅普诺夫分析技术的出现，社区现在很好地将注意力集中在同时实现不确定非线性系统的最优性和稳定性上。开发的控制器的学习能力将使分析最优控制解决方案比目前可能的更广泛的工程系统。优化控制系统的性能以及所需的控制工作将提高效率，从而及时节省经济和环境成本。智力优势：很少有数学工具能够综合具有模型不确定性和未建模干扰的非线性系统的控制器。在现有的少数工具中，开发的控制器要么需要不连续反馈，要么在残余误差的意义上表现出退化的稳态性能。最近的发展产生了一类新的连续控制器，它可以通过非线性微分方程隐式地学习这种干扰。这一进展为非线性系统学界重新关注一般系统的对偶稳定性和最优性问题开辟了新的可能性。在这个项目的努力寻求探索如何这种隐式学习控制器（和潜在的排列）可以用来产生不同的最优控制问题的分析解决方案。将所提出的隐式学习控制器（以及与其他自适应和学习技术集成的此类控制器）与最优控制方法集成的能力是一个尚未探索的概念。新的闭环误差系统开发，稳定性分析和最优分析方法将需要确定最优性，学习能力和鲁棒性的相互作用。这些目标的结果可能为增强控制器将最优性纳入设计过程的新方法提供了一条途径。广泛影响：理论发现有望对不确定非线性系统的最优控制方法产生变革性影响。解决当前最优控制问题的一种方法是使用仅提供局部最优结果（充其量）的数值方法，通常没有稳定性或最优性的证明，并且通常是开环的。此外，数值方法是黑盒方法，因此设计师无法直观地了解系统参数对最优性的影响。这些问题促使人们需要分析方法。然而，开发分析解决方案的挑战在于，它们通常不能优化实际的工程问题，因为可以分析检查的系统类别很窄。这个项目的预期结果是新的数学工具来开发分析稳定性和最优解的广泛类别的非线性系统。通过将研究成果纳入教育和推广工作，将实现进一步的广泛影响。将努力将研究成果传播给工业工程师、研究人员和从小学到研究生的学生，重点关注代表性不足的群体。研究成果将通过以下渠道传播给这些群体：同行评审出版物、会议研讨会、课程开发、工业控制工程师新证书课程的开发、本科荣誉？他的论文研究，佛罗里达大学现有的高中生和代表性不足的学生项目，以及一个面向小学生的机器人夏令营。A -

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-