An Optimization Approach for Nonlinear Optimal Feedback Control Design and Uncertainty Propagation

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The multi-resolution articulation abilities of the next-generation autonomous systems, used in advanced robotics, rehabilitation, tele-operation, manufacturing and infrastructure applications are made possible by unique designs and complex mechanisms. These next generation robotic systems can benefit greatly by the development of computationally efficient tools to synthesize optimal feedback control laws to achieve specified output signal statistics in presence of model and sensing uncertainties. To support commercial applications such as package delivery, it is imperative that the robotic systems operating in uncertain environments negotiate specific waypoints with prescribed tolerance. This unique challenge of routing the vehicles operating in uncertain environments involves a strong coupling between the uncertainty propagation and control. Successful negotiation of this challenge is contingent on the development of stable optimal feedback control laws, while accounting for the uncertainties that pervade dynamical systems. The fundamental formalisms that underpin this research are widely applicable to the control of next generation robotic systems and unmanned vehicles. The project includes plans to integrate research into educational efforts involving graduate and undergraduate students, including expanding efforts to reach under-represented populations in science and engineering. Outreach activities include the technical interchange meetings with local school teachers and instituting robotics related tutorial courses to motivate middle and high school students.The focus of the research is to investigate a unified approach to stable optimal feedback control laws and uncertainty propagation methods by developing computationally efficient solutions to the Hamilton Jacobi Bellman (HJB) and Fokker-Planck-Kolmogorov (FPK) partial differential equations (PDEs), respectively. Recent advances in sparse approximation and non-product quadrature rules are exploited to solve the HJB and FPK equations. The intellectual merits are drawn from advancing the state of knowledge in uncertainty quantification, optimal control theory, and numerical analysis, and integrating them effectively to realize a scalable framework for study of dynamical systems. The Conjugate Unscented Transformation (CUT) technique in conjunction with sparse approximation methods forms an enabling tool to drive the uncertainty propagation and optimal feedback control realization processes. This research culminates in the development of nonlinear stable feedback control laws for uncertain dynamic systems, impacting various estimation and control problems in engineering. The proposed research will be demonstrated on several benchmark problems, along with two key applications involving optimal momentum transfer in control of gyroscopic systems and surveillance of a ground region with the help of unmanned vehicles.