New Theoretical and Applied Methods in Optimal Control

最优控制的新理论和应用方法最优控制是应用动力学和控制的所有努力的基础：如果任何系统被强迫做某事，它可以被强迫以最优方式去做。历史上，还没有单一的、系统的程序来解决非线性最优反馈控制律，可以应用于各种边界条件下的给定最优控制问题。当系统应用不同的边界或终端条件时，最优反馈控制律的性质会发生剧烈变化，且无明显规律。这是一个基本的困难，并且意味着给定动力系统的最优控制律必须作为系统变化的边界条件和目标被“重新解决”。我们提出的研究直接解决了这一限制。从导出Hamilton-Jacobi-Bellman方程的相同基本基础出发，我们开发了一种解决最优反馈控制问题的新方法，克服了这些障碍，实现了真正的可重构控制。利用经典的正则变换理论（对应于由最优性必要条件导出的哈密顿系统的解流），我们能够对动态系统上具有任意边界条件的最优控制问题提出形式化的解。这些形式化的结果已经被证明是富有成效的，因为我们已经能够开发一个显式的解决过程，为一类一般问题找到非线性最优反馈控制律的解析形式。此外，我们的方法可以提供一种明确的算法来重新配置最优反馈控制，以处理边界条件和终端约束的变化，只要成本函数和动力学（即哈密顿函数）保持不变。我们将继续发展我们的方法，并探索将我们的应用方法应用于更大类的控制问题，包括那些具有控制约束、状态约束、欠驱动控制和非分析成本函数的控制问题。这项研究的主要成果将是解决和分析最优控制问题的新理论形式，以及可以为一般系统生成最优反馈控制律的计算工具。我们的新形式主义和这个工具都将在教育和研究环境中发挥重要作用，并将提供给密歇根大学最优控制研究生课程的学生。

New Theoretical and Applied Methods in Optimal ControlOptimal control is fundamental to all endeavors that apply dynamics and control: if any system is to be forced to do something, it can be forced to do so in an optimal way. Historically there has been no single, systematic procedure for the solution of non-linear optimal feedback control laws that can be applied to a given optimal control problem across a variety of boundary conditions. As different boundary or terminal conditions are applied to the system, the nature of the optimal feedback control laws can change drastically and with no apparent pattern. This is a fundamental difficulty, and implies that the optimal control law for a given dynamical system must be "re-solved" as the boundary conditions and targets for the system change. The research we are proposing directly addresses this limitation. Starting from the same basic foundations from which the Hamilton-Jacobi-Bellman equation is derived, we have developed a new approach to solving optimal feedback control problems that overcome some of these barriers to truly reconfigurable control. Using the classical theory of canonical transformations (corresponding to the solution flow of the Hamiltonian system derived from the necessary conditions for optimality) we are able to pose a formal solution to the optimal control problem with arbitrary boundary conditions placed on the dynamical system. These formal results have proven to be fruitful, as we have been able to develop an explicit solution procedure that finds an analytical form for the non-linear optimal feedback control law for a general class of problems. Furthermore, our approach can provide an explicit algorithm for reconfiguring optimal feedback controls to deal with changes in boundary conditions and terminal constraints, so long as the cost function and dynamics (i.e., the Hamiltonian function) remains the same.We will continue the development of our approach and explore the application of our applied methodology to larger classes of control problems, including those with control constraints, state constraints, under-actuated controls, and non-analytic cost functions. The main outcomes of this research will be a new theoretical formalism for solving and analyzing optimal control problems and a computational tool that can generate optimal feedback control laws for a general class of systems. Both our new formalism and this tool will be of great use in educational and research settings, and will be made available to students taking graduate courses in optimal control at Michigan.