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所有应用动力学和控制的努力的基础：如果任何系统被迫做某事，它可以被迫以最优的方式这样做。历史上，一直没有一个单一的，系统的程序的解决方案的非线性最优反馈控制律，可以适用于一个给定的最优控制问题，跨越各种边界条件。当不同的边界条件或终端条件应用于系统时，最优反馈控制律的性质可能会急剧变化，并且没有明显的模式。 这是一个基本的困难，并意味着最优控制律为一个给定的动态系统必须“重新解决”的边界条件和目标系统的变化。我们提出的研究直接解决了这一限制。 从相同的基本基础，从其中的Hamilton-Jacobi-Bellman方程推导出，我们已经开发出一种新的方法来解决最优反馈控制问题，克服了这些障碍，真正的可重构控制。 使用经典的正则变换理论（对应于从最优性的必要条件导出的Hamilton系统的解流），我们能够对动力系统上的任意边界条件的最优控制问题提出正式的解决方案。 这些正式的结果已被证明是富有成效的，因为我们已经能够开发出一个明确的解决方案的过程中，找到一个分析形式的非线性最优反馈控制律的一般类的问题。此外，我们的方法可以提供一个明确的算法，用于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，只要成本函数和动态（即，我们将继续发展我们的方法，并探索我们的应用方法在更大类别的控制问题中的应用，包括具有控制约束、状态约束、欠驱动控制和非解析成本函数的控制问题。这项研究的主要成果将是一个新的理论形式主义的解决和分析最优控制问题和计算工具，可以生成最优反馈控制律的一般类系统。 我们的新形式主义和这个工具将在教育和研究环境中有很大的用处，并将提供给在密歇根大学攻读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.