New approach to nonlinear control theory by Hamiltonian mechanics

本文提出了一种新的求解Hamilton-Jacobi方程的近似方法，该方程是非线性控制理论中最优控制和H ∞控制中最重要的方程之一。我们还开发了计算程序，并将其应用于许多问题。内容包括数值计算问题、飞行器姿态控制和磁悬浮系统控制实验。我们的研究结果解决了40多年来一直是控制理论中的瓶颈问题。所提出的方法是适合于计算机实现和实验验证的控制器的Hamilton-Jacobi方程是，据我们所知，在这一领域的非常少的重要成就之一。工程方面的具体问题如下。飞机在大迎角、磁悬浮系统、输入饱和系统和与驾驶员诱发振荡抑制有关的输入速率饱和系统的最优镇定问题。所提出的算法是在计算机程序中实现的数值方法，这种方法在应用中的优势，非解析的非线性，如饱和，可以处理。

In this research we proposed a new approximate solution method to the Hamilton-Jacobi equation, which is one of the most important equations in nonlinear control theory such as optimal control and H-infinity control. We also developed computation programs and applied them to numerous problems. They include numerical problems, aircraft attitude control and control of a magnetic levitation system with experiments. Our results solve the longstanding problem for over 40 years that has been a bottle neck in control theory. The proposed method is suitable for computer implementation and the experimental verification of controllers by the Hamilton-Jacobi equation is, to the best of our knowledge, one of very few important achievements in this field. The concrete problems from engineering are the following. Optimal stabilization problem of an aircraft at high angle-of-attack, magnetic levitation system, systems with input saturation, and systems with input rate saturation related to Pilot-Induced-Oscillation suppression. The proposed algorithm is implemented in computer program in a numerically way and this approach has advantages in application in that non-analytic nonlinearities, such as saturations, can be handled.