Special Topics in Dynamical Systems: A New Mathematical Framework for the Design of Switching and Continuous Control Strategies

动态系统专题：切换和连续控制策略设计的新数学框架研究人员经常通过操纵外部参数来改变物理过程以获得他们想要的响应。很难创建一个可以连续改变外部参数以获得所需结果的控制器，因此控制器通常只有有限的选项。这种有限的选择系统称为切换控制。例如，在防抱死制动系统中，控制器可以完全打开或关闭阀门来释放制动器中的压力，而不是通过程度。虽然这两个阶段的数学计算很简单，但设计两个阀门之间的相互作用是一个具有挑战性的数学问题。该项目提供了工程师在设计开关控制策略时将使用的知识，这些策略可以将动力学移动到所需的鲁棒状态。这个项目不仅会改进像防抱死制动系统这样的简单系统的设计，还会改进像用两条腿走路的机器人这样的复杂系统的设计。当机器人用两条腿行走时，它必须能够改变其质心，腿部和膝盖交替运动，并解释足部与地面几乎无限次的重复碰撞。除了机器人和汽车工业的意义之外，该项目中发展的理论与流体力学（风和河流，其中控制是连续的，但不是平滑的），生态学（放牧管理）和医学（间歇性疗法，如激素疗法）相关，如机器人，系统是复杂的。该项目为有兴趣与其他领域的研究人员合作的数学学生提供培训。该项目将研究具有不可微右手项的微分方程动力学的本质非光滑（即新）方面。如果右侧的不连续点（开关）形成不连续流形，如果轨迹在有限时间内具有无限数量的不连续点（碰撞）（抖振），如果轨迹随着时间发展成曲线漏斗（即使长期预测不可能），或者如果平滑模型的不连续点（即引入慢速快动力学），当前的理论无法预测复杂振荡的出现（分岔）。这四种机制描述了该项目的四个目标，因为它们是在非光滑控制应用中发生本质上非光滑吸引机制的四个中心途径。为了实现这一目标，该领域的降维、范式和分岔理论等章节将得到显著的发展。

Special Topics in Dynamical Systems: A New Mathematical Framework for the Design of Switching and Continuous Control StrategiesResearchers often change a physical process by manipulating an external parameter to get the response they desire. It is hard to create a controller that can continuously change the external parameter to get the desired results, so often controllers have only a limited amount of options. This limited option system is called a switching control. For example, in an anti-lock braking system, the controller can either open or close valves completely to relieve pressure in the brakes rather than by degrees. While the mathematics of each of these stages is simple, the design of interplay between the two valves is a challenging mathematical problem. This project provides the knowledge that engineers will use when designing switching control strategy that move the dynamics toward a required robust regime. Not only will this project improve designs for simple systems like anti-lock braking systems, but also in complex systems like a robot walking on two legs. When a robot walks on two legs, it must be able to change its center of mass, alternate leg and knee motion and account for the nearly infinite number of repeated collisions the feet have with the ground. In addition to implications in the robotics and automotive industry, the theory developed in this project is relevant in fluid mechanics (wind and river flow, where the control is continuous, but nonsmooth), ecology (grazing management), and medicine (intermittent therapies, such as hormone therapies) where, as in robotics, the systems are complex. This project offers training for mathematics students interested in collaboration with researchers in other fields. The project will investigate the intrinsically nonsmooth (i.e. new) aspects of the dynamics of differential equations with nondifferentiable right-hand terms. The current theory is incapable to predict the emergence (bifurcation) of complex oscillations, if the points of discontinuity of the right-hand side (switchings) form discontinuous manifolds, if the trajectories feature infinite number of discontinuities (collisions) in finite time (chattering), if a trajectory develops into a funnel of curves over time (i.e. makes long-term prediction impossible), or if one smooths the discontinuities of the model (i.e. introduces the slow-fast dynamics). These four mechanisms describe the four objectives of the project because they are the four central routes for intrinsically nonsmooth attractive regimes to occur in nonsmooth control applications. To achieve the goal, such chapters of the field as dimension reduction, normal forms and bifurcation theory will be significantly advanced.