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更改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。这种有限的选项系统被称为开关控制。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刹车中的压力，而不是逐步释放。虽然每个阶段的数学计算都很简单，但两个阀门之间相互作用的设计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切换控制策略时将使用的知识，这些策略将使动态向所需的健壮机制移动。该项目不仅将改进防抱死制动系统等简单系统的设计，还将改进机器人用两条腿行走等复杂系统的设计。当机器人用两条腿行走时，它必须能够改变重心，交替腿部和膝盖的运动，并解释脚与地面几乎无限次的重复碰撞。除了对机器人和汽车行业的影响外，该项目中开发的理论还与流体力学(风和河流流动，其中控制是连续的，但不是平滑的)、生态学(放牧管理)和医学(间歇疗法，如激素疗法)有关，在这些领域，与机器人学一样，系统很复杂。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内在非光滑(即新的)方面。目前的理论不能预测复杂振荡的出现(分叉)，如果右侧(切换)的不连续点形成不连续的流形，如果轨迹在有限时间内具有无限数量的不连续(碰撞)(抖动)，如果轨迹随着时间发展成曲线漏斗(即使长期预测变得不可能)，或者如果一个人平滑了模型的不连续(即引入了慢速动力学)。这四种机制描述了项目的四个目标，因为它们是本质上非平稳的有吸引力的制度在非平稳控制应用中发生的四条主要途径。为了实现这一目标，该领域的降维、范式和分叉理论等章节将得到显著的进步。

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.