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

动力系统专题：开关和连续控制系统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被称为切换控制。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以释放制动器中的压力，而不是逐步释放。虽然这些阶段中的每一个的数学是简单的，但是两个阀之间的相互作用的设计是具有挑战性的数学问题。该项目提供的知识，工程师将使用时，设计切换控制策略，朝着所需的鲁棒政权的动态。该项目不仅将改进防抱死制动系统等简单系统的设计，还将改进两条腿走路的机器人等复杂系统的设计。当机器人用两条腿走路时，它必须能够改变它的质心，交替的腿和膝盖运动，并考虑到脚与地面几乎无限次的重复碰撞。除了在机器人和汽车行业的影响，在这个项目中开发的理论是相关的流体力学（风和河流，其中控制是连续的，但不平滑），生态学（放牧管理）和医学（间歇性治疗，如激素治疗），在机器人，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内在非光滑（即新）方面。目前的理论无法预测（分叉）复杂的振荡，如果点的不连续的右手边（切换）形成不连续流形，如果轨迹具有无限数量的不连续性（碰撞）在有限时间内（喋喋不休），如果一个轨迹随着时间的推移发展成一个漏斗形的曲线，（即，使得长期预测不可能），或者如果平滑模型的不连续性（即，引入慢-快动态）。这四个机制描述了该项目的四个目标，因为它们是四个中心路线的内在非光滑吸引机制发生在非光滑控制应用。为了实现这一目标，降维，规范形和分歧理论等章节的领域将显着推进。

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.