Mathematical Sciences: Feedback Design for Nonlinear Systems

9634395林本研究计划将开发一个系统的方法，用于非线性动力系统的鲁棒控制反馈机制的设计，在非平衡环境中运行。 非线性反馈定律提供了利用真正的非线性效应的潜力，例如跟踪或衰减极限环的影响。 这项研究将涉及发展标准的存在极限环的非线性反馈系统，根据综合的方法得出的代数和微分拓扑和动力系统理论，以及发展技术的反馈稳定的极限环和更一般的吸引。 这些技术将基于以前的工作的非线性系统的非平衡情况下的零动态的扩展。 这一工作也将包括被动系统理论的扩展，使其适用于控制变量不一定是仿射的系统。 动力系统的复杂性有多种原因;例如，在使用微芯片技术设计用于感测或控制飞机、汽车或高速列车中的位置和速度的设备时，复杂性会增加。 由于作用于系统上的物理力的复杂性，物理和工业系统的数学模型也会出现复杂性。这种现象的一个例子是由空气微下击暴流引起的风切变，这可以在商用飞机起飞和降落时经历。 风切变力与使飞机获得升力所需的力之间的相互作用是相当复杂的，常常对飞行员诱导力产生非直观的反应。 这项研究的目的是确定如何利用物理力与物理系统的复杂相互作用所引起的潜在的戏剧性的，虽然非直观的影响。 ***

9634395 Lin This research program will develop a systematic methodology for the design of feedback mechanisms for the robust control of nonlinear dynamical systems, operating in a nonequilibrium environment. Nonlinear feedback laws offer the potential to take advantage of genuinely nonlinear effects, for example to track or to attenuate the effects of, limit cycles. The research will involve developing criteria for the existence of limit cycles in nonlinear feedback systems, based on a synthesis of methods drawn from algebraic and differential topology and from dynamical systems theory, as well as the development of techniques for feedback stabilization about limit cycles and more general attractors. These techniques will be based on an extension of previous work on zero dynamics for nonlinear systems to the nonequilibrium case. This wopk will also include an extension of passive system theory to systems which are not necessarily affine in the control variables. %%% Dynamical systems can exhibit complexity for a variety of reasons; for example, complexity arises in using microchip technologies to design devices for sensing, or controlling, the position and speed in an airplane, an automobile, or a high-speed train. Complexity can also arise in mathematical models of physical and industrial systems because of the complex nature of the physical forces acting on a system. An example of this phenomenon is windshear, caused by a microburst of air, which can be experienced during take-off and landing of commercial aircraft. The interaction of windshear forces with the forces required to achieve lift an aircraft is quite complex, often producing nonintuitive reactions to pilot induced forces. The purpose of this research is to determine ways to take advantage of the potentially dramatic, though nonintuitive, effects caused by the complex interactions of physical forces with physical systems. ***