Mathematical Sciences: Feedback Design For Nonlinear Systems

9623435本研究计划将为在非平衡环境中运行的非线性动力系统的鲁棒控制设计反馈机制开发一种系统的方法。非线性反馈律提供了利用真正的非线性效应的可能性，例如跟踪或减弱极限环的影响。这项研究将基于对代数、微分拓扑学和动力系统理论方法的综合，以及关于极限环和更一般吸引子的反馈镇定技术的发展，来建立非线性反馈系统存在极限环的判据。这些技术将基于以前关于非线性系统零动态的工作扩展到非平衡情况。这项工作还将包括将被动系统理论扩展到控制变量不一定是仿射的系统。由于各种原因，动力系统可能会表现出复杂性；例如，在使用微芯片技术设计感测或控制飞机、汽车或高速列车中的位置和速度的设备时，会出现复杂性。由于作用在系统上的物理力的复杂性，物理和工业系统的数学模型也可能出现复杂性。这种现象的一个例子是由空气微爆发引起的风切变，在商用飞机的起飞和降落过程中可能会经历这种情况。风切变力与实现飞机升力所需的力之间的相互作用相当复杂，通常会对飞行员诱导的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量与物理系统的复杂相互作用所产生的潜在的戏剧性的、尽管不直观的影响。***

9623435 Byrnes 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 work 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. ***