Mathematical Sciences: Feedback Design for Nonlinear Systems

本研究计划将发展一套系统的方法，用于设计在非平衡环境下运行的非线性动力系统的鲁棒控制反馈机制。非线性反馈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。该研究将涉及发展非线性反馈系统中极限环存在的准则，基于从代数和微分拓扑以及从动力系统理论中得出的综合方法，以及关于极限环和更一般吸引子的反馈稳定技术的发展。这些技术将基于先前非线性系统零动力学工作的扩展到非平衡情况。这项工作还将包括将被动系统理论扩展到在控制变量中不一定是仿射的系统。由于各种原因，动力系统可以表现出复杂性；例如，在使用微芯片技术设计用于感知或控制飞机、汽车或高速列车的位置和速度的设备时，会出现复杂性。由于作用于系统的物理力的复杂性，复杂性也可能出现在物理和工业系统的数学模型中。这种现象的一个例子是风切变，它是由空气微爆引起的，在商用飞机起飞和降落时可以经历。风切变力与飞机升力之间的相互作用非常复杂，通常会对飞行员诱导的力产生非直觉的反应。这项研究的目的是确定利用物理力与物理系统之间复杂的相互作用所产生的潜在的戏剧性（尽管不是直观的）效应的方法。＊＊＊

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. ***