Mathematical Sciences: Feedback Design for Nonlinear Systems

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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. ***