Disturbance Attenuation in Nonlinear Control Systems

对于具有复杂非线性动态行为的系统，反馈律的设计是一个极具挑战性的研究课题，近年来引起了人们越来越多的兴趣。非线性控制理论的进步将极大地改进轻量化机械臂、自动驾驶仪和高度机动飞机的制导系统的设计，增加化工厂监管的灵活性和能效，便于在发生故障后对控制系统进行实时重新配置。在必须面对的许多问题中，最重要的是设计反馈律，以实现对规定的外部命令的跟踪和对外部干扰的抑制。在控制系统中跟踪/拒绝外源输入问题的方法取决于所选择的模型来描述预期影响系统的外源输入族。如果没有可用的模型，一个吸引人的策略--在线性系统的情况下显示出自己非常强大和有吸引力--是基于最糟糕的可能情况进行设计。例如，在频域中，反馈律被设计为最小化对外部输入的频率响应的最大幅度。这一研究项目将有助于将这些设计方法扩展到由非线性数学模型描述的系统。利用控制系统几何理论、微分对策理论和动力系统定性理论的方法和结果，研究的目的是建立关于反馈控制律的存在性和实际设计的充分一般性的结果，以实现对未建模的外部干扰的衰减。

The design of feedback laws for systems characterized by complicated nonlinear dynamical behavior is a challenging research task which has attracted an increasing interest in recent years. Advances in nonlinear control theory will substantially improve the design of control systems for light-weighted robot arms, for autopilots and guidance systems of highly maneuverable aircrafts, increase the flexibility and energy efficiency in the regulation of chemical plants, facilitate the real-time reconfiguration of a control system after the occurrence of a failure. Among the many problems which must be confronted, a most important one is the desing of feedback laws achieving tracking of prescribed exogenous commands and rejection of exogenous disturbances. The approaches to the problem of tracking/rejection of exogenous inputs in a control system depend upon the model chosen to describe the family of exogenous inputs which are expected to affect the system. If no model is available, an appealing strategy -which revealed itself very powerful and attractive in the case of linear systems- is to base the design on the worst possible situation. For instance, in the frequency domain, the feedback laws are designed so as to minimize the maximal amplitude of the frequency response to the exogenous inputs. This research project would contribute to the extension of these design methodologies to systems described by nonlinear mathematical models. Using methods and results from the geometric theory of control systems, the theory of differential games and the qualitative theory of dynamical system, the purpose of the research is to establish sufficiently general results about existence and actual design of feedback control laws achieving attenuation of unmodeled external disturbances.