Analysis and Control of Complex Behavior: Linear Transfer Operator Approach

本文提出了一种用于分析和控制具有复杂动力学特性的非线性系统的新框架。该框架的基本思想是研究相空间中集合的演化，而不是传统的逐点方法。该方法的优点是集合的演化是线性的，并由一个线性转移Perron-Frobenius算子来描述。该框架的线性特性允许我们将直觉从线性系统（一个成熟的研究领域）转移到非线性系统。一个完美的例子是李亚普诺夫测度的引入作为验证集合论概念的新工具？几乎所有地方?非线性系统的稳定性。证明了李雅普诺夫测度是李雅普诺夫函数的对偶，并提出了利用有限维近似的Perron-Frobenius算子计算李雅普诺夫测度的线性规划。提出的研究利用框架的线性度，在具有复杂动力学的系统中进行模型降阶和参数识别，并在喷气发动机中应用。该框架还为复杂动力学控制提供了基于线性规划的系统程序。本研究将此方法应用于流体流动混合控制问题，在分析大尺度海洋和大气流动的输运和混合特性方面具有潜在的应用前景。这项研究的成功将产生一套新的工具和方法，这些工具和方法足够通用，可以应用于不同的领域，如流体动力学和机械系统。喷气发动机的降阶模型将用于更好的控制设计，最终达到降低燃油消耗和延长发动机寿命的目的。改进的混合控制算法将使制药、石化、运输和发电等行业受益。π吗?我们与联合技术研究中心的合作将有助于将技术诀窍从工业界转移到学术界，反之亦然。作为教育计划的一部分，PI将在本科和研究生阶段引入非线性动力学的跨学科课程。通过担任机器人技术特殊兴趣小组的教师导师，PI将有助于增加本科生对研究的参与。

This proposal introduces a novel framework for the analysis and control of nonlinear systems exhibiting complex dynamics. The underlying idea of the framework is to study the evolution of sets in the phase space as opposed to the conventional point-wise approach. The advantage of the approach is that the evolution of sets is linear and is described by a linear transfer Perron-Frobenius operator. The linear nature of the framework allows us to carry our intuition from linear systems, a mature area of research, to nonlinear systems. A perfect example is the introduction of Lyapunov measure as a new tool to verify a set-theoretic notion of ?almost everywhere? stability in nonlinear systems. Lyapunov measure is shown to be dual to the Lyapunov function and a linear program using the finite dimensional approximation of the Perron-Frobenius operator is proposed for its computation. The proposed research exploits the linearity of the framework for the model order reduction and parameter identification in systems exhibiting complex dynamics with applications in jet engines. The framework also provides a systematic procedure based on the linear programming for the control of complex dynamics. The proposed research uses this procedure for the problem of control of mixing in fluid flows with potential application in analyzing the transport and mixing properties of large-scale oceanographic and atmospheric flows. The success of this research will result in a new set of tools and methods that are general enough to be applied in diverse fields, such as fluid dynamics and mechanical systems. The reduced order model in jet engines will be used in better control design, which ultimately results in less fuel consumption and increased life of engines. The improved control algorithm for mixing will benefit the pharmaceutical, petrochemical, transportation, and power generation industries among others. The PI?s collaboration with United Technologies Research Center will help in transferring the know-how from industry to academia, and vice versa. As the part of the education plan, the PI will introduce interdisciplinary courses on nonlinear dynamics at the undergraduate and graduate levels. By serving as a faculty mentor for the special interest group in robotics, the PI will help increase the involvement of undergraduate students in research.