Analysis and Control of Complex Behavior: Linear Transfer Operator Approach

该建议介绍了一种新的框架，表现出复杂的动态非线性系统的分析和控制。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，并且由线性转移Perron-Frobenius算子描述。框架的线性性质使我们能够将我们的直觉从线性系统（一个成熟的研究领域）带到非线性系统。一个完美的例子是引入李雅普诺夫测度作为一种新的工具，以验证一套理论的概念？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与李雅普诺夫函数是对偶的，并提出了一个线性规划，利用Perron-Frobenius算子的有限维近似来计算李雅普诺夫测度。所提出的研究利用线性的框架模型降阶和参数识别系统表现出复杂的动态与应用在喷气发动机。该框架还提供了一个系统的过程，基于线性规划的控制复杂的动态。拟议的研究使用此程序的控制问题的混合在流体流动与潜在的应用分析大规模的海洋和大气流动的运输和混合特性。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，最终导致更少的燃油消耗和增加发动机的寿命。改进的混合控制算法将使制药、石化、交通和发电等行业受益。私家侦探？与联合技术研究中心的合作将有助于将技术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参与研究。

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.