Feedback Control Approaches to Uncertain Nonlinear Structured Population Models

像工程系统一样，人口系统通常涉及非线性反馈回路、参数不确定性和外部（通常是持续的）干扰。例如，在植物种群中，幼苗的招募通常是结构化种群的非线性反馈。另一个例子是，通过分散耦合的种群自然地被闭环反馈系统描述。本项目将开发一类具有不确定性和干扰的非线性结构人口模型的分析和控制技术。任何生态系统都受到各种自然干扰和参数不确定性的影响，在种群管理中，对这些干扰和不确定性对种群的影响进行鲁棒性分析是至关重要的。本研究项目旨在获得关于稳定性分析、跟踪控制器、观测器和基于观测器的控制器以及这些系统的鲁棒性的新的数学结果。最优控制，即成本函数最小化，在生态学中也经常被使用。比例-积分-导数控制器是观察到的测量的反馈，并且已知具有良好的鲁棒性。这项工作将鲁棒反馈控制理论的思想引入到植物和动物种群的管理中，特别强调濒危植物。这项研究项目的结果将对环境、经济和社会政策领域产生潜在影响。反馈回路的鲁棒性、小增益、绝对稳定性和输入到状态稳定性的概念非常适合于不确定、结构化、非线性种群动态的建模和分析。反馈控制可以鲁棒地应用，而不需要知道不确定性和干扰的概率分布，因此它可以应用于一些不可能进行随机分析的模型。我们研究一类可被描述为线性系统和非线性系统之间的反馈互连的Lure系统。我们在离散时间或连续时间，有限维状态空间或无限维状态空间中考虑这些系统。我们在模型中加入了不确定性和干扰。我们建议发展数学，将系统理论技术和概念扩展到描述结构化种群的Lure系统。一个关键特征是线性部分和非线性部分之间的相互作用，这将需要经典工程输入输出技术和数学状态空间分析的结合。我们的主要目标是：确定非控制连续时间系统的渐近稳定结构；为了研究跟踪控制器的效果，跟踪控制器是一种反馈控制器，它使用对系统的观察（而不是整个状态）作为输入；并为Lure系统构建和分析一个观察者和基于观察者的控制器。观测器是一个动力系统，它以原系统的观测值作为输入，渐近地重构原系统的整个状态。我们将把这些成果和技术应用于人口管理问题。

Like engineering systems, population systems often involve nonlinear feedback loops, parameter uncertainty, and external (often persistent) disturbances. For instance, in plant populations, seedling recruitment is often a nonlinear feedback of the structured population. As another example, populations coupled via dispersal are naturally described by a closed-loop feedback system. This project will develop techniques for the analysis and control of a class of nonlinear structured population models with uncertainties and disturbances. Any ecological system is subject to a wide range of natural disturbances and parameters uncertainties, and in population management it is critical to do a robustness analysis of the effect of these disturbances and uncertainties on the population. This research project aims to obtain new mathematical results about stability analysis, tracking controllers, and observers and observer-based controllers, and robustness for these systems. Optimal control, where a cost function is to be minimized, has also been used frequently in ecology. Proportional-integral-derivative controllers are feedbacks of observed measurements, and are known to have good robustness properties. This work introduces ideas from robust feedback control theory into management of plant and animal populations, with a particular emphasis on endangered plants. The results of this research project will have potential impact in environmental, economic, and social policy areas. The concepts of robustness of feedback loops, small gains, absolute stability, and input-to-state stability are well suited to modeling and analysis of uncertain, structured, nonlinear population dynamics. Feedback control can be applied robustly without knowledge of the probability distributions of the uncertainties and disturbances, so that it can be applied to some models for which a stochastic analysis is not possible. We study the class of Lure systems, which can be described as a feedback interconnection between a linear system and a nonlinearity. We consider these systems in either discrete time or continuous time, and with either a finite dimensional state space or an infinite dimensional state space. We include uncertainties and disturbances in the model. We propose to develop mathematics that will extend systems theory techniques and concepts to Lure systems that describe structured populations. A key feature is the interplay between the linear part and nonlinear part, and will require a combination of classical engineering input-output techniques and mathematical state space analysis. Our key goals are: to determine the asymptotic stability structure of the uncontrolled continuous time system; to study the effect of a tracking controllers, which are feedback controllers that uses an observation of the system (instead of the whole state) as their input; and to construct and analyze an observer and observer-based controller for Lure systems. An observer is a dynamical system that takes the observation of the original system as its input, and asymptotically reconstructs the whole state of the original system. We will apply these results and techniques to population management problems.