Feedback Control Approaches to Uncertain Nonlinear Structured Population Models

像工程系统一样，人口系统经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，幼苗招募通常是结构化种群的非线性反馈。作为另一个例子，通过扩散耦合的种群自然地被描述为闭环反馈系统。这个项目将开发一类具有不确定性和扰动的非线性结构种群模型的分析和控制技术。任何生态系统都会受到各种各样的自然扰动和参数不确定性的影响，在种群管理中，对这些扰动和不确定性对种群的影响进行稳健性分析是至关重要的。该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。在生态学中，最优控制也被频繁地使用，其中代价函数是最小化的。比例积分导数控制器是观测值的反馈，具有良好的鲁棒性。这项工作将稳健反馈控制理论的思想引入到动植物种群的管理中，特别强调濒危植物。这一研究项目的结果将在环境、经济和社会政策领域产生潜在影响。反馈回路的稳健性、小增益、绝对稳定性和输入-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。反馈控制可以在不知道不确定性和扰动的概率分布的情况下稳健地应用，因此它可以应用于一些不可能进行随机分析的模型。我们研究了一类诱饵系统，它可以描述为线性系统和非线性系统之间的反馈互连。我们在离散时间或连续时间，在有限维状态空间或无限维状态空间中考虑这些系统。我们在模型中考虑了不确定性和扰动。我们建议发展数学，以扩展系统理论、技术和概念，以引诱描述结构化人口的系统。一个关键特征是线性部分和非线性部分之间的相互作用，需要结合经典的工程输入输出技术和数学状态空间分析。我们的主要目标是：确定不受控连续时间系统的渐近稳定结构；研究跟踪控制器的效果；跟踪控制器是以系统的观测值(而不是整个状态)作为输入的反馈控制器；以及构造和分析诱饵系统的观测器和基于观测器的控制器。观测器是一种动力系统，它以对原始系统的观测为输入，渐进地重构原始系统的整个状态。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.