Feedback Control Approaches to Uncertain Nonlinear Structured Population Models

与工程系统一样，总体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，幼苗的补充通常是结构化种群的非线性反馈。作为另一个例子，通过扩散耦合的群体自然地由闭环反馈系统描述。该项目将开发用于分析和控制一类具有不确定性和扰动的非线性结构总体模型的技术。任何生态系统都会受到广泛的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稳健性分析至关重要。该研究项目旨在获得有关稳定性分析、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数学结果。最优控制（即成本函数最小化）在生态学中也经常使用。比例积分微分控制器是观测测量的反馈，并且已知具有良好的稳健性。这项工作将稳健反馈控制理论的思想引入植物和动物种群的管理中，特别强调濒临灭绝的植物。该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。反馈环鲁棒性、小增益、绝对稳定性和输入状态稳定性等概念非常适合对不确定的、结构化的、非线性总体动态进行建模和分析。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无法进行随机分析的模型。我们研究一类诱饵系统，它可以被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中考虑这些系统，并且具有有限维状态空间或无限维状态空间。我们在模型中包含不确定性和干扰。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群体的诱饵系统。一个关键特征是线性部分和非线性部分之间的相互作用，并且需要结合经典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：确定非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，跟踪控制器是使用系统观察（而不是整个状态）作为输入的反馈控制器；并构建和分析诱饵系统的观察者和基于观察者的控制器。观察者是一个动态系统，它将原系统的观察结果作为输入，并渐进地重构原系统的整个状态。我们将把这些结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.