Sampled-Data and Discrete-Time Control of Infinite Dimensional Systems

在可用于控制器设计的信息有限的情况下，考虑了两类控制问题。在一类问题中，输出数据仅在离散时间可用，因此采用采样数据控制而不是连续时间控制。项目包括如何设计偏微分方程组的采样数据控制器；分析哪些系统可以通过简单的采样数据控制器稳定；如何修改连续时间反馈控制器使其成为有效的采样数据控制器；分析采样数据控制器的性能与相关连续时间控制器的性能比较；以及设计即使在有关对象的信息非常少的情况下也能跟踪外部参考信号的采样数据控制器。在这些项目中，人们感兴趣的不仅是采样数据控件设计可以做些什么，还在于它的局限性。由于离散时间控制器只能在有限的频率响应下工作，而无限维系统往往具有不可忽略的高频效应，因此无限维系统的采样和离散时间控制是非常精细的。此外，还考虑了由无穷维离散时间系统所模拟的种群动力学问题，如积分-差分方程组。种群生态学中的大多数这样的问题都有高度不确定的数据，我们将分析数据不确定性是如何影响增长或衰退的。采样数据控制的发展部分是由数字电子技术的进步推动的，这导致了越来越多的采样数据控制算法的设计和实现。此外，在许多实际应用中，系统的输出数据往往只在离散时间而不是连续时间可用，因此了解如何处理这类数据具有实用价值。所提出的研究将解决关于无限维系统的采样数据控制设计能力的基本问题。种群生态学中的保护问题通常具有非常不确定的数据，濒危或入侵物种的生长或衰退可能会受到这种不确定性的极大影响。无限维模型用于模拟人口的空间分布，或在生长阶段连续分布的情况下。这项研究将开发新的技术，并使用鲁棒控制理论的技术来分析不确定数据对长期人口增长的影响。这些技术将可用于解决类似的人口问题。本科生和研究生研究人员将被纳入这项研究计划。

Two types of control problems are considered where the information available for controller design is limited. In one type of problem, output data is available only at discrete times, so sampled-data control is used instead of continuous-time control. Projects include how to design sampled-data controllers for partial differential equations; an analysis of which systems can be stabilized by a simple sampled-data controller; how to modify continuous-time feedback controllers so that they are effective sampled-data controllers; an analysis of how the performance of a sampled-data controller compares to that of related continuous time controllers; and the design of sampled-data controllers which track an external reference signals, even when very little information about the plant is available. In these projects, the interest lies not only in what can be done with sampled-data control design, but also its limitations. Since a discrete time controller can operate with only limited frequency response, and infinite-dimensional systems often have high frequency effects which cannot be ignored, sampled-data and discrete-time control of infinite-dimensional systems is very delicate. In addition, population dynamics problems modeled by infinite-dimensional discrete-time systems, such as integro-difference equations, are considered. Most such problems in population ecology have highly uncertain data, and we will analyze how growth or decay is affected by the data uncertainties.The development of sampled-data control is motivated in part by advances in digital electronics, which has led increasingly to sampled-data design and implementation of control algorithms. Furthermore, in many practical applications output data from a system is often only available at discrete times rather than in continuous time, so it is of practical value to know how to deal with such data. The proposed research will address basic questions about the capabilities of sampled-data control design for infinite-dimensional systems. Conservation problems in population ecology typically have very uncertain data, and the growth or decay of an endangered or invasive species can be dramatically affected by this uncertainty. Infinite-dimensional models are used to model spatial spread of a population or in cases where the growth stages are continuously distributed. This research will develop new techniques, and use techniques from robust control theory, to analyze the effect of uncertain data on long-term population growth. These techniques will be available and accessible for use on similar population problems. Undergraduate and graduate researchers will be incorporated into this research program.