Mathematical Sciences: "Robust and Sampled Data Control of Distributed Parameter Systems"

9623392 Rebarber这项研究将解决与分布参数系统的稳定性相关的几个重要问题。主要问题涉及在缺乏关于系统的完整信息的情况下稳定系统的可能性。例如，使系统稳定的反馈控制可能在反馈回路中具有未建模的小时滞，这对系统的稳定性产生不利影响。因此，重要的是要确定这种情况发生的时间和方式，并确定需要在反馈回路中进行哪些修改来缓解问题。一个相关的问题涉及这样的系统，在该系统中，可用于控制设计的唯一数据是对实际系统的采样数据观测，或该系统的数值近似。这项研究将根据这样有限的信息确定未知系统何时可能镇定。所采用的方法是系统论的，即对一大类抽象状态空间系统和频域系统的结果进行发展，然后将其应用于特定系统的振动控制问题。许多问题需要结合偏微分方程技术、频域技术和状态空间技术的方法。振动系统稳定反馈设计的主要问题之一是基于系统不完全信息的反馈控制设计问题。这种系统的一个重要范例出现在结构声学中。在这个问题中，声波存在于空腔内，例如飞机机舱，并且空腔边界的一部分受到振动的影响，例如，由于安装在机舱其中一面墙附近的飞机发动机。墙体振动既可以放大声波，也可以衰减声波，本研究的应用之一就是利用墙体振动来实现声波的衰减。用来实现这一点的现代技术使用的是当电压施加到振动墙上时，使其产生弯矩的压电片。数值实验表明，反馈回路中自然出现的非常小的时间延迟可能会使其中一些稳定方案变得无效。这样的延迟无法精确确定，因此确定需要在反馈环路中进行哪些修改以缓解问题非常重要。这个项目将在基于系统的不完全信息或基于采样数据的无限维振动系统的镇定的一般背景下准确地解决这样的问题。***

9623392 Rebarber This research will address several important problems associated with the stabilization of distributed parameter systems. The main issue concerns the possibility of stabilizing a system in the absence of complete information about the system. For instance, a feedback control which stabilizes a system might have unmodeled small time delays in the feedback loop that adversely affect the stability of the system. It is therefore important to determine when and how this occurs, and to decide what modifications in the feedback loop are needed to alleviate the problem. A related issue concerns systems in which the only data available for use in a control design is a sampled-data observation of the actual system, or a numerical approximation of that system. This research will determine when stabilization of an unknown system is possible based on such limited information. The approach taken is system theoretic in the sense that results are developed for a large class of abstract state space systems and frequency domain systems, and then applied to vibration control problems for specific systems. Many of the problems require approaches which combine partial differential equation techniques, frequency domain techniques and state space techniques. %%% One of the principal issues in the design of stabilizing feedbacks for vibrating systems is that of design of a feedback control based on incomplete information of the system. An important paradigm of such a system occurs in structural acoustics. In this problem, sound waves are present within a cavity, such as an airplane cabin, and part of the boundary of the cavity is subject to vibrations due, for example, to an aircraft engine mounted near one of the walls of the cabin. Wall vibrations may either amplify or attenuate the sound waves, and one of the applications of this research is to use the wall vibrations to accomplish the latter. Modern techniques used to accomplish this use piezoceramic patches th at impart bending moments on the vibrating wall when voltages are applied to them. Numerical experiments suggest that some of these stabilization schemes may be rendered ineffective by very small time delays that naturally arise in the feedback loop. Such delays cannot be precisely determined, so it is important to decide what modifications in the feedback loop are needed to alleviate the problem. This project is will address precisely such issue in the general context of stabilization of infinite- dimensional vibrating systems based on incomplete information of the system, or based on sampled data. ***