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

9623392 Rebarber本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 在缺乏系统完整信息的情况下稳定系统的可能性。例如，稳定系统的反馈控制可能在反馈回路中具有未建模的小时间延迟，这对系统的稳定性产生不利影响。 因此，重要的是要确定何时以及如何发生这种情况，并决定需要在反馈回路中进行哪些修改以缓解问题。 一个相关的问题涉及的系统中，唯一可用的数据用于控制设计是实际系统的采样数据观察，或该系统的数值近似。这项研究将确定一个未知系统的稳定时，可能基于这样的有限信息。所采取的方法是系统理论的意义上，结果开发了一大类抽象的状态空间系统和频域系统，然后应用到振动控制问题的具体系统。 许多问题需要结合联合收割机偏微分方程技术，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。 这种系统的一个重要范例出现在结构声学中。 在该问题中，声波存在于诸如飞机机舱的空腔内，并且空腔的边界的一部分例如由于安装在机舱的壁之一附近的飞机发动机而经受振动。 壁面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壁面振动来实现声波的衰减。 用于实现这一点的现代技术使用压电陶瓷贴片，当电压施加到压电陶瓷贴片上时，压电陶瓷贴片在振动壁上施加弯曲力矩。 数值实验表明，这些稳定计划中的一些可能会变得无效的反馈回路中自然出现的非常小的时间延迟。 这种延迟无法精确确定，因此重要的是要决定需要对反馈回路进行哪些修改以缓解问题。 这个项目将解决正是这样的问题，在一般情况下，稳定的无限维振动系统的不完全信息的基础上，系统，或基于采样数据。 ***

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. ***