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

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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. ***