This project will focus on developing techniques for sampled-data feedback for infinite dimensional systems. A discrete time controller can operator with only limited frequency response, and infinite dimensional systems often have high frequency effects which cannot be ignored, so there is not only interest in what can be done with sampled-data control design, but also in its limitations. The following question is basic: can a given continuous-time controller be replaced by a related sampled-data controller, while maintaining the desired response of the closed-loop system? Both idealized and generalized sample-and-hold will be considered. Generalized hold can be used as a design parameter, and generalized sampling can be used when the output is not sufficiently smooth to accommodate point evaluations in time. The PI will characterize as completely as possible those continuous-time feedback systems which do not lose their closed-loop stability when a sampled data scheme (with sufficiently small sampling time) is applied to the feedback, and will determine whether the performance of the sampled-data system can approximate the continuous time performance. The performance measures considered are closed-loop growth rate and stability radius. Tracking techniques for infinite dimensional systems will also be studied and developed. Suppose a system has an external disturbance term which is to be rejected, or an external reference term to be tracked. One common approach to doing this is by a low-gain controller suggested by the internal model principle. The effectiveness of such a controller for a wide class of systems will be studied, as well as its sensitivity to frequency variations in the external signal. Sampled-data versions of tracking controllers will also be considered. These sampled-data and tracking results will be applied to PDEs in more than one space variable, especially coupled PDE models with at least one hyperbolic component. Due to the fact that these models involve two different of PDEs, coupled via highly unbounded operators, the analysis has features which are distinct from the analysis for uncoupled systems. For this problem a central concern for output feedback design is the analysis of the input-output map, i.e. the map from the control to the observation.
Advances in digital technology have led to an emphasis on sampled-data design in control engineering, but the development of sampled data control for infinite-dimensional systems such as PDEs has been limited. In many applications output data is available in discrete time rather than continuous time, and a feedback controller for such a system should be designed to take discrete data as its input, but act in continuous time. Since there is already an enormous literature on continuous time stabilization of PDEs, the project will involve the investigation of how to modify continuous time controllers to obtain sampled data controllers, while maintaining system performance. Also of interest are techniques for designing effective sampled-data controllers without reference to continuous time design. Another topic to be considered is the design of active feedback control tracking external signals and rejecting noise. As an application, these methods will be used to design a controller to reject noise in a PDE model which describes the interaction between sound waves in a cavity (for instance, an airplane cockpit) and the motion of a flexible wall of the cavity. Suppose that there is an external noise source, such as engine noise, which is to be rejected, and active feedback control is to be applied to smart material actuators on the cavity walls. Then a properly designed low-gain controller (either continuous time feedback or sampled-data) will attenuate the sound pressure at and near finitely many points of the cavity.
