Sampled Data Stabilization and Tracking for Partial Differential Equations

0206951RebarberThis项目将集中于开发无限维系统的采样数据反馈技术。 离散时间控制器只能在有限的频率响应下工作，而无穷维系统往往具有不可忽略的高频效应，因此，人们不仅对采样控制器的设计感兴趣，而且对它的局限性也很感兴趣。 下面的问题是基本的：一个给定的连续时间控制器可以被相关的采样数据控制器所取代，同时保持闭环系统的期望响应？ 理想化和广义采样保持都将被考虑。广义保持可以用作设计参数，并且当输出不足够平滑以及时适应点评估时，可以使用广义采样。 PI将尽可能完整地表征那些连续时间反馈系统，这些系统在采样数据方案（具有足够小的采样时间）应用于反馈时不会失去其闭环稳定性，并将确定采样数据系统的性能是否可以近似连续时间性能。 考虑的性能指标是闭环增长率和稳定半径。还将研究和开发无限维系统的跟踪技术。 假设一个系统有一个要被拒绝的外部干扰项，或一个要被跟踪的外部参考项。 一种常见的方法是通过内模原理建议的低增益控制器来实现。 将研究这样的控制器的有效性，为广泛的一类系统，以及其对外部信号的频率变化的敏感性。采样数据版本的跟踪控制器也将被考虑。 这些采样数据和跟踪结果将被应用到多个空间变量的偏微分方程，特别是耦合偏微分方程模型与至少一个双曲分量。 由于这些模型涉及两个不同的偏微分方程，耦合通过高度无界的运营商，分析的特点是不同于非耦合系统的分析。 对于这个问题，输出反馈设计的一个中心问题是输入输出映射的分析，即从控制到观测的映射。数字技术的进步导致了控制工程中对采样数据设计的重视，但对于无限维系统（如偏微分方程）的采样数据控制的发展受到限制。 在许多应用中，输出数据是在离散时间而不是连续时间，这样的系统的反馈控制器应设计为离散数据作为其输入，但在连续时间的行为。 由于已经有大量的文献连续时间稳定的偏微分方程，该项目将涉及如何修改连续时间控制器，以获得采样数据控制器，同时保持系统性能的调查。 也感兴趣的是设计有效的采样数据控制器，而不参考连续时间设计的技术。 另一个要考虑的问题是设计跟踪外部信号和抑制噪声的主动反馈控制。作为一个应用，这些方法将被用来设计一个控制器，以拒绝在一个PDE模型，它描述了在一个空腔（例如，飞机驾驶舱）和一个灵活的腔壁的运动之间的相互作用的声波噪声。 假设有一个外部噪声源，如发动机噪声，这是要拒绝，主动反馈控制是要施加到智能材料驱动器的空腔壁。 然后，适当设计的低增益控制器（连续时间反馈或采样数据）将衰减腔体的许多点处和附近的声压。

0206951RebarberThis 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.