Robust Stabilization Design via Delayed Feedback Control and Its Applications

本研究的目的是揭示延迟反馈控制在混沌控制领域中的能力，并建立其设计方法，以期将其应用于各种工程问题。延迟反馈在稳健性方面有一定的优点，但其适用性长期以来一直不清楚。证明了当不稳定极点的个数为偶数时，系统可以真正镇定；如果允许使用补偿器，具有不稳定极点的一阶补偿器就足以使具有奇数个不稳定极点的系统镇定。在这个项目中，我们证明了在较温和的条件下，用差逼近镇定导数控制器可以得到镇定延迟反馈控制器。然而，也有研究表明，这种替换对于一类特殊的多输入多输出系统不起作用。此外，我们还得到了判定这种差分近似是否成功地保持闭环系统稳定性的准则。作为应用，我们采用了在曲面上运动的小车-摆系统作为应用。设计的指导思想是首先设计一个稳定的导数反馈，然后用一个差分反馈来代替它，并调整所涉及的延迟以获得良好的整体性能，其中LMI方法是有效的。虽然模拟是有利的，但在实验中我们遇到了一些小车运动的振荡，原因仍然不确定。

The purpose of the present research is to disclose the ability of the delayed feedback control cited in the area of chaos control, and to establish the design method with a view to applying it to various engineering problems. The delayed feedback has some merit in robustness, but its applicability has been long unclear. We have shown that it can really stabilize if the associated unstable poles are of even number, and if it is allowed to use a compensator, first-order compensator with an unstable pole is sufficient for stabilizing a system with an odd number of unstable poles. In this project, we have shown under a mild condition that approximating a stabilizing derivative controller by a difference can yield a stabilizing delayed feedback controller. However, it has been also shown that such replacement does not work for a special class of multi-input multi-output systems. Further we have obtained criteria which determine if such a difference approximation is successful to preserve the closed-loop stability. It is also clarified that the difference approximation is superior in the closed-loop stability to other alternate means such as realization by a sampled-data system and the derivative feedback with a small delay.As an application, we have adopted a cart-pendulum system moving on a curving surface. A guide of design is to firstly design a stabilizing derivative feedback and secondly replace it by a difference counterpart and adjust the delay involved to obtain a good overall performance where an LMI approach is effective. While simulation is favorable, in the experiment we have faced with some oscillations of cart motion, and still the cause is undetermined.