Controller Synthesis Subject to Structure and Order Constraints

本研究的主要目的是为定序控制器的设计发展一个系统的理论。具体地说，我们计划将重点放在描述一组给定顺序的所有控制器的特征上，这些控制器稳定给定的工厂，然后在该稳定集上优化各种设计标准。上述问题源于我们早期的研究和最近与工业界的互动，可能是自动控制领域最突出的未解决问题之一。在过去的四十年里，控制理论取得了巨大的理论进步——20世纪40年代和50年代的经典设计程序已经让位于H2、Hoo、L1和u等优雅的控制综合技术的发展。然而，这些设计方法中的大多数不能应用于许多工业情况，例如在过程控制中，低阶和固定结构控制器（如PID）是首选控制器。当然，考虑到设计最优和鲁棒控制器的现代方法不能将控制器结构和/或顺序约束纳入其设计方法，这并不奇怪。更糟糕的是，目前工业中使用的一些临时设计方法，而不是现代方法，可能导致设计危险地接近不稳定。因此，迫切需要发展一种系统的理论来研究受控制器顺序和结构约束的最优鲁棒控制器设计。我们相信，这里提出的研究将满足这一需求。初步结果表明了明确的研究方向。具体地说，我们得到了一个推广经典赫米特-比勒定理的基本结果。此外，利用这一结果，在解决固定顺序和结构稳定的一些具体情况和设计问题方面取得了重大进展。基于这些，我们有信心提出的研究是非常有前途的，并将导致基础理论的发展，立即适用于工业环境。除了上面概述的主要推力外，我们还计划将我们早期的无约束、单输入单输出的自适应内部模型控制结果扩展到多变量系统、外线性系统和受约束系统的领域。这个话题是独立感兴趣的，特别是过程控制行业。然而，它很自然地适用于我们项目的广泛范围，因为内部模型控制本身就是设计PID控制器的流行策略。

The principal aim of the research outlined in this proposal is to develop a systematic theory for the design of fixed order controllers. Specifically, we plan to focus on characterizing the set of all controllers of a prescribed order that stabilize a given plant and then optimize various design criteria over that stabilizing set. The above problem which is motivated from our earlier research and recent interactions with industry, is probably one of the most outstanding unsolved problems in the area of automatic control. Control Theory has witnessed tremendous theoretical advances during the last four decades-the largely adhoc classical design procedures of the 1940's and 50's have given way to the development of elegant control synthesis techniques such as H2, Hoo, L1, and u. However, most of these design methods cannot be applied to many industrial situations, such as in process control, where low order and fixed structure controllers such as PID's are the preferred controllers of choice. This, of course, is not surprising given the fact that the modern approaches for designing optimal and robust controllers cannot incorporate the controller structure and/or order constraints into their design methods. To make matters worse, some of the adhoc design methods currently used in industry, in lieu of the modern approaches, can lead to designs that are dangerously close to instability. Thus there is a very critical need for developing a systematic theory for optimal and robust controller design subject to constraints on the controller order and structure. We believe that the research being proposed here will fulfill this need. Preliminary results, indicating a clear research direction, have already been obtained. Specifically, we have obtained a fundamental result generalizing the classical Hermit-Biehler Theorem. Furthermore, using this result, significant strides have been made in solving some specific cases of the fixed order and structure stabilization and design problems. Based on these, we are confident that the proposed research is very promising and will lead to the development of fundamental theory that is immediately applicable in an industrial setting. In addition to the main thrust outlined above, we also plan to focus on extending our earlier unconstrained, single-input single-output results on adaptive internal model control to the domain of multivariable systems, outlinear systems and systems subject to constraints. This topic is of independent interest, especially to the process control industries. Nevertheless, it fits in very naturally within the broad scope of our projects since internal model control is itself a popular strategy for designing PID controllers.