SGER: New Approaches to the Design of Fixed Order Controllers

控制理论正处于一个特殊的十字路口。在过去43年里发展起来的许多优雅的理论已经走到了尽头，现在吸引越来越少的学者，而它几乎没有引起工业界的兴趣，而且大多数与工程设计无关。另一方面，控制工程师面临的实际设计问题仍未得到研究，缺乏理论基础，具有基础性和理论上极具挑战性，迫切需要分析和计算机辅助设计方法。我们的建议试图解决这一差距，并努力解决这一差距。这里描述的探索性研究涉及设计的新方法的发展。混合顺序和结构控制器。本研究的需要源于以下几点：a)自1960年以来由控制理论家集中发展的时间控制理论导致高阶控制器是基于模型的，该理论不能处理的约束。混合顺序，这在工业应用中一直存在；这使得现代理论甚至不适用于诸如asPID和超前/滞后等广泛使用的控制器。b)时间和高阶控制器可能有严重的结构和灵敏度问题，正如Keel和Bhattacharyya在1997年的论文“鲁棒，脆弱或最优？”h（IEEE t - ac1997年8月）中所显示的那样。这表明，采用H2、H”、H”、l和技术设计的控制器对控制器内部的扰动非常敏感。客户端使其无法使用。c)在实际设计问题中，一个重要的需求是有一组可行的解决方案，满足各种目标，如稳定性、鲁棒性、时延容忍等，以便设计贸易。S可以被明智地比较。的最新理论。混合顺序控制器远不能回答这些问题——事实上，在大多数情况下，它甚至不能确定是否有可能实现名义上的稳定。在上述背景的激励下，我们提出的研究开始探索一些新的方法来解决这个问题。杂序控制器设计问题。这个。在这个RST步骤中，必然是稳定集的确定。这是di。控制理论中的邪教和长期未解决的问题。我们最近在两个特殊案例中取得了突破。具体地说，我们得到了一个混合解析/计算的方法来确定a。一阶或PID控制器可以稳定给定的植物。我们的解可以确定稳定控制器的完整集合，并证明该集合不凸甚至不连通。考虑到这一点，性能约束是如何实现的还不清楚。两架。此外，我们在这些情况下使用的解决方法似乎并不适用于一般情况。对于一般情况，我们打算探讨两种方法。这个。RST将是一个近似的尝试。通过凸集的并集实现一般不连通的混合阶控制器稳定集。我们将探索使用稳定性的交错条件来生成这些，并且看起来我们可以得到稳定集作为凸集的并集的良好而严密的近似。这可以打开大门，通过计算易于处理的方法，如线性规划和凸优化，性能约束的位置和他们的满足，尽管复杂的稳定集的理论。我们打算探索的第二种方法是使用混合的奈奎斯特/ d -分解技术，我们发现它在简单的情况下很有用。该方法具有额外的潜在优势，即可以仅从频域数据确定稳定集，从而使设计完全基于实验数据并且基本上“无模型”。在这种情况下，还应该开发设计对数据的鲁棒性。总之，本文提出的探索性研究解决了基于经典和现代性能的固定但任意顺序，鲁棒，非脆弱控制器的设计标准。这是并排的。邪教开放控制问题。然而，从理论和实践的角度来看，这是一个普遍需要解决的问题。希望进步。在提案中，我们描述了一些看似有希望的新方法，这些方法明显偏离了主流的优化和基于模型的方法。目前还没有。我们打算探索这些方法在设计中可以发展到什么程度。各自方案。并我们认为这些程序将会受益。控制工程实践和学术研究，有助于缩小我们这个领域迫在眉睫的理论与实践差距。

Control theory is at a eculiar crossroad.Much of the elegant theory developed over the last 43 years hasrun its course and now appeals to a dwindling number of academics,while it has generated little interestfrom industry and is mostly irrelevant for engineering design.On the other hand the real design roblemsfaced by control engineers remain unresearched,lack theoretical foundation,are of a fundamental natureand theoretically extremely challenging,and there is a crying need for analytical and computer aideddesign ap roaches.Our roposal attem ts to address this gap and to work towards its resolution.The exploratory research described here deals with the develo ment of new approaches to the design of.xed order and structure controllers.The need for this research stems from the following:a)The o timalcontrol theory developed intensively since 1960 by control theorists result in high order controllers thatare model-based and the theory cannot handle the constraint of .xed order,which is ever present inindustrial applications;this makes modern theory inapplicable to even such widely used controllers asPID and lead/lag,for instance.b)O timal and high order controllers can have severe structural andsensitivity roblems as shown dramatically in the 1997 paper "Robust,Fragile or Optimal?"h(IEEE T-ACAugust 1997)by Keel and Bhattacharyya.This a er showed that controllers designed by H2,H ",l1and techniques could be acutely sensitive to erturbations in the controller coe .cients rendering themunusable.Since then no systematic solution to the fragility roblem has emerged despite signi .cantactivity in this area.c)An important need in practical design roblems is to have sets of feasiblesolutions meeting various objectives such as stability,robustness,time-delay tolerance etc so that designtradeo .s can be intelligently compared.The resent theory of .xed order controllers is far from answeringsuch questions-indeed it cannot in most cases determine if even nominal stabilization is ossible at all.With the above background as motivation our roposed research sets out to explore some new ap-proaches to solve the .xed order controller design roblem.The .rst step in this,necessarily,is thedetermination of the stabilizing set.This is a di .cult and longstanding unsolved roblem in controltheory.We have recently obtained a breakthrough in two s ecial cases.Speci .cally we obtained a mixedanalytical/com utational solution to the roblem of determining whether a .rst order or PID controllercan stabilize a given lant.Our solution can determine the complete set of stabilizing controllers andshows that this set is not convex or even connected.In view of this it is not clear how erformanceconstraints can be e .ectively imposed.Moreover the solution technique used by us in these cases doesnot seem to extend to the general case.We intend to explore two approaches for the general case.The .rst will be an attempt to approximatethe .xed order controller stabilizing sets,which are disconnected in general,through unions of convexsets.We will explore the use of the interlacing conditions for stability to generate these and it appearsossible that we can get a good and tight approximation to the stabilizing sets as unions of convexsets.This can open the door to the im osition of erformance constraints and their satisfaction throughcomputationally tractable methods such as linear rogramming and convex optimization,despite thecomplicated to ology of the stabilizing sets.The second approach that we intend to explore is theuse of a hybrid Nyquist/D-Decomposition technique that we have found to be useful in simple cases.This method has the additional otential advantage that it may be ossible to determine stabilizing setsfrom frequency domain data only thus making the design based exclusively on experimental data andessentially "gmodel-free "h.In this setting achieving robustness of the design to the data should also bedevelo ed.In summary the exploratory research pro osed here addresses the design of fixed but arbitrary or-der,robust,non-fragile controllers based on classical as well as modern erformance criteria.This is alongstanding and di .cult open control roblem.Nevertheless it needs to be solved in general for thetheory and ractice of the .eld to rogress.In the roposal we describe some seemingly romising newapproaches which are signi .cant departures from the optimization and model based approaches thatdominate the .eld at resent.We intend to explore how far these approaches can be develo ed intodesign e .ective rocedures.We believe that such procedures will bene .t control engineering ractice aswell as academic research and hel close the looming theory-practice gap in our .eld.