Set-theoretic methods in control
Set-theoretic methods in control
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DOI：
10.1007/978-3-319-17933-9
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发表时间：
2007-11
期刊：
影响因子：
--
通讯作者：
F. Blanchini;S. Miani
中科院分区：
文献类型：
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作者：
F. Blanchini;S. Miani
许多控制问题可以在集合论背景下自然地表述、分析和解决。当考虑控制系统设计中至关重要的三个方面时，集合自然会出现：约束、不确定性和设计规范。此外，集合是指定多个系统性能的最合适的语言，例如当我们有兴趣确定吸引域、测量反馈环路中持续噪声的影响或限制估计算法的误差时。从概念的角度来看，本书所介绍材料的独特之处在于，集合不仅是表述的术语，而且在问题的解决中也发挥着积极的作用。一般来说，在控制论背景下，所有理论上基于状态空间子集的某些性质的技术都可以称为集合论方法。最流行和最明显的联系是与李雅普诺夫理论和正不变性的联系。李亚普诺夫函数是状态变量的正定能量型函数，具有随时间递减的性质，是保证稳定性的基本工具。此外，它们的子水平集是正不变的，因此它们的形状对于表征系统动力学非常有意义，这也是本书将要阐明的一个关键点。不变性将被证明是处理饱和控制、噪声抑制、模型预测控制等问题的基础。本书的主要目的是从理论和实践的角度描述动态系统控制和分析的集合论方法。书中提供的材料仅部分归功于作者的工作。其中大部分源自现有文献，这些文献始于 70 年代初有关特殊动态游戏的一些开创性著作。就其本质而言，本书与控制理论的其他领域有许多交叉点，包括约束控制、鲁棒控制、干扰抑制和鲁棒估计。这些都没有被完全涵盖，但对于每一个我们只会提出一个特定的观点。但必要时，读者会参考专业文献进行补充阅读。七
Many control problems can be naturally formulated, analyzed, and solved in a set-theoretic context. Sets appear naturally when three aspects, which are crucial in control systems design, are considered: constraints, uncertainties, and design specifications. Furthermore, sets are the most appropriate language to specify several system performances, for instance when we are interested in determining the domain of attraction, in measuring the effect of a persistent noise in a feedback loop or in bounding the error of an estimation algorithm. From a conceptual point of view, the peculiarity of the material presented in this book lies in the fact that sets are not only terms of the formulation, but they play an active role in the solution of the problems as well. Generally speaking, in the control theory context, all the techniques which are theoretically based on some properties of subsets of the state-space could be referred to as set-theoretic methods. The most popular and clear link is that with the Lyapunov theory and positive invariance. Lyapunov functions are positive-definite energy-type functions of the state variables which have the property of being decreasing in time and are a fundamental tool to guarantee stability. Besides, their sublevel sets are positively invariant and thus their shape is quite meaningful to characterize the system dynamics, a key point which will be enlightened in the present book. The invariance property will be shown to be fundamental in dealing with problems such as saturating control, noise suppression, model-predictive control, and many others. The main purpose of this book is to describe the set-theoretic approach for the control and analysis of dynamic systems from both a theoretical and practical standpoint. The material presented in the book is only partially due to the authors’ work. Most of it is derived from the existing literature starting from some seminal works of the early 70s concerning a special kind of dynamic games. By its nature, the book has many intersections with other areas in control theory including constrained control, robust control, disturbance rejection, and robust estimation. None of these is fully covered, but for each of them we will present a particular view only. However, when necessary, the reader will be referred to specialized literature for a complementary reading. vii