Quantitative analysis of simple and interconnected systems: Stability, boundedness, and trajectory behavior
Quantitative analysis of simple and interconnected systems: Stability, boundedness, and trajectory behavior
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简单和互连系统的定量分析：稳定性、有界性和轨迹行为
DOI：
10.1109/tct.1970.1083119
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发表时间：
1970
期刊：
影响因子：
--
通讯作者：
A. Michel
中科院分区：
文献类型：
--
作者：
A. Michel
在实践中，人们不仅对从动态系统的稳定性（在李亚普诺夫意义上）获得的定性信息感兴趣，而且对定量数据感兴趣，例如特定的轨迹边界和特定的瞬态行为。例如，一个系统可能是稳定的，但仍然完全无用，因为它可能表现出不希望的瞬态特性（例如，它可能超过强加在轨迹边界上的某些限制）。为了发展一个有意义的定量理论来分析动态系统，稳定性在这里被定义为状态空间的子集，这些子集在给定的问题中是预先指定的，通常，可能是时变的。这些子集的性质不仅提供了关于系统稳定性的信息，而且还提供了轨迹界和轨迹行为的估计。这里发展的理论足够普遍，包括自治和非自治系统，线性和非线性系统，简单系统和互联系统。所考虑的组合系统或相互连接的系统根据其子系统进行分析和处理。在陈述了稳定性和不稳定性的各种定义之后，陈述并证明了产生稳定性和不稳定性充分条件的定理。这些定理涉及类李雅普诺夫函数的存在性，这些函数一般不具备V和V的通常确定性要求。为了证明已发展的理论，考虑了几个例子。
In practice one is not only interested in the qualitative type of information obtainable from the stability (in the Lyapunov sense) of a dynamic system, but also in quantitative data, such as specific trajectory bounds and specific transient behavior. A system could, for example, be stable and still be completely useless because it may exhibit undesirable transient characteristics (e.g., it may exceed certain limits imposed on the trajectory bounds). In order to develop a meaningful quantitative theory for the analysis of dynamic systems, stability is defined here in terms of subsets of the state space that are prespecified in a given problem and, in general, may be time varying. The properties of these subsets yield not only information about the stability of a system, but they also yield estimates of trajectory bounds and of trajectory behavior. The theory developed here is general enough to include autonomous and nonautonomous systems, linear and nonlinear systems, simple systems and interconnected systems. The composite, or interconnected systems considered are analyzed and treated in terms of their subsystems. After stating various definitions of stability and instability, theorems that yield sufficient conditions for stability and instability are stated and proved. These theorems involve the existence of Lyapunov-like functions which in general do not possess the usual definiteness requirements on V and V. In order to demonstrate the developed theory, several examples are considered.