一类具有时滞状态依赖扰动的切换系统的界与不变集
Bounds and Invariant Sets for a Class of Switching Systems with Delayed-state-dependent Perturbations
AI总结：
提出一种分量方法，用于计算具有非线性时滞状态依赖扰动的切换连续时间线性系统的分量瞬态界、最终界和不变区域，并给出局部或半全局性质的条件以及仿射依赖情形下的全局有效界。
摘要：
我们提出了一种新颖的方法，用于计算一类具有扰动界的切换连续时间线性系统的分量瞬态界、最终界和不变区域，其中扰动界可以非线性地依赖于时滞状态。该方法的主要优势在于其分量特性，即它允许扰动向量的每个分量具有独立的界，并且所得到的界和集也是以分量形式给出的。这种分量方法不使用范数来界定扰动或状态向量，避免了缩放不同状态向量分量以获得有用结果的需要，并且在某些情况下可以减少保守性。我们给出了所推导的界具有局部或半全局性质的条件。此外，我们处理了扰动界对时滞状态的依赖为仿射形式的情况，作为非线性依赖的一个特例，并证明了在这种情况下所推导的界是全局有效的。还给出了实际稳定性的充分条件。本文建立在作者先前关于时滞状态依赖扰动的切换系统的结果之上并进行了扩展。从这个意义上说，贡献有三方面：推导上述扩展；阐明所提方法适用的切换线性系统类与那些允许共同二次李雅普诺夫函数的系统类之间的精确关系（这是我们先前工作中遗留的问题）；以及推导一种技术，用于计算具有由状态向量分量绝对值仿射函数分量有界扰动的切换线性系统的共同二次李雅普诺夫函数。
英文摘要：
We present a novel method to compute componentwise transient bounds, ultimate bounds, and invariant regions for a class of switching continuous-time linear systems with perturbation bounds that may depend nonlinearly on a delayed state. The main advantage of the method is its componentwise nature, i.e. the fact that it allows each component of the perturbation vector to have an independent bound and that the bounds and sets obtained are also given componentwise. This componentwise method does not employ a norm for bounding either the perturbation or state vectors, avoids the need for scaling the different state vector components in order to obtain useful results, and may also reduce conservativeness in some cases. We give conditions for the derived bounds to be of local or semi-global nature. In addition, we deal with the case of perturbation bounds whose dependence on a delayed state is of affine form as a particular case of nonlinear dependence for which the bounds derived are shown to be globally valid. A sufficient condition for practical stability is also provided. The present paper builds upon and extends to switching systems with delayed-state-dependent perturbations previous results by the authors. In this sense, the contribution is three-fold: the derivation of the aforementioned extension; the elucidation of the precise relationship between the class of switching linear systems to which the proposed method can be applied and those that admit a common quadratic Lyapunov function (a question that was left open in our previous work); and the derivation of a technique to compute a common quadratic Lyapunov function for switching linear systems with perturbations bounded componentwise by affine functions of the absolute value of the state vector components.
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