In recent years, the ability to accommodate various nonlinearities has become even more important to support systems design and analysis in a broad area of engineering and science. In this line of research, this paper discusses usefulness of the notion of integral input-to-state stability (iISS) in assessing and establishing system properties through interconnection of component systems. The focus is to construct Lyapunov functions which explain mechanism and provide estimate of stability and robustness of interconnected systems. Unique issues arising in dealing with iISS systems are reviewed in comparison with interconnections of input-to-state stable (ISS) systems. The max-separable Lyapunov function and the sum-separable Lyapunov function which are popular for ISS and iISS, respectively, are revisited. The max-separable function cannot be qualified as a Lyapunov function when component systems are not ISS. Level sets of the max-separable function are rectangles, and the rectangles cannot be expanded to encompass the entire state space in the presence of non-ISS components. The sum-separable function covers iISS components which are not ISS. However, it has practical limitations when stability margins are small. To overcome the limitations, this paper brings in a new idea emerged recently in the literature, and proposes a new type of construction looking at level sets of a Lyapunov function. It is shown how an implicit function allows us to draw chamfered rectangles based on fictitious gain functions of component systems so that they provide reasonable estimates of forward invariant sets producing a Lyapunov function applicable to both iISS and ISS systems equally.

近年来，适应各种非线性的能力对于在广泛的工程和科学领域中支持系统设计和分析变得更加重要。在这一研究领域中，本文讨论了积分输入到状态稳定性（iISS）概念在通过组件系统的互连来评估和建立系统特性中的有用性。重点是构造Lyapunov函数，这些函数解释机制并提供互连系统的稳定性和鲁棒性估计。与输入到状态稳定（ISS）系统的互连比较，回顾了在处理iISS系统时出现的独特问题。再次讨论了分别在ISS和iISS上流行的最大可分Lyapunov函数和和可分Lyapunov函数。当组件系统不是ISS时，max-separable函数不能被视为Lyapunov函数。max-separable函数的级别集是矩形，并且在存在非ISS组件的情况下，矩形不能扩展为包含整个状态空间。可和的功能涵盖了不是ISS的iISS组件。但是，当稳定性裕度较小时，它具有实际的局限性。为了克服这些局限性，本文引入了一种新的思想，该思想最近出现在文献中，并提出了一种新的构造类型，它研究了Lyapunov函数的水平集。