State-Dependent Scaling Problems and Stability of Interconnected iISS and ISS Systems
State-Dependent Scaling Problems and Stability of Interconnected iISS and ISS Systems
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DOI：
10.1109/tac.2006.882930
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发表时间：
2006-10
影响因子：
6.8
通讯作者：
H. Ito
中科院分区：
文献类型：
--
作者：
H. Ito
本文研究非线性关联系统的稳定性问题。本文介绍了状态相关标度问题的一种数学形式，它的解直接提供李雅普诺夫函数，以统一的方式证明了关联耗散系统的稳定性。稳定性准则被解释为状态相关标度问题解存在的充分条件。计算解决方案的问题是直接的经典稳定性准则所涵盖的系统。然而，这对于具有强非线性的系统来说可能太困难了。本文的主要目的是通过关注关联积分输入到状态稳定（iISS）系统和输入到状态稳定（ISS）系统来证明形式适用性之外的有效性。本文从状态相关标度公式出发，导出了含iISS系统的关联系统的小增益型定理。本文给出了解决方案和李雅普诺夫函数明确。新的框架无缝地推广了ISS小增益定理和经典的稳定性准则，如Lp小增益定理，无源性定理，圆和波波夫准则。标度的状态依赖性对于有效处理本质非线性是至关重要的，而常数对于经典非线性是足够的
This paper addresses the problem of establishing stability of nonlinear interconnected systems. This paper introduces a mathematical formulation of the state-dependent scaling problems whose solutions directly provide Lyapunov functions proving stability properties of interconnected dissipative systems in a unified manner. Stability criteria are interpreted as sufficient conditions for the existence of solutions to the state-dependent scaling problems. Computing solutions to the problems is straightforward for systems covered by classical stability criteria. It, however, could be too difficult for systems with strong nonlinearity. The main purpose of this paper is to demonstrate the effectiveness beyond formal applicability by focusing on interconnected integral input-to-state stable (iISS) systems and input-to-state stable (ISS) systems. This paper derives small-gain-type theorems for interconnected systems involving iISS systems from the state-dependent scaling formulation. This paper provides solutions and Lyapunov functions explicitly. The new framework seamlessly generalizes the ISS small-gain theorem and classical stability criteria such as the Lp small-gain theorem, the passivity theorems, the circle, and Popov criteria. State-dependence of the scaling is crucial for effective treatment of essential nonlinearities, while constants are sufficient for classical nonlinearities