Small-gain theory for stability and control of dynamical networks: A Survey

Zhong Ping Jiang, Tengfei Liu

Research output: Contribution to journalReview articlepeer-review


This paper provides a personal account of the small-gain theory as a tool for stability analysis, control synthesis, and robustness analysis for interconnected uncertain systems. A milestone in modern control theory is the development of a transformative stability criterion known as the classical small-gain theorem proposed by George Zames in 1966, that surpasses Lyapunov theory in that there is no need to construct Lyapunov functions for the finite-gain stability of feedback systems. Under the small-gain framework, a feedback system composed of two finite-gain stable subsystems remains finite-gain stable if the loop gain is less than one. Despite its apparent simplicity at first sight, Zames's small-gain theorem plays a crucial role in the development of linear robust control theory. Borrowing techniques in modern nonlinear control, especially Sontag's notion of input-to-state stability (ISS), the first generalized, nonlinear ISS small-gain theorem proposed by one of the authors in 1994 overcomes the two shortcomings of Zames's small-gain theorem. First, the use of nonlinear gains allows to consider strongly nonlinear, interconnected systems. Second, the role of initial conditions is made explicit so that both internal Lyapunov stability and external input-output stability can be studied in a unified framework. In this survey paper, we first review early developments in the nonlinear small-gain theory for interconnected systems of various types such as continuous-time systems, discrete-time systems, hybrid systems and time-delay systems, along with applications in robust nonlinear control. Then, we describe how to obtain a network small-gain theory for large-scale dynamical networks that are comprised of more than two interacting nonlinear systems. Constructive methods for the generation of Lyapunov functions for the total network are presented as well. Finally, this paper discusses how the network/nonlinear small-gain theory can be applied to obtain innovative solutions to quantized and event-based nonlinear control problems, that are important for the development of a complete theory of controlling cyber-physical systems subject to communications and computation constraints.

Original languageEnglish (US)
Pages (from-to)58-79
Number of pages22
JournalAnnual Reviews in Control
StatePublished - 2018


  • Input-to-state stability (ISS)
  • Interconnected systems
  • Large-scale dynamical networks
  • Nonlinear small-gain theorem

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering


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