The purpose of this paper is to construct Lyapunov functions to prove the key fundamental results of linear system theory, namely, the small gain (bounded real), positivity (positive real), circle, and Popov theorems. For each result a suitable Riccati-like matrix equation is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time-invariant system and a memoryless nonlinearity. Lyapunov functions for the small gain and positivity results are also constructed for the interconnection of two transfer functions. A multivariable version of the circle criterion, which yields the bounded real and positive real results as limiting cases, is also derived. For a multivariable extension of the Popov criterion, a Lure-Postnikov Lyapunov function involving both a quadratic term and an integral of the nonlinearity, is constructed. Each result is specialized to the case of linear uncertainty for the problem of robust stability. In the case of the Popov criterion, the Lyapunov function is a parameter-dependent quadratic Lyapunov function.

Explicit construction of quadratic lyapunov functions for the small gain, positivity, circle, and popov theorems and their application to robust stability. part II: Discrete-time theory

A large-scale system is considered as a system constituted of subsystems which may be connected or disconnected from each other during operation. A new concept of connective stability is introduced by which a large-scale system is regarded as stable if it remains stable (in the sense of Lyapunov) under structural perturbations produced by the on-off participation of the subsystems. Algebraic conditions are developed that guarantee exponential connective stability of large-scale systems which may be composed of linear and nonlinear time-varying subsystems coupled by linear or nonlinear connections.

Stability of Large-Scale Systems under Structural Perturbations

Lyapunov function proofs of sufficient conditions for asymptotic stability are given for feedback interconnections of bounded real and positive real transfer functions. Two cases are considered: a proper bounded real (resp., positive real) transfer function with a bounded real (resp., positive real) time-varying memoryless nonlinearity; and two strictly proper bounded real (resp., positive real) transfer functions. A similar treatment is given for the circle and Popov theorems. Application of these results to robust stability with time-varying bounded real, positive real, and sector-bounded uncertainty is discussed. >

Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle and Popov theorems and their application to robust stability

Concepts of Stability and Resilience in Predator-Prey Models

Stability of Large-Scale Systems

Nonlinear discrete systems absolute stability exponential property proved by modified frequency condition, applying analog transformation to Szego-Kalman equations

Exponential absolute stability of discrete systems

Fur die absolute exponentielle Stabilitat nichtlinearer diskreter Systeme vom Lurje-Typ werden hinreichende Bedingungen formuliert. Durch einen Grenzprozes werden sie auf diejenigen fur die absolute exponentielle Stabilitat stetiger Systeme zuruckgefuhrt.



Sufficient conditions are formulated under which nonlinear discrete systems of the Lur'e type are exponentially absolutely stable. By a limiting process the obtained conditions are reduced to those for exponential absolute stability of continuous systems.

C. K. Sun

Papers

Exponential absolute stability of discrete systems

Exponential Absolute Stability of Discrete Systems