One presents a survey of the investigations in the theory of Volterra integral equations, reviewed in Ref. Zh. “Mat.” between 1966–1976.

Volterra integral equations

Nonlinear time varying discrete feedback systems input-output properties, deriving stability criteria by generalized small gain and passivity theorems

https://ntrs.nasa.gov/api/citations/19690026528/downloads/19690026528.pdf

Input—output properties of multiple-input, multiple-output discrete systems: part I☆

The purpose of this letter is to initiate a study of the question of L p (p≥1) stability of a larger class of nonlinear feedback systems whose forward loop is represented by a truncated Volterra series of the form (Ax)(t) = Σ n = 1 m 0 t∫ 0 tK n (t; τ 1 , ... τ n ) Π i = 1 nx(τ i )dτ i . It is demonstrated that under suitable conditions the open-loop system A is continuous and boundaed, and maps L p [0, ∞) into L p [0, ∞). It is shown that if the corresponding (unity gain) feedback system has a solution in some Lebesgue class L p (0 > p ≥ ∞), then the output of the feedback system belongs to the same class as the input.

http://ieeexplore.ieee.org/iel5/5/31125/01449348.pdf

On the L p (p&#8805;1) stability of a class of nonlinear systems

The bounded-input bounded-output stability, finite time stability and settling time of a single-loop feedback system consisting of a nonlinear time-varying gain followed by a linear time-invariant system are investigated via a nonlinear integral inequality. The gain has the form k 0 +k 1 (t)+k 2 (t)g(bd) where g(bd) is a monotonic increasing function. The system is bounded-input bounded-output stable provided the time-varying gains are L 1 ( 0 , t8) functions and is finite time stable for bounded gains. The nonlinear integral inequality, which is used to obtain explicit and useful bounds on the output of the system, is also employed to determine the settling time.

Stabilities and Settling Times of Nonlinear and Time-varying Feedback Systems

The bounded-input bounded-output stability of feedback control systems with time-varying gain is studied, and both continuous and sampled-data feedback systems are considered. It is shown that if the total gain deviation from a constant is finite, and if the feedback system with that constant gain is stable, then the output of the time-varying feedback system due to any bounded input is bounded; furthermore, the response of the time-varying system tends to that of the constant gain system. Similar results, somewhat weaker, are obtained for the time-varying sampled-data feedback system.

Bounded input bounded-output stability of linear time-varying feedback systems

Based on the requirement of stability and causality, this paper presents a new method of designing IIR digital filters. The method is essentially a combination of a time-domain and a frequency-domain technique. The result of this method is identical to the one in (5) which was derived without any physical interpretation. Because of this new derivation, the method can be readily extended to two or more dimensional noncausal digital filters.

Padé approximants of noncausal digital filters

This paper presents a complete algorithm which transforms a square matrix into the Jordan normal form.

Chi-Tsong Chen

Papers

Bounded input bounded-output stability of linear time-varying feedback systems

Padé approximants of noncausal digital filters

An Algorithm for Jordan Form Transformation