On the Interval Stability of Weak-Nonlinear Control Systems with Aftereffect.

Andriy Shatyrko,Denys Khusainov

doi:10.1155/2016/6490826

Abstract

Sufficient conditions of interval absolute stability of nonlinear control systems described in terms of systems of the ordinary differential equations with delay argument and also neutral type are obtained. The Lyapunov-Krasovskii functional method in the form of the sum of a quadratic component and integrals from nonlinearity is used at construction of statements.

Highlights

The actuality of absolute interval stability problem of the dynamical systems, mentioned in the present paper, proves to be true as a lot of interesting reports at the international congresses and conferences and set of foreign publications, for example, [1,2,3,4,5,6,7]
Interesting fundamental necessary and sufficient conditions of interval stability of the linear differential equations with uncertainty parameters have been obtained at papers of Kharitonov [9,10,11,12]
We will consider the direct control system described by the differential equations with deviating argument of neutral type and with interval given coefficients of linear part: d dt

Summary

Introduction

The actuality of absolute interval stability problem of the dynamical systems, mentioned in the present paper, proves to be true as a lot of interesting reports at the international congresses and conferences and set of foreign publications, for example, [1,2,3,4,5,6,7]. Another alternative approach which has had development in works by Barbashin et al is the Lyapunov second (direct) method with function type of “quadratic form plus integral from nonlinearity” [20,21,22,23] Distribution of this method on systems with delay and neutral type has been obtained in Shatyrko and Khusainov works [24,25,26,27] and numerous works of Chinese scientists (e.g., [5, 28]). The condition of Razumikhin facilitates solving the investigation problem of sign-definiteness of Lyapunov function total derivative along the system solution. By means of this approach it is possible to estimate influence of aftereffect, that is, to obtain the conditions of absolute interval stability depending on delay. The matrix of the quadratic form of a total derivative has twice the big dimension

Direct Control Systems with Time-Delay Argument

Direct Control Systems of Neutral Type

Conclusion and Prospects