Стійкість та позитивність лінійних динамічних систем відносно світлового конуса

Abstract

Викладено методику дослідження стійкості та позитивності систем лінійних диференціальних та різницевих рівнянь відносно світлового конуса. Встановлено умови інваріантності світлового конуса у вигляді алгебраїчних нерівностей. Наведені умови отримано на основі спектральних оцінок для матричних коефіцієнтів. Розглянута методика дає підхід до розв’язування задачі позитивної стабілізації відносно класу світлових конусів.

We propose a method for the investigation of differential and difference models of dynamic objects whose phase spaces contain invariant sets, cones in particular. Positivity and monotony sets for such systems are usually caused by the nature of the object of study or structure of designed control system. The construction of such sets is calculated and used in qualitative research methods of dynamical systems, for analysis of stability and control. Classes of positive and monotone systems also arise in the theory of stability in the application of method for the comparison of differential systems. For analysis the stability of dynamical systems special methods were developed that are based on spectral properties of positive and positive invertible operators. We formulate conditions for the stability and the invariance of the cones in linear dynamic systems. Special attention is paid to the ellipsoidal and the light cone. Necessary and sufficient conditions of stability and positivity for linear differential and difference systems with respect to ellipsoidal cones are given in the form of matrix inequalities. Sufficient conditions for the asymptotic stability and the invariance of the light cones for differential and difference systems are the result of the allegations and the general theorems on the stability of positive linear systems. The proposed stability and positivity conditions of the linear systems are stated in terms of the solutions of set of algebraic inequalities. Their evidences are obtained by means of spectral estimations of matrix coefficients. Our method allows to specify an approach to the problem solving of positive stabilization relative to the class of light cones, that is, to build a control in the form of feedback or dynamic compensator, which provids both positivity relative to a given cone and the asymptotic stability of the closed system. This result can be used for the second-order differential systems. Numerical example of application of the proposed method to first-order differential systems is presented