On finite time delay dependent stability of linear discrete delay systems: Numerical solution approach

Dragutin Debeljkovic,Milan Misic,Aleksandar Cvetkovic,Vladimir Jankovic,Ivan Buzurovic

doi:10.5937/str1503039d

Abstract

In this paper, a possible solution of the basic nonlinear quadratic matrix equation was proposed. The solution is crucial in the formulation of the particular criteria for the delay-dependent finite time stability of discrete time delay systems represented as x(k+1)=A0(k)+A1x(k-h). The time delay-dependent criteria have been derived. In addition, the significance of the nonlinear discrete polynomial matrix equation is explained. With the use of the mathematical formalism based on the Traub and Bernoulli's algorithms, it was concluded that the computation of the dominant solvent of the matrix polynomial equation does not guarantee a necessary convergence in all cases, unlike in the traditional numerical procedures. In this paper, we presented one particular and one general solution valid in the case when the discrete matrix equation was presented in its factorial form. The numerical computations are performed to illustrate the suggested results.

Highlights

TO investigate the stability of a control system, the Lyapunov method was widely used in the control system community
This stability concept was introduced in the era of modern control systems [1,2,3,4,5,6], and it is still widely used nowadays
We present the concept of the finite time stability of discrete time-delay systems previously investigated in [7]

Summary

Introduction

TO investigate the stability of a control system, the Lyapunov method was widely used in the control system community. The Lyapunov stability is insufficient to describe the dynamical behavior of the special classes of the systems or to give satisfactory conclusions about the different types of stability This is the case for the finite time stability in which the requirements are set on the states of the system. The system is stable if the states of the system do not exceed the predefined boundaries on some fixed time interval This stability concept was introduced in the era of modern control systems [1,2,3,4,5,6], and it is still widely used nowadays. We present the concept of the finite time stability of discrete time-delay systems previously investigated in [7]. The novel sufficient stability conditions were presented in a form of algebraic inequalities with a necessity of solving the particular quadratic discrete matrix equation with the defined matrix A

Notations and preliminaries

Problem formulation

Numerical solution

Modification of the Newton method

Conclusion