Stability of linear discrete time delay systems

S.B.Stojanovic

University of Nis

Bulevar Oslobodjenja, 124, 16000, Leskovac, Serbia

D. Lj.Debeljkovic

University of Belgrade

11000, Belgrade, Serbia

In this paper, we have established new, necessary and sufficient, conditions for the asymptotic stability of a particular class of linear discrete time delay systems. The time-dependent criteria are derived by Lyapunov's direct method and are exclusively based on the maximal and dominant solvents of particular matrix polynomial equation. It can be shown that these solvents exist only under some conditions, which, in a sense, limits the applicability of the method proposed. The solvents can be calculated using generalized Traub`s or Bernoulli`s algorithms. Both of them possess significantly smaller number of flops counts than the standard algorithm. However, for large time delays within the system generalized Bernoulli's algorithm has shown very large ET. ET of Traub's algorithm is many times smaller than ET of Bernoulli's algorithm, but is somewhat larger than ET of standard algorithm.

Improving the converging properties of used algorithms for these purposes, may be a particular research topic in the future.