Stability by averaging of linear discrete-time systems

Recently, a constructive approach to averaging-based stability was proposed for linear continuous-time systems with small parameter ε > 0 and rapidly-varying almost periodic coefficients. The present paper extends this approach to discrete-time linear systems with rapidly-varying periodic coefficients. We consider linear systems with state delays, where results on the stability via averaging are missing. Differently from the continuous-time, our linear matrix inequalities (LMIs) are feasible for any delay (i.e. the system is exponentially stable) provided
$\mathcal{E}$
is small enough. We introduce an efficient change of variables that leads to a perturbed averaged system, and employ Lyapunov analysis to derive LMIs for finding maximum values of the small parameter ε > 0 and delay that guarantee the exponential stability. Numerical example illustrates the effectiveness of the proposed approach.