Kybernetika 58 no. 4, 593-625, 2022

Stability and stabilization of one class of three time-scale systems with delays

Valery Y. GlizerDOI: 10.14736/kyb-2022-4-0593


A singularly perturbed linear time-invariant time delay controlled system is considered. The singular perturbations are subject to the presence of two small positive multipliers for some of the derivatives in the system. These multipliers (the parameters of singular perturbations) are of different orders of the smallness. The delay in the slow state variable is non-small (of order of $1$). The delays in the fast state variables are proportional to the corresponding parameters of singular perturbations. Three much simpler parameters-free subsystems are associated with the original system. It is established that the exponential stability of the unforced versions of these subsystems yields the exponential stability of the unforced version of the original system uniformly in the parameters of singular perturbations. It also is shown that the stabilization of the parameters-free subsystems by memory-free state-feedback controls yields the stabilization of the original system by a memory-free state-feedback control uniformly in the parameters of singular perturbations. Illustrative examples are presented.


exponential stability, time delay system, linear controlled system, three time-scale singularly perturbed system, memory-free state-feedback stabilization


93C23, 93C70, 93D15, 93D23


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