Stability and stabilisability of the singularly perturbed system with delay on time scales: a decomposition approach

ABSTRACT

Sufficient conditions for the exponential stability and stabilisability of linear singularly perturbed control systems with the small parameter defined on homogeneous time scale are studied. Stability conditions are formulated in terms of a spectrum of two parameter-free subsystems of lower dimensions than the original one: slow and fast subsystems. These conditions, being parameter-free, provide the exponential stability of the original singularly perturbed system with delay for all sufficiently small positive values of parameter, i.e. robust with respect to the parameter of singular perturbation. A stabilisation problem for a given class of control systems is solved by a design of a linear well-conditioned parameter-free composite state-feedback control which is a sum of a stabilising slow and fast control.

KEYWORDS:

• Singularly perturbed system
• time-delay
• decomposition
• time scales
• stability
• stabilisability

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For the purpose of these studies, we use a different notation than the standard in the time scales theory, namely the (forward) graininess function is denoted as κ instead of the standard used μ. μ later on will serve as a parameter.

Additional information

Funding

The work of Ewa Pawluszewicz has been supported by grant No. WZ/WM-IIM/1/2019 of Bialystok University of Technology. The work of Olga Tsekhan was partially supported by Education Ministry of Republic of Belarus under the State program of scientific research “Convergence-2020”: Task 1.3.02.