On the Stabilization through Linear Output Feedback of a Class of Linear Hybrid Time-Varying Systems with Coupled Continuous/Discrete and Delayed Dynamics with Eventually Unbounded Delay

Manuel De la Sen
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Manuel De la Sen: Department of Electricity and Electronics, Faculty of Science and Technology, Institute of Research and Development of Processes, University of the Basque Country (UPV/EHU), 48940 Leioa, Bizkaia, Spain

Mathematics, 2022, vol. 10, issue 9, 1-23

Abstract: This research studies a class of linear, hybrid, time-varying, continuous time-systems with time-varying delayed dynamics and non-necessarily bounded, time-varying, time-differentiable delay. The considered class of systems also involves a contribution to the whole delayed dynamics with respect to the last preceding sampled values of the solution according to a prefixed constant sampling period. Such systems are also subject to linear output-feedback time-varying control, which picks-up combined information on the output at the current time instant, the delayed one, and its discretized value at the preceding sampling instant. Closed-loop asymptotic stabilization is addressed through the analysis of two “ad hoc” Krasovskii–Lyapunov-type functional candidates, which involve quadratic forms of the state solution at the current time instant together with an integral-type contribution of the state solution along a time-varying previous time interval associated with the time-varying delay. An analytic method is proposed to synthesize the stabilizing output-feedback time-varying controller from the solution of an associated algebraic system, which has the objective of tracking prescribed suited reference closed-loop dynamics. If this is not possible—in the event that the mentioned algebraic system is not compatible—then a best approximation of such targeted closed-loop dynamics is made in an error-norm sense minimization. Sufficiency-type conditions for asymptotic stability of the closed-loop system are also derived based on the two mentioned Krasovskii–Lyapunov functional candidates, which involve evaluations of the contributions of the delay-free and delayed dynamics.

Keywords: hybrid dynamic systems; time-varying delay; sampled systems; asymptotic stability; asymptotic stabilization; linear output feedback (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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