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

• Published in: Mathematics (Basel) Vol. 10; no. 9; p. 1424
• Main Author: De la Sen, Manuel
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.05.2022
• Subjects: Algebra; Approximation; Asymptotic properties; asymptotic stability; asymptotic stabilization; Delay; Design; Dynamics; Feedback control; hybrid dynamic systems; Hybrid systems; linear output feedback; Mathematics; Output feedback; Partial differential equations; Quadratic forms; sampled systems; Sampling; Stabilization; Time varying control; Time varying control systems; time-varying delay
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math10091424

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.