Adaptive stabilisation for a class of non-linear state time-varying delay systems with unknown time-delay bound

In this study, the authors address the problem of adaptive backstepping control for a class of single-input and single-output (SISO) non-linear time-delay systems in triangular structure. Both the parameters of the system to be controlled and the upper bounds of the time delays and their derivatives are assumed to be unknown. A direct adaptive controller for this class of uncertain non-linear systems is proposed. The assumption on delay-related non-linearities is further relaxed. An appropriate Lyapunov-Krasovskii functional is constructed, and the backstepping technique is used. It is shown that all the closed-loop signals are bounded, while the plant states converge to zero asymptotically. A simulation example is provided to demonstrate the design procedure and performance of the proposed method.