On Sontag’s formula for the sampled-data observer-based stabilization of nonlinear time-delay systems

In this paper, a new methodology for the design of sampled-data dynamic output feedback (DOF) stabilizers for control-affine nonlinear time-delay systems is proposed. In particular, the methodology here developed is based on the Artstein’s theory of control Lyapunov functions and related Sontag’s formula, here extended, for the first time in the literature, to the case of DOF controllers. Firstly, the new notion of Steepest Descent Error Dynamics (SDED) is introduced and used to suitably re-formulate the well-known Sontag’s universal formula for the case of DOF controllers. Then, it is shown that, under suitably fast sampling, a proposed sampled-data DOF controller based on the new provided version of Sontag’s universal formula ensures the semiglobal practical stability of the related sampled-data closed-loop system. Time-varying sampling intervals, as well as the intersampling system behavior, are taken into account. In the theory here developed, the case of delay-free systems is addressed as a special case. Applications are presented for the validation of the proposed theoretical results.