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Conventional and Extended Time-delayed Feedback Controlled Zero-Crossing Digital Phase Locked Loop

This article investigates the effect of the conventional and extended time-delayed feedback control techniques of chaos control on a first-order positive zero-crossing digital phase locked loop (ZC1-DPLL) using local stability analysis, two-parameter bifurcation studies and two-parameter Lyapunov exponent spectrum. Starting from the nonlinear dynamics of a ZC1-DPLL, we at first explore the time-delayed feedback control (TDFC) algorithm on a ZC1-DPLL in the parameter space. A condition for the optimum value of the system control parameter is derived analytically for a TDFC based ZC1-DPLL. Next, the extended time-delayed feedback control (ETDFC) technique on a ZC1-DPLL is described. It is observed that the application of the delayed feedback control (DFC) technique on the sampled values of the incoming signal inside the loop finally results in the nonlinear DFC of the phase error dynamics. We prove that, for some suitably chosen control parameters, an ETDFC based ZC1-DPLL has a broader stability zone in comparison with a ZC1-DPLL and its TDFC version.