Carleman framework filtering of nonlinear noisy phase-locked loop system

Abstract

Abstract In control and communications, the phase-locked loop (PLL) is regarded as the demodulator. Under the presence of small noise, the PLL system fails to accomplish the locking condition resulting in the stochastic phase difference. As a result of this, the PLL becomes a nontrivial nonlinear stochastic system. To circumvent the curse of dimensionality and nonlinearity, we exploit the method of linearization in the Carleman framework in combination with the finite closure for the stochastic system considered here. We show that the Carleman linearization has proven useful to preserve the nonlinearity via bilinearization. The Carleman setup of the nonlinear stochastic differential system has the Markov property and the terms are manageable. Then, we filter the states of the PLL using the filtering theory of the homogeneous Markov process. Finally, the numerical simulations reveal the superiority of the proposed filtering in Carleman setting in contrasts with the celebrated extended Kalman filtering framework.