Design and convergence analysis of stochastic frequency estimator using contraction theory

This study investigates the design and analysis of an estimator for unknown frequencies of a sinusoid in the presence of additive noise. A dynamic stochastic estimator is proposed to ensure simultaneous globally convergent estimation of the state and the frequencies of a sinusoid comprising multiple frequencies. Approach given in this study exploits the results of stochastic contraction theory and the observers. The concept of contraction theory related to semi-contracting systems is used to show the asymptotic convergence of the proposed non-linear estimator. The boundedness and convergence of the state and frequencies estimates for all initial conditions and frequency values has been shown analytically. The proposed estimator is generalised to estimate n-unknown frequencies of a given noisy sinusoid. Numerical simulations of estimator are presented for different combinations of frequencies to justify the claim.