Theoretical Research On Parameter Estimation Of Sinusoidal Signals Based On A Class Of Nonlinear Luenberger Observers

The parameter estimation of sinusoidal signals has been widely used in many fields such as military,electric power,biomedicine,etc.Therefore,it has been highly valued by researchers and a large variety of algorithms have been proposed.The classical algorithms such as fast Fourier transform(FFT)method have been applied in actual application and obtained excellent results.In recent years,the parameter estimation of sinusoidal signals has also received more and more attention in the field of control.For example,when dealing with disturbance rejection of linear(non-linear)systems or vibration suppression of flexible robots,etc.,frequency estimators with asymptotical-convergence properties and certain stability are often essential tools.Thus,many researchers try to use nonlinear control theory methods to give theoretical analysis and research on the problem of parameter estimation of sinusoidal signals,and have obtained a large variety of results such as global asymptotic convergence and global(semi-global)exponential convergence.However,in the field of control,there are relatively few research results on the convergence speed and robustness of estimation algorithms,especially for discrete-time sinusoidal signals,no researchers have tried to use nonlinear control theory to analyze the effect of noise on the estimation algorithm.Therefore,how to improve the convergence speed of the estimation algorithm and characterize the impact of noise on the estimation algorithm qualitatively or quantitatively still has important theoretical research value.With the development of nonlinear theory,especially the development of nonlinear observation theory and input-to-state stability(ISS)theory provides a new theoretical basis for the analysis of parameter estimation of sinusoidal signal,because the problem of parameter estimation of sinusoidal signal can be transformed into a state estimation problem of a class of nonlinear systems,and when noise is used as input,the ISS theory can describe the specific impact of noise on the estimation algorithm in detail.Therefore,this article attempts to design new nonlinear estimation algorithms,and to analyze the convergence and robustness of the given algorithms with the help of nonlinear parameter estimation theory,nonlinear observation theory,and ISS theory.The main contents include:(1)For a noisy continuous-time sinusoidal signal,we first consider a class of adaptive identifier algorithms.In terms of the nonlinear observation theory,we prove the exponential convergence of parameter estimation and show that the parameter estimation error is strong integral input-state stable(Strong-i ISS)with respect to noise.Furthermore,we consider a class of reduced-order adaptive identifier algorithms.In terms of the nonlinear observation theory,we prove the exponential convergence of parameter estimation and show the Strong-i ISS property of the parameter estimation error with respect to noise.Finally,numerical simulation examples show the robustness of the adaptive identifier algorithm.(2)For a noisy continuous-time sinusoidal signal,we first introduce a class of nonlinear estimation algorithms.It is proved that the designed algorithms can achieve arbitrary exponential convergence of parameter estimation by adjusting the designed parameters.Second,we prove that the estimation error is Strong-i ISS with respect to noise.Moreover,it is proved that for any bounded noise,the estimator can still be effectively operated by adjusting the design parameters,and the estimation error is bounded.Finally,several numerical simulation examples are given to show the effectiveness of the proposed estimation algorithm.(3)For a noisy discrete-time sinusoidal signal.We first design a class of nonlinear parameter estimator and show the asymptotic convergence of the parameter estimation.Second,we show the i ISS properties of the parameter estimation with respect to noise.Finally,several numerical simulation examples are given to show the effectiveness of the proposed estimation algorithm.