Singular Values Correction Methods With Double Parameters For Ill Conditioned Problems And Their Applications

The ill-conditioned problems exist in many researching and engineering fields,such as computational mathematics,statistics,economics,geodetic measurement,satellite orbit determination and so on.Gauss-Markov models and discrete dynamic systems are static models and dynamic models respectively,favored by scientists and technicians due to their high agreement with practical problems and being easy to be solved.LS estimation and Kalman filter are the most common methods for solving Gauss-Markov models and discrete dynamic systems.However,when the Gauss-Markov models and the discrete dynamic systems are ill-conditioned,the estimation of LS and the Kalman filter are unstable,which leads to the unreliability of the estimation results.Biased estimations,regularization methods and iterative algorithms are main methods to solve ill-conditioned problems,but these methods fail to fully exploit ill-conditioning information,sometimes resulting in unsatisfactory estimation results.Aiming at the ill-conditioned problems in the Gauss-Markov models and the discrete dynamic systems,this paper is to combine the diagnostic with processing of ill-conditioning.This paper introduces the concept of signal-to-noise ratio(SNR),and applies statistical methods to diagnose the harm of ill-conditioning to parameter estimations,and then proposes three singular values correction methods with double parameters for ill conditioned problems.The partial singular value correction estimation with double parameters and the double truncated singular value estimation are proposed to solve the ill conditioned problems in Gauss-Markov models,and the double-parameter ridge type Kalman filter method is proposed for the ill-conditioned problems in discrete dynamic systems.The main contents and creations are as follows:1.The SNR method is introduced to mine the key ill-conditioning information,and to test the effect of ill-conditioning on the estimation of parameters,thereby obtaining a more comprehensive understanding of ill-conditioning,providing the basis for the subsequent processing of ill-conditioned linear model and discrete dynamic system.2.For the ill-conditioned problem in the Gauss-Markov model,using signal to noise ratio to obtain the key ill-conditioning information,this paper improves the ridge estimation and accordingly proposes a method named partial singular value correction estimation with double parameters based on signal-to-noise ratio test,improves the truncated singular value estimation and accordingly proposes the double truncated singular value estimation based on signal-to-noise ratio test.The regularization method is unified in form.3.For the ill-conditioning problem in discrete dynamic system,this paper first deeply analyses the mechanism that how ill-conditioning influences the state estimation,and then introduces the signal-to-noise ratio method into discrete dynamic system to mine ill-conditioning information,and proposes double-parameter ridge-type Kalman filter based on signal-to-noise ratio test.4.There are ill-conditioned problems in the fields of control network adjustment,airborne gravimetry data downward continuation and satellite orbit determination.By analyzing the ill-conditioned mechanism of their models,this paper applies the proposed double parameter correction methods based on SNR test in these fields to improve the accuracy of parameter estimations.