Parameter Estimation And Its Convergence Of A Special Linear System Model

System identification mainly studies some problems how mathematical model and parameters of system are determined . It is a widely used subject . It involves how predict dynamic mathematical model of system that best describes the data according to input and output information . It is a chief problem to estimate model parameters in the course of determining mathematical model of system .The hidden markov model(HMM) is a sort of statistical signal model. It has been widely applied to many fields such as speech recognition , image processing and signal processing etc . There are three fundamental problems needed to be considered for HMM , namely: training , recognizing and decoding . Parameter estimation is the core problem of it. Therefore , it is a suitable idea to solve parameter estimation by using a combined method connecting the system identification with the HMM theory . This idea is shown to be very effective in treating the recursive parameter estimation problems of this paper .The paper mainly illustrates parameter estimation and convergence of a special linear system model . Some improved parameter estimation algorithms are presented , their convergent properties are proved by using the martingale convergence theorem . The detailed discussion is organized as follows.1. The parameters of discrete-time linear system model are estimated by the least-squares algorithm . The result that parameter estimator convergent to the true value is proved . The proof is quite strict and it is based on the martingale theory and Kronecker lemma.2. The convergent property and convergent rate of parameter estimation error are analyzed . Some sufficient conditions are given to guarantee the asymptotic normality of parameter estimation error.3. When adaptive control system is described by extended ARM A time series model , an adaptive prediction algorithm is established . The key features of the algorithm are (1) the prediction errors are used in the regression vector , (2) an arbitrary feedback is allowed between output and input . In the end , the adaptive prediction algorithm is proved to be globally convergent by using the extended form of Martingales Convergence Theorem .