Maximum Likelihood Parameter Identification For Input Nonlinear Systems

In field of industrial control,there always exists some nonlinear characteristics in the real control systems.Parameter identification of nonlinear systems is also the basis of controller design.However,the structure of nonlinear systems is more complex than that of linear systems.Traditional parameter identification methods cannot be directly applied to the identification of nonlinear systems.Therefore,the problem of parameter identification of nonlinear systems has more practical meaning.In this paper,the problem of the input nonlinear system which is consisted by a static nonlinear block followed by a linear dynamic subsystem is discussed,and this kind of system has been widely used to describe the dynamic systems which have input nonlinearities.This paper mainly focuses on the parameter identification of the controlled autoregressive moving average model with hard nonlinearity based on the maximum likelihood principle.The main results are summarized as follows.1.For the nonlinear systems with polynomial nonlinearity,the recursive least squares based on maximum likelihood algorithm is used to identify the parameters of the controlled autoregressive moving average model.Based on the idea of the maximum likelihood principle,a novel maximum likelihood based stochastic gradient algorithm is proposed.The latest input and output data of the system is used to update the current likelihood function and to obtain the estimates of the parameters,so as to reduce the computational burden of the identification process.The algorithm is further improved by utilizing the convergence factor.The numerical simulation demonstrates the effectiveness of the proposed algorithms.2.For the equation error controlled autoregressive systems with saturation hard nonlinearity,the parameter identification algorithm based on gradient iteration is deduced by means of the key term separation technique.Furthermore,for the equation error controlled autoregressive moving average systems with saturation hard nonlinearity,the maximum likelihood based least squares auxiliary variable algorithm and the maximum likelihood based bias compensation gradient iterative algorithm are proposed.The unbiased estimates of the model can be obtained by both of these algorithms.The maximum likelihood based least squares auxiliary variable algorithm has the characteristics of fewer iterations and faster calculation speed,but it can not get the parameters of the colored noise.However,the maximum likelihood based bias compensation gradient iterative algorithm can effectively identify all the parameters of the model with colored noise.Two numerical simulation examples are applied to demonstrate the effectiveness of the proposed algorithms.3.Considering the equation error controlled autoregressive systems with piecewise hard nonlinearity,an Aitken based stochastic gradient algorithm is presented.For the controlled autoregressive moving average systems with piecewise hard nonlinearity,two algorithms are proposed based on the idea of the maximum likelihood principle.One is the forgetting factor maximum likelihood based stochastic gradient algorithm,the other is the Aitken maximum likelihood based stochastic gradient algorithm.The variance of estimate error of the Aitken maximum likelihood based stochastic gradient algorithm is smaller than that of the forgetting factor maximum likelihood based stochastic gradient algorithm,which means the estimate of the previous algorithm is more stable.The numerical simulation of the algorithms verifies the effectiveness of the derived algorithms.4.For the Output Error model with random time delay,the finite impulse response method is used to approximate the model to the FIR model.Since the information vector of the model contains unobservable hidden variables,the Expectation-maximization algorithm is used to first calculate the expectation step to obtain the implicit estimated values,and then maximize the expectation in the previous step to obtain the unknown parameters of the model.Two steps alternate iteratively to complete the identification of the model,and the effectiveness of the algorithm is verified by the convergence analysis and numerical simulations of the algorithm.