Parameter Estimation Of Transfer Functions

Transfer functions play an important role in the classical control theory. The classical PID control algorithm is based on the transfer function model of systems. The existing statistical identi?cation methods are based on the discrete-time model of systems. In this paper, the nonlinear optimization principles of statistical identi?cation methods are used to study the parameter estimation of transfer functions for continuous-time systems. The study subject has theoretical signi?cance and practical value. The main works of this paper are as follows.1. The transfer functions of linear dynamical systems are rational fraction functions.The coe?cients of the numerator and denominator polynomial of the transfer functions are de?ned as the system parameters. It is well known that the impulse response and the step response of systems are highly nonlinear functions with respect to the system parameters and time. In terms of the rolling optimization problem of the nonlinear function, this paper proposes a recursive parameter estimation scheme based on the gradient search and the Newton search utilizing the recursive identi?cation idea. Using the discrete observation data by the pulse or step response test, this paper derives the stochastic gradient algorithm, the multi-innovation stochastic gradient algorithm and the Newton recursive algorithm in the condition of the dynamical datum length. Furthermore, in order to reduce the computation load and enhance the stability of the algorithm, the Newton recursive algorithm based on decomposition is deduced.2. In terms of the optimization problem of the general nonlinear function for the measured datum of ?nite length, the iterative parameter estimation schemes based on gradient search and Newton search are proposed according to the iterative principle. Using the discrete observation data based system the pulse or step responses, this paper derives the iterative parameter estimation methods based gradient and the Newton iterative parameter estimation method of the ?nite data. Meanwhile, the iterative parameter estimation method based gradient of sliding datum window. Furthermore, based on the hierarchical identi?cation principle, the Newton iterative parameter estimation method based on decomposition is proposed for the aim of reducing the computation load. In order to improve the stability of the Newton algorithm, this paper presents some modi?ed measures by adding step size or modi?ed factors. Moreover, according to the quasi-Newton idea,the Gauss-Newton algorithm parameter estimation algorithm and the Davidon-FletcherPowell parameter estimation algorithm are proposed for improving the stability of the Newton algorithms. The simulation results show that the proposed method can reduce the computation load and improve the stability of algorithms.3. The above-mentioned parameter estimation algorithms are based on the impulse or step signal excitation. On this basis, the paper studies the parameter estimation method of transfer functions under the sinusoidal excitation signal. From simple to complex, by applying the single frequency sine excitation signal, the double different frequency sine excitation signal and multi-frequency combination sine excitation signal, this paper studies the problem of parameter identi?ability, i.e., the relationship between the parameter number and the signal frequency number. For the characteristic parameter estimation problem of the sine excitation signals, this paper proposes the algebraic solution parameter estimation method, the least square parameter estimation algorithm and the algorithm based gradient. Furthermore, this paper proposes the algebraic solution parameter estimation method, the least square parameter estimation algorithm, the parameter estimation algorithm based gradient iteration, the Gauss-Newton parameter estimation algorithm and the damping iterative parameter estimation method for the parameter estimation of transfer function using the frequency responses. The results of the simulation examples and the comparison analysis show that the proposed methods can estimate the parameters of transfer functions effectively.