Instrument variable method based on nonlinear transformed instruments for Hammerstein system identification

Abstract

An extended instrumental variable (EIV) method is considered for the stochastic Hammerstein system (ARMAX and general model structure). The EIV method provides consistent parameter estimates by eliminating noise-induced bias in the least square (LS) method. To estimate the parameters, the Hammerstein model is formulated using the bilinear parameterization. The bilinear model is identified by introducing the nonlinear instrumental variables obtained from transformed delayed outputs using nonlinear mapping and polynomial basis of delayed inputs. These instruments are analyzed in full generality by computing the bounds on expected relationship between instruments and noise for the general noise disturbance structure. Then, a specific case with hyperbolic tangent (tanh) transformation is considered. Comparative performance analysis of the proposed IV method with the existing IV method, the data filtering-based LS methods, and the extended LS method shows improvement in the statistical properties of parameters estimates.

Regularized estimation of Hammerstein systems using a decomposition-based iterative instrumental variable method

“…To formulate the IV vector, collect the obtained auxiliary outputs in (17) and rearrange in the vector form with delay to give…”

Identification of nonlinear block-oriented systems with backlash and saturation

“…In the case of saturation, the slopes m R2 and m L2 are equal to zero. The three-block cascade model description can be created by consecutive substitutions of (4) into (8) and then the result of this substitution into (11). However, this will result in a very complex equation with crossmultiplications of parameters.…”

“…However, this will result in a very complex equation with crossmultiplications of parameters. Therefore, the so-called key-term separation principle [10,25] will be applied to transform the system described by (4), (8) and 11into a pseudo-linear representation without the products of parameters. Moreover, the parameterization of the cascade connection of three blocks is not unique, as many combinations of block parameters can be found.…”

“…To get a unique parameterization, one parameter has to be fixed in at least two blocks. For application of the key term separation principle, it is appropriate to choose a 1 = 1 in (8), and m R1 = 1 in (11).…”

“…The identification techniques mainly distinguish themselves in the way the static nonlinearity is represented and in the form of optimization problem that is finally obtained. Known approaches for different types of nonlinearities and estimation methods can be found eg in [1][2][3][4][5][6][7][8][9][10][11][12][13]. The Wiener model has a linear dynamic block followed by a nonlinear static block and many techniques have been proposed to solve the identification problems typically encountered in using this model, see, eg [14][15][16][17][18][19][20][21][22][23][24][25][26].…”