Filtering Based Recursive Least Squares Algorithm for Multi-Input Multioutput Hammerstein Models

This paper considers the parameter estimation problem for Hammerstein multi-input multioutput finite impulse response (FIRMA) systems. Filtered by the noise transfer function, the FIR-MA model is transformed into a controlled autoregressive model. The key-term variable separation principle is used to derive a data filtering based recursive least squares algorithm.The numerical examples confirm that the proposed algorithm can estimate parameters more accurately and has a higher computational efficiency compared with the recursive least squares algorithm.

The least squares algorithm is a fundamental method [20][21][22] and many methods such as the iterative algorithm [23,24] and the gradient algorithm [25] are widely used in the parameter estimation.In the field of Hammerstein system identification, several methods have been developed [26].For example, a least squares based iterative algorithm and an auxiliary model based recursive least squares algorithm have been presented, respectively, for Hammerstein nonlinear ARMAX systems and Hammerstein output error systems [27,28]; a Newton recursive algorithm and a Newton iterative algorithm for Hammerstein controlled autoregressive systems are presented in [29].
Consider a multi-input multioutput (MIMO) Hammerstein finite impulse response (FIR) system depicted by y () = B () u () +  () k () , where u() := [ 1 (),  2 (), . . .,   ()]  ∈ R  is the nonlinear system input vector with zero mean and unit variances, y() ∈ R  is the measurement of x() := B()u() but is corrupted by w() := ()k(), k() ∈ R  is the white noise vector with zero mean, and B() and () are polynomials in the unit backward shift operator  −1 [ −1 y() = y( − 1)]: ( It is obvious that the relation between sizes  and  would influence the model identification of this multi-input multioutput Hammerstein system.For example, the dimension of the output vector is not less than that of the input vector if  ⩾ ; otherwise, when  < , the output size is smaller compared with that of the input vector.In this paper, we discuss the identification problem with  ⩾ .

Mathematical Problems in Engineering
The nonlinear block in the Hammerstein model is a linear combination of the known basis f := ( 1 ,  2 , . . .,   ): where the superscript  denotes the matrix transpose.The function   (  ()) in ( 3) is a nonlinear function of a known basis ( 1 ,  2 , . . .,    ): where the coefficients c := ( 1 ,  2 , . . .,    ) are unknown.Substituting (4) into (3) yields Assume that u() = 0, y() = 0, and k() = 0 for  ⩽ 0, and the orders   ,   , and   are known but can be obtained by trial and error.In general, the orders of the Hammerstein model should be large when the nonlinear system is used in prediction; otherwise, the orders should be small if the system is applied for control.The objective of this paper is to estimate the unknown parameter matrices: B  ,   ,   from the available input-output data {u(), y()} of the multivariable Hammerstein finite impulse response moving average (FIR-MA) models [30].
Recently, the filtering idea has received much attention [31][32][33].Xiao and Yue studied input nonlinear dynamical adjustment models and presented a recursive generalized least squares algorithm and a filtering based least squares algorithm by replacing the unknown terms in the information vectors with their estimates [34].The overparameterization method in [34] leads to a redundant estimated product of the nonlinear systems and requires extra computation.Differing from the work in [30,34,35], this paper discusses the estimation problem of the MIMO Hammerstein systems using the data filtering idea and transfers the FIR-MA system to controlled autoregressive model by means of the keyterm variable separate principle in [36][37][38].The proposed algorithm used in this paper can extend to study parameter estimation problems of dual-rate/multirate sampled systems [39][40][41][42] and other linear or nonlinear systems [43][44][45][46].
Briefly, the rest of this paper is recognized as follows.Section 2 discusses a recursive least squares algorithm for the Hammerstein systems.Section 3 presents a filtering based recursive least squares algorithm by transferring an FIR-MA model to a controlled autoregressive model.Section 4 provides an illustrative example.Finally, some concluding remarks are offered in Section 5.

The MRLS Algorithm
For comparison, the MRLS algorithm is listed in Section 2.
Here we introduce some notations. represents the current time in this paper and " =: " or " := " means that " is defined as ".The symbol I × represents an identity matrix of size  followed by a null matrix of the last  −  rows when  ⩾  and vice versa.The norm of a matrix (or a column vector) X is defined by ‖X‖ 2 := tr[XX  ]; ⊗ denotes the Kronecker product or direct product is supposed to be the vector formed by the column of the matrix X: From (1)-( 5), the intermediate variables x() and w() and output of the system y() can be expressed as Note that the subscripts (Roman)  and  denote the first letters of "system" and "noise" for distinguishing the types of the unknown parameter vectors or matrices, respectively.Define the parameter matrix  1 , the parameter vectors  2 ,   , and the information vectors  1 (),  2 (), and   () as Then, we have Distinguished from the hierarchical identification methods, we reparameterize the model in ( 7) by using the Kronecker product to get a parameter matrix  and by gathering the input information vectors  1 () and  2 () and output information matrix   () into one information matrix Ψ() to obtain an information matrix Ψ() as follows: Thus, we obtain Equation ( 11) is the identification model of the multivariable Hammerstein FIR-MA system.Defining and minimizing the cost function and using the least squares search principle, we obtain the following recursive least squares algorithm [35] to obtain parameter estimates θ(): Since the information matrix Ψ() in ( 13 Define the parameter estimation matrices By replacing the parameters   ( = 1, 2, . . .,   ) in ( 4) with ĉ (), the output of the proposed auxiliary model û() is given by From (11), we obtain k() = y() − Ψ().Replacing Ψ() and  with Ψ() and θ( − 1), the residual k() can be written as k() = y() − Ψ() θ( − 1).
To summarize, we conclude the following recursive least squares algorithm for multivariable Hammerstein FIR-MA models (the MRLS algorithm for short): (

The F-MRLS Algorithm
The convergence rate of the MRLS algorithm in Section 2 is slow because the noise information intermediate variables w() contain unmeasurable time-delay noise k(−).The solution here is to present a filtering based recursive least squares algorithm (the F-MRLS algorithm) for the multivariable Hammerstein models by filtering the rational function () and transferring the FIR-MA model in (1) into a controlled autoregressive (CAR) model.Multiplying both sides of (1) by  −1 () yields or where Thus, ( 21) can be rewritten as Define the filtered information matrices: Since the polynomial () is unknown and to be estimated, it is impossible to use u  () to construct  1 () in (25).Here, we adopt the principle of the MRLS algorithm in Section 2 and replace the unmeasurable variables and vectors with their estimates to derive the following algorithm.By using the parameter estimates θ1 () and θ (), the estimates of polynomials B() and () at time  can be constructed as ( The estimate of U  () can be computed by ( Define the estimate of   () by and construct the estimate of Ψ  () with φ1 () and φ2 () as follows: The filtered model in ( 21) can be rewritten in a matrix form: or Based on the MRLS search principle, we can obtain the estimate of   by the following algorithm: The estimate û() can be computed by Filter û() by 1/ N(, ) to obtain the estimate û (): Replacing k(), y  (), Ψ  (), and   in (37) with their estimates k(), ŷ (), Ψ (), and θ () at time , the noise vector can be computed by v () = ŷ () − Ψ () θ () .
To illustrate the advantages of the proposed algorithm, the numbers of multiplications and additions for each step of the F-MRLS algorithm and the MRLS algorithm are listed in Table 5.From Tables 1-5, Figures 1-4, we can draw the following conclusions.
(i) The parameter estimation errors are getting smaller with  increasing, which proves that the proposed algorithms are effective.(ii) The F-MRLS algorithm is more accurate than the MRLS algorithm, which means the proposed F-MRLS algorithm has a better performance compared with the MRLS algorithm.
(iii) The parameter estimates given by the F-MRLS algorithm have faster convergence than those given by the MRLS algorithm.

Conclusions
This paper presents a data filtering based recursive least squares algorithm for MIMO nonlinear FIR-MA systems.
The simulation results show that the proposed data filtering based recursive least squares algorithm is more accurate and reduces computational burden compared with the recursive least squares algorithm.
) contains the unknown intermediate variables u( − ) and the unmeasurable terms k( − ), the recursive algorithm in (13)-(15) cannot compute the parameter estimate θ().The solution here is replacing the unknown intermediate variables u(−) and the unmeasurable terms k(−) in Ψ() with the variable estimates (or the outputs of the auxiliary model) û( − ) and the estimates k( − ) based on the auxiliary model identification idea.The replaced information matrices are defined as

Table 5 :
Comparison of the computational efficiency of the F-MRLS and MRLS algorithms.