Separate block-based parameter estimation method for Hammerstein systems

Different from the output–input representation-based identification methods of two-block Hammerstein systems, this paper concerns a separate block-based parameter estimation method for each block of a two-block Hammerstein CARMA system, without combining the parameters of two parts together. The idea is to consider each block as a subsystem and to estimate the parameters of the nonlinear block and the linear block separately (interactively), by using two least-squares algorithms in one recursive step. The internal variable between the two blocks (the output of the nonlinear block, and also the input of the linear block) is replaced by different estimates: when estimating the parameters of the nonlinear part, the internal variable between the two blocks is computed by the linear function; when estimating the parameters of the linear part, the internal variable is computed by the nonlinear function. The proposed parameter estimation method possesses property of the higher computational efficiency compared with the previous over-parametrization method in which many redundant parameters need to be computed. The simulation results show the effectiveness of the proposed algorithm.

Hammerstein systems consist of a nonlinear static block followed by a linear dynamic block, see figure 1.
Previous identification methods are based on an output-input representation of Hammerstein systems.
The investigated method in this paper is based on two separate block representations of Hammerstein systems.
The model of the nonlinear block, and the model of the linear block, We perform two least-squares algorithms on these two models, respectively. When identifying Model 1, the internal variable x(t) is replaced with its estimate, computed from the linear block (Model 2); when identifying Model 2, the internal variable x(t) is replaced with its estimate, computed from the nonlinear block (Model 1). The rest of the paper is organized as follows. Section 2 shows the separate block-based least-squares identification algorithms for two blocks of the Hammerstein system. Section 3 provides a numerical example for the proposed algorithms. Finally, the concluding remarks are provided in §4.

The separate block-based least-squares identification method
Let us introduce some notation. 'A =: B' stands for 'A is defined as B'; the symbol I (I n ) stands for an identity matrix of appropriate size (n × n); z represents a unit forward shift operator: z −1 x(t) = x(t − 1); z(t) stands for the estimate of z at time t; the superscript T denotes the matrix/vector transpose; tr [M] represents the trace of a square matrix M.
The input nonlinear and output linear functions of a Hammerstein CARMA system in figure 2 are expressed as and where u(t) and y(t) are the system input and output,  Figure 2. The Hammerstein CARMA system. polynomials in the unit backward shift operator z −1 (z −1 y(t) = y(t − 1)), and defined by A(z) := 1 + a 1 z −1 + a 2 z −2 + · · · + a n a z −n a , Assume the orders n a , n b , n d are known and y(t) = 0, u(t) = 0 and v(t) = 0 for t ≤ 0.

The model of the linear block
Let y 1 (t) = y(t) − x(t), then from the y − x linear relationship in (2.2), we get Define a least-squares quadratic criterion function The internal variable x(t) and the noise term v(t) in the information vector ϕ h (t) are unknown, the solution is to replace them with their estimates, the estimate of x(t) is computed from the nonlinear part, by replacing c i with its estimateĉ i (t),

The model of the nonlinear block
From the x − u nonlinear relationship in (2.1), we get Define a least-squares quadratic criterion function x(t) as an output of the nonlinear block, can be derived from the function (2.2) of the linear block, replacing A(z), B(z) and D(z) with their estimatesÂ(t, z),B(t, z) andD(t, z), and v(t) withv(t − 1) in (2.7); then from the linear block, the estimate of x(t) can be written as,

The separate block-based least-squares method for two blocks
By using a standard least-squares method, minimizing the cost functions J 1 (θ) with respect to θ and replacing y 1 (i) and ϕ h (i) with their estimatesŷ 1 (i) andφ h (i) in J 1 (θ), we can obtain the separate blockbased least-squares algorithm for the linear block of the Hammerstein CARMA system as follows: and v(t) Figure 3. The flowchart of computing the estimatesθ(t) andĉ(t).
The flowchart of computing the parameter estimatesθ(t) andĉ(t) using the separate block-based leastsquares algorithm in (2.9)-(2.22) is shown in figure 3.

Results
Consider the following Hammerstein CARMA system x(t) = c 1 u(t) + c 2 u 2 (t) + c 3 u 3 (t) = 0.65u(t) + 0.90u 2 (t) + 1.25u 3 (t), In simulation, the input {u(t)} is taken as an uncorrelated persistently excited signal vector sequence with zero mean and unit variance, and {v(t)} is taken as a white noise sequence with zero mean and variances σ 2 = 1.00 2 and σ 2 = 3.00 2 . Applying the proposed separate block-based least-squares algorithm to estimate the parameters of this system, the parameter estimates and their errors are shown in table 1. The estimation error of the parameters is From table 1 we can get: (1) the parameter estimation errors become (generally) smaller and smaller with the recursion t increasing; (2) the parameter estimates converge to their true values as the noise variance becomes small.

Conclusion
This paper concerns a separate block-based parameter identification method for each block of the Hammerstein CARMA system, without forming the whole output-input representation of the system. The idea is to consider each block separately as a subsystem in the Hammerstein system and to estimate the parameters of the nonlinear block and the parameters of the linear block separately (interactively), by using two least-squares algorithm in one recursive step. The internal variable between the two blocks is replaced by different estimates: for estimating the parameters of the nonlinear part, the internal variable is computed by the linear function; for estimating the parameters of the linear part, the internal variable is computed by the nonlinear function.
The proposed simple parameter estimation method possesses property of the higher computational efficiency compared with the previous over-parametrization method in which many redundant parameters need to be computed, and can be extended to systems with more blocks [38] and networked dynamic systems [39,40], etc.