# Recursive Least-Squares Estimation for Hammerstein Nonlinear Systems with Nonuniform Sampling

Authors: Xiangli Li; Lincheng Zhou; Ruifeng Ding; Jie Sheng
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper focuses on the identification problem of Hammerstein nonlinear systems with nonuniform sampling. Using the key-term separation principle, we present a discrete identification model with nonuniform sampling input and output data based on the frame period. To estimate parameters of the presented model, an auxiliary model-based recursive least-squares algorithm is derived by replacing the unmeasurable variables in the information vector with their corresponding recursive estimates. The simulation results show the effectiveness of the proposed algorithm.


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## Body

## 1. Introduction

In actual industrial processes, there exist widely nonlinear systems which are described by block-oriented nonlinear systems [1–3]. Block-oriented nonlinear models are in general divided into Hammerstein systems and Wiener systems [4]. A Hammerstein system, which consists of a static nonlinear subsystem followed by a linear dynamic subsystem, can represent some nonlinear systems [5]. Many publications have been reported for the identification of the Hammerstein systems [6, 7]. For example, Chen et al. studied identification problems for the Hammerstein systems with saturation and dead-zone nonlinearities by choosing an appropriate switching function [8]; Ding et al. presented the projection, the stochastic gradient, and the Newton recursive and the Newton iterative identification algorithms for the Hammerstein nonlinear systems, and then they analyzed and compared the performances of these approaches by numerical examples [9]. Li et al. derived a least-squares based iterative algorithm for the Hammerstein output error systems with nonuniform sampling by using the overparameterization model [10].The different input-output updating period (or called multirate sampling) is inevitable in discrete-time systems [11–13]. The identification of multirate sampled systems have attracted much attention of many researchers. Recently, Liu et al. proposed a novel hierarchical least-squares algorithm for a class of nonuniformly sampled systems based on the hierarchical identification principle [14]. Shi et al. presented a crosstalk identification algorithm for multirate xDSL FIR systems [15]. Han et al. gave state-space models for multirate multi-input sampled-data systems and derived an auxiliary model-based recursive least-squares algorithm for identifying the parameters of multirate systems [16].The recursive least-squares algorithm is a class of basic parameter estimation approaches which are suitable for online applications. In this literature, Wang adopted a filtering auxiliary model-based recursive least-squares identification algorithm for output error moving average systems [17]. Differing from the work in [14, 16], this paper discusses the parameter estimation problem for nonuniformly sampled Hammerstein nonlinear systems. The basic idea is, to combine the auxiliary model identification idea [18–24] and the key-term separation principle to derive the auxiliary model-based recursive least-squares algorithm for the Hammerstein nonlinear systems with nonuniform sampling.The rest of this paper is organized as follows. Section2 establishes the identification model of the Hammerstein nonlinear systems with nonuniform sampling. Section 3 derives a recursive least-squares parameter estimation algorithm based on the auxiliary model identification idea. Section 4 provides an example to illustrate the effectiveness of the proposed algorithm. The conclusions of the paper are summarized in Section 5.


## 2. The Identification Model

Let us introduce some notations. The superscriptT denotes the matrix transpose; I stands for an identity matrix of appropriate sizes; 1n represents an n-dimensional column vector whose elements are 1; “X:=A” stands for “A is defined as X”; and z-1 is a unit backward shift operator; that is, z-1x(t)=x(t-1).Consider a Hammerstein nonlinear system with nonuniform sampling shown in Figure1, where Hτ is a nonuniform zero-order hold with irregularly updating intervals {τ1,τ2,…,τr}, dealing with a discrete-time signal u(kT+ti) and producing the input u(t) of the nonlinear subsystem f(·); u-(t) is the output of the nonlinear subsystem; Pc is a continuous-time process; y(t) is the true output of Pc but is unmeasurable; ST is a sampler that produces a discrete-time signal y(kT) with period T=τ1+τ2+⋯+τr; and y1(kT) is the system output but is corrupted by the additive noise v(kT).Figure 1
Hammerstein systems with nonuniform sampling.Assuming that the inputu(t) has the updating intervals {τ1,τ2,…,τr}, we have [11, 25]
(1)u(t)={u(kT),kT⩽t<kT+t1,u(kT+t1),kT+t1⩽t<kT+t2,⋮u(kT+tr-1),kT+tr-1⩽t<(k+1)T,
where T:=τ1+τ2+⋯+τr is the frame period. The nonlinear subsystem f(·) in the Hammerstein nonlinear system is a polynomial of a known order:
(2)u-(t)=f(u(t))=γ1u(t)+γ2u2(t)+⋯+γnγunγ(t),
where nγ is the polynomial order.Suppose thatPc has the following state-space representation:
(3)x˙(t)=Acx(t)+Bcu-(t),y(t)=Cx(t)+Du-(t),
where x(t)∈ℝn is the state vector, u-(t) and y(t) are the input and output of the continuous-time process, respectively, and Ac, Bc, C, and D are matrices of appropriate sizes. Referring to [25] and discretizing (3) with the frame period T, we have
(4)x(kT+T)=eAcTx(kT)+∑i=1r‍eAc(T-ti)∫0τi‍eActdtBcu-(kT+ti-1)=Ax(kT)+∑i=1r‍Biu-(kT+ti-1),
where
(5)A≔eAcT∈ℝn×n,Bi≔eAc(T-ti)∫0τi‍eActdtBc∈ℝn×n.
The output y(t) at the sampling instant t=kT can be expressed as
(6)y(kT)=Cx(kT)+Du-(kT).
Hence, the system output y1(kT) is written as
(7)y1(kT)=y(kT)+v(kT).
Referring to [26] and from (4) and (6), we have
(8)y(kT)=∑i=1r‍z-nCadj[zI-A]Biz-ndet[zI-A]gi×u-(kT+ti-1)+Du-(kT)=1α(z)∑i=1r‍βi(z)u-(kT+ti-1),
where
(9)α(z)≔z-ndet[zI-A]=1+α1z-1+α2z-2+⋯+αnz-n,β1(z)≔z-nCadj[zI-A]B1+Dα(z)=β10+β11z-1+β12z-2+⋯+β1nz-n,βi(z)≔z-nCadj[zI-A]Bi=βi1z-1+βi2z-2+⋯+βinz-n,=+βi21+βi2z-2i=1,2,3,…,r.
Equation (8) can be transformed into
(10)y(kT)=[1-α(z)]y(kT)+∑i=1r‍βi(z)u-(kT+ti-1).
Substituting (10) into (7), the system output y1(kT) can be expressed as
(11)y1(kT)=[1-α(z)]y(kT)+∑i=1r‍βi(z)u-(kT+ti-1)+v(kT).
Equation (11) can be rewritten equivalently as
(12)y1(kT)=∑j=1n‍αjy(kT-jT)+β10u-(kT)+∑j=1n‍∑i=1r‍βiju-(kT+ti-1-jT)+v(kT).
Here, substituting (2) into (12) results in a complex expression containing the products of parameters. To solve this problem, we use the key-term separation principle presented in [27], and let β10=1. Then, the identification model of the proposed system is as follows:
(13)y1(kT)=∑j=1n‍αjy(kT-jT)+∑i=1nγ‍γiui(kT)+∑j=1n‍∑i=1r‍βiju-(kT+ti-1-jT)+v(kT).
The objective of this paper is to develop a recursive least-squares algorithm for estimating the parameters of the nonuniformly sampled Hammerstein systems by using the auxiliary model identification idea in [11].


## 3. The Recursive Least-Squares Algorithm

In this section, we derive the recursive least-squares estimation algorithm for the Hammerstein nonlinear systems with nonuniform sampling, referring to the method in [1].Define the information vectors and the parameter vectors as(3)φy(kT)∶=[-y(kT-T),-y(kT-2T),…,-y(kT-nT)]∈ℝn,φu(kT)≔[(kT-nT+tr-1)u-(kT-T),u-(kT-2T),…,u-(kT-nT),u-(kT-T+t1),u-(kT-2T+t1),…,u-(kT-nT+t1),…,u-(kT-T+tr-1),u-(kT-2T+tr-1),…,u-(kT-nT+tr-1)]T∈ℝrn,φγ(kT):=[u(kT),u2(kT),…,unγ(kT)]T∈ℝnγ,φ(kT):=[φy(kT)φu(kT)φγ(kT)]∈ℝn0,n0:=(r+1)n+nγ,θy:=[α1,α2,…,αn]T∈ℝn,θu≔[β11,β12,…,β1n,β21,β22,…,β2n,…,Dβr1,βr2,…,βrn]T∈ℝrn,θγ:=[γ1,γ2,…,γnγ]T∈ℝnγ,θ:=[θyθuθγ]∈ℝn0.
Equation (13) can be written in a regressive form as
(15)y1(kT)=φT(kT)θ+v(kT).
Define a quadratic criterion function as
(16)J(θ):=∑i=1k‍[y1(iT)-φT(iT)θ]2.
Let θ^(kT) be the estimate of θ at time kT. Minimizing J(θ) gives the following recursive least-squares algorithm:
(17)θ^(kT)=θ^(kT-T)+P(kT)φ(kT)×[y1(kT)-φT(kT)θ^(kT-T)],(18)P(kT)=P(kT-T)-P(kT-T)φ(kT)φT(kT)P(kT-q)1+φT(kT)P(kT-T)φ(kT).
Note that the information vector φ(kT) in (17) contains unknown inner variables y(kT-jT) and u-(kT+ti-1-jT); the parameter vector θ cannot be estimated by the standard least-squares method. The solution is based on the auxiliary model identification idea [11]: to replace the unmeasurable term y(kT-jT) in φ(kT) with its estimate
(19)y^(kT)=φ^T(kT)θ^(kT).
Replacing γi in (2) with its estimate γ^i(kT), we can obtain the estimate u^(kT+ti-1) of u-(kT+ti-1) as follows:
(20)u^(kT+ti-1)=γ^1(kT)u(kT+ti-1)+γ^2(kT)u2(kT+ti-1)+⋯+γ^nγ(kT)unγ(kT+ti-1),=uu(kT+ti-1)+1i=1,2,…,r.Define the estimate ofφ(kT) as
(21)φ^(kT):=[φ^y(kT)φ^u(kT)φ^γ(kT)]∈ℝn0,φ^y(kT):=[-y^(kT-T),-y^(kT-2T),…,-y^(kT-nT)]∈ℝn,φ^u(kT)≔[tr-1u^(kT-T),u^(kT-2T),…,u^(kT-nT),u^(kT-T+t1),u^(kT-2T+t1),…,u^(kT-nT+t1),…,u^(kT-T+tr-1),u^(kT-2T+tr-1),…,u^(kT-nT+tr-1)]T∈ℝrn.
Using φ^(kT) in place of φ(kT) in (17) and (18), we have
(22)θ^(kT)=θ^(kT-T)+P(kT)φ^(kT)×[y1(kT)-φ^T(kT)θ^(kT-q)],P(kT)=P(kT-T)-P(kT-T)φ^(kT)φ^T(kT)P(kT-T)1+φ^T(kT)P(kT-T)φ^(kT).
Equations (19) to (22) form the AM-RLS algorithm for the not uniformly sampled Hammerstein nonlinear systems, which can be summarized as
(23)θ^(kT)=θ^(kT-T)+P(kT)φ^(kT)[y1(kT)-φ^T(kT)θ^(kT-T)],P(kT)=P(kT-T)-P(kT-T)φ^(kT)φ^T(kT)P(kT-T)1+φ^T(kT)P(kT-T)φ^(kT),φ^(kT)=[φ^y(kT)φ^u(kT)φ^γ(kT)],φ^y(kT)=[-y^(kT-T),-y^(kT-2T),…,-y^(kT-nT)]T,φ^u(kT)=[tr-1u^(kT-T),u^(kT-2T),…,u^(kT-nT),u^(kT-T+t1),u^(kT-2T+t1),…,u^(kT-nT+t1),…,u^(kT-T+tr-1),u^(kT-2T+tr-1),…,u^(kT-nT+tr-1)]T,φγ(kT)=[u(kT),u2(kT),…,unγ(kT)]T,u^(kT+ti-1)=γ^1(kT)u(kT+ti-1)+γ^2(kT)u2(kT+ti-1)+⋯+γ^nγ(kT)unγ(kT+ti-1),y^(kT)=φ^T(kT)θ^(kT).
To initialize the algorithm, we take θ^(0) to be a small real vector; for example, θ^(0)=1n0/p0 and P(0)=p0I with p0 normally a large positive number (e.g., p0=106).


## 4. Example

An example is given to demonstrate the feasibility of the proposed algorithm. Assume that the dynamical linear subsystemPc has the following state-space representation:
(24)x˙(t)=[-0.3-0.210]x(t)+[10]u-(t),y(t)=[-0.4,0.2]x(t)+u-(t),
and the static nonlinear subsystem is denoted by
(25)u-(t)=f(u(t))=γ1u(t)+γ2u2(t)+γ3u3(t)=u(t)+0.5u2(t)+0.25u3(t).
Let r=2, τ1=1s, and τ2=2s; that is, t1=τ1=1s and t2=t1+τ2=T=3s. Discretizing Pc with the frame period T, we obtain
(26)x(kT+T)=[0.1274-0.28351.41770.5527]x(kT)+[0.29811.11961.29780.9388][u(kT)u(kT+t1)],y(kT)=[0.50,0.21]x(kT).
Thus, the corresponding input-output expression is given by
(27)y1(kT)=(-0.68000z-1+0.47241z-2)×y(kT)+u-(kT)+(-0.52674z-1+0.73948z-2)u-(kT)+(-0.25070z-1+0.66221z-2)×u-(kT+t1)+v(kT).In simulation, the inputs{u(kT)} and {u(kT+t1)} are taken as persistent excitation signal sequences with zero mean and unit variance; {v(kT)} is a white noise with zero mean and variance σ2. Applying the proposed algorithm to estimate the parameters of this system, the estimates of θ and their errors with different noise variances are shown in Tables 1 and 2, and the parameter estimation errors δ:=∥θ^(kT)-θ∥/∥θ∥ versus k are shown in Figure 2. When σ2=2.002 and σ2=0.502, the corresponding signal-to-noise ratios (the square root of the ratio of output and noise variances) are SNR=1.0342 and SNR=4.1367, respectively.Table 1
The parameter estimates and their errors (σ2=2.002).
k
α
1
α
2
β
1
β
2
β
3
β
4
γ
1
γ
2
γ
3
δ (%)
100
−0.03122
−0.05592
0.02344
0.39192
−0.08169
0.14407
1.52981
0.68962
0.02254
73.36523
500
−0.53614
0.31821
−0.43136
0.63245
−0.30871
0.53689
1.30985
0.69341
0.10358
26.81256
1000
−0.55071
0.44816
−0.41426
0.78764
−0.24332
0.62087
1.12565
0.56690
0.22360
12.86475
2000
−0.57852
0.42159
−0.43335
0.71714
−0.22310
0.63026
1.04567
0.49863
0.23534
8.87717
3000
−0.60057
0.39569
−0.44560
0.68205
−0.23745
0.60417
1.09320
0.49337
0.20626
10.44562
4000
−0.61945
0.41573
−0.44904
0.68292
−0.24136
0.60799
1.03387
0.50662
0.24914
7.82199
5000
−0.66253
0.42857
−0.49385
0.68309
−0.26702
0.62299
1.00550
0.50873
0.25535
5.03493
6000
−0.67482
0.46019
−0.52100
0.71556
−0.26552
0.64442
0.99935
0.53599
0.26184
2.87761
True values
−0.68000
0.47241
−0.52674
0.73948
−0.25070
0.66221
1.00000
0.50000
0.25000Table 2
The parameter estimates and their errors (σ2=0.502).
k
α
1
α
2
β
1
β
2
β
3
β
4
γ
1
γ
2
γ
3
δ (%)
100
−0.69605
0.48092
−0.59484
0.83269
−0.30638
0.59981
1.17677
0.54701
0.18587
13.23523
500
−0.68503
0.51969
−0.54542
0.81494
−0.30331
0.68632
1.05813
0.54706
0.22836
7.30117
1000
−0.65503
0.49678
−0.50768
0.79029
−0.26915
0.68531
1.01708
0.51876
0.25274
4.13831
2000
−0.64233
0.46233
−0.49229
0.74295
−0.25140
0.66911
1.00467
0.50079
0.25083
2.89926
3000
−0.64733
0.45069
−0.49445
0.72782
−0.25286
0.65764
1.01820
0.49893
0.24231
3.07166
4000
−0.65262
0.45294
−0.49564
0.72445
−0.25257
0.65608
1.00431
0.50179
0.25233
2.68214
5000
−0.66319
0.45510
−0.50654
0.72291
−0.25789
0.65842
0.99803
0.50215
0.25341
2.01651
6000
−0.66697
0.46155
−0.51409
0.72921
−0.25706
0.66243
0.99700
0.50881
0.25474
1.45453
True values
−0.68000
0.47241
−0.52674
0.73948
−0.25070
0.66221
1.00000
0.50000
0.25000Figure 2
The estimation errorsδ versus k.From Tables1 and 2 and Figure 2, we can draw the following conclusions.(i)
The parameter estimation errors of the AM-RLS algorithm become (generally) smaller ask increases; see the estimation errors of the last columns of Tables 1 and 2 and Figure 2.(ii)
Under different noise levels, the parameter estimates can converge to the true value, and a lower noise level results in a faster convergence rate of the parameter estimates to the true parameters; see the error curves in Figure2 and the estimation errors in Tables 1 and 2.(iii)
The proposed recursive algorithm differs from the iterative identification approach in [10] and can be used as an online identification.


## 5. Conclusions

In this paper, we have established the identification model of the Hammerstein nonlinear systems with nonuniform sampling by using the key-term separation principle. To estimate the parameters of the proposed model, the recursive least-squares parameter estimation algorithm is derived based on the auxiliary model identification idea. The proposed algorithm can simultaneously estimate the parameters of the linear and nonlinear subsystems of the Hammerstein nonlinear systems with nonuniform sampling. The simulation results show that the parameters of the Hammerstein systems with nonuniform sampling can be estimated effectively by the proposed algorithm. Although the algorithm is presented for a class of nonuniformly sampled Hammerstein nonlinear systems, the basic idea can also be extended to identify other linear and nonlinear systems [28, 29] and can combine the hierarchical identification methods [30–34], the multi-innovation identification methods [35–44], and other identification methods [45–58] to present new identification algorithms for linear or nonlinear and scalar or multivariable systems [59].



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