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.
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. Section 2 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 superscript T 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 Figure 1, 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).
Hammerstein systems with nonuniform sampling.
Assuming that the input u(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 that Pc 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 of φ(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 subsystem Pc 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.
The parameter estimates and their errors (σ2=2.002).
k
α1
α2
β1
β2
β3
β4
γ1
γ2
γ3
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
0.25000
The parameter estimates and their errors (σ2=0.502).
k
α1
α2
β1
β2
β3
β4
γ1
γ2
γ3
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
0.25000
The estimation errors δ versus k.
From Tables 1 and 2 and Figure 2, we can draw the following conclusions.
The parameter estimation errors of the AM-RLS algorithm become (generally) smaller as k increases; see the estimation errors of the last columns of Tables 1 and 2 and Figure 2.
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 Figure 2 and the estimation errors in Tables 1 and 2.
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].
Acknowledgments
This work was supported by the Fundamental Research Funds for the Central Universities (JUDCF11042 and JUDCF12031) and the PAPD of Jiangsu Higher Education Institutions and the 111 Project (B12018).
DingF.ZhouL. C.LiX. L.PanF.Gradient-based iterative identification for MISO Wiener nonlinear systems: application to a glutamate fermentation processWangD.DingF.Least squares based and gradient based iterative identification for Wiener nonlinear systemsCherifI.AbidS.FnaiechF.Nonlinear blind identification with three-dimensional tensor analysisWangD.DingF.ChuY.Data filtering based recursive least squares algorithm for Hammerstein systems using the key-term separation principleWangD. Q.DingF.Hierarchical least squares estimation algorithm for Hammerstein-Wiener systemsDingF.Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modelingChenJ.WangX.DingR.Gradient based estimation algorithm for Hammerstein systems with saturation and dead-zone nonlinearitiesDingF.LiuX. P.LiuG.Identification methods for Hammerstein nonlinear systemsLiX. L.DingR. F.ZhouL. C.Least-squares-based iterative identification algorithm for Hammerstein nonlinear systems with non-uniform samplingLiuY.XieL.DingF.An auxiliary model based on a recursive least-squares parameter estimation algorithm for non-uniformly sampled multirate systemsLiuX. G.LuJ.Least squares based iterative identification for a class of multirate systemsWuY.LiuY.ZhangW.A discrete-time chattering free sliding mode control with multirate sampling method for flight simulatorLiuY.DingF.ShiY.Least squares estimation for a class of non-uniformly sampled systems based on the hierarchical identification principleShiY.DingF.ChenT.Multirate crosstalk identification in xDSL systemsHanL.ShengJ.DingF.ShiY.Auxiliary model identification method for multirate multi-input systems based on least squaresWangD. Q.Least squares-based recursive and iterative estimation for output error moving average systems using data filteringDingF.ChenT.Combined parameter and output estimation of dual-rate systems using an auxiliary modelDingF.ChenT.Parameter estimation of dual-rate stochastic systems by using an output error methodDingF.ShiY.ChenT.Auxiliary model-based least-squares identification methods for Hammerstein output-error systemsDingF.DingJ.Least-squares parameter estimation for systems with irregularly missing dataDingF.ChenT.Identification of dual-rate systems based on finite impulse response modelsDingF.GuY.Performance analysis of the auxiliary model-based least-squares identification algorithm for one-step state-delay systemsDingF.GuY.Performance analysis of the auxiliary model-based stochastic gradient parameter estimation algorithm for state-space systems with one-step state delayDingF.QiuL.ChenT.Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systemsXieL.LiuY. J.YangH. Z.DingF.Modelling and identification for non-uniformly periodically sampled-data systemsVörösJ.Identification of Hammerstein systems with time-varying piecewise-linear characteristicsZhouL. C.LiX. L.PanF.Gradient based iterative parameter identification for Wiener nonlinear systemsZhouL.LiX.PanF.Least-squares-based iterative identification algorithm for Wiener nonlinear systemsDingF.ChenT.Hierarchical gradient-based identification of multivariable discrete-time systemsDingF.ChenT.Hierarchical least squares identification methods for multivariable systemsDingF.ChenT.Hierarchical identification of lifted statespace
models for general dual-rate systemsDingJ.DingF.LiuX. P.LiuG.Hierarchical least squares identification for linear SISO systems with dual-rate sampled-dataWangD.DingR.DongX.Iterative parameter estimation for a class of multivariable systems based on the hierarchical identification principle and the gradient searchDingF.ChenT.Performance analysis of multi-innovation gradient type identification methodsDingF.LiuP. X.LiuG.Multiinnovation least-squares identification for system modelingDingF.LiuP. X.LiuG.Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noisesDingF.Several multi-innovation identification methodsDingF.ChenH.LiM.Multi-innovation least squares identification methods based on the auxiliary model for MISO systemsHanL.DingF.Multi-innovation stochastic gradient algorithms for multi-input multi-output systemsWangD.DingF.Performance analysis of the auxiliary models based multi-innovation stochastic gradient estimation algorithm for output error systemsLiuY.YuL.DingF.Multi-innovation extended stochastic gradient algorithm and its performance analysisLiuY.XiaoY.ZhaoX.Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary modelHanL.DingF.Identification for multirate multi-input systems using the multi-innovation identification theoryDingF.LiuG.LiuX. P.Partially coupled stochastic gradient identification methods for non-uniformly sampled systemsDingF.Coupled-least-squares identification for multivariable systemsDingF.ChenT.Performance bounds of forgetting factor
least-squares algorithms for time-varying systems with finite
meaurement dataLiuY.ShengJ.DingR.Convergence of stochastic gradient estimation algorithm for multivariable ARX-like systemsDingF.LiuG.LiuX. P.Parameter estimation with scarce measurementsDingF.ChenT.QiuL.Bias compensation based recursive
least-squares identification algorithm for MISO systemsDingF.YangH.LiuF.Performance analysis of stochastic gradient algorithms under weak conditionsDingF.LiuP. X.YangH.Parameter identification and intersample output estimation for dual-rate systemsWangW.DingF.DaiJ.Maximum likelihood least squares identification for systems with autoregressive moving average noiseDingF.LiuX. G.ChuJ.Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principleDingF.LiuY.BaoB.Gradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systemsDingF.LiuP. X.LiuG.Gradient based and least-squares based iterative identification methods for OE and OEMA systemsDingF.Decomposition based fast least squares algorithm for output error systemsDingF.Two-stage least squares based iterative estimation algorithm for CARARMA system modelingDingF.DuanH. H.Two-stage parameter estimation algorithms for Box-Jenkins systems