# Multi-Innovation Stochastic Gradient Identification Algorithm for Hammerstein Controlled Autoregressive Autoregressive Systems Based on the Key Term Separation Principle and on the Model Decomposition

Authors: Huiyi Hu; Xiao Yongsong; Rui Ding
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

An input nonlinear system is decomposed into two subsystems, one including the parameters of the system model and the other including the parameters of the noise model, and a multi-innovation stochastic gradient algorithm is presented for Hammerstein controlled autoregressive autoregressive (H-CARAR) systems based on the key term separation principle and on the model decomposition, in order to improve the convergence
speed of the stochastic gradient algorithm. The key term separation principle can simplify the identification model of the input nonlinear system, and the decomposition technique can enhance computational efficiencies of identification algorithms. The simulation results show that the proposed algorithm is effective for estimating the parameters of IN-CARAR systems.


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## Body

## 1. Introduction

There exist many nonlinear systems in process control [1–3]. A nonlinear system can be modeled by input nonlinear systems [4] and output nonlinear systems [5], input-output nonlinear systems [6], feedback nonlinear systems [7], and so on. Input nonlinear systems, which are called Hammerstein systems [8], include input nonlinear equation error type systems and input nonlinear output error type systems. Recently, many identification algorithms have been developed for input nonlinear systems, such as the iterative methods [9–11], the separable least squares methods [12, 13], the blind methods [14], the subspace methods [15], and the overparameterization methods [16, 17]. Some methods require paying much extra computation.The stochastic gradient (SG) algorithm is widely applied to parameter estimation. For example, Wang and Ding presented an extended SG identification algorithm for Hammerstein-Wiener ARMAX systems [18], but it is well known that the SG algorithm has slower convergence rates. In order to improve the convergence rate of the SG algorithm, Xiao et al. presented a multi-innovation stochastic gradient parameter estimation algorithm for input nonlinear controlled autoregressive (IN-CAR) models using the over-parameterization method [19]; Chen et al. proposed a modified stochastic gradient algorithm by introducing a convergence index in order to improve the convergence rate of the parameter estimation [20]; Han and Ding developed a multi-innovation stochastic gradient algorithm for multi-input single-output systems [21]; Liu et al. studied the performance of the stochastic gradient algorithm for multivariable systems [22].The decomposition identification techniques include matrix decomposition and model decomposition. Hu and Ding presented a least squares based iterative identification algorithm for controlled moving average systems using the matrix decomposition [23]; Ding derived an iterative least squares algorithm to estimate the parameters of output error systems, and the matrix decomposition can enhance computational efficiencies [24]. Ding also divided a Hammerstein nonlinear system into two subsystems based on the model decomposition and presented a hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear systems [25].This paper discusses identification problems of input nonlinear controlled autoregressive autoregressive (IN-CARAR) systems or Hammerstein controlled autoregressive autoregressive (H-CARAR) systems, which is one kind of input nonlinear equation error type systems. The basic idea is using the key term separation principle [26] and the decomposition technique [24] to derive a multi-innovation stochastic gradient identification algorithm, which is different from the work in [19, 21, 25].The rest of this paper is organized as follows. Section2 gives the identification model for the IN-CARAR systems. Section 3 introduces the SG algorithm for the IN-CARAR system. Section 4 deduces a multi-innovation SG algorithm for IN-CARAR system using the decomposition technique. Section 5 provides a numerical example to show the effectiveness of the proposed algorithm. Finally, Section 6 offers some concluding remarks.


## 2. The System Identification Model

The paper focuses on the parameter estimation of a Hammerstein nonlinear controlled autoregressive autoregressive (H-CARAR) system, that is, an input nonlinear controlled autoregressive autoregressive (IN-CARAR) system, which consists of a nonlinear block and a linear dynamic subsystem. It is worth noting that Xiao and Yue discussed a data filtering based recursive least squares algorithm for H-CARAR systems [27] and a multi-innovation stochastic gradient parameter estimation algorithm for input nonlinear controlled autoregressive (IN-CAR) systems.An IN-CARAR system shown as Figure1 is expressed as [27]
(1)A(z)y(t)=B(z)u-(t)+1C(z)v(t),(2)w(t)=1C(z)v(t),(3)u-(t)=f(u(t))γ,
where u(t) and y(t) are the system input and output, u-(t) is the output of the nonlinear part, and v(t) is an uncorrelated stochastic noise with zero mean. Here, u-(t) is expressed as [28]
(4)u-(t)=f(u(t))γ=∑i=1nγ‍γifi(u(t))=γ1f1(u(t))+γ2f2(u(t))+⋯+γnγfnγ(u(t)),
where f(u(t)):=[f1(u(t)),f2(u(t)),…,fnγ(u(t))]∈ℝ1×nγ and γ:=[γ1,γ2,…,γnγ]T∈ℝnγ is the parameters vector of the nonlinear part.Figure 1
The IN-CARAR system.In (1), A(z), B(z), and C(z) are the polynomials, of known orders na, nb, and nc, in the unit backward shift operator z-1 [z-1y(t)=y(t-1)], defined by
(5)A(z):=1+a1z-1+a2z-2+⋯+anaz-na,B(z):=1+b1z-1+b2z-2+⋯+bnbz-nb,C(z):=1+c1z-1+c2z-2+⋯+cncz-nc.
Assume y(t)=0, u(t)=0, and v(t)=0 for t⩽0. ai, bi, and ci are the parameters to be estimated from measured input–output data {u(t),y(t)}.Define the parameter vectors:(6)θ:=[θsT,θnT]T∈ℝna+nb+nγ+nc,θs:=[aT,bT,γT]T∈ℝna+nb+nγ,a:=[a1,a2,…,ana]T∈ℝna,b:=[b1,b2,…,bnb]T∈ℝnb,θn:=[c1,c2,…,cnc]T∈ℝnc,
and the information vectors:
(7)φ(t)≔[φs(t)φn(t)]∈ℝna+nb+nγ+nc,φs(t)≔[-y(t-1),-y(t-2),…,-y(t-na),u-(t-1),…,u-(t-nb),f(u(t))]T∈ℝna+nb+nγ,φn(t)≔[-w(t-1),-w(t-2),…,-w(t-nc)]T∈ℝnc.
Equation (2) can be written as
(8)w(t)=[1-C(z)]w(t)+v(t)=-∑i=1nc‍ciw(t-i)+v(t)=φnT(t)θn+v(t).
Using the key term separation principle [26], (1) can be written as
(9)y(t)=[1-A(z)]y(t)+[B(z)-1]u-(t)+u-(t)+w(t)=-∑i=1na‍aiy(t-i)+∑i=1nb‍biu-(t-i)+∑i=1nγ‍γifi(u(t))+w(t)=φsT(t)θs+w(t)(10)=φsT(t)θs+φnT(t)θn+v(t)(11)=φT(t)θ.
This is the identification model of the IN-CARAR system.


## 3. The Stochastic Gradient Algorithm

According to [1] and based on the identification model in (11), we can obtain the stochastic gradient (SG) algorithm:
(12)θ^(t)=θ^(t-1)+φ^(t)r(t)e(t),e(t)=y(t)-φ^T(t)θ^(t-1),r(t)=r(t-1)+∥φ^(t)∥2,r(0)=1,φ^(t)=[φ^s(t)φ^n(t)],φ^s(t)=[u-^(t-2)-y(t-1),-y(t-2),…,-y(t-na),u-^(t-1),u-^(t-2),…,u-^(t-nb),f(u(t))]T,φ^n(t)=[-w^(t-1),-w^(t-2),…,-w^(t-nc)]T,u-^(t)=f(u(t))γ^(t),w^(t)=y(t)-φ^sT(t)θ^s(t),f(u(t))=[f1(u(t)),f2(u(t)),…,fnr(u(t))],θ^(t)=[θ^n(t)θ^s(t)],θ^s(t)=[a^T(t),b^T(t),γ^T(t)]T,a^(t)=[a^1(t),a^2(t),…,a^na(t)]T,b^(t)=[b^1(t),b^2(t),…,b^nb(t)]T,γ^(t)=[γ^1(t),γ^2(t),…,γ^nγ(t)]T,θ^n(t)=[c^1(t),c^2(t),…,c^nc(t)]T,
where X^(t) represents the estimate of X at time t; for example, θ^(t)=[θ^n(t)θ^s(t)]∈ℝna+nb+nγ+nc is the estimate of θ=[θsθn] at time t.


## 4. The Multi-Innovation Stochastic Gradient Algorithm

This section deduces the multi-innovation stochastic gradient identification algorithm for the IN-CARAR system using the decomposition technique [1].Define two intermediate variables,(13)y1(t):=y(t)-φnT(t)θn,y2(t):=y(t)-φsT(t)θs.
From (10), we have
(14)y1(t):=φsT(t)θs+v(t),y2(t):=φnT(t)θn+v(t).
These two subsystems include the parameter vectors θs and θn, respectively. θs contains the parameters od the system model and θn contains the parameters od the noise model.Define the stacked information matrices and the stacked output vectors:(15)Y(p,t):=[y(t),y(t-1),…,y(t-p+1)]T∈ℝp,Y1(p,t):=[y1(t),y1(t-1),…,y1(t-p+1)]T∈ℝp,Y2(p,t):=[y2(t),y2(t-1),…,y2(t-p+1)]T∈ℝp,Φs(p,t):=[φs(t),φs(t-1),…,φs(t-p+1)]T∈ℝp×(na+nb+nc),Φ^n(p,t):=[φ^n(t),φ^n(t-1),…,φ^n(t-p+1)]T∈ℝp×nγ,Es(p,t):=[es(t),es(t-1),…,es(t-p+1)]T∈ℝp,En(p,t):=[en(t),en(t-1),…,en(t-p+1)]T∈ℝp.
According to the multi-innovation identification theory [29–41], we expand the scalar innovations:
(16)es(t)=y1(t)-φsT(t)θ^s(t-1),en(t)=y2(t)-φ^nT(t)θ^n(t-1),
to the innovation vectors,
(17)Es(p,t)=Y^1(p,t)-ΦsT(p,t)θ^s(t-1),En(p,t)=Y^2(p,t)-Φ^nT(p,t)θ^n(t-1).
Define two criterion functions,
(18)J1(θs):=∥Y1(p,t)-ΦsT(p,t)θs∥2,J2(θn):=∥Y2(p,t)-Φ^nT(p,t)θn∥2.
The gradients of J1 and J2 with respect to θs and θn, respectively, are
(19)grad[J1(θs)]=∂J1(θs)∂θn=-2Φs(p,t)[Y1(p,t)-ΦsT(p,t)θs],grad[J2(θn)]=∂J2(θn)∂θn=-2Φ^n(p,t)[Y2(p,t)-Φ^nT(p,t)θn].
Minimizing J1(θs) and J2(θn) using the negative gradient search, we can obtain the multi-innovation stochastic gradient algorithm (MISG) for the IN-CARAR system:
(20)θ^s(t)=θ^s(t-1)+Φ^s(p,t)r1(t)Es(p,t),Es(p,t)=Y^1(p,t)-Φ^sT(p,t)θ^s(t-1)=Y(p,t)-Φ^sT(p,t)θ^s(t-1)-Φ^nT(p,t)θ^n(t-1),r1(t)=r1(t-1)+∥Φ^s(p,t)∥2,r1(0)=1,Y(p,t)=[y(t),y(t-1),…,y(t-p+1)]T,Φ^s(p,t)=[φ^s(t),φ^s(t-1),…,φ^s(t-p+1)]T,θ^n(t)=θ^n(t-1)+Φ^n(p,t)r2(t)En(p,t),En(p,t)=Y^2(p,t)-Φ^nT(p,t)θ^n(t-1)=Y(p,t)-Φ^sT(p,t)θ^s(t-1)-Φ^nT(p,t)θ^n(t-1),r2(t)=r2(t-1)+∥Φ^n(p,t)∥2,r2(0)=1,Φ^n(p,t)=[φ^n(t),φ^n(t-1),…,φ^n(t-p+1)]T,φ^s(t)=[u-^(t-nb)-y(t-1),-y(t-2),…,-y(t-na),u-^(t-1),u-^(t-2),…,u-^(t-nb),f(u(t))]T,φ^n(t)=[-w^(t-1),-w^(t-2),…,-w^(t-nc)]T,u-^(t)=f(u(t))γ^(t),w^(t)=y(t)-φ^sT(t)θ^s(t),f(u(t))=[f1(u(t)),f2(u(t)),…,fnγ(u(t))],θ^(t)=[θ^n(t)θ^s(t)],θ^s(t)=[a^T(t),b^T(t),γ^T(t)]T,a^(t)=[a^1(t),a^2(t),…,a^na(t)]T,b^(t)=[b^1(t),b^2(t),…,b^nb(t)]T,γ^(t)=[γ^1(t),γ^2(t),…,γ^nγ(t)]T,θ^n(t)=[c^1(t),c^2(t),…,c^nc(t)]T.
The initial values can be taken to be θ^(0)=1na+nb+nγ+nc/p0, w^(i)=1/p0, i⩽0, and p0=106.


## 5. Numerical Examples

Consider the following IN-CARAR system:(21)A(z)y(t)=B(z)u-(t)+1C(z)v(t),A(z)=1+a1z-1+a2z-2=1+1.80z-1+0.80z-2,B(z)=1+b1z-1+b2z-2=1+0.50z-1+0.65z-2,C(z)=1+c1z-1+c2z-2=1+0.30z-1+0.20z-2,u-(t)=f(u(t))γ=γ1f1(u(t))+γ2f2(u(t))+γ3f3(u(t))=1.00u(t)+0.50u2(t)+0.25u3(t),θ=[a1,a2,b1,b2,γ1,γ2,γ3,c1,c2]T=[1.80,0.80,0.5,0.65,1.00,0.50,0.25,0.30,0.20]T.
In this example, the input {u(t)} is taken as a persistent excitation signal sequence with zero mean and unit variance and {v(t)} as a white noise sequence with zero mean and variance σ2=0.502. Applying the SG algorithm and the MISG algorithm to estimate the parameters of this IN-CARAR system, the parameter estimates and their estimation errors are shown in Tables 1 and 2 with the data length L=3000 and the estimation error δ:=∥θ^(t)-θ∥/∥θ∥ versus t being shown in Figures 2 and 3.Table 1
The SG parameter estimates and errors.
t
a
1
a
2
b
1
b
2
γ
1
γ
2
γ
3
c
1
c
2
δ (%)
100
0.49020
−0.49380
0.01083
0.01505
0.00905
0.01482
0.02428
−0.00163
−0.00411
95.19650
200
0.49438
−0.49611
0.01052
0.01544
0.00908
0.01488
0.02437
−0.00159
−0.00415
95.15025
500
0.50037
−0.50067
0.01024
0.01579
0.00910
0.01491
0.02443
−0.00156
−0.00418
95.11481
1000
0.50136
−0.50130
0.01017
0.01588
0.00911
0.01492
0.02444
−0.00156
−0.00418
95.10595
2000
0.50017
−0.50007
0.01016
0.01589
0.00911
0.01492
0.02444
−0.00156
−0.00418
95.10507
3000
0.50042
−0.50031
0.01016
0.01589
0.00911
0.01492
0.02444
−0.00156
−0.00418
95.10474
True values
1.80000
0.80000
0.50000
0.65000
1.00000
0.50000
0.25000
0.30000
0.20000Table 2
The MISG parameter estimates and errors.
t
a
1
a
2
b
1
b
2
γ
1
γ
2
γ
3
c
1
c
2
δ (%)
100
1.79885
0.80074
0.49998
0.65013
0.99968
0.50004
0.24953
0.21240
0.36032
7.46185
200
1.79813
0.80153
0.49999
0.65014
0.99955
0.50000
0.24927
0.37423
0.25902
3.87441
500
1.79820
0.80154
0.50001
0.65020
0.99927
0.49994
0.24876
0.32965
0.06371
5.69751
1000
1.79998
0.79973
0.50000
0.65023
0.99910
0.49990
0.24844
0.32849
0.19041
1.23012
2000
1.79979
0.79992
0.50000
0.65023
0.99909
0.49990
0.24842
0.28734
0.20472
0.55706
3000
1.79989
0.79982
0.50000
0.65023
0.99909
0.49990
0.24842
0.28556
0.20496
0.62823
True values
1.80000
0.80000
0.50000
0.65000
1.00000
0.50000
0.25000
0.30000
0.20000Figure 2
The SG estimation errorδ versus t.Figure 3
The MISG estimation errorδ versus.From Tables1 and 2 and Figures 2 and 3, we can draw the conclusions. The parameters estimation errors become smaller with the data length t increasing. The estimation errors given by the MISG algorithm are much smaller than that of the SG algorithm. The convergence speed of the multi-innovation SG algorithm is faster than those of the SG algorithm. These indicate that the MISG algorithm has better performance than the SG algorithm.


## 6. Conclusions

The gradient and least squares algorithms are two different kinds of important identification methods. It is well known that the gradient algorithm has poor convergence rates. This paper studies the multi-innovation SG identification methods for IN-CARAR systems. The numerical examples show that the proposed MISG algorithm can estimate effectively the parameters of input nonlinear systems and indicate that increasing the innovation length can improve parameter estimation accuracy of the multi-innovation identification algorithm because the algorithm uses more information in each recursion for a large innovation length. The proposed method can be applied to nonlinear output error systems. Although the algorithm is presented for the IN-CARAR systems, the basic idea can be extended to other linear or nonlinear systems with colored noises [42–61].



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