# Adaptive Gradient-Based Iterative Algorithm for Multivariable Controlled Autoregressive Moving Average Systems Using the Data Filtering Technique

Authors: Jian Pan; Hao Ma; Xiao Jiang; Wenfang Ding; Feng Ding
Journal: Complexity
 (2018)

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## Abstract

The identification problem of multivariable controlled autoregressive systems with measurement noise in the form of the moving average process is considered in this paper. The key is to filter the input–output data using the data filtering technique and to decompose the identification model into two subidentification models. By using the negative gradient search, an adaptive data filtering-based gradient iterative (F-GI) algorithm and an F-GI with finite measurement data are proposed for identifying the parameters of multivariable controlled autoregressive moving average systems. In the numerical example, we illustrate the effectiveness of the proposed identification methods.


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## Body

## 1. Introduction

Parameter estimation plays an important role in system control [1–4], system analysis [5–8], and signal processing [9–13]. Parameter estimation is significant in system modeling [14, 15]. Multi-input multi-output systems widely exist in industrial control areas, which are also called multivariate systems or multivariable systems [16–18]. They are more complex in model structures than single-input single-output systems and always have high dimensions and numerous parameters, which make the parameter estimation more difficult. In this literature, Ding et al. proposed a filtering decomposition-based least squares iterative algorithm for multivariate pseudolinear ARMA systems [19]. Ma et al. studied the parameter estimation problem of multivariate Hammerstein systems and presented a modified Kalman filter-based recursive least squares algorithm to give the parameter estimates [20]. Pan et al. proposed a filtering-based multi-innovation extended stochastic gradient algorithm for multivariable systems [21].The data filtering technique is an important approach in system identification [22] and state estimation. Chen and Ding applied the data filtering technique to identify the multi-input and single-output system based on the maximum likelihood recursive least squares algorithm [23]. Mao et al. derived an adaptive filtering-based multi-innovation stochastic gradient algorithm for the input nonlinear system with autoregressive noise [24]. They introduced a linear filter to filter the input and output signals and decomposed the identification model into two subidentification models (i.e., a noise model and a system filtered model), which can improve the convergence rate and computation efficiency [25]. The identification methods can be applied to many areas [26–29].The gradient search is useful for identification as an optimization method [30, 31]. Many gradient-based algorithms, including the stochastic gradient algorithms [32–34] and the gradient-based iterative algorithms, have been developed using the multi-innovation identification theory, the maximum likelihood estimation methods [35, 36], the key-term separation principle [37, 38], and the data filtering theory. For example, Ma et al. presented an iterative variational Bayesian method to identify the Hammerstein varying systems with parameter uncertainties. Chen et al. studied the identification problem of bilinear-in-parameter systems and presented a gradient-based iterative algorithm by using the hierarchical identification principle and the gradient search [39]. Deng and Ding developed a Newton iterative identification method for an input nonlinear finite impulse response system with moving average noise [40]. Other methods can be referred as to the transfer function identification [41–45], linear system identification [46–51], and nonlinear system identification [52–59].This paper uses the hierarchical identification principle to study the data filtering-based iterative identification methods for a multivariable controlled autoregressive moving average (M-CARMA) system. The basic idea is to introduce a linear filter to decompose the original identification model into two subidentification models and then obtain the parameter estimates using the negative gradient search. The main contributions are as follows:(i)
A filtering-based gradient iterative (F-GI) algorithm is proposed using the data filtering technique and the gradient search.
(ii)
A filtering-based gradient iterative algorithm with finite measurement data is developed to obtain the parameter estimates.The layout of the remainder of this paper is as follows. Section2 derives the identification model for the M-CARMA system. In Section 3, we derive a data filtering-based gradient iterative algorithm based on the data filtering technique. A filtering-based gradient iterative algorithm with finite measurement data is developed to estimate the unknown parameters in Section 4. A numerical example is shown in Section 5 to illustrate the benefits of the proposed methods in this paper. Finally, some concluding remarks are given in Section 6.


## 2. The Problem Formulation

Some notation is introduced for convenience:θ^t denotes the estimate of θ at time t; “A≕X” or “X≔A” stands for“A is defined as X”; the symbol I (In) represents an identity matrix of appropriate size (n×n); the symbol 1n represents an n-dimensional column vector whose elements are 1; z denotes a unit forward shift operator like zxt=xt+1 and z−1xt=xt−1 ; the superscript T symbolizes the vector/matrix transpose; and the norm of a matrix X is defined by ∥X∥2≔trXXT.The following multivariable controlled autoregressive moving average system in Figure1 is considered,
(1)Azyt=Bzut+Dzυt,where ut∈ℝr is the system input vector, yt∈ℝm is the system output vector, vt∈ℝm is a white noise vector with zero mean, Az and Bz are the matrix polynomials in the unit backward shift operator z−1, and Dz is the polynomial in z−1.
(2)Az≔I+A1z−1+A2z−2+⋯+Anaz−na,Ai∈ℝm×m,Bz≔B1z−1+B2z−2+⋯+Bnbz−nb,Bi∈ℝm×r,Dz≔1+d1z−1+d2z−2+⋯+dndz−nd,di∈ℝ.Figure 1
A multivariable controlled autoregressive moving average system.Assume that the ordersna, nb, and nd are known, and ut=0, yt=0, and vt=0 for t≤0. The intermediate variable is defined as
(3)wt≔Dzυt∈ℝm.The system information vectorφst, the noise information vector ψt, the system parameter matrix θs, and the noise parameter vector θn are defined as
(4)φst≔−yTt−1,−yTt−2,…,−yTt−na,uTt−1,uTt−2,…,uTt−nbT∈ℝmna+rnb,ψt≔υt−1,υt−2,…,υt−nd∈ℝm×nd,θsT≔A1,A2,…,Ana,B1,B2,…,Bnb∈ℝm×mna+rnb,θn≔d1,d2,…,dndT∈ℝnd.Equations (3) and (1) can be written as
(5)wt=1+d1z−1+d2z−2+⋯+dndz−ndυt=d1υt−1+d2υt−2+⋯+dndυt−nd+υt=ψtθn+υt,(6)yt=I−Azyt+Bzut+Dzυt=θsTφst+wt=θsTφst+ψtθn+υt.Equation (5) is the noise identification model. For the M-CARMA system in (1), choose the polynomial Lz≔1/Dz as a filter. Define the filtered input vector uft, the filtered output vector yft, and the filtered information vector φft as
(7)uft≔Lzut=1Dzut∈ℝr,yft≔Lzyt=1Dzyt∈ℝm,φft≔Lzφst=1Dzφst=−yfTt−1,−yfTt−2,…,−yfTt−na,ufTt−1,ufTt−2,…,ufTt−nbT∈ℝmna+rnb.Multiplying both sides of (1) by Lz obtains
(8)AzLzyt=BzLzut+υt,or
(9)Azyft=Bzuft+υt.Then we have(10)yft=I−Azyft+Bzuft+υt=θsTφft+υt.Equations (10) and (5) form the filtered identification models of the M-CARMA system.


## 3. The F-GI Algorithm

In this section, a linear filterLz is applied to deal with the moving average noise. A gradient-based iterative identification algorithm is proposed for M-CARMA systems by using the data filtering technique [60–63].Considering the newestp data from i=t−p+1 to i=t, the stacked filtered output matrix Yfp,t, the stacked filtered information matrix Φfp,t, the stacked noise vector Wp,t, and the stacked noise information matrix Φnp,t are defined as
(11)Yfp,t≔yft,yft−1,…,yft−p+1∈ℝm×p,Φfp,t≔φft,φft−1,…,φft−p+1∈ℝmna+rnb×p,Wp,t≔wt,wt−1,…,wt−p+1T∈ℝpm,Φnp,t≔ψt,ψt−1,…,ψt−p+1T∈ℝpm×nd.Define a quadratic criterion function:(12)J1θs=Yfp,t−θsTΦfp,t2,J2θn=Wp,t−Φnp,tθn2.Letk=1,2,3,… be an iterative variable. Let θ^s,kt and θ^n,kt be the estimates of θs and θn at iteration k. Minimizing J1θs and J2θn and using the negative gradient search will give the following iterative relations for obtaining the parameter estimates of θs and θn:
(13)θ^s,kt=θ^s,k−1t−μ1,kt2gradJ1θs=θ^s,k−1t+μ1,ktΦfp,tYft−θ^s,k−1TtΦfp,tT,(14)θ^n,kt=θ^n,k−1t−μ2,kt2gradJ2θn=θ^n,k−1t+μ2,ktΦnTp,tWt−Φnp,tθ^n,k−1t,where μ1,kt≥0 and μ2,kt≥0 are the iterative step size or the convergence factor. However, the difficulty is that the noise information matrix Φnp,t (i.e., ψt) contains the unmeasured vector vt−i. So the gradient-based iterative algorithm in (13) and (14) cannot give the parameter estimate θ^n,kt directly. The solution is to use the hierarchical identification principle and to replace the unknown variable vt−i with its corresponding estimates υ^k−1t−i at iteration k−1, and to define the estimate of ψt as
(15)ψ^kt≔υ^k−1t−1,υ^k−1t−2,…,υ^k−1t−nd∈ℝm×nd.Usingψ^kt to construct the estimate of Φn,kp,t obtains
(16)Φ^n,kp,t≔ψ^kt,ψ^kt−1,…,ψ^kt−p+1T∈ℝpm×nd.Replacingt in (6) with t−i gives
(17)wt−i=yt−i−θsTφst−i.Replacingθs in (17) with θ^s,k−1t obtains the estimate of wt−i at iteration k:
(18)w^kt−i=yt−i−θ^s,k−1Ttφst−i.From (6), we have
(19)υt−i=yt−i−θ^sTφst−i−ψt−iθn.Replacingθs, θn, and ψt−i with θ^s,kt, θ^n,kt, and ψ^kt−i obtains the iterative estimate of vt−i at iteration k:
(20)υ^kt−i=yt−i−θ^s,kTtφst−i−ψ^kt−iθ^n,kt.Then, usingw^kt to construct the iterative estimate of Wp,t at iteration k gives
(21)W^kp,t=w^kt,w^kt−1,…,w^kt−p+1T∈ℝpm.Useθ^n,kt=d^1,kt,d^2,kt,…,d^nd,ktT to construct the estimate of Dt,z as
(22)D^kt,z=1+d^1,ktz−1+d^2,ktz−2+⋯+d^nd,ktz−nd.UsingD^kt,z to filter yt and ut gives the filtered estimates y^f,kt and u^f,kt of yft and uft:
(23)u^f,kt=ut−d^1,ktu^f,kt−1−d^2,ktu^f,kt−2−⋯−d^nd,ktu^f,kt−nd,y^f,kt=yt−d^1,kty^f,kt−1−d^2,kty^f,kt−2−⋯−d^nd,kty^f,kt−nd.Furthermore, we usey^f,kt to construct the estimate of Yfp,t, use y^f,kt and u^f,kt to construct the estimate of φft, and use φ^f,kt to construct the estimate of Φfp,t at iteration k:
(24)Y^f,kp,t≔y^f,kt,y^f,kt−1,…,y^f,kt−p+1∈ℝm×p,φ^f,kt≔−y^f,kTt−1,…,−y^f,kTt−na,u^f,kTt−1,…,u^f,kTt−nbT∈ℝmna+rnb,Φ^f,kp,k≔φ^f,kt,φ^f,kt−1,…,φ^f,kt−p+1∈ℝmna+rnb×p.From the above derivation, we can summarize a filtering-based multi-innovation gradient iterative identification algorithm:(25)θ^s,kt=θ^s,k−1t+μ1,ktΦ^f,ktY^f,kt−θ^s,k−1TtΦ^f,ktT,(26)Y^f,kp,t=y^f,kt,y^f,kt−1,…,y^f,kt−p+1,(27)Φf,kp,t=φ^f,kt,φ^f,kt−1,…,φ^f,kt−p+1,(28)θ^n,kt=θ^n,k−1t+μ2,ktΦ^n,kTtW^kt−Φ^n,ktθ^n,kt,(29)W^kp,t=w^kt,w^kt−1,…,w^kt−p+1T,(30)Φ^n,kp,t=ψ^kt,ψ^kt−1,…,ψ^kt−p+1T,(31)φ^f,kt=−y^f,kTt−1,…,−y^f,kTt−na,u^f,kTt−1,…,u^f,kTt−nbT,(32)ψ^kt=υ^k−1t−1,υ^k−1t−2,…,υ^k−1t−nd,(33)φst=−yTt−1,…,−yTt−na,uTt−1,…,uTt−nbT,(34)u^f,kt=ut−d^1,ktu^f,kt−1−d^2,ktu^f,kt−2,−⋯−d^nd,ktu^f,kt−nd,(35)y^f,kt=yt−d^1,kty^f,kt−1−d^2,kty^f,kt−2,−⋯−d^nd,kty^f,kt−nd,(36)w^kt−i=yt−i−θ^s,k−1Ttφst−i,(37)υ^kt−i=yt−i−θ^s,kTtφst−i−ψ^kt−iθ^n,kt,(38)θ^n,kt=d^1,kt,d^2,kt,…,d^nd,ktT,(39)μ1,kt≤2Φ^f,kt2−1,(40)μ2,kt≤2Φ^n,kt2−1.The identification steps of the algorithm in (25), (26), (27), (28), (29), (30), (31), (32), (33), (34), (35), (36), (37), (38), (39), and (40) to compute θ^s,kt and θ^n,kt are listed as follows:
(1)
Set the initial values: lett=1, give the data length p, and give a small positive number ε. Set the initial values θ^s0=1mna+rnb×m/p0, θ^n0=1nd/p0, p0=106.
(2)
Collect the input–output dataut and yt and construct φst using (33).
(3)
Letk=1 and set the initial values w^0t−i=1m/po,υ^0t−i=1m/p0,y^f,0t−i=1m/p0,,u^f,0t−i=1r/p0,i=1,2,…,maxna,nb,nd.
(4)
Constructψ^kt and φ^f,kt using (31) and (32).
(5)
Computew^kt using (36) and form the stacked noise vector W^kp,t using (29) and the stacked noise matrix Φn,kp,t using (30).
(6)
Chooseμ2,kt using (40) and update the noise parameter estimates θ^n,kt using (28).
(7)
Readd^i,kt from θ^n,kt in (38). Compute u^f,kt and y^f,kt using (34) and (35).
(8)
ConstructΦf,kp,t using (27) and Y^f,kp,t using (26).
(9)
Chooseμ1,kt using (39) and update the system parameter estimates θ^s,kt using (25).
(10)
Computeυ^kt−i using (37).
(11)
Compareθ^s,kt with θ^s,k−1t and compare θ^n,kt with θ^n,k−1t: if ∥θ^s,kt−θ^s,k−1t∥>ε and ∥θ^n,kt−θ^n,k−1t∥>ε, increase k by 1 and turn to Step 4; otherwise, obtain k and the parameter estimate vector θ^s,kt and θ^n,kt, let θ^s,0t+1≔θ^s,kt, θ^n,0t+1≔θ^n,kt, increase t by 1 and turn to Step 2.The flowchart of computingθ^s,kt and θ^n,kt from the F-GI algorithm is shown in Figure 2.Figure 2
The flowchart of computing the F-GI parameter estimates.


## 4. The F-GI Algorithm with Finite Measurement Data

Consider the data fromt=1 to t=L and define the stacked filtered output matrix YfL, the stacked filtered information matrix ΦfL, the stacked noise vector WL, and the stacked noise information matrix ΦnL as
(41)YfL≔yfL,yfL−1,…,yf1∈ℝm×L,ΦfL≔φfL,φfL−1,…,φf1∈ℝmna+rnb×L,WL≔wL,wL−1,…,w1T∈ℝLm,ΦnL≔ψL,ψL−1,…,ψ1T∈ℝLm×nd.Note thatYfL and ΦfL contain all the measured data ut,yt:t=1,2,…,L.The two gradient criterion functions are defined as(42)J3θs≔YfL−θsTΦfL2,J4θn≔WL−ΦnLθn2.Similarly, minimizingJ3θs and J4θn, we can derive a filtering-based gradient iterative (F-GI) algorithm with the data length L for the M-CARMA system:
(43)θ^s,k=θ^s,k−1+μ1,ktΦ^f,kLY^f,kL−θ^s,k−1TΦ^f,kLT,(44)Y^f,kL=y^f,kL,y^f,kL−1,…,y^f,k1,(45)Φf,kL=φ^f,kL,φ^f,kL−1,…,φ^f,k1,(46)θ^n,k=θ^n,k−1+μ2,ktΦ^n,kTLW^kL−Φ^n,kLθ^n,k−1,(47)W^kL=w^kL,w^kL−1,…,w^k1T,(48)Φn,kL=ψ^kL,ψ^kL−1,…,ψ^k1T,(49)φ^f,kt=−y^f,kTt−1,…,−y^f,kTt−na,u^f,kTt−1,…,u^f,kTt−nbT,(50)ψ^kt=υ^k−1t−1,υ^k−1t−2,…,υ^k−1t−nd,(51)φst=−yTt−1,…,−yTt−na,uTt−1,…,uTt−nbT,(52)u^f,kt=ut−d^1,ktu^f,kt−1−d^2,ktu^f,kt−2−⋯−d^nd,ktu^f,kt−nd,(53)y^f,kt=yt−d^1,kty^f,kt−1−d^2,kty^f,kt−2−⋯−d^nd,kty^f,kt−nd,(54)w^kt=yt−θ^s,k−1Tφst,(55)υ^kt=yt−θ^s,kTφst−ψ^ktθ^n,k,(56)θ^n,k=d^1,kt,d^2,kt,…,d^n,d,ktT,(57)μ1,kt≤2Φf,kL2−1,(58)μ2,kt≤2Φn,kL2−1.The identification steps of the F-GI algorithm with finite measurement data in (43), (44), (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), and (58) to compute θ^s,k and θ^n,k are listed as follows.
(1)
Collect the input–output dataut,yt,t=1,2,…,Land give a small positive number ε. Construct φst using (51).
(2)
Letk=1 and set the initial values θ^s,0=1mna+rnb×m/p0,θ^n,0=1nd/p0,w^0t=1m/p0,υ^0t=1m/p0,y^f,0t=1m/p0,u^f,0t=1r/p0,p0=106.
(3)
Constructψ^kt and φ^f,kt using (49) and (50).
(4)
Computew^kt using (54), construct W^kL using (47), and construct Φ^n,kL using (48).
(5)
Chooseμ2,kt using (58) and update θ^n,k using (46).
(6)
Readd^i,kt from θ^n,k in (56) and compute u^f,kt and y^f,kt using (52) and (53).
(7)
ConstructΦ^f,kL using (45) and construct Y^f,kL using (44).
(8)
Chooseμ1,kt using (57) and update θ^s,k using (43).
(9)
Computeυ^kt using (55).
(10)
Compareθ^s,k with θ^s,k−1 and compare θ^n,k with θ^n,k−1: if ∥θ^s,k−θ^s,k−1∥>ε and ∥θ^n,k−θ^n,k−1∥>ε, increase k by 1 and turn to Step 3; otherwise, obtain k and the parameter estimate vector θ^s,k and θ^n,k.The flowchart of computingθ^s,k and θ^n,k from the F-GI algorithm with finite measurement data is shown in Figure 3.Figure 3
The flowchart of computing the F-GI parameter estimates with finite measurement data.The algorithm in (43), (44), (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), and (58) for multivariable CARMA systems is based on the filtering technique and the gradient search and can be extended to more complex multivariable systems with cooled noises.


## 5. Numerical Example

Consider a two-input two-output CARMA system:(59)Azyt=Bzut+Dzυt,where
(60)yt=y1ty2t,ut=u1tu2t,υt=υ1tυ2t,Az=I+A1z−1=1001+a11a12a21a22z−1=1+0.65z−10.60z−1−0.30z−11+1.00z−1,Bz=B1z−1=b11b12b21b22z−1=0.45z−10.35z−10.38z−1−0.40z−1,Dz=1+d1z−1=1−0.60z−1,θsT=A1,B1,θn=d1.In simulation,u1t and u2t are taken as two persistent excitation signal sequences with zero mean and unit variance, and v1t and v2t as two white noise sequences with σ12=σ22. Take the data length L=1000 and L=2000 and apply the F-GI algorithm with finite measurement data in (43), (44), (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), and (58) to estimate the parameters of this M-CARMA system. The parameter estimates and errors are shown in Table 1 with L=1000, σ12=σ22=0.52, and σ12=σ22=1.22, and the parameter estimation errors versus t are shown in Figure 4. For comparison with the different data length L=2000, the simulation results are shown in Table 2 and Figure 5.Table 1
The F-GI parameter estimates and errors (L=1000).
σ2
k
a11
a12
b11
b12
a21
a22
b21
b22
d1
δ%
0.52
1
0.14294
0.09843
0.00868
0.00915
0.00296
0.08637
0.00937
−0.00661
−0.27864
86.50364
10
0.42310
0.34081
0.15604
0.12072
−0.02498
0.29864
0.20824
−0.15361
−0.70856
56.91872
50
0.63701
0.59049
0.44414
0.30015
−0.21622
0.83432
0.36331
−0.32901
−0.77686
16.06327
200
0.65476
0.60674
0.46465
0.31821
−0.31707
1.02684
0.38855
−0.39407
−0.56444
3.58517
500
0.65477
0.60673
0.46466
0.31824
−0.31710
1.02689
0.38857
−0.39418
−0.56441
3.58655
1.22
1
0.14514
0.09935
0.00228
0.00199
0.00410
0.08912
0.00243
−0.00127
−0.28184
86.71589
10
0.52198
0.41436
0.13005
0.04549
−0.05221
0.40777
0.16182
−0.15682
−0.91038
54.85474
50
0.65005
0.60460
0.37591
0.17468
−0.29039
0.97368
0.33485
−0.29105
−0.60764
13.34916
200
0.65652
0.60385
0.46751
0.25681
−0.31942
1.02762
0.38992
−0.36323
−0.56348
6.73257
500
0.65706
0.60351
0.48224
0.27468
−0.31970
1.02832
0.39775
−0.37791
−0.56279
5.95620
True values
0.65000
0.60000
0.45000
0.35000
−0.30000
1.00000
0.38000
−0.40000
−0.60000Figure 4
The CARMA-FGI estimation errorsδ with different noise variances L=1000.Table 2
The F-GI parameter estimates and errors (L=2000).
σ2
k
a11
a12
b11
b12
a21
a22
b21
b22
d1
δ%
0.52
1
0.14346
0.09808
0.01067
0.00762
0.00511
0.08406
0.01113
−0.00704
−0.27896
86.56006
10
0.39890
0.32288
0.22390
0.15247
−0.01745
0.27349
0.21267
−0.15857
−0.68297
57.02995
50
0.63521
0.59550
0.44755
0.31132
−0.21207
0.82865
0.34932
−0.32757
−0.77919
16.41676
200
0.65416
0.60726
0.45998
0.33687
−0.31265
1.01818
0.37903
−0.40383
−0.58336
1.98279
500
0.65417
0.60725
0.45998
0.33691
−0.31269
1.01824
0.37904
−0.40396
−0.58332
1.98657
1.22
1
0.14560
0.09899
0.00315
0.00157
0.00466
0.08794
0.00329
−0.00156
−0.28174
86.73735
10
0.51092
0.41373
0.17192
0.06708
−0.05673
0.42188
0.14417
−0.12991
−0.91458
53.92889
50
0.64851
0.60875
0.38838
0.20796
−0.28513
0.96729
0.30507
−0.28172
−0.62099
12.59343
200
0.65613
0.60516
0.46036
0.30206
−0.31371
1.01919
0.36807
−0.38499
−0.58216
3.60147
500
0.65684
0.60452
0.47158
0.31970
−0.31423
1.02001
0.37658
−0.40343
−0.58126
2.91324
True values
0.65000
0.60000
0.45000
0.35000
−0.30000
1.00000
0.38000
−0.40000
−0.60000Figure 5
The CARMA-FGI estimation errorsδ with different noise variances L=2000.From Tables1–3 and Figures 4 and 5, we can draw the following conclusions.
(1)
The parameter estimation errors obtained by the presented algorithms gradually become smaller with the iterative variablek increasing. Thus, the proposed algorithms for M-CARMA systems are effective.
(2)
The system parameter estimates converge to their true values with the increasing of the data length.
(3)
Under the same data length, a smaller noise variance leads to higher parameter estimation accuracy and a faster convergence rate.
(4)
A longer data lengthL leads to a smaller estimation error under the same noise level.Table 3
The F-GI parameter estimates and errors (k=500).
σ2
L
a11
a12
b11
b12
a21
a22
b21
b22
d1
δ%
0.52
1000
0.65477
0.60673
0.46466
0.31824
−0.31710
1.02689
0.38857
−0.39418
−0.56441
3.58655
2000
0.65417
0.60725
0.45998
0.33691
−0.31269
1.01824
0.37904
−0.40396
−0.58332
1.98657
1.22
1000
0.65706
0.60351
0.48224
0.27468
−0.31970
1.02832
0.39775
−0.37791
−0.56279
5.95620
2000
0.65684
0.60452
0.47158
0.31970
−0.31423
1.02001
0.37658
−0.40343
−0.58126
2.91324
True values
0.65000
0.60000
0.45000
0.35000
−0.30000
1.00000
0.38000
−0.40000
−0.60000


## 6. Conclusions

An F-GI algorithm and an F-GI algorithm with finite measurement data are proposed for identifying the multivariable controlled autoregressive system with measurement noise in this paper. The linear filter is introduced to filter the input–output data, and the hierarchical identification principle is applied to decompose the identification model into two subidentification models. The simulation results show that the proposed algorithms can generate accurate estimates. The proposed approaches in the paper can combine other mathematical tools [64–69] and statistical strategies [70–75] to study the performances of some parameter estimation algorithms and can be applied to other multivariable systems with different structures and disturbance noises and other literature [76–86] such as system identification [87–92].



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*Source: 9598307-2018-07-24.xml*