df2010_mcm52(5-6)_hls algorithms for simo systems based on the auxiliary model

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Mathematical and Computer Modelling 52 (2010) 918–924

Contents lists available at ScienceDirect

Mathematical and Computer Modelling

journal homepage: www.elsevier.com/locate/mcm

Hierarchical least squares algorithms for single-input multiple-outputsystems based on the auxiliary model$

Lili Xiang a, Linbo Xie a, Yuwu Liao b, Ruifeng Ding a,∗

a School of Communication and Control Engineering, Jiangnan University, Wuxi 214122, PR Chinab Department of Physics and Electronics Information Technology, Xiangfan University, Xiangfan 441053, PR China

a r t i c l e i n f o

Article history:

Received 5 February 2010

Received in revised form 21 May 2010

Accepted 25 May 2010

Keywords:

Least squares

Parameter estimation

Auxiliary model identification

Hierarchical identification

a b s t r a c t

This paper presents an auxiliary model based hierarchical least squares algorithm to

estimate the parameters of single-input multi-output system modelling by combining

the auxiliary model identification idea and the hierarchical identification principle. A

numerical example is given to show the performance of the proposed algorithm.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Parameter estimation is very important in system modelling and identification, signal processing, and adaptive control,e.g.,[1–12]. Two typical estimation methods are the least squares methods [13–18] andstochastic gradient methods [19–22]for system identification. Other methods include the multi-innovation stochastic gradient type identification algorithms[23–30] and multi-innovation least squares type algorithms [30–33], the hierarchical stochastic gradient algorithm [34],hierarchical least squares algorithms [35,36], and the auxiliary model based algorithms [24,37–42].

Recently, Sun and Wu studied the consistency of the regularized least-square regression in a general reproducing kernelHilbert space [43]; Han et al. proposed an auxiliary model identification method for multirate multi-input systems basedon least squares [41]; Ding, Han and Chen presented an estimation algorithm for time series AR modeling with missingobservations based on the polynomial transformation [44].

The auxiliary model identification method is effective for solving the identification problem with the unknown variablesin the information vector, and the hierarchical identification principle is based on the decomposition and can deal withparameter estimation for multivariable systems. In this literature, Ding and Chen proposed an auxiliary model basedrecursive least squares algorithm for dual-rate sampled-data systems [37]; Wang proposed an auxiliary model basedextended least squares algorithm for the output error moving average model [ 45]; Ding and Chen developed a hierarchicalstochastic gradient algorithm and a hierarchical least squares algorithm for multi-input multi-output systems [ 34,35].

By combining the auxiliary model identification idea [37,38,45,46] and the hierarchical identification principle [34,35],this paper studies and presents an auxiliary model based hierarchical identification method for single-input multiple-outputsystems.

$ This work was supported by the National Natural Science Foundation of China (No. 60804013).∗ Corresponding author.

E-mail addresses: [email protected] (L. Xiang), [email protected] (L. Xie), [email protected] (Y. Liao), [email protected],

[email protected] (R. Ding).

0895-7177/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2010.05.025

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L. Xiang et al. / Mathematical and Computer Modelling 52 (2010) 918–924 919

The paper is organized as follows. Section 2 describes the system formulation related to the single-input multiple-output(SIMO) systems. Section 3 derives an auxiliary model based hierarchical identification method of SIMO systems. Section 4provides an illustrative example for the results in this paper. Finally, concluding remarks are given in Section 5.

2. Problem formulation

Consider a discrete-time single-input multiple-output (SIMO) system described by the following state space model

[16,31]: x(t + 1) = Ax(t ) + bu(t ), y (t ) = Cx(t ) + v (t ),

(1)

where x(t ) ∈ Rn is the state vector, u(t ) ∈ R is the input variable, y (t ) = [ y1(t ), y2(t ) , . . . , ym(t )]T ∈ R

m is the outputvector, v (t ) = [v1(t ), v2(t ) , . . . , vm(t )]T ∈ R

m is a white noise vector with a zero mean, A ∈ Rn×n, b ∈ R

n, and C ∈ Rm×n

are unknown constant matrices or vector.Let I be an identity matrix of an appropriate size and z−1 represent a unit backward shift operator: z−1 x(t ) = x(t − 1)

and z x(t ) = x(t + 1). The SIMO system in (1) has the following input–output relationship:

y (t ) = C ( z I − A)−1bu(t ) + v(t )

=z−nC adj[ z I − A]b

z−n det[ z I − A]u(t ) + v (t )

=:β( z)

α( z)u(t ) + v (t ), (2)

where

α( z) := z−n det[ z I − A] = 1 + α1 z−1 + α2 z

−2 + · · · + αn z−n ∈ R,

β( z) := z−nC adj[ z I − A]b = β1 z−1 + β2 z

−2 + · · · + βn z−n ∈ R

m.

Assume that the order n is known and u(t ) = 0 and y (t ) = 0 for t 0.The objective of this paper is to present a new identification method to estimate the parameters (αi,βi) for the the SIMO

system in (2) by combining the auxiliary model identification idea [37] and the hierarchical identification principle [34,35],and to evaluate the estimation accuracy under different noise variances.

3. The auxiliary model based hierarchical least squares algorithm

The idea of the auxiliary model identification is to construct an auxiliary model using the measured data and to replacethe unmeasurable variables in the information matrix with the outputs of the auxiliary model. The following discusses theauxiliary model based hierarchical least squares algorithm.

Define an intermediate variable (i.e., the true output or noise-free output of the system):

s(t ) :=β( z)

α( z)u(t ). (3)

Then (2) can be written as

y (t ) = s(t ) + v (t ). (4)

Expanding (3) gives

s(t ) = −α1s(t − 1) − α2s(t − 2) − · · · − αns(t − n) + β1u(t − 1) + β2u(t − 2) + · · · + βnu(t − n). (5)

Define the parameter vector α, the parameter matrix θ , the input information vector ϕ(t ) and the information matrix ψ(t )as

α := [α1, α2, . . . , αn]T ∈ Rn,

θ T := [β1,β2, . . . ,βn] ∈ Rm×n,

ϕ(t ) := [u(t − 1), u(t − 2) , . . . , u(t − n)]T ∈ Rn,

ψ(t ) := [−s(t − 1), −s(t − 2) , . . . ,−s(t − n)] ∈ Rm×n.

Using these definitions, (5) and (4) can be respectively written as

s(t ) = ψ(t )α + θ Tϕ(t ), (6)

y (t ) = ψ(t )α + θ Tϕ(t ) + v (t ). (7)

Eq. (7) is the identification model for the SIMO system in (2) which contains the parameter vector α and the parametermatrix θ . Here we use the hierarchical identification principle and derive a hierarchical algorithm to estimate them.



920 L. Xiang et al. / Mathematical and Computer Modelling 52 (2010) 918–924

According to the hierarchical identification principle, we identify the parameter vectorα ∈ Rn and the parameter matrix

θ ∈ Rn×m, respectively, and thus define two quadratic cost functions:

J 1(α) :=

t j=1

y ( j) − ψ( j)α − θ Tϕ( j)2,

J 2(θ ) :=

t

j=1

y ( j) − ψ( j)α − θ Tϕ( j)2,

where X 2 := tr[ XX T].

Let α(t ) be the estimate of α at time t , and θ (t ) be the estimate of θ at time t . Minimizing J 1(α) and J 2(θ ) yields thefollowing recursive algorithm [36,47]:

α(t ) = α(t − 1) + L1(t )[ y (t ) − θ Tϕ(t ) − ψ(t )α(t − 1)], (8)

L1(t ) = P 1(t )ψT(t ) =P 1(t − 1)ψT(t )

1 + ψ(t ) P 1(t − 1)ψT(t ), (9)

P 1(t ) = [ I − L1(t )ψ(t )] P 1(t − 1), (10)

θ (t ) = θ (t − 1) + L2(t )[ y (t ) − ψ(t )α − θ T(t − 1)ϕ(t )]T, (11)

L2(t ) = P 2(t )ϕ(t ) =P 2(t − 1)ϕ(t )

1 + ϕT(t ) P 2(t − 1)ϕ(t ), (12)

P 2(t ) = [ I − L2(t )ϕT(t )] P 2(t − 1). (13)

A difficulty is that Eqs. (8) and (11) contain the unknown θ and α, respectively. The solution is using the idea of the Jacobi

iteration for Ax = b [48], replacing the unknown θ in (8) and α in (11) with their preceding estimates θ (t − 1) and α(t − 1),respectively, Eqs. (8)–(13) give

α(t ) = α(t − 1) + L1(t )[ y (t ) − ψ(t )α(t − 1) − θ T(t − 1)ϕ(t )], (14)

L1(t ) = P 1(t )ψT(t ) =P 1(t − 1)ψT(t )

1 + ψ(t ) P 1(t − 1)ψT(t ), (15)

P 1(t ) = [ I − L1(t )ψ(t )] P 1(t − 1), (16)

θ (t ) = θ (t − 1) + L2(t )[ y (t ) − ψ(t )α(t − 1) − θ T(t − 1)ϕ(t )]T, (17)

L2(t ) = P 2(t )ϕ(t ) =P 2(t − 1)ϕ(t )

1 + ϕT(t ) P 2(t − 1)ϕ(t ), (18)

P 2(t ) = [ I − L2(t )ϕT(t )] P 2(t − 1). (19)

However, the information vector ψ(t ) contains the unknown variables s(t − i), the algorithm in (14)–(19) cannot beimplemented. The solution here is based on the auxiliary model identification idea to construct an auxiliary model [ 37,38]:the unknown s(t − i) in ψ(t ) are replaced with the outputs s(t ) of the auxiliary model. Define the estimate of ψ(t ) as

ψ(t ) := [−s(t − 1), −s(t − 2) , . . . , −s(t − n)] ∈ Rm×n.

Replacing α, θ and ψ(t ) in (6) with the estimates α(t ) and θ (t ) and ψ(t ), the output s(t ) of the auxiliary model can becomputed by

s(t ) = ψ(t )α(t ) + θ T(t )ϕ(t ). (20)

Replacing ψ(t ) in (14)–(18) with ψ(t ), we can obtain the auxiliary model based hierarchical least squares (AM-HLS)algorithm for estimating the parameter vector α and the parameter matrix θ as follows:

α(t ) = α(t − 1) + L1(t )[ y (t ) − θ T(t − 1)ϕ(t ) − ψ(t )α(t − 1)], (21)

L1(t ) = P 1(t )ψT(t ) =

P 1(t − 1)ψT(t )

1 + ψ(t ) P 1(t − 1)ψT(t )

, (22)

P 1(t ) = [ I − L1(t )ψ(t )] P 1(t − 1), P 1(0) = p0 I , (23)




Fig. 1. The flowchart of computing the AM-HLS estimates α(t ) and θ (t ).

θ (t ) = θ (t − 1) + L2(t )[ y (t ) − θ T(t − 1)ϕ(t ) − ψ(t )α(t − 1)]T, (24)

L2(t ) = P 2(t )ϕ(t ) =P 2(t − 1)ϕ(t )

1 + ϕT(t ) P 2(t − 1)ϕ(t ), (25)

P 2(t ) = [ I − L2(t )ϕT(t )] P 2(t − 1), P 2(0) = p0 I , (26)

ϕ(t ) = [u(t − 1), u(t − 2) , . . . , u(t − n)]

T

, (27)ψ(t ) = [−s(t − 1), −s(t − 2) , . . . , −s(t − n)], (28)

s(t ) = ψ(t )α(t ) + θ T(t )ϕ(t ). (29)

L1(t ) ∈ Rn and L2(t ) ∈ R

n are two gain vectors and P 1(t ) ∈ Rn×n and P 2(t ) ∈ R

n×n are two covariance matrices.

The steps of computing the estimates α(t ) and θ (t ) in the AM-HLS algorithm are listed below.

1. Let t = 1, set the initial values s( j) = 1m/ p0 ( j 0), α(0) = 1n/ p0, θ (0) = 1n×m/ p0, where 1i represents an i-dimensionalcolumn vector whose elements are 1, 1n×m stands for an n × m-dimensional matrix whose elements are 1, P 1(0) = p0 I , P 2(0) = p0 I , p0 is a large constant, e.g., p0 = 106.

2. Collect the input–output data u(t ) and y (t ), construct the information vectors ϕ(t ) by (27) and ψ(t ) by (28).3. Compute L1(t ) by (22), P 1(t ) by (23), and L2(t ) by (25) and P 2(t ) by (26).

4. Update the parameter estimates α(t ) by (21) andˆθ (t ) by (24).5. Compute s(t ) by (29).

6. Increase t by 1 and go to step 2.

The flowchart of computing the parameter estimates α(t ) and θ (t ) for the AM-HLS algorithm in (21)–(29) is shown inFig. 1.

4. Simulation example

Consider the following 1-input 2-output error system,

y (t ) =β( z)

α( z)u(t ) + v (t ),

α( z) = 1 + α1 z−1 + α2 z

−2 = 1 − 0.50 z−1 + 0.60 z−2,

β( z) = β1 z−1 = [2.00, 1.00]T z−1,




Table 1

The parameter estimates and errors.

σ 2 t α1 α2 β11 β12 δ (%)

0.502 100 −0.51153 0.62574 1.88913 1.01356 4.86363

200 −0.49515 0.61473 1.97069 1.05485 2.70607

500 −0.49139 0.60135 2.00549 1.04255 1.84846

1000 −0.49531 0.59140 2.01978 1.00397 0.94691

2000 −0.49850 0.59938 2.01415 0.99818 0.60634

3000 −0.49660 0.59979 1.99915 0.99962 0.149124000 −0.49512 0.59877 2.00660 1.00313 0.37438

1.002 100 −0.51995 0.61860 1.88313 1.08236 6.14512

200 −0.48446 0.60880 1.98880 1.13467 5.75521

500 −0.48104 0.59456 2.03214 1.09620 4.36244

1000 −0.48953 0.57851 2.05042 1.01360 2.42483

2000 −0.49597 0.59602 2.03397 0.99932 1.45440

3000 −0.49248 0.59768 2.00206 1.00120 0.34709

4000 −0.48969 0.59610 2.01594 1.00770 0.88064

2.002 100 −0.54175 0.57464 1.94439 1.26277 11.52598

200 −0.46072 0.58221 2.05684 1.31163 13.49741

500 −0.45992 0.57624 2.10015 1.21168 10.08082

1000 −0.47710 0.55002 2.11911 1.03697 5.75429

2000 −0.48985 0.58662 2.07753 1.00375 3.35291

3000 −0.48354 0.59151 2.01046 1.00577 0.93067

4000 −0.47838 0.58941 2.03653 1.01788 1.99531

True values −0.50000 0.60000 2.00000 1.00000

Fig. 2. The parameter estimation errors versus t with different noise variances.

α = [α1, α2]T = [−0.50, 0.60]T,

θ T = β1 = [2.00, 1.00]T.

The input {u(t )} is taken as an uncorrelated persistent excitation signal sequence with a zero mean and unit variance, andv (t ) = [v1(t ), v2(t )]T as a white noise vector sequence with zero mean and variances σ 21 for v1(t ) and σ 22 for v2(t ). Changing

the values of σ 21 and σ 22 , one can adjust the noise-to-signal ratio δns(1) and δns(2) of the two output channels. Set the initial

values α(0) = 12×1/ p0, θ (0) = 11×2/ p0, p0 = 106. Applying the AM-HLS algorithm to estimate the parameters of thissystem, the parameter estimates and their errors are shown in Table 1 with different variances and data length, and theestimation errors

δ :=

α(t ) − α2 + θ (t ) − θ 2

α2 + θ 2

versus t are shown in Fig. 2, when the noise variances σ 21 = σ 22 = σ 2 = 0.502, the corresponding noise-to-signal ratios

are δns(1) = 35.00% and δns(2) = 70.00%; when σ 2 = 1.002, the noise-to-signal ratios are δns(1) = 70.00% and δns(2) =140.00%; when σ 2 = 2.002, the noise-to-signal ratios are δns(1) = 140.00% and δns(2) = 280.00%.

From Table 1 and Fig. 2, we can get the following conclusions: the parameter estimation errors become (generally) large

with the noise variances increasing and the parameter estimation errors become (generally) small with the data lengthincreasing.




5. Conclusions

The AM-HLS algorithm is presented for estimating the parameters of the single-input multiple-output systems bycombining the auxiliary model identification idea and the hierarchical identification principle. The AM-HLS approach canbe combined with other methods (e.g., iterative algorithms) to study new identification algorithms [49–53]. The simulationresults show that the proposed algorithm is effective.

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