Systems & Control Letters 56 (2007) 373–380www.elsevier.com/locate/sysconle

Auxiliary model-based least-squares identification methods forHammerstein output-error systems�

Feng Dinga,∗,1, Yang Shib, Tongwen Chenc

aControl Science and Engineering Research Center, Southern Yangtze University, Wuxi 214122, P.R. ChinabDepartment of Mechanical Engineering, University of Saskatchewan, Saskatoon, Canada S7N 5A9

cDepartment of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada T6G 2V4

Received 28 August 2004; received in revised form 26 October 2006; accepted 30 October 2006Available online 2 January 2007

Abstract

The difficulty in identification of a Hammerstein (a linear dynamical block following a memoryless nonlinear block) nonlinear output-errormodel is that the information vector in the identification model contains unknown variables—the noise-free (true) outputs of the system. Inthis paper, an auxiliary model-based least-squares identification algorithm is developed. The basic idea is to replace the unknown variables bythe output of an auxiliary model. Convergence analysis of the algorithm indicates that the parameter estimation error consistently converges tozero under a generalized persistent excitation condition. The simulation results show the effectiveness of the proposed algorithms.© 2006 Elsevier B.V. All rights reserved.

Keywords: Recursive identification; Parameter estimation; Least squares; Multi-innovation identification; Hierarchical identification; Auxiliary model;Convergence properties; Stochastic gradient; Hammerstein models; Wiener models; Martingale convergence theorem

1. Introduction

Two typical classes of nonlinear systems—linear time-invariant blocks following (or followed by) static nonlinearblocks—are Hammerstein and Wiener (H–W) nonlinear sys-tems, which are common in industry, e.g., the valve saturationnonlinearities, dead-zone nonlinearities and linear systemsequipped with nonlinear sensors [35]. In general, existingidentification approaches for H–W models can be roughlydivided into two categories: the iterative and the recursivealgorithms. In order to distinguish on-line from off-line calcu-lation, we use iterative for off-line algorithms, and recursivefor on-line ones. We imply that a recursive algorithm canbe on-line implemented, but an iterative one cannot. For a

� This research was supported by the Natural Sciences and EngineeringResearch Council of Canada and the National Natural Science Foundation ofChina (No. 60574051 and 60528007).

∗ Corresponding author. Tel.: 86510 88637783; fax: +86510 85910652.E-mail addresses: dingf@tsinghua.edu.cn, fding@sytu.edu.cn (F. Ding),

tchen@ece.ualberta.ca (T. Chen).1 F. Ding is also an Adjunct Professor in the College of Automation at

the Nanjing University of Technology, Nanjing 210009, P.R. China.

0167-6911/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2006.10.026

recursive algorithm, new information (input and/or output data)is always used in the algorithm which recursively computes theparameter estimates every step as time increases.

Some iterative and/or off-line algorithms of H–W mod-els were discussed in [2,1,4,6,7,22,23,25,28,30,36,11,20] andother recursive and/or on-line algorithms were studied in, e.g.,[35,11,20,32,33,3,5,34]. In the identification area of nonlinearsystems, Bai reported a two-stage identification algorithm forHammerstein–Wiener nonlinear systems based on singularvalue decomposition (SVD) and least squares [1] and studiedidentification problem of systems with hard input nonlineari-ties of known structure [2]; Vörös presented a half-substitutionalgorithm to identify Hammerstein systems with two-segmentnonlinearities and with multisegment piecewise-linear charac-teristics, but no convergence analysis was carried out [32,33];Cerone and Regruto analyzed parameter error bounds in theHammerstein models by using the output measurement errorbound [6]. Also, Pawlak used the series expansion approachto study the identification of Hammerstein nonlinear output-error (state-space) models [29]. Recently, an iterative least-squares and a recursive least-squares identification methodswere reported in [11], and an iterative gradient and a recursive

374 F. Ding et al. / Systems & Control Letters 56 (2007) 373–380

f (·)B(z)A(z)

+

v(t)

u(t) u(t) x(t) y(t)

Fig. 1. The Hammerstein nonlinear output-error system.

stochastic gradient algorithms were developed in [20] fornonlinear ARMAX models based only on the availableinput–output data, and the convergence properties of the recur-sive algorithms involved were proved.

However, most of the identification approaches in the liter-ature assume that the systems under consideration are nonlin-ear ARX models, or equation-error-like models [35,2,1,7,3].That is, each element of the information vector consisting ofinput–output data is measured. In this paper, we focus on theidentification problem of a class of Hammerstein output-error-type nonlinear systems and present an auxiliary-model least-squares (AMLS) algorithm, which is different from the onesmentioned above in that the information vector in our identifi-cation model contains unknown variables (namely, unavailablenoise-free outputs), and we adopt an auxiliary model or refer-ence model to estimate these unknown variables and further usethe outputs of the auxiliary model instead of the unknown noise-free outputs to identify the system parameters. The basic ideais to extend the Landau’s output-error method to study identifi-cation problem of nonlinear systems [21,31]. To the best of ourknowledge, few publications addressed identification methodsof Hammerstein nonlinear output-error systems, especially theconvergence problem of the algorithms involved, which are thefocus of this work.

The objective of this paper is, by means of the auxiliarymodel identification principle, to derive an algorithm to esti-mate the system parameters of the nonlinear output-error mod-els based on the available input–output data {u(t), y(t)}, andto study the properties of the algorithm involved.

Briefly, the paper is organized as follows. Section 2 describesthe identification algorithms related to the Hammerstein sys-tems. Section 3 analyzes the properties of the proposed stochas-tic algorithm. Section 4 provides an illustrative example to showthe effectiveness of the algorithm proposed. Finally, we offersome concluding remarks in Section 5.

2. The algorithm description

Consider the Hammerstein output-error system shown inFig. 1 that consists of a nonlinear memoryless element fol-lowed by a linear output-error model [6,29], where the trueoutput (namely, the noise-free output) x(t) and the innervariable u(t) (namely, the output of the nonlinear block) areunmeasurable, u(t) is the system input, y(t) is the measure-ment of x(t), v(t) is an additive noise with zero mean. Thenonlinear part in the Hammerstein model is a polynomialof a known order in the input [7,22,28], or, more gener-ally, a nonlinear function of a known basis (�1, �2, . . . , �m)

as follows [6,11,20]:

u(t) = f (u(t)) = c1�1(u(t)) + c2�2(u(t))

+ · · · + cm�m(u(t)) =m∑

j=1

cj �j (u(t)). (1)

Then the Hammerstein nonlinear output-error model in Fig. 1may be expressed as

x(t) = B(z)

A(z)u(t) = B(z)

A(z)[c1�1(u(t)) + c2�2(u(t))

+ · · · + cm�m(u(t))],y(t) = x(t) + v(t).

Here, A(z) and B(z) are polynomials in the shift operator z−1

[z−1y(t) = y(t − 1)] with

A(z) = 1 + a1z−1 + a2z

−2 + · · · + anz−n,

B(z) = b1z−1 + b2z

−2 + b3z−3 + · · · + bnz

−n.

Notice that for the Hammerstein model shown in Fig. 1, f (u)

and G(z) := B(z)/A(z) are actually not unique. Any pair(�f (u), G(z)/�) for some nonzero and finite constant � wouldproduce identical input and output measurements. In otherwords, any identification scheme cannot distinguish between(f (u), G(z)) and (�f (u), G(z)/�). Therefore, to get a uniqueparameterization, without loss of generality, one of the gainsof f (u) and G(z) has to be fixed. There are several ways tonormalize the gains [1,6,22]. Here, we adopt the assumption[22,3]: the first coefficient of the function f (·) equals 1; i.e.,c1 = 1 [11,20].

Eq. (2) can be rewritten as a recursive form

x(t) = −n∑

i=1

aix(t − i) +n∑

i=1

biu(t − i)

= −n∑

i=1

aix(t − i) +n∑

i=1

bi

m∑j=1

cj �j (u(t − i)).

Define the parameter vector � and information vector �0(t) as

� =

⎡⎢⎢⎢⎢⎣

ac1bc2b...

cmb

⎤⎥⎥⎥⎥⎦ ∈ Rn0 ,

�0(t) =

⎡⎢⎢⎢⎢⎣

−x(t − 1)

−x(t − 2)...

−x(t − n)

�(t)

⎤⎥⎥⎥⎥⎦ ∈ Rn0 , n0 := (m + 1)n, (2)

a = [a1, a2, . . . , an]T ∈ Rn, b = [b1, b2, . . . , bn]T ∈ Rn,

c = [c2, c3, . . . , cm]T ∈ Rm−1,

�(t) = [�T1 (t), �T

2 (t), . . . , �Tm(t)]T ∈ Rmn, (3)

F. Ding et al. / Systems & Control Letters 56 (2007) 373–380 375

�j (t) = [�j (u(t − 1)), �j (u(t − 2)),

. . . , �j (u(t − n))]T ∈ Rn, j = 1, 2, . . . , m. (4)

Here, the superscript T denotes the matrix transpose. Then weobtain the identification model of the HOE systems:

x(t) = �T0 (t)�, y(t) = �T

0 (t)� + v(t). (5)

The systems in (5) by parameterization in (2) contain moreparameters than actually needed, but the benefit to do this isthat a linear regression form can be obtained. However, thedisadvantage is that it requires extra computation, which canbe approached using the hierarchical estimation methods basedon the hierarchical identification principle [8,12–15,18].

From (5), we can see that the information vector �0(t) con-sists of unknown noise-free outputs x(t − i) and system inputs.Let p be the data length (p?n0), and define

Y(p, t) :=

⎡⎢⎢⎣

y(t)

y(t − 1)...

y(t − p + 1)

⎤⎥⎥⎦ ,

�0(p, t) :=

⎡⎢⎢⎢⎣

�T0 (t)

�T0 (t − 1)

...

�T0 (t − p + 1)

⎤⎥⎥⎥⎦ ,

V(p, t) :=

⎡⎢⎢⎣

v(t)

v(t − 1)...

v(t − p + 1)

⎤⎥⎥⎦ . (6)

From (5) and (6), we have

Y(p, t) = �0(p, t)� + V(p, t). (7)

Because �0(p, t) contains unmeasured inner variablesx(t − i), i = 1, 2, . . . , n, the standard least-squares methodcannot be applied directly to obtain the least-squares estimate[�T

0 (p, t)�0(p, t)]−1�T0 (p, t)Y(p, t) of the parameter vector

� by minimizing the cost function ‖Y(p, t)−�0(p, t)�‖2 evenif V(p, t) is a “white” noise vector with zero mean, where‖X‖2 = tr[XXT].

If these unknowns x(t−i) are replaced by the outputs xa(t−i)

of an auxiliary (or reference) model,

xa(t) = Pa(z)f (u(t)), or xa(t) = �T(t)�a,

then the identification problem of � can be solved using xa(t)

instead of x(t). Here, Pa(z), �(t) and �a are the transfer func-tion, information vector and parameter vector of the auxiliarymodel, respectively. If we use the estimate of B(z)/A(z) as anauxiliary model Pa(z), namely, take �a to be the estimate �(t)

of �, and �(t) to be the regressive vector of xa(t) and u(t), anduse �(t) as �0(t), then the identification algorithms based onthis idea are called the auxiliary (or reference) model identifi-cation method or output-error method [21,31]. Of course, thereare other ways to choose auxiliary models, e.g., using the finiteimpulse response model [9,10].

According to the auxiliary model identification principle: theunknown variables x(t − i) in �0(t) are replaced by the outputxa(t − i) of the auxiliary model, �0(t) by �(t) and �0(p, t) by�(p, t); then it is easy from (7) to get a recursive pseudo-linearregression least-squares identification algorithm based on theauxiliary model as follows:

�(t) = �(t − 1) + P(t)�T(p, t)[Y(p, t) − �(p, t)�(t − 1)],(8)

P−1(t) = P−1(t − 1) + �T(p, t)�(p, t), P(0) = p0I, (9)

xa(t − i) = �T(t − i)�(t), i = 0, 1, 2, . . . , n, (10)

�(p, t) =

⎡⎢⎢⎣

�T(t)

�T(t − 1)...

�T(t − p + 1)

⎤⎥⎥⎦ ,

�(t) =

⎡⎢⎢⎢⎢⎣

−xa(t − 1)

−xa(t − 2)...

−xa(t − n)

�(t)

⎤⎥⎥⎥⎥⎦ ∈ Rn0 . (11)

Here, p0 is a large positive number, e.g., p0 =106, and we referto xa(t) as the estimate of x(t). Eqs. (8)) to (11) is also calledthe AMLS identification algorithm of estimating � in (7) ofthe Hammerstein output-error model, HOE-AMLS algorithmfor short. Note that the algorithm is combined in the sense theparameters and true outputs are estimated simultaneously, andcan be implemented on-line.

Note c1 = 1, the estimates a = [a1, a2, . . . , an]T and b =[b1, b2, . . . , bn]T of a and b can be read from the first andsecond n entries of �, respectively. Let �i be the ith elementof �, referring to the definition of �, then the estimates of cj ,j = 2, 3, . . . , m, may be computed by

cj = �jn+i/bi , j = 2, 3, . . . , m; i = 1, 2, . . . , n.

Since we do not need such n estimates cj , we may calculatethe c-parameters by a least-squares fit or take their average likein [11,20] as the estimate of cj , i.e.,

cj = 1

n

n∑i=1

�jn+i

bi

, j = 2, 3, . . . , m.

3. The performance analysis

Let us introduce some notation first. The symbol I standsfor an identity matrix of appropriate sizes; |X| = det[X] rep-resents the determinant of the square matrix X; �max[X] and�min[X] represent the maximum and minimum eigenvalues ofthe symmetric matrix X, respectively; for g(t)�0, we writef (t) = O(g(t)) if there exists positive constants �1 and t0 suchthat |f (t)|��1g(t) for t � t0.

376 F. Ding et al. / Systems & Control Letters 56 (2007) 373–380

Define

P−10 (t) =

t∑i=1

�T0 (p, i)�0(p, i) + 1

p0I;

r(t) = tr[P−1(t)]; r0(t) = tr[P−10 (t)].

Hence, we easily get

|P−1(t)|�rn0(t); r(t)�n0�max[P−1(t)];ln |P−1(t)| = O(ln r(t)). (12)

The stochastic process theory [27,16] and the stochastic mar-tingale theory [11,10,17,24] are the main tool of analyzing per-formance of identification algorithms. Here, we prove the mainconvergence results of the HOE-AMLS algorithm by using themartingale convergence theorem. To do this, some mathemati-cal preliminaries are required.

Lemma 1. For each i (i = 0, 1, . . . , p − 1), the following in-equalities hold:

〈1〉t∑

j=1

�T(j − i)P(j)�(j − i)� ln |P−1(t)| + n0 ln p0, a.s.,

〈2〉∞∑t=1

�T(t − i)P(t)�(t − i)

[ln |P−1(t)|]� < ∞, a.s., for any � > 1.

Proof. From the definition of P(t) in (9), we have

P−1(t − 1) = P−1(t) − �T(p, t)�(p, t)

�P−1(t) − �(t − i)�T(t − i)

= P−1(t)[I − P(t)�(t − i)�T(t − i)].Taking determinants on both sides and using the formula det[I+DE] = det[I + ED] yields

|P−1(t − 1)|� |P−1(t)||I − P(t)�(t − i)�T(t − i)|= |P−1(t)|[1 − �T(t − i)P(t)�(t − i)].

Hence

�T(t − i)P(t)�(t − i)� |P−1(t)| − |P−1(t − 1)||P−1(t)| . (13)

Replacing t with j and summing for j from 1 to t yields (notingthat |P−1(t)| is a nondecreasing function of t)

t∑j=1

�T(j − i)P(j)�(j − i)

�t∑

j=1

|P−1(j)| − |P−1(j − 1)||P−1(j)|

=t∑

j=1

∫ |P−1(j)|

|P−1(j−1)|dx

|P−1(j)|

�∫ |P−1(t)|

|P−1(0)|dx

x= ln |P−1(t)| − ln |P−1(0)|

= ln |P−1(t)| + n0 ln p0, a.s.

Dividing (13) by [ln |P−1(t)]� and summing for t give

∞∑t=1

�T(t − i)P(t)�(t − i)

[ln |P−1(t)|]�

�∞∑t=1

|P−1(t)| − |P−1(t − 1)||P−1(t)|[ln |P−1(t)|]�

�∫ |P−1(∞)|

|P−1(0)|dx

x(ln x)�= −1

� − 1

1

(ln x)�−1

∣∣∣∣|P−1(∞)|

|P−1(0)|

= 1

� − 1

(1

[ln |P−1(0)|]�−1− 1

[ln |P−1(∞)|]�−1

)< ∞, a.s. �

Define the parameter estimation error vector �(t) and a non-negative definite function W(t) as

�(t) = �(t) − �, (14)

W(t) = �T(t)P−1(t)�(t). (15)

Lemma 2. For the system in (7) and the HOE-AMLS algorithmin (8)–(11), assume the noise sequence {v(t)} with zero meanand bounded time-varying variance satisfies [24]:

(A1) E[v(t)|Ft−1] = 0, a.s.,(A2) E[v2(t)|Ft−1] = �2

v(t)� �2v < ∞, a.s.,

(A3) H(z) = 1A(z)

− 12 is strictly positive real,

where {v(t),Ft } is a martingale sequence defined on a prob-ability space {�,F, P } and {Ft } is the � algebra sequencegenerated by {v(t)}. Then the following inequality holds:

E[W(t) + S(t)|Ft−1]�W(t − 1) + S(t − 1)

+ 2p

p−1∑i=0

�T(t − i)P(t)�(t − i)�2v, a.s.,

where

S(t) = 2t∑

i=1

UT(i)Y(i), a.s., (16)

Y(t) = 12 �(p, t)�(t)

+ [Y(p, t) − �(p, t)�(t) − V(p, t)], (17)

U(t) = −�(p, t)�(t). (18)

Here, (A3) guarantees that S(t)�0.

The proof can be done in a similar way in [11,19] and isomitted here.

Next, we give prove the main results of this paper.

Theorem 1. For the system in (7), assume that (A1)–(A3) hold,and A(z) is stable, i.e., all zeros of A(z) are inside the unitcircle. Then for any � > 1, the parameter estimation error by

F. Ding et al. / Systems & Control Letters 56 (2007) 373–380 377

the HOE-AMLS algorithm in (8)–(11) satisfies:

‖�(t) − �‖2 = O

([ln r0(t)]�

�min[P−10 (t)]

), a.s.

Proof. From the definition of W(t), we have

‖�(t)‖2 � �T(t)P−1(t)�(t)

�min[P−1(t)] = W(t)

�min[P−1(t)] . (19)

Let

Z(t) = W(t) + S(t)

[ln |P−1(t)|]� .

Since ln |P−1(t)|] is nondecreasing, according to Lemma 2, wehave

E[Z(t)|Ft−1]

�Z(t − 1) + 2p

p−1∑i=0

�T(t − i)P(t)�(t − i)

[ln |P−1(t)|]� �2v, a.s. (20)

Using Lemma 1 and applying the martingale convergence the-orem (Lemma D.5.3 in [24]) to (20), we conclude that Z(t)

converges a.s. to a finite random variable, say, Z0; i.e.,

Z(t) = W(t) + S(t)

[ln |P−1(t)|]� → Z0 < ∞, a.s.,

or

W(t) = O([ln |P−1(t)|]�), a.s.,

S(t) = O([ln |P−1(t)|]�), a.s. (21)

Since H(z) is a strictly positive real function, from the defini-tion of S(t) and referring to [11], we have

t∑i=1

‖U(i)‖2 = O([ln |P−1(t)|]�).

From (19), (21) and (12), we have

‖�(t)‖2 = O

([ln |P−1(t)|]��min[P−1(t)]

)

= O

([ln r(t)]�

�min[P−1(t)]

), a.s. for any � > 1. (22)

Like in [11,19], it follows that there exist positive constants k1and k2 such that

t∑i=1

‖�(i) − V(p, i)‖2 �k1

t∑i=1

‖U(i)‖2 + k2

= O([ln |P−1(t)|]�) = O([ln r(t)]�). (23)

Using the similar way in [11,19], it is easy to get r(t)=O(r0(t)),�min[P−1(t)] = O(�min[P−1

0 (t)] + O([ln r0(t)]�). Thus, from

(22), we have

‖�(t) − �‖2 = O

([ln r0(t)]�

�min[P−10 (t)] + [ln r0(t)]�

)

= O

([ln r0(t)]�

�min[P−10 (t)]

), a.s. for any � > 1.

This proves Theorem 1. �

From Theorem 1, we can see that if [ln r0(t)]� =o(�min[P−1

0 (t)]) (namely, the generalized persistent excitation(PE) condition [10], the weak PE condition [10] or the strongPE condition [16] holds), then the estimation error �(t) − �

converges to zero. This requires that G(z) is stable, u(t) is aPE signal, and so is u(t) = f −1(u(t)). The positive real con-dition in (A3) depends on the unknown model parameters andis a more restrictive assumption than A(z) being stable [27].

As p = 1, e(t) := y(t) − �T(t)�(t − 1) ∈ R1 is called theinnovation [26], and E(t) := [Y(p, t) − �(p, t)�(t − 1)] ∈Rp is referred as to the innovation vector. p here may be alsoknown as the innovation length, the algorithm in (8)–(11) isalso called the auxiliary model multi-innovation least-squaresidentification algorithm for Hammerstein output-error systems.When p = 1, we get a simple HOE-AMLS algorithm:

�(t) = �(t − 1) + P(t)�(t)[y(t) − �T(t)�(t − 1)], (24)

P(t) = P(t − 1) − P(t − 1)�(t)�T(t)P(t − 1)

1 + �T(t)P(t − 1)�(t), (25)

xa(t) = �T(t)�(t), (26)

�(t) =

⎡⎢⎢⎢⎢⎣

−xa(t − 1)

−xa(t − 2)...

−xa(t − n)

�(t)

⎤⎥⎥⎥⎥⎦ ∈ Rn0 . (27)

4. Example

An example is given to demonstrate the effectiveness of theproposed algorithms. Consider the following system:

y(t) = B(z)

A(z)u(t) + v(t),

A(z) = 1 + a1z−1 + a2z

−2 = 1 − 1.60z−1 + 0.80z−2,

B(z) = b1z−1 + b2z

−2 = 0.85z−1 + 0.65z−2,

u(t) = f (u(t)) = c1u(t) + c2u2(t) + c3u

3(t)

= u(t) + 0.5u2(t) + 0.25u3(t),

�s = [a1, a2, b1, b2, c2, c3]T.

{u(t)} is taken as a PE signal sequence with zero mean andunit variance �2

u = 1.002, and {v(t)} as a white noise sequencewith zero mean and constant variance �2

v = 0.502 and �2v =

2.002. Apply the HOE-AMLS algorithm with p = 1 to esti-mate the parameters of this system, the parameter estimates� = [1, 2, . . . , 8]T and �s and their errors with different

378 F. Ding et al. / Systems & Control Letters 56 (2007) 373–380

Table 1The estimates of � (�2

v = 0.502)

t 1 2 3 4 5 6 7 8 � (%)

100 −1.56901 0.77630 0.68233 1.04526 0.37656 0.40992 0.21864 0.13271 20.43326200 −1.57996 0.78122 0.79989 0.91609 0.35609 0.37159 0.22293 0.15234 13.14011300 −1.57992 0.78325 0.83494 0.85387 0.36760 0.39116 0.20882 0.15149 10.34029500 −1.59002 0.78999 0.87510 0.79772 0.37839 0.39249 0.19782 0.15553 7.939381000 −1.58361 0.78623 0.85321 0.71349 0.37916 0.39854 0.20294 0.17042 5.085421500 −1.59141 0.79217 0.83909 0.69136 0.37660 0.38461 0.21061 0.17118 4.108872000 −1.59375 0.79512 0.86519 0.69088 0.38268 0.38439 0.20713 0.16963 3.956822500 −1.59131 0.79218 0.85338 0.68343 0.38296 0.38079 0.21037 0.16826 3.625983000 −1.59376 0.79477 0.85300 0.66855 0.38739 0.37451 0.21052 0.16821 3.03148

True values −1.60000 0.80000 0.85000 0.65000 0.42500 0.32500 0.21250 0.16250

Table 2The estimates of (ai , bi , ci ) (�2

v = 0.502)

t a1 a2 b1 b2 c2 c3 � (%)

100 −1.56901 0.77630 0.68233 1.04526 0.47202 0.22370 20.05581200 −1.57996 0.78122 0.79989 0.91609 0.42540 0.22250 13.13781300 −1.57992 0.78325 0.83494 0.85387 0.44918 0.21376 9.97809500 −1.59002 0.78999 0.87510 0.79772 0.46220 0.21051 7.419681000 −1.58361 0.78623 0.85321 0.71349 0.50149 0.23835 3.155391500 −1.59141 0.79217 0.83909 0.69136 0.50256 0.24930 2.057652000 −1.59375 0.79512 0.86519 0.69088 0.49934 0.24247 2.083522500 −1.59131 0.79218 0.85338 0.68343 0.50296 0.24636 1.662603000 −1.59376 0.79477 0.85300 0.66855 0.50717 0.24920 1.00617

True values −1.60000 0.80000 0.85000 0.65000 0.50000 0.25000

Table 3The estimates of � (�2

v = 2.002)

t 1 2 3 4 5 6 7 8 � (%)

100 −1.54443 0.75677 0.35049 1.37101 0.41764 0.43136 0.25470 0.10459 41.00730200 −1.55876 0.76487 0.67966 1.14544 0.33138 0.36260 0.26717 0.13651 24.88620300 −1.55951 0.76570 0.77079 1.03161 0.37081 0.42585 0.21899 0.12295 18.98097500 −1.58706 0.78914 0.85335 0.93939 0.35883 0.43890 0.18798 0.13346 14.787861000 −1.56785 0.77375 0.81059 0.78027 0.33976 0.47286 0.20004 0.17329 10.274591500 −1.58573 0.78720 0.77961 0.76794 0.32739 0.43761 0.22060 0.16684 9.396542000 −1.59032 0.79314 0.87038 0.77592 0.34143 0.44251 0.20796 0.16391 8.897192500 −1.58242 0.78315 0.83949 0.76061 0.33394 0.43979 0.21446 0.15866 8.555833000 −1.59036 0.79298 0.84170 0.71035 0.34846 0.42285 0.21475 0.16187 6.40464

True values −1.60000 0.80000 0.85000 0.65000 0.42500 0.32500 0.21250 0.16250

Table 4The estimates of (ai , bi , ci ) (�2

v = 2.002)

t a1 a2 b1 b2 c2 c3 � (%)

100 −1.54443 0.75677 0.35049 1.37101 0.75310 0.40149 43.00408200 −1.55876 0.76487 0.67966 1.14544 0.40206 0.25613 24.82545300 −1.55951 0.76570 0.77079 1.03161 0.44694 0.20165 18.52710500 −1.58706 0.78914 0.85335 0.93939 0.44386 0.18118 14.049421000 −1.56785 0.77375 0.81059 0.78027 0.51259 0.23443 6.658121500 −1.58573 0.78720 0.77961 0.76794 0.49490 0.25011 6.430232000 −1.59032 0.79314 0.87038 0.77592 0.48129 0.22509 6.109212500 −1.58242 0.78315 0.83949 0.76061 0.48800 0.23203 5.364743000 −1.59036 0.79298 0.84170 0.71035 0.50463 0.24150 2.91094

True values −1.60000 0.80000 0.85000 0.65000 0.50000 0.25000

F. Ding et al. / Systems & Control Letters 56 (2007) 373–380 379

0 500 1000 1500 2000 2500 3000

0

0.2

0.4

0.6

0.8

1

�v

2

= 0.502

�v

2

= 2.002

t

�

Fig. 2. The parameter estimation errors � vs. t.

0 500 1000 1500 2000 2500 3000

0

0.2

0.4

0.6

0.8

1

v

2

= 0.502

�v

2 = 2.00

2

t

� s

�

Fig. 3. The parameter estimation errors �s vs. t.

noise variances are shown in Tables 1–4, and the parameterestimation errors � and �s vs. t are shown in Figs. 2 and 3,�=‖�(t)−�‖/‖�‖, �s =‖�s(t)−�s‖/‖�‖, �s(t), is the estimateof �s .

From Tables 1–4 and Figs. 2 and 3, we can draw the followingconclusions:

• A high noise level results in a slow rate of convergence ofthe parameter estimates to the true parameters.

• It is clear that the errors � and �s are becoming smaller (ingeneral) as t increases. This confirms the proposed theorem.

5. Conclusions

A recursive AMLS algorithm based on replacing unavailablevariables (noise-free outputs) by their estimates is derived for

Hammerstein output-error models. The analysis using the mar-tingale convergence theorem indicates that the proposed algo-rithm can give consistent parameter estimation.

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