September 21, 2006 21:54 International Journal of Control ”A Blind Approach to Closed-Loop ID of Hammerstein SYS”

International Journal of Control

Vol. 00, No. 00, DD Month 200x, 1–18

A Blind Approach to Closed-Loop Identification of Hammerstein Systems

Jiandong Wanga, Akira Sanob, David Shookc, Tongwen Chena, ∗, and Biao Huangd

a Dept. of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada T6G 2V4.b Dept. of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Japan 223-8522.

c Matrikon Inc., Suite 1800, 10405 Jasper Ave., Edmonton, Canada T6N 4A3.d Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada T6G 2G6.

(Received 00 Month 200x; In final form 00 Month 200x)

This paper is a continuing study of the blind approach for the Hammerstein identification in Sun, Liu and Sano (1999) and Bai and Fu(2002). In the framework of a closed-loop sampled-data system, the output is sampled faster than the updating period of the input. Theparameters of the linear dynamics are consistently estimated from the information of the output only, after which the unmeasurable innersignal is uniquely reconstructed. The noise effect is explicitly considered in both the parameter and inner signal estimation. The estimationof the system orders and time delay are studied on the basis of two groups of basic equations obtained by polyphase decomposition. Theproposed blind approach is validated and illustrated by a simulated numerical example.

Keywords: Hammerstein systems, nonlinear identification, blind identification, output fast-sampling

1 Introduction

Hammerstein systems form a class of block-oriented nonlinear models, where a static nonlinearity precedesa linear dynamic subsystem – see Chapter 1 of Janczak (2005) for a recent overview on the identificationof block-oriented nonlinear systems. Many real-time processes can be well represented by Hammersteinmodels, such as distillation columns, heat exchangers (Eskinat, Johnson and Luyben 1991), electrical drives(Balestrino et al. 2001), thermal microsystems (Sung 2002), and sticky valves (Srinivasan 2005).

One possible way to classify the identification of Hammerstein systems is based on whether the structureof the nonlinearity is known a prior. If the nonlinearity is parameterized by a polynomial model or otherknown basic functions, many methods have been devised (see, e.g., Chapter 5 in Janczak (2005)). Thispaper studies the other case where the structure is unknown. Such a case arises when the nonlinearity hasmany possible structures or is hard to be represented by parametric models. In particular, one real-timeapplication is to capture the nonlinearities of actuators in feedback control systems. It has been found thatcontrol valves account for about one third of control-loop oscillations (Biakowski 1992; Desborough andMiller 2001). The nonlinearity of an actuator has a variety of possible structures, e.g., deadband, saturation,backlash and hysteresis (Choudhury et al. 2005). Srinivasan et al. (2005) demonstrated the potentiality ofexploiting the Hammerstein identification in diagnosing valve stiction; however, their approach was basedon a separable least-squares identification algorithm proposed by Bai (2002) and applicable to only thenonlinearity with known structure and one single unknown parameter.

When the structure of the nonlinearity is unknown, the relay feedback approach and the blind approachcould be applied. The relay feedback approach was first proposed in Astrom and Hagglund (1984) andfurther developed in Luyben and Eskinat (1994), Balestrino et al. (2001), Sung (2002), Park et al. (2004)and Bai (2004). Its essence is to have a binary-valued input by the user design in open-loop systems or by a

∗Corresponding author. Telephone: (780)492-3940; Fax: (780)492-1811; E-mail: tchen@ece.ualberta.ca.

International Journal of Control

ISSN 0020-7179 print/ ISSN 1366-5820 online c©2005 Taylor & Francis Ltd

http://www.tandf.co.uk/journals

DOI: 10.1080/00207170xxxxxxxxxxxx

September 21, 2006 21:54 International Journal of Control ”A Blind Approach to Closed-Loop ID of Hammerstein SYS”

2

relay feedback-controller in closed-loop systems, so that identification of the linear dynamics is decoupledfrom that of the nonlinearity. The blind approach was formulated in Sun, Liu and Sano (1999) and furtherstudied in Bai and Fu (2002). The basic idea is as follows. First, the process output is sampled faster thanthe updating period of the input, which is piece-wise constant because of the user design or more often owingto the zero-order hold (ZOH) in sampled-data systems. Second, the linear dynamics is estimated solelyfrom the output measurement. Next, the unmeasurable inner signal between the nonlinearity and lineardynamics is estimated from the output measurement and the inverse of the identified linear dynamics1.Finally, the nonlinearity is captured from a graph of the measured input and the estimated inner signal.

This paper is a continuing study of the blind approach for the Hammerstein identification in the frame-work of closed-loop systems. Its main contributions are: (i) We consider the noise-corrupted cases insteadof the noise-free ones in Bai and Fu (2002). The battle against noises leads to a new series of staticerrors-in-variables (EIV) systems. By contrast to Sun, Liu and Sano (1999), the realization of static EIVsystems significantly reduces the complexity of estimating the numerator parameters (Section 3). (ii) Theestimation of the process orders and the time delay, ignored in Sun, Liu and Sano (1999) and Bai and Fu(2002), is resolved in a simple way on the basis of two groups of equations (Eqs. (10) and (11)) formed bythe polyphase decomposition. (iii) Bai and Fu (2002) estimated the inner signal by taking a direct inverseor by exploiting Bezout identity for minimum and non-minimum phase linear dynamics, respectively; itsdrawback is the propagation of the output noise into the estimates. The counterpart in Sun, Liu and Sano(1999) is unnecessary complicated due to the way in estimating the numerator parameters. We estimatethe inner signal differently by a least-squares method borrowed from the blind equalization; by doing so,the noise effect is reduced. On the other hand, as a continuing study, the proposed blind approach inheritssome technical features from those in Sun, Liu and Sano (1999) and Bai and Fu (2002), e.g., AssumptionsA1, A2 and A4 in Section 2 and the estimation of the denominator parameters in Section 3.2.

The rest of the paper is organized as follows. Section 2 describes the problem and gives some necessaryassumptions. Section 3 estimates the parameters of the linear dynamics, the numerator and denominatororders, and the time delay. With the identified linear dynamics, the inner signal is estimated in Section 4.The consistency of parameter estimation and the uniqueness of inner signal estimation are proved inSection 5. Section 6 illustrates the proposed blind approach by a simulated numerical example. Someconcluding remarks are provided in Section 7.

2 Problem Description

Consider a sampled-data closed-loop Hammerstein system depicted in Figure 1. The process Gc(s) is alinear time-invariant causal continuous-time dynamic system. Its input u is an unmeasurable inner signalbetween Gc(s) and the static nonlinearity fc. The structure of fc is unknown for the reasons discussed inSection 1. The output x of a discrete-time controller C(z) is updated with period T via the ZOH. No otherinformation of C(z) is required in the proposed blind approach, as long as the whole feedback loop is stableand has at least one-sample time delay to avoid algebraic loops. Contaminated additively by noise v, theprocess output y is sampled faster with the period h := T/p to yield y(t). Here p is a positive integer,usually named as the fast-sampling ratio.

•y

u v+

++

−cf

Controller ProcessNonlinearity

ZOH

TT/p h

( )cG s( )C z

( )y t( )x n( )r n

Figure 1. A sampled-data closed-loop Hammerstein system with output fast-sampling

1A special treatment is necessary if the linear dynamics is non-minimum phase.

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3

Owing to the ZOH, a fast-rate version of the controller output x(n) is available at the sampling periodh, denoted as x(t). By taking x(t) and y(t) as the input and output, respectively, the one in the dot-dashbox of Figure 1 is equivalent to a discrete-time Hammerstein system with the sampling period h, depictedin Figure 2. The objective of the Hammerstein identification is to estimate G(q) and f in Figure 2, thediscrete-time counterparts of Gc and fc, respectively.

( )G q( )x t( )v t+

+f

�������

( )u t

���������

( )y t

Figure 2. An equivalent discrete-time Hammerstein system at the sampling period h

Because the ZOH works at the period T = ph, x(t) is constant during one period of T , i.e., x(t)−x(t−1) =0, for (kp + 1) ≤ t < (kp + p), ∀k ∈ Z+, Z+ being the set of nonnegative integers. The nonlinearity f isstatic, i.e., u(t) = f(x(t)); thus, the inner signal u(t) inherits the same property:

u(t) − u(t − 1) = 0, for (kp + 1) ≤ t < (kp + p), ∀k ∈ Z+. (1)

The property is essential for the blind approach. In fact, f does not have to be static in order to pass thepiece-wise constant property of x(t) to u(t), e.g., the backlash nonlinearity in the example of Section 6.Considering a well-known fact that a high-order ARX model is capable of approximating any linear systemarbitrarily well (Ljung (1999) (Page 336)), we describe the linear process and noise dynamics at thesampling period h as

y (t) =B (q)

A (q)u (t − τ) +

1

A (q)e (t) , (2)

where

A (q) = 1 + a1q−1 + a2q

−2 + · · · + anaq−na ,

B (q) = b1q−1 + b2q

−2 + · · · + bnbq−nb .

Here q−1 appeared earlier is the backward shift operator: q−1u(t) = u(t− 1); the noise source e(t) is whitewith zero mean and variance σ2. The blind approach consists of two steps: first, the parameters ai and bj

in A(q) and B(q) are identified together with the orders na and nb and the time delay τ from the collected

output data {y(t)}Nt=1 := {y(1), y(2), · · · , y(N)}; second, the estimated process G(q) and y(t) provide the

estimates of the unmeasurable inner signal u(t) or its slow-rate version U(n) := u(pn), which, along withx(n), yields an operating graph of f .

We make the following assumptions throughout the paper:

A1. The reference signal r is persistently excited (PE) and independent to the noise v.A2. The upper bound n0

b of the order nb is known a prior, and the fast-sampling ratio p is no less thann0

b + 1, i.e., p ≥ n0b + 1.

A3. The portion of the time delay τ , being an integer multiple of p, is known a prior.A4. B(q) does not have the zero at 1, i.e.,

∑nb

k=1 bk 6= 0.

Assumption A1 is standard in closed-loop identification, where a PE external signal is necessary to achievethe identifiability. For Hammerstein systems, “identifiability” is understood with a gain ambiguity betweenf and G(q); the ambiguity can be removed by letting b1 = 1. Assumption A2 is inherent for all the blindapproaches in Sun, Liu and Sano (1999) and Bai and Fu (2002); in fact, Theorem 2.1 in Bai, Li andDasgupta (2002) has shown that Assumption A2 is a sufficient and necessary condition for G(q) to beblindly identifiable. Assumption A3 is reasonable, since the information of y(t) only (without input) cannotdistinguish time delays τ1 = k1p and τ2 = k2p for k1 6= k2; however, fast-sampling output is capable of

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estimating τ ∈ [0, p). Assumption A4 makes the estimation of the inner signal unique (see Theorem 5.3),and is a mild assumption satisfied by most systems.

3 Identification of Linear Dynamics

First, two groups of equations are obtained by the polyphase decomposition of involved signals. Based onthem, the parameters in the linear dynamics, the orders na, nb and the time delay τ are estimated. Finally,we propose the instrumental variable method as an alternative for parameter estimation.

3.1 Two Groups of Basic Equations

For the time being, the orders na and nb and the time delay τ are assumed to be known; thus, τ becomeszero after shifting data properly. We will return to the estimation of na, nb and τ later in Section 3.4. Bydenoting w(t) := A (q) y (t), (2) becomes

w (t) = B(q)u (t) + e (t)

=

nb∑

j=1

bju (t − j) + e (t) . (3)

Subtracting two consecutive samples w(t) and w(t − 1) yields

w (t) − w (t − 1) =

nb∑

j=1

bj (u (t − j) − u (t − j − 1)) + e (t) − e (t − 1) . (4)

Define the difference signals ∆w (t) := w (t) − w (t − 1), and ∆u (t) and ∆e (t) likewise. Eq. (4) can berewritten as

∆w (t) =

nb∑

j=1

bj∆u (t − j) + ∆e (t) ,

whose z-transformation is

∆w (z) =

nb∑

j=1

bjz−j∆u (z) + ∆e (z) . (5)

The polyphase decomposition of ∆w (z) for the factor of p is (Vaidyanathan, 1993; Fliege, 1994)

∆w (z) =∑

t

∆w (t) z−t

=∑

k

p∑

l=1

∆w (kp + l) z−(kp+l)

=

p∑

l=1

z−l∑

k

∆w (kp + l) (zp)−k

=:

p∑

l=1

z−l∆(l)w (zp) . (6)

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Similarly, the polyphase decomposition of ∆u (z) is

∆u (z) =

p∑

l=1

z−l∆(l)u (zp) , (7)

where

∆(l)u (zp) :=

∑

k

∆u (kp + l) (zp)−l .

Thanks to the property of u (t) in (1), ∆u (t) is nonzero only at t = kp, ∀k ∈ Z+. Thus, we have

∆(l)u (zp) = 0, l = 1, 2, · · · , p − 1,

∆(p)u (zp) 6= 0.

Eq. (7) reduces to

∆u (z) = z−p∆(p)u (zp) . (8)

Substituting (6) and (8) into (5) yields

p∑

l=1

z−l∆(l)w (zp) =

nb∑

l=1

blz−lz−p∆(p)

u (zp) +

p∑

l=1

z−l∆(l)e (zp) . (9)

With Assumption A2, i.e., p ≥ (n0b + 1) > nb, (9) implies

∆(nb+1)w (zp) = ∆(nb+1)

e (zp) , (10a)

∆(nb+2)w (zp) = ∆(nb+2)

e (zp) , (10b)

...

∆(p)w (zp) = ∆(p)

e (zp) , (10c)

and

∆(1)w (zp) = b1z

−p∆(p)u (zp) + ∆(1)

e (zp) , (11a)

∆(2)w (zp) = b2z

−p∆(p)u (zp) + ∆(2)

e (zp) , (11b)

...

∆(nb)w (zp) = bnb

z−p∆(p)u (zp) + ∆(nb)

e (zp) . (11c)

The two groups of equations (10) and (11) are the bases to estimate the parameters in A(q) and B(q).Moreover, they make the estimation of the numerator order nb and the time delay τ possible. The idea ofexploiting the polyphase decomposition is inspired by Bai and Fu (1999) where only noise-free cases wereconsidered. A timing diagram of polyphase decomposition of signals is available in many textbooks, e.g.,Figure 1.4 in Fliege (1994). The two groups of equations (10) and (11) can also be seen from Figure 3appeared later.

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3.2 Estimation of A(q)

In the time domain, (10) implies

∆w (kp + l) = ∆e (kp + l) , l = nb + 1, nb + 2, · · · , p, ∀k ∈ Z+. (12)

Define ∆y analogously to ∆w and ∆e, i.e., ∆y (kp + l) := y (kp + l)− y (kp + l − 1). Eq. (12) is written interms of ∆y as

∆y (kp + l) =

na∑

i=1

−ai∆y (kp + l − i) + ∆e (kp + l)

= φTy (k) θa + ∆e (kp + l) , l = nb + 1, nb + 2, · · · , p, ∀k ∈ Z+, (13)

where

θa =[

a1 a2 · · · ana

]T,

φy (k) =[

−∆y (kp + l − 1) −∆y (kp + l − 2) · · · −∆y (kp + l − na)]T

.

Eq. (13) is linear in the parameter ai. However, if the ordinary least-squares method (LSM) is applied to(13) with the collected data {y(t)}N

t=1, i.e.,

θa =

[

1

K

K−1∑

k=0

φy (k)φTy (k)

]−11

K

K−1∑

k=0

φy (k) ∆y (kp + l) ,

the resulted estimate θa is biased. Here K is the largest integer less than or equal to N/p. The bias arisefrom the correlation between the noise term ∆e(kp + l) with the first regressor ∆y(kp + l − 1), which canbe resolved by a bias-compensated LSM. The difference between θa and θa is

θa − θa =

[

1

K

K−1∑

k=0

φy (k)φTy (k)

]−11

K

K−1∑

k=0

φy (k) ∆e (kp + l) .

As K → ∞,

limK→∞

1

K

K−1∑

k=0

φy (k) ∆e (kp + l) = E {φy (k) ∆e (kp + l)} =[

σ2 0 · · · 0]T

1×na

.

Expression E{ · } denotes expectation. Recall that σ2 is the variance of the noise source e(t). Thus, θa canbe estimated without bias by explicitly compensating the noise effect, i.e.,

θ(l)a =

[

1

K

K−1∑

k=0

φy (k)φTy (k)

]−1

[

1

K

K−1∑

k=0

φy (k)∆y (kp + l)

]

−

σ2

0...0

. (14)

A consistent estimate of σ2 was developed in Theorem 1 of Kagiwada et al. (1998): σ2 is the smaller oneof the roots (x1, x2) of a quadratic equation in x, i.e.,

σ2 = min (x1, x2) . (15)

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The quadratic equation in x is

0.5g11x2 − x + σ2 = 0,

where

g11 =

[

1

K

K−1∑

k=0

φy (k)φTy (k)

]−1

11

,

σ2 =1

K

K−1∑

k=0

(

∆y (kp + l) − φTy (k) θa

)2.

Here {·}11 stands for the first-column and first-row element of the operand matrix.

3.3 Estimation of B (q)

Let b1 = 1 to remove a scalar ambiguity between the process G (q) and the nonlinearity f ; thus, (11a)gives

z−p∆(p)u (zp) = ∆(1)

w (zp) − ∆(1)e (zp) . (16)

Substituting (16) into the other equations in (11) yields

∆(l)w (zp) = bl

(

∆(1)w (zp) − ∆(1)

e (zp))

+ ∆(l)e (zp) , l = 2, 3, · · · , nb,

which implies that in the time domain,

∆w (kp + l) = bl (∆w (kp + 1) − ∆e (kp + 1)) + ∆e (kp + l) , l = 2, 3, · · · , nb, ∀k ∈ Z+. (17)

Taking ∆w (kp + 1) and ∆w (kp + l) as the input and output, respectively, (17) is a static errors-in-variables(EIV) system with the input and output noises, ∆e (kp + 1) and ∆e (kp + l), respectively. It is straightfor-ward to derive the following properties of ∆e (kp + 1) and ∆e (kp + l) by considering the facts that p ≥ 2and e(t) is white noise with variance σ2:

(i) Both ∆e (kp + 1) and ∆e (kp + l) are white noises, having the same variance 2σ2.(ii) If l = 2, ∆e (kp + 1) and ∆e (kp + l) are correlated; their correlation is equal to σ2δ (k), where δ (·)

denotes the Dirac delta function.(iii) If l = 3, 4, · · · , nb, ∆e (kp + 1) and ∆e (kp + l) are mutually independent.

Due to the second property, some of the existing identification methods for EIV systems, e.g., the totalleast-squares method, cannot be applied directly to estimate b2. In parallel to Section 3.2, we propose anew bias-compensated LSM to estimate bl,

bl =

[

1

K

K−1∑

k=0

∆2w (kp + 1) − 2σ2

]−1 [

1

K

K−1∑

k=0

∆w (kp + 1)∆w (kp + l) + σ2δ (l − 2)

]

. (18)

Here σ2 has been obtained in (15). Under Assumptions A1-A3, Theorem 5.2 in Section 5 proves that bl in(18) is a consistent estimate.

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3.4 Determination of na, nb and τ

This subsection briefly describes the principles to determine the orders na, nb and the time delay τ ; theseprinciples are implemented in a slightly complicated way in Section 6 and are also illustrated by an exampletherein.

As θ(l)a in (14) is based on the linear regression in (13), the determination of na is rather standard by

the model structure determination methods in Section 11.5 in Soderstrom and Stoica (1989) and Section16.4 in Ljung (1999). For instance, if the so-called Akaike information criterion (AIC) is adopted as thecriterion related to the prediction error of (13),

V (l) (na) =

(

1 + 2na

K

)

1

K

K−1∑

k=0

(

∆y (kp + l) − φTy (k) θ(l)

a

)2, l = nb + 1, nb + 2, · · · , p, (19)

na would be the integer associated with the minimum value of V (l) (na).The time delay τ is determined by a careful observation of the two groups of equations in (10) and (11).

If τ is nonzero, to let the counterpart of (4) reach one of the equations in (10) needs two inequalities:kp ≤ t − τ − nb − 1 and t − τ − 1 ≤ kp + p − 1 for some integer k ∈ Z+. The inequalities say that twotime delays τ1 = k1p + τ0 and τ2 = k2p + τ0 for k1 6= k2 and τ0 ∈ [0, p) cannot be distinguished from theinformation of y(t) only. Hence, the time delay τ in (2) is assumed in the range of [0, p) without loss ofgenerality under Assumption A3. In this case, (10) and (11) respectively hold for

nb + 1 + τ ≤ l ≤ p + τ and 1 + τ ≤ l ≤ nb + τ.

They imply that V (p) (na) defined in (19) has a non-zero contribution from ∆u extra to that from ∆e for nb

times, and a sole contribution from ∆e for (p−nb) times, when the output is consecutively shifted forwardby l = 0, 1, · · · , p − 1 samples, i.e., y(t) = y(t + l). Therefore, τ is the largest number of the consecutiveshifts resulting (p − nb) equivalent smallest numbers among all the p values of V (p) (na). In practice, theestimation of na and τ are lumped together by looking at V (p) (na)’s for different combinations of na andτ — see the example in Section 6.

The two groups of equations in (10) and (11) also tell the order nb once na, τ and A(q) have been

estimated. Due to the contribution from ∆u, the first nb ∆(l)w ’s have the larger variances than the rest,

∆(nb+1)w , ∆

(nb+2)w , · · ·, ∆

(p)w , which have the same variance 2σ2; thus, nb is determined as the difference

between p and the number of equivalent smallest ∆(l)w ’s. In fact, nb can also be determined from the

number of equivalent V (p) (na)’s; however, we would prefer to decouple the estimation of na and τ fromthat of nb, which is found in simulations to be easier and more robust.

3.5 Instrumental Variable Method

If an external signal being independent with e(t) is available, e.g., the reference signal r in Figure 1, wecould use the instrumental variable method (IVM) to estimate parameters in A(q) and B(q). Based on(13), the parameters in A (q) are estimated by the (extended) IVM,

θ(l)a =

[

1

K

K−1∑

k=0

ζ (k) φTy (k)

]†

1

K

K−1∑

k=0

ζ (k)∆y (kp + l) , l = nb + 1, nb + 2, · · · , p, (20)

where

ζ (k) =[

r (k − 1) r (k − 2) · · · r (k − m)]T

, m ≥ na.

Superscript (†) represents Moore-Penrose pseudoinverse. Note that the elements in ζ (k) cannot be r(k+j)for j ∈ Z+ in order to make ζ (k) correlated to φT

y (k). From (17), bl is estimated by the (extended) IVM

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as

bl =

[

1

K

K−1∑

k=0

ζ (k)∆w (kp + 1)

]†

1

K

K−1∑

k=0

ζ (k) ∆w (kp + l) , l = 2, 3, · · · , nb, (21)

where

ζ (k) =[

r (k − 1) r (k − 2) · · · r (k − m)]T

, m ≥ 1.

Accompanying with their simplicity, the IVMs have the drawback of being sub-optimal, i.e., the variancesof the estimated parameters are usually larger than those of the estimates from the bias-compensatedLSMs — see the example in Section 6; however, the IVM could play a role in cross-checking or validation,i.e., we would have more confidence on the estimates from the the bias-compensated LSM, if they matchthe counterparts from the IVM.

4 Inner Signal Estimation

With A(q) and B(q) in hand, the second step of the blind Hammerstein identification is to estimate theunmeasurable inner signal u(t), or equivalently its slow-rate version U (n) := u(pn). We first connectsthe special finite impulse-response (FIR) system in (3) with its equivalent single-input multiple-output(SIMO) counterpart, and then estimate U (n) by a method borrowed from the blind equalization (see, e.g.,Abed-Meraim, Qiu and Hua (1997)). The method has no differentiation on minimum or non-minimumphase systems.

In general, a fast-rate FIR model like (3) with the sampling period h can be described as

w (t) =∞

∑

j=1

h (j) u (t − j) + e (t) . (22)

Owing to the property in (1), (22) is equivalent to a slow-rate SIMO FIR model with the sampling periodT ,

W (n) =∞

∑

k=0

H (k)U (n − k) + E (n) , (23)

where

W (n) =

w1 (n)w2 (n)

...wp (n)

:=

w (pn + 1)w (pn + 2)

...w (pn + p)

,

E (n) =

e1 (n)e2 (n)

...ep (n)

:=

e (pn + 1)e (pn + 2)

...e (pn + p)

,

U (n) := u (pn) .

By substituting (22) into (23) and exploiting the property in (1), the impulse response of the SIMO model

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1wU1

2

, ,0,0,bn

kk

b b=

∑ �

++

1e

2w+

+2e

pe

�

1 23

, ,0,0,bn

kk

b b b=

+ ∑ �

�

1bnw −

++1

1

, ,0,0,b

b

n

k nk

b b−

=

∑ �

bnw+

+

1

,0,0,0,bn

kk

b=

∑ �

pw+

+

1

,0,0,0,bn

kk

b=

∑ �

1bne −

bne

�

�

Figure 3. An equivalent slow-rate SIMO model and its impulse responses

is connected with that of the fast-rate model in (22) as

H (k) =

h1 (k)h2(k)

...hp(k)

=

∑p−1l=0 h (kp + 1 − l)

∑p−1l=0 h (kp + 2 − l)

...∑p−1

l=0 h (kp + p − l)

. (24)

We now return to the special fast-rate FIR model in (3) that has the impulse responses,

h(0) = 0, h(1) = b1, h(2) = b2, · · · , h(nb) = bnb, h(nb+1) = 0, · · · , h(p) = 0, · · ·

Its equivalent slow-rate SIMO model according to (24) has only two non-zero impulse responses, i.e.,

H (0) =

h1 (0)h2 (0)

...hnb

(0)hnb+1 (0)

...hp (0)

=

b1

b1 + b2...

∑nb

k=1 bk∑nb

k=1 bk

...∑nb

k=1 bk

, H (1) =

h1 (1)h2 (1)

...hnb

(1)hnb+1 (1)

...hp (1)

=

∑nb

k=2 bk∑nb

k=3 bk

...bnb

0...0

, (25)

and H(k) = 0p×1, ∀k ≥ 2. For clarity, the SIMO model is depicted in Figure 3. The data of the i-th outputwi (n) are associated with those of the unknown input U(n) as

wi =[

wi (1) wi (2) · · · wi (K − 1)]T

=[

w (p + i) w (2p + i) · · · w ((K − 1) p + i)]T

September 21, 2006 21:54 International Journal of Control ”A Blind Approach to Closed-Loop ID of Hammerstein SYS”

11

=

hi (1) hi (0) 0 · · · 0

0 hi (1) hi (0). . .

......

. . .. . .

. . . 00 · · · 0 hi (1) hi (0)

U (0)U (1)

...U (K − 1)

+

ei (0)ei (1)

...ei (K − 1)

=: HiU + Ei,

where hi(0) and hi(1) are given in (25). Since e1, e2, · · · , ep are mutually independent and have the samevariance, it is reasonable to stack all yi’s together, i.e.,

W =

w1

w2...

wp

=

H1

H2...

Hp

U+

E1

E2...

Ep

=: HU + E. (26)

Based on (26) with w(t) = A(q)y(t) and bl, a least-squares estimate is obtained,

U =(

HT H)−1

HTW, (27)

which is also a maximum-likelihood estimate if el is white and Gaussian noise.

5 Theoretical Analysis

This section analyzes the consistency of the estimated parameters ai, bj and the uniqueness of the innersignal estimation in Sections 3 and 4, respectively.

Lemma 5.1 Under Assumptions A1-A3, the matrix

limK→∞

1

K

K−1∑

k=0

φy (k) φTy (k)

is positive definite.

Proof of Lemma 5.1: It follows with some modifications from Lemma 1 in Sun, Liu and Sano (1999) bynoticing that the input u (t) always has a PE contribution from the reference signal r that is independentto the noise source e (t). ¤

Theorem 5.2 Under Assumptions A1-A3, the estimated parameters θ(l)a in (14) and bl in (18) are con-

sistent, i.e., θ(l)a → θ(l) and bl → bl, as K → ∞.

Proof of Theorem 5.2: Based on Lemma 5.1, the consistency of θ(l)a can be proved analogously to the

counterpart proof of Theorem 1 in Sun, Liu and Sano (1999). We only provide the proof for the consistency

of bl. Under Assumptions A1-A3, (17) holds once na, nb and τ are obtained as shown in Section 3.4.Assuming the noise ∆e (t) available and applying the ordinary LSM to (17) yields

bl =

[

1

K

K−1∑

k=0

(∆w (kp + 1) − ∆e (kp + 1))2]−1

·1

K

K−1∑

k=0

(∆w (kp + 1) − ∆e (kp + 1)) (∆w (kp + l) − ∆e (kp + l)) . (28)

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12

The condition that

1

K

K−1∑

k=0

(∆w (kp + 1) − ∆e (kp + 1))2 6= 0

is always fulfilled under Assumption A1. As K → ∞, A (q) converges into A (q) so that w (t) → A (q) y (t) =B (q) u (t) + e (t). The time-domain expression of (11) is

∆w (kp + l) = bl∆u (kp − p) + ∆e (kp + l) , l = 1, 2, · · · , nb, ∀k ∈ Z+. (29)

Since there is at least one sample time delay in the feedback loop, u (t) is only possibly correlated withe (t − d) for d ≥ 1. As a result, ∆u (kp − p) and ∆e (kp + l) for l = 1, 2, · · · , nb are mutually independent,which, together with (29), implies

E {∆w (kp + 1)∆e (kp + 1)} = 2σ2,

E {∆w (kp + 1) ∆e (kp + l)} = −σ2δ (l − 2) ,

E {∆e (kp + 1)∆w (kp + l)} = −σ2δ (l − 2) ,

E {∆w (kp + 1) ∆e (kp + l)} = −σ2δ (l − 2) .

Therefore, as K → ∞, (28) becomes

bl =[

E{

(∆w (kp + 1) − ∆e (kp + 1))2}]−1

·E {(∆w (kp + 1) − ∆e (kp + 1)) (∆w (kp + l) − ∆e (kp + l))}

=[

E{

∆2w (kp + 1)

}

− 2σ2]−1 [

E {∆w (kp + 1)∆w (kp + l)} + σ2δ (l − 2)]

. (30)

Comparing (18) with (30), we have the consistency of bl, i.e., limK→∞ bl = bl. ¤

The estimates θ(l)a in (20) and bl in (21) are also consistent under Assumptions A1-A3. The proof is trivial,

since the instrumental variables, being composed from the reference signal r, are apparently correlated to∆y and ∆w, but independent to the noise e(t).

Theorem 5.3 Under Assumption A4, the estimated inner signal U(n) is uniquely determined in (27) fora given realization.

Proof of Theorem 5.3: The uniqueness of U(n) in (27) requires that the p(K − 1) × K matrix H hasfull-column rank, which is true if and only if all the channels Hi(q) for i = 1, 2, · · · , p do not share anycommon zero except at infinity (Lemma 2 in Hua and Wax (1996) and Corollary 3.1 in Bai and Ding(2000)). Specifically, in this context,

Hi(q) = hi(0) + hi(1)q−1.

To make all the channels Hi(q) for i = 1, 2, · · · , p do not share any common zero except at infinity is

September 21, 2006 21:54 International Journal of Control ”A Blind Approach to Closed-Loop ID of Hammerstein SYS”

13

equivalent to the condition that the matrix

h1 (0) h1 (1)h2 (0) h2 (1)

......

hp (0) hp (1)

=

b1∑nb

k=2 bk

b1 + b2∑nb

k=3 bk

......

∑nb−1k=1 bk bnb

∑nb

k=1 bk 0...

...∑nb

k=1 bk 0

has a trivial null space, which is satisfied by Assumption A4, i.e.,∑nb

k=1 bk 6= 0. ¤

6 Algorithm and Simulation

This section summarizes the detailed steps of the proposed blind approach and presents a simulatednumerical example to illustrate them.

Algorithm:

(i) The order na and the time delay τ are obtained first as discussed in Section 3.4 by looking at V (p) (na)’sin (19) with l = p for different combinations of na and τ . Two sets of V (p) (na)’s can be obtained by

the estimates θ(p)a from the bias-compensated LSM in (14) and the IVM in (20), for the cross-checking

purpose, i.e., if the two sets are in consistency, we would have more confidence on the estimates.(ii) The output y(t) is shifted properly according to τ to make (3) hold. Two groups of denominator

parameters are estimated with na = na, nb = n0b and θ

(l)a from the bias-compensated LSM in (14) and

the IVM in (20). The group having a smaller integrated AIC is selected.

(iii) The order nb is obtained on the basis of the filtered output w(t) = A(q)y(t), as discussed in Section 3.4.If nb 6= n0

b , the estimation of denominator parameters in the second step could be performed again foran improvement by taking nb in place of n0

b .(iv) Two groups of parameters b2, b3, · · ·, bnb

are estimated by the bias-compensated LSM in (18) and the

IVM in (21), from which the one having a smaller generalized prediction error V(l)EIV is selected. Here

the generalized prediction error is defined as (see e.g., Definition 3 in Sima and Van Huffel (2004))

V(l)EIV =

1K

∑K−1k=0

(

∆w (kp + l) − bl∆w (kp + 1))2

b2l + 1

.

(v) From ai and bj , the unmeasurable inner signal U(n) is estimated in (27). The nonlinearity f can be

seen from a graph of U(n) v.s. the controller output x(n); some data shifting of U(n) according to τ

may be necessary to make U(n) and x(n) synchronize.

Example: In Figure 1, the process Gc(s) is the same as that in Bai and Fu (2002) except that it hasan additional time delay 0.36 sec and works in a feedback loop with a pure gain controller, i.e.,

Gc (s) =0.4095s + 1.0921

s2 + 0.32s + 0.02e−0.36s, C(z) = 0.1.

Here the updating period of the ZOH is T = 0.6 sec. The nonlinearity fc is a backlash with dead-band 0.1. The process noise is generated by passing zero-mean white noise having variance σ2 through1/

(

s2 + 0.32s + 0.02)

. The upper bound of the order nb is known a prior as n0b = 4. The fast-sampling

ratio is chosen as p = n0b + 1 = 5. Thus, the fast-rate process G(q) at the sampling period h = T/p = 0.12

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14

sec is

G (q) = 0.05597q−3 q−1 − 0.7244q−2

1 − 1.962q−1 + 0.9623q−2.

The reference signal r is a random binary sequence with frequency band [0, 0.5] and values ±1. Thesimulation duration is 500 sec and the slow-rate input U(n) has around 800 data points to be estimated.

Now we illustrate the estimation of na = 2, nb = 2 and τ = 3. First, na and τ are obtained together bylooking at the AIC V (p) (na) defined in (19) for different combinations of na and τ . One typical realization

(σ2 = 0.001) gives the AICs (×10−7) in Table 1, where the subscript “LSM” or “IV M” denotes that θ(l)a

is obtained from the bias-compensated LSM in (14) or the IVM in (20). By looking at the columns with

τ 0 1 2 3 4

V(p)LSM (1) 12.010 11.235 9.941 8.948 82.172

V(p)IV M (1) 12.448 11.725 10.324 9.215 81.864

V(p)LSM (2) 4.619 0.992 0.982 1.073 152.020

V(p)IV M (2) 47.556 0.980 0.980 1.067 76.116

V(p)LSM (3) 1.543 1.012 1.735 2.093 152.37

V(p)IV M (3) 5.273 1.270 1.103 1.070 77.205

V(p)LSM (4) 1.256 1.111 1.620 1.181 151.480

V(p)IV M (4) 3.284 1.432 2.709 1.550 77.039

V(p)LSM (5) 1.148 1.172 1.304 1.204 151.800

V(p)IV M (5) 2.681 1.306 1.553 0.954 77.089

Table 1. AIC for different combinations of na and τ

τ = 1, 2, 3 in Table 1, the AICs do not have significant improvement after na = 2; by looking at therows with na = 2, the AICs are almost the same for the consecutive shifts 1, 2, 3 and are smaller thanthe rest two, i.e., τ = 3. The parsimony principle rules out two other possible pairs (na = 3, τ = 3) and(na = 3, τ = 4). In fact, the pair (na = 3, τ = 3) yields the almost same inner signal estimation as ourchoice (na = 2, τ = 3), while τ = 4 in the other pair is inconsistent with nb = 2 obtained next in terms ofthe number of equivalent AICs. Second, after shifting y(t) by τ and estimating the parameters in A(q) as

described in the second step of Algorithm, the filtered output w(t) = A(q)y(t) is formed; then, the true

order nb is obtained, i.e., nb = 2, because p = 5 and the variances of ∆(l)w ’s have three almost same smallest

numbers (×10−7) in Table 2.

l 1 2 3 4 5

V ar{

∆(l)w

}

31.591 59.353 0.99745 0.99204 0.94626

Table 2. The variance of ∆(l)w

Remark: Some modifications are perhaps necessary in estimating τ . For instance, if τ = 1, one realization(σ2 = 0.001) gives the AICs (×10−7) in Table 3. The smallest V (p) (na)’s in Table 3 are at three inconsec-

τ 0 1 2 3 4

V(p)LSM (na = 2) 0.99475 1.0182 122.03 3.2380 1.0146

V(p)IV M (na = 2) 0.99809 1.0193 46.08 12.927 1.0134

Table 3. AICs for time delay estimation

utive shifts 0, 1, and 4; thus, the shift 4 needs to be regarded as −1, because the sole information of y (t)cannot tell the difference between τ and (τ + kp), k ∈ Z+.

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σ2 × 10−2 SNR a1 = −1.9620 a2 = 0.9623 b2 = 0.7244

0 ∞−1.9620−1.9620

0.96230.9623

0.72440.7244

0.01 16.2627−1.9623 ± 0.0018−1.9619 ± 0.0022

0.9626 ± 0.00190.9622 ± 0.0023

0.7240 ± 0.00270.7243 ± 0.0064

0.05 7.2287−1.9634 ± 0.0038−1.9627 ± 0.0045

0.9638 ± 0.00380.9630 ± 0.0045

0.7242 ± 0.00530.7216 ± 0.0151

0.1 5.1003−1.9651 ± 0.0054−1.9633 ± 0.0063

0.9656 ± 0.00550.9636 ± 0.0063

0.7247 ± 0.00850.7171 ± 0.0188

0.5 2.4956−1.9723 ± 0.0112−1.9647 ± 0.0170

0.9735 ± 0.01130.9650 ± 0.0171

0.7252 ± 0.02400.7353 ± 0.0519

1 1.8889−1.9781 ± 0.0150−1.9635 ± 0.0241

0.9799 ± 0.01510.9639 ± 0.0240

0.7232 ± 0.03480.7325 ± 0.0663

Table 4. Estimated parameters and their standard deviations from the biased-compensated LSM (upper) and IVM (lower)

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Controller output x(n)

Est

imat

ed in

ner

sign

al U

(n)

Figure 4. A graph of the controller output x(n) v.s. the estimated inner signal U(n)

Next, we investigate the performance of the parameter and inner signal estimation by multiple MonteCarlo simulations. Table 4 presents the averaged estimates of the parameters a1, a2 and b2 and theirstandard deviations for different noise levels; 100 Monte Carlo simulations are performed for each non-zeronoise level. Here the signal-to-noise ratio (SNR) is defined as SNR = ‖y0 (t)‖2 / ‖v (t)‖2 , where y0 (t) isthe noise-free component of y (t) and ‖·‖2 denotes the Euclidean norm. The upper and lower estimatesin each noise level are obtained by the bias-compensated LSM and the IVM, respectively. As expected,the estimates are consistent; the sub-optimal IVM results in estimates with larger variances than thosefrom the bias-compensated LSM. Using the estimated parameters, the unmeasurable inner signal U(n)

is obtained in (27); a graph of the controller output x(n) v.s. U(n) from one typical realization withσ2 = 0.001 is shown in Figure 4. As a comparison, Figure 5 shows the graph of x(n) v.s. the true innersignal U(n). Since the gain of G(q) is impossible to get in practice from the information of the outputonly, the vertical axes of Figs. 4 and 5 have different scales. Nevertheless, the nonlinearities in Figs. 4and 5 have a good match in terms of the backlash structure; the deadband is read from the graph to beapproximately 0.1.

For the purpose of comparison, the gain of G(q) is assumed to be known, and U(n) is scaled properly.The error between the two signals is measured numerically by a fitness in % (see ‘compare’ command in

September 21, 2006 21:54 International Journal of Control ”A Blind Approach to Closed-Loop ID of Hammerstein SYS”

16

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Controller output x(n)

Tru

e in

ner

sign

al U

(n)

Figure 5. A graph of the controller output x(n) v.s. the true inner signal U(n)

σ2 × 10−2 SNR FLSM F 0LSM FINV F 0

INV

0 ∞ 100 100 100 1000.01 16.2627 95.1989 ± 1.6295 97.2922 93.8494 ± 1.4933 95.60090.05 7.2287 89.4944 ± 3.8574 93.8981 86.4244 ± 3.5078 90.10670.1 5.1003 85.0520 ± 5.2490 91.3634 80.7704 ± 4.7868 86.04970.5 2.4956 62.1737 ± 16.5220 80.9620 53.0911 ± 15.5598 69.28151 1.8889 49.0576 ± 24.8790 73.2331 36.0619 ± 24.0173 57.4234

Table 5. Averaged fitnesses and their standard deviations: the standard deviations of F 0LSM

and F 0INV

are omitted

Matlab System Identification Toolbox),

F = 100

1 −

∥

∥

∥Us (n) − U (n)

∥

∥

∥

2

‖U (n) − E {U (n)}‖2

.

Here Us(n) stands for the estimated inner signal after scaling. Table 5 lists the averaged fitnesses andtheir standard deviations from the same Monte Carlo simulations as those in Table 4. FLSM is the fitnessbetween U(n) and Us(n) obtained from the LSM in (27). The upper bound of FLSM , denoted by F 0

LSM ,is calculated by using the true parameters ai and bj in (27); the standard deviation of F 0

LSM is relativelysmall and is omitted here. In this example, G(q) is minimum phase so that U(n) can also estimated by

passing y(t) through the direct inverse of G(q) and downsampling the resulted signal u(t) by p = 5. Thecorresponding fitness and its upper bound are denoted by FINV and F 0

INV , respectively. Eq. (27) reducesthe noise effect in the inner signal estimation, as F 0

LSM and FLSM are always larger than F 0INV and FINV ,

respectively. The difference between F 0LSM and FLSM is getting larger as the SNR decreases; this is due

to the propagation of the errors in parameter estimation into the inner signal estimation.

7 Conclusion

This paper continued the study of the blind approach to Hammerstein identification in Sun, Liu and Sano(1999) and Bai and Fu (2002). It would be interesting to see the performance of the proposed approachin the real world. Our intended application is to diagnose the various nonlinearities of actuators that may

September 21, 2006 21:54 International Journal of Control ”A Blind Approach to Closed-Loop ID of Hammerstein SYS”

17

be the root causes of control-loop oscillations. Along with its merits, the proposed blind approach has aninherent limitation (Bai, Li and Dasgupta (2002)) that fast-sampling output has to be possible. On theother hand, it is a common practice in identification to sample signals as fast as possible with “cheap” dataacquisition (Ljung (1999) Page 452), but there are few studies on how to use the fast-sampled signals. Inthis sense, the blind approach to Hammerstein identification is one of the directions worthy of exploration.

Acknowledgment

This research was supported by the Natural Sciences and Engineering Research Council of Canada, theJapan Society for the Promotion of Science, the Alberta Ingenuity Fund, the Informatics Circle of ResearchExcellence, and the Izaak Walton Killam Trusts.

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