Closed-loop identification via output fast sampling

Journal of Process Control 14 (2004) 555–570

www.elsevier.com/locate/jprocont

Closed-loop identification via output fast sampling

Jiandong Wang a, Tongwen Chen a,*, Biao Huang b

a Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada T6G 2V4b Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G6

Received 21 July 2003; received in revised form 26 September 2003; accepted 26 September 2003

Abstract

This paper addresses a challenge: Is a closed-loop system without external excitation identifiable? The so-called fast-sampling

direct approach provides a positive answer. It removes a traditional restrictive identifiability condition for linear output feedback

closed-loop systems, i.e., an external persistently exciting test signal is not required. Identifiability is analyzed using the lifting

technique, the bifrequency map and bispectrum concepts. The proposed approach is further investigated and evaluated by simu-

lation.

� 2003 Elsevier Ltd. All rights reserved.

Keywords: Closed-loop identification; Fast-sampling direct approach; Cyclo-stationarity; Bifrequency maps; Bispectra

1. Introduction

Closed-loop identification has long been of prime

interest both in academic research and in industrial

applications for many reasons. In some cases, closed-

loop identification may be necessary due to physical

constraints, e.g., inherent feedback mechanisms exist, oropen-loop systems are unstable and have to be con-

trolled for production or safety reasons [10]. Sometimes,

it owes to some specific objective of identification, e.g., if

a linearized dynamic model around a nominal operating

point is desired for control, then it is natural and ben-

eficial to achieve this by closed-loop identification with

process working around the nominal operating point

[11]. Moreover, it has also been shown that if there areconstraints in the input/output variance, it would be

better off to implement identification using closed-loop

experiments [8].

A variety of approaches for closed-loop identifica-

tion have been studied in numerous references and

conventionally categorized into three groups: the direct

approach, the indirect approach, and the joint input–

output approach [6,9,13]. The direct approach, ignoringany possible feedback and identifying the open-loop

system using measurements of the closed-loop process

*Corresponding author. Tel.: +1-780-492-3940; fax: +1-780-492-

1811.

E-mail address: [email protected] (T. Chen).

0959-1524/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jprocont.2003.09.009

input and output, is regarded as a prime choice for its

simplicity and optimality [5,13]. We will show soon that

the proposed fast-sampling direct approach is closely

related to the direct approach, which is henceforth called

‘‘the traditional direct approach’’ to avoid confusion.

In this paper, we consider a typical single-input and

single-output (SISO) closed-loop system depicted in Fig.1, where Gp is a continuous-time linear time-invariant

(LTI) and causal process; Nc is a continuous-time LTI

noise system; F is a discrete-time LTI controller; HT and

ST are the zero-order hold and the sampler, respectively,

both with period T . For such a system, an identification

objective is to estimate a discrete-time model of

GT :¼ STGpHT .

One of the kernel topics in closed-loop identificationis identifiability. It is well known that identifiability is

ensured by adding an external persistently exciting (PE)

test signal [10], e.g., the reference signal r. However,

adding such a signal is often undesirable or too expen-

sive, since it disturbs the process operation and usually

affects product qualities [29]. Most often, the available

data are collected under the nominal operating condi-

tion and are only partially excited, e.g., a piecewise stepreference (load) variation. Here we consider a worst case

problem: Is it possible to achieve identifiability without

external excitation? Using the traditional direct ap-

proach, the answer is positive only if some restrictive

conditions on the orders of the true process and con-

troller are satisfied [15]; whereas, a so-called output

mail to: [email protected]

TH

++

-

+pG TS

F

cN

cuTur cy

ce

Ty

Fig. 1. Block diagram of a typical closed-loop system.

556 J. Wang et al. / Journal of Process Control 14 (2004) 555–570

intersampling approach [22,23] may remove these re-

strictions.

The fast-sampling direct approach 1 defined in our

context is formally stated as the following: For a closed-

loop system depicted in Fig. 1, the output yc is fast-

sampled with period h :¼ T=p to obtain y, where p is apositive integer. A fast-sampled version of the input ucwith period h, say u, is measured directly or is available

from the known process input uT because of the zero-

order hold. Based on the fast-rate signals y and u, ap-plying the traditional direct approach gives a fast-rate

process model bGh with period h, i.e., an estimate of

Gh :¼ ShGpHh. Next, from bGh we can uniquely determine

a slow-rate process model bGT , which is the objective ofidentification.

Clearly, the fast-sampling direct approach is a simple

modification of the traditional direct approach in terms

of the computation. However, as shown later, analyzing

the approach and proving its feasibility pose some great

challenges. First, faster sampling introduces co-existence

of multirate signals: the signals used in identification are

sampled faster than those in the feedback loop. Thus,some multirate tools are needed for the analysis, e.g., the

lifting technique. Second, one type of nonstationary

signal, cyclo-stationary signals arise, which invalidates

an underlying assumption in identifiability analysis for

the traditional direct approach. Therefore, the current

identifiability theory requires a nontrivial extension.

In [23], identifiability was proven to be achieved by

analyzing time-domain properties of the identificationalgorithms, e.g., Jacobian and Hessian of the prediction

error. However, the time-domain viewpoint has a clear

limitation on insightful analysis of the identifiability

conditions, such as finding out the factors that may af-

fect identifiability. On the contrary, our work is devel-

oped along the line of frequency-domain analysis and is

different from [23] in at least two main aspects: We

develop a generalized frequency-domain expression of

1 We feel that the name is more transparent than the output

intersampling approach, even though the idea is same.

asymptotic parameter estimation, and answer an iden-

tifiability question:

Why does the fast-sampling direct approach make

the closed-loop system without external excitation

identifiable?

Moreover, results in our identifiability analysis are in-

sightful so that another important question can be

answered:

What are the factors that affect the fast-sampling

direct approach?

The rest of the paper is organized as follows. To give

a clear background, we review the conventional concept

of identifiability and a frequency-domain expression for

the analysis of identifiability in Section 2. Section 3

proposes an idea that leads to the subsequent identifi-

ability analysis. In Section 4, linear periodically time-

varying (LPTV) systems are introduced and equivalent

closed-loop systems are found using the lifting tech-nique. The presence of LPTV systems introduces the

concept of cyclo-stationarity of signals; this is discussed

in Section 5. To describe statistical properties of cyclo-

stationary signals, bifrequency maps and bispectra are

introduced in Section 6. Using these tools, identifiability

of the fast-sampling direct approach is analyzed in

Section 7. Based on a new frequency-domain expression,

the proposed approach is further evaluated and inves-tigated in Section 8. Finally, we conclude in Section 9.

2. Preliminary

Since the terminology ‘‘identifiability’’ and a fre-

quency-domain expression will be used extensively, it is

worthy to introduce them in advance. First, we quote

some words about identifiability from [13]:

Identifiability is a concept that is central in identifi-

cation problems. Loosely speaking, the problem is

++

TG

F

TN

Tu

Te

Ty

-

Fig. 2. A closed-loop system in the traditional direct approach.

J. Wang et al. / Journal of Process Control 14 (2004) 555–570 557

whether the identification procedure will yield a uni-

que value of the parameter, and/or whether the re-

sulting model is equal to the true system. . . . Theissue involves aspects on whether the data set (theexperimental condition) is informative enough to

distinguish between different models as well as

properties of the model structure itself . . .

Henceforth, we interpret identifiability simply as a

concept of whether the resulting model is equal to the

true system [13] and assume that other aspects, e.g., the

model structure, satisfy the required conditions and donot cause difficulties in obtaining the true system. SincebGT is obtained uniquely via bGh in the fast-sampling di-

rect approach, our identifiability question in Section 1

can be rephrased as: Does the fast-sampling direct ap-

proach give a fast-rate model bGh such that bGh ¼ Gh as

the data length L goes to infinity, if there is no external

excitation?

Second, we review a frequency-domain expression forthe analysis of identifiability (see Chapter 8 in [13]). The

frequency-domain approach has also proven to be use-

ful for the analysis of roles of the noise models and data

prefilters [15]. If the true discrete-time system is de-

scribed as

yðtÞ ¼ GðqÞuðtÞ þ NðqÞeðtÞ;then we estimate bG and bN in a model description

yðtÞ ¼ bGðqÞuðtÞ þ bN ðqÞeðtÞby minimizing the predication error e defined as

eðt; hÞ ¼ bN �1ðqÞ½ðGðqÞ � bGðqÞÞuðtÞ þ NðqÞeðtÞ�:Here t enumerates the sampling instant. h denotes the

parameter set to be estimated, and q is the forward shift

operator, quðtÞ ¼ uðt þ 1Þ. If the prediction error method

(PEM) is used with the most-common criterion

hL ¼ argminh

1

L

XLt¼1

1

2e2ðt; hÞ; ð1Þ

we have a frequency-domain description of the para-

meter estimation as the data length L goes to infinity [24]:

h ¼ argminh

1

4p

Z p

�p

GðejxÞ þ bBðejx; hÞ � bGðejx; hÞ�� 2UuðejxÞ

bN ðejx; hÞ�� 2

2664

þNðejxÞ � bN ðejx; hÞ�� 2 k� UueðejxÞj j2

UuðejxÞ

� �bN ðejx; hÞ�� 2 þ k

3775dx; ð2Þ

where

bBðejx; hÞ ¼ NðejxÞ � bN ðejx; hÞ� �

UeuðejxÞUuðejxÞ

: ð3Þ

Here k is the variance of the white noise e; UuðejxÞ is thepower spectrum of the input signal u; UeuðejxÞ and

UueðejxÞ are the cross spectra of u and e.

3. The main idea

Before touching the identifiability question for the

fast-sampling direct approach, let us first investigate the

reason that identifiability is lost for the traditional direct

approach if there is no external excitation. If r ¼ 0, Fig. 1

is simplified to Fig. 2, where NT is the discrete-timecounterpart of Nc with period T , i.e., NT :¼ STNcHT . Note

that the continuous-time white noise ec is fictitiously

sampled into eT with period T , the same sampling period

as that of yT . We have two equations in Fig. 2

yT ðtÞ ¼ GT ðqÞuT ðtÞ þ NT ðqÞeT ðtÞ; ð4ÞuT ðtÞ ¼ �F ðqÞyT ðtÞ: ð5ÞIn the sequel, the arguments t and q are dropped for

short notation unless confusion may arise.

When the traditional direct approach is applied to yTand uT , identification likely converges to (5) instead of

(4), since the former gives a smaller or even non pre-

diction error. Such a viewpoint can be formally captured

by analyzing the frequency-domain expression (2) as

follows. The transfer function from eT to uT is

uT ¼ �FS0NT eT ; ð6Þwhere S0 is the sensitivity function, S0 :¼ ð1þ GTF Þ�1

.

Since F , S0 and NT are SISO LTI systems, we have [13]

UuT ðejxÞ ¼ F ðejxÞS0ðejxÞNT ðejxÞ�� 2kT ; ð7Þ

UuT eT ðejxÞ ¼ �F ðejxÞS0ðejxÞNT ðejxÞkT ; ð8Þ

where kT is the variance of eT . Thus, the second

weighting term in (2) becomes

k�UueðejxÞ�� 2UuðejxÞ

¼ kT �F ðejxÞS0ðejxÞNT ðejxÞ�� 2k2TF ðejxÞS0ðejxÞNT ðejxÞj j2kT

¼ 0:

ð9Þ


The zero weighting implies that the bias bB in (3) does

not vanish, as in general bNT 6¼ NT if NT 6¼ 1. As a con-

sequence, bGT 6¼ GT , i.e. identifiability is lost. Note that

here GT , NT , bGT and bNT play the same roles as G, N , bGand bN in (2) respectively.

Now, if the fast-sampling direct approach is applied,

there will be an LPTV controller instead of an LTI one

(to be clarified later). Intuitively, an LPTV controller

brings possibility that identification could converge to a

system equation like (4), because data are fitted by LTI

models. Taking an alternative viewpoint, sampling sig-

nals faster alleviates the main problem in the closed-loopidentification, namely, the correlation between the out-

put noise and the process input. This alleviation will

certainly have a reflection in the frequency domain. So,

the idea is to start the identifiability analysis from

finding the variations caused by fast sampling, and de-

velop a parallel frequency-domain analysis as that in

(7)–(9).

4. Equivalent closed-loop systems

To perform an analysis similar to that in Section 3,

we first need to find the counterpart of (6) in the fast-

sampling direct approach framework. The counterpartcan be obtained by the lifting technique in several steps.

Other methods may be able to do this; however, using

the lifting technique is clearer and more systematic.

Our starting point is Fig. 1, where r ¼ 0 if no external

excitation exists. Let Sp be the discrete-time down-sam-

pler by a factor of p and Hp be the discrete-time zero-

order hold by a factor of p. Since T ¼ ph, there are two

useful identities [2]

ST ¼ SpSh; HT ¼ HhHp:

From these identities and linearities of Sp and Sh, Fig. 3results. Here ec in Fig. 1 is fictitiously sampled with

period h into e; u is the fast-rate process input with pe-

riod h; y is the fast-rate output with period h. Thus, wereach the fast-rate process Gh ¼ ShGpHh and the fast-rate

hH pG

FTu

ehH

pH

u

-

cN

Fig. 3. A closed-loop system with

noise system Nh :¼ ShNcHh. Note that multirate signals,

e.g., y and yT , u and uT , co-exist in Fig. 3.

Now we introduce the lifting technique [12]. Let x be

a discrete-time signal defined on Zþ, the set of nonneg-ative integers. The p-fold discrete lifting operator Lp is

defined as the mapping from x to x, where underlining

denotes lifting:

xð0Þ; xð1Þ; . . .f g7!

xð0Þxð1Þ...

xðp � 1Þ

2666437775;

xðpÞxðp þ 1Þ

..

.

xð2p � 1Þ

2666437775; . . .

8>>><>>>:9>>>=>>>;:

ð10Þ

If the underlying period of x is h, that of the lifted signal

x is ph. The inverse of the lifting operator, L�1p , is defined

as the reverse operation of (10), i.e., L�1p Lp ¼ I and

LpL�1p ¼ I , where I denotes the identity system.

The beauty of the lifting technique is its ability to dealwith LPTV systems elegantly by associating them with

some LTI systems, the so-called lifted systems. By

properties LpL�1p ¼ I and L�1

p Lp ¼ I , Fig. 4 is equivalent

to Fig. 3, where we notice a fact about the lifted sys-

tem F .

Fact 1. F :¼ LpHpFSpL�1p is a p-input and p-output LTI

system; it has a state-space representation

if F has a state-space representation :¼DF þ

CF ðqI � AF Þ�1BF .

Proof. To see that F is LTI, it suffices to show thateUFU ¼ U , where eU and U are the discrete-time unit

time-advance and time-delay operators, respectively. By

properties [2]

++ y

Ty

hS

hS

pS

multirate blocks Hp and Sp.

++

hG

F

hN

y

e

pH pS

u

pL1-pL pL1-

pL

-

F

Fig. 4. A lifted closed-loop system No. 1.

++

hG

F

y

e

u

pL

e

-

hN

Fig. 5. A lifted closed-loop system No. 2.


eULp ¼ LpeU p; L�1

p U ¼ UpL�1p ;eUpHp ¼ Hp

eU ; SpUp ¼ USp;

we have

eUFU ¼ eULpHpFSpL�1p U ¼ Lp

eUpHpFSpUpL�1p

¼ LpHpeUFUSpL�1

p ¼ LpHpFSpL�1p ¼ F :

The transfer matrix of F comes from the facts that the

transfer matrices of LpHp and SpL�1p are ½I I � � � I �T and

½I 0 � � � 0�, respectively [2]. h

Remark. HpFSp has been shown p-periodic in the proof

( eU pHpFSpUp ¼ HpFSp). In other words, an LPTV con-

troller arises from sampling signals faster. As discussedin Section 3, the presence of the LPTV controller makes

identifiability attainable even though no external exci-

tation exists.

Next, we simplify Figs. 4 and 5 by the identity

L�1p Lp ¼ I and linearities of Lp and L�1

p . In Fig. 5, Gh and

Nh are LTI lifted systems [12]

Gh :¼ LpGhL�1p ; Nh :¼ LpNhL�1

p ;

u, y and e are lifted signals

u ¼ Lpu; y ¼ Lpy; e ¼ Lpe:

Therefore, we form an LTI closed-loop system depictedwithin the dash-dot rectangle in Fig. 5, where two

equations exist

y ¼ Ghuþ Nhe; ð11Þu ¼ �F y: ð12Þ

From (11) and (12), we have

u ¼ � F ðI þ GhF Þ�1Nh|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}M

e: ð13Þ

The transfer function M between u and e has a special

structure stated in Fact 2.

Fact 2. M :¼ �F ðI þ GhF Þ�1Nh has a special transfermatrix

M :¼

M0 M1 � � � Mp�1

M0 M1 � � � Mp�1

..

. ... ..

.

M0 M1 � � � Mp�1

2666437775; ð14Þ

where

Mi :¼ �F 1

"þ

Xp�1

l¼0

G0;l

!F

#�1

N0;i:

Here Gi;l ðNi;lÞ is the ði; lÞth subsystem of Gh ðNhÞ, fori ¼ 0; 1; . . . ; p � 1 and l ¼ 0; 1; . . . ; p � 1.

Proof. See Appendix A. h

++

-hG

pp FSH

hN

u

e

y

Fig. 6. A closed-loop system in the fast-sampling direct approach.


The special structure in (14) is a natural result be-cause each subsignal of u is identical. Eq. (13) implies

that

u ¼ L�1p ½�F ðI þ GhF Þ�1Nh�Lp|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

M 0

e: ð15Þ

The new system M 0 between u and e is LPTV, stated in

Fact 3.

Fact 3. M 0 :¼ L�1p ½�F ðI þ GhF Þ�1Nh�Lp is a p-periodic

LPTV system, abbreviated as ðLPTVÞp.

Proof. It suffices to show that eUpM 0Up ¼ M 0, which can

be done using the techniques in the proof of Fact 1. h

In summary, sampling signals faster brings a corre-

sponding closed-loop system depicted in Fig. 6, having

an LPTV controller HpFSp instead of an LTI one. The

relationship between u and e is an (LPTV)p system M 0 in

(15). Note that a statement in [23] that parts of u and ework in open loop is incorrect.

5. Cyclo-stationary signals

There is an underlying assumption in the fundamen-

tal frequency-domain expression (2), namely, signals are

quasi-stationary [13]. However, the presence of LPTVsystems, e.g., M 0 in (15), implies that the underlying

assumption does not hold. The purpose of this section is

to check the assumption. Before doing so, let us first

review some terminologies.

A. Wide-sense-stationary signals

A discrete-time signal x is said to be wide-sense-stationary (WSS) if its mean is constant

Eðxðt þ lÞÞ ¼ EðxðtÞÞ; ð16Þand its autocorrelation depends only on the time dif-

ference

EðxðtÞxyðt � lÞÞ ¼ RxxðlÞ; ð17Þ

for all integers t and l [17]. Superscript y denotes the

conjugate transpose. Note that here signals are normally

assumed to be zero mean. Two discrete-time signals xand y are said to be jointly wide-sense-stationary if the

vectorxy

� �is WSS [17].

Remark. Conditions in (16) and (17) are the relaxed

definitions of the familiar stationarity that is usually

defined in terms of density functions [18]. Quasi-sta-

tionarity is a generalization of wide-sense stationarity

[13] and is often used in the identification literature in

order to put stochastic and deterministic signals underthe same roof. In our context, we do not distinguish

them strictly.

B. Cyclo-wide-sense-stationary signals

Let Rxxðt1; t2Þ ¼ Eðxðt1Þxyðt2ÞÞ. A discrete-time signal xis said to be cyclo-wide-sense-stationary (CWSS) with

period p, abbreviated as (CWSS)p, if

Eðxðt þ lpÞÞ ¼ EðxðtÞÞ;Rxxðt1 þ lp; t2 þ lpÞ ¼ Rxxðt1; t2Þ;for all integers t1, t2 and l [17]. If p ¼ 1, then x is WSS.

Two discrete-time signals x and y are said to be jointly

cyclo-wide-sense-stationary with period p, abbreviated

as (JCWSS)p, if the vectorxy

� �is (CWSS)p [1]. If p ¼ 1,

(JCWSS)p is synonymous with jointly WSS.

Next, we introduce a very useful theorem describingthe statistical relationship between a signal and its lifted

version.

Theorem 1. Let x, y be scalar discrete-time signals, and x,y be the lifted versions of x, y, i.e., x ¼ Lpx, y ¼ Lpy. Then,x is (CWSS)p if and only if (iff) x is WSS; x is WSS iff x isWSS with pseudocirculant power spectrum matrix; x, yare (JCWSS)p iff x, y are jointly WSS.

Proof. The proof can be collected from various places in

[19] and is summarized in [1]. h

Remark. Pseudocirculant matrices often occur in thecontext of multirate systems (see Chapter 5 in [26]).

Take a 3 · 3 pseudocirculant matrix as an example,

which has a form

P0ðejxÞ P1ðejxÞ P2ðejxÞe�jxP2ðejxÞ P0ðejxÞ P1ðejxÞe�jxP1ðejxÞ e�jxP2ðejxÞ P0ðejxÞ

24 35:Let us apply

Theorem 1 to analyze the signals appearing in the fast-

sampling direct approach framework, namely, the pro-

cess input u and the prediction error e.


From the definition of the lifting operation (10), we

know that the lifted version of a WSS signal is WSS. As

the white noise e is WSS, e is WSS. Since u and e are

connected by an LTI system M in (13), u is WSS; u and eare jointly WSS. Now we use a special case to show that

the power spectrum of u, UuðejxÞ, is not pseudocirculant.If p ¼ 2, from (14) and (13), we have

UuðejxÞ ¼ MðejxÞUeðejxÞM yðejxÞ

¼ k0 jM0ðejxÞj2�

þ jM1ðejxÞj2� I I

I I

� �;

where k0 is the power spectrum of e; UeðejxÞ follows

from (10) and the independent property of the white

noise:

UeðejxÞ ¼k0 0

0 k0

� �:

Clearly, UuðejxÞ is not pseudocirculant. Applying The-

orem 1 gives that u is (CWSS)p, but not WSS; u and e are(JCWSS)p, but not jointly WSS.

Remark. We could also conclude that u is not WSS

through another theorem in [1]: a rational (LPTV)pscalar system produces WSS outputs for all WSS inputs

iff it is either a rational LTI system, an exponential

(LPTV)p modulator (whose i-channel system is a con-stant multiplier ej2pl=p for some integer l), or a cascade of

these.

If we denote bGh and bNh as the estimates of fast-rate

systems Gh and Nh respectively, then the prediction error

e, defined as

e ¼ bN �1h ½ðGh � bGhÞuþ Nhe�; ð18Þ

is (CWSS)p, but not WSS. To see that it is (CWSS)p, we

use the lifting technique to form a lifted system

Lpe ¼ LpbN �1h ðGh

h� bGhÞL�1

p Lpuþ NhL�1p Lpe

i;

i.e.,

e ¼ LpbN �1h ðGh � bGhÞL�1

p|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}G1

uþ LpNhL�1p|fflfflfflffl{zfflfflfflffl}

G2

e:

Since G1 and G2 are LTI [12], and u and e have been

shown to be WSS, e is WSS. Applying Theorem 1 gives

that e is (CWSS)p. The fact that e is not WSS can be seenby checking its power spectrum or more directly from

definitions in (16) and (17).

2 The output of G1 is the input of G2.

6. Bifrequency maps and bispectra

In Section 5, u and e are shown to be CWSS, intro-

duced by the fast sampling. As a consequence, we need

to know counterparts of power spectra in (7) and (8) for

CWSS signals, before making an identifiability analysis

analogous to that in Section 3. Therefore, some concepts

are required to describe statistical properties of CWSS

signals, namely, bifrequency maps and bispectra.

6.1. Bifrequency maps

Bifrequency maps are frequency-domain descriptions

of system characteristics. We start with bifrequency

maps for general linear time-varying (LTV) systems.

Next, we focus on a special class of LTV systems––

LPTV systems.

6.1.1. Bifrequency maps of general LTV systems

A discrete-time LTV system with input x and output yis fully specified in the time domain [4]

yðmÞ ¼X1n¼�1

kðm; nÞxðnÞ; ð19Þ

where kðm; nÞ, called the Green’s function, gives the

system response at time m to an impulse at time n (m and

n are integers). The LTV system is also fully specified in

the frequency domain [4]

Y ðejx0 Þ ¼Z p

�pK ejx

0; ejx

� �X ðejxÞdx;

where

K ejx0; ejx

� �¼ 1

2p

X1m¼�1

X1n¼�1

kðm; nÞe�jx0mejxn:

Here Y ðejx0 Þ and X ðejxÞ are the discrete-time Fourier

transformations of y and x, respectively. The so-called

bifrequency function Kðejx0; ejxÞ describes a mapping of

the input frequency x onto the output frequency x0.

Hence, it is called as a bifrequency map [14].

Cascading two LTV systems G1 and G2 with Green’s

functions k1ðm; nÞ and k2ðm; nÞ, and bifrequency func-

tions K1ðejx0; ejxÞ and K2ðejx

0; ejxÞ respectively in this

order, 2 gives a new LTV system specified by [3,4]

kðm; nÞ ¼X1l¼�1

k2ðm; lÞk1ðl; nÞ; ð20Þ

K ejx0; ejx

� �¼Z p

�pK2 ejx

0; ejx

00� �

K1 ejx00; ejx

� �dx00: ð21Þ

6.1.2. Bifrequency maps of LPTV systems

The bifrequency map Kðejx0; ejxÞ of an (LPTV)p sys-

tem is characterized by [1,14]

K ejx0; ejx

� �¼ F ejx

0; ejx

� � X1l¼�1

d x

�� x0 þ 2pl

p

�; ð22Þ

0.8

1

1.2

1.4

Bifrequency Maps


where dð�Þ is the Dirac-delta function and F ðejx0; ejxÞ is

a finite two-dimensional function to be defined later.

Eq. (22) shows that Kðejx0; ejxÞ lies on the lines

x� x0 þ 2plp ¼ 0, for all integers l. The shape of the

bifrequency map on the lth line is given by

FlðejxÞ :¼ F ejx; ej x�2plpð Þ

� �: ð23Þ

Now, we find Fl as follows. The Green’s function of an

(LPTV)p system has a property

kðmþ p; nþ pÞ ¼ kðm; nÞ; 8m; n:

An equivalent and useful representation can be ob-tained by setting bðn;mÞ ¼ kðnþ m; nÞ. For an (LPTV)psystem, bðn;mÞ is periodic in the index n with period p,i.e.,

bðnþ p;mÞ ¼ bðn;mÞ; 8m; n: ð24Þ

If we define an LTI system as

BnðqÞ ¼X1

m¼�1bðn;mÞq�m; ð25Þ

from (24) we have Bn ¼ Bnþp for all integers n. Then, an(LPTV)p system can always be represented as a maxi-

mally decimated filter bank in Fig. 7 [16,26], where # p is

a down-sampler by a factor of p and " p is an up-sam-

pler by a factor of p.A p-channel maximally decimated filter bank has a

relation between its input x and its output y [26]

Y ðejxÞ ¼Xp�1

l¼0

AlðejxÞX ej x�2plpð Þ

� �:

In Fig. 7, Al is determined by the nth-channel analysisbank filter qn and rth-channel synthesis bank filter

q�nBnðqÞ

AlðejxÞ ¼1

p

Xp�1

n¼0

ej x�2plpð Þ

� �nðejxÞ�nBnðejxÞ

¼ 1

p

Xp�1

n¼0

e�j2plp nBnðejxÞ: ð26Þ

If (26) is written in a matrix form, we have

Fig. 7. Filter bank representation of general LPTV systems.

A0ðejxÞA1ðejxÞ

..

.

Ap�1ðejxÞ

2666437775 ¼ 1

pWp

B0ðejxÞB1ðejxÞ

..

.

Bp�1ðejxÞ

2666437775; ð27Þ

where Wp is the Discrete Fourier Transform (DFT)matrix [26]. It is a p � p symmetric matrix whose entry at

the mth row and nth column is equal to e�j2pmn=p for

m ¼ 0; 1; . . . ; p � 1 and n ¼ 0; 1; . . . ; p � 1. In addition,

it has been proven in [1] that

FlðejxÞ ¼ AlðejxÞ; l ¼ 0; 1; . . . ; p � 1;

FlðejxÞ ¼ FlþpðejxÞ; for all integers l:ð28Þ

Thus, the bifrequency map of an (LPTV)p system is fully

characterized by the first p functions of Fl that areconnected with the impulse response of the LPTV sys-

tem through (28), (27) and (25) (see Example 4 in Sec-

tion 8).

Example 1. Let us look at the bifrequency map of the

(LPTV)p system M 0 in (15). If p ¼ 2, the bifrequencymap for x0 2 ½�p; p� and x 2 ½�p; p� has a form as that

depicted in Fig. 8, which is actually the result of Ex-

ample 2 in Section 8. The projection of the bifrequency

map on the x0–x plane within the square area

(x0 2 ½�p; p�, x 2 ½�p; p�) consists of three lines shown

in Fig. 9(b). The cases p ¼ 3 and p ¼ 4 are shown in

Figs. 9(c) and (d), respectively. If p ¼ 1, i.e., the system

is indeed LTI, the projection has only one line in thearea, shown in Fig. 9(a).

4

2

0

2

4

10

5

0

5

100

0.2

0.4

0.6

ωω′

|K|

Fig. 8. Bifrequency maps for p ¼ 2.

w

'w

w

'w

w

'w

w

'w

0F1F2F

1F

2F 0F1F2F3F

3F

2F

1F

0F 0F

1F

1F

(a) (b)

(c) (d)

π

π− π

π−

π

π− π

π−

π

π− π

π−

π

π− π

π−

Fig. 9. Projection of bifrequency maps on the x0–x plane: (a) p ¼ 1,

(b) p ¼ 2, (c) p ¼ 3 and (d) p ¼ 4.


6.2. Bispectra

Bispectra and cross bispectra, defined for nonsta-

tionary signals, are counterparts of power spectra andcross power spectra. Note that there exist other parallel

choices for cyclo-stationary signals, e.g., the cyclic

spectrum in [7]. Here we adopt the bispectrum, first

proposed in [28], for it is geometrically clear [1].

6.2.1. (Cross) Bispectra of nonstationary signals

The bispectrum Uxðejx0; ejxÞ of a nonstationary signal

x is defined as the double Fourier transform of the au-tocorrelation Rxðm; nÞ [17]

Ux ejx0; ejx

� �¼ 1

2p

X1m¼�1

X1n¼�1

Rxðm; nÞe�jx0mejxn; ð29Þ

where Rxðm; nÞ ¼ EðxðmÞxyðnÞÞ. The cross bispectrum

Uyxðejx0; ejxÞ of two nonstationary signals x and y is

defined as [17]

Uyx ejx0; ejx

� �¼ 1

2p

X1m¼�1

X1n¼�1

Ryxðm; nÞe�jx0mejxn;

( )ωω ′+ jj eeK , ( ωjx ee ,′Φ

Fig. 10. Schematic illustration of transform

where the cross correlation Ryxðm; nÞ ¼ EðyðmÞxyðnÞÞ.Uxyðejx

0; ejxÞ is defined analogously.

6.2.2. (Cross) Bispectra of cyclo-wide-sense-stationary

signals

Similar to the bifrequency map of an LPTV system,

the bispectrum of a (CWSS)p signal consists of parallel

lines. Indeed, the bispectrum has an explicit expression

similar to (22) [1]. Instead of offering the expression, we

will next present the bispectrum in terms of the trans-

formed bispectrum through linear systems.

It is well known that when a WSS signal x with powerspectrum UxðejxÞ is passed through an LTI system hav-

ing a transfer function GðqÞ, the output y is also a WSS

signal with power spectrum [13]

UyðejxÞ ¼ GðejxÞUxðejxÞGyðejxÞ: ð30Þ

When a nonstationary signal x with bispectrum

Uxðejx0; ejxÞ is passed through an LTV system with bi-

frequency function Kðejx0; ejxÞ, the output y is generally

nonstationary. Its bispectrum Uyðejx0; ejxÞ is the bifre-

quency function of the cascade shown in Fig. 10 [1].

Note that Fig. 10 is a schematic illustration only, i.e.

Uy ejx0; ejx

� �6¼ K ejx

0; ejx

� �Ux ejx

0; ejx

� �Kyðejx; ejx0 Þ;

instead, (21) is applicable to the serial cascade. In par-

ticular, if the LTV system reduces to an LTI system withtransfer function GðqÞ, then [1] shows that

Uy ejx0; ejx

� �¼ G ejx

0� �

Ux ejx0; ejx

� �GyðejxÞ: ð31Þ

As shown later in Section 7, we are particularly in-terested in a configuration: LPTV systems with WSS

inputs. Let us assume that there is an (LPTV)p system

with bifrequency function Kðejx0; ejxÞ characterized in

(22) and the input x is WSS with power spectrum

UxðejxÞ. Under such a configuration, the output y in

general is (CWSS)p according to the discussion in Sec-

tion 5. Then, we have a fact describing the transformed

bispectra through LPTV systems. Note that y and x arescalar and superscript � denotes the conjugate.

Fact 4. The bispectrum of the CWSS signal y is

Uy ejx0;ejx

� �¼Xp�1

l¼0

P ly ejx

0� � X1

m¼�1d x

��x0 þ 2pl

pþ 2pm

�;

ð32Þ

( )ωω jj eeK ,′)ωj

ation of bispectra by linear systems.


where

P ly ejx

0� �

¼Xp�1

n¼0

Fnþl ej x0þ2plpð Þ

� �F �n ejx

0� �

Ux ej x0�2pnpð Þ

� �:

The cross bispectra of y and x are

Uyx ejx0; ejx

� �¼Xp�1

l¼0

FlðejxÞUx ej x0�2plpð Þ; ejx

� �; ð33Þ

Uxy ejx0; ejx

� �¼Xp�1

l¼0

Ux ejx0; ej x�2pl

pð Þ� �

F �l ðejxÞ: ð34Þ

Proof. The bispectrum part was proven in [1]; the proof

of the cross bispectra part is in Appendix B. h

Eq. (32) implies that the bispectrum of y lies onparallel lines

x� x0 þ 2plp

þ 2pm ¼ 0; 8l:

The lth line is characterized by Ply , which is a combi-

nation of power spectra of x and its shifted versions. In

particular, when p ¼ 1, i.e., y is WSS, the bispectrum is

connected with the power spectrum as

Uy ejx0; ejx

� �¼ Uy ejx

X1l¼�1

dðx� x0 þ 2plÞ; ð35Þ

where UyðejxÞ comes from (30). Eq. (35) says that there

is only one line in the square area (x0 2 ½�p; p�, x 2½�p; p�) on the x0–x plane. Eqs. (33) and (34) have the

similar discussion by noticing that the bispectra of xtherein are connected with UxðejxÞ in forms similar to

(35).

3 Either the controller or both Gh and bGh contain a delay so that the

output y depends only on past process input u values.

7. Identifiability analysis

We are now ready to analyze identifiability of the

fast-sampling direct approach. A new frequency-domain

expression analogous to (2) will be derived. Instead of a

zero weighting term (9), there will be a new nonzero

term, which implies that identifiability is achieved in the

fast-sampling direct approach. The whole analysis con-

sists of three steps.Step 1: The loss function in terms of bispectra: The

prediction error e in (18) has been shown to be (CWSS)pin Section 5. Since e is real and scalar in our context,

E e2ðt; hÞ

¼ Eðeðt; hÞeyðt; hÞÞ ¼ Reðt; tÞ: ð36Þ

From (29), a 2-D inverse Fourier transformation [17]

gives

Reðt; tÞ ¼1

2p

Z p

�p

Z p

�pUe ejx

0; ejx

� �ejx

0te�jxt dxdx0: ð37Þ

Both Re and Ue are also functions of h, even though the

argument h is dropped for short notation. If a PEM

algorithm (1) is used to estimate parameters h, the loss

function as the data length L goes to infinity is [13]

V ðhÞ ¼ limL!1

1

L

XLt¼1

1

2E e2ðt; hÞ

:

From (36) and (37), V ðhÞ can be represented in the

frequency domain

V ðhÞ ¼ limL!1

1

L

XLt¼1

1

4p

Z p

�p

Z p

�pUe ejx

0; ejx

� �� ejx

0te�jxt dxdx0

¼ limL!1

1

L

XLt¼1

1

4p

Z p

�p

Z p

�pUe ejx

0; ejx

� �� e�j2plp t dxdx0; where l ¼ 0; 1; . . . ; p � 1

¼ limL!1

1

L

XLt¼1

1

4p

Z p

�p

Z p

�pUe ejx

0; ejx

� �dxdx0;

where x ¼ x0: ð38Þ

Here the second equality follows from a fact that thebispectrum of a (CWSS)p signal lies on the lines

x� x0 ¼ 2plp , where l ¼ 0; 1; . . . ; p � 1 (see Section 6.2),

and the last equality follows from an identity [17]

limL!1

1

L

XLt¼1

e�j2plp t ¼ 1; l ¼ 0;0; l ¼ 1; 2; . . . ; p � 1:

�Eq. (38) says that V ðhÞ depends only on the diagonal

part of Ueðejx0; ejxÞ, namely, that on the line x ¼ x0.

Step 2: A generalized frequency-domain expression:

We now have a characterization of estimation in the

frequency domain

h ¼ argminhV ðhÞ

¼ argminh

1

4p

Z p

�p

Z p

�pUe ejx

0; ejx

� �dxdx0;

where x ¼ x0: ð39Þ

Let us partition e into two parts

e ¼ bN �1h ½ðGh � bGhÞuþ Nhe�

¼ bN �1h Gh � bGh Nh � bNh

� ue

� �þ e: ð40Þ

Under a standard assumption 3 to avoid algebraic loops

[21], the two parts in (40) are uncorrelated. Since Gh, bGh,

Nh and bNh are LTI, (31) gives

4 Matlab programs are available online: http://www.ece.ualberta.ca/

~jwang/research.htm.


Ue ejx0; ejx

� �¼ 1bNh ejx0 Gh ejx

0� �

� bGh ejx0

� �Nh ejx

0� �

� bNh ejx0

� �h i

�Uu ejx

0; ejx

� �Uue ejx

0; ejx

� �Ueu ejx

0; ejx

� �Ue ejx

0; ejx

� �264

375�

G�hðejxÞ � bG�

hðejxÞ

N �h ðejxÞ � bN �

h ðejxÞ

" #1bNhðejxÞ

þ Ue ejx0; ejx

� �:

By Schur complementary factorization of

Uu ejx0; ejx

� �Uue ejx

0; ejx

� �Ueu ejx

0; ejx

� �Ue ejx

0; ejx

� �264

375and a bias term defined as

bBhðejxÞ ¼NhðejxÞ � bNhðejxÞ� �

Ueu ejx0; ejx

� �Uu ejx0 ; ejx ; ð41Þ

we obtain (similar to Section 8.5 in [13])

Ue ejx0; ejx

� �¼

Gh ejx0

� �� bGh ejx

0� �

þ bBh ejx0

� �bNh ejx0 Uu ejx

0; ejx

� �

� GhðejxÞ � bGhðejxÞ þ bBhðejxÞbNhðejxÞþNh ejx

0� �

� bNh ejx0

� �bNh ejx0

� Ue ejx0; ejx

� �0@ �Uue ejx

0; ejx

� �Ueu ejx

0; ejx

� �Uu ejx0 ; ejx

1A� NhðejxÞ � bNhðejxÞbNhðejxÞ

þ Ue ejx0; ejx

� �: ð42Þ

Substituting (42) into (39) gives a frequency-domain

description of asymptotic parameter estimation for the

fast-sampling direct approach. The new description can

be regarded as a generalized version of (2) for identifi-

cation of LTI systems with nonstationary inputs.

Therefore, many ideas based on (2), e.g., those aboutbias distribution in [13], are also applicable here by

giving components new interpretations. If the input u is

WSS, the new description reduces to (2), as expected.

Step 3: A nonzero weighting term: Since only the di-

agonal part of Ueðejx0; ejxÞ plays a role in (39), we can

focus our attention on the diagonal parts of Uuðejx0; ejxÞ,

Uueðejx0; ejxÞ and Ueuðejx

0; ejxÞ. Eq. (32) gives the bispec-

trum of u on the diagonal line x ¼ x0

Uu ejx0; ejx

� �¼ P 0

u ðejxÞ ¼Xp�1

n¼0

FnðejxÞF �n ðejxÞk0: ð43Þ

Eqs. (33) and (34) give the cross bispectra of u and e onthe diagonal line

Uue ejx0; ejx

� �¼ F0ðejxÞk0; ð44Þ

Ueu ejx0; ejx

� �¼ k0F �

0 ðejxÞ: ð45Þ

From (43)–(45), we have an explicit expression of the

new second weighting term in (42)

Ue ejx0; ejx

� ��Uue ejx

0; ejx

� �Ueu ejx

0; ejx

� �Uu ejx0 ; ejx

¼Pp�1

n¼0 FnðejxÞF �n ðejxÞk

20

� �� F0ðejxÞF �

0 ðejxÞk20Pp�1

n¼0 FnðejxÞF �n ðejxÞk0

¼Pp�1

n¼1 jFnðejxÞj2Pp�1

n¼0 jFnðejxÞj2k0: ð46Þ

Eq. (46) will be nonzero if FnðejxÞ 6¼ 0, for n ¼1; 2; . . . ; p � 1, which is true in the fast-sampling direct

approach framework. Therefore, identifiability is

achieved.

Remark. FnðejxÞ, for n ¼ 1; 2; . . . ; p � 1, are usually

called aliasing components in signal processing [1,19],

where a desirable action is to remove them as much as

possible. However, in our context, the aliasing compo-

nents provide necessary information to achieve identi-

fiability.

8. Examples and analysis

In this section, we use examples 4 and the previous

theoretical results to analyze the fast-sampling direct

approach. First, Example 2 shows the effectiveness of

the approach. However, a variation of Example 2 re-

veals a problem that motivates a further investigation.Finally, we find out the factors affecting the proposed

approach and summarize its properties.

8.1. Effectiveness and a problem

Example 2. We consider a SISO closed-loop system

without external excitation, depicted in Fig. 1 with

r ¼ 0, where the true process, the noise system and the

controller are

GcðsÞ ¼1

2sþ 1; NcðsÞ ¼

sþ 1:2

sþ 0:6; F ðqÞ ¼ 1:

Here T ¼ 0:4 s and the variance of the white noise ec is0.1. To use the fast-sampling direct approach, we let

http://www.ece.ualberta.ca/~jwang/research.htm

http://www.ece.ualberta.ca/~jwang/research.htm


p ¼ 2, i.e., the fast-sampling period h ¼ T=p ¼ 0:2 s. Theidentification procedure is

(1) The output yc is sampled with period h into y. A fast-rate input u with period h is measured directly or is

available from uT because of the zero-order hold.

Thus, the sampling rates of y and u are twice faster

than those of yT and uT .(2) Taking y and u as the output and the input, we esti-

mate fast-rate process models and noise models

using the traditional direct approach.

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4 Step responses

Time (sec)

Am

plitu

de

Fig. 12. Step responses and Bode plots of GT (solid) and bGT

100

10-1

100

Bode plots of actual process and estimated models

Mag

nitu

de (

dB)

Frequency (rad/sec)

Fig. 11. Bode plots of GT (solid) and bGT obtained by the fast-sampling

direct approach (small-dotted), the traditional direct approach (dash-

dot), and the traditional direct approach with external excitation (big-

dotted).

(3) The slow-rate models with period T are computed

uniquely from the fast-rate models with period h.

Bode plots of the true process and the slow-rateprocess model are shown in Fig. 11. As a comparison, a

model obtained directly by the traditional direct ap-

proach using the slow-rate signals yT and uT is presented

too. It is clear that if there is no external excitation, the

traditional direct approach loses identifiability, while the

fast-sampling direct approach gives a good estimation.

To be more comprehensive, we also present a model

obtained by the traditional direct approach with exter-nal excitation (the variance of r is 1), which achieves

identifiability as expected; see also examples in [15].

Remark. In Example 2, we use a simple (low-order)

closed-loop system for the sake of a clearer analysis

later. See [22,23] for examples of high-order closed-loop

system simulations.

The fast-sampling direct approach works very well in

Example 2. However, this approach may not always give

satisfactory results. We use a variation of Example 2 to

see the problem.

Example 3. We have the same configuration as that in

Example 2 except a different controller with a smallergain F ðqÞ ¼ 0:5. Following the same procedure as that

in Example 2, a slow-rate process model with period T is

estimated and presented in Fig. 12. Unfortunately, the

true dynamics is not captured well. Before giving an

explicit explanation of the problem, let us discuss an

intuitive viewpoint as that in Section 3. In Fig. 6, we

have two equations

101 100 10110-2

10-1

100

101Bode plots

Mag

nitu

de (

dB)

Frequency (rad/sec)

obtained by the fast-sampling direct approach (dotted).


y ¼ Ghuþ Nhe; ð47Þu ¼ ð�HpFSpÞy: ð48ÞSince u and y satisfy both equations, identification may

converge to (48) in practice if the error caused by ap-

proximating the LPTV controller with an LTI model is

less significant than the prediction error in (47).

8.2. Frequency-domain analysis

To explain the problem shown in Example 3, we shall

now use the results in the identifiability analysis to in-

vestigate the fast-sampling direct approach. From (42),

we know that bGh is mainly determined by the bias term

(41) and bNh is determined by the second weighting term

(46). Here we use an example to show how to express the

bispectra in (46) and (41) explicitly.

Example 4. We focus on the case p ¼ 2; the general case

follows similarly. Let the true fast-rate systems have

state-space representations

Fact 4 gives the diagonal parts of the bispectrum and the

cross bispectra

Uu ejx0; ejx

� �¼ jF0ðejxÞj2k20 þ jF1ðejxÞj2k20; ð49Þ

Uue ejx0; ejx

� �¼ k0F0ðejxÞ; Ueu ejx

0; ejx

� �¼ k0F �

0 ðejxÞ:

ð50Þ

Eq. (28) says that FlðejxÞ ¼ AlðejxÞ, while Al comes from

(27)

A0ðqÞA1ðqÞ

� �¼ 1

2

B0ðqÞ þ B1ðqÞB0ðqÞ � B1ðqÞ

� �:

Appendix C shows that B0 and B1 can be represented in

terms of the fast-rate systems and the controller as

B0ðqÞ ¼ ð1þ q�1ÞM0ðq2Þ; B1ðqÞ ¼ ð1þ qÞM1ðq2Þ;

where M0 and M1 are given by Fact 2

�1
M0 ¼ �F 1½ þ ðG00 þ G01ÞF � N00;
W ðqÞ ¼ 1� jð1þ q�1ÞN00ðq2Þ þ ð1þ qÞN01ðq2Þj2

jð1þ q�1ÞN00ðq2Þ þ ð1þ qÞN01ðq2Þj2 þ jð1þ q�1ÞN00ðq2Þ � ð1þ qÞN01ðq2Þj2h i ; ð54Þ

�1
M1 ¼ �F 1½ þ ðG00 þ G01ÞF � N01:
Here G00, G01, N00 and N01 are subsystems of the lifted

systems Gh and Nh, respectively, (see Fact 2 and Chapter

8 in [2])

Thus, we can associate F0 and F1 with the fast-rate sys-

tems and the controller as

F0ðqÞ ¼1

2

ð1þ q�1ÞN00ðq2Þ þ ð1þ qÞN01ðq2Þ½ �F ðq2Þ1þ G00ðq2Þ þ G01ðq2Þð ÞF ðq2Þ ;

ð51Þ

F1ðqÞ ¼1

2

ð1þ q�1ÞN00ðq2Þ � ð1þ qÞN01ðq2Þ½ �F ðq2Þ1þ G00ðq2Þ þ G01ðq2Þð ÞF ðq2Þ :

ð52Þ

Substituting (51) and (52) into (49) and (50) gives ex-plicit expressions of the bispectrum and the cross bi-

spectra.

With the help of those explicit expressions, we can

find out the factors affecting the fast-sampling direct

approach. Since the quality of bNh will affect that of bGh

through bBh (see (42)), we first investigate the second

weighting term (46), from which we could define anindex function

W ðejxÞ :¼ 1� jF0ðejxÞj2Pp�1

n¼0 jFnðejxÞj2: ð53Þ

Thus, W is a real function with range ð0; 1Þ. A larger

weighting will give a better estimation of the noise sys-

tem. It is equivalent to preference that F0 occupies a

small portion in all Fn, which is reasonable as aliasingcomponents Fn for n 6¼ 0 makes the closed-loop system

identifiable.

Furthermore, taking p ¼ 2 with results (51) and (52)

as an example (the general case follows similarly), W can

be expanded as

from which we conclude that W is only related to the

characteristics of the noise system. For instance, if


p ¼ 2, we prefer a large difference between N00 and N01,

which is achieved, e.g., when the noise system has a

small time constant.

Next, we check the bias term (41). Substituting (43)and (45) into (41), we have

bBhðejxÞ ¼ NhðejxÞ�

� bNhðejxÞ� F �

0 ðejxÞPp�1

n¼0 jFnðejxÞj2: ð55Þ

If (55) is expanded as we did for W in (54), we can

conclude that bBh is related to all dynamics in the closed-

loop system: the noise system, the process and thecontroller. Since they will affect the bias distribution in a

complicated way, an affirmative conclusion is hard to

make. However, if the gain of the controller is too small

or the time constant of the process is too large, the ex-

pansion of (55) reveals that bBh will be large and the

estimation may be unsatisfactory. Here we use an ex-

ample to verify these conclusions.

Example 5. We list identification results of Example 2

and its three variations in a table, where the varying

parts are in the first column. H2-norm and H1-norm

are used to measure the model mismatch.

In the second row, we decrease the time constant of the

noise system by half and keep its gain unchanged. Asexpected, the estimation is poorer (as W is smaller). In

the third and the fourth rows, where the process time

constant and the controller gain are decreased by half

respectively, qualities of bGh become poorer, but those ofbNh are not affected. The simulation results are consistent

with the previous analysis. Besides the above numerical

analysis, it is worthy to point out that closed-loop data

has less information than the open-loop data due to theexistence of feedback [13]; such a problem appears even

severer in the systems dealt with by the fast-sampling

direct approach since the whole system is driven by the

noise source only.

We can also use the frequency-domain results to see

the roles of two other components in the closed-loop

system: the variance k0 of the white noise and the fast-

sampling ratio p.

• The bias term (55) and the weighting terms in (42)

reveal that k0 has no effect on the estimation and

jGh�bGhj2jNh�bNhj2

jGh�bGhj1jNh�bNhj1

Example 2 0.0144 0.0589 0.0475 0.2029

NcðsÞ ¼ sþ0:6sþ0:3

0.0465 0.0977 0.1862 0.4927

GcðsÞ ¼ 14sþ1

0.0891 0.0657 0.6497 0.1938

F ðqÞ ¼ 0:5 0.0955 0.0577 0.4361 0.1460

works only as a scaling factor. This is intuitively

natural since the whole system is driven by the white

noise only and there are no other external signals to

be compared with.• There is a tradeoff in the role of p. If p increases, F0

will occupy a smaller portion in all Fn, bBh in (55)

will be smaller and W in (53) will be larger, i.e., es-

timation will be better. On the other hand, h will be

smaller if p is increased. It is equivalent to increas-

ing time constants of the systems, which deterio-

rates the estimation. A simulation demonstrating

the tradeoff was given in [23] with an implicit expla-nation.

In summary, the main advantage of the fast-sam-

pling direct approach is that external excitation is not

required, which means that it could be a useful ap-

proach when external excitation is not permitted or the

requirement of normal operation is stringent [13]. The

main drawback of the approach is that it may not al-ways be able to give a satisfactory estimation. Another

drawback inherits from the traditional direct approach:

a simultaneous good estimation of a noise model is

required to accompany the estimation of the plant

model [27].

9. Conclusion

In this paper, we have studied a so-called fast-sam-

pling direct approach in an instructive and insightful

way. The proposed approach removes a traditional

identifiability condition for closed-loop systems with

linear output feedback, i.e., an external persistentlyexciting test signal is not required. Based on a gen-

eralized frequency-domain expression, some character-

istics of the approach have been realized: Theoretically,

identifiability is achieved without external excitation;

practically, a satisfactory estimation is confined by the

dynamics of the closed-loop system; even so, the ap-

proach has a great potential to explore, e.g., it may

achieve smaller asymptotic variances than the tradi-tional direct approach. In this study, the lifting tech-

nique, bifrequency maps and bispectra have been

shown as powerful tools when LPTV systems and

cyclo-stationary signals are involved. In the same time,

some gaps of bifrequency maps, bispectra and LPTV

systems are filled.

There are still quite a few open problems about the

fast-sampling direct approach. We would like to men-tion two of them. First, we have seen that the approach

sometimes may not give a satisfactory estimation. So, is

it possible to give a quantitive index indicating when the

approach will work well? Second, what are the explicit


asymptotic variance expressions of the approach? These

are left to the future studies.

Appendix A

We shall prove Fact 2 through a special case when

p ¼ 2 in (14); the general case follows similarly.

M ¼ �F ðI þ GhF Þ�1Nh ¼�F 0

�F 0

" #1þ ðG00 þ G01ÞF 0

ðG10 þ G11ÞF 1

" #�1 N00 N01

N10 N11

" #

¼�F 0

�F 0

" #½1þ ðG00 þ G01ÞF ��1

0

�ðG10 þ G11ÞF ½1þ ðG00 þ G01ÞF ��11

24 35 N00 N01

N10 N11

" #

¼�F ½1þ ðG00 þ G01ÞF ��1N00 �F ½1þ ðG00 þ G01ÞF ��1N01

�F ½1þ ðG00 þ G01ÞF ��1N00 �F ½1þ ðG00 þ G01ÞF ��1N01

24 35 ¼:M0 M1

M0 M1

" #;

where the transfer matrix of F comes from Fact 1 and

the third equality follows from the matrix inversion

lemma [20]. h

Appendix B

This appendix serves to prove (33) and (34). Let x andy be, respectively, the input and output of an

(LPTV)p system with Green’s function kðm; nÞ. Using

(19), we have

Ryxðm; nÞ ¼ EðyðmÞxyðnÞÞ ¼ EX1l¼�1

kðm; lÞxðlÞxyðnÞ !

¼X1l¼�1

kðm; lÞRxðl; nÞ: ðB:1Þ

By comparing with (20), (B.1) is the Green’s function of

the cascade of two systems with Green’s func-

tions Rxðm; nÞ and kðm; nÞ in this order. From (21), wehave

Uyx ejx0; ejx

� �¼Z p

�pK ejx

0; ejx

00� �

Ux ejx00; ejx

� �dx00

¼Z p

�pF ejx

0; ejx

00� � X1

l¼�1d x00�

� x0

þ 2plp

�!Ux ejx

00; ejx

� �dx00

¼Xp�1

l¼0

F ejx0; ej x0�2pl

pð Þ� �

Ux ej x0�2plpð Þ; ejx

� �¼Xp�1

l¼0

FlðejxÞUx ej x0�2plpð Þ; ejx

� �;

where the second equality follows from (22); the third

equality uses the sifting property of the dð�Þ function; the

last equality is the definition in (23). Eq. (34) can be

proven similarly (see [25]). h

Appendix C

Section 6.1 shows that an LPTV system can always be

represented as a filter bank in Fig. 7. We also know that

the lifting technique can associate an LPTV system with

an LTI lifted system [12]. This appendix shows the way

that Bn, defined in (25), in the filter bank representationis connected with the lifted system.

Fact 5. Suppose that G is the lifted version of an ðLPTVÞpsystem G, then Bn in the filter bank representation de-picted in Fig. 7 are

BnðqÞ ¼Xp�1

i¼0

qn�iGi;nðqpÞ; for n ¼ 0; 1; . . . ; p; ðC:1Þ

whereGi;n is the ði; nÞth element of the transfer matrix ofG.

Proof. To see (C.1), we take p ¼ 2; the general case

follows similarly. Let G be represented by a state-spacemodel

By the system matrices of L2 and L�12 [2], the system

matrix ½G� is

ðC:2Þ


From (C.2) and the definition of Bn in (25), we obtain

B0ðqÞ ¼ D00 þ D10q�1 þ C0B0q�2 þ C1B0q�3

þ C0AB0q�4 þ C1AB0q�5 þ � � �¼ ðD00 þ C0B0q�2 þ C0AB0q�4 þ � � �Þ

þ q�1ðD10 þ C1B0q�2 þ C1AB0q�4 þ � � �Þ¼ G00ðq2Þ þ q�1G10ðq2Þ; ðC:3Þ

and similarly

B1ðqÞ ¼ qG01ðq2Þ þ G11ðq2Þ: ðC:4ÞGeneralizing (C.3) and (C.4) in a compact form gives

(C.1). h

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