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
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
558 J. Wang et al. / Journal of Process Control 14 (2004) 555–570
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.
J. Wang et al. / Journal of Process Control 14 (2004) 555–570 559
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.
560 J. Wang et al. / Journal of Process Control 14 (2004) 555–570
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.
J. Wang et al. / Journal of Process Control 14 (2004) 555–570 561
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
562 J. Wang et al. / Journal of Process Control 14 (2004) 555–570
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.
J. Wang et al. / Journal of Process Control 14 (2004) 555–570 563
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.
564 J. Wang et al. / Journal of Process Control 14 (2004) 555–570
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.
J. Wang et al. / Journal of Process Control 14 (2004) 555–570 565
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
566 J. Wang et al. / Journal of Process Control 14 (2004) 555–570
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).
J. Wang et al. / Journal of Process Control 14 (2004) 555–570 567
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
568 J. Wang et al. / Journal of Process Control 14 (2004) 555–570
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
J. Wang et al. / Journal of Process Control 14 (2004) 555–570 569
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
570 J. Wang et al. / Journal of Process Control 14 (2004) 555–570
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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