In this paper, an efficient decentralized iterative learning tracker is proposed to improve the dynamic performance of the unknown controllable and observable sampled-data interconnected large-scale state-delay system, which consists of NN multi-input multi-output (MIMO) subsystems, with the closed-loop decoupling property. The off-line observer/Kalman filter identification (OKID) method is used to obtain the decentralized linear models for subsystems in the interconnected large-scale system. In order to get over the effect of modeling error on the identified linear model of each subsystem, an improved observer with the high-gain property based on the digital redesign approach is developed to replace the observer identified by OKID. Then, the iterative learning control (ILC) scheme is integrated with the high-gain tracker design for the decentralized models. To significantly reduce the iterative learning epochs, a digital-redesign linear quadratic digital tracker with the high-gain property is proposed as the initial control input of ILC. The high-gain property controllers can suppress uncertain errors such as modeling errors, nonlinear perturbations, and external disturbances (Guo et al., 2000) [18]. Thus, the system output can quickly and accurately track the desired reference in one short time interval after all drastically-changing points of the specified reference input with the closed-loop decoupling property.

1. ISA Transactions 51 (2012) 81–94
Contents lists available at SciVerse ScienceDirect
ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans
Efficient decentralized iterative learning tracker for unknown sampled-data
interconnected large-scale state-delay system with closed-loop
decoupling property
Jason Sheng-Hong Tsaia,∗
, Fu-Ming Chena
, Tze-Yu Yua
, Shu-Mei Guob,∗∗
, Leang-San Shiehc
a
Control System Laboratory, Department of Electrical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC
b
Department of Computer Science and Information Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC
c
Department of Electrical Engineering, University of Houston, University Park, Houston, TX 77204-4005, USA
a r t i c l e i n f o
Article history:
Received 18 May 2011
Received in revised form
8 July 2011
Accepted 6 August 2011
Available online 27 August 2011
Keywords:
Decentralized control
Digital redesign
Linear quadratic tracker
Iterative learning control
a b s t r a c t
In this paper, an efficient decentralized iterative learning tracker is proposed to improve the dynamic
performance of the unknown controllable and observable sampled-data interconnected large-scale state-
delay system, which consists of N multi-input multi-output (MIMO) subsystems, with the closed-loop
decoupling property. The off-line observer/Kalman filter identification (OKID) method is used to obtain
the decentralized linear models for subsystems in the interconnected large-scale system. In order to get
over the effect of modeling error on the identified linear model of each subsystem, an improved observer
with the high-gain property based on the digital redesign approach is developed to replace the observer
identified by OKID. Then, the iterative learning control (ILC) scheme is integrated with the high-gain
tracker design for the decentralized models. To significantly reduce the iterative learning epochs, a digital-
redesign linear quadratic digital tracker with the high-gain property is proposed as the initial control input
of ILC. The high-gain property controllers can suppress uncertain errors such as modeling errors, nonlinear
perturbations, and external disturbances (Guo et al., 2000) [18]. Thus, the system output can quickly and
accurately track the desired reference in one short time interval after all drastically-changing points of
the specified reference input with the closed-loop decoupling property.
© 2011 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
In the recent years, the decentralized control of interconnected
large-scale systems has been one of the popular research topics in
control theory. Large-scale systems have very complex dynamic
models due to the uncertain environment, the interconnected
structure of the system, and the varying system parameters. All
these issues make the stabilization of such large-scale systems a
difficult control problem. Examples of practical systems including
power systems, chemical processes, and communications systems
are shown in [1,2]. Many researches focusing on the above issues
have appeared in [3], and various methods have been proposed
to deal with those problems. Decentralized adaptive control work
started long before 1996 but Ioannou and Sun [4] is a good
literature review up to that date. Ref. [4] also shows that even weak
interconnections can make a decentralized adaptive controller
∗ Corresponding author. Tel.: +886 6 2757575x62360; fax: +886 6 2345482.
∗∗ Corresponding author. Tel.: +886 6 2757575x62525; fax: +886 6 2747076.
E-mail addresses: shtsai@mail.ncku.edu.tw (J.S.-H. Tsai),
guosm@mail.ncku.edu.tw (S.-M. Guo).
unstable. In the main, the decentralized adaptive control methods
need to utilize real system information to modify the model so
as to improve the model-base optimum [5]. So, when the system
information could not be obtained or measured at some time, the
previous methods could not be used.
Iterative learning control (ILC) was firstly proposed by Arimoto
et al. [6,7]. The main advantage of the iterative learning control
strategy requires less prior knowledge about the system dynamics
and less computational effort than many other types of control
strategies. Furthermore, the controller can exhibit the perfect
control performance according to less prior knowledge of the
system. Due to its advantages which present higher efficiency
and more convenience, many researchers discuss ardently the
subjects of ILC and usually apply the controller in several physical
systems such as industrial robots [8], robotic manipulators [9],
computer numerical control (CNC) machines [10], and antilock
braking systems (ABS) [11], etc. An optimization paradigm of
the iterative learning control that concentrated on linear systems
and the potential of the use of optimization methods to achieve
effective control has been proposed by Owens’s research group at
Sheffield University [12]. In recent years, decentralized iterative
learning controls [13,14] have been developed maturely and
0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.isatra.2011.08.001

2. 82 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 1. The decentralized observer-based digital-redesign trackers for the unknown sampled-data interconnected large-scale system (for N = 2) with the state-delay terms.
embedded to many industrial processes, such as petrochemical
processes and metallurgical processes [14,15]. The aim to embed
decentralized ILC into the above procedures is to improve the
dynamic performance of the transient response which limits
to stable systems only. However, in general, the decentralized
iterative learning control law is high-order and more complex for
a class of large-scale interconnected dynamical systems.
In this paper, an efficient decentralized iterative learning
tracker is proposed to improve the dynamic performance of the un-
known sampled-data interconnected large-scale state-delay sys-
tem, which consists of N MIMO subsystems, with the closed-loop
decoupling property. Yi et al. [16] derived the criteria for point-
wise controllability and observability and developed a method to
use the matrix Lambert W function-based solution form for the lin-
ear time delay system with one state-delay. According to controlla-
bility and observability of time delay system with the multi-state
delays and interconnected state delays, the literature is not pre-
sented so far.
The purpose of this paper is to propose an efficient decen-
tralized iterative learning tracker for the unknown sampled-data
interconnected large-scale state-delay system with closed-loop
decoupling property. First, the appropriate (low order) decen-
tralized linear observers are determined by the off-line OKID
method [17] for a class of (unknown) controllable and ob-
servable sampled-data interconnected large-scale system with
state-delay. The OKID method is a time-domain technique that
identifies a discrete input–output mapping in the general coordi-
nate form by using known input–output sampled data, through an
extension of the eigensystem realization algorithm (ERA), so that
the order-determination problem existing in the system identifica-
tion problem can be solved. In order to get over the effect of mod-
eling error on the identified linear model of each subsystem, an
improved observer with high-gain property based on the digital
redesign approach will be developed to replace the identified ob-
server based on OKID. Then, the iterative learning control scheme is
integrated with the high-gain tracker design for the decentralized
models. To significantly reduce the iterative learning epochs, we
propose the improved observer-based digital redesign tracker with
the high-gain property [18] to generate the initial control input of
ILC. Furthermore, it can suppress the uncertain errors such as non-
linear perturbations and external disturbances as well as make the
system output quickly and accurately track the desired reference in
one short time interval after all drastically-changing points of the
specified reference input. Finally, an example is given to demon-
strate the high-performance trajectory tracking with the closed-
loop decoupling property by the proposed methodology.
2. Problem description
Consider the unknown controllable and observable system
consisting of N interconnected MIMO subsystems with state delay
shown as follows
Si : ˙xdi(t) =
αi−
j=0
Aijxdi(t − τij)
+
N−
k=1,k̸=i
βk−
j=0
εikj
¯Aikjxdk(t − ¯τikj) + Biudi(t), (1a)
ydi(t) = Cixdi(t). (1b)
Notation Si in (1a) denotes the ith subsystem of the interconnected
large-scale system, where i = 1, 2, . . . , N. The first term presented
by summation from j = 0 to αi in the right-hand side of (1a)
denotes various linear combinations of the internal state delays of
Subsystem Si. The second term presented by double summations
in the right-hand side of (1a) denotes various linear combinations
of the external state delays from Subsystem Sk to Subsystem Si.
Notation τij is the jth the internal delay of Subsystem Si, ¯τikj is
the jth external delay from Subsystem Sk to Subsystem Si, εikj is
the jth weighting external disturbance gain from Subsystem Sk to
Subsystem Si, and Aij and ¯Aikj are system matrices and external
delay system matrices, respectively. Notation xdi(t) ∈ ℜni is the
state vector of the Subsystem Si, udi(t) ∈ ℜmi is the input, and
ydi(t) ∈ ℜpi is the output.
The design procedure of this paper is then briefly described as
follows. First, applying the off-line OKID method, the appropriate
(low order) decentralized linear observers for the interconnected
sampled-data large-scale system can be determined. Then, in order
to overcome the effect of modeling error on the identified linear
model of each subsystem, an improved observer with the high-
gain property based on the digital redesign approach will be
developed to take the place of the observer determined by the
OKID. Subsequently, the decentralized digital redesign trackers
with the high-gain property shown in Fig. 1 will be proposed, so
that the closed-loop system has a good tracking performance and
the decoupling property. To improve the dynamic performance
of the transient response, the ILC scheme is embedded in the
decentralized models. The digital redesign linear quadratic tracker
with the high-gain property is then applied to generate the
first input of ILC to significantly reducing the iterative learning
epochs. All the detailed design procedures will be presented in the
following sections, respectively.

3. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 83
Fig. 2. The traditional OKID-based modeling for unknown stochastic sampled-data systems.
3. Traditional OKID-based modeling for unknown sampled-
data systems
The traditional decentralized iterative learning control law is
high-order and complicated for a class of large-scale intercon-
nected dynamical systems with state delay. The appropriate (low
order) decentralized linear observer for the sampled-data inter-
connected large-scale linear system is then desired to be deter-
mined by the off-line OKID [17] method.
The discrete-time state-space model of a multi-variable linear
system can be represented in the following general form
x(k + 1) = Gx(k) + Hu(k), (2a)
y(k) = Cx(k), (2b)
where x(k) ∈ ℜn
, u(k) ∈ ℜm
and y(k) ∈ ℜp
are state, output,
and control input vectors, respectively, and G ∈ ℜn×n
, H ∈ ℜn×m
and C ∈ ℜp×n
are system, input, and output matrices, respectively.
The following result is summarized from the approach in [17,19].
Note that the Hankel matrix obtained from the combined Markov
parameters is associated with the system and the observer as
follows
¯H(k − 1) =




Υk Υk+1 · · · Υk+β−1
Υk+1 Υk+2 · · · Υk+β
...
...
...
...
Υk+α−1 Υk+α · · · Υk+α+β−2



 , (3)
where α and β are two sufficiently large but otherwise arbitrary in-
tegers, and Υk = [CGk−1
H CGk−1
F], and F is the observer gain to be
determined based on input and output measurements. When the
combined Markov parameters are determined, the eigensystem re-
alization algorithm (ERA) is used to obtain the desired appropri-
ate (low order) n∗
and the discrete system and observer realization
[ˆG, ˆH, ˆC, F] through the singular value decomposition (SVD) of the
Hankel matrix.
The ERA processes a factorization of the block data matrix in
(3), starting from k = 1, using the singular value decomposition
formula ¯H(0) = V
∑
ST
, where the columns of matrices V and S
are orthonormal and
∑
is a rectangular matrix of the form
−
=
−
˜n
0
0 0

, (4)
where
∑
˜n = diag[σ1, σ2, . . . , σn∗ , σn∗+1, · · · , σ˜n] contains mono-
tonically non-increasing entries σ1 ≥ σ2 ≥ · · · ≥ σn∗ > σn∗+1 ≥
· · · ≥ σ˜n > 0. Here, some singular values σn∗+1, . . . , σ˜n are rel-
atively small (σn∗+1 ≪ σn∗ ). In order to construct the low order
observer of the system, let us define
∑
n∗ = diag [σ1, σ2, . . . , σn∗ ].
The realizations of the system parameters and observer parame-
ters by the ERA are given as
ˆG =
−1/2−
n∗
VT
n∗ ¯H(1)Sn∗
−1/2−
n∗
, (5a)

ˆH F

= First (m + p) columns of
1/2−
n∗
ST
n∗ , (5b)
ˆC = First p rows of Vn∗
1/2−
n∗
. (5c)
For system identification, SVD is very useful in determining the sys-
tem order. In practice, the primary purpose of applying the OKID
method is that the constructed observer satisfies the least-squares
solution or acts the input–output map same as a Kalman filter. If
the data length is sufficiently long and the order of the observer is
sufficiently large, the truncation error is negligible.
Now, we show the relationship between the identified observer
and the Kalman filter. Let (2a) and (2b) be extended to include
process and measurement noises described as
x(k + 1) = ˆGx(k) + ˆHu(k) + w(k), (6a)
y(k) = ˆCx(k) + v(k), (6b)
where the process noise w(k)is a Gaussian, zero-mean white signal
with covariance matrix Q, and the measurement noise satisfies the
same assumption as w(k) with a different covariance matrix R. The
sequences w(k) and v(k) are independent of each other. Then, a
typical Kalman filter can be written as
ˆx(k + 1) = ˆGˆx(k) + ˆHu(k) + Kεr (k), (7a)
ˆy(k) = ˆC ˆx(k), (7b)
where ˆx(k) is the estimated state and εr (k) is defined as the
difference between the real measurement y(k) and the estimated
measurement ˆy(k). Therefore, when the residual εr (k) is a white
sequence of the Kalman filter residual, the observer gain F
converges to the steady-state Kalman filter gain for F = −K, where
K is the Kalman filter gain. The traditional OKID-based modeling
for the unknown stochastic sampled-data system is shown in Fig. 2.
4. Prediction-based digital redesign observer for unknown
deterministic sampled-data systems
4.1. Digital redesign of the observer-based linear quadratic analog
tracker
The conventional observer has the property that ˆxd (kTs) is
reconstructed from yd (kTs − Ts) , yd (kTs − 2Ts) , . . . . It is also

4. 84 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 3. Observer-based linear quadratic analog tracker.
possible to derive an observer, which also uses yd (kTs) to estimate
ˆxd (kTs), as follows. The digital estimator gain Ld will be indirectly
designed in this section, rather than being directly estimated from
the identified discrete-time parameters of the OKID model. The
following result is summarized from [18].
First, consider the linear continuous-time deterministic system
described as
˙xc (t) = Axc (t) + Buc (t), xc (0) = x0 (8a)
yc (t) = Cxc (t), (8b)
which is assumed to be both controllable and observable, where
xc (t) ∈ ℜn
, uc (t) ∈ ℜm
, and yc (t) ∈ ℜp
. One can take advantage
of the observer to estimate the unmeasured system state. Let the
linear observer with respect to the continuous-time system (8a)
and (8b) be presented by
˙ˆxc (t) = Aˆxc (t) + Buc (t) + Lc [yc (t) − C ˆxc (t)], (9)
where ˆxc (t) is the estimate of xc (t) and Lc ∈ ℜn×p
is the observer
gain [20], where
Lc = PobCT
R−1
ob , (10)
in which Pob is the symmetric and positive definite solution of the
following Riccati equation
APob + PobAT
− PobCT
R−1
ob CPob + Qob = 0, (11)
where Qob ≥ 0 and Rob > 0 with appropriate dimensions.
The observer-based linear quadratic analog tracker is shown in
Fig. 3.
The digital redesign-based observer for the deterministic
sampled-data system is then given by
˙ˆxd(kTs + Ts) = Gd ˆxd(kTs) + Hdud(kTs) + Ldyd(kTs + Ts) (12a)
and
ˆyd(kTs + Ts) = C ˆxd(kTs + Ts), (12b)
where
Ld = (G − In)A−1
Lc (Im + C(G − In)A−1
Lc )−1
, (13)
Gd = G − LdCG, (14)
Hd = (In − LdC)H, (15)
G = eATs
, (16)
H = (G − In)A−1
B. (17)
In view of practical implementation, the alternative discrete
observer utilizes the current output yd(kTs) and the previously
estimated state ˆxd(kTs − Ts) to estimate the current state ˆxd(kTs)
as follows
˙ˆxd(kTs) = Gd ˆxd(kTs − Ts) + Hdud(kTs − Ts) + Ldyd(kTs), (18a)
ˆyd(kTs) = C ˆxd(kTs). (18b)
The observer-based digital tracker for the deterministic sampled-
data system is shown in Fig. 4.
4.2. Linear quadratic analog tracker design
Consider the linear analog system given in (8). The optimal
state-feedback control law for the linear quadratic tracker is to
minimize the following performance index
J =
∫ ∞
0

[Cxc (t) − r(t)]T
Q [Cxc (t) − r(t)] + uT
c (t)Ruc (t)

dt, (19)
with Q ≥ 0 and R > 0. This optimal control [18] is given by
uc (t) = −Kc xc (t) + Ec r(t), (20)
where the analog feedback gain Kc ∈ ℜm×n
and the forward gain
Ec ∈ ℜm×m
for m = p are
Kc = R−1
BT
P, (21)
Ec = −R−1
BT
[(A − BKc )−1
]T
CT
Q . (22)
Then, the resulting closed-loop system becomes
˙xc (t) = (A − BKc )xc (t) + BEc r(t). (23)
Here, r(t) is a reference input or desired trajectory, and P is the
positive definite and symmetric solution of the following Riccati
equation as
AT
P + PA − PBR−1
BT
P + CT
QC = 0. (24)
The closed-loop system (23) is asymptotically due to the property
of LQR design (20).
4.3. Digital redesign of the linear quadratic analog tracker
Let the continuous-time state-feedback controller be
uc (t) = −Kc xc (t) + Ec r(t), (25)
where Kc ∈ ℜm×n
and Ec ∈ ℜn×m
have been designed to satisfy
some specified goals, and r(t) ∈ ℜm
is a desired reference input
vector. Thus, the analogously controlled system is
˙xc (t) = Ac xc (t) + BEc r(t), xc (0) = xc0 = x0, (26)
where Ac = A − BKc . Let the state equation of a corresponding
discrete-time equivalent model be
˙xd(t) = Axd(t) + Bud(t), xd(0) = xd0 = x0, (27)
where ud(t) ∈ Rm
is a piecewise-constant input vector, satisfying
ud(t) = ud(kTs), for kTs ≤ t < (k + 1)Ts, (28)
and Ts > 0 is the sampling period. Then, the discrete-time state-
feedback controller is given by [18] as
ud(kTs) = −Kdxd(kTs) + Edr∗
(kTs), (29)
where

5. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 85
Fig. 4. OKID-based digital tracker and observer for the deterministic sampled-data system.
Fig. 5. A memory based ILC.
Fig. 6. Comparison between the actual output and estimated output obtained by OKID: (a) outputs Yid11(kTs) and Yo11(kTs), (b) outputs Yid12(kTs) and Yo12(kTs), (c) outputs
Yid21(kTs) and Yo21(kTs), (d) outputs Yid22(kTs) and Yo22(kTs).

6. 86 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 7. Errors between: (a) outputs Yid11(kTs) and Yo11(kTs), (b) outputs Yid12(kTs) and Yo12(kTs), (c) outputs Yid21(kTs) and Yo21(kTs), (d) outputs Yid22(kTs) and Yo22(kTs).
Kd = (Im + Kc H)−1
Kc G, (30a)
Ed = (Im + Kc H)−1
Ec , (30b)
r∗
(kTs) = r(kTs + Ts), (30c)
G = eATs
, (30d)
H = (G − In)A−1
B for nonsingular A, (30e)
H =
[
Ts + A
(Ts)2
2!
+ A2 (Ts)3
3!
+ · · ·
]
B for singular A, (30f)
Kd ∈ ℜm×n
is a digital state-feedback gain, Ed ∈ ℜm×m
is a digital
feed-forward gain, and r∗
(kTs) ∈ ℜm
is a piecewise-constant
reference input vector determined in terms of r(kTs) for tracking
purposes.
It is well known that the high-gain (analog) controller/observer
induces a high quality performance on trajectory tracking
design/state estimation, and it can also suppress system uncer-
tainties such as nonlinear perturbations, parameter variations,
modeling errors and external disturbances. For these reasons, the
sub-optimal analog controller and observer with a high-gain prop-
erty is adopted in our approach. The high-gain property controller
can be obtained by choosing a sufficiently high ratio of Q to R in
(19) so that the system output can closely track a pre-specified
trajectory. However, the high-gain property of the analog tracker
usually yields large control signals, which might cause the system
actuator to saturate and give an unsatisfactory system response. To
overcome this difficulty, the tracker is redesigned based on the ad-
vanced digital redesign technique equipped with a suitably large
sampling period and zero hold, which yields an equivalent dig-
ital controller but with a low-gain, without possibly losing the
high quality performance. However, a large sampling period usu-
ally induces a degradation of the tracking performance. Therefore,
in general, a suitable compromise between the pre-specified per-
formance and the selections of the sampling time Ts, weighting ma-
trices (Qo, Ro) in (11) and (Q , R) in (19) should be considered. For
simplicity in discussion, we neglect the actuator saturation prob-
lem in this paper.
5. Iterative learning control
In recent years, there have been many subjects and approaches
in the field of ILC. The basic idea of ILC is that a system performs
the same task repeatedly, and the control performance of the
system can be improved by learning from previous iterations.
Many existing control methods are not able to fulfill such a task,
because they only warrant an asymptotic convergence, and being
more essential, they are unable to learn from previous control
trails, whether those succeeded or failed. Without learning, a
control system can only produce the same performance without
improvement, even if the task repeats consecutively. On the
other hand, an initial state error occurs when the initial state
of the system is different from the initial state that is implicitly
given by the reference trajectory. It is shown that under this
sufficient condition [21–23], the iterative learning control can
ensure the system output converge to desired trajectories with
bounded tracking errors. Besides, the proposed iterative learning
controller [21,22] works with a reduced sampling rate that ensures

7. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 87
Fig. 8. Comparison between the actual output and its observer-based output by OKID: (a) outputs Yid11(kTs) and Yo11(kTs), (b) outputs Yid12(kTs) and Yo12(kTs), (c) outputs
Yid21(kTs) and Yo21(kTs), (d) outputs Yid22(kTs) and Yo22(kTs).
the reduction of an appropriate norm of the error trajectory
from cycle to cycle. Notice that the convergence of ILC is
directly influenced by the initial control input. To accelerate the
convergence of ILC, the digital-redesign linear quadratic digital
tracker with the high-gain property is proposed as the initial
control input of ILC.
5.1. The design of the ILC controller
ILC has earned a lot of interest in the current years. The basic
concept of the ILC is to use the control information of the previous
iterations to improve the control performance of the present
iteration. This is realized through the memory based learning
scheme which is shown in Fig. 5.
Consider a discrete time system
x(k + 1) = Gx(k) + Hu(k), x(0) = x0, (31a)
y(k) = Cx(k), (31b)
where G = eAc Ts and H =
 T
0
eAc Ts Bdt. The above equations are the
zero order holder equivalent of continuous time systems described
by x(t) ∈ ℜn
, Ac ∈ ℜn×n
, Bc ∈ ℜn×m
, and C ∈ ℜp×n
. For the
iterative learning cycle l with l = 0, 1, . . . , Nl, the system can be
described by [21,22].
yl(0) = Cx0, (32)
yl = L0x0 + Lul, (33)
yl = [yl(1)yl(2)yl(3) · · · yl(Nl)]T
, (34)
ul = [ul(0)ul(1)ul(2) · · · ul(Nl − 1)]T
, (35)
with
L0 =






CG
CG2
CG3
...
CGNl






, (36)
and
L =






CH 0 · · · 0
CGH CH
...
...
...
...
...
...
CGNl−1
H · · · CGH CH






(37)
where CH is assumed to be non-singular.
5.2. Problem formulation and discrete ILC updating law
According to the literature [21,22], the update learning law is
presented as
ul+1(k) = ul(k) + Γ el (k + 1) , (38)
el(k) = yd(k) − yl(k), (39)

8. 88 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 9. Errors between: (a) outputs Yid11(kTs) and Yo11(kTs), (b) outputs Yid12(kTs) and Yo12(kTs), (c) outputs Yid21(kTs) and Yo21(kTs), (d) outputs Yid22(kTs) and Yo22(kTs).
where el is the error term, yd is the tracking trajectory, and Γ is the
learning control gain. Then, substituting (38) into (33) yields the
following error function
El+1 = LeEl, (40)
where El = [eT
l (1)eT
l (2)eT
l (3) · · · eT
l (N)]T
, and the matrix Le is
presented as
Le =






I − CHΓ 0 · · · 0
−CGHΓ I − CHΓ
...
...
...
...
... 0
−CGNl−1
HΓ · · · −CGHΓ I − CHΓ






. (41)
While we assign an appropriate learning gain Γ to induce the
1-norm of the matrix Le to be less than 1, i.e. ‖Le‖1 < 1, the tracking
error el converges to zero as l approaches infinity. To assure
the condition ‖Le‖1 < 1 can be satisfied, the literature [21,22]
suggests reducing the matrix norm ‖G‖ via choosing a large sample
time Ts.
Moreover, ILC takes a lot of iterative learning epochs to update
the input of the system owing to the input of the initial iterative
learning epoch being set to zero, which is commonly used in
literature. To significantly reduce the iterative learning epochs and
greatly promote the tracking performance, we use the observer-
based digital redesign linear quadratic analog tracker shown in
Fig. 4, which has been shown [18] to be a high performance
approach, to set up the primary cycle input of the control
system.
6. Design procedure
The design procedure is summarized as follows:
Step 1: Perform the off-line observer/Kalman filter identification
method to determine the appropriate (low order) decen-
tralized linear system/observer models from the unknown
sampled-data interconnected large-scale linear system
with state delay.
Step 2: Transform the obtained discrete-time linear system/obser-
ver models to continuous-time linear system/observer
models with the pre-specified sampling times.
Step 3: Select appropriate weighting matrices (Q , R)and (Qob, Rob)
for tracker and observer designs by choosing sufficiently
high ratios of Q to R and Qob to Rob, so that the tracker and
observer have the high-gain property.
Step 4: Embed the iterative learning control scheme in the decen-
tralized models. Extract the control input determined in
Step 3 as the first generation control input of ILC.
Step 5: Repeatedly apply the ILC algorithm ul(k), for l = 1, 2,
3, . . . , until it reaches the good performance tracking
object.

9. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 89
Fig. 10. Input responses via the proposed method at the 10th generation: (a) Input responses of Subsystem S1: input ud11(t). (b) Input responses of Subsystem S1: input
ud12(t). (c) Input responses of Subsystem S2: input ud21(t). (d) Input responses of Subsystem S2: input ud22(t).
7. An illustrative example
To show the effectiveness of the newly proposed method, an ex-
ample is given to explain the high performance trajectory tracking
with the closed-loop decoupling property as following. From the
academic study point of view, consider an unknown controllable
and observable deterministic large-scale linear system which con-
tains two interconnected 2-input–2-output subsystems with state
delays as follows
Si : ˙xd1(t) = A10xd1(t − τ10) + A11xd1(t − τ11)
+ A12xd1(t − τ12) + ε120
¯A120xd2(t − ¯τ120)
+ ε121
¯A121xd2(t − ¯τ121)
+ ε122
¯A122xd2(t − ¯τ122) + B1ud1(t),
S2 : ˙xd2(t) = A20xd2(t − τ20) + A21xd2(t − τ21)
+ A22xd1(t − τ22) + ε210
¯A210xd1(t − ¯τ210)
+ ε211
¯A211xd1(t − ¯τ211)
+ ε212
¯A212xd2(t − ¯τ212) + B2ud2(t),
where
ud1(t) =

ud1,1(t)
ud1,2(t)

, ud2(t) =

ud2,1(t)
ud2,2(t)

,
xd1(t) =

xd1,1(t)
xd1,2(t)

, xd2(t) =

xd2,1(t)
xd2,2(t)

,
A10 =

−2.1172 0.0377
0.6958 −1.8828

,
A11 =

−0.1672 −0.0184
−0.1198 −0.1328

,
A12 =

−0.0331 −0.0074
−0.0479 −0.0469

,
¯A120 =

−0.0072 0.0068
−0.0220 −0.0403

,
¯A121 =

−0.0200 −0.0019
0.0063 −0.0104

,
¯A122 =

−0.0215 −0.0032
0.0103 −0.0060

,
A20 =

−1.8994 0.2636
0.3433 −1.1006

,
A21 =

−0.2264 −0.1994
−0.0097 −0.2736

,
A22 =

−0.0153 −0.0399
−0.0019 −0.0247

,
¯A210 =

−0.0430 −0.0625
0.0015 −0.0196

,
¯A211 =

−0.0343 −0.0500
0.0013 −0.0157

,
¯A212 =

−0.0332 −0.0600
0.0015 −0.0108

,

10. 90 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 11. Comparison between the proposed method and the traditional ILC at the 10th generation for output responses: (a) Output responses of Subsystem S1: outputs
Yd11(t) and reference r11(t). (b) Output responses of Subsystem S1: outputs Yd12(t) and reference r12(t). (c) Output responses of Subsystem S2: outputs Yd21(t) and reference
r21(t). (d) Output responses of Subsystem S2: outputs Yd22(t) and reference r22(t).
B1 =

−0.15 0.3
0.2 0.4

, B2 =

−0.2 0.2
0.21 0.15

,
C1 =

0.1 0.1
−0.4 0.1

, C2 =

0.1 0.2
−0.2 0.1

,
and the initial condition xd1(0) =

0.1 0.5
T
, xd2(0) =

0.5 0.1
T
. The time delays of the nonlinear interconnected
terms are τ10 = ¯τ210 = 0, τ11 = ¯τ211 = 0.6Ts1, τ12 =
¯τ212 = 1.2Ts1, τ20 = ¯τ120 = 0, τ21 = ¯τ121 = 1.6Ts2, and
τ22 = ¯τ122 = 2Ts2, where Ts1 = 0.05 s, Ts2 = 0.05 s and
the simulation time for the off-line OKID is set as 3 s. Notations
ε120 = ε121 = ε122 = 0.5, ε210 = ε211 = ε212 = 0.4. The eigenval-
ues of A10 (denoted as σ(A10)) are {−2.1999, −1.8001} , σ(A11) =
{−0.2, −0.1} , σ(A12) = {−0.0199, −0.0601} , σ(A20) =
{−2, −1} , σ(A21) = {−0.2001, −0.2999} , σ(A22) = {−0.0101,
−0.0299}.
Since the given system model is unknown, it is desired to
construct the Hankel matrix based on the off-line OKID to de-
termine the appropriate order as 4, where
∑
1 = diag[8.6588,
3.5468, 0.4780, 0.0107, 0.0000, . . .],
∑
2 = diag[8.4041, 5.7287,
0.6776, 0.0592, 0.0000, . . .], and the system is excited by the
white noise signal udi(t) =

udi1(t) udi2(t)
T
for i = 1, 2 with
zero mean and covariance diag

cov (udi1(t)) cov (udi2(t))

=
diag

0.2 0.2

, for Subsystem S1 and S2, respectively. The iden-
tified system matrices and observer gains for Subsystem S1 and
Subsystem S2 are respectively given as
ˆG1 =



0.9522 −0.0133 0.0827 0
0.0236 0.9056 −0.100 −0.0254
0.2293 −0.0289 0.5991 −0.0133
0.0038 0.0425 0.0157 0.1155


 ,
ˆH1 =



0.0025 −0.0032
−0.0015 −0.0025
0.0021 −0.0022
0 0


 ,
ˆC1 =
[
−0.2755 −0.8358 −0.1531 −0.0884
1.0030 −0.1978 0.4942 −0.0218
]
,
F1 =



0.3047 −1.7287
1.1671 0.2419
−0.0677 0.5082
−0.0789 −0.0107


 ,
ˆG2 =



0.9431 −0.0106 −0.0927 −0.0105
0.0860 0.9639 0.0645 −0.0325
0.2802 −0.0031 0.6257 0.0879
−0.0098 0.0799 −0.1007 0.2615


 ,
ˆH2 =



0.0022 0
−0.0064 −0.0014
0.0018 0
0 −0.0002


 ,

11. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 91
Fig. 12. Comparison between the proposed method and the traditional ILC at the 30th generation for output responses: (a) Output responses of Subsystem S1: outputs
Yd11(t) and reference r11(t). (b) Output responses of Subsystem S1: outputs Yd12(t) and reference r12(t). (c) Output responses of Subsystem S2: outputs Yd21(t) and reference
r21(t). (d) Output responses of Subsystem S2: outputs Yd22(t) and reference r22(t).
ˆC2 =
[
−0.9912 −0.4815 −0.5800 −0.0290
−0.4390 −0.7261 0.0605 −0.2081
]
,
F2 =



1.402 −1.0201
−0.9319 1.0871
−0.4670 −0.3851
0.1014 −0.1590


 .
The OKID-based output compared with the actual system
output for Subsystem S1 and Subsystem S2 are shown in Figs. 6 and
7, respectively.
To overcome the effect of modeling error, an improved observer
with the high-gain property based on the digital redesign approach
has been used, where
Gd1 =

In − Ld1
ˆC1

ˆG1
=



0.1516 −0.0027 −0.2440 0.0053
0.0058 0.0231 −0.0113 −0.0124
−0.2674 0.0041 0.4990 −0.0106
−0.0055 −0.0136 0.0140 0.1164


 ,
Hd1 = (In − Ld1
ˆC1) ˆH1 =



−0.0001 0
0 −0.00005
−0.0004 −0.0003
0.0001 0


 ,
Ld1 =



−0.1790 0.6993
−1.0877 −0.3079
−0.0786 −0.4435
−0.0711 −0.0125


 ,
with Qob1 = 106
× I4, Rob1 = I2,
Gd2 =

In − Ld2
ˆC2

ˆG2
=



0.1500 −0.0334 −0.2094 −0.0469
−0.1057 0.0735 0.1883 −0.0411
−0.3134 0.0998 0.5178 0.0606
0.0079 −0.0768 −0.0711 0.2623


 ,
Hd2 = (In − Ld2
ˆC2) ˆH2 =



0.0002 0.00002
−0.0004 −0.0001
−0.0005 −0.0001
0.0003 0.0002


 ,
Ld2 =



−0.5708 −0.4158
0.2789 −1.0629
−0.4837 −0.1807
0.0868 −0.1616


 ,
for Qob2 = 106
× I4, Rob2 = I2. Then, the comparisons between the
actual outputs and their observer-based outputs by digital redesign
for two subsystems are shown in Figs. 8 and 9, respectively.
The simulation result shows the proposed observer by digital
redesign significantly improves the performance of the OKID-
based observer.
Here, we would like to point out that in this paper, for simplicity
in design, an experience approach is utilized for selecting weight-
ing matrices Qob and Rob. A more complicated and sophisticated
approach for selecting the weighting matrices Qob and Rob can be
found in [24–27].

12. 92 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 13. Errors of the traditional ILC and the well-initialized ILC at each iteration: (a) errors
∑Nf
k=1[|r11(k) − Yd11(k)|] at each iteration, (b) errors
∑Nf
k=1[|r12(k) − Yd12(k)|] at
each iteration, (c) errors
∑Nf
k=1[|r21(k) − Yd21(k)|] at each iteration, (d) errors
∑Nf
k=1[|r22(k) − Yd22(k)|] at each iteration.
To demonstrate that the proposed ILC-based high-gain tracker
is superior to the traditional ILC tracker, a digital-redesign linear
quadratic digital tracker with the high-gain property as the initial
control input of the modified ILC is proposed in this paper. The
reference input is given by
r(t) =

r11(t) r12(t) r21(t) r22(t)
T
,
where
r11(t) =

0.5 cos(πt) 0 ≤ t < 1
0.5 sin(πt) 1 ≤ t < 2
−0.5 − 0.5 sin(πt) 2 ≤ t < 4,
r12(t) =

0.5 sin(πt) 0 ≤ t < 1.5
0.5 sin(πt), 1 ≤ t < 2.5
−0.5 − 0.5 sin(πt) 2.5 ≤ t < 4,
r21(t) =

0.5 sin(πt) 0 ≤ t < 0.8
0.5 sin(πt) 0.8 ≤ t < 2.3
1 + 0.5 sin(πt) 2.3 ≤ t < 4
r22(t) =

0.5 cos(πt) 0 ≤ t < 0.5
0.5 sin(πt) 0.5 ≤ t < 1.8
−1 − 0.5 sin(πt) 1.8 ≤ t < 4.
In the following, we apply the digital-redesign method to design
an observer-based linear quadratic tracker for setting up the initial
control input of the iterative learning control system. The feedback
gain Kd and feed-forward gain Ed of the observer-based digital
tracker for Subsystems S1 and S2 are respectively given as
Kd1 =
[
161.9073 −227.8525 31.0387 3.1829
−105.3106 −182.3413 −19.9290 3.9588
]
,
Ed1 =
[
244.1982 255.6082
251.7290 −23.2788
]
,
Kd2 =
[
117.6028 −88.0957 30.6766 5.1789
−622.3522 −225.8409 −36.0860 −19.1845
]
,
Ed2 =
[
−127.6567 37.7943
347.3539 554.8802
]
,
with Q1 = 107
× I2, R1 = I2, Q2 = 107
× I2, R2 = I2. Fig. 10 shows
the control input with well-initialized ILC at the 10th generation.
The simulation results of the novel iterative learning tracker and
the traditional ILC at the 10th generation are shown in Fig. 11.
The simulation results of the novel iterative learning tracker and
the traditional ILC at the 30th generation are shown in Fig. 12.
The comparison learning errors of the every iteration between the
traditional ILC and the novel iterative learning tracker are shown in
Fig. 13. From simulations, it shows the system outputs quickly and
accurately track the desired reference in one short time interval
after all drastically-changing points of the specified reference input
via the proposed method.
To further show that the newly proposed ILC-based high-gain
tracker can improve the transient response and decrease the Q /R
ratio of the controlled system under the traditional digital tracker,
we consider the same system given in this example. When the

13. J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94 93
Fig. 14. Comparison between the proposed method and the traditional digital redesign tracker for output responses: (a) Output responses of Subsystem S1: outputs Yd11(t)
and reference r11(t). (b) Output responses of Subsystem S1: outputs Yd12(t) and reference r12(t). (c) Output responses of Subsystem S2: outputs Yd21(t) and referencer21(t).
(d) Output responses of Subsystem S2: outputs Yd22(t) and reference r22(t).
Q /R ratio is sufficiently high, the newly proposed iterative learning
tracker and the traditional digital redesign tracker have a good
tracking performance in both the transient response and steady
state response. However, when the Q /R ratio is not sufficiently
high, the traditional digital redesign tracker has a poor tracking
performance in both the transient response and steady state
response. To overcome the above problem, the ILC-based high-gain
tracker is newly proposed to improve the transient and steady state
response. The simulation results (Q1 = 104
× I2, R1 = I2, Q2 =
104
× I2 and R2 = I2) of the traditional digital redesign tracker and
the novel iterative learning tracker are shown in Fig. 14.
To show the robustness of the proposed method, let the
tracker have a good performance in the beginning, but the first
subsystem input is artificially reduced to 30% of the determined
input by external factor in 2.5–3.0 s without the fault-tolerant
control. Fig. 15 shows that the decentralized controller induces a
good robustness on the decoupling of the closed-loop controlled
system. When the inputs of parts of the system are broken, the
others are not influenced entirely, so the other digitally controlled
systems still follow the reference inputs with quite a satisfactory
performance.
8. Conclusions
The efficient decentralized iterative learning tracker is pro-
posed to improve the dynamic performance of the unknown
sampled-data interconnected large-scale state-delay system,
which consists of N multi-input multi-output subsystems, with the
closed-loop decoupling property in this paper, which is regarded
as an open problem in literature. The appropriate (low order) de-
centralized linear observer for the sampled-data linear system is
determined by the off-line OKID method. In order to get over the
effect of modeling error on the identified linear models of each sub-
system, an improved observer with high-gain property based on
the digital redesign approach is developed to replace the identified
observer based on OKID. Then, each subsystem of the large-scale
decentralized systems is identified as a linear model; applying the
PD-type ILC method trains the dynamics of decentralized models
to trace the desired trajectory as fast as possible. In order to pursue
a faster learning performance, the digital redesign linear quadratic
tracker with the high-gain property is constructed to generate
the initial control input for ILC. Indeed, the proposed technique
greatly promotes the tracking performance and decreases the it-
erative epochs thanks to its well-selected initial iterative learning
epoch.
Acknowledgment
This work was supported by the National Science Council of
Republic of China under contract NSC99-2221-E-006-206-MY3
and NSC98-2221-E-006-159-MY3.

14. 94 J.S.-H. Tsai et al. / ISA Transactions 51 (2012) 81–94
Fig. 15. Outputs via the proposal method without the fault-tolerant control, where the first subsystem inputs are all artificially reduced to 30% of the determined inputs
due to some unexpected external factors in 2.5–3 s: (a) Output 1 of Model 1. (b) Output 2 of Model 1. (c) Output 1 of Model 2. (d) Output 2 of Model 2.
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