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IEEE TRANSACTIONS ON CYBERNETICS 1
Iterative Learning Control With
Data-Driven-Based Compensation
Shaoying He , Wenbo Chen, Dewei Li , Yugeng Xi ,Senior Member, IEEE,
Yunwen Xu ,Member, IEEE, and Pengyuan Zheng, Member, IEEE
Abstract—The robust iterative learning control (RILC) can
deal with the systems with unknown time-varying uncertainty to
track a repeated reference signal. However, the existing robust
designs consider all the possibilities of uncertainty, which makes
the design conservative and causes the controlled process converg-
ing to the reference trajectory slowly. To eliminate this weakness,
a data-driven method is proposed. The new design intends to
employ more information from the past input–output data to
compensate for the robust control law and then to improve
performance. The proposed control law is proved to guaran-
tee convergence and accelerate the convergence rate. Ultimately,
the experiments on a robot manipulator have been conducted to
verify the good convergence of the trajectory errors under the
control of the proposed method.
Index Terms—Data-driven method, iterative learning control
(ILC), robot manipulator, robust.
I. INTRODUCTION
AS THE name suggests, iterative learning control (ILC) is
an algorithm that imitates the human learning process to
improve the performance of the controller. It aims to improve
the tracking performance and promote the precision of the
repeated task under the same executing conditions by learning
from the previous experiences. The strategy of iterative learn-
ing for repeating a given task with high precision was first
proposed by Uchiyama [1] and was later mathematically for-
mulated by Arimoto et al. [2]. Since the ILC task is to track
a specific command trajectory repeatedly [3], [4], the systems
under the ILC framework all have repetitive operations [5],
such as trajectory tracking [6], [7]; robot manipulator [8];
subway train system [9]; agent formation [10]; chemical pro-
cess [11]; turbine control [12]; motor control [13]; and so
Manuscript received July 16, 2020; revised November 5, 2020; accepted
November 24, 2020. This work was supported in part by the National Key
Research and Development Project under Grant 2018YFB1305902; in part by
the National Science Foundation of China under Grant 61973214 and Grant
61963030; and in part by the Natural Science Foundation of Shanghai under
Grant 19ZR1476200. This article was recommended by Associate Editor
Z.-G. Hou. (Corresponding authors: Yunwen Xu.)
Shaoying He, Wenbo Chen, Dewei Li, Yugeng Xi, and Yunwen Xu are
with the Department of Automation, Shanghai Jiao Tong University, Shanghai
200240, China (e-mail: shaoyinghe@foxmail.com; chenwenbo860@126.com;
dwli@sjtu.edu.cn; ygxi@sjtu.edu.cn; willing419@sjtu.edu.cn).
Pengyuan Zheng is with the College of Automation Engineering,
Shanghai University of Electric Power, Shanghai 200090, China (e-mail:
pyzheng@shiep.edu.cn).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TCYB.2020.3041705.
Digital Object Identifier 10.1109/TCYB.2020.3041705
on. By making full use of the repeated properties in the
iteration process, ILC can keep updating the control inputs
and eliminate the tracking error gradually [14].
The typical ILC usually updates the control law by using
the information from the previous error and control sequence
so that the output trajectory can converge asymptotically to
the reference. Hillenbrand and Pandit [15] introduced an ILC
law for the linear time-invariant system model, which can
guarantee an exponential rate of convergence by reducing
the sampling rate. Longman [16] presented general iteration
control laws with the method of tuning the parameters eas-
ily for the practicing control engineer. The design strategy
for ILC based on the optimal control theory was designed
in [17]. Xiong et al. [18] proposed an optimal ILC with model
predictive control (MPC) for the linear time-varying model
without modeling error. These above strategies have achieved
great success, but the uncertainty of the model was not consid-
ered. If some uncertainty existed in the model, the convergence
of the system cannot be rigorously guaranteed.
To eliminate the effect of the model uncertainty on con-
vergence, Gao et al. [19] proposed a robust ILC (RILC) to
reject the bounded uncertainty and disturbance by the robust
selection of the weighting matrices. An ILC with the singular
value decomposition of the lifted system matrix and the guide-
lines how to tune the learning gains to reduce the model errors
were introduced in [20]. Tayebi and Zaremba [21] proposed an
RILC design procedure for the uncertain linear time-invariant
system by the μ-Analysis and synthesis approach. Later, Shi
et al. [22] formulated the robust design as matrix inequality
conditions, which can be solved by an algorithm based on the
linear matrix inequality (LMI). Meng et al. [23] proposed an
RILC for the time-delay systems with uncertainty by a LMI
approach. Besides, an RILC for the polytopic uncertain system
was proposed in [24]. For the above methods, the control laws
were just designed to meet the feasibility not the optimality.
Thus, the performance weighting in the control law might be
selected to be large and the performance is limited.
For better performance, an RILC design frame with an
efficient combination of LMI and an appropriate parame-
ter optimization was proposed in [25], which extended the
LMI approach by the minimization of a suitable cost func-
tion and led to an improved convergence rate. Considering
that ILC can be treated as a special class of 2-D model,
which is advantageous to investigate ILC, Wang et al. [26]
designed a robust iterative learning MPC based on the 2-D
model and Hladowski et al. [27] designed an RILC for a 2-D
2168-2267 c
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2IEEE TRANSACTIONS ON CYBERNETICS
model with the robustness analysis. The robust norm-optimal
ILC design was introduced in [28]. It takes the robust conver-
gence conditions as the constraints in the optimal problem,
which can maximize available performance and ensure the
convergence. However, the existing robust ILC designs consid-
ered all the possibilities of uncertainty, which makes the design
conservative and causes the controlled process to converge to
the reference trajectory slowly.
Since the input–output data of the system can directly reflect
the system characteristics, the data-driven method, such as the
identification method, can obtain more information than the
robust method to reduce model uncertainty. The identification
methods were used in a lot of research to compensate the
ILC and improve the design performance. Janssens proposed
a data-driven norm-optimal ILC in [29] and the key contri-
bution was the estimation of the system impulse response for
the design controller using input–output data from previous
iterations, but the full rank of the past trial input sequences
was required. Chi used the past input–output to identify the
lifted model and proposed a convergent condition for model
updating [30], [31], but the identification result is sensitive
to initial parameters, and the choice of initial parameters was
very important. The above two methods can obtain an accurate
model with the increasing number of iterations and past data,
but the system identification algorithms need a large amount
of past trial input–output data, which made the system con-
verge slowly. Thus, Li et al. [32] proposed a novel data-driven
method for the ILC combined with MPC in which past data
were directly used to update the control output for avoiding
identification.
To solve the existing disadvantages of RILC and adaptive
ILC, we design a data-driven method for ILC by employ-
ing the special optimal linear combination of the past input
sequences to compensate for a traditional RILC law, which
can be developed based on a rough model. Since the proposed
method has no need to do the model identification, we call
it the ILC with data-driven-based compensation (ILC-DDC).
The main contribution in this article is that the data-driven
module is embedded into the traditional RILC to overcome
the conservation and improve the performance. Besides, the
convergence of the algorithm proposed in this article can be
guaranteed strictly in theory, and it can also be proved that
the proposed data-driven method accelerates the convergence
rate of the classical RILC.
This article is organized as follows. Section II introduces the
considered system model and the classical RILC. Section III
presents the design procedure of the proposed data-driven
compensation method for ILC in detail. Subsequently, the
proof about the monotonous convergence of the proposed
method and the analysis of the convergence rate compared
with RILC are both described in Section IV. Finally, the exper-
iments on the 6-freedom manipulator to test the proposed
method are given in Section V.
II. PROBLEM FORMULATION
In this section, we first present the considered system model
and then introduce the classical RILC method as well as its
TAB LE I
NOTATIONS OF SYMBOLS
convergence condition. The notations of the symbols used in
this article are presented in Table I.
Consider a discrete linear time-variant system model in the
repeated iterative process as follows:
x(k,t+1)=A(t)x(k,t)+B(t)u(k,t)(1)
y(k,t)=Cx(k,t)(2)
where kand tare the trial number and the discrete-time instant,
respectively. The discretization time of each trial is denoted by
t∈{0,1,...,N−1}, where Nis the number of samples in a
trial. x(k,t)∈Rnis the measurable system state, u(k,t)∈Rmis
the control input, and y(k,t)∈Rh(h≤m) is the system output
in the time t,atthekth trial. A(t)and B(t)are system dynamic
matrices in the time t, but they are irrelevant to the times of
trial. Cis a constant matrix. Here, we make an assumption
that (A,C)and (A,B)are observability and controllability. A
and Bcan be indicated as the following:
A(t)=A0(t)+A(t)(3)
B(t)=B0(t)+B(t)(4)
where A0(t)and B0(t)are known model matrices of the
system, and A(t)and B(t)are unknown and time-varying
within one trial. Note that A(t)and B(t)are iteration invari-
ants. The control goal is to steer the system output to track a
reference trajectory −→
γ(k). In addition, the initial conditions of
all repetition trials are the same, and without loss of generality,
we set x(1,0)=···= x(k,0)and y(1,0)= ···=y(k,0).
According to the system model (1)–(4), the states of the
trial process at the kth trial can be characterized by
x(k,1)=A(0)x(k,0)+B(0)u(k,0)
.
.
.
x(k,i+1)=A|i
0x(k,0)+B(i)u(k,i)
+
i
j=1
A|i
jB(j−1)u(k,j−1)
.
.
.(5)
and the corresponding outputs are
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HE et al.: ITERATIVE LEARNING CONTROL WITH DATA-DRIVEN-BASED COMPENSATION 3
y(k,1)=CA(0)x(k,0)+CB(0)u(k,0)
.
.
.
y(k,i+1)=CA|i
0x(k,0)+CB(i)u(k,i)
+C
i
j=1
A|i
jB(j−1)u(k,j−1)
.
.
.(6)
where A|i
j=i
g=jA(g).
Therefore, the output sequence at the kth trial can be
expressed in the form of vectors as follows:
−→
Y(k)=Px(k,0)+G−→
u(k)
=Px(k,0)+(G0+G)−→
u(k)(7)
where −→
Y(k)[yT(k,1), yT(k,2), ··· ,yT(k,N)]T∈RNh,
−→
u(k)[uT(k,0), uT(k,1), ··· ,uT(k,N−1)]T∈RNm
P⎡
⎢
⎢
⎢
⎣
CA|0
0
CA|1
0
.
.
.
CA|N−1
0
⎤
⎥
⎥
⎥
⎦
G⎡
⎢
⎢
⎢
⎢
⎣
CB(0)O··· O
CA|1
1B(0)CB(1)....
.
.
.
.
..
.
....O
CA|N−1
1B(0)··· ··· CB(N−1)
⎤
⎥
⎥
⎥
⎥
⎦
G0⎡
⎢
⎢
⎢
⎢
⎣
CB0(0)O··· O
CA0|1
1B0(0)CB0(1)....
.
.
.
.
..
.
....O
CA0|N−1
1B0(0)··· ··· CB0(N−1)
⎤
⎥
⎥
⎥
⎥
⎦
A0|i
j=i
g=jA0(g),P∈RNh×nand G∈RNh×Nm are the
system matrices, G0∈RNh×Nm is the certain part in Gwhich
are determined by A0(t)and B0(t),GG−G0is the uncer-
tainty in the model which is affected by A(t)and B(t).
Without loss of generality, for the uncertain matrix G,we
give the following assumption which is also used in [25].
Assumption 1: The uncertain matrices A(t)and B(t)are
assumed to belong to a convex bounded uncertainty domain
Das follows:
D=⎧
⎨
⎩[A(t), B(t)]=
L
j=1
aj[Aj,Bj]
L
j=1
aj=1;aj≥0⎫
⎬
⎭(8)
where Ajand Bjare the vertices, and Ldenotes the
corresponding number of vertices.
Therefore, the uncertain system matrix Gbelongs to a
larger convex bounded uncertainty domain
Das follows:
D=⎧
⎨
⎩G=
L
j=1
bjGj;
L
j=1
bj=1;bj≥0⎫
⎬
⎭(9)
where Gjare the vertices,
Ldenotes the corresponding num-
ber of vertices, and
Lis much larger than L. The polytope
D
makes the uncertainty Gbelong to a certain range.
The reference trajectory can be defined as −→
γ(k)=
[γT(k,1), γ T(k,2),...,γT(k,N)]Tand the error at time tof
kth trial can be defined as e(k,t)=γ(k,t)−y(k,t). Since the
initial output y(k,0)is fixed and the aim is to track −→
γ(k)from
1toN, the initial error can be set as e(k,0)=0. Referring
to (7), the trajectory tracking error of the system in the kth
trial can be denoted as follows:
−→
e(k)=−→
γ(k)−−→
Y(k)
=−→
γ(k)−Px(k,0)−(G0+G)−→
u(k)(10)
where −→
e(k)[eT(k,1), eT(k,2),...,eT(k,N)]T.
Due to the repeatability of −→
γ(k), that is, −→
γ(k+1)=−→
γ(k),
Px(k+1,0)=Px(k,0)and (10), the error model in (k+
1)th trial and the input increment sequence between the two
adjacent trials can be written as
−→
e(k+1)=−→
γ(k+1)−−→
Y(k+1)
=−→
e(k)−(G0+G)−→
u(k)(11)
−→
e(k)−→
e(k+1)−−→
e(k)=−(G0+G)−→
u(k)
(12)
where −→
u(k)−→
u(k+1)−−→
u(k)is the updating control
law at the k+1th trial. It is obvious that the updating control
−→
u(k)leads to the variety of the error −→
e(k).
During the iteration process, the classical ILC aims at reduc-
ing the error at the next time trial between the output and the
reference using the current error information, so at the next
time trial, the next control input sequences are usually obtained
as the following [5]:
−→
u(k+1)=−→
u(k)+−→
u(k)=−→
u(k)+K−→
e(k)(13)
where K∈RNm×Nh is the feedback gain matrix for the cur-
rent error −→
e(k). Many robust methods, such as the linear
matrix inequality or norm-optimal method, have been utilized
to design the robust feedback gain Kof RILC, such as [13],
[24], and [28]. In the literature, the robust monotonic con-
vergence conditions of the RILC control law are similar and
defined as follows:
||I−(G0+G)K|| <I,∀G,G∈DG.(14)
In this article, we choose the method introduced in [24] to
design Kas follows. The RILC with polytopic uncertainty
can be first given by
u(k+1,t)=u(k,t)+kt(e(k,t+1)−e(k,t)) (15)
where t=0,...,N−1 and kt∈Rm×h. Writing the above
equations in the matrix form, we have
−→
u(k)=−→
u(k+1)−−→
u(k)=K−→
e(k)(16)
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4IEEE TRANSACTIONS ON CYBERNETICS
where
K=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
k0O··· ··· O
−k1k1.......
.
.
O.........O
.
.
..........O
O··· O−kN−1kN−1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(17)
According to [24], the learning gain ktcan be designed such
that
maxj∈[1,L]||I−C(B0(t)+Bj)kt|| <I(18)
where Bjis the vertices of the convex bounded uncer-
tainty domain D, and Lis the number of the vertices. The
condition (18) can be further converted to LMIs as follows:
I(I−C(B0(t)+Bj)kt)T
∗I>0 (19)
where t∈{0,...,N−1}and j∈{1,...,L}. By solving (19),
the feedback law Kcan be obtained by kt, which satisfies the
convergent condition (14). There exists ktand (19) is feasible
if the system matrix CB(t)has full-row rank [24].
III. DATA-DRIVEN COMPENSATION METHOD FOR ILC
Although the robust control law with (14) can ensure the
convergence of the RILC, the robust law is conservative, and
the convergence rate is slow because all possible uncertainties
are considered in the design. Considering that the relationship
between the output and input is caused by the model with
uncertainty, we employ the past input–output data to improve
the performance in this article.
From (12), the relation between the past data and model in
past trials can be obtained as
−→
e(k−1)=−(G0+G)−→
u(k−1)
−→
e(k−2)=−(G0+G)−→
u(k−2).
.
.
.(20)
Since G0and Gare fixed, according to the data-driven
method in [32], the following general relation can be obtained:
λ−→
e(i)=−(G0+G)λ−→
u(i)(21)
where iis less than k, and λis a scalar, which will be
determined by the controller.
During the iteration process, we can obtain the past data of
the update control law −→
uand the corresponding −→
e.So
we define that
U(k−1)[−→
u(k−1);··· ;−→
u(k−l)]
and
E(k−1)[−→
e(k−1);··· ;−→
e(k−l)]
where U(k−1)∈RNm×l,E(k−1)∈RNh×land lis the
length of moving window, and if k−i<0(i∈1,...,l), set
−→
u(k−i)and −→
e(k−i)as zeros. Then, according to (21),
it is obvious that
E(k−1)−→
λk−1=−(G0+G)U(k−1)−→
λk−1(22)
where −→
λk−1=[λk−1,...,λ
k−l]∈Rlis a vector.
As for the kth trial, since the past data of the update con-
trol law as −→
u(k−1),−→
u(k−2),−→
u(k−3)··· can be
obtained, Li et al. [32] adopted these past data to design the
control law of the ILC with MPC. In this article, we intend
to employ the linear combination of the past update control
data to compensate the original robust iterative learning update
control law. That is, we use a moving window with a fixed
length to acquire the past data near the current trial for com-
pensating the original robust control law (13). In the kth trial,
the update control law −→
u(k)can be denoted by
−→
u(k)ˆu(k)+
l
j=1
λk−j−→
u(k−j)(23)
where ˆu(k)represents the robust update-feedback control law
from (13), another term l
j=1λk−j−→
u(k−j)represents the
update law from the combination of the past update control
data, ldenotes the length of the moving window (i.e., the
number of the past data selected to compensate the control),
and λk−jis the linear combination parameter of the past data,
which will be determined by the controller.
According to the increment error model (12) and the
update control law (23), the tracking error increment can be
transformed as follows:
−→
e(k)=−(G0+G)−→
u(k)
=−(G0+G)(ˆu(k)+
l
j=1
λk−j−→
u(k−j)).
(24)
For brevity, the update control law (23) and the error
model (24) will be written in the vector or matrix form as
follows:
−→
u(k)ˆu(k)+U(k−1)−→
λk−1(25)
−→
e(k)=−(G0+G)(ˆu(k)+U(k−1)−→
λk−1).
(26)
Then the error in the k+1th trial can be described by
−→
e(k+1)=−→
e(k)−(G0+G)−→
u(k)
=−→
e(k)−(G0+G)(ˆu(k)
+U(k−1)−→
λk−1). (27)
According to (22), the relation between the vector of the
past update control law U(k−1)and the error increment
E(k−1)can be written as follows:
−(G0+G)U(k−1)−→
λk−1=E(k−1)−→
λk−1.(28)
Therefore, substituting (28) into (27), the error in the k+1th
trial can be obtained as follows:
−→
e(k+1)=−→
e(k)+E(k−1)−→
λk−1−(G0+G)ˆu(k).
(29)
Then, in order to guarantee the convergence of the system,
ˆu(k)will be designed as the previous RILC method and
required to satisfy (14). Define ˆe(k)−→
e(k)+E(k−
1)−→
λk−1,so−→
e(k+1)can be rewritten as follows:
−→
e(k+1)=ˆe(k)−(G0+G) ˆu(k). (30)
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HE et al.: ITERATIVE LEARNING CONTROL WITH DATA-DRIVEN-BASED COMPENSATION 5
The update robust control law of the error feedback is similar
with (13) as follows:
ˆu(k)=Kˆe(k)(31)
where Kis the RILC law designed for satisfying the robust
convergence condition (14).
Considering that the optimal target of the ILC-DDC is to
reduce the next trial error ||−→
e(k+1)||2, the optimal problem
can be redefined as follows:
J=min
−→
λk−1
||−→
e(k+1)||2
=min
−→
λk−1
||−→
e(k)+E(k−1)−→
λk−1−(G0+G)ˆu(k)||2
(32)
where the linear combination parameter of past data −→
λk−1
is the optimization vector, −→
e(k)is the current error, and
E(k−1)and U(k−1)are the past error increment and con-
trol increment directly acquired from past data. The solution
of (32) can be directly obtained as
−→
λ∗
k−1=−(ET(k−1)E(k−1))†ET(k−1)
−→
e(k)−(G0+G)ˆu(k).(33)
The optimality solution −→
λ∗
k−1contains the uncertain part
Gin (G0+G)ˆu(k), which provides the difficulty in
both calculation and theoretical analysis. Thus, we remove
this uncertain term (G0+G)ˆu(k)in (33) and simplify the
solution (33) to obtain the solution as
−→
λk−1=−(ET(k−1)E(k−1))†ET(k−1)−→
e(k).
(34)
After removing the uncertain part, −→
λk−1can be easily cal-
culated by the past data. Then, the ultimate data-driven-based
robust update control law (25) can be rewritten as follows:
−→
u(k)=ˆu(k)+U(k−1)−→
λk−1
=K(−→
e(k)+E(k−1)−→
λk−1)+U(k−1)−→
λk−1.
(35)
The convergence of the control law (35) with the compensation
coefficients −→
λk−1will be discussed in Section IV.
In the iteration process, with the increase of iteration times,
the error −→
e(k)and the E(k−1)increment error of the system
will become smaller and smaller and close to 0, which will
make ρmax(ET(k−1)E(k−1)) close to 0 and the pseudo-
inverse of the ET(k−1)E(k−1)not exist. In the system
that E(k−1)is close to 0 implies that the iteration process
is coming to an end. Therefore, in the iteration process, when
ρmax(ET(k−1)E(k−1)) is less than εand εis set as a
very small value, we directly set the −→
λk−1to be 0.
From the proposed ILC-DDC strategy, one can obtain the
block diagram of how the ILC-DDC controller works in Fig. 1,
and the complete algorithm steps of the ILC-DDC are given in
Algorithm 1. Compared with the work in [32], our proposed
data-driven method in Algorithm 1 can compensate all kinds
of ILC law and improve its performance. In addition, it can
be proved that the ILC with the data-driven compensation is
Fig. 1. ILC-DDC.
Algorithm 1 ILC-DDC
1: Construct the feedback law Kwhich satisfies (14) and set
the initial values l,E(0)=0, −→
λ0=0, U(0)=0 and
−→
u(0)=0 offline.
2: At the begin of trial k, compute the solution −→
λk−1by
(34). If ρmax(ET(k−1)E(k−1)) < ε (εis a very
small value), set −→
λk−1=0.
3: Calculate ˆe(k)=−→
e(k)+e(k−1)−→
λk−1and −→
u(k)=
Kˆe(k)+
l
j=1
λk−j−→
u(k−j), then implement −→
u(k)=
−→
u(k−1)+−→
u(k)on the controlled process.
4: At the end of trial k, add −→
u(k)and −→
e(k)into memory
unit and obtain new U(k)and E(k).Letk=k+1 and
return to Step 2.
convergent and the convergent rate is faster than the original
algorithm without data-driven compensation.
Remark 1: The main computation of the above data-driven-
based compensation method in the iterative process is (34),
which is an inverse matrix operation and can be solved by a
standard matrix inverse software package. The computational
burden of (34) depends on the data length l. The LMIs problem
for obtaining K(19) can be solved offline by a standard LMI
software package and does not affect the computations in the
iterative process.
IV. CONVERGENCE AND CONVERGENCE RATE ANALYSIS
In this section, the convergence of Algorithm 1 will be ana-
lyzed. In addition, it is proved that the convergence rate of
Algorithm 1 is faster than RILC. The main results of this
article are as follows.
Theorem 1: Consider system model (7) with polytopic
uncertainty (8) controlled by the control input generated from
Algorithm 1. When the original RILC control law Kmeets the
convergence condition (14), the system with the data-driven
design proposed in Algorithm 1 is convergent.
Proof: First, the traditional RILC law Kcan be designed for
satisfying the convergent condition (14) by the method from
the literature [24].
Define the e(k)as follows:
e(k)−E(k−1)−→
λk−1.(36)
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6IEEE TRANSACTIONS ON CYBERNETICS
Fig. 2. Decomposition of e(k).
Then, according to the definition of ˆe(k)−→
e(k)+E(k−
1)−→
λk−1,−→
e(k)can be divided into the following two parts:
−→
e(k)=ˆe(k)+e(k). (37)
In addition, substituting (34) into (36), the expression of
e(k)can be shown in the following:
e(k)=E(k−1)(ET(k−1)
E(k−1))†ET(k−1)−→
e(k)=Z−→
e(k)(38)
which implies e(k)is the projection of −→
e(k)on E(k−1).
Meanwhile, we define the singular value decomposition of the
E(k−1)as E(k−1)=LoeLc. The following equation
can be obtained:
Z=E(k−1)(ET(k−1)E(k−1))†ET(k−1)
=LoeLc((LoeLc)TLoeLc)†(LoeLc)T
=LoImLT
o
ZTZ=(LoImLT
o)TLoImLT
o=LoImLT
o(39)
where Im=I
Oand Ihas same dimension as the matrix e.
Then, the product between ˆeand e(k)can be obtained as
follows:
ˆeT(k)e(k)=(−→
e(k)−e(k))Te(k)
=−→
e(k)TZ−→
e(k)−−→
e(k)TZTZ−→
e(k)
=0.(40)
Therefore, ˆe(k)and e(k)have the vertical correlation in
multidimensional space as shown in Fig. 2. ˆeT(k)e(k)=0is
the key for the later theoretical demonstration.
Next, the error model in the k+1th trial (27) can also be
divided and rewritten by the following two subterms:
−→
e(k+1)=ˆe(k+1)+e(k+1). (41)
From (29), the subterms −→
e(k+1)can be separately denoted
by the following two equations:
ˆe(k+1)=ˆe(k)−(G0+G)ˆu(k)(42)
e(k+1)=e(k)−(G0+G)U(k)−→
λ(k). (43)
For (43), due to (28) and (36), e(k+1)=0 can be obtained.
Thus, from (41), (42), and precondition (14), we can obtain
the relationship between ||−→
e(k+1)||2and ||ˆe(k)||2
||−→
e(k+1)||2=ˆeT(k+1)ˆe(k+1)
=[ˆe(k)−(G0+G)ˆu(k)]T
[ˆe(k)−(G0+G)ˆu(k)]
=[ˆe(k)−(G0+G)Kˆe(k)]T
[ˆe(k)−(G0+G)Kˆe(k)]
=ˆeT(k)[I−(G0+G)K]T
[I−(G0+G)K]ˆe(k)
<||ˆe(k)||2.(44)
Ultimately, from the error in the kth trial and (40), we can
obtain another relation between ||−→
e(k)||2and ||ˆe(k)||2
||−→
e(k)||2=(ˆe(k)+e(k))T(ˆe(k)+e(k))
=ˆeT(k)ˆe(k)+eT(k)e(k)+2ˆeT(k)e(k)
=ˆeT(k)ˆe(k)+eT(k)e(k)
>||ˆe(k)||2.(45)
Therefore, utilizing the intermediate variable ||ˆe(k)||2,wehave
||−→
e(k)||2>||ˆe(k)||2>||−→
e(k+1)||2.(46)
In conclusion, ||−→
e(k)||2>||−→
e(k+1)||2for ∀k∈
{0,1,...,}can be proved, which means that the error of the
trial tracking is monotonically decreasing while the number of
iterations increases. Therefore, the iteration strategy proposed
in this article is monotonic convergent.
Theorem 2: Consider system model (7) with polytopic
uncertainty (8) controlled by the control input generated from
Algorithm 1. In the kth trial, when the ILC-DDC has the
same error situation −→
e(k)as RILC, that is, ||−→
e(k)RILC||2=
||−→
e(k)ILC−DDC||2, the error of ILC-DDC is smaller than RILC
in the next trial, that is, ||−→
e(k+1)RILC||2≥||
−→
e(k+
1)ILC−DDC||2, where −→
e(k+1)RILC and −→
e(k+1)ILC−DDC
are the error of the RILC and ILC-DDC in the k+1th trial,
respectively.
Proof: To compare −→
e(k+1)RILC and −→
e(k+1)ILC−DDC,the
precondition in the kth trial, such as the error −→
e(k),system
model G0+Gand the feedback control law Kshould be
kept the same in the RILC or ILC-DDC. In this situation, the
errors of the RILC and ILC-DDC in the k+1th trial after the
action of the control (13) or (25) can be written separately as
follows:
−→
e(k+1)RILC =−→
e(k)−(G0+G)K−→
e(k)(47)
−→
e(k+1)ILC−DDC =−→
e(k)−(G0+G)
(ˆu(k)+U(k−1)−→
λk−1). (48)
Considering −→
e(k)=ˆe(k)+e(k),ˆu(k)=Kˆe(k), and
−(G0+G)U(k−1)=E(k−1), (47) and (48) can be
transformed as follows:
−→
e(k+1)RILC =ˆe(k)+e(k)
−(G0+G)K(ˆe(k)+e(k))
=(I−(G0+G)K)ˆe(k)+(I−(G0
+G)K)e(k)(49)
−→
e(k+1)ILC−DDC =ˆe(k)+e(k)−(G0+G)( ˆu(k)
+U(k−1)−→
λk−1)
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HE et al.: ITERATIVE LEARNING CONTROL WITH DATA-DRIVEN-BASED COMPENSATION 7
=(I−(G0+G)K)ˆe(k)+e(k)
−(G0+G)U(k−1)−→
λk−1
=(I−(G0+G)K)ˆe(k)+e(k)
+e(k−1)−→
λk−1
=(I−(G0+G)K)ˆe(k). (50)
Therefore, the norm of the RILC error and ILC-DDC error
can be written as
|−→
e(k+1)RILC||2= ||(I−(G0+G)K)ˆe(k)||2
+||(I−(G0+G)K)e(k)||2
+2[(I−(G0+G)K)ˆe(k)]T
(51)
|−→
e(k+1)ILC−DDC||2= ||(I−(G0+G)K)ˆe(k)||2.(52)
According to (40), [(I−(G0+G)K)ˆe(k)][(I−(G0+
G)K)e(k)]T=0 can be obtained. Thus, the relation between
the error norms of the RILC and ILC-DDC can be shown as
follows:
||−→
e(k+1)RILC||2≥||
−→
e(k+1)ILC−DDC||2.(53)
Remark 2: The quantification of the performance improve-
ment in our method can be calculated by Q=||
−→
e(k+
1)RILC||2−||
−→
e(k+1)ILC−DDC||2. The formula can convert to
Q=||(I−(G0+G)K)e(k)||2. When ||−→
e(k+1)RILC||2=
||−→
e(k+1)ILC−DDC||2if and only if Qis 0. From (34) and (36),
it can be obtained that Q=0 means that the ET(k−1)=0
or −→
e(k)=0. When the increment error or the error becomes
0, it means that the iteration process has come to an end. Thus,
at the end of the iteration, the two ILCs will have an equiva-
lent performance, while in other situations, ||−→
e(k+1)RILC||2
is always larger than ||−→
e(k+1)ILC−DDC||2.
In conclusion, ||−→
e(k+1)RILC||2≥||
−→
e(k+1)ILC−DDC||2
means that the convergence rate of ILC-DDC is faster than
RILC and the data-driven-based compensation can accelerate
the convergence rate of RILC.
V. EXPERIMENTAL STUDY
In the application, the traditional tasks of industrial manip-
ulators include welding, stacking, and spraying on assembly
lines, which have repetitive characteristics and high-precision
requirements. Thus, in this section, we design experiments on
a platform of a 6-freedom manipulator to move repeatedly for
verifying the proposed ILC-DDC and making a comparison
with the existing approaches. In experiments, the manipulator
is designed to track the repetitive reference trajectory.
The robot employed is shown in Fig. 3. For this object, the
repeated position accuracy of joint motors is 0.01 degrees. In
order to keep the model uncertainty in a certain range, the
range of the joint movement needs to be limited in a small
region. Thus, the upper boundary of the joint angle is selected
as follows:
Joint =[20,75,100,5,80,25]◦(54)
and the lower limit of joint angle is selected as
Joint =[−5,20,50,−5,20,−25]◦.(55)
Fig. 3. Six degree-of-freedom manipulator.
TAB LE I I
D-H MODEL OF MANIPULATOR
Under the influence of the joint motor capability, the upper
boundary of joint angle speed is
Speed =[20,20,20,20,20,20]◦/s(56)
and the lower boundary of joint angle speed is
Speed =[−20,−20,−20,−20,−20,−20]◦/s.(57)
A simple saturation method is just utilized to protect the
manipulator in actual experiments. Due to the proper feed-
back law, the control inputs do not exceed the constraints
in the experiment. The corresponding modified Denavit–
Hartenberg model [33] can be subsequently established. The
corresponding D-H parameters of the manipulator are listed in
Table II.
According to the Jacobian matrix of the robot manipu-
lator [34], the object kinematic model can be obtained as
follows:
˙s=J(θ ) ˙
θ(58)
where ˙s=[˙x,˙y,˙z]Tis the end-effector velocity of the manip-
ulator, ˙
θis the angular velocity of the manipulator joints, and
J(θ) is the Jacobian matrix.
By discretizing model (58) according to the Euler forward
difference rule [35], the discrete manipulator model can be
obtained as
s(k,t+1)=CA(t)s(k,t)+CB(t)˙
θ(k,t)
s(k,t+1)=s(k,t)+B(t)˙
θ(k,t)(59)
where B(t)=TJ(θ (t)),A(t)=I,C=I, and Tis the sampling
time of the discrete system. Then, the lifted system can be
described as
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−→
S(k)=Ps(k,0)+G−→
u(k)
=Ps(k,0)+(G0+G)−→
u(k)(60)
where −→
S(k)=[s(k,1)T,...,s(k,N)T]Tis the end-effector
position in the ktrial, P=[I,...,I]T, and u(k)=
[˙
θ(k,0)T,..., ˙
θ(k,N−1)T]Tis the joint velocity in the kth
trial. The state manipulator and the joint position can be
measured in the trial.
In experiments, the initial joint angle is set as follows:
θ(0)=[14.74,66.37,58.92,0.00,54.71,14.74]◦.(61)
Herein, the initial joint angle θ(0)is selected randomly within
the range of the joint boundary, which has no effect on the
control performance. The discretization time is set as T=0.1.
The total time period is set as N=50. The initial position of
the manipulator end-effector is set as follows:
s(k,0)=[0.38,0.1,0.15]T.(62)
In the experiments, the certain model, which can be known
in the design phase, is B(θ(0)). Thus, we set B0=B0(0)=
···=B0(N−1)=B(θ (0)) as the certain model and B(t)=
B(t)−B0is the uncertain part. The certain matrix G0can be
set as follows:
G0⎡
⎢
⎢
⎢
⎢
⎣
B0O··· O
B0B0....
.
.
.
.
..
.
....O
B0B0··· B0
⎤
⎥
⎥
⎥
⎥
⎦
.(63)
The corresponding system matrices Gwith uncertainty can be
written as follows:
G⎡
⎢
⎢
⎢
⎢
⎣
B(0)O··· O
B(0)B(1)....
.
.
.
.
..
.
....O
B(0)B(1)··· B(N−1)
⎤
⎥
⎥
⎥
⎥
⎦
.(64)
Since the joint angles in the working process are in a cer-
tain boundary, we use the boundary points Joint and Joint to
describe the convex hull of uncertainty. According to Joint and
Joint, there are 26=64 different combinations for six joints as
θj(j=1,...,64). The convex vertices of B(t)are constructed
as Bj=B(θj). Thus, the uncertain model B(t)=B(t)−B0
belongs to the convex hull {B=64
j=1ajBj,64
j=1aj=1,
aj≥0}, where Bj=Bj−B0is the vertex of B(t).
By using the method in [24], the feedback-control law K
can be obtained for satisfying the condition (14). Since in
experiments B0and the convex hull of B(t)are the same
at every moment, the ktin Kare the same. Based on this
feedback control law Kof RILC, we use the proposed data-
driven method in Algorithm 1 to compensate it.
In order to reduce the computational burden, the data length
lis set as 1, which makes the computational burden of −→
λk−1
in (34) be a simple algebraic computation. The computational
complexity is simple in the iteration process.
To evaluate the performance of our method, we compare our
proposed ILC-DDC with the RILC [24] and the data-driven
optimal ILC algorithm (DDOILC) [31] in the experiments. In
Fig. 4. Error variation curves of the ILCs for the line function.
Fig. 5. Tracking trajectories in the x-axis for the line function.
the control process, we only need to send the joint positions
calculated by the above ILCs to the manipulator in each trial,
and the manipulator can detect the current joint position, cal-
culate, and feedback the posture of the end effector by the
robot forward kinematics [33].
A. Case 1: Tracking Straight Line Repeatedly
In this experiment, the manipulator end effector is to track
the repeated reference constructed by a straight line from
the point [0.38,0.1,0.15]Tto point [0.30,0.02,0.15]T, whose
length is 0.1131 m.
The Euclidean error norm of the considered algorithms is
shown in Fig. 4, which indicates that all algorithms can achieve
the convergence, but ILC-DDC and DDOILC converge faster
than RILC. This is because ILC-DDC and DDOILC utilize
the past data to enhance either the system model or the con-
trol inputs, resulting in faster convergence rates. Comparing
ILC-DDC and DDOILC carefully, we can find that DDOILC
provides a slower convergence rate than ILC-DDC since
DDOILC utilizes the model identified by a large amount of
past data (data from one trial) for control, while ILC-DDC
directly utilizes the data with length l(l=1) for the design
of the control law. In the first trial, ILC-DDC has no past data
for control, so its trajectories are the same with DDOILC and
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HE et al.: ITERATIVE LEARNING CONTROL WITH DATA-DRIVEN-BASED COMPENSATION 9
Fig. 6. Tracking trajectories in the y-axis for the line function.
Fig. 7. Tracking trajectories in 3-D space at different trials for the line
function.
RILC. But with the increase of the number of trials, the tra-
jectories of ILC-DDC outperform the DDOILC and RILC.
Figs. 5 and 6 show the actual trajectories of algorithms in
the x- and y-axis. Fig. 7 provides the actual trajectory of the
algorithms in the 3-D coordinate system corresponding to dif-
ferent trials. We can see that in the 6th trial, ILC-DDC has
tracked the path better than the RILC and DDOILC. Due to
the direct utilization of past data for constructing the control,
the convergence rate of ILC-DDC is faster than RILC and
DDOILC.
B. Case 2: Tracking Sine Curve Repeatedly
Without loss of generality, the experiment for the sine
function reference is also designed to verify the adaptabil-
ity of the proposed algorithm in different tasks. In this
experiment, the manipulator end effector is to track the ref-
erence constructed by a sine curve from [0.38,0.1,0.15]T
to [0.3035,0.0745,0.15]T. The sine function is y=
0.01sin(255(x−0.38)).
The curve of tracking errors in the different trials for the sine
function is shown in Fig. 8. It can be also seen that all algo-
rithms can converge, and the convergence rate of ILC-DDC is
Fig. 8. Error variation curves of the ILCs for the sine function.
Fig. 9. Tracking trajectories in the x-axis for the sine function.
Fig. 10. Tracking trajectories in the y-axis for the sine function.
still faster than RILC and DDOILC as case 1. Figs. 9 and 10
show the actual trajectories of algorithms in the x- and y-axis.
Fig. 11 gives the actual tracking trajectories in the 3-D coordi-
nate system under different algorithms in the 6th, 12th, 18th,
and 49th trials. It can be obviously seen that ILC-DDC pro-
vides less tracking error than RILC and DDOILC from the 6th
trial and completes the tracking task faster than them, while
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10 IEEE TRANSACTIONS ON CYBERNETICS
Fig. 11. Tracking trajectories in 3-D space at different trials for the sine
function.
Fig. 12. Joint trajectories of ILC-DDC at different trials for the line function.
RILC and DDOILC almost eliminate the tracking error in the
49th trial. Figs. 12 and 13 show that the joint trajectories of
ILC-DDC at different trials for both line and sine functions are
smooth and within the constraints. The saturation method is
just utilized for protecting the manipulator in the actual exper-
iment. However, due to the proper feedback law, the control
inputs do not exceed the constraints.
In Fig. 14, the manipulator uses the control law of the ILC-
DDC in the 49th trial to draw the two trajectories in a white
paper, which indicates that the proposed method can achieve
good results in the actual environment.
In conclusion, the above two experiments illustrate that the
ILC algorithms can all achieve good tracking performance ulti-
mately. Compared with the RILC algorithm, the ILC-DDC and
DDOILC with the aid of past data have better performance.
The DDOILC needs more data from the controlled plant to
identify the more exact model, which limits the convergence
rate. Unlike DDOILC, the ILC-DDC has no requirement on
the amount of data and avoids identifying the model by the
Fig. 13. Joint trajectories of ILC-DDC at different trials for the sine function.
Fig. 14. Drawing the results in the environment.
Fig. 15. Error curves of the ILC-DDC with the different data lengths.
past data. It directly uses the optimized combination of the
past control law to compensate the current control law by
increasing the proportion of the past control law which makes
the tracking error drop faster. Thus, ILC-DDC can obtain the
better convergence rates as shown in Figs. 4 and 8.
C. Case 3: The Effect of the Data Length l in ILC-DDC
To show the influence of the data lengths lon the
performance of ILC-DDC, we plot the error curves for
both line and sine functions with different data lengths in
Fig. 15, where the data lengths are set as 1,2,3,4,5, and
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HE et al.: ITERATIVE LEARNING CONTROL WITH DATA-DRIVEN-BASED COMPENSATION 11
TABLE III
AVERAGE COST TIME WITH DIFFERENT l
Fig. 16. Error variation curves of the ILCs for “α.”
Fig. 17. Tracking trajectories in 3-D space at different trials for “α.”
10, respectively. From the result, we can see the ILC-DDC
with the larger ltends to have better control performance. But
after the lrises to 4, the increase of lhas little effect on the
performance improvement. This is because with a larger l,the
ILC-DDC can obtain more information about the system for
the compensation of the control, while too large value of l
instead tends to cause the information redundancy, reducing
the control performance to some extent.
To investigate the computation with different data lengths
l, we calculate the main computational cost in ILC-DDC as
O(l3+l2·Nh +l·((Nh)2+Nh)), that is, calculating −→
λk−1
by (34), where Nis the number of samples in each trial and h
is the system output dimension. In the experiment, the average
cost time for calculating −→
λk−1as lgoes from 1 to 10 is listed
in Table III, which is performed on MATLAB2019a with core
i5-8300. It can be seen that the increasing of lincreases the
cost time of −→
λk−1. Thus, the large data length lwill bring the
large computational burden.
D. Case 4: Real-World Application for Writing Character
In the case, the manipulator learns to follow the reference
curve and write the character “α” by ILC-DDC, RILC, and
DDOILC. The curve describing the handwriting “α”isset
as the reference. The results of error variation curves under
different methods are shown in Fig. 16, and the trajectories
for tracking “α” in the 6th, 12th, 18th, and 49th trials are
shown in Fig. 17, respectively. It is obvious that the system
under ILC-DDC achieves the best convergence performance
and the manipulator can complete the tracking task faster than
DDOILC and RILC.
VI. CONCLUSION
ILC-DDC for the linear systems with unknown time-varying
uncertainty was proposed in this article. This strategy consists
of two terms. One is the inputs from the error-feedback law
of the RILC and the other is the special optimal combination
of the past data. This design overcame the conservation of the
RILC, improved the performance, and accelerated the conver-
gence rate. For the proposed ILC-DDC, it was proved that the
convergence can be guaranteed and the convergence rate can
be accelerated. Finally, experiments on a 6-freedom manipu-
lator also showed that the proposed design can provide a great
performance.
The 2-D model can be used to design a real-time feedback
ILC controller. Since the trial number and the one along the
trial were used to describe the system, the dimensions of the
considered system were much less. Thus, we will make efforts
to extend the data-drive method to 2-D system in future work.
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Shaoying He received the B.Eng. degree from
Central South University, Changsha, China, in 2016,
and the M.Eng. degree from Shanghai Jiao Tong
University, Shanghai, China, in 2019, where he
is currently pursuing the D.Eng. degree with the
Department of Automation.
His research interests include predictive control
and robot.
Wenbo Chen received the B.S. degree from the
East China University of Science and Technology,
Shanghai, China, in 2011. He is currently pursu-
ing the Ph.D. degree with the Automation Institute,
Shanghai Jiao Tong University, Shanghai.
His research interest covers the algorithm and
application of model predictive control.
Dewei Li received the B.S. and Ph.D. degrees in
automation from Shanghai Jiao Tong University,
Shanghai, China, in 1993 and 2009, respectively.
He is a Professor with the Department of
Automation, Shanghai Jiao Tong University, where
he worked as a Postdoctoral Researcher from 2009
to 2010. His research interests include predictive
control, robust control, and the related applications.
Yugeng Xi (Senior Member, IEEE) was born in
Shanghai, China. He received the Dr.-Ing. degree
in electrical engineering from Technical University
Munich, Munich, Germany, in 1984.
Since then, he has been with the Department
of Automation, Shanghai Jiao Tong University,
Shanghai, and as a Professor since 1988. He has
authored or coauthored three books and more than
300 journal papers. His research interests include
model-predictive control, optimization and con-
trol of large-scale network systems, and intelligent
robotic systems.
Prof. Xi is currently an Advisory Committee Member of the Asian Control
Association and an Honorary Council Member of the Chinese Association of
Automation.
Yunwen Xu (Member, IEEE) received the B.S.
degree in automation from the Nanjing University of
Science and Technology, Nanjing, China, in 2012,
and the M.S. and Ph.D. degrees in control sci-
ence and engineering from Shanghai Jiao Tong
University, Shanghai, China, in 2014 and 2019,
respectively.
She is currently a Postdoctoral Researcher with
the Department of Automation, Shanghai Jiao Tong
University. Her research interests include model-
predictive control, urban traffic modeling, and intel-
ligent control of complex systems.
Pengyuan Zheng (Member, IEEE) received the
B.Sc. degree in electrical engineering and automa-
tion from the North University of China, Taiyuan,
China, in 2000, the M.Sc. degree in measurement
technology and instrumentation from the University
of Shanghai for Science and Technology, Shanghai,
China, in 2005, and the Ph.D. degree in control
theory and control engineering from Shanghai Jiao
Tong University, Shanghai, in 2010.
He was a Postdoctoral Research Fellow with
Shanghai Jiao Tong University from 2012 to 2014.
Since 2014, he has been an Associate Professor with the College of
Automation Engineering, Shanghai University of Electric Power, Shanghai.
His research interests include predictive control and optimization for
microgrids.
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