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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 1
Computationally Efficient Data-Driven Higher
Order Optimal Iterative Learning Control
Ronghu Chi , Zhongsheng Hou, Senior Member, IEEE, Shangtai Jin, and Biao Huang, Fellow, IEEE
Abstract— Based on a nonlifted iterative dynamic linearization
formulation, a novel data-driven higher order optimal iterative
learning control (DDHOILC) is proposed for a class of non-
linear repetitive discrete-time systems. By using the historical
data, additional tracking errors and control inputs in previous
iterations are used to enhance the online control performance.
From the online data, additional control inputs of previous time
instants within the current iteration are utilized to improve
transient response. The data-driven property of the proposed
method implies that no model information except for the I/O data
is utilized. The computational complexity is reduced by avoiding
matrix inverse operation in the proposed DDHOILC approach
due to the nonlifted linear formulation of the original model.
The asymptotic convergence is proved rigorously. Furthermore,
the convergence property is analyzed and evaluated via three
performance indexes. By elaborately selecting the higher order
factors, the higher order learning control law outperforms the
lower order one in terms of convergence performance. Simulation
results verify the effectiveness of the proposed approach.
Index Terms—Computational efficiency, convergence evalua-
tion, data driven, higher order learning law, nonlifted iterative
dynamic linearization.
I. INTRODUCTION
IN PRACTICAL industries, many processes repetitively
perform the same task. To improve the tracking accuracy
of such processes, iterative learning control (ILC) [1] was
proposed with the ability of learning from previous executions.
Amann et al. [2] pioneered the optimization-based ILC for
linear systems. Since then, many alternative approaches of
optimal ILC have been explored with successful applica-
tions [3]–[8] because all the tracking errors, as well as the
constraints on the input difference between trials, input effort,
Manuscript received November 20, 2016; revised November 25, 2017 and
February 25, 2018; accepted March 5, 2018. This work was supported in part
by the National Science Foundation of China under Grant 61374102, Grant
61573054, and Grant 61433002 and in part by the Taishan Scholar Program of
Shandong Province of China. (Corresponding author: Ronghu Chi.)
R. Chi is with the School of Automation and Electronic Engineering,
Qingdao University of Science and Technology, Qingdao 266061, China
(e-mail: ronghu_chi@hotmail.com).
Z. Hou and S. Jin are with the Advanced Control Systems Lab-
oratory, School of Electronics and Information Engineering, Beijing
Jiaotong University, Beijing 100044, China (e-mail: zhshhou@bjtu.edu.cn;
shtjin@bjtu.edu.cn).
B. Huang is with the Department of Chemical and Materials Engi-
neering, University of Alberta, Edmonton, AB T6G 2G6, Canada (e-mail:
bhuang@ualberta.ca).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNNLS.2018.2814628
and system output can be easily considered through the objec-
tive function. By properly selecting weight matrices to satisfy
the convergence condition, the optimal ILC can make tracking
error converge asymptotically along the iteration direction,
which is most desired in practical applications. However, two
major problems still remain that hamper the application of
optimal ILC methods to practical problems.
The first is the computation efficiency problem of the
optimal ILC methods. Due to the supervector representation
used in norm-optimal ILCs, the lifted system matrix dimension
is increasing exponentially with the batch length and may have
several million elements in the case of the robotic applica-
tions [9], which turns out to be computationally infeasible.
Therefore, a computationally efficient optimal ILC is more
attractive in real industries. Recently, several works have been
done to address the issue of computation efficiency of the
lifted optimal ILC [10]–[13]. But, all the above optimal ILC
approaches [3]–[6], [10]–[13] are limited to exactly known
linear system models or linearly approximated models. This
is the second problem of optimal ILCs preventing them from
real applications.
Although Volckaert et al. [7] and Axelsson et al. [8] have
discussed the optimal ILC design for nonlinear systems by
adding model estimation, an explicit linearized model is still
required to approximate the original nonlinear plant, whose
mathematical model should be known exactly. Therefore,
a negative influence on the stability and robustness will occur
due to the model mismatch and model complexity.
Moreover, with the increasingly large scale and complexity
in real production processes, it is very difficult to acquire their
accurate mathematical models, no matter linear or nonlinear,
by first principle or system identification [14]. Consequently,
data-driven control [14] becomes an interesting and attractive
topic in recent years, where no explicit model is required for
the controller design and analysis. Recently, several develop-
ments [15], [16] under the term of “data-driven ILC” have
been reported by estimating a system representation using
I/O measurements. However, these approaches are designed
and analyzed in a linear system framework. In [17], a data-
driven optimal ILC is presented for nonlinear systems based on
a lifted representation, where the issue of efficient computation
is still open.
In order to achieve a better control performance, the higher
order ILC was originally proposed in [18] to employ control
information of previous iterations. Subsequently, many alterna-
tive higher order ILC methods [19]–[22] have been proposed.
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2IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
The basic idea of the works in [18]–[22] is to utilize more
historical data collected from previous iterations to enhance
the online control performance since the historical data con-
tain system information. However, it is worth pointing out
that online data in the current iteration, compared with the
historical data in the previous iterations, can reflect varying
characteristics of a process and the process disturbances in real
time. Therefore, whenever possible, one should incorporate the
historical as well as online data to achieve a better control
performance. Furthermore, the existing higher order methods
mainly focused on PID-type and optimization-based schemes,
and their design and analysis is also limited to linear systems.
Convergence speed is one of the most important factors to
consider in the ILC. The convergence speed is defined in the
frequency domain in [23] and the tradeoff in normal optimal
ILC in terms of robustness, convergence speed, and steady
state error has been described. In [24], the relationship between
the learning gains and the convergencerate has been discussed.
Son et al. [25] discussed the compromise between robustness
and convergence speed in the robust ILC design.
Higher order ILC [26]–[30] offers the possibility of a higher
error convergence speed by utilizing more control information
from several previous trials than the standard ILC that only
uses the immediate previous one trial. The effects of the mem-
ory length on the error convergence rate have been explored
and evaluated in [26]–[29]; however, up to now, there is little
work done to guarantee in theory that the higher order ILC
outperforms the lower order one with a faster convergence.
In this paper, a novel historical and online data-driven
higher order optimal iterative learning control (DDHOILC) is
proposed based on a nonlifted iterative dynamic linearization
for the nonlinear discrete-time system. The nonlifted iterative
dynamic linearization used in this paper does not require an
exactly known mathematical model of the controlled plant
and results in a model equivalent to the original nonlinear
plant without an approximation. Thus, the proposed approach
is data-driven and no process model is explicitly required.
Owing to the use of additional historical and online data,
that is, more data related to the tracking errors from the
previous iterations and more data from control inputs at
previous time instants within the current iteration, a better
control performance is achieved by applying the proposed
DDHOILC. Rigorous mathematical analysis is provided to
show the asymptotic convergence of the tracking error and to
evaluate the convergence property of the proposed DDHOILC.
It is concluded that under certain conditions, the convergence
speed of the higher order learning law can be faster than that
of the lower order one by selecting the higher order factors
and controller parameters properly. Furthermore, due to the
avoidance of the matrix inverse calculation in the control law,
an efficient computation can be achieved even though the
number of samples in the trial and the trajectory length is
larger. Simulations in this paper verify the derived theoretical
results.
The remainder of this paper is structured as follows.
Section II formulates a nonlifted dynamical linearization for
nonlinear systems. Section III is the controller design of
DDHOILC. Section IV shows the asymptotic convergence
of the proposed method. Section V evaluates convergence
property of the proposed method with rigorous derivations.
Two examples are considered in Section VI to verify the
effectiveness of the proposed approach. Section VII provides
the conclusion.
II. NONLIFTED ITERATIVE DYNAMIC LINEARIZATION
OF REPETITIVE NONLINEAR SYSTEMS
Consider a class of repeatable nonaffine nonlinear discrete-
time systems with unknown orders
yk(t+1)=f(yk(t),...,yk(t−ny), uk(t),...,uk(t−nu))
(1)
where yk(t)and uk(t)are the system output and input,
respectively, and yk(t)=0anduk(t)=0forallt<0,
f(·)is an unknown real nonlinear function and continuously
differentiable, and f(0·0)=0, nyand nuare the orders of
system output and input, respectively, t∈{0,...,N}, with N
being the endpoint of the finite time interval, and kdenotes
the index of iteration.
By following the similar steps in [17], the system output can
be expressed by initial states and the control input series as:
yk(t+1)=gt(yk(0), uk(0),...,uk(t)) (2)
where gt(·),t=0,...,N−1, is a proper nonlinear function
and is also continuously differentiable.
In the following discussion, two assumptions [17] are made.
Assumption 1: The initial value yk(0)is unchanged for all
iterations, i.e., yk(0)=c0,∀k,andc0is a constant.
Assumption 2: Nonlinear function gt(·)is globally
Lipschitz, i.e.,
|gt(x1,u1)−gt(x2,u2)|≤Lx|x1−x2|+Luu1−u2
where Lxand Luare the two positive Lipschitz constants.
Remark 1: Because the original controlled plant (1) is
nonlinear, nonaffine, and completely unknown, the globally
Lipschitz condition (Assumption 2) is required in the follow-
ing analysis as a tradeoff. The way of relaxing Assumption 2
to locally Lipschitz nonlinearity is to transfer the original
nonaffine system into an affine form with respect to control
input by referring the existing works of contraction-mapping-
based ILC with locally Lipschitz nonlinearities [31], [32].
It is not a trivial work to extend the proposed ILC to locally
Lipschitz nonlinear systems because the original controlled
plant considered in this paper is nonlinear and nonaffine.
On the other hand, some recent works have shown that the
contraction-mapping-basedILC approach can deal with locally
Lipschitz nonlinearity, where the nonlinear system considered
is affine to the control input. Therefore, the extension may be
possible by dividing the original plant into two parts: locally
Lipschitz nonlinearity and affine control input.
The iterative dynamic linearization in [17] can be easily
extended to a nonlifted form, summarized as follows.
Lemma 1: For the nonlinear system (2), which is a reformu-
lation of system (1) with respect to system output, initial state,
and control input, satisfying assumptions 1 and 2, according
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CHI et al.: COMPUTATIONALLY EFFICIENT DDHOILC 3
to the mean value Theorem, there exists an optimal gradient
vector θt
k(t)such that
yk(t+1)=yk−1(t+1)+θt
k(t)uk(t)(3)
and θt
k(t)≤Lu,whereuk(t)=uk(t)−uk−1(t),uk(t)=
[uk(0), uk(1),...,uk(t)]T, whose dimension varies with time
instants, e.g., uk(t−1)=[uk(0), uk(1),...,uk(t−1)]T;and
θt
k(t)=θt
k(0), θ t
k(1),...,θt
k(t)
=∂gt(·,·)
∂uk(0),∂gt(·,·)
∂uk(1),...,∂gt(·,·)
∂uk(t)
whose dimension also varies with time instant, denotes the
optimal partial derivatives of gt(·,·)with respect to uk(t)in
the interval of [uk(t), uk−1(t)].
Readers can refer to the literature [17] for the detail proof.
Note that the above nonlifted iterative dynamic lineariza-
tion (3) is completely equivalent to the original nonlinear
system (2), which is equivalent to (1) in turn.
III. HIGHER ORDER DATA -DRIVEN OPTIMAL ILC
Assume that yd(t),t∈{0,...,N}is a target trajectory and
is bounded for all time instants. The control objective is to
make the tracking error ek(t)=yd(t)−yk(t)converge to
zero asymptotically along the iterations. Since θt
k(t)in (3) is
unknown and slowly iteration and time varying, a modified
projection algorithm [17] is utilized for its iterative estimate
ˆ
θt
k(t)=ˆ
θt
k−1(t)
+η(yk−1(t+1)−ˆ
θt
k−1(t)uk−1(t))uT
k−1(t)
µ+uk−1(t)2(4)
ˆ
θt
k(t)=ˆ
θt
0(t), if sgnˆ
θt
k(i)= sgnˆ
θt
0(i)or
ˆ
θt
k(t)
≤ε
(5)
where i=0,...,t,ˆ
θt
k(t)is the estimation of θt
k(t),µ>0
denotes a weighting factor, η∈(0,2)is a step-size factor, and
εis a small positive scalar. The initial value of ˆ
θt
0(t)should
be selected such that all its elements have the same signs as
that of θt
k(t), whose elements’ signs can be identified by using
the I/O data sampled from the controlled plant.
Remark 2: In this paper, we mainly focus on the controlled
systems with relatively insignificant measurement noise, and
thus, the direction of θt
k(t)can be obtained by the experimental
trials along with other priori knowledge. The cases with
significant random data noise will be addressed in our future
work followingsimilar line as the stochastic ILC methods [33].
However, it would be difficult for most of the existing control
methods to deal with the case that θt
k(t)converges to 0 because
the control action becomes weaker and weaker.
Consider an objective function with higher order factors
J(uk(t), α)=M
m=1
αmek−m+1(t+1)2
+λ(uk(t)−uk−1(t))2(6)
where λ>0 is a weighting factor, and α=[α1,...,α
M]
denotes the higher order factors and M
m=1αm=1,
0<α
m≤1, α1+α2−M
m=3αm=¯α>0, where Mis
a positive integer.
Remark 3: Since the controlled plant (1) is a nonlinear
unknown process and the system uncertainties are inevitable,
according to [25] theSterm used in the classical optimal ILC
has not been included in the cost function (6) to enhance the
robustness of the designed control system.
Rewrite (3) as
yk(t+1)=yk−1(t+1)
+θt
k(t−1), θ t
k(t)uT
k(t−1), uk(t)T(7)
where θt
k(t−1)=[θt
k(0), θ t
k(1),...,θt
k(t−1)].
Substituting (7) into (6), yields
J(uk(t), α)
=α1ek−1(t+1)−
t−1
i=0
θt
k(i)uk(i)−θt
k(t)uk(t)
+
M
m=2
αmek−m+1(t+1)2
+λ|uk(t)−uk−1(t)|2
(8)
where uk(i)=uk(i)−uk−1(i),i=0,1,...,t.
Minimizing objective function (6) with respect to uk(t)by
replacing the unknown θt
k(t)with ˆ
θt
k(t), a learning control law
is derived as
uk(t)=uk−1(t)−ρα2
1ˆ
θt
k(t)t−1
i=0ˆ
θt
k(i)uk(i)
λ+α2
1ˆ
θt
k(t)2
+
ρα1ˆ
θt
k(t)α1ek−1(t+1)+M
m=2αmek−m+1(t+1)
λ+α2
1ˆ
θt
k(t)2
(9)
where ρ>0 is a positive factor.
Remark 4: From the proposed DDHOILC (4), (5), and (9),
it is clearly seen that: 1) there is no matrix inverse calculation
included, so an improved computational efficiency can be
achieved; 2) the unknown parameters in (9) are updated in
terms of (4) and (5) such that the proposed method is more
flexible for the modifications/or expansions of the controlled
plant; and 3) there is no model information used except for
the input and output measurements, so the proposed method
is data driven and suitable to complex nonlinear processes in
practice.
IV. CONVERGENCE ANALYSIS
Before proceeding with the analysis, three lemmas are given
as follows.
Lemma 2 [34]: Let
A=⎡
⎢
⎢
⎢
⎢
⎢
⎣
010··· 0
001··· 0
.
.
..
.
..
.
.....
.
.
000··· 1
a1a2a3··· at
⎤
⎥
⎥
⎥
⎥
⎥
⎦
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4IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
and s(A)is the spectral radius of A.Ift
i=1|ai|<d,0<
d<1, then s(A)<d.
Lemma3[35]:A∈Cn×n,ands(A)is the spectral radius
of A. Then, for any δ>0, there always exists a proper matrix
norm ·
von the normed vector space v, such that
Av<s(A)+δ.
Lemma4[35]:Let Avand Aµbe the proper matrix
norm of Aon the normed vector spaces νand µ, respectively.
Then, there must exist γ≥σ>0suchthat
γAv≥Aµ≥σAv,∀A∈Cm×n.
The boundedness of ˆ
θt
k(t)has been shown in [17], so we
have |ˆ
θt
k(i)|≤bθ,i=0,1,...,t,wherebθis a positive
constant. Then, the convergence property of the proposed
DDHOILC (4), (5), and (9) is summarized in the following
theorem.
Theorem 1: Consider nonlinear system (1) satisfying
assumptions 1 and 2. Applying the proposed DDHOILC (4),
(5), and (9) with proper controller parameters λand ρsuch
that
λ>max ρ2α2
1b2
θN2,γ2ρ2
4,ρ2L2
u
4,γ2ρ2L2
u
4
where γis a positive constant, it is guaranteed that (t1) the
tracking error ek(t+1)converges to zero asymptotically and
iteratively; (t2) the control system is bounded-input-bounded-
output stable.
Remark 5: From Theorem 1, the bounds bθ,Lu,and γ
should be known for a proper selection of λ. For a practical
control process in which the above bounds are known, one can
select a larger λconservatively to guarantee the asymptotical
convergence of the tracking error. The tradeoff is that the
convergence speed may become slower. Alternatively, one
can estimate these bounds by using experiments and then
determine the range of λby using the estimated values.
Proof: The two matrices Au
k(t)and Bu
k(t)are defined in
the equation given at the bottom of this page.
Define Cu=[0,...,0,1]T∈Rt+1,and¯ek−1(t+1)=
[ek−M+1(t+1), ek−M+2(t+1),...,ek−1(t+1)]T∈RM−1.
Furthermore, we define an expended vector with fixed (t+1)
dimensions as ¯uk(t)=uk(t)=[uk(0)uk(1)··· uk(t)]T,then
¯uk(t−1)=[uk(−1)uk(0)··· uk(t−1)]T, where the input
signal uk(t)is set as zero for t<0.
Then, according to (9), one has
¯uk(t)=Au
k(t)¯uk(t−1)+CuBu
k(t)¯ek−1(t+1). (10)
Note that the following inequality holds:
t−1
i=0
ρα2
1ˆ
θt
k(t)ˆ
θt
k(i)
λ+α2
1ˆ
θt
k(t)
2≤
t−1
i=0
ρα2
1ˆ
θt
k(t)ˆ
θt
k(i)
2√λα1ˆ
θt
k(t)
≤
t−1
i=0
ρα1ˆ
θt
k(i)
2√λ.(11)
Since tis finite over {0,1,...,N}, by properly select-
ing λand ρ,suchλ>(ρα
1bθN)2, there exists a series
of positive constants at time instant tof the kth iteration
0<d1(k,t)<0.5where0<d1(k,t)<0.5isdefinedin
the following equation such that for each fixed time instant t
and fixed iteration number k, the following inequality
holds, i.e.,
0≤
t−1
i=0
ρα2
1ˆ
θt
k(t)ˆ
θt
k(i)
λ+α2
1ˆ
θt
k(t)
2≤
t−1
i=0
ρα1ˆ
θt
k(i)
2√λ
≤ρα1bθN
2√λ=d1(k,t)<0.5 (12)
which implies that the spectral radius s(Au
k(t)) < d1(k,t)
holds for all tand kin view of Lemma 2. Then, according
to Lemma 3, there exist a series of arbitrary small positive
constants δ1(k,t)such that
Au
k(t)
v≤s(Au
k(t)) +δ1(k,t)≤d1(k,t)+δ1(k,t)
=d2(k,t)≤d2<0.5 (13)
where Au
k(t)vdenotes a proper matrix norm of Au
k(t)on the
normed vector space vat thetth time instant of the kth iteration,
and 0 <d2(k,t)<0.5 is a series of positive constants.
d2=supk,td2(k,t). It is obvious that 0 <d2<0.5 because
0<d2(k,t)<0.5 holds for all time instants tand iterations k,
and the condition of 0 <d2<0.5 will be used in the following
analysis.
Taking norm on both sides of (10), yields
¯uk(t)v≤
Au
k(t)
v¯uk(t−1)v
+Cuv
Bu
k(t)
v¯ek−1(t+1)v.(14)
According to Lemma 4, there exists a constant γsuch that
Bu
k(t)
v≤γ
Bu
k(t)
2.
Au
k(t)=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
01 0··· 0
00 1··· 0
.
.
..
.
..
.
.....
.
.
00 0··· 1
0−ρα2
1ˆ
θt
k(t)ˆ
θt
k(0)
λ+α2
1ˆ
θt
k(t)
2−ρα2
1ˆ
θt
k(t)ˆ
θt
k(1)
λ+α2
1ˆ
θt
k(t)
2··· −ρα2
1ˆ
θt
k(t)ˆ
θt
k(t−1)
λ+α2
1ˆ
θt
k(t)
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(t+1)×(t+1)
Bu
k(t)=ρα1ˆ
θt
k(t)αM
λ+α2
1ˆ
θt
k(t)
2
ρα1ˆ
θt
k(t)αM−1
λ+α2
1ˆ
θt
k(t)
2··· ρα1ˆ
θt
k(t)α3
λ+α2
1ˆ
θt
k(t)
2
ρα1ˆ
θt
k(t)(α1+α2)
λ+α2
1ˆ
θt
k(t)
21×(M−1)
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CHI et al.: COMPUTATIONALLY EFFICIENT DDHOILC 5
So, the following inequality can be derived directly since
M
m=1αm=1, 0 <α
m≤1:
Bu
k(t)
2
v≤γ2
Bu
k(t)
2
2
≤γ2ρα1ˆ
θt
k(t)2(α1+α2)2+α2
3+···+α2
M
4λα2
1ˆ
θt
k(t)2
≤γ2ρα1ˆ
θt
k(t)2(α1+α2+···+αM)2
4λα2
1ˆ
θt
k(t)2=γ2ρ2
4λ.
(15)
One can select λproperly such that λ>γ
2ρ2/4, then
Bu
k(t)
v≤γ
Bu
k(t)
2≤γρ/(2√λ) =d3<1 (16)
where 0 <d3<1 is a positive constant.
Hence, one can derive from (13)–(16) that
¯uk(t)v≤d2¯uk(t−1)v+d3¯ek−1(t+1)v
≤···≤d3
t
i=0
dt−i
2¯ek−1(i+1)v.(17)
By virtue of (3) and (10), one has
ek(t+1)=ek−1(t+1)−θt
k(t)CuBu
k(t)¯ek−1(t+1)
−θt
k(t)Au
k(t)¯uk(t−1). (18)
The second error term in (18) can be further expanded as
θt
k(t)CuBu
k(t)¯ek−1(t+1)
=θt
k(t)Bu
k(t)¯ek−1(t+1)
=ρα1θt
k(t)ˆ
θt
k(t)(α1+α2)
λ+α2
1ˆ
θt
k(t)2ek−1(t+1)
+
M
m=3
ρα1θt
k(t)ˆ
θt
k(t)αm
λ+α2
1ˆ
θt
k(t)2ek−m+1(t+1). (19)
Let ϑk,1(t)=ρα1θt
k(t)ˆ
θt
k(t)(α1+α2)/(λ +α2
1ˆ
θt
k(t)2),and
ϑk,j(t)=ρα1θt
k(t)ˆ
θt
k(t)αj+1/(λ +α2
1ˆ
θt
k(t)2),j=2,...,
M−1, for notational convenience. According to the reset algo-
rithm (5), it is easy to obtain that θt
k(t)ˆ
θt
k(t)>0. Since
0<α
m≤1,m=1,...,M, one can get ϑk,j(t)>0,
j=1,...,M−1and
0<ϑ
k,1(t)≤ρθt
k(t)
2√λand 0 <ϑ
k,j(t)≤ρθt
k(t)
2√λ.(20)
Since θt
k(t)≤Lu, by properly selecting λand ρsuch
that λ>ρ
2L2
u/4, one can guarantee that
0<ϑ
k,1(t)≤ρLu
2√λ<1and0<ϑ
k,j(t)≤ρLu
2√λ<1.(21)
Define
Ae
k(t)=⎡
⎢
⎢
⎢
⎢
⎢
⎣
01 0··· 0
00 1··· 0
.
.
..
.
..
.
.....
.
.
00 0··· 1
−ϑk,M(t)−ϑk,M−1(t)−ϑk,M−2(t)···1−ϑk,1(t)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
and Ce=[0··· 01]T∈RM−1. In terms of (18), we have
¯ek(t+1)=Ae
k(t)¯ek−1(t+1)−Ceθopt
k(t)Au
k(t)¯uk(t−1).
(22)
From (21)
|ϑk,M(t)|+|ϑk,M−1(t)|+···+|ϑk,2(t)|+|1−ϑk,1(t)|
=1−(ϑk,1(t)−ϑk,2(t)−ϑk,3(t)−···−ϑk,M(t))
=1−
ρα1α1+α2−M
m=3αmθt
k(t)ˆ
θt
k(t)
λ+α2
1ˆ
θt
k(t)2.(23)
According to the condition 0 <α
1+α2−M
m=3αm=
¯α<1, as long as λ>ρ
2L2
u/4, the following inequality holds:
0<M1≤
ρα1α1+α2−M
m=3αmθt
k(t)ˆ
θt
k(t)
λ+α2
1ˆ
θt
k(t)2
≤ρα1¯αθt
k(t)ˆ
θt
k(t)
2α1√λˆ
θt
k(t)
<ρLu
2√λ=d4<1 (24)
where 0 <M1<1and0<d4<1 are two positive constants.
Therefore, it is obtained from (23) and (24) that
1−d4<|ϑk,M(t)|+···+|ϑk,2(t)|+|1−ϑk,1(t)|
≤1−M1<1.(25)
In terms of Lemma 2 and inequality (25), one can obtain
that s(Ae
k(t)) < 1−M1<1. Consequently, one can find an
arbitrarily small positive constant δ2such that
Ae
k(t)
v≤sAe
k(t)+δ2≤1−M1+δ2≤d5<1 (26)
where 0 <d5<1 is a positive constant.
By virtue of (13), (17), and (26), one can derive that
¯ek(t+1)
v≤
Ae
k(t)
v¯ek−1(t+1)v
+Cev
θt
k
v
Au
k(t)
v¯uk(t−1)v
≤d5¯ek−1(t+1)v+Lud2¯uk(t−1)v
≤d5¯ek−1(t+1)v
+Lud3
t
t=0
dt−i+1
2¯ek−1(i+1)v.(27)
Assume that ¯ek(τ +1)v=maxt∈{0,...,N−1}
{¯ek(t+1)v}=¯emax
kis attained at time instant τ+1,
where ¯emax
kdenotes the maximum value at the kth iteration.
From (27), we have
¯emax
k=¯ek(τ +1)v
≤d5+Lud3d2
1−d2¯ek−1(τ +1)v≤d6¯emax
k−1(28)
where d6=d5+(Lud3d2)/(1−d2).Sinceλ > ((γ 2ρ2L2
u)/4),
one has 0 <Lud3<1. Furthermore, 0 <d5<1and0<
d2/(1−d2)<1 due to 0 <d2<0.5, so by selecting λ
properly, one can get 0 <d6<1. Then inequality (28) implies
that
¯emax
k≤d6¯emax
k−1≤···≤dk
6¯emax
0.(29)
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6IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
Since the initial value ¯emax
0is bounded, we have
0≤lim
k→∞ ¯ek(t+1)v≤lim
k→∞ ¯emax
k=0.(30)
So, the asymptotic convergence of tracking error is proved.
The boundedness of both of yd(t+1)and ek(t+1)means
that yk(t+1)is bounded. It is also clear that
¯uk(t)v≤
k
j=1¯uj(t)v+¯u0(t)v.(31)
According to (17) and (31), we have
¯uk(t)v≤d3
k
j=1
t
i=0
dt−i
2¯ej−1(i+1)v+¯u0(t)v.(32)
Since ¯ek(t+1)v≤¯emax
k, one can derive that
¯uk(t)v≤d31−dt
2
1−d2
k
j=1¯emax
k+¯u0(t)v.(33)
Then, according to (29)
¯uk(t)v≤M3dk
6+dk−1
6+···+d0
6¯emax
0+¯u0(t)v
≤M31
1−d6¯emax
0+¯u0(t)v(34)
where M3=d3/(1−d2)is bounded. Because ¯emax
0and
¯u0(t)vare bounded, then (34) means that the control input
is bounded at all-time instants and iterations.
Remark 6: The spectral radius analysis is used to achieve the
convergence of the proposed method by extending the similar
results in [36] from time domain into iteration domain. On the
other hand, one can also prove the convergence via linear
matrix inequality technique similar to the existing work in
[37], but the selectable range of the controller parameters may
become indistinct.
V. CONVERGENCE PROPERTY EVA L U AT I O N
The convergence speed is of interest for practical applica-
tions. However, the quantification of the convergence speed is
very difficult and little related work can be found in [23]–[25].
In this paper, according to (14) and (27), it is clear that
the convergence performance is affected mainly by the norm
values of Au
k(t)vBu
k(t)v,andAe
k(t)v. Therefore, one can
define three indexes as follows to evaluate the convergence
property of the proposed higher order learning control law:
S1(α)=
Au
k(t)
v,S2(α)=
Bu
k(t)
v,S3(α)=
Ae
k(t)
v.
All the three indices reflect the convergence rate qualita-
tively in the same direction, that is, the smaller the indexes
are, the faster the convergence rate is.
Therefore, in order to guarantee that higher order algorithm
outperforms the lower order one, it is required that Sn(αH)<
Sn(αL),n=1,2,3, where αHdenotes the weighting vector
for the higher order algorithm, and αLis for the lower order
algorithm.
Convergence Index 1: Provided that αL
1>α
H
1,then
t−1
i=0
ραH
12ˆ
θt
k(t)ˆ
θt
k(i)
λ+αH
12ˆ
θt
k(t)2≤
t−1
i=0
ραL
12ˆ
θt
k(t)ˆ
θt
k(i)
λ+αL
12ˆ
θt
k(t)2
.(35)
According to (12) and (13), inequality (35) implies that
S1(αH)<S1(αL). So, one can select the higher order factors
such that αL
1>α
H
1to make the higher order learning law
have a better performance than the lower order one.
Convergence Index 2: Because αH
1+αH
2+···+αH
H=1,
αL
1+αL
2+···+αL
L=1, α1+α2−M
m=3αm=¯α>0, and
H>L, it is obvious that
ραH
1ˆ
θt
k(t)2αH
1+αH
22+αH
32+···+αH
H2
λ+αH
12ˆ
θt
k(t)2
<ραH
1ˆ
θt
k(t)2αL
1+αL
22+αL
32+···+αL
L2
λ+αH
12ˆ
θt
k(t)22.(36)
According to (15), in order to warrant S2(αH)<S2(αL),
it is required that
αH
1
λ+αH
12ˆ
θt
k(t)2<αL
1
λ+αL
12ˆ
θt
k(t)2.(37)
Solving (37) subject to the condition of αL
1>α
H
1,
we get λ>α
H
1αL
1ˆ
θt
k(t)2. That is, the conditions guaranteeing
S2(αH)<S2(αL)are λ>α
H
1αL
1b2
θand αL
1>α
H
1.
Convergence Index 3: Define ¯αH=(α H
1+αH
2−
H
h=3αH
h),¯αL=(αL
1+αL
2−L
l=3αL
h).IfαL
1¯αL<α
H
1¯αH,
then
λαL
1¯αL−αH
1¯αH<α
L
1αH
1αH
1¯αH−αL
1¯αLˆ
θopt
k(t)
2(38)
which means
αH
1¯αH
λ+αH
12ˆ
θt
k(t)2>αL
1¯αL
λ+αL
12ˆ
θopt
k(t)
2.(39)
Hence, according to (23)–(26), inequality (39) implies that
S3(αH)<S3(αL).
Note that the condition of αL
1¯αL<α
H
1¯αHis not applicable
to the cases of first-order and second-order algorithms where
¯α1=¯α2=1 are fixed and cannot be manipulated further.
So, for the first-order and second-order algorithms, one can
only use S1(α)and S2(α)to evaluate the convergence property
of the tracking error in theory.
Remark 7: As a summary of the above three cases, for the
higher order algorithms with more than third orders, one can
select αproperly such that αL
1>α
H
1,λ>α
H
1αL
1b2
θ,and
αL
1¯αL<α
H
1¯αHto guarantee an improved performance of the
higher order learning control law.
Remark 8: Note that the controller parameters must be
selected to guarantee the convergence of the proposed method
at first, and then, the convergence property is evaluated in the
sequel. So a faster convergence by selecting the higher order
factors properly does not neglect the tracking performance but
actually enhances it.
Remark 9: In general, one can fix the value of λand
then tune the value of ρby trials such that they satisfy the
condition given in Theorem 1. If the parameters are selected
differently under the given condition, the convergence can still
be guaranteed but the convergence rate may be different. The
larger values of ρand smaller values of λmay generally
produce a faster convergence speed.
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CHI et al.: COMPUTATIONALLY EFFICIENT DDHOILC 7
Fig. 1. Tracking errors with respect to iterations in Example 1.
VI. SIMULATION EXAMPLES
Example 1: For comparison with traditional optimal ILC,
a linear time-varying (LTV) system is adopted from [38].
When applying the traditional lifted optimal ILC, the linear
model is required to be known exactly. However, it is con-
sidered to be unknown and only serves for generating the I/O
data for evaluating the proposed DDHOILC. The LTV system
is shown as follows:
x(t+1)=A(t)x(t)+B(t)u(t)
y(t)=C(t)x(t)(40)
where
A(t)=01
−0.5−10−3t−0.5−10−3t,B(t)=0
1
and C(t)=[10];t∈{0,...,200}. The control task is to
track the following desired trajectory:
yd(t)=10−6(t−1)3(4−0.03(t−1)), t∈{0,...,200}.
(41)
In the simulation, the initial states in all iterations are set
as 0 and the input of the first iteration is 0.
The controller parameters are selected as: ˆ
θt
0(t)=0.9,
ρ=1, η=1, λ=0.5, and µ=0.1. For comparison, the
first-order, second-order, third-order, and fourth-order forms
of the proposed DDHOILC are applied respectively with the
same simulation conditions except that the higher order factors
are selected differently: α1=[1]for the first order, α2=
[0.9,0.1]for the second-order, and α3=[0.8,0.14,0.06]
for the third-order, and α4=[0.75,0.23,0.01,0.01]for the
fourth order. Note that α1
1>α
2
1>α
3
1>α
4
1and α3
1¯α3=
0.704 <α
4
1¯α4=0.72 satisfy the conditions derived from the
convergence indexes.
It should be noted that in order to compare the proposed
DDHOILC and the traditional OILC under the same simula-
tion conditions, all the learning control laws start from the
third iteration because the calculation of the control input in
the fourth-order DDHOILC uses the error information from
the previous three iterations.
Fig. 2. Computation time in Example 1.
The simulation results are shown in Figs. 1 and 2. Fig. 1
demonstrates the asymptotic convergence of the proposed
nonlifted DDHOILC, where the y-axis is the mean absolute
value of tracking error emean (k)=200
t=1|yd(t)−yk(t)|/200.
Obviously, the asymptotic convergence of tracking error can
be guaranteed by the proposed DDHOILC. Meanwhile, it is
demonstrated that a faster convergence can be achieved by
selecting the higher order factors according to the conditions
given in this paper.
Fig. 2 shows the computation time of the proposed
DDHOILC with third-order learning control law as an exam-
ple. The horizontal axis denotes the operating length of the
controlled process within each batch. The unit of the horizontal
axis is 200 instants per batch. The vertical axis denotes the
calculation time and the unit is seconds.
The calculation is performed in MATLAB on a Lenovo
ThinkPad laptop computer with a 2.40 GHz Intel Core i3
process and 2 GB of RAM. The batch length is changed
by selecting different sampling rate. It is verified that the
computation time of the proposed DDHOILC via nonlifted
iterative dynamic linearization increases very slowly with
the increasing size of data sets because no matrix inverse
operation is required for the implementation of the learning
control law.
For a comparison, an optimal ILC law is selected from [13]
Uk+1=(GTQG +R+S)−1(GTQG +R)Uk
+(GTQG +R+S)−1GTQEk(42)
where Q,R,andSare real-valued symmetric positive definite
matrices. By selecting Q=I,R=0.2I,S=0the simulation
results are also shown in Figs. 1 and 2.
From Fig. 1, it is seen that the control performance of
the proposed DDHOILC is better than that of the traditional
OILC because more historical and online control information
is employed. Only the historical control information in the
immediate previous one iteration is used in the traditional
OILC.
Fig. 2 shows that the computational time of the tradi-
tional OILC approach (42) increases dramatically as the
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8IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
TAB L E I
PARAMETERS OF A CSTR SYSTEM
trial length increases. The matrix inverse operation in lifted
OILC becomes impractical with increasing dimension and data
points.
Example 2: In chemical industries, the continuously stirred
tank reactor (CSTR) is a highly nonlinear process with a com-
plex dynamic behavior [39]–[41]. In this paper, we consider
the concentration control of the CSTR, and the extensively
used model is shown as follows:
˙
CA=F
V(CA0−CA)−k0e−E/RTRCA+σ1(k)(43)
˙
TR=F
V(TA0−TR)+−H
ρCpk0e−E/RTRCA+Qσ
ρCpV(44)
where CAis the concentration of Ain the reactor (kmol/m3),
TRis the reactor temperature (K),Qσ=2.789 ×104denotes
the removed heat from the reactor (kJ/min),Vdenotes the
reactor volume (m3),k0is a pre-exponential constant (min−1),
Eis the activation energy (kJ/kmol),His the reaction
enthalpy (kJ/kmol),Cpis the heat capacity (kJ/kgK),andρ
denotes fluid density (kg/m3). The parameter values are listed
in Table I [39]. 0 <CA0<2kmol/m3is the control input.
The control objective is to drive the system to an unstable
steady state Cas =0.57 kmol/m3where CA0is the control
input. The finite batch time is 3 min and the sampling time is
h=0.01 min.
In practice, a process can be affected by various distur-
bances. So, the measurement error σ1(k)of concentration
CAis considered in the simulation as shown in (43), which
randomly varies with both the time instants and the batch
numbers over the interval of [−0.01 0.01]. Besides, the ini-
tial concentration fluctuation in the inlet flow CA0is also
considered, that is, xk(0)=(0.47 +σ2(k))kmol/m3,where
kdenotes the batch number of the CSTR process, and σ2(k)
also randomly varies over the interval of [−0.01 0.01].
In the simulation, the initial input signal at the first
batch is selected as u0(t)=0.05 for all the time instants
t∈{0,1,...,299}; the controller parameters are chosen as
ρ=2, η=0.00001, λ=0.001, and µ=0.1. The
initial value ˆ
θt
0(t)=0.1. Select the higher order weighting
factors as α1=[1],α2=[0.9,0.1],α3=[0.7,0.2,0.1],
and α4=[0.6,0.38,0.015,0.005]for the corresponding first-
order, second-order, third-order and fourth-order algorithms,
respectively. Apparently, α1
1>α
2
1>α
3
1>α
4
1and α3
1¯α3=
Fig. 3. Tracking errors with respect to iterations in Example 2.
TAB L E I I
VALUES OF THE EVAL U AT E INDEX WITH DIFFERENT
ORDERS OF THE PROPOSED DDHOILC
0.56 <α
4
1¯α4=0.576 also satisfy the conditions derived
in Section V.
The mean tracking error, emean(k)=300
t=1
|Cas −CA,k(t)|/300, is shown in Fig. 3 by using the
proposed DDHOILC.
From Fig. 3, the convergence of the tracking error is clearly
seen. Note that the proposed DDHOILC is executed by using
the input and output measurements only, without requiring any
model information of the controlled nonlinear system. In this
sense, it is data driven and is applicable to complex nonlinear
industrial processes.
Furthermore, to evaluate the control performance of higher
order control laws numerically, a numerical index is defined as
JIE =
100
k=1
emean(k)
where emean(k)is the mean tracking error defined earlier.
Applying the above four different order algorithms,
the numerical indices are shown in Table II. It is clear
that the higher order algorithm can achieve a better control
performance than the lower order one.
It should be noted that it is difficult to apply the traditional
OILC [13] to the CSTR process because of its strong nonlin-
earities and strong uncertainties. Comparatively, the proposed
DDHOILC in this paper is data driven and can be directly
applied to such a nonlinear uncertain process.
VII. CONCLUSION
In this paper, a new historical and online DDHOILC is
proposed for a class of nonlinear and nonaffine discrete-time
systems, which repetitively run over a finite time interval.
A nonlifted iterative dynamic linearization is constructed for
the nonlinear plant at first, and then the DDHOILC is proposed
by designing a control objective function with higher order
factors introduced. More historical and online measurements in
previous iterations and in previous time instants of the current
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CHI et al.: COMPUTATIONALLY EFFICIENT DDHOILC 9
iteration, respectively, are fully utilized to enhance the control
performance. It is shown theoretically that the convergence
property of the higher order learning control law can be better
than that of the lower order one by properly selecting the
higher order factors. Furthermore, the computational complex-
ity of the proposed DDHOILC is reduced significantly since no
matrix inversion is included in the proposed learning control
law. The proposed DDHOILC does not require any explicit
model information except for the I/O data and thus can be
more applicable to complex nonlinear processes in practice.
It is worth pointing out that a basic premise of ILC is that
the desired task should be performed under restricted repetitive
conditions such as identical initial condition and identical trial
length for all iterations. However, these assumptions can be
violated in many practical applications. Therefore, the results
will be further extended to the systems with nonrepetitive
uncertainties in our future work.
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This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
10 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
Ronghu Chi received the Ph.D. degree from
Beijing Jiaotong University, Beijing China, in 2007.
From 2011 to 2012, he was a Visiting Scholar
with Nanyang Technological University, Singapore.
From 2014 to 2015, he was a Visiting Professor with
the University of Alberta, Edmonton, AB, Canada.
In 2007, he joined the Qingdao University of Science
and Technology, Qingdao, China, where he is cur-
rently a Full Professor with the School of Automa-
tion and Electronic Engineering. He has published
over 100 papers in important international journals
and conference proceedings. His current research interests include iterative
learning control, data-driven control, intelligent transportation systems, and
so on.
Dr. Chi has also served as a Council Member of the Shandong Institute
of Automation and a Committee Member of the Data-Driven Control, Learn-
ing and Optimization Professional Committee, and so on. He received the
Taishan Scholarship in 2016. He served as various positions in international
conferences. He was an Invited Guest Editor of the International Journal of
Automation and Computing.
Zhongsheng Hou (SM’13) received the bachelor’s
and master’s degrees from the Jilin University of
Technology, Changchun, China, in 1983 and 1988,
respectively, and the Ph.D. degree from Northeastern
University, Shenyang, China, in 1994.
From 1995 to 1997, he was a Post-Doctoral Fellow
with the Harbin Institute of Technology, Harbin,
China. From 2002 to 2003, he was a Visiting
Scholar with Yale University, New Haven, CT, USA.
In 1997, he joined Beijing Jiaotong University,
Beijing, China, where he is currently a Distinguished
Professor and the Founding Director of the Advanced Control Systems
Laboratory, and the Head of the Department of Automatic Control. He is also
the Founding Director of the Technical Committee on Data-Driven Control,
Learning and Optimization, Chinese Association of Automation. His current
research interests include data-driven control, model-free adaptive control,
learning control, and intelligent transportation systems.
Dr. Hou is an IFAC Technical Committee Member on Adaptive and Learning
Systems and Transportation Systems. His original contribution in model-free
adaptive control has been recognized by over 152 different field applications.
He has also done a lot pioneering work in Data-Driven Control and Learning
Control and organized many special issues on data-driven control in the
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
in 2011, the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS in 2016,
and so on.
Shangtai Jin received the bachelor’s, master’s, and
Ph.D. degrees from Beijing Jiaotong University,
Beijing, China, in 1999, 2004, and 2009,
respectively.
He is currently a Lecturer with Beijing Jiaotong
University. His current research interests include
model-free adaptive control, data-driven control,
learning control, and intelligent transportation
systems.
Biao Huang (M’97–SM’11–F’18) received the
B.Sc. and M.Sc. degrees in automatic control from
the Beijing University of Aeronautics and Astronau-
tics, Beijing, China, in 1983 and 1986, respectively,
and the Ph.D. degree in process control from the
University of Alberta, Edmonton, AB, Canada, in
1997.
In 1997, he joined the Department of Chemical
and Materials Engineering, University of Alberta,
as an Assistant Professor, where he is currently a
Professor. He has applied his expertise extensively in
industrial practice. His current research interests include process control, data
analytics, system identification, control performance assessment, Bayesian
methods, and state estimation.
Dr. Huang is a fellow of the Canadian Academy of Engineering and the
Chemical Institute of Canada. He is the Industrial Research Chair in control
of oil sands processes with the Natural Sciences and Engineering Research
Council of Canada and the Industry Chair of process control with Alberta
Innovates. He is a recipient of a number of awards, including the Alexander
von Humboldt Research Fellowship from Germany, the Best Paper Award
from the IFAC Journal of Process Control, the APEGA Summit Award in
Research Excellence, and the Bantrel Award in Design and Industrial Practice,
and so on.
