Adaptive Backstepping Repetitive Learning Control
for Discrete-time Periodic Systems
Mingxuan Sun Hongbo Bi Haigang He
College of Information Engineering
Zhejiang University of Technology
Hangzhou, China
Email: mxsun@zjut.edu.cn
Sheng Zhu
Zhejiang University City College
Hangzhou, China
Email: zhus@zucc.edu.cn
Abstract—In this paper, the problem of adaptive backstepping
repetitive learning control is addressed for a class of periodically
time-varying discrete-time strict-feedback systems. A repetitive
learning least squares algorithm is applied for parameter estima-
tion, where the lower bound for the control gain is introduced to
avoid the potential singularity. An iteration-domain key technical
lemma is given for the purpose of performance analysis, which is
a slight modification of the key technical lemma used for analysis
of discrete adaptive systems. It is shown that the zero-error
convergence can be achieved as the iteration increases, while the
variables of the closed-loop system undertaken are bounded.
Index Terms—repetitive leaning control, adaptive control,
backstepping, least squares, periodically time-varying discrete-
time systems.
I. INTRODUCTION
Periodic trajectory tracking or periodic disturbance rejection
is a control problem that has attracted increasing attention in
recent years, among which repetitive learning control (RLC)
techniques are applied to tackle this problem in continuous-
time domain [1-12]. The advantageous feature of the reported
control schemes is that no initial resetting is required, while
perfect tracking performance can be achieved as time increases.
There are potential applications for RLC such as robotic
motions, HDD/CD servos, power electronics, etc.
Backstepping is among the most important tools in design
and analysis of nonlinear control techniques, and significant
results have been reported in continuous-time domain (see,
e.g., [13], and the references therein). A wide variety of
discrete-time adaptive control schemes for systems with para-
metric uncertainties, have been developed, and the majority
of convergence results are established on the basis of the
key technical lemma [14]. In [15-18], as preliminary studies,
the backstepping technique was employed to design adaptive
controllers for nonlinear discrete-time systems. More recent
developments can be found in [19,20]. In [19], a backstepping
adaptive controller is designed to overcome the overparameter-
ization problem in adaptive control of nonlinear discrete-time
systems. Output feedback adaptive control is dealt with, in
[20], for systems with unknown control gains.
Repetitive learning can be considered to be an extension of
the conventional adaptive mechanisms, to address the control
problem of periodically time-varying dynamical systems [21,
22]. The studies have been carried out under the assumption
that the uncertainty of the systems undertaken satisfies the
linear growing condition [21]. In [22], the linear growing
condition was examined, by which the boundedness and con-
vergence can be established for the systems with input-output
description. Key technical lemma plays a fundamental role for
analysis of discrete adaptive control systems[14], while the
slightly-modified iteration-domain one is found to be useful
for learning systems[23, 24].
In this paper, the method of adaptive backstepping RLC is
presented for discrete-time periodic systems in strict-feedback
form. The time-varying parameters are unknown but periodic
with known period. Least squares periodic learning algorithms
are used for estimation of the unknowns, where a lower
bound of control gain is introduced to avoid the potential
singularity. As for the RLC scheme, the reference trajectories
are not required to be repetition-independent, and no initial
repositioning is required. A key technical lemma in iteration
domain is given for the performance analysis of the proposed
RLC scheme.
II. P
ROBLEM FORMULATION AND PRELIMINARIES
Consider the following class of discrete-time periodic sys-
tems in strict-feedback form
x
j
(k, t + 1) = θ
T
j
(t)φ
j
(x
j
(k, t), t) + b
j
(t)x
j+1
(k, t)
j = 1, 2, ···, n −1
x
n
(k, t + 1) = θ
T
n
(t)φ
n
(x
n
(k, t), t) + b
n
(t)u(k, t)
y(k, t) = x
1
(k, t) (1)
where t = 0, 1, ···, N − 1 is the time, and N − 1 > n, k =
0, 1, ···, is the period index, x
j
(k, t) = [x
1
(k, t), x
2
(k, t), ···,
x
j
(k, t)]
T
is the vector of state , u(k, t) and y(k, t) are
the scalar system input and output, respectively, θ
j
(t), j =
1, 2, ···, n are the unknown parameter vectors, b
j
(t), j =
1, 2, ···, n, are unknown control gains, satisfying that b
j
(t) >
0, and φ
j
(x
j
(k, t), t), j = 1, 2, ···, n, represent the known
vector-valued nonlinear functions. In the sequel, for the sake
of convenience, let us denote φ
j
(k, t) = φ
j
(x
j
(k, t), t), j =
1, 2, ···, n.
The reference trajectory y
d
(k, t), is given a priori, satis-
fying y
d
(k, 0) = y
d
(k − 1, N). The control objective of this
paper is to find an input profile u(k, t) such that the system
2013 International Conference on Mechatronic Sciences, Electric Engineering and Computer (MEC)
Dec 20-22, 2013, Shenyang, China
978-1-4799-2565-0/13/$31.00 ©2013 IEEE
263
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