Comments on Discrete-Time Adaptive Backstepping Nonlinear Control via High-Order Neural Networks

Abstract

The purpose of this comment is to point out some mistakes in the above paper. It is shown that the main results of the paper cannot stand in general. Also, it is pointed out that after some corrections, the proposed control algorithm is still applicable to a more simple system. For simplicity, all the symbols in this comment are the same as those in the above paper.

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IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 5, MAY 2009897
Comments and Replies
Comments on “Discrete-Time Adaptive Backstepping
Nonlinear Control via High-Order Neural Networks”
Weisheng Chen
Abstract—The purpose of this comment is to point out some mistakes in
the above paper. It is shown that the main results of the paper cannot stand
in general. Also, it is pointed out that after some corrections, the proposed
control algorithm is still applicable to a more simple system. For simplicity,
all the symbols in this comment are the same as those in the above paper.
Index Terms—Backstepping, discrete-time systems, high-order neural
networks.
I. MISTAKES
In [1], the authors consider the tracking control problem for a class
of discrete-time systems with the block strict feedback form [1, eq.
(12)], and claim that the system (12) as a one-step ahead predictor can
be transformed into an equivalent maximum ?-step ahead one [1, eq.
(13)].
Unfortunately, this transformation cannot stand. The system (13) in
[1] should be
???? ? ?? ? ??????????????
? ?????????????????? ? ? ? ?? ? ???? ? ? ? ??
...
?????? ? ?? ? ??????????????????
? ?????????????????????? ? ?? ? ?????? ? ??
???? ? ?? ? ????????? ???????????? ? ?????
???? ? ?????
(1)
where ????? ? ?????????????????????????????????????????
???????????? ? ? ??????? ? ? with
????? ? ? ? ? ? ?? ? ??????????? ? ??????????? ? ? ? ? ? ???
????? ? ? ? ? ? ?? ? ??????????? ? ??????????? ? ? ? ? ? ???
...
????????? ? ?? ? ??????????????????? ? ????
It should be emphasized that this mistake is not a type error. There are
some essential differences between system (1) and system (13) in [1].
For example, in [1], ????? and ????? are functions of partial states
? ?????, but it can be seen from (1) that in fact, they should be functions
of all states ???? and disturbances ?????, which will directly affect
Manuscript received August 21, 2008; revised January 19, 2009; accepted
February 24, 2009. First published March 27, 2009; current version published
May 01, 2009. This work was supported by the National Natural Science Foun-
dation of China under Grant 60804021.
The author is with the Department of Applied Mathematics, Xidian Univer-
sity, Xi’an 710071, China (e-mail: wshchen@126.com).
Digital Object Identifier 10.1109/TNN.2009.2016757
the number of neural network (NN) inputs and destroy the NN approx-
imation capability due to the appearance of disturbances ?????. Since
the control design procedure is based on system (13) in [1], the ob-
tained control algorithm for system (12) in [1] cannot stand.
For clarity, we consider a single-input–single-output second-order
system to demonstrate the error in this transformation. Consider the
system
???? ? ?? ? ???? ?????? ? ???? ???????????? ?????
???? ? ?? ? ????????? ????????????? ?????
???? ? ?????
(2)
where ???? ? ??????????????. From the first equation in (2), we
have
???? ? ??
? ???? ???? ? ??? ? ???? ???? ? ??????? ? ?? ? ???? ? ??
? ?????? ?????? ? ???? ???????????? ?????
??????? ??????????? ????????????????? ???????????????
? ??????????????? ?????????????????? ? ?? ? ???? ? ??
(3)
where
?????????????? ? ??????? ?????? ? ???? ???????????? ??????
?????????????? ? ??????? ?????? ? ???? ???????????? ???????
Thatis, system(2)canbe transformed intoamaximumtwo-step-ahead
predictor as follows:
????????????????????????????????????? ???????????????
???? ? ???????????? ????????????? ?????
???????????
(4)
It is obvious that system (4) has the form (1) instead of [1, eq. (13)].
Moreover, even provided that [1, eq. (14)] holds, [1, eq. (15)] seems
to be also wrong. It should be
?????? ?? ???? ? ? ? ?? ? ???? ?????????? ? ???
?????? ?? ???? ? ? ? ?? ? ???? ?????????????
...
???????? ?? ???? ? ?? ? ?????? ?????????????????
????? ? ?????????????????
???? ? ??????
(5)
This is also not a type error, and it makes the error in [1, eq. (24)]
needed to be redefined. So, the controller must be reconstructed
correspondingly.
II. CORRECTIONS
In this section, motivated by [2]–[4], we will point out that the main
result cannot hold for system (12) in [1], but after some corrections,
the control algorithm of [1] is still applicable to a more simple system
1045-9227/$25.00 © 2009 IEEE
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898 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 5, MAY 2009
in which the disturbances appear only in the last equation. In this case,
system (12) in [1] should be updated by
??????? ? ???? ??????????? ???????? ??????
???? ? ?? ? ????????? ???????????? ? ?????
???? ? ?????
? ? ???????????
(6)
which as a one-step-ahead predictor, and it can be transformed into an
equivalent maximum ?-step-ahead one as follows:
???? ? ?? ? ????????? ???????????? ? ? ? ??
...
?????? ? ?? ? ??????????? ?????????????? ? ??
???? ? ?? ? ????????? ????????????? ?????
???? ? ??????
(7)
That is, systems (13) and (14) in [1] should be replaced by the above
system. Then, the control algorithm is designed based on system (7),
and some adjustments must be done correspondingly. For example,
? ??????? ? ??????? ? ? in [1, eq. (15)] should be updated as ????,
andtheNNinputs?????in[1,eq.(16)]shouldbe correctedas????? ?
?????????? ? ????and ????? ? ????????????????? ? ?????????.
For the more detailed design procedure, the reader may be referred to
[2]–[4].
III. CONCLUSION
In this comment, we point out some mistakes in [1] and demonstrate
these mistakes by a second-order system. We also give some correc-
tions to improve the main results.
REFERENCES
[1] A. Y. Alanis, E. N. Sanchez, and A. G. Loukianov, “Discrete-time
adaptive backstepping nonlinear control via high-order neural net-
works,” IEEE Trans. Neural Netw., vol. 18, no. 4, pp. 1185–1195, Jul.
2007.
[2] S. S. Ge, G. Y. Li, and T. H. Lee, “Adaptive NN control for a class of
strict-feedback discrete-time nonlinear systems,” Automatica, vol. 39,
no. 5, pp. 807–819, 2003.
[3] S. S. Ge, G. Y. Li, and T. H. Lee, “Correction to ‘adaptive NN control
for a class of strict-feedback discrete-time nonlinear systems’,” Auto-
matica, vol. 44, no. 7, pp. 1930–1931, 2008.
[4] S. S. Ge, J. Zhang, and T. H. Lee, “Adaptive neural network control
for a class of MIMO nonlinear systems with disturbances in discrete-
time,” IEEE Trans. Syst. Man Cybern. B, Cybern., vol. 34, no. 4, pp.
1630–1645, Aug. 2004.
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