Abstract
This paper investigates the backstepping control problem for nonlinear pure-feedback systems with time-varying delays. A virtual controller is designed to counteract the effects caused by the state perturbation of time delay, and improve the stability of the system. The assumption on delay-dependent nonlinearities is further relaxed by a backstepping auxiliary controller and a Lyapunov–Krasovskii functional. A suitable coordinate transformation is introduced to reduce the complexity of computation caused by nonaffine structures. The globally uniform boundedness of the closed-loop signals and the asymptotical stability of the state are proved by Lyapunov–Krasovskii stability theory. Finally, the effectiveness of our method is demonstrated by two illustrations.
Issue Section:
Technical Briefs
References
1.
Cao
,
L.
,
Xiao
,
B.
, and
Golestani
,
M.
, 2020
, “
Robust Fixed-Time Attitude Stabilization Control of Flexible Spacecraft With Actuator Uncertainty
,” Nonlinear Dyn.
,
100
(3
), pp. 2505
–2515
.10.1007/s11071-020-05596-52.
Xiao
,
B.
,
Cao
,
L.
,
Xu
,
S.
, and
Liu
,
L.
, 2020
, “
Robust Tracking Control of Robot Manipulators With Actuator Faults and Joint Velocity Measurement Uncertainty
,” IEEE/ASME Trans. Mechatronics
,
25
(3
), pp. 1354
–1365
.10.1109/TMECH.2020.29751173.
Zhang
,
S.
,
Cui
,
W.
, and
Alsaadi
,
F. E.
, 2019
, “
Adaptive Backstepping Control Design for Uncertain Non-Smooth Strict Feedback Nonlinear Systems With Time-Varying Delays
,” Int. J. Control, Autom. Syst.
,
17
(9
), pp. 2220
–2233
.10.1007/s12555-019-0046-04.
Li
,
Z.
,
Li
,
T.
,
Feng
,
G.
,
Zhao
,
R.
, and
Shan
,
Q.
, 2018
, “
Neural Network-Based Adaptive Control for Pure-Feedback Stochastic Nonlinear Systems With Time-Varying Delays and Dead-Zone Input
,” IEEE Trans. Syst., Man, Cybern. Syst.
(in press).10.1109/TSMC.2018.28724215.
Yoshimura
,
T.
, 2015
, “
Design of a Simplified Adaptive Fuzzy Backstepping Control for Uncertain Discrete-Time Nonlinear Systems
,” Int. J. Syst. Sci.
,
46
(5
), pp. 763
–775
.10.1080/00207721.2014.9734686.
Ma
,
J.
,
Park
,
J. H.
,
Xu
,
S.
,
Cui
,
G.
, and
Yang
,
Z.
, 2020
, “
Command-Filter-Based Adaptive Tracking Control for Nonlinear Systems With Unknown Input Quantization and Mismatching Disturbances
,” Appl. Math. Comput.
,
377
, p. 125161
.10.1016/j.amc.2020.1251617.
Chen
,
M.
,
Ge
,
S.
, and
Ren
,
B.
, 2011
, “
Adaptive Tracking Control of Uncertain MIMO Nonlinear Systems With Input Constraints
,” Automatica
,
47
(3
), pp. 452
–465
.10.1016/j.automatica.2011.01.0258.
Zhang
,
T.
,
Zhu
,
Q.
, and
Yang
,
Y.
, 2012
, “
Adaptive Neural Control of Non-Affine Pure-Feedback Non-Linear Systems With Input Nonlinearity and Perturbed Uncertainties
,” Int. J. Syst. Sci.
,
43
(4
), pp. 691
–706
.10.1080/00207721.2010.5190609.
Morawiec
,
M.
,
Strankowski
,
P.
,
Lewicki
,
A.
,
Guzinski
,
J.
, and
Wilczynski
,
F.
, 2020
, “
Feedback Control of Multiphase Induction Machines With Backstepping Technique
,” IEEE Trans. Ind. Electron.
,
67
(6
), pp. 4305
–4314
.10.1109/TIE.2019.293123610.
Zhou
,
J.
,
Wen
,
C.
,
Wang
,
W.
, and
Yang
,
F.
, 2019
, “
Adaptive Backstepping Control of Nonlinear Uncertain Systems With Quantized States
,” IEEE Trans. Autom. Control
,
64
(11
), pp. 4756
–4763
.10.1109/TAC.2019.290693111.
Krstic
,
M.
,
Kokotovic
,
P. V.
, and
Kanellakopoulos
,
I.
, 1995
, Nonlinear and Adaptive Control Design
, John Wiley and Sons
, New York
.12.
Polycarpou
,
M. M.
, 1996
, “
Stable Adaptive Neural Control Scheme for Nonlinear Systems
,” IEEE Trans. Autom. Control
,
41
(3
), pp. 447
–451
.10.1109/9.48664813.
Wang
,
M.
,
Wang
,
X.
,
Chen
,
B.
, and
Tong
,
S.
, 2007
, “
Robust Adaptive Fuzzy Tracking Control for a Class of Strict-Feedback Nonlinear Systems Based on Backstepping Technique
,” J. Control Theory Appl.
,
5
(3
), pp. 317
–322
.10.1007/s11768-005-5298-y14.
Zhang
,
T.
,
Ge
,
S.
, and
Hang
,
C.
, 2000
, “
Adaptive Neural Network Control for Strict-Feedback Nonlinear Systems Using Backstepping Design
,” Automatica
,
36
(12
), pp. 1835
–1846
.10.1016/S0005-1098(00)00116-315.
Tong
,
S.
,
Li
,
Y.
,
Li
,
Y.
, and
Liu
,
Y.
, 2011
, “
Observer-Based Adaptive Fuzzy Backstepping Control for a Class of Stochastic Nonlinear Strict-Feedback Systems
,” IEEE Trans. Syst., Man, Cybern., Part B
,
41
(6
), pp. 1693
–1704
.10.1109/TSMCB.2011.215926416.
Tong
,
S.
, and
Li
,
Y.
, 2012
, “
Adaptive Fuzzy Output Feedback Tracking Backstepping Control of Strict-Feedback Nonlinear Systems With Unknown Dead Zones
,” IEEE Trans. Fuzzy Syst.
,
20
(1
), pp. 168
–180
.10.1109/TFUZZ.2011.217118917.
Li
,
Y.
,
Tong
,
S.
, and
Li
,
T.
, 2015
, “
Composite Adaptive Fuzzy Output Feedback Control Design for Uncertain Nonlinear Strict-Feedback Systems With Input Saturation
,” IEEE Trans. Cybern.
,
45
(10
), pp. 2299
–2308
.10.1109/TCYB.2014.237064518.
Li
,
H.
,
Wang
,
L.
,
Du
,
H.
, and
Boulkroune
,
A.
, 2017
, “
Adaptive Fuzzy Backstepping Tracking Control for Strict-Feedback Systems With Input Delay
,” IEEE Trans. Fuzzy Syst.
,
25
(3
), pp. 642
–652
.10.1109/TFUZZ.2016.256745719.
Park
,
J. H.
,
Park
,
T. S.
, and
Kim
,
S. H.
, 2019
, “
Approximation-Free Output-Feedback Non-Backstepping Controller for Uncertain SISO Nonautonomous Nonlinear Pure-Feedback Systems
,” Mathematics
,
7
(5
), p. 456
.10.3390/math705045620.
Gao
,
T.
,
Liu
,
Y.
,
Liu
,
L.
, and
Li
,
D.
, 2018
, “
Adaptive Neural Network-Based Control for a Class of Nonlinear Pure-Feedback Systems With Time-Varying Full State Constraints
,” IEEE/CAA J. Autom. Sin.
,
5
(5
), pp. 923
–933
.10.1109/JAS.2018.751119521.
Li
,
Y.
, and
Yang
,
G.
, 2018
, “
Adaptive Neural Control of Pure-Feedback Nonlinear Systems With Event-Triggered Communications
,” IEEE Trans. Neural Networks Learn. Syst.
,
29
(12
), pp. 6242
–6251
.10.1109/TNNLS.2018.282814022.
Shi
,
X.
,
Xu
,
S.
,
Li
,
Y.
,
Chen
,
W.
, and
Chu
,
Y.
, 2017
, “
Robust Adaptive Control of Strict-Feedback Nonlinear Systems With Unmodelled Dynamics and Time-Varying Delays
,” Int. J. Control
,
90
(2
), pp. 334
–347
.10.1080/00207179.2016.117881023.
Niculescu
,
S. I.
, 2001
, Delay Effects on Stability: A Robust Control Approach
,
Springer Science & Business Media
, Berlin
.24.
Yi
,
Y.
,
Guo
,
L.
, and
Wang
,
H.
, 2009
, “
Adaptive Statistic Tracking Control Based on Two-Step Neural Networks With Time Delays
,” IEEE Trans. Neural Networks
,
20
(3
), pp. 420
–429
.25.
Ma
,
Q.
, and
Xu
,
S.
, 2020
, “
Consensus Switching of Second-Order Multiagent Systems With Time Delay
,” IEEE Trans. Cybern.
(in press).10.1109/TCYB.2020.301144826.
Kumar Jain
,
A.
, and
Bhasin
,
S.
, 2019
, “
A Robust Delay Compensator for Uncertain Nonlinear Systems With Unknown Time-Varying Input Delay
,” ASME J. Dyn. Syst. Meas. Control
,
141
(11
), p. 114501.10.1115/1.404417927.
Ren
,
B.
,
Ge
,
S.
,
Lee
,
T. H.
, and
Su
,
C.
, 2009
, “
Adaptive Neural Control for a Class of Nonlinear Systems With Uncertain Hysteresis Inputs and Time-Varying State Delays
,” IEEE Trans. Neural Networks
,
20
(7
), pp. 1148
–1164
.10.1109/TNN.2009.201695928.
Khanesar
,
M. A.
,
Kaynak
,
O.
,
Yin
,
S.
, and
Gao
,
H.
, 2015
, “
Adaptive Indirect Fuzzy Sliding Mode Controller for Networked Control Systems Subject to Time-Varying Network-Induced Time Delay
,” IEEE Trans. Fuzzy Syst.
,
23
(1
), pp. 205
–214
.10.1109/TFUZZ.2014.236254929.
Wang
,
C.
, and
Lin
,
Y.
, 2015
, “
Adaptive Dynamic Surface Control for MIMO Nonlinear Time-Varying Systems With Prescribed Tracking Performance
,” Int. J. Control
,
88
(4
), pp. 832
–843
.10.1080/00207179.2014.98186030.
Wang
,
Y.
,
Xu
,
L.
, and
Wu
,
H.
, 2018
, “
Adaptive Robust Backstepping Output Tracking Control for a Class of Uncertain Nonlinear Systems Using Neural Network
,” ASME J. Dyn. Syst. Meas. Control
,
140
(7
), p. 071014
.10.1115/1.403915131.
Wang
,
M.
,
Ge
,
S.
, and
Hong
,
K.
, 2010
, “
Approximation-Based Adaptive Tracking Control of Pure-Feedback Nonlinear Systems With Multiple Unknown Time-Varying Delays
,” IEEE Trans. Neural Networks
,
21
(11
), pp. 1804
–1816
.10.1109/TNN.2010.207371932.
Zhang
,
T.
, and
Zhu
,
Q.
, 2011
, “
Adaptive Neural Network Control of Nonlinear Time-Varying Delay Systems
,” Control Decision
,
26
(2
), pp. 263
–270
.33.
Wang
,
J.
, and
Sun
,
C.
, 2018
, “
Delay-Dependent Exponential Stabilization for Linear Distributed Parameter Systems With Time-Varying Delay
,” ASME J. Dyn. Syst. Meas. Control
,
140
(5
), p. 051003
.10.1115/1.403837434.
Gao
,
D.
,
Sun
,
Z. Q.
, and
Liu
,
J.
, 2012
, “
Dynamic Inversion Control for a Class of Pure-Feedback Systems
,” Asian J. Control
,
14
(2
), pp. 605
–611
.10.1002/asjc.33135.
Liu
,
Y.
, 2014
, “
Dynamic Surface Control for Pure-Feedback Nonlinear Systems
,” Control Theory Appl.
,
31
(9
), pp. 1262
–1267
.10.7641/CTA.2014.3110136.
Jia
,
F.
,
Wang
,
X.
, and
Zhou
,
X.
, 2019
, “
Robust Adaptive Prescribed Performance Control for a Class of Nonlinear Pure-Feedback Systems
,” Int. J. Robust Nonlinear Control
,
29
(12
), pp. 3971
–3987
.10.1002/rnc.459237.
Zhang
,
Z.
,
Xu
,
S.
, and
Chu
,
Y.
, 2010
, “
Adaptive Stabilisation for a Class of Non-Linear State Time-Varying Delay Systems With Unknown Time-Delay Bound
,” IET Control Theory Appl.
,
4
(10
), pp. 1905
–1913
.10.1049/iet-cta.2009.041438.
Liu
,
Y.
, 2016
, “
Adaptive Tracking Control for a Class of Uncertain Pure-Feedback Systems
,” Int. J. Robust Nonlinear Control
,
26
(5
), pp. 1143
–1154
.10.1002/rnc.335039.
Ioannou
,
P. A.
, and
Sun
,
J.
, 2012
, Robust Adaptive Control
, Prentice Hall, Upper Saddle River
, NJ
.40.
Min
,
H.
,
Xu
,
S.
,
Gu
,
J.
,
Zhang
,
B.
, and
Zhang
,
Z.
, 2020
, “
Further Results on Adaptive Stabilization of High-Order Stochastic Nonlinear Systems Subject to Uncertainties
,” IEEE Trans. Neural Networks Learn. Syst.
,
31
(1
), pp. 225
–234
.10.1109/TNNLS.2019.290033941.
Ding
,
Z.
, 2013
, Nonlinear and Adaptive Control Systems
,
Institution of Engineering and Technology
, London
.42.
Zhong
,
Q.
, 2006
, Robust Control of Time-Delay Systems
,
Springer Science and Business Media
, London
.Copyright © 2021 by ASME
You do not currently have access to this content.
