In this paper, the tracking-control problem of multiple-integrator (MI) systems is considered and investigated by combining Zhang dynamics (ZD) and gradient dynamics (GD). Several novel types of Zhang-gradient (ZG) controllers are proposed for the tracking control of MI systems (e.g., triple-integrator (TI) systems). As an example, the design processes of ZG controllers for TI systems with a linear output function (LOF) and/or a nonlinear output function (NOF) are presented. Besides, the corresponding theoretical analyses are elaborately given to guarantee the convergence performance of both z3g0 controllers (ZG controllers obtained by utilizing the ZD method thrice) and z3g1 controllers (ZG controllers obtained by utilizing the ZD method thrice and the GD method once) for TI systems. Numerical simulations concerning the tracking control of MI systems with different types of output functions are further performed to substantiate the feasibility and effectiveness of ZG controllers for tracking-control problems solving. Besides, comparative simulation results of the tracking control for MI systems with NOFs (e.g., y=cos(x1),y=x12+x22) substantiate that controllers of zmg1 type can resolve the singularity problem effectively with m being the times of using the ZD method.

References

References
1.
Rao
,
V. G.
, and
Bernstein
,
D. S.
,
2001
, “
Naive Control of the Double Integrator
,”
IEEE Control Syst. Mag.
,
21
(
5
), pp.
86
97
.
2.
Marchand
,
N.
,
2003
, “
Further Results on Global Stabilization for Multiple Integrators with Bounded Controls
,” 42nd
IEEE
Conference on Decision and Control
, Dec. 9–12, pp.
4440
4444
.
3.
Kaliora
,
G.
, and
Astolfi
,
A.
,
2004
, “
Nonlinear Control of Feedforward Systems With Bounded Signals
,”
IEEE Autom. Control
,
49
(
11
), pp.
1975
1990
.
4.
Zhou
,
B.
,
Duan
,
G.
, and
Li
,
Z.
,
2008
, “
On Improving Transient Performance in Global Control of Multiple Integrators System by Bounded Feedback
,”
Syst. Control. Lett.
,
57
(
10
), pp.
867
875
.
5.
Zhang
,
Y.
,
Zhai
,
K.
,
Wang
,
Y.
,
Chen
,
D.
, and
Peng
,
C.
,
2014
, “
Design and Illustration of ZG Controllers for Linear and Nonlinear Tracking Control of Double-Integrator System
,”
33rd Chinese Control Conference
, Nanjing, China, July 28–30, pp.
3462
3467
.
6.
Xu
,
D.
, and
Wang
,
W.
,
2010
, “
Research on Applications of Linear System Theory in Economics
,” 8th
IEEE
International Conference on Control and Automation
, Xiamen, China, June 9–11, pp.
1843
1847
.
7.
Mizoguchi
,
T.
, and
Yamada
,
I.
,
2014
, “
An Algebraic Translation of Cayley–Dickson Linear Systems and Its Applications to Online Learning
,”
IEEE Trans. Signal Proces.
,
62
(
6
), pp.
1438
1453
.
8.
Jarzebowska
,
E. M.
,
2008
, “
Advanced Programmed Motion Tracking Control of Nonholonomic Mechanical Systems
,”
IEEE Trans. Rob.
,
24
(
6
), pp.
1315
1328
.
9.
Li
,
W.
,
2010
, “
Tracking Control of Chaotic Coronary Artery System
,”
Int. J. Syst. Sci.
,
43
(
1
), pp.
21
30
.
10.
Zhang
,
Y.
,
Yu
,
X.
,
Yin
,
Y.
,
Peng
,
C.
, and
Fan
,
Z.
,
2014
, “
Singularity-Conquering ZG Controllers of Z2G1 Type for Tracking Control of the IPC System
,”
Int. J. Control
,
87
(
9
), pp.
1729
1746
.
11.
Zhang
,
Y.
,
Yang
,
Y.
,
Zhao
,
Y.
, and
Wen
,
G.
,
2013
, “
Distributed Finite-Time Tracking Control for Nonlinear Multi-Agent Systems Subject to External Disturbances
,”
Int. J. Control
,
86
(
1
), pp.
29
40
.
12.
Bialy
,
B. J.
,
Pasiliao
,
C. L.
,
Dinh
,
H. T.
, and
Dixon
,
W. E.
,
2014
, “
Tracking Control of Limit Cycle Oscillations in an Aero-Elastic System
,”
ASME J. Dyn. Syst., Meas., Control
,
136
(
6
), p.
064505
.
13.
Dontchev
,
A. L.
,
Krastanov
,
M. I.
,
Rockafellar
,
R. T.
, and
Veliov
,
V. M.
,
2014
, “
Neural Network-Based Tracking Control of Underactuated Autonomous Underwater Vehicles With Model Uncertainties
,”
ASME J. Dyn. Syst., Meas., Control
,
137
(
2
), p.
021004
.
14.
Hirschorn
,
R. M.
,
2001
, “
Output Tracking Through Singularities
,”
SIAM J. Control Optim.
,
40
(
4
), pp.
993
1010
.
15.
Kevin Jui
,
C. K.
, and
Qiao
,
S.
,
2005
, “
Path Tracking of Parallel Manipulators in the Presence of Force Singularity
,”
ASME J. Dyn. Syst., Meas., Control
,
127
(
4
), pp.
550
563
.
16.
Teel
,
A. R.
,
1992
, “
Global Stabilization and Restricted Tracking for Multiple Integrators With Bounded Controls
,”
Syst. Control Lett.
,
18
(
3
), pp.
165
171
.
17.
Marchand
,
N.
, and
Hably
,
A.
,
2005
, “
Global Stabilization of Multiple Integrators With Bounded Controls
,”
Automatica
,
41
(
12
), pp.
2147
2152
.
18.
Zhou
,
B.
, and
Duan
,
G.
,
2007
, “
Global Stabilization of Multiple Integrators Via Saturated Controls
,”
IET Control Theory A.
,
1
(
6
), pp.
1586
1593
.
19.
Zhou
,
B.
, and
Duan
,
G.
,
2008
, “
A Novel Nested Nonlinear Feedback Law for Global Stabilization of Linear Systems With Bounded Controls
,”
Int. J. Control
,
81
(
9
), pp.
1352
1363
.
20.
Zhou
,
B.
, and
Duan
,
G.
,
2009
, “
Global Stabilization of Linear Systems Via Bounded Controls
,”
Syst. Control Lett.
,
58
(
1
), pp.
54
61
.
21.
Zhang
,
Y.
, and
Yi
,
C.
,
2011
,
Zhang Neural Networks and Neural-Dynamic Method
,
Nova Science Publishers
,
New York
.
22.
Xiao
,
L.
, and
Zhang
,
Y.
,
2011
, “
Zhang Neural Network Versus Gradient Neural Network for Solving Time-Varying Linear Inequalities
,”
IEEE Trans. Neural Network
,
22
(
10
), pp.
1676
1684
.
23.
Zhang
,
Y.
,
Yi
,
C.
,
Guo
,
D.
, and
Zheng
,
J.
,
2011
, “
Comparison on Zhang Neural Dynamics and Gradient-Based Neural Dynamics for Online Solution of Nonlinear Time-Varying Equation
,”
Neural Comput. Appl.
,
20
(
1
), pp.
1
7
.
24.
Zhang
,
Y.
,
Yang
,
Y.
, and
Ruan
,
G.
,
2011
, “
Performance Analysis of Gradient Neural Network Exploited for Online Time-Varying Quadratic Minimization and Equality-Constrained Quadratic Programming
,”
Neurocomputing
,
74
(
10
), pp.
1710
1719
.
25.
Zhang
,
Y.
,
Yin
,
Y.
,
Wu
,
H.
, and
Guo
,
D.
,
2012
, “
Zhang Dynamics and Gradient Dynamics With Tracking-Control Application
,”
5th International Symposium on Computational Intelligence and Design
(
ISCID
), Hangzhou, China, Oct. 28–29, pp.
235
238
.
26.
Zhang
,
Y.
,
Li
,
F.
,
Yang
,
Y.
, and
Li
,
Z.
,
2012
, “
Different Zhang Functions Leading to Different Zhang-Dynamics Models Illustrated Via Time-Varying Reciprocal Solving
,”
Appl. Math. Modell.
,
36
(
9
), pp.
4502
4511
.
27.
Zhang
,
Z.
, and
Zhang
,
Y.
,
2013
, “
Design and Experimentation of Acceleration-Level Drift-Free Scheme Aided by Two Recurrent Neural Networks
,”
IET Control Theory Appl.
,
7
(
1
), pp.
25
42
.
28.
Abramowitz
,
M.
, and
Stegun
,
I. A.
,
1972
,
Handbook of Mathematical Function With Formulas, Graphs, and Mathematical Tables
, Dover Publications, New York.
29.
Chu
,
S.
, and
Metcalf
,
F.
,
1967
, “
On Gronwall's Inequality
,”
Proc. Amer. Math. Soc.
,
18
(
3
), pp.
439
440
.
30.
Zhang
,
Y.
,
Luo
,
F.
,
Yin
,
Y.
,
Liu
,
J.
, and
Yu
,
X.
,
2013
, “
Singularity-Conquering ZG Controller for Output Tracking of a Class of Nonlinear Systems
,”
32nd Chinese Control Conference
,
Xi'an, China
, pp.
477
482
.
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