Abstract

This paper formulates and solves new problems of inverse optimal stabilization and inverse optimal stabilization with gain assignment for nonlinear systems by Wiener processes. First, a theorem is developed to design inverse optimal stabilizers (i.e., covariance matrix multiplied by variance of Wiener processes), where it does not require to solve a Hamilton–Jacobi–Belman equation. Second, another theorem is developed to design inverse optimal stabilizers with gain assignment for nonlinear systems perturbed by both nonvanishing deterministic and stochastic (Wiener processes) disturbances without having to solve a Hamilton–Jacobi–Isaacs equation.

References

References
1.
Mao
,
X.
,
2007
,
Stochastic Differential Equations and Applications
, 2nd ed.,
Woodhead Publishing
,
Cambridge, UK
.
2.
Hasminskii
,
R. Z.
, 2012,
Stochastic Stability Differential Equations
, Springer, Berlin.
3.
Arnold
,
L.
,
Crauel
,
H.
, and
Wihstutz
,
V.
,
1983
, “
Stabilization of Linear Systems by Noise
,”
SIAM J. Control Optim.
,
21
(
3
), pp.
451
461
.10.1137/0321027
4.
Mao
,
X.
,
1994
, “
Stochastic Stabilisation and Destabilization
,”
Syst. Control Lett.
,
23
, pp.
279
290
.10.1016/0167-6911(94)90050-7
5.
Appleby
,
J. A. D.
,
Mao
,
X.
, and
Rodkina
,
A.
,
2008
, “
Stabilization and Destabilization of Nonlinear Differential Equations by Noise
,”
IEEE Trans. Autom. Control
,
53
(
3
), pp.
683
691
.10.1109/TAC.2008.919255
6.
Hoshino
,
K.
,
Nishimura
,
Y.
,
Yamashita
,
Y.
, and
Tsubakino
,
D.
,
2016
, “
Global Asymptotic Stabilization of Nonlinear Deterministic Systems Using Wiener Processes
,”
IEEE Trans. Autom. Control
,
61
(
8
), pp.
2318
2323
.10.1109/TAC.2015.2495622
7.
Huang
,
L.
,
2013
, “
Stochastic Stabilization and Destabilization of Nonlinear Differential Equations
,”
Syst. Control Lett.
,
62
, pp.
163
169
.10.1016/j.sysconle.2012.11.008
8.
Liu
,
L.
, and
Shen
,
Y.
,
2012
, “
Noise Suppresses Explosive Solutions of Differential Systems With Coefficients Satisfying the Polynomial Growth Condition
,”
Automatica
,
48
(
4
), pp.
619
624
.10.1016/j.automatica.2012.01.022
9.
Song
,
S.
, and
Zhu
,
Q.
,
2015
, “
Noise Suppresses Explosive Solutions of Differential Systems: A New General Polynomial Growth Condition
,”
J. Math. Anal. Appl.
,
431
(
1
), pp.
648
661
.10.1016/j.jmaa.2015.05.066
10.
Wu
,
F.
, and
Hu
,
S.
,
2009
, “
Suppression and Stabilisation of Noise
,”
Int. J. Control
,
82
(
11
), pp.
2150
2157
.10.1080/00207170902968108
11.
Zhu
,
S.
,
Sun
,
K.
,
Zhou
,
S.
, and
Shi
,
Y.
,
2017
, “
Stochastic Suppression and Almost Surely Stabilization of Non-Autonomous Hybrid System With a New General One-Sided Polynomial Growth Condition
,”
J. Franklin Inst.
,
354
(
15
), pp.
6550
6566
.10.1016/j.jfranklin.2017.08.007
12.
Hu
,
Y.
,
Wu
,
F.
, and
Huang
,
C.
,
2009
, “
Robustness of Exponential Stability of a Class of Stochastic Functional Differential Equations With Infinite Delay
,”
Automatica
,
45
(
11
), pp.
2577
2584
.10.1016/j.automatica.2009.07.007
13.
Wu
,
F.
, and
Hu
,
S.
,
2011
, “
Stochastic Suppression and Stabilization of Delay Differential Systems
,”
Int. J. Robust Nonlinear Control
,
21
(
5
), pp.
488
500
.10.1002/rnc.1606
14.
Yin
,
R.
,
Zhu
,
Q.
,
Shen
,
Y.
, and
Hu
,
S.
,
2016
, “
The Asymptotic Properties of the Suppressed Functional Differential System by Brownian Noise Under Regime Switching
,”
Int. J. Control
,
89
(
11
), pp.
2227
2239
.10.1080/00207179.2016.1152400
15.
Do
,
K. D.
,
2019
, “
Stabilization of Dynamical Systems by Wiener Processes
,”
J. Math. Anal. Appl.
, In Press.
16.
Sepulchre
,
R.
,
Jankovic
,
M.
, and
Kokotovic
,
P.
,
1997
,
Constructive Nonlinear Control
,
Springer
,
New York
.
17.
Krstic
,
M.
, and
Deng
,
H.
,
1998
,
Stabilization of Nonlinear Uncertain Systems
,
Springer
,
London
.
18.
Deng
,
H.
, and
Krstic
,
M.
,
1997
, “
Stochastic Nonlinear Stabilization—Part II: Inverse Optimality
,”
Syst. Control Lett.
,
32
(
3
), pp.
151
159
.10.1016/S0167-6911(97)00067-4
19.
Deng
,
H.
,
Krstic
,
M.
, and
Williams
,
R.
,
2001
, “
Stabilization of Stochastic Nonlinear Systems Driven by Noise of Unknown Covariance
,”
IEEE Trans. Autom. Control
,
46
(
8
), pp.
1237
1253
.10.1109/9.940927
20.
Do
,
K. D.
,
2015
, “
Global Inverse Optimal Stabilization of Stochastic Nonholonomic Systems
,”
Syst. Control Lett.
,
75
, pp.
41
55
.10.1016/j.sysconle.2014.11.003
21.
Do
,
K. D.
, and
Lucey
,
A. D.
,
2019
, “
Inverse Optimal Control of Evolution Systems and Its Application to Extensible and Shearable Slender Beams
,”
IEEE/CAA J. Autom. Sin.
,
6
(
2
), pp.
395
409
.10.1109/JAS.2019.1911381
22.
Do
,
K. D.
,
2019
, “
Inverse Optimal Control of Stochastic Systems Driven by Lévy Processes
,”
Automatica
,
107
, pp.
539
550
.10.1016/j.automatica.2019.06.016
23.
Chen
,
H.
,
2014
, “
Robust Stabilization for a Class of Dynamic Feedback Uncertain Nonholonomic Mobile Robots With Input Saturation
,”
Int. J. Control Autom. Syst.
,
12
(
6
), pp.
1216
1224
.10.1007/s12555-013-0492-z
24.
Chen
,
H.
,
Zhang
,
J.
,
Chen
,
B.
, and
Li
,
B.
,
2013
, “
Global Practical Stabilization for Non-Holonomic Mobile Robots With Uncalibrated Visual Parameters by Using a Switching Controller
,”
IMA J. Math. Control Inf.
,
30
(
4
), pp.
543
557
.10.1093/imamci/dns044
25.
Do
,
K. D.
,
2015
, “
Global Output-Feedback Path-Following Control of Unicycle-Type Mobile Robots: A Level Curve Approach
,”
Rob. Auton. Syst.
,
74
, pp.
229
242
.10.1016/j.robot.2015.07.019
26.
Do
,
K. D.
, and
Pan
,
J.
,
2009
,
Control of Ships and Underwater Vehicles
,
Springer, London
.
27.
Zhu
,
L.
,
Yang
,
T.
, and
Pan
,
J.
,
2019
, “
Design of Nonlinear Active Noise Control Earmuffs for Excessively High Noise Level
,”
J. Acoust. Soc. Am.
,
146
(
3
), pp.
1547
1555
.10.1121/1.5124472
28.
Khalil
,
H.
,
2002
,
Nonlinear Systems
,
Prentice Hall
, Upper Saddle River, NJ.
29.
Do
,
K. D.
,
2019
, “
Stochastic Control of Drill-Heads Driven by Lévy Processes
,”
Autom. Press
,
103
, pp.
36
45
.10.1016/j.automatica.2019.01.016
30.
Do
,
K. D.
, and
Nguyen
,
H. L.
,
2018
, “
Almost Sure Exponential Stability of Dynamical Systems Driven by Lévy Processes and Its Application to Control Design for Magnetic Bearings
,”
Int. J. Control
(in press).10.1080/00207179.2018.1482502
31.
Hardy
,
G.
,
Littlewood
,
J. E.
, and
Polya
,
G.
,
1989
,
Inequalities
, 2nd ed.,
Cambridge University Press
,
Cambridge, UK
.
32.
Krstic
,
M.
, and
Li
,
Z. H.
,
1998
, “
Inverse Optimal Design of Input-to-State Stabilizing Nonlinear Controllers
,”
IEEE Trans. Autom. Control
,
43
(
3
), pp.
336
350
.10.1109/9.661589
33.
Astrom
,
K. J.
,
1970
,
Introduction to Stochastic Control Theory
,
Academic Press
,
New York
.
You do not currently have access to this content.