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.
Issue Section:
Research Papers
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
.10.1137/03210274.
Mao
,
X.
, 1994
, “
Stochastic Stabilisation and Destabilization
,” Syst. Control Lett.
,
23
, pp. 279
–290
.10.1016/0167-6911(94)90050-75.
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.2008.9192556.
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.2015.24956227.
Huang
,
L.
, 2013
, “
Stochastic Stabilization and Destabilization of Nonlinear Differential Equations
,” Syst. Control Lett.
,
62
, pp. 163
–169
.10.1016/j.sysconle.2012.11.0088.
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.2012.01.0229.
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.2015.05.06610.
Wu
,
F.
, and
Hu
,
S.
, 2009
, “
Suppression and Stabilisation of Noise
,” Int. J. Control
,
82
(11
), pp. 2150
–2157
.10.1080/0020717090296810811.
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.2017.08.00712.
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.2009.07.00713.
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.160614.
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
.10.1080/00207179.2016.115240015.
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)00067-419.
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
.10.1109/9.94092720.
Do
,
K. D.
, 2015
, “
Global Inverse Optimal Stabilization of Stochastic Nonholonomic Systems
,” Syst. Control Lett.
,
75
, pp. 41
–55
.10.1016/j.sysconle.2014.11.00321.
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.2019.191138122.
Do
,
K. D.
, 2019
, “
Inverse Optimal Control of Stochastic Systems Driven by Lévy Processes
,” Automatica
,
107
, pp. 539
–550
.10.1016/j.automatica.2019.06.01623.
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-z24.
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/dns04425.
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.2015.07.01926.
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
.10.1121/1.512447228.
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.2019.01.01630.
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).10.1080/00207179.2018.148250231.
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
.10.1109/9.66158933.
Astrom
,
K. J.
, 1970
, Introduction to Stochastic Control Theory
,
Academic Press
,
New York
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