This paper presents a method for designing and quantifying the performance of feedback stochastic controls for nonlinear systems. The design makes use of the method of stochastic averaging to reduce the dimension of the state space and to derive the Ito^ stochastic differential equation for the response amplitude process. The moment equation of the amplitude process closed by the Rayleigh approximation is used as a means to characterize the transient performance of the feedback control. The steady state and transient response of the amplitude process are used as the design criteria for choosing the feedback control gains. Numerical examples are studied to demonstrate the performance of the control.

1.
Brockett, R. W., and Liberzon, D., 1998, “On Explicit Steady-State Solutions of Fokker-Planck Equations for a Class of Nonlinear Feedback Systems,” Proceedings of American Control Conference, pp. 264–268.
2.
Wang, R., and Yasuda, K., 1999, “Exact Stationary Response Solutions of Nine Classes of Nonlinear Stochastic Systems Under Stochastic Parametric and External Excitations,” Proceedings of 1999 ASME Design Engineering Technical Conferences, pp. 1–15.
3.
Caughey
,
T. K.
, and
Ma
,
F.
,
1982
, “
Exact Steady-State Solution of a Class of Non-Linear Stochastic Systems
,” International Journal of Non-Linear Mechanics,
Int. J. Non-Linear Mech.
,
17
, pp.
137
142
.
4.
Caughey
,
T. K.
, and
Ma
,
F.
,
1982
, “
Steady-State Response of a Class of Dynamical Systems to Stochastic Excitation
,”
ASME J. Appl. Mech.
,
49
, pp.
629
632
.
5.
Wang
,
R.
, and
Yasuda
,
K.
,
1997
, “
Exact Stationary Probability Density for Second Order Non-Linear Systems Under External White Noise Excitation
,”
J. Sound Vib.
205
, pp.
647
655
.
6.
Roberts
,
J. B.
, and
Spanos
,
P. D.
,
1986
, “
Stochastic Averaging: An Approximate Method of Solving Random Vibration Problems
,”
Int. J. Non-Linear Mech.
21
, pp.
111
134
.
7.
Lin, Y. K., 1967, Probabilistic Theory of Structural Dynamics, Robert E. Krieger Pub. Co, Malabar, Florida.
8.
Lin, Y. K., and Cai, G. Q., 1995, Probabilistic Structural Dynamics, Advanced Theory and Applications, McGraw-Hill, New York.
9.
Socha
,
L.
, and
Soong
,
T. T.
,
1991
, “
Linearization in Analysis of Nonlinear Stochastic Systems
,”
Appl. Mech. Rev.
44
, pp.
399
422
.
10.
Socha
,
L.
,
1994
, “
Some Remarks on Exact Linearization of a Class of Stochastic Dynamical Systems
,”
IEEE Trans. Autom. Control
,
39
, pp.
1980
1984
.
11.
Socha
,
L.
,
1999
, “
Statistical and Equivalent Linearization Techniques with Probability Density Criteria
,”
Journal of Theoretical and Applied Mechanics
,
37
, pp.
369
382
.
12.
Socha
,
L.
,
1999
, “
Probability Density Equivalent Linearization and Non-Linearization Techniques
,”
Arch. Mech.
,
51
, pp.
587
507
.
13.
Sun
,
J. Q.
, and
Hsu
,
C. S.
,
1989
, “
Cumulant-Neglect Closure Methods for Asymmetric Non-Linear Systems Driven by Gaussian White Noise
,”
J. Sound Vib.
,
135
, pp.
338
345
.
14.
Florchinger
,
P.
,
1997
, “
Feedback Stabilization of Affine in the Control Stochastic Differential Systems by the Control Lyapunov Function Methods
,”
SIAM J. Control Optim.
,
35
, pp.
500
511
.
15.
Bensoubaya
,
M.
,
Ferfera
,
A.
, and
Iggidr
,
A.
,
2000
, “
Jurdjevic-Quinn-type theorem for stochastic nonlinear control systems
,”
IEEE Trans. Autom. Control
,
45
, pp.
93
98
.
16.
Boulanger
,
C.
,
2000
, “
Stabilization of a Class of Nonlinear Stochastic Systems
,”
Nonlinear Analysis, Theory, Methods and Applications
,
41
, pp.
277
286
.
17.
Housner
,
G. W.
,
Bergman
,
L. A.
,
Caughey
,
T. K.
,
Chassiakos
,
A. G.
,
Claus
,
R. O.
,
Masri
,
S. F.
,
Skelton
,
R. E.
,
Soong
,
T. T.
,
Spencer
,
B. F.
, and
Yao
,
J. T. P.
,
1997
, “
Structural Control: Past, Present, and Future
,”
J. Eng. Mech.
,
123
, pp.
897
971
.
18.
Crespo, L. G., and Sun, J. Q., 2001, “Control of Nonlinear Stochastic Systems via Stationary Probability Distributions,” Probab. Eng. Mech., In Press.
19.
Bratus
,
A.
,
Dimentberg
,
M.
,
Iourtchenko
,
D.
, and
Noori
,
M.
,
2000
, “
Hybrid Solution Method for Dynamic Programming Equations for MDOF Stochastic Systems
,”
Dyn. Control
,
10
, pp.
107
116
.
20.
Khasminskii
,
R. Z.
,
1966
, “
A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides
,”
Theor. Probab. Appl.
,
11
, pp.
390
405
.
21.
Zhu
,
W. Q.
,
Ying
,
Z. G.
, and
Song
,
T. T.
,
2001
, “
Optimal Nonlinear Feedback Control Strategy for Randomly Excited Structural Systems
,”
Rev. Int. Hautes Temp. Refract.
,
24
, pp.
31
51
.
22.
Wu
,
W. F.
, and
Lin
,
Y. K.
,
1984
, “
Cumulant-Neglect Closure for Non-Linear Oscillators Under Parametric and External Excitations
,”
Int. J. Non-Linear Mech.
,
9
, pp.
349
362
.
23.
Sun
,
J. Q.
, and
Xu
,
Q.
,
1998
, “
Response Variance Reduction of a Nonlinear Mechanical System via Sliding Mode Control
,”
ASME J. Vibr. Acoust.
,
120
, pp.
801
805
.
24.
Sun
,
J. Q.
, and
Hsu
,
C. S.
,
1987
, “
Cumulant-Neglect Closure Methods for Nonlinear Systems Under Random Excitations
,”
ASME J. Appl. Mech.
,
54
, pp.
649
655
.
25.
Sun
,
J. Q.
, and
Hsu
,
C. S.
,
1989
, “
Generalized Cell Mapping Method in Nonlinear Random Vibration Based Upon Short-Time Gaussian Approximation
,”
ASME J. Appl. Mech.
,
57
, pp.
1018
1025
.
26.
Slotine, J. J., and Li, W., 1991, Applied Nonlinear Control, Prentice Hall, New York.
You do not currently have access to this content.