A wavelet domain forward differential Ricatti formulation is proposed in this paper for control of linear time-varying (LTV) systems. The control feedback gains derived are time-frequency dependent, and they can be appropriately tuned for each wavelet scale or frequency band. The gains in the proposed forward formulation are functions of the present and past states and hence lead to a nonlinear controller. This nonlinear controller does not require information or approximation about future system matrices. The proposed controller is suitable for systems with time-varying (TV) system matrices and also for controlling transient dynamics. The performance of the proposed controller is compared with two other control strategies, namely, a TV linear quadratic regulator (LQR) based on a backward formulation of the differential Ricatti equation (DRE) and a multiscale wavelet-LQR controller based on asymptotic assumptions. Two numerical examples demonstrate promising results on the performance of the controller.

References

References
1.
Den Hartog
,
J. P.
,
1956
,
Mechanical Vibrations
,
McGraw-Hill Book Company
,
New York
.
2.
Ibrahim
,
R. A.
,
1985
,
Parametric Random Vibration
,
Wiley
,
New York
.
3.
Dimentberg
,
M. F.
,
1988
,
Statistical Dynamics of Non-Linear and Linear Time-Varying Systems
,
Research Studies Press
,
Taunton, UK
.
4.
Nagarajaiah
,
S.
,
2009
, “
Adaptive Passive, Semiactive, Smart Tuned Mass Dampers: Identification and Control Using Empirical Mode Decomposition, Hilbert Transform, and Short-Term Fourier Transform
,”
J. Struct. Control Health Monit.
,
16
(
7–8
), pp.
800
841
.
5.
Vu
,
L.
, and
Liberzon
,
D.
,
2011
, “
Supervisory Control of Uncertain Linear Time-Varying Systems
,”
IEEE Trans. Autom. Control
,
56
(
1
), pp.
27
42
.
6.
Pan
,
Y.
,
Kumar
,
K. D.
,
Liu
,
G.
, and
Furuta
,
K.
,
2009
, “
Design of Variable Structure Control System With Nonlinear Time-Varying Sliding Sector
,”
IEEE Trans. Autom. Control
,
54
(
8
), pp.
1981
1986
.
7.
Mohammadpour
,
J.
, and
Scherer
,
C. W.
,
2012
,
Control of Linear Parameter Varying Systems With Applications
,
Springer-Verlag
,
New York
.
8.
Becker
,
G.
, and
Packard
,
A. K.
,
1994
, “
Robust Performance of Linear Parametrically Varying Systems Using Parametrically-Dependent Linear Feedback
,”
Syst. Control. Lett.
,
23
(
3
), pp.
205
215
.
9.
Briat
,
C.
,
2015
,
Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering and Control
,
Springer-Verlag
,
Berlin
.
10.
Wu
,
F.
, and
Grigoriadis
,
K. M.
,
2001
, “
LPV Systems With Parameter-Varying Time Delays: Analysis and Control
,”
Automatica
,
37
(
2
), pp.
221
229
.
11.
Kalman
,
R. E.
,
1964
, “
When is a Linear Control System Optimal?
ASME J. Basic Eng.
,
86
(
1
), pp.
51
60
.
12.
Kwakernaak
,
H.
, and
Sivan
,
R.
,
1972
,
Linear Optimal Control Systems
, Vol.
1
,
Wiley
,
New York
.
13.
Chen
,
M. S.
, and
Kao
,
C.
,
1997
, “
Control of Linear Time-Varying Systems Using Forward Riccati Equation
,”
ASME J. Dyn. Syst. Meas. Control
,
119
(
3
), pp.
536
540
.
14.
Staszewski
,
W. J.
,
1998
, “
Wavelet Based Compression and Feature Selection for Vibration Analysis
,”
J. Sound Vib.
,
211
(
5
), pp.
735
760
.
15.
Goggins
,
J.
,
Broderick
,
B. M.
,
Basu
,
B.
, and
Elghazouli
,
A. Y.
,
2007
, “
Investigation of the Seismic Response of Braced Frames Using Wavelet Analysis
,”
J. Struct. Control Health Monit.
,
14
(
4
), pp.
627
648
.
16.
Zhong
,
S.
, and
Oyadiji
,
S. O.
,
2011
, “
Crack Detection in Simply Supported Beams Using Stationary Wavelet Transform of Modal Data
,”
J. Struct. Control Health Monit.
,
18
(
2
), pp.
169
190
.
17.
Beskhyroun
,
S.
,
Oshima
,
T.
, and
Mikami
,
S.
,
2010
, “
Wavelet-Based Technique for Structural Damage Detection
,”
J. Struct. Control Health Monit.
,
17
(
5
), pp.
473
494
.
18.
He
,
H. X.
, and
Yan
,
W. M.
,
2007
, “
Structural Damage Detection With Wavelet Support Vector Machine: Introduction and Applications
,”
J. Struct. Control Health Monit.
,
14
(
1
), pp.
162
176
.
19.
Jiang
,
X.
, and
Mahadevan
,
S.
,
2008
, “
Bayesian Wavelet Methodology for Structural Damage Detection
,”
J. Struct. Control Health Monit.
,
15
(
7
), pp.
974
991
.
20.
Staszewski
,
W. J.
,
1998
, “
Identification of Non-Linear Systems Using Multi-Scale Ridges and Skeletons of the Wavelet Transform
,”
J. Sound. Vib.
,
214
(
4
), pp.
639
658
.
21.
Sureshbabu
,
N.
, and
Farrell
,
J. A.
,
1999
, “
Wavelet-Based System Identification for Nonlinear Control
,”
IEEE Trans. Autom. Control
,
44
(
2
), pp.
412
417
.
22.
Min
,
Z. H.
, and
Sun
,
L. M.
,
2013
, “
Wavelet-Based Structural Modal Parameter Identification
,”
J. Struct. Control Health Monit.
,
20
(
2
), pp.
121
138
.
23.
Kijewski
,
T.
, and
Kareem
,
A.
,
2003
, “
Wavelet Transforms for System Identification in Civil Engineering
,”
Comput. Aided Civ. Infrastruct. Eng.
,
18
(
5
), pp.
339
355
.
24.
Hou
,
Z.
,
Noori
,
M.
, and
St-Amand
,
R.
,
2000
, “
Wavelet-Based Approach for Structural Damage Detection
,”
J. Eng. Mech. ASCE
,
126
(
7
), pp.
677
683
.
25.
Basu
,
B.
, and
Nagarajaiah
,
S.
,
2008
, “
A Wavelet-Based Time-Varying Adaptive LQR Algorithm for Structural Control
,”
Eng. Struct.
,
30
(
9
), pp.
2470
2477
.
26.
Girard
,
A.
,
2006
, “
Towards a Multiresolution Approach to Linear Control
,”
IEEE Trans. Autom. Control
,
51
(
8
), pp.
1261
1270
.
27.
Basu
,
B.
, and
Nagarajaiah
,
S.
,
2010
, “
Multiscale Wavelet-LQR Controller for Linear Time Varying Systems
,”
J. Eng. Mech. ASCE
,
136
(
9
), pp.
1143
1151
.
28.
Basu
,
B.
, and
Gupta
,
V. K.
,
1999
, “
On Equivalent Linearization Using Wavelet Transform
,”
ASME J. Vib. Acoust.
,
121
(
4
), pp.
429
432
.
29.
Basu
,
B.
, and
Gupta
,
V. K.
,
2000
, “
Stochastic Seismic Response of Single-Degree-of-Freedom Systems Through Wavelets
,”
Eng. Struct.
,
22
(
12
), pp.
1714
1722
.
30.
Chen
,
C. T.
,
1984
,
Linear System Theory and Design, Holt, Rinehart, and Winston
,
New York
.
31.
Chan
,
Y. T.
,
1995
,
Wavelet Basics
,
Kluwer Academic Publisher
,
Boston, MA
.
32.
Daubechies
,
I.
,
1992
,
Ten Lectures on Wavelets
,
Society of Industrial and Applied Mathematics (SIAM)
,
Philadelphia, PA
.
You do not currently have access to this content.