We study the control design problem for uncertain nonlinear systems. A new matching condition is presented. The main idea is to explore the route through which the (worst case) uncertainty may affect the stability. This route is then used to establish the new matching condition. Compared with the previous case, the current matching condition prescribes the route nonlinearly while the early matching condition is a special case of the linear description. A class of robust controls, which guarantees practical stability, can be designed based on this new matching condition. The size of the uniform ultimate boundedness ball can be made arbitrarily small by an appropriate choice of a design parameter.

Issue Section:
Technical Papers
Topics:
Design, Robust control, Stability, Nonlinear systems, Uncertainty
1.
Barmish
B. R.
,
1985
, “
Necessary and Sufficient Conditions for Quadratic Stabilizability of an Uncertain System
,”
Journal of Optimization Theory and Applications
, Vol.
46
, pp.
399
–
408
.
2.
Barmish
B. R.
,
Corless
M.
, and
Leitmann
,
1983
, “
A New Class of Stabilizing Controllers for Uncertain Dynamical Systems
,”
SIAM Journal of Control and Optimization
, Vol.
21
, pp.
246
–
255
.
3.
Barmish
B. R.
, and
Leitmann
G.
,
1982
, “
On Ultimate Boundedness Control of Uncertain Systems in the Absence of Matching Conditions
,”
IEEE Transactions on Automatic Control
, Vol.
27
, pp.
153
–
157
.
4.
Chen, Y. H., 1994, “Robust Control Design for a Class of Mismatched Uncertain Nonlinear Systems,” American Control Conference, Baltimore, pp. 796–800.
5.
Chen, Y. H., and J. S. Chen, 1991, “Robust Control of Uncertain Systems with Time-Varying Uncertainty—An Optimization Setting,” Mechanics and Control, J. M. Skowronski, H. Flasher, and R. S. Guttalu, eds., pp. 97–114, Springer-Verlag, New York.
6.
Chen
Y. H.
, and
Leitmann
G.
,
1987
, “
Robustness of Uncertain Systems in the Absence of Matching Assumptions
,”
International Journal of Control
, Vol.
45
, pp.
1527
–
1542
.
7.
Corless
M. J.
, and
Leitmann
G.
,
1981
, “
Continuous State Feedback Guaranteeing Uniform Ultimate Boundedness for Uncertain Dynamic Systems
,”
IEEE Transactions on Automatic Control
, Vol.
26
, pp.
1139
–
1144
.
8.
Corless, M., and Leitmann, G., 1990, “Deterministic Control of Uncertain Systems: A Lyapunov Theory Approach,” Variable Structure Control Systems, A. Zinober, ed., IEE Control Engineering Series 40, Peter Peregrinus Ltd., London.
9.
Ehrhard
P.
, and
Muller
U.
,
1990
, “
Dynamic Behavior of Natural Convection in a Single-Phase Loop
,”
Journal of Fluid Mechanics
, Vol.
217
, pp.
487
–
518
.
10.
Freeman, R. A., and Kokotovic, P. V., 1992, “A New Lyapunov Function for the Backstepping of Softer Robust Nonlinear Control Laws,” Nonlinear Control System Design Symposium, Bordeaux, France.
11.
Gavel
D. T.
, and
Siljak
D. D.
,
1989
, “
Decentralized Adaptive Control: Structural Conditions for Stability
,”
IEEE Transactions on Automatic Control
, Vol.
129
, pp.
413
–
426
.
12.
Geromel
J. C.
,
Peres
P. L.
, and
Bernussou
J.
,
1991
, “
On a Convex Parameter Space Method for Linear Control Design of Uncertain Systems
,”
SIAM Journal of Control and Optimization
, Vol.
29
, pp.
381
–
402
.
13.
Gu
K.
,
Chen
Y. H.
,
Zhody
M. A.
, and
Loh
N. K.
,
1991
, “
Quadratic Stabilizability of Uncertain Systems: A Two Level Optimization Setup
,”
Automatica
, Vol.
27
, pp.
161
–
165
.
14.
Kanellakopoulos, I., Kokotovic, P. V., and Middleton, R. H., 1990, “Observer-Based Adaptive Control of Nonlinear Systems Under Matching Conditions,” Proceedings of the 1990 American Control Conference, San Diego, pp. 549–555.
15.
Lorenz
E. N.
,
1963
, “
Deterministic Nonperiodic Flow
,”
Journal of The Atmospheric Sciences
, Vol.
20
, pp.
130
–
141
.
16.
Ott
E.
,
Grebogi
C.
, and
Yorke
J. A.
,
1990
, “
Controlling Chaos
,”
Physical Review Letters
, Vol.
64
, pp.
1196
–
1199
.
17.
Petersen
I. R.
, and
Hollot
C. V.
,
1986
, “
A Riccati Equation Approach to the Stabilization of Uncertain Linear Systems
,”
Automatica
, Vol.
22
, pp.
397
–
411
.
18.
Salam
F. M. A.
, and
Bai
S.
,
1988
,
IEEE Transactions on Circuits and Systems
, Vol.
35
, pp.
842
–
849
.
19.
Schmitendorf
W. E.
,
1988
, “
Designing Stabilizing Controllers for Uncertain Systems Using the Riccati Equation Approach
,”
IEEE Transactions on Automatic Control
, Vol.
33
, pp.
376
–
379
.
20.
Singer
J.
,
Wang
Y-Z.
, and
Bau
H. H.
,
1991
, “
Controlling a Chaotic System
,”
Physical Review Letters
, Vol.
66
, pp.
1123
–
1125
.
21.
Swei, S. M., Rotea, M. A., and Corless, M., 1994, “System Order Reduction in Robust Stabilization Problems,” International Journal of Control, to appear.
22.
Thorpe
J. S.
, and
Barmish
B. R.
,
1981
, “
On Guaranteed Stability of Uncertain Systems via Linear Control
,”
Journal of Optimization Theory and Applications
, Vol.
35
, pp.
559
–
579
.
23.
Tsitras
P.
, and
Kelley
H. J.
,
1991
, “
Drag-Law Effects in the Goddard Problem
,”
Automatica
, Vol.
27
, pp.
481
–
490
.
24.
Vincent
T. L.
, and
Yu
J.
,
1991
, “
Control of a Chaotic System
,”
Dynamics and Control: An International Journal
, Vol.
1
, pp.
35
–
52
.
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