Robust internal stabilization is a strong notion of stabilization, whereby stability is maintained regardless of small disturbances, noises, and uncertainties. In this paper, simple tools are developed for achieving robust internal stabilization of a rather large family of nonlinear systems. The main notion is that of a strict observer function, a function characterized by the following feature: subtracting a strict observer function from the differential equation of the controlled system results in an asymptotically stable differential equation. Strict observer functions are relatively easy to derive, and they directly yield robust asymptotic observers; the latter can be combined with robust state feedback controllers to achieve robust internal stabilization.

References

References
1.
Hammer
,
J.
,
2014
, “
State Feedback Control of Nonlinear Systems: A Simple Approach
,”
Int. J. Control
,
87
(
1
), pp.
143
160
.10.1080/00207179.2013.824612
2.
Hammer
,
J.
,
2013
, “
A Simple Approach to Nonlinear State Feedback Design
,” Asian
Control Conference
, Istanbul, Turkey, June 23–26.10.1109/ASCC.2013.6606370
3.
Astolfi
,
A.
, and
Ortega
,
R.
,
2003
, “
Immersion and Invariance: A New Tool for Stabilization and Adaptive Control of Nonlinear Systems
,”
IEEE Trans. Autom. Control
,
48
(
4
), pp.
590
606
.10.1109/TAC.2003.809820
4.
Hammer
,
J.
,
1988
, “
Assignment of Dynamics for Nonlinear Recursive Feedback Systems
,”
Int. J. Control
,
48
(
3
), pp.
1183
1212
.10.1080/00207178808906242
5.
Kalman
,
R.
, and
Bucy
,
R.
,
1961
, “
New Results in Linear Filtering and Prediction Theory
,”
J. Basic Eng.
,
83
(
3
), pp.
95
108
.10.1115/1.3658902
6.
Luenberger
,
D. G.
,
1966
, “
Observers for Multivariable System
,”
IEEE Trans. Autom. Control
,
11
(
2
), pp.
190
197
.10.1109/TAC.1966.1098323
7.
Song
,
Y.
, and
Grizzle
,
J.
,
1995
, “
The Extended Kalman Filter as a Local Asymptotic Observer for Discrete-Time Nonlinear Systems
,”
J. Math. Syst. Estim. Control
,
5
(
1
), pp.
59
78
.
8.
Freedman
,
L.
,
Shtessel
,
Y.
,
Edwards
,
C.
, and
Yan
,
X.-G.
,
2007
, “
Higher-Order Sliding-Mode Observer for State Estimation and Input Reconstruction in Nonlinear Systems
,”
Int. J. Robust Nonlinear Control
,
18
(
4–5
), pp.
399
412
.10.1002/rnc.1198
9.
Lasalle
,
J.
, and
Lefschetz
,
S.
,
1961
,
Stability by Liapunov's Direct Method With Applications
,
Academic Press
,
New York
.
10.
Lefschetz
,
S.
,
1965
,
Stability of Nonlinear Control Systems
,
Academic Press
,
New York
.
11.
Hammer
,
J.
,
1984
, “
Nonlinear Systems: Stability and Rationality
,”
Int. J. Control
,
40
(
1
), pp.
1
35
.10.1080/00207178408933254
12.
Hammer
,
J.
,
1985
, “
Nonlinear Systems, Stabilization, and Coprimeness
,”
Int. J. Control
,
42
(
1
), pp.
1
20
.10.1080/00207178508933342
13.
Hammer
,
J.
,
1989
, “
Robust Stabilization of Nonlinear Systems
,”
Int. J. Control
,
49
(
2
), pp.
629
653
.10.1080/00207178908559657
14.
Hammer
,
J.
,
1994
, “
Internally Stable Nonlinear Systems With Disturbances: A Parametrization
,”
IEEE Trans. Autom. Control
,
39
(
2
), pp.
300
314
.10.1109/9.272325
15.
Desoer
,
C.
, and
Kabuli
,
M.
,
1988
, “
Right Factorization of a Class of Nonlinear Systems
,”
IEEE Trans. Autom. Control
,
33
(
8
), pp.
755
756
.10.1109/9.1292
16.
Chen
,
G.
, and
de Figueiredo
,
R.
,
1990
, “
Construction of the Left Coprime Fractional Representations for a Class of Nonlinear Control Systems
,”
Syst. Control Lett.
,
14
(
4
), pp.
353
361
.10.1016/0167-6911(90)90057-2
17.
Paice
,
A.
, and
Moore
,
J.
,
1990
, “
Robust Stabilization of Nonlinear Plants Via Left Coprime Factorizations
,”
Syst. Control Lett.
,
15
(
2
), pp.
121
129
.10.1016/0167-6911(90)90027-R
18.
Sandberg
,
I.
,
1993
, “
Uniform Approximation and the Circle Criterion
,”
IEEE Trans. Autom. Control
,
38
(
10
), pp.
1450
1458
.10.1109/9.241560
19.
Paice
,
A.
, and
van der Schaft
,
A.
,
1994
, “
Stable Kernel Representation and the Youla Parametrization for Nonlinear Systems
,” 33rd
IEEE
Conference on Decision and Control, Lake Buena Vista, FL, Dec. 14–16, pp.
781
786
.10.1109/CDC.1994.410856
20.
Baramov
,
L.
, and
Kimura
,
H.
,
1995
, “
Nonlinear Coprime Factorizations and Parametrization of a Class of Stabilizing Controllers
,” 1995
IEEE
Conference on Decision and Control, Lake Buena Vista, FL, Dec. 14–16 pp.
981
986
.10.1109/CDC.1994.410856
21.
Georgiou
,
T.
, and
Smith
,
M.
,
1997
, “
Robustness Analysis of Nonlinear Feedback Systems: An Input–Output Approach
,”
IEEE Trans. Autom. Control
,
42
(
9
), pp.
1200
1221
.10.1109/9.623082
22.
Logemann
,
H.
,
Ryan
,
E.
, and
Townley
,
S.
,
1999
, “
Integral Control of Linear Systems With Actuator Nonlinearities: Lower Bounds for the Maximal Regulating Gain
,”
IEEE Trans. Autom. Control
,
44
(
6
), pp.
1315
1319
.10.1109/9.769399
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