An invariance principle for a class of ordinary differential equations with discontinuous right-hand side is developed. Based on this principle, asymptotic stability of one-degree-of-freedom mechanical oscillators with Coulomb friction is studied. The system is shown to be asymptotically stabilizable via a static feedback of the position, unlike those systems with no friction, whose stabilization requires a dynamic feedback when the position is the only available measurement. Along with this development, a velocity observer is proposed. Theoretical results of the paper are supported by some numerical simulations which, in addition, carry out a finite-time convergence of the controller and the observer proposed. [S0022-0434(00)00804-2]

1.
Branicky
,
M. S.
,
1998
, “
Multiple Lyapunov functions and other analysis tools for switched and hybrid systems
,”
IEEE Trans. Autom. Control
,
43
, No.
4
, pp.
475
482
.
2.
Clarke
,
F. H.
,
Ledyaev
,
Y. S.
,
Sontag
,
E. D.
, and
Subbotin
,
A. I.
,
1997
, “
Asymptotic controllability implies feedback stabilization
,”
IEEE Trans. Autom. Control
,
42
, No.
10
, pp.
1394
1407
.
3.
Filippov, A. F., 1988, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic, Publishers, Dordrecht.
4.
Johansson
,
K. H.
,
Rantzer
,
A.
, and
Astrom
,
K. J.
,
1999
, “
Fast switches in relay feedback systems
,”
Automatica
,
35
, pp.
539
552
.
5.
Skafidas
,
E.
,
Evans
,
J. R.
,
Savkin
,
A. V.
, and
Petersen
,
I. R.
,
1999
, “
Stability results for switched controller systems
,”
Automatica
,
35
, pp.
553
546
.
6.
Utkin, V. I., 1992, Sliding Modes in Control Optimization, Springer Verlag, Berlin.
7.
Michel, A. N., and Wang, K., 1995, Qualitative Theory of Dynamical Systems, the Role of Stability Preserving Mappings, Marcel Dekker, New York.
8.
Berghuis
,
H.
, and
Nijmeijer
,
H.
,
1993
, “
Global regulation of robots using only position measurements
,”
Syst. Control Lett.
,
21
, pp.
289
293
.
9.
Hahn, W., 1967, Stability of Motion, Springer, Berlin.
10.
Henry, D., 1981, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Springer-Verlag, Berlin.
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