This paper deals with the output feedback $H∞$ control design problem for continuous-time switched linear systems. More specifically, the main goal is to design a switching rule together with a dynamic full order linear controller to satisfy a prespecified $H∞$ level defined by the $L2$ gain from the input to the output signal. Initially, the state feedback version of this problem is solved in order to put in evidence the main difficulties we have to face toward the solution of the output feedback control design problem. The results reported in this paper are based on the so called Lyapunov–Metzler inequalities, which express a sufficient condition for switched linear systems global stability. The solution of the previously mentioned output feedback control design problem through a linear matrix inequality based method is the main contribution of the present paper. An academic example borrowed from literature is used for illustration.

1.
Decarlo
,
R. A.
,
Branicky
,
M. S.
,
,
S.
, and
Lennartson
,
B.
, 2000, “
Perspectives and Results on the Stability and Stabilizability of Hybrid Systems
,”
Proc. IEEE
0018-9219,
88
, pp.
1069
1082
.
2.
Liberzon
,
D.
, and
Morse
,
A. S.
, 1999, “
Basic Problems in Stability and Design of Switched Systems
,”
IEEE Control Syst. Mag.
0272-1708,
19
, pp.
59
70
.
3.
Liberzon
,
D.
, 2003,
Switching in Systems and Control
,
Birkhäuser
,
Boston
.
4.
Sun
,
Z.
, and
Ge
,
S. S.
, 2005,
Switched Linear Systems: Control and Design
,
Springer
,
London
.
5.
Branicky
,
M. S.
, 1998, “
Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems
,”
IEEE Trans. Autom. Control
0018-9286,
43
, pp.
475
482
.
6.
Ji
,
Z.
,
Guo
,
X.
,
Wang
,
L.
, and
Xie
,
G.
, 2006, “
Robust H∞ Control and Stabilization of Uncertain Switched Linear Systems: A Multiple Lyapunov Function Approach
,”
ASME J. Dyn. Syst., Meas., Control
0022-0434,
128
, pp.
696
700
.
7.
Peleties
,
P.
, and
DeCarlo
,
R. A.
, 1991, “
Asymptotic Stability of M-Switched Systems Using Lyapunov-Like Functions
,”
Proceedings of the American Control Conference
, pp.
1679
1684
.
8.
Zhao
,
J.
, and
Hill
,
D. J.
, 2008, “
On stability, L2 Gain and H∞ Control for Switched Systems
,”
Automatica
0005-1098,
44
, pp.
1220
1232
.
9.
Deaecto
,
G. S.
, 2007, “
Control Synthesis for Dynamic Switched Systems
,” MS thesis, University of Campinas—FEEC/UNICAMP, in Portuguese.
10.
Geromel
,
J. C.
, and
Colaneri
,
P.
, 2006, “
Stability and Stabilization of Continuous-Time Switched Linear Systems
,”
SIAM J. Control Optim.
0363-0129,
45
, pp.
1915
1930
.
11.
Geromel
,
J. C.
,
Colaneri
,
P.
, and
Bolzern
,
P.
, 2008, “
Dynamic Output Feedback Control of Switched Linear Systems
,”
IEEE Trans. Autom. Control
0018-9286,
53
, pp.
720
733
.
12.
Geromel
,
J. C.
, and
Deaecto
,
G. S.
, 2009, “
Switched State Feedback Control for Continuous-Time Uncertain Systems
,”
Automatica
0005-1098,
45
, pp.
593
597
.
13.
Hu
,
T.
,
Ma
,
L.
, and
Lin
,
Z.
, 2006, “
On Several Composite Quadratic Lyapunov Functions for Switched Systems
,”
Proceedings of the 45th IEEE Conference on Decision and Control
, pp.
113
118
.
14.
Ji
,
Z.
,
Wang
,
L.
, and
Xie
,
G.
, 2005, “
,”
Int. J. Syst. Sci.
0020-7721,
36
, pp.
395
404
.
15.
Savkin
,
A. V.
,
Petersen
,
I. R.
,
Skafidas
,
E.
, and
Evans
,
R. J.
, 1996, “
Hybrid Dynamical Systems: Robust Control Synthesis Problems
,”
Syst. Control Lett.
0167-6911,
29
, pp.
81
90
.
16.
Savkin
,
A. V.
,
Skafidas
,
E.
, and
Evans
,
R. J.
, 1999, “
Robust Output Feedback Stabilizability via Controller Switching
,”
Automatica
0005-1098,
35
, pp.
69
74
.
17.
Skafidas
,
E.
,
Evans
,
R. J.
,
Savkin
,
A. V.
, and
Petersen
,
I. R.
, 1999, “
Stability Results for Switched Controller Systems
,”
Automatica
0005-1098,
35
, pp.
553
564
.
18.
Garg
,
K. M.
, 1998,
Theory of Differentiation: A Unified Theory of Differentiation via New Derivate Theorems and New Derivatives
,
Wiley-Interscience
,
New York
.
19.
Lasdon
,
L. S.
, 1970,
Optimization Theory for Large Systems
,
Macmillan
,
New York
.
20.
Geromel
,
J. C.
,
Korogui
,
R. H.
, and
Bernussou
,
J.
, 2007, “
On H2 and H∞ Robust Output Feedback Control for Continuous-Time Polytopic Systems
,”
IET Control Theory and Applications
,
1
, pp.
1541
1549
.
21.
Xie
,
G.
, and
Wang
,
L.
, 2005, “
Periodical Stabilization of Switched Linear Systems
,”
J. Comput. Appl. Math.
0377-0427,
181
, pp.
176
187
.