G
0
0
0
0
λ−
⎢
i
i 2 X 2
"
*
⎥
⎢
#
#
#
#
#
#
%
*
⎥
⎢
⎥
1
⎢
G
0
0
0
0
0
0 λ−
⎥
⎣
i
iN XN ⎦
is satisfied, where
T
T
Θ = − X + ε [ M M + M M ]
i
i
i
ai
ai
bi
bi ,
then the state feedback gain matrices can be given by (4) with
1
Ki
i
Y i
G−
=
(20)
356
Discrete Time Systems
and the corresponding switching rule is given by
T
1
σ ( x( k)) = arg min{ x ( k) X− x( k)}
i
(21)
i∈Ξ
Proof. By theorem 1, the closed-loop system (5) is asymptotically stable for all admissible
uncertainties if that there exist G , X ,
λ
i
i
i
Y and ij such that
T
⎡∑λ ( G + G − X ) * * ⎤
ij
i
i
i
⎢
⎥
j∈Ξ
⎢
⎥
⎢
A G + B Y
Θ
* ⎥ < 0 (22)
i i
i i
i
⎢
0
⎥
Γ
Φ
⎢
i
i ⎥
⎢⎣
⎥⎦
where Γ = [ G G " G ]T
, ,
i
i
i
i
,
Φ = diag{1 /λ
λ
λ
λ
",1 /λ iNXN}
1 X 1 ,1 /
2 X 2 ,",
i
i
i
1 / ( −1) X −1,1 / ( +1) X +1,
i i
i
i i
i
,
which can be rewritten as
T T
T
A + M F ( k) N + N F ( k) M < 0
i
i i
i
i i
i
where
T
⎡∑λ ( G + G − X ) * * ⎤
ij
i
i
i
⎢
⎥
j∈Ξ
⎡ 0
0 ⎤
⎢
⎥
⎢
⎥
A = ⎢
A G + B Y
Θ
* ⎥ , M = M
M
,
i
i i
i i
i
i
⎢ ai
bi ⎥
⎢
0
⎥
Γ
Φ
⎢⎣ 0
0 ⎥
i
i
⎢
⎥
⎦
⎣
⎦
F ( k) = diag( F ( k), F ( k)),
⎡ NaiGi
⎤
i
ai
bi
0 0
N =
i
⎢ N K 0 0⎥
⎣ bi i
⎦
It follows from Lemma 1 and T
T
F ( t) F ( t) ≤
i
i
I that
T
T
A + M M + N N < 0
i
i
i
i
i
(23)
By virtue of the properties of the Schur complement lemma, inequality (19) can be rewritten
as
T
⎡∑λ ( G + G − X ) * *
*
* ⎤
ij
i
i
i
⎢
⎥
j∈Ξ
⎢
⎥
⎢
A G + B Y
Θ
*
*
* ⎥
i i
i i
i
⎢
⎥ < 0 i
∀ ∈ Ξ (24)
Γ
0
Φ
*
*
i
i
⎢
⎥
⎢
N G
0
0
ε
− I
*
ai i
i
⎥
⎢
N Y
0
0
0
ε ⎥
−
⎣
bi i
iI ⎦
It is obvious that inequality (24)is equal to inequality (19), which finished the proof.
Let the scalars λ = 0
=
=
ij
and Xi Xj X , it is easily to obtain the condition for robust stability
of the closed-loop system (5) under arbitrary switching as follows.
Robust Stabilization for a Class of Uncertain Discrete-time Switched Linear Systems
357
Corollary 1: The closed-loop system (5) is asymptotically stable for all admissible
uncertainties under arbitrary switching if there exist a symmetric positive definite matrix
n n
X
×
∈ R
ε >
i
, matrices
m n
i
G
×
∈ R
,
m n
i
Y
×
∈ R
, scalars
0
i
and such that
T
⎡− G − G + X
*
*
* ⎤
i
i
i
⎢
⎥
A G + B Y
Θ
⎢
*
*
i i
i i
i
⎥ < 0 i
∀ ∈ Ξ (25)
⎢
N G
0
ε I
* ⎥
−
ai i
i
⎢
⎥
⎢⎣
N Y
0
0
ε
−
bi i
iI ⎥⎦
is satisfied, where
T
T
Θ = − X + ε [ M M + M M ]
i
i
i
ai
ai
bi
bi , then the state feedback gain matrices can
be given by (4) with
1
Ki
i
Y i
G−
=
(26)
4. Example
Consider the uncertain discrete-time switched linear system (1) with N =2. The system
matrices are given by
⎡1.5 1.5 ⎤
⎡1⎤
⎡0.5⎤
⎡0.3⎤
=
=
=
=
=
=
1
A
⎢
⎥ , 1
B
⎢ ⎥ , Ma 1 ⎢
⎥ , Na 1 [0.4 0.2], Mb 1 ⎢
⎥ , Nb 1 [0.2],
0
1
−
⎣
.2⎦
⎣0⎦
⎣0.2⎦
⎣0.4⎦
⎡1.2
0 ⎤
⎡0⎤
⎡0.3⎤
⎡0.3⎤
=
=
=
=
=
=
2
A
⎢
⎥ , 2
B
⎢ ⎥ , Ma 2 ⎢
⎥ , Na 2 [0.3 0.2], Mb 2 ⎢
⎥ , Nb 2 [0.1].
⎣0.6 1.2⎦
⎣1⎦
⎣0.4⎦
⎣0.3⎦
Obviously, the two subsystems are unstable, and it is easy to verify that neither subsystem
can be individually stabilized via state feedback for all admissible uncertainties. Thus it is
necessary to design both switching rule and feedback control laws to stabilize the uncertain
switched system. Letting λ = −
λ = −
12
10 and 21
10 , the inequality (19) in Theorem 1 is
converted into LMIs. Using the LMI control toolbox in MATLAB, we get
⎡41.3398
8.7000
−
⎤
⎡38.1986
8
− .6432⎤
X =
=
1
⎢
⎥ , X 2
8.7000 86.6915
⎢ 8.6432 93.8897⎥
−
−
⎣
⎦
⎣
⎦
⎡41.3415
8.6
−
656⎤
⎡−51.2846 T
⎤
=
=
1
G
⎢
⎥ , 1
Y
⎢
⎥ ,
8.7540
−
86.4219
26.5670
−
⎣
⎦
⎣
⎦
⎡38.1665
8.6003
−
⎤
⎡ 44.3564 T
−
⎤
=
=
2
G
⎢
⎥ , 2
Y
⎢
⎥ ,
8.61
−
⎣
86 93.6219⎦
⎣ 54.4478 ⎦
ε =
ε =
1
56.6320, 1 24.3598
With
1
Ki
i
Y i
G−
=
, the switched state feedback controllers are
K = −
−
= −
1
[ 1.4841 1.1505], K 2 [ 1.0527 0.4849].
358
Discrete Time Systems
It is obvious that neither of the designed controllers stabilizes the associated subsystem.
Letting that the initial state is x = −
0
[ 3,2] and the time-varying uncertain
F ( k) = F ( k) = f ( k)
ia
ib
( i = 1,2) as shown in Figure 1 is random number between -1 and 1, the
simulation results as shown in Figure 2, 3 and 4 are obtained, which show that the given
uncertain switched system is stabilized under the switched state feedback controller
together with the designed switching rule.
1
0.5
k)
0
f(
-0.5
-10
5
10
15
20
k/step
Fig. 1. The time-varying uncertainty f(k)
2
x1
x2
1
0
k)x( -1
-2
-30
5
10
15
20
k(step)
Fig. 2. The state response of the closed-loop system
5. Conclusion
This paper focused on the robust control of switched systems with norm-bounded
time-varying uncertainties with the help of multiple Lyapunov functions approach and
Robust Stabilization for a Class of Uncertain Discrete-time Switched Linear Systems
359
matrix inequality technique. By the introduction of additional matrices, a new condition
expressed in terms of matrices inequalities for the existence of a state-based switching
strategy and state feedback control law is derived. If some scalars parameters are selected in
advance, the conditions can be dealt with as LMIs for which there exists efficient numerical
software available. All the results can be easily extended to other control problems
( H 2 , H∞ control, etc.).
2
iganl
ng S
itchiwS
1
0
5
10
15
20
k(step)
Fig. 3. The switching signal
2
1
0
x2
-1
-2
-3-3
-2
-1
0
1
x1
Fig. 4. The state trajectory of the closed-loop system
6. Acknowledgment
This paper is supported by the National Natural Science Foundation of China (60674043).
360
Discrete Time Systems
7. References
Liberzon, D. & Morse, A.S. (1999). Basic problems in stability and design of switched
systems, IEEE Control Syst. Mag., Vol 19, No. 5, Oct. 1999, pp. 59-70
DeCarlo, R. A.; Branicky, M. S.; Pettersson, S. & Lennartson, B. (2000). Perspectives and
results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE,
Vol 88, No. 7, Jul. 2000, pp. 1069-1082.
Sun, Z. & Ge, S. S. (2005). Analysis and synthesis of switched linear control systems,
Automatica, Vol 41, No 2, Feb. 2005, pp. 181-195.
Branicky, M. S. (1998). Multiple Lyapunov functions and other analysis tools for switched
and hybrid systems, IEEE Transactions on Automatic Control, Vol 43, No.4, Apr. 1998,
pp. 475-482.
G. S. Zhai, D. R. Liu, J. Imae, (1998). Lie algebraic stability analysis for switched systems
with continuous-time and discrete-time subsystems, IEEE Transactions on Circuits
and Systems II-Express Briefs, Vol 53, No. 2, Feb. 2006, pp. 152-156.
Margaliot, M. & Liberzon, D. (2006). Lie-algebraic stability conditions for nonlinear switched
systems and differential inclusions, Systems and Control Letters, Vol 55, No. 1, Jan.
2006, pp. 8-16.
Akar, M.; Paul, A.; & Safonov, M. G. (2006). Conditions on the stability of a class of second-
order switched systems, IEEE Transactions on Automatic Control, Vol 51, No. 2, Feb.
2006, pp. 338-340.
Daafouz, J.; Riedinger, P. & Iung, C. (2002). Stability analysis and control synthesis for
switched systems: A switched Lyapunov function approach, IEEE Transactions on
Automatic Control, Vol 47, No. 11, Nov. 2002, pp. 1883-1887.
Xie, D.; Wang, Hao, L. F. & Xie, G. (2003). Robust stability analysis and control synthesis for
discrete-time uncertain switched systems, Proceedings of the 42nd IEEE Conference on
Decision and Control, Maui, HI, Dec. 2003, pp. 4812-4817.
Ji, Z.; Wang, L. and Xie, G. (2003). Stabilizing discrete-time switched systems via observer-
based static output feedback, IEEE Int. Conf. SMC, Washington, D.C, October 2003,
pp. 2545-2550.
Li, Z. G.; Wen, C. Y. & Soh, Y. C. (2003). Observer based stabilization of switching linear
systems, Automatica. Vol. 39 No. 3, Feb. 2003, pp:17-524.
Ji, Z. & Wang, L. (2005). Robust H∞ control and quadratic stabilization of uncertain discrete-
time switched linear systems, Proceedings of the American Control Conference.
Portland, OR, Jun. 2005, pp. 24-29.
Zhang, Y. & Duan, G. R. (2007). Guaranteed cost control with constructing switching law of
uncertain discrete-time switched systems, Journal of Systems Engineering and
Electronics, Vol 18, No. 4, Apr. 2007, pp. 846-851.
Ji, Z.; Wang, L. & Xie, G. (2004). Robust H∞ Control and Stabilization of Uncertain Switched
Linear Systems: A Multiple Lyapunov Functions Approach, The 16th Mathematical
Theory of Networks and Systems Conference. Leuven, Belgium, Jul. 2004, pp. 1~17.
Boyd, S.; Ghaoui, L.; Feron, E. & Balakrishnan, V. (1994). Linear Matrix Inequalities in System
and Control Theory, SIAM, Philadelphia.
Part 5
Miscellaneous Applications
21
Half-overlap Subchannel Filtered
MultiTone Modulation and Its Implementation
Pavel Silhavy and Ondrej Krajsa
Department of Telecommunications, Faculty of Electrical Engineering and
Communication, Brno University of Technology,
Czech Republic
1. Introduction
Multitone modulations are today frequently used modulation techniques that enable
optimum utilization of the frequency band provided on non-ideal transmission carrier
channel (Bingham, 2000). These modulations are used with especially in data transmission
systems in access networks of telephone exchanges in ADSL (asymmetric Digital Subscriber
Lines) and VDSL (Very high-speed Digital Subscriber Lines) transmission technologies, in
systems enabling transmission over power li