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20
Robust Stabilization for a Class of Uncertain
Discrete-time Switched Linear Systems
Songlin Chen, Yu Yao and Xiaoguan Di
Harbin Institute of Technology
P. R. China
1. Introduction
Switched systems are a class of hybrid systems consisting of several subsystems (modes of
operation) and a switching rule indicating the active subsystem at each instant of time. In
recent years, considerable efforts have been devoted to the study of switched system. The
motivation of study comes from theoretical interest as well as practical applications.
Switched systems have numerous applications in control of mechanical systems, the
automotive industry, aircraft and air traffic control, switching power converters, and many
other fields. The basic problems in stability and design of switched systems were given by
(Liberzon & Morse, 1999). For recent progress and perspectives in the field of switched
systems, see the survey papers (DeCarlo et al., 2000; Sun & Ge, 2005).
The stability analysis and stabilization of switching systems have been studied by a number
of researchers (Branicky, 1998; Zhai et al., 1998; Margaliot & Liberzon, 2006; Akar et al.,
2006). Feedback stabilization strategies for switched systems may be broadly classified into
two categories in (DeCarlo et al., 2000). One problem is to design appropriate feedback
control laws to make the closed-loop systems stable for any switching signal given in an
admissible set. If the switching signal is a design variable, then the problem of stabilization
is to design both switching rules and feedback control laws to stabilize the switched
systems. For the first problem, there exist many results. In (Daafouz et al., 2002), the switched
Lyapunov function method and LMI based conditions were developed for stability analysis
and feedback control design of switched linear systems under arbitrary switching signal.
There are some extensions of (Daafouz et al., 2002) for different control problem (Xie et al.,
2003; Ji et al., 2003). The pole assignment method was used to develop an observer-based
controller to stabilizing the switched system with infinite switching times (Li et al., 2003).
It is should be noted that there are relatively little study on the second problem, especially
for uncertain switched systems. Ji had considered the robust H∞ control and quadratic
stabilization of uncertain discrete-time switched linear systems via designing feedback
control law and constructing switching rule based on common Lyapunov function approach
(Ji et al., 2005). The similar results were given for the robust guaranteed cost control problem
of uncertain discrete-time switched linear systems (Zhang & Duan, 2007). Based on multiple
Lyapunov functions approach, the robust H∞ control problem of uncertain continuous-time
switched linear systems via designing switching rule and state feedback was studied (Ji et
al., 2004). Compared with the switching rule based on common Lyapunov function
approach (Ji et al., 2005; Zhang & Duan, 2007), the one based on multiple Lyapunov
352
Discrete Time Systems
functions approach (Ji et al., 2004) is much simpler and more practical, but discrete-time case
was not considered.
Motivated by the study in (Ji et al., 2005; Zhang & Duan, 2007; Ji et al., 2004), based on the
multiple Lyapunov functions approach, the robust control for a class of discrete-time
switched systems with norm-bounded time-varying uncertainties in both the state matrices
and input matrices is investigated. It is shown that a state-depended switching rule and
switched state feedback controller can be designed to stabilize the uncertain switched linear
systems if a matrix inequality based condition is feasible and this condition can be dealt
with as linear matrix inequalities (LMIs) if the associated scalar parameters are selected in
advance. Furthermore, the parameterized representation of state feedback controller and
constructing method of switching rule are presented. All the results can be considered as
extensions of the existing results for both switched and non-switched systems.
2. Problem formulation
Firstly, we consider a class of uncertain discrete-time switched linear systems described by
⎧ (
x k + 1) = ( Aσ + Δ
+
+ Δ
( k)
Aσ( k)) (
x k) ( Bσ( k)
Bσ( k)) (
u k)
⎪⎨
A
(1)
σ ( k )
Bσ ( k)
⎪
y( k) = C
⎩
σ ( k) (
x k)
where ( )
n
x k ∈ R is the state, ( )
m
u k ∈ R is the control input, ( )
q
y k ∈ R is the measurement
output and σ ( k)∈ Ξ = {1,2,"Ν} is a discrete switching signal to be designed. Moreover,
σ ( k) = i means that the ith subsystem ( A , B , C )
i
i
i is activated at time k (For notational
simplicity, we may not explicitly mention the time-dependence of the switching signal
below, that is, σ ( k) will be denoted as σ in some cases). Here i
A , i
B and Ci are constant
matrices of compatible dimensions which describe the nominal subsystems. The uncertain
matrices
i
A
Δ and
i
B
Δ are time-varying and are assumed to be of the forms as follows.
A
Δ
= M F ( k) N B
Δ = M F ( k)
i
ai ai
ai
i
bi bi
Nbi (2)
where Mai , Nai , Mbi , Nbi are given constant matrices of compatible dimensions which
characterize the structures of the uncertainties, and the time-varying matrices
( )
ai
F k and
( )
bi
F k satisfy
T
T
T
F ( k) F ( k) ≤ I, F ( k) F ( k) ≤ I ∀ ∈ Ξ
ai
ai
bi
bi
i
(3)
where I is an identity matrix.
We assume that no subsystem can be stabilized individually (otherwise the switching
problem will be trivial by always choosing the stabilized subsystem as the active
subsystem). The problem being addressed can be formulated as follows:
For the uncertain switched linear systems (1), we aim to design the switched state feedback
controller
(
u k) = Kσ x( k) (4)
and the state-depended switching rule σ ( x( k)) to guarantee the corresponding closed-loop
switched system
Robust Stabilization for a Class of Uncertain Discrete-time Switched Linear Systems
353
x( k + 1) = [ Aσ + A
Δ σ + ( Bσ + B
Δ σ ) Kσ ] x( k) (5)
is asymptotically stable for all admissible uncertainties under the constructed switching
rule.
3. Main results
In order to derive the main result, we give the two main lemmas as follows.
Lemma 1: (Boyd, 1994) Given any constant ε and any matrices M, N with compatible
dimensions, then the matrix inequality
T T
T
T
1
T
MFN N F M
ε MM
ε −
+
<
+
N N
holds for the arbitrary norm-bounded time-varying uncertainty F satisfying T
F F ≤ I .
Lemma 2: (Boyd, 1994) (Schur complement lemma) Let S, P, Q be given matrices such that
T
=
,
T
Q Q P = P , then
T
⎡ P S ⎤
T
1
⎢
⎥ < 0 ⇔ Q < 0, P − S Q− S < 0.
⎢⎣ S Q ⎥⎦
A sufficient condition for existence of such controller and switching rule is given by the
following theorem.
Theorem 1: The closed-loop system (5) is asymptotically stable when A
Δ
= B
Δ = 0
i
i
if there
exist symmetric positive definite matrices
n n
X
×
∈ R
i
, matrices
n n
i
G
×
∈ R
,
m n
i
Y
×
∈ R
, scalars
ε > 0
λ <
∈ Ξ λ = −
i
( i ∈Ξ) and scalars
0
ij
( i, j
,
1)
ii
such that
T
⎡∑λ ( G + G − X ) *
*
*
"
*
⎤
ij
i
i
i
⎢
⎥
j∈Ξ
⎢
⎥
⎢
A G + B Y
− X
*
*
"
*
i i
i i
i
⎥
⎢
⎥
1
G
0
λ−
⎢
i
i 1 X 1
*
"
*
⎥ < 0 i
∀ ∈ Ξ (6)
⎢
1
G
0
0
λ−
⎥
i
i 2 X 2
"
*
⎢
⎥
⎢
#
#
#
#
%
*
⎥
⎢
⎥
1
⎢⎣
G
0
0
0
0 λ−
i
iN XN ⎥⎦
is satisfied ( ∗ denotes the corresponding transposed block matrix due to symmetry), then
the state feedback gain matrices can be given by (4) with
1
Ki
i
Y i
G−
=
(7)
and the corresponding switching rule is given by
T
1
σ ( x( k)) = arg min{ x ( k) X− x( k)}
i
(8)
i∈Ξ
Proof. Assume that there exist G , X , Y ,ε
λ
i
i
i
i and ij such that inequality (6) is satisfied.
By the symmetric positive definiteness of matrices Xi , we get
T
1
( G
X ) X−
−
( G − X ) ≥ 0
i
i
i
i
i
354
Discrete Time Systems
which is equal to
T
1
−
T
≥
+
−
i
G Xi i
G
i
G
i
G
Xi
It follows from (6) and λ < 0
ij
that
T
1
⎡∑λ G X− G
*
* ⎤
ij i
i
i
⎢
⎥
j∈Ξ
⎢
⎥
⎢ A G + B Y
− X
* ⎥ < 0
i i
i i
i
(9)
⎢
0
⎥
Γ
Φ
i
i
⎢
⎥
⎣
⎦
where Γ = [ G G " G ]T
, ,
Φ =
λ
λ
"
λ
λ
"
λ
i
i
i
i
,
diag{1/ 1 1
X ,1/ 2 2
X , ,1/ ( −1) X −1,1/ ( +1) X +1, ,1/
i
i
i
i i
i
i i
i
iN N
X }
Pre- and post- multiplying both sides of inequality (9) by
1
−
T
diag{
, , }
−
i
G
I I and
1
diag{
, , }
i
G
I I ,
we get
1
⎡∑λ X−
*
* ⎤
ij i
⎢
⎥
j∈Ξ
⎢
⎥
⎢ A + B K
− X
* ⎥ < 0
i
i i
i
(10)
⎢
0
⎥
Π
Φ
i
i
⎢
⎥
⎣
⎦
where Π = [
" ]T
, ,
i
I I
I .
By virtue of the properties of the Schur complement lemma, inequality (10) is equal to
1
−
1
−
--1
⎡− X + ∑ λ ( X − X ) * ⎤
i
ij