Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 131, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
H∞ CONTROL OF SWITCHED LINEAR PARABOLIC SYSTEMS
LEPING BAO, SHUMIN FEI, LIN CHAI
Abstract. The H∞ control problem of a class of switched linear parabolic
systems is considered. By applying the multiple Lyapunov function method
and the average dwell time scheme, sufficient conditions for exponential stability and the H∞ control synthesis are established in terms of LMIs and a family
of switching signals. The advantage in this work lies in the fact that sufficient
conditions completely depend on the system parameters and the system can
be analyzed by using numerical softwares. At the end of the paper, an example
is given to illustrate the obtained result.
1. Introduction
During the previous decade, the study of switched systems has attracted considerable attention because of its significance in both theoretical research and practical
applications [1, 3, 6, 10, 11, 15, 18, 20, 22, 23]. A switched system is a dynamical system described by a family of continuous-time subsystems and a rule that governs the
switching among them. In many realistic cases, switched systems can be described
by partial differential equations (PDE) or a combination of ordinary differential
equations (ODE) and PDE such as in chemical industry process and biomedical
engineering. We refer to these switched systems as distributed parameter switched
systems (DPSS) or infinite dimensional switched systems [5, 13]. However, there are
very few works concerning DPSS (see, eg. [1, 7, 9, 14, 16, 19, 21] and the references
cited therein). For example, Sasane generalized the results presented in [15] to
infinite dimensional Hilbert spaces [21], and showed when all subsystems are stable
and commutative pairwise, the switched linear system is stable under an arbitrary
switching via the common Lyapunov function. Michel and Sun provided the stability conditions for switched nonlinear systems on Banach spaces under constrained
switching [14]. Hante and Sigalotti gave necessary and sufficient conditions in term
of the existence of common Lyapunov functions for DPSS [1, 7]. It seems that the
majority of works deal with the stability of DPSS.
The H∞ control is an interesting research topic in the field of switched systems.
Up to now, most of results in the literature are regarding the H∞ control of switched
2000 Mathematics Subject Classification. 34K20, 35R15, 34D10.
Key words and phrases. Switched systems; parabolic systems; H∞ control;
linear matrix inequalities; average dwell time.
c
2014
Texas State University - San Marcos.
Submitted August 16, 2013. Published June 10, 2014.
1
2
L. BAO, S. FEI, L. CHAI
EJDE-2014/131
systems which are described by ODEs [6, 12, 23]. For example, the stability, the L2 gain analysis and the H∞ control for switched systems via the multiple Lyapunovlike function methods is considered in [23]. The dynamic output feedback in H∞
control design of switched linear systems are studied in terms of linear matrix
inequalities (LMIs) in [6]. To the best of our knowledge, the H∞ control has not
been investigated for DPSS.
Motivated by the above consideration, we study the H∞ control synthesis for
switched linear parabolic systems in this paper. The main contributions of the
present paper can be summarized as follows: Firstly, the concept of H∞ control is
extended to DPSS. Secondly, sufficient conditions for the exponential stabilization
and the H∞ control synthesis of DPSS are developed in terms of LMIs and a
class of switching signals. Compared with the work in [7], our sufficient conditions
completely depend on the system parameters.
n diIn this article, L2 (Ω, Rn ) denotes the Hilbert space of square integrable
R
mensional vector-valued functions ν(x), x ∈ Ω with the norm kνkL2 = Ω ν T νdx.
L2 [t0 , ∞; L2 (Ω, Rn )) is the Hilbert space of square integrable functions ν(t, ·) ∈
L2 [t0 , ∞) with values ν(·, x) ∈ L2 (Ω, Rn ). H 2 (Ω, Rn ) and H02 (Ω, Rn ) denote the
∂2ν
n
classical Sobolev spaces defined by H 2 (Ω, Rn ) = {ν ∈ L2 (Ω, Rn ) : ∂x
2 ∈ L2 (Ω, R )}
∂ν
and H01 (Ω, Rn ) = {ν ∈ L2 (Ω, Rn ) : ∂x
∈ L2 (Ω, Rn ), ν(∂Ω, t) = 0} respectively.
γM (P )(γm (P )) denotes the largest (smallest) eigenvalue of P . The symmetric elements of the matrix will be denoted by T .
2. Problem formulation and preliminaries
Consider the switched linear parabolic systems
∂ν(x, t)
= Dσ(t) ∆ν(x, t) + Aσ(t) ν(x, t) + Bσ(t) u(x, t) + Cσ(t) ω(x, t)
∂t
y(x, t) = Eσ(t) ν(x, t) + Fσ(t) ω(x, t)
(2.1)
ν(t0 ) = ν0
ν(x, t) = 0,
(x, t) ∈ ∂Ω × [t0 , +∞)
with the static state feedback control
u(x, t) = Kσ(t) ν(x, t)
n
(2.2)
where ν(x, t) ∈ L2 [t0 , ∞; L2 (Ω, R )) is a vector-valued function representing the
state of the process, u(x, t) ∈ L2 [t0 , ∞; L2 (Ω, Rs )) denotes the manipulated input,
q
ω(x, t) ∈ L2 [t0 , ∞; L2 (Ω, Rp )) is the disturbance, and y(x, t) ∈ L
√2 [t0 , ∞;√L2 (Ω, R2 ))
denotes the measured output with (x, t) ∈ Ω×[t0 , +∞). Ω = [0, 2]×[0, 2] ⊂ R is
a bounded domain with the smooth boundary. ∆ denotes the Laplace operator; i.e.,
P2
∂2
∆ = k=1 ∂x
2 . Di = diag(di1 , di2 , . . . , din ) represent positive diagonal matrices,
k
and Ai , Bi , Ci , Ei , Fi (i = 1, 2, . . . , m) represent constant matrices of compatible
dimensions. σ(t) : [t0 , ∞) → Θ is the switching signal mapping time to some finite
index set Θ = {1, 2, . . . , m}, and the switching signal σ(t) is a piecewise continuous
(from the right) function depending on time or state or both. The discontinuities of
σ(t) are called switching times or switching instants. The integer m is the number
of models (called subsystems) of the switched system.
The objective of this article is to establish sufficient conditions of the H∞ control
for the system (2.1)–(2.2). That is, we look for controller gain matrices Ki (i ∈ Θ)
and a class of switching signals σ(t), such that
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CONTROL OF SWITCHED SYSTEMS
3
1. When ω = 0, the system (2.1) is exponentially stabilized by the state
feedback control (2.2).
2. The system (2.1) is exponentially stabilized by (2.2) with the H∞ disturbance level γ > 0, i.e., for a prescribed scalar γ > 0, the performance index
is
Z ∞Z
[y T (x, s)y(x, s) − γ 2 ω T (x, s)ω(x, s)] dx ds ≤ 0
J(ω) =
t0
Ω
for all non-zero ω(x, t) ∈ L2 [t0 , ∞; L2 (Ω, Rp )) under the zero initial condition.
The following is the definition of average dwell time (ADT) [10].
Definition 2.1. Given some family of switching signals σ(t) ∈ Θ, for each σ(t)
and each t > s ≥ t0 , let Nσ (s, t) denote the number of switching of σ(t) in the open
interval (s, t). If Nσ (s, t) ≤ N0 + t−s
τa holds for τa > 0 and N0 > 0, then the positive
constant τa is called the ADT and N0 is the chatter bound.
Lemma 2.2 (Poincare’s inequality [4]). Let the scalar function u ∈ H01 (Ω̄, R) with
Ω ⊆ Ω1 , then we have
Z
Z X
Z
n
∂u 2
2
2
2
u dx ≤ γ
(
) dx = γ
| ∇u |2 dx
(2.3)
Ω
Ω i=1 ∂xi
Ω
where Ω1 : 0 ≤ xi ≤ δ(i = 1, 2, . . . , n), γ =
√δ ,
n
∂
, ∂ , . . . , ∂x∂n ).
and ∇ = ( ∂x
1 ∂x2
3. Stabilization analysis of switched parabolic systems
In this section, we consider the exponential stabilization problem of the switched
linear parabolic system
∂ν(x, t)
= Dσ(t) ∆ν(x, t) + Aσ(t) ν(x, t) + Bσ(t) u(x, t)
∂t
(3.1)
ν(t0 ) = ν0
ν(x, t) = 0,
(x, t) ∈ ∂Ω × [t0 , +∞)
with the static state feedback control (2.2).
We assume that the state of system does not jump at switching instants; i.e., the
state trajectory is continuous and the switching signal σ(t) has the finite switching
number in any finite time interval [10].
We start with the well-posedness problem of the closed-loop system (2.2)-(3.1).
Define the state function z(t) as z(t) = ν(·, t) on the Hilbert space H = L2 (Ω, Rn )
with the norm k · kL2 , then the equation of closed-loop (2.2) and (3.1) can be
rewritten as
eσ(t) z(t) + fσ(t) (t), t ≥ t0
ż(t) = A
(3.2)
ei (A
ei x = Di ∆x) have the dense domain
in H, where the infinitesimal operators A
ei ) = {ν ∈ H 2 (Ω, Rn ) T H 1 (Ω, Rn ) : ν(x) = 0, x ∈ ∂Ω}, fσ(t) (t) =
W = D(A
0
[Aσ(t) + Bσ(t) Kσ(t) ]z(t).
ei generate analytical semigroups Ti (t)
As we know, the infinitesimal operators A
[17]. Because the state of system (2.2)-(3.1) does not jump at switching instants,
for every initial value ν0 ∈ W , there exists a unique solution for system (2.2)-(3.1).
Thus, the initial problem (2.2)-(3.1) turns out to be well-posed on the time interval
[t0 , ∞).
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L. BAO, S. FEI, L. CHAI
EJDE-2014/131
Lemma 3.1. For the given scalar λ > 0, if there exist positive constants αi , βi , a
constant µ ≥ 1, and continuous functions Vi ∈ C(H × [t0 , +∞), R+ ) such that the
functions Vi (t) = Vi (ν, t) are absolutely continuous along the solutions ν of system
(2.2)-(3.1) and satisfy
αi kν(t)kL2 ≤ Vi (t) ≤ βi kν(t)kL2
(3.3)
V̇i (t) + λVi (t) ≤ 0.
(3.4)
Furthermore, the Lyapunov function of the system satisfies
Vi (t) ≤ µVj (t),
∀i, j ∈ Θ.
(3.5)
Then the closed-loop system (2.2)-(3.1) is exponentially stable for the arbitrary
switching signal σ(t) with the ADT τa > lnµ
λ .
Proof. For t ∈ [tk , tk+1 )(k = 0, 1 . . . ), from (3.4) it follows that
Vσ(t) (t) ≤ e−λ(t−tk ) Vσ(tk ) (tk ).
This, together with (3.5) gives
−λ(t−tk )
Vσ(t) (t) ≤ µVσ(t− ) (t−
.
k )e
k
It is easy to show that
−λ(t−tk )
Vσ(t) (t) ≤ µVσ(t− ) (t−
≤ µVσ(tk−1 ) (tk−1 )e−λ(t−tk ) e−λ(tk −tk−1 )
k )e
k
2
≤µ
−λ(t−tk−1 )
Vσ(t− ) (t−
k−1 )e
k−1
≤ . . . · · · ≤ µk Vσ(t0 )(t0 )e−λ(t−t0 )
for all t ≥ t0 and a constant µ ≥ 1.
Note that when kτa ≤ t − t0 , we have
lnµ
Vσ(t) (t) ≤ e−λ(t−t0 ) eklnµ Vσ(t0 )(t0 ) ≤ e−(λ− τa
)(t−t0 )
Vσ(t0 )(t0 ).
(3.6)
Combing (3.3) and (3.6), we obtain
kν(t)kL2 ≤
lnµ
Vσ(t) (t)
1
≤ e−(λ− τa )(t−t0 ) Vσ(t0 )(t0 ).
α
α
Thus we have
kν(t)kL2 ≤
1 −(λ− lnµ
γ −(λ− lnµ
τa )(t−t0 ) · γkν k
τa )(t−t0 ) kν k
e
e
0 L2 ≤
0 L2
α
α
where γ = maxi∈Θ {βi } and α = mini∈Θ {αi }. Let h = λ − lnµ
τa > 0, and we have
lnµ
τa > λ . It is obvious that the system (2.2)-(3.1) is exponentially stable for the
arbitrary switching signal σ(t) with the ADT τa > lnµ
λ .
Theorem 3.2. For the given scalar λ > 0, if there exist diagonal matrices Xi > 0,
and matrices Yi > 0 such that
Πi = −2Di Xi + Ai Xi + XiT ATi + Bi Yi + YiT BiT + λXi < 0.
(3.7)
Then system (3.1) can be exponentially stabilized by the state feedback control (2.2)
with Ki = Yi Xi−1 for the arbitrary switching signal σ(t) with the ADT τa > ln(µ)/λ,
where µ is determined by
γM (Xk ) µ = max
.
(3.8)
∀k,l∈Θ γm (Xl )
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CONTROL OF SWITCHED SYSTEMS
Proof. Choose the multiple Lyapunov function for the system (2.2)-(3.1)
Z
V (t) = Vσ(t) (t) =
ν T (x, t)Pσ(t) ν(x, t) dx
5
(3.9)
Ω
with constant diagonal matrices Pi > 0. It is not difficult to see that there exist
positive numbers αi , βi and a constant µ ≥ 1 such that inequalities (3.3) and (3.5)
(Pk )
} [22].
hold. For inequalities (3.5), we can choose µ = max∀k,l∈Θ { γγM
m (Pl )
Differentiating V (t) with respect to t along the trajectory of the closed-loop
system (2.2)-(3.1), we have
V̇i (t) + λVi (t)
Z
Z
= [∆ν(x, t)]T Di Pi ν(x, t)dx +
ν T (x, t)Pi Di ∆ν(x, t)dx
Ω
Ω
Z
+
ν T (x, t)[Pi Ai + Pi Bi Ki ]ν(x, t)dx
Ω
Z
Z
+
ν T (x, t)[ATi Pi + KiT BiT Pi ]ν(x, t)dx +
ν T (x, t)λPi ν(x, t)dx.
Ω
(3.10)
Ω
Because Di and Pi are positive diagonal matrices, we find that Pi Di = Di Pi . Thus
we have
Z
Z
[∆ν(x, t)]T Di Pi ν(x, t)dx +
ν T (x, t)Pi Di ∆ν(x, t)dx
Ω
Ω
Z
T
≤ 2λmax (Pi Di )
∆ν (x, t)Iν(x, t)dx
Ω


Z
ν1 (x, t)
≤ 2λmax (Pi Di ) [∆ν1 (x, t), . . . , ∆νn (x, t)]  . . .  dx
Ω
νn (x, t)
Z
≤ −2λmax (Pi Di ) [ν1 (x, t)∆ν1 (x, t) + · · · + νn (x, t)∆νn (x, t)]dx.
Ω
According to Gauss’s divergence theorem, Poincare’s inequality (2.3) and taking
into account the boundary condition in (3.1), we obtain
Z
Z
[∆ν(x, t)]T Di Pi ν(x, t)dx +
ν T (x, t)Pi Di ∆ν(x, t)dx
Ω
Ω
Z
2
≤ −2λmax (Pi Di ) [ν1 (x, t) + · · · + νn (2 x, t)]dx
ZΩ
(3.11)
T
≤ −2λmax (Pi Di )
ν (x, t)Iν(x, t)dx
Ω
Z
≤
ν T (x, t)(−2Pi Di )ν(x, t)dx < 0.
Ω
Substituting (3.11) into (3.10) yields
Z
V̇i (t) + λVi (t) ≤
ν T (x, t)Γi ν(x, t) dx
Ω
where
Γi = −2Pi Di + Pi Ai + ATi Pi + Pi Bi Ki + KiT BiT Pi + λPi .
When
Γi = −2Pi Di + Pi Ai + ATi Pi + Pi Bi Ki + KiT BiT Pi + λPi < 0,
(3.12)
6
L. BAO, S. FEI, L. CHAI
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we have V̇i (t) + λVi (t) < 0 (for all ν(x, t) 6= 0). If the switching signal σ(t) satisfies
the ADT τa > ln(µ)/λ, all conditions in Lemma 3.1 hold. Hence, the closed-loop
system (2.2)-(3.1) is exponentially stable.
Left- and right- multiplying (3.12) by Pi−1 and letting Xi = Pi−1 and Yi =
Ki Pi−1 , it is not difficult to see that equality (3.12) is equivalent to (3.7) and
(Pk )
µ = max∀k,l∈Θ { γγM
} leads to (3.8) immediately. Consequently, the proof is
m (Pl )
completed.
4. H∞ control synthesis
In the section, we consider the H∞ control problem for system (2.1)–(2.2).
Lemma 4.1. For given scalars λ > 0 and γ > 0, if there exist diagonal matrices
Pi > 0 such that
Γi + EiT Ei
Pi Ci + EiT Fi
< 0,
(4.1)
CiT Pi + FiT Ei −γ 2 I + FiT Fi
then for any t ∈ [tk , tk+1 ), along the trajectory of system (2.1)-(2.2), we have
Z tZ
Vi (t) ≤ e−λ(t−tk ) Vi (tk ) −
e−λ(t−s) Υ(x, s) dx ds
(4.2)
tk
Ω
where
Γi = −2Pi Di + Pi Ai + ATi Pi + Pi Bi Ki + KiT BiT Pi + λPi ,
Υ(x, s) = y T (x, s)y(x, s) − γ 2 ω T (x, s)ω(x, s).
Proof. Differentiating Vi (t) with respect to t along the trajectory of the closed-loop
system (2.1)–(2.2) and using the similar argument described in the previous section,
we have
Z
Γ
Pi Ci
V̇i (t) + λVi (t) ≤
η T (x, t) T i
η(x, t)dx,
(4.3)
Ci Pi
0
Ω
where η T (x, t) = [ν(x, t), ω(x, t)]T . It follows from (4.1) and (4.3) that
T
Z
Z
E E
EiT Fi
V̇i (t) + λVi (t) < −
η T (x, t) iT i
η(x,
t)dx
=
−
Υ(x, s)dx.
Fi Ei −γ 2 I + FiT Fi
Ω
Ω
By calculation, we have
d λt
(e Vi (t)) < −eλt
dt
Integrating leads to (4.2).
Z
Υ(x, s)dx.
(4.4)
Ω
Theorem 4.2. For given scalars λ > 0 and γ > 0, if there exist diagonal matrices
Pi > 0 such that
Γi + EiT Ei
Pi Ci + EiT Fi
< 0.
(4.5)
CiT Pi + FiT Ei −γ 2 I + FiT Fi
Then, the system (2.1) can be exponentially stabilized by the state feedback control
(2.2) with the H∞ disturbance level γ > 0 for the arbitrary switching signal σ(t)
γM (Pk )
with the ADT τa > lnµ
λ , where µ is determined by µ = max∀k,l∈Θ { γm (Pl ) }.
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CONTROL OF SWITCHED SYSTEMS
Proof. It follows from (4.5) that
Pi Ci + EiT Fi
Γi + EiT Ei
Γ
= i
∗
CiT Pi + FiT Ei −γ 2 I + FiT Fi
T
Pi Ci
E E
+ iT i
−γ 2 I
Fi Ei
7
EiT Fi
< 0.
FiT Fi
Since
T
Ei EiT Fi
Ei
=
T
FiT
Fi Fi
T
Ei Ei
FiT Ei
Fi ≥ 0,
we have
Γi
∗
Pi Ci
< 0.
−γ 2 I
According to the Schur complement [2], we have Γi < 0. By virtue of the proof of
Theorem 3.1, the closed-loop system (2.1)–(2.2) is exponentially stable when ω = 0.
On the other hand, combining (3.5) and (4.2), we obtain
Z tZ
−
−λ(t−tk )
Vi (t) ≤ µe
Vi (tk ) −
e−λ(t−s) Υ(x, s) dx ds.
tk
Ω
Following [22] and repeating the above procedure, we obtain
Vi (t)
≤ µk e−λt V (ν0 ) − µk
t1
Z
t0
− µk−1
t2
Z
Z
Z
e−λ(t−s) Υ(x, s) dx ds
Ω
e−λ(t−s) Υ(x, s) dx ds − · · · −
Ω
t1
= µk e−λt V (ν0 ) −
Z tZ
tk
Z tZ
t0
e−λ(t−s) Υ(x, s) dx ds
(4.6)
Ω
e−λ(t−s)+Nσ (s,t)lnµ Υ(x, s) dx ds.
Ω
Because Vi (t) > 0, the zero initial condition implies V (ν0 ) = 0. Using (4.6) yields
Z tZ
e−λ(t−s)+Nσ (s,t)lnµ Υ(x, s) dx ds ≤ 0.
t0
It follows from e
Z tZ
t0
Ω
Nσ (s,t)lnµ
≥ 1(µ ≥ 1, Nσ (s, t) > 0) that
e−λ(t−s) [y T (x, s)y(x, s) − γ 2 ω T (x, s)ω(x, s)] dx ds ≤ 0.
(4.7)
Ω
Notice that since Nσ (t0 , s) ≤ N0 +
s−t0
τa ,
N0 > 0, and τa >
lnµ
λ ,
we derive that
Nσ (t0 , s)lnµ ≤ N0 lnµ + λ(s − t0 ).
Multiplying both sides of (4.7) by e−[N0 lnµ+λ(s−t0 )] gives
Z tZ
e−λ(t−s)−[N0 lnµ+λ(s−t0 )] y T (x, s)y(x, s) dx ds
t0
Ω
(4.8)
Z tZ
≤
e
t0
−λ(t−s)−[N0 lnµ+λ(s−t0 )] 2
T
γ ω (x, s)ω(x, s) dx ds.
Ω
Thus we obtain
Z tZ
Z tZ
e−λt y T (x, s)y(x, s) dx ds ≤
e−λt γ 2 ω T (x, s)ω(x, s) dx ds.
t0
Ω
t0
Ω
(4.9)
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L. BAO, S. FEI, L. CHAI
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Integrating both sides from t = t0 to ∞ gives
Z ∞Z
Z ∞Z
γ 2 ω T (x, s)ω(x, s) dx ds
y T (x, s)y(x, s) dx ds ≤
t0
Ω
t0
(4.10)
Ω
i.e., J(ω) ≤ 0. This completes the proof.
The matrix inequalities (4.5) are not LMIs. Left- and right- multiplying (4.5) by
diag{Pi−1 , I}. Let Xi = Pi−1 , Yi = Ki Pi−1 . It follows from the Schur complement
that (4.5) is equivalent to the LMIs


Xi
0
Πi Ci + Xi EiT Fi
 ∗ −γ 2 I + F T Fi
0
0
i
 < 0,

(4.11)
−1
T
∗
0
∗
(−Ei Ei )
∗
∗
∗
−I
where Πi = −2Di Xi + Ai Xi + XiT ATi + Bi Yi + YiT BiT + λXi . Next, we show a
result which can be obtained using matlab software.
Theorem 4.3. For given scalars λ > 0 and γ > 0, if there exist diagonal matrices
Xi > 0 and matrices Yi > 0, such that the LMIs (4.11) are feasible. Then the
system (2.1) can be exponentially stabilized by the state feedback control (2.2) with
Ki = Yi Xi−1 with the H∞ disturbance level γ > 0 for the arbitrary switching signal
σ(t) with the ADT τa > lnµ
λ , where µ is determined by (3.8).
Example 4.4. Consider the switched parabolic equations (2.1) under the state
feedback (2.2). Suppose there are two subsystems with parameters
D1 = [1 0; 0 2],
D2 = [2 0; 0 1],
A1 = [1 3; 2 3],
A2 = [1 2; 3 1],
B1 = [6 7; 5 1],
B2 = [5 2; 3 1],
C1 = [3 5; 4 2],
C2 = [5 2; 3 2],
E1 = [1 0; 0 1],
E2 = [1 0; 0 1],
F1 = [1 2; 2 1],
F2 = [2 1; 1 2].
Set λ = 0.6, γ = 0.8, using Theorem 4.3, by resolving LMIs (4.11), we obtain
X1 = [489.9209 0; 0 490.7482],
The state feedback matrices are
−0.1487 −6.4763
K1 =
,
−4.0296 3.3322
X2 = [195.4787 0; 0 193.5938].
1.8293
K2 =
−5.6515
From (3.8) we obtain that µ = 2.5349 and τa >
be exponentially stabilized.
lnµ
λ
−3.8292
9.3622
= 1.5503. So system (2.1) can
Conclusion. By using multiple Lyapunov function and ADT method, we establish
some new criteria for the exponential stabilization and the H∞ control synthesis of
switched linear parabolic systems via the state feedback. All the results are given in
terms of LMIs and a class of signals which can be easily tested by matlab software.
Acknowledgments. This work is supported by the National Science Foundation
of China (No. 61273119, 61374038), and by the Natural Science Foundation of
Jiangsu Province (BK2011253).
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CONTROL OF SWITCHED SYSTEMS
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Leping Bao
School of Automation, Southeast University, Nanjing, Jiangsu 210096, China.
Department of Automation, Taiyuan Institute of Technology, Taiyuan, Shangxi 030008,
China
E-mail address: jsll68@126.com
Shumin Fei
School of Automation, Southeast University, Nanjing, Jiangsu 210096, China
E-mail address: smfei@seu.edu.cn
10
L. BAO, S. FEI, L. CHAI
EJDE-2014/131
Lin Chai
School of Automation, Southeast University, Nanjing, Jiangsu 210096, China
E-mail address: Chailin-1@163.com