1. Art and Diseño

Noname manuscript No.
(will be inserted by the editor)
On homogeneous parameter dependent quadratic
Lyapunov function for robust H∞ filtering design in
switched linear discrete-time systems with polytopic
uncertainties
Guochen Pang · Kanjian Zhang
Received: date / Accepted: date
Abstract This article is concerned with the problem of robust H∞ filter
design for switched linear discrete-time systems with polytopic uncertainties.
The condition of robustly asymptotically stable for uncertain switched system
and a less conservative H∞ noise-attenuation level bounds are obtained by
homogeneous parameter dependent quadratic Lyapunov function. Moreover,
a more feasible and effective against the variations of uncertain parameter
robust switched linear filter is designed under the given arbitrary switching
signal. Lastly, simulation results are used to illustrate the effectiveness of our
method.
Keywords switched linear systems · robust H∞ filter · homogeneous
polynomial function · polytopic uncertainties
1 Introduction
Switched systems are a class of hybrid systems that consist of a finite number
of subsystems and a logical rule orchestrating switching between the subsystems. Since this class of systems have numerous applications in the control of
mechanical systems, the automotive industry, aircraft and air traffic control,
switching power converters and many other fields, the problems of stability
analysis and control design for switched systems have received wide attention
Guochen Pang
Key Laboratory of Measurement and Control of CSE Ministry of Education, School of
Automation, Southeast University, Nanjing 210096, China.
Tel.: +86 02583794766
Fax: +86 02583794766
E-mail: [email protected]
Kanjian Zhang
Key Laboratory of Measurement and Control of CSE Ministry of Education, School of
Automation, Southeast University, Nanjing 210096, China.
2
Guochen Pang, Kanjian Zhang
during the past two decades [1–15]. [2] proposed the H∞ weight learning law
for switched Hopfield neural networks with time-delay under parametric uncertainty. [8] dealt with the delay-dependent exponentially convergent state
estimation problem for delayed switched neural networks. Some criteria for
exponential stability and asymptotic stability of a class of nonlinear hybrid
impulsive and switching systems have been established using switched Lyapunov functions in [9]. [14] investigated the problem of designing a switching
compensator for a plant switching amongst a family of given configurations.
On the other hand, within robust control theory scheme, the H∞ noiseattention level is an important index for the influence of external disturbance
on system stability[17–20]. However, the uncertainties which generally exist
in many practical plants and environments may result in significant changes
in robust H∞ noise-attention level. In order to suppress the conservativeness,
many new methods have been considered. Among these methods, homogeneous polynomial parameter-dependent quadratic Lyapunov function is one of
the most effective methods. The main feature of this function is that they are
quadratic Lyapunov functions whose dependence on the uncertain parameters
is expressed as a polynomial homogeneous form. It is firstly introduced to study
robust stability of polynomial systems in [21]. Most results have been presented in [23–27]. In [20], homogeneous parameter-dependent quadratic Lyapunov
functions were used to establish tightness in robust H∞ analysis. [23] presented some general results concerning the existence of homogeneous polynomial
solutions to parameter-dependent linear matrix inequalities whose coefficients
are continuous functions of parameters lying in the unit simplex. [25] investigated the problems of checking robust stability and evaluating robust H2
performance of uncertain continuous-time linear systems with time-invariant
parameters lying in polytropic domains. [27] introduced Gram-tight forms,
i.e. forms whose minimum coincides with the lower bound provided by LMI
optimizations based on SOS(sum of squares of polynomials) relaxations.
Thus, in order to suppress the influence of uncertainties on the system’s robust H∞ control, the homogeneous parameter-dependent quadratic Lyapunov
functions are used to desgin robust H∞ filter in this paper. We first consider
the system’s anti-interference of external disturbance. Along this direction, the
robust H∞ filters for switched linear discrete-time systems are designed. Lastly, through the comparison, we have that homogeneous parameter-dependent
quadratic Lyapunov functions can suppress conservativeness which the uncertainties bring.
The rest of this article is organised as follows. We state the problem formulation in Section 2. The main results are presented in Section 3. Section 4
illustrates the obtained result by numerical examples, which is followed by the
conclusion in Section 5.
Notation: Rn denotes the n-dimension Euclidean space and Rn×m is the
real matrices with dimension n × m; Rn0 means Rn /{0}; The notation X ≥ Y
(respectively X > Y ) where X and Y are symmetric matrices, represents that
the matrix X − Y is positive semi-definite (respectively, positive definite); AT
denotes the transposed matrix of A; sq(x) represents (x21 , · · · , x2n ) with x ∈ Rn ;
Robust H∞ Filtering
3
he(X) means X + X T with X ∈ Rn×n ; x ⊗ y denotes the Kronecker product of
vectors x and y. ∥ · ∥ denotes Euclidean norm for vector or the spectral norm
of matrices.
2 Problem Statement
Consider a class of uncertain switched linear discrete-time systems which were
given in [1].
x(k + 1) = Ai (λ)x(k) + Bi (λ)ω(k)
y(k) = Ci (λ)x(k) + Di (λ)ω(k)
z(k) = Hi (λ)x(k) + Li (λ)ω(k)
(1)
where x(k) ∈ Rn is state vector, ω(k) ∈ Rl is disturbance input which belongs
to l2 [0, +∞), y(k) is the measurement output, z(k) is objective signal to be
attenuated, i is switching rule, which takes its value in the finite set Π :=
{1, · · · , N }. As in reference [1], the switching signal i is unknown a priori,
but its instantaneous value is available in real time. As an arbitrary discrete
time k, the switching signal i is dependent on k or x(k), or both or other
switching rules.
∑s λ is an uncertain parameter vector supposed to satisfy λ ∈ Λ =
{λj ≥ 0,
j=1 = 1}. The vector λ represents the time-invariant parametric
uncertainty which affects linearly the system dynamics. The vector λ can take
any value in Λ, but it is known to be constant in time.
The matrices of each subsystem have appropriate dimensions and are assumed to belong to a given convex-bounded polyhedral domain described by
s vertices in the ith subsystem, i.e.
(Ai (λ) Bi (λ) Ci (λ) Di (λ) Hi (λ) Li (λ) ) ∈ Γi
(2)
where
Γi := {(Ai (λ) Bi (λ) Ci (λ) Di (λ) Hi (λ) Li (λ) )
s
s
∑
∑
=
λj (Aij Bij Cij Dij Hij Lij ); λj ≥ 0
λj = 1.
j=1
j=1
Hence, we are interested in designing an estimator or filter of the form
xf (k + 1) = Af i xf (k) + Bf i y(k)
zf (k) = Cf i xf (k) + Df i y(k)
(3)
where xf (k) ∈ Rn , zf (k) ∈ Rq and Af i Bf i Cf i Df i are the parameterized
filter matrices to be determined. The filter with the above structure may be
called switched linear filter, in which the switching signal i is also assumed unknown a priori but available in real-time and homogeneous with the switching
signal in system (1).
4
Guochen Pang, Kanjian Zhang
Augmenting the model of (1) to include the state of the filer (3), we obtain
the filtering error system:
xe (k + 1) = Aei (λ)xe (k) + Bei (λ)ω(k)
e(k) = Cei (λ)xe (k) + Dei (λ)ω(k)
(4)
where
(
)T
xe (k) = x(k) xf (k) , e(k) = z(k) − zf (k)
(
)
(
)T
Ai (λ)
0
Aei (λ) =
, Bei (λ) = Bi (λ) Bf i Di (λ) ,
Bf i Ci (λ) Af i
(
)
Cei (λ) = Hi (λ) − Df i Ci (λ) −Cf i , Dei (λ) = Li (λ) − Df i Di (λ).
Based on [1], the robust H∞ filtering problem addressed in this paper can
be formulated as follows: found a prescribed level of noise attention γ > 0,
and determining a robust switched linear filter (3) such that the filtering error
system is robustly asymptotically stable and
∥ e ∥2 < γ 2 ∥ ω ∥2 .
(5)
This problem has been received a great deal of attention. In order to get
a better level of noise attention γ > 0, we will use homogeneous polynomial
functions which have been demonstrated non-conservative result for serval
problems. In this paper, we extend these methods to design robust switched
linear filter.
The following preliminaries are given, which are essential for later developments. We firstly recall the homogeneous polynomial function from chesi et
al. [16].
Definition 1 The function h : Rn → R is a form of degree d in n scalar
variables if
∑
h(x) =
aq xq
(6)
where Ln,s = {q ∈ N n :
monomial xq .
∑n
q∈Ln,s
i=1 qi
= s}, x ∈ Rn and aq ∈ R is coefficient of the
The set of forms of degree s in n scalar variables is defined as
Ξn,s = {h : Rn → R : (6) holds}.
(7)
Definition 2 The function f : Rn → R is a polynomial of degree less than
or equal to s, in n scalar variables, if
f (x) =
s
∑
i=0
where x ∈ Rn and hi ∈ Ξn,i , i = 1, · · · , s.
hi (x)
(8)
Robust H∞ Filtering
5
Definition 3 Consider the vector x ∈ Rn , x = [x1 , · · · , xn ]T . The power
transformation of degree m is a nonlinear change of coordinates that forms a
new vector xm of all integer powered monomials of degree m that can be made
from the original x vector,
m
m
j1
jn
xm
x2 j2 · · · xm
, mji ∈ 1, · · · , n,
n
j = cj x1
n
∑
(n + m − 1)!
mji = m, j = 1, · · · , d(n,m) , d(n,m) =
.
(n − 1)!m!
i=1
Usually we take cj = 1, then with m > 0,
 m 
T
x1 (x1 , x2 , x3 , · · · , xn )m−1
x1
T

 x2 
 x2 (x2 , x3 , · · · , xn )m−1
 
 ..  = 
..

 . 

.



;


(9)
T
xn
xn (xn )m−1
otherwise

m
x1
 x2 
 
 ..  = 1.
 . 
(10)
xn
For example,
n = 2, m = 2, =⇒ d(n,m) = 3,
(
x=
x1
x2
)

x21
=⇒ x2 =  x1 x2  .
x22

(11)
Definition 4 The function M : Rn → Rr×r is a homogeneous parameter
dependent matrix of degree m in n scalar variables if
Mi,j ∈ Ξn,m ,
∀i, j = 1, · · · , r.
(12)
We denote the set of r × r homogeneous parameter dependent matrices of
degree m in n scalar variables as
#
Ξn,m,r
= {M : Rn → Rr×r : (12) holds}
(13)
and the set of symmetric matrix forms as
#
Ξn,m,r = {M ∈ Ξn,m,r
: M (x) = M T (x)
∀x ∈ Rn }.
(14)
Definition 5 Let M ∈ Ξn,2m,r and H ∈ Srd(n,m) such that
M (x) = Φ(H, xm , r)
(15)
where Φ(H, x , r) = (x ⊗ Ir ) H(x ⊗ Ir ). Then (15) is called a Square Matrical Representation (SMR) of M (x) with respect to xm ⊗ Ir . Moreover, H is
called a SMR matrix of M (x) with respect to xm ⊗ Ir .
m
m
T
m
6
Guochen Pang, Kanjian Zhang
Lemma 1 (Chesi et al. [16]) Let M ∈ Ξn,2m,r . Then H = {H ∈ Srd(n,m) :
(15) holds} is an affine space. Moreover,
H(M ) = {H + L : H ∈ Srd(n,m) satisf ies (15), L ∈ Ln,m,r }
(16)
where Ln,m,r is linear space
Ln,m,r = {L ∈ Srd(n,m) : Φ(L, xm , r) = 0r×r
∀x ∈ Rn }
(17)
whose dimension is given by
ω(n, m, r) =
r
((d(n,m) (rd(n,m) + 1)) − (r + 1)d(n,2m) .
2
(18)
Lemma 2 (Chesi et al. [16]) Let M ∈ Ξn,s,r . Then
M (x) > 0, ∀x ∈ γq = {x ∈ Rq : xi ≥ 0,
q
∑
xi = 1}
(19)
i=1
holds if and only if
M (sq(x)) > 0, ∀x ∈ Rn0 .
Lemma 3 (Boyd et al. [28]) The linear matrix inequalities
)
(
S11 S12
<0
S=
T
S22
S12
(20)
(21)
T
T
is equivalent to
and S22 = S22
where S11 = S11
−1
T
S11 < 0, S22 − S12
S11
S12 < 0.
(22)
3 Main Result
In this section, the sufficient condition for existence of robust H∞ filter for uncertain switched systems are formulated. For this purpose, we firstly consider
the anti-interference of system (1) for disturbance.
Theorem 1 For a given scalar γ > 0. Consider the system (1), if there exist
matrices Pij > 0 and matrices Rij such that

 

Ψ11 (P¯ij , Rij ) 0 Ψ13 (Rij , Aij ) Ψ14 (Rij , Bij )
L11 L12 L13 L14



∗
−Ic Ψ23 (Cij )
Ψ24 (Dij ) 

 +  ∗ L22 L23 L24  < 0



∗
∗
Ψ33 (Pij )
0
∗ ∗ L33 L34 
2
∗
∗
∗
−γ Ic
∗ ∗ ∗ L44
∏
i, ¯i ∈
, j ∈ [1, s]
(23)
Robust H∞ Filtering
7
with
ΦP¯i Ri (λm+1 ) =
s
∑
λj (P¯i (λm ) − Ri (λm ) − RiT (λm ));
j=1
m+1
) = Ri (λm )Ai (λ); ΦRi Bi (λm+1 ) = Ri (λm )Bi (λ);
s
s
∑
∑
λj )m Ci (λ); ΦDi (λm+1 ) = (
λj )m Di (λ);
ΦCi (λm+1 ) = (
ΦRi Ai (λ
j=1
j=1
ΦI (λm+1 ) = (
s
∑
λj )m+1 I; ΦPi (λm+1 ) =
j=1
m+1
ΦP¯i Ri (sq(λ
s
∑
λj Pi (λm );
j=1
T
m+1
)) = ϖ (λ
)Ψ11 (P¯ij , Rij )ϖ(λm+1 );
ΦRi Ai (sq(λm+1 )) = ϖT (λm+1 )Ψ13 (Rij , Aij )ϖ(λm+1 );
ΦRi Bi (sq(λm+1 )) = ϖT (λm+1 )Ψ14 (Rij , Bij )ϖ(λm+1 );
ΦCi (sq(λm+1 )) = ϖT (λm+1 )Ψ23 (Cij )ϖ(λm+1 );
ΦDi (sq(λm+1 )) = ϖT (λm+1 )Ψ24 (Dij )ϖ(λm+1 );
ΦPi (sq(λm+1 )) = ϖT (λm+1 )Ψ33 (Pij )ϖ(λm+1 );
ΦI (sq(λm+1 )) = ϖT (λm+1 )Ic ϖ(λm+1 );
ϖ(λm+1 ) = λm+1 ⊗ I;
L11 · · · L44 ∈ Ls,m,n
where Ψ (·) is a matrix which is made up of Aij , Bij , Cij , Dij , Pij , Rij and
the set Ls,m,n is defined in Lemma 1, then system (1) is robustly asymptotically
stable with H∞ performance γ for any switching signal.
Proof : Consider the following homogeneous parameter dependent quadratic
Lyapunov function
V (k, x(k)) = xT (k)Pi (λm )x(k).
(24)
Then, along the trajectory of system (1), we have
△V = V (k + 1, x(k + 1)) − V (k, x(k))
= xT (k)[ATi (λ)P¯i (λm )Ai (λ) − Pi (λm )]x(k)
+2xT (k)[ATi (λ)P¯i (λm )Bi (λ)]ω(k)
+ω T (k)[BiT (λ)P¯i (λm )Bi (λ)]ω(k).
(25)
When i = ¯i, the switched system is described by the ith mode. When i ̸= ¯i,
it represents the switched system being at the switching times from mode ¯i to
mode i.
Pre- and post-multiply inequality (23) by {λm+1 ⊗ I}T and {λm+1 ⊗ I},
we have


ΦP¯i Ri (sq(λm+1 ))
0
ΦRi Ai (sq(λm+1 )) ΦRi Bi (sq(λm+1 ))

∗
−ΦI (sq(λm+1 )) ΦCi (sq(λm+1 )) ΦDi (sq(λm+1 )) 

 < 0.


∗
∗
ΦPi (sq(λm+1 ))
0
2
m+1
∗
∗
∗
−γ ΦI (sq(λ
))
8
Guochen Pang, Kanjian Zhang
From Lemma 2, one has


ΦP¯i Ri (λm+1 )
0
ΦRi Ai (λm+1 ) ΦRi Bi (λm+1 )

∗
−ΦI (λm+1 ) ΦCi (λm+1 ) ΦDi (λm+1 ) 

<0


∗
∗
ΦPi (λm+1 )
0
∗
∗
∗
−γ 2 ΦI (λm+1 )
which implies

P¯i (λm ) − Ri (λm ) − RiT (λm )

∗


∗
∗
0
−I
∗
∗
(26)

Ri (λm )Ai (λ) Ri (λm )Bi (λ)

Ci (λ)
Di (λ)
 < 0. (27)
m

Pi (λ )
0
2
∗
−γ I
Under the disturbance ω(k) = 0, we get
△V = xT (k)[ATi (λ)P¯i (λm )Ai (λ) − Pi (λm )]x(k).
(28)
ATi (λ)P¯i (λm )Ai (λ) − Pi (λm ) < 0, ∀i, ¯i ∈ Π
(29)
If
then △V < 0. It implies the system (1) is robust asymptotic stable.
In terms of Lemma 3, the condition (29) is equivalent to
(
)
−P¯i (λm ) P¯i (λm )Ai (λ)
< 0.
∗
Pi (λm )
(30)
Since (23), it follows that
P¯i (λm ) − Ri (λm ) − RiT (λm ) < 0
(31)
which implies the matrices Ri (λm ) are non-singular for each i. Then, we have
(P¯i (λm ) − Ri (λm ))P¯i−1 (λm )(P¯i (λm ) − Ri (λm ))T ≥ 0
(32)
which is equivalent to
P¯i (λm ) − Ri (λm ) − RiT (λm ) ≥ −Ri (λm )P¯i−1 (λm )RiT (λm ).
(33)
Hence, it can be readily established that (27) is equivalent to


−Ri (λm )P¯i−1 (λm )RiT (λm ) 0 Ri (λm )Ai (λ) Ri (λm )Bi (λ)


Di (λ)
∗
−I
Ci (λ)

 < 0.(34)
m


0
∗
∗
−Pi (λ )
2
−γ I
∗
∗
∗
Pre- and post-multiplying by diag{−Ri−T (λm )P¯i (λm ), I, I, I} to above inequality, we obtain


−P¯i (λm ) 0 P¯i (λm )Ai (λ) P¯i (λm )Bi (λ)


∗
−I
Ci (λ)
Di (λ)

 < 0.
m


∗
∗
−Pi (λ )
0
2
∗
∗
∗
−γ I
Robust H∞ Filtering
9
which implies (30) hold. Then, the stability of system (1) can be deduced.
On other hand, let
J=
∞
∑
[y T (k)y(k) − γ 2 ω T (k)ω(k)]
(35)
k=0
as performance index to establish the H∞ performance of system (1). When
the initial condition x(0) = 0, we have V (k, x(k))|k=0 = 0 which implies
J<
=
∞
∑
k=0
∞
∑
k=0
[y T (k)y(k) − γ 2 ω T (k)ω(k) + △V ]
(
(xT (k) ω T (k))
Θ11 Θ12
T
Θ12
Θ22
)(
x(k)
ω(k)
)
(36)
where
Θ11 = ATi (λ)P¯i (λm )Ai (λ) − Pi (λm ) + CiT (λ)Ci (λ);
Θ12 = ATi (λ)P¯i (λm )Bi (λ) + CiT (λ)Di (λ);
Θ22 = −γ 2 I + BiT (λ)P¯i (λm )Bi (λ) + DiT (λ)Di (λ).
(37)
In term of Lemma 3, we have J < 0 which means that ∥ y ∥2 < γ ∥ ω ∥2 .
Then, the proof is completed.
2
Remark 1 Within robustly H∞ performance scheme, the basic idea is to get a
upper bound of H∞ noise-attention level. Since model uncertainties may result
in significant changes in H∞ noise-attention level, we should effectively reduce
this effect. For this purpose, the homogeneous parameter-dependent quadratic
Lyapunov function is exploited in Theorem 1.
Remark 2 In Theorem 1, Ψ (·) is a matrix which is made up of Aij , Bij , Cij , Dij ,
Pij , Rij . For example, when n = 2 and m = 1, the condition (23) can be written as


T
0
0
P¯i1 − Ri1

 −Ri1


T


∗
−Ri1 − Ri1
0


T
,

−Ri2 − Ri2
Ψ11 (P¯ij , Rij ) = 



+P¯i1 + P¯i2



∗
∗
P¯i2 − Ri2 
T
−Ri2


Ri1 Ai1
0
0
Ri1 Ai2 + Ri2 Ai1
0 ,
Ψ13 (Rij , Aij ) =  ∗
∗
∗
Ri2 Ai2


Ri1 Bi1
0
0
Ri1 Bi2 + Ri2 Bi1
0 ,
Ψ14 (Rij , Bij ) =  ∗
∗
∗
Ri2 Bi2
10
Guochen Pang, Kanjian Zhang




Ci1
0
0
Di1
0
0
Ψ23 (Cij ) =  ∗ Ci1 + Ci2 0  , Ψ24 (Cij ) =  ∗ Di1 + Di2 0  ,
∗
∗
Ci2
∗
∗
Di2


−Pi1
0
0
Ψ33 (Pij ) =  ∗ −Pi1 − Pi2 0  .
∗
∗
−Pi2
(38)
Remark 3 About the calculation of matrices L11 · · · L44 , the following algorithm is presented in [16]:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
choose xm and x2m as in Definition 3 where x ∈ Rn
set A = 0d(n,2m) d(r,2) ×3 and b = 0 and define the variable α ∈ Rω(n,m,r)
for i = 1, . . . , d(n,m) and j = 1, . . . , r
set c = r(i − 1) + j
for k = 1, . . . , d(n,m) and l = max{1, j + r(i − k)}, . . . , r
set d = r(k − 1) + l and f = ind((xm )i (xm )k , x2m )
set g = ind(yj yl , y 2 ) and a = f dr,2 + g and Aa,1 = Aa,1 + 1
if Aa,1 = 1
set Aa,2 = c and Aa,3 = d
else
set b = b + 1 and G = 0rd(n,m) ×rd(n,m) and Gc,d = 1
set h = Aa,2 and p = Aa,3 and Gh,p = Gh,p − 1
set L = L + αb G
endif
endfor
endfor
set L = 0.5he(L)
Next, the condition for robust H∞ filter are formulated. For convenience,
we define n = 2, m = 1 and s = 2.
¯f i , C¯f i , D
¯ f i , xij
Theorem 2 Given a constant γ > 0, if there exist matrices P2ij , yi , ζij , A¯f i , B
and positive definite matrices P1ij , P3ij and scalar εi such that

Λ11
 ∗

 ∗

 ∗

 ∗
∗
Λ12
Λ22
∗
∗
∗
∗
0
0
Λ33
∗
∗
∗
Λ14
Λ24
Λ34
Λ44
∗
∗
Λ15
Λ25
Λ35
Λ45
Λ55
∗
 
Λ16
L11
 ∗
Λ26 
 

Λ36 
+ ∗

0  
 ∗
0   ∗
Λ66
∗
L12
L22
∗
∗
∗
∗
L13
L23
L33
∗
∗
∗
L14
L24
L34
L44
∗
∗
L15
L25
L35
L45
L55
∗

L16
L26 

L36 
<0
L46 

L56 
L66
(39)
Robust H∞ Filtering
with

Λ11
Λ12
Λ14
Λ15
Λ16
Λ22
Λ24

P1¯i1 − xi1
0
0
 −xTi1



T


0
−x
−
x
0
i1
i1


T
,
−x
−
x
= 
i2
i2




+P
+
P
¯
¯
1i1
1i2



0
0
P1¯i2 − xi2 
−xTi2


P2i1 − εi yi
0
0

 −ζi1




0
P2i1 + P2i2
0



,
−ζi1 − ζi2
= 



−2εi yi



0
0
P2i1 − εi yi 
−ζi2


xi1 Ai1
0
0
¯f i Ci1
 +εi B



¯


0
xi1 Ai2 + εi Bf i Ci1
0
,
= 
¯


+xi2 Ai1 + εi Bf i Ci2



0
0
xi2 Ai2 
¯f i Ci2
+εi B


εi A¯f i
0
0

0 2εi A¯f i 0  ,
=
0
0 εi A¯f i


xi1 Bi1
0
0
¯f i Di1
 +εi B



¯


0
xi1 Bi2 + εi Bf i Di1
0
,
= 
¯


+xi2 Bi1 + εi Bf i Di2



0
0
xi2 Ai2 
¯f i Di2
+εi B


P3i1 − yi
0
0
 −yiT





0
P3i1 + P3i2
0
,
= 
T


−2yi − 2yi



0
0
P3i2 − yi 
−yiT


ζi1 Ai1
0
0
¯f i Ci1
 +B



¯


0
ζi1 Ai2 + Bf i Ci1
0
,
= 
¯


+ζi2 Ai1 + Bf i Ci2



0
0
ζi2 Ai2 
¯f i Ci2
+B
11
12
Guochen Pang, Kanjian Zhang

Λ25
Λ26
A¯f i 0

0 2A¯f i
=
0
0

ζi1 Bi1
¯f i Di1
 +B


0
= 



0
Λ33 =
Λ34 =
Λ35 =
Λ36 =
Λ44 =
Λ55 =
L11 · · ·

0
0 ,
¯
Af i
0
0



¯

ζi1 Bi2 + Bf i Di1
0
,
¯

+ζi2 Bi1 + Bf i Di2

0
ζi2 Ai2 
¯f i Di2
+B


−I1×1
0
0
 0 −2I1×1 0  ,
0
0
−I1×1


¯
Hi1 − Df i Ci1
0
0
¯ f i Ci1


0
Hi1 − D
0

,
¯


+Hi2 − Df i Ci2
¯
0
0
Hi2 − Df i Ci2


¯
−Cf i 0
0
 0 −2C¯f i 0  ,
0
0 −C¯f i


¯ f i Di1
Li1 − D
0
0
¯ f i Di1


0
Li1 − D
0

,
¯


+Li2 − Df i Di2
¯
0
0
Li2 − Df i Di2




−P1i1
0
0
−P2i1
0
0
 0 −P1i1 − P1i2 0  , Λ45 =  0 −P2i1 − P2i2 0 
0
0
−P1i2
0
0
−P2i2




2
−P3i1
0
0
−γ I1×1
0
0
 0 −P3i1 − P3i2 0  , Λ66 = 
,
0
−2γ 2 I1×1
0
2
0
0
−P3i2
0
0
−γ I1×1
L66 ∈ L2,2,2 .
Then, there exist a robust switched linear filter in form of (3) such that, for all
admissible uncertainties, the filter error system (4) is robustly asymptotically
stable and performance index holds for any non-zero ω ∈ l2 [0, ∞) where Af i =
¯f i , Cf i = C¯f i , Df i = D
¯ f i.
yi−1 A¯f i , Bf i = yi−1 B
Proof : Let
(
Pi (λ) =
P1i (λ) P2i (λ)
∗ P3i (λ)
)
(
, Ri (λ) =
xi (λ) εi yi
ζi (λ) yi
where
(
P1i (sq(λ)) = {λ ⊗ I}T
P1i1 0
0 P1i2
)
{λ ⊗ I}
)
Robust H∞ Filtering
13
(
P2i (sq(λ)) = {λ ⊗ I}T
(
P2i1 0
0 P2i2
)
{λ ⊗ I}
)
P3i1 0
{λ ⊗ I}
0 P3i2
(
)
xi1 0
xi (sq(λ)) = {λ ⊗ I}T
{λ ⊗ I}
0 xi2
(
)
ζi1 0
ζi (sq(λ)) = {λ ⊗ I}T
{λ ⊗ I}
0 ζi2
P3i (sq(λ)) = {λ ⊗ I}T
(40)
From Theorem 1, the filter error system is robust asymptotically stable with
a prescribed H∞ noise-attenuation level bound γ if the following matrix inequality hold:


P1i (λ) − xi (λ) P2i (λ) − εi yi 0
xi (λ)Ai (λ) εi A¯f i xi (λ)Bi (λ)
¯f i Ci (λ)
¯f i Di (λ) 
 −xi (λ)T
−ζi (λ)
+εi B
+εi B


¯


∗
P
−
y
0
ζ
(λ)A
(λ)
A
ζ
(λ)B
3i
i
i
i
fi
i
i (λ) 

T
¯f i Ci (λ)
¯f i Di (λ) 

−y
+
B
+
B
i


<0

∗
∗
−I
Hi (λ)
C¯f i
Li (λ)


¯ f i Ci (λ)
¯ f i Di (λ) 

−
D
−
D




∗
∗
∗
−P1i (λ) −P2i (λ)
0




∗
∗
∗
∗
−P3i (λ)
0
∗
∗
∗
∗
∗
−γ 2 I
(41)
By (40) and Lemma 2, one can obtain that the inequality (39) is equivalent
to (41). Thus, if (39) holds, the filter error system is robustly asymptotically
stable with an H∞ noise-attenuation level bound γ > 0. Then, the proof is
completed.
4 Examples
The following example exhibits the effectiveness and applicability of the proposed method for robust H∞ filtering problems with polytopic uncertainties.
Consider the following uncertain discrete-time switched linear systems (1)
consisting of two uncertain subsystems which is given in [1]. There are two
groups of vertex matrices in subsystem 1:
(
)
(
)
0.82 0.10
0
A11 = ρ
, B11 = ρ
,
−0.06 0.77
0.1
(
)
(
)
C11 = ρ 1 0 , D11 = ρ, H11 = ρ 1 0 , L11 = 0,
(
A12 = ρ
0.82 0.10
−0.06 −0.75
)
(
, B12 = ρ
0
−0.1
)
,
14
Guochen Pang, Kanjian Zhang
(
)
(
)
C12 = ρ 1 0.2 , D12 = 0.8ρ, H12 = ρ 1 0 , L12 = 0
and two groups of vertex matrices in subsystem 2:
(
)
)
(
0.82 0.06
0.1
A21 = ρ
, B21 = ρ
,
−0.10 0.77
0
(
)
(
)
C21 = ρ 0 −1 , D21 = −ρ, H21 = ρ 1 0 , L21 = 0,
(
A22 = ρ
0.82 0.06
−0.10 −0.75
)
(
, B22 = ρ
−0.1
0
)
,
(
)
(
)
C22 = ρ 0.2 −1 , D22 = −0.8ρ, H22 = ρ 1 0 , L22 = 0.
Moreover, we define the disturbance
ω(k) = 0.001e−0.003k sin(0.002πk).
(42)
Firstly, consider the problem of robust asymptotically stable with an H∞
noise-attenuation for the uncertain switched system which was given in Theorem 1. The different minimum H∞ noise-attenuation level bounds γ can be
obtained by different methods. Table 1 lists the different calculation results.
From Tabler 1, it can be clearly seen that the results which are gotten by
Table 1 Different minimum γ for uncertain switched system
ρ
1
1.1
1.2
[1]
1.2799
1.6985
6.2048
Theorem 1
1.2799
1.6984
4.0988
homogeneous parameter dependent quadratic Lyapunov functions are better.
In addition, for given ρ = 1.2, and initial condition x(0) = [0.1, − 0.3]T ,
Fig. 1 and Fig. 2 show systems (1) are robustly asymptotically stable with an
H∞ noise-attenuation level bound γ = 4.0988.
Next, we consider the problem of robust H∞ filtering. In order to get
H∞ noise-attenuation level bound γ, we define ε1 = ε2 = 1. By solving the
corresponding convex optimization problems in Theorem 2, we obtain the
minimum H∞ noise-attenuation level bounds γ. Table 2 lists our results and
[1]’s results. Moreover, When ρ = 1.2, the admissible filter parameters can be
obtained according to Theorem 2 as
(
)
(
)
(
)
0.8770 0.0505
−0.0852
Af 1 =
, Bf 1 =
, Cf 1 = −1.2057 0.0112 ,
−0.7334 0.3356
−0.1077
(
)
(
)
(
)
1.1457 0.6743
−0.2593
Af 2 =
, Bf 1 =
, Cf 2 = −0.8695 −0.1525 ,
−1.2492 −0.8883
0.5929
Df 1 = −0.0073, Df 2 = −0.0045.
Robust H∞ Filtering
15
0.2
x1(t)
x2(t)
0.15
Magnitude
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
0
50
100
t/step
150
200
Fig. 1 The states responses corresponding to uncertain parameter λ1 = 0.4, λ2 = 0.6.
−3
3
x 10
y(t)
γw(t)
−γw(t)
2
Magnitude
1
0
−1
−2
−3
0
100
200
300
400
500
t/step
Fig. 2 H∞ noise-attenuation level bound γ = 4.0988
By comparison, using homogeneous parameter dependent quadratic Lyapunov
function for the existence of a robust switched linear filter in Theorem 2 are
better.
By giving H∞ noise-attenuation level bound γ = 4.2892 and initial condition xe (0) = [0.1 − 0.3 0 0]T , Fig. 3 shows the filtering error system is robustly
asymptotically stable and Fig. 4 shows the error response of the resulting filtering error system by applying above filter. It is clearly that the method in
Theorem 2 is feasible and effective against the variations of uncertain parameter.
16
Guochen Pang, Kanjian Zhang
Table 2 Different minimum γ for robust switched linear filter
ρ
1
1.1
1.2
[1]
0.5375
1.1414
4.4493
Theorem 2
0.5148
1.0826
4.2892
0.5
x1(t)
x2(t)
xf1(t)
xf2(t)
0.4
0.3
Magnitude
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
0
50
100
t/step
150
200
Fig. 3 The states responses corresponding to uncertain parameter λ1 = 0.4, λ2 = 0.6.
5 Conclusions
In this paper, the problems of robustly asymptotically stable with an H∞
noise-attenuation level bounds γ and switched linear filter design for uncertain switched linear system are studied by homogeneous parameter dependent
quadratic Lyapunov functions. By using this method, the less conservative H∞
noise-attenuation level bounds are obtained. Moreover, we also get a more feasible and effective against the variations of uncertain parameter under the given arbitrary switching signal. Numerical examples illustrate the effectiveness
of our method.
Acknowledgements This project is supported by Major Program of National Natural
Science Foundation of China under Grant 11190015, National Natural Science Foundation
of China under Grant 61374006 and Graduate Student Innovation Foundation of Jiangsu
province under Grant 3208004904.
References
1. L. Zhang, P. Shi, C. Wang and H. Gao, Robust H∞ filtering for switched linear discretetime systems with polytopic uncertainties, International journal of adaptive conrol and
Robust H∞ Filtering
17
0.15
e(t)
0.1
Magnitude
0.05
0
−0.05
−0.1
0
50
100
t/step
150
200
Fig. 4 Filtering error response corresponding to uncertain parameter λ1 = 0.4, λ2 = 0.6.
signal processing, 20, 291-304 (2006)
2. C.K. Ahn, An H∞ approach to stability analysis of switched Hopfield neural networks
with time-delay, Nonlinear Dynamics, 60, 703-711 (2010)
3. C.K. Ahn, Passive learning and input-to-state stability of switched Hopfield neural networks with time-delay, Information Sciences, 180, 4582-4594 (2010)
4. C.K. Ahn, M.K. Song, L2 − L∞ filtering for time-delayed switched Hopfield neural networks, International journal of innovative computing, information and control, 7, 18311844 (2011)
5. C.K. Ahn, Switched exponential state estimation of neural networks based on passivity
theory, Nonlinear Dynamics, 67, 573-586 (2012)
6. C.K. Ahn, Linear matrix inequality optimization approach to exponential robust filtering
for switched Hopfield neural networks, Journal of Optimization Theory and Applications,
154, 573-587 (2012).
7. C.K. Ahn, Switched exponential state estimation of neural networks based on passivity
theory, Nonlinear Dynamics, 67, 573-586 (2012)
8. C.K. Ahn, Exponentially convergent state estimation for delayed switched recurrent neural networks, The European Physical Journal E, 34, 122 (2011)
9. Z.H. Guan, D.J. Hill, X. Shen, On hybrid impulsive and switching systems and application
to nonlinear control, IEEE Transactions on Automatic Control, 50, 1058-1062 (2005)
10. W. Ni, D. Cheng, Control of switched linear systems with input saturation, International
Journal of Systems Science, 41, 1057-1065 (2010)
11. Y. Chen, S. Fei, K. Zhang, Stabilization of impulsive switched linear systems with
saturated control input, Nonlinear Dynamics, 69, 793-804 (2012)
12. Y. Chen, S. Fei, K. Zhang, Z. Fu, Control Synthesis of Discrete-Time Switched Linear
Systems with Input Saturation Based on Minimum Dwell Time Approach, Circuits System
Signal Process, 31, 779-795 (2012)
13. H. Mahboubi, B. Moshiri, S.A. Khaki, Hybrid modeling and optimal control of a twotank system as a switched system, World Academy of Science, Engineering and Technology,
11, 319- 324 (2005)
14. F. Blanchini, S. Miani, F. Mesquine, A separation principle for linear switching systems and parametrization of all stabilizing controllers, IEEE Transactions on Automatic
Control, 54, 279-292 (2009)
15. J. Lia, G. Yan, Fault detection and isolation for discrete-time switched linear systems
based on average dwell-time method, International Journal of Systems Science, 44, 23492364 (2013)
18
Guochen Pang, Kanjian Zhang
16. G. Chesi, A. Tesi, A. Garulli, A. Vicino, Homogneous Polynomial Forms for Robustness
Analysis of Uncertain Systems, 1-132. Springer-Verlag, Berlin Heidelberg (2009)
17. D. Ding, G. Yang, Finite frequency H∞ filtering for uncertain discrete-time switched
linear systems, Progress in Natural Science, 19, 1625-1633 (2009)
18. D. Ding, G. Yang, H∞ static output feedback control for discrete-time switched linear
systems with average dwell time, IET Control Theory and Applications, 4, 381-390 (2010)
19. H. Zhang, X. Xie, S. Tong, Homogenous Polynomially Parameter-Dependent H∞ Filter Designs of Discrete-Time Fuzzy Systems, IEEE Transactions On Systems, Man, And
Cyberneticspart B: Cybernetics, 41, 1313-1322 (2011)
20. G. Chesi, A. Garulli, A. Tesi, A. Vicino, Polynomially Parameter-Dependent Lyapunov
Functions For Robust H∞ Performance analysis, 16th Triennial World Congress, Prague,
Czech Republic, 457-462 (2005)
21. G. Chesi, A. Garulli, A. Tesi, A. Vicino, Robust stability for polytopic systems via
polynomial parameter-dependent Lyapunov functions, 42nd IEEE confereness On Decision
and Control, Maui, Hawaii, 4670-4675 (2003)
22. G. Chesi. On the non-conservatism of a novel LMI relaxation for robust analysis of
polytopic systems, Automatica, 44, 2973-2976 (2008)
23. P.-A. Bliman, R.C.L.F. Oliveira, V.F. Montagner, P.L.D. Peres, Existence of homogeneous polynomial solutions for parameter-dependent linear matrix inequalities with parameters in the simplex, IEEE Conf. on Decision and Control, San Diego, 1486-1491 (2006)
24. R.C.L.F. Oliveira, P.L.D. Peres, Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations, IEEE Trans. on Automatic Control, 52, 1334-1340 (2007)
25. R.C.L.F. Oliveira, M.C. de Oliveira, P.L.D. Peres, Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent
Lyapunov functions, Systems and Control Letters, 57, 680-689 (2008)
26. G. Chesi, Establishing stability and instability of matrix hypercubes, Systems and Control Letters, 54, 381-388 (2005)
27. G. Chesi, Tightness conditions for semidefinite relaxations of forms minimizations. IEEE
Trans. on Circuits and Systems II, 55, 1299-1303 (2008)
28. S. Boyd, L. EI Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems
and Contril Theorey. SIAM: Philadelphia, PA, (1994)