Delay dependent exponential \({H {\infty}}\) filter for uncertain nonlinear singular time delay systems through T S fuzzy model

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R E S E A R C H

Open Access

Delay-dependent exponential

H

_∞

ﬁlter

for uncertain nonlinear singular time-delay

systems through T-S fuzzy model

Yuechao Ma and Menghua Chen

*

*_{Correspondence:}

[email protected] College of science, Yanshan University, 438 Hebeidajie Road, Qinhuangdao, P.R. China

Abstract

In this paper, the problem of delay-dependent exponentialH_∞ﬁlter is investigated for uncertain nonlinear singular systems with interval time-varying delay through T-S fuzzy model. A new Lyapunov-Krasovskii functional (LKF) is constructed by

considering triple integral and the left and right endpoints of the time-delay to obtain a less conservative condition, which can guarantee that the error system is not only exponentially admissible but also satisfying a prescribedH_∞performance. Then, by solving the linear matrix inequalities (LMIs), the filter parameter matrices can be obtained. In the end, some numerical simulations are given to illustrate the effectiveness and the benefits of the methods.

Keywords: nonlinear singular systems; interval time-varying delay; exponentially stable; T-S fuzzy model;H_∞ﬁlter

1 Introduction

Filter design is one of the fundamental problems in various engineering applications, espe-cially in the ﬁelds of signal processing and control communities. In order to obtain the de-sired signal and eliminate interference, the method ofH∞ﬁlter is very appropriate to some

practical applications, and there has been considerable attention toH∞ﬁlter problems for

dynamic systems. Non-fragileH∞ﬁlter was designed for discrete-time fuzzy systems with multiplicative gain variations in []. RobustH∞ﬁltering for a class of uncertain linear

sys-tems with time-varying delay was studied in []. Delay-dependent robustH∞ﬁlter for T-S

fuzzy time-delay systems with exponential stability was proposed in []. ExponentialH∞

filtering for time-varying delay systems was studied through Markovian approach in []. Singular systems have extensive applications in electrical circuits, power systems, eco-nomics and other areas, so the filtering of a singular system is especially important and has been extensively studied [, ]. ExponentialH∞filtering for singular systems with

Marko-vian jump parameters was put forward in [].H∞ ﬁltering for a class of discrete-time singular Markovian jump systems with time-varying delays was designed in []. In [], the problem of exponentialH∞ ﬁltering was studied for discrete-time switched singular

time-delay systems.

On the other hand, time delay is frequently one of the main causes of instability and poor performance of systems, and it is encountered in a variety of engineering systems such as

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[–, –]. But many of the results are about constant time-delay, in practice, which is conservative to some extent. Recently, interval time-varying delay has been identiﬁed in practical engineering systems. Interval time-varying delay is a time delay that varies in an interval in which the lower bound is not restricted to be zero. For more details, we refer readers to [–].

In the past two decades, the T-S fuzzy model has been used to represent many nonlin-ear systems. And practice has proved that the T-S fuzzy model is very eﬀective to simplify many complex systems. In [], the ﬁlter was designed through discrete-time singularly perturbed T-S fuzzy model. For T-S fuzzy discrete-time systems,H∞descriptor fault

de-tection ﬁlter was designed in []. New results on H∞ ﬁltering for fuzzy systems with

interval time-varying delays were obtained in []. However, to the authors’ knowledge, the problem of delay-dependent exponentialH∞ﬁlter for nonlinear singular systems with

interval time-varying delay through T-S fuzzy model has rarely been reported.

Motivated by the above discussions, the problem of delay-dependent exponentialH∞

filter for nonlinear singular systems with interval time-varying delay through T-S fuzzy mode is considered in this paper. The main constructions of this paper can be concluded as follows. () The triple integral was taken into account when constructing the LKF, which can reduce the conservatism effectively. () The use of Lemma  refines the delay interval, which is the other reason why the result is less conservative. () The parameters of the filter can be obtained by solving the LMIs. () The comparisons with other references are listed in the form of tables, which can clearly show the advantage of the proposed method in this paper.

2 Preliminaries

Consider the following nonlinear system with interval time-varying delay which could be approximated by a T-S singular model withrplant rules.

Plant rulei: Ifξ(t) isMiandξ(t) isMi· · · ξp(t) isMpi, then

Ex˙(t) = (Ai+Ai)x(t) + (Ai+Ai)x

t–d(t)+ (Bi+Bi)ω(t), y(t) = (Ci+Ci)x(t) + (Ci+Ci)x

t–d(t)+ (Di+Di)ω(t), z(t) = (Li+Li)x(t) + (Li+Li)x

t–d(t),

x(t) =φ(t), t∈[–d, ],

()

whereMijdenotes the fuzzy sets;rdenotes the number of model rules;x(t)∈Rnis the

input vector;y(t)∈Rq_{is the measurement;}_z₍_t₎_∈_Rp_{is the signal to be estimated;}_w₍_t₎_∈_Rm

is the noise signal vector;φ(t) is a compatible vector valued initial function;Eis a singular matrix and assume thatrankE=r≤n;Ai,Ai,Bi,Ci,Ci,Di,Li,Liare constant matrices

with appropriate dimensions;ξ(t),ξ(t), . . . ,ξp(t) are the functions of state variables;d(t) is

time-varying delay and satisﬁes

≤d≤d(t)≤d, d˙(t)≤μ;

Ai,Ai,Bi,Ci,Ci,Di,Li,Liare unknown matrices describing the model

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unknown real time-varying matrix satisfying

F_iTFi≤I. ()

The parametric uncertaintiesAi,Ai,Bi,Ci,Ci,Di,Li,Liare said to

be admissible if both () and () hold.

Letε(t) = [ε(t)ε(t)· · ·εp(t)]T, by using a center-average defuzziﬁer, product fuzzy

in-ference and singleton fuzziﬁer, the global model of the system is as follows:

Ex˙(t) =

In this paper, we consider the following delay-dependent fuzzy ﬁlter:

Ex˙ˆ(t) =

the ﬁlter matrices with appropriate dimensions, which are to be designed.

Remark  WhenE=IandCﬁ= , the fuzzy ﬁlter () was studied in [], which is

time-independent ﬁlter. The simulations will demonstrate the advantage in this paper. The ﬁltering error system from () and () can be described by

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where

System () is simply denoted by the following form:

¯

Definition  The filtering error system () is said to be exponentially admissible if it sat-isfies the following:

(a) The augmented system () withw(t) = is said to be regular and impulse-free if the pair(E¯,A˜),(E¯,A˜+A˜d)is regular and impulse-free;

(b) The ﬁltering error system () is said to be exponentially stable if there exist scalars δ> andλ> such thatx(t) ≤δe–λt_φ₍_θ₎_d

, whereφ(θ)d=sup–d≤θ≤|φ(θ)| and| · |is the Euclidean norm inRn_{. When the above inequality is satisﬁed,}_λ_,_σ _and e–λt_φ₍_θ₎_d

 are called the decay rate, the decay coeﬃcient and an upper bound of

the state trajectories, respectively.

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Lemma  For any symmetric positive-deﬁnite constant matrices M> and any constant matrix E∈Rn×n_,_{and scalars d}

>d> ,if there exists a vector functionx˙(·) : [–d, ]→Rn

such that the following integration is well deﬁned,then it holds that

–d

 –d



–d

–d

t

t+θ

Ex˙(t)TMEx˙(t)ds dθ

≤

x(t)

t–d t–d x(s)ds

T

–d

ETME dETME

dETME –ETME

x(t)

t–d t–d x(s)ds

.

Proof Notice that

(Ex˙(t))TMEx˙(t) (Ex˙(t))T

Ex˙(t) M–

=

(Ex˙(t))TM/ _

M–/ 

M/_E_x_˙₍_t₎ _M–/

 

≥.

Hence,

–d

–d

t

t+θ(Ex˙(t))

T

MEx˙(t)ds dθ –d

–d

t

t+θ(Ex˙(t))

T ds dθ –d

–d

t

t+θ(Ex˙(t))ds dθ

–d

–d

t t+θM

–_{ds d}_θ

≥.

Using Schur complements and some simple manipulation, the lemma can be obtained.

Lemma ([]) Suppose that a positive continuous function f(t)satisﬁes f(t)≤ς sup

t–τ≤s≤t

f(s) +ςe–εt,

whereε> ,ς< ,ς> andτ> .Then f(t)satisﬁes

f(t)≤ sup –τ≤s≤

f(s)e–ξt₊ ςe

–ξt

 –ςeξτ

,

whereξ=min{ε,ξ}and <ξ< –(/τ)lnς.

Lemma ([]) Given a set of suited dimension real matrices Q,H,E,Q is a symmetric matrix such that Q+HFE+ET_FT_HT _{< }_{for all F satisﬁes F}T_F_≤_{I if and only if there} existsε> such that

Q+εHHT+ε–ETE< .

Aims of this paper The exponentialH∞ﬁlter problem to be addressed in this paper is

for-mulated as follows: given the uncertain time-delay T-S fuzzy system () and a prescribed level of noise attenuationγ > , determine an exponentially stable ﬁlter in the form of () such that the following requirements are satisﬁed:

(a) The ﬁltering error system () is exponentially admissible;

(b) Under zero initial conditions, the ﬁltering error system () satisﬁes

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3 Main results

3.1 Exponential stability andH_∞performance analysis

Theorem  For prescribed scalar≤μ< ,system()is exponentially admissible with H∞ performance γ for any time delay d(t)satisfying ≤d≤d(t)≤d if there exist

Proof Firstly, we prove that system () is regular and impulse-free. We choose two non-singular matricesUandV, note that

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On the other hand, via the Schur complement, it can be seen from () that< ,

pre-multiplying and post-pre-multiplying it, by [I I I I] and its transposition, we can get

PT_(A˜+A˜d) + _˜

AT_+A˜T_d_P+Z< ,

together withZ> , which impliesA˜andA˜dare non-singular. Hence, the pair (E¯,A˜) and

(E¯,A˜+A˜d) is regular and impulse-free. According to Deﬁnition , system () is regular and

impulse-free.

Next, we show the exponential stability of system (). Choose the following Lyapunov functional

V(x˜t) =



i=

Vi(x˜t), ()

where

V(x˜t) =x˜T(t)E¯ T

Px˜(t),

V(x˜t) =

t

t–d ˜

xT(s)Qx˜(s)ds+ t–d

t–d ˜

xT(s)Qx˜(s)ds+ t–d

t–d(t)

˜

xT(s)Qx˜(s)ds,

V(x˜t) =d 

–d

t

t+θ

˙˜

xT(s)E¯TZE¯x˙˜(s)ds dθ+d –d

–d

t

t+θ

˙˜

xT(s)E¯TZE¯x˙˜(s)ds dθ,

V(x˜t) =d 

–d

t

t+θ

˜

xT(s)Zx˜(s)ds dθ+d –d

–d

t

t+θ

˜

xT(s)Zx˜(s)ds dθ,

V(x˜t) = d





–d



θ

t

t+λ

˙˜

xT(s)ER¯ E¯x˙˜(s)ds dλdθ

+ds

–d

–d



θ

t

t+λ

˙˜

xT(s)E¯TRE¯x˙˜(s)ds dλdθ,

wherex˜t=x˜(t+α), –d≤α≤.

Then the time-derivative ofV(x˜t) along the solution of system () gives ˙

V(x˜t) = x˜T(t)E¯TPx˙˜(t), ˙

V(x˜t)≤ ˜xT(t)Qx˜(t) +x˜T(t–d)(–Q+Q+Q)x˜(t–d)

–x˜T(t–d)Qx˜(t–d) – ( –μ)x˜T

t–d(t)Qx˜

t–d(t),

˙

V(x˜t) =x˙˜ T

(t)E¯Td_Z+dZ _¯

Ex˙˜(t) –d t

t–d ˙˜

xT(s)E¯TZE¯x˙˜(s)ds

–d t–d

t–d ˙˜

xT(s)E¯TZE¯x˙˜(s)ds,

˙

V(x˜t) =x˜T(t)

d_Z+dZ

˜ x(t) –d

t

t–d ˜

xT(s)Zx˜(s)ds

–d t–d

t–d ˜

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˙

Taking into account the lemma (Jensen’s integral inequality), Lemma  and Lemma , we get

Then we have

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=

Now, we have

d

Integrating both sides from  toT>  gives

eεTV(x˜T) –V(x˜) = (εδ–δ)

By interchanging the integration sequence, we can get that

T

there exists a scalarκ>  such that

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It is not diﬃcult to see that, for anyT> ,

V(x˜T)≤κe–εTφ(θ)_d

. ()

Note that the regularity and the absence of impulses of the pair (E,A˜) imply that there always exist two non-singular matricesMandNsuch that

MEN¯ =

Ir 

 

, MAN˜ =

A 

 In–r

, ()

MA˜dN=

Ad Ad

Ad Ad

, M–T_PN₌

P P

P P

,

NTQN=

Q Q

QT

 Q

.

()

Substituting the partition into () yields thatP> ,P= .

Deﬁne

ε(t) =

ε(t) ε(t)

=N–x˜(t). ()

Using the expressions in (), () and (), system () can be written as

˙

ε(t) =Aε(t) +Adε

t–d(t)+Adε

t–d(t),  =ε(t) +Adε

t–d(t)+Adε

t–d(t).

()

It is easy to show that

V(x˜t,t)≥ ˜xT(t)E¯TPx˜(t) =x˜T(t)N–T

NTE¯TMTM–TPNN–x˜(t)

=εT(t)

Ir 

 

P P

P P

ε(t) =εT_(t)Pε(t)≥



P– ε(t)



.

Hence, for anyt≥d,

ε(t)≤P–κe–εTφ(θ)_d

. ()

Deﬁneς(t) =Adε(t–d(t)), then from () a scalarm>  can be found such that for

anyt> ,ς(t)≤me–εt_φ₍_θ₎

d.

To study the exponential stability ofε(t), we construct a function as

J(t) =ε_T(t)Qε(t) –εT

t–d(t)Qε

t–d(t). ()

By pre-multiplying the second equation of () withεT_(t)PT, we obtain that

 =εT_(t)P_Tε(t) +εT(t)PTAdε

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Adding () in () and using Lemma  yields that

J(t) =ε_T(t)PT_+P+Q

ε(t) + εT(t)PTAdε

t–d(t) –εT_t–d(t)Qεt–d(t)+ ε_T(t)PT_ς(t)

≤

ε(t) ε(t–d(t))

T PT

+P+Q PTAd

∗ –Q

ε(t) ε(t–d(t))

+γεT(t)ε(t) +γ–ςT(t)PTPς(t), ()

whereγis any positive scalar.

Pre-multiplying and post-multiplying

 

∗ 

< ,

by

N 

 N

T

and

N 

 N

,

respectively, a scalarγ>  can be found such that

PT

+P+Q PTAd

∗ –Q

≤–

γI 

 

.

Then

J(t)≤(γ–γ)εT(t)ε(t) +γ–ςT(t)PPTς(t).

On the other hand, sinceγcan be chosen arbitrarily,γcan be chosen small enough

such thatγ–γ> . Then a scalarγ>  can always be found such that

Q– (γ–γ)I≥γQ. ()

It follows from (), () and () that

εT_(t)Qε(t)≤γ–εT

t–d(t)Qε

t–d(t)+ (γγ)–ςT(t)PPTς(t),

which infers f(t)≤γ_–sup_t_–_d

≤s≤tf(s) +τe

–σt_{, where  <}_σ _< _min_{_ε_,_d–_ln_γ

}, f(t) = εT_(t)Qε(t) andτ = (γγ)–mPφ(t)d.

Therefore, applying Lemma , we can obtain that

ε(t)≤λ–min(Q)λmax(Q)e–σtε(t)+

λ–_min(Q)τe–σt

 –γ_–eσd ,

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Finally, we show that for any nonzero ω(t)∈L[,∞), system () satisﬁes e(t) < γω(t)under the zero initial condition.

Observing (), we have

˙

V(x˜t) +eT(t)e(t) –γωT(t)ω(t) < .

Integrating both sides from  toT>  gives

V(T) –V() +

T



e(t)dt–

T



γω(t)dt< .

SinceV() = , whenT→ ∞, we have

∞



e(t)dt<

∞



γω(t)dt,

which implies e(t)<γω(t) for any nonzero ω(t)∈L[,∞). This completes the

proof.

Remark  It should be pointed out that LKF () is chosen here based on triple integral and both left and right endpoints of the time-varying delay interval, which will decrease the conservatism.

Remark  In [],_–_h_tt₊_θξ˙T(s)Zξ˙(s)ds dθ was estimated as

hξ˙T(t)Zξ˙(t) –

t

t–d(t)

˙

ξT(s)Zξ˙(s)ds,

and the term –_tt_––_hd(t)ξ˙T(s)Zξ˙(s)dswas ignored, which may lead to considerable conser-vativeness. In this paper, by using Lemma , the shortcoming can be avoided and a less conservative result can be obtained.

3.2 Delay-dependent ﬁlter design

Based on the suﬃcient conditions above, the design problem of robustH∞ﬁlter can be

transformed into a problem of LMIs.

Theorem  Given a ﬁnite scalarγ > ,the exponential H∞ ﬁlter problem is solved for system()if there exist constant scalarsεij> ,symmetric matrix

Q,

Q,

Q,

R,

R,

Z,

Z,and matrix P=diag(P,P),P> ,such that

ETP=PTE≥, ETP=PTE≥, ()

ii< , i= , , . . . ,r,

ij+ji< , i<j,i,j= , , . . . ,r,

()

where

ij=

⎡ ⎢ ⎣

ij  εijT

∗ –εijI  ∗ ∗ –εijI

⎤ ⎥

⎦, ij=

⎡ ⎢ ⎣

ij ij ij ∗ ij  ∗ ∗ ij

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Then we have

and other items are deﬁned in Theorem . Based on () and from Lemma , we obtain that

ij=ηFiη+ηTFiTηT≤ε–ijηηT+εijηTη,

Via the Schur complement, we obtain that

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It can be easily shown that inequality () equals the following condition:

r

i=

h_iε(t)ii+ r

i<j hi

ε(t)hj

ε(t)(ji+ij) < .

LetQ=Wˆ–_,_M

i=PTAfi,Mi=PTBfi,Mi=PTCfi, and then it leads to LMIs (). This

completes the proof.

The parameters of theH∞ﬁlter are given by

Aﬁ=

P_–TMi, Bﬁ=

P–_ TMi, Cﬁ=

P_–TMi, Lﬁ.

4 Numerical value examples

Example  In order to show the advantage of the proposed method in this paper, we consider the time-delay T-S fuzzy systems as the system shown in Example  in [] with two rules:

IFxisWi(i= , ), THEN

Ex˙(t) = (Ai+Ai)x(t) + (Adi+Adi)x

t–d(t)+ (Bi+Bi)ω(t), y(t) = (Ci+Ci)x(t) + (Cdi+Cdi)x

t–d(t)+ (Di+Di)ω(t), z(t) = (Li+Li)x(t) + (Ldi+Ldi)x

t–d(t),

where

A=

 . –. 

, A=

 . –. 

, Ad=

 . –. .

,

Ad=

 .  .

, B=

 

, B=

 

, C= [ ],

Cd= [–. .], C= [. –.], Cd= [–. .],

L=D= [–. .], L=D= [–. ],

H=

. –.

, H=H=

. –.

, H=

–. .

,

H=H=

–. .

, H=H=

–. .

,

E=E=E= [–. .], E= [–. .],

E=E= [. –.], E= [–. –.],

E= [–. .], E= [–. .], E= [. –.].

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Then consider thatEis singular, suppose

E=

 . . .

,

and letμ= ,d= ,d= ,γ= . The ﬁlter parameters can be obtained as follows:

Af=

–. . –. –.

, Af=

–. . –. –.

,

Bf=

. .

, Bf=

–. .

,

Cf=

–. –. –. –.

, Cf=

. . –. –.

,

Lf= [. .], Lf= [–. –.].

However, it cannot be solved by criterion in [] because of the emergence of the impulse phenomenon. Based on the discussion above, the results obtained in this paper are more widely applicable and less conservative.

Example  Consider the example in [] with two fuzzy rules and without uncertainties: IFxisWi(i= , ), THEN

˙

x(t) =Aix(t) +Adix

t–d(t)+Biω(t), y(t) =Cix(t) +Cdix

t–d(t)+Diω(t), z(t) =Eix(t) +Edix

t–d(t),

where

A=

–. .  –

, A=

–.  –. –.

, Ad=

–. . –. –.

,

Ad=

–.  –. –.

, B=

 –.

, B=

. .

, C= [ ],

C= [. –.], Cd= [–. .], Cd= [–. ],

D= ., D= –., E= [ –.], E= [–. .],

Ed= [. ], Ed= [ .].

Let the fuzzy weighting functions beh(θ(t)) =sin(t) andh(θ(t)) =cos(t). Forσ=  in

[], Table  gives diﬀerent values ofdand yields minimumγ ford= ,μ= .. From

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Table 1 The minimumγ with differentd2(Example 2)

d2 0.5 0.6 0.8 1

[17] 0.38 0.43 0.83 2.22 [19] 0.27 0.31 0.58 1.57 Theorem 2 0.14 0.27 0.42 1.44

Table 2 Comparison of minimumγ(Example 3)

Method [21] [19] [20] This paper

γ 0.7 0.4338 0.3449 0.2748

Figure 1 Tunnel diode circuit (Example 4).

Example  Consider TS fuzzy system () without uncertainties and the fuzzy ruler=  in [], the parameters are as follows:

A=

–. .  –

, A=

–. . –. –.

, A=

–.  –. –.

,

A=

–.  –. –.

, B=

 –.

, B=

. .

, C= [ ],

C= [–. .], C= [. –.], C= [–. ],

D= ., D= –., L= [ –.], L= [–. .],

L= [. ], L= [ .].

Setd= ,d= .,μ= .. Table  lists the comparison results of the minimum ofγ

with [–] (δ= ).

From the comparison, it is clear that the minimumγ obtained in this paper is smaller. It has to be mentioned that the value ofγ depends on the parameterδin [–]; however,

δis not easy to determine. In this paper, this drawback can be avoided.

To sum up, the method in this paper is less conservative and easily computed.

Example  Consider a tunnel diode circuit shown in Figure , whose fuzzy model was done in [].x(t) =vc(t),x(t) =iL(t),ω(t) is the disturbance noise input,y(t) is the

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Figure 2 Error response ofe(t) (Example 4).

Plant rulei(i= , ): IFx(t) isMi(x(t)), THEN

˙

x(t) =Aix(t) +Biω(t), y(t) =Cix(t) +Diω(t), z(t) =Lix(t),

where

A=

–.  – –

, A=

–.  – –

, B=B=

 

,

C=C= [ ], D=D= , L=L= [ ].

The fuzzy membership function is assumed as follows:

h(t) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x(t)+

 , –≤x(t)≤,

, x(t) < –, –x(t)

 , ≤x(t)≤,

, x(t) > ,

h(t) =  –h(t).

Give theH∞ performanceγ = , the following ﬁlter parameter matrices could be ob-tained:

Af=

–. –. –. –.

, Bf=

–. –.

, Lf= [. .],

Af=

–. –. –. –.

, Bf=

. .

, Lf= [. .].

We assume the disturbanceω(t) = .e.t_{, using this ﬁlter, Figure  plots the response}

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5 Conclusion

In this paper, based on the T-S fuzzy model, we have treated the delay-dependent exponen-tialH∞ﬁlter for a class of nonlinear singular systems with interval time-varying delay. In

order to obtain a less conservative result, a new filter and new LKF has been constructed. The newH∞filter guarantees the error system to be regular, impulse-free, exponentially stable and satisfies a prescribedH∞performance. Three numerical examples have

demon-strated the eﬀectiveness and superiority of the proposed approach.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MC carried out the main part of this paper, YM participated in the discussion and revision of the paper. All authors read and approved the ﬁnal manuscript.

Acknowledgements

This work was supported by the National Natural Science Foundation of China No. 61273004 and the Natural Science Foundation of Hebei province No. F2014203085. The authors would like to thank the editor and anonymous reviewers for their many helpful comments and suggestions to improve the quality of this paper.

Received: 16 March 2015 Accepted: 17 June 2015

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