On energy-to-peak filtering for semi-Markov jump singular systems with unideal measurements

Signal Processing 144 (2018) 127–133

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On energy-to-peak filtering for semi-Markov jump singular systems with unideal measurementsR Hao Shen a,b,∗, Shiyu Jiao b, Shicheng Huo b, Mengshen Chen b, Jianning Li a, Bo Chen c a

School of Automation, Hangzhou Dianzi University, Hangzhou 310018, China School of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan 243002, China c College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China b

a r t i c l e

i n f o

Article history: Received 18 December 2016 Revised 5 October 2017 Accepted 8 October 2017 Available online 9 October 2017 Keywords: Markov jump models Singular semi-Markov jump systems Energy-to-peak filtering Unideal measurements

a b s t r a c t This paper focuses on the energy-to-peak filtering issue for a class of singular semi-Markov jump systems with unideal measurements. Some network-induced phenomena, such as sensor nonlinearity and packet dropouts caused by the unideal measurements are considered. The occurrence of sensor nonlinearity is described in a random way and obeyed a Bernoulli distribution. In the framework of the Lyapunov–Krasovskii stability theory, some sufficient conditions are given to ensure that the considered error system is stochastically mean-square stable and guarantees an energy-to-peak (or called L2 − L∞ ) performance level. On the basis of these conditions, an available design method to the desired filter is proposed drawing support from an improved matrix decoupling approach. For showing the effectiveness and superiority of the proposed method, we finally provide two illustrated examples.

1. Introduction Singular systems (SSs) are regard as implicit systems, differential-algebraic systems, generalized systems, descriptor systems or semi-state systems. Owing to their approximate description of some physical systems compared with the state-space systems, SSs have been applied in quantities of areas, such as, power systems, electrical circuits, chemical processes and others. As a consequence, it is no wonder that SSs have attracted particular research attention during the last several decades and tremendous research progress has been generated, (see [1–4], and the references therein). Furthermore, it is certainly worth pointing out that a kind of stochastic SSs, i.e., Markov jump singular systems (MJSSs), has been seen as a highlight during the development of SSs [5]. In fact, MJSSs represent the stochastic switched systems composed of a number of sub-systems where switching among themselves is governed by a Markov chain. Because of the above light spot of Markov jump systems (MJSs) [6–9], MJSSs have the

R This work was supported by the Project supported by Open Foundation of first level Zhejiang key in key discipline of Control Science and Engineering; the National Natural Science Foundation of China under Grant 61304066,617030 04,615030 02,61473171, the Natural Science Foundation of Anhui Province under Grant 1708085MF165. ∗ Corresponding author at: School of Automation, Hangzhou Dianzi University, Hangzhou 310018, China. E-mail address: [email protected] (H. Shen).

https://doi.org/10.1016/j.sigpro.2017.10.013 0165-1684/© 2017 Elsevier B.V. All rights reserved.

© 2017 Elsevier B.V. All rights reserved.

unique advantage for modelling the SSs with abrupt changes in their structures. Therefore, quite a few research efforts have been devoted to the study of MJSSs. For instance, some design methods to solving the H∞ filtering and control issues for continuous-time MJSSs were proposed in [10] and [11], respectively. In the context of discrete-time MJSSs, the H∞ filtering problem was addressed in [12]. Although the MJSSs get the favour of many researchers for their better ability to model systems, it still has a certain degree of boundedness in some scenarios due to the inherent imperfections of MJSs. A truism restriction for MJSs is that the sojourn-time existed in two consecutive jumps is required to follow the exponential distribution which is a memoryless distribution [13]. Such a requirement on MJSs is too harsh, which not only is unreasonable in some practical applications but also brings some conservative. So as to relax the restriction, a new class of MJSs called semi-Markov jump systems (SMJSs) [14–17] was proposed. Very recently, due to the fact that SMJSs have maintained momentum not only in practice but also in theory, quite a few efforts have been assembled into this topic [18,19]. Unfortunately, a drawback of the above-mentioned works is lack of considering the feature of singular systems in the study of SMJSs [20,21]. Fortunately, Wang et. al. originally investigated the study for continuous-time semiMarkov jumping singular systems (SMJSSs) with uncertainty and nonlinearity in [22], where a sufficient condition was presented to successfully ensure that the nonlinear SMJSSs were stochastically admissible. It should be also pointed out that the network induced

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H. Shen et al. / Signal Processing 144 (2018) 127–133

phenomena were not fully considered in [22]. Such a regret provides the first motivation of our recent work. Different from [22], we focus on the issue of energy-to-peak filtering for SMJSSs, where the phenomena of sensor nonlinearity and packet dropouts occur simultaneously. In parallel, the robust filtering problem is a permanent topic in the study of SSs with external noises. In the practical situation, some system states are often difficult to be measurable, or even unmeasurable due to technical constraints and other factors [23–30] . How to estimate the unmeasurable states becomes an interesting question. Mainly for the above reason, the concept of filtering has been put forward. With tremendous attention from researchers, the filtering technique has developed rapidly, from the conventional Kalman filtering which has long enjoyed a good reputation for handling the systems with white noise to energy-to-peak filtering and H∞ filtering which are only required the noise signal is the ∞-norm and energy-bounded. More precisely, for the issue of energy-to-peak filtering, which includes of minimizing the peak error value, the filter design with ensured the fixed energy-to-peak performance of uncertain systems was discussed in [31,32]. The results in [33] coped perfectly with such a problem for a class of MJSSs. It is worth remarkable that the results in [31,33] were obtained based on an ideal network transmission environment, that is, any network induced phenomenon does not occur. However, such an assumption could be hardly guaranteed due to the complication of environments. Therefore, it is necessary to consider the filtering problem for systems with unreliable links. Above all, how to address this intriguing question is the another motivation of this work. Summarizing the discussions mentioned above, this paper addresses the issue of energy-to-peak filtering for SMJSSs. A construction method of the mode-dependent filter is given to ensure the resulting error system is stochastically mean-square stable with an energy-to-peak disturbance rejection attenuation level. There are three main contributions in this paper: 1) The issue of energy-to-peak filtering for a class of quite comprehensive system models (i.e. SMJSSs) is investigated for the first time; 2) The unideal measurements case is fully taken into account, and accordingly the occurrence of some network induced phenomena include sensor nonlinearity and packet dropouts caused by the unideal measurements are described in a stochastic way. 3) An improved matrix decoupling approach, which may provide more free degree and flexibility to structure the desired filter than that in [10], is introduced. 2. Problem formulation Considering the following class of linear continue-time singular system ( )

E x˙ (t ) = A(α (t ) )x(t ) + B(α (t ) )ω (t ),

(1)

y(t ) = δ (α (t ), t )C (α (t ) )x(t ) + (1 − δ (α (t ), t ) )(x(t ) ),

(2)



ability matrix



= {πmn ( )} is described by



Pr {α (t +  ) = n|α (t ) = m } =

πmn ( ) + o( ), m = n , 1 + πmn  + o( ), m = n

(4)

where  > 0 is the sojourn time, lim→0 (o( )/ ) = 0 and π mn () ≥ 0, for n = m, is the transition rate from mode m at time t to mode n at time t +  and



πmm ( ) = −

πmn ( ).

n∈S ,n=m

Remark 1. As stated in [18], the transition rate of semi-Markov jump process is often partly available. Therefore, that  we assume  the transition rate π mn () is in the range of

i ,π j πmn mn and can

be naturally rewritten as follows

πmn ( ) = and

πmn,h =



H 

H 

h=1

h=1

βh πmn,h ,

βh = 1, βh  0,

i πmn,h + (h − 1 )

π

j mn,h

− (h − 1 )

j i πmn,h −πmn,h

(5)

H−1

,

n = m,

n∈S

H−1

,

n = m,

n∈S

j i πmn,h −πmn,h

.

(6)

Specifically, the nonlinear function  ( · ) is assumed to satisfy the following condition for each x, y ∈ Ra [34]

[(x ) − (y ) −Y1 (x −y )] [(x ) − (y ) −Y2 (x −y )]  0, (0 ) = 0, T

(7) where Y1 and Y2 are appropriate dimensional constant matrices known beforehand. δ (α (t), t) are mode-dependent random variables taking values 0 or 1. They obey the following probability distribution laws

Pr {δ (α (t ), t ) = 1} = E {δ (α (t ), t )} = δ¯ (α (t ) ), Pr {δ (α (t ), t ) = 0} = 1 − δ¯ (α (t ) ), where δ¯ (α (t ) ) ∈ [0, 1] are known beforehand constants. Clearly, for the stochastic variables δ (α (t), t), one has

E



δ (α (t ), t ) − δ¯ (α (t ) ) = 0, 

 E |δ (α (t ), t ) − δ¯ (α (t ) )|2 = δ¯ (α (t ) ) 1 − δ¯ (α (t ) ) .

(8)

Remark 2. For modeling the randomly occurring of sensor nonlinearity and packet dropouts, the mode-dependent random variables δ (α (t), t) are taken into account, which are obeyed the Bernoulli distribution. Form the description of measurement output y(t) in (2), we can easily discover the following two facts: 1) under the condition of δ (α (t ), t ) = 0 , the equality of (2) will degrades to y(t ) = (x(t ) ), which only has the sensor nonlinear and denotes the packet dropout; 2) under the condition of δ (α (t ), t ) = 1 , the equality of (2) will changes to y(t ) = C (α (t ))x(t ), which only has the normal measurement output and expresses the normal case. In this paper, we are interested in designing a filter as follow

z (t ) = D(α (t ) )x(t ),

(3)

where x(t ) ∈ Ra , y(t ) ∈ Rb and z (t ) ∈ Rc are the system state and the measurement output and the controlled output, respectively. ω (t ) ∈ Rd is the disturbance. E ∈ Ra×a may be a singular matrix, and A(α (t)) , B(α (t)), C(α (t)) and D(α (t)) are fixed real constant matrices with appropriate dimensions. The random variable {α (t)} stands for a semi-Markov jump process with right continuous trajectories, which is homogeneous, finite-state and takes discrete values in a fixed set S = {1, 2, . . . , r }. Furthermore, its transition prob-

E x˙ f (t ) = A f (α (t ) )x f (t ) + B f (α (t ) )y(t ), z f (t ) = C f (α (t ) )x f (t ),

(9)

(10)

where xf (t) is the filter state vector, zf (t) is the estimate of z(t), Af (α (t)), Bf (α (t)) and Cf (α (t)) are constant real matrices of filter to be determined. From convenience point of view, we denote Am = A(α (t ) ) and Afm = A f (α (t ) ) for each α (t ) = m ∈ S, and the other symbols are similar denoted.

H. Shen et al. / Signal Processing 144 (2018) 127–133

129

Augmenting the continue-time system ( ) to the  filter (9)-(10), ˜ we can get the following filtering error system 



 δm (t ) − δ¯m B¯ m H1 ξ (t ) − δm (t ) − δ¯m B¯ f m (x(t ) )

 + 1 − δ¯m B¯ f m (x(t ) ) + C¯m ω (t ), (11)

E¯ ξ˙ (t ) = A¯ m ξ (t ) +



e(t ) = D¯ m ξ (t ), where



E 0

E¯ =

A¯ m =



0 x(t ) , ξ (t ) = , e(t ) = z (t ) − z f (t ), E x f (t )

Am δ¯m B f mCm

B¯ m =

(12)



0 , Afm





 0 0 Bm ¯ , B¯ f m = , C¯m = , Dm = Dm B f mCm Bfm 0



−C f m .

Before further proceeding, the following definition and lemmas are given first. 1. [35] Given a scalar β > 0, the filtering error system

Definition  ˜ is stochastically mean-square stable with an energy-to-peak 

 ˜ is stochastically meanperformance level β , if the system  square stable and under zero initial condition, the following is true for all nonzero ω(t):

 supE eT (t )e(t ) < β 2



t>0

∞



0

 ωT (t )ω (t ) dt.

H 

βh = 1,

h=1

β h ≥ 0, matrices Ah , B, if the following condition holds for each h = 1, 2, . . . , H:

∗ ∗

m 



1 − δ¯m PmT B¯ f m + μH1T Y¯ −2μI ∗

PmT C¯m 0 −I

 < 0,

(15)

E¯ T Pm ∗

D¯ Tm  0, β 2I

where

¯T

11 m  2Am Pm +



(16)

 πmn,h E¯ T Pn − μH1T Yˆ H1 , H1T  I



0 ,

n∈S





βh Ah + B < 0.

The addressed problem of energy-to-peak filtering for SMJSSs is, therefore, stated as follows: find a mode-depended filter design method such that

 ˜ is stochastically stable; (1) the filtering error system  (2) under the zero initial condition, the energy-to-peak performance ∞

T

t>0



˜ with Proof. At the beginning, let us explain that the system  ω = 0 is regular and causal. Owing to rank E = a  n, there exist non-singular matrices N and V such that

h=1

 supE e (t )e(t ) < β 2

m 

 11

m

(14)

˜ is stochastically meanthen the considered filtering error system  square stable with ensured a fixed energy-to-peak performance β .

then one can get that



E¯ T Pm = PmT E¯  0,

Y¯  Y1 + Y2 , Yˆ  Y1T Y2 + Y2T Y1 ,

Ah + B < 0, H 

Theorem 1. For given scalars δ¯m , β , and matrices Y1 , Y2 , if there exist a scalar μ and real matrices Pm > 0 so that following conditions hold for each m ∈ S

(13)

T Lemma 1.

[36] For a given symmetrical matrix W = W = W1 W2 , the following three conditions are equivalent: ∗ W3 1) W < 0; 2) W1 < 0, W3 − W2T W1−1W2 < 0; 3) W2 < 0, W1 − W2T W3−1W2 < 0.

Lemma 2. For scalars β h , h = 1, 2, . . . , H, which satisfy

Fig. 1. The possible time sequences of the mode jumps for α (t ).

0



 ω (t )ω (t ) dt



I N E¯V = a+a 0 Then, denote



N A¯ mV =

T

is ensured for any ω(t) = 0.

A¯ 1m A¯ 3m

0 . 0

(17)



A¯ 2m P¯ , V −1 Pm N = 1m ¯ P¯3m A4m

(18)

From (14), (17) and (18), we can get that for each m ∈ S

P¯2m = 0. 

3. Main results In such a section, our main aim is to decide the parameters of the filter in the form  of (9)-(10) to ensure the filtering error dy˜ is stochastically mean-square stable and satnamics system  isfies a performance level in the form of energy-to-peak. In the following, we will firstly establish the performance analysis conditions as follows.

P¯2m . P¯4m

11 m

VT

by and V, respectively, one Pre- and post-multiplying has that A¯ T4m P¯4m + P¯4m A¯ 4m < 0. Hence, for each m ∈ S, A¯ 4m are non

 singular and thus the pair E¯ , A¯ m is regular and causal. And in

 ˜ with ω = 0 is reglight of the Definition 1 in [33], the system 



˜ ular and causal. Next position, we will prove that the system  is stochastically mean-square stable.

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H. Shen et al. / Signal Processing 144 (2018) 127–133

Fig. 2. The state response of e(t).



˜ , the Lyapunov function for analyzing perFor the system  formance conditions is chosen as

V (ξ (t ), m, t ) = ξ

T

(t )E¯ T Pm ξ (t ).



E {LV (ξ (t ), m, t )} = E 2ξ˙ T (t )E¯ T Pm ξ (t ) +



 πmn ( )ξ

T

(t )E¯ T Pn ξ (t ) .

 ⎡ ˆ 11 ˆ 12

⎤

m m 1 − δ¯m (d2 − d1 )B˜ f m + μY¯ XmT Bm − d1VmT Bm ˆ 22 ⎥ ∗

− 1 − δ¯m d1 B˜ f m XmT Bm ˆm  ⎢ m

⎣ ⎦ ∗ ∗ −2μI 0 ∗

(20)

In light of (7), it is straightforward that for a positive scalar μ

μ

T

x(t ) (x(t ) )

Yˆ ∗

−Y¯ 2I





x(t )  0. (x(t ) )

(21)

By combining Lemma 2, (5), (15) and (21), we obtain

E {LV (xt , m, t )} < ωT (t )ω (t ). Further, under the condition of ω ≡ 0, one could yield that:

(23)

And under zero initial condition, one has

V (xt , m, t ) <



t 0

ωT (s )ω (s )ds.

∗

−I

m 

E T Xm ∗ ∗

(29) −d1 E T Vm d2 E T Vm ∗

DTm −C˜Tf m β 2I

  0,

(30)

where

(22)

E {LV (xt , m, t )} < 0.

∗

< 0,

n∈S



(28)

(19)

˜ Taking the time derivative along the trajectory of system  yields



E T Vm = VmT E  0,

(24)

T T T ˜T ¯ ˆ 11 ˆ

m  2Am Xm − 2d1 AmVm + 2δm (d2 − d1 )Cm B f m − μY   T  + πmn,h E Xn + (d2 − d1 )E T Vn , n∈S T T T ˜T ¯ ¯ ˆ 12 ˜T ˜

m  2Am Xm −d1Vm Am + δm (d2 −d1 )B f mCm −d1 A f m − δm d1Cm B f m 

T  + πmn,h E Xn − d1 E T Vn , n∈S



Moreover, by using Lemma 1, the condition (16) can be rewritten as

T T ˜T ¯ ˆ 22

m  2Am Xm − 2δm d1Cm B f m +

β 2 E¯ T Pm − D¯ Tm D¯ m  0.

 ˜ is stochastically mean-square stathen the filtering error system  ble with a prescribed energy-to-peak disturbance attenuation level β .

(25)

Therefore, one has

= ξ T (t )D¯ Tm D¯ m ξ (t )  β 2 ξ T (t )E¯ T Pm ξ (t ) < β 2

 0

t

ωT (s )ω (s )ds. (26)

Thus, in accordance with Definition 1, the considered system ˜ is stochastically mean-square stable and satisfies an ensured  energy-to-peak performance level β . This completes the proof. Theorem 2. For given scalars d1 , d2 , δ¯m , β , and matrices Y1 , Y2 , if there exist a scalar μ, real matrices Xm > 0, Vm > 0, A˜ f m , B˜ f m and C˜ f m so that the following conditions hold for each m ∈ S

E Xm =

XmT E

n∈S

In this case, the filter gains Afm , Bfm , and Cfm can be given by

eT (t )e(t )

T

πmn,h E T Xn ,

 0,

(27)

A f m  Vm−T A˜ f m , B f m  Vm−T B˜ f m , C f m  C˜ f m . Proof. Define



Pm 

Xm −d1Vm



E Then noting E¯ = 0



E¯ T Pm =



−d1Vm (Xm − Vm )−1 , Km  d2Vm (Xm − Vm )−1

E T Xm −d1 E T Vm

I . 0

0 , we have E

−d1 E T Vm = PmT E¯ . d2 E T Vm

(31)

H. Shen et al. / Signal Processing 144 (2018) 127–133

131

Inspired by [10], pre- and post-multiplying (29) by  diag (Xm − Vm )−T , I, I, I and its transpose, respectively. Then





pre- and post-multiplying the above result by diag K −T , I, I and its transpose, respectively, we can get the condition (15) holds. Further, the condition (30) can ensure that the conditions (14) and (16) are satisfied. This completes the proof.  Remark 3. Different with [10], the positive matrices Pm 

Xm −d1Vm in the Lyapunov–Krasovskii function is more −d1Vm d2Vm general in this paper, because of the introduction of d1 and d2 . Specially, when d1 = d2 = 1, the matrices Pm degrade to the same in [10] . And according to appropriately adjusting the values of d1 and d2 , the results in this paper can be less conservatism than [10], which will be illustrated in Example 2. Remark 4. In this paper, the time delays are not considered in the investigation of the addressed issue. In fact, such phenomena usually exist in many practical systems [37–43] . Accordingly, one future work is concerned to employ the proposed method in this work to cope with the problem of energy-to-peak filtering for semi-Markov jump singular delayed systems with unideal measurements.

Fig. 3. Comparisons of minimum allowed β min for different decoupling methods.

Under the above  obtained gains, we  suppose that the initial condition x1,0 = 1 is

ω (t ) = T

4. Illustrative examples In the section, with the help of two numerical examples, we will illustrate the effectiveness and improvement of our results, respectively. Example 1. In such an example, we consider the SMJSSs with two modes and the following system parameters



A1 =

B1 =

−3 0.3



1 −2.5 , A2 = −2.5 0.1



0 −0.6 , B2 = 1 0

1 0



C1 = C2 =

0.8 0.3



0.5 , −3.5

0 , 0.5

 0.3 , D1 = D2 = −0.5 0.5





−0.1 , E =

1 0

0 . 0

We suppose that the energy-to-peak performance level is β = 1, the scalars for optimizing results are set to d1 = d2 = 1, the transition probabilities of semi-Markov chain α (t) are π 12 () ∈ [0.2, 0.3] and π 21 () ∈ [0.7, 0.8], the probabilities of δ1 (t ) = 1 and δ2 (t ) = 1 are δ¯1 = 0.5 and δ¯2 = 0.6, respectively. The nonlinear function is chosen as

(x(t ) ) = 0.5((Y1 + Y2 )x(t ) + (Y1 − Y2 ) cos (t )x(t ) ),



0.16 0 0.18 0 , Y2 = . By solving the con0 0.18 0 0.16 ditions in Theorem 2, we can get the filter gains as follows where Y1 =



xT0 = x1,0

⎧ ⎪ ⎨0.1



0.1 ,

−0.1

⎪ ⎩0





−0.1 ,

0 ,

x2,0

T

and the external disturbance

5  t  10s 15  t  20s . otherwise

Then, the possible time sequences of the mode jumps for α (t) and the state response of e(t) are shown in Fig. 1 and Fig. 2, respectively. From the Fig. 1, we can see that the value of α (t) switches back and forth between 1 and 2 corresponded the above consideration of system mode number. And one could observe from Fig. 2 that the trajectory of filtering error e(t) reaches zero at about the simulation time 20s and then stabilizes at zero at the rest of simulation time. Therefore, the effectiveness of our method has been demonstrated by Fig. 2. Example 2. In this example, we want to illustrate the superiority of our decoupling methods compared with [10], because of the introduction of d1 and d2 . Firstly, the following parameters are considered



A1 =

A2 =















−0.7 0.5 −0.8 0 0.9 1.3 −0.1 , B2 = , C2 = , DT2 = , 0.1 −1.1 0 0.5 0.4 0.5 −0.3

E =

−1 0.5 1.5 0 0.6 0.3 −0.7 , B1 = , C1 = , DT1 = , 0.3 −1.3 0 0.4 1.2 0.5 −0.4

1 0

0 , π12 ( ) ∈ [0.2, 0.3], π21 ( ) ∈ [0.7, 0.8]. 0

And, the nonlinear function is given as the same with Example 1. Beside, the concept of optimizing rate (OR) is given for showing the improvement of our method evidently. The expression of OR can be shown as follow

βmin (using the decoupling method in [10]) − βmin (using our decoupling method) βmin (using the decoupling method in [10])

OR =

Af1 =

−11.7077 3.3112



C Tf 1 =

Af2 =





2.9620 −2.6127 , Bf1 = −10.8711 0.8835

−0.1895 , −0.0660

−7.5238 1.6710

C Tf 2





1.5629 −1.6741 , Bf2 = −7.5569 0.4757

−0.2756 = . −0.0767

0.1291 , −1.9053

0.0195 , −1.3472

By calculating the LMI-based conditions in Theorem 2, we can get the following table and Fig. 3 under different conditions: From the landscape of Table 1 and Fig. 3, one can observe that the smaller value of β min can be obtained in our results with choosing the different scalars d1 and d2 than those by using the decoupling method in [10] (i.e. under the condition of d1 = 1 and d2 = 1). That is to say, when we introduce the two scalars d1 and d2 , and appropriately adjust them, the value of β min can get smaller. Therefore, the introduction of d1 and d2 can optimize the energy-to-peak performance level to a certain degree and make our results have less conservatism than [10], which mentioned in

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H. Shen et al. / Signal Processing 144 (2018) 127–133

Table 1 Comparisons of minimum allowed β min for different decoupling methods. method

[10]

Theorem 2

Theorem 2

β min

d1 = 1, d2 = 1

d1 = 0.7, d2 = 3/d1 = 0.5, d2 = 6

OR

δ¯1 = δ¯2 = 0.2 δ¯1 = δ¯2 = 0.3 δ¯1 = δ¯2 = 0.4 δ¯1 = δ¯2 = 0.5 δ¯1 = δ¯2 = 0.6

0.2976 0.2520 0.2225 0.1993 0.1773

0.1592/0.1107 0.1548/0.1073 0.1520/0.1057 0.1498/0.1047 0.1479/0.1040

46.5%%/62.8%% 38.6%%/57.4%% 31.7%%/52.5%% 24.8%%/47.5%% 16.6%%/41.3%%

Remark 3. On the other hand, from the portrait of Table 1, it is easy to see that the value of β min decreases with the increasing of δ¯1 and δ¯2 . As a result, the above table can further testify the fact that the part of nonlinearity has adverse effect on system performance. 5. Conclusions In this paper, the energy-to-peak filtering problem over imperfect communication network links has been addressed. A class of reasonably comprehensive system models, more precisely, semiMarkov jump singular systems, has been introduced. Besides, some network induced phenomena, i.e., sensor nonlinearity and packet dropouts, are considered simultaneously, which occur in a stochastic way. By means of stochastic analysis and an improved matrix decoupling approach, sufficient conditions, which can guarantee that the considered system is stochastically mean-square stable with a prescribed energy-to-peak performance level, have been established. Based on these conditions, the explicit expression of the desired filter have been obtained. Two examples are finally presented to explain the superiority and effectiveness of our method. It should be remarkable that the data-driven filtering issue is a hot topic to achieve more industrial oriented results [44,45]. Therefore, a direct future work would be the investigation of the data-driven filter design methods. References [1] S. L. Campbell, Singular systems of differential equations, 1980, San Francisco: Pitman. [2] Z. Feng, J. Lam, Robust reliable dissipative filtering for discrete delay singular systems., Signal Proc. 92 (12) (2012) 3010–3025. [3] M. Kchaou, H. Gassara, A. El-Hajjaji, Robust observer-based control design for uncertain singular systems with time-delay., Int. J. Adapt. Control Signal Proc. 28 (2) (2014) 169–183. [4] S. Xu, J. Lam, Robust Control and Filtering of Singular Systems, Berlin: Springer-Verlag, 2006. [5] L. Wu, X. Su, P. Shi, Sliding mode control with bounded L2 gain performance of markovian jump singular time-delay systems., Automatica 48 (2012) 1929–1933. [6] X. Song, Y. Men, J. Zhou, J. Zhao, H. Shen, Event-triggered H∞ control for networked discrete-time Markov jump systems with repeated scalar nonlinearities., Appl. Math. Comput. 298 (2017) 123–132. [7] K. Mathiyalagan, J.H. Park, R. Sakthivel, S.M. Anthoni, Robust mixed H∞ and passive filtering for networked Markov jump systems with impulses., Signal Proc. 101 (2014) 162–173. [8] H. Shen, S. Xu, J. Zhou, J. Lu, Fuzzy H∞ filtering for nonlinear Markovian jump neutral systems., Int. J. Syst. Sci. 42 (5) (2011) 767–780. [9] L. Wu, P. Shi, H. Gao, State estimation and sliding-mode control of Markovian jump singular systems., IEEE Trans. Autom. Control 55 (5) (2010) 1213–1219. [10] Z. Wu, H. Su, J. Chu, Delay-dependent H∞ filtering for singular Markovian jump time-delay systems, Signal Proc. 90 (6) (2010) 1815–1824. [11] J. Wang, H. Wang, A. Xue, R. Lu, Delay-dependent H∞ control for singular markovian jump systems with time delay., Nonlinear Anal. Hybrid. Syst. 8 (2013) 1–12. [12] J. Lin, S. Fei, J. Shen, Delay-dependent H∞ filtering for discrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilities., Signal Proc. 91 (2) (2011) 277–289. [13] Z.-G. Wu, P. Shi, Z. Shu, H. Su, R. Lu, Passivity-based asynchronous control for Markov jump systems., IEEE Trans. Autom. Control 62 (4) (2016) 2020–2025. [14] F. Li, L. Wu, P. Shi, C.C. Lim, State estimation and sliding mode control for semi-Markovian jump systems with mismatched uncertainties., Automatica 51 (2015) 385–393.

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