- Email: [email protected]
Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
Available online at www.sciencedirect.com
Journal of the Franklin Institute xxx (xxxx) xxx www.elsevier.com/locate/jfranklin
Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channelsR Jing Wang a, Liang Shen a, Jianwei Xia b, Zhen Wang c,∗, Xiangyong Chen d,∗ a School
of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan 243002, China b School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, PR China c College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China d School of Automation and Electrical Engineering, Linyi University, Linyi 276005, China Received 16 November 2018; received in revised form 5 August 2019; accepted 19 September 2019 Available online xxx
Abstract The fuzzy asynchronous dissipative filtering issue for Markov jump discrete-time nonlinear systems subject to fading channels is discussed in this paper, where the Rice fading model is employed to characterize the fading channels phenomenon in the system measurements for the first time. The attention is focused on developing an available asynchronous filter, which can ensure that the underlying error system is dissipative. In this regard, several important performances can be investigated conveniently by introducing adjustment matrices. By means of the stochastic analysis theory and the network control technique, some sufficient conditions for the solvability of the addressed problem are presented, simultaneously, the gains of the filter desired are determined correspondingly. An illustrative example is finally exploited to explain the utilizability of the developed approach. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
R This work was supported by the National Natural Science Foundation of China under Grants 61703004, 61573008, 61973199, the National Natural Science Foundation of Anhui Province under Grant 1808085QA18. ∗ Corresponding authors. E-mail addresses: [email protected] (Z. Wang), [email protected] (X. Chen).
https://doi.org/10.1016/j.jfranklin.2019.09.031 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
2
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
1. Introduction The Takagi–Sugeno (T–S) fuzzy model, consisting of a set of local linear subsystems, was firstly proposed in 1985 [1]. In such a model, the local linear subsystems are smoothly connected by fuzzy membership functions, and the local input–output relationship of the nonlinear systems is represented by fuzzy IF-THEN rules [2,3]. For fuzzy systems, if the fuzzy reasoning system based on IF-THEN rules is used, the qualitative aspects of reasoning process and human knowledge can be modeled without precise quantitative analysis. Therefore, in recent years, especially in the field of control and estimation of nonlinear systems, it is hardly surprising that the T–S fuzzy model has been widely concerned and many important research results have emerged. For uncertain continuous-time T–S fuzzy systems, a novel integral sliding mode fuzzy control method on the basis of dissipation principle was proposed in [4], by which an improved adaptive fuzzy integral sliding mode control rate was designed. For continuous-time nonlinear systems, a fuzzy resilient energy-to-peak filter was designed in [5]. In the investigation of T–S fuzzy systems, sudden changes in their structure and parameters cannot be ignored. Such a case frequently encounters in many practical systems, due perhaps to a variety of reasons including internal subsystems faults, environmental conditions, sudden disturbances, changes in the correlation between subsystems and so on [6,7]. In this regard, one needs to introduce a kind of stochastic hybrid system models, namely, Markov jump systems (MJSs) [8–11]. It is known that MJSs play a vital role in modeling the systems subjected to sudden changes. Based on such models, many control problems have been well investigated [12–15]. Remarkably, scholars are, in most instances, taking particular aim at MJSs with normal channels. By comparison, there is less energy devoted to nonlinear MJSs, not to mention nonlinear ones subject to fading channels, which is one of the driving forces of the current work. The case of fading channels is a typical network-induced phenomenon when signals are transmitted over networks experiencing some obstacles or path losses. The main reasons of above-mentioned may be summed up as reflection, diffraction and scattering, which have a great influence on signal power. In recent years, many researchers have devoted to the investigation of fading channels [16–18]. Thereinto, a fundamental question, how to find an appropriate mathematical model to describe it, has become a hot topic. In order to response such an issue, the Rice fading model was introduced in [19]. Since then, many meaningful results have been put forward successively [20,21]. On the other hand, robust filtering is, roughly speaking, one of the essential topics in information and control theory [22,23]. For MJSs, there exist many results on Kalman filtering, H∞ filtering, passive filtering and dissipative filtering [24,25]. However, it should be noted that most of the developed filters are mode-dependent. This may limit their applications in some complex network environments [26]. One solution is to design asynchronous filters [27–29]. As we know, such a question still has no answers for nonlinear MJSs subject to fading channels. How to find a solution is, naturally, our main purpose in this work. In view of the above statements, the asynchronous dissipative filtering problem is discussed in this paper for nonlinear MJSs, where the fading channels phenomenon is fully taken into account. A T-S fuzzy system model approach is utilized to dispose of the nonlinearity in systems. By employing some stochastic analysis tools and fuzzy system theory, a fuzzy asynchronous filter is presented, and some sufficient conditions for ensuring the strict dissipativity property of the resulting error system are obtained. At length, an example is shown to exPlease cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
3
Fig. 1. Filtering for T–S fuzzy systems.
plain the availability of the presented approach. The main contributions of this paper can be summarized as follows: (1) For T-S fuzzy systems, incomplete measurements due to fading channels described by stochastic processes are considered. The property of such a phenomenon is well reflected in the Rice fading model used in this paper, which improves the reliability of the design method. (2) The extended dissipative performance is employed to evaluate the robustness of the system. By introducing adjustment matrices, several important performances can be investigated conveniently. (3) The filter is designed based on an asynchronous strategy since it is hard to implement real-time synchronization of mode information, which is more practical in the case of incomplete measurements. In addition, the notations are standard presented, which are similar as those in [17]. 2. Problem formulation In the work, the considered nonlinear MJSs under fading channels driven by T–S fuzzy model approach, are described in Fig. 1. According to Fig. 1, the measurement signals received by the filter are transmitted over fading channels. 2.1. Nonlinear MJSs based on T–S fuzzy model Firstly, consider a nonlinear discrete-time Markov jump system described by a T–S fuzzy model: Plant Rule ϱ: IF g1 (t ) is ψ ϱ1 , g2 (t ) is ψ ϱ2 , ..., gδ (t ) is ψ ϱδ , THEN x (t + 1 ) = A (α(t ))x(t ) + B (α(t )) f (x(t ) ) + C (α(t ))ω(t ),
(1)
y(t ) = D (α(t ))x(t ) + E (α(t ))ω(t ),
(2)
z(t ) = G (α(t ))x(t ),
(3)
where x(t) ∈ Rm , y(t) ∈ Rn and z(t) ∈ Rp are the state, the measured output and the signal to be estimated, respectively; ψ ϱi and gi (t ) are presented as a fuzzy set and the premise variable for = 1, 2, . . . , g, i = 1, 2, . . . , δ, respectively, and the scalar g is the number of IF-THEN rules; Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
4
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
ω(t) ∈ Rs is the disturbance input belonging to l2 [0, ∞); Aϱ (α(t)), Bϱ (α(t)), Cϱ (α(t)), Dϱ (α(t)), Eϱ (α(t)), and Gϱ (α(t)), are known matrices with appropriate dimensions. α(t) is the concerned discrete-time homogeneous Markov process taking values in a finite set L = {1, 2, . . . , L } with its transition probability matrix {γ ϖq } given by γ q = Pr {α(t + 1) = q|α(t ) = }, where 0 ≤ γ ϖq ≤ 1, ∀ , q ∈ L, and Lq=1 γ q = 1. For convenience, let α(t ) = , the system matrices can be denoted as Aϱ (ϖ), Bϱ (ϖ), Cϱ (ϖ), Dϱ (ϖ), Eϱ (ϖ), and Gϱ (ϖ), respectively. Assumption 1. [30] The nonlinear function f(•) in (1) satisfies f a´ + f b` ≤ a´ + b` , ∀a´, b` ∈ Rn .
(4)
The fuzzy basis functions in this paper are given by δ ψi (gi (t ) ) h (g(t ) ) g i=1 , δ i=1 ψi (gi (t ) ) =1 where ψi (gi (t ) )( = 1, 2, . . . , g, i = 1, 2, .g. . , δ) is the grade of membership of the premise parameter gi (t ) in ψ ϱi . It follows that =1 h (g (t ) ) = 1, h (g (t ) ) ∈ [0, 1]. By using the standard fuzzy blending method, the fuzzy mode of system (1)–(3) can be inferred as follows: x (t + 1 ) =
g
h (g(t ) ) A ( )x(t ) + B ( ) f (x(t ) ) + C ( )ω(t ) ,
(5)
=1
y(t ) =
g
h (g(t ) ) D ( )x(t ) + E ( )ω(t ) ,
(6)
h (g(t ) ) G ( )x(t ) .
(7)
=1
z(t ) =
g =1
2.2. Fading channels As we know, in some circumstances, the phenomenon of fading measurement is inescapable when the system measurements transmit to the filter through the channels with limited bandwidth. This may result in the difference occur between the measured output from the sensor and the actually measured output the filter received. To reveal this difference, the lth-order Rice fading model [31] is considered in this paper which is shown as follows: lt y˜(t ) = θs (t )y (t − s ) + v (t ), (8) s=0
where lt = min {l, t }, v (t ) is an external disturbance, and θ s (t) are the channel coefficients that are mutually independent random variables taking values of 0 and 1 with Pr {θs (t ) = 1} = ε {θs (t )} = θ¯s , Pr {θs (t ) = 0} = 1 − θ¯s . Remark 1. For Markov jump system (1), usually, the change of the system mode that results in the change of the received signal quantity of the filter. So introducing the channel coefficient here which is mode-independent. As a random fading model widely used in the field of signal Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
5
processing, the lth-order Rice fading model can initially deal with the initial phenomenon of fading channels for some multi-path transmission and shadowing problems. As far as we know, the lth-order Rice fading model is not only appropriate to point-to-point communication systems, but also can consider time delay, packet loss and fading channels, and it is usually assumed that its coefficients are Gauss random variables, being mutually independent, with the same distribution. Remark 2. As we know, the phenomenon of fading channels can be divided into large-scale fading and small-scale fading, while large-scale fading can be divided into path loss and shadow effect. The rain attenuation studied in [32,33] belongs to the shadow effect. As the name suggests, it refers to the attenuation caused by the wave entering the rain layer. The research on this aspect is not deeply enough, and it is worthy of further research in the future. From above descriptions, the asynchronous fuzzy filter can be expressed as follows: x¯(t + 1 ) =
g
h (g(t ) ) A f (β(t ) )x¯(t ) + B f (β(t ) )y˜(t ) ,
(9)
=1
z¯(t ) =
g
h (g(t ) ) G f (β(t ) )x¯(t ) ,
(10)
=1
where x¯(t ) ∈ Rm is the filter state and z¯(t ) ∈ R p is output vector. Afϱ (β(t)), Bfϱ (β(t)) and Gfϱ (β(t)) are the filter gains to be determined, where β(t) is assumed to be described by a non-stationary Markov chain, which takes values in a finite state set U = {1, 2, . . . , U } with Pr {β(t + 1) = n|β(t ) = ι} = πιnα(t+1) , U α(t+1) where πιnα(t+1) ∈ [0, 1], and = 1 for ∀ι, n ∈ U. For convenience, denoting n=1 πιn Afϱ (ι)Afϱ (β(t)), Bfϱ (ι)Bfϱ (β(t)) and Gfϱ (ι)Gfϱ (β(t)) for each β(t ) = ι ∈ U.
T From the conditions (5)–(10), let x˘(t ) = x T (t ) x¯T (t ) , z˘(t ) = z(t ) − z¯(t ), ωv (t ) = T
T
T ω (t ) vT (t ) , f (x˘(t )) = f T (x(t )) f T (x¯(t )) , and θ˜s (t ) = θs (t ) − θ¯s , respectively, the filtering error system can be obtained as follows: x˘(t + 1) = A( , ι)x˘(t ) + B( ) f (x˘(t ) ) + C( , ι) (t ) + θ¯0 D( , ι)x˘(t ) + θ˜0 (t )D( , ι)x˘(t ) +
lt
θ¯s D( , ι)x˘(t − s) +
s=1
+
lt
lt
θ˜s (t )D( , ι)x˘(t − s) + θ˜0 (t )E( , ι)ωv (t )
s=1
θ¯s E( , ι)ωv (t − s ) +
s=0
lt
θ˜s (t )E( , ι)ωv (t − s ),
(11)
s=1
z˘(t ) = G( , ι)x˘(t ),
(12)
where A( , ι)
g g =1 ρ=1
C( , ι)
g g =1 ρ=1
h (g(t ) )hρ (g(t ) )A˘ ρ ( , ι), B( )
g g
h (g(t ) )hρ (g(t ) )B˘ ( ),
=1 ρ=1
h (g(t ) )hρ (g(t ) )C˘ρ ( , ι), D( , ι)
g g
h (g(t ) )hρ (g(t ) )D˘ ρ ( , ι),
=1 ρ=1
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
6
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
E( , ι)
g g
h (g(t ) )hρ (g(t ) )E˘ρ ( , ι), G( , ι)
=1 ρ=1
g g
h (g(t ) )hρ (g(t ) )G˘ ρ ( , ι),
=1 ρ=1
with
A˘ ρ ( , ι) diag A ( ), A f ρ (ι) , B˘ ( ) diag B ( ), 0 ,
0 0 ˘ ˘ , Cρ ( , ι) diag C ( ), B f ρ (ι) , Dρ ( , ι) B f ρ (ι)D ( ) 0
0 0 , G˘ ρ ( , ι) G ( ) −G f ρ (ι) . E˘ρ ( , ι) B f ρ (ι)E ( ) 0 Lemma 1 [34]. Given an appropriate dimensional matrix Rρ , one has Rss
g g
h (g(t ) )hρ (g(t ) )Rρ < 0,
=1 ρ=1
if R < 0, ∀ = 1, 2, . . . , g, and 2 R + Rρ + Rρ < 0, ∀, ρ = 1, 2, . . . , g, = ρ. g−1 Definition 1 [35,36]. For given real matrices J1 = J1T ≤ 0, J2 , J3 = J3T > 0, and J4 = J4T > 0, a known constant φ = 0, 1. For any ϖ(t) ∈ l2 [0, ∞), the following condition holds:
ε ϒ (J1 , J2 , J3 , t ) ≥ φ sup ε z˘T (t )J4 z˘(t ) , ∀t > 0, (13) 0≤t≤
t=0
where
ϒ (J1 , J2 , J3 , t ) (1−φ) z˘ (t )J1 z˘(t ) + sym T
lt s=0
z˘ (t )J2 ωv (t −s ) T
+
lt
ωvT (t −s )J3 ωv (t −s ),
s=0
then, under the zero initial conditions, the filtering error system is said to be extended stochastically dissipative. Remark 3. As we know, the inequality (13) is considered to be an extended dissipative performance indicator. If φ = 0, let J1 = −J¯1 J¯1T < 0, J2 = 0 and J3 = ϑ 2 I > 0, the inequality (13) turns into the standard dissipative performance index; if φ = 0 and J2 = 0, let J1 = J1T < 0 and J3 = ϑ 2 I > 0, the inequality (13) becomes the H∞ performance index; if φ = 0 and J1 = 0, let J2 = J2T > 0 and J3 = ϑ 2 I > 0, inequality (13) transforms into the standard form of the passive performance index; if φ = 1, let J3 = ϑ 2 I > 0 and J4 = J4T > 0, inequality (13) is in line with l2 − l∞ performance index. Remark 4. Dissipativity property was firstly proposed by Willems [37]. The purpose of the use of this property is to link the common ideas of system theory and control skills. It is now a good tool for system analysis and synthesis. 3. Filter performance analysis In this section, the main purpose is to design an asynchronous filter to enable the system (11) and (12) to meet stochastic stability and extended (J1 , J2 , J3 )-dissipativity. A useful lemma is given before presenting the main results. Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
7
Lemma 2 [30,38]. Given a positive diagonally dominant matrix P > 0(P ≥ 0), and nonlinear function f( · ) satisfying Eq. (4), then the following inequality can be obtained that f T ( ζ )P f ( ζ ) ≤ ζ T P ζ .
(14)
In order to simplify expressions, the following relations are given:
T x˘∗ (t ) x˘T (t − 1) x˘T (t − 2) . . . x˘T (t − l ) ,
T ωv∗ (t ) ωvT (t ) ωvT (t − 1) . . . ωvT (t − l ) ,
I0 I 0 . . . 0 , 1 θ¯1 I θ¯2 I . . . θ¯l I ,
0 θ¯0 I θ¯1 I . . . θ¯l I , Tς diag θ˜12 , θ˜22 , . . . , θ˜l2 , ˘ ς diag θ˜1 , θ˜2 , . . . , θ˜l . T Theorem 1. Given scalars θ¯ ∈ [0, 1], φ = 0, 1, matrices J1 , J2 , J3 and J4 , if there exists matrix Ql , for each = 1, 2, . . . , g, ρ = 1, 2, . . . , g, ∈ L, ι ∈ U, the following inequalities hold that: ˘ 2 ( , ι) ˘ 1 ( , ι) ˘ (15) ( , ι) ˘ 3 ( , ι) , −( , ι) + GT ( , ι)J4 G( , ι) < 0, where
(16)
⎡
⎤ ˘ 11 0 −(1 − φ)GT ( , ι)J2 1 ( , ι) ˘ 1 ( , ι) ⎣ ⎦, ˘ 22 0 1 ( , ι) −J3 ⎡ ⎤T T T ˘ 11 ˘ 31 D( , ι)1 2 ( , ι) 2 ( , ι) ⎢ ⎥ ˘ 12T ˘ 32T 0 2 ( , ι) 2 ( , ι)⎥ ⎢ 13T ⎢ ˘ 0 C( , ι)I0 ⎥ 2 ( , ι) ⎢ ⎥ ⎢ ˘ 2 ( , ι) ⎢ 0 0 E( , ι)0 ⎥ ⎥ , √ ⎢− 1 − φ J¯1 G( , ι) ⎥ 0 0 ⎢ ⎥ ⎣ ⎦ 0 D( , ι)1 0 27 T 37 T ˘ 2 ( , ι) ˘ 2 ( , ι) 0
−1 −1 −1 ˘ ( , ι), − ˘ ( , ι), − ˘ ( , ι), − ˘ −1 ( , ι), −I , ˘ 3 ( , ι) diag − ˘ −1 ( , ι), − ˘ −1 ( , ι) , −
with ˘ 11 1 ( , ι) −( , ι) +
lt
˘ , ι)B( ), Qv + 4BT ( )(
v=1
˘ 22 1 ( , ι) −diag{Q1 , Q2 , . . . , Ql }, T ¯ T ˘ 11 ˘ 12 2 ( , ι) A ( , ι) + θ0 D ( , ι), 2 ( , ι)
θ¯0 1 − θ¯0 DT ( , ι),
T T ˘ ¯ T ˘ 13 ˘ 27 2 ( , ι) A ( , ι) + θ0 D ( , ι), 2 ( , ι) Tς D ( , ι),
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
8
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
T T T T ˘ 31 ˘ 32 2 ( , ι) I0 C ( , ι) + 0 E ( , ι), 2 ( , ι)
˘ 37 ˘ ς E( , ι) T , T 2 ( , ι) 0
θ¯0 1 − θ¯0 I0T ET ( , ι),
then, the stochastic dissipativity of the filtering error system under consideration is ensured. Proof. See the Appendix. 4. Fuzzy filter design Through the above derivation, we can guarantee that the error system (11) and (12) meets the dissipativity performance, but we can see that conditions (15) and (16) are not linear matrix inequalities (LMIs), so we need to convert them into the strict LMIs that can be easily solved by software. Then, the problem of the dissipativity-based filter design will be solved via the following Theorem. Theorem 2. Given scalars θ¯ ∈ [0, 1], φ = 0, 1, d1 , d2 , matrices ( , ι) = diag{1 ( , ι), 2 ( , ι)}, J1 = −J¯1TJ¯1 , J2 , J3 , J4 , if there exist matrices Aˆ f ρ(ι), Bˆ f ρ (ι), 1 2 ˘ ˘ 1 ( ,ι) d1 M(ι) Q1l Q2l ,ι ,ι ˘ , ι) , Ql , ( , for each Gˆ f ρ (ι), ( , ι) 3 ˘3 2 ( ,ι)
d2 M(ι)
Ql
,ι
= 1, 2, . . . , g, ρ = 1, 2, . . . , g, ∈ L, ι ∈ U, the following inequalities hold that: ˜ 2 ( , ι) ˜ 1 ( , ι) ˜ ( , ι) ˜ 3 ( , ι) < 0, 2 ˜ ˜ ρ ( , ι) + ˜ ρ ( , ι), ( , ι) + g−1 ⎡ 1 T 2 T ˘ ˘ − − ,ι + φG ( )J4 G ( ) ,ι − φG ( )J4 G ( ) 3 ⎣ ˘ 0> − ,ι 0>
where
(17) (18)
√
⎤ 0 φGTf ρ (ι)⎦, −J4
(19)
⎡
⎤ ˜ 11 0 −(1 − φ)GTρ ( , ι)J2 ρ1 ( , ι) ˜ ρ1 ( , ι) ⎣ ⎦, ˜ 22 0 ρ1 ( , ι) −J3 ⎡ ⎤T T T T ˜ 11 ˜ 21 ˜ 31 ρ2 ( , ι) ρ2 ( , ι) ρ2 ( , ι) ⎢ ⎥ ˜ 12T ˜ 32T ⎢ ⎥ 0 ρ2 ( , ι) ρ2 ( , ι)⎥ ⎢ ⎢ ⎥ T ˜ 11 ˜ 34T 0 ⎢ ρ2 ( , ι) ρ2 ( , ι)⎥ ⎢ ⎥ T ⎥ ˜ ρ2 ( , ι) ⎢ ˜ 35 0 0 ρ2 ( , ι)⎥ , ⎢ √ ⎢ ⎥ ⎢− 1 − φ J¯1 Gρ ( , ι) ⎥ 0 0 ⎢ ⎥ ⎢ ⎥ 21 T ˜ ρ2 ( , ι) 0 0 ⎣ ⎦ 0
T ˜ 27 ρ2 ( , ι)
T ˜ 37 ρ2 ( , ι)
1 ˜ ρ3 ( , ι), ˜ ρ3 ( , ι) diag ˜ 1ρ3 ( , ι), ˜ 1ρ3 ( , ι), ˜ 1ρ3 ( , ι), −I , ˜ 1ρ3 ( , ι), ˜ 1ρ3 ( , ι) , Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
9
with
d1 DT ( )Bˆ Tf ρ (ι) d2 DT ( )Bˆ Tf ρ (ι) T T ˜ 21 , ρ2 ( , ι) 1 ρ ( , ι), 0 0 d1 ET ( )Bˆ Tf ρ (ι) d2 ET ( )Bˆ Tf ρ (ι) T ¯ ˜ 22 , ρ ( , ι) ρ1 ( , ι) −diag{Q1 , Q2 , . . . , Ql }, 0 0 T T C ( )T1 ( , ι) CT ( )T2 ( , ι) T 12 ˇ ρ ˜ , θ¯0 1−θ¯0 ρ , ι) ( , ι) ( ( , ι), T T ρ2 ˆ ˆ d1 B f ρ (ι) d2 B f ρ (ι) T A ( )T1 ( , ι) AT ( )T2 ( , ι) 11 T ˜ + θ¯0 ρ ρ2 ( , ι) ( , ι), Aˆ Tf ρ (ι) Aˆ Tf ρ (ι) T ρ ( , ι)
˜ 11 ρ1 ( , ι) −( , ι) +
lt
˘ , ι)B( ), Qv + 4BT ( )(
v=1 T T ˇT T ¯T ˘ ˜ 27 ˜ 31 ρ2 ( , ι) Tς ρ ( , ι), ρ2 ( , ι) = I0 ρ ( , ι) + 0 ρ ( , ι), T T ˇT ˜ 32 ¯ ρ ˜ 34 θ¯0 1 − θ¯0 I0T ( , ι), ρ2 ( , ι) ρ2 ( , ι) I0 ρ ( , ι), T ¯T ˜ 35 ρ2 ( , ι) = 0 ρ ( , ι),
˜ 37 ρ2 ( , ι)
0
˜ 11
T 1 ˜ ρ3 ( , ι) ρ3 ( , ι) ˘ς ¯ ρ ( , ι) , T
˜ 12 ρ3 ( , ι) , ˜ 13 ρ3 ( , ι)
T T T ˜ 11 ˘1 ρ3 ( , ι) 1 ( , ι) ,ι 1 ( , ι) − 1 ( , ι)1 ( , ι) − 1 ( , ι)1 ( , ι), T T T ˜ 12 ˘2 ρ3 ( , ι) 1 ( , ι) ,ι 2 ( , ι) − 1 ( , ι)2 ( , ι) − d1 M (ι)2 ( , ι), T T T ˜ 13 ˘3 ρ3 ( , ι) 2 ( , ι) ,ι 2 ( , ι) − d2 2 ( , ι)M (ι) − d2 M (ι)2 ( , ι),
then, the error system is extended stochastically dissipative. The filter gains can be given as follows: A f ρ (ι) M−1 (ι)Aˆ f ρ (ι), B f ρ (ι) M−1 (ι)Bˆ f ρ (ι), G f ρ (ι) Gˆ f ρ (ι). Proof. First, we define the following matrices: 1 ˘ ˘2 1 ( , ι) d1 M(ι) ˘ , ι) ,ι ,ι , , ( ( , ι) 3 ˘ 2 ( , ι) d2 M(ι) ,ι 1 Ql Q2l , ( , ι) diag{1 ( , ι), 2 ( , ι)}, Ql Q3l Aˆ f ρ (ι) M(ι)A f ρ (ι), Bˆ f ρ (ι) M(ι)B f ρ (ι), Gˆ f ρ (ι) G f ρ (ι). According to Lemma 1, with the conditions (17) and (18), the following inequality is recognized g g
˜ ρ ( , ι) < 0. h (g(t ) )hρ (g(t ) )
(20)
=1 ρ=1
−1 ˘ , ι) − ( , ι) ˘ ( , ι) ( ˘ , ι) − ( , ι) T ≥ 0, one can get that Since (
˘ , ι)T − T ( , ι) − ( , ι)T . ˘ −1 ( , ι)T ( , ι) ≤ ( −( , ι)
(21)
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
10
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
Pre- and post-multiplying diag{I, I, I, (ϖ, ι), (ϖ, ι), (ϖ, ι), (ϖ, ι), I, (ϖ, ι), (ϖ, ι)} and its transpose to the condition (15), and combining with Eq. (21), one can get the condition (17). This completes the proof. 5. A numerical example An example in the section is given to illustrate the availability of the developed design approach. Consider the aforementioned Markov jump fuzzy system and the system matrices are presented as below: −0.1300 −0.3270 0.1500 0.1500 −0.1900 0.1200 , B1 (1 )= , C1 (1 )= , A1 (1 )= −0.4300 0.0200 −0.1800 0.1600 0.3000 0.1200 0.4200 0.1300 −0.2200 0.3600 , G1 ( 1 ) = , D1 (1 )=diag{0.5920, 0.0360}, E1 (1 )= 0.0300 −0.3300 0.1300 0.0100 0.1330 −0.1900 0.2250 −0.1200 0.1400 0.1100 , B1 (2 )= , C1 (2 )= , A1 (2 )= −0.0320 0.0200 −0.1500 0.1200 −0.2300 0.4200 0.1300 0.1800 0.0200 0.3500 , G1 ( 2 ) = , D1 (2 ) = diag{0.4380, 0.0640}, E1 (2 ) = 0.1700 0.1700 0.1700 −0.0900 0.1490 A2 (1 )= 0.0200
−0.1450 0.1400 , B2 (1 )= 0.0390 −0.1300 −0.2300 D2 (1 )=diag{0.3630, 0.0340}, E2 (1 )= 0.2100 −0.4500 0.1650 −0.1500 , B2 (2 )= A2 ( 2 ) = 0.0410 0.0560 −0.0500 0.2600 D2 (2 ) = diag{0.2270, 0.0300}, E2 (2 )= 0.3200
−0.1300 0.1300 0.2500 , C2 (1 )= , 0.1200 −0.3200 0.2000 0.3200 0.3100 −0.1200 , G2 (1 )= , −0.1600 0.2600 0.3300 0.1400 −0.1500 0.1500 , C2 (2 )= , 0.0200 0.1100 0.1600 0.1900 0.1100 0.2200 , G2 ( 2 ) = . 0.1200 0.3800 −0.1200
The nonlinear function f(ϕ) and the membership functions are chosen as follows: f (ϕ ) = − tanh (0.4ϕ ) + 0.2 tanh (ϕ ), 1 + sin (x1 (t ) ) 1 − sin (x1 (t ) ) h1 (x1 (t ) ) = , h2 (x1 (t ) ) = . 2 2 1 2 ¯ ¯ The parameters d1 = 0.5, d2 = 1, lt = 2, θ¯0 = 0. 55, θ1 = 0.75, θ2 = 0.85, ι = ι = T ¯ ¯ diag{1, 1}. Let φ = 0, J1 = −J1 J1 = −5, J2 = 0.5 0.5 0.5 0.5 0.5 0.5 , J3 = ϑ 2 I = 5, J4 = 1. The transition probability matrices of Markov chain α(t) and β(t) are chosen as follows respectively: 0.65 0.35 0.3 0.7 0.15 0.85 1 2 , = , = . = 0.25 0.75 0.6 0.4 0.55 0.45 According to Definition 2, by solving the LMIs (17)–(19), the following filter gains can be obtained that 0.1256 0.0449 −0.0072 −0.0151 A f 1 (1 ) = , B f 1 (1 ) = , −0.1886 −0.0992 −0.0261 0.0033 Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
System mode
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
11
3 2 1 0
0
5
10
15
20
15
20
Filter mode
Time (t) 3 2 1 0
0
5
10
Time (t) Fig. 2. The mode jump of α(t) and β(t).
G f 1 (1 ) = A f 1 (2 ) = G f 1 (2 ) = A f 2 (1 ) = G f 2 (1 ) = A f 2 (2 ) = G f 2 (2 ) =
−0.0512 0.3290 0.0466 −0.2011 −0.0293 0.3584 −0.2026 0.0547 0.2121 0.7490 −0.2249 0.0515 0.2187 0.8426
0.4277 , 0.0628 0.1280 −0.0070 , B f 1 (2 ) = −0.0973 −0.0281 0.4533 , 0.0581 0.0497 −0.0243 , B f 2 (1 ) = 0.0367 0.0106 −0.0614 , 0.0455 0.0710 −0.0277 , B f 2 (2 ) = 0.0398 0.0103 −0.0838 . −0.0075
−0.0146 , 0.0030 0.0241 , −0.0090 0.0253 , −0.0083
For simulation, choosing the initial values x (0 ) = 0.5 the possible external disturbance is assumed as follows: ω (t ) = v (t ) = 0.4 sin (1.5t ) exp (−2t )
T −0.5 , x¯(0 ) = 0.5
T −0.5 ,
T 0.4 sin (1.5t ) exp (−2t ) .
In the simulation, the possible time sequences of the mode jumps for α(t) and β(t) are shown in Fig. 2. Fig. 3 shows the state error. And the filtering errors are shown in Fig. 4. The simulation results verify the availability of the developed asynchronous filter. By applying the bisection method, the minimum value of the performance in this case can be estimated as ϑmin = 0.8224, which may provide a benchmarks for the evaluation of the system robustness. Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
12
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
0.5
0
-0.5
0
5
10
15
Time (t)
20
Fig. 3. The response of state error.
0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3
0
2
4
6
8
10
12
14
16
18
Time (t) Fig. 4. The filtering error.
6. Conclusion In this paper, the extended dissipative asynchronous filtering problem for Markov jump nonlinear systems with fuzzy rules and fading channels has been discussed. Some sufficient conditions which can ensure the stochastic dissipativity of the filtering error system have been obtained. In view of the convex optimization, the expected asynchronous filter with fading channels has been designed. A numerical example has been finally shown to explain the availability of the developed approach. In the future, because of the intensive study of hidden Markov model, our work will be focused on how to combine the asynchronous fuzzy filter designed in this paper with the hidden Markov model. Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
13
Appendix A Choose the following Lyapunov function: V (t ) = V1 (t ) + V2 (t ),
(22)
where V1 (t ) x˘T (t )( , ι)x˘(t ), lt t−1 V2 (t ) x˘T (s )Qv x˘(s ). v
s=t−v
Then, the following condition can be derived that ε {V (t )} ε {V (t + 1 ) − V (t )}
˘ , ι)A( , ι)x˘(t ) + sym x˘T (t )AT ( , ι)( ˘ , ι)B( ) f (x˘(t ) ) = x˘T (t )AT ( , ι)( ˘ , ι)C( , ι)I0 ωv∗ (t ) + θ¯0 x˘T (t )AT ( , ι)( ˘ , ι)D( , ι)x˘(t ) + x˘T (t )AT ( , ι)(
˘ , ι)D( , ι)1 x˘∗ (t ) + x˘T (t )AT ( , ι)( ˘ , ι)E( , ι)0 ωv∗ (t ) + x˘T (t )AT ( , ι)( ˘ , ι)B( ) f (x˘(t ) ) + sym f T (x˘(t ) )BT ( )( ˘ , ι)C( , ι)I0 ωv∗ (t ) + f T (x˘(t ) )BT ( )( ˘ , ι)D( , ι)x˘(t ) + f T (x˘(t ) )BT ( )( ˘ , ι)D( , ι)1 x˘∗ (t ) + θ¯0 f T (x˘(t ) )BT ( )( ˘ , ι)E( , ι)0 ωv∗ (t ) + ωv∗T (t )I0T CT ( , ι)( ˘ , ι)C( , ι)I0 ωv∗ (t ) + f T (x˘(t ) )BT ( )( ˘ , ι)D( , ι)x˘(t ) + ∗T (t )I0T CT ( , ι)( ˘ , ι) + sym θ¯0 ωv∗T (t )I0T CT ( , ι)( ˘ , ι)E( , ι)0 ∗ (t ) × D( , ι)1 x˘∗ (t ) + ωv∗T (t )I0T CT ( , ι)( ˘ , ι)D( , ι)x˘(t ) + sym θ¯0 x˘T (t )DT ( , ι)( ˘ , ι) + θ¯02 x˘T (t )DT ( , ι)( ˘ , ι)E( , ι)0 ωv∗ (t ) × D( , ι)1 x˘∗ (t ) + θ¯0 x˘T (t )DT ( , ι)( ˘ , ι)E( , ι)I0 ωv∗ (t ) + θ¯0 1 − θ¯0 x˘T (t )DT ( , ι) + sym θ¯0 1 − θ¯0 x˘T (t )DT ( , ι)( ˘ , ι)D( , ι)x˘(t ) + x˘∗T (t )T1 DT ( , ι)( ˘ , ι)D( , ι)1 x˘∗ (t ) × ( ˘ , ι)E( , ι)0 ωv∗ (t ) + θ¯0 1 − θ¯0 ∗T (t )ET ( , ι) + sym x˘∗T (t )T1 DT ( , ι)( ˘ , ι)E( , ι)ωv∗ (t ) + sym x˘∗T (t ) 0 Tς DT ( , ι)( ˘ , ι)E( , ι) ωv∗ (t ) × ( ˘ , ι)D( , ι) x˘∗ (t ) + ωv∗T (t )T0 ET ( , ι)( ˘ , ι)E( , ι) + x˘∗T (t ) Tς DT ( , ι)( ˘ , ι)E( , ι) ωv∗ (t ) − x˘T (t )( , ι)x˘(t ) × 0 ωv∗ (t ) + ωv∗T (t ) 0 Tς ET ( , ι)( + x˘T (t )
lt
Qv x˘(t ) − x˘∗T (t )diag{Q1 , Q2 , . . . , Ql }x˘∗ (t ),
v=1
˘ , ι) = where (
q∈L
n∈U
(23)
γ q πιnq (q, n ).
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
14
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
By using the same way from [39], the following inequalities can be obtained that ˘ , ι) A( , ι)x˘(t ) + C( , ι)I0 ωv∗ (t ) + θ¯0 D( , ι)x˘(t ) sym f T (x˘(t ) )BT ( )( ˘ , ι)B( ) f (x˘(t ) ) + x˘T (t )AT ( , ι)( ˘ , ι)A( , ι)x˘(t ) ≤ f T (x˘(t ) )BT ( )( T T ∗ T ˘ , ι)C( , ι)I0 ωv (t ) + θ¯0 x˘ (t )AT ( , ι)( ˘ , ι)D( , ι)x˘(t ) + sym x˘ (t )A ( , ι)( ˘ , ι)D( , ι)x˘(t ) + sym θ¯0 ωv∗T (t )I0T CT ( , ι)( ˘ , ι)D( , ι)x˘(t ) + ωv∗T (t )I0T CT ( , ι)( ˘ , ι)C( , ι)I0 ωv∗ (t ), + θ¯02 x˘T (t )DT ( , ι)( (24) ˘ , ι)D( , ι)1 x˘∗ (t ) sym f T (x˘(t ) )BT ( )( ˘ , ι)B( ) f (x˘(t ) ) + x˘∗T (t )T1 DT ( , ι)( ˘ , ι)D( , ι)1 x˘∗ (t ), ≤ f T (x˘(t ) )BT ( )( (25) ˘ , ι)E( , ι)0 ωv∗ (t ) sym f T (x˘(t ) )BT ( )( ˘ , ι)B( ) f (x˘(t ) ) + ωv∗T (t )T0 ET ( , ι)( ˘ , ι)E( , ι)0 ωv∗ (t ). ≤ f T (x˘(t ) )BT ( )( (26) According to Lemma 2, and combining with the conditions (23)–(26), the following inequality obtains that l l t t T T T ε V (t ) − (1 − φ) z˘ (t )J1 z˘(t ) + sym z˘ (t )J2 ωv (t −s ) − ωv (t − s )J3 ωv (t − s ) s=0
s=0
≤ T (t )( , ι) (t ),
T where (t ) = x˘T (t ) x˘∗T (t ) ωv∗T (t ) , and ⎡ ⎤ 11 ( , ι) 12 ( , ι) 13 ( , ι) 22 ( , ι) 23 ( , ι)⎦, ( , ι) ⎣ 33 ( , ι) with ˘ , ι)A( , ι) + θ¯0 AT ( , ι)( ˘ , ι)D( , ι) 11 ( , ι) AT ( , ι)( ˘ , ι)D( , ι) + θ¯0 1 − θ¯0 DT ( , ι)( ˘ , ι)D( , ι) +θ¯02 DT ( , ι)( −( , ι) +
lt
˘ , ι)B( ) − (1 − φ)GT ( , ι)J1 G( , ι), Qv + 4BT ( )(
v=1
˘ , ι)D( , ι)1 + θ¯0 DT ( , ι)( ˘ , ι)D( , ι)1 , 12 ( , ι) A ( , ι)( T T ˘ , ι)C( , ι)I0 + A ( , ι)( ˘ , ι)E( , ι)0 13 ( , ι) 2A ( , ι)( T T ˘ , ι)C( , ι)I0 + θ¯0 D ( , ι)( ˘ , ι)E( , ι)0 +2θ¯0 D ( , ι)( T ¯ ¯ ˘ +θ0 1 − θ0 D ( , ι)( , ι)E( , ι)I0 − (1 − φ)GT ( , ι)J2 , T
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
15
˘ , ι)D( , ι)1 + Tς DT ( , ι)( ˘ , ι)D( , ι) 22 ( , ι) T1 DT ( , ι)( −diag{Q1 , Q2 , . . . , Ql }, ˘ , ι)C( , ι)I0 + T1 DT ( , ι)( ˘ , ι)E( , ι)0 23 ( , ι) T1 DT ( , ι)( ˘ , ι)E( , ι) , + 0 Tς DT ( , ι)( ˘ , ι)C( , ι)I0 + I0T CT ( , ι)( ˘ , ι)E( , ι)0 33 ( , ι) 2I0T CT ( , ι)( T ˘ , ι)E( , ι) + 2T0 ET ( , ι)( ˘ , ι)E( , ι)0 +θ¯0 1 − θ¯0 E ( , ι)( ˘ , ι)E( , ι) − J3 . + 0 Tς ET ( , ι)( ˘ , ι) < 0, and it By using the Schur complement to condition (15), one can find that ( is obviously found that lt lt T T T ε V (t )−(1 − φ) z˘ (t )J1 z˘(t ) + 2 z˘ (t )J2 ωv (t − s ) − ωv (t − s )J3 ωv (t −s ) < 0. s=0
s=0
According to Definition 1, when φ = 0, inequality (13) goes to a (J1 , J2 , J3 )-dissipative constraint condition. Hence, the following inequality may be, under the zero initial condition, satisfied that t t −ε ϒ (J1 , J2 , J3 , t ) [ϒ (J1 , J2 , J3 , t )] ≤ V (t ) − ε k=0
≤ε
k=0 t
[V (t ) − ϒ (J1 , J2 , J3 , t )]
k=0
< 0.
(27)
Therefore, when φ = 0, the condition (13) is satisfied. On the other hand, from Definition 1, if φ = 1, condition (13) goes to a l2 − l∞ performance condition. Hence, for any 0 < t < λ, the inequality holds as below ⎫ ⎧ t−1 ⎨
⎬ V (g) − ϒ (J1 , J2 , J3 , g) ε ⎩ ⎭ g=0 ⎧ ⎫ lt t−1 ⎨ ⎬ = ε V (t ) − ωvT (g − s )J3 ωv (g − s ) < 0, ⎩ ⎭ g=0 s=0
and one further has
ε x˘T (t )( , ι)x˘(t ) < ε {V (t )} < ε ≤ε
lt
⎧ lt t−1 ⎨ ⎩
g=0
⎫ ⎬ ωvT (g − s )J3 ωv (g − s ) ⎭ s=0
ωvT (t − s )J3 ωv (t − s ) .
(28)
s=0
According to conditions (12) and (16), the following inequality is ensured ε z˘T (t )J4 z˘(t ) − x˘T (t )( , ι)x˘(t ) < 0.
(29)
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
16
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
Combining with Eqs. (28) and (29), one can find that lt
T T T ε z˘ (t )J4 z˘(t ) < ε x˘ (t )( , ι)x˘(t ) + ωv (t − s )J3 ωv (t − s ) s=0
< ε V (t ) + <ε
lt s=0
lt
ωvT (t
− s )J3 ωv (t − s )
(t − s )J3 ωv (t − s ) . T
s=0
Obviously, the condition (13) can be obtained. In a word, the error system can achieve the extended (J1 , J2 , J3 )-dissipativity based on Definition 1. References [1] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern. (1) (1985) 116–132. [2] M. Chadli, H.R. Karimi, P. Shi, On stability and stabilization of singular uncertain Takagi–Sugeno fuzzy systems, J. Frankl. Inst. 351 (3) (2014) 1453–1463. [3] D. Ye, N.-N. Diao, X.-G. Zhao, Fault-tolerant controller design for general polynomial-fuzzy-model-based systems, IEEE Trans. Fuzzy Syst. 26 (2) (2018) 1046–1051. [4] Y. Wang, H. Shen, H.R. Karimi, D. Duan, Dissipativity-based fuzzy integral sliding mode control of continuous-time T–S fuzzy systems, IEEE Trans. Fuzzy Syst. 26 (3) (2018) 1164–1176. [5] X.H. Chang, J.H. Park, P. Shi, Fuzzy resilient energy-to-peak filtering for continuous-time nonlinear systems, IEEE Trans. Fuzzy Syst. 25 (6) (2017) 1576–1588. [6] Z. Wang, L. Shen, J. Xia, H. Shen, J. Wang, Finite-time non-fragile l2 − l∞ control for jumping stochastic systems subject to input constraints via an event triggered mechanism, J. Frankl. Inst. 355 (14) (2018) 6371–6389. [7] J. Wang, T. Ru, J. Xia, Y. Wei, Z. Wang, Finite-time synchronization for complex dynamic networks with semi-Markov switching topologies: An H∞ event-triggered control scheme, Appl. Math. Comput. 356 (2019) 235–251. [8] B. Zhang, W.X. Zheng, S. Xu, Filtering of Markovian jump delay systems based on a new performance index, IEEE Trans. Circuits Syst. I, Reg. Papers 60 (5) (2013) 1250–1263. [9] M. Dai, J. Xia, X. Huang, H. Shen, Event-triggered passive synchronization for Markov jump neural networks subject to randomly occurring gain variations, Neurocomputing 331 (2019) 403–411. [10] H. Shen, F. Li, H. Yan, H.R. Karimi, H.-K. Lam, Finite-time event-triggered H∞ control for T–S fuzzy Markov jump systems, IEEE Trans. Fuzzy Syst. 26 (5) (2018) 3122–3135. [11] M. Dai, Z. Huang, J. Xia, B. Meng, J. Wang, H. Shen, Non-fragile extended dissipativity-based state feedback control for 2-D Markov jump delayed systems, Appl. Math. Comput. 362 (2019) 124571. [12] X. Hu, J. Xia, Y. Wei, B. Meng, H. Shen, Passivity-based state synchronization for semi-Markov jump coupled chaotic neural networks with randomly occurring time delays, Appl. Math. Comput. 361 (2019) 32–41. [13] J.H. Park, H. Shen, X.-H. Chang, T.H. Lee, Recent Advances in Control And Filtering of dynamic Systems With Constrained signals,, Springer, 2018. [14] H. Shen, F. Li, S. Xu, V. Sreeram, Slow state variables feedback stabilization for semi-Markov jump systems with singular perturbations, IEEE Trans. Autom. Control 63 (8) (2018) 2709–2714. [15] H. Shen, Y. Men, Z. Wu, J. Cao, G. Lu, Network-based quantized control for fuzzy singularly perturbed semi– Markov jump systems and its application, IEEE Trans. Circuits Syst. I, Reg. Papers 66 (3) (2019) 1130–1140. [16] H. Yan, H. Zhang, F. Yang, X. Zhan, C. Peng, Event-triggered asynchronous guaranteed cost control for Markov jump discrete-time neural networks with distributed delay and channel fading, IEEE Trans. Neural Netw. Learn. Syst. 29 (8) (2018) 3588–3598. [17] L. Shen, X. Yang, J. Wang, J. Xia, Passive gain-scheduling filtering for jumping linear parameter varying systems with fading channels based on the hidden Markov model, Proc. Inst. Mech. Eng. Part I J Syst. Control Eng. 233 (1) (2019) 67–79. Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
JID: FI
ARTICLE IN PRESS
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
17
[18] K. An, M. Lin, J. Ouyang, W.-P. Zhu, Secure transmission in cognitive satellite terrestrial networks, IEEE J. Selected Areas Commun. 34 (11) (2016) 3025–3037. [19] N. Elia, Remote stabilization over fading channels, Syst. Control Lett. 54 (3) (2005) 237–249. [20] X. Ren, J. Wu, K.H. Johansson, G. Shi, L. Shi, Infinite horizon optimal transmission power control for remote state estimation over fading channels, IEEE Trans. Autom. Control 63 (1) (2018) 85–100. [21] F.F. Digham, M.-S. Alouini, M.K. Simon, On the energy detection of unknown signals over fading channels, IEEE Trans. Commun. 55 (1) (2007) 21–24. [22] L. Delle Monache, T. Nipen, Y. Liu, G. Roux, R. Stull, Kalman filter and analog schemes to postprocess numerical weather predictions, Mon. Weather Rev. 139 (11) (2011) 3554–3570. [23] K. Arakawa, Median filter based on fuzzy rules and its application to image restoration, Fuzzy Sets Syst. 77 (1) (1996) 3–13. [24] P. Shi, E.-K. Boukas, R.K. Agarwal, Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters, IEEE Trans. Autom. Control 44 (8) (1999) 1592–1597. [25] M. Xing, J. Xia, J. Wang, B. Meng, H. Shen, Asynchronous H∞ filtering for nonlinear persistent dwell-time switched singular systems with measurement quantization, Appl. Math. Comput. 362 (2019) 124578. [26] Z. Huang, J. Xia, J. Wang, Y. Wei, Z. Wang, J. Wang, Mixed h∞ /l2 − l∞ state estimation for switched genetic regulatory networks subject to packet dropouts: a persistent dwell-time switching mechanism, Appl. Math. Comput. 355 (2019) 198–212. [27] Z.-G. Wu, P. Shi, H. Su, J. Chu, Asynchronous l2 − l∞ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities, Automatica 50 (1) (2014) 180–186. [28] M. Dai, J. Xia, J.H. Park, X. Huang, H. Shen, Asynchronous dissipative filtering for Markov jump discrete-time systems subject to randomly occurring distributed delays, J. Frankl. Inst. 356 (4) (2019) 2395–2420. [29] Y. Men, X. Huang, Z. Wang, H. Shen, B. Chen, Quantized asynchronous dissipative state estimation of jumping neural networks subject to occurring randomly sensor saturations, Neurocomputing 291 (2018) 207–214. [30] Y.-C. Chu, K. Glover, Bounds of the induced norm and model reduction errors for systems with repeated scalar nonlinearities, IEEE Trans. Autom. Control 44 (3) (1999) 471–483. [31] N. Elia, Remote stabilization over fading channels, Syst. Control Lett. 54 (3) (2005) 237–249. [32] K. An, T. Liang, G. Zheng, X. Yan, Y. Li, S. Chatzinotas, Performance limits of cognitive uplink FSS and terrestrial fs for ka-band, IEEE Trans. Aerospe. Electronic Syst. (2019), doi:10.1109/TAES.2018.2886611. [33] W. Lu, K. An, T. Liang, Robust beamforming design for sum secrecy rate maximization in multibeam satellite systems, IEEE Trans. Aerospace. Electronic Syst. 55 (3) (2019) 1568–1572. [34] H.D. Tuan, P. Apkarian, T. Narikiyo, Y. Yamamoto, Parameterized linear matrix inequality techniques in fuzzy control system design, IEEE Trans. Fuzzy Syst. 9 (2) (2001) 324–332. [35] H. Shen, S. Huo, J. Cao, T. Huang, Generalized state estimation for Markovian coupled networks under round-robin protocol and redundant channels, IEEE Trans. Cybern. 49 (4) (2019) 1292–1301. [36] J. Xia, H. Gao, M. Liu, G. Zhuang, B. Zhang, Non-fragile finite-time extended dissipative control for a class of uncertain discrete time switched linear systems, J. Frankl. Inst. 355 (6) (2018) 3031–3049. [37] J.C. Willems, Dissipative dynamical systems part I: General theory, Arch. Ration. Mech. Anal. 45 (5) (1972) 321–351. [38] J. Wang, S. Huo, J. Xia, J.H. Park, X. Huang, H. Shen, Generalised dissipative asynchronous output feedback control for Markov jump repeated scalar non-linear systems with time-varying delay, IET Control Theory Appl. 13 (13) (2019) 2114–2121. [39] J. Song, Y. Niu, J. Lam, H.-K. Lam, Fuzzy remote tracking control for randomly varying local nonlinear models under fading and missing measurements, IEEE Trans. Fuzzy Syst. 26 (3) (2018) 1125–1137.
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
JID: FI
[m1+;October 19, 2019;11:50]
Available online at www.sciencedirect.com
Journal of the Franklin Institute xxx (xxxx) xxx www.elsevier.com/locate/jfranklin
Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channelsR Jing Wang a, Liang Shen a, Jianwei Xia b, Zhen Wang c,∗, Xiangyong Chen d,∗ a School
of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan 243002, China b School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, PR China c College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China d School of Automation and Electrical Engineering, Linyi University, Linyi 276005, China Received 16 November 2018; received in revised form 5 August 2019; accepted 19 September 2019 Available online xxx
Abstract The fuzzy asynchronous dissipative filtering issue for Markov jump discrete-time nonlinear systems subject to fading channels is discussed in this paper, where the Rice fading model is employed to characterize the fading channels phenomenon in the system measurements for the first time. The attention is focused on developing an available asynchronous filter, which can ensure that the underlying error system is dissipative. In this regard, several important performances can be investigated conveniently by introducing adjustment matrices. By means of the stochastic analysis theory and the network control technique, some sufficient conditions for the solvability of the addressed problem are presented, simultaneously, the gains of the filter desired are determined correspondingly. An illustrative example is finally exploited to explain the utilizability of the developed approach. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
R This work was supported by the National Natural Science Foundation of China under Grants 61703004, 61573008, 61973199, the National Natural Science Foundation of Anhui Province under Grant 1808085QA18. ∗ Corresponding authors. E-mail addresses: [email protected] (Z. Wang), [email protected] (X. Chen).
https://doi.org/10.1016/j.jfranklin.2019.09.031 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
2
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
1. Introduction The Takagi–Sugeno (T–S) fuzzy model, consisting of a set of local linear subsystems, was firstly proposed in 1985 [1]. In such a model, the local linear subsystems are smoothly connected by fuzzy membership functions, and the local input–output relationship of the nonlinear systems is represented by fuzzy IF-THEN rules [2,3]. For fuzzy systems, if the fuzzy reasoning system based on IF-THEN rules is used, the qualitative aspects of reasoning process and human knowledge can be modeled without precise quantitative analysis. Therefore, in recent years, especially in the field of control and estimation of nonlinear systems, it is hardly surprising that the T–S fuzzy model has been widely concerned and many important research results have emerged. For uncertain continuous-time T–S fuzzy systems, a novel integral sliding mode fuzzy control method on the basis of dissipation principle was proposed in [4], by which an improved adaptive fuzzy integral sliding mode control rate was designed. For continuous-time nonlinear systems, a fuzzy resilient energy-to-peak filter was designed in [5]. In the investigation of T–S fuzzy systems, sudden changes in their structure and parameters cannot be ignored. Such a case frequently encounters in many practical systems, due perhaps to a variety of reasons including internal subsystems faults, environmental conditions, sudden disturbances, changes in the correlation between subsystems and so on [6,7]. In this regard, one needs to introduce a kind of stochastic hybrid system models, namely, Markov jump systems (MJSs) [8–11]. It is known that MJSs play a vital role in modeling the systems subjected to sudden changes. Based on such models, many control problems have been well investigated [12–15]. Remarkably, scholars are, in most instances, taking particular aim at MJSs with normal channels. By comparison, there is less energy devoted to nonlinear MJSs, not to mention nonlinear ones subject to fading channels, which is one of the driving forces of the current work. The case of fading channels is a typical network-induced phenomenon when signals are transmitted over networks experiencing some obstacles or path losses. The main reasons of above-mentioned may be summed up as reflection, diffraction and scattering, which have a great influence on signal power. In recent years, many researchers have devoted to the investigation of fading channels [16–18]. Thereinto, a fundamental question, how to find an appropriate mathematical model to describe it, has become a hot topic. In order to response such an issue, the Rice fading model was introduced in [19]. Since then, many meaningful results have been put forward successively [20,21]. On the other hand, robust filtering is, roughly speaking, one of the essential topics in information and control theory [22,23]. For MJSs, there exist many results on Kalman filtering, H∞ filtering, passive filtering and dissipative filtering [24,25]. However, it should be noted that most of the developed filters are mode-dependent. This may limit their applications in some complex network environments [26]. One solution is to design asynchronous filters [27–29]. As we know, such a question still has no answers for nonlinear MJSs subject to fading channels. How to find a solution is, naturally, our main purpose in this work. In view of the above statements, the asynchronous dissipative filtering problem is discussed in this paper for nonlinear MJSs, where the fading channels phenomenon is fully taken into account. A T-S fuzzy system model approach is utilized to dispose of the nonlinearity in systems. By employing some stochastic analysis tools and fuzzy system theory, a fuzzy asynchronous filter is presented, and some sufficient conditions for ensuring the strict dissipativity property of the resulting error system are obtained. At length, an example is shown to exPlease cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
3
Fig. 1. Filtering for T–S fuzzy systems.
plain the availability of the presented approach. The main contributions of this paper can be summarized as follows: (1) For T-S fuzzy systems, incomplete measurements due to fading channels described by stochastic processes are considered. The property of such a phenomenon is well reflected in the Rice fading model used in this paper, which improves the reliability of the design method. (2) The extended dissipative performance is employed to evaluate the robustness of the system. By introducing adjustment matrices, several important performances can be investigated conveniently. (3) The filter is designed based on an asynchronous strategy since it is hard to implement real-time synchronization of mode information, which is more practical in the case of incomplete measurements. In addition, the notations are standard presented, which are similar as those in [17]. 2. Problem formulation In the work, the considered nonlinear MJSs under fading channels driven by T–S fuzzy model approach, are described in Fig. 1. According to Fig. 1, the measurement signals received by the filter are transmitted over fading channels. 2.1. Nonlinear MJSs based on T–S fuzzy model Firstly, consider a nonlinear discrete-time Markov jump system described by a T–S fuzzy model: Plant Rule ϱ: IF g1 (t ) is ψ ϱ1 , g2 (t ) is ψ ϱ2 , ..., gδ (t ) is ψ ϱδ , THEN x (t + 1 ) = A (α(t ))x(t ) + B (α(t )) f (x(t ) ) + C (α(t ))ω(t ),
(1)
y(t ) = D (α(t ))x(t ) + E (α(t ))ω(t ),
(2)
z(t ) = G (α(t ))x(t ),
(3)
where x(t) ∈ Rm , y(t) ∈ Rn and z(t) ∈ Rp are the state, the measured output and the signal to be estimated, respectively; ψ ϱi and gi (t ) are presented as a fuzzy set and the premise variable for = 1, 2, . . . , g, i = 1, 2, . . . , δ, respectively, and the scalar g is the number of IF-THEN rules; Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
4
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
ω(t) ∈ Rs is the disturbance input belonging to l2 [0, ∞); Aϱ (α(t)), Bϱ (α(t)), Cϱ (α(t)), Dϱ (α(t)), Eϱ (α(t)), and Gϱ (α(t)), are known matrices with appropriate dimensions. α(t) is the concerned discrete-time homogeneous Markov process taking values in a finite set L = {1, 2, . . . , L } with its transition probability matrix {γ ϖq } given by γ q = Pr {α(t + 1) = q|α(t ) = }, where 0 ≤ γ ϖq ≤ 1, ∀ , q ∈ L, and Lq=1 γ q = 1. For convenience, let α(t ) = , the system matrices can be denoted as Aϱ (ϖ), Bϱ (ϖ), Cϱ (ϖ), Dϱ (ϖ), Eϱ (ϖ), and Gϱ (ϖ), respectively. Assumption 1. [30] The nonlinear function f(•) in (1) satisfies f a´ + f b` ≤ a´ + b` , ∀a´, b` ∈ Rn .
(4)
The fuzzy basis functions in this paper are given by δ ψi (gi (t ) ) h (g(t ) ) g i=1 , δ i=1 ψi (gi (t ) ) =1 where ψi (gi (t ) )( = 1, 2, . . . , g, i = 1, 2, .g. . , δ) is the grade of membership of the premise parameter gi (t ) in ψ ϱi . It follows that =1 h (g (t ) ) = 1, h (g (t ) ) ∈ [0, 1]. By using the standard fuzzy blending method, the fuzzy mode of system (1)–(3) can be inferred as follows: x (t + 1 ) =
g
h (g(t ) ) A ( )x(t ) + B ( ) f (x(t ) ) + C ( )ω(t ) ,
(5)
=1
y(t ) =
g
h (g(t ) ) D ( )x(t ) + E ( )ω(t ) ,
(6)
h (g(t ) ) G ( )x(t ) .
(7)
=1
z(t ) =
g =1
2.2. Fading channels As we know, in some circumstances, the phenomenon of fading measurement is inescapable when the system measurements transmit to the filter through the channels with limited bandwidth. This may result in the difference occur between the measured output from the sensor and the actually measured output the filter received. To reveal this difference, the lth-order Rice fading model [31] is considered in this paper which is shown as follows: lt y˜(t ) = θs (t )y (t − s ) + v (t ), (8) s=0
where lt = min {l, t }, v (t ) is an external disturbance, and θ s (t) are the channel coefficients that are mutually independent random variables taking values of 0 and 1 with Pr {θs (t ) = 1} = ε {θs (t )} = θ¯s , Pr {θs (t ) = 0} = 1 − θ¯s . Remark 1. For Markov jump system (1), usually, the change of the system mode that results in the change of the received signal quantity of the filter. So introducing the channel coefficient here which is mode-independent. As a random fading model widely used in the field of signal Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
5
processing, the lth-order Rice fading model can initially deal with the initial phenomenon of fading channels for some multi-path transmission and shadowing problems. As far as we know, the lth-order Rice fading model is not only appropriate to point-to-point communication systems, but also can consider time delay, packet loss and fading channels, and it is usually assumed that its coefficients are Gauss random variables, being mutually independent, with the same distribution. Remark 2. As we know, the phenomenon of fading channels can be divided into large-scale fading and small-scale fading, while large-scale fading can be divided into path loss and shadow effect. The rain attenuation studied in [32,33] belongs to the shadow effect. As the name suggests, it refers to the attenuation caused by the wave entering the rain layer. The research on this aspect is not deeply enough, and it is worthy of further research in the future. From above descriptions, the asynchronous fuzzy filter can be expressed as follows: x¯(t + 1 ) =
g
h (g(t ) ) A f (β(t ) )x¯(t ) + B f (β(t ) )y˜(t ) ,
(9)
=1
z¯(t ) =
g
h (g(t ) ) G f (β(t ) )x¯(t ) ,
(10)
=1
where x¯(t ) ∈ Rm is the filter state and z¯(t ) ∈ R p is output vector. Afϱ (β(t)), Bfϱ (β(t)) and Gfϱ (β(t)) are the filter gains to be determined, where β(t) is assumed to be described by a non-stationary Markov chain, which takes values in a finite state set U = {1, 2, . . . , U } with Pr {β(t + 1) = n|β(t ) = ι} = πιnα(t+1) , U α(t+1) where πιnα(t+1) ∈ [0, 1], and = 1 for ∀ι, n ∈ U. For convenience, denoting n=1 πιn Afϱ (ι)Afϱ (β(t)), Bfϱ (ι)Bfϱ (β(t)) and Gfϱ (ι)Gfϱ (β(t)) for each β(t ) = ι ∈ U.
T From the conditions (5)–(10), let x˘(t ) = x T (t ) x¯T (t ) , z˘(t ) = z(t ) − z¯(t ), ωv (t ) = T
T
T ω (t ) vT (t ) , f (x˘(t )) = f T (x(t )) f T (x¯(t )) , and θ˜s (t ) = θs (t ) − θ¯s , respectively, the filtering error system can be obtained as follows: x˘(t + 1) = A( , ι)x˘(t ) + B( ) f (x˘(t ) ) + C( , ι) (t ) + θ¯0 D( , ι)x˘(t ) + θ˜0 (t )D( , ι)x˘(t ) +
lt
θ¯s D( , ι)x˘(t − s) +
s=1
+
lt
lt
θ˜s (t )D( , ι)x˘(t − s) + θ˜0 (t )E( , ι)ωv (t )
s=1
θ¯s E( , ι)ωv (t − s ) +
s=0
lt
θ˜s (t )E( , ι)ωv (t − s ),
(11)
s=1
z˘(t ) = G( , ι)x˘(t ),
(12)
where A( , ι)
g g =1 ρ=1
C( , ι)
g g =1 ρ=1
h (g(t ) )hρ (g(t ) )A˘ ρ ( , ι), B( )
g g
h (g(t ) )hρ (g(t ) )B˘ ( ),
=1 ρ=1
h (g(t ) )hρ (g(t ) )C˘ρ ( , ι), D( , ι)
g g
h (g(t ) )hρ (g(t ) )D˘ ρ ( , ι),
=1 ρ=1
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
6
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
E( , ι)
g g
h (g(t ) )hρ (g(t ) )E˘ρ ( , ι), G( , ι)
=1 ρ=1
g g
h (g(t ) )hρ (g(t ) )G˘ ρ ( , ι),
=1 ρ=1
with
A˘ ρ ( , ι) diag A ( ), A f ρ (ι) , B˘ ( ) diag B ( ), 0 ,
0 0 ˘ ˘ , Cρ ( , ι) diag C ( ), B f ρ (ι) , Dρ ( , ι) B f ρ (ι)D ( ) 0
0 0 , G˘ ρ ( , ι) G ( ) −G f ρ (ι) . E˘ρ ( , ι) B f ρ (ι)E ( ) 0 Lemma 1 [34]. Given an appropriate dimensional matrix Rρ , one has Rss
g g
h (g(t ) )hρ (g(t ) )Rρ < 0,
=1 ρ=1
if R < 0, ∀ = 1, 2, . . . , g, and 2 R + Rρ + Rρ < 0, ∀, ρ = 1, 2, . . . , g, = ρ. g−1 Definition 1 [35,36]. For given real matrices J1 = J1T ≤ 0, J2 , J3 = J3T > 0, and J4 = J4T > 0, a known constant φ = 0, 1. For any ϖ(t) ∈ l2 [0, ∞), the following condition holds:
ε ϒ (J1 , J2 , J3 , t ) ≥ φ sup ε z˘T (t )J4 z˘(t ) , ∀t > 0, (13) 0≤t≤
t=0
where
ϒ (J1 , J2 , J3 , t ) (1−φ) z˘ (t )J1 z˘(t ) + sym T
lt s=0
z˘ (t )J2 ωv (t −s ) T
+
lt
ωvT (t −s )J3 ωv (t −s ),
s=0
then, under the zero initial conditions, the filtering error system is said to be extended stochastically dissipative. Remark 3. As we know, the inequality (13) is considered to be an extended dissipative performance indicator. If φ = 0, let J1 = −J¯1 J¯1T < 0, J2 = 0 and J3 = ϑ 2 I > 0, the inequality (13) turns into the standard dissipative performance index; if φ = 0 and J2 = 0, let J1 = J1T < 0 and J3 = ϑ 2 I > 0, the inequality (13) becomes the H∞ performance index; if φ = 0 and J1 = 0, let J2 = J2T > 0 and J3 = ϑ 2 I > 0, inequality (13) transforms into the standard form of the passive performance index; if φ = 1, let J3 = ϑ 2 I > 0 and J4 = J4T > 0, inequality (13) is in line with l2 − l∞ performance index. Remark 4. Dissipativity property was firstly proposed by Willems [37]. The purpose of the use of this property is to link the common ideas of system theory and control skills. It is now a good tool for system analysis and synthesis. 3. Filter performance analysis In this section, the main purpose is to design an asynchronous filter to enable the system (11) and (12) to meet stochastic stability and extended (J1 , J2 , J3 )-dissipativity. A useful lemma is given before presenting the main results. Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
7
Lemma 2 [30,38]. Given a positive diagonally dominant matrix P > 0(P ≥ 0), and nonlinear function f( · ) satisfying Eq. (4), then the following inequality can be obtained that f T ( ζ )P f ( ζ ) ≤ ζ T P ζ .
(14)
In order to simplify expressions, the following relations are given:
T x˘∗ (t ) x˘T (t − 1) x˘T (t − 2) . . . x˘T (t − l ) ,
T ωv∗ (t ) ωvT (t ) ωvT (t − 1) . . . ωvT (t − l ) ,
I0 I 0 . . . 0 , 1 θ¯1 I θ¯2 I . . . θ¯l I ,
0 θ¯0 I θ¯1 I . . . θ¯l I , Tς diag θ˜12 , θ˜22 , . . . , θ˜l2 , ˘ ς diag θ˜1 , θ˜2 , . . . , θ˜l . T Theorem 1. Given scalars θ¯ ∈ [0, 1], φ = 0, 1, matrices J1 , J2 , J3 and J4 , if there exists matrix Ql , for each = 1, 2, . . . , g, ρ = 1, 2, . . . , g, ∈ L, ι ∈ U, the following inequalities hold that: ˘ 2 ( , ι) ˘ 1 ( , ι) ˘ (15) ( , ι) ˘ 3 ( , ι) , −( , ι) + GT ( , ι)J4 G( , ι) < 0, where
(16)
⎡
⎤ ˘ 11 0 −(1 − φ)GT ( , ι)J2 1 ( , ι) ˘ 1 ( , ι) ⎣ ⎦, ˘ 22 0 1 ( , ι) −J3 ⎡ ⎤T T T ˘ 11 ˘ 31 D( , ι)1 2 ( , ι) 2 ( , ι) ⎢ ⎥ ˘ 12T ˘ 32T 0 2 ( , ι) 2 ( , ι)⎥ ⎢ 13T ⎢ ˘ 0 C( , ι)I0 ⎥ 2 ( , ι) ⎢ ⎥ ⎢ ˘ 2 ( , ι) ⎢ 0 0 E( , ι)0 ⎥ ⎥ , √ ⎢− 1 − φ J¯1 G( , ι) ⎥ 0 0 ⎢ ⎥ ⎣ ⎦ 0 D( , ι)1 0 27 T 37 T ˘ 2 ( , ι) ˘ 2 ( , ι) 0
−1 −1 −1 ˘ ( , ι), − ˘ ( , ι), − ˘ ( , ι), − ˘ −1 ( , ι), −I , ˘ 3 ( , ι) diag − ˘ −1 ( , ι), − ˘ −1 ( , ι) , −
with ˘ 11 1 ( , ι) −( , ι) +
lt
˘ , ι)B( ), Qv + 4BT ( )(
v=1
˘ 22 1 ( , ι) −diag{Q1 , Q2 , . . . , Ql }, T ¯ T ˘ 11 ˘ 12 2 ( , ι) A ( , ι) + θ0 D ( , ι), 2 ( , ι)
θ¯0 1 − θ¯0 DT ( , ι),
T T ˘ ¯ T ˘ 13 ˘ 27 2 ( , ι) A ( , ι) + θ0 D ( , ι), 2 ( , ι) Tς D ( , ι),
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
8
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
T T T T ˘ 31 ˘ 32 2 ( , ι) I0 C ( , ι) + 0 E ( , ι), 2 ( , ι)
˘ 37 ˘ ς E( , ι) T , T 2 ( , ι) 0
θ¯0 1 − θ¯0 I0T ET ( , ι),
then, the stochastic dissipativity of the filtering error system under consideration is ensured. Proof. See the Appendix. 4. Fuzzy filter design Through the above derivation, we can guarantee that the error system (11) and (12) meets the dissipativity performance, but we can see that conditions (15) and (16) are not linear matrix inequalities (LMIs), so we need to convert them into the strict LMIs that can be easily solved by software. Then, the problem of the dissipativity-based filter design will be solved via the following Theorem. Theorem 2. Given scalars θ¯ ∈ [0, 1], φ = 0, 1, d1 , d2 , matrices ( , ι) = diag{1 ( , ι), 2 ( , ι)}, J1 = −J¯1TJ¯1 , J2 , J3 , J4 , if there exist matrices Aˆ f ρ(ι), Bˆ f ρ (ι), 1 2 ˘ ˘ 1 ( ,ι) d1 M(ι) Q1l Q2l ,ι ,ι ˘ , ι) , Ql , ( , for each Gˆ f ρ (ι), ( , ι) 3 ˘3 2 ( ,ι)
d2 M(ι)
Ql
,ι
= 1, 2, . . . , g, ρ = 1, 2, . . . , g, ∈ L, ι ∈ U, the following inequalities hold that: ˜ 2 ( , ι) ˜ 1 ( , ι) ˜ ( , ι) ˜ 3 ( , ι) < 0, 2 ˜ ˜ ρ ( , ι) + ˜ ρ ( , ι), ( , ι) + g−1 ⎡ 1 T 2 T ˘ ˘ − − ,ι + φG ( )J4 G ( ) ,ι − φG ( )J4 G ( ) 3 ⎣ ˘ 0> − ,ι 0>
where
(17) (18)
√
⎤ 0 φGTf ρ (ι)⎦, −J4
(19)
⎡
⎤ ˜ 11 0 −(1 − φ)GTρ ( , ι)J2 ρ1 ( , ι) ˜ ρ1 ( , ι) ⎣ ⎦, ˜ 22 0 ρ1 ( , ι) −J3 ⎡ ⎤T T T T ˜ 11 ˜ 21 ˜ 31 ρ2 ( , ι) ρ2 ( , ι) ρ2 ( , ι) ⎢ ⎥ ˜ 12T ˜ 32T ⎢ ⎥ 0 ρ2 ( , ι) ρ2 ( , ι)⎥ ⎢ ⎢ ⎥ T ˜ 11 ˜ 34T 0 ⎢ ρ2 ( , ι) ρ2 ( , ι)⎥ ⎢ ⎥ T ⎥ ˜ ρ2 ( , ι) ⎢ ˜ 35 0 0 ρ2 ( , ι)⎥ , ⎢ √ ⎢ ⎥ ⎢− 1 − φ J¯1 Gρ ( , ι) ⎥ 0 0 ⎢ ⎥ ⎢ ⎥ 21 T ˜ ρ2 ( , ι) 0 0 ⎣ ⎦ 0
T ˜ 27 ρ2 ( , ι)
T ˜ 37 ρ2 ( , ι)
1 ˜ ρ3 ( , ι), ˜ ρ3 ( , ι) diag ˜ 1ρ3 ( , ι), ˜ 1ρ3 ( , ι), ˜ 1ρ3 ( , ι), −I , ˜ 1ρ3 ( , ι), ˜ 1ρ3 ( , ι) , Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
9
with
d1 DT ( )Bˆ Tf ρ (ι) d2 DT ( )Bˆ Tf ρ (ι) T T ˜ 21 , ρ2 ( , ι) 1 ρ ( , ι), 0 0 d1 ET ( )Bˆ Tf ρ (ι) d2 ET ( )Bˆ Tf ρ (ι) T ¯ ˜ 22 , ρ ( , ι) ρ1 ( , ι) −diag{Q1 , Q2 , . . . , Ql }, 0 0 T T C ( )T1 ( , ι) CT ( )T2 ( , ι) T 12 ˇ ρ ˜ , θ¯0 1−θ¯0 ρ , ι) ( , ι) ( ( , ι), T T ρ2 ˆ ˆ d1 B f ρ (ι) d2 B f ρ (ι) T A ( )T1 ( , ι) AT ( )T2 ( , ι) 11 T ˜ + θ¯0 ρ ρ2 ( , ι) ( , ι), Aˆ Tf ρ (ι) Aˆ Tf ρ (ι) T ρ ( , ι)
˜ 11 ρ1 ( , ι) −( , ι) +
lt
˘ , ι)B( ), Qv + 4BT ( )(
v=1 T T ˇT T ¯T ˘ ˜ 27 ˜ 31 ρ2 ( , ι) Tς ρ ( , ι), ρ2 ( , ι) = I0 ρ ( , ι) + 0 ρ ( , ι), T T ˇT ˜ 32 ¯ ρ ˜ 34 θ¯0 1 − θ¯0 I0T ( , ι), ρ2 ( , ι) ρ2 ( , ι) I0 ρ ( , ι), T ¯T ˜ 35 ρ2 ( , ι) = 0 ρ ( , ι),
˜ 37 ρ2 ( , ι)
0
˜ 11
T 1 ˜ ρ3 ( , ι) ρ3 ( , ι) ˘ς ¯ ρ ( , ι) , T
˜ 12 ρ3 ( , ι) , ˜ 13 ρ3 ( , ι)
T T T ˜ 11 ˘1 ρ3 ( , ι) 1 ( , ι) ,ι 1 ( , ι) − 1 ( , ι)1 ( , ι) − 1 ( , ι)1 ( , ι), T T T ˜ 12 ˘2 ρ3 ( , ι) 1 ( , ι) ,ι 2 ( , ι) − 1 ( , ι)2 ( , ι) − d1 M (ι)2 ( , ι), T T T ˜ 13 ˘3 ρ3 ( , ι) 2 ( , ι) ,ι 2 ( , ι) − d2 2 ( , ι)M (ι) − d2 M (ι)2 ( , ι),
then, the error system is extended stochastically dissipative. The filter gains can be given as follows: A f ρ (ι) M−1 (ι)Aˆ f ρ (ι), B f ρ (ι) M−1 (ι)Bˆ f ρ (ι), G f ρ (ι) Gˆ f ρ (ι). Proof. First, we define the following matrices: 1 ˘ ˘2 1 ( , ι) d1 M(ι) ˘ , ι) ,ι ,ι , , ( ( , ι) 3 ˘ 2 ( , ι) d2 M(ι) ,ι 1 Ql Q2l , ( , ι) diag{1 ( , ι), 2 ( , ι)}, Ql Q3l Aˆ f ρ (ι) M(ι)A f ρ (ι), Bˆ f ρ (ι) M(ι)B f ρ (ι), Gˆ f ρ (ι) G f ρ (ι). According to Lemma 1, with the conditions (17) and (18), the following inequality is recognized g g
˜ ρ ( , ι) < 0. h (g(t ) )hρ (g(t ) )
(20)
=1 ρ=1
−1 ˘ , ι) − ( , ι) ˘ ( , ι) ( ˘ , ι) − ( , ι) T ≥ 0, one can get that Since (
˘ , ι)T − T ( , ι) − ( , ι)T . ˘ −1 ( , ι)T ( , ι) ≤ ( −( , ι)
(21)
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
10
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
Pre- and post-multiplying diag{I, I, I, (ϖ, ι), (ϖ, ι), (ϖ, ι), (ϖ, ι), I, (ϖ, ι), (ϖ, ι)} and its transpose to the condition (15), and combining with Eq. (21), one can get the condition (17). This completes the proof. 5. A numerical example An example in the section is given to illustrate the availability of the developed design approach. Consider the aforementioned Markov jump fuzzy system and the system matrices are presented as below: −0.1300 −0.3270 0.1500 0.1500 −0.1900 0.1200 , B1 (1 )= , C1 (1 )= , A1 (1 )= −0.4300 0.0200 −0.1800 0.1600 0.3000 0.1200 0.4200 0.1300 −0.2200 0.3600 , G1 ( 1 ) = , D1 (1 )=diag{0.5920, 0.0360}, E1 (1 )= 0.0300 −0.3300 0.1300 0.0100 0.1330 −0.1900 0.2250 −0.1200 0.1400 0.1100 , B1 (2 )= , C1 (2 )= , A1 (2 )= −0.0320 0.0200 −0.1500 0.1200 −0.2300 0.4200 0.1300 0.1800 0.0200 0.3500 , G1 ( 2 ) = , D1 (2 ) = diag{0.4380, 0.0640}, E1 (2 ) = 0.1700 0.1700 0.1700 −0.0900 0.1490 A2 (1 )= 0.0200
−0.1450 0.1400 , B2 (1 )= 0.0390 −0.1300 −0.2300 D2 (1 )=diag{0.3630, 0.0340}, E2 (1 )= 0.2100 −0.4500 0.1650 −0.1500 , B2 (2 )= A2 ( 2 ) = 0.0410 0.0560 −0.0500 0.2600 D2 (2 ) = diag{0.2270, 0.0300}, E2 (2 )= 0.3200
−0.1300 0.1300 0.2500 , C2 (1 )= , 0.1200 −0.3200 0.2000 0.3200 0.3100 −0.1200 , G2 (1 )= , −0.1600 0.2600 0.3300 0.1400 −0.1500 0.1500 , C2 (2 )= , 0.0200 0.1100 0.1600 0.1900 0.1100 0.2200 , G2 ( 2 ) = . 0.1200 0.3800 −0.1200
The nonlinear function f(ϕ) and the membership functions are chosen as follows: f (ϕ ) = − tanh (0.4ϕ ) + 0.2 tanh (ϕ ), 1 + sin (x1 (t ) ) 1 − sin (x1 (t ) ) h1 (x1 (t ) ) = , h2 (x1 (t ) ) = . 2 2 1 2 ¯ ¯ The parameters d1 = 0.5, d2 = 1, lt = 2, θ¯0 = 0. 55, θ1 = 0.75, θ2 = 0.85, ι = ι = T ¯ ¯ diag{1, 1}. Let φ = 0, J1 = −J1 J1 = −5, J2 = 0.5 0.5 0.5 0.5 0.5 0.5 , J3 = ϑ 2 I = 5, J4 = 1. The transition probability matrices of Markov chain α(t) and β(t) are chosen as follows respectively: 0.65 0.35 0.3 0.7 0.15 0.85 1 2 , = , = . = 0.25 0.75 0.6 0.4 0.55 0.45 According to Definition 2, by solving the LMIs (17)–(19), the following filter gains can be obtained that 0.1256 0.0449 −0.0072 −0.0151 A f 1 (1 ) = , B f 1 (1 ) = , −0.1886 −0.0992 −0.0261 0.0033 Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
System mode
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
11
3 2 1 0
0
5
10
15
20
15
20
Filter mode
Time (t) 3 2 1 0
0
5
10
Time (t) Fig. 2. The mode jump of α(t) and β(t).
G f 1 (1 ) = A f 1 (2 ) = G f 1 (2 ) = A f 2 (1 ) = G f 2 (1 ) = A f 2 (2 ) = G f 2 (2 ) =
−0.0512 0.3290 0.0466 −0.2011 −0.0293 0.3584 −0.2026 0.0547 0.2121 0.7490 −0.2249 0.0515 0.2187 0.8426
0.4277 , 0.0628 0.1280 −0.0070 , B f 1 (2 ) = −0.0973 −0.0281 0.4533 , 0.0581 0.0497 −0.0243 , B f 2 (1 ) = 0.0367 0.0106 −0.0614 , 0.0455 0.0710 −0.0277 , B f 2 (2 ) = 0.0398 0.0103 −0.0838 . −0.0075
−0.0146 , 0.0030 0.0241 , −0.0090 0.0253 , −0.0083
For simulation, choosing the initial values x (0 ) = 0.5 the possible external disturbance is assumed as follows: ω (t ) = v (t ) = 0.4 sin (1.5t ) exp (−2t )
T −0.5 , x¯(0 ) = 0.5
T −0.5 ,
T 0.4 sin (1.5t ) exp (−2t ) .
In the simulation, the possible time sequences of the mode jumps for α(t) and β(t) are shown in Fig. 2. Fig. 3 shows the state error. And the filtering errors are shown in Fig. 4. The simulation results verify the availability of the developed asynchronous filter. By applying the bisection method, the minimum value of the performance in this case can be estimated as ϑmin = 0.8224, which may provide a benchmarks for the evaluation of the system robustness. Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
12
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
0.5
0
-0.5
0
5
10
15
Time (t)
20
Fig. 3. The response of state error.
0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3
0
2
4
6
8
10
12
14
16
18
Time (t) Fig. 4. The filtering error.
6. Conclusion In this paper, the extended dissipative asynchronous filtering problem for Markov jump nonlinear systems with fuzzy rules and fading channels has been discussed. Some sufficient conditions which can ensure the stochastic dissipativity of the filtering error system have been obtained. In view of the convex optimization, the expected asynchronous filter with fading channels has been designed. A numerical example has been finally shown to explain the availability of the developed approach. In the future, because of the intensive study of hidden Markov model, our work will be focused on how to combine the asynchronous fuzzy filter designed in this paper with the hidden Markov model. Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
13
Appendix A Choose the following Lyapunov function: V (t ) = V1 (t ) + V2 (t ),
(22)
where V1 (t ) x˘T (t )( , ι)x˘(t ), lt t−1 V2 (t ) x˘T (s )Qv x˘(s ). v
s=t−v
Then, the following condition can be derived that ε {V (t )} ε {V (t + 1 ) − V (t )}
˘ , ι)A( , ι)x˘(t ) + sym x˘T (t )AT ( , ι)( ˘ , ι)B( ) f (x˘(t ) ) = x˘T (t )AT ( , ι)( ˘ , ι)C( , ι)I0 ωv∗ (t ) + θ¯0 x˘T (t )AT ( , ι)( ˘ , ι)D( , ι)x˘(t ) + x˘T (t )AT ( , ι)(
˘ , ι)D( , ι)1 x˘∗ (t ) + x˘T (t )AT ( , ι)( ˘ , ι)E( , ι)0 ωv∗ (t ) + x˘T (t )AT ( , ι)( ˘ , ι)B( ) f (x˘(t ) ) + sym f T (x˘(t ) )BT ( )( ˘ , ι)C( , ι)I0 ωv∗ (t ) + f T (x˘(t ) )BT ( )( ˘ , ι)D( , ι)x˘(t ) + f T (x˘(t ) )BT ( )( ˘ , ι)D( , ι)1 x˘∗ (t ) + θ¯0 f T (x˘(t ) )BT ( )( ˘ , ι)E( , ι)0 ωv∗ (t ) + ωv∗T (t )I0T CT ( , ι)( ˘ , ι)C( , ι)I0 ωv∗ (t ) + f T (x˘(t ) )BT ( )( ˘ , ι)D( , ι)x˘(t ) + ∗T (t )I0T CT ( , ι)( ˘ , ι) + sym θ¯0 ωv∗T (t )I0T CT ( , ι)( ˘ , ι)E( , ι)0 ∗ (t ) × D( , ι)1 x˘∗ (t ) + ωv∗T (t )I0T CT ( , ι)( ˘ , ι)D( , ι)x˘(t ) + sym θ¯0 x˘T (t )DT ( , ι)( ˘ , ι) + θ¯02 x˘T (t )DT ( , ι)( ˘ , ι)E( , ι)0 ωv∗ (t ) × D( , ι)1 x˘∗ (t ) + θ¯0 x˘T (t )DT ( , ι)( ˘ , ι)E( , ι)I0 ωv∗ (t ) + θ¯0 1 − θ¯0 x˘T (t )DT ( , ι) + sym θ¯0 1 − θ¯0 x˘T (t )DT ( , ι)( ˘ , ι)D( , ι)x˘(t ) + x˘∗T (t )T1 DT ( , ι)( ˘ , ι)D( , ι)1 x˘∗ (t ) × ( ˘ , ι)E( , ι)0 ωv∗ (t ) + θ¯0 1 − θ¯0 ∗T (t )ET ( , ι) + sym x˘∗T (t )T1 DT ( , ι)( ˘ , ι)E( , ι)ωv∗ (t ) + sym x˘∗T (t ) 0 Tς DT ( , ι)( ˘ , ι)E( , ι) ωv∗ (t ) × ( ˘ , ι)D( , ι) x˘∗ (t ) + ωv∗T (t )T0 ET ( , ι)( ˘ , ι)E( , ι) + x˘∗T (t ) Tς DT ( , ι)( ˘ , ι)E( , ι) ωv∗ (t ) − x˘T (t )( , ι)x˘(t ) × 0 ωv∗ (t ) + ωv∗T (t ) 0 Tς ET ( , ι)( + x˘T (t )
lt
Qv x˘(t ) − x˘∗T (t )diag{Q1 , Q2 , . . . , Ql }x˘∗ (t ),
v=1
˘ , ι) = where (
q∈L
n∈U
(23)
γ q πιnq (q, n ).
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
14
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
By using the same way from [39], the following inequalities can be obtained that ˘ , ι) A( , ι)x˘(t ) + C( , ι)I0 ωv∗ (t ) + θ¯0 D( , ι)x˘(t ) sym f T (x˘(t ) )BT ( )( ˘ , ι)B( ) f (x˘(t ) ) + x˘T (t )AT ( , ι)( ˘ , ι)A( , ι)x˘(t ) ≤ f T (x˘(t ) )BT ( )( T T ∗ T ˘ , ι)C( , ι)I0 ωv (t ) + θ¯0 x˘ (t )AT ( , ι)( ˘ , ι)D( , ι)x˘(t ) + sym x˘ (t )A ( , ι)( ˘ , ι)D( , ι)x˘(t ) + sym θ¯0 ωv∗T (t )I0T CT ( , ι)( ˘ , ι)D( , ι)x˘(t ) + ωv∗T (t )I0T CT ( , ι)( ˘ , ι)C( , ι)I0 ωv∗ (t ), + θ¯02 x˘T (t )DT ( , ι)( (24) ˘ , ι)D( , ι)1 x˘∗ (t ) sym f T (x˘(t ) )BT ( )( ˘ , ι)B( ) f (x˘(t ) ) + x˘∗T (t )T1 DT ( , ι)( ˘ , ι)D( , ι)1 x˘∗ (t ), ≤ f T (x˘(t ) )BT ( )( (25) ˘ , ι)E( , ι)0 ωv∗ (t ) sym f T (x˘(t ) )BT ( )( ˘ , ι)B( ) f (x˘(t ) ) + ωv∗T (t )T0 ET ( , ι)( ˘ , ι)E( , ι)0 ωv∗ (t ). ≤ f T (x˘(t ) )BT ( )( (26) According to Lemma 2, and combining with the conditions (23)–(26), the following inequality obtains that l l t t T T T ε V (t ) − (1 − φ) z˘ (t )J1 z˘(t ) + sym z˘ (t )J2 ωv (t −s ) − ωv (t − s )J3 ωv (t − s ) s=0
s=0
≤ T (t )( , ι) (t ),
T where (t ) = x˘T (t ) x˘∗T (t ) ωv∗T (t ) , and ⎡ ⎤ 11 ( , ι) 12 ( , ι) 13 ( , ι) 22 ( , ι) 23 ( , ι)⎦, ( , ι) ⎣ 33 ( , ι) with ˘ , ι)A( , ι) + θ¯0 AT ( , ι)( ˘ , ι)D( , ι) 11 ( , ι) AT ( , ι)( ˘ , ι)D( , ι) + θ¯0 1 − θ¯0 DT ( , ι)( ˘ , ι)D( , ι) +θ¯02 DT ( , ι)( −( , ι) +
lt
˘ , ι)B( ) − (1 − φ)GT ( , ι)J1 G( , ι), Qv + 4BT ( )(
v=1
˘ , ι)D( , ι)1 + θ¯0 DT ( , ι)( ˘ , ι)D( , ι)1 , 12 ( , ι) A ( , ι)( T T ˘ , ι)C( , ι)I0 + A ( , ι)( ˘ , ι)E( , ι)0 13 ( , ι) 2A ( , ι)( T T ˘ , ι)C( , ι)I0 + θ¯0 D ( , ι)( ˘ , ι)E( , ι)0 +2θ¯0 D ( , ι)( T ¯ ¯ ˘ +θ0 1 − θ0 D ( , ι)( , ι)E( , ι)I0 − (1 − φ)GT ( , ι)J2 , T
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
15
˘ , ι)D( , ι)1 + Tς DT ( , ι)( ˘ , ι)D( , ι) 22 ( , ι) T1 DT ( , ι)( −diag{Q1 , Q2 , . . . , Ql }, ˘ , ι)C( , ι)I0 + T1 DT ( , ι)( ˘ , ι)E( , ι)0 23 ( , ι) T1 DT ( , ι)( ˘ , ι)E( , ι) , + 0 Tς DT ( , ι)( ˘ , ι)C( , ι)I0 + I0T CT ( , ι)( ˘ , ι)E( , ι)0 33 ( , ι) 2I0T CT ( , ι)( T ˘ , ι)E( , ι) + 2T0 ET ( , ι)( ˘ , ι)E( , ι)0 +θ¯0 1 − θ¯0 E ( , ι)( ˘ , ι)E( , ι) − J3 . + 0 Tς ET ( , ι)( ˘ , ι) < 0, and it By using the Schur complement to condition (15), one can find that ( is obviously found that lt lt T T T ε V (t )−(1 − φ) z˘ (t )J1 z˘(t ) + 2 z˘ (t )J2 ωv (t − s ) − ωv (t − s )J3 ωv (t −s ) < 0. s=0
s=0
According to Definition 1, when φ = 0, inequality (13) goes to a (J1 , J2 , J3 )-dissipative constraint condition. Hence, the following inequality may be, under the zero initial condition, satisfied that t t −ε ϒ (J1 , J2 , J3 , t ) [ϒ (J1 , J2 , J3 , t )] ≤ V (t ) − ε k=0
≤ε
k=0 t
[V (t ) − ϒ (J1 , J2 , J3 , t )]
k=0
< 0.
(27)
Therefore, when φ = 0, the condition (13) is satisfied. On the other hand, from Definition 1, if φ = 1, condition (13) goes to a l2 − l∞ performance condition. Hence, for any 0 < t < λ, the inequality holds as below ⎫ ⎧ t−1 ⎨
⎬ V (g) − ϒ (J1 , J2 , J3 , g) ε ⎩ ⎭ g=0 ⎧ ⎫ lt t−1 ⎨ ⎬ = ε V (t ) − ωvT (g − s )J3 ωv (g − s ) < 0, ⎩ ⎭ g=0 s=0
and one further has
ε x˘T (t )( , ι)x˘(t ) < ε {V (t )} < ε ≤ε
lt
⎧ lt t−1 ⎨ ⎩
g=0
⎫ ⎬ ωvT (g − s )J3 ωv (g − s ) ⎭ s=0
ωvT (t − s )J3 ωv (t − s ) .
(28)
s=0
According to conditions (12) and (16), the following inequality is ensured ε z˘T (t )J4 z˘(t ) − x˘T (t )( , ι)x˘(t ) < 0.
(29)
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
ARTICLE IN PRESS
JID: FI
16
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
Combining with Eqs. (28) and (29), one can find that lt
T T T ε z˘ (t )J4 z˘(t ) < ε x˘ (t )( , ι)x˘(t ) + ωv (t − s )J3 ωv (t − s ) s=0
< ε V (t ) + <ε
lt s=0
lt
ωvT (t
− s )J3 ωv (t − s )
(t − s )J3 ωv (t − s ) . T
s=0
Obviously, the condition (13) can be obtained. In a word, the error system can achieve the extended (J1 , J2 , J3 )-dissipativity based on Definition 1. References [1] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern. (1) (1985) 116–132. [2] M. Chadli, H.R. Karimi, P. Shi, On stability and stabilization of singular uncertain Takagi–Sugeno fuzzy systems, J. Frankl. Inst. 351 (3) (2014) 1453–1463. [3] D. Ye, N.-N. Diao, X.-G. Zhao, Fault-tolerant controller design for general polynomial-fuzzy-model-based systems, IEEE Trans. Fuzzy Syst. 26 (2) (2018) 1046–1051. [4] Y. Wang, H. Shen, H.R. Karimi, D. Duan, Dissipativity-based fuzzy integral sliding mode control of continuous-time T–S fuzzy systems, IEEE Trans. Fuzzy Syst. 26 (3) (2018) 1164–1176. [5] X.H. Chang, J.H. Park, P. Shi, Fuzzy resilient energy-to-peak filtering for continuous-time nonlinear systems, IEEE Trans. Fuzzy Syst. 25 (6) (2017) 1576–1588. [6] Z. Wang, L. Shen, J. Xia, H. Shen, J. Wang, Finite-time non-fragile l2 − l∞ control for jumping stochastic systems subject to input constraints via an event triggered mechanism, J. Frankl. Inst. 355 (14) (2018) 6371–6389. [7] J. Wang, T. Ru, J. Xia, Y. Wei, Z. Wang, Finite-time synchronization for complex dynamic networks with semi-Markov switching topologies: An H∞ event-triggered control scheme, Appl. Math. Comput. 356 (2019) 235–251. [8] B. Zhang, W.X. Zheng, S. Xu, Filtering of Markovian jump delay systems based on a new performance index, IEEE Trans. Circuits Syst. I, Reg. Papers 60 (5) (2013) 1250–1263. [9] M. Dai, J. Xia, X. Huang, H. Shen, Event-triggered passive synchronization for Markov jump neural networks subject to randomly occurring gain variations, Neurocomputing 331 (2019) 403–411. [10] H. Shen, F. Li, H. Yan, H.R. Karimi, H.-K. Lam, Finite-time event-triggered H∞ control for T–S fuzzy Markov jump systems, IEEE Trans. Fuzzy Syst. 26 (5) (2018) 3122–3135. [11] M. Dai, Z. Huang, J. Xia, B. Meng, J. Wang, H. Shen, Non-fragile extended dissipativity-based state feedback control for 2-D Markov jump delayed systems, Appl. Math. Comput. 362 (2019) 124571. [12] X. Hu, J. Xia, Y. Wei, B. Meng, H. Shen, Passivity-based state synchronization for semi-Markov jump coupled chaotic neural networks with randomly occurring time delays, Appl. Math. Comput. 361 (2019) 32–41. [13] J.H. Park, H. Shen, X.-H. Chang, T.H. Lee, Recent Advances in Control And Filtering of dynamic Systems With Constrained signals,, Springer, 2018. [14] H. Shen, F. Li, S. Xu, V. Sreeram, Slow state variables feedback stabilization for semi-Markov jump systems with singular perturbations, IEEE Trans. Autom. Control 63 (8) (2018) 2709–2714. [15] H. Shen, Y. Men, Z. Wu, J. Cao, G. Lu, Network-based quantized control for fuzzy singularly perturbed semi– Markov jump systems and its application, IEEE Trans. Circuits Syst. I, Reg. Papers 66 (3) (2019) 1130–1140. [16] H. Yan, H. Zhang, F. Yang, X. Zhan, C. Peng, Event-triggered asynchronous guaranteed cost control for Markov jump discrete-time neural networks with distributed delay and channel fading, IEEE Trans. Neural Netw. Learn. Syst. 29 (8) (2018) 3588–3598. [17] L. Shen, X. Yang, J. Wang, J. Xia, Passive gain-scheduling filtering for jumping linear parameter varying systems with fading channels based on the hidden Markov model, Proc. Inst. Mech. Eng. Part I J Syst. Control Eng. 233 (1) (2019) 67–79. Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031
JID: FI
ARTICLE IN PRESS
[m1+;October 19, 2019;11:50]
J. Wang, L. Shen and J. Xia et al. / Journal of the Franklin Institute xxx (xxxx) xxx
17
[18] K. An, M. Lin, J. Ouyang, W.-P. Zhu, Secure transmission in cognitive satellite terrestrial networks, IEEE J. Selected Areas Commun. 34 (11) (2016) 3025–3037. [19] N. Elia, Remote stabilization over fading channels, Syst. Control Lett. 54 (3) (2005) 237–249. [20] X. Ren, J. Wu, K.H. Johansson, G. Shi, L. Shi, Infinite horizon optimal transmission power control for remote state estimation over fading channels, IEEE Trans. Autom. Control 63 (1) (2018) 85–100. [21] F.F. Digham, M.-S. Alouini, M.K. Simon, On the energy detection of unknown signals over fading channels, IEEE Trans. Commun. 55 (1) (2007) 21–24. [22] L. Delle Monache, T. Nipen, Y. Liu, G. Roux, R. Stull, Kalman filter and analog schemes to postprocess numerical weather predictions, Mon. Weather Rev. 139 (11) (2011) 3554–3570. [23] K. Arakawa, Median filter based on fuzzy rules and its application to image restoration, Fuzzy Sets Syst. 77 (1) (1996) 3–13. [24] P. Shi, E.-K. Boukas, R.K. Agarwal, Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters, IEEE Trans. Autom. Control 44 (8) (1999) 1592–1597. [25] M. Xing, J. Xia, J. Wang, B. Meng, H. Shen, Asynchronous H∞ filtering for nonlinear persistent dwell-time switched singular systems with measurement quantization, Appl. Math. Comput. 362 (2019) 124578. [26] Z. Huang, J. Xia, J. Wang, Y. Wei, Z. Wang, J. Wang, Mixed h∞ /l2 − l∞ state estimation for switched genetic regulatory networks subject to packet dropouts: a persistent dwell-time switching mechanism, Appl. Math. Comput. 355 (2019) 198–212. [27] Z.-G. Wu, P. Shi, H. Su, J. Chu, Asynchronous l2 − l∞ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities, Automatica 50 (1) (2014) 180–186. [28] M. Dai, J. Xia, J.H. Park, X. Huang, H. Shen, Asynchronous dissipative filtering for Markov jump discrete-time systems subject to randomly occurring distributed delays, J. Frankl. Inst. 356 (4) (2019) 2395–2420. [29] Y. Men, X. Huang, Z. Wang, H. Shen, B. Chen, Quantized asynchronous dissipative state estimation of jumping neural networks subject to occurring randomly sensor saturations, Neurocomputing 291 (2018) 207–214. [30] Y.-C. Chu, K. Glover, Bounds of the induced norm and model reduction errors for systems with repeated scalar nonlinearities, IEEE Trans. Autom. Control 44 (3) (1999) 471–483. [31] N. Elia, Remote stabilization over fading channels, Syst. Control Lett. 54 (3) (2005) 237–249. [32] K. An, T. Liang, G. Zheng, X. Yan, Y. Li, S. Chatzinotas, Performance limits of cognitive uplink FSS and terrestrial fs for ka-band, IEEE Trans. Aerospe. Electronic Syst. (2019), doi:10.1109/TAES.2018.2886611. [33] W. Lu, K. An, T. Liang, Robust beamforming design for sum secrecy rate maximization in multibeam satellite systems, IEEE Trans. Aerospace. Electronic Syst. 55 (3) (2019) 1568–1572. [34] H.D. Tuan, P. Apkarian, T. Narikiyo, Y. Yamamoto, Parameterized linear matrix inequality techniques in fuzzy control system design, IEEE Trans. Fuzzy Syst. 9 (2) (2001) 324–332. [35] H. Shen, S. Huo, J. Cao, T. Huang, Generalized state estimation for Markovian coupled networks under round-robin protocol and redundant channels, IEEE Trans. Cybern. 49 (4) (2019) 1292–1301. [36] J. Xia, H. Gao, M. Liu, G. Zhuang, B. Zhang, Non-fragile finite-time extended dissipative control for a class of uncertain discrete time switched linear systems, J. Frankl. Inst. 355 (6) (2018) 3031–3049. [37] J.C. Willems, Dissipative dynamical systems part I: General theory, Arch. Ration. Mech. Anal. 45 (5) (1972) 321–351. [38] J. Wang, S. Huo, J. Xia, J.H. Park, X. Huang, H. Shen, Generalised dissipative asynchronous output feedback control for Markov jump repeated scalar non-linear systems with time-varying delay, IET Control Theory Appl. 13 (13) (2019) 2114–2121. [39] J. Song, Y. Niu, J. Lam, H.-K. Lam, Fuzzy remote tracking control for randomly varying local nonlinear models under fading and missing measurements, IEEE Trans. Fuzzy Syst. 26 (3) (2018) 1125–1137.
Please cite this article as: J. Wang, L. Shen and J. Xia et al., Asynchronous dissipative filtering for nonlinear jumping systems subject to fading channels, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.031