Finite-time dissipative filtering for uncertain discrete-time systems with state and disturbance-dependent noise over fading channels

Accepted Manuscript Finite-time dissipative filtering for uncertain discrete-time systems with state and disturbance-dependent noise over fading channels Huaxiang Han, Xiaohua Zhang, Weidong Zhang

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S0019-0578(18)30424-5 https://doi.org/10.1016/j.isatra.2018.10.045 ISATRA 2946

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ISA Transactions

Received date : 2 May 2018 Revised date : 18 September 2018 Accepted date : 29 October 2018 Please cite this article as: Han H., Zhang X. and Zhang W. Finite-time dissipative filtering for uncertain discrete-time systems with state and disturbance-dependent noise over fading channels. ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.10.045 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Finite-time dissipative filtering for uncertain discrete-time systems with state and disturbance-dependent noise over fading channels I Huaxiang Hana,c , Xiaohua Zhangb , Weidong Zhangc,∗ a College of Engineering Science and Technology, ShangHai Ocean University, Shanghai, 201306, P. R. China of Information Engineering, North China University of Water Resources and Electric Power, Zhengzhou, 450045, P. R. China c Depatment of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai, 200240, P. R. China b School

Abstract This paper concerns with the finite-time exponential dissipative filtering problem for a class of discrete stochastic system subject to randomly occurring uncertainties and channel fadings. A modified Lth-order Rice fading model is presented to better characterize the multipath fading phenomena in real wireless communication environment. Our objective is to design a filter such that the filtering error system is finite-time stochastic bounded with a prespecified exponential dissipative performance. With the aid of the auxiliary function, a new set of sufficient conditions is derived for the existence of an acceptable filter which can degrade into the conditions for filtering with H∞ performance. Then, the filter gains are obtained by solving these inequality sufficient conditions. Finally, an illustrative example is given to demonstrate the validity of the proposed approach. Keywords: Finite-time dissipative filter, Channel fadings, State and disturbance-dependent noise 1. Introduction

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Different from the conventional asymptotic stability, which defines the convergence of system states over an infinite-time interval, the finite-time stability focuses on the quantitative features of system states over a finite-time interval. Actually, in many real engineering projects, the system states are not allowed to exceed a certain bound and the satisfactory transient behaviours are extremely desired. Therefore, great efforts have been recently devoted to the finite-time stability and control issues of networked control systems(NCSs)[19, 21, 23–26]. By contrast, there are few researches on the finite-time filtering or state estimation problem for NCSs. [27] designed a finite-time H∞ state estimator for Markovian jump systems with quantizations and randomly happening non-linearities based on event trigger mechanism. In [28], the finite-time asynchronous H∞ filtering problem was considered for a class of discrete Markov jump system over an unreliable communication link where packet losses, time delays, and sensor non-linearity were supposed to coexist. By the obtained filter, both the finite-time boundedness and the specified H∞ performance were ensured. To the authors’ knowledge, most investigations on the finitetime filter design for NCSs are mainly based on the H∞ framework.

Over the years, Filtering problem has been a research hotspot in the fields of control, communication and signal processing[1–5]. With the growing popularity of network com30 munication, more and more control and signal processing algorithms are implemented through communication links[6–9]. In networked systems, the limited bandwidth of the communication channel always brings about all kinds of networkinduced phenomena such as transmission delays[6, 10, 11], 35 packet losses[6, 10], signal quantization[12, 13] and random occurring non-linearities[7, 14]. Specially, when a signal is transmitted in a wireless communication channel, it is always reflected or diffracted several times because of many stumbling blocks in its path [15, 16]. This phenomenon is called multi40 path fading, which reduces the quality and accuracy of the signal to great extent [11, 17]. Fading is usually modelled by some random time-varying mathematics models, of which Rice fading model is a representative one. To overcome its negative effects, some results has been reported on the stability, control 45 and filtering issues involving the fading channel, see [7, 18, 19] and the references therein. But the investigation on channel fading is still at its early stage and not enough despite the fact that It is also worth noticing that dissipativity theory, which not the wireless channel is susceptible to fading effect[7]. only unifies H∞ and passivity performance but also introduces On another frontier, finite-time stability or boundedness initiated by Doratohas drawn great research attention [14, 20–22]. 50 a more general and less conservative robust design criterion because it can make a good trade-off between gain and phase ∗ Corresponding author at: Depatment of Automation, Shanghai Jiao Tong performance. Dissipative filtering has been discussed in many University, and Key Laboratory of System Control and Information Processing, works[29–36]. For example,[30] has discussed dissipativityMinistry of Education of China, Shanghai, 200240, P. R. China based state estimation for uncertain discrete-time Markov jump Email addresses: [email protected] (Huaxiang Han), 55 neural networks with mixed time delays; [36] has focused on [email protected] (Xiaohua Zhang), [email protected] (Weidong Zhang) the dissipative filtering problem for T-S fuzzy system with mulPreprint submitted to ISA Transactions

October 31, 2018

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tiple time delays. But most works considered the system dissipativity in an infinite time interval and they didn’t look into transient performance. And there still lacks of a fully research115 on the finite-time dissipative filtering problem. Especially, [19] proposed a novel concept of finite-time stochastic exponential dissipative and designed a finite-time dissipative controller for a class of discrete stochastic systems subject to randomly occurring uncertainties and fading measurements. But the corresponding finite-time dissipative filtering problem has not been discussed, which stimulates us to make up for this gap. Different from the existing works, our goal is to design a finite-time exponential dissipative filter by fully taking into account the uncertainties of the system parameters and the channel fading measurements. Through such a filter, both the meansquare finite-time stochastic boundedness and exponential dissipative performance of the filtering error system can be guaranteed. The main contributions of this study can be emphasized as follows: (1) Distinguished from the existing Rice fading model, in this paper a modified and more compressive one, which contains both disturbance and disturbance-dependent noise, is introduced to better characterize the channel fading in the wireless communication. And the technique treatment is done for their impact on the proposed filter design. (2) A finite-time exponential dissipative filter design method is presented for a class of discrete time stochastic system suffering randomly occurring uncertainties and channel fading measures synchronously. And the designed dissipative filter can reduce into finite-time H∞ filter in a special case. (3) Sufficient conditions are established to ensure the filtering error system finite-time exponential dissipative and a method is given to solve the filter gains. The structures of this paper are stated as follows. In section 2, the research problem is presented and some preliminaries are given. In section 3, by virtue of the auxiliary function and some coping techniques , sufficient conditions are established to enable the filtering error system finite-time exponential dissipative, then the solving method of the corresponding filter parameters is given. In section 4, a numerical example is provided to demonstrate the effectiveness of the obtained filter. Finally, some conclusions are drawn in section 5. Notation: The notation is fairly standard. Rn refers to the ndimensional Euclidean space, Rn×m denotes the set of all n × m real matrices. (Ω, F , Prob) denotes a probability space, where Ω is sample space, F is the σ-algebra of subsets of the sample space and Prob is the probability measure with total mass 1. The notation X > 0 (or X ≥ 0) means the matrix X is real symmetric positive definite (or semi-definite). I and 0 are separately the unit matrix and zero matrix with appropriate di-120 mensions. The superscript ‘T’ and ‘-1’ denote the transpose and inverse of a matrix. The asterisk ∗ refer to the symmetric elements in a symmetric matrix. E {.} and D {.} represent the mathematical expectation and variance of a random variable, respectively. The shorthand diag{.} stands for a block-diagonal125 matrix. λmin (.) and λmax (.) are respectively the minimum and maximum eigenvalue of a matrix. 2

2. Problem formulations and preliminaries Consider the following discrete uncertain system   x (k + 1) = [A +α (k) ∆A (k)] x (k) + B d1 (k) + [Aw x (k)       + Dw d1 (k)] w1 (k)      y (k) = C x (k) + D d2 (k) (1)       z (k) = F x (k)       x (s) = ϕ (s) , s = −L, −L + 1, . . . , 0

where x(k) ∈ Rn , y(k) ∈ Rm , z(k) ∈ Rq are the state vector, the measurement output, the signal to be estimated, respectively. d1 (k) ∈ Rh is the disturbance input of the system dynamics; d2 (k) ∈ R p is the disturbance in measurement process; w1 (k) ∈ R is a scalar Wiener process defined on a{ given}probability space (Ω, F , Prob) with E {w1 (k)} = 0, E w21 (k) = 1, { } E wT1 (i)w1 ( j) = 0 for i , j. The system matrices A, B, Aw , Dw , C , D, F are constant matrices with compatible dimensions. ϕ (s) is the given initial conditions of the system with the constraint ϕT (s) ϕ (s) ≤ Γ, s = −L, −L + 1, . . . , 0 and L is a positive integer to be specified in (4). The matrix ∆A (k) quantifies the parameter variation defined as follows ∆A (k) = Ma ∆a (k) Na

(2)

where Ma and Na are known constant matrices; ∆a (k) is an unknown time-varying matrix satisfying ∆Ta (k) ∆a (k) ≤ I. The random variable α (k) is used to describe the randomly occurring parameter uncertainties, which is assumed to obey the following Bernoulli distribution Prob {α (k) = 1} = α, ¯

Prob {α (k) = 0} = 1 − α, ¯

(3)

where α¯ ∈ [0, 1] is a known Define α˜ (k) = α (k) − α, ¯ { constant. } 2 ¯ = α∗ . then E {α˜ (k)} = 0 and E α˜ (k) = α¯ (1 − α) In this paper, it’s considered that the measurements of sensors are transmitted to the filter over a wireless communication network, where the channel fading is often unavoidable. To deal with the impact of the channel fading, the following stochastic L-th Rice fading model is developed to characterize the actual received signal of the filter: y f (k) = λ0 (k) y (k)+

L ∑ τ=1

λτ (k) y (k − τ)+Ed3 (k)+ Mw d3 (k)w2 (k),

(4) where L is the given amount of the paths. λτ (k) ∈ R (τ = 1, 2, . . . , L) are the channel coefficients which are mutually independent random variables and they take values on the interval [0, 1] with E{λτ (k)} = λ¯ τ and D{λτ (k)} = λ∗τ , which are employed to depict the random amplitude fading in practice. d3 (k) ∈ Ru is the exogenous disturbance in the communication channel, w2 (k) ∈ R is an independent scalar Wiener process defined on space}(Ω, F , Prob) with { a given } probability { 2 T E {w2 (k)} = 0, E w2 (k) = 1, E w2 (i) w2 ( j) = 0 for i , j. E and Mw are known constant matrices with proper dimension.

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Remark 1. It should be noted that the majority of the existing Rice fading model considered the exogenous disturbance in an additive form [15, 16, 18]. But in fact, the exogenous disturbance may influence the channel transmission in a multiplicative form. Based on this, [19] introduces a novel Rice fading model with the disturbance-dependent noise. But as yet, there is still no literature that has taken into account both the two forms. Actually, the external disturbances always affect the actual receivers in a more complex form. For this, we consider a more comprehensive stochastic Rice fading model (4), which contains both two forms of the disturbance synchronously, to145 better depict the realistic complex fading phenomena in wireless communication channels. For the discrete system (1) and the received signal model (4), we try to design the following filter    x f (k + 1) = A f x f (k) + B f y f (k)   z f (k) = C f x f (k) n

m

(5)

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{ } { 2 } Obviously, E λ˜ τ (k) = 0 and E λ˜ τ (k) = λ∗τ , τ = 0, 1, . . . , L. For simplicity, we assume that the random variables α(k), λτ (k)(τ = 0, 1, . . . , L), w1 (k) and w2 (k) are mutually independent. In addition, the external disturbance is assumed to satisfy N ∑ k=0

dT (k)d(k) ≤ δ

(7)

where δ is a known constant and N is a given positive integer. Remark 2. Remarkably, the formula (7) indicates the exogenous disturbance d(k) belongs to l2 [0, N]. As mentioned in [19] , it’s more general than the one belonging to l2 [0, +∞). Before going further, we first give the following several definitions. Definition 1 ([28]). For two scalars 0 < c1 < c2 , a positive integer N and a matrix R > 0, the filtering error system (6) is said to be finite-time stochastic stable with respect to (c1 , c2 , N, R), if when the external disturbance d(k) = 0 and d˜ (k) = 0, the following condition holds { } { } E ηT (0) Rη (0) ≤ c1 ⇒ E ηT (k) Rη (k) < c2 , ∀k ∈ {1, 2, . . . , N} .

q

where x f (k) ∈ R , y f (k) ∈ R and z f (k) ∈ R are the states of the filter, the input of the filter and the estimation of z(k), respectively. A f , B f and C f are the filter gains to be designed. By defining [ ]T η (k) = xT (k) xTf (k) , ze (k) = z(k) − z f (k),

Definition 2 ( [19, 37, 38]). For two scalars 0 < c1 < c2 , a positive integer N and a matrix R > 0, the filtering error system (6) is said to be finite-time stochastic bounded (FTSB) with respect to (c1 , c2 , N, R), if for the external disturbance d(k) satisfying (7), the following condition holds { } { } T T E η (0) Rη (0) ≤ c1 ⇒ E η (k) Rη (k) < c2 , ∀k ∈ {1, 2, . . . , N} .

[ ]T ξ (k) = ηT (k − 1) ηT (k − 2) . . . ηT (k − L) ,

]T [ d (k) = d1T (k) d2T (k) d3T (k) , [ ]T d˜ (k) = dT (k − 1) dT (k − 2) . . . dT (k − L) ,

and combining (1), (4) and (5), we can get the following filtering error system

Definition 3 ( [19]). For two scalars γ > 1, β∗ > 0 and any nonzero d(k) ∈ l2 [0, N], the filtering error system (6) is said to be finite-time stochastic exponential dissipative (FTSED), if under the zero initial condition, the estimation error ze (k) satisfies  N   ]   ∑γ−k [ T  T T ze (k) Q ze(k)+2 ze (k) Sd (k)+ d (k) Rd(k)  E     k=0   N      ∑ T (k) d(k) , ≥ β∗ E (8) d     

  η (k + 1) = A¯ η (k) + B¯ d (k) + λ˜ 0 A˜ η (k) + λ˜ 0 B˜ d (k)       ˜ ˜ d(k)  + α˜ (k) ∆A¯ (k) η (k) + Λ¯ L C˜ ξ(k) + Λ¯ L D    ( )   (6) + Λ˜ L C˜ ξ(k) + A¯ w η (k) + D¯ w d (k) w1 (k)        ˜ +M ˜ d(k) ¯ w d(k) w2 (k)  + Λ˜ L D      z (k) =Fη ¯ (k) , e

where

] [ ] [ 0 0 B A + α¯ ∆A¯ (k) 0 ¯ ¯ , B= A= , λ¯ 0 B f C 0 λ¯ 0 B f D B f E Af ] [ ] [ ] [ 0 0 0 ¯ 0 0 ˜ Dw 0 0 ˜ , B= A= , Dw = , 0 Bf D 0 0 0 0 Bf C 0 [ ] [ ] 0 ¯ 0 0 0 A ˜ · · · , A˜ }, A¯ w = w , Mw = , C˜ = diag{A, | {z } 0 0 0 0 B f Mw L [ ] ∆A (k) 0 ˜ = diag{ B, ˜ · · · , B˜ }, ∆A¯ (k) = , D | {z } 0 0 L [ ] ˜ ˜ ˜ ˜ ΛL = [λ1 (k) I 2n , λ2 (k) I 2n , · · · , λL (k) I2n ], F¯ = F −C f ,

Λ¯ L = [λ¯ 1 I2n , λ¯ 2 I2n , · · · , λ¯ L I2n ], λ˜ τ (k) = λτ (k) − λ¯ τ , τ = 1, . . . , L.

k=0

where Q, S and R are the known real matrices with Q and R symmetric. In general, we assume Q ≤ 0 and denote Q− = (−Q)1/2 . 155

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Remark 3. It is worth noting that when γ = 1, the abovementioned exponential dissipative performance can degrade into a general performance index. In addition, H∞ performance can also be included as a special case: ( ) (i) when γ = 1, Q = −I, S = 0, R = µ2 +β I, the forgoing dissipative performance can degrade into finite-time H∞ performance[39–41].

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 ¯ ˜  Λ¯ L C˜ √ B¯ Λ¯ L D  √ A   0 0  , λ∗0 B˜ Ω21 =  λ∗0 A˜  ¯  Aw 0 D¯ w 0 ] [ ¯w 0 0 0 M √ , Ω41 = α∗ ∆A¯ (k) 0 0 0 [ ] ˜ , Ω31 = 0 ΓL C˜ 0 ΓL D √ √ √ ΓL = diag{ λ∗1 I2n , λ∗2 I2n , · · · , λ∗L I2n }, } } { { σP = λmax R−1/2 P R−1/2 , σP = λmin R−1/2 P R−1/2 ,

Therefore, the goal of this paper is to design the filter (5) for the uncertain discrete-time system (1) with state and disturbance-dependent noise subject to the fading measurements (4). Specially, we attempt to obtain the filter parameters A f , B f and C f to make the following two requirements met simultaneously: (R1) The filtering error system (6) is FTSB with respect to (c1 , c2 , N, R, δ).

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(R2) Under the zero initial condition, for any non-zero d(k) ∈ l2 [0, N], the estimation error ze (k) satisfies (8).

then the filtering error system (5) is FTSB with regard to (c1 , c2 , N, R, δ).

3. Main results

Proof. First, for the matrix PL > 0 in (9), we have { T T E{Λ˜ L P Λ˜ L } = E [λ˜ 1 (k) I2n , λ˜ 2 (k) I2n , · · · , λ˜ L (k) I2n ] P [ ] × λ˜ 1 (k) I2n , λ˜ 2 (k) I2n , · · · , λ˜ L (k) I2n

Firstly, we give the following lemma that will be used in the sequel. Lemma 1 ([6]). Given the proper-dimensional matrices L = LT , H, E and F satisfying FF T ≤ I. The condition L + HFE + E T F T H T < 0 holds, if and only if there exists a scalar ε > 0 such that L + ε−1 HH T + εE T E < 0, or equivalently    L H εE T    T −εI 0  < 0. H  εE 0 −εI 175

180

= diag{λ∗1 P, λ∗2 P, · · · , λ∗L P} {√ } √ √ = diag λ∗1 I2n , λ∗2 I2n , · · · , λ∗L I2n

(11) × diag{P, P, · · · , P} | {z } L {√ } √ √ ∗ × diag λ1 I2n , λ∗2 I2n , · · · , λ∗L I2n

Next, three theorems are given to provide sufficient conditions which guarantee the two requirements mentioned in the previous section can be satisfied simultaneously. And the filter gains are derived in the third theorem.

Choose the following Lyapunov-Krasovskii function

3.1. Finite time stochastic boundedness

where

V(k) = V1 (k) + V2 (k),

In this subsection, the finite time stochastic bounded performance of the filtering error system (6) will be discussed.

V1 (k) = ηT (k) Pη (k) , V2 (k) =

σ ¯ P c1 +Γ

∗ − P−1 3 0 0 L ∑

s=1

∗ ∗ − P−1 L 0

 ∗   ∗   < 0, ∗  − P−1 2

σP

E{V1 (k + 1)} { } = E ηT (k + 1) Pη (k + 1) {[ = E A¯ η (k)+ B¯ d (k)+ λ˜ 0 A˜ η (k)+ λ˜ 0 B˜ d (k)+ α˜ (k) ∆A¯ (k) η (k) ( ˜ ˜ ˜ d(k)+ ˜ d(k)+ + Λ¯ L C˜ ξ(k) + Λ¯ L D Λ˜ L C˜ ξ(k)+ Λ˜ L D A¯ w η (k) ) ¯ w d(k) w2 (k) ]T P[ A¯ η (k) + B¯ d (k) +D¯ w d (k) w1 (k) + M

(9)

(10)

where Ω11

  L      ∑ , = diag  W s − γ P, −γ WL , −γI, −γI      s=1

P3 = diag {P, P, P} ,

, · · · , P}, PL = diag{| P, P{z } L

WL = diag{W1 , W2 , · · · , WL }, P2 = diag {P, P} ,

ηT (i) W s η (i) ,

P > 0, W s > 0 (s = 1, 2, . . . , L). According to the filtering error system model (6), the following can be deduced

s λmax (W s ) + (L + 1) δ

< γ−N c2 ,

L ∑ k−1 ∑

(13)

s=1 i=k−s

Theorem 1. Consider the discrete-time system (1). Given scalars c2 > c1 > 0, Γ > 0, δ > 0, an integer N > 0 and matrix R > 0 and suppose the filter parameters in (5) are given. If there exist a scalar γ > 1, matrices P > 0 and W s > 0 (s = 1, 2, . . . , L) such that the following matrix inequalities hold  Ω11 Ω21  Ω31 Ω41

(12)

= ΓL PL ΓL .

4

+ λ˜ 0 A˜ η (k) + λ˜ 0 B˜ d (k) + α˜ (k) ∆A¯ (k) η (k) + Λ¯ L C˜ ξ(k) ( ) ˜ + Λ˜ L C˜ ξ(k) + Λ˜ L D ˜ + A¯ w η (k) + D¯ w d (k) ˜ d(k) ˜ d(k) + Λ¯ L D } ¯ w d(k) w2 (k) ] × w1 (k) + M [ ˜ ]T P[ A¯ η (k)+ B¯ d (k) ˜ d(k) = A¯ η (k)+ B¯ d (k)+ Λ¯ L C˜ ξ(k) + Λ¯ L D ]T [ [ ˜ ˜ ]d(k)+ × Λ¯ L C˜ ξ(k) + Λ¯ L D λ∗0 A˜ η (k) + B˜ d (k) P A˜ η (k) ] [ ] [ + B˜ d (k) + A¯ w η (k) + B¯ d (k) + D¯ w d (k) T P A¯ w η (k) ] ˜ ]T ΓTL ˜ d(k) ¯ w d(k) + [ C˜ ξ(k) + D ¯ Tw P M + D¯ w d (k) + dT (k) M

[ ˜ ] + α∗ ηT (k) ∆ A¯ T (k)P∆A(k)η ˜ d(k) ¯ (k) × PL ΓL C˜ ξ(k) + D ( ) = XT (k) ΩT21 P3 Ω21 + ΩT31 PL Ω31 + ΩT41 P2 Ω41 X(k) = XT (k)ΥX(k), [

By conducting successive substitution of (19), we have E{V (k)} < γ V (k − 1) + γ dT (k − 1) d (k − 1) T

+ γ d˜ (k − 1) d˜ (k − 1)

(14)

< γ2 V (k − 2) + γ2 dT (k − 2) d (k − 2)

]T T (k) ξ (k) d (k) d˜ (k) . where X(k) = Also, it follows that T

ηT

E{V2 (k + 1)} =

L ∑

k ∑

T

T + γ dT (k − 1) d (k − 1) + γ2 d˜ (k − 2) d˜ (k − 2)

.. .

ηT (i) W s η (i) .

s=1 i=k+1−s

(15)

Combining (14) and (15), we get E{V (k + 1)} = X(k)T ΥX(k)+

L ∑

< γk V (0) +

ηT (i) W s η (i) .

E{V (k)} < γ N V (0) +

−γ

+

ηT (i) W s η (i) <

ηT (i) W s η (i)

i=k+1−s

dT ( j) d ( j) + γ N

j=0

T d˜ ( j) d˜ ( j)

j=0

  T d˜ ( j)d˜ ( j)

s=1

γN

  L N−1 ∑ ∑   T σ s λmax (W s ) + δ + d˜ ( j) d˜ ( j)  ¯ P c1 +Γ j=0

j=0 s=1

  L L N−1−s ∑ ∑ ∑   ¯ P c1 +Γ = γ N σ s λmax (W s )+δ + dT (i) d (i) s=1 s=1 i=−s   L N−1−s L ∑ ∑ ∑   T N  s λmax (W s )+δ + ¯ P c1 +Γ = γ σ d (i) d (i) s=1 s=1 i=0   L ∑    N  γ ¯ P c1 +Γ s λmax (W s ) + (L + 1) δ . < (21) σ

T

s=1

T − γ ξ (k) WL ξ (k) − γ dT (k) d (k) − γ d˜ (k) d˜ (k)

(18)

s=1

Moreover, in light of (13), we obtain { } { } E{V (k)} > E ηT (k) Pη (k) ≥ σP E ηT (k) Rη (k) .

Applying Schur complement lemma to Eq. (9), it follows that

(22)

From (21) and (23), we can get ( ) L ∑ γN σ c ( ) (L ¯ +Γ s + + 1) δ W λ P 1 max s { } s=1 E ηT (k) Rη (k) < . σP (23) { } Obviously, it can be available from (10) that E ηT (k) Rη (k) ≤ c2 holds for all k ∈ {1, 2, . . . , N}. This completes the proof.

Ω11 + ΩT21 P3 Ω21 + ΩT31 PL Ω31 + ΩT41 P2 Ω41 ≤ 0. From (14) and (18), we can easily know that Υ = ˜ = Υ + Ω11 . Thus, the ΩT21 P3 Ω21 + ΩT31 PL Ω31 + ΩT41 P2 Ω41 ,Υ ˜ above inequality means Υ ≤ 0, that is to say, J(k) < 0. According to (17), it can be inferred that T E{V (k + 1)} < γ V (k) + γ dT (k) d (k) + γ d˜ (k) d˜ (k) .

N−1 ∑

s=1

T

185

γ N− j d˜T ( j) d˜ ( j)

 N−1∑ L ∑  + δ+ dT( j − s)d( j − s)

]

= X(k)T (Υ + Ω11 ) X(k) ˜ = X(k)T ΥX(k).

N−1 ∑ j=0

N−1 ∑

s=1

  L  ∑  ≤ X(k) ΥX(k) + η (k)  W s − γP η (k) T

γ N− j dT ( j) d ( j) +

 L ∑  N ¯ P c1 +Γ s λmax (W s ) = γ σ

s=1

ηT (i) W s η (i)

(20)

N−1 ∑ j=0

s=1 i=k+1−s

= X(k) ΥX(k) − γη (k)T Pη (k) − γ dT (k) d (k) L [ ∑ T ηT (k) W s η (k)−γ ηT (k− s) W s η(k− s) − γ d˜ (k) d˜ (k)+ + (1−γ)

γk− j d˜T ( j) d˜ ( j) .

 L ∑  N σ γ ¯ P ηT (0) Rη (0)+Γ s λmax (W s ) + δ <

s=1 i=k−s T

k−1 ∑

N−1 ∑

≤ γ N V (0) + γ N

T

J(k) = X(k)T ΥX(k) − γη (k)T Pη (k) − γ dT (k) d (k)

L ∑ k−1 ∑

k−1 ∑ j=0

j=0

J(k) = E{V (k + 1)} − γ V (k) − γ d (k) d (k) − γ d˜ (k) d˜ (k) . (17) From (16), it has

L k ∑ ∑

γk− j dT ( j) d ( j) +

Noting k ∈ [1, 2, . . . , N] and γ > 1, it follows

s=1 i=k+1−s

T

T

k−1 ∑ j=0

k ∑

(16) Now, let us analyse the finite time stochastic boundedness for the filtering error system (6). Employ the following auxiliary function:

− γ d˜ (k) d˜ (k) +

T + γ d˜ (k − 1) d˜ (k − 1)

(19) 5

190

3.2. Finite time stochastic exponential dissipativity Next, we continue to analyse the finite time stochastic dissipative performance of the filtering error system (6).

N ∑ ] γ−k dT (k) d(k) + dT (k) Rd(k) − Lβ

Theorem 2. For the discrete system (1), suppose the filter parameters in (5)are given. If there exist scalars γ > 1, β > 0, matrices P > 0 and W s > 0 (s = 1, 2, . . . , L) such that the following inequalities hold  ¯ ∗ ∗ ∗  Ω11  Ω21 − P−1 ∗ ∗  3  < 0,  (24) ∗  0 − P−1 Ω31 L 0 0 − P−1 Ω41 2

+β

k=0

¯ 11 Ω

∗

−γ WL 0

∗

∗ −R + (L + 1)βI 0

Based on N ∑

β γN

N ∑

k=0

dT (k) d(k) < β

γ−k d˜T (k) d˜ (k) = =

 ∗   ∗  ,  ∗  −βI

=

− γ ξ (k) WL ξ (k) −

T ¯ (k) − 2 ηT (k) F¯ T S (k) F¯ QFη T

× d (k) − dT (k) (R − (L + 1) βI) d (k) − β d˜ (k) d˜ (k) = X(k)T (Ω¯ 11 +Υ)X(k)

ˆ = X(k)T ΥX(k),

k=0

γ−k dT (k) d(k), and

L N ∑ ∑

L ∑ N−s ∑

γ−k dT (k − s)d(k − s) γ−(s+ j) dT ( j)d( j)

L ∑ N−s ∑

γ−(s+ j) dT ( j)d( j)

s=1 j=0

<

L ∑ N ∑

γ− j dT ( j)d( j)

s=1 j=0

=L

N ∑

γ−k dT (k)d(k),

(30)

k=0

we draw from (29) N N ∑ [ β ∑ T γ−k zTe (k) Q ze (k)+ 2 zTe (k) Sd (k) (k) d(k) < d γN k=0 k=0 ] + dT (k) Rd(k) . (31)

s=1

ηT

(29)

s=1 j=−s

Proof. Introduce the following auxiliary function [ H(k) = E{V (k + 1)} − γ V (k) − zTe (k) Q ze (k) + 2 zTe (k) Sd (k) ] T (25) + dT (k) (R − (L + 1)βI)d(k) + β d˜ (k) d˜ (k) .

T

N ∑

k=0 s=1

k=0

matrices Ω21 , Ω31 , Ω41 , P3 , PL and P2 are defined in Theorem 1, thus the filtering error system (5) is FTSED.

Follow the similar line of (19), we have   L  ∑ T T  H(k) < X(k) ΥX(k) + η (k)  W s − γP η (k)

γ−k d˜T (k) d˜ (k) .

k=0

where

∑  L ¯T ¯  s=1 W s − γ P − F QF  0 =   − S T F¯  0

N ∑

195

Set the dissipation rate β∗ in (8) as γβN . Then, the filtering error system (5) can reach the requirement in Definition 3. This ends the proof. Remark 4. In the above two theorems, we provide the sufficient conditions for the filtering error system (5) to be finite-time stochastic bounded and exponential dissipative, respectively. −1 −1 But the existence of P−1 2 , PL and P3 make these conditions difficult to be solved, that is, it is hard to obtain the filter parameters. In the subsequent subsection, this problem is handled and the explicit expressions of the filter gains are presented.

(26)

ˆ < According to Schur complement, it follows from (24) that Υ 200 0. Therefore, we can get [ E{V (k + 1)} < γ V (k) + zTe (k) Q ze (k)+ 2 zTe (k) Sd (k) ] T + dT (k) (R − (L + 1)βI)d(k) − β d˜ (k) d˜ (k) . Remark 5. In the above two theorems, some inequalities are (27) 205 employed to derive the sufficient conditions of the existence of Under the zero initial condition, conducting successive substithe desired filter, which may cause some conservativeness of tution to (27), one has the results. A further reduction of the conservativeness still deserves concern in the future work. N ∑ [ N−k T T γ ze (k) Q ze (k)+ 2 ze (k) Sd (k) 0 < E{V (N + 1)} < 3.3. Finite time dissipative filter design k=0 ] T So far, we have analysed the finite-time stochastic bounded+ dT (k) (R − (L + 1)βI)d(k) + β d˜ (k) d˜ (k) 210. ness and dissipativity of the the filtering error system (5). In (28) this subsection, the filter design problem is addressed. It stems from (28) that Theorem 3. For given scalars c2 > c1 > 0, Γ > 0, δ > 0, an N N ∑ ∑ [ integer N > 0 and matrix R > 0, if there exist scalars γ > 1, −k zT T −k T γ γ β d (k) d(k) < e (k) Q ze (k)+ 2 ze (k) Sd (k) β > 0, ε1 > 0, ε2 > 0, υ2 > υ1 > 0, ρ s > 0(s = 1, 2, . . . , L), k=0 k=0 6

matrices P > 0, W s > 0(s = 1, 2, . . . , L), G1 , G2 , G3 , A¯ f , B¯ f and C¯ f such that the following inequalities hold υ1 R < P < υ2 R,

(32)

W s < ρ s I,

(33)

− γ−N c2 υ1 + υ2 c1 +Γ

L ∑

s ρ s + (L + 1) δ < 0,

Π4 =

[

0 Π6 = 0

(34)

s=1

  ∗ ∗ ∗ ∗   Ω  Π T ∗ ∗ ∗   P3 − G3 − G3   Ξ ∗ ∗  < 0, 0 PL − GL − GTL   F ∗  0 0 P2 − G2 − GT2   K 0 Ψ − ε 1 I2 (35)   ∗ ∗ ∗ ∗ ∗   Σ  T  Π P3 − G3 − G3 ∗ ∗ ∗ ∗     Ξ ∗ ∗ ∗  0 PL − GL − GTL <0  0 0 ∗ ∗  P2 − G2 − GT2  F   Z K 0 Ψ − ε2 I2 ∗   M 0 0 0 0 −I (36) where P3 , PL , P2 and Ω = Ω11 are defined in Theorem 1, and G3 = diag {G, G, G} , GL = diag{G, G, · · · , G}, G2 = diag {G, G} , | {z } L

[ G G= 1 G3

 Π1 G2  , Π = Π5 G3  Π7 ]

Π2 0 0

∑  L ∗  s=1 W s − γ P  0 −γ WL Σ =   − T F¯ 0 S  0 0 [ Ξ = 0 Ξ1 [

0 Z= Z1 Π1 =

0

∗

∗ −R + (L + 1)βI 0

[ ] 0 0 0 , F = 0 0

] [ 0 0 0 K , K= 1 0 0 0 0

[ T G1 A +λ¯ 0 d1 B¯ f C GT2 A +λ¯ 0 d2 B¯ f C

[( λ¯ 1 d1 B¯ f C Π2 = ¯ λ1 d2 B¯ f C Π3 =

Ξ2

 Π4   0  ,  0

Π3 Π6 Π8

F1 0

 ∗   ∗  ,  ∗  −βI

[ T G1 Dw GT2 Dw

] [ 0 0 , K1 = α¯ MaT G1 0 0

)] λ¯ L d1 B¯ f D 0 , λ¯ L d2 B¯ f D 0

] 0 , 0

] α¯ MaT G2 ,

{[ √ ∗ ] [√ λ1 d1 B¯ f C 0 √λ∗2 d1 B¯ f C √ Ξ1 = diag , λ∗1 d2 B¯ f C 0 λ∗2 d2 B¯ f C ]} [√ ∗ λ d B¯ C 0 , · · · , √ ∗L 1 ¯ f λ L d2 B f C 0 Ξ2 = diag

{[ 0 0

√ ∗ ¯ √λ1∗ d1 B f D λ1 d2 B¯ f D [ √ 0 √λ∗L d1 B¯ f ··· , 0 λ∗ d2 B¯ f L

F1 =

[ 0 0

Ψ1 =

[√

0 0

] [ ] √ 0 0 √λ∗2 d1 B¯ f D 0 , , 0 0 λ∗2 d2 B¯ f D 0 ]} D 0 , D 0

] [ d1 B¯ f Mw , Z1 = ε N a ¯ d2 B f Mw √

α∗ MaT G1

] 0 , 0

] 0 , I2 = diag{I, I},

] [ α∗ MaT G2 , M = Q− F

] − C¯ f ,

¯ B f = G−T 3 Bf ,

C f = C¯ f .

(37)

The dissipativity rate in (8) is given by β∗ =

β . γN

(38)

Proof. First, in light of (32) and (33), it can be easily learnt that the condition (34) is a sufficient condition for (10). For the condition (35), by Schur complement, it is equivalent to   ∗ ∗ ∗  Ω11   Π P3 − G3 − GT ∗ ∗ 3  + ε1 Σ1T Σ1  T ∗ 0 PL − GL − GL   Ξ F 0 0 P2 − G2 − GT2

] 0 0 , 0 0

] [ T [√ ∗ ¯ G1 B λ¯ 0 d1 B¯ f D d1 B¯ f E √λ0∗ d1 B f C , = Π 5 T ¯ ¯ ¯ λ0 d2 B¯ f C G2 B λ0 d2 B f D d2 B f E

) ( λ¯ 2 d1 B¯ f D 0 0 ··· λ¯ 2 d2 B¯ f D 0 0

[ T ] √ ∗ ¯f D 0 λ B G A d 1 0 √ ∗ , Π7 = 1T w λ0 d2 B¯ f D 0 G2 Aw

¯ A f = G−T 3 Af ,

] 0 , 0

) ( λ¯ d B¯ C 0 ··· ¯L 1 ¯ f λL d2 B f C 0

Π8 =

)( λ¯ 1 d1 B¯ f D 0 0 λ¯ 1 d2 B¯ f D 0 0

thus the filtering error system (6) is both FTSB and FTSED with regard to (c1 , c2 , N, R, δ), and the corresponding filter parameters in (5) can be obtained by

] [ ] 0 Ψ1 d1 A¯ f , Ψ = , 0 0 d2 A¯ f ,

)( 0 λ¯ 2 d1 B¯ f C 0 λ¯ 2 d2 B¯ f C

[( 0 0

T + ε−1 1 Σ2 Σ2 < 0

)] 0 , 0 ] 0 , 0

(39)

where matrix Ω11 is given in Theorem 1, and ) ( ) ( [( 0 0 0 0 0 0 0 0 Σ1 = ε−1 1 Z1

7

[( Σ2 = 0

0

0

0

) (

K1

0 0

)

0

( 0

)] 0 ,

)] Ψ1 .

By lemma 1, the inequality (39) is further equivalent to   ∗ ∗ ∗ Ω11   Π P3 − G3 − GT  ∗ ∗ 3   T  Ξ  0 ∗ PL − GL − GL   F 0 0 P2 − G2 − GT2 +Σ1T ∆Ta Σ2 + Σ2T ∆a Σ1 < 0

Define the following variables A¯ f = GT3 A f ,

215

B¯ f = GT3 B f ,

C¯ f = C f .

220

Remark 6. It is noteworthy that the matrix inequalities (34), (40)225 (35) and (36) in Theorem 3 are hard to solve because the scalar γ > 1 is unknown. According to (34), we can have

(41)

After some routine matrix operations, it can be obtained from (40) that   ∗ ∗ ∗  Ω11  GT  T ∗ ∗  3 Ω21 P3 − G3 − G3  < 0, T T  GL Ω31 0 ∗ PL − GL − GL   T T 0 0 G2 Ω41 P2 − G2 − G2 (42) where Ω21 , Ω31 and Ω41 are defined in Theorem 1. T Due to P2 − G2 − GT2 ≥ − GT2 P−1 ≥ 2 G2 , P3 − G3 − G3 T −1 T T −1 − G3 P3 G3 and PL − GL − GL ≥ − GL PL GL , it follows from (42) that   230 ∗ ∗ ∗   Ω11 T −1  GT Ω  ∗ ∗  3 21 − G3 P s,3 G3  < 0. T −1 GT Ω31  0 − ∗ G G P L s,L L  TL  0 0 − GT2 P−1 G G2 Ω41 s,2 2 (43) } { −T −T and its trans, , Pre- and post-multiplying diag I, G−T G G L 2 3 pose on both sides of the above inequality, we have (9). Thus, the condition (35) is a sufficient condition of (9). Similarly, recalling Schur complement, the condition (36) is equivalent to  ¯ ∗ ∗ ∗  Ω11   Π P3 − G3 − GT ∗ ∗ 3  + ε2 Σ1T Σ1  T 0 ∗ PL − GL − GL   Ξ F 0 0 P2 − G2 − GT2 T + ε−1 2 Σ2 Σ2 < 0

γ N c1 υ < υ1 . c2 2

(47)

Furthermore, the condition (32) implies that 0 < υ1 < υ2 . Hence we can deduce that ( )1/N c2 . (48) γ< c1 This indicates that the scalar γ, which can make the conditions (32)-(36) Theorem 3 feasible, should take value in the range ( ( )in ) c2 1/N of 1, c1 .

Therefore, we can find the feasible solution( of the problem in ( c )1/N ) 2 , which the Theorem 3 by searching on the interval 1, c1 is shown and verified in the subsequent numerical example. 4. A numerical example In this subsection, we give an example to illustrate the correctness of the result. Consider a tunnel diode circuit in [17, 42] displayed in Figure 1. This nonlinear system can be represented by the discrete uncertain model (1) with the following parameters: [ ] [ ] [ ] [ ] 0.51 0 0.1 0.1 −0.2 0.12 A= , B= , Aw = , Dw = , 0 0.37 0.1 0 0.08 0.23

(44)

where Ω¯ 11 is given in Theorem 2, and Ω¯ 11 = Σ + MT M. According to the lemma 1, the inequality (44) can be further expressed as  ¯ ∗ ∗ ∗  Ω11   Π P3 − G3 − GT ∗ ∗ 3  + Σ1T ∆Ta Σ2  T   Ξ ∗ 0 PL − GL − GL   F 0 0 P2 − G2 − GT2 +Σ2T ∆a Σ1 < 0

Pre- { and post-multiplying both sides of (46) by } −T −T −T diag I, G3 , GL , G2 and its transpose, we get (24). Therefore, the condition (36) is a sufficient condition for (24). This ends the proof.

d2 (k) = 0.4 sin (0.8k).

] 0.6 sin (0.8k) 1.1 , d1 (k) = , 1+k

Assume the system is affected by randomly occurring parameter uncertainties with the probability α¯ = 0.57, and the uncertain parameters are [ Ma = 1

(45)

. Using variable replacement in (41), after some arrangement, the above inequality can be represented as   ¯ ∗ ∗ ∗   Ω11  GT Ω21 P3 − G3 − GT ∗ ∗  < 0. 3 3  T T  GL Ω31 ∗ 0 PL − GL − GL   T T 0 0 G2 Ω41 P2 − G2 − G2 (46)

[ ] [ C = 1 0 , D = 1, F = 0.8

]T [ ] 0.6 sin (2k) 0.7 , Na = 0.09 0.1 , ∆a (k) = √ . 1+ k

The stochastic Lth-order Rice fading model parameters in (4) are given as L = 2, λ¯ 0 = 0.8, λ¯ 1 = 0.3174, λ¯ 2 = 0.5, λ∗0 = 0.0133, λ∗1 = 0.0194, λ∗2 = 0.01, E = 1, Mw = 0.8, d3 (k) = 0.2 sin (2k).

8

[ ] C f = −0.2390 −0.3935 ,

L

i

i

L

D 235

i R

C

V

V

C

D

240

! Figure 1: Tunnel diode circuit

245

Given the finite-time parameters are c1 = 1, c2 = 10, Γ = 1, N = 20, δ = 16, R = diag {0.1, 0.1, 0.1, 0.1}. The dissipativity matrices are chosen as Q = −1,

S = [1 2

1],

1 0.8

R = diag {10, 20, 10} .

0.6 0.4

According to (48), we should check the feasibility of the conditions (32)-(36) with regard to the scalar γ over the range (1, 1.122). On this interval, searching by the fixed step size 0.001, we can obtain that when γ ∈ [1, 1.08], there exist the feasible solutions for the conditions (32)-(36). Choose γ = 1.05, by solving the linear matrix inequalities (32)-(36), we have   0.6197   7.5648 0.40321 1.2774  −0.4032 6.87430 0.4898 1.4678  , P =  5.8359 −0.3797  1.27745 0.4898 0.6197 1.4678 −0.3797 6.1847   0.7456 −0.4234 W1 =   0.4216 0.1872

  0.7214 −0.4017 W2 =   0.3716 0.1790 [

−0.4234 0.9214 0.0039 0.2663

−0.4017 0.8847 0.0194 0.2314 ]

0.4216 0.0039 2.1422 −0.3278

0.3716 −0.0194 1.9318 −0.2565

0.2 0 -0.2 -0.4 -0.6 0

[ 12.0695 −2.4361 −3.3235 G1 = , G2 = −2.8198 11.2008 2.7511

5

10

15

20

25

Figure 2: Measurements and received signals

 0.1872   0.2663  , −0.3278 2.1954

 0.1790   0.2314  , −0.2565 1.9773

and the solved dissipativity rate is β∗ = 0.7011. Take the initial states of the system and the filter as x(0) = [ ]T x f (0) = 0 0 . Conducting simulations with the obtained filter, the results are shown in Figure 2 and Figure 3. Among them, Figure 2 depicts the measurements y(k) of the sensors and the actual received signal y f (k) of the filter, which reveals the fading channels result in large distortions and fluctuations of the transmitted signals. This is what exactly inspired us to study the channel fadings. Figure 3 displays the system signal to be estimated z(k) and its estimation z f (k), in which the good tracking performance illustrates the effectiveness of the designed dissipative filter.

0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

]

-0.1

3.0288 , −3.6115

0

4

6

8

10

12

14

16

18

20

Figure 3: The outputs z(k) and its estimation z f (k) in dissipative case

[

] 8.8395 −3.4384 G3 = , υ1 = 41.0525, υ2 = 86.2906, −3.6381 8.2369

Besides, as aforementioned in Remark 3, when γ = 1, Q = ( ) −I, S = 0, R = µ2 +β I, the finite dissipative performance (8) can reduce to the finite-time H∞ performance. Choose the following matrix parameters

ε1 = 50.8483, ε2 = 3, ρ1 = 10, ρ2 = 11, β = 1.8602. In the light of (37), the corresponding dissipative filter parameters are [ ] [ ] 0.1641 0.1321 −0.0779 Af = , Bf = , 0.1006 0.1107 −0.0326

2

Q = −1,

S = [0

0

0],

R = diag {3, 3, 3} .

Through solving the matrix inequalities (32)-(36), we derive 9

5. Conclusions

0.1 0.08 255

0.06 0.04 0.02 260

0 -0.02 -0.04 -0.06

265

-0.08 -0.1 0

2

4

6

8

10

12

14

16

18

20 270

Figure 4: The outputs z(k) and its estimation z f (k) in H∞ case

  6.5176 −0.1840 P =   1.0984 0.4460

  0.7179 −0.4215 W1 =   0.3701 0.1178

  0.6921 −0.3961 W2 =   0.3260 0.1168

G1 =

[

−0.1840 5.9161 0.3151 1.2009 −0.4215 0.9697 0.0411 0.2379 −0.3961 0.9293 0.0207 0.2052 ]

References

 1.0984 0.4760   0.3151 1.2009  , 5.0220 −0.2536 −0.2536 5.3471 0.3701 0.0411 1.8284 −0.2267

0.3260 0.0207 1.6702 −0.1788 [

275

 0.1178   0.2379  , −0.2267 1.8992

 0.1168   0.2052  , −0.1788 1.7310

7.8093 −0.4374 −3.5452 , G2 = −0.7575 7.4920 1.8173

280

285

290

]

2.0250 , −3.7019

[

] 5.8387 −1.4067 G3 = , υ1 = 36.8677, υ2 = 74.0571, −1.5098 5.8248 ε1 = 40.1343, ε2 = 3, ρ1 = 11, ρ2 = 12, β = 0.4450. On the grounds of (37), the obtained H∞ filter gains are [ ] [ ] 0.1079 0.0809 −0.0608 Af = , Bf = , 0.0672 0.0700 −0.0371 [ ] C f = −0.3236 −0.4491 , 250

In this paper, we have addressed the finite-time stochastic exponential dissipative filtering problem for a class of discrete system with state and disturbance-dependent noise suffering randomly occurring uncertainties and channel fadings. A modified Lth-order Rice fading model is supplied to better depict the multipath fading phenomena in realistic wireless communication channels. With resort to the auxiliary function method and some treatment tricks, we give sufficient conditions for the existence of the desired finite-time exponential dissipative filter, and the method for acquiring the filter gains. Finally, an illustrative example is presented to display the effectiveness of the results. In our succedent research, the extensions of the proposed method to the analysis and synthesis of the other stochastic nonlinear systems in the networked environment will be considered. Besides, applying the obtained theoretical results to the physical systems is also one aspect of our future research efforts.

295

300

305

and the corresponding H∞ performance index is µ2 = 2.555. 310 Suppose the initial states of the system and the filter are [ ]T x(0) = x f (0) = 0 0 . Simulating with the above-obtained filter, we can see the signal z(k) and its estimation z f (k) in the finite-time H∞ filtering case as shown in Figure 4, which con-315 forms the filter performs very well. 10

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*Highlights (for review)

Highlights: 

A finite-time exponential dissipative filter design method is presented for a class of discrete stochastic system subject to randomly occurring uncertainties and channel fading measures synchronously. And the designed dissipative filter can reduce into finite-time H  filter in a

 

special case. A modified Rice fading model is introduced to better characterize the channel fading in the wireless communication. A new result for the finite-time exponential dissipative of the filtering error system.