Finite-horizon H∞ filtering for switched time-varying stochastic systems with random sensor nonlinearities and packet dropouts

Signal Processing 138 (2017) 138–145

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Finite-horizon H∞ filtering for switched time-varying stochastic systems with random sensor nonlinearities and packet dropoutsR Yonggang Chen a, Zidong Wang b,c,∗, Wei Qian d, Fuad E. Alsaadi e a

School of Mathematical Sciences, Henan Institute of Science and Technology, Xinxiang 453003, China College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China c Department of Computer Science, Brunel University London, Uxbridge, Middlesex, UB8 3PH, United Kingdom d College of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo 454000, China e Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia b

a r t i c l e

i n f o

Article history: Received 11 December 2016 Revised 10 February 2017 Accepted 6 March 2017 Available online 22 March 2017 Keywords: Finite-horizon H∞ filtering Switched time-varying systems Stochastic disturbances Random sensor nonlinearities Successive packet dropouts

a b s t r a c t This paper is concerned with the finite-horizon H∞ filtering problem for a class of switched time-varying systems with state-dependent stochastic disturbances. The system outputs are subject to randomly occurring sensor nonlinearities and successive packet dropouts. Attention is focused on the design of a mode-dependent asynchronous time-varying filter such that the prescribed weighted H∞ performance requirement can be achieved under the average dwell-time switching. By utilizing the piecewise function approach and stochastic analysis technique, sufficient conditions are first established to ensure the existence of the desired finite-horizon asynchronous H∞ filter. Then, the explicit characterization of the filter gains is presented in terms of the solutions to certain recursive linear matrix inequalities. Finally, the effectiveness of the proposed filtering scheme is illustrated via a simulation example.

1. Introduction Over the past two decades, the H∞ filtering problem has become a focus of research in control and signal processing communities owing to its clear engineering background in handling disturbance rejection/attenuation issues. Accordingly, a great number of H∞ filtering schemes have been proposed for various systems subject to energy-bounded disturbances, see e.g. [1,4,9,10,36] and the references therein. The main objective of the H∞ filter design is to guarantee the H∞ performance index of the filtering error dynamics. Compared with the celebrated Kalman filtering algorithm [13,17,35], a distinctive feature of the H∞ filter is that the exogenous disturbance is no longer required to be the Gaussian noise of known statistics. On the other hand, in practical applications, due to various reasons such as sensor temporal failures and unreliable communication networks, system measurements may contain missing observations (also called packet dropouts or packet

R This work was supported in part by the Royal Society of the UK, the Research Fund for the Taishan Scholar Project of Shandong Province of China, the National Natural Science Foundation of China under Grants 61304061, 61329301 and 61573130, the Science and Technology Innovation Talents Project of Henan Province (No. 164100510004) and the Alexander von Humboldt Foundation of Germany. ∗ Corresponding author. E-mail address: [email protected] (Z. Wang).

http://dx.doi.org/10.1016/j.sigpro.2017.03.004 0165-1684/© 2017 Elsevier B.V. All rights reserved.

© 2017 Elsevier B.V. All rights reserved.

losses) [3,16,27,41,43]. As such, the H∞ filtering problem with missing measurements has been extensively investigated in the literature, see e.g. [26,33,39], where the missing measurement has been typically assumed to obey a Bernoulli distributed white sequence taking values on 0 or 1. In many engineering and industrial applications, the system outputs are often measured from physically saturated sensors within harsh environments. As a typical kind of nonlinearities, the sensor saturation is a commonly encountered phenomenon because of physical or technological constraints on power, spectrum, speed, capacity and network bandwidth, etc. [25,29]. Such sensor nonlinearities, if ignored when designing filters or estimators, would seriously degrade the corresponding filter performance. Therefore, during the past years, much research attention has been paid to the filtering/estimation problems with respect to sensor nonlinearities/saturations during the past years, see e.g. [7,24,32,34,37]. Furthermore, it is quite common that the sensor nonlinearities/saturations occur in a probabilistic way in practical engineering such as networked control systems where the fluctuation of network load is typically random [7,32,37]. For example, in [7,32], both randomly occurring sensor saturations and randomly occurring missing measurement have been taken into account in the filter design for networked systems. Switched systems have become an issue of recurring research interest for over 20 years and a rich body of literature has been

Y. Chen et al. / Signal Processing 138 (2017) 138–145

available in the literature, see e.g. [2,5,11,15,40,47–49]. Compared with the other systems, switched systems comprise a finite number of subsystems with a switching rule and such a specific feature makes the corresponding H∞ filtering problem quite challenging. So far, some initial efforts have been made in this regard [8,14,23,28,30,38,42,45,46]. For example, under arbitrary switching, the H∞ filtering problem has been studied in [8] for discrete-time switched delay systems. Under average dwell time (ADT) switching, an exponential H∞ filter has been designed in [45] for uncertain discrete-time switched linear systems, and a fault detection filter has been proposed in [28] for nonlinear switched stochastic systems. Considering the inconsistency between the filter switching and the subsystem switching, the asynchronous H∞ filtering problems have been studied in [14,46] for switched delay systems and switched linear systems, respectively. As for switched delay systems with missing measurements, an H∞ filter has been designed in [42] under arbitrary switching, and an l2 − l∞ filter has been designed in [38] under ADT switching. Nevertheless, all the aforementioned literature has been focused on switched systems with linear measurements and the effects of sensor nonlinearities have not been taken into account despite their practical significance. So far, in most existing literature concerning H∞ filter design problem for switched systems, it has been implicitly assumed that the underlying system is time-invariant and the filter is implemented on an infinite time-horizon where the steady-state filter performance (e.g. asymptotic stability) is evaluated. In reality, however, such an assumption is not always true as most realtime dynamical systems are time-varying and the transient behaviors over a finite-horizon are of more interests [18]. As such, it makes more practical sense to investigate the filtering problems for time-varying systems over a finite-horizon [6,21,22,31,44]. For example, in [6], the finite-horizon robust H∞ filtering problem has been considered for a class of Markovian jump systems with sensor saturation in terms of recursive linear matrix inequalities (RLMIs). In [31], the finite-horizon H∞ filter has been designed for discrete time-varying systems with missing measurements and quantization effects. Nonetheless, a literature review has revealed that the finite-horizon H∞ filtering problem has not been fully investigated yet for switched time-varying systems, not to mention the case where stochastic disturbances, sensor nonlinearities and packet dropouts are simultaneously involved. It is, therefore, the main motivation of this paper to shorten such a gap. Based on the above discussions, in this paper, we aim to investigate the finite-horizon H∞ filtering problem for switched timevarying stochastic systems with randomly occurring sensor nonlinearities and successive packet dropouts. The proposed filter is mode-dependent and is also asynchronous with switched systems. By applying the piecewise function approach and stochastic analysis technique, sufficient conditions are first proposed under ADT switching that guarantee the existence of a desired finite-horizon asynchronous H∞ filter. Subsequently, the explicit characterization of the filter gains is obtained by means of the solution to a set of RLMIs. Finally, a simulation example is given to show the effectiveness of the proposed filtering scheme. The main contributions of this paper are summarized as follows: (1) a novel measurement model is proposed that accounts for both the phenomena of randomly occurring sensor nonlinearities and successive packet dropouts; (2) a novel weighted H∞ performance constraint is proposed on the disturbance rejection/attenuation level of the filter error dynamics against the energy-bounded exogenous disturbance; and (3) the finite-horizon H∞ filter design algorithm is designed, for the first time, for switched time-varying stochastic systems with random sensor nonlinearities and packet dropouts.

139

Notation. The superscript “T ” stands for the matrix transposition. Rn denotes the n-dimensional Euclidean space, and x denotes the Euclidean norm of the vector x. l2 [0, N] is the space of square summable vector functions over an interval [0, N]. Prob{·} means the occurrence probability of the event “·”. E{x} stands for the expectation of the stochastic variable x. Real symmetric matrix P > 0 ( ≥ 0) denotes that P is a positive definite (positive semi-definite) matrix. I denotes an identity matrix with proper dimension. The symmetric terms in a symmetric matrix are denoted by ∗ . Matrices, if not explicitly stated, are assumed to have compatible dimensions. 2. Problem formulation Consider the following discrete-time switched time-varying stochastic system defined on k ∈ [0, N]:



x(k + 1 ) = Akσ x(k ) + Bkσ ω (k ) + gkσ (x(k ))v(k ), y˜(k ) = λk ψ (Cσk x(k )) + (1 − λk )Cσk x(k ) + Dkσ ω (k ), z(k ) = Lkσ x(k ),

(1)

where x(k ) ∈ Rn is the system state, y˜(k ) ∈ Rm denotes the sensor output, z(k ) ∈ R p is the output to be estimated, ω (k ) ∈ Rq denotes the exogenous disturbance belonging to l2 [0, N], v(k) is a scalar Wiener process (Brownian Motion) defined on a compete probability space with E{v(k )} = 0, E{v2 (k )} = 1 and E{v(i )v( j )} = 0 (i = j ), λk is a Bernoulli distributed white sequence taking values on 0 or 1. σ denotes the switching signal, which is a piecewise constant function of the time k, and takes its values in the set [1, M] (M > 1 is the number of subsystems). For any switching time sequence k0 < k1 <  < kl <  < N, σ is assumed to be continuous from the right everywhere and, when k ∈ [kl , kl+1 ), the σ (kl ) − th subsystem is activated. Aki , Bki , Cik , Dki and Lki (i ∈ [1, M] ) are known time-varying matrices with appropriate dimensions. The nonlinear vector-valued function gki (x(k )) satisfies the following condition:

gki (x(k )2 ≤ Gki x(k )2 ,

(2)

where Gki is a known time-varying matrix. In addition, the nonlinear vector-valued function ψ (·) represents the sensor nonlinearities which satisfies the following sector condition:

(ψ (u ) − K1 u )T (ψ (u ) − K2 u ) ≤ 0,

(3)

where K1 and K2 are diagonal matrices with K2 > K1 ≥ 0. Noting that the sensor saturation is a special kind of sensor nonlinearity, it is clear that the function ψ (·) can be used to represent the sensor saturations if one sets K1 = 0 and K2 = I in the sector condition (3). Letting φ (u )  ψ (u ) − K1 u and K  K2 − K1 , it follows from (3) that

φ T (u )(φ (u ) − Ku ) ≤ 0.

(4)

In this paper, λk is a Bernoulli distributed white sequence governing the random occurrence of the sensor saturations and y˜(k ) is the corresponding intermediate measurement. The actual measurement output y(k ) ∈ Rm has the following from:

y(k ) = (1 − θk )y˜(k ) + θk y(k − 1 ),

(5)

where β k is another Bernoulli distributed white sequence taking values on 0 or 1. In the addressed system model (1)–(5), the stochastic variables λk , θ k and v(k) are mutually uncorrelated, and

Prob{λk = 1} = λ, Prob{λk = 0} = 1 − λ, Prob{θk = 1} = θ , Prob{θk = 0} = 1 − θ ,

140

Y. Chen et al. / Signal Processing 138 (2017) 138–145

where λ and θ are known constants. Denoting θ − θk  δk and (1 − θ )λ − (1 − θk )λk  νk , by some simple calculations, one can obtain E{δk } = 0, E{νk } = 0 and

E{δk2 } =

θ ( 1 − θ )  ρ1 ,

E{νk2 } = (1 − θ )λ − (1 − θ )2 λ2  ρ2 , E{δk νk } = (1 − θ )θ λ  ρ3 . Remark 1. In this paper, the stochastic variables λk and θ k are introduced in the measurement output (5) to describe the randomly occurring sensor nonlinearity and successive packet dropouts, respectively. Different from the measurement models in [7,32], the proposed measurement model (5) covers the case where the data subjected to sensor nonlinearity/saturation can also be lost during the signal transmission. Compared with the measurement model in [37], the model (5) incorporates the phenomenon of packet dropouts and is therefore more practical in a networked environment. Throughout this paper, we are interested in designing the following mode-dependent time-varying filter



xˆ(k + 1 ) = Aˆ σkˆ xˆ(k ) + Bˆσkˆ y(k ), zˆ(k ) = Lˆσkˆ xˆ(k ),

(6)

where xˆ(k ) ∈ Rn is the state estimate, z(k ) ∈ R p is the estimated output, σˆ is the switching signal of the filter that is not necessarily the same with that of the system (1), and Aˆ kj , Bˆkj , Lˆkj (j ∈ [1, M]) are the filter gains. Compared with the mode-independent filter, the modedependent filter can utilize the information of the current mode and then select the corresponding filter, and thus can achieve better filter performance requirements for switched systems. However, when applying the mode-dependent filter, it may take some time to identify the active subsystem. In this paper, we assume that the switching time sequence over the finite-horizon [0, N + 1] is described by k0  0 < k1 < · · · < kl < N + 1 ≤ kl+1 and the active subsystem over the initial interval [k0 , k1 ) is known in advance. Also, we introduce the scalar Tr as the time to identify the active subsystem over the interval [kr , kr+1 ) and set TM = max∀r∈[1,l] {Tr }. Without loss of generality, the scalar Tr is assumed to satisfy 0 < Tr < kr+1 − kr (r = 1, 2, · · · , l − 1 ) and Tl < N + 1 − kl . It is clear that the interval [kr , kr+1 ) can be divided into two subintervals, i.e., asynchronous interval [kr , k¯ r ) and synchronous interval [k¯ r , kr+1 ), where k¯ r = kr + Tr . Noting φ (u )  ψ (u ) − K1 u, it follows from (1) and (5)–(6) that

⎧ η (k + 1 ) = [A¯ kσ σˆ + ρ0 Aˆ kσ σˆ ]η (k ) + ρ0 B¯ σkˆ φ (Cσk x(k )) ⎪ ⎨ +D¯ kσ σˆ ω (k ) + g¯kσ v(k ) + δk [Aˇ kσ σˆ η (k ) + Dˇ kσ σˆ ω (k )] k k ˆk ⎪ ⎩ +νk [Ak σ σˆ η (k ) + B¯ σˆ φ (Cσ x(k ))], e(k ) = L¯ σ σˆ η (k ),

(7)

where η (k ) = [xT (k ) xˆT (k ) yT (k − 1 )]T , e(k ) = z(k ) − zˆ(k ), ρ0 = (1 − θ )λ and

⎡

Akσ k ˜ ¯ ⎣ ˆ Aσ σˆ = θ Bσkˆ Cσk θ˜Cσk

0 Aˆ σkˆ 0

0 Bˆσkˆ K˜1Cσk K˜1Cσk

0 0 0



Aˆ kσ σˆ =

Aˇ kσ σˆ =



0 Bˆσkˆ Cσk Cσk

L¯ kσ σˆ = Lkσ

0 0 0 −Lˆσkˆ

⎤

0 θ Bˆσkˆ ⎦, B¯ σkˆ = θI

⎡









0 , g¯kσ = [gkσ (x(k ))]T

As commonly used in the literature, we choose the chatter bound as Q0 = 0 for simplicity. In this paper, we have assumed that the switching among subsystems of the system (1) is slow. In this case, it is obvious that the ADT switching is more appropriate for performance analysis. Of course, the arbitrary switching can also be applicable. However, the results based on the arbitrary switching will be more conservative. The main objective of this paper is to design the finite-horizon asynchronous filter (6) such that, for given scalars 0 < α < 1, γ > 0 and the positive definite weighting matrix S > 0, under the ADT switching, the augmented system (7) satisfies the following weighted H∞ performance constraint:



J =E

N  

e(k )2 − α −k−1 γ 2 ω (k )2

k=0





− γ 2 η T ( 0 )S η ( 0 ) < 0.

(8)

Remark 2. It is clear that the performance constraint (8) proposed in this paper involves the term α −k−1 , which makes it different from the H∞ performance indices defined in [6,31]. In fact, when analyzing the switched systems under ADT switching, the induced H∞ performance level γ is often referred to as the weighted H∞ performance level, which is no longer the conventional one as explained in, e.g. [28] and [45]. If the filter (6) is designed under the arbitrary switching, the performance constraint (8) should be modified by setting α = 1. On the other hand, it is worth mentioning that, different from the infinite-horizon case, the system performance over a finite-horizon depends on the initial condition and, thus the term γ 2 ηT (0)Sη(0) should be incorporated in (8). 3. Main results This section first discusses the problem of weighted H∞ performance analysis for the augmented system (7). Theorem 1. Let the scalars 0 < α < 1, β > 1, γ > 0, μ ≥ 1, TM ≥ ≤N ≤N , {Bˆki }01≤k , 1, the matrix S > 0, and the filter parameters {Aˆ ki }01≤k ≤i≤M ≤i≤M 0≤k≤N k ∗ ˆ {Li }1≤i≤M be given. Under ADT switching with τa > τa = −[lnμ + TM ln(β /α )]/lnα , the augmented system (7) satisfies the weighted H∞ performance constraint (8) for all nonzero ω(k) if, with the initial condition {Pi0 ≤ γ 2 S}1≤i≤M , there exist a family of positive definite ≤N {ϕik }01≤k such that, for 1 ≤ i, j ≤ M (i = j), the following RLMIs: ≤i≤M



⎤

0 Bˆσkˆ Dkσ Dkσ

Definition 1 [15]. For any k > s ≥ k0 , let Qσ (s, k) be the switching number of σ over [s, k). If Qσ (s, k ) ≤ Q0 + (k − s )/τa holds for any given τ a > 0 and Q0 ≥ 0, then τ a and Q0 are called the average dwell time (ADT) and the chatter bound, respectively.

≤N+1 ≤N matrices {Pik }01≤k and two families of positive scalars {εik }01≤k , ≤i≤M ≤i≤M

0 Bˆσkˆ , I

Bkσ 0 k ˜ ¯ ⎣ ˆ 0 , Dσ σˆ = θ Bσkˆ Dkσ ⎦, 0 θ˜ Dkσ 0 −Bˆσkˆ , Dˇ kσ σˆ = −I

with θ˜ = 1 − θ and K˜1 = K1 − I. In what follows, when σ = i, σˆ = j (i, j ∈ [1, M]), the matrix A¯ kσ σˆ reads A¯ ki j . For the matrices Aˆ kσ σˆ , B¯ σkˆ , D¯ kσ σˆ , Aˇ kσ σˆ , Dˇ kσ σˆ , L¯ kσ σˆ , and the vector g¯kσ , we have the similar arguments.

⎡

,

0

I

T

0 ,

0

1ki ⎢ 2kii ⎢ ⎢ρ¯ 1 3kii ⎣ρ¯  k 2 4ii 5kii





0 Pik+1 I

∗ −Qik+1 0 0 0

0

0

T

∗ ∗ −Qik+1 0 0

≤ εik I,

∗ ∗ ∗ −Qik+1 0

(9)

⎤

∗ ∗⎥ ⎥ ∗ ⎥ < 0, ∗⎦ −I

(10)

Y. Chen et al. / Signal Processing 138 (2017) 138–145

⎡

 ⎢  ⎢ ⎢ρ  ⎢ ⎣ρ  

∗ −Qik+1 0 0 0

k 1i k 2i j ¯ 1 3ki j ¯ 2 4ki j k 5i j

≤μ

Pik+1

∗ ∗ −Qik+1 0 0

∗ ∗ ∗ −Qik+1 0

Pjk+1 ,



−α Pik + εik (Gki R )T Gki R  = ϕik KCik R 0

∗ −2ϕik I 0

−β Pik + εik (Gki R )T Gki R  = ϕik KCik R 0

∗ −2ϕik I 0

k 1i



k 1i

 = 

k 2i j

=

  

A¯ kii

+ρ

A¯ ki j

+ρ

ˆk 0 Aii ˆk 0 Ai j

ρ 

¯k 0 Bi

ρ

D¯ kii

¯k 0B j



D¯ ki j

(11)

k 4ii

∗ ∗

−γ 2 I ∗ ∗



, B¯ ki

,

0

−γ 2 I

Letting (18) and (19) that

,





R = [I 0 0].

≤α

×

+ ρ2 [Aˆ kii η (k ) + B¯ ki φ (Cik x(k ))]T Pik+1

−

(14)

η (k ) +

Dˇ kii

−

N 

α N−k E{(k )}

k=k¯ l

α N−k (β /α )T

+

(k+1,N+1 )

kl −1



α N−k (β /α )T

+

(k+1,N+1 )

E{(k )}

(g¯ki )T Pik+1 (g¯ki ),

α N−k (β /α )T

+

(k+1,N+1 )

E{(k )}

+ ≤ · · · ≤ μQσ (0,N+1) α N+1 (β /α )T (0,N+1) E{Vσ (0) (0 )}

−

it follows from (2) and (9) that

(16)

On the other hand, it is clear from (4) that there exists a scalar

ϕik > 0 such that

  (k )) φ (Cik x(k )) − KCik x(k ) ≥ 0.

N  k=kl

(15)

(g¯ki )T Pik+1 (g¯ki ) ≤ εik xT (k )(Gki )T Gki x(k ).

φ (

N 

−μ

ω ( k )]

× [Aˆ kii η (k ) + B¯ ki φ (Cik x(k ))].

−2ϕ

β k¯ l −k−1 E{(k )} −

k=kl−1

+ [Aˆ kii η (k ) + B¯ ki φ (Cik x(k ))]T Pik+1

Cik x

¯

E{Vσ (kl ) (kl )} − α N−kl +1

k=kl

× [Aˆ kii η (k ) + B¯ ki φ (Cik x(k ))]

T



β

k¯ l −kl

+ ≤ μα N−kl−1 +1 (β /α )T (kl−1 ,N+1) E{Vσ (kl−1 ) (kl−1 )}

2[Aˇ kii η (k ) + Dˇ kii ω (k )]T Pik+1

k i

α N−k E{(k )}

+  α N−kl +1 (β /α )T (kl ,N+1) E{Vσ (kl ) (kl )}

Using the well-known fact 2aT b ≤ aT a + bT b where a and b are vectors of appropriate dimensions, we have the following inequality:

For the term

N 

k=kl

× [Aˆ kii η (k ) + B¯ kj φ (Cik x(k ))] + 2ρ3 [Aˇ kii η (k ) + Dˇ kii ω (k )]T

ω ( k )]

N−k¯ l +1 k¯ l −1

+ ρ1 [Aˇ kii η (k ) + Dˇ kii ω (k )]T Pik+1 [Aˇ kii η (k ) + Dˇ kii ω (k )]

η (k ) +

(22)

k=k¯ l

× Pik+1 [(A¯ kii + ρ0 Aˆ kii )η (k ) + ρ0 B¯ ki φ (Cik x(k )) + D¯ kii ω (k )]

Pik+1 [Aˇ kii

(21)

E{Vσ (kl ) (N + 1 )} ¯

=E [(A¯ kii + ρ0 Aˆ kii )η (k ) + ρ0 B¯ ki φ (Cik x(k )) + D¯ kii ω (k )]T

T

from

In the sequel, we will show that the weighted H∞ performance constraint (8) is satisfied under the initial condition Pi0 ≤ γ 2 S. Using (20)–(22) recursively, we obtain

≤ α N−kl +1 E{Vσ (kl ) (k¯ l )} −



Dˇ kii

follows

(20)

Vi (k + 1 ) ≤ μV j (k + 1 ), k ∈ [0, N].



+ (g¯ki )T Pik+1 g¯ki − αηT (k )Pik η (k ) .

it

In addition, it can be seen from (12) that the following inequality holds:

0 ,



 (k ),

E{Vi (k + 1 )} < β E{Vi (k )} − E{(k )}.



(13)

× Pik+1 [Aˆ kii η (k ) + B¯ ki φ (Cik x(k ))]

(19)

For any k ∈ [kr , k¯ r ) (r = 1, 2, · · · , l ), it is clear that σ = σˆ . Without loss of generality, we denote σ = i ∈ [1, M] and σˆ = j ∈ [1, M]. Similar to the above deductions, one can infer from (9) and (11) that

For any k ∈ [k0 , k1 ) ∪ [k¯ r , kr+1 ) (r = 1, 2, · · · , l ), noting σ = σˆ  i ∈ [1, M], we have

≤[Aˇ kii

− γ 2 ω T ( k )ω ( k )

E{Vi (k + 1 )} < α E{Vi (k )} − E{(k )}.

0 ,

Vσ (k ) =xT (k )Pσk x(k ).



(18)

where ζ (k ) = [ηT (k ) φ T (Cik x(k )) ωT (k )]T , and 1ki , 2kii , 3kii , 4kii , 5kii , ρ¯ 1 , ρ¯ 2 are defined in the statement of this theorem. For the matrix inequality (10), it follows from the well-known Schur complement that

eT ( k )e ( k )

Proof. Choose the following piecewise function:

E Vi (k + 1 ) − αVi (k )



ζ T (k )[1ki + (2kii )T Pik+1 2kii + ρ¯ 12 (3kii )T Pik+1 3kii  +ρ¯ 22 (4kii )T Pik+1 4kii + (5kii )T 5kii ]ζ (k ) , ≤E

+ρ¯ 22 (4kii )T Pik+1 4kii + (5kii )T 5kii < 0.

Dˇ ki j ,

0



1ki + (2kii )T Pik+1 2kii + ρ¯ 12 (3kii )T Pik+1 3kii

,



 = Aˇ kii 0 Dˇ kii ,  = Aˆ kii    5kii = L¯ kii 0 0 , 3ki j = Aˇ ki j    4ki j = Aˆ ki j B¯ kj 0 , 5ki j = L¯ ki j   ρ¯ 1 = ρ1 + ρ3 , ρ¯ 2 = ρ2 + ρ3 , k 3ii



E Vi (k + 1 ) − αVi (k ) + eT (k )e(k ) − γ 2 ωT (k )ω (k )

(12)

hold, where Qik+1 = (Pik+1 )−1 and

k 2ii

Adding the left-hand side of (17) to E{Vi (k + 1 ) − αVi (k )} and using (14)–(16), one obtains

⎤

∗ ∗⎥ ⎥ ∗ ⎥ < 0, ⎥ ∗⎦ −I

141

N 

μQσ (k,N+1) α N−k (β /α )T

+

(k+1,N+1 )

E{(k )},

(23)

k=0

where T + (k, N + 1 ) represents the overall time length of the asynchronous intervals over [k, N + 1]. Noting Vσ (kl ) (N + 1 ) ≥ 0 and Pi0 ≤ γ 2 S, it follows from (23) that N 

(17) k=0

μQσ (k,N+1) α N−k (β /α )T

+

(k+1,N+1 )

E{(k )} ≤

142

Y. Chen et al. / Signal Processing 138 (2017) 138–145

μQσ (0,N+1) α N+1 (β /α )T

+

(0,N+1 )

γ 2 E{η T ( 0 )S η ( 0 )}.

(24)

Multiplying both sides of the inequality (24) by the scalar + μ−Qσ (0,N+1) α −N−1 (β /α )−T (0,N+1) yields N 

μ−Qσ (0,k) α −k−1 (β /α )−T

+

(0,k+1 )



¯ 3kii

E{(k )}

≤ γ 2 E{η T ( 0 )S η ( 0 )}.

(25)

Notice the facts that μQσ (0,k ) ≤ μQσ (0,k+1 ) , T + (0, k + 1 ) ≤ Qσ (0, k + 1 )TM and Qσ (0, k + 1 ) ≤ (k + 1 )/τa . Under the assumption τa > τa∗ = −[ln(μ ) + TM ln(β /α )]/lnα , one obtains

μ−Qσ (0,k) α −k−1 (β /α )−T ≥ {[μ(β /α )TM ]

1 τa

α}−k−1 > 1.

From (25) and (26), and noting (β /α ) 1, we have N 

N 

E{eT ( k )e ( k )} −

k=0

≤ 1, μ−Qσ (0,k ) ≤

Remark 3. In Theorem 1, two scalars 0 < α < 1 and β > 1 are introduced to describe the different characteristics of the piecewise function (13) over different running intervals. As for the asynchronous filter design of switched systems, it is generally recognized that the piecewise function should have different characteristics over the asynchronous interval [kr , k¯ r ) and the synchronous interval [k¯ r , kr+1 ) [46]. Next, let us handle the finite-horizon H∞ filter design problem in the following Theorem. Theorem 2. Let the scalars 0 < α < 1, β > 1, γ > 0, μ ≥ 1, TM ≥ 1, and the matrix S > 0 be given. Under ADT switching with τa > τa∗ = −[lnμ + TM ln(β /α )]/lnα , the augmented system (7) satisfies the weighted H∞ performance constraint (8) for all nonzero ω(k) if, with the initial condition {Pi0 ≤ γ 2 S}1≤i≤M , there exist a family of

≤N+1 ≤N positive definite matrices {Pik }01≤k , families of matrices {Xik }01≤k , ≤i≤M ≤i≤M 0≤k≤N 0≤k≤N 0≤k≤N 0≤k≤N 0≤k≤N 0≤k≤N k k k k k k ˆ {Y } , {Z } , {U } , {V } , {W } , {L } , 1≤i≤M

i

1≤i≤M

i

i

1≤i≤M

i 1≤i≤M

≤N ≤N and families of positive scalars {εik }01≤k , {ϕik }01≤k such that, for 1 ≤i≤M ≤i≤M ≤ i, j ≤ M (i = j), the RLMIs (9) and (12) as well as the following RLMIs:

⎡

1ki ⎢ ¯ 2kii ⎢ ¯k ⎢ρ¯ 1 3ii ⎣ρ¯ ¯ k 2 4ii 5kii

⎡

1ki ⎢ ¯ 2ki j ⎢ ⎢ρ¯ 1 ¯ 3ki j ⎢ ⎣ρ¯ 2 ¯ 4ki j 5ki j hold, where

⎡

∗

6kii

∗ ∗

0 0 0

6kii 0 0

6kii

∗

∗ ∗

0 0 0

6ki j

∗ ∗ ∗

6ki j

1ki ,

¯ 2kii11 ¯ 2kii = ⎣¯ 2kii21 ¯ 2kii31

Uik Uik 0

∗ ∗⎥ ⎥ ∗ ⎥ < 0, ⎦ ∗ −I

0

0 0

1ki ,

⎤

∗ ∗ ∗

6ki j 0

5kii ,

5ki j

θ Vik θ Vik θ Wik

are defined in Theorem 1 and

ρ0Vik ρ0Vik ρ0Wik

θ

⎤ ¯ 2kii15 ¯ 2kii25 ⎦,

˜W k D k i i

0 0 , 0

0 0 0 0

−V jk −V jk −W jk

0 0 0

0 0 0

V jk V jk W jk

6kii12 6kii22

0 0

0

0 0 0

0

6ki11j k12 T ⎣ ( ) =− 6i j

6ki12j 6ki22j

0

0



⎤ ¯ 2ki15j ¯ 2ki25j ⎦, ˜ θ W jk Dki ⎤ k

V jk Di V jk Dki ⎦, W jk Dki

⎤

0 0⎦, 0

k33 6ii

0 0

6ki33j

+ Pik+1 ,

⎤ ⎦ + Pik+1 ,

with

¯ 2kii11 =Xik Aki + θ˜VikCik + ρ0Vik K˜1Cik , ¯ 2kii15 =Xik Bki + θ˜Vik Dki , ¯ 2kii21 =Zik Aki + θ˜VikCik + ρ0Vik K˜1Cik , ¯ 2kii25 =Zik Bki + θ˜Vik Dki , ¯ 2kii31 =θ˜WikCik + ρ0Wik K˜1Cik , ¯ 2ki11j =X jk Aki + θ˜V jkCik + ρ0V jk K˜1Cik , ¯ 2ki15j =X jk Bki + θ˜V jk Dki , ¯ 2ki21j =Z kj Aki + θ˜V jkCik + ρ0V jk K˜1Cik , ¯ 2ki25j =Z kj Bki + θ˜V jk Dki , ¯ 2ki31j =θ˜W jkCik + ρ0W jk K˜1Cik , 6kii11 =Xik + (Xik )T , 6kii12 = Yik + (Zik )T , 6kii22 =Yik + (Yik )T , 6kii33 = Wik + (Wik )T , 6ki11j =X jk + (X jk )T , 6ki12j = Y jk + (Z kj )T , 6ki22j =Y jk + (Y jk )T , 6ki33j = W jk + (W jk )T .

(28)

⎤

∗ ∗⎥ ⎥ ∗ ⎥ < 0, ⎥ ∗⎦ −I

6ki j

Vik Vik Wik

ρ0V jk ρ0V jk ρ0W jk

V jk K˜1Cik = ⎣ V jk K˜1Cik W jk K˜1Cik

⎡

(27)

0 0 0

Vik Dki Vik Dki , Wik Dki

θ V jk θ V jk θ W jk

6kii11  = − (6kii12 )T

which implies that weighted H∞ performance constraint (8) is satisfied. The proof is now complete. 

i 1≤i≤M

V j Ci = ⎣ V jkCik W jkCik

⎡

¯ 4ki j

0 0 0

U jk U jk

k 6ii

k=0

1≤i≤M

k 3i j

¯ 2ki11j ¯ k21 ⎣ =  2i j ¯ 2ki31j ⎡ k k

−Vik −Vik −Wik 0 0 0



α −k−1 γ 2 E{ωT (k )ω (k )}

≤ γ 2 E{η T ( 0 )S η ( 0 )},

i

¯ 2ki j

(26) −T + (0,k+1 )

Vik K˜1Cik Vik K˜1Cik Wik K˜1Cik

⎡

¯

(0,k+1 )

0 0 0



¯ 4kii =

k=0

+

VikCik = VikCik WikCik

Furthermore, if feasible solutions exist, then the other two parameters of the filter parameters are given by Aˆ ki = (Yik )−1Uik and Bˆki = (Yik )−1Vik . Proof. Define Jik = diag{Hik , Wik } where Hik = [

(29)

Xik Zik

Yik ]. Recalling Yik

that Uik = Yik Aˆ ki and Vik = Yik Bˆki , the RLMI (28) can be rewritten as

⎡

1ki ⎢ Jik 2kii ⎢ ⎢ρ¯ 1 Jik 3kii ⎣ρ¯ J k  k 2 i 4ii 5kii Noting

∗

6kii 0 0 0

6kii 0 0

∗ ∗ ∗

6kii 0

⎤

∗ ∗⎥ ⎥ ∗ ⎥ < 0. ∗⎦ −I

(30)

6kii = Pik+1 − Jik − (Jik )T and the inequality ≤ 6kii resulting from (Jik − Pik+1 )Qik+1 (Jik −

that

−Jik Qik+1 (Jik )T

∗ ∗

Y. Chen et al. / Signal Processing 138 (2017) 138–145

Pik+1 )T ≥ 0, one obtains from (30) that

⎡

1ki ⎢ Jik 2kii ⎢ ⎢ρ¯ 1 Jik 3kii ⎣ρ¯ J k  k 2 i 4ii 5kii where

∗ −Qki 0 0 0

∗ ∗ −Qki 0 0

Qki = Jik Qik+1 (Jik )T .

∗ ∗ ∗ −Qki 0 If

⎤

∗ ∗⎥ ⎥ ∗ ⎥ < 0, ∗⎦ −I the

143

RLMI

(31)

(28)

is

feasible,

it is clear from the term 6kii < 0 that the matrix Jik is invertible. Pre- and post-multiplying the RLMI (31) by diag{I, (Jik )−1 , (Jik )−1 , (Jik )−1 , I} and its transpose, we have the RLMI (10). Using the matrix inequality −J jk Qik+1 (J jk )T ≤

6ki j = Pik+1 − J jk − (J jk )T , and similar to the above deductions,

one can obtain from RLMI (29) that the RLMI (11) holds. The rest of the proof follows directly from Theorem 1 and the proof is complete.  Remark 4. As for the synchronous filter design case, the RLMI (29) should be removed, and the ADT constraint should be revised as τa > τa∗ = −lnμ/lnα due to the fact TM = 0. In addition, noticing that T + (0, k + 1 ) = 0, the inequality (26) be−Qσ (0,k ) α −k−1 > 1/α . Then, we obtain from (25) that Jˇ = comes  N μ   2 −k 2 2 − αγ 2 η T (0 )Sη (0 ) < 0. If we E k=0 e (k ) − α γ ω (k )

remove the RLMI (9) and the term εik (Gki R )T Gki R in RLMIs (28) and (29), Theorem 2 becomes the special case without stochastic disturbances. Remark 5. The addressed system model (1)-(5) is fairly comprehensive that covers a few practically motivated complexities including time-varying parameters, switched behavior, H∞ performance constraints, state-dependent stochastic disturbances, randomly occurring sensor nonlinearities as well as successive packet dropouts. In Theorem 2, all of the information on the considered complexities is reflected in the established sufficient conditions so as to facilitate the design of the finite-horizon filter. The algorithm for filter design is given in terms of the feasibility of a series of RLMIs that are suitable for online implementation. The RLMI-based filter algorithm is implemented recursively for N + 1 steps and, at each step, we need to solve 2M2 LMIs with (6n2 + 1.5m2 + 3mn + pn + n + 0.5m + 2 )M scalar variables. If the parameters in system model (1)-(5) are time-invariant, the infinite-horizon filter can be readily obtained, by removing the initial condition {Pi0 ≤ γ 2 S}1≤i≤M , setting Pik+1 = Pik  Pi (∀k ≥ 0, i ∈ [1, M]) and specifying other variables involved as time-invariant in Theorem 2.

Fig. 1. Switching signals σ and σˆ .





C1k = 0.9

0.5sin(5k ) , Dk1 = −0.3sin(3k ),

C2k = 1.3

0.2sin(k ) , Dk2 = −0.2sin(4k ),





Lk1 = [0.3sin(2k ) 0.7], Lk2 = [0.4sin(2k ) 0.2],

(x ) = gk2 (x ) = [ f 1 (x ) f 2 (x )]T , λ = 0.2, θ = 0.15, ω (k ) = sin(2k )/(k + 1 ), K1 = 0.6, K2 = 0.8,

gk1

with

f 1 (x ) = − 0.05|x1 |sin(x21 + x22 ), f 2 (x ) =0.05|x2 |sin(x21 + x22 ). For this example, part of the system parameters given above is taken from [6]. According to the definitions of the functions gk1 (x ) and gk2 (x ), it can be seen that Gk1 = Gk2 = 0.05I. In addition, the nonlinear function ψ (u) is selected as ψ (u ) = K2 −K1 2 sin (u ).

K1 +K2 2 u

+

Remark 6. Over the past several decades, stability and filtering (or state estimation) problems have also been widely investigated for Markovian jump systems [6,12,20,34]. It is well known Markovian jump system consists of a set of subsystems and a stochastic Markov process that coordinates the switching among subsystems. Noting that the switching signals discussed in this paper are deterministic, it is clear that the switched stochastic system (1) considered in this paper is essentially different from the systems in [6,12,20,34].

It is easy to verify that the function ψ (u) satisfies the sector condition (3). Let α = 0.9, β = 1.1, γ = 1.5, μ = 1.24, S = diag{1.5, 1.5, 1.5, 1.5, 1}, and then choose P10 = P20 = diag{3, 3, 3, 3, 2} such that the initial condition P10 = P20 ≤ γ 2 S is satisfied. Setting N = 60 and solving the RLMIs (9), (12), (28), (29) in Theorem 2 by Matlab toolbox, one can obtain the desired filter parameters, part of which are listed in Table 1. In the simulation, the initial values of the system and filter are chosen as x(0 ) = [0.2 0.1] and xˆ(0 ) = [0 0], and the asynchronous time is set as Tr ≡ TM = 1 (r = 1, 2, . . . , l ). Fig. 1 plots the switching signals of the system and filter, where ADT is chosen as τa = 4 > −[ln(μ ) + TM ln(β /α )]/lnα = 3.9463. Under the switching signals depicted in Fig. 1, Figs. 2 and 3 plot the output z(k) and its estimate zˆ(k ), and the estimation error e(k), respectively. It is clear from Figs. 2 and 3 that the designed filter performs very well over a finite horizon.

4. Numerical example

5. Conclusions

Example 1. Consider the discrete-time switched stochastic system described by (1) and (5), where



Ak1 =

 Ak2 =













0.2 1.1sin(5k )

0.2sin(k ) 0.1sin(3k ) , Bk1 = , 0.5 −0.3

0.3sin(k ) 1.3

0.1 0.1 , Bk2 = , 0.5sin(5k ) 0.4sin(3k )

In this paper, we have investigated the finite-horizon H∞ filtering problem for switched time-varying stochastic systems with randomly occurring sensor nonlinearities and successive packet dropouts. Under ADT switching, and combining with the piecewise function approach and stochastic analysis technique, RLMI-based sufficient conditions have been proposed to ensure the existence of a desired finite-horizon asynchronous H∞ filter, and its effectiveness has been shown by a numerical example. Compared with

144

Y. Chen et al. / Signal Processing 138 (2017) 138–145 Table 1 The filter parameters.

acteristic of switched time-varying systems. Therefore, the filtering scheme in this paper may be less conservative than the existing ones. In this paper, the filtering scheme has based on the uncorrelation of the stochastic variables λk , θ k and v(k). As for the case that λk , θ k and v(k) are correlated, the corresponding analysis becomes more complex, which is our further research. On the other hand, it is worth mentioning that the proposed filtering scheme in this paper can be applied to some practical systems, such as the two-tank system [2] and power converters [46], and the specific applications are our future work. In addition, it should be pointed out that the analysis techniques in this paper remain conservative to some extent, the combined switching signal and state matrix analysis techniques proposed in [47] can be utilized to improve the present filter algorithm. In addition, it is worth mentioning that the considered model in this paper is linear. As for switched nonlinear systems, the corresponding filtering scheme can also be devolved by combining deep neural networks [19] or the T-S fuzzy approach [28]. Fig. 2. The output z(k) and its estimate zˆ(k ).

Fig. 3. The estimation error e(k).

the existing results [7,32,37], this paper has adopted a new measurement model, which can better reflect the actual measurement in a networked environment. In particular, it is worth mentioning that our proposed filtering scheme can be applied to the switched time-varying systems. Moreover, different from the existing filtering schemes [8,14,23,28,30,38,42,45,46], the proposed filter in this paper is time-varying, and can better capture the time-varying char-

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