Variance-constrained resilient H∞ filtering for time-varying nonlinear networked systems subject to quantization effects

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Variance-constrained resilient H∞ filtering for time-varying nonlinear networked systems subject to quantization effects Ming Lyu a,∗, Yuming Bo b a b

Simulation Equipment Business Department, North Information Control Group Co., Ltd., Nanjing 211153, China School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China

a r t i c l e

i n f o

Article history: Received 24 March 2017 Revised 26 May 2017 Accepted 2 June 2017 Available online xxx Communicated by Prof. Wei Guoliang Keywords: Time-varying systems Error variance constraints H∞ filtering Recursive linear matrix inequalities Randomly occurring nonlinearity Quantization effects

a b s t r a c t This paper deals with the resilient variance-constrained H∞ finite-horizon filtering problem for a class of discrete time-varying nonlinear networked system with quantization effects. The nonlinearity enters the system in a probabilistic way that is characterized by a binary sequence with known distribution. The system parameters under investigation are all time-varying and the randomly occurring filter gain variations are modeled by utilizing a random variable obeying prespecified binary distribution which is uncorrelated with other stochastic variables. The nonlinearities and exogenous disturbances we adopt are non-zero mean, which makes the variance analysis become more difficult. Furthermore, the quantization effects are also taken into account to describe the unavoidable constraints imposed on the signal during the transmission in networked systems. Sufficient conditions are established for the finite-horizon filter guaranteeing the constraints imposed on both estimation error variance and H∞ specification. By means of the recursive linear matrix inequality approach, the algorithm for computing the desired filtering gains is provided. Finally, a numerical illustrative example is used to verify the effectiveness of the proposed design method. © 2017 Elsevier B.V. All rights reserved.

1. Introduction During the past few decades, filtering or state estimation problem has been playing a pivotal role in signal processing and other relevant areas which has therefore attracted numerous research interests from a wide variety fields of science and engineering. Up to now, quite a few effective filtering theories have been developed, including but are not limited to Kalman filtering and its variants [10,14,23,28,33], H∞ filtering [9,11,41], variance-constrained filtering [1,21], reduced-order filtering [34] and event-based filtering [40], to name but a few. The celebrated Kalman filtering is a minimum-variance state estimation technique for linear systems subject to Gaussian noises which was firstly proposed around 60s in last century. Since then, the Kalman filtering approach has been extensively applied in many practical engineering. Based on the Kalman filtering theory, several variants have been exploited to deal with the nonlinear dynamics [22,27]. Nevertheless, when the system contains parameter uncertainties and noises with only partially known information, the Kalman filtering and its variants will be no longer applicable. In these cases, the H∞ filtering is developed to grantee that the system’s l2 -norm gain defined from ∗

Corresponding author. E-mail address: [email protected] (M. Lyu).

the noises to the estimation error is not more than a prespecified level. Here, noise signals under consideration are required to be energy bounded. Some representative publications can be found in [1,5,37,42,43] and the references therein. It should be pointed out that the Riccati matrix equation and linear matrix inequality (LMI) approaches have been widely utilized in H∞ filtering/control problems. In various real-world engineering, the filtering requirements are usually characterized by the upper-bounds on the estimation error variances. There have been so far a multitude of research fruits available in the literature concerning the variance-constrained filtering/control for stochastic systems, see, e.g. [1,17–19,30,35]. For instance, in [30], the filtering problem has been solved for the system with norm-bounded parameter uncertainties, ensuring the estimation error variances are not more than the prescribed values. With the variance constraints, the nonlinear phenomenon and exogenous disturbance signals, which are zero mean, have been considered in [1,17,18] and the non-Gaussian noises case has been examined in [35]. However, to the best of the authors’ knowledge, there have been so far no existing results on filtering problem with error variance constraints in the case where the nonlinearities and external signals are with non-zero mean. On another research frontier, it is now well recognized that in comparison to analog systems, digital systems have many

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Please cite this article as: M. Lyu, Y. Bo, Variance-constrained resilient H∞ filtering for time-varying nonlinear networked systems subject to quantization effects, Neurocomputing (2017), http://dx.doi.org/10.1016/j.neucom.2017.06.001

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advantages, such as small size, high precision and strong antiinterference. On the other hand, in today’s networked systems, the extensive adoption of digital components will inevitably give rise to certain unanticipated phenomena, for example, the quantization effects [8,12,13,29,38]. The error between the original value and quantized value cannot be negligible, and accordingly, quite a few techniques have been proposed to analyze and mitigate the quantization effects, among which the most common way is to treat the quantization errors as uncertainties. Generally, there have been mainly two usually adopted quantized communication models adopted in the literature, i.e., the uniform quantizer [7,16] and the logarithmic quantizer. In [25,31], the logarithmic quantizer has been adopted and the quantizing uncertainties are elegantly transformed into the sector-bounded nonlinearity that can be easily dealt by resorting to certain existing techniques. It is worth mentioning that the aforementioned results have not taken into account the impact from, in the implementation of the filter, some possible slight changes of the filter gains that may be brought by the unknown noise, finite word length and so on. The resilient filtering/control (also known as non-fragile filtering/control) whose aim is to design a filter insensitive to certain errors with respect to its gain, has been studied widely for decades, resulting in many results published in the literature, see, e.g., [2– 4,6,36,44,45]. In [36], the interval type of uncertainty is adopted to replace the norm-bounded one to depict the uncertain information caused by the finite word length (FWL) effects. In [2–4], the problems of designing non-fragile filters for fuzzy systems have been addressed. Nevertheless, despite the fact that there are lots of results about non-fragile filtering problem, few has been concerned with the nonlinear stochastic systems, not to mention the situation when the quantization effects are also involved. Summarizing the above discussions, the purpose of this work is to propose the variance-constrained resilient H∞ finite-horizon filtering scheme for a class of nonlinear discrete time-varying stochastic system with quantization effects. The stochastic nonlinearity and exogenous disturbance signals are assumed to be nonzero mean, which is common in practice but rarely discussed on the filtering problems with variance constraints. All parameters of the designed filter have randomly occurring gain variations that are described by mutually uncorrelated Bernoulli sequences with known distribution. We also adopt a logarithmic quantizer to describe quantization effects. The main difficulty of the addressed problem lies in the cross coupling between the time-varying parameters and the uncertain filtering gains. Moreover, the nonlinear dynamics and quantization effects taken into consideration would also result in substantial challenges in the performance analysis and filter design. Sufficient conditions for the existence of the finite-horizon filter, which satisfies the H∞ performance index and error variance constraints simultaneously, are obtained by solving a set of recursive linear matrix inequalities (RLMIs). Finally, a numerical example will be used to demonstrate the applicability of the proposed method. Notation: the notation used here is fairly standard except where otherwise stated. Rn denotes the n-dimensional Euclidean space. l2 [0, N] is the space of square-summable vector functions over [0, N]. E{x} means expectation of the stochastic variable x. ||x|| describes the Euclidean norm of a vector x. AT represents the transpose of A. I and 0 denote the identity matrix and zero matrix of compatible dimensions, respectively. The notation X ≥ Y (respectively, X > Y), where X and Y are symmetric matrices, means that X − Y is positive semi-definite (respectively, positive definite). tr(A) represents the trace of a matrix A. diag{F1 , F2 , . . . , Fn } stands for a block-diagonal matrix whose diagonal blocks are given by F1 , F2 , . . . , Fn . The symbol ∗ in a matrix means that the corresponding term of the matrix can be obtained by symmetric property.

2. Problem formulation and preliminaries In this paper, we consider a discrete time-varying stochastic nonlinear system defined on k ∈ [0, N] as follows:

⎧ x ( k + 1 ) = A ( k )x ( k ) + A1 ( k )x ( k )ω ( k ) ⎪ ⎪ ⎨ + r (k ) f (k, x(k )) + D1 (k )w(k ) y ( k ) = B ( k )x ( k ) + D2 ( k )v ( k ) ⎪ ⎪ ⎩ z ( k ) = C ( k )x ( k )

(1)

where x(k ) ∈ Rn represents the state vector, y(k ) ∈ Rr is the process output, z(k ) ∈ Rm is the signal to be estimated, ω(k) is a onedimensional, zero mean Gaussian white noise sequence on a probability space (, F , P rob) with E{ω2 (k )} = 1, and A(k), A1 (k), B(k), C(k), D1 (k), D2 (k) are known matrices of compatible dimensions. The stochastic variable r(k) satisfies the Bernoulli distribution with

Prob{r (k ) = 1} = E{r (k )} = r¯

(2)

Prob{r (k ) = 0} = 1 − r¯ where r¯ ∈ [0, 1] is a known constant. The nonlinear function f(k, x(k)) is assumed to satisfy

 f (k, x(k ))2 ≤ θ (k )G(k )x(k )2

(3)

for all k ∈ [0, N] with θ (k) > 0 and G(k) being a known constant scalar and a known matrix, respectively. The exogenous disturbance signals w(k ) ∈ R p and v(k ) ∈ Rq belong to l2 [0, N] and satisfy





w (k ) v (k )



vT ( k ) ≤ W

wT ( k )

(4)

where W is a positive definite matrix. In this paper, the quantization effects are taken into account. Denote the quantizer by

h ( · ) = [h1 ( · )

h2 ( · )

···

h r ( · )] T

which is symmetric, i.e., h j (−y ) = −h j (y ) ( j = 1, 2, . . . , r ). The map of the quantization process is described as



h(y(k )) = h1 (y(1) (k ))

h2 (y(2) (k ))

···

T

hr (y(r ) (k ))

where the quantizer h( · ) is assumed to be of the logarithmic type. Specifically, for each hj ( · ) (1 ≤ j ≤ r), the set of quantization levels is described by



( j)

( j)

U j = ±μi , μi



= χ ji μ0( j ) , i = 0, ±1, ±2, · · · ∪ {0}

0 < χ j < 1,

μ0( j ) > 0

where χ j ( j = 1, 2, . . . , r ) is called the quantization density. Each of the quantization level corresponds to a segment such that the quantizer maps the whole segment to this quantization level. In addition, the logarithmic quantizer is defined as

⎧ 1 1 ( j) ( j) ( j) ( j) ⎪ ⎪ ⎨ μi , 1 + δ j μi ≤ y ( k ) ≤ 1 − δ j μi h j (y( j ) (k )) = 0, y ( j ) (k ) = 0 ⎪ ⎪ ⎩ − h j (−y( j ) (k )), y( j ) (k ) < 0

Please cite this article as: M. Lyu, Y. Bo, Variance-constrained resilient H∞ filtering for time-varying nonlinear networked systems subject to quantization effects, Neurocomputing (2017), http://dx.doi.org/10.1016/j.neucom.2017.06.001

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where

δj =

A¯1 (k ) =

1 − χj 1 + χj



It follows readily that hj (y(j) (k)) can be expressed as h j (y( j ) (k )) = (1 + ( j ) (k ))y( j ) (k ) for certain (j) (k) satisfying |(j) (k)| ≤ δ j . In the following, we shall see that the quantization phenomenon discussed above can be characterized by the so-called sector-bounded uncertainty. ¯ = By denoting (k ) = diag{(1 ) (k ), (2 ) (k ), . . . , (r ) (k )},  ¯ −1 , we can easily see that diag{δ1 , δ2 , . . . , δr }, and F (k ) = (k ) F(k)FT (k) ≤ I. Then, the measurements after quantization are expressed by

h(y(k )) =(I + (k ))y(k ) =(I + (k ))(B(k )x(k ) + D2 (k )v(k ))

(5)

In this paper, we adopt the following filter structure for system (1):

⎧ ⎨ xˆ(k + 1 ) = (A f (k ) + α (k )A f (k ))xˆ(k ) + (B f (k ) + β (k )B f (k ))h(y(k )) ⎩ zˆ(k ) = (C f (k ) + γ (k )C f (k ))xˆ(k )

D¯ (k ) =

 C¯1 (k ) =

Prob{α (k ) = 1} = E{α (k )} = α¯

0 B f (k )(I + (k ))B(k ) D1 ( k ) 0



A¯2 (k ) = A¯1 (k ) =

0 0



0



D¯ 22 (k )



−C f (k )

0

h(k, x¯ (k )) = [ f T (k, x(k )) 0 0

0

0]T



A f ( k )   A1 ( k ) 0 0

0

The state covariance matrix of the augmented system (8) can be defined as



X¯ (k ) = E x¯ (k )x¯T (k )



=E

(6)

where xˆ(k ) ∈ Rn and zˆ(k ) ∈ Rm represent the state estimate and the estimated output, respectively, Af (k), Bf (k) and Cf (k) are filter parameters to be designed. Here, Af (k), Bf (k), Cf (k) are uncertainties defined as: A f (k ) = HA (k )A (k )EA (k ), B f (k ) = HB (k )B (k )EB (k ), C f (k ) = HC (k )C (k )EC (k ). In this case, for the subscript o = A, B, C, there are Ho (k), Eo (k) known matrices of compatible dimensions and o (k) uncertain matrices bounded such as: To (k )o (k ) ≤ I. The stochastic variables α (k), β (k) and γ (k) are mutually independent sequences obeying Bernoulli distribution that are introduced to characterize the phenomena of randomly occurring filter gain variations. Assume that

3



x (k ) xˆ(k )



T

(9)

x (k ) xˆ(k )

It is our objective in this paper to seek a time-varying filtering strategy of the form of (6) to achieve the following two requirements at the same time: (Q1) Given a scalar γ > 0, a matrix S > 0 and the initial state x¯ (0 ). The following H∞ performance criterion defined as

J :=E



z¯ (k )2[0,N−1] − γ 2 w¯ (k )2[0,N−1]



(10)

− γ 2 (x(0 ) − xˆ(0 ))T S(x(0 ) − xˆ(0 )) < 0

is satisfied despite the existence of parameter uncertainties and nonlinearities. Here,

z¯ (k )2[0,N−1] =

N−1 

|z¯ (k )|2

k=0

Prob{α (k ) = 0} = 1 − α¯

w¯ (k )2[0,N−1] =

Prob{β (k ) = 1} = E{β (k )} = β¯

|w¯ (k )|2

k=0

(7)

Prob{β (k ) = 0} = 1 − β¯

N−1 

Prob{γ (k ) = 1} = E{γ (k )} = γ¯

(Q2) Given a family of positive definite matrices { (k)}0 < k ≤ N . At each time step k, the estimation error covariance achieves

Prob{γ (k ) = 0} = 1 − γ¯

E (x(k ) − xˆ(k ))(x(k ) − xˆ(k ))T ≤ (k ),



where α¯ ∈ [0, 1], β¯ ∈ [0, 1] and γ¯ ∈ [0, 1] are known constants. Setting x¯ (k ) = [xT (k ) xˆT (k )]T , we acquire

⎧ x¯ (k + 1 ) = (A¯ (k ) + (β (k ) − β¯ )A¯1 (k ) + (α (k ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − α¯ )A¯2 (k ))x¯ (k ) + (D¯ (k ) ⎨ + (β (k ) − β¯ )D¯1 (k ))w¯ (k ) + A¯1 (k ) ⎪ ⎪ ⎪ × x¯ (k )ω (k ) + r (k )h(k, x¯ (k )) ⎪ ⎪ ⎪ ⎩ z¯ (k ) = (C¯ (k ) + (γ (k ) − γ¯ )C¯1 (k ))x¯ (k ) where

w¯ (k ) = [wT (k )

vT (k )]T , z¯ (k ) = z(k ) − zˆ(k ) ¯ D22 (k ) = (B f (k ) + β¯ B f (k ))(I + (k ))D2 (k ) A¯ 21 (k ) = (B f (k ) + β¯ B f (k ))(I + (k ))B(k )





−(C f (k ) + γ¯ C f (k ))

C¯ (k ) = C (k )



D¯1 (k ) =

 A¯ (k ) =

0 0



0 B f (k )(I + (k ))D2 (k )

A (k ) A¯ 21 (k )



0 A f (k ) + α¯ A f (k )

(8)



∀k

(11)

Remark 1. It follows from the given hypothesis that the random variables α (k), β (k) and γ (k) satisfy E{α (k )} = α¯ , E{α (k ) − α} ¯ = 0, ¯ = 0, E{(α (k ) − α¯ )2 } = α¯ (1 − α¯ ), E{β (k )} = β¯ , E{β (k ) − β} E{(β (k ) − β¯ )2 } = β¯ (1 − β¯ ), E{γ (k )} = γ¯ , E{γ (k ) − γ¯ } = 0, E{(γ (k ) − γ¯ )2 } = γ¯ (1 − γ¯ ). These random variables are used to model the probability distribution of randomly occurring filtering gain variations. 3. Analysis of H∞ and covariance performances First of all, we introduce the following lemmas which will be used in this paper. Lemma 1. (Schur Complement Equivalence [1]) For real-valued matrices S1 , S2 and S3 where S1 = S1T and 0 < S2 = S2T , the inequality S1 + S3T S2−1 S3 < 0 holds if and only if



S1 S3



S3T <0 −S2



or

−S2 S3T



S3 <0 S1

(12)

Lemma 2. (S-procedure) Given real-valued matrices N = N T , H and E with appropriate dimensions. Let FT (t)F(t) ≤ I. Then, inequality

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+ 2r¯hT (k, x¯ (k ))Q (k + 1 )A¯ (k )x¯ (k )} ¯ ξ ( k )} = E{ξ T ( k )

N + HF E + (HF E )T < 0 is true if and only if there exists a positive scalar ε such that N + ε H H T + ε −1 E T E < 0 or, equivalently,



N εH T E

εH −ε I

ET 0 −ε I

0



<0

(13)

Lemma 3. For any real vector a ∈ Rn , we have

aaT ≤ tr(aaT )I

(14)

Lemma 4. For any real vectors a, b ∈ Rn , we have

abT + baT ≤ aaT + bbT

(15)

where

¯ 31 = r¯Q (k + 1 )A¯ (k ),  ¯ 21 = D¯ T (k )Q (k + 1 )A¯ (k )  ⎡ ⎤ ¯1  ∗ ∗ ¯ = ⎣ ⎦ ¯ 21  2 ∗ ¯ ¯ 31 r¯Q (k + 1 )D(k ) r¯Q (k + 1 )  T T ξ (k ) = x¯ (k ) w¯ T (k ) hT (k, x¯ (k ))

3.1. H∞ performance

Next, it can be easily acquired that

Theorem 1. Consider system (1). Given the filter parameters Af (k), Bf (k), Cf (k), Af (k), Bf (k) and Cf (k) in (6). Let γ > 0 and S > 0



T 



be given. If, under the initial condition Q (0 )  γ 2 I −I S I −I , there exist a family of matrices {Q(k) > 0}1 ≤ k ≤ N and a set of scalars {τ1 (k ) > 0}0≤k≤N−1 such that the following recursive matrix inequalities:



1  = 21 31

(18)

∗ 2 − γ 2 I

32

∗ ∗

33



<0

(16)

E{J ( k )} = E where

  ˜ ξ (k ) − z¯T (k )z¯ (k ) + γ 2 w¯ T (k )w¯ (k ) ξ T ( k )

⎡

˜ 11  ˜ = ⎣ ˜ 21  ˜ 31 

∗ 2 − γ 2 I ˜ 32 

(19)

⎤

∗ ⎦ ∗ r¯Q (k + 1 )

T ˜ 11 =  ¯ 1 + C¯ T (k )C¯ (k ) + γ¯ (1 − γ¯ )C¯1 (k )C¯1 (k ) 

˜ 31 = r¯Q (k + 1 )A¯ (k ), 

˜ 32 = r¯Q (k + 1 )D¯ (k ) 

˜ 21 = D¯ T (k )Q (k + 1 )A¯ (k ) 

where T

¯ 1 + C¯ T (k )C¯ (k ) + γ¯ (1 − γ¯ )C¯1 (k )C¯1 (k ) 1 = 

From (3), we have

h(k, x¯ (k ))2 ≤ θ (k )G¯ (k )x¯ (k )2 .

+ τ1 (k )θ (k )G (k )G¯ (k ) ¯T

T

Then,

¯ 1 = β¯ (1 − β¯ )A¯1 (k )Q (k + 1 )A¯1 (k ) − Q (k ) 

˜ ξ (k ) − τ1 (k )(hT (k, x¯ (k )) E{J ( k )} ≤ E{ξ T ( k ) × h(k, x¯ (k )) − θ (k )x¯T (k )G¯ T (k )

T

+ α¯ (1 − α¯ )A¯2 (k )Q (k + 1 )A¯2 (k ) + A¯ T1 (k )Q (k + 1 )A¯1 (k )

× G¯ (k )x¯ (k )) − z¯T (k )z¯ (k )

+ A¯ T (k )Q (k + 1 )A¯ (k )

32 = r¯Q (k + 1 )D¯ (k ),

G¯ (k ) = [G(k )

γ 2 w¯ T (k )w¯ (k )} = E{ξ T (k )ξ (k ) − z¯T (k )z¯ (k ) + γ 2 w¯ T (k )w¯ (k )} +

0]

T 2 = β¯ (1 − β¯ )D¯1 (k )Q (k + 1 )D¯1 (k )

+ D¯ T (k )Q (k + 1 )D¯ (k )

N−1 



E{J (k )} = E x¯T (N )Q (N )x¯ (N ) − x¯T (0 )Q (0 )x¯ (0 )

k=0

31 = r¯Q (k + 1 )A¯ (k )

≤E

hold, then the H∞ performance specification defined in (10) is satisfied for all nonzero w¯ (k ).

J (k ) := x¯T (k + 1 )Q (k + 1 )x¯ (k + 1 ) − x¯T (k )Q (k )x¯ (k ), and then substituting (8) into (17), we obtain

E{J (k )} = E{x¯ (k )A (k )Q (k + 1 )A¯ (k )x¯ (k ) + (1 − β¯ ) ¯T

T × β¯ x¯T (k )A¯1 (k )Q (k + 1 )A¯1 (k )x¯ (k )

T

× w¯ (k )D¯1 (k )Q (k + 1 )D¯1 (k )w¯ (k ) T

+ x¯ (k )A¯1 (k )Q (k + 1 )A¯1 (k )x¯ (k ) + w¯ (k ) × D¯ T (k )Q (k + 1 )D¯ (k )w¯ (k ) T

T

+ r¯hT (k, x¯ (k ))Q (k + 1 )h(k, x¯ (k )) + 2w¯ T (k ) × D¯ T (k )Q (k + 1 )A¯ (k )x¯ (k ) + 2r¯hT (k, x¯ (k ))Q (k + 1 )D¯ (k )w¯ (k )

ξ (k )ξ (k ) T



−E

N−1 

(z¯ (k )z¯ (k ) − γ w¯ (k )w¯ (k )) T

2

T

(22)

k=0

(17)

Subsequently, we can find that the H∞ criterion defined in (10) can be expressed by



J≤E

T + α¯ (1 − α¯ )x¯T (k )A¯2 (k )Q (k + 1 )A¯2 (k )x¯ (k ) − x¯T (k )Q (k )x¯ (k ) + β¯ (1 − β¯ )

N−1 



k=0

Proof. Defining

T

(21)

Consequently, one has

21 = D¯ T (k )Q (k + 1 )A¯ (k ) 33 = r¯Q (k + 1 ) − τ1 (k )I

T

(20)

N−1 



 ξ (k )ξ (k ) − E x¯T (N )Q (N )x¯ (N ) T

k=0



+ x¯T (0 )(Q (0 ) − γ 2 I

−I

T



×S I



−I )x¯ (0 )

It follows from  < 0, Q(N) > 0 and Q (0 )  that J < 0, which completes the proof. 

γ 2[

I

(23) −I

]T S[

I

−I]

3.2. Variance analysis Theorem 2. Consider the system (1). Given the filter parameters Af (k), Bf (k), Cf (k), Af (k), Bf (k) and Cf (k). Let X¯ (0 ) be given.

Please cite this article as: M. Lyu, Y. Bo, Variance-constrained resilient H∞ filtering for time-varying nonlinear networked systems subject to quantization effects, Neurocomputing (2017), http://dx.doi.org/10.1016/j.neucom.2017.06.001

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If, under the initial condition P (0 ) = X¯ (0 ), there exists a family of matrices {P (k + 1 ) > 0}0≤k≤N such that the following matrix inequality:

P (k + 1 )  (P (k ) )

(24)

5 T

+A¯1 (k )X¯ (k )A¯1 (k ) = (X¯ (k ))

(28)

The rest of the proof is performed by induction. It is obvious that P (0 )  X¯ (0 ). Suppose that P (k )  X¯ (k ), we now acquire that

where

P (k + 1 )  (P (k ))  (X¯ (k ))  X¯ (k + 1 )

(P (k )) := (2 + r¯ )A¯ (k )P (k )A¯ T (k ) + β¯ (1 − β¯ )A¯1 (k ) T

which indicates that the inequality holds at the step k + 1. The proof is now finished. 

T

× P (k )A¯1 (k ) + A¯1 (k )P (k )A¯1 (k )





+ 3r¯θ (k )tr G¯ (k )P (k )G¯ T (k ) I¯

Remark 2. So far, the variance constraints imposed on the filtering error have been investigated. The sufficient condition has been given in terms of a series of recursive linear matrix inequalities. It should be noted that in the derivation, Lemmas 3 and 4 have played paramount roles, which have converted the constraints on nonlinearity and disturbances to certain matrix inequalities that can be combined in the theoretical framework. Such a manipulation will inevitably induce some conservatism of the obtained condition, which could be reduced by introducing certain weighting coefficients and appropriately selecting their values.

T + α¯ (1 − α¯ )A¯2 (k )P (k )A¯2 (k ) T

+ β¯ (1 − β¯ )D¯1 (k )W D¯1 (k ) + (2 + r¯ )D¯ (k )W D¯ T (k )



I¯ =



I 0

0 0

holds, then P (k )  X¯ (k ) (∀k ∈ {1, 2, . . . , N + 1}). Proof. It can be inferred from (9) that X¯ (k ) satisfied the following matrix equation:

X¯ (k + 1 )



= E x¯ (k + 1 )x¯T (k + 1 )



Based on Theorem 2, we can easily obtain the following corollary: Corollary 1. The inequality holds





E (x(k ) − xˆ(k ))(x(k ) − xˆT (k ))

= E A¯ (k )x¯ (k )x¯T (k )A¯ T (k ) + A¯1 (k )x¯ (k )



T

×x¯T (k )A¯1 (k ) + r¯D¯ (k )w¯ (k )hT (k, x¯ (k )) +r¯h(k, x¯ (k ))w¯ T (k )D¯ T (k ) +r¯h(k, x¯ (k ))h (k, x¯ (k )) + D¯ (k )w¯ (k ) ×w¯ T (k )D¯ T (k ) + r¯A¯ (k )x¯ (k )hT (k, x¯ (k ))

+β¯ (1 − β¯ )A¯1 (k )x¯ (k )x¯ (k )A¯1 (k )



T

T +β¯ (1 − β¯ )D¯1 (k )w¯ (k )w¯ T (k )D¯1 (k )



(25)





θ (k )x¯T (k )G¯ T (k )G¯ (k )x¯ (k )I¯  = θ (k )tr G¯ (k )x¯ (k )x¯T (k )G¯ T (k ) I¯ ≤

(26)

By using lemma 4, we know that

A¯ (k )x¯ (k )w¯ T (k )D¯ T (k ) + D¯ (k )w¯ (k )x¯T (k )A¯ T (k ) ≤ A¯ (k )x¯ (k ) ×x¯T (k )A¯ T (k ) + D¯ (k )w¯ (k )w¯ T (k )D¯ T (k ) r¯A¯ (k )x¯ (k )hT (k, x¯ (k )) + r¯h(k, x¯ (k ))x¯T (k )A¯ T (k ) ≤ r¯A¯ (k )x¯ (k ) ×x¯T (k )A¯ T (k ) + r¯h(k, x¯ (k ))hT (k, x¯ (k )) r¯D¯ (k )w¯ (k )hT (k, x¯ (k )) + r¯h(k, x¯ (k ))w¯ T (k )D¯ T (k ) ≤ r¯D¯ (k )w¯ (k ) ×w¯ T (k )D¯ T (k ) + r¯h(k, x¯ (k ))hT (k, x¯ (k )) (27) Consequently,





X¯ (k + 1 ) ≤ 3r¯θ (k )tr G¯ (k )X¯ (k )G¯ T (k ) I¯ T +β¯ (1 − β¯ )A¯1 (k )X¯ (k )A¯1 (k ) T

+α¯ (1 − α¯ )A¯2 (k )X¯ (k )A¯2 (k ) T +β¯ (1 − β¯ )D¯1 (k )W D¯1 (k ) +(2 + r¯ )A¯ (k )X¯ (k )A¯ T (k )

+(2 + r¯ )D¯ (k )W D¯ T (k )



−I

≤ I

−I P (k ) I

−I ,

T

∀k



Q (0 ) ≤ γ 2 I P (0 ) = X¯ (0 )

T 

−I S I

−I



(30)

there exist sequences of matrices {Q (k ) > 0}1kN+1 , {P (k ) > 0}1kN+1 , and constants {τ 1 (k) > 0}0 ≤ k ≤ N satisfying the following matrix inequalities:

From lemma 3, we obtain that

E h(k, x(k ))hT (k, x(k ))



T

−I X¯ (k ) I

Theorem 3. Consider system (1). Given the filter parameters Af (k), Bf (k), Cf (k), Af (k), Bf (k) and Cf (k). Given γ > 0 and S > 0, if, under the initial condition

T

+α¯ (1 − α¯ )A¯2 (k )x¯ (k )x¯T (k )A¯2 (k )





In the following, we proceed to give a theorem to take into account both the H∞ performance criterion and the covariance constraint by means of the recursive matrix inequality approach.

+r¯h(k, x¯ (k ))x¯T (k )A¯ T (k ) + A¯ (k )x¯ (k ) ×w¯ T (k )D¯ T (k ) + D¯ (k )w¯ (k )x¯T (k )A¯ T (k ) T



= I



T



(29)



11 21

−22

¯ 11  ¯ 21 

∗ ¯ 22 −

where

11 =

∗

 111 0 0



<0

(31)

≤0

(32)



∗

−γ I 0

∗ ∗

2



−τ1 (k )I

111 = τ1 (k )θ (k )G (k )G¯ (k ) − Q (k ) ⎧ ⎫ ⎨ ⎬ 22 = diag Q −1 (k + 1 ), . . . , Q −1 (k + 1 ), I, I   ⎭ ⎩ ¯T



6

21 = 211 212 213  T 211 = 1211 (k ) 2211 (k ) 3211 (k )  ! T 1 ¯ ¯ ¯ 1211 = A¯ T (k ) 0 β (1 − β )A (k )

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2211 =



T

T A¯1 (k )

α¯ (1 − α¯ )A¯2 (k )



! T 1 ¯ γ¯ (1 − γ¯ )C (k )

3211 = C¯T (k )  212 = D¯ T (k )

0

0

0

0

T

 213 = r¯I

0

0

r¯ (1 − r¯ )I

0

0





0

4



¯ 21 =  ¯1  21

¯2 

21

2 + r¯A¯ (k )

0



0

T



−I]T ,

0

know −I]

1111

that

J<0 P ( k )[ I

and −I]T ,





∗ Pˆ2 (k + 1 )

=

∀k ∈ {0, 1, . . . , N + 1}, which completes the

 ⎧ Q (0 ) − γ 2 S ∗ ⎪ ⎪ 1 ≤0 ⎪ ⎪ ⎨ Q3 ( 0 ) + γ 2 S Q2 ( 0 ) − γ 2 S E{(x(0 ) − xˆ(0 ))(x(0 ) − xˆ(0 ))T } ⎪ ⎪ ⎪ = P (0 ) + P2 (0 ) − P3 (0 ) − P3T (0 ) ⎪ ⎩ 1 ≤ (0 )

(33)

∗ Qˆ2 (k + 1 )

P1 (k + 1 ) P3 (k + 1 )

∗ P2 (k + 1 )

−1 (37)



   111 ∗ 1111 ∗ 1 ,  = 11 0 311 211 1222 ⎡ ⎤ 1 ∗ ∗ ∗ ∗ ∗ ⎢−Q (k ) 2 ⎥ =⎣ 3 ⎦ 0 0 −γ 2 I ∗ 3 0 0 4 − γ 2 I

¯ T ¯ B (k ) 1 = τ1 (k )θ (k )GT (k )G(k ) + 1 (k )BT (k ) T T + εB ( k )B ( k )EB ( k )EB ( k )B ( k ) − Q1 ( k ) 2 = εA EAT (k )EA (k ) + εC (k )ECT (k )EC (k ) − Q2 (k ) ¯ T ¯ D2 ( k ) + εB ( k ) DT ( k ) × E T ( k ) EB ( k ) D2 ( k ) 4 = 1 (k )DT2 (k ) B ⎧ ⎫ 2 ⎨ ⎬ 211 = diag 111 , . . . , 111 , −I, −I   ⎩ ⎭ 6   −Qˆ1 (k + 1 ) ∗ 111 = −Qˆ3 (k + 1 ) −Qˆ2 (k + 1 )  311 = T1 T2 T3 T4 0 0 0 T 0 T5 0 0 0 T6 0

1222 = diag{−τ1 (k )I, −τ1 (k )I}  1 = A(k ) 0 D1 (k ) 0 r¯I  2 = B f ( k ) B ( k ) A f ( k ) 0 B f ( k )D2 ( k )

α

¯ HAT

(k )



0 ··0  · 

7

(34)



12

··0 ¯2 = 0  ·  



r¯I

0

0

 3 = 0 0 0 0 r¯ (1 − r¯ )I 0  4 = 0 0 0 0 0 r¯ (1 − r¯ )I  5 = A 1 ( k ) 0 0 0 0 0  6 = C (k ) −C f (k ) 0 0 0 0  T T ¯ ¯T ¯T ¯T ϒ21 =     1 2 3 4  ¯1 =  ¯ 11  ¯ 12  ¯ 13   ¯ 11 = εB (k )E T (k )EB (k )B(k ) 0 0  B  ¯ 12 = εB (k )E T (k )EB (k )D2 (k ) 0 0  B   0 BTf (k ) 0 · · · 0 ¯    13 = 

such that the following recursive LMIs:

<0

−1

Qˆ1 (k + 1 ) Qˆ3 (k + 1 )

¯ T ¯ B ( k ) + εB ( k )DT ( k )E T ( k ) × EB ( k )B ( k ) 3 = 1 (k )DT2 (k ) 2 B

Theorem 4. Given γ > 0, S > 0 and a family of prespecified variance upper bounds {(k ) > 0}0≤k≤N+1 . The design objectives (Q1) and (Q2) are simultaneously satisfied if there exist sequences of positive definite matrices {Qˆ1 (k )}1≤k≤N+1 , {Qˆ2 (k )}1≤k≤N+1 , {P1 (k )}1≤k≤N+1 , {P2 (k )}1≤k≤N+1 , positive scalars {τ 1 (k)}0 ≤ k ≤ N , { 1 (k)}0 ≤ k ≤ N , { 2 (k)}0 ≤ k ≤ N , { 3 (k)}0 ≤ k ≤ N , {ε A (k)}0 ≤ k ≤ N , {ε B (k)}0 ≤ k ≤ N , {ε C (k)}0 ≤ k ≤ N , {ε¯A (k )}0≤k≤N , {ε¯B (k )}0≤k≤N , and sequences of realvalued matrices {Qˆ3 (k )}1≤k≤N+1 , {P3 (k )}1≤k≤N+1 , {Af (k)}0 ≤ k ≤ N , {Bf (k)}0 ≤ k ≤ N , {Cf (k)}0 ≤ k ≤ N , under initial conditions

ϒ22



ϒ11 =

21

In this section, we shall propose an algorithm to handle the filter design problem considered for system (1). It will be shown that the filtering gains can be computed by solving a series of recursive linear matrix inequalities.

∗



∗ = Q2 ( k + 1 )

Pˆ1 (k + 1 ) Pˆ3 (k + 1 )

where

4. Robust finite horizon filter design

ϒ11 ϒ21

Q1 ( k + 1 ) Q3 ( k + 1 )



Proof. By Schur complement lemma, it is easy to find out that the inequality (31) is equivalent to (16), and the inequality (32) holds if and only if (24) is true. With the initial conditions (30), and noting Theorems 1, 2 and Corollary 1, we can easily acquire that the H∞ specification defined in (10) achieves J < 0 and, meanwhile,  the error covariance satisfies E (x(k ) − xˆ(k ))(x(k ) − xˆ(k ))T  [I −I]P (k )[I proof. 

(35)

(36)

T

then, for system (8),  we

E (x(k ) − xˆ(k ))(x(k ) − xˆ(k ))T [I ∀k ∈ {0, 1, . . . , N + 1}.

∗ <0 ϒˆ 22

are satisfied with parameters updated by

β¯ (1 − β¯ )A¯1 (k )  ¯2 =  α¯ (1 − α¯ )A¯2 (k ) A¯1 (k ) 21  ¯ 3 = √2 + r¯D¯ (k )  β¯ (1 − β¯ )D¯1 (k ) 21 ¯1 =  21

√

¯3 



ϒˆ 11 ϒˆ 21

P1 (k + 1 ) + P2 (k + 1 ) − P3 (k + 1 ) − P3T (k + 1 ) − (k + 1 ) ≤ 0

!T

β¯ (1 − β¯ )D¯1 (k ) 0

! 0

¯ 11 = −P (k + 1 ) + 3r¯θ (k )tr G¯ (k )P (k )G¯ T (k ) I¯  ⎧ ⎫ ⎨ ⎬ ¯ 22 = diag P −1 (k ), . . . , P −1 (k ), W −1 , W −1    ⎩ ⎭



[m5G;July 20, 2017;23:6]

5

α¯ (1 − α¯ )

HAT

(k )

0 ··0  · 



6

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  ¯ ¯ ¯ ¯ 3 = 31 32 33   ¯ T ··0 ¯ 31 = 0  ·  β HB (k ) 0  β¯ (1 − β¯ )H T (k )

¯ 32 = 

5



¯4 = 

 ϒˆ 11 =

β¯ (1 − β¯ )HBT (k ) 18

¯1  11 ¯3 

¯1 

111

¯1 

221

∗ ¯2 

0



 ¯1 =  11

,

11

11

¯1  ⎣ = √−P3 (k +T 1 ) 2 + r¯A (k ) ⎡ √

∗

0

⎡

−Pˆ2 (k ) 0 0

ˆ2 = 0 

∗ ∗ −Pˆ2 (k )

¯4  −Pˆ3 (k )

∗



 ˆ 32 = 0  ⎤

ˆ 33 = 

⎦

ˆ 34 = 

⎤ ϒˆ 22

⎦

0

0

⎡ ¯2  111 2 ¯ ⎣ 11 = 0  ¯2 =  111

 ¯2 =  222

∗

¯2 

222

0

0 −Pˆ1 (k ) −Pˆ3 (k ) −Pˆ1 (k ) −Pˆ3 (k )

⎡ ¯2 =⎢  ⎣ 333

ε3 ( k )

0

ˆ1 −W ˆ3 −W 0 0

∗ ∗

DT2

⎤

−2 (k )I ⎢ 0 =⎣ 0 β¯ EB (k )

∗ −Pˆ2 (k )

 Q (k ) =



¯ T ¯ B (k ) ( k )

Q −1 (k ) =



∗

−3 (k )I 0 ϒˆ 442

6

0

0 ··0  · 



6



∗ ∗

−ε¯A (k )I 0

∗ ∗ ∗

⎤ ⎥ ⎦

−ε¯B (k )I

∗ P2 (k )

Pˆ1 (k ) Pˆ3 (k )

∗ Pˆ2 (k )



Q1 ( k ) Q3 ( k )

∗



Q2 ( k )

Qˆ1 (k ) Qˆ3 (k )



∗

Qˆ2 (k ) ∗ ˆ2 W



Thus, it is obvious that (33) is equivalent (30). To deal with (k) in (31), we rewrite it in the following form:

0



P1 (k ) P3 (k )

ˆ1 W ˆ3 W

W −1 =

N (k ) + H (k )F (k )E (k ) + (H (k )F (k )E (k ))T < 0



where



N (k ) =

⎦

N11 (k ) N21 (k )



−22 ∗

(38)



∗

N111 (k ) 0 0

N11 (k ) =



∗ ˜1 ˆ2 +  −W 0 0

0 ··0  · 

β¯ (1 − β¯ EB (k )



¯2  333 

∗ −Pˆ2 (k )

0



β¯ (1 − β¯ )EB (k )D2 (k )



(k )

¯ T ¯ B (k ) 0 0 0 ε2 (k )(2 + r¯ )DT2 (k )



⎡

P −1 (k ) =

 T ¯3 = 0 0  ˜T ˜T ˜T ˜T  0   0  11 1 2 3 4  ˜ 1 = AT ( k ) 0 0 0 0 0  1 √ ˜2 =  2 + r¯DT1 (k ) 0 0 0 0 0  √ ˜3 = 0 2 + r¯DT2 (k )BTf (k )  ˜4 = 0 

$

2 + r¯β¯ EB (k )D2 (k )

P (k ) =

¯ 3 = −Pˆ1 (k ) + 2 (k )(2 + r¯ )BT (k ) ¯ T ¯ B (k )  T T ¯ ¯ ¯ ˆ 4 = −P1 (k ) + 3 (k )B (k ) B(k ) 2 + r¯B (k )

0

Proof. The proof is based on Theorem 3. We suppose that the variables P(k), Q(k) and W can be decomposed as follows

¯ 2 = −P2 (k + 1 ) + ε¯A HA (k )H T (k )  A + ε¯B (k )HB (k )HBT (k )

BTf

β¯ (1 − β¯ )EB (k )B(k )

√

$

EBT

T

0

¯1  222

ϒ22 = diag − 1 (k )I + εB (k ) (k )EB (k ),  − εA (k )I, −εB (k )I, −εC (k )I  ¯ 1 = −P1 (k + 1 ) + 3r¯θ (k )tr G(k )P1 (k )GT (k ) I 

¯ 32 = 

0

 ˆ3 =  ˆ 31  ˆ 32  ˆ 33  ˆ 34   √ ˆ 31 = 0 0  2 + r¯β¯ EB (k )B(k )

ϒˆ 442 =



0

√ 2 + r¯α¯ EA (k )

α¯ (1 − α¯ )EA (k )

¯1  111 ¯ 1221 ⎤

−Pˆ3 (k ) 0 0

∗

¯1 =⎣  222

12





∗ ∗ ⎦ ¯3 

¯2  ¯ 32

2 + r¯ATf (k ) 0 0

0

= ⎣0



 − γ¯ (1 − γ¯ )HCT (k )

−γ¯ HCT (k )

0 ··0  · 

⎡

0

˜ 2 = 3 (k )DT (k ) ¯ T ¯ D2 ( k )  2  T ˆT ˆT ˆT ϒˆ 21 =  0   1 2 3   0 BTf (k ) 0 · ··0 ˆ   1 =



0 ··0  · 

B



0

7

 ¯ 33 = 

0

7

−γ I 0 2



∗ ∗

−τ1 (k )I

N111 (k ) = τ1 (k )θ (k )G¯ T (k )G¯ (k ) − Q (k )



∗ ∗ ˆ1 −W ˆ3 −W

˜ 1 = 2 (k )(2 + r¯ )DT (k ) ¯ T ¯ D2 ( k )  2

∗ ∗ ∗ ˜2 ˆ2 +  −W

⎤ ⎥ ⎦



N21 (k ) = N211 (k )

N212 (k )

213

1 N211 (k ) = N211 (k )

2 N211 (k )

3 N211 (k )





T

! T β¯ (1 − β¯ )A˜1 (k )  ! T T 2 2 ¯ ¯ N211 (k ) = α¯ (1 − α¯ )A (k ) A1 (k )

1 N211 (k ) = A˜ T (k )

0

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3 N211 (k ) = 0



N212 (k ) = D˜ T (k )

!

T

γ¯ (1 − γ¯ )C¯1 (k )

C¯ T (k ) 0

0

0

0

T

β¯ (1 − β¯ )D˜1 (k )   A (k ) 0 A˜ (k ) = ˜ A21 (k ) A˜ 22 (k )



!T

0



D˜1 (k ) =

 A˜1 (k ) =

 D˜ (k ) =

0 0

0

0 0

D1 ( k ) 0

H (k ) = 0

 E (k ) =



H2T

0

(k )

E2 ( k )

0

0

∗

−1 (k )I 0

<0

−1 (k )I



∗

(39)



∗ ∗

−1 (k )I 0

+ H¯ A (k ) × A (k )E¯A (k )

−1 (k )I

+ (H¯ A (k )A (k )E¯A (k ))T + H¯ B (k )B (k )E¯B (k ) + (H¯ B (k ) × B (k )E¯B (k ))T + H¯ C (k )C (k )E¯C (k ) + (H¯ C (k )C (k )E¯C (k ))T < 0



N11 (k ) N¯ 21 (k )

−22

N¯ 21 (k ) = N¯ 211 (k )

N¯ 212 (k )



213

N¯ 211 (k ) = Aˆ T (k )

0

0

0

N¯ 212 (k ) = Dˆ T (k )

0

0

0





Aˆ (k ) =

 Dˆ (k ) =



A (k ) B f ( k )B ( k ) D1 ( k ) 0

C˜(k ) = C (k )

(40)



∗

A¯1 (k )

0 A f (k )

0



0 B f ( k )D2 ( k )

−C f (k )



T



E¯B1 (k )

E¯B2 (k )

0 0

0 ··0  · 

H¯ BT2 (k ) 0

0 EB ( k )

0

0

T 0



9

T β¯ HBT (k )

T β¯ (1 − β¯ )HBT (k )  E¯B1 (k ) = EB (k )B(k ) 0  E¯B2 (k ) = 0 EB (k )D2 (k )  T 0 · · · 0 −γ¯ HCT (k ) H¯ C1 (k ) H¯ C (k ) =    9  ¯ HC1 (k ) = − γ¯ (1 − γ¯ )HCT (k ) 0 0   ··0 E¯C1 (k ) 0  ·  E¯C (k ) =

It is obviously that there only exist the parameter uncertainties rewrite it as

H¯ BT2 (k )

0





∗ ∗

H¯ BT1 (k )

H¯ B2 (k ) = 0

A (k), B (k) and C (k) in (35). In order to eliminate them, we



0

H¯ B1 (k ) = 0

Then, by means of S-procedure, we can obtain

N1 (k ) =

0



T (B f (k ) + β¯ B f (k ))T $  T H2 (k ) = β¯ (1 − β¯ ) 0 BTf (k )  ¯ B (k ) 0 E1 ( k ) =   ¯ D2 ( k ) E2 ( k ) = 0 

where

H¯ B (k ) = 0

T

9

N1 (k ) H¯ T (k ) 1 (k )E (k )

E¯A1 (k ) = 0

E¯B (k ) =

0

6

T α¯ (1 − α¯ )HAT (k ) EA ( k )



 0 ··0  · 

0 ··0  · 

T (k )



H2T (k )

T

α



0

0

12

H¯ A2 (k ) = 0



H¯ AT1 (k )

¯ HAT





0 ··0  ·  7

 0 ··0  · 

E¯A1 (k )

H¯ A1 (k ) = 0

H1T (k )

0

E1 ( k )

N (k ) H T (k ) 1 (k )E (k )

0



H1 (k ) = 0



0

D˜ 22 (k )

0 0

H¯ A (k ) = 0

E¯A (k ) =

D˜ 22 (k ) = (B f (k ) + β¯ B f (k ))D2 (k )



T (k )





0

H¯ 1 (k ) = 0

BTf

H¯ AT2 (k )

0 B f ( k ) D 2 ( k )

B f ( k ) B ( k )

0



0

A˜ 21 (k ) = (B f (k ) + β¯ B f (k ))B(k ) A˜ 22 (k ) = A f (k ) + α¯ A f (k )

0



T

H¯ 1T (k )

0

H¯ (k ) =

0

0 0

!T C¯ T (k ) T

0



E¯C1 (k ) = 0

EC (k )

12

Then, by virtue of Schur Complement Lemma and Sprocedure, it is seen that (34) is true if and only if (31) is true. Likewise, we can also know that (35) holds if and only if (32) holds. As such, according to Theorem 3, we arrive at J < 0 and E{(x(k ) − xˆ(k ))(x(k ) − xˆ(k ))T }  [I − I]P (k )[I − I]T , ∀k ∈ {0, 1, . . . , N + 1}. Moreover, from (36), it is easily seen that E{(x(k ) − xˆ(k ))(x(k ) − xˆ(k ))T }  [I − I]P (k )[I − I]T ≤ (k ), ∀k ∈ {0, 1, . . . , N}. Thus, the requirements (Q1) and (Q2) are simultaneously satisfied. The proof is now complete.  According to Theorem 4, we can summarize the nonfragile variance-constrained H∞ finite-horizon filter design algorithm (NFD) as follows. Algorithm

NFD

Step 1.

Let the H∞ performance index γ > 0, the positive definite matrix S > 0, the error initial condition x(0 ) − xˆ(0 ) and positive definite matrix (0 ) be given. Choose the initial values for matrices {Q1 (0 ), Q2 (0 ), Q3 (0 ), P1 (0 ), P2 (0 ), P3 (0 )} to satisfy the condition (33) and set k = 0. Calculate the values of matrices {Qˆ1 (k + 1 ), Qˆ2 (k + 1 ), Qˆ3 (k + 1 ), P1 (k + 1 ), P2 (k + 1 ), P3 (k + 1 )} and the required filter parameters A f (k ), B f (k ), C f (k ) for the time instant k by solving the LMIs (34)–(36). Set k = k + 1 and obtain {Q1 (k ), Q2 (k ), Q3 (k ), Pˆ1 (k ), Pˆ2 (k ), Pˆ3 (k )} by (37). If k < N, then go to Step 2, else go to Step 5. Stop.

Step 2.

Step 3. Step 4. Step 5.

Please cite this article as: M. Lyu, Y. Bo, Variance-constrained resilient H∞ filtering for time-varying nonlinear networked systems subject to quantization effects, Neurocomputing (2017), http://dx.doi.org/10.1016/j.neucom.2017.06.001

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9

Remark 3. So far, we have investigated the resilient filtering problem for a class of time-varying quantized nonlinear networked system to satisfy both variance and H∞ performance constraints. The existence of the desired filter has been cast into the solvability of a series of recursive linear matrix inequalities. It should be mentioned that, within our proposed framework, multiple other performance specifications could be considered systematically, such as the H∞ and l2 − l∞ criterion [39], probability constraints [32] and so on.

are given as follows:

Remark 4. Note that the existence of the desired filter has been characterized by the feasibility of a set of recursive linear matrix inequalities. It is well known that one of the major disadvantages of the LMI-based algorithm is the conservatism. As this paper is concerned, the conservatism mainly result from the handling of the parameter uncertainties and the nonlinearities. It is should be pointed out that, unfortunately, the conservatism cannot be eliminated thoroughly in the proposed framework but could be largely reduced by utilizing certain techniques. For instance, when dealing with the nonlinearity, we can introduce some coefficients and choose their values appropriately. On the other hand, notice that the RLMI algorithm proposed in this paper is based on LMI approach. As discussed in [20], the computational complexity of an LMI system is bounded by O(PQ3 log(U/ε )) where P represents the row size, Q stands for the number of scalar decision variables, Q is a data-dependent scaling factor and ε is relative accuracy set for algorithm. It can be seen that the computational complexity of our proposed RLMI algorithm depends linearly on the length of time interval k. It is worth mentioning that the study on LMI optimization is very active in recent years within the communities of applied mathematics, control science and signal processing. We can expect substantial speedups in the near future.

 0.1 θ ( k ) = 1, G ( k ) = 0   0.2 0 W (k ) =



f (k, x(k )) =

In this section, we are going to verify the effectiveness of the proposed filtering scheme by presenting a numerical illustrative example. The parameters of the addressed system (1) is given as follows:

 ⎧ 0.2 + 0.1 sin (3k ) ⎪ ⎪ x (k + 1 ) = ⎪ ⎪ −0.1 ⎪ ⎪ ⎪  ⎪ ⎪ 0.05 ⎪ ⎪ x ( k )ω ( k ) ⎪ ⎪ 0.1 + 0.01 sin(0.2k ) ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ 0.3 + 0.1 sin (3k ) 0.2 ⎪ ⎪ + x (k ) ⎪ 0 0.4 ⎪ ⎪ ⎪   ⎪ ⎨ 0.2 − 0.1 sin (0.3k ) + w (k ) −0.15 ⎪ ⎪ ⎪ ⎪ ⎪ + r (k ) f (k, x(k )) ⎪ ⎪   ⎪ ⎪ ⎪ 0 . 5 0 . 3 ⎪ ⎪ y (k ) = x (k ) ⎪ ⎪ 0 0.2 − 0.05 sin (k ) ⎪ ⎪ ⎪   ⎪ ⎪ 0.1 ⎪ ⎪ + v (k ) ⎪ ⎪ 0.2 ⎪ ⎪ ⎪  ⎩ z(k ) = 0.1 0.25 + 0.1 sin(0.2k ) x(k ) with the state initial value x(0 ) = [0.26 − 0.4]T and xˆ(0 ) = [0.2 − 0.16]. The zero mean Gaussian white noise sequence ω(k) are generated that obey the standard normal distribution. The nonlinear functions f(k, x(k)) and the exogenous disturbance signals w(k), v(k)

0.1x2 (k ) sin(x1 (k ))

w(k ) = 0.2 exp(−0.1k ) sin(0.1k )

v(k ) = −0.1 exp(−0.2k ) Then, it is easy to check that the constraints (3) and (4) can be satisfied with



0 0.1

0.2

0

The probability of stochastic variable r(k), α (k), β (k), γ (k) are taken as r¯ = 0.6, α¯ = 0.4, β¯ = 0.5, γ¯ = 0.6. The parameters of logarithmic quantizer h( · ) are taken as χ1 = 0.3, χ2 = 0.6 and uncertain parameter F(k) satisfies FT (k)F(k) ≤ I. The known matrices of the filter gain variations Ho (k), Eo (k) (o = A, B, C ) are given as follows:



HA (k ) =





0.2 , 0.15



HB (k ) =



0.2 0



EA ( k ) = 0.1

0.2 + 0.3 sin(k )

EB ( k ) = 0.1

0.2 + 0.1 sin(k )

 



EC (k ) = 0.1 + 0.2exp(−k )

0



HC (k ) = 0.2 and

5. Numerical simulations



0. 1x 1 ( k ) (x22 (k )+1 )

the

uncertain

parameter

To (k )o (k ) ≤ I.

o (k ) (o = A, B, C )

satisfies

For given positive scalar γ = 0.5 and positive definite matrices S = diag{8, 8}, (0 ) = [1.4036, −0.0144; −0.0144, 1.4036], (k ) = diag{0.2, 0.2} (k = 1, . . . , N ), initial values Q1 (0 ) = diag{1, 1}, Q2 (0 ) = diag{1, 1}, Q3 (0 ) = 0 and



P1 (0 ) =

 P2 (0 ) =

 P3 (0 ) =



0.1676 −0.1040

−0.1040 0.1676

0.1400 −0.0320

−0.0320 0.1400

0.0520 −0.0800

−0.0800 0.0520

 

are chosen to satisfy the initial condition (33). Based on the proposed NFD algorithm, the set of time-varying LMIs in Theorem (4) can be solved recursively under the given initial conditions. Table 1 lists the positive scalar variables τ 1 (k),  1 (k),  2 (k),  2 (k), εA (k), εB (k), εC (k), ε¯A (k ), ε¯B (k ), positive matrix variables Q1 (k), Q2 (k), Q1 (k), P1 (k), P2 (k), P3 (k) and the desired parameters of filter Af (k), Bf (k), Cf (k) from the time k = 0 to k = 2. The simulation figures are shown in Figs. 1–6. Figs. 1–4 give the state variables x1 (k), x2 (k) and their estimation xˆ1 (k ), xˆ2 (k ), their estimation errors x1 (k ) − xˆ1 (k ), x2 (k ) − xˆ2 (k ), respectively, and Fig. 5 plots the output z(k) and its estimation zˆ(k ), while the estimation error z(k ) − zˆ(k ) is shown in Fig. 6. The H∞ performance index is J = −0.1236 and (x(k ) − xˆ(k ))(x(k ) − xˆ(k ))T < (k ) are satisfied for all k = 0, . . . , N. From the simulation results, we can see that the desired design objects are both achieved, which indicates that the proposed filtering algorithm is effective.

Please cite this article as: M. Lyu, Y. Bo, Variance-constrained resilient H∞ filtering for time-varying nonlinear networked systems subject to quantization effects, Neurocomputing (2017), http://dx.doi.org/10.1016/j.neucom.2017.06.001

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M. Lyu, Y. Bo / Neurocomputing 000 (2017) 1–12 Table 1 Recursive PROCESS. k

0

1

2



τ 1 (k)  1 (k)  2 (k)  3 (k) ε A (k) ε B (k) ε C (k) ε¯A (k ) ε¯B (k )

4 2.2188 12 13 0.3454 0.6441 0.9878 14 15   1 0 0 1

4 5.3094 12 13 0.4253 0.8340 2.0794 14 15  2.3354 −0.5671

4 6.8245 12 13 0.6899 1.1061 2.5268 14 15  5.6878 0.7788

        

0.7788 5.6878

5.1201 1.2053

1.2053 5.1201

Q1 (k)



Q2 (k)

 Q3 (k)

 P1 (k)

 P2 (k)

 P3 (k)

 Af (k)

 Bf (k) Cf (k)





1 0

0 1

0 0

0 0



3.1051 0.1647



 

















0.1676 −0.1040

−0.1040 0.2600

0.0800 −0.0300

−0.0300 0.0917

0.0520 −0.0416

−0.0800 0.0640

0.2878 −0.0739

−0.0524 0.2354

0.4019 −0.0764

−0.0982 0.6548

0.5627



0.5190

−0.7442 0.0554



0.0836 0.0125

0.0800 −0.0300 0.0342 0.0036





−0.5671 2.3354



0.1647 3.1051





−0.0863 −0.1067





−0.0300 0.0917



−0.2364 1.0149

0.1095 −0.0642 0.0423 0.0125







−1.4117 −0.3491





−0.0642 0.1497





0.0184 0.0383





0.1117 −0.0481

−0.2211 0.1698

0.6519 −0.2955

−0.2561 0.9816

1.3609





0.0492 0.1029





0.4959 −0.2506

1.0581

0.0832 0.0492





−0.2611 0.2797

1.2473

−2.8904 −0.6491



0.2844 −0.2072













0.0125 0.1420

−0.0153 0.0528





1.2151



  

Fig. 1. State x1 (k) and its estimate.

Fig. 2. Estimation error x1 (k ) − xˆ1 (k ).

In this paper, we have investigated the variance-constrained H∞ finite-horizon filtering problem for a class of discrete timevarying stochastic nonlinear system with quantization effects and randomly occurring nonlinearity. The randomly occurring nonlinearity is modeled by a binary sequence that obeys the Bernoulli distribution with known statistical information. The nonlinearity and the exogenous disturbance signals are non zero mean but covariance bounded, which have been rarely discussed on the filtering problems with variance constraint. By introducing a lemma, the product terms among them and state variables are decoupled. Logarithmic quantizer has been taken to transform the quantization

effects of measurements into the sector-bounded uncertainties. The filter gain variations are governed by the mutually uncorrelated Bernoulli distributed white sequences taking values on 0 or 1. Theoretical framework for designing the desired finite-horizon filter to satisfy both the estimation error variance constraints and the prescribed H∞ performance requirement has been established by resorting to the recursive linear matrix inequality method. Simulation results have shown the effectiveness of the proposed design method. Based on the proposed framework in our paper, we could extend our work to some potential areas such as neural networks [15], systems with missing measurements and correlated noises [26], large-scale interconnected systems with topologies [20], systems subject to Markovian jump parameters [24] and so on.

6. Conclusions

Please cite this article as: M. Lyu, Y. Bo, Variance-constrained resilient H∞ filtering for time-varying nonlinear networked systems subject to quantization effects, Neurocomputing (2017), http://dx.doi.org/10.1016/j.neucom.2017.06.001

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11

Fig. 6. Estimation error z(k ) − zˆ(k ).

Fig. 3. State x2 (k) and its estimate.

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Fig. 4. Estimation error x2 (k ) − xˆ2 (k ).

Fig. 5. Output z(k) and its estimate.

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[39] J. Zhang, L. Ma, Y. Liu, Y. Bo, H∞ and l2 − l∞ finite-horizon filtering with randomly occurring gain variations and quantization effects, Appl. Math. Comput. 298 (2017) 171–187. [40] W. Zhang, Z. Wang, Y. Liu, D. Ding, F.E. Alsaadi, Event-based state estimation for a class of complex networks with time-varying delays: a comparison principle approach, Phys. Lett. A 381 (1) (2017) 10–18. [41] Y. Zhang, G. Cheng, C. Liu, Finite-time unbiased H∞ filtering for discrete jump time-delay systems, Appl. Math. Model. 38 (13) (2014) 3339–3349. [42] D. Zhang, Q.-G. Wang, L. Yu, Q.K. Shao, H∞ filtering for networked systems with multiple time-varying transmissions and random packet dropouts, IEEE Trans. Ind. Inform. 9 (3) (2013) 1705–1716. [43] M. Zhong, D. Zhou, S.X. Ding, On designing H∞ fault detection filter for linear discrete time-varying systems, IEEE Trans. Autom. Control 55 (7) (2010) 1689–1695. [44] Y. Yuan, H. Yuan, L. Guo, H. Yang, S. Sun, Resilient control of networked control system under DoS attacks: a unified game approach, IEEE Trans. Ind. Inform. 12 (5) (2016) 1786–1794. [45] Y. Yuan, F. Sun, H. Liu, Resilient control of cyber-physical systems against intelligent attacker: a hierarchal Stackelberg game approach, Int. J. Syst. Sci. 47 (9) (2016) 2067–2077. Ming Lyu was born in Taizhou, China, in June 1980. She received her B.Sc. degree in Automatic Control in 2002, her M.Sc. degree in Automatic Control in 2004 and her Ph.D. degree in Control Theory and Control Engineering in 2007, all from Nanjing University of Science and Technology, Nanjing, China. She is currently a senior engineer in the Department of Simulation Equipment Business, North Information Control Institute Group Co., Ltd. She current research interests include stochastic filtering, networked control systems and stochastic control systems.

Yuming Bo received his Ph.D. degree from the Nanjing University of Science and Technology, China. He worked as an assistant professor, associate professor and professor, respectively, in School of Automation in Nanjing University of Science and Technology, China. Professor Bo is a member of the Chinese Association of Automation and Vice Chairman of Jiangsu Branch. He is a standing council member of China Command and Control Society. Professor Bo’s research interests are focused on filtering and system optimization. He was granted a secondary prize of Natural science from the Ministry of Education of China in 2005 and a secondary prize of technology promotion from Shandong Province in 2012.

Please cite this article as: M. Lyu, Y. Bo, Variance-constrained resilient H∞ filtering for time-varying nonlinear networked systems subject to quantization effects, Neurocomputing (2017), http://dx.doi.org/10.1016/j.neucom.2017.06.001