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Resilient Asynchronous H∞ Filtering for Markov Jump Neural Networks With Unideal Measurements and Multiplicative Noises Lixian Zhang, Senior Member, IEEE, Yanzheng Zhu, Peng Shi, Fellow, IEEE, and Yuxin Zhao

Abstract—This paper is concerned with the resilient H∞ filtering problem for a class of discrete-time Markov jump neural networks (NNs) with time-varying delays, unideal measurements, and multiplicative noises. The transitions of NNs modes and desired mode-dependent filters are considered to be asynchronous, and a nonhomogeneous mode transition matrix of filters is used to model the asynchronous jumps to different degrees that are also mode-dependent. The unknown time-varying delays are also supposed to be mode-dependent with lower and upper bounds known a priori. The unideal measurements model includes the phenomena of randomly occurring quantization and missing measurements in a unified form. The desired resilient filters are designed such that the filtering error system is stochastically stable with a guaranteed H∞ performance index. A monotonicity is disclosed in filtering performance index as the degree of asynchronous jumps changes. A numerical example is provided to demonstrate the potential and validity of the theoretical results. Index Terms—Asynchronous jumps, missing measurements, multiplicative noises, quantization, resilient filter, time-varying delays.

I. I NTRODUCTION In the past decades, neural networks (NNs) have been gaining persistent attention due to their extensive applications in a broad range of areas such as associative memory, Manuscript received March 3, 2014; revised October 2, 2014 and December 10, 2014; accepted December 23, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 61021002 and Grant 61322301, in part by the Australian Research Council under Grant DP140102180 and Grant LP140100471, in part by the 111 Project under Grant B12018, in part by the Fundamental Research Funds for the Central Universities, China, under Grant HIT.BRETIII.201211 and Grant HIT.BRETIV.201306, in part by the Deanship of Scientific Research, in part by King Abdulaziz University under Grant 81-130-35-HiCi. This paper was recommended by Associate Editor Z. Zeng. L. Zhang is with the School of Astronautics, Harbin Institute of Technology, Harbin 150080, China, and also with the Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia (e-mail: [email protected]). Y. Zhu is with the School of Astronautics, Harbin Institute of Technology, Harbin 150080, China, and also with the Department of Electrical and Computer Engineering, the Ohio State University, Columbus, OH 43210, USA (e-mail: [email protected]; [email protected]). P. Shi is with the School of Electrical and Electronic Engineering, University of Adelaide, Adelaide, SA 5005, Australia, and also with the College of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia (e-mail: [email protected]). Y. Zhao is with the College of Automation, Harbin Engineering University, Harbin 150001, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2387203

repetitive learning, classification of patterns, and image processing, etc., (see [1], [12], [19] and the references therein). Often, the NNs display a feature of network modes jumps and such jumps are commonly considered to be determined by a time homogeneous Markov chain. With the aid of analysis and synthesis methodologies in the area of Markov jump linear systems (MJLSs) [9], [10], [17], the resulting Markov jump NNs (MJNNs) have been also studied in [13]. Besides, time delays are often encountered in the MJNNs due to the finite switching speed of amplifiers and the inherent communication time of neurons, which has been regarded as a crucial source of negative effects such as periodic oscillation, divergence and even chaos. So far, a great number of studies have been carried out for MJNNs with diverse delays (see [15], [20]–[22], [35]), to mention a few. On the other hand, complex dynamics such as multiplicative noises, data missing, and quantization are commonly unavoidable in various types of NNs. For instance, the multiplicative noises are involved with a class of feedforward NNs and the influences of which on network performance are analyzed in [25]. In [15], the effects of state-multiplicative noises have been considered when the MJNNs are used to model human stick balancing. In addition, the NNs with missing data have been broadly treated and the applications of NNs with missing data have been reported during the last two decades (see [11], [12], [29] and the references therein). Also, one of the typical problems encountered in the implementation of artificial NNs is determining how many bits are appropriate to represent the physical states, parameters, or variables. Thus, the quantization of neurons data is generally needed before being trained, which has significant influence on the learning and generalization performances of NNs [5], [7], [8], [31]. However, to date, limited work has been available in the literature on a wide range of issues for MJNNs in the simultaneous presence of multiplicative noises, quantization and missing measurements. On another research frontier, much progress has been made in the study on H∞ filtering problem for dynamic systems (see [4], [6], [14], [16], [30]). For the situation where inaccuracies or uncertainties occur in the implementation of a designed filter due to the numerical roundoff errors, fixed word length and finite resolution instrumentation, etc., the so-called resilient H∞ filtering problem for uncertain systems, has been intensively investigated (see [3], [18], [24]). In addition, depending on the scenarios

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that the system modes are completely available to designers or not, the filtering technique concerned with MJLSs can be classified into the mode-dependent [2], [27] and mode-independent filtering [23], [33], [34]. It is noticeable that, in [6], an asynchronous mode-dependent filtering problem for MJLSs with partial information on mode jumps has been studied, which verified that the designed filter is effective and better-preformed in noise attenuation, compared with the mode-independent filter in the concern of achieving a better filtering performance index. Nonetheless, to the best of the authors’ knowledge, no results have been reported so far on the resilient asynchronous H∞ filtering problem for discrete-time MJNNs, not to mention including the aforesaid complex dynamics, such as timevarying delays, multiplicative noises, quantization and missing measurements. In response to the above discussions, in this paper, we investigate the resilient asynchronous H∞ filtering problem for a class of discrete-time MJNNs with time-varying delays, multiplicative noises and unideal measurements. An unideal measurement model is established, which is capable of covering both quantization and missing measurements phenomena in a unified way by using two Bernoulli distributed white sequences with known conditional probabilities. A sufficient condition is presented under which the filtering error system is stochastically stable and the H∞ performance requirement is satisfied. A monotonicity is revealed in solving a better filtering performance index while decreasing the degrees of asynchronous jumps based on the NNs modes. The remainder of this paper is organized as follows. In Section II, the model description of the underlying MJNNs is introduced and some preliminaries are given for problem formulation. The analysis and design problems of resilient asynchronous H∞ filter are solved in Section III. Then, a numerical example is given to demonstrate the effectiveness of the proposed approach in Section IV, and this paper is concluded in Section V. Notations: The notations used in this paper are standard. The superscript “T” stands for matrix transposition, and the notation · refers to the Euclidean vector norm. Rn and Z+ denote the n dimensional Euclidean space and set of nonnegative integers, respectively. l2 [0, ∞) is the space of summable infinite sequence over [0, ∞) and for v = {v(k)} ∈ l2 [0, ∞),   2 its norm is given by ||v||2 = E{ ∞ k=0 |v(k)| }. For notation (, F, P),  represents the sample space, F is the σ -algebra of subsets of the sample space, and P is the probability measure on F. Pr{·} means the occurrence probability of the event “·.” E {·} stands for the mathematical expectation and for sequences e = {e(k)} ∈ l2 ((, F, P), [0, ∞)),   2 its norm is given by ||e||E2 = E{ ∞ k=0 |e(k)| }. In addition, in symmetric block matrices or long matrix expressions, we use  to denote the terms which can be easily reproduced by symmetry. diag{· · · } stands for a blockdiagonal matrix, and cov(X) is the covariance matrix of X. I and 0 represent, respectively, identity matrix and zero matrix.

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II. P RELIMINARIES A. Discrete-Time MJNNs Consider a class of discrete-time MJNNs with modedependent time-varying delays, multiplicative noises, and unideal measurements, on a probability space (, F, P) ⎧ x(k + 1) = Aσ (k) x(k) + Bσ (k),1 f (x(k)) ⎪ ⎪ ⎪ ⎪ + Bσ (k),2 g(x(k − τσ (k) (k))) + Eσ (k),1 v(k) ⎪ ⎪ ⎨ y(k) = δ1 (k)Cσ (k) x(k) + (1 − δ1 (k))δ2 (k)q(Cσ (k) x(k)) (1) + Eσ (k),2 v(k) ⎪ ⎪ ⎪ ⎪ z(k) = Fσ (k) x(k) + Lσ (k) v(k) ⎪ ⎪ ⎩ x() = ψ(),  = −τmax , −τmax + 1, . . . 0 n x x with Aσ (k)  Aσ (k),0 + px =1 Aσ (k),px ωpx (k), Cσ (k)  ny y Cσ (k),0 + py =1 Cσ (k),py ωpy (k), nx , ny ∈ Z+ , where x(k)  T [x1 (k), x2 (k), . . . , xn (k)] ∈ Rn is the neural state vector with n neurons, y(k) ∈ Rm is the measured output, z(k) ∈ Rp is the linear combination of the neuron states to be estimated, and v(k) ∈ Rq is the additive noise belonging to l2 [0, ∞). τσ (k) (k) denotes the mode-dependent time-varying delays satisfying τmin ≤ τσ (k) (k) ≤ τmax , where τmin and τmax are constant positive scalars representing the lower and upper y bounds of delays, respectively. ωsx (k) and ω (k) represent the state- and measurement-multiplicative noises, respectively. Aσ (k),0 = diag{a1σ (k),0 , a2σ (k),0 , . . . , anσ (k),0 } > 0 is a diagop nal matrix with aσ (k),0 representing the rate in which the pth neuron will reset its potential to the resting state in isolation when disconnected from the networks and exterp,q nal inputs, where p = 1, . . . , n. Bσ (k),1 = [bσ (k),1 ]n×n p,q and Bσ (k),2 = [bσ (k),2 ]n×n are, respectively, the connection weight matrix and the delayed connection weight matrix. Both f (x(k))  [f1 (x1 (k)), f2 (x2 (k)), . . . , fn (xn (k))]T ∈ Rn and g(x(k))  [g1 (x1 (k)), g2 (x2 (k)), . . . , gn (xn (k))]T ∈ Rn denote the nonlinear neuron activation functions. q(·) is the round-off function representing the quantization effect, and ψ() describes the initial condition. Let {σ (k), k ≥ 0} be a right-continuous Markov chain taking values in a finite state space L = {1, 2, . . . , N} with mode transition probabilities (TPs) Pr{σ (k + 1) = j|σ (k) = i} = πij  where πij ≥ 0, ∀i, j ∈ L, and N j=1 πij = 1 for each mode i. Then the transition probability matrix (TPM) of system (1) can be further defined by ⎡ ⎤ π11 π12 · · · π1N ⎢ π21 π22 · · · π2N ⎥ ⎢ ⎥ (2) π =⎢ . .. .. ⎥. . . . ⎣ . . . . ⎦ πN1 πN2 · · · πNN For each possible value of σ (k) = i ∈ L, the system matrices in (1) are indicated by Ai,0 , Ai,1 , . . ., Ai,nx , Bi,1 , Bi,2 , Ci,0 , Ci,1 , . . ., Ci,ny , Ei,1 , Ei,2 , Fi , and Li , which are known constant matrices with appropriate dimensions. The two stochastic variables δ1 (k) and δ2 (k) here are mutually independent, which are used to model the missing measurements and quantization of real signals in a random way.

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TABLE I D IFFERENT C ASES OF M EASUREMENTS O UTPUT FOR D IFFERENT δ1 (k) AND δ2 (k)

Hence, we get from (1) and (4) that x

n 

Aσ (k),px ωpxx (k)

=

px =1

Cσ (k),py ωpyy (k) =

py =1

Both variables are Bernoulli distributed white sequences taking values on 0 or 1 with the following probabilities: Pr{δl (k) = 1} = αl , Pr{δl (k) = 0} = 1 − αl where αl ∈ [0, 1], are known constants for ∀l = 1, 2. Furthermore, we have E{δl (k)} = αl , E{(δl (k) − αl )2 } = αl (1 − αl ). Remark 1: The proposed measurement model in (1) provides a unified representation to describe the phenomenon of either missing measurements or quantization at each sampling instant. The different outputs of measurements are given in Table I when stochastically choosing values of δ1 (k) and δ2 (k). In addition, the measurement model in (1) can be further extended to consider more types of unideal measurements, such as randomly varying nonlinearities, senor saturation and communication delay, etc. Assumption 1 [33]: 1) The state-multiplicative noises {ωpxx (k), px = 1, . . . , nx } and the measurement-multiplicative y noises {ωpy (k), py = 1, . . . , ny } are both zero-mean independent sequences of random variables with variance equal y y to 1 and E{ωpx (k)ωqx (k)} = 0, E{ωp (k)ωq (k)} = 0 for all k and p = q. 2) The mutual correlation between ωpxx (k) y y and ωpy (k) is given as E{ωpxx (k)ωpy (k)} = ρ px ,py . 3) {ωpxx (k), y x px = 1, . . . , n }, {ωpy (k), py = 1, . . . , ny }, v(k) and ψ(k) are all mutual independent. 4) The Markov chain {σ (k), k ≥ 0} y is independent of {ωpxx (k), px = 1, . . . , nx }, {ωpy (k), py = y 1, . . . , n }, ψ(k) and v(k). The following lemma is needed to give a concise form for the multiplicative noises contained in both the state model and the measurement model in (1). Lemma 1 [26]: Consider the linear transformation of a random vector X, i.e., the random p-vector Y of the form Y = GX + b

(3)

where G is a p × n matrix and b is a p-vector. If Y is any finite variance random p-vector, for some n ≤ p, there are a random n-vector X and a vector b such that cov(X) = In , then one has (3) holds. Suppose that ωx (k)  [ ω1x (k); . . .; ωnxx (k) ], ωy (k)  y y [ ω1 (k); . . .; ωny (k) ], w(k) ¯  [ w¯ 1 (k); . . .; w¯ n¯ (k) ], G x  x ׯ y y x n n y and G  [γpy ,q ] ∈ Rn ׯn . By Lemma 1, [γpx ,q ] ∈ R there exists a null-mean random vector w(k) ¯ with dimension ¯ = In¯ , such that n¯ ≤ nx + ny , and cov(w(k)) 

  x ωx (k) G w(k). ¯ = Gy ωy (k)

(4)

A¯ σ (k),s w¯ s (k)

(5)

C¯ σ (k),s w¯ s (k)

(6)

s=1

y

n 

n¯ 

n¯  s=1

n x x ¯ where A¯ σ (k),s   px =1 Aσ (k),px γpx ,s and Cσ (k),s ny y C γ . py =1 σ (k),py py ,s The quantization denoted by q(·) in (1) is a source that has significant impact on the achievable performance of the dynamical systems. Generally, the quantizer is given as q(·)  [ q1 (·) q2 (·) . . . qj (·) ]T , which is symmetric, i.e., qr (−ℵ) = −qr (ℵ) (r = 1, 2, . . . j ). Specifically, for each qr (·) (1 ≤ r ≤ j ), the set of quantization levels is described by the following: (r) ι (r) ι = 0, ±1, ±2, . . .} ∪ {0} Ur = {±u(r) ι , uι = χr u0 (r)

for 0 < χrι < 1 and u0 > 0, where χr (r = 1, 2, . . . j ) is called the quantization density. In addition, a logarithmic quantizer is considered as follows: ⎧ (r) (r) (r) 1 1 ⎪ if 1+ξ uι < y(r) (k) < 1−ξ uι ⎨ uι r r (r) (r) qr (y (k)) = 0 if y (k) = 0 ⎪ ⎩ −q (−y(r) (k)) if y(r) (k) < 0 r where ξr  (1 − χr )/(1 + χr ). It follows from [28] that: (r) qr (y(r) (k)) = (1 + (r) q )y (k)

(7)

≤ ξr . where |(r) q | (j ) (1) (2) diag{q , q , . . . q }, the

Then, defining q  quantization effects can be transformed into the following sector-bounded uncertainties:   q (˜y(k)) = I + q y˜ (k) (8)

where y˜ (k)  Cσ (k) x(k). Furthermore, consider K  diag{κ1 , κ2 , . . . , κj }, then, denote E  q K−1 satisfying EET = ET E ≤ I, and K  (I + EK), (8) is equivalent to   q Cσ (k) x(k) = KCσ (k) x(k). (9) B. Resilient Asynchronous Filter The filter to be considered with additive gain perturbations in this paper is of the following full-order form: ⎧   f f f ⎪ x (k + 1) = A + A (k) xf (k) + Bη(k) y(k) ⎪ f η(k) η(k) ⎨   f f (10) zf (k) = Lη(k) + Lη(k) (k) xf (k), ⎪ ⎪ ⎩ xf () = 0,  = −τmax , −τmax + 1, . . . 0 where xf (k) ∈ Rn is the estimate of state x(k), zf (k) ∈ Rp is the f f f output of filter. The matrices Aη(k) , Bη(k) , and Lη(k) are the filter gains to be designed, and η(k) is supposed to be described by a Markov chain that is defined on a probability space takf ing values in a finite set M = {1, 2, . . . , M}. Aη(k) (k) and f

Lη(k) (k) are unknown matrices representing the additive gain perturbations of the form     f f Aη(k) (k) Mη(k),1 f (11) η(k) (k)Nη(k)  f f Lη(k) (k) Mη(k),2

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f

f

f

where Mη(k),1 , Mη(k),2 , and Nη(k) are known, real, constant matrices of appropriate dimensions for fixed filter mode, and η(k) (k) is an unknown time-varying matrix satisfying Tη(k) (k)η(k) (k) ≤ Iη(k) , ∀η(k) ∈ M. Remark 2: In general, the H∞ filter design is based on an implicit assumption that the filter will be implemented by exactly using the obtained filter gains. However, the inaccuracies or uncertainties, which are originated from the roundoff errors in numerical computation and the need of the safe-tuning margins provided for designers, do occur in the implementation of a designed filter. Hence, in this paper, the resilient H∞ filter is designed for a given discrete-time MJNNs with time-varying delays and multiplicative noises such that the designed filter is insensitive to certain errors with respect to its gains, which has not been fully investigated yet and remains challenging due to the complexity and difficulty of this problem in the area of neural networks. In this paper, we suppose that the presented filter is governed by a mode-dependent piecewise homogeneous Markov chain, which leads to the jumps among neuron state modes and those among the filter/estimator to be designed are asynchronous. Before proceeding further, the following definition is introduced, and the more details of which can be seen in [32]. Definition 1 [2]: A finite Markov chain α(k) ∈ M = {1, 2, . . . , M} is said to be homogeneous (respectively, nonhomogeneous) if for all k ≥ 0 the TP satisfies Pr{α(k + 1) = j|α(k) = i} = p(i, j) (respectively, Pr{α(k + 1) = j|α(k) = i} = p(k, i, j)), where p(i, j) (or p(k, i, j)) denotes a probability function. By Definition 1, the Markov chain {η(k), k ≥ 0} in (10) is nonhomogeneous, which is dependent on σ (k + 1), and takes values in another finite set M = {1, 2, . . . , M} with the following TPs: σ (k+1) (12) Pr{η(k + 1) = n|η(k) = m} = νmn M σ (k+1) σ (k+1) in which νmn ≥ 0 and = 1, ∀m ∈ M. n=1 νmn Correspondingly, the TPM for filter mode jumps is given by ⎤ ⎡ σ (k) σ (k) σ (k) ν11 ν12 · · · ν1M ⎢ σ (k) σ (k) σ (k) ⎥ ⎥ ⎢ν ν22 · · · ν2M ⎥ ⎢ 21 σ (k) . (13) =⎢ . ν .. .. ⎥ .. ⎢ .. . . . ⎥ ⎦ ⎣ σ (k) σ (k) σ (k) νM1 νM2 · · · νMM

It is clearly demonstrated by Definition 1 that the Markov chain σ (k) is homogeneous, while the Markov chain η(k) is nonhomogeneous but finite piecewise homogeneous because ν σ (k) is time-varying but invariant for the same σ (k). Assumption 2: The Markov chain σ (k) is assumed to be independent of Fk−1 = σ {η(1), η(2), . . . , η(k − 1)}, where Fk−1 is a σ -algebra generated by {η(1), η(2), . . . , η(k − 1)}. Then, consider the two different Markov chains σ (k) and η(k) to be associated with system modes and filter modes, respectively, the asynchronous jumps between system modes and desired mode-dependent filters can be effectively modeled. Besides, as the TPs of filter modes are nonhomogeneous and dependent on σ (k + 1), different asynchronous degrees can be

also regulated by giving certain values to the TPs in (13) for different modes of (1). C. Filtering Error Systems Denote x˜ (k)  [ xT (k) xfT (k) ]T , e(k)  z(k) − zf (k), the filtering error system can be obtained from (1) and (10) as follows: ⎧ ˜ i,m v(k) D ⎨ x˜ (k + 1) = A˜ i,m x˜ (k) + B˜ i,1 f (Hx(k)) +  ˜ + Bi,2 g(Hx(k − τi (k))) + 3h=1 ςh (k) (14) ⎩ e(k) = L˜ i,m x˜ (k) + Li v(k) where A˜ i,m B˜ i,1 L˜ i,m ˜ i,m D

   0 0 Ai,0 0 +  f f f f 0 Mm,1 m (k)Nm B˜ m Ci,0 Am     Bi,1 ˜ Bi,2 , Bi,2   0 0     f f f + 0 −Mm,2  Fi −Lm m (k)Nm   T T   f T ET  Ei,1 ,H  I 0 i,2 Bm 

1 2 3 x˜ (k), ς2 (k)  i,m x(k), ς3 (k)  i,m x(k) ς1 (k)  i,m     n¯ ¯ Ai,s 0 0 1 ω¯ s (k), H1   i,m ˜ mf C¯ i,s 0 s=1 B I   2  H1 Bmf δ˜1 (k)Ci,0 + δ˜2 (k)KCi,0 i,m 3 i,m 

n¯     H1 Bmf δ˜1 (k)C¯ i,s + δ˜2 (k)K C¯ i,s ω¯ s (k) s=1

δ˜1 (k)  δ1 (k) − α1 , B˜ mf  α1 Bmf + (1 − α1 )α2 Bmf K δ˜2 (k)  (1 − δ1 (k))δ2 (k) − (1 − α1 )α2 . Now, the following definitions for system (14) are restated to give the main objective of this paper more precisely. Definition 2: System (14) is said to be stochastically stable if for v(k) ≡ 0 and any initial condition x(0) ∈ Rn , σ (0) ∈ L and η(0) ∈ M, the following inequality holds: ∞   E ||˜x(k)||2 |x(0), σ (0), η(0) < ∞. k=0

Definition 3: Given a scalar γ > 0, (14) is said to be stochastically stable and has an H∞ noise attenuation performance index γ if it is stochastically stable and under zero initial condition, ||e(k)||E2 < γ ||v||2 holds for all nonzero v(k) ∈ l2 [0, ∞). The aim in this paper is to consider (1) subject to nonhomogeneous TPM (13) of filter mode jumps, design a resilient asynchronous filter (10) such that the resulting filtering error system (14) is stochastically stable with a prescribed H∞ noise attenuation performance index. D. Necessary Lemmas Lemma 2 [20]: Suppose that  = diag{λ1 , λ2 , . . . , λn } > 0 and  = diag{ θ1 , θ2 , . . . θn } > 0. For z = 0, z ∈ R, p = 1, 2, . . . , n, the neuron activation functions satisfy fp (z) gp (z) ≤ lp+ , p− ≤ ≤ p+ lp− ≤ z z

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where lp− , lp+ , p− , and p+ are constant scalars, then   T T T H 1 H −H 2 1 ≤ 0 1 ∗    T H ϒ1 H −H T ϒ2 2 < 0 2T ∗ 

with  T  1,2 1,3 1,4 1,5 1,6 ϒ1,2,1  L˜ i,m 1,3  α12 [H1 Bfm KCi,0 H]T P˜ i,m

 [ x˜ T (k) f T (H x˜ (k)) ]T, 2  where 1 T T [ x˜ (k − τi (k)) g (H x˜ (k − τi (k))) ]T, H  [ I 0 ]   1  diag l1+ l1− , l2+ l2− , . . . , ln+ ln−   l1+ + l1− l2+ + l2− ln+ + ln− 2  diag − ,− ··· ,− 2 2 2  + − + −  + − ϒ1  diag 1 1 , 2 2 , . . . , n n   1+ + 1− 2+ + 2− n+ + n− ϒ2  diag − ,− ,...,− . 2 2 2 Remark 3: As discussed in [20], the constants lp− , lp+ , and p+ are allowed to be positive, negative, or zero. Therefore, the resulting neuron activation functions could be nonmonotonic and more general than the usual sigmoidtype functions and Lipschitz-type conditions. Such a description facilitates obtaining less conservative results since it restricts the lower and upper bounds of the neuron activation functions. Lemma 3 [18]: For any real matrices X, Y, and P with appropriate dimensions and PT P ≤ I, it follows that: p− ,

XPY + Y T PT X T ≤  −1 XX T + Y T Y, ∀ > 0.

44  (τmax − τmin + 1)Q2 , 22  − H T ϒ1 H − Q1 11  −Pi,m − H T 1 H + (τmax − τmin + 1)Q1   qjmn πij Pj,n , 13  − H T 2 P˜ i,m  n∈M

1,2 

1,6  57  α12  1,4

ϒ2,2 ϒ1,2

s=1 n¯ 

 T α12 H1 Bmf K C¯ i,s H P˜ i,m

s=1 P˜ Ti,m B˜ Ti,2 , 77



˜ Ti,m  −P˜ i,m , 67  P˜ Ti,m D

(1 − α1 )α2 − (1 − α1 )2 α22 T n¯  ¯  Ai,s 0  P˜ i,m f B˜ m C¯ i,s 0 s=1

then the filtering error system (14) is stochastically stable and has a prescribed H∞ performance index γ . Proof: Consider the following Lyapunov–Krasovskii functional: V(˜x(k), σ (k), η(k)) =

5 

V (˜x(k), σ (k), η(k))

=1

The following theorem presents sufficient conditions under which the filtering error system (14) is stochastically stable and achieves a prescribed H∞ performance index. Theorem 1: Consider (1) with resilient asynchronous filters (10) subject to nonhomogeneous TPM (13) of filter mode jumps and let 0 ≤ α1 ≤ 1, 0 ≤ α2 ≤ 1, γ > 0, τmax ≥ τmin ≥ 0 be given constants. If there exist matrices Pi,m , Q1 , Q2 , two diagonal matrices  > 0 and  > 0, such that for any i ∈ L, m ∈ M   ϒ1,1 ϒ1,2