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Resilient fault-tolerant anti-synchronization for stochastic delayed reaction–diffusion neural networks with semi-Markov jump parameters
Journal Pre-proof Resilient fault-tolerant anti-synchronization for stochastic delayed reaction–diffusion neural networks with semi-Markov jump parameters Jianping Zhou, Yamin Liu, Jianwei Xia, Zhen Wang, Sabri Arik
PII: DOI: Reference:
S0893-6080(20)30069-1 https://doi.org/10.1016/j.neunet.2020.02.015 NN 4413
To appear in:
Neural Networks
Received date : 23 August 2019 Revised date : 16 February 2020 Accepted date : 24 February 2020 Please cite this article as: J. Zhou, Y. Liu, J. Xia et al., Resilient fault-tolerant anti-synchronization for stochastic delayed reaction–diffusion neural networks with semi-Markov jump parameters. Neural Networks (2020), doi: https://doi.org/10.1016/j.neunet.2020.02.015. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Resilient fault-tolerant anti-synchronization for stochastic delayed
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reaction-diffusion neural networks with semi-Markov jump parameters Jianping Zhou†, Yamin Liu†, Jianwei Xia‡, Zhen Wang§, Sabri Arik¶∗ February 16, 2020
Abstract
This paper deals with the anti-synchronization issue for stochastic delayed reaction-diffusion neural networks subject to semi-Markov jump parameters. A resilient fault-tolerant controller is utilized to ensure the anti-synchronization in the presence of actuator failures as well as gain perturbations, simultaneously. Firstly, by means of the Lyapunov functional and stochastic analysis methods, a mean-square exponential stability criterion is derived for the resulting error system. It is shown the obtained criterion improves a previously reported result. Then, based on the present analysis result and using several decoupling techniques, a strategy for designing the desired resilient fault-tolerant controller is proposed. At last, two numerical examples are given to illustrate the superiority of the present stability analysis method and the applicability of the proposed resilient fault-tolerant anti-synchronization control strategy, respectively. Keywords: Anti-synchronization; semi-Markov process; Fault-tolerant control; Resilient control; Neural networks.
1
Introduction
†
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Neural network is a complex nonlinear system composed of numerous neurons, which possesses distinguished information storage capacity and associative memory function. Recent years have witnessed a rapidly growing interest in neural networks because of their potential applications in system identification [1], medical diagnosis [2], data mining [3], and many other fields. Generally, time delay is unavoidable in most applications of neural networks, and its existence is usually one of the main sources of oscillation, divergence, and chaos. Therefore, the study of delayed neural networks (DNNs) is of practical significance, and a great amount of impressive results have been proposed in the literature; see, e.g., [4–8]. For nonlinear systems, one of the most interesting phenomena is anti-synchronization, which depicts a dynamic behavior that the state curves of coupled systems possess the same amplitude but opposite phase. It has been shown that the anti-synchronization capability of chaotic nonlinear systems can be successfully applied to secure encryption [9] and secure communication [10]. Over the past ten years, much attention has been directed to the anti-synchronization control of DNNs. For example, a class of DNNs with unknown parameters and external disturbances were studied in [11], and an adaptive H∞ control scheme was proposed in terms School of Computer Science & Technology, Anhui University of Technology, Ma’anshan 243032, PR China School of Mathematics Science, Liaocheng University, Liaocheng 252000, PR China § College of Mathematics & Systems Science, Shandong University of Science & Technology, Qingdao 266590, PR China ¶ Department of Computer Engineering, Faculty of Engineering, Istanbul University-Cerrahpasa, Istanbul 34320, Turkey ∗ Corresponding author. Email: [email protected] ‡
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of linear matrix inequalities (LMIs). Chaotic DNNs with stochastic perturbations were examined in [12], where a memory controller was employed to guarantee the exponential anti-synchronization by combining the Halanay inequality with stochastic analysis techniques. When both time delays and reaction-diffusion factors are involved, non-fragile and pinning anti-synchronization control strategies were proposed in [13] and [14], respectively. In practical engineering applications, neural networks sometimes need to be modeled as switched systems consisting of finite subsystems and a switching signal. Generally speaking, switched systems can be divided into arbitrary switched systems and restricted switched ones based on the switching signal. As a class of typical restricted switched systems, Markov jump systems have drawn particular attention from various fields and been recognized to be very suitable for describing practical systems with a sudden change of the structure or parameters. In the context of DNNs with Markov jump parameters, many results have been reported during the past two decades; see, e.g., [15–21]. However, an assumption for obtaining these results is that the sojourn time needs to obey the exponential distribution, which is a memoryless distribution. In other words, the transition probability is assumed to be only related to the current state of the system, but not to the past or the future state. This imposes a certain limitation on the applications of the DNNs with Markov jump parameters. To deal with the problem, a few research activities have been focused on semi-Markov DNNs, where the sojourn time is not limited to the exponential distribution and the transition probability of the system varies with time. Particularly, the synchronization problems of DNNs subject to semi-Markov jump parameters have been examined in [22–24], where some sufficient conditions on the synchronization analysis and control synthesis in mean square were proposed. However, to the authors’ knowledge, anti-synchronization related results of delayed reaction-diffusion neural networks (DRDNNs) with semi-Markov jump parameters have not been reported yet. In many actual control systems, actuator failures occur frequently and they may cause unpredictable adverse consequences, such as degrading the controller performance and even destroying the controller components. As a practical means of settlement, fault-tolerant control scheme has been introduced to the control community over the past few decades. It was shown in [25, 26] that such a control scheme can automatically compensate for the effects of failures to keep the system stability and recover the system performance as much as possible so that the system works stably and reliably. In 2016, a fault-tolerant controller was designed in [27] for the fault estimation issue of a class of Markov jump systems with time delay and Lipschitz nonlinearity. Later, the fault-tolerant passive control issue of fuzzy systems subject to semi-Markov parameters was studied in [28] on the basis of event-triggered mechanisms. On the other hand, in the hardware implementation, a digital controller often has gain uncertainties to some extent because of restrictions in accessible process memory, imprecisions of analog-digital conversion, rounding errors of digital calculators, etc [29]. It is thus necessary to design resilient (or non-fragile) controllers to ensure the control effects. A type of nonlinear multi-agent system with both stochastic disturbances and Markovian switching topologies was evaluated in [30], where a resilient dynamic output-feedback controller was designed for the L2 − L∞ consensus. In [31], the resilient estimation issue for a class of semi-Markov jump descriptor systems was investigated, and an approach for ensuring the H∞ stability of the estimation error system was developed. For the controller designs of DRDNNs with Markov jump parameters, it is noteworthy that actuator failures and gain perturbations have not been considered simultaneously. This constitutes another motivation for our research. Based on the above discussions, the anti-synchronization control issue is investigated for semi-Markov jump DRDNNs subject to possible actuator failures and gain perturbations under the master-slave synchro2
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2
Preliminaries
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nization configuration. As [32], the slave neural network is assumed to be disturbed by stochastic noises and modeled by stochastic functional differential equations. The aim is to design a resilient fault-tolerant controller to ensure that the master and slave DRDNNs are exponentially anti-synchronized in mean square. The main contributions of this work are as follows: 1. The considered master-slave DRDNN models, which not only involve the semi-Markov jump parameters but also the stochastic disturbances, are more general than the common DRDNN ones. 2. A mean-square exponential stability condition is presented for the resulting anti-synchronization error system with the application of the Lyapunov functional and some stochastic analysis methods. It is shown that the analysis result improves an existing result given in [33]. 3. A strategy for the design of the desired resilient fault-tolerant controller is proposed via eliminating the higher-order nonlinearities caused by the coupling of the Lyapunov matrices, the gain-perturbation matrices, and the uncertainties of the actuator failures. It is shown that the required control gains can be obtained via solving a few LMIs, which are able to be checked easily by the popular computing software MATLAB. The organization of this paper is as follows: In Section 2, the master-slave DRDNN models with semiMarkov jump parameters are proposed and some necessary preliminaries are prepared. In Section 3, a mode-dependent Lyapunov functional (LF) and some stochastic analysis methods are used to analyze the exponential stability for the resulting anti-synchronization error system. In Section 4, the design strategy of the desired resilient fault-tolerant anti-synchronization controller is presented. In Section 5, two numerical examples are given to illustrate the validity and superiority of the present analysis and synthesis methods. Finally, a conclusion is given in Section 6. Notation: In the present study, we use E {·} to denote the mathematical expectation operator, |·| to represent the Euclidean vector norm, N, N+ , Rn , and Rm×n to represent the set of non-negative integers, the set of positive integers, the normal n-dimensional Euclidean space, and the family of m × n real matrices, respectively. Denote by I and 0 the unity and zero matrices, respectively. For a square real matrix M , denote by tr(M ) its trace, by S (M ) the sum of M and M T , by ∗ the symmetry blocks, and by λmax (M ) and λmin (M ) its maximum and minimum eigenvalues respectively. Let C m (Z, Rn ) be the set of m-times continuously-differentiable functions which map Z into Rn , and L2F0 ([−¯ τ , 0] × Ω, Rn ) be the n set of all F0 − measurable C ([−¯ τ , 0] × Ω, R ) − random variables χ = {χ(s, x) : − τ¯ ≤ s ≤ 0} such that 2 sup−¯τ ≤s≤0 E |χ(s, x)| < ∞, where Ω = {x = [x1 · · · xq ]T | φi ≤ xi ≤ ψi , φi , ψi ∈ R, i = 1, · · · , q} is a q closed set in R . The boundary of Ω is defined as ∂Ω.
Consider the following DRDNN with semi-Markov jump parameters: q
dz1 (t, x) X ∂ = dt ∂xk
∂z1 (t, x) Dk ∂xk
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k=1
− Aγ(t) z1 (t, x) + W1γ(t) f (z1 (t, x)) + W2γ(t) f (z1 (t − τ (t), x)),
(1)
where z1 (t, x) = [z11 (t, x), · · · , z1n (t, x)]T with z1i (t, x) ∈ C 2 (R × Ω, R) denoting the state variable of the ith neuron at t and x = [x1 , · · · , xq ]T ; Dk = diag{D1k , · · · , Dnk } with Dik ≥ 0 denoting the transmission diffusion coefficient along the ith neuron, Aγ(t) = diag{A1γ(t) , · · · , Anγ(t) } with Aiγ(t) > 0 being the rate at which the ith neuron resets the potential to the resting state in the case when disconnected from the network ij ij ij ij and external inputs, W1γ(t) = (w1γ(t) )n×n and W2γ(t) = (w2γ(t) )n×n with w1γ(t) and w2γ(t) representing the connection weights of neurons; f (z1 (t, x)) = [f1 (z11 (t, x)) · · · fn (z1n (t, x))] with fj (z1j (t, x)) representing the activation function of the jth neuron which is odd for realizing anti-synchronization as [34, 35], and 3
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τ (t) is a variant time delay subject to 0 < τ (t) ≤ τ¯, τ˙ (t) ≤ µ < 1, where both τ¯ and µ are known nonnegative scalars; γ(t)t≥0 is a continuous-time semi-Markov process taking values in a limited set M = {1, · · · , m|m ∈ N+ }; tl (l ∈ N) represents the l-th transition point of γ(t); for l ∈ N+ , γl stands for the indicator of the system mode and hl = tl − tl−1 the corresponding sojourn time with density function eγ(t) (h). The evolution of the semi-Markov process is governed by [36]: ( P r{γl+1 = j, hl+1 ≤ h + θ|γl = i, hl+1 > h} = ξij (h)θ + o(θ), i 6= j, (2) P r{γl+1 = j, hl+1 > h + θ|γl = i, hl+1 > h} = 1 + ξii (h)θ + o(θ), i = j, where o(θ) stands for the little-o notation, which is defined by limθ→0 (o(θ)/θ) = 0; ξij (h) corresponds to the transition rate from mode i at time t to mode j at time t + h with ξij (h) ≥ 0 (for i 6= j) and m P ξij (h). ξii (h) = − j=1,j6=i
Remark 1 In a continuous-time Markov process, the sojourn time is exponentially distributed and the transition rates are constants, while for a semi-Markov process, the sojourn time can obey the Weibull distribution, which includes exponential distribution as its special case. It is noteworthy that the transition rates of a semi-Markov process vary with time, which leads to difficulties in the anti-synchronization analysis. Throughout this paper, the master-slave synchronization scheme shall be adopted. We take DRDNN (1) to be a master system, and the slave system is modeled by the following stochastic functional differential equations: " q X ∂ ∂z2 (t, x) dz2 (t, x) = Dk − Aγ(t) z2 (t, x) + W1γ(t) f (z2 (t, x)) + W2γ(t) f (z2 (t − τ (t), x)) ∂xk ∂xk k=1 i +Bγ(t) ufγ(t) (t, x) dt + σ(t, z1 (t, x) + z2 (t, x), z1 (t − τ (t), x) + z2 (t − τ (t), x))dω(t), (3)
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where z2 (t, x) = [z21 (t, x), · · · , z2n (t, x)]T with z2i (t, x) representing the ith neuron state; σ(t, z1 (t, x) + z2 (t, x), z1 (t − τ (t), x) + z2 (t − τ (t), x)) denotes a noise intensity matrix satisfying σ(t, 0, 0) = 0; ω(t) stands for a vector Brownian motion [37] with E {dω(t)} = 0 and E dω 2 (t) = dt; for any γ(t) ∈ M, Bγ(t) ∈ Rn×p is a constant matrix, and ufγ(t) (t, x) corresponds to the control input with possible actuator failures, which is given as follows: ufγ(t) (t, x) = Gγ(t) uγ(t) (t, x), (4) where Gγ(t) is the actuator fault matrix [38], which is defined as Gγ(t) = diag{g1γ(t) , · · · , gpγ(t) } with 0 ≤ gˇjγ(t) ≤ gjγ(t) ≤ gˆjγ(t) ≤ 1 (j = 1, · · · , p). Here gˇjγ(t) and gˆjγ(t) (j = 1, · · · , p) are known constants.
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Remark 2 Note that Gγ(t) reflects actuator effectiveness. When Gγ(t) = I (i.e., ufγ(t) (t, x) = uγ(t) (t, x)), there is no failure in the γ(t)-th actuator. When 0 ≤ Gγ(t) < I, there is a loss of efficiency in the γ(t)-th actuator, and moreover, if Gγ(t) = 0, then ufγ(t) (t, x) is outage or stuck. Let us define
[γ(t)]
G0γ(t) = diag{g01
[γ(t)]
, · · · , g0p
[γ(t)]
},
G1γ(t) = diag{g11
[γ(t)]
, · · · , g1p
}
[γ(t)] [γ(t)] with g0j = gˆjγ(t) + gˇjγ(t) /2 and g1j = gˆjγ(t) − gˇjγ(t) /2 (j = 1, · · · , p). Then, fault matrix Gγ(t) can be rewritten as Gγ(t) = G0γ(t) + G1γ(t) Λγ(t) , Λγ(t) = diag λ1γ(t) , · · · , λpγ(t) , −1 ≤ λjγ(t) ≤ 1, j = 1, · · · , p. (5) 4
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In the present study, the case of gain perturbations will also be considered. Without loss of generality [29], it is assumed that ¯ γ(t) (z1 (t, x) + z2 (t, x)) , uγ(t) (t, x) = K (6)
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¯ γ(t) = Kγ(t) +∆Kγ(t) (t), Kγ(t) is the control gain that needs to be determined and ∆Kγ(t) (t) denotes where K the possible gain perturbation, which is described as ∆Kγ(t) (t) = Lγ(t) Fγ(t) (t)Mγ(t) . Here, Fγ(t) (t) is an T (t)F unknown matrix function which meets Fγ(t) γ(t) (t) ≤ I; Lγ(t) and Mγ(t) are known constant matrices. Now we are in a position to give the boundary conditions and initial conditions of systems (1) and (3), which are as follows: zi (t, x) = 0,
(t, x) ∈ [0, +∞) × ∂Ω,
(s, x) ∈ [−τ , 0] × Ω,
zi (s, x) = ϕi (s, x),
(7) (8)
where ϕi (s, x) ∈ L2F0 ([−¯ τ , 0] × Ω, Rn ), i = 1, 2. Set z1 (t, x) + z2 (t, x) = ζ(t, x) (= [ζ1 (t, x), · · · , ζn (t, x)]T ). Then we are able to write the antisynchronization error system as Pl ∂ζ(t,x) ∂ − Aγ(t) − Bγ(t) Gγ(t) Kγ(t) + ∆Kγ(t) (t) ζ(t, x) dζ(t, x) = k=1 ∂xk Dk ∂xk (9) +W1γ(t) g(ζ(t, x)) + W2γ(t) g(ζ(t − τ (t), x)) dt + σ (t, ζ(t, x), ζ (t − τ (t), x)) dω(t), where g (ζ(t, x)) = f (ζ(t, x) − z1 (t, x)) + f (z1 (t, x)). The activation functions and the noise intensify matrix are assumed to meet the following conditions, which are widely used in the literature (see, e.g., [39–41]). Assumption 1 For any α1 , α2 ∈ R, α1 6= α2 , activation function fi (·) (i = 1, · · · , n) satisfies β1i ≤
fi (α1 ) − fi (α2 ) ≤ β2i , α1 − α2
where β1i and β2i are known constant scalars.
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Assumption 2 There exists positive scalars σ1 and σ2 such that tr σ T (t, ζ(t, x), ζ (t − τ (t), x)) σ (t, ζ(t, x), ζ (t − τ (t), x)) ≤ σ1 ζ T (t, x)ζ(t, x) + σ2 ζ T (t − τ (t), x) ζ (t − τ (t), x) .
(10)
Noticing that f (z(t, x)) is an odd function, one can obtain from Assumption 1 that β1i ≤
gi (ζi (t, x)) ≤ β2i , i = 1, · · · , n. ζi (t, x)
(11)
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At the end of this section, let us give the definition on the exponential anti-synchronization and prepare several useful lemmas: Definition 1 Semi-Markov jump DRDNNs (1) and (3) are said to be exponentially anti-synchronized in mean square if the resulting error system in (9) is mean-square exponentially stable, i.e., there exist positive constants κ and H such that Z E |ζ(t, x)|2 dx ≤ He−κt . Ω
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Lemma 1 [42] Let Wa and Wb be two symmetric matrices of the same dimension. If Wb ≥ 0, then λmin (Wa )tr(Wb ) ≤ tr(Wa Wb ) ≤ λmax (Wa )tr(Wb ). Lemma 2 [43] Let z(x) ∈ C 1 (Ω, R) with z(x)/∂Ω = 0, then Ω
2
z (x)dx ≤
ψk − φk π
2 Z
∂z(x) ∂xk
2
dx,
k = 1, · · · , q.
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Z
Ω
Lemma 3 [44] Suppose that (11) is satisfied. Then
ζ T (t, x)T1 RT2 ζ(t, x) − ζ T (t, x)R(T1 + T2 )g(ζ(t, x)) + g T (ζ(t, x))Rg(ζ(t, x)) ≤ 0
holds true for any diagonal matrix R > 0, where T1 = diag (β11 , · · · , β1n ) and T2 = diag (β21 , · · · , β2n ). Lemma 4 [45] For any real matrices Wa and Wb of suitable dimensions, there exists a constant ε > 0 such that S (Wa Wb ) ≤ ε−1 Wa WaT + εWbT Wb . Lemma 5 [46] Suppose there are a real constant ς and real matrices Θ, Ui , Vi , and Si (i = 1, · · · , p) satisfying " # Θ UV < 0, ∗ S where UV = [U1 + ςV1 , · · · , Up + ςVp ], and S =diag{S (−ςS1 ), · · · , S (−ςSp )}. Then, one can obtain Θ+
p X i=1
S (Ui Si−1 ViT ) < 0.
Lemma 6 [47] Let Θ, F, Wa , and Wb be real matrices of suitable dimensions. Then Θ + S (Wa FW b ) < 0
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holds for F T F ≤ I if and only if there is a scalar ε > 0 such that
Θ + ε−1 Wa WaT + εWbT Wb < 0.
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Mean-square stability analysis
When uγ(t) (t, x) ≡ 0, system (9) can be rewritten as the following stochastic semi-Markov jump system:
where
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dζ(t, x) = hγ(t) (t, ζ(t, x), ζ(t − τ (t), x))dt + σ (t, ζ(t, x), ζ (t − τ (t), x)) dω(t),
hγ(t) (t, ζ(t, x), ζ(t − τ (t), x)) q X ∂ ∂ζ(t, x) = Dk − Aγ(t) ζ(t, x) + W1γ(t) g(ζ(t, x)) + W2γ(t) g(ζ(t − τ (t), x)). ∂xk ∂xk k=1
For system (12), we give a sufficient condition in the following theorem: 6
(12)
(13)
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Theorem 1 System (12) is exponentially stable in mean square, if, for any i ∈ M, there exist scalars ρi > 0, matrices Q1 > 0, Q2 > 0, diagonal matrices Pi , R1 > 0, and R2 > 0 such that the following LMIs hold:
where [i]
Ξ11 = −2Pi Dπ − S (Pi Ai ) + [i]
[i]
ξ¯ij Dπ
< 0,
m X j=1
(14)
(15)
ξ¯ij Pj + Q1 + ρi σ1 I − T1 R1 T2 ,
R1 (T1 + T2 ) , 2 = (µ − 1)Q1 − T1 R2 T2 + ρi σ2 I, Z ∞ = E {ξij (h)} = ξij (h)ei (h)dh, ) ( q 0 2 2 q X X π π D1k , · · ·, Dnk . = diag ψk − φk ψk − φk
Ξ13 = Pi W1i + Ξ22
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Pi ≤ ρi I, [i] [i] Ξ 0 Ξ13 Pi W2i 11 R2 (T1 +T2 ) ∗ Ξ[i] 0 22 2 Ξi = ∗ ∗ Q − R1 0 2 ∗ ∗ ∗ (µ − 1)Q2 − R2
k=1
k=1
Proof. For any t ≥ 0, by Lemma 1 one can write that tr σ T (t, ζ(t, x), ζ (t − τ (t), x)) Pi σ (t, ζ(t, x), ζ (t − τ (t), x)) = tr Pi σ (t, ζ(t, x), ζ (t − τ (t), x)) σ T (t, ζ(t, x), ζ (t − τ (t), x)) ≤ λmax (Pi ) tr σ T (t, ζ(t, x), ζ (t − τ (t), x)) σ (t, ζ(t, x), ζ (t − τ (t), x)) .
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It follows by (10) and (14) that tr σ T (t, ζ(t, x), ζ (t − τ (t), x)) Pi σ (t, ζ(t, x), ζ (t − τ (t), x))
≤ ρi σ1 ζ T (t, x)ζ(t, x) + ρi σ2 ζ T (t − τ (t), x) ζ (t − τ (t), x) , i = 1, · · · , n.
(16)
Utilizing the integration by parts formula and noticing the Dirichlet boundary condition in (7), for any i ∈ {1, · · · , n} and k ∈ {1, · · · , q}, one has Z Z ∂ ∂ζi (t, x) ∂ζi (t, x) 2 Dik dx = − Dik dx, ζi (t, x) ∂xk ∂xk ∂xk Ω
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Ω
which, together with Lemma 2, suggests 2 Z Z ∂ ∂ζi (t, x) π ζi (t, x) Dik dx ≤ − Dik ζi2 (t, x)dx. ∂xk ∂xk ψk − φk Ω
Ω
Thus, for diagonal matrix Pi , the following inequality can also be established: Z Z q X ∂ζ(t, x) ∂ 2 ζ T (t, x)Pi Dk dx ≤ −2 ζ T (t, x)Pi Dπ ζ(t, x)dx. ∂xk ∂xk Ω
k=1
Ω
7
(17)
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Furthermore, let Γ = diag{β1 , · · · , βn } with βi = max{|β1i |, |β2i |}, i = 1, · · · , n. Then, from (11) one has
T
g T (ζ(t, x)) g (ζ(t, x)) ≤ ζ T (t, x)Γ2 ζ(t, x),
T
(18)
ζ (t, x)T1 R1 T2 ζ(t, x) − ζ (t, x)R1 (T1 + T2 ) g (ζ(t, x)) T
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+g T (ζ(t, x)) R1 g (ζ(t, x)) ≤ 0,
(19)
ζ (t − τ (t), x)T1 R2 T2 ζ(t − τ (t), x)
−ζ T (t − τ (t), x)R2 (T1 + T2 ) g (ζ(t − τ (t), x))
+g T (ζ(t − τ (t), x)) R2 g (ζ(t − τ (t), x)) ≤ 0,
(20)
where the second and third inequalities follow from Lemma 3. Next, define a mode-dependent LF as
Vγ(t) (t, ζ(t, x)) = V1γ(t) (t, ζ(t, x)) + V2γ(t) (t, ζ(t, x)), where V1γ(t) (t, ζ(t, x)) =
Z
ζ T (t, x)Pγ(t) ζ(t, x)dx,
Ω
V2γ(t) (t, ζ(t, x)) =
Z
(21)
Zt
T
ζ (ν, x)Q1 ζ(ν, x)dνdx +
Ω t−τ (t)
Then, it follows that
λmin (Pγ(t) )
Z
Zt
g T (ζ(ν, x)) Q2 g (ζ(ν, x)) dνdx.
Ω t−τ (t)
Z
ζ T (t, x)ζ(t, x)dx
Ω
≤ Vγ(t) (t, ζ(t, x)) Z ≤ λmax (Pγ(t) ) ζ T (t, x)ζ(t, x)dx
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Ω
+λmax (Q1 )
Z
Zt
ζ T (ν, x)ζ(ν, x)dνdx
Z
Zt
g T (ζ(ν, x)) g (ζ(ν, x)) dνdx.
Ω t−τ (t)
+λmax (Q2 )
(22)
Ω t−τ (t)
where
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With the aid of the extended Itˆo’s formula [48], one can write E{eκt Vγ(t) (t, ζ(t, x))} = E{Vγ(t) (0, ζ(0, x))} t Z +E eκs [κVγ(s) (s, ζ(s, x)) + LVγ(s) (s, ζ(s, x))]ds , 0
1 E Vγ(s+θ) (s + θ, ζ(s + θ, x))| (s, γ(s), ζ(s, x)) − Vγ(s) (s, ζ(s, x)) . LVγ(s) (s, ζ(s, x)) = lim θ→0+ θ 8
(23)
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According to (2), for γ(s) = i one has LV1i (s, ζ(s, x)) ( m X 1 = lim Pr{γl+1 = j, hl+1 ≤ h + θ|γl = i, hl+1 > h}V1j (s + θ, ζ(s + θ, x)) + θ→0 θ
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j=1,j6=i
)
+ Pr{γl+1 = i, hl+1 > h + θ|γl = i, hl+1 > h}V1i (s + θ, ζ(s + θ, x)) − V1i (s, ζ(s, x))
1 = lim θ→0+ θ
(
m X
[ξij (h)θ + o(θ)] V1j (s + θ, ζ(s + θ, x))
j=1,j6=i
)
+ [1 + ξii (h)θ + o(θ)] V1i (s + θ, ζ(s + θ, x)) − V1i (s, ζ(s, x)) =
m X
ξij (h)V1j (s, ζ(s, x)) + lim
θ→0+
j=1
Noting that
1 V1i (s + θ, ζ(s + θ, x)) − V1i (s, ζ(s, x)) . θ
(24)
1 E V1i (s + θ, ζ(s + θ, x)) − V1i (s, ζ(s, x)) θ ∂ ∂ = V1i (s, ζ(s, x)) + V1i (ζ(s, x))hi (s, ζ(s, x), ζ(s − τ (s), x)) ∂s ∂ζ 1 ∂2 + tr σ T (s, ζ(s, x), ζ (s − τ (s), x)) 2 V1i (ζ(s, x))σ (s, ζ(s, x), ζ (s − τ (s), x)) , 2 ∂ζ lim
θ→0+
one obtains from (13) and (24) that
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E LV1i (s, ζ(s, x)) Z X Z q m X ∂ζ(s, x) ∂ Dk dx = ξ¯ij ζ T (s, x)Pj ζ(s, x)dx + 2 ζ T (s, x)Pi ∂xk ∂xk j=1 k=1 Ω Ω Z T +2 ζ (s, x)Pi − Ai ζ(s, x) + W1i g(ζ(s, x)) + W2i g(ζ(s − τ (s), x)) dx ZΩ + tr σ T (s, ζ(s, x), ζ (s − τ (s), x)) Pi σ (s, ζ(s, x), ζ (s − τ (s), x)) dx. Ω
By (16) and (17), it follows that
E LV1i (s, ζ(s, x)) ≤
Z
Ω
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m X ξ¯ij Pj + ρi σ1 I ζ(s, x)dx ζ T (s, x) − 2Pi Dπ − 2Pi Ai +
+2
+2
Z
j=1
ζ T (s, x)Pi W1i g (ζ(s, x)) dx
Ω
Z
Ω
+ρi σ2
ζ T (s, x)Pi W2i g(ζ(s − τ (s), x))dx Z
ζ T (s − τ (s), x)ζ(s − τ (s), x)dx.
Ω
9
(25)
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Similarly, the following inequality can be derived: Z Z T E LV2i (s, ζ(s, x)) = ζ (s, x)Q1 ζ(s, x)dx − (1 − τ˙ (s)) ζ T (s − τ (s), x) Q1 ζ (s − τ (s), x) dx Ω
Z
+
g T (ζ (s, x)) Q2 g (ζ (s, x)) dx
Ω
Z
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Ω
− (1 − τ˙ (s))
g T (ζ (s − τ (s), x)) Q2 g (ζ (s − τ (s), x)) dx.
(26)
Ω
From (19)-(21), (25), and (26) one can write E LVi (s, ζ(s, x)) ≤
Z
Ω
+
m X ζ T (s, x) − 2Pi Dπ − 2Pi Ai + ξ¯ij Pj + Q1 + ρi σ1 I − T1 R1 T2 ζ(s, x)dx
Z
Ω
+2 +
Z
ZΩ
Ω
+
Z
j=1
ζ T (s, x) 2Pi W1i + R1 (T1 + T2 ) g (ζ(s, x)) dx ζ T (s, x)Pi W2i g(ζ(s − τ (s), x))dx
ζ T (s − τ (s), x) ρi σ2 I + (µ − 1) Q1 − T1 R2 T2 ζ (s − τ (s), x) dx ζ T (s − τ (s), x) R2 (T1 + T2 ) g (ζ (s − τ (s), x)) dx
Ω
+
Z
g T (ζ (s, x)) (Q2 − R1 ) g (ζ (s, x)) dx
Ω
+ =
Z
Z
Ω
g T (ζ (s − τ (s), x)) (µ − 1) Q2 − R2 g (ζ (s − τ (s), x)) dx
¯ x)dx ζ¯T (s, x)Ξi ζ(s, Z
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Ω
≤ λmax (Ξi )
¯ x)dx, ζ¯T (s, x)ζ(s,
(27)
Ω
where
T ζ¯T (s, x) = ζ T (s, x) ζ T (s − τ (s), x) g T (ζ (s, x)) g T (ζ (s − τ (s), x)) .
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Now, by (18), (22), (23), and (27), one has (Z κt
min λmin (Pi )e E
|ζ(t, x)| dx
1≤i≤n
≤
Ω
(
max E λmax (Pi )
1≤i≤n
(
+E λmax (Q2 )
Z
T
)
(
ζ (0, x)ζ(0, x)dx + E λmax (Q1 )
Ω
Z
2
Z0
)
g T (ζ(ν, x)) g (ζ(ν, x)) dνdx
Ω −τ (0)
10
Z
Z0
Ω −τ (0)
T
)
ζ (ν, x)ζ(ν, x)dνdx
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+ max E 1≤i≤n
+E
( Z Zt
κs
)
T
e κλmax (Pi )ζ (s, x)ζ(s, x)dsdx
Ω 0
( Z Zt Zs
κs
)
T
e κλmax (Q1 )ζ (ν, x)ζ(ν, x)dνdsdx
Ω 0 s−τ (s) κs
T
e κλmax (Q2 )g (ζ(ν, x)) g (ζ(ν, x)) dνdsdx
Ω 0 s−τ (s)
+ max λmax (Ξi ) E 1≤i≤n
≤
max H1i E
1≤i≤n
Z Zt Ω 0
)
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+E
( Z Zt Zs
( Z Zt Ω 0
κs
e
eκs ζ T (s, x)ζ(s, x)dsdx
2
)
|ζ(s, x)| dsdx + max H2i E
where
1≤i≤n
(Z
Ω
2
)
sup |ϕ1 (ν, x) + ϕ2 (ν, x)| dx ,
−¯ τ ≤ν≤0
H1i = κλmax (Pi ) + (eκ¯τ − 1) λmax (Q1 ) + λmax (Q2 )Γ2 + λmax (Ξi ) ,
H2i = λmax (Pi ) + τ¯λmax (Q1 ) + τ¯λmax (Q2 )Γ2 + λmax (Q1 ) + λmax (Q2 )Γ2
(28)
1 κ¯τ (e − 1)(1 − e−κ¯τ ). κ
Noting (15), one can choose κ > 0 small enough such that H1i < 0 (i = 1, · · · , m). Let R sup |ϕ1 (ν, x) + ϕ2 (ν, x)|2 dx max1≤i≤n H2i E τ ≤ν≤0 Ω −¯ H0 = . min1≤i≤n λmin (Pi ) Then, (28) yields
E
Z
|ζ(t, x)|2 dx
Ω
≤ H0 e−κt .
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Therefore, by Definition 1, system (12) is exponentially stable in mean square. In this way, the proof is completed.
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Remark 3 Compared with the reported results on semi-Markovian jump systems, the criterion given in Theorem 1 can be applied to test the stability of stochastic partial functional differential equations. It is also worth mentioning that, in the proof of Theorem 1, the derivation of infinitesimal operator LVγ(s) is directly based on the transition probabilities of the semi-Markov process, which is more clear and concise than existing ways in the literature (see, e.g., [22–24]). When the sojourn time obeys exponential distribution, the transition rates are independent of h (i.e., ξij (h) ≡ ξij ). In this case, the semi-Markov process reduces to a normal Markov process. And we can write the following result: Corollary 1 System (12) with Markov jump parameters is exponentially stable in mean square, if, for any i ∈ M, there are scalars ρi > 0, matrices Q1 > 0, Q2 > 0, diagonal matrices Pi > 0, R1 > 0, and R2 > 0, such that (14) and (15) hold. 11
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4
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Remark 4 The stochastic stability of system (12) with Markov jump parameters was also considered in [33]. Comparing the criterion in Corollary 1 with that in [33] (see Theorem 1 therein), by adopting similar derivation procedures as those in [49] it is not difficult to show that, the former, which involves fewer decision variables, is a necessary but not sufficient condition for the latter. Thus, the present analysis method is theoretically less conservative than that reported in [33].
Anti-synchronization control synthesis
Based on Theorem 1, one can develop a strategy for the resilient fault-tolerant anti-synchronization controller design, which is given by the following theorem: Theorem 2 Given a constant ς > 0, semi-Markov jump DRDNNs (1) and (3) are exponentially antisynchronized in mean square under the resilient fault-tolerant control, if, ∀i ∈ M, there exist scalars ρi > 0, ε1 > 0, ε2 > 0, matrices Q1 > 0, Q2 > 0, Xi , Yi , diagonal matrices Pi > 0, R1 > 0, R2 > 0, such that (14) and the following LMI hold: [i] [i] [i] [i] [i] Φ11 0 Φ13 Pi W2i Φ15 Φ16 Φ17 [i] ∗ Φ22 0 Φ24 0 0 0 ∗ ∗ Φ33 0 0 0 0 (29) ∗ ∗ Φ44 0 0 0 < 0, ∗ [i] T ∗ ∗ ∗ ∗ Φ55 ε2 Li 0 [i] ∗ ∗ ∗ ∗ ∗ Φ66 0 ∗ ∗ ∗ ∗ ∗ ∗ −ε2 I where [i]
Φ11 = −2Pi Dπ − S (Pi Ai − Bi G0i Yi ) + [i]
Φ44 ξ¯ij
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[i]
j=1
R1 (T1 + T2 ) [i] [i] , Φ15 = Pi Bi G0i Li , Φ16 = Pi Bi G0i − Bi G0i Xi + ςYiT , 2 R2 (T1 + T2 ) [i] = Pi Bi G1i , Φ22 = (µ − 1)Q1 − T1 R2 T2 + ρi σ2 I, Φ24 = , Φ33 = Q2 − R1 , 2 [i] [i] = (µ − 1)Q2 − R2 , Φ55 = −ε1 I + ε2 LiT Li , Φ66 = −S (ςXi ) + ε2 I, ) ( 2 2 Z ∞ q q X X π π D1k , · · · , Dnk . = E {ξij (h)} = ξij (h)ei (h)dh, Dπ = diag ψk − φk ψk − φk 0
Φ13 = Pi W1i + Φ17
m X ξ¯ij Pj + Q1 + ρi σ1 I − T1 R1 T2 + ε1 MiT Mi ,
k=1
k=1
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Moreover, the desired feedback gain of (6) can be given by Ki = Xi−1 Yi , i ∈ M.
(30)
Proof : By Theorem 1, the mean-square exponential anti-synchronization of system (1) and system (3) under the resilient fault-tolerant control is able to be realized if (14) and [i] [i] ¯ Ξ 0 Ξ13 Pi W2i 11 R2 (T1 +T2 ) ∗ Ξ[i] 0 22 2 <0 (31) ∗ ∗ Q2 − R1 0 ∗ ∗ ∗ (µ − 1)Q2 − R2 12
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are satisfied, where ¯ [i] = Ξ[i] + S (Pi Bi Gi (Ki + ∆Ki (t))) . Ξ 11 11 Noting ∆Ki (t) = Li Fi (t)Mi and FiT (t)Fi (t) ≤ I, one has from Lemma 4 that
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T T S (Pi Bi Gi ∆Ki (t)) ≤ ε−1 1 (Pi Bi Gi Li ) (Pi Bi Gi Li ) + ε1 Mi Mi .
(32)
In addition, in the light of (30), one can write
S (Pi Bi Gi Ki ) = S Bi G0i Yi + (Pi Bi Gi −Bi G0i Xi ) Xi−1 Yi .
(33)
By (32) and (33), it is obvious that (31) holds true if
[i] Ξi + diag{Πi , 0, 0, 0} + S Ui Xi−1 ViT < 0,
where [i]
Πi
Ui
(34)
T = S (Bi G0i Yi ) + ε1 MiT Mi + ε−1 1 (Pi Bi Gi Li ) (Pi Bi Gi Li ) , T = (Pi Bi Gi −Bi G0i Xi )T , 0, 0, 0 ,
Vi = [Yi , 0, 0, 0] T .
Let
¯ [i] = Pi Bi Gi Li , Φ 15
¯ [i] = Pi Bi Gi − Bi G0i Xi + ςY T . Φ i 16
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Then, according to Lemma 5 and Schur’s complement, (34) is ensured by [i] [i] ¯ [i] ¯ [i] Φ11 0 Φ13 Pi W2i Φ Φ 15 16 ∗ Φ[i] 0 Φ24 0 0 22 ∗ ∗ Φ33 0 0 0 < 0, ∗ ∗ ∗ Φ 0 0 44 ∗ ∗ ∗ ∗ −ε1 I 0 ∗ ∗ ∗ ∗ ∗ −S (ςXi )
which, by (5), can be re-expressed as
Φi + S (Wai Λi Wbi ) < 0,
=
Φi
Wai Wbi
[i]
Φ11 ∗ ∗ ∗ ∗ ∗
0
[i]
[i]
Φ13 Pi W2i
Φ22 0 ∗ Φ33 ∗ ∗ ∗ ∗ ∗ ∗
Φ24 0 Φ44 ∗ ∗
[i]
Φ15
[i]
Φ16
0 0 0 0 0 0 −ε1 I 0 ∗ −S (ςXi ) h iT = , (Pi Bi G1i )T , 0, 0, 0, 0, 0 h i = 0, 0, 0, 0, Li , I .
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where
(35)
13
,
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Noting ΛTi Λi ≤ I, by Lemma 6, (35) holds true if the following inequality is satisfied: T T Φi + ε2−1 Wai Wai + ε2 Wbi Wbi < 0,
which, by Schur’s complement again, can be re-organized as (29). This completes the proof of the theorem.
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Remark 5 During the design of the desired resilient fault-tolerant anti-synchronization controller, highorder nonlinearities appear as a result of the couplings of the Lyapunov matrices, the gain perturbation matrices, and the uncertainties of the actuator failures. How to deal with such high-order nonlinearities is not easy. In this paper, based on the analysis result and using several decoupling approaches, a strategy is developed in Theorem 2, where the desired gains can be obtained via solving several LMIs, which are checked easily by the computing software MATLAB. Considering the actuator failure problem and the gain perturbation problem separately, the following results can be established: Corollary 2 Suppose that there are no actuator failures; i.e., Gi = G0i = I (i = 1, · · · , n). Then, for a given constant ς > 0, semi-Markov jump DRDNNs (1) and (3) are exponentially anti-synchronized in mean square under the resilient control, if, ∀i ∈ M, there exist scalars ρi > 0, ε1 > 0, matrices Q1 > 0, Q2 > 0, Xi , Yi , diagonal matrices Pi > 0, R1 > 0, R2 > 0, such that (14) and the following LMI hold: [i] [i] [i] Φ16 Φ11 0 Φ13 Pi W2i Pi B2i Li ∗ Φ[i] 0 Φ24 0 0 22 ∗ ∗ Φ33 0 0 0 < 0, ∗ ∗ ∗ Φ44 0 0 ∗ ∗ ∗ ∗ −ε1 I 0 ∗ ∗ ∗ ∗ ∗ −S (ςXi ) [i]
[i]
[i]
[i]
where Φ11 , Φ13 , Φ16 , Φ22 , Φ24 , Φ33 , and Φ44 are the same as those in Theorem 2. In this case, the required feedback gain can be provided by Ki = Xi−1 Yi (i ∈ M).
where
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Corollary 3 Suppose that there are no gain perturbations; i.e., ∆Ki (t) = 0 (i = 1, · · · , n). Then, for a given constant ς > 0, semi-Markov jump DRDNNs (1) and (3) are exponentially anti-synchronized in mean square under the fault-tolerant control, if, ∀i ∈ M, there exist scalars ρi > 0, ε1 > 0, matrices Q1 > 0, Q2 > 0, Xi , Yi , diagonal matrices Pi > 0, R1 > 0, R2 > 0, such that (14) and the following LMI hold: [i] [i] [i] ˜ Φ 0 Φ13 Pi W2i Pi Bi G1i Φ16 11 ∗ Φ[i] 0 Φ24 0 0 22 ∗ ∗ Φ33 0 0 0 < 0, ∗ ∗ ∗ Φ 0 0 44 ∗ ∗ ∗ ∗ −ε1 I 0 [i] ˜ ∗ ∗ ∗ ∗ ∗ Φ66 ˜ [i] = −2Pi Dπ − S (Pi Ai − Bi G0i Yi ) + Φ 11 [i] Φ13 ,
[i] Φ16 ,
m X j=1
˜ [i] = −S (ςXi ) + ε1 I, ξ¯ij Pj + Q1 + ρi σ1 I − T1 R1 T2 , Φ 66
[i] Φ22 ,
and Φ24 , Φ33 , Φ44 , and ξ¯ij are the same as those in Theorem 2. In this case, the required feedback gain can be provided by Ki = Xi−1 Yi (i ∈ M). 14
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5
Numerical Examples
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In the section, two examples are given to illustrate the superiority of the stability analysis method and applicability of the resilient fault-tolerant anti-synchronization control strategy, respectively. Example 1. Consider unforced Markov jump system (9) (i.e., system (12) ) with two modes, where the parameters are provided by " # " # " # 0.5 0.5 0.5 0 0.4 0 D = , A1 = , A2 = , 0.3 0.7 0 0.5 0 0.4 " # " # 0.5 0.3 0.1 0.2 W11 = , W21 = , 0.2 0.1 0.3 0.4 " # " # 0.5 0.6 0.2 0.3 W12 = , W22 = , 0.4 0.2 0.4 0.5 ξ11 = −0.6, ξ12 = 0.6, ξ21 = 0.4, ξ22 = −0.4, 1 (|ζi + 1| − |ζi − 1|) (i = 1, 2), Ω = {x|x ∈ [−1, 1]} . gi (ζi ) = 2
Set µ = 0.05. Then, for the case that σ1 = 1.6, it is found that the criterion in Theorem 1 in [33] fails to check whether the system is stable or not since the LMIs therein have no feasible solutions ∀σ2 > 0, while by Corollary 1, it can be concluded that the system is exponentially stable in mean square for σ2 ∈ (0, 0.037]. And for the case that σ1 = 0.8, the maximum allowed σ2 ensuring the stability is calculated as 0.755 by Theorem 1 in [33] and 0.189 by Corollary 1, respectively. More detail comparisons between Theorem 1 in [33] and Corollary 1 in this paper for different values of σ1 are provided in Table 1, by which one can see the present analysis method is much less conservative than that proposed in [33], which further confirms the statement in Remark 4. Table 1: Comparisons of maximum allowed σ2 for different methods σ2 σ1 = 0.1 σ1 = 0.2 σ1 = 0.4 σ1 = 0.8 σ1 = 1.6 1.408 0.854
1.314 0.759
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Corollary 1 Theorem 1 of [33]
1.127 0.569
0.755 0.189
0.037 no solution
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Example 2. Consider two-mode semi-Markov jump DRDNNs given by (1) and (3) with " # " # 0.1 0 1.0 0 D = , A1 = , 0 0.1 0 1.0 " # " # 0.9 0 −2 1 A2 = , B1 = B2 = , 0 0.9 0 −2 " # " # 2.0 −0.1 −1.6 −0.1 W11 = , W21 = , −5.0 2.8 −0.3 −2.5 " # " # 0.5 0.6 0.2 0.3 W12 = , W22 = , 0.4 0.2 0.4 0.5 fi (·) = tanh(·) (i = 1, 2), Ω = {x|x ∈ [−2, 2]} , τ (t) = 1, σ(t, z1 (t, x) + z2 (t, x), z1 (t − τ (t), x) + z2 (t − τ (t), x))
= diag{z1 (t, x) + z2 (t, x), z1 (t − τ (t), x) + z2 (t − τ (t), x)}. 15
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The initial conditions are taken as z1 (s, x) =
"
1 −1
#
, z2 (s, x) =
"
−1 2
#
, (s, x) ∈ [−1, 0] × Ω,
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and the boundary conditions are set as the type of Dirichlet. Assume the sojourn time obeys the Weibull distribution. Specifically, let hl ∼ Weibull(2, 2) for i = 1 (i.e., e1 (h) = 0.5h exp(−(0.5h)2 )) and hl ∼ Weibull (1, 3) for i = 2 (i.e., e2 (h) = 3h2 exp(−h3 )) , respectively. Then we have " # " # ξ11 (h) ξ12 (h) −0.5h 0.5h = . ξ21 (h) ξ22 (h) 3h2 −3h2 By calculating the mathematical expectation we can get the transition rates as " # −0.8862 0.8862 E {ξ(h)} = . 2.7082 −2.7082 3
2.5
γ(t)
2
1.5
1
0.5
0
0
50
100
150
t
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Figure 1: Semi-Markov jump signal
Figure 2: State trajectories of the anti-synchronization error system without control 16
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Figure 3: State trajectories of the controlled anti-synchronization error system
Let us consider the following parameters concerning the actuator failures as well as gain perturbations: " # " # 0.2 0.1 0.2 0.1 L1 = , L2 = , 0 0.1 0 0.1 " # " # 0.02 0 0.01 0.1 M1 = , M2 = , 0 0.01 0 0.02 Fi (t) = sin(t), gˇji = 0.2, gˆji = 0.8 (i, j = 1, 2).
Then, by setting ς = 0.01 and utilizing MATLAB to solve the LMIs in Theorem 2, the desired control gain matrices can be obtained as " # " # 95.1945 4.3730 48.9157 25.2136 K1 = , K2 = . −55.3191 101.1654 −36.9121 78.3478
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5 4 3
z12 (t, −1.5)
2 1 0
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−1 −2 −3 −4
−5 −1
−0.5
0 z11 (t, −1.5)
0.5
Figure 4: Chaos behaviour
17
1
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15 z11 (t, −1.5)
10
z21 (t, −1.5)
5 0 −5 −10 −15
0
20
40
60
80
100
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t
20
z12 (t, −1.5)
10 0 −10 −20 0
20
40
60
z22 (t, −1.5)
80
100
t
Figure 5: State trajectories of the unforced master-slave DRDNNs
The semi-Markov jump signal is given in Figure 1. Under the switching signal, the state trajectories of the anti-synchronization error system without control are depicted in Figure 2, while, with the application of the designed controller, the desired fast convergence of the anti-synchronization error system is shown in Figure 3. 2
z11 (t, −1.5)
1 0 −1 −2
0
20
40
60
z21 (t, −1.5)
80
100
t
10
z12 (t, −1.5)
5
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0
z22 (t, −1.5)
−5
−10
0
20
40
60
80
100
t
Figure 6: State trajectories under control
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In order to illustrate the control effect more clearly, let us consider the case that x = −1.5. In such a case, the phase-plane plot of the master DRDNN is given in Figure 4, which shows a chaotic behavior. The state trajectories for the unforced master-slave DRDNNs and the controlled master-slave DRDNNs are depicted in Figure 5 and Figure 6, respectively. From the simulations, it can be observed that the designed resilient fault-tolerant controller works well for the anti-synchronization purpose.
18
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6
Conclusions
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The anti-synchronization control problem for stochastic DRDNNs with semi-Markov jump parameters has been studied in the presence of actuator failures as well as gain perturbations. First, by using a modedependent LF and some stochastic analysis methods, a sufficient condition has been derived to ensure the resulting anti-synchronization error system to be exponentially stability in mean square. It has been shown that the obtained criterion improves a previously reported result. Then, based on the present analysis result and employing some decoupling techniques, a strategy for the design of a resilient fault-tolerant anti-synchronization controller has been developed. Finally, two numerical examples have been given to verify the superiority of the stability analysis method and applicability of the resilient fault-tolerant antisynchronization control strategy, respectively.
References
[1] Stephen A Billings. Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains. West Sussex: John Wiley & Sons, 2013. [2] Hubert Kordylewski, Daniel Graupe, and Kai Liu. A novel large-memory neural network as an aid in medical diagnosis applications. IEEE Transactions on Information Technology in Biomedicine, 5(3):202– 209, 2001. [3] Yashpal Singh and Alok Singh Chauhan. Neural networks in data mining. Journal of Theoretical & Applied Information Technology, 5(1): 37–42, 2009. [4] Yong He, Meng-Di Ji, Chuan-Ke Zhang, and Min Wu. Global exponential stability of neural networks with time-varying delay based on free-matrix-based integral inequality. Neural networks, 77:80–86, 2016. [5] Tae H. Lee, Myeong-Jin Park, Ju H. Park, Oh-Min Kwon, and Sang-Moon Lee. Extended dissipative analysis for neural networks with time-varying delays. IEEE Transactions on Neural Networks and Learning Systems, 25(10):1936–1941, 2014.
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[6] Qiankun Song, Qinqin Yu, Zhenjiang Zhao, Yurong Liu, and Fuad E. Alsaadi. Boundedness and global robust stability analysis of delayed complex-valued neural networks with interval parameter uncertainties. Neural Networks, 103:55–62, 2018. [7] Xian-Ming Zhang, Qing-Long Han, and Zhigang Zeng. Hierarchical type stability criteria for delayed neural networks via canonical bessel–legendre inequalities. IEEE Transactions on Cybernetics, 48(5):1660– 1671, 2017.
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[8] Zhilian Yan, Xia Huang, Yingjie Fan, Jianwei Xia, and Hao Shen. Threshold-function-dependent quasisynchronization of delayed memristive neural networks via hybrid event-triggered control. IEEE Transactions on Systems, Man, and Cybernetics: Systems, in press, doi: 10.1109/TSMC.2020.2964605, 2020. ˇ [9] Volodymyr Lynnyk and Sergej Celikovsk´ y. On the anti–synchronization detection for the generalized Lorenz system and its applications to secure encryption. Kybernetika, 46(1):1–18, 2010. [10] Emad E. Mahmoud and S. M. Abo-Dahab. Dynamical properties and complex anti synchronization with applications to secure communications for a novel chaotic complex nonlinear model. Chaos, Solitons & Fractals, 106:273–284, 2018. [11] Choon Ki Ahn. Adaptive H∞ anti-synchronization for time-delayed chaotic neural networks. Progress of Theoretical Physics, 122(6):1391–1403, 2009. 19
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Conflict of Interest
Conflicts of Interest Statement
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Manuscript title: Resilient fault-tolerant anti-synchronization for stochastic delayed reactiondiffusion neural networks with semi-Markov jump parameters
The authors whose names are listed immediately below certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.
Author names: Jianping Zhou Yamin Liu Jianwei Xia Zhen Wang
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Sabri Arik
PII: DOI: Reference:
S0893-6080(20)30069-1 https://doi.org/10.1016/j.neunet.2020.02.015 NN 4413
To appear in:
Neural Networks
Received date : 23 August 2019 Revised date : 16 February 2020 Accepted date : 24 February 2020 Please cite this article as: J. Zhou, Y. Liu, J. Xia et al., Resilient fault-tolerant anti-synchronization for stochastic delayed reaction–diffusion neural networks with semi-Markov jump parameters. Neural Networks (2020), doi: https://doi.org/10.1016/j.neunet.2020.02.015. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Manuscript
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Resilient fault-tolerant anti-synchronization for stochastic delayed
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reaction-diffusion neural networks with semi-Markov jump parameters Jianping Zhou†, Yamin Liu†, Jianwei Xia‡, Zhen Wang§, Sabri Arik¶∗ February 16, 2020
Abstract
This paper deals with the anti-synchronization issue for stochastic delayed reaction-diffusion neural networks subject to semi-Markov jump parameters. A resilient fault-tolerant controller is utilized to ensure the anti-synchronization in the presence of actuator failures as well as gain perturbations, simultaneously. Firstly, by means of the Lyapunov functional and stochastic analysis methods, a mean-square exponential stability criterion is derived for the resulting error system. It is shown the obtained criterion improves a previously reported result. Then, based on the present analysis result and using several decoupling techniques, a strategy for designing the desired resilient fault-tolerant controller is proposed. At last, two numerical examples are given to illustrate the superiority of the present stability analysis method and the applicability of the proposed resilient fault-tolerant anti-synchronization control strategy, respectively. Keywords: Anti-synchronization; semi-Markov process; Fault-tolerant control; Resilient control; Neural networks.
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Introduction
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Neural network is a complex nonlinear system composed of numerous neurons, which possesses distinguished information storage capacity and associative memory function. Recent years have witnessed a rapidly growing interest in neural networks because of their potential applications in system identification [1], medical diagnosis [2], data mining [3], and many other fields. Generally, time delay is unavoidable in most applications of neural networks, and its existence is usually one of the main sources of oscillation, divergence, and chaos. Therefore, the study of delayed neural networks (DNNs) is of practical significance, and a great amount of impressive results have been proposed in the literature; see, e.g., [4–8]. For nonlinear systems, one of the most interesting phenomena is anti-synchronization, which depicts a dynamic behavior that the state curves of coupled systems possess the same amplitude but opposite phase. It has been shown that the anti-synchronization capability of chaotic nonlinear systems can be successfully applied to secure encryption [9] and secure communication [10]. Over the past ten years, much attention has been directed to the anti-synchronization control of DNNs. For example, a class of DNNs with unknown parameters and external disturbances were studied in [11], and an adaptive H∞ control scheme was proposed in terms School of Computer Science & Technology, Anhui University of Technology, Ma’anshan 243032, PR China School of Mathematics Science, Liaocheng University, Liaocheng 252000, PR China § College of Mathematics & Systems Science, Shandong University of Science & Technology, Qingdao 266590, PR China ¶ Department of Computer Engineering, Faculty of Engineering, Istanbul University-Cerrahpasa, Istanbul 34320, Turkey ∗ Corresponding author. Email: [email protected] ‡
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of linear matrix inequalities (LMIs). Chaotic DNNs with stochastic perturbations were examined in [12], where a memory controller was employed to guarantee the exponential anti-synchronization by combining the Halanay inequality with stochastic analysis techniques. When both time delays and reaction-diffusion factors are involved, non-fragile and pinning anti-synchronization control strategies were proposed in [13] and [14], respectively. In practical engineering applications, neural networks sometimes need to be modeled as switched systems consisting of finite subsystems and a switching signal. Generally speaking, switched systems can be divided into arbitrary switched systems and restricted switched ones based on the switching signal. As a class of typical restricted switched systems, Markov jump systems have drawn particular attention from various fields and been recognized to be very suitable for describing practical systems with a sudden change of the structure or parameters. In the context of DNNs with Markov jump parameters, many results have been reported during the past two decades; see, e.g., [15–21]. However, an assumption for obtaining these results is that the sojourn time needs to obey the exponential distribution, which is a memoryless distribution. In other words, the transition probability is assumed to be only related to the current state of the system, but not to the past or the future state. This imposes a certain limitation on the applications of the DNNs with Markov jump parameters. To deal with the problem, a few research activities have been focused on semi-Markov DNNs, where the sojourn time is not limited to the exponential distribution and the transition probability of the system varies with time. Particularly, the synchronization problems of DNNs subject to semi-Markov jump parameters have been examined in [22–24], where some sufficient conditions on the synchronization analysis and control synthesis in mean square were proposed. However, to the authors’ knowledge, anti-synchronization related results of delayed reaction-diffusion neural networks (DRDNNs) with semi-Markov jump parameters have not been reported yet. In many actual control systems, actuator failures occur frequently and they may cause unpredictable adverse consequences, such as degrading the controller performance and even destroying the controller components. As a practical means of settlement, fault-tolerant control scheme has been introduced to the control community over the past few decades. It was shown in [25, 26] that such a control scheme can automatically compensate for the effects of failures to keep the system stability and recover the system performance as much as possible so that the system works stably and reliably. In 2016, a fault-tolerant controller was designed in [27] for the fault estimation issue of a class of Markov jump systems with time delay and Lipschitz nonlinearity. Later, the fault-tolerant passive control issue of fuzzy systems subject to semi-Markov parameters was studied in [28] on the basis of event-triggered mechanisms. On the other hand, in the hardware implementation, a digital controller often has gain uncertainties to some extent because of restrictions in accessible process memory, imprecisions of analog-digital conversion, rounding errors of digital calculators, etc [29]. It is thus necessary to design resilient (or non-fragile) controllers to ensure the control effects. A type of nonlinear multi-agent system with both stochastic disturbances and Markovian switching topologies was evaluated in [30], where a resilient dynamic output-feedback controller was designed for the L2 − L∞ consensus. In [31], the resilient estimation issue for a class of semi-Markov jump descriptor systems was investigated, and an approach for ensuring the H∞ stability of the estimation error system was developed. For the controller designs of DRDNNs with Markov jump parameters, it is noteworthy that actuator failures and gain perturbations have not been considered simultaneously. This constitutes another motivation for our research. Based on the above discussions, the anti-synchronization control issue is investigated for semi-Markov jump DRDNNs subject to possible actuator failures and gain perturbations under the master-slave synchro2
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2
Preliminaries
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nization configuration. As [32], the slave neural network is assumed to be disturbed by stochastic noises and modeled by stochastic functional differential equations. The aim is to design a resilient fault-tolerant controller to ensure that the master and slave DRDNNs are exponentially anti-synchronized in mean square. The main contributions of this work are as follows: 1. The considered master-slave DRDNN models, which not only involve the semi-Markov jump parameters but also the stochastic disturbances, are more general than the common DRDNN ones. 2. A mean-square exponential stability condition is presented for the resulting anti-synchronization error system with the application of the Lyapunov functional and some stochastic analysis methods. It is shown that the analysis result improves an existing result given in [33]. 3. A strategy for the design of the desired resilient fault-tolerant controller is proposed via eliminating the higher-order nonlinearities caused by the coupling of the Lyapunov matrices, the gain-perturbation matrices, and the uncertainties of the actuator failures. It is shown that the required control gains can be obtained via solving a few LMIs, which are able to be checked easily by the popular computing software MATLAB. The organization of this paper is as follows: In Section 2, the master-slave DRDNN models with semiMarkov jump parameters are proposed and some necessary preliminaries are prepared. In Section 3, a mode-dependent Lyapunov functional (LF) and some stochastic analysis methods are used to analyze the exponential stability for the resulting anti-synchronization error system. In Section 4, the design strategy of the desired resilient fault-tolerant anti-synchronization controller is presented. In Section 5, two numerical examples are given to illustrate the validity and superiority of the present analysis and synthesis methods. Finally, a conclusion is given in Section 6. Notation: In the present study, we use E {·} to denote the mathematical expectation operator, |·| to represent the Euclidean vector norm, N, N+ , Rn , and Rm×n to represent the set of non-negative integers, the set of positive integers, the normal n-dimensional Euclidean space, and the family of m × n real matrices, respectively. Denote by I and 0 the unity and zero matrices, respectively. For a square real matrix M , denote by tr(M ) its trace, by S (M ) the sum of M and M T , by ∗ the symmetry blocks, and by λmax (M ) and λmin (M ) its maximum and minimum eigenvalues respectively. Let C m (Z, Rn ) be the set of m-times continuously-differentiable functions which map Z into Rn , and L2F0 ([−¯ τ , 0] × Ω, Rn ) be the n set of all F0 − measurable C ([−¯ τ , 0] × Ω, R ) − random variables χ = {χ(s, x) : − τ¯ ≤ s ≤ 0} such that 2 sup−¯τ ≤s≤0 E |χ(s, x)| < ∞, where Ω = {x = [x1 · · · xq ]T | φi ≤ xi ≤ ψi , φi , ψi ∈ R, i = 1, · · · , q} is a q closed set in R . The boundary of Ω is defined as ∂Ω.
Consider the following DRDNN with semi-Markov jump parameters: q
dz1 (t, x) X ∂ = dt ∂xk
∂z1 (t, x) Dk ∂xk
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k=1
− Aγ(t) z1 (t, x) + W1γ(t) f (z1 (t, x)) + W2γ(t) f (z1 (t − τ (t), x)),
(1)
where z1 (t, x) = [z11 (t, x), · · · , z1n (t, x)]T with z1i (t, x) ∈ C 2 (R × Ω, R) denoting the state variable of the ith neuron at t and x = [x1 , · · · , xq ]T ; Dk = diag{D1k , · · · , Dnk } with Dik ≥ 0 denoting the transmission diffusion coefficient along the ith neuron, Aγ(t) = diag{A1γ(t) , · · · , Anγ(t) } with Aiγ(t) > 0 being the rate at which the ith neuron resets the potential to the resting state in the case when disconnected from the network ij ij ij ij and external inputs, W1γ(t) = (w1γ(t) )n×n and W2γ(t) = (w2γ(t) )n×n with w1γ(t) and w2γ(t) representing the connection weights of neurons; f (z1 (t, x)) = [f1 (z11 (t, x)) · · · fn (z1n (t, x))] with fj (z1j (t, x)) representing the activation function of the jth neuron which is odd for realizing anti-synchronization as [34, 35], and 3
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τ (t) is a variant time delay subject to 0 < τ (t) ≤ τ¯, τ˙ (t) ≤ µ < 1, where both τ¯ and µ are known nonnegative scalars; γ(t)t≥0 is a continuous-time semi-Markov process taking values in a limited set M = {1, · · · , m|m ∈ N+ }; tl (l ∈ N) represents the l-th transition point of γ(t); for l ∈ N+ , γl stands for the indicator of the system mode and hl = tl − tl−1 the corresponding sojourn time with density function eγ(t) (h). The evolution of the semi-Markov process is governed by [36]: ( P r{γl+1 = j, hl+1 ≤ h + θ|γl = i, hl+1 > h} = ξij (h)θ + o(θ), i 6= j, (2) P r{γl+1 = j, hl+1 > h + θ|γl = i, hl+1 > h} = 1 + ξii (h)θ + o(θ), i = j, where o(θ) stands for the little-o notation, which is defined by limθ→0 (o(θ)/θ) = 0; ξij (h) corresponds to the transition rate from mode i at time t to mode j at time t + h with ξij (h) ≥ 0 (for i 6= j) and m P ξij (h). ξii (h) = − j=1,j6=i
Remark 1 In a continuous-time Markov process, the sojourn time is exponentially distributed and the transition rates are constants, while for a semi-Markov process, the sojourn time can obey the Weibull distribution, which includes exponential distribution as its special case. It is noteworthy that the transition rates of a semi-Markov process vary with time, which leads to difficulties in the anti-synchronization analysis. Throughout this paper, the master-slave synchronization scheme shall be adopted. We take DRDNN (1) to be a master system, and the slave system is modeled by the following stochastic functional differential equations: " q X ∂ ∂z2 (t, x) dz2 (t, x) = Dk − Aγ(t) z2 (t, x) + W1γ(t) f (z2 (t, x)) + W2γ(t) f (z2 (t − τ (t), x)) ∂xk ∂xk k=1 i +Bγ(t) ufγ(t) (t, x) dt + σ(t, z1 (t, x) + z2 (t, x), z1 (t − τ (t), x) + z2 (t − τ (t), x))dω(t), (3)
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where z2 (t, x) = [z21 (t, x), · · · , z2n (t, x)]T with z2i (t, x) representing the ith neuron state; σ(t, z1 (t, x) + z2 (t, x), z1 (t − τ (t), x) + z2 (t − τ (t), x)) denotes a noise intensity matrix satisfying σ(t, 0, 0) = 0; ω(t) stands for a vector Brownian motion [37] with E {dω(t)} = 0 and E dω 2 (t) = dt; for any γ(t) ∈ M, Bγ(t) ∈ Rn×p is a constant matrix, and ufγ(t) (t, x) corresponds to the control input with possible actuator failures, which is given as follows: ufγ(t) (t, x) = Gγ(t) uγ(t) (t, x), (4) where Gγ(t) is the actuator fault matrix [38], which is defined as Gγ(t) = diag{g1γ(t) , · · · , gpγ(t) } with 0 ≤ gˇjγ(t) ≤ gjγ(t) ≤ gˆjγ(t) ≤ 1 (j = 1, · · · , p). Here gˇjγ(t) and gˆjγ(t) (j = 1, · · · , p) are known constants.
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Remark 2 Note that Gγ(t) reflects actuator effectiveness. When Gγ(t) = I (i.e., ufγ(t) (t, x) = uγ(t) (t, x)), there is no failure in the γ(t)-th actuator. When 0 ≤ Gγ(t) < I, there is a loss of efficiency in the γ(t)-th actuator, and moreover, if Gγ(t) = 0, then ufγ(t) (t, x) is outage or stuck. Let us define
[γ(t)]
G0γ(t) = diag{g01
[γ(t)]
, · · · , g0p
[γ(t)]
},
G1γ(t) = diag{g11
[γ(t)]
, · · · , g1p
}
[γ(t)] [γ(t)] with g0j = gˆjγ(t) + gˇjγ(t) /2 and g1j = gˆjγ(t) − gˇjγ(t) /2 (j = 1, · · · , p). Then, fault matrix Gγ(t) can be rewritten as Gγ(t) = G0γ(t) + G1γ(t) Λγ(t) , Λγ(t) = diag λ1γ(t) , · · · , λpγ(t) , −1 ≤ λjγ(t) ≤ 1, j = 1, · · · , p. (5) 4
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In the present study, the case of gain perturbations will also be considered. Without loss of generality [29], it is assumed that ¯ γ(t) (z1 (t, x) + z2 (t, x)) , uγ(t) (t, x) = K (6)
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¯ γ(t) = Kγ(t) +∆Kγ(t) (t), Kγ(t) is the control gain that needs to be determined and ∆Kγ(t) (t) denotes where K the possible gain perturbation, which is described as ∆Kγ(t) (t) = Lγ(t) Fγ(t) (t)Mγ(t) . Here, Fγ(t) (t) is an T (t)F unknown matrix function which meets Fγ(t) γ(t) (t) ≤ I; Lγ(t) and Mγ(t) are known constant matrices. Now we are in a position to give the boundary conditions and initial conditions of systems (1) and (3), which are as follows: zi (t, x) = 0,
(t, x) ∈ [0, +∞) × ∂Ω,
(s, x) ∈ [−τ , 0] × Ω,
zi (s, x) = ϕi (s, x),
(7) (8)
where ϕi (s, x) ∈ L2F0 ([−¯ τ , 0] × Ω, Rn ), i = 1, 2. Set z1 (t, x) + z2 (t, x) = ζ(t, x) (= [ζ1 (t, x), · · · , ζn (t, x)]T ). Then we are able to write the antisynchronization error system as Pl ∂ζ(t,x) ∂ − Aγ(t) − Bγ(t) Gγ(t) Kγ(t) + ∆Kγ(t) (t) ζ(t, x) dζ(t, x) = k=1 ∂xk Dk ∂xk (9) +W1γ(t) g(ζ(t, x)) + W2γ(t) g(ζ(t − τ (t), x)) dt + σ (t, ζ(t, x), ζ (t − τ (t), x)) dω(t), where g (ζ(t, x)) = f (ζ(t, x) − z1 (t, x)) + f (z1 (t, x)). The activation functions and the noise intensify matrix are assumed to meet the following conditions, which are widely used in the literature (see, e.g., [39–41]). Assumption 1 For any α1 , α2 ∈ R, α1 6= α2 , activation function fi (·) (i = 1, · · · , n) satisfies β1i ≤
fi (α1 ) − fi (α2 ) ≤ β2i , α1 − α2
where β1i and β2i are known constant scalars.
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Assumption 2 There exists positive scalars σ1 and σ2 such that tr σ T (t, ζ(t, x), ζ (t − τ (t), x)) σ (t, ζ(t, x), ζ (t − τ (t), x)) ≤ σ1 ζ T (t, x)ζ(t, x) + σ2 ζ T (t − τ (t), x) ζ (t − τ (t), x) .
(10)
Noticing that f (z(t, x)) is an odd function, one can obtain from Assumption 1 that β1i ≤
gi (ζi (t, x)) ≤ β2i , i = 1, · · · , n. ζi (t, x)
(11)
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At the end of this section, let us give the definition on the exponential anti-synchronization and prepare several useful lemmas: Definition 1 Semi-Markov jump DRDNNs (1) and (3) are said to be exponentially anti-synchronized in mean square if the resulting error system in (9) is mean-square exponentially stable, i.e., there exist positive constants κ and H such that Z E |ζ(t, x)|2 dx ≤ He−κt . Ω
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Lemma 1 [42] Let Wa and Wb be two symmetric matrices of the same dimension. If Wb ≥ 0, then λmin (Wa )tr(Wb ) ≤ tr(Wa Wb ) ≤ λmax (Wa )tr(Wb ). Lemma 2 [43] Let z(x) ∈ C 1 (Ω, R) with z(x)/∂Ω = 0, then Ω
2
z (x)dx ≤
ψk − φk π
2 Z
∂z(x) ∂xk
2
dx,
k = 1, · · · , q.
lP repro of
Z
Ω
Lemma 3 [44] Suppose that (11) is satisfied. Then
ζ T (t, x)T1 RT2 ζ(t, x) − ζ T (t, x)R(T1 + T2 )g(ζ(t, x)) + g T (ζ(t, x))Rg(ζ(t, x)) ≤ 0
holds true for any diagonal matrix R > 0, where T1 = diag (β11 , · · · , β1n ) and T2 = diag (β21 , · · · , β2n ). Lemma 4 [45] For any real matrices Wa and Wb of suitable dimensions, there exists a constant ε > 0 such that S (Wa Wb ) ≤ ε−1 Wa WaT + εWbT Wb . Lemma 5 [46] Suppose there are a real constant ς and real matrices Θ, Ui , Vi , and Si (i = 1, · · · , p) satisfying " # Θ UV < 0, ∗ S where UV = [U1 + ςV1 , · · · , Up + ςVp ], and S =diag{S (−ςS1 ), · · · , S (−ςSp )}. Then, one can obtain Θ+
p X i=1
S (Ui Si−1 ViT ) < 0.
Lemma 6 [47] Let Θ, F, Wa , and Wb be real matrices of suitable dimensions. Then Θ + S (Wa FW b ) < 0
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holds for F T F ≤ I if and only if there is a scalar ε > 0 such that
Θ + ε−1 Wa WaT + εWbT Wb < 0.
3
Mean-square stability analysis
When uγ(t) (t, x) ≡ 0, system (9) can be rewritten as the following stochastic semi-Markov jump system:
where
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dζ(t, x) = hγ(t) (t, ζ(t, x), ζ(t − τ (t), x))dt + σ (t, ζ(t, x), ζ (t − τ (t), x)) dω(t),
hγ(t) (t, ζ(t, x), ζ(t − τ (t), x)) q X ∂ ∂ζ(t, x) = Dk − Aγ(t) ζ(t, x) + W1γ(t) g(ζ(t, x)) + W2γ(t) g(ζ(t − τ (t), x)). ∂xk ∂xk k=1
For system (12), we give a sufficient condition in the following theorem: 6
(12)
(13)
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Theorem 1 System (12) is exponentially stable in mean square, if, for any i ∈ M, there exist scalars ρi > 0, matrices Q1 > 0, Q2 > 0, diagonal matrices Pi , R1 > 0, and R2 > 0 such that the following LMIs hold:
where [i]
Ξ11 = −2Pi Dπ − S (Pi Ai ) + [i]
[i]
ξ¯ij Dπ
< 0,
m X j=1
(14)
(15)
ξ¯ij Pj + Q1 + ρi σ1 I − T1 R1 T2 ,
R1 (T1 + T2 ) , 2 = (µ − 1)Q1 − T1 R2 T2 + ρi σ2 I, Z ∞ = E {ξij (h)} = ξij (h)ei (h)dh, ) ( q 0 2 2 q X X π π D1k , · · ·, Dnk . = diag ψk − φk ψk − φk
Ξ13 = Pi W1i + Ξ22
lP repro of
Pi ≤ ρi I, [i] [i] Ξ 0 Ξ13 Pi W2i 11 R2 (T1 +T2 ) ∗ Ξ[i] 0 22 2 Ξi = ∗ ∗ Q − R1 0 2 ∗ ∗ ∗ (µ − 1)Q2 − R2
k=1
k=1
Proof. For any t ≥ 0, by Lemma 1 one can write that tr σ T (t, ζ(t, x), ζ (t − τ (t), x)) Pi σ (t, ζ(t, x), ζ (t − τ (t), x)) = tr Pi σ (t, ζ(t, x), ζ (t − τ (t), x)) σ T (t, ζ(t, x), ζ (t − τ (t), x)) ≤ λmax (Pi ) tr σ T (t, ζ(t, x), ζ (t − τ (t), x)) σ (t, ζ(t, x), ζ (t − τ (t), x)) .
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It follows by (10) and (14) that tr σ T (t, ζ(t, x), ζ (t − τ (t), x)) Pi σ (t, ζ(t, x), ζ (t − τ (t), x))
≤ ρi σ1 ζ T (t, x)ζ(t, x) + ρi σ2 ζ T (t − τ (t), x) ζ (t − τ (t), x) , i = 1, · · · , n.
(16)
Utilizing the integration by parts formula and noticing the Dirichlet boundary condition in (7), for any i ∈ {1, · · · , n} and k ∈ {1, · · · , q}, one has Z Z ∂ ∂ζi (t, x) ∂ζi (t, x) 2 Dik dx = − Dik dx, ζi (t, x) ∂xk ∂xk ∂xk Ω
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Ω
which, together with Lemma 2, suggests 2 Z Z ∂ ∂ζi (t, x) π ζi (t, x) Dik dx ≤ − Dik ζi2 (t, x)dx. ∂xk ∂xk ψk − φk Ω
Ω
Thus, for diagonal matrix Pi , the following inequality can also be established: Z Z q X ∂ζ(t, x) ∂ 2 ζ T (t, x)Pi Dk dx ≤ −2 ζ T (t, x)Pi Dπ ζ(t, x)dx. ∂xk ∂xk Ω
k=1
Ω
7
(17)
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Furthermore, let Γ = diag{β1 , · · · , βn } with βi = max{|β1i |, |β2i |}, i = 1, · · · , n. Then, from (11) one has
T
g T (ζ(t, x)) g (ζ(t, x)) ≤ ζ T (t, x)Γ2 ζ(t, x),
T
(18)
ζ (t, x)T1 R1 T2 ζ(t, x) − ζ (t, x)R1 (T1 + T2 ) g (ζ(t, x)) T
lP repro of
+g T (ζ(t, x)) R1 g (ζ(t, x)) ≤ 0,
(19)
ζ (t − τ (t), x)T1 R2 T2 ζ(t − τ (t), x)
−ζ T (t − τ (t), x)R2 (T1 + T2 ) g (ζ(t − τ (t), x))
+g T (ζ(t − τ (t), x)) R2 g (ζ(t − τ (t), x)) ≤ 0,
(20)
where the second and third inequalities follow from Lemma 3. Next, define a mode-dependent LF as
Vγ(t) (t, ζ(t, x)) = V1γ(t) (t, ζ(t, x)) + V2γ(t) (t, ζ(t, x)), where V1γ(t) (t, ζ(t, x)) =
Z
ζ T (t, x)Pγ(t) ζ(t, x)dx,
Ω
V2γ(t) (t, ζ(t, x)) =
Z
(21)
Zt
T
ζ (ν, x)Q1 ζ(ν, x)dνdx +
Ω t−τ (t)
Then, it follows that
λmin (Pγ(t) )
Z
Zt
g T (ζ(ν, x)) Q2 g (ζ(ν, x)) dνdx.
Ω t−τ (t)
Z
ζ T (t, x)ζ(t, x)dx
Ω
≤ Vγ(t) (t, ζ(t, x)) Z ≤ λmax (Pγ(t) ) ζ T (t, x)ζ(t, x)dx
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Ω
+λmax (Q1 )
Z
Zt
ζ T (ν, x)ζ(ν, x)dνdx
Z
Zt
g T (ζ(ν, x)) g (ζ(ν, x)) dνdx.
Ω t−τ (t)
+λmax (Q2 )
(22)
Ω t−τ (t)
where
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With the aid of the extended Itˆo’s formula [48], one can write E{eκt Vγ(t) (t, ζ(t, x))} = E{Vγ(t) (0, ζ(0, x))} t Z +E eκs [κVγ(s) (s, ζ(s, x)) + LVγ(s) (s, ζ(s, x))]ds , 0
1 E Vγ(s+θ) (s + θ, ζ(s + θ, x))| (s, γ(s), ζ(s, x)) − Vγ(s) (s, ζ(s, x)) . LVγ(s) (s, ζ(s, x)) = lim θ→0+ θ 8
(23)
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According to (2), for γ(s) = i one has LV1i (s, ζ(s, x)) ( m X 1 = lim Pr{γl+1 = j, hl+1 ≤ h + θ|γl = i, hl+1 > h}V1j (s + θ, ζ(s + θ, x)) + θ→0 θ
lP repro of
j=1,j6=i
)
+ Pr{γl+1 = i, hl+1 > h + θ|γl = i, hl+1 > h}V1i (s + θ, ζ(s + θ, x)) − V1i (s, ζ(s, x))
1 = lim θ→0+ θ
(
m X
[ξij (h)θ + o(θ)] V1j (s + θ, ζ(s + θ, x))
j=1,j6=i
)
+ [1 + ξii (h)θ + o(θ)] V1i (s + θ, ζ(s + θ, x)) − V1i (s, ζ(s, x)) =
m X
ξij (h)V1j (s, ζ(s, x)) + lim
θ→0+
j=1
Noting that
1 V1i (s + θ, ζ(s + θ, x)) − V1i (s, ζ(s, x)) . θ
(24)
1 E V1i (s + θ, ζ(s + θ, x)) − V1i (s, ζ(s, x)) θ ∂ ∂ = V1i (s, ζ(s, x)) + V1i (ζ(s, x))hi (s, ζ(s, x), ζ(s − τ (s), x)) ∂s ∂ζ 1 ∂2 + tr σ T (s, ζ(s, x), ζ (s − τ (s), x)) 2 V1i (ζ(s, x))σ (s, ζ(s, x), ζ (s − τ (s), x)) , 2 ∂ζ lim
θ→0+
one obtains from (13) and (24) that
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E LV1i (s, ζ(s, x)) Z X Z q m X ∂ζ(s, x) ∂ Dk dx = ξ¯ij ζ T (s, x)Pj ζ(s, x)dx + 2 ζ T (s, x)Pi ∂xk ∂xk j=1 k=1 Ω Ω Z T +2 ζ (s, x)Pi − Ai ζ(s, x) + W1i g(ζ(s, x)) + W2i g(ζ(s − τ (s), x)) dx ZΩ + tr σ T (s, ζ(s, x), ζ (s − τ (s), x)) Pi σ (s, ζ(s, x), ζ (s − τ (s), x)) dx. Ω
By (16) and (17), it follows that
E LV1i (s, ζ(s, x)) ≤
Z
Ω
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m X ξ¯ij Pj + ρi σ1 I ζ(s, x)dx ζ T (s, x) − 2Pi Dπ − 2Pi Ai +
+2
+2
Z
j=1
ζ T (s, x)Pi W1i g (ζ(s, x)) dx
Ω
Z
Ω
+ρi σ2
ζ T (s, x)Pi W2i g(ζ(s − τ (s), x))dx Z
ζ T (s − τ (s), x)ζ(s − τ (s), x)dx.
Ω
9
(25)
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Similarly, the following inequality can be derived: Z Z T E LV2i (s, ζ(s, x)) = ζ (s, x)Q1 ζ(s, x)dx − (1 − τ˙ (s)) ζ T (s − τ (s), x) Q1 ζ (s − τ (s), x) dx Ω
Z
+
g T (ζ (s, x)) Q2 g (ζ (s, x)) dx
Ω
Z
lP repro of
Ω
− (1 − τ˙ (s))
g T (ζ (s − τ (s), x)) Q2 g (ζ (s − τ (s), x)) dx.
(26)
Ω
From (19)-(21), (25), and (26) one can write E LVi (s, ζ(s, x)) ≤
Z
Ω
+
m X ζ T (s, x) − 2Pi Dπ − 2Pi Ai + ξ¯ij Pj + Q1 + ρi σ1 I − T1 R1 T2 ζ(s, x)dx
Z
Ω
+2 +
Z
ZΩ
Ω
+
Z
j=1
ζ T (s, x) 2Pi W1i + R1 (T1 + T2 ) g (ζ(s, x)) dx ζ T (s, x)Pi W2i g(ζ(s − τ (s), x))dx
ζ T (s − τ (s), x) ρi σ2 I + (µ − 1) Q1 − T1 R2 T2 ζ (s − τ (s), x) dx ζ T (s − τ (s), x) R2 (T1 + T2 ) g (ζ (s − τ (s), x)) dx
Ω
+
Z
g T (ζ (s, x)) (Q2 − R1 ) g (ζ (s, x)) dx
Ω
+ =
Z
Z
Ω
g T (ζ (s − τ (s), x)) (µ − 1) Q2 − R2 g (ζ (s − τ (s), x)) dx
¯ x)dx ζ¯T (s, x)Ξi ζ(s, Z
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Ω
≤ λmax (Ξi )
¯ x)dx, ζ¯T (s, x)ζ(s,
(27)
Ω
where
T ζ¯T (s, x) = ζ T (s, x) ζ T (s − τ (s), x) g T (ζ (s, x)) g T (ζ (s − τ (s), x)) .
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Now, by (18), (22), (23), and (27), one has (Z κt
min λmin (Pi )e E
|ζ(t, x)| dx
1≤i≤n
≤
Ω
(
max E λmax (Pi )
1≤i≤n
(
+E λmax (Q2 )
Z
T
)
(
ζ (0, x)ζ(0, x)dx + E λmax (Q1 )
Ω
Z
2
Z0
)
g T (ζ(ν, x)) g (ζ(ν, x)) dνdx
Ω −τ (0)
10
Z
Z0
Ω −τ (0)
T
)
ζ (ν, x)ζ(ν, x)dνdx
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+ max E 1≤i≤n
+E
( Z Zt
κs
)
T
e κλmax (Pi )ζ (s, x)ζ(s, x)dsdx
Ω 0
( Z Zt Zs
κs
)
T
e κλmax (Q1 )ζ (ν, x)ζ(ν, x)dνdsdx
Ω 0 s−τ (s) κs
T
e κλmax (Q2 )g (ζ(ν, x)) g (ζ(ν, x)) dνdsdx
Ω 0 s−τ (s)
+ max λmax (Ξi ) E 1≤i≤n
≤
max H1i E
1≤i≤n
Z Zt Ω 0
)
lP repro of
+E
( Z Zt Zs
( Z Zt Ω 0
κs
e
eκs ζ T (s, x)ζ(s, x)dsdx
2
)
|ζ(s, x)| dsdx + max H2i E
where
1≤i≤n
(Z
Ω
2
)
sup |ϕ1 (ν, x) + ϕ2 (ν, x)| dx ,
−¯ τ ≤ν≤0
H1i = κλmax (Pi ) + (eκ¯τ − 1) λmax (Q1 ) + λmax (Q2 )Γ2 + λmax (Ξi ) ,
H2i = λmax (Pi ) + τ¯λmax (Q1 ) + τ¯λmax (Q2 )Γ2 + λmax (Q1 ) + λmax (Q2 )Γ2
(28)
1 κ¯τ (e − 1)(1 − e−κ¯τ ). κ
Noting (15), one can choose κ > 0 small enough such that H1i < 0 (i = 1, · · · , m). Let R sup |ϕ1 (ν, x) + ϕ2 (ν, x)|2 dx max1≤i≤n H2i E τ ≤ν≤0 Ω −¯ H0 = . min1≤i≤n λmin (Pi ) Then, (28) yields
E
Z
|ζ(t, x)|2 dx
Ω
≤ H0 e−κt .
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Therefore, by Definition 1, system (12) is exponentially stable in mean square. In this way, the proof is completed.
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Remark 3 Compared with the reported results on semi-Markovian jump systems, the criterion given in Theorem 1 can be applied to test the stability of stochastic partial functional differential equations. It is also worth mentioning that, in the proof of Theorem 1, the derivation of infinitesimal operator LVγ(s) is directly based on the transition probabilities of the semi-Markov process, which is more clear and concise than existing ways in the literature (see, e.g., [22–24]). When the sojourn time obeys exponential distribution, the transition rates are independent of h (i.e., ξij (h) ≡ ξij ). In this case, the semi-Markov process reduces to a normal Markov process. And we can write the following result: Corollary 1 System (12) with Markov jump parameters is exponentially stable in mean square, if, for any i ∈ M, there are scalars ρi > 0, matrices Q1 > 0, Q2 > 0, diagonal matrices Pi > 0, R1 > 0, and R2 > 0, such that (14) and (15) hold. 11
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4
lP repro of
Remark 4 The stochastic stability of system (12) with Markov jump parameters was also considered in [33]. Comparing the criterion in Corollary 1 with that in [33] (see Theorem 1 therein), by adopting similar derivation procedures as those in [49] it is not difficult to show that, the former, which involves fewer decision variables, is a necessary but not sufficient condition for the latter. Thus, the present analysis method is theoretically less conservative than that reported in [33].
Anti-synchronization control synthesis
Based on Theorem 1, one can develop a strategy for the resilient fault-tolerant anti-synchronization controller design, which is given by the following theorem: Theorem 2 Given a constant ς > 0, semi-Markov jump DRDNNs (1) and (3) are exponentially antisynchronized in mean square under the resilient fault-tolerant control, if, ∀i ∈ M, there exist scalars ρi > 0, ε1 > 0, ε2 > 0, matrices Q1 > 0, Q2 > 0, Xi , Yi , diagonal matrices Pi > 0, R1 > 0, R2 > 0, such that (14) and the following LMI hold: [i] [i] [i] [i] [i] Φ11 0 Φ13 Pi W2i Φ15 Φ16 Φ17 [i] ∗ Φ22 0 Φ24 0 0 0 ∗ ∗ Φ33 0 0 0 0 (29) ∗ ∗ Φ44 0 0 0 < 0, ∗ [i] T ∗ ∗ ∗ ∗ Φ55 ε2 Li 0 [i] ∗ ∗ ∗ ∗ ∗ Φ66 0 ∗ ∗ ∗ ∗ ∗ ∗ −ε2 I where [i]
Φ11 = −2Pi Dπ − S (Pi Ai − Bi G0i Yi ) + [i]
Φ44 ξ¯ij
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[i]
j=1
R1 (T1 + T2 ) [i] [i] , Φ15 = Pi Bi G0i Li , Φ16 = Pi Bi G0i − Bi G0i Xi + ςYiT , 2 R2 (T1 + T2 ) [i] = Pi Bi G1i , Φ22 = (µ − 1)Q1 − T1 R2 T2 + ρi σ2 I, Φ24 = , Φ33 = Q2 − R1 , 2 [i] [i] = (µ − 1)Q2 − R2 , Φ55 = −ε1 I + ε2 LiT Li , Φ66 = −S (ςXi ) + ε2 I, ) ( 2 2 Z ∞ q q X X π π D1k , · · · , Dnk . = E {ξij (h)} = ξij (h)ei (h)dh, Dπ = diag ψk − φk ψk − φk 0
Φ13 = Pi W1i + Φ17
m X ξ¯ij Pj + Q1 + ρi σ1 I − T1 R1 T2 + ε1 MiT Mi ,
k=1
k=1
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Moreover, the desired feedback gain of (6) can be given by Ki = Xi−1 Yi , i ∈ M.
(30)
Proof : By Theorem 1, the mean-square exponential anti-synchronization of system (1) and system (3) under the resilient fault-tolerant control is able to be realized if (14) and [i] [i] ¯ Ξ 0 Ξ13 Pi W2i 11 R2 (T1 +T2 ) ∗ Ξ[i] 0 22 2 <0 (31) ∗ ∗ Q2 − R1 0 ∗ ∗ ∗ (µ − 1)Q2 − R2 12
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are satisfied, where ¯ [i] = Ξ[i] + S (Pi Bi Gi (Ki + ∆Ki (t))) . Ξ 11 11 Noting ∆Ki (t) = Li Fi (t)Mi and FiT (t)Fi (t) ≤ I, one has from Lemma 4 that
lP repro of
T T S (Pi Bi Gi ∆Ki (t)) ≤ ε−1 1 (Pi Bi Gi Li ) (Pi Bi Gi Li ) + ε1 Mi Mi .
(32)
In addition, in the light of (30), one can write
S (Pi Bi Gi Ki ) = S Bi G0i Yi + (Pi Bi Gi −Bi G0i Xi ) Xi−1 Yi .
(33)
By (32) and (33), it is obvious that (31) holds true if
[i] Ξi + diag{Πi , 0, 0, 0} + S Ui Xi−1 ViT < 0,
where [i]
Πi
Ui
(34)
T = S (Bi G0i Yi ) + ε1 MiT Mi + ε−1 1 (Pi Bi Gi Li ) (Pi Bi Gi Li ) , T = (Pi Bi Gi −Bi G0i Xi )T , 0, 0, 0 ,
Vi = [Yi , 0, 0, 0] T .
Let
¯ [i] = Pi Bi Gi Li , Φ 15
¯ [i] = Pi Bi Gi − Bi G0i Xi + ςY T . Φ i 16
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Then, according to Lemma 5 and Schur’s complement, (34) is ensured by [i] [i] ¯ [i] ¯ [i] Φ11 0 Φ13 Pi W2i Φ Φ 15 16 ∗ Φ[i] 0 Φ24 0 0 22 ∗ ∗ Φ33 0 0 0 < 0, ∗ ∗ ∗ Φ 0 0 44 ∗ ∗ ∗ ∗ −ε1 I 0 ∗ ∗ ∗ ∗ ∗ −S (ςXi )
which, by (5), can be re-expressed as
Φi + S (Wai Λi Wbi ) < 0,
=
Φi
Wai Wbi
[i]
Φ11 ∗ ∗ ∗ ∗ ∗
0
[i]
[i]
Φ13 Pi W2i
Φ22 0 ∗ Φ33 ∗ ∗ ∗ ∗ ∗ ∗
Φ24 0 Φ44 ∗ ∗
[i]
Φ15
[i]
Φ16
0 0 0 0 0 0 −ε1 I 0 ∗ −S (ςXi ) h iT = , (Pi Bi G1i )T , 0, 0, 0, 0, 0 h i = 0, 0, 0, 0, Li , I .
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where
(35)
13
,
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Noting ΛTi Λi ≤ I, by Lemma 6, (35) holds true if the following inequality is satisfied: T T Φi + ε2−1 Wai Wai + ε2 Wbi Wbi < 0,
which, by Schur’s complement again, can be re-organized as (29). This completes the proof of the theorem.
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Remark 5 During the design of the desired resilient fault-tolerant anti-synchronization controller, highorder nonlinearities appear as a result of the couplings of the Lyapunov matrices, the gain perturbation matrices, and the uncertainties of the actuator failures. How to deal with such high-order nonlinearities is not easy. In this paper, based on the analysis result and using several decoupling approaches, a strategy is developed in Theorem 2, where the desired gains can be obtained via solving several LMIs, which are checked easily by the computing software MATLAB. Considering the actuator failure problem and the gain perturbation problem separately, the following results can be established: Corollary 2 Suppose that there are no actuator failures; i.e., Gi = G0i = I (i = 1, · · · , n). Then, for a given constant ς > 0, semi-Markov jump DRDNNs (1) and (3) are exponentially anti-synchronized in mean square under the resilient control, if, ∀i ∈ M, there exist scalars ρi > 0, ε1 > 0, matrices Q1 > 0, Q2 > 0, Xi , Yi , diagonal matrices Pi > 0, R1 > 0, R2 > 0, such that (14) and the following LMI hold: [i] [i] [i] Φ16 Φ11 0 Φ13 Pi W2i Pi B2i Li ∗ Φ[i] 0 Φ24 0 0 22 ∗ ∗ Φ33 0 0 0 < 0, ∗ ∗ ∗ Φ44 0 0 ∗ ∗ ∗ ∗ −ε1 I 0 ∗ ∗ ∗ ∗ ∗ −S (ςXi ) [i]
[i]
[i]
[i]
where Φ11 , Φ13 , Φ16 , Φ22 , Φ24 , Φ33 , and Φ44 are the same as those in Theorem 2. In this case, the required feedback gain can be provided by Ki = Xi−1 Yi (i ∈ M).
where
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Corollary 3 Suppose that there are no gain perturbations; i.e., ∆Ki (t) = 0 (i = 1, · · · , n). Then, for a given constant ς > 0, semi-Markov jump DRDNNs (1) and (3) are exponentially anti-synchronized in mean square under the fault-tolerant control, if, ∀i ∈ M, there exist scalars ρi > 0, ε1 > 0, matrices Q1 > 0, Q2 > 0, Xi , Yi , diagonal matrices Pi > 0, R1 > 0, R2 > 0, such that (14) and the following LMI hold: [i] [i] [i] ˜ Φ 0 Φ13 Pi W2i Pi Bi G1i Φ16 11 ∗ Φ[i] 0 Φ24 0 0 22 ∗ ∗ Φ33 0 0 0 < 0, ∗ ∗ ∗ Φ 0 0 44 ∗ ∗ ∗ ∗ −ε1 I 0 [i] ˜ ∗ ∗ ∗ ∗ ∗ Φ66 ˜ [i] = −2Pi Dπ − S (Pi Ai − Bi G0i Yi ) + Φ 11 [i] Φ13 ,
[i] Φ16 ,
m X j=1
˜ [i] = −S (ςXi ) + ε1 I, ξ¯ij Pj + Q1 + ρi σ1 I − T1 R1 T2 , Φ 66
[i] Φ22 ,
and Φ24 , Φ33 , Φ44 , and ξ¯ij are the same as those in Theorem 2. In this case, the required feedback gain can be provided by Ki = Xi−1 Yi (i ∈ M). 14
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5
Numerical Examples
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In the section, two examples are given to illustrate the superiority of the stability analysis method and applicability of the resilient fault-tolerant anti-synchronization control strategy, respectively. Example 1. Consider unforced Markov jump system (9) (i.e., system (12) ) with two modes, where the parameters are provided by " # " # " # 0.5 0.5 0.5 0 0.4 0 D = , A1 = , A2 = , 0.3 0.7 0 0.5 0 0.4 " # " # 0.5 0.3 0.1 0.2 W11 = , W21 = , 0.2 0.1 0.3 0.4 " # " # 0.5 0.6 0.2 0.3 W12 = , W22 = , 0.4 0.2 0.4 0.5 ξ11 = −0.6, ξ12 = 0.6, ξ21 = 0.4, ξ22 = −0.4, 1 (|ζi + 1| − |ζi − 1|) (i = 1, 2), Ω = {x|x ∈ [−1, 1]} . gi (ζi ) = 2
Set µ = 0.05. Then, for the case that σ1 = 1.6, it is found that the criterion in Theorem 1 in [33] fails to check whether the system is stable or not since the LMIs therein have no feasible solutions ∀σ2 > 0, while by Corollary 1, it can be concluded that the system is exponentially stable in mean square for σ2 ∈ (0, 0.037]. And for the case that σ1 = 0.8, the maximum allowed σ2 ensuring the stability is calculated as 0.755 by Theorem 1 in [33] and 0.189 by Corollary 1, respectively. More detail comparisons between Theorem 1 in [33] and Corollary 1 in this paper for different values of σ1 are provided in Table 1, by which one can see the present analysis method is much less conservative than that proposed in [33], which further confirms the statement in Remark 4. Table 1: Comparisons of maximum allowed σ2 for different methods σ2 σ1 = 0.1 σ1 = 0.2 σ1 = 0.4 σ1 = 0.8 σ1 = 1.6 1.408 0.854
1.314 0.759
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Corollary 1 Theorem 1 of [33]
1.127 0.569
0.755 0.189
0.037 no solution
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Example 2. Consider two-mode semi-Markov jump DRDNNs given by (1) and (3) with " # " # 0.1 0 1.0 0 D = , A1 = , 0 0.1 0 1.0 " # " # 0.9 0 −2 1 A2 = , B1 = B2 = , 0 0.9 0 −2 " # " # 2.0 −0.1 −1.6 −0.1 W11 = , W21 = , −5.0 2.8 −0.3 −2.5 " # " # 0.5 0.6 0.2 0.3 W12 = , W22 = , 0.4 0.2 0.4 0.5 fi (·) = tanh(·) (i = 1, 2), Ω = {x|x ∈ [−2, 2]} , τ (t) = 1, σ(t, z1 (t, x) + z2 (t, x), z1 (t − τ (t), x) + z2 (t − τ (t), x))
= diag{z1 (t, x) + z2 (t, x), z1 (t − τ (t), x) + z2 (t − τ (t), x)}. 15
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The initial conditions are taken as z1 (s, x) =
"
1 −1
#
, z2 (s, x) =
"
−1 2
#
, (s, x) ∈ [−1, 0] × Ω,
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and the boundary conditions are set as the type of Dirichlet. Assume the sojourn time obeys the Weibull distribution. Specifically, let hl ∼ Weibull(2, 2) for i = 1 (i.e., e1 (h) = 0.5h exp(−(0.5h)2 )) and hl ∼ Weibull (1, 3) for i = 2 (i.e., e2 (h) = 3h2 exp(−h3 )) , respectively. Then we have " # " # ξ11 (h) ξ12 (h) −0.5h 0.5h = . ξ21 (h) ξ22 (h) 3h2 −3h2 By calculating the mathematical expectation we can get the transition rates as " # −0.8862 0.8862 E {ξ(h)} = . 2.7082 −2.7082 3
2.5
γ(t)
2
1.5
1
0.5
0
0
50
100
150
t
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Figure 1: Semi-Markov jump signal
Figure 2: State trajectories of the anti-synchronization error system without control 16
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Figure 3: State trajectories of the controlled anti-synchronization error system
Let us consider the following parameters concerning the actuator failures as well as gain perturbations: " # " # 0.2 0.1 0.2 0.1 L1 = , L2 = , 0 0.1 0 0.1 " # " # 0.02 0 0.01 0.1 M1 = , M2 = , 0 0.01 0 0.02 Fi (t) = sin(t), gˇji = 0.2, gˆji = 0.8 (i, j = 1, 2).
Then, by setting ς = 0.01 and utilizing MATLAB to solve the LMIs in Theorem 2, the desired control gain matrices can be obtained as " # " # 95.1945 4.3730 48.9157 25.2136 K1 = , K2 = . −55.3191 101.1654 −36.9121 78.3478
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5 4 3
z12 (t, −1.5)
2 1 0
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−1 −2 −3 −4
−5 −1
−0.5
0 z11 (t, −1.5)
0.5
Figure 4: Chaos behaviour
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15 z11 (t, −1.5)
10
z21 (t, −1.5)
5 0 −5 −10 −15
0
20
40
60
80
100
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t
20
z12 (t, −1.5)
10 0 −10 −20 0
20
40
60
z22 (t, −1.5)
80
100
t
Figure 5: State trajectories of the unforced master-slave DRDNNs
The semi-Markov jump signal is given in Figure 1. Under the switching signal, the state trajectories of the anti-synchronization error system without control are depicted in Figure 2, while, with the application of the designed controller, the desired fast convergence of the anti-synchronization error system is shown in Figure 3. 2
z11 (t, −1.5)
1 0 −1 −2
0
20
40
60
z21 (t, −1.5)
80
100
t
10
z12 (t, −1.5)
5
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0
z22 (t, −1.5)
−5
−10
0
20
40
60
80
100
t
Figure 6: State trajectories under control
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In order to illustrate the control effect more clearly, let us consider the case that x = −1.5. In such a case, the phase-plane plot of the master DRDNN is given in Figure 4, which shows a chaotic behavior. The state trajectories for the unforced master-slave DRDNNs and the controlled master-slave DRDNNs are depicted in Figure 5 and Figure 6, respectively. From the simulations, it can be observed that the designed resilient fault-tolerant controller works well for the anti-synchronization purpose.
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6
Conclusions
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The anti-synchronization control problem for stochastic DRDNNs with semi-Markov jump parameters has been studied in the presence of actuator failures as well as gain perturbations. First, by using a modedependent LF and some stochastic analysis methods, a sufficient condition has been derived to ensure the resulting anti-synchronization error system to be exponentially stability in mean square. It has been shown that the obtained criterion improves a previously reported result. Then, based on the present analysis result and employing some decoupling techniques, a strategy for the design of a resilient fault-tolerant anti-synchronization controller has been developed. Finally, two numerical examples have been given to verify the superiority of the stability analysis method and applicability of the resilient fault-tolerant antisynchronization control strategy, respectively.
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Conflict of Interest
Conflicts of Interest Statement
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Manuscript title: Resilient fault-tolerant anti-synchronization for stochastic delayed reactiondiffusion neural networks with semi-Markov jump parameters
The authors whose names are listed immediately below certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.
Author names: Jianping Zhou Yamin Liu Jianwei Xia Zhen Wang
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Sabri Arik