The paper considers the issue of fault-tolerant output-feedback stabilization for complex-valued neural networks with both time delay and actuator failures. The aim is to design a fault-tolerant output-feedback controller to ensure the network to be asymptotically stable. By using the discretized Lyapunov-Krasovskii functional method as well as the free-weighting matrix approach, a delay-dependent stability criterion is proposed. Then, with the aid of some decoupling techniques, a method for the design of desired output-feedback fault-tolerant controller is developed. Finally, a numerical example is given to verify the effectiveness of the present stabilizing method.

In the last few decades, neural networks have received extensive attention owing to their potential applications in a variety of engineering fields including system identification, signal classification, optimization, and sequence recognition. For some applications such as optimization, the stability of the neural networks is the most fundamental issue.1 Time delays are always inevitable in the engineering implementation of a neural network, and their existence is found to be one of the sources of oscillation and instability.2–4 Therefore, stability analysis and stabilization design of time delays neural networks have been hot topics of research in the past few decades. A considerable number of results on these two topics have been reported in the literature.5–13 

Complex-valued neural networks (CVNNs) are a more general class of neural networks that allow complex-valued states, outputs, connection weights, and activation functions. When a neural network is designed to deal with ill-posed issues to reconstruct physical fields such as electromagnetic fields, sound fields, and light fields, the inputs and solution data of the network need to be expressed by complex values.14 In this case, it is suitable to introduce the CVNN models. In addition, with a wide range of applications of analytical signals, CVNNs have demonstrated the advantages in some areas where the signals need to be analyzed and handled in time, frequency, and phase space.15 It is therefore no surprise that many efforts have been devoted into the research of CVNNs with time delays. For example, Bao et al.16 considered the issue of synchronization for fractional-order CVNNs with time delays and developed a method for designing linear time delays feedback controllers. By using the stability theory of fractional-order differential equations and Laplace transforms, Wang et al.17 proposed new stability criteria for fractional-order time delays complex-valued single neuron model. For recent reports concerning the analysis and synthesis of time delays CVNNs, one can refer to Refs. 18–24.

It appears from the literature that most of the studies on synthesis are based on an assumption that all states of the time delays complex-valued neural networks are available, whereas it is known that a complete measurement of the state variables is often inconvenient or even infeasible in practice.25 Besides, it is found that the problem of actuator failures has not received enough attention. However, as noted in Refs. 26–28, actuator failures can happen for various causes, ranging from component malfunctions to structural faults, and they may be the main reason for some accidents.

Motivated by the above observations, in this paper we consider the issue of fault-tolerant output-feedback stabilization for a class of CVNNs with time delay as well as actuator failures. By using the discretized Lyapunov-Krasovskii functional (DLKF) method and combining with the free-weighting matrix approach, we propose a delay-dependent sufficient condition for ensuring the network to be asymptotically stable. Then, with the aid of some decoupling techniques, we develop a method for the design of an effective output feedback fault-tolerant controller. It is worth pointing out that the control gains can be achieved via solving a series of linear matrix inequalities, which is numerically efficient via the popular mathematical software Matlab. Finally, we give a numerical example to verify the effectiveness of the proposed stabilizing method. It is shown that the higher the degree of discretization, the lower the conservativeness of the results obtained.

Notation: Rn and Rn×n represent n-dimensional real vectors and n × n real matrices, respectively. Cn and Cn×n represent n-dimensional complex vectors and n × n complex matrices, respectively. For xCn, the real and imaginary parts of x are represented by x1 and x2, respectively. For f(x)Cn, f1x and f2x denote the real and imaginary parts of fx, respectively. For XCn×n, X1 and X2 denote the real and imaginary parts of matrix X, respectively. The minimum eigenvalue of matrix X is denoted as λminX. denote the Euclidean norm. τ represents time delay, and N and h represent number of divisions of the interval [−τ, 0] in discretization and length of each division, respectively. For the convenience of writing, we use HeX to represent the sum of matrix X and its transpose, and * to represent the transpose caused by the block matrix.

Consider a class of delayed CVNNs as follows:

(1)

And the controller to be designed is given by the form of output feedback:29–32 

(2)

In (1) and (2), xt=[x1t,x2t,,xnt]TCn is the state vector, yt=[y1t,y2t,,ynt]TCn is the output vector, and ut=[u1t,u2t,,unt]TCn is the control input vector; fxt=f1x1t,f2x2t,,fnxntTCn denotes the vector-valued activation function; ARn×n, W1Cn×n; W2Cn×n, DRn×n are parameter matrices; τ denotes the time delay. The actuator failure model is described as33 

where ujt is the input signal of the jth actuator, and FjR is an unknown scalar. Note the Fj reflects the actuator effectiveness.

Denote

Then uFt can be expressed by a concise form as

(3)

Assumption 1
The neuron activation functionfj.satisfies fj(0) = 0 and the following Lipschitz condition:34–36 
where lj > 0 is a constant, j = 1, 2, …, n.

Lemma 1
(Ref. 37) Suppose that the following condition concerning a real constant ρ and real matrices Φ, Uj, VjandWjj=1,2,,nholds:
Then one can obtain

Lemma 2
(Ref. 38) For any vector x,yRnand positive real constant κ, the following matrix inequality holds:
Now, we transform (1) into a real-valued neural network. Let x(t)=x1t+ix2t,yt=y1t+iy2t,W1=W11+iW12,W2=W21+iW22,fxt=f1x1t,x2t+if2x1t,x2t,fxtτ=f1x1tτ,x2tτ+if2x1tτ,x2tτ; and ut=u1t+iu2t, where i represents the imaginary unit, i.e., i=1. For the sake of brevity, we denote xj=xjt,yj=yjt,uFj=uFjt,uj=ujt,xjτ=xjtτ(j=1,2). Then, (1) and (2) can be separated into real and imaginary parts, respectively, as follows
(4)
(5)
By substituting (5) into system (4), one can get the closed-loop system as
(6)
where

Theorem 1
Under Assumption 1, system(6)is asymptotically stable if there are constants ηj > 0 and matrices Mj1, Mj2, Pj > 0, Qpj, Spj=SpjT,Rpqj=(Rpqj)T, j = 1, 2, p, q = 0, 1, …, N such that
(7)
(8)
where

Proof
Let us consider a Lyapunov-Krasovskii functional as
where Qjζ, Sjζ=SjT(ζ), Rjζ,η=RjTζ,η, hj = ẋj are continuous functions. The derivative of Vt along the system is given by
According to Assumption 1, we have
for η1, η2 > 0. We apply the DLKF method in Ref. 39: divide the delay interval τ,0 into N segments θp,θp1 of equal length h = τ/N, where θp=php=1,2,,N. This divides the square θp,θp1×θq,θq1 and each small square is further divided into two triangles. The continuous matrix functions Qjζ and Sjζ are taken to be linear within each segment and the continuous matrices Rjζ,η are taken to be linear within each triangular region. They can be rewritten in terms of their values at the dividing points by the linear interpolation approach, i.e.,
for 0 ≤ α ≤ 1, 0 ≤ β ≤ 1, j = 1, 2, p, q = 1, 2, …, N. Thus the DLKF is completely determined by Pj, Qpj, Spj, Rpj, j = 1, 2, p, q = 1, 2, …, N. Similar to Ref. 39, the DLKF condition Vt,xtj=12(μjxjt2) is satisfied for μj=λminPj>0, if Spj>0(j=1,2,p=0,1,,N) and (7) hold true. We note that for θp < ζ < θp−1, θq < η < θq−1,
For any n × n matrices Mj1 and Mj2(j = 1, 2), the following equations hold:
Thus
or
(9)
where
From (9), it is easy to verify that
(10)
where
for arbitrary matrix Uj(j = 1, 2) satisfying
(11)
Then, using Jensen’s inequality, we can have
(12)
Combining (10), (11) and (12), we can get
where
By (8), it is not difficult to write
Thus the system is asymptotically stable by means of the Lyapunov stability theory, and the proof is complete.

Theorem 2
Let ρ1, ρ2be given positive scalars. Then system(6)is asymptotically stable under Assumption 1, if there are positive constants ηj,εjand matrices Mj1, Mj2, Pj > 0,Qpj, Spj=SpjT, Rpqj=(Rpqj)T, j = 1, 2, p, q = 0, 1, …, N and X, Y such that inequality(7)and the following
(13)
hold true, where
and Δ12is the same as that in Theorem 1. In such a case, the desired gain is
(14)

Proof
Clearly, (8) can be rearranged as
(15)
where
According to (14), the following are established
(16)
Substituting (16) into (15), we can get
(17)
where
According to Lemma 1, (17) can be guaranteed by the following inequlity
(18)
where
By Lemma 2 and (3), it can be obtained that
(19)
where
Using (19), it can be seen that (18) is satisfied if the following holds true
which can be rewritten as (13). Therefore, it can be concluded that (13) guarantees (8). This completes the proof of the theorem.

By Lemmas 1 and 2, Theorem 2 presents a design method of the output-feedback fault-tolerant controller to stabilize the CVNN in (1) in terms of linear matrix inequalities. Recently, master-slave synchronization of nonlinear chaotic systems has drawn considerable attention due to their successful application in secure communication.40–42 CVNN with time delays, which are able to generate complicated chaotic attractors,43 can serve as suitable system models to implement the master-slave synchronization. The proposed stabilizing method can be used to ensure the asymptotic stability of the resulting synchronization error system in the presence of actuator failures.

Example 1.
Consider CVNN (1) with
Clearly, Assumption 1 is satisfied with L=0.39000.39. Fig. 1 shows the state of networks trajectories without control. It can be found that the unforced system is unstable.
Suppose the fault-tolerant controller is subject to F^=0.8000.8, F̌=0.2000.2. Then, it can be acquired that F0=0.5000.5, H=0.6000.6. Taking ρ1 = ρ2 = 0.15, N = 3, and solving the LMIs in Theorem 2, we can obtain
Fig. 2 depicts the state of networks trajectories under control. It is not difficult to find that the states rapidly converge to 0 under the controller. Therefore, the example verifies the effectiveness of the controller designed in the paper.

Furthermore, applying Theorem 2, one can estimate the maximum allowable time delay such that the CVNN retains stability. Table I shows the results for different N. It can be seen that, as N increases, the maximum allowable time delay that the CVNN can maintain asymptotic stability is also increasing.

FIG. 1.

State trajectories without control.

FIG. 1.

State trajectories without control.

Close modal
FIG. 2.

State trajectories with control.

FIG. 2.

State trajectories with control.

Close modal
TABLE I.

Upper bound of delay (ρ1 = ρ2 = 0.15).

N123456
τ 0.508 0.557 0.630 0.666 0.689 0.702 
N123456
τ 0.508 0.557 0.630 0.666 0.689 0.702 

The problem of fault-tolerant output-feedback stabilization for CVNNs with time delay and actuator failures has been studied in the paper. A delay-dependent stability criterion has been proposed by combining the DLKF method with the free-weighting matrix approach. On the basis of this, a constructive approach to the design of a fault-tolerant output feedback controller has been developed. The effectiveness of the developed stabilizing method has been verified by a numerical example. Future work in the field of stabilization for complex-valued networks will focus on finite-time control strategies44–48 with consideration of time-varying or neutral-type time delays.49–53 

This work was supported in part by the National Natural Science Foundation of China under Grant number 61503002, the Excellent Youth Talent Support Program of Universities in Anhui Province under Grant number GXYQZD2019021, and the Natural Science Foundation of the Anhui Higher Education Institutions under grant numbers KJ2015A130 and KJ2017A064.

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