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.
I. INTRODUCTION
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: and represent n-dimensional real vectors and n × n real matrices, respectively. and represent n-dimensional complex vectors and n × n complex matrices, respectively. For , the real and imaginary parts of x are represented by x1 and x2, respectively. For , and denote the real and imaginary parts of , respectively. For , X1 and X2 denote the real and imaginary parts of matrix X, respectively. The minimum eigenvalue of matrix X is denoted as . 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 to represent the sum of matrix X and its transpose, and * to represent the transpose caused by the block matrix.
II. MODEL DESCRIPTIONS
Consider a class of delayed CVNNs as follows:
And the controller to be designed is given by the form of output feedback:29–32
In (1) and (2), is the state vector, is the output vector, and is the control input vector; denotes the vector-valued activation function; , ; , are parameter matrices; τ denotes the time delay. The actuator failure model is described as33
where is the input signal of the jth actuator, and is an unknown scalar. Note the Fj reflects the actuator effectiveness.
Denote
Then can be expressed by a concise form as
III. MAIN RESULTS
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.
IV. SIMULATION EXAMPLE
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.
V. CONCLUSIONS
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
ACKNOWLEDGMENTS
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.