Abstract
This paper considers exponential stabilization for a class of coupled hybrid stochastic delayed bidirectional associative memory neural networks (HSD-BAM-NN) with reaction-diffusion terms. A periodically intermittent controller is proposed to exponentially stabilize such an unstable HSD-BAM-NN, and sufficient conditions of the closed-loop BAM-NN system with exponential stabilization are derived by using Lyapunov-Krasovskii functional method, stochastic analysis techniques, and integral inequality property, which decide the basic parameters of the proposed controller. Furthermore, a framework to establish simulation algorithm with sampled states is presented to implement the stabilization controller. With a HSD-BAM-NN model of power synchronization in a photovoltaic (PV) array field, we illustrate numerical simulation results to verify the correctness and effectiveness of the proposed controller.
1. Introduction
Spurred by pioneering works on the neural networks models with BAM in Kosko [1–3], much attention has been paid to BAM-NN owning to its various applications in image processing, pattern recognition, automatic control [4, 5], associate memory, parallel computing [6], and optimization [7]. It is worth noting that the results of global stability for BAM models obtained in [1–3] require severe constraint conditions of symmetric connection weight matrix. In the neural networks with very large scale circuits, it is difficult for a practical NN system to satisfy the absolutely symmetric conditions in BAM models. Recently, the stability analysis of BAM neural networks has been paid considerable attention, and many stability conditions of such NN models have been reported in the published literature [8–22]. Since diffusion effects cannot be avoided in the neural networks, in a physical sense, when electrons are moving in asymmetric electromagnetic fields, it is more valuable to consider the alternative activation of neurons in an available space as well as in a given time interval. Therefore, the model of BAM neural networks could be formulated as partial differential equations (PDE) instead of only ordinary differential equations (ODE) [9, 11]. Based on PDE models, many contributions have been published focusing on the stability of BAM neural networks (BAM-NN) with reaction-diffusion terms by using Lyapunov functional method and LMI techniques [7, 9, 11–15, 17, 18, 23, 24].
Haykin [25] proposed that in real neural networks systems, synaptic transmission is a stochastic process with noise released by random electronic fluctuations of the neurotransmitters and other disturbance. Following the experimental conclusions, Hossain and Anagnostou [26] further stated that noise posed a basic problem for information processing which affected all aspects of nervous system function in a nervous system. Hence, it is also important to study the effects of noise perturbations existing in neutral networks dynamics. Literature [8] investigated the stability of stochastic NN for the first time, which guided followers exploring novel results of such problems, especially for stochastic NN with reaction-diffusion terms, and some striking published contributions can be found in [7, 11–15, 17, 18, 27].
It is worth mentioning that the hybrid BAM-NN driven by continuous-time Markov chains have been used to model many practical systems where they may experience abrupt changes in their structure and parameters. To consider the issues of system structures’ abrupt changes, hybrid disturbance, and unreliable subsystem interconnections, the evolution dynamics of BAM-NN could be modeled as jump systems [28, 29]. As Markovian jump system was first introduced in [10], two fundamental components have formed original ideas of considering a jumping system as continuous state described by differential equations, and discrete-time state described by a continuous-time finite-state Markovian process [28–32]. For theory and techniques recently developed to analyze BAM-NN with Markovian jump factors, here we mention [11, 22, 24, 28, 29].
In recent literature, a variety of approaches have been published for the stabilization control of BAM neural networks with or without delays and reaction-diffusion terms which include feedback control [9, 21, 23, 32, 33], impulsive control [22, 24, 34], and intermittent control [35–38]. The type of control considered in this paper is intermittent control, which was first introduced to control nonlinear dynamical systems in [39] and has aroused much interest of researchers due to its merits in engineering applications. Different from continuous control approaches, intermittent control is more effective because the system output is measured intermittently rather than continuously [35]. Some novel contributions of intermittent control [35, 36, 38] based on the stability analysis of BAM-NN have been achieved in recent years. Nevertheless, the complex effects on BAM-NN with stimulation time-varying delays and stochastic reaction-diffusion terms have not been considered to use intermittent control in published results.
Motivated by the above discussion, the main purpose of this paper is to investigate BAM-NN with delays and reaction-diffusion terms and focus on its exponential stabilization by designing a periodically intermittent controller [35, 36, 38]. The main novelties in this paper can be highlighted as follows: Firstly, the model under consideration covers the frequently investigated models which often are characterized by special cases in structures or systematic functions. Secondly, most available results have been concerned with the stability problems, and in this technical note, stochastic stabilization is taken into account and a periodically intermittent controller with more flexible conditions is first proposed for such BAM-NN with stochastic reaction-diffusion terms. Finally, a numerical simulation method is designed to simulate the behavior of the time-varying delays stochastic hybrid partial differential equations, which enriches the theory of delayed stochastic partial differential equation.
Throughout the paper, we take as a complete probability space with a filtration satisfying the usual conditions; i.e., is right-continuous and contains all -null sets. Let , , be one-dimensional Brownian motion defined on the probability space. Let , , be right-continuous Markov chain on the probability space taking values in a finite-state space with generator . And the transition probability from model at time to model at time is where and . Here, , , is the transition probability from model to model and . We assume that the Markovian chain is independent of the Brownian motion . It is well known that almost every sample path of is right-continuous step function with a finite number of simple jumps in any finite subinterval .
2. Preliminaries and Problem Formulation
In this paper, we consider the hybrid stochastic BAM neural network model with reaction-diffusion terms [11–13, 15], which are formulated as a couple of neurons’ dynamics indexed by a pair superscripts of where denote the index of a couple of neuron networks (NN) with and neurons, respectively; i.e., or . The notations and are, respectively, the indices of the neurons in the th NN and the th NN. is the dimension of space variable vector , and , where is a compact set and measurable with smooth boundary in space , . The definition of is the same as and we let . and are the states of the th neuron in th NN and the th neuron in th NN at time and in space of , , respectively. For simplicity, we define , , and to denote the state vectors of the th, th, and whole coupled neural networks, respectively. denotes the time delay satisfying with constant . denotes the activation function of the th neuron in th NN stimulating the th neurons in the th NN. denotes changing rate of the th neuron under the condition that neural network is disconnected and no external additional activation exists. is the connection weight of neurons in coupled NN and denotes the corresponding delayed connection weight. The smooth function is the transmission diffusion operator of neurons, is the stochastic disturbance function of neurons, and is the Brownian motions as noise acting on the transmission from the th neuron to the th neuron.
According to [9, 10], we present the following initial value condition for (2):which is with Dirichlet boundary value.For convenience, we denote all the possible models of BAM-NN as , , , and .
Definition 1. A stochastic vector is the solution of system (2)-(4) if it satisfies the following conditions:
(i) is adapted to ,
(ii) for , (iii) for ,
Definition 2. The HSD-BAM-NN (2)-(4) is exponentially stable in th moment if, for arbitrary model combinations of , , , the states of the system satisfy where , . To ensure the existence and uniqueness of the solution to system (2)-(4), we suppose the following assumptions.
Assumption 3. For , arbitrary , the activation function is bounded with and where are constants.
Remark 4. In the above assumption, could be positive, negative, or zero. Such an assumption is weaker than the one in [14] where it demands .
Assumption 5. For , there exits nonnegative constant satisfying the following. As noted in [6, 10], even if the parameters or time-varying delay in neural networks are appropriately chosen, neural networks may lead to some phenomena such as instability, divergence, oscillation, and chaos [10, 11, 40]. In order to stabilize the BAM-NN system (2)-(4), we introduce the following periodically intermittent controllerwhere denotes the control period, is the control periods number, is called the control time width, and are the control gains.
Under the periodically intermittent controller (10), the closed-loop systems of (2)-(4) can be described as follows.For the th moment of , we cite the following lemma to derive the stability conditions of the system.
Lemma 6 (see [36]). Let be a super cuboid set and , , . If is a real-valued continuous derivable function as , , then
3. Main Results
In this section, we present the sufficient conditions for stability of the controlled HDS-BAM-NN (11). For simplicity, the following notations are used to state and prove the main results.
In the above equations, , , , , , , and other variable parameters are given as and satisfy that .
In order to establish the sufficient conditions, the following two assumptions are further introduced.
Assumption 7. The following inequalities holds: where .
Here we define a functionwhere . For being continuous derivable, by intermediate value theorem, we could conclude that there exists satisfying with . Similarly, choose and to have the following function. Let ; we have the uniform inequality as , .
Assumption 8. The following inequality holds: , where , .
Theorem 9. Under Assumptions 3–5 and 7-8, the closed-loop system of (2)-(4) with the periodically intermittent controller (10) is exponentially stable in th moment.
Proof. Choose a candidate average Lyapunov-Krasovskii function [41] withwhere .
By the generalized Itô formula [42], we have the following.For short expression, denote , , . By Lemma 2.7 in [10], we can get the following.By employing the absolute value inequality and noticing that , we can obtain the following.Applying the fundamental inequality, i.e., , , yields the following.Substituting (25)-(28) into (24) leads to the following.Letand by (22), we get the following inequality.According to Dirichlet boundary conditions, by Lemma 6, we can derive the integral part of evolution dynamics as follows.Using It’s formula , we obtainwhere .
Similarly, when , we havewhere , .
By using the Gronwall inequality to (34) it yields the following.From (33)-(35), we can conclude that
(I) for and , by (33), we can, respectively, have and(II) for and , by (35), we can, respectively, have andRepeating above procedure (I)-(II), for , then and and for , then and Hence, for arbitrary , By calculation, we obtainsoUnder Assumption 8, Theorem 9 holds.
In Theorem 9, if we let , then , , , , , , , , , , . Thus, we can obtain the following corollary.
Corollary 10. Under Assumptions 3–5, system (2)-(4) with periodically intermittent controllers (10) is exponentially stable in th moment if the following conditions hold:
(I) .
(II) , where .
In Theorem 9, letting , which means that , , , and , we can get the following.We can also have a further extended corollary.
Corollary 11. Under Assumptions 3 to 5, the BAM-NN system (2)-(4) with periodically intermittent controllers (10) is exponentially stable in th moment if the following conditions hold:
(I) .
(II) , .
(III) , where ,
Remark 12. Unlike the result in [22], which only considered the mean square exponential stability of stochastic BAM-NN with time-varying delays and reaction-diffusion terms, we design a periodically intermittent controller to exponentially stabilize the unstable neural network in th moment. Moreover, the controller (10) is linear which can be easily implemented in practice.
Algorithm 13. For the periodically intermittent controller (10) of the BAM-NN system (2)-(4), we summarize the following algorithm to implement the controller.
(1) By instigating a practical system’s structure and dynamics’ characterization with its disturbance, we can develop a model of the BAM-NN system, i.e., , . Supposing and examining the transition probability as well as the main parameters of each mode, we can have , , and other relations of their states.
(2) By using (13)-(16) and Corollary 11, we can calculate the values of stabilization indices of the system and the controller gains with the parameters of BAM-NN.
(3) Choose a numerical solution to the stochastic partial differential equation (SPDE) to simulate the sample states of BAM-NN. Here we use a so-called estimation-correction method, which is based on the following main steps [43].
Firstly, with given steps of simulation time interval and space , we can get the grids of time and space, i.e., , , , , by which we define a numerical solution to BAM-NN as . Then we denote , where is an integer times of to .
Secondly, according to the basic formulation of SPDE (2) with its discretized time and space, we have an estimation formula for the numerical solution where , , is an identity matrix with suitable dimensions. By the boundary function (4), we can calculate the values of and . Then we can solve three diagonal linear equations from (47) to get an estimated value of .
Finally, based on the estimated value of , we have a further correction formula as follows.Similarly, we also transfer formulation (48) to three diagonal linear equations and calculate the numerical solution with permitted calculation errors.
(4) Comparing the sampled data profiles of the simulation states from (1)-(3) in Algorithm 13 with the measured data profiles from the aimed practical system, we can identify the validness of the model and performance of the controlled BAM-NN system. Furthermore, we can improve the parameters and control gains to obtain a better model by repeating steps (1)-(3).
4. Numerical Simulation of an Illustrative Application
For a grid-connected photovoltaic (PV) power generation system with series-connected inverters, which is illustrated by it is equivalent topology structure in Figure 1, every branch-circuit with a group of PV panels penetrates power into utility grid via a inverter. Since the current/power generated by the th group of PV panels is strongly dependent on operating conditions and field factors, such as sun geometric locations, their irradiation levels () of the sun and the ambient temperature () stochastically fluctuate with the environmental factors of PV power fields. Thus it is technically necessary to maintain power synchronization of series-connected inverters to improve their output power and standard voltage and current [44]. For this purpose, we need to develop a model with power and current difference between every couple-connected PV inverter based on basic photovoltaic model of PV panels [44, 45], which can be modeled as a HSD-BAM-NN ((2)-(4)), and the power/current difference () is taken as states with irradiation levels () as the space variable . It is assumed that the temperature difference can be ignored. By [44, 45], we formulate every part of BAM-NN ((2)-(4)) and calculate the parameters in the model; i.e., the modeled BAM-NN (2) has the same coefficient matrix, which is denoted as follows.
Firstly, we get the initial value conditions where , . And the boundary value functions are taken as follows. Denote , ; the generator of the Markovian chain is as follows.
The activation functions are , , and the stochastic disturbed functions are , .
According to the given structure and parameters of the BAM-NN (2)-(4), we can perform the numerical simulations in instability and stabilization cases.
(1) Instability. Using Algorithm 13 in Section 3, the sample states of BAM-NN (2) are calculated, and the surfaces of versus are shown in Figures 2 and 3, while the versus profile curves are shown in Figures 4 and 5. These figures show the instability behavior.
(2) Stabilization. Let . By (13)-(16), we can calculate that , , , , . The gain coefficients of periodically intermittent controllers (10) are obtained as follows by sufficient conditions Furthermore, we can get the indices’ values of the stable conditions of HSD-BAM-NN with , , , , . Also, we have . From (III) in Corollary 11, we know that . Letting , we can calculate the sample stabilized states of the controlled system. The trend surfaces of versus are shown in Figures 6 and 7 with the profile curves of versus in Figures 8 and 9. From the states’ trends, we can see that an unstable system (2)-(4) can be stabilized by the controller (10) with appropriate parameters, which theoretically shows that the output power of connected PV panels can be synchronized to a given level with its BAM-NN model, even though the power stochastically fluctuate.
5. Conclusion
In this paper, a hybrid stochastic delayed BAM neural network is considered for its stabilization problem and a periodically intermittent controller is designed to stabilize an unstable HSD-BAM-NN with an exponential convergence property. The sufficient conditions of exponential stabilization of HSD-BAM-NN are derived by Lyapunov-Krasovskii functional method, stochastic analysis techniques, and integral inequality. And the framework is established to give a solution algorithm to the sufficient conditions. The simulation results of the grid-connected photovoltaic (PV) power generation system verify the effectiveness of the proposed controller.
Data Availability
The data underlying the research and Matlab program code used in numerical simulation are available by email to the first author (pengyj@scut.edu.cn).
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grants 61573154 and 60904032 and Science & Technology Planning Project of Guangdong Province under Grant 2015A010106003.