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In this paper, we present an implementation scheme of memristor-based multilayer feedforward small-world neural network (MFSNN) inspirited by the lack of the hardware realization of the MFSNN on account of the need of a large number of electronic neurons and synapses. More specially, a mathematical closed-form charge-governed memristor model is presented with derivation procedures and the corresponding Simulink model is presented, which is an essential block for realizing the memristive synapse and the activation function in electronic neurons. Furthermore, we investigate a more intelligent memristive PID controller by incorporating the proposed MFSNN into intelligent PID control based on the advantages of the memristive MFSNN on computation speed and accuracy. Finally, numerical simulations have demonstrated the effectiveness of the proposed scheme.

In 1971, Professor Chua theoretically formulated and defined the memristor and described that the memristance (short for resistor of a memristor) is characterized by the relationship between the electrical charge _{2} memristor model on account of its simplified expressions and the same ideal physical behaviors.

Brain neural network emerges from the interactions of dozens, perhaps hundreds, of brain regions, each containing millions of neurons [

Notably, Watts and Strogatz revealed a significant effect that is in common among complex networks. They pointed out that the real architecture of network is nearly a middle model between regular connection and random connection and defined it as small-world network (WS model) in 1998 [

This paper is organized as follows. In Section

A memristor or memristive device is essentially a two-terminal passive electronic element with memory capacity. Its memristance state depends on the amplitude, polarity, and duration of the external applied power. The physical model of the HP memristor from [_{2} sandwiched between two platinum electrodes. One of the layers, which is described as _{2} layer owning an insulating property has a perfect 2 : 1 oxygen-to-titanium ratio, and this layer is referred to the undoped region. Generally, an external excitation

Schematic model of the HP memristor.

The total resistance of the memristor,

When

The movement speed of the boundary between the doped and undoped regions depends on the resistance of doped area, the passing current, and other factors according to the state equation:

Figure

The influence of different values of the integer

Substituting (

Assume

The initial value of the state variable can be expressed as

Then, the expression of

Combining (

Giving a sine stimulus to the memristor, we get the simulation results using MATLAB software. It is noteworthy that the memristor is a two-terminal element with polarity, which is shown in Figure ^{2}s^{−1}V^{−1}. Moreover, the simulation results in Figure

The nonlinear memristor model and its characteristic curves. (a) The equivalent circuit of the memristor and its 3D symbol. (b) The relationship between memristance and the charge.

For the sake of analyzing the characteristics of the memristor model comprehensively, a Simulink model is built upon (

The Simulink model of the nonlinear memristor.

The simulation results are exhibited in Figure

The results of the memristor Simulink model. (a) The input current source. (b) Relationship between the current

Generally, small-world phenomenon indicates that a network has highly concentrated local connections and also includes a few random long connections. In real world, a large number of networks have the small-world effect, such as disease transmission network, social network, and the food chain network [

Set the network has

Set each layer has

For

For

While

For

The construction progress of the multilayer feedforward WS small-world neural network. (a) The regular network (

As shown in Figure

More specially, we set the network connection matrix as

The nanoscale memristor has high potential of information storage on account of the non-volatility with respect to long periods of power-down, so it can be used as electric synapse in the artificial neural networks, and the primary reasons are manifold. Firstly, as a kind of analog component, this device can realize weight updating continuously. Moreover, the memristor possesses the capacity of information storage due to the nonvolatility. This feature is consistent with the memory ability of the neurons in human’s brain. Additionally, the memristive neural network can be further integrated in crossbar array which has significant advantages in better information processing capacity and huger storage.

According to the nonlinear memristor model in Section

Differentiating (

The relationship curve between the rate of the memristive conductance change and the current.

In the standard MFSNN, the activation function for each neuron is usually the nonlinear Sigmoid function. Particularly, the activation function of the hidden layer adopts bipolar Sigmoid function, but the output layer activation function is unipolar Sigmoid function.

Based on the constitutive relationship of the memristor, a lot of nonlinear curves can be simulated and substituted [

The simplified memristor model. (a) The simplified Simulink model of the memristor. (b) The package of the memristor.

Then, the behavior of the output curve can be adjusted efficiently by the parameter control module, gain module, and internal operation module, which is crucial to implement the activation function in the neural network. Here, we set the activation functions for the hidden layer and output layer neurons of the memristive MFSNN as

Figure ^{2}s^{−1}V^{−1}. The input signal is a sinusoidal current with an amplitude of 0.5 mA and a frequency of 1 Hz. Notably, the polarity of the voltage applied into the memristor is opposite to the polarity of the memristor itself; that is, the current flows through the memristor from the negative polar to the positive polar. Figure

The principle diagram of the memristive activation function in the hidden layer. (a) The Simulink model of the memristive activation function in the hidden layer. (b) The curve of the memristive activation function

Similarly, Figure

The principle diagram of the memristive activation function of the output layer. (a) The Simulink model of the memristive activation function of the output layer. (b) The curve of the memristive activation function

So far, the PID control has found widespread applications in the modern control field. By adjusting the control action of the proportion, integration, and differentiation, we get an interactive nonlinear relationship among these control variables. The neural network has the ability of expressing the nonlinearity, which can be used in the PID control for implementing the optimal nonlinear relationship among control variables. In this work, we build up a more intelligent PID controller with the parameters (

According to the literature [

In Figure

The structure diagram of the memristive neural network PID controller.

Based on the novel neural network presented in Section

The input and output vectors of the first hidden layer can be expressed as

By that analogy, the input and output vectors of the

Finally, the input and output vectors of the output layer can be obtained as

From [

In this section, some numerical simulations of the memristive multilayer feedforward small-world neural network PID controller have been executed on MATLAB software. The mathematical model of the controlled plant is given as

The memristive neural network under investigation is constituted by seven layers with four neurons in the input layer, three neurons in the output layer, and five in each of the five hidden layers. The learning rate of the network ^{2}s^{−1}V^{−1},

Figure

The simulation results of the memristive neural network PID controller

In order to verify the superior performance of the memristive small-world neuronal networks and figure out the optimal structure, we conducted a series of simulations to observe the convergence performance of the proposed network under different

The convergence performance of the memristive neural network under different

Notably, the mathematical function of this system has the local minimum, for getting out of the local minimum, we define the maximum allowable iteration times to be 10000, as previously mentioned for each

A mathematical closed-form charge-governed memristor model is recalled firstly and the corresponding Simulink model is presented. Using the change rule of memconductance, a memristive realization scheme for synaptic weight is proposed. Moreover, the activation functions in electric neurons are also implemented based on the single-input and double-output package of the memristor. Combining the proposed memristive synapse and activation functions, a memristor-based MFSNN is addressed. It exhibits advantages in computation speed and accuracy over the traditional multilayer neural networks by considering the small-world effect. Meanwhile, it has potential of hardware realization of the neural network because of the nanoscale size of the memristive synapse. These superior properties can further improve the application of the neural networks, such as in the intelligent controller design. Motivated by this, we apply the memristor-based MFSNN to classical PID control, and the proposed memristive PID controller may possess the following superiorities. (i) Its nanoscale physical implementation could promote the development of the microcontroller. (ii) Because of the participation of the memristive neural network, the proposed PID controller can realize the parameters self-adjustment. (iii) The control speed and accuracy are improved. Eventually, extensive numerical simulations justify the effectiveness and efficiency of the memristive PID controller over the regular neural network PID controller. This work may provide a theoretical reference to physically realize the small-world neural networks and further promote the development of modern intelligent control technology.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work was supported by Program for New Century Excellent Talents in University (Grant nos.