Finite time fractional order delay coupling competitive neural network synchronous learning method

Abstract

The invention provides a finite time fractional order delay coupling competitive neural network synchronous learning method, and belongs to the technical field of iterative learning control. The technical problems that a fractional-order time-delay coupling competitive neural network is difficult to realize complete synchronization on a time axis and the structural design of a controller is complicated are solved. According to the technical scheme, the method comprises the following steps: S1, establishing a single competitive neural network model with time delay; s2, matrix lifting is carried out on the data, and the data are written into a compact form; s3, considering a coupling competitive neural network formed by a plurality of time-delay competitive neural networks, and describing communication topology between the networks by means of a graph theory; s4, designing an input sharing fractional order quantitative learning control protocol, and writing the input sharing fractional order quantitative learning control protocol into a matrix form; and S5, carrying out convergence proving on the designed control protocol. The beneficial effects of the invention are that the synchronization speed of the fractional-order time-delay coupling competitive neural network is obviously improved through the input shared control cooperation.

Description

Technical Field

[0001] The present invention relates to the technical field of iterative learning control, and in particular to a finite-time fractional-order time-delay coupled competitive neural network synchronous learning method. Background Art

[0002] In the past few decades, a large number of neural network models and their dynamic characteristics have been widely used in many fields, such as signal processing, image recognition, and optimization algorithms. Fractional-order competitive neural networks enhance the dynamic characteristics of traditional neural networks. They have the ability to describe the memory and forgetting effects of the system, enabling the network to better capture complex time-dependent and nonlinear behaviors. On the other hand, coupled neural networks consisting of many interconnected nodes are another important class of complex neural networks. Extensive research has been conducted on the communication topology and dynamic behavior of convolutional neural networks, focusing on consistent synchronization, which is one of the most valuable dynamic characteristics of convolutional neural networks. So far, substantial progress has been made in studying the synchronization of various neural networks, including time-varying, impulse behaviors, and non-identical network models.

[0003] The synchronization research of this type of neural network has gradually become a hot topic and has great application value. For example, in the article "Stability and Application of Mittag-Leffler Delayed Fractional-Order Competitive Neural Networks", the authors used operator pairs to test and evaluate the coexistence criteria of equilibrium points of delayed fractional-order complex neural networks. This comparison principle established the Mittag-Leffler stability of delayed CNNs and improved the evaluation strategy of the stability of fractional-order neural networks. However, this method can only achieve gradual synchronization of the network on the time axis, and cannot achieve complete synchronization within a limited time. It is also not easy to transfer to repetitive industrial application environments, because these scenarios often require the system to remain completely synchronized within a limited time interval.

[0004] Iterative learning control has obvious advantages in systems with strong coupling, nonlinearity and model agnosticism, and it can achieve complete synchronization of the system on the time axis through iteration within a limited time. To this end, a fractional-order iterative learning control protocol is designed to synchronize coupled fractional-order neural networks within a limited time. An input sharing strategy is added to improve the control protocol. The coupled fractional-order competitive neural networks share the control input information between each network to accelerate the consistency synchronization of the network. Summary of the invention

[0005] The purpose of the present invention is to provide a finite time fractional order time-delay coupled competitive neural network synchronization learning method, which solves the technical problems that the fractional order time-delay coupled competitive neural network is difficult to achieve complete synchronization on the time axis and the controller structure design is complex, so that it can achieve complete synchronization on the time axis within a limited time. In addition, the input sharing control coordination significantly improves the synchronization speed of the fractional order time-delay coupled competitive neural network.

[0006] In order to achieve the above-mentioned invention object, the technical solution adopted by the present invention is specifically: a fractional-order coupled competitive neural network synchronization method with time delay within a finite time, comprising the following steps:

[0007] Step 1: Establish a single competitive neural network model with time delay;

[0008] Step 2: Perform matrix enhancement and write it in compact form;

[0009] Step 3: Consider a coupled competitive neural network with time delay composed of multiple time-delay competitive neural networks, and use graph theory to describe the communication topology between each network;

[0010] Step 4: Design a fractional-order quantized learning control protocol with input sharing and write it in matrix form;

[0011] Step 5: Using the compression mapping principle and with the help of the Bellman-Gronwald inequality and the Mittag-Leffler function, the convergence of the control protocol designed above is proved.

[0012] Furthermore, in step 1, the fractional-order competitive neural network model with a single time delay can be expressed as:

[0013]

[0014] Where i = 1, 2, ..., n is the number of nodes in the network, and α∈(0,1) denotes the Caputo fractional derivative; x i (t) and y i (t) is the neuron state and synaptic efficiency; a ij and b ij is the weight, J i is the external input, d i 、c i and b i is a positive number; τ>0 is a time delay.

[0015] If -τ≤s≤0 then

[0016] Where f(·) represents the activation function, which satisfies the following Lipschitz condition

[0017]

[0018] Furthermore, in step 2, the single time-delay fractional-order coupled competitive neural network in step 1 is subjected to matrix lifting, as follows:

[0019]

[0020] in

[0021] Furthermore, in step 3, multiple coupled fractional-order competitive neural networks with time delays in step 2 are considered, and the communication topological relationship between the networks is characterized by using graph theory, as shown below:

[0022]

[0023] where z i (t) = [z i1 (t), z i2 (t), ..., z i2n (t)] T , is the control input, is the input matrix; c>0 is the coupling strength; is a positive definite inner coupling matrix.

[0024] For the N FOCNNs above, the communication topology between them is composed of a directed graph Describes the expected synchronization state z for a fixed time interval [0, T] d (t), z d (t) and u d The relationship between (t) can be expressed by the following equation:

[0025]

[0026] Therefore, the problem is transformed into finding a suitable control input u for t∈[0,T] when the number of iterations k tends to infinity. d (t) to achieve the synchronization of each follower state, that is,

[0027]

[0028] Furthermore, in step 4, a quantized fractional-order iterative learning control protocol with input sharing is designed to synchronize the above-mentioned coupled fractional-order neural network. The specific control protocol is as follows:

[0029]

[0030] in represents the distributed error, is the learning gain, b i is the connection weight between the i-th fractional-order neural network and the leader.

[0031] The first term in the formula represents the input sharing of the neural network. The shared neural network input consists of the input weight of itself and the input weight of the adjacent neural network, where is a logarithmic quantizer.

[0032] From the theorem in the claims, it can be seen that the quantized fractional-order input shared iterative learning control protocol can be converted to:

[0033]

[0034] The quantization error is described as

[0035]

[0036] Therefore, the fractional-order quantized iterative learning control protocol with input sharing is The input sharing control protocol can be written as

[0037]

[0038] Furthermore, in step 5, the above-mentioned The convergence of the input sharing control protocol is proved as follows:

[0039] For the delayed fractional-order coupled competitive neural network, use The coupled neural network will be fully synchronized with the increase of the number of iterations when the control gain Γ satisfies the following conditions:

[0040]

[0041] First, three assumptions should be met:

[0042] First, the activation function satisfies the global Lipschitz condition, that is,

[0043]

[0044] Second, for each extended graph The leader is its root, that is, the expansion graph contains a spanning tree. Third, the fractional-order coupled competitive neural network with time delay satisfies the initial value condition in iterative learning, which is a common condition in the field of iterative learning control, namely

[0045]

[0046] The convergence of the delayed fractional-order coupled competitive neural network that satisfies the above basic assumptions is proved as follows: First, because Then from formula (3) we can get

[0047]

[0048] Arrangement available

[0049]

[0050] Where δu k (t) = u d (t)-u k (t),δf(z k (t)) = f(z d (t))-f(z k (t))

[0051] Taking the norm of both ends of equation (4) and combining the Lipschitz condition, we note that |Δ(k, t)|≤β, so we have

[0052] In the formula,

[0053]

[0054] According to the definition of the λ norm, combined with the above formula, taking the λ norm on both sides at the same time, we can deduce

[0055]

[0056] Subtracting equation (1) from equation (2) and using the Kronecker product, we can get

[0057]

[0058] By performing fractional integration from 0 to t on both sides of the above equation, we can obtain the following equation

[0059]

[0060] Taking the norm of both ends of equation (6), we can get

[0061]

[0062] In the formula In order to solve the influence of state lag, the fractional-order integral is discussed in a classified manner, specifically:

[0063]

[0064] if but

[0065]

[0066] if Then the above integral can be divided into two parts, and it can be deduced that

[0067]

[0068] Therefore, based on the above discussion, we can conclude that

[0069]

[0070] Combining the above discussion results, and taking the λ norm on both ends of equation (7), we can get

[0071] To apply the Bellman-Gronwald inequality to the above equation, we first set up two functions, namely

[0072]

[0073] According to the two-parameter function definition of Mittag-Leffler, for λ>0, we have

[0074]

[0075] Since its derivative is greater than zero, we know

[0076] is an increasing function.

[0077] Applying the Bellman-Gronwald inequality to equation (8) yields

[0078]

[0079] The parameters in formula (9) are replaced as follows:

[0080]

[0081] Substituting formula (10) into formula (9), we can get

[0082] ||δz k || λ ≤ρ1||δu k || λ , (11)

[0083] In the formula

[0084] Substituting formula (11) into formula (5), we can get

[0085] ||δu k+1 || λ ≤ρ2||δu k || λ +(c4+l f c5+e λT l f c6)(1+β)ρ1||δu k || λ

[0086] From the convergence condition, we can know that we can choose a sufficiently large λ to satisfy the following equation:

[0087]

[0088] therefore Can be launched Substituting formula (12) into formula (11), we can get

[0089]

[0090] Note that ||δz k (t)||≤e λT ||δz k || λ , so there is

[0091]

[0092] Furthermore, considering δz k (t) is in compact form, and we expand it, so

[0093]

[0094] And because |δz k (t)|, so

[0095] ‖z i,k (t)-z j,k (t)‖≤‖δz i,k (t)‖+‖δz j,k (t)‖

[0096] (14) Combining equation (14) and equation (13), we can get

[0097]

[0098] The convergence of the time-delayed fractional-order coupled competitive neural network of the present invention is verified.

[0099] Iterative learning control provides a new idea for the synchronization of complex dynamic coupled neural networks. Through reasonable control strategy design, a fractional-order distributed iterative learning control protocol is proposed to achieve complete synchronization of the network on the time axis. In order to speed up the synchronization, input sharing is added to the learning control protocol to share information between neural networks, so that they can achieve synchronization faster, which has broad practical application value and research prospects.

[0100] Compared with the prior art, the present invention has the following beneficial effects:

[0101] 1. This paper designs a fractional-order quantized iterative learning control protocol and applies it to the synchronization problem of fractional-order coupled competitive neural networks with time delay. Different from the gradual synchronization along the time axis studied in previous work, this method enables it to achieve complete synchronization on the time axis within a limited time and can be applied to various repetitive scenarios. The controller is designed using a logarithmic quantizer, which reduces the use of communication resources.

[0102] 2. The present invention combines the fractional-order quantized iterative learning control protocol with the input sharing strategy, and uses the weights of each neural network to determine the proportion of shared information, so that the neural networks can share their respective control inputs, thereby accelerating the synchronization speed of the time-delayed fractional-order coupled competitive neural network, and has good practical application value.

[0103] 3. The present invention generalizes the fixed communication topology between networks to a topology that can be switched arbitrarily with the number of iterations, and proves the convergence of the synchronization error with the help of the compression mapping method and the Mittag-Leffler function, and gives the sufficient conditions for the synchronization of the delayed fractional-order coupled competitive neural network. BRIEF DESCRIPTION OF THE DRAWINGS

[0104] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. They are used to explain the present invention together with the embodiments of the present invention and do not constitute a limitation of the present invention.

[0105] Figure 1 This is a flow chart of the time-delayed fractional-order coupled neural network synchronization control protocol in the present invention.

[0106] Figure 2 This is a communication topology diagram between the time-delayed fractional-order coupled neural networks in the present invention.

[0107] Figure 3 It is a schematic diagram of the synchronization of four neural network states x1 and x2 in the present invention.

[0108] Figure 4 It is a schematic diagram of synchronization of four neural network states y1 and y2 in the present invention.

[0109] Figure 5 This is a comparison diagram of errors with and without input sharing in the present invention.

[0110] Figure 6 This is the quantized output graph after 80 iterations of the neural network in the present invention.

[0111] Figure 7 is the switching function that randomly switches the communication topology along the iteration number.

[0112] Figure 8 There are four communication topologies in the switching topology case.

[0113] Fig. 9 Schematic diagram of the synchronization of four neural networks with random switching of topology along the iteration number. DETAILED DESCRIPTION

[0114] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. Of course, the specific embodiments described here are only used to explain the present invention and are not used to limit the present invention.

[0115] Example 1

[0116] See also Figure 1 and Figure 6 The technical solution provided in this embodiment is a finite-time fractional-order time-delay coupled competitive neural network synchronous learning method. The proposed controller is verified by numerical simulation. The specific steps are as follows:

[0117] Step 1: Establish a mathematical model of a fractional-order coupled competitive neural network with a single time delay, as follows

[0118]

[0119] Where i = 1, 2, ..., n is the number of nodes in the network, and α∈(0,1) denotes the Caputo fractional derivative; x i (t) and y i (t) is the neuron state and synaptic efficiency; a ij and b ij is the weight, J i is the external input, d i 、c i and b i is a positive number; τ>0 is a time delay. If -τ≤s≤0, then

[0120] Step 2: Use the matrix to represent it in a compact form and perform matrix lifting on it. The result is as follows

[0121]

[0122] in

[0123] Step 3: Consider a coupled time-delay fractional-order competitive neural network composed of multiple neural networks. Using graph theory to represent the communication topology between networks, its model can be described as

[0124]

[0125] where z i (t) = [z i1 (t), z i2 (t), ..., z i2n (t)] T , is the control input, is the input matrix; c>0 is the coupling strength; is a positive definite internal coupling matrix

[0126] Step 4: Design a quantized fractional-order iterative learning control protocol for the above system, as follows

[0127]

[0128] in represents the distributed error. The matrix form of the control protocol can be written as:

[0129]

[0130] Step 5: Using the compression mapping principle, and with the help of the Bellman-Gronwald inequality and the Mittag-Leffler function, prove its convergence and perform simulation verification.

[0131] The specific simulation process and parameters are as follows:

[0132] Consider the following topological network with 4 nodes, each node represents a neural network, and there is a coupling relationship between the networks. Each neural network contains 2 neurons, and each neuron has 2 states, divided into states x and y. The details are as follows

[0133]

[0134] The specific parameters selected in the formula are t∈[0,1.2], τ=0.2, c=0.3, α=0.85

[0135] The main matrices are selected as follows

[0136]

[0137] The expected trajectory of selecting a virtual leader is

[0138] z d (t) = [0.3te t ,0.8log(1+sint),cos(3πt)-1,sin(3πt)] T .

[0139] The activation function of the fractional-order coupled competitive neural network with time delay is

[0140]

[0141] Set the topological relationship between neural networks as Figure 2 As shown, and select the Laplace matrix and The matrix is ​​as follows

[0142]

[0143] Select the learning gain that meets the convergence condition, select Γ = [0.55, 0.51, 0.48, 0.55], so that

[0144]

[0145] Result description:

[0146] Figure 3 and Figure 4 The four states of the time-delayed fractional-order coupled competitive neural network are designed Tracking effect after k iterations under the iterative learning control protocol, where each line type represents a network.

[0147] Figure 5 It is the quantized output diagram of the four states of the time-delayed fractional-order coupled competitive neural network (FOCNNs). The figure shows the synchronization effect after 80 iterations. It can be seen from the figure that the synchronous learning effect is better.

[0148] Example 2

[0149] Consider the following topological network with 4 nodes, each node represents a neural network, and there is a coupling relationship between the networks. Each neural network contains 2 neurons, and each neuron has 2 states, divided into states x and y. The details are as follows

[0150]

[0151] The specific parameters selected in the formula are t∈[0,1.2],τ=0.2,c=0.3,α=0.85.

[0152] A quantized fractional-order input sharing iterative learning control protocol designed for the above system is added with an input sharing strategy, as follows

[0153]

[0154] in represents the distributed error, is the learning gain matrix, b i is the connection weight between the ith neural network and the leader.

[0155] Its matrix form is:

[0156]

[0157] In order to compare with the results of Example 1, under the input sharing control protocol, the system parameters are selected to be the same as those of Example 1 to ensure the effectiveness of the comparison.

[0158] Result description:

[0159] Figure 6 This is a comparison chart of the convergence speed of the fractional-order controller after adding input sharing and the controller without input sharing. The logarithmic coordinate axis is selected. The dotted line in the figure represents the control protocol without input sharing, and the solid line represents the control protocol with input sharing. Obviously, the controller with input sharing converges faster than the controller without input sharing.

[0160] Example 3

[0161] Consider the following topological network with 4 nodes, each node represents a neural network, and there is a coupling relationship between the networks. Each neural network contains 2 neurons, and each neuron has 2 states, divided into states x and y. The details are as follows

[0162]

[0163] The selection of system parameters and activation function is the same as in Example 1.

[0164] The communication topology of the fractional-order coupled competitive neural network with time delay switches periodically and randomly with the number of iterations k. The fractional-order quantized iterative learning control protocol is designed as follows

[0165]

[0166] in, and denote the Laplacian matrix and the connection matrix with respect to iteration k, respectively. mark is diag{b1(k),b2(k),...,b N(k)}.Select as follows:

[0167]

[0168] Select Γ = diag{0.69, 0.73, 0.75, 0.62}, so that:

[0169] Obviously, the convergence condition is satisfied.

[0170] In this example, the communication topology will change periodically with the number of iterations, i.e., k = 4i + h. like Figure 8 shown.

[0171] Furthermore, considering the case where the communication topology switches randomly with the number of iterations, the random switching function is as follows: Figure 7 shown.

[0172] Fig. 9 The figure shows the synchronization error of the fractional-order coupled competitive neural network with time delay under the fractional-order quantized iterative learning control protocol on the logarithmic axis. The dotted line in the figure represents the control protocol without input sharing, and the solid line represents the control protocol with input sharing. It can be seen from the figure that the fractional-order quantized iterative learning control protocol with input sharing still shows good rapidity when the topology is randomly switched with the number of iterations.

[0173] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A finite-time fractional-order time-delay coupled competitive neural network synchronous learning method, characterized in that: The following steps are involved: Step 1: Establish a single competitive neural network model with time delay; Step 2: Perform matrix enhancement and write it in compact form; Step 3: Consider a coupled competitive neural network with time delay composed of multiple time-delay competitive neural networks, and use graph theory to describe the communication topology between each network; Step 4: Design a fractional-order quantized learning control protocol with input sharing and write it in matrix form; Step 5: Using the compression mapping principle and with the help of the Bellman-Gronwald inequality and the Mittag-Leffler function, the convergence of the control protocol designed above is proved.

2. The finite-time fractional-order time-delay coupled competitive neural network synchronous learning method according to claim 1 is characterized in that: In the step 1, the fractional-order competitive neural network model with a single time delay is expressed as: Where i = 1, 2, ..., n is the number of nodes in the network, and α∈(0,1) denotes the Caputo fractional derivative; x i (t) and y i (t) is the neuron state and synaptic efficiency; a ij and b ij is the weight, J i is the external input, d i 、c i and b i is a positive number; τ>0 is time delay; like but Where f(·) represents the activation function, which satisfies the following Lipschitz condition 3. The finite-time fractional-order time-delay coupled competitive neural network synchronous learning method according to claim 1 is characterized in that: In the step 2, matrix lifting is performed on the single time-delay fractional-order coupled competitive neural network in step 1, specifically as follows: in 4. The finite-time fractional-order time-delay coupled competitive neural network synchronous learning method according to claim 1, characterized in that: In step 3, multiple coupled fractional-order competitive neural networks with time delays in step 2 are considered, and the communication topology relationship between networks is characterized by using graph theory, as shown below: in is the control input, is the input matrix; c>0 is the coupling strength; is a positive definite internal coupling matrix; For the N FOCNNs above, the communication topology between them is composed of a directed graph describe: For a fixed time interval [0, T], when the desired synchronization state z is given d (t), z d (t) and u d The relationship between (t) is expressed by the following equation: When the number of iterations k tends to infinity, find a suitable control input u for t∈[0,T] d (t) to achieve the synchronization of each follower state, that is, 5. The finite-time fractional-order time-delay coupled competitive neural network synchronous learning method according to claim 1, characterized in that: In the step 4, the control input is shared by using the weighted average of the control input of the fractional-order competitive neural network and the control input of its neighboring fractional-order competitive neural network; Therefore, the control protocol with input sharing is designed as follows in represents the distributed error, is the learning gain, b i is the connection weight between the i-th fractional-order neural network and the leader; The first term in the formula represents the input sharing of the neural network. The shared neural network input consists of the input weight of itself and the input weight of the adjacent neural network, where is the logarithmic quantizer; The quantized fractional-order input sharing iterative learning control protocol is transformed into: The quantization error is described as Therefore, the fractional-order quantized iterative learning control protocol with input sharing is The input sharing control protocol is written as 6. The finite-time fractional-order time-delay coupled competitive neural network synchronous learning method according to claim 1, characterized in that: In step 5, the above-mentioned The convergence of the input sharing control protocol is proved as follows: For the delayed fractional-order coupled competitive neural network, use The coupled neural network will be fully synchronized with the increase of the number of iterations when the control gain Γ satisfies the following conditions: First, three assumptions should be met: First, the activation function satisfies the global Lipschitz condition, that is, Second, for each extended graph With the leader as its root, the extended graph contains a spanning tree; Third, the fractional-order coupled competitive neural network with time delay satisfies the initial value condition in iterative learning, which is a common condition in the field of iterative learning control, namely The convergence of the time-delayed fractional-order coupled competitive neural network that meets the above basic assumptions is proved as follows: First, because Then from formula (3) we get where δu k (t) = u d (t) - u k (t), δf(z k (t)) = f(z d (t)) - f(z k (t)). Taking the norm of both ends of equation (4) and combining it with the Lipschitz condition, we note that ||Δ(k, t)||≤γ, so we have In the formula, According to the definition of the λ norm, combined with the above formula, taking the λ norm on both sides, we get ||duh k+1 || λ ≤ρ2||δu k || λ +(c4+l f c5+e λT l f c6)(1+γ)||δz k || λ (5) Subtract equation (1) from equation (2) and use the Kronecker product to obtain Performing fractional integration from 0 to t on both sides of the above equation simultaneously, we obtain the following equation Taking the norm of both ends of equation (6), we get In the formula In order to solve the influence of state delay, the fractional order integral is classified and discussed, and it is obtained Combining the above discussion results, we then take the λ norm on both ends of equation (7) and obtain To apply the Bellman-Gronwald inequality to the above equation, combined with the function definition of the two-parameter form by Mittag-Leffler, for λ>0, we have Substituting the parameters in equation (9), we get ||δz k || λ ≤ρ1||δu k || λ ,(10) In the formula Substituting formula (10) into formula (5), we get ||duh k+1 || λ ≤ρ2||δu k || λ +(c4+l f c5+e λT l f c6)(1+β)ρ1||δu k || λ According to the convergence condition, we can choose a large enough λ to satisfy the following equation: therefore Since ||δz k (t)||≤e λT ||δz k || λ , combined with |δz k (t)|, we get

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