Fractional-order memristive hopfield neural network state feedback control method and device

Abstract

The application provides a state feedback control and device of a fractional order memristive Hopfield neural network, and the method comprises the following steps: constructing a fractional order memristive Hopfield neural network model; calculating an equilibrium point; performing dissipation analysis on the fractional order memristive Hopfield neural network model; taking a coupling strength coefficient as a bifurcation parameter, and performing stability analysis on the fractional order memristive Hopfield neural network model at the equilibrium point; designing a fractional order state feedback controller, and constructing a fractional order memristive Hopfield neural network control model; taking the coupling strength coefficient as the bifurcation parameter, and performing stability analysis on the fractional order memristive Hopfield neural network control model; and determining the control parameter of the fractional order state feedback controller according to the analysis result, so as to realize feedback control on the fractional order memristive Hopfield neural network. Finally, simulation verification shows that the fractional order state feedback controller enables the original system to realize stability while retaining the characteristic that the original system can generate chaos, and good control effect is achieved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of memristor neural network control, and particularly relates to a state feedback control method and device for a fractional-order memristor Hopfield neural network. BACKGROUND

[0002] Chaotic theory was born in the field of mathematics and physics in the 1960s, and it is honored as one of the three great scientific revolutions in the 20th century together with the theory of relativity and quantum mechanics. Chaos is a very complex, random-like behavior of a system, especially a nonlinear system, which cannot be determined according to the initial conditions of the system. The research on chaos has developed in both theory and practice. Nonlinear phenomenon is the most common phenomenon in nature and production practice. In recent years, with the development of science and technology, a large number of nonlinear dynamic systems have been proposed, and scientists have found a large number of complex nonlinear phenomena in these system models, such as bifurcation, chaos, fractal, etc. For some completely determined nonlinear systems, when a parameter μ of the system continuously changes to a critical value μc, the global behavior (qualitative properties, topological properties, etc.) of the system will suddenly change. μc is called the bifurcation value or branch value of the parameter μ. This phenomenon is called bifurcation, which is an important nonlinear phenomenon. And continuous bifurcation is a precursor to chaotic phenomenon. There are many types of bifurcation: saddle-node bifurcation, period-doubling bifurcation, Hopf bifurcation, and fork bifurcation. Unlike linear systems, nonlinear systems cannot use the superposition principle, and their stability is more complex. Nonlinear systems can have multiple equilibrium points, and at different equilibrium points, the system can be stable or bifurcated. Stability is a basic property of a system. When the system is disturbed, it may deviate from the equilibrium state. The relationship between the stability condition of the system and the parameters needs to be analyzed to determine the stability of the system, which is more conducive to subsequent research.

[0003] However, there is little research on the stability control of fractional-order memristor Hopfield neural networks at present, and the control method is relatively complex, the control effect is not obvious, and the stability and life characteristics of the actual system cannot be better controlled. SUMMARY

[0004] The main purpose of the present application is to solve the technical problem that the control method in the prior art is relatively complex, the control effect is not obvious, and the stability and life characteristics of the actual system cannot be better controlled.

[0005] To achieve the above purpose, the present application provides a state feedback control method for a fractional-order memristor Hopfield neural network, comprising the following steps:

[0006] constructing a fractional-order memristor Hopfield neural network model;

[0007] calculating an equilibrium point of the fractional-order memristive Hopfield neural network model;

[0008] calculating a divergence of the fractional-order memristive Hopfield neural network model, and determining whether the fractional-order memristive Hopfield neural network model can generate a chaotic attractor according to a value of the divergence;

[0009] performing stability analysis on the fractional-order memristive Hopfield neural network model at the equilibrium point by taking the coupling strength coefficient as a bifurcation parameter, to obtain a first stability analysis result;

[0010] designing a fractional-order state feedback controller for the fractional-order memristive Hopfield neural network model according to the first stability analysis result, to construct a fractional-order memristive Hopfield neural network control model;

[0011] performing stability analysis on the fractional-order memristive Hopfield neural network control model at the equilibrium point by taking the coupling strength coefficient as a bifurcation parameter, to obtain a second stability analysis result;

[0012] determining a control parameter of the fractional-order state feedback controller according to the second stability analysis result, to realize feedback control on the fractional-order memristive Hopfield neural network.

[0013] Further, a mathematical expression of the fractional-order memristive Hopfield neural network model is as follows:

[0014]

[0015] where x1, x2, x3 and x4 represent four neuron states, which are state variables changing with time, s 11 ,s 22 are self-connection synaptic weights of neurons 1 and 2, s 12 ,s 21 ,s 31 ,s 41 correspond to mutual connection weights of neurons 1 to 2, neurons 2 to 1, neurons 3 to 2 and neurons 4 to 1 respectively, tanh(·) is a neuron activation function, k represents a coupling strength coefficient of a memristor, and q ∈ (0, 1] represents an order of Caputo fractional derivative.

[0016] Further, the calculating the equilibrium point of the fractional-order memristive Hopfield neural network model comprises:

[0017] Let the right side of the fractional-order memristive Hopfield neural network model be equal to zero, that is:

[0018]

[0019] The equilibrium point is calculated Where c1, c2 are constants.

[0020] Further, the step of calculating the divergence of the fractional-order memristive Hopfield neural network model, and determining whether the fractional-order memristive Hopfield neural network model can appear a chaotic attractor according to the divergence value, comprises:

[0021] The divergence of the memristive Hopfield neural network model is calculated as:

[0022]

[0023] Where sech(·) is the hyperbolic secant function, and 0≤sech(·)≤1, the system parameters are set as s 11 =0.1, s 12 =2.8, s 21 =3, s 22 =4, and ∇·V=-2-0.1sech 2 (x1)+4sech 2 (x2), therefore, there must exist a solution (x1, x2, x3, x4) that can satisfy ∇·V less than 0. For the fractional-order memristive Hopfield neural network model, when the time tends to infinity, each unit state volume element of the fractional-order memristive Hopfield neural network model converges to 0, and finally tends to a chaotic attractor.

[0024] Further, the mathematical expression of the fractional-order memristive Hopfield neural network control model is:

[0025]

[0026] Where k a is the state feedback gain coefficient.

[0027] Further, the step of taking the coupling strength coefficient as a bifurcation parameter, and performing stability analysis on the fractional-order memristive Hopfield neural network model at the equilibrium point to obtain a first stability analysis result, specifically comprises:

[0028] The characteristic equation of the fractional-order memristive Hopfield neural network model is calculated as:

[0029] L0s 4α +L1s 3α+L2s 2α = 0

[0030] wherein

[0031] L0= 1

[0032] L1= 2-s 22 +s 11

[0033] L2= 1-s 22 +s 11 -s 11 s 22 +s 21 s 12 -s 21 A1-s 12 A2+A1A2

[0034]

[0035]

[0036] Setting system parameter s 11 = 0.1, s 12 = 2.8, s 21 = 3, s 22 = 4, thus L1= -1.9 < 0, according to Routh criterion, the system is unstable, and chaos phenomenon is generated, and each state variable waveform oscillates and cannot converge.

[0037] Further, the step of taking the coupling strength coefficient as a bifurcation parameter, and performing stability analysis on the fractional-order memristive Hopfield neural network control model at the equilibrium point to obtain a second stability analysis result, specifically comprises:

[0038] calculating eigenvalues of the fractional-order memristive Hopfield neural network control model:

[0039] λ0= 0

[0040]

[0041] wherein a1= -(k a +1.9),

[0042] According to two eigenvalues λ ± , performing stability analysis on the fractional-order memristive Hopfield neural network control model to obtain a second stability analysis result as follows:

[0043] Case 1: when a2= 0 and k≠ 0, the eigenvalues λ ± satisfy λ + = 0, λ_ = -a1= -ka -1.9, the equilibrium point is a fold bifurcation point;

[0044] Case 2: when a1=0, i.e., k a =-1.9, it can be found that the real part of the eigenvalue is 0, and the imaginary part has two roots, on the imaginary axis, at this time the equilibrium point is a Hopf bifurcation point;

[0045] Case 3: when a1>0, a2>0, and ± The real part of the eigenvalue is a pair of negative conjugate complex roots on the negative half axis, or two negative roots, at this time the equilibrium point is stable;

[0046] Case 4: when a1<0, a1 2 -4a2<0, the eigenvalue ± is a pair of complex conjugate roots with positive real part, at this time the equilibrium point is an unstable node, which can theoretically produce chaos;

[0047] Case 5: when a1<0, a1 2 -4a2>0, the eigenvalue ± has at least one positive real root, at this time the system is unstable.

[0048] Further, after the step of determining the control parameter of the fractional order state feedback controller according to the second stability analysis result to realize feedback control of the fractional order memristive Hopfield neural network, the method further comprises:

[0049] Numerical simulation is performed on the fractional order memristive Hopfield neural network model and the fractional order memristive Hopfield neural network control model respectively, and simulation graphs are obtained for comparison of control effect.

[0050] In addition, in order to achieve the above purpose, the application further provides a state feedback control device for a fractional order memristive Hopfield neural network, comprising the following units:

[0051] A first modeling unit is configured to construct a fractional order memristive Hopfield neural network model.

[0052] A first calculation unit is configured to calculate the equilibrium point of the fractional order memristive Hopfield neural network model.

[0053] A second calculation unit is configured to calculate the divergence of the fractional order memristive Hopfield neural network model, and determine whether the fractional order memristive Hopfield neural network model can appear a chaotic attractor according to the divergence value.

[0054] The first analysis unit is configured to perform stability analysis on the fractional-order memristive Hopfield neural network model at the equilibrium point by taking the coupling strength coefficient as a bifurcation parameter, and obtain a first stability analysis result.

[0055] The second modeling unit is configured to design a fractional-order state feedback controller for the fractional-order memristive Hopfield neural network model according to the first stability analysis result, and construct a fractional-order memristive Hopfield neural network control model.

[0056] The second analysis unit is configured to perform stability analysis on the fractional-order memristive Hopfield neural network control model at the equilibrium point by taking the coupling strength coefficient as a bifurcation parameter, and obtain a second stability analysis result.

[0057] The parameter determination unit is configured to determine a control parameter of the fractional-order state feedback controller according to the second stability analysis result, so as to realize feedback control on the fractional-order memristive Hopfield neural network.

[0058] Further, the device further comprises:

[0059] The numerical simulation unit is configured to perform numerical simulation on the fractional-order memristive Hopfield neural network model and the fractional-order memristive Hopfield neural network control model respectively, and obtain simulation diagrams for comparison of control effects.

[0060] The technical scheme provided by the present application has the following beneficial effects:

[0061] The present application mainly aims at stability research and control based on a four-dimensional fractional-order memristive Hopfield neural network. First, a fractional-order memristive Hopfield neural network model is established, a memristive coupling strength coefficient is selected as a bifurcation parameter, and the stability of the system equilibrium point is researched. According to theoretical derivation, when the memristive coupling strength coefficient changes in a given range of-5-1, the fractional-order memristive Hopfield neural network model will enter a double-vortex hyperchaotic state, a single-vortex hyperchaotic state and a limit cycle state in turn, and then diverge. Through numerical simulation and stability analysis, it is known that the fractional-order memristive Hopfield neural network model is unstable. In addition, a fractional-order state feedback controller is designed according to the stability analysis result. The fractional-order state feedback controller enables the original system to be stable while retaining the original system's ability to generate chaos, and has good control effect. Finally, the correctness of the theoretical reasoning is verified through numerical simulation. BRIEF DESCRIPTION OF DRAWINGS

[0062] The present application will be further described below in combination with the drawings and embodiments. In the drawings:

[0063] The present application will be further described below in combination with the drawings and embodiments. In the drawings:Figure 1 A flow chart of a state control method of a fractional order memristive Hopfield neural network according to an embodiment of the application;

[0064] Figure 2 A structure diagram of a state feedback control device of a fractional order memristive Hopfield neural network according to an embodiment of the application;

[0065] Figure 3 A bifurcation diagram of a fractional order memristive Hopfield neural network model according to an embodiment of the application, in which the bifurcation parameter is the memristive coupling strength coefficient k;

[0066] Figure 4 When k = -1.865, the x1 and x3 phase trajectory diagram and the waveform diagram of the state variable x1 changing with time of the fractional order memristive Hopfield neural network model according to an embodiment of the application;

[0067] Figure 5 When k = -0.8, the x1 and x3 phase trajectory diagram and the waveform diagram of the state variable x1 changing with time of the fractional order memristive Hopfield neural network model according to an embodiment of the application;

[0068] Figure 6 When k = 0.1, the x1 and x3 phase trajectory diagram and the waveform diagram of the state variable x1 changing with time of the fractional order memristive Hopfield neural network model according to an embodiment of the application;

[0069] Figure 7 When k = 0.5, the x1 and x3 phase trajectory diagram and the waveform diagram of the state variable x1 changing with time of the fractional order memristive Hopfield neural network model according to an embodiment of the application;

[0070] Figure 8 When the control feedback gain coefficient k a = -1.3 and the memristive coupling strength coefficient k = -0.4, the x1 and x3 phase trajectory diagram and the waveform diagram of the state variable x1 changing with time of the fractional order memristive Hopfield neural network control model according to an embodiment of the application;

[0071] Figure 9 When the control feedback gain coefficient k a = -1.3 and the memristive coupling strength coefficient k = -0.25, the x1 and x3 phase trajectory diagram and the waveform diagram of the state variable x1 changing with time of the fractional order memristive Hopfield neural network control model according to an embodiment of the application;

[0072] Figure 10 When the control feedback gain coefficient k aThe x1-x3 trajectory diagram and the waveform diagram of the state variable x1 changing with time of the fractional-order memristive Hopfield neural network control model under the condition that the feedback gain coefficient k is 0.5 and the memristive coupling strength coefficient k is -0.1.

[0073] Figure 11 For the embodiment of the present application, when the feedback gain coefficient k is a The x1-x3 trajectory diagram and the waveform diagram of the state variable x1 changing with time of the fractional-order memristive Hopfield neural network control model under the condition that the feedback gain coefficient k is 0.5 and the memristive coupling strength coefficient k is -0.1. DETAILED DESCRIPTION

[0074] In order to have a clearer understanding of the technical features, objectives and effects of the present application, the specific embodiments of the present application will be described in detail with reference to the drawings.

[0075] Reference Figure 1 The embodiment provides a state control method of a fractional-order memristive Hopfield neural network, and specifically comprises the following steps:

[0076] S1: constructing a fractional-order memristive Hopfield neural network model;

[0077] S2: calculating an equilibrium point of the fractional-order memristive Hopfield neural network model;

[0078] S3: calculating the divergence of the fractional-order memristive Hopfield neural network model, and determining that the fractional-order memristive Hopfield neural network model can appear a chaotic attractor according to the divergence value;

[0079] S4: taking a coupling strength coefficient as a bifurcation parameter, performing stability analysis on the fractional-order memristive Hopfield neural network model at the equilibrium point, and obtaining a first stability analysis result;

[0080] S5: according to the first stability analysis result, designing a fractional-order state feedback controller for the fractional-order memristive Hopfield neural network model, and constructing a fractional-order memristive Hopfield neural network control model;

[0081] S6: taking the coupling strength coefficient as a bifurcation parameter, performing stability analysis on the fractional-order memristive Hopfield neural network control model at the equilibrium point, and obtaining a second stability analysis result;

[0082] S7: determining a control parameter of the fractional-order state feedback controller according to the second stability analysis result, so as to realize feedback control on the fractional-order memristive Hopfield neural network;

[0083] S8: numerical simulation is respectively performed on the fractional order memristive Hopfield neural network model and the fractional order memristive Hopfield neural network control model, simulation graphs are obtained, and control effects are compared.

[0084] In the traditional Hopfield neural network, the model of a memristor is introduced, such a method has a precedent in the field, and the method can enhance the memory capacity of the neural network and enhance the computing ability of the neuron. However, the type of such a memristor is relatively common, and generally it is a quadratic, cubic nonlinear segmented magnetic controlled memristor. The model has discrete points at the segmented points in theory, so it is difficult to truly realize the model when physically realizing. Even if it is realized, it needs to spend a relatively expensive cost. Therefore, the type of the memristor is changed on the basis of the traditional Hopfield neural network, and a new type of exponential smooth magnetic controlled memristor is designed, the model of which is a smooth curve and does not have discrete points, so as a neuron synapse simulation element, the new type of exponential smooth magnetic controlled memristor can not only enhance the memory and computing ability of the model, but also is easier to realize in physics. The expression of the new type of exponential smooth magnetic controlled memristor is as follows:

[0085]

[0086] Here, represents the magnetic flux, represents the memristor, v(t) represents the voltage applied to the memristor, and i(t) represents the current flowing through the memristor after the voltage is applied. The magnetic flux is considered to be generated by adding neurons, and the memristor is coupled with the neural network.

[0087] Based on the above new type of exponential smooth magnetic controlled memristor, the mathematical expression of the fractional order memristive Hopfield neural network model constructed is as follows:

[0088]

[0089] Where x1, x2, x3, x4 represent the states of four neurons respectively, which are state variables changing with time, s 11 , s 22 , s 12 , s 21 , s 31 , s 41 respectively correspond to the mutual connection weight of neuron 1 to neuron 2, the mutual connection weight of neuron 2 to neuron 1, the mutual connection weight of neuron 3 to neuron 2, the mutual connection weight of neuron 4 to neuron 1, tanh(·) is a neuron activation function, k represents the coupling strength coefficient of the memristor, and q∈(0,1] represents the order of Caputo fractional derivative; it should be noted that q=0.8 in the following cases where the order is not changed.

[0090] Equilibrium point of fractional-order memristive Hopfield neural network model satisfy the following equation group:

[0091]

[0092] The equilibrium point of formula (1) is solved, and the third and fourth equations of formula (1) can obtain x1=x2=0, and then substituted into the first two equations, the equilibrium point set E={(x1,x2,x3,x4|x1=x2=0,x3=c1,x4=c2)} can be solved, c1,c2 are constants. Therefore, let the equilibrium point In addition, under the condition that the initial value of each state variable is 1, the MATLAB calculation can obtain

[0093] In this embodiment, first, the dissipation of the fractional-order memristive Hopfield neural network model is analyzed, and the stability and bifurcation are analyzed:

[0094] Stability is a basic performance of a system. When the system is disturbed, it may deviate from the equilibrium state, and the relationship between the stability condition of the system and each parameter needs to be analyzed to determine the stability performance of the system, so as to be more beneficial to subsequent research.

[0095] In fact, dissipation is a concept of generalized Lyapunov stability. It refers to the characteristics of being able to continuously exchange energy and matter with the environment, and being able to continue to maintain the balance of the open system. Therefore, by studying the dissipative system, many previously inexplicable phenomena in nature can be explained, so the research on the dissipative system is of great research value from both the theoretical and practical perspectives.

[0096] Any nonlinear system can appear chaotic phenomenon, so the divergence of the system is The divergence of the memristive Hopfield neural network model is obtained by calculating the divergence of formula (2):

[0097]

[0098] sech(·) is the hyperbolic secant function, and 0≤sech(·)≤1, since the system parameters are set as s 11 =0.1, s 12 =2.8, s 21 =3, s 22 =4, so there is ▽·V=-2-0.1sech 2 (x1)+4sech 2(x2), therefore, there must exist a solution (x1, x2, x3, x4) that can satisfy ▽·V is less than 0. At this time, the memristive Hopfield neural network model is dissipative, in this case, the system converges exponentially, the fractional order memristive Hopfield neural network model is the same, that is, when t tends to infinity, each unit state volume element of the system converges to 0. Finally, it tends to a attractor, that is, the system can theoretically appear chaotic attractor.

[0099] According to the equilibrium point The Jacobian matrix of the fractional order memristive Hopfield neural network model is obtained as follows:

[0100]

[0101] Wherein each parameter is as follows:

[0102]

[0103] The characteristic equation is obtained as follows:

[0104]

[0105] The above characteristic equation (5) is equivalent to:

[0106]

[0107] Written in equation form is:

[0108] L0s 4α +L1s 3α +L2s 2α =0 (7)

[0109] Wherein

[0110]

[0111] In the above formula, the uncertain parameter is the coupling strength coefficient k of the memristor, therefore, take this parameter as the bifurcation parameter to analyze the stability. Set the parameter values of the fractional order memristive Hopfield neural network model as s 11 = 0.1, s 12 = 2.8, s 21 = 3, s 22 = 4, therefore, there is ·

[0112]

[0113] According to the Routh-Hurwitz stability criterion, it can be known that when L0>0, L1<0, the characteristic equation has zero points in the right half plane, which is an unstable equilibrium point, and from L0→L1, the sign changes from positive to negative, so there is at least one root in the right half plane, if L2>0, the sign changes once, there are two roots in the right half plane, if L2<0, there is only one unstable equilibrium point. In summary, the system is unstable at the equilibrium point, and the following numerical simulation verifies the theoretical derivation. Figure 3 is the bifurcation diagram of the fractional-order memristive Hopfield neural network model with the coupling strength coefficient k of the memristor as the bifurcation parameter, in which Figure 3 , the cross-section graph of the system state variable can be found to exist double-scroll hyperchaos, single-scroll hyperchaos, and limit cycle phenomenon, and the phase trajectory graph can further prove that the fractional-order memristive Hopfield neural network model satisfies:

[0114] (1) When-5<k<-1.026, the system based on the equilibrium point appears double-scroll hyperchaos, and the system is unstable;

[0115] (2) When-1.026<k<-0.305, the system appears single-scroll hyperchaos, and the system is unstable;

[0116] (3) When-0.305<k<0.365, the system appears limit cycle phenomenon, and the system is unstable;

[0117] (4) When-0.365<k<1, the system diverges.

[0118] In the following, the corresponding x1 and x3 phase trajectory graphs and the waveform graphs of the state variable x1 changing with time are taken respectively. Wherein, let q=0.8.

[0119] Figure 4 is the x1 and x3 phase trajectory graph and the waveform graph of the state variable x1 changing with time when k=-1.865;

[0120] Figure 5 is the x1 and x3 phase trajectory graph and the waveform graph of the state variable x1 changing with time when k=-0.8;

[0121] Figure 6 is the x1 and x3 phase trajectory graph and the waveform graph of the state variable x1 changing with time when k=0.1;

[0122] Figure 7 is the x1 and x3 phase trajectory graph and the waveform graph of the state variable x1 changing with time when k=0.5;

[0123] In summary, the fractional-order memristive Hopfield neural network model can produce chaotic phenomenon both in theory and in practice, but the time series of each state variable, i.e. the waveform diagram, does not converge, which means that the system is unstable. Therefore, a state feedback controller is designed to control the system, which can make the system stable while keeping the chaotic characteristics.

[0124] In this embodiment, a fractional-order state feedback controller is added to the nonlinear constant system based on the memristor (fractional-order memristive Hopfield neural network model), which can make the system converge and become stable while keeping the chaotic characteristics of the original system. After adding the state feedback controller, the mathematical expression of the fractional-order memristive Hopfield neural network control model is as follows:

[0125]

[0126] where k a is the feedback gain coefficient, and the meanings of the other parameters are the same as those of the fractional-order memristive Hopfield neural network model (2).

[0127] The Jacobian matrix of the fractional-order memristive Hopfield neural network control model is calculated as follows:

[0128]

[0129] where A1, A2, B1, and B2 are the same as those in equation (4).

[0130] The characteristic equation corresponding to the fractional-order memristive Hopfield neural network control model is calculated as follows:

[0131]

[0132] Equation (11) can be simplified as follows:

[0133]

[0134] which can be written in the form of an equation as follows:

[0135]

[0136] In the above equation, a1=-(k a +1.9),

[0137] The characteristic value of the fractional-order memristive Hopfield neural network control model is obtained as follows:

[0138]

[0139] According to the two characteristic values λ ±The analysis yielded the parameter k corresponding to different ranges of eigenvalues. a The impact of k on system stability is discussed below. Stability analysis of the fractional memristor Hopfield neural network control model is performed for some key cases to find a control effect that allows the system to converge at a certain point while exhibiting chaos, thus achieving system stability.

[0140] Case 1: When a2 = 0, the calculated value corresponding to the equilibrium point is...

[0141] When a2=0 and k≠0, that is

[0142]

[0143] Since a2 = 0, the eigenvalue λ ± for

[0144] λ + =0,λ _ =-a1=-k a -1.9 (16)

[0145] Therefore, the above equation indicates that this is a folding bifurcation point.

[0146] Case 2: When a1 = 0, i.e., k a If the value is -1.9, we can find that the real part of the eigenvalue is 0 and the imaginary part has two roots on the imaginary axis. The system is a Hopf bifurcation point at this point.

[0147] Case 3: When a1 > 0 and a2 > 0, that is...

[0148]

[0149] Therefore, a1 > 0 indicates that λ ± The real part of the eigenvalue is less than 0, λ ± It is a pair of negative conjugate complex roots on the negative half-axis, or two negative roots. Therefore, in this case, the equilibrium point is a stable node.

[0150] Case 4: When a1 < 0, a1 2 -4a2 < 0, that is

[0151]

[0152] because So 16k 2 The inequality above can only be satisfied if -30.89k-210 < 0. In this case, -2.785 < k < 4.715, and the eigenvalue λ is... ± If a pair of complex conjugate roots has a positive real part, then the equilibrium point is an unstable node.

[0153] Case 5: when a1< 0, a1 2 -4a2> 0, as above analysis, the eigenvalue λ ± There is at least one positive real root. At this time, the system can occur chaotic phenomenon.

[0154] To make the system exist chaotic phenomenon, therefore k a The value of the case is case 4 or 5, that is, k a > -1.9. It is known that the memristive coupling coefficient k takes any value, the uncontrolled system is unstable, so as a control, take the memristive coupling coefficient k = -0.1, the state quantity waveform of the original fractional order Hopfield neural network model with memristive must not converge. At this time, if the state feedback gain k a = -1.3. Then based on the following lemma 1, 2, and has been found:

[0155]

[0156]

[0157] Therefore, for λ ± , k = -0.1, k a = -1.3, c1 = 0.8463, c2 = 0.5623 into the above formula can be solved λ ± = 0.3 ± 2.22i,

[0158] So

[0159]

[0160] Therefore, the lemma 2 is proved. So theoretically, take the feedback gain coefficient k a = -1.3, the fractional order Hopfield neural network control model with memristive is stable at k = -0.1 (the original fractional order Hopfield neural network model with memristive is unstable at this point), the waveform should be convergent. From Figure 11 It can be verified.

[0161] The above stability analysis method references the following theoretical theorems:

[0162] Lemma 1: The sufficient and necessary condition for the above linear autonomous fractional order system to be asymptotically stable is:

[0163]

[0164] Where arg(·) represents the argument principal value, Λ(A) is the set of all eigenvalues λ of matrix A;

[0165] Corollary 1.1: When 0 < α < 1, the sufficient and necessary condition for the above linear autonomous fractional order system to be asymptotically stable is equivalent to:

[0166]

[0167] where Re(·) and Im(·) denote the real and imaginary parts of a complex number, respectively.

[0168] For nonlinear fractional-order systems such as fractional-order Hopfield neural networks with memory, Lemma 1 is also applicable to analyze the local stability of the system. For example, consider the following nonlinear autonomous fractional-order system:

[0169]

[0170] where x(t) ∈ R n is the state vector, f: R n → R n is a nonlinear function. For nonlinear fractional-order systems, the stability of equilibrium points is mainly studied. Therefore, the definition of equilibrium points is given.

[0171] Definition: If there exists a constant vector such that the above nonlinear fractional-order system satisfies then is called an equilibrium point of the system.

[0172] Lemma 2: Assume that the above nonlinear fractional-order system has an equilibrium point then the sufficient condition for the local asymptotic stability of the equilibrium point is:

[0173]

[0174] where is the Jacobin matrix of f at , i.e.,

[0175]

[0176] Lemma 3: Routh-Hurwitz criterion: Consider the following characteristic polynomial equation:

[0177] a n λ n +a n-1 λ n-1 +…+a1λ+a0=0

[0178] Assume that a n > 0, and determine the position of the zero of the above polynomial by constructing the Routh table. The Routh tables for n being even and odd are as follows, respectively:

[0179]

[0180]

[0181] where a -1 ,a -2 To make 0, other parameters are as follows:

[0182]

[0183]

[0184] If the first column elements of the Routh table are all greater than 0, it is equivalent to that all zero points of the polynomial are located in the left half plane of the negative half axis, and the system belongs to the stable category. If one of the elements is negative, the number of zero points in the right half plane is equal to the number of times of positive-negative transformation before and after the cumulative negative.

[0185] In summary, taking the feedback gain coefficient k a =-1.3, the system can have stability while maintaining the original fractional-order memristive Hopfield neural network model capable of generating chaos.

[0186] In addition, in order to realize the state feedback control method of the fractional-order memristive Hopfield neural network, the embodiment also provides a state feedback control device of a fractional-order memristive Hopfield neural network. Figure 2 The device comprises the following units:

[0187] A first modeling unit 1 is configured to construct a fractional-order memristive Hopfield neural network model.

[0188] A first calculation unit 2 is configured to calculate an equilibrium point of the fractional-order memristive Hopfield neural network model.

[0189] A second calculation unit 3 is configured to calculate a divergence of the fractional-order memristive Hopfield neural network model, and determine whether the fractional-order memristive Hopfield neural network model can have a chaotic attractor according to the divergence value.

[0190] A first analysis unit 4 is configured to take a coupling strength coefficient as a bifurcation parameter, analyze stability of the fractional-order memristive Hopfield neural network model at the equilibrium point, and obtain a first stability analysis result.

[0191] A second modeling unit 5 is configured to design a fractional-order state feedback controller for the fractional-order memristive Hopfield neural network model according to the first stability analysis result, and construct a fractional-order memristive Hopfield neural network control model.

[0192] The second analysis unit 6 is configured to perform stability analysis on the fractional-order memristive Hopfield neural network control model at the equilibrium point by taking the coupling strength coefficient as a bifurcation parameter, and obtain a second stability analysis result.

[0193] The parameter determination unit 7 is configured to determine the control parameter of the fractional-order state feedback controller according to the second stability analysis result, so as to realize feedback control on the fractional-order memristive Hopfield neural network.

[0194] The numerical simulation unit 8 is configured to respectively perform numerical simulation on the fractional-order memristive Hopfield neural network model and the fractional-order memristive Hopfield neural network control model, and obtain simulation graphs for comparison of control effects.

[0195] Further, in order to verify the control effect of the fractional-order state feedback controller in the embodiment, the following simulation experiment is performed:

[0196] The given fractional-order memristive Hopfield neural network model is as follows:

[0197]

[0198] The values of the system parameters are as follows:

[0199] s 11 = 0.1, s 12 = 2.8, s 21 = 3, s 22 = 4, and the equilibrium point of the system at this time is calculated as The following uses MATLAB drawing software to perform numerical simulation.

[0200] Figure 4 When k = -1.865, the x1-x3 trajectory graph of the fractional-order memristive Hopfield neural network model and the waveform graph of the state variable x1 changing with time are as follows:

[0201] Figure 5 When k = -0.8, the x1-x3 trajectory graph of the fractional-order memristive Hopfield neural network model and the waveform graph of the state variable x1 changing with time are as follows:

[0202] Figure 6 When k = 0.1, the x1-x3 trajectory graph of the fractional-order memristive Hopfield neural network model and the waveform graph of the state variable x1 changing with time are as follows:

[0203] Figure 7 When k = 0.5, the x1-x3 trajectory graph of the fractional-order memristive Hopfield neural network model and the waveform graph of the state variable x1 changing with time are as follows:

[0204] The above is an example simulation of the fractional-order memristive Hopfield neural network model. The system can produce interesting chaotic phenomena, but from the waveform graph, it can be observed that the system does not converge, and the waveform is always oscillating, indicating that the system is unstable.

[0205] In this embodiment, the fractional-order memristive Hopfield neural network control model obtained by adding a fractional-order state feedback controller is also simulated as follows:

[0206] Figure 8 For the case where the control feedback gain coefficient k a = -1.3 and the memristive coupling strength coefficient k = -0.4, the x1 and x3 phase trajectory graph and the waveform graph of the state variable x1 changing with time of the fractional-order memristive Hopfield neural network control model produce similar double-scroll chaotic phenomena, and the waveform is relatively smooth, which can verify the improvement effect of the added controller on the original system.

[0207] Figure 9 For the case where the control feedback gain coefficient k a = -1.3 and the memristive coupling strength coefficient k = -0.25, the x1 and x3 phase trajectory graph and the waveform graph of the state variable x1 changing with time of the fractional-order memristive Hopfield neural network control model produce single-scroll chaotic phenomena.

[0208] Figure 10 For the case where the control feedback gain coefficient k a = -1.3 and the memristive coupling strength coefficient k = -0.175, the x1 and x3 phase trajectory graph and the waveform graph of the state variable x1 changing with time of the fractional-order memristive Hopfield neural network control model produce limit cycles, and the system is asymptotically stable.

[0209] Figure 11 For the case where the control feedback gain coefficient k a = -1.3 and the memristive coupling strength coefficient k = -0.1, the x1 and x3 phase trajectory graph and the waveform graph of the state variable x1 changing with time of the fractional-order memristive Hopfield neural network control model converge, and the waveform tends to be stable. Compared with the fractional-order memristive Hopfield neural network model, the added controller has good control effect.

[0210] It should be noted that, in the present document, the terms "comprising", "comprising" or any other variant thereof are intended to cover non-exclusive inclusions, so that a process, method, article or system that includes a list of elements not only includes those elements, but also includes other elements not explicitly listed, or inherent to such a process, method, article or system. Without more limitations, the element defined by the statement "comprising a" does not exclude the presence of other identical elements in the process, method, article or system that includes the element.

[0211] The above-mentioned embodiment numbers of the present application are only for description, and do not represent the advantages and disadvantages of the embodiments. In the unit claims of several devices, several of these devices can be embodied by the same hardware item. The use of the words first, second, and third does not represent any order, and these words can be interpreted as identification.

[0212] The above is only the preferred embodiment of the present application, and does not limit the patent scope of the present application. Any equivalent structure or equivalent process transformation, or direct or indirect application in other related technical fields, is also included in the patent protection scope of the present application.

Claims

1. A state feedback control method for a fractional-order memristor Hopfield neural network, characterized in that, Includes the following steps: Construct a fractional-order memristor Hopfield neural network model; Calculate the equilibrium point of the fractional-order memristor Hopfield neural network model; Calculate the divergence of the fractional-order memristor Hopfield neural network model, and determine whether the fractional-order memristor Hopfield neural network model can exhibit chaotic attractors based on the divergence value; Using the coupling strength coefficient as a bifurcation parameter, stability analysis is performed on the fractional-order memristor Hopfield neural network model at the equilibrium point to obtain the first stability analysis result. Based on the first stability analysis results, a fractional-order state feedback controller is designed for the fractional-order memristor Hopfield neural network model, and a fractional-order memristor Hopfield neural network control model is constructed. Using the coupling strength coefficient as a bifurcation parameter, stability analysis is performed on the fractional-order memristor Hopfield neural network control model at the equilibrium point to obtain a second stability analysis result. The control parameters of the fractional-order state feedback controller are determined based on the second stability analysis results to achieve feedback control of the fractional-order memristor Hopfield neural network. The mathematical expression of the fractional-order memristor Hopfield neural network model is as follows; in These represent the states of four neurons, all of which are state variables that change over time. These are the self-connection synaptic weights of neurons 1 and 2. These correspond to the interconnection weights of neuron 1 to neuron 2, neuron 2 to neuron 1, neuron 3 to neuron 2, and neuron 4 to neuron 1, respectively. It is the neuron activation function. This represents the coupling strength coefficient of the memristor. This indicates the order of the Caputo fractional derivative.

2. The state feedback control method for a fractional-order memristor Hopfield neural network according to claim 1, characterized in that, The calculation of the equilibrium point of the fractional-order memristor Hopfield neural network model includes: Set the right-hand side of the fractional-order memristor Hopfield neural network model to zero, that is: ， The equilibrium point was calculated. ,in It is a constant.

3. The state feedback control method for a fractional-order memristor Hopfield neural network according to claim 1, characterized in that, The step of calculating the divergence of the fractional-order memristor Hopfield neural network model and determining whether the fractional-order memristor Hopfield neural network model can exhibit chaotic attractors based on the divergence value includes: The divergence of the memristor Hopfield neural network model is calculated as follows: Among them, among them, It is a hyperbolic secant function, and has The system parameters are set to have Therefore, a solution must exist. Able to meet If the value is less than 0, for a fractional-order memristor Hopfield neural network model, as time approaches infinity, each unit state volume element of the fractional-order memristor Hopfield neural network model converges to 0, eventually tending to a chaotic attractor.

4. The state feedback control method for a fractional-order memristor Hopfield neural network according to claim 1, characterized in that, The mathematical expression for the fractional-order memristor Hopfield neural network control model is: in, It is the state feedback gain coefficient.

5. The state feedback control method for a fractional-order memristor Hopfield neural network according to claim 2, characterized in that, The step of using the coupling strength coefficient as a bifurcation parameter to perform stability analysis on the fractional-order memristor Hopfield neural network model at the equilibrium point to obtain the first stability analysis result specifically includes: The characteristic equation of the fractional-order memristor Hopfield neural network model is calculated as follows: in Set system parameters therefore According to the Routh criterion, the system is unstable and exhibits chaotic phenomena, with the waveforms of all state variables oscillating and unable to converge.

6. The state feedback control method for a fractional-order memristor Hopfield neural network according to claim 3, characterized in that, The step of using the coupling strength coefficient as a bifurcation parameter to perform stability analysis on the fractional-order memristor Hopfield neural network control model at the equilibrium point to obtain a second stability analysis result specifically includes: Calculate the eigenvalues ​​of the fractional-order memristor Hopfield neural network control model: in Based on two eigenvalues The stability analysis of the fractional-order memristor Hopfield neural network control model yielded the following second stability analysis result: Scenario 1: When and Then the eigenvalues satisfy At this point, the equilibrium point is a folding bifurcation point; Scenario 2: When ,Right now Then we can find that the real part of the eigenvalue is 0 and the imaginary part has two roots on the imaginary axis. At this time, the equilibrium point is a Hopf bifurcation point. Scenario 3: When , The real part of the eigenvalue is a pair of negative conjugate complex roots on the negative half-axis, or two negative roots, in which case the equilibrium point is stable; Scenario 4: When eigenvalues If a pair of complex conjugate roots have positive real parts, then the equilibrium point is an unstable node that can theoretically generate chaos. Scenario 5: When eigenvalues There is at least one positive real root, at which point the system is unstable.

7. The state feedback control method for a fractional-order memristor Hopfield neural network according to claim 1, characterized in that, After the step of determining the control parameters of the fractional-order state feedback controller based on the second stability analysis result to achieve feedback control of the fractional-order memristor Hopfield neural network, the method further includes: Numerical simulations were performed on the fractional-order memristor Hopfield neural network model and the fractional-order memristor Hopfield neural network control model, and the simulation results were compared to assess the control effectiveness.

8. A state feedback control device for a fractional-order memristor Hopfield neural network, characterized in that, Includes the following units: The first modeling unit is used to construct a fractional-order memristor Hopfield neural network model; The first computing unit is used to calculate the equilibrium point of the fractional-order memristor Hopfield neural network model; The second calculation unit is used to calculate the divergence of the fractional-order memristor Hopfield neural network model, and to determine whether the fractional-order memristor Hopfield neural network model can exhibit chaotic attractors based on the divergence value. The first analysis unit is used to perform stability analysis on the fractional-order memristor Hopfield neural network model at the equilibrium point, using the coupling strength coefficient as a bifurcation parameter, to obtain the first stability analysis result. The second modeling unit is used to design a fractional-order state feedback controller for the fractional-order memristor Hopfield neural network model based on the first stability analysis results, and to construct a fractional-order memristor Hopfield neural network control model. The second analysis unit is used to perform stability analysis on the fractional-order memristor Hopfield neural network control model at the equilibrium point, using the coupling strength coefficient as a bifurcation parameter, to obtain the second stability analysis result. The parameter determination unit is used to determine the control parameters of the fractional-order state feedback controller based on the second stability analysis result, so as to realize the feedback control of the fractional-order memristor Hopfield neural network. The mathematical expression of the fractional-order memristor Hopfield neural network model is as follows; in These represent the states of four neurons, all of which are state variables that change over time. These are the self-connection synaptic weights of neurons 1 and 2. These correspond to the interconnection weights of neuron 1 to neuron 2, neuron 2 to neuron 1, neuron 3 to neuron 2, and neuron 4 to neuron 1, respectively. It is the neuron activation function. This represents the coupling strength coefficient of the memristor. This indicates the order of the Caputo fractional derivative.

9. The state feedback control device for a fractional-order memristor Hopfield neural network according to claim 8, characterized in that, Also includes: The numerical simulation unit is used to perform numerical simulations on the fractional-order memristor Hopfield neural network model and the fractional-order memristor Hopfield neural network control model respectively, and to obtain simulation diagrams for comparison of control effects.

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