A control method and system of a dual-neuron neural network model PD controller

Abstract

The application provides a control method and system of a PD controller of a double-neuron neural network model under the influence of time delay and diffusion, and the method comprises the following steps: establishing a time delay reaction diffusion double-neuron neural network model described by a partial differential equation; applying a PD controller on the basis of the time delay reaction diffusion double-neuron neural network model without control to obtain a time delay reaction diffusion double-neuron neural network model with the PD controller; obtaining a characteristic equation of a controlled network after linearization by taking the time delay reaction diffusion double-neuron neural network model under the action of the PD controller; selecting a communication time delay as a bifurcation parameter, and performing stability analysis and bifurcation analysis on the characteristic equation of the controlled network after linearization, and further selecting appropriate control parameters so that the network is locally asymptotically stable near an equilibrium point. The application can better realize the regulation and control of network stability, the control parameters are flexible to adjust, the actual operation is simple and easy to implement, and the control effect is obvious.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of controller, and particularly relates to a control method and system of a PD controller of a time-delay reaction-diffusion double-neuron neural network model. BACKGROUND

[0002] In the 1940s, W. Mcculloch and W. Pitts described the topological structure of neural networks according to biological systems, and first obtained a mathematical model of neural networks. Based on the model, subsequent scholars proposed diversified topological structures to better describe biological neural networks. In the 1980s, an American scholar J. Hopfield proposed a Hopfield neural network model and introduced an energy function. Studies have shown that the relationship between neural networks and dynamics can be explored by a series of methods of dynamics of nonlinear systems. Nowadays, with the development of artificial intelligence and deep learning technology, neural network research based on brain science and cognitive science has received extensive attention. In the neural network model, the movement of electrons and the transmission of neurotransmitters may cause differences in the state of neurons in space and time. At present, only a few scholars have studied this situation.

[0003] In the related research of neural networks, Hopf bifurcation is a common dynamic behavior, which usually shows that when a certain parameter in the neural network exceeds the bifurcation threshold, the network changes from stable to unstable. In order to regulate the dynamic characteristics of the neural network model, scholars have introduced some control strategies, including time-delay feedback control, mixed control, PI control and PD control, etc. PD control has the nature of advance, which can improve the stability of the neural network to a certain extent, reduce the maximum deviation and residual error, speed up the control process, and improve the control quality. SUMMARY

[0004] The main purpose of the present application is to provide a novel control method and system of a PD controller. The PD controller of the present application optimizes and regulates the spatiotemporal dynamic behavior of a double-neuron neural network under the condition of considering the diffusion effect and the existence of time delay, so as to effectively increase the stable domain of the network and delay the occurrence of Hopf bifurcation.

[0005] To achieve the above purpose, the present application provides a control method of a PD controller of a double-neuron neural network model, comprising the following steps:

[0006] Step S1: establishing a time-delay reaction-diffusion double-neuron neural network model without control;

[0007] Step S2: introducing a PD controller on the basis of the established time-delay reaction-diffusion double-neuron neural network model, and calculating the equilibrium point of the network;

[0008] Step S3: linearizing the PD-controlled time-delay reaction-diffusion two-neuron neural network model at the equilibrium point to obtain a characteristic equation of the linearized controlled network;

[0009] Step S4: selecting the time delay as a Hopf bifurcation parameter, and adjusting and selecting appropriate controller parameters k d 、k p , so that the neural network remains asymptotically stable near the equilibrium point and improves the stability threshold of the network.

[0010] Further improvements of the present application are that in step S1, the mathematical expression of the uncontrolled time-delay reaction-diffusion two-neuron neural network model (1) is:

[0011]

[0012]

[0013]

[0014]

[0015] wherein u(t,x) and v(t,x) represent the states of the two neurons at time t; Ω=(0,π) is a bounded domain; Δ represents the Laplace operator on R n ; τ is the time required for information transmission in the neural network; c i (i=1,2) represent the self-feedback parameters of the neurons; a i (i=1,2) are the connection weights between the neurons; b i (i=1,2) are the self-connection weights of the two neurons; d1 and d2 represent the self-diffusion coefficients of the two neurons, respectively; f i (·) (i=1,2) is an activation function, and satisfies f i (0) (i=1,2)=0 and f i ′(0) (i=1,2)≠0.

[0016] Further improvements of the present application are that the partial differential PD controller is defined as The controlled time-delay reaction-diffusion two-neuron neural network model (2) is obtained as:

[0017]

[0018]

[0019]

[0020]

[0021] The equilibrium point of the system is the origin O=(0,0) T ; Next, the communication delay τ will be selected as the bifurcation parameter, and the parameter condition for Hopf bifurcation of the network will be derived; Then, the two controller parameters k d , k p are adjusted to ensure that the network remains locally asymptotically stable at the origin and to expand the stable region of the network to delay the occurrence of bifurcation.

[0022] Further improvement of the present application is that, in step S3, the linearization process at the equilibrium point of the controlled model obtains the characteristic equation of the controlled system:

[0023] det(λI-M k -G1-G2e -λτ )=0,

[0024] Where -k 2 (k∈N0) is the eigenvalue of Laplace operator Δ, and I is the second-order unit matrix; and

[0025]

[0026] b 11 =b1f1′(0), a 11 =a1f2′(0),

[0027] b 22 =b2f3′(0), a 22 =a2f4′(0).

[0028] Then the characteristic equation of the controlled system is:

[0029] λ 2 +A(k)λ+B(k)+Ce -2λτ =0,

[0030] Where,

[0031]

[0032]

[0033]

[0034] Further improvement of the present application is that, step S4 further comprises:

[0035] (1) When the network has no time delay (τ=0), the characteristic equation can be rewritten as:

[0036] λ 2 +A(k)λ+B(k)+C=0,

[0037] Discuss and study the sufficient condition of the existence of negative real part of the above equation to ensure the stability of the network under the condition of no time delay;

[0038] (2) When the network has communication delay (τ>0), explore the condition of Hopf bifurcation of the network, and calculate the bifurcation threshold point τ0; by comparing the size of the bifurcation threshold point τ0 and the communication delay τ, it is determined whether the network will occur Hopf bifurcation.

[0039] In order to achieve the above object of the application, the application further provides a control system of a dual-neuron neural network model PD controller, which is used for executing the foregoing method.

[0040] The beneficial effects of the application are as follows:

[0041] 1. The time-delay reaction-diffusion neural network model proposed in the application not only considers the communication delay between neurons, but also considers the diffusion phenomenon in the neural network caused by the diffusion of electrons and the concentration difference of neurotransmitters.

[0042] 2. The PD controller designed in the application has many adjustable parameters and good applicability, and the control method can be widely applied to most complex dynamic network models.

[0043] 3. Compared with other control methods, the controller proposed in the application can effectively reduce the maximum deviation and residual error, speed up the control process, and improve the control quality. DETAILED DESCRIPTION

[0044] Figure 1 The method flowchart described in the application;

[0045] Figure 2 The time and space waveform diagram of the stable network under the condition of the uncontrolled network (12) d1=0.8, d2=0.01, τ=0.112;

[0046] Figure 3 The time and space waveform diagram of the stable neural network under the condition of the uncontrolled network (12) d1=0.8, d2=0.01, τ=0.112;

[0047] Figure 4 The time and space waveform of the unstable neural network under the condition of the uncontrolled network (12) d1=0.8, d2=0.01, τ=0.143;

[0048] Figure 5 The time and space waveform of the unstable neural network under the condition of the uncontrolled network (12) d1=0.8, d2=0.01, τ=0.143;

[0049] Figure 6For the controlled network (13) with parameters d1 = 0.8, d2 = 0.01, τ = 0.143, k p =-0.05,k d =-0.8, the stable spatiotemporal waveform of the neural network;

[0050] Figure 7 For the controlled network (13) with parameters d1 = 0.8, d2 = 0.01, τ = 0.143, k p =-0.05,k d =-0.8, the stable spatiotemporal waveform of the neural network;

[0051] Figure 8 For the controlled network model, d1 = 0.8, d2 = 0.01, k p =-0.05 when another control parameter k d The relationship diagram between it and the bifurcation threshold τ0;

[0052] Figure 9 For the controlled network model, d1 = 0.8, d2 = 0.01, k d =-0.05 when another control parameter k p Relationship diagram with bifurcation threshold τ0. DETAILED DESCRIPTION

[0053] In order to make the objectives, technical solutions and advantages of the present invention more clear, the present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0054] It should be emphasized that, in the process of describing the present invention, various formulas and constraints are distinguished by using consistent labels, but it is not excluded that different labels are used to mark the same formulas and / or constraints. The purpose of this setting is to more clearly illustrate the characteristics of the present invention.

[0055] like Figure 1 As shown, the present invention is the design and implementation of a dual-neuron neural network model PD controller under time lag and diffusion induction. The specific design steps are as follows:

[0056] Step 1: Transform the uncontrolled model and apply a controller to it.

[0057] The uncontrolled time-delay reaction-diffusion dual-neuron neural network model is:

[0058]

[0059]

[0060]

[0061]

[0062] where u(t, x) and v(t, x) represent the states of two neurons at time t, respectively; Ω = (0, π) is a bounded domain; Δ represents the Laplace operator on R n ; τ is the time required for information transmission in the neural network; c i (i = 1, 2) represent the self-feedback parameters of neurons; a i (i = 1, 2) are the connection weights between neurons; b i (i = 1, 2) are the self-connection weights of two neurons; d1and d2represent the self-diffusion coefficients of two neurons, respectively. f i (·) (i = 1, 2) are activation functions, and satisfy f i (0) (i = 1, 2) = 0 and f′ i (0) (i = 1, 2) ≠ 0.

[0063] Subsequently, the partial differential PD controller is defined as follows:

[0064]

[0065] where k p , k d are two adjustable controller parameters, and e(x, t) is the difference between the current state and the stable state. Therefore, the model after applying the partial differential PD controller can be described as:

[0066]

[0067]

[0068]

[0069]

[0070] Since the PD control does not change the equilibrium point of the system, it is not difficult to obtain that the network has an equilibrium point o(0, 0) T .

[0071] Step two: linearize the model (2) at the origin, and obtain the linearized system as:

[0072]

[0073] where

[0074]

[0075] b 11 = b1f′1(0), a 11=a1f′2(0),

[0076] b 22 =b2f′3(0),a 22 =a2f′4(0).

[0077] -k 2 (k∈N0) is the characteristic root of Δ. Further, the characteristic equation of the controlled system (2) is obtained as:

[0078] det(λI-M k -G1-G2e -λτ )=0, (4)

[0079] Right now

[0080] λ 2 +A(k)λ+B(k)+Ce -2λτ =0, (5)

[0081] in

[0082]

[0083]

[0084] g1=b 11 -c1,g2=b 22 -c2.

[0085] Step 3: Select the system bifurcation parameters and perform stability analysis on the controlled system.

[0086] When the network communication delay τ = 0, the characteristic equation (5) can be rewritten as:

[0087] λ 2 +A(k)λ+B(k)+C=0, (6)

[0088] The following will discuss the conditions under which the roots of Equation (6) have negative real parts. According to the Routh-Hurwitz criterion, when

[0089] When A(k)>0, B(k)+C>0, all roots of Equation (6) have negative real parts. Therefore, we can see that when the inequalities A(k)>0, B(k)+C>0 hold, the network without time delay is always in a stable state.

[0090] Furthermore, we can draw the following conclusion:

[0091] When the controller parameters satisfy the above inequality, the network is stable without time delay.

[0092] When there is a communication delay (τ>0) in the network, substituting λ = iω into the characteristic equation (5) gives

[0093] - ω 2 + A(k) iω + B(k) + Ce -2iωτ = 0, (7)

[0094] Separating the real and imaginary parts of equation (7) gives:

[0095]

[0096] Adding the squares of the two equations in (8) gives:

[0097] ω 4 + (A(k) 2 - 2B(k)) ω 2 + B(k) 2 - C 2 = 0, (9)

[0098] where:

[0099]

[0100]

[0101] To simplify the symbolic operations, we let hi = A(k) 2 - 2B(k), h2 = B(k) 2 - C 2 , and define

[0102] ω 4 + hi ω 2 + h2 = 0. (10)

[0103] When h2 < 0, equation (10) has at least one positive root ω0, which corresponds to the solvable time delay τ0 at this time

[0104]

[0105] At this time, τ0 is a critical point, and further discussion of the crossing condition is needed to prove whether it is a bifurcation threshold point. The bifurcation point is a point of sudden change from stable to unstable for a system, and for the characteristic equation of the network, it is usually reflected in the process of the characteristic root crossing from the left half-plane to the right half-plane with the change of the key parameter. In order to ensure that this crossing process can occur, it should be ensured that the condition that the real part of the derivative of the characteristic equation with respect to the "key parameter" is greater than zero is met, which is also called the "crossing condition".

[0106] Taking the derivative of both sides of the characteristic equation (5) with respect to τ gives

[0107]

[0108] Substituting λ=iω0,τ=τ0, we can further obtain the real part of the derivative:

[0109] in

[0110]

[0111] in

[0112] T1=2ω0cos2ω0τ0-A(k)sin2ω0τ0,

[0113] T2=2ω0C.

[0114] like It is established, and it can be seen that the crossing condition is met at τ0. Therefore, we can conclude that τ0 is the bifurcation point of the original controlled system. In summary, the following conclusion can be drawn:

[0115] When the lag is selected to satisfy τ∈[0,τ0), the controlled neural network is at the origin o(0,0) T is locally asymptotically stable;

[0116] When the time lag satisfies τ=τ0, the network is at the origin o(0,0) T Hopf bifurcation occurs around it, and when τ is greater than τ0, the system produces a set of periodic solutions.

[0117] The present invention is further described below using an example. The present invention is verified using Matlab simulation.

[0118] Step 1: Select the uncontrolled time-delay reaction-diffusion dual-neuron neural network model. The specific mathematical expression is as follows:

[0119]

[0120]

[0121]

[0122]

[0123] where f i (·)(i=1,2,3,4)=tanh(·). The calculation program shows that the bifurcation point of model (12) with ω0=1.311 is τ0=0.1274.

[0124] like Figure 2 As shown in Figure 3, when the communication delay is selected as τ=0.112<τ0, the uncontrolled model (12) is asymptotically stable at the equilibrium point and no Hopf bifurcation occurs.

[0125] As Figure 4 shown in Fig. 5, when the leakage time delay τ = 0.143 > τ0is selected, the uncontrolled network (12) loses stability, produces oscillation, and Hopf bifurcation occurs around the equilibrium point.

[0126] Step 3: The partial differential PD controller is applied to the two-neuron neural network model under the influence of time delay and diffusion, and the controller parameters k p = -0.05, k d = -0.8. At this time, the specific mathematical expression of the controlled network is as follows:

[0127]

[0128]

[0129]

[0130]

[0131] From the calculation program, it can be obtained that the bifurcation point of the uncontrolled system is τ0= 0.1274.

[0132] As Figure 6 shown in Fig. 7, when the communication time delay τ = 0.143 > τ0is selected, the controlled network is still asymptotically stable at the equilibrium point.

[0133] As Figure 8 shown, when k p = -0.05 is fixed, the bifurcation threshold τ0gradually becomes smaller with the increase of the control parameter k d , which indicates that the stability of the network is deteriorating, and the Hopf bifurcation will occur in advance.

[0134] As Figure 9 shown, when k d = -0.05 is fixed, the bifurcation threshold τ0gradually becomes smaller with the increase of the control parameter k p , which indicates that the stability of the network is gradually deteriorating, and the Hopf bifurcation will occur in advance.

[0135] It should be noted that in this paper, relational terms such as first and second are used merely to distinguish one entity or action from another entity or action, and do not necessarily require or imply that there is any such actual relationship or order between these entities or actions.

[0136] The above examples are only used to illustrate the technical solutions of the present application but not limit the present application. Although the present application is described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present application can be modified or equivalent replaced without departing from the spirit and scope of the technical solutions of the present application.

Claims

1. A control method of a dual-neuron neural network model PD controller, characterized in that: The following steps are involved: Step S1: Establish an uncontrolled time-delay reaction-diffusion dual-neuron neural network model; Step S2: Based on the established time-delay reaction-diffusion dual-neuron neural network model, a PD controller is introduced to calculate the equilibrium point of the network; Step S3: linearize the time-delay reaction-diffusion two-neuron neural network model controlled by PD at the equilibrium point to obtain the characteristic equation of the linearized controlled network; Step S4: Select the time delay as the Hopf bifurcation parameter, perform Hopf bifurcation analysis on the characteristic equation of the linearized controlled network, and adjust and select the appropriate controller parameter k d 、k p , so that the neural network remains asymptotically stable near the equilibrium point and increases the stability threshold of the network; Wherein, in step S1, the mathematical expression of the uncontrolled time-delay reaction-diffusion dual-neuron neural network model (1) is: Among them, u(t, x) and v(t, x) represent the states of the two neurons at time t respectively; Ω = (0, π) is a bounded domain; Δ represents R n Laplace operator on ; τ is the time required for information transmission in the neural network; c i (i=1,2) represents the self-feedback parameter of the neuron; a i (i=1,2) is the connection weight between neurons; b i (i=1, 2) is the self-connection weight of the two neurons; d1 and d2 represent the self-diffusion coefficients of the two neurons respectively; f i (·)(i=1,2) is the activation function and satisfies f i (0)(i=1,2)=0 and f i ′(0)(i=1,2)≠0; Among them, the partial differential PD controller is defined as The controlled time-delay reaction-diffusion dual-neuron neural network model (2) can be obtained as: The equilibrium point of the system is the origin O = (0, 0) T Next, we select the communication delay τ as the bifurcation parameter and derive the parameter conditions for the network to undergo Hopf bifurcation. Then, we adjust the two controller parameters k d 、k p , to ensure that the network remains locally asymptotically stable at the origin and to expand the stable region of the network to delay the occurrence of bifurcation; In step S3, the controlled model is linearized at the equilibrium point to obtain the characteristic equation of the controlled system: it(λI-M k -G1-G2e -λτ )=0, Among them, -k 2 (k∈N0) is the characteristic root of the Laplace operator Δ, I is the second-order identity matrix; and b 11 =b1f1′(0),a 11 =a1f2′(0), b 22 =b2f3′(0),a 22 =a2f4′(0). Then the characteristic equation of the controlled system is: λ 2 +A(k)λ+B(k)+Ce -2λτ =0, in, 2. The control method of the dual-neuron neural network model PD controller according to claim 1, characterized in that: Step S4 further comprises: (1) When the network has no time delay (τ = 0), the characteristic equation can be rewritten as: l 2 +A(k)λ+B(k)+C=0, Discuss and study the sufficient conditions for the existence of negative real parts in the above equations to ensure the stability of the network under time-delay-free conditions; (2) When there is a communication delay in the network (τ>0), the conditions for the occurrence of Hopf bifurcation in the network are explored and the bifurcation threshold point τ0 is calculated; by comparing the bifurcation threshold point τ0 and the communication delay τ, it is determined whether the network will have a Hopf bifurcation.

3. A control system of a dual-neuron neural network model PD controller, characterized by: The control method according to any one of claims 1 to 2 can be executed.

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