Dynamic boundary window function design method for mathematical model of memristive system

Abstract

The invention discloses a dynamic boundary window function design method for a memristive system mathematical model, and belongs to the technical field of memristive systems. The memristor mathematical model comprises a state equation and a state-based volt-ampere equation; the state equation comprises a state change function for reflecting a state variable change rule under the action of a signal by the memristor and a window function for limiting a state variable change range for the state variable of which the boundary can change dynamically; the window function comprises two independent variables which are respectively used for reflecting the state change corresponding to the state equation and the direction of the state change. According to the method, the window function is used for defining the memristor mathematical model, the definition interval of the state variable of the memristor mathematical model is more flexible, the state equation can describe the memristive system behavior of which the definition interval dynamically changes, and model design of the memristive system with complex dynamic behaviors is facilitated.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of memristive system, and particularly relates to a dynamic boundary window function design method for a mathematical model of a memristive system. BACKGROUND

[0002] Memristor is considered as the fourth basic passive circuit element besides resistor, capacitor and inductor. As the distribution density of components inside integrated circuits gradually approaches the physical limit, the memory wall problem between processing units and storage units in the traditional von Neumann computing architecture is increasingly prominent, and the computing architecture of memory and calculation integrated by using memristor is considered as an important means to solve this problem. This new computing architecture has ideal characteristics such as low energy consumption, three-dimensional integration, high speed and compatibility with CMOS technology, and can effectively break through the "von Neumann bottleneck", which has attracted high attention from the academic and industrial circles.

[0003] The mathematical model of memristor is an important tool for completing the research analysis and application design of memristive circuit system. The common mathematical model of memristive system usually includes output equation and state equation two parts, wherein the output equation is usually an algebraic equation describing the relationship between the output variable of the system and the input variable and the state variable of the system, and the state equation is a differential equation describing the variation law of the state variable under the action of the input signal. Generally, the state variable of the memristive system is a variable defined on a bounded closed interval, and a window function can usually be used in the state equation to ensure the boundedness of the state variable variation process.

[0004] In the reported experimental research of memristor, the state boundary of some resistance variation behaviors of the memristor will change after experiencing a certain electric signal excitation, however, the change boundary of the state variable described by the existing window function is usually a fixed value, and the definition interval of most existing window functions is [0, 1]. Such existing window function cannot describe the case that the boundary may change dynamically, which limits the description ability and application range of the existing window function.

[0005] Therefore, designing a window function with a dynamically changing boundary is an important work to further improve the description ability of the mathematical model of the memristive system. SUMMARY

[0006] In order to facilitate the design of a memristor model with a state variable having a dynamically changing boundary, the present application provides a dynamic boundary window function design method for a mathematical model of a memristor system, and belongs to the technical field of memristor systems. The mathematical model of the memristor includes a state equation and a state-based volt-ampere equation; for a state variable with a dynamically changing boundary, the state equation includes a state change function reflecting the change rule of the state variable under the action of a memristor reflection signal and a window function limiting the change range of the state variable; the window function includes two independent variables, which are used to reflect the state change corresponding to the state equation and the direction of the state change, respectively. The present application uses a window function for the mathematical model of the memristor, and the definition interval of the state variable of the mathematical model of the memristor is more flexible. The state equation can describe the behavior of the memristor system with a dynamically changing definition interval, and facilitates the model design of the memristor system with complex dynamic behavior.

[0007] A dynamic boundary window function design method for a mathematical model of a memristor system, the mathematical model of the memristor including a state equation and a state-based volt-ampere equation; the state-based volt-ampere equation reflects the state-dependent volt-ampere characteristics of the memristor system, and the state equation reflects the change rule of the state variable under the action of the input signal of the memristor system; for a state variable with a dynamically changing boundary, the state equation includes a state change function f x (x,u) reflecting the change rule of the state variable under the action of the reflection signal of the memristor and a window function f w (x,k w ) limiting the change range of the state variable.

[0008] The window function f w (x,k w ) ensures that the dynamic change process of the state variable always remains within the definition interval; the expression of the window function f w (x,k w ) is:

[0009]

[0010] When x max >x minn , the first independent variable of the window function f w (x,k w ) is the state variable x∈[x min ,x max ], which reflects the state change corresponding to the state equation, and the value of the state variable x is determined according to the specific form of the state equation; the second independent variable k w of the window function f w (x,k w ) reflects the change direction of the state variable x; J, α and p are all constant parameters, and satisfy J=[1*(1*α) p ] *1, a [0, 1], p > 0; the value of the function s(k w ) is related to the second independent variable k w , and the expression of the function s(k w ) is:

[0011]

[0012] When x = x max = x min , the function f x (x, u) is a state change function under the action of the input signal u in the state equation, which describes the change rule of the state variable x under the action of the input signal u; the window function f w (x, k w ) in the state equation of the mathematical model of the memristor is to ensure that the change of the state variable x does not exceed the definition interval of the state variable x, and at the same time, describes the nonlinear characteristics of the state variable x in the change process; the value range of the window function f w (x, k w ) is [0, 1];

[0013] Therefore, the expression of the state equation corresponding to the window function f w (x, k w ) is:

[0014]

[0015] When x max >x min , the change rule of the state variable x is described by the functions f x (x, u) and f w (x, k w );

[0016] When x = x max = x min , the change rule of the state variable x is consistent with the change rule of the lower limit x min of the state variable x.

[0017] Further, the definition interval of the state variable x is [x min , x max ], the boundary x min and the boundary x max take a constant or a variable or one constant and one variable at the same time, and the values of x min and x max should always satisfy x min ≤ x ≤ x max .

[0018] Further, the input signal u represents the voltage across the memristor or the current flowing through the memristor.

[0019] Further, the second independent variable k w is taken as f x (x,u) or the variable that always keeps consistent with f x (x,u).

[0020] In addition to the state equation in the above form, the memristor model using the designed window function should also include a state-based voltage-current equation for describing the state-based voltage-current characteristics of the memristor device.

[0021] The present application has the beneficial effects that the present application designs the mathematical model of the memristor using the window function, the definition interval of the state variable of the mathematical model of the memristor is more flexible, the state equation can describe the behavior of the memristor system that the definition interval may dynamically change, and the model design of the memristor system with complex dynamic behavior is facilitated. BRIEF DESCRIPTION OF DRAWINGS

[0022] In order to clearly illustrate the technical scheme of the method of the present application, the following will briefly introduce the drawings needed to be used in the embodiment or the prior art description.

[0023] Figure 1 is the window function curve when the parameter alpha is 0.2 and the value of p is 1, 10 and 30 respectively;

[0024] Figure 2 is the window function curve when the parameter p is 10 and the value of alpha is 0.1, 0.3 and 0.9 respectively;

[0025] Figure 3 is the component structure of the mathematical model of the memristor system;

[0026] Figure 4 is the formation curve of the memory and the long-term memory in the learning process under different learning rates;

[0027] Figure 5 is the forgetting curve of the forgetting stage with different amounts of long-term memory. DETAILED DESCRIPTION

[0028] The present application will be further described below in combination with the drawings and embodiments.

[0029] As Figure 1 shown, the greater the value of the parameter p, the greater the change of the window function f w (x,k w ), the greater the change of the state equation, and the greater the boundary change amplitude; as Figure 2 shown, the greater the value of the parameter alpha, the greater the change of the window function f w (x,k wThe greater the change amount of the state variable M is, the greater the change amount of the state equation is, and the greater the change range of the boundary is; by setting the values of the parameters p and a, the required dynamic characteristics can be obtained through the mathematical model of the memristor. Figure 3 As shown in FIG. 1, the mathematical model of the memristor includes a state-based volt-ampere equation and a state equation, wherein the state-based volt-ampere equation describes the state-related volt-ampere characteristics of the memristor system, and the state equation describes the change rule of the state variable under the action of the system input signal. For the state variable with a dynamically changing boundary, the state equation includes a function describing the change rule of the state variable under the action of the input signal, and a window function limiting the change range of the state variable.

[0030] The memristor has the characteristics of simulating human memory and learning behavior; the increase and decrease of the resistance of the memristor are defined as the disappearance and formation of the memory of the memristor device, respectively; the memory of the memristor includes short-term memory and long-term memory of human memory, and the short-term memory formed in the learning process will be forgotten relatively quickly after the learning excitation signal applied to the memristor is removed, while the long-term memory has a relatively long duration.

[0031] In this embodiment, the mathematical model of the memristor is used to simulate the characteristics of human memory and learning behavior, the total memory formed in the learning process of the memristor is described by a state variable M, i.e., the total memory M includes long-term memory and short-term memory, the long-term memory of the memristor is described by a state variable L, and the short-term memory of the memristor is (M-L); when there is no memory formation, M=0, and when the memory is completely formed, M=1, then M and L should satisfy 0≤L≤M≤1, i.e., the amount of long-term memory cannot exceed the total amount of memory formed, and the total amount of memory should be less than or equal to the maximum amount of memory that can be formed; it can be seen that the lower bound of the state variable L is a constant, and the upper bound of the state variable L is a state variable, i.e., the upper bound of L will dynamically change; the lower bound of the state variable M is a state variable, and the upper bound of the state variable M is a constant, i.e., the lower bound of M will dynamically change.

[0032] The mathematical model of the memristor with long-term memory behavior and short-term memory behavior is designed as follows:

[0033] i=(1-M)f off (v)+Mf on (v) (1)

[0034]

[0035] wherein equation (1) is a state-based volt-ampere equation, i is the current flowing through the memristor, v represents the voltage across the memristor, the functions f off (v) and f on (v) represent the volt-ampere characteristics of the high-resistance state and the low-resistance state of the memristor, respectively; and respectively, and F M and F L are respectively expressed as:

[0036]

[0037] F L = k L f E (v) (5)

[0038] wherein the function f E (v) is a monotonically increasing function and satisfies f E (0) = 0, τ and k L are both positive parameters; f wM (M, F M ) represents a window function of the total memory M, which ensures that the variation range of the total memory M is always kept within the interval [L, 1], and the expression of f wM (M, F M ) is:

[0039]

[0040] wherein, α M ∈ [0, 1], p M > 0, and the value of the function s M (F M ) is related to the current variation direction of the total memory M, and the expression of s M (F M ) is:

[0041]

[0042] f wL (L, F L ) represents a window function of the long-term memory L, which ensures that the variation range of the long-term memory L is always kept within the interval [0, M], and the expression of f wL (L, F L ) is:

[0043]

[0044] wherein, α L ∈ [0, 1], p L > 0, and the value of the function s L (F L ) is related to the current variation direction of the long-term memory L, and the expression of s L (F L ) is:

[0045]

[0046] A set of simulation curves of the memristor mathematical model with different long-term memory growth rates during the learning phase are shown in Figure 2. Figure 4 As shown, during the simulation, p M =p L =10,α M =α L =0.9, f E (v) = 30, the initial values ​​of total memory M and long-term memory L are both 0, the simulation time is 0.3s, and in each simulation, k L The values ​​of k are 0.1, 0.2, 0.3, 1 and 5 respectively. It can be seen from the simulation results that as k L As k increases, the growth rate of long-term memory in the learning process will first increase significantly. When the growth rate of long-term memory L is close to the growth rate of total memory M, the growth rate of long-term memory L will no longer increase with k. L That is, the growth rate of long-term memory L will not exceed the growth rate of its upper bound M.

[0047] Figure 5 The simulation curves of the forgetting stage with different amounts of long-term memory are given. During the simulation, k L =0.3,f E (v) = 0, the simulation duration is 2 seconds, the initial value of M is 1, and the initial values ​​of L are 0.1, 0.3, 0.5, 0.7, and 0.9, respectively. The simulation results show that during the forgetting process, long-term memory L remains essentially unchanged, while total memory M decreases significantly at the initial moment of forgetting. As M approaches L, the rate of decrease of total memory M gradually slows, eventually becoming equal to L. In other words, the long-term memory L described by the memristor mathematical model is a lower bound for total memory.

[0048] It can be seen from the simulation results that the mathematical model of the memristor using the designed window function can effectively limit the range of change of the state variables whose boundaries may change dynamically, so that they always remain within a meaningful range.

[0049] Memristors can simulate human memory and learning behavior. When establishing a mathematical model of a memristor, the definition interval of the state variable will change with the learning and forgetting process. In the state equation of the state variable, the window function designed by the present invention can be used to describe the state variable whose definition interval will change.

[0050] The above is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with this technical field, within the technical scope disclosed by the present invention, can make equivalent replacements or changes based on the technical solution and inventive concept of the present invention, which should be covered by the scope of protection of the present invention.

Claims

1. A method for designing a dynamic boundary window function for a mathematical model of a memristor system, characterized in that: The mathematical model of the memristor includes a state equation and a state-based volt-ampere equation; for state variables whose boundaries change dynamically, the state equation includes a state change function f that reflects the change law of the state variable under the action of the signal. x (x,u) and the window function f that limits the range of changes in state variables w (x,k w ); The window function f w (x,k w ) ensures that the dynamic change process of the state variable always remains within the defined range; the window function f w (x,k w ) is: When x max >x minn When the window function f w (x,k w ) is the state variable x∈[x min ,x max ], reflecting the state change corresponding to the state equation, the value of the state variable x is determined according to the specific form of the state equation; the window function f w (x,k w )'s second argument k w Reflects the direction of change of the state variable x; J, α and p are all constant parameters, and satisfy J = [1*(1-α) p ] -1 ,α∈[0,1],p>0; function s(k w ) and the value of the second independent variable k w Related, function s(k w ) is expressed as: When x=x max =x min When the function f x (x,u) is the state change function under the action of input signal u in the state equation, which describes the change law of state variable x under the action of input signal u; the window function f in the state equation used in the mathematical model of memristor w (x,k w ) is to ensure that the change of the state variable x does not exceed the definition interval of the state variable x, and at the same time, it describes the nonlinear characteristics of the state variable x in the process of change; the window function f w (x,k w ) has a value range of [0,1]; Therefore, the window function f w (x,k w )The corresponding state equation is: When x max >x min When the state variable x changes, the function f x (x,u) and f w (x,k w )Common description; When x=x max =x min When the state variable x changes, the change law of the state variable x is consistent with the lower limit x of the state variable x. min The changing pattern remains consistent.

2. A method for designing a dynamic boundary window function for a memristor system mathematical model according to claim 1, characterized in that: The definition interval of the state variable x is [x min ,x max ], boundary x min and the boundary x max Take constants or variables or a constant and a variable at the same time, x min and x max The value should always satisfy x min ≤x≤x max .

3. The method for designing a dynamic boundary window function for a memristor system mathematical model according to claim 1, wherein: The input signal u represents the voltage across the memristor or the current flowing through the memristor.

4. The method for designing a dynamic boundary window function for a memristor system mathematical model according to claim 1, wherein: The second independent variable k w The value is f x (x,u) or take the sign and f x (x,u) are variables that always remain the same.

5. An electronic device, characterized in that: include: one or more processors; Memory; One or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, and the one or more programs are configured to perform the method according to any one of claims 1 to 4.

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