Adaptive gradient neural dynamics optimization method and its application in AOA positioning

Abstract

The present invention relates to the field of neural network and wireless sensor positioning technology, and discloses an adaptive gradient neural dynamics optimization method. The actual engineering application requirements are converted into linear optimization problems and modeled; the model designs adaptive parameters under the premise of satisfying the Lyapunov stability theorem, and solves the problem through the ADRNN model based on gradient descent so that the solved value approaches the optimal solution, thereby further reducing the loss value of the objective function and achieving higher accuracy, while improving the ability to resist noise interference. In addition, the effectiveness and superiority of the model are verified through theoretical verification and simulation experiments. Finally, the proposed ADRNN model is applied to AOA positioning to verify whether the positioning accuracy can be improved. Compared with other methods, the present invention avoids the improper setting of adaptive parameters from the root, improves the convergence speed and the ability to resist noise interference on the basis of achieving global convergence, and can be applied to solving complex-valued problems; in application, the accuracy of AOA positioning is improved.

Description

Technical Field

[0001] The present invention relates to the field of software communication technology, and in particular to an adaptive gradient neural dynamics optimization method and its application in AOA positioning. Background Art

[0002] Transforming real-world problems into mathematical ones has been widely applied across various disciplines and industries, including control engineering, machine learning, and artificial intelligence. Without considering constraints, these problems can be transformed into a linear system; when constraints are considered, they are transformed into a linear system of projected equations. In general, these problems often boil down to solving optimization problems, making the development of efficient solvers crucial.

[0003] The linear time-varying equation S(t)X(t)=W(t) has long been a research focus due to its universal applicability. Furthermore, equations such as the Sylvester and Lyapunov equations can be converted to S(t)X(t)=W(t) via the Kronecker product operation. Furthermore, when W(t)=I, solving the time-varying equation S(t)X(t)=W(t) becomes an inverse operation.

[0004] The current research focus is on how to create an effective and high-performance optimization algorithm. With the continuous development of neural networks, neural networks have been widely used in many fields such as pattern recognition, automatic control, signal processing, decision support, artificial intelligence, etc. because of their strong robustness, ability to process systems adaptively, and speed up computing speed through parallel processing. The inputs and outputs of traditional neural networks (Convolutional Neural Networks, CNN, etc.) are independent of each other, while recurrent neural networks (RNN) are fully connected neural networks with feedback functions, and messages are transmitted between neurons. In recent years, researchers have applied recursive neural networks to more fields, such as robotic arm control and pattern recognition. Conventional Gradient Neural Network (CGNN), as a type of recursive neural network, constructs an energy function and then iterates along the direction of the gradient to generate new values ​​x1, x2,…, x t , and expect x t Finally converges to the expected optimal answer x *To achieve the goal of minimizing the energy function and thus solving the problem. In recent years, researchers have proposed different optimization solvers based on improvements in learning rates, including Adagrad, AdaDelta, RMSprop, etc. Studies have found that the CGNN model performs well on time-invariant tasks, but cannot handle time-varying tasks. Compared with the CGNN model, the conventional zeroing neural network (CZNN) is an effective method for solving time-varying problems, but its convergence efficiency needs to be further improved. In addition, current methods such as CtFZNN add a fuzzy factor to make the model converge quickly, or such as PC-CFZNN achieve the purpose of convergence through a time-varying convergence factor, but the convergence factor of few algorithms is related to the problem itself.

[0005] Research has shown that there is a strong correlation between model quality and noise immunity. In real-world applications, we encounter many types of noise, and a model's noise immunity determines its effectiveness in these applications. The CGNN and CZNN models described above were both considered in the absence of noise interference, and their noise immunity performance is poor.

[0006] Generally speaking, most models are based on real-valued problems, while existing technologies have relatively little research on complex-valued problems. Furthermore, many models that solve real-valued problems perform poorly on complex-valued problems. Neural network models that can handle complex-valued problems have many applications in speech enhancement and image classification, making the study of complex-valued problems crucial.

[0007] Angle of Arrival (AOA) positioning, one of the primary indoor positioning technologies, provides precise location information for object tags. It can also be used to track human activity and control indoor fixed assets. Furthermore, it can be used to locate industrial robots, plan their routes, and control their operations. Because AOA has broad commercial potential, improving its positioning accuracy is crucial. Summary of the Invention

[0008] (1) Technical problems solved

[0009] To address the shortcomings of existing technologies, the present invention provides an adaptive gradient neural dynamics optimization method and its application in AOA positioning. Based on traditional gradient neural networks, this method designs a noise-resistant neural network model with adaptive coefficients based on Lyapunov stability theory from the control field to solve the time-varying linear equation S(t)X(t)=W(t). Compared with traditional methods, this method utilizes control theory to design the iterative step size, further reducing the model's loss, resulting in higher accuracy and enabling global convergence. The addition of a time-varying exponential parameter enhances the algorithm's noise resistance. Compared with other methods, this model exhibits excellent convergence performance for both real-valued and complex-valued problems, addressing the relatively limited research on complex-valued problems in existing technologies. Furthermore, many models for real-valued problems perform poorly for complex-valued problems. However, neural network models that can handle complex-valued problems have numerous applications in speech enhancement and image classification, making the study of complex-valued problems crucial.

[0010] (2) Technical solution

[0011] To achieve the above, a neural network model with adaptive coefficients that can resist noise interference is designed based on the traditional gradient neural network and the Lyapunov stability theory in the control field to solve the time-varying linear equation S(t)X(t)=W(t). Compared with traditional methods, this method uses the theory of the control field to design the iterative step size, so that the loss value of the model is further reduced, thereby having a higher accuracy rate and being able to achieve global convergence; and by adding a time-varying exponential parameter, the algorithm's ability to resist noise interference is enhanced. Compared with other methods, this model shows excellent convergence performance in solving both real-valued and complex-valued problems.

[0012] The present invention provides the following technical solution: an adaptive gradient neural dynamics optimization method, comprising the following steps:

[0013] S1. First, we take the CGNN model and CZNN model to solve the time-varying equation S(t)X(t)=W(t) as an example;

[0014] S11. The design process of the CGNN model based on solving S(t)X(t)=W(t) is as follows: First, construct the energy function ε(t), then Iteratively update X(t) to minimize the value of ε(t), and finally obtain the CGNN model as

[0015]

[0016] S12. The design process of the CZNN model to solve S(t)X(t)=W(t) is as follows: First, define the error function as: E(X(t),t)=S(t)X(t)-W(t); Based on the design form of CZNN The CZNN model can be obtained as

[0017] S2. Based on the CGNN model and the CZNN model, an adaptive gradient robust negative feedback neural network is defined. Then, according to the Lyapunov stability theorem, the adaptive coefficient of the model based on the error function is defined. Then, the convergence of the adaptive recurrent neural network (ADRNN) model in real-valued and complex-valued problems is theoretically analyzed.

[0018] S3. Inject noise into the model and discuss the robust performance of the model under noise interference;

[0019] S4. Set parameters, conduct numerical experiments and analysis, and present the residual results of the calculated solution and the actual solution when the model implements real-valued examples under various models and different noise interferences. In addition, provide a comparison chart and residual results of the actual trajectory and theoretical trajectory obtained when the present invention implements complex-valued examples. Finally, convert AOA positioning into a mathematical optimization problem and apply the proposed model to AOA positioning. Give examples of positioning to verify the positioning effect of the model.

[0020] Preferably, in step S2, an adaptive gradient robust negative feedback neural network is defined based on the CGNN model and the CZNN model, and then the adaptive coefficient of the model based on the error function is defined according to the Lyapunov stability theorem. Finally, the convergence of the adaptive recursive neural network (ADRNN) model in the real-valued problem is theoretically analyzed. The specific steps are as follows:

[0021] S21. Define an adaptive gradient robust negative feedback neural network as:

[0022]

[0023] S22. The robust negative feedback neural network with adaptive gradient for solving the time-varying equation S(t)X(t)=W(t) is:

[0024]

[0025] S23. For real-valued problems, define the Lyapunov candidate function as

[0026]

[0027] S24, convergence of ADRNN model in real-valued problems;

[0028] It is obvious that Ω(t) in step B3 is positive definite;

[0029] set up as well as Then the time derivative of Ω(t) can be described as:

[0030]

[0031] when|(e i (t))|≥1,Ψ(·) ps can be expressed as when|(e i (t))|<

[0032] 1,Ψ(·) ps can be expressed as get for

[0033]

[0034] definition because

[0035] get

[0036]

[0037] Since μ(t)>0, we set the right side of the above inequality to 0, and get μ(t)

[0038]

[0039] In order to make We add a time-varying parameter p t , the adaptive coefficient based on Lyapunov stability theory is:

[0040]

[0041] Where p>1, w and p are constants and wp t >1;

[0042] According to Lyapunov's stability theorem, since Ω(t) is positive definite, is negative definite, so the ADRNN model converges in the real-valued case.

[0043] Preferably, the processing of the complex value problem in step S2 is converted into a triangular form by Euler's formula, and the global convergence in the complex value problem is verified by theoretical analysis. The specific steps are as follows:

[0044] S25. For complex-valued problems, define the Lyapunov candidate function as

[0045]

[0046] S26, Convergence of ADRNN model in complex-valued problems;

[0047] Obviously, Ω(t) in step S2 is positive definite;

[0048] According to Euler's formula, the complex value matrix in the activation function is converted into the form of trigonometric function.

[0049] ψ(γe iθ )=Ψ(γ(t)(cosθ+isinθ));

[0050] The time derivative of Ω(t) can be described as:

[0051]

[0052] When γ(t)=|e mn (t)|=γ(t)≥1, according to the activation function It can be simplified to:

[0053]

[0054] when|e mn When (t)|=γ(t)∈(0,1), is abbreviated to:

[0055]

[0056] Where θ∈(-π,π], the range of cosθ is (-1,1]. Since |(e i (t))|≥1, when ωp t Obviously when it is big enough therefore is negatively definite;

[0057] According to Lyapunov's stability theorem, since Ω(t) is positive definite, is negative definite, so the ADRNN model converges in the complex valued case.

[0058] Preferably, in step S3, noise is injected into the model, that is, noise exists in the solution process, and then the anti-noise interference performance of the model is verified from a theoretical analysis. The specific steps are as follows:

[0059] S31. The ADRNN model with injected noise is designed as follows:

[0060]

[0061] where Γ(t)∈R m×p represents ordinary external random noise;

[0062] S32. Define the Lyapunov candidate function:

[0063]

[0064] S33, robust performance determination of ADRNN model;

[0065] According to the properties of inequalities, we can get the following inequalities:

[0066] trace(MN)≤|M| F |N| F ;

[0067] make The time derivative of γ(t) is as follows:

[0068]

[0069] Obviously, when there is no noise interference, is negative definite. When there is noise interference, according to the lemma, the above formula can be rewritten as

[0070]

[0071] Since p>1 and ωp t >1, when ωp t →∞,

[0072] According to Lyapunov's stability theorem, since γ(t) is positive definite, is negative definite, so the ADRNN model under noise interference is globally convergent.

[0073] Preferably, in step S4, firstly, numerical results of the model when solving real-valued examples under various models and different noise interferences are given; in addition, numerical results obtained by the present invention when solving complex-valued examples are given;

[0074] S41. Verify the effectiveness and robustness of the model for given real-valued examples;

[0075] S411. Given a real-valued example:

[0076]

[0077] S412. Adjust the values ​​of ω and p and substitute them into the ADRNN model; similarly, adjust the values ​​of η and α and substitute them into the CGNN model and CZNN model respectively;

[0078] S413. Verify the performance of the ADRNN model under no noise, constant noise, linear noise, and bounded random noise conditions.

[0079] Another technical problem to be solved by the present invention is to provide an application of adaptive gradient neural dynamics optimization in AOA positioning, wherein the AOA positioning is any of the above-mentioned methods, converting the angle of arrival AOA positioning into a mathematical optimization problem and solving it through the ADRNN model to verify the practicality and superiority of the model;

[0080] S42, applying the ADRNN model to 3D AOA positioning;

[0081] S421. According to the principle of AOA positioning, the AOA positioning problem is converted into an optimization problem for solution;

[0082] When the antenna array is installed on the BS, the antenna array obtains the angles of incidence α1, α2, β1, and β2 according to the signal sent by the MS. The intersection of the angles of incidence is the position of the MS. Let the position coordinates of the MS be (x, y, z) and the position coordinates of the BS be (x i ,y i ,z i ), where i=1, ... n is the number of BSs. If n=2, then the two existing base stations must satisfy the following relationship:

[0083]

[0084] The process of solving the MS position becomes solving the equation S(t)X(t)=W(t). The positions of the two base stations are set to (-6, -6, -2) and (6, 6, 2) respectively. The ADRNN model is used to solve the position of the MS (x, y, z) to verify the practicality and superiority of the ADRNN model.

[0085] S422, considering the positioning performance of the ADRNN model when measurement errors and noise interference exist;

[0086] Since the error in the measurement process cannot be ignored in practical applications, the angle measurement result BS is obtained. i for where n i is additive Gaussian noise, which obeys zero mean distribution, that is, n i ~N(0,σ 2 ), verifying the positioning effect of the model when there are errors in the measurement process;

[0087] In addition to the measurement error, we also considered the case of noise interference in the positioning process. When adding sinusoidal noise Γ(t) = sin(4πt+π), we verified the positioning effect of the model in the presence of noise interference.

[0088] (3) Beneficial effects

[0089] Compared with the existing technology, the present invention provides an adaptive gradient neural dynamics optimization method and its application in AOA positioning, which has the following beneficial effects:

[0090] 1. This adaptive gradient neural dynamics optimization method and its application in AOA positioning. By designing the convergence parameters of the model based on the Lyapunov stability theorem, the model can achieve global convergence and avoid improper parameter adjustment in practical applications.

[0091] 2. This adaptive gradient neural dynamics optimization method and its application in AOA positioning converts the complex numbers in the activation function into triangular form through the Euler formula. Through theoretical analysis, the global convergence of the ADRNN model under complex-valued problems is analyzed, which makes up for the defect that some models cannot solve complex-valued problems.

[0092] 3. The adaptive gradient neural dynamics optimization method and its application in AOA positioning, through the time-varying exponential parameter p t The time-varying exponential parameter can be changed according to the noise level. On the basis of achieving global convergence, increasing the parameters p and ω can further improve the convergence efficiency and noise resistance. In practical applications, the values ​​of ω and p are more flexible and variable depending on the noise level.

[0093] 4. This adaptive gradient neural dynamics optimization method and its application in AOA positioning transform three-dimensional AOA positioning into a mathematical optimization problem and solve it through the ADRNN model. Experimental results show that the ADRNN model has better positioning effect and higher accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0094] Figure 1 A flowchart of an adaptive gradient neural dynamics optimization method and its application in AOA positioning according to the present invention;

[0095] Figure 2 A comparison chart of the actual and theoretical trajectories of the ADRNN model, CGNN model, and CZNN model when solving real-valued examples in the present invention;

[0096] Figure 3 The residual results of the calculated solution and the actual solution when solving the real-valued example of the present invention under the interference of no noise, random noise, linear noise, and bounded random noise;

[0097] Figure 4 This is a comparison diagram of the actual trajectory and theoretical trajectory obtained by the ADRNN model when solving complex-valued examples in the present invention;

[0098] Figure 5 The residual result of the calculated solution and the actual solution obtained by the ADRNN model when solving the complex-valued example of the present invention;

[0099] Figure 6 This is a schematic diagram of the principle of 3D AOA positioning;

[0100] Figure 7 The effect diagram of CGNN model, CZNN model and CGNN model as AOA positioning task solver;

[0101] Figure 8 This is the effect diagram of the ADRNN model in the presence of measurement errors and noise interference in AOA positioning. DETAILED DESCRIPTION

[0102] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0103] See also Figure 1-8 , the present invention provides the following technical solutions:

[0104] An adaptive gradient neural dynamics optimization method and its application in AOA positioning. The algorithm can be used to solve the real-valued and complex-valued time-varying equations S(t)X(t)=W(t), including the following steps:

[0105] In order to more clearly present the technical solution of the present invention, the technical solution in the embodiment of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiment of the present invention.

[0106] S1. First, establish a mathematical model to solve the time-varying equation S(t)X(t)=W(t);

[0107] S11. The CGNN model design process for solving S(t)X(t)=W(t) is as follows:

[0108] S111. First, construct the energy function as follows:

[0109]

[0110] in‖·‖ F is the Frobenius norm.

[0111] S112. Update X(t) iteratively by the following formula to minimize the value of ε(t):

[0112]

[0113] Where η is a positive parameter that adjusts the iteration step size.

[0114] S113, then the CGNN model is:

[0115]

[0116] The matrix S T (t) represents the transpose of S(t).

[0117] S12. The design process of the CZNN model for solving S(t)X(t)=W(t) is as follows:

[0118] S121. First, define the error function as:

[0119] E(X(t),t)=S(t)X(t)-W(t);

[0120] The design form of S122 and CZNN is

[0121]

[0122] Where α is a positive parameter that modifies the convergence rate, and Ψ(·) is the activation function.

[0123] S123, then the CZNN model is:

[0124]

[0125] S2. Define the adaptive coefficient of the model based on the error function according to Lyapunov stability theorem, and discuss the convergence of the adaptive recurrent neural network (ADRNN) model in real-valued and complex-valued problems;

[0126] S21. Define an adaptive gradient robust negative feedback neural network as:

[0127]

[0128] S22. The robust negative feedback neural network with adaptive gradient for solving the time-varying equation S(t)X(t)=W(t) is:

[0129]

[0130] S23. For real-valued problems, define the Lyapunov candidate function as

[0131]

[0132] S24. Convergence of ADRNN model in real-valued problems.

[0133] It is obvious that Ω(t) in step B3 is positive definite.

[0134] set up as well as Then the time derivative of Ω(t) can be described as:

[0135]

[0136] when|(e i (t))|≥1,Ψ(·) ps can be expressed as when|(e i (t))|<1,Ψ(·) ps can be expressed as get for

[0137]

[0138] definition because get

[0139]

[0140] Since μ(t)>0, we set the right side of the above inequality to 0, and get μ(t)

[0141]

[0142] In order to make We add a time-varying parameter p t , the adaptive coefficient based on Lyapunov stability theory is:

[0143]

[0144] Where p>1, w and p are constants and wp t >1.

[0145] According to Lyapunov's stability theorem, since Ω(t) is positive definite, is negative definite, so the ADRNN model converges in the real-valued case.

[0146] S25. For complex-valued problems, define the Lyapunov candidate function as

[0147]

[0148] S26. Convergence of ADRNN model in complex-valued problems.

[0149] Obviously, Ω(t) in step S25 is positive definite.

[0150] According to Euler's formula, the complex matrix in the activation function is converted into the form of trigonometric function, Ψ(γe iθ )=Ψ(γ(t)(cosθ+isinθ));

[0151] The time derivative of Ω(t) can be described as:

[0152]

[0153] When γ(t)=|e mn (t)|=γ(t)≥1, according to the activation function It can be simplified to:

[0154]

[0155] when|e mn When (t)|=γ(t)∈(0,1), is abbreviated to:

[0156]

[0157] Where θ∈(-π,π], the range of cosθ is (-1,1]. Since |(e i (t))|≥1, when ωp t Obviously when it is big enough therefore It is negatively definite.

[0158] According to Lyapunov's stability theorem, since Ω(t) is positive definite, is negative definite, so the ADRNN model converges in the complex valued case.

[0159] S3. Inject noise into the model and discuss the robust performance of the model under noise interference;

[0160] S31. The ADRNN model with injected noise is designed as follows:

[0161]

[0162] where Γ(t)∈R m×p represents ordinary external random noise.

[0163] S32. Define the Lyapunov candidate function:

[0164]

[0165] S33. Robust performance determination of ADRNN model.

[0166] According to the properties of inequalities, we can get the following inequalities:

[0167] trace(MN)≤|M| F |N| F ;

[0168] make The time derivative of Υ(t) is as follows:

[0169]

[0170] Obviously, when there is no noise interference, is negative. When there is noise interference, according to the lemma, the above formula can be rewritten as

[0171]

[0172] Since p>1 and ωp t >1, when ωp t →∞,

[0173] According to Lyapunov's stability theorem, since Υ(t) is positive definite, is negative, so the ADRNN model is convergent under noise interference.

[0174] S4. Set parameters, conduct numerical experiments and analysis, and give application examples of AOA.

[0175] S41. Verify the effectiveness and robustness of the model for given real-valued examples.

[0176] S411. Given a real-valued example:

[0177]

[0178] S412. Adjust the values ​​of ω and p and substitute them into the ADRNN model. Similarly, adjust the values ​​of η and α and substitute them into the CGNN model and CZNN model respectively.

[0179] S413. Verify the performance of the ADRNN model under no noise, constant noise, linear noise, and bounded random noise conditions.

[0180] The specific numerical experiment process is as follows:

[0181] Given a real-valued example:

[0182]

[0183] Adjust ω = 2, p = 1.25, and substitute the values ​​of ω and p into the ADRNN model for numerical experiments.

[0184] Substitute η=2 and α=2 into the CGNN model and CZNN model respectively and perform numerical experiments.

[0185] Verify the performance of the ADRNN model under no noise, constant noise, linear noise, and bounded random noise conditions.

[0186] The state trajectory diagram and error diagram of the numerical experimental results are shown in Figure 2. Figure 2 and Figure 3 As shown. Among them, Figure 2 Comparison diagram between theoretical trajectories and actual trajectories of real-valued examples in ADRNN model, CGNN model and CZNN model, Figure 3 Neutron (a) is the residual graph of the real-valued example without noise interference. Figure 3 Figure (b) shows the residual graph of a real-valued example with constant noise Γ(t) = 1.5, Figure (c) shows the residual graph of a real-valued example with linear noise Γ(t) = 0.5t, and Figure (d) shows the residual graph of a real-valued example with bounded random noise Γ(t) = [1, 2].

[0187] like Figure 2 As shown in Figure 2, compared with the CGNN model and the CZNN model, the convergence performance of the ADRNN model is superior, which can make the model loss value lower and the accuracy value higher. In addition, as Figure 3 As shown, the influence of various noises can be suppressed by adjusting the parameters involved in the algorithm. The actual solution of the equation can globally converge to the theoretical solution under the interference of constant noise, linear noise, and bounded random noise. Figure 2 and Figure 3 The effectiveness, superiority and robustness of the ADRNN model were verified.

[0188] S42. For given complex-valued examples, verify the effectiveness of the model in complex-valued problems.

[0189] S421. Given a complex value example:

[0190]

[0191] S411. Adjust the values ​​of ω and p and substitute them into the ADRNN model.

[0192] The specific numerical experiment process is as follows:

[0193] Given a real-valued example:

[0194]

[0195] Set ω = 6, p = 1.25;

[0196] Substitute the values ​​of ω and p into the ADRNN model for numerical experiments.

[0197] The state trajectory diagram and error diagram of the numerical experimental results are shown in Figure 2. Figure 4 and Figure 5 As shown. Among them, Figure 4 This is a comparison diagram between the theoretical trajectory and the actual trajectory of the complex-valued example in the ADRNN model. Figure 5 Plot of the residuals of the calculated solution to the complex-valued example with random initial values ​​and the theoretical solution.

[0198] according to Figure 4 and Figure 5 ,In the complex value example, the calculated solution can quickly converge to the ,theoretical solution, and the residual between the calculated and ,theoretical solutions can quickly converge to zero, verifying the ,effectiveness of the ADRNN model in complex-valued problems.

[0199] S43. Applying the ADRNN model to 3D AOA positioning

[0200] S431. According to the principle of AOA positioning, the AOA positioning problem is converted into an optimization problem for solution.

[0201] like Figure 6 As shown in FIG, in the AOA positioning technology, the arrival angle between the mobile station (MS) and each base station (BS) is first measured, and the position of the MS can be determined by the rays in the direction of the incident angle with each base station as the starting point.

[0202] When the antenna array is installed on the BS, the antenna array obtains the angles of incidence α1, α2, β1, and β2 according to the signal sent by the MS. The intersection of the angles of incidence is the position of the MS. Let the position coordinates of the MS be (x, y, z) and the position coordinates of the BS be (x i ,y i ,z i ), where i=1, ...n is the number of BSs.

[0203] according to Figure 6 The geometric meaning of the two existing base stations must satisfy the following relationship:

[0204]

[0205] The process of solving the MS position becomes solving the equation S(t)X(t)=W(t). Assume the positions of the two base stations are (-6,-6,-2) and (6,6,2) respectively. Set ω=3 and p=1.25, and use the ADRNN model to solve the position of the MS (x,y,z). The experimental results are as follows: Figure 7 shown.

[0206] from Figure 7 As can be seen in a, the CGNN model cannot solve the AOA positioning problem because the trajectory generated by the CGNN model cannot follow the actual trajectory. Figure 7 In b, the actual trajectory solved by the CZNN model has a slight deviation from the expected trajectory, but they can eventually coincide. Figure 7 In Figure c, the state trajectory generated by the ADRNN model is consistent with the expected trajectory. This proves that the ADRNN model can be used as a localization task solver and is superior to other solvers.

[0207] S432. Consider the positioning performance of the ADRNN model when measurement errors and noise interference exist.

[0208] The above experiments were conducted without noise interference. However, AOA positioning technology still faces various challenges in practical applications, such as angle measurement errors and signal reflection interference.

[0209] Since the error in the measurement process cannot be ignored in practical applications, the angle measurement result BS is obtained. i for where n i is additive Gaussian noise, which obeys zero mean distribution, that is, n i ~N(0,σ 2 ).exist Figure 8 In a, the actual trajectory obtained by the ADRNN model is consistent with the target trajectory.

[0210] In addition to the measurement error, we also consider the case of noise interference in the positioning process. In the case of additional sinusoidal noise Γ(t) = sin(4πt+π), we have Figure 8 b It can be seen that the actual trajectory obtained by the ADRNN model converges to the expected trajectory.

[0211] While embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions, and variations may be made to these embodiments without departing from the principles and spirit of the invention, and that the scope of the invention is defined by the appended claims and their equivalents.

Claims

1. Adaptive gradient neural dynamics optimization method, characterized by: The following steps are involved: S1. First, we take the CGNN model and CZNN model to solve the time-varying equation S(t)X(t)=W(t) as an example; S11. The design process of the CGNN model based on solving S(t)X(t)=W(t) is as follows: First, construct the energy function ε(t), then Iteratively update X(t) to minimize the value of ε(t), and finally obtain the CGNN model as S12. The design process of the CZNN model to solve S(t)X(t)=W(t) is as follows: First, define the error function as: E(X(t),t)=S(t)X(t)-W(t); Based on the design form of CZNN The CZNN model is obtained as S2. Based on the CGNN model and the CZNN model, an adaptive gradient robust negative feedback neural network is defined. Then, according to the Lyapunov stability theorem, the adaptive coefficient of the model based on the error function is defined. Then, the convergence of the adaptive recurrent neural network (ADRNN) model in real-valued and complex-valued problems is theoretically analyzed. Based on the CGNN model and the CZNN model, an adaptive gradient robust negative feedback neural network is defined. Then, according to the Lyapunov stability theorem, the adaptive coefficient of the model based on the error function is defined. Finally, the convergence of the adaptive recurrent neural network (ADRNN) model in real-valued problems is theoretically analyzed. The specific steps are as follows: S21. Define an adaptive gradient robust negative feedback neural network as: S22. The robust negative feedback neural network with adaptive gradient for solving the time-varying equation S(t)X(t)=W(t) is: S23. For real-valued problems, define the Lyapunov candidate function as S24, convergence of ADRNN model in real-valued problems; It is obvious that Ω(t) in step B3 is positive definite; set up as well as Then the time derivative of Ω(t) can be described as: when|(e i (t))|≥1,Ψ(·) ps can be expressed as when|(e i (t))|<1,Ψ(·) ps can be expressed as get for definition because get Since μ(t)>0, let the right side of the above inequality be 0, and we get μ(t) In order to make We add a time-varying parameter p t , the adaptive coefficient based on Lyapunov stability theory is: Where p>1, w and p are constants and wp t >1; According to Lyapunov's stability theorem, since Ω(t) is positive definite, is negative, so the ADRNN model converges in the real-valued case; S3. Inject noise into the model and discuss the robust performance of the model under noise interference; S4. Set parameters, conduct numerical experiments and analysis, and give the residual results of the calculated solution and the actual solution under various models and different noise interference when implementing the real-valued example. In addition, give the comparison diagram and residual results of the actual trajectory and theoretical trajectory obtained when implementing the complex-valued example of the present invention. Finally, convert the three-dimensional angle of arrival (AOA) positioning into a mathematical optimization problem and apply the proposed model to AOA positioning. Give an example positioning example to verify the positioning effect of the model.

2. The adaptive gradient neural dynamics optimization method according to claim 1, characterized in that: The complex value problem in step S2 is processed by Euler's formula to convert the complex number in the activation function into a triangular form, and the global convergence in the complex value problem is verified by theoretical analysis. The specific steps are as follows: S25. For complex-valued problems, define the Lyapunov candidate function as S26, Convergence of ADRNN model in complex-valued problems; Obviously, Ω(t) in step S2 is positive definite; According to Euler's formula, the complex value matrix in the activation function is converted into the form of trigonometric function. ψ(γe iθ )=Ψ(γ(t)(cosθ+isinθ)); The time derivative of Ω(t) can be described as: When γ(t)=|e mn (t)|=γ(t)≥1, according to the activation function It can be simplified to: when|e mn When (t)|=γ(t)∈(0,1), is abbreviated to: Where θ∈(-π,π], the range of cosθ is (-1,1]. Since |(e i (t))|≥1, when ωp t Obviously when it is big enough therefore is negatively definite; According to Lyapunov's stability theorem, since Ω(t) is positive definite, is negative definite, so the ADRNN model converges in the complex valued case.

3. The adaptive gradient neural dynamics optimization method according to claim 1, characterized in that: In step S3, noise is injected into the model, that is, noise exists in the solution process, and then the anti-noise interference performance of the model is verified from a theoretical analysis. The specific steps are as follows: S31. The ADRNN model with injected noise is designed as follows: where Γ(t)∈R m×p represents ordinary external random noise; S32. Define the Lyapunov candidate function: S33, robust performance determination of ADRNN model; According to the properties of inequalities, we can get the following inequalities: trace(MN)≤|M| F |N| F ; make The time derivative of γ(t) is as follows: Obviously, when there is no noise interference, is negative definite. When there is noise interference, according to the lemma, the above formula can be rewritten as Since p>1 and ωp t >1, when ωp t →∞, According to Lyapunov's stability theorem, since γ(t) is positive definite, is negative definite, so the ADRNN model under noise interference is globally convergent.

4. The adaptive gradient neural dynamics optimization method according to claim 1, characterized in that: In step S4, the numerical results of the model when solving real-valued examples under various models and different noise interferences are first given; in addition, the numerical results obtained by the present invention when solving complex-valued examples are given; S41. Verify the effectiveness and robustness of the model for given real-valued examples; S411. Given a real-valued example: S412. Adjust the values ​​of ω and p and substitute them into the ADRNN model; similarly, adjust the values ​​of η and α and substitute them into the CGNN model and CZNN model respectively; S413. Verify the performance of the ADRNN model under no noise, constant noise, linear noise, and bounded random noise conditions.

5. Application of adaptive gradient neural dynamics optimization in AOA positioning, characterized by: The AOA positioning is any one of claims 1-4, converting the arrival angle AOA positioning into a mathematical optimization problem and solving it through the ADRNN model to verify the practicality and superiority of the model; S42, applying the ADRNN model to 3D AOA positioning; S421. According to the principle of AOA positioning, the AOA positioning problem is converted into an optimization problem for solution; When the antenna array is installed on the BS, the antenna array obtains the angles of incidence α1, α2, β1, and β2 according to the signal sent by the MS. The intersection of the angles of incidence is the position of the MS. Let the position coordinates of the MS be (x, y, z) and the position coordinates of the BS be (x i ,y i ,z i ), where i=1, ... n is the number of BSs. If n=2, then the two existing base stations must satisfy the following relationship: The process of solving the MS position becomes solving the equation S(t)X(t)=W(t). The positions of the two base stations are set to (-6, -6, -2) and (6, 6, 2) respectively. The ADRNN model is used to solve the position of the MS (x, y, z) to verify the practicality and superiority of the ADRNN model. S422, considering the positioning performance of the ADRNN model when measurement errors and noise interference exist; Since the error in the measurement process cannot be ignored in practical applications, the angle measurement result BS is obtained. i for where n i is additive Gaussian noise, which obeys zero mean distribution, that is, n i ~N(0,σ 2 ), verifying the positioning effect of the model when there are errors in the measurement process; In addition to the measurement error, we also considered the case of noise interference in the positioning process. When adding sinusoidal noise Γ(t) = sin(4πt+π), we verified the positioning effect of the model in the presence of noise interference.

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