Passive time difference positioning method based on improved Kepler optimization algorithm

Abstract

The invention discloses a passive time difference positioning method based on an improved Kepler optimization algorithm, and the method comprises the steps: firstly converting a passive time difference positioning equation solving problem into an optimization problem which takes the real position of a positioning target as a design variable, and designing a target function; secondly, using Logistic chaotic mapping to improve initialization of positions of candidate solutions, calculating fitness of the candidate solutions by using an objective function according to the positions of the candidate solutions, and taking the candidate solution with the optimal fitness as a current optimal solution; according to the positions of the candidate solution and the optimal solution, the gravitation between the candidate solution and the optimal solution is calculated, and the speed of the candidate solution is calculated; and finally, generating a random number, updating the position of a candidate solution, calculating fitness by using an objective function, updating global optimum, and finally outputting a position corresponding to an optimal solution to realize target positioning. The method is higher in global search and local search capability, higher in convergence speed, better in positioning effect and suitable for complex positioning scenes.

Description

Technical Field

[0001] The invention relates to the technical field of passive positioning, and in particular to a passive time difference positioning method based on an improved Kepler optimization algorithm. Background Art

[0002] Passive positioning is an important technical means of modern electronic warfare. That is, the positioning device does not actively emit signals to the outside, but locates the target by receiving signals radiated by the target.

[0003] Time difference positioning is a commonly used passive positioning method. At present, passive time difference positioning methods can be divided into two categories: closed algebraic solutions and iterative methods. Closed algebraic solutions avoid convergence problems and achieve global optimality, but are very sensitive to TDOA measurement errors. Iterative methods are divided into deterministic iterative methods and random optimization methods. The former has the risk of non-convergence and has high requirements for the setting of initial values, while the latter has the problem of local optimal solutions. Therefore, it is of great value to study passive time difference positioning methods that are less sensitive to measurement errors, have better convergence performance, and are easier to obtain global optimal solutions.

[0004] In recent years, meta-heuristic algorithms such as particle swarm optimization (PSO) and grey wolf optimization (GWO) have been applied to passive time difference positioning. Kepler optimization algorithm (KOA) is a new meta-heuristic algorithm proposed by Mohamed Abdel-Basset et al. Inspired by Kepler's laws of planetary motion, KOA uses parameters such as the position, mass, gravity and orbital velocity of candidate solutions to control the update process of candidate solutions, and has better interpretability in the global optimization problem. In KOA, candidate solutions are randomly updated according to the position of the best solution to achieve the optimization process. However, the performance of KOA is highly dependent on the characteristics of the problem and the choice of parameters. Therefore, how to reasonably improve the algorithm is also of great research value. Summary of the invention

[0005] In view of the above problems, the present invention proposes a passive time difference positioning method based on an improved Kepler optimization algorithm, which transforms the passive time difference positioning equation solving problem into an optimization problem of seeking the optimal solution of the objective function, making the solution process simpler and the positioning accuracy higher.

[0006] The technical solution of the present invention is:

[0007] Step 1: Convert the problem of solving the passive time difference positioning equation into an optimization problem with the real position of the positioning target as the design variable, and determine the objective function of the Kepler optimization algorithm;

[0008] Step 2: Initialize the Kepler algorithm parameters, use Logistic chaotic mapping to improve the initialization of the candidate solution position, calculate the fitness of the candidate solution based on the position of the candidate solution using the objective function in step 1, and take the candidate solution with the best fitness as the current optimal solution.

[0009] Step 3: Based on the positions of the candidate solution and the optimal solution in step 2, calculate the gravitational force between them.

[0010] Step 4: Calculate the speed of the candidate solution.

[0011] Step 5: Generate random numbers r and r 1 , select different solutions according to the size relationship between the two, and update the candidate solution position in combination with the parameters obtained in steps 3 and 4. In this process, the linear decreasing factor η is improved to a nonlinear decreasing factor, and finally an elite strategy is used to ensure the best position of the candidate solution and the optimal solution.

[0012] Step 6: According to the new position obtained in step 5, use the objective function in step 1 to calculate the fitness, update the global optimum, and return to step 3 until the termination time is reached. The position corresponding to the optimal solution finally output is the target position, thereby achieving the positioning of the target.

[0013] The present invention proposes a passive time difference positioning method based on an improved Kepler optimization algorithm. By converting the problem of solving the positioning equation into an optimization problem, the improved Kepler optimization algorithm is used to obtain the global optimal solution, thereby realizing passive time difference positioning.

[0014] Furthermore, the above step 1 specifically includes:

[0015] Assume that there are at least three receiving stations, i.e., M ≥ 3, and the locations of the receiving stations are known, denoted by S i (x i ,y i ,z i ), i=1,2,...,M. The position of the unknown target is T(x,y,z). The distance from the receiving station to the unknown target is defined as

[0016]

[0017] Taking receiving station 1 as the reference station and the other stations as observation stations, the difference between the distance of the unknown target to observation station i and the distance to the reference station is defined as

[0018] R i1 =R i -R 1 ,i=2,3,...,M (2)

[0019] The arrival time difference between the reference station and the observation station is expressed as

[0020]

[0021] Where c is the signal propagation speed, n i1 represents the measurement noise of the arrival time difference between the i-th observation station and the reference station, and its known variance is

[0022] Assumptions

[0023]

[0024] Then there is

[0025] ΔR=RR 1 +n (5)

[0026] Assuming that the maximum likelihood method is used to estimate the unknown target, the maximum likelihood function is:

[0027]

[0028] Then the problem of solving the passive time difference positioning equation can be transformed into finding the optimal Make L the largest, that is

[0029]

[0030] Therefore, the objective function is set to

[0031]

[0032] Furthermore, the above step 2 includes:

[0033] During the initialization process, the Kepler optimization algorithm will randomly distribute N candidate solutions. The candidate solution positions of the standard Kepler optimization algorithm are initialized according to the following formula:

[0034]

[0035] in, represents the j-th dimension variable of the i-th candidate solution in the search space; N represents the number of candidate solutions in the search space, also known as the population size; d represents the dimension of the problem to be optimized; and They represent the upper and lower limits of the j-th dimension variable of the i-th candidate solution; rand [0,1] is a random number between 0 and 1.

[0036] The present invention uses Logistic chaotic mapping to improve the initialization of the candidate solution position of the Kepler optimization algorithm. The mapping principle is simple and has strong randomness and ergodicity. The improved candidate solution position initialization is as follows:

[0037]

[0038] where z i is the Logistic chaos factor, a is the control parameter, and takes values ​​in (0,4]. The larger a is, the higher the chaos is. When a=4, it is in a completely chaotic state. Finally, z i The value range of is still between [0,1].

[0039] Candidate solution eccentricity parameter e i The initialization definition is as follows:

[0040] e i =rand [0,1] ,i=1,2,...,N (11)

[0041] The cycle initialization is defined as:

[0042] T i =|r|,i=1,2,...,N (12)

[0043] Where r is a random number generated according to a normal distribution.

[0044] Finally, the fitness is calculated according to formula (8) using the initialized candidate solution positions, and the candidate solution with the current best fitness is selected as the optimal solution.

[0045] Furthermore, the above step 3 includes:

[0046] Optimal Solution X S and candidate solution X i The gravitational force between them is defined as follows

[0047]

[0048] in and Respectively represent X S and X i The quality of M S and m i The normalized value of M S and m i are given by (15) and (16) respectively; ε is the minimum value to prevent division by 0 error, ζ is the gravitational constant, and r2 is a random number between 0 and 1 to allow more changes in the gravitational value during the optimization process; Yes X S and X i The Euclidean distance R i The normalized value, R i The definition of

[0049]

[0050] Among them, ||X S (t)-X i (t)|| 2 Represents X S and X i The Euclidean distance between variables of the same dimension. The quality of the optimal solution and candidate solution i at time t is simply calculated by the fitness evaluation as shown below

[0051]

[0052]

[0053]

[0054]

[0055] Among them, fit s (t) represents the fitness of the optimal solution, fit k (t) represents the fitness of the kth candidate solution, best(t) represents the current best fitness, worst(t) represents the current worst fitness, r 2 is a random number between 0 and 1 to distribute the quality values ​​of each candidate solution. The above fitness is calculated according to formula (8). μ(t) is a function that decreases exponentially with time t to control the search accuracy, defined as follows

[0056]

[0057] where γ is a constant, μ 0 is the initial value, t and T max are the current number of iterations and the maximum number of iterations respectively.

[0058] Furthermore, in the above step 4, the speed of calculating the candidate solution includes:

[0059] According to the Kepler algorithm, candidate solutions approach the optimal solution within a certain period of time, and then move away from the optimal solution. This algorithm simulates this behavior through two phases: exploration and development phase. It explores candidate solutions that are far from the optimal solution to update the candidate solution position, and at the same time makes better use of candidate solutions that are close to the optimal solution to develop new positions near the optimal solution.

[0060] The speed of the candidate solution is calculated by formula (20)

[0061]

[0062]

[0063]

[0064] M=(r3 ×(1-r 4 )+r 4 ) (twenty three)

[0065]

[0066]

[0067]

[0068]

[0069]

[0070]

[0071] Among them, l, M, L, U 2 are all parameters used in the calculation process, V i (t) represents the velocity of candidate solution i at time t, represents candidate solution i, r 3 and r 4 is a random number between [0,1], and is a random vector between [0,1]. and represents a solution randomly selected from the population, M S and m i Represent the optimal solution X S and candidate solution X i The quality of R i (t) represents the optimal solution and candidate solution X at time t i The distance between them; F is used as a sign to change the search direction, a i (t) represents the semi-major axis parameter of candidate solution i at time t, which is defined as follows

[0072]

[0073] In this algorithm, it is assumed that the semi-major axis of candidate solution i gradually decreases with iteration, that is, the candidate solution moves to the area where the optimal solution may be found. i-norm (t) represents the S and X i The Euclidean distance is normalized and defined as follows

[0074]

[0075] Among them, min(R(t)) and max(R(t)) represent the minimum and maximum distances between the optimal solution and the candidate solutions, respectively.

[0076] Further, the step 5 comprises:

[0077] Generate random numbers r and r 1 , when r ≥ r 1 , update the position of the candidate solution by the following formula

[0078]

[0079] When r<r 1 , update the candidate solution position by the following formula

[0080]

[0081] Where h is an adaptive factor that controls the distance between the optimal solution at time t and the current candidate solution, and is defined as follows:

[0082]

[0083] where r is a randomly generated number according to a normal distribution, and η is a linearly decreasing factor from 1 to -2, defined as follows:

[0084] η=(a 2 -1)×r 4 +1 (35)

[0085] a 2 is a loop control parameter. During the entire optimization process, a 2 go through The cycle gradually decreases from -1 to -2 and is defined as follows:

[0086]

[0087] When the value of h is large, the exploration operator is used to control the candidate solutions away from the optimal solution; on the contrary, when the value is small, the development operator is used to explore the area around the current optimal solution.

[0088] However, the linear decrease of η will lead to uneven changes in h, which is not conducive to the early large-scale search. Therefore, the present invention defines η as a nonlinear decrease and improves the original algorithm as follows

[0089]

[0090] The elite strategy in the Kepler optimization algorithm can be summarized by the following formula

[0091]

[0092] Where f represents the objective function defined in equation (8).

[0093] Further, the step 6 comprises:

[0094] According to the new position obtained in step 5, the fitness of the candidate solution and the optimal solution is calculated using formula (8), the global optimum is updated, and it is iterated until the termination time is reached. The position of the optimal solution is output to achieve the positioning of the target.

[0095] The present invention proposes a passive time difference positioning method based on an improved Kepler optimization algorithm. Compared with the traditional positioning algorithm, the method has stronger global search and local search capabilities, faster convergence speed, better positioning effect, and is more suitable for complex positioning scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0096] Figure 1 is a flow chart of the present invention;

[0097] Figure 2 It is a comparison chart of the convergence speed of the algorithm proposed in the present invention, the standard KOA algorithm, and the particle swarm algorithm (PSO). DETAILED DESCRIPTION

[0098] The present invention is further described below in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and cannot be used to limit the protection scope of the present invention.

[0099] The present invention proposes a passive time difference positioning method based on an improved Kepler optimization algorithm. By improving the initialization and linear decreasing factor of the algorithm, the algorithm has stronger global optimization and local search capabilities, faster convergence speed and higher positioning accuracy.

[0100] Reference Figure 1 The specific process of the present invention can be seen:

[0101] Step 1: Convert the problem of solving the passive time difference positioning equation into an optimization problem with the real position of the positioning target as the design variable, and determine the objective function of the optimization algorithm;

[0102] Step 2: Initialize the Kepler algorithm parameters, use the Logistic chaotic map to improve the initialization of the candidate solution position, calculate the fitness of the candidate solution based on the position of the candidate solution using the objective function in step 1, and take the candidate solution with the best fitness as the current optimal solution;

[0103] Step 3: Combine the positions of the candidate solution and the optimal solution in step 2 and calculate the gravitational force between them;

[0104] Step 4: Calculate the speed of the candidate solution;

[0105] Step 5: Generate random numbers r and r 1, select different solutions according to the size relationship between the two, and update the candidate solution position in combination with the parameters obtained in steps 3 and 4. In this process, the linear decreasing factor η is improved to a nonlinear decreasing factor, and finally an elite strategy is used to ensure the best position of the candidate solution and the optimal solution;

[0106] Step 6: According to the new position obtained in step 5, use the objective function in step 1 to calculate the fitness of the candidate solution and the optimal solution, update the global optimum, and return to step 3 until the termination time is reached. The position corresponding to the optimal solution is finally output, that is, the target position, thereby achieving the positioning of the target.

[0107] The specific embodiments are:

[0108] 1. Establish a passive time difference positioning model and determine the objective function of the optimization algorithm, specifically:

[0109] There are 4 observation stations with known locations, i.e., M=4, and their coordinates are shown below: 1 (50,0,50),S 2 (0,0,50),S 3 (50,50,50),S 4 (0,100,50), unit is km, the same below.

[0110] The noise has zero mean and variance The real position of the target is expressed as T(200,200,200). According to the formula, the passive time difference positioning model is established, and the objective function of the optimization algorithm is determined. The algorithm calculates the fitness of the candidate solution and the optimal solution according to the objective function.

[0111] 2. Use Logistic chaotic mapping to improve the initialization of candidate solution positions, initialize algorithm parameters at the same time, calculate fitness based on the initialized positions, and determine the current optimal solution.

[0112] Let the maximum number of iterations T max =500, the number of planets N = 100, μ 0 =0.1,γ=15. The upper and lower bounds of the global search are ub=(500,500,500) and lb=(-500,-500,0).

[0113] Finally, the fitness is calculated according to formula (8) using the initialized candidate solution positions, and the candidate solution with the current best fitness is selected as the optimal solution.

[0114] 3. Calculate the gravitational force between the optimal solution and the candidate solution, as well as the velocity of the candidate solution.

[0115] 4. According to the random numbers r and r 1The size of the candidate solution is updated, and the linear decreasing factor η is improved to a nonlinear decreasing factor. Finally, an elite strategy is used to ensure that the positions of the candidate solution and the optimal solution are both optimal.

[0116] 5. According to the current candidate solution and the optimal solution, the fitness is calculated using formula (8), the global optimum is updated and iterated until the termination time is reached, and the optimal solution is finally output as the location of the positioning target. In this example, the optimal fitness is the maximum value of the objective function, and its corresponding position is recorded as

[0117] 6. Analyze the positioning results, specifically:

[0118] (1) The convergence speed of the algorithm (IKOA) proposed in this invention is compared with that of the traditional particle swarm optimization algorithm (PSO) and the standard Kepler optimization algorithm (KOA). The results are as follows: Figure 2 As shown, it can be seen that the algorithm proposed in the present invention has a faster convergence speed.

[0119] (2) In order to verify the robustness of the algorithm proposed in this invention in TDOA positioning, 1000 Monte Carlo simulations were performed to statistically analyze the positioning accuracy of IKOA, KOA and PSO under different target positions. If the error with T(x,y,z) exceeds 20m, it is considered a positioning error. The statistical results are shown in the following table.

[0120] Table 1 Positioning accuracy of different algorithms

[0121]

[0122]

[0123] It can be seen from Table 1 that the IKOA proposed in the present invention has the best robustness, followed by KOA, and PSO has the worst robustness.

[0124] (3) In order to verify the positioning accuracy of IKOA in TDOA positioning, Monte Carlo simulation is performed under the same target position, and the root mean square error RMSE of positioning is statistically calculated.

[0125] The statistical results are shown in the following table

[0126] Table 2 Root mean square error of positioning of different algorithms

[0127] Algorithm Name RMSE IKOA 0.00332 KOA 0.00866 PSO 0.0112

[0128] As shown in Table 2, the IKOA proposed in the present invention has the highest positioning accuracy. Although the positioning accuracy of KOA is higher than that of PSO, the effects of the two are very close.

[0129] In summary, the passive time difference positioning method based on the improved Kepler optimization algorithm proposed in the present invention has faster convergence speed, stronger robustness and higher positioning accuracy than the standard KOA algorithm and PSO algorithm, and thus can achieve better positioning effect.

[0130] The above is only a preferred embodiment of the present invention and does not limit the present invention in any way. Any simple modification, change and equivalent change made to the above embodiment according to the technical essence of the invention still falls within the protection scope of the technical solution of the present invention.

Claims

1. A passive time difference positioning method based on an improved Kepler optimization algorithm, characterized in that: The following steps are involved: Step 1: Convert the problem of solving the passive time difference positioning equation into an optimization problem with the real position of the positioning target as the design variable, and determine the objective function of the Kepler optimization algorithm; Step 2: Use Logistic chaotic mapping to improve the initialization of the candidate solution position. According to the position of the candidate solution, use the objective function to calculate the fitness of the candidate solution, and take the candidate solution with the best fitness as the current optimal solution. Step 3: Calculate the gravitational force between the candidate solution and the optimal solution according to the positions of the candidate solution and the optimal solution in step 2; Step 4: Calculate the speed of the candidate solution; Step 5: Generate random numbers r and r1, and update the candidate solution position based on the parameters obtained in steps 3 and 4; Step 6: According to the new position obtained in step 5, use the objective function to calculate the fitness, update the global optimum, and return to step 3 until the termination time is reached. The position corresponding to the optimal solution finally output is the target position, which realizes the positioning of the target.

2. The passive time difference positioning method based on the improved Kepler optimization algorithm according to claim 1, characterized in that: The specific process of determining the objective function of the optimization algorithm is as follows: Assume that there are at least three receiving stations, i.e., M ≥ 3, and the locations of the receiving stations are known, denoted by S i (x i ,y i ,z i ), i = 1, 2, ..., M; the position of the unknown target is T (x, y, z); the distance from the receiving station to the unknown target is defined as: Taking receiving station 1 as the reference station and the other stations as observation stations, the difference between the distance of the unknown target to observation station i and the distance to the reference station is defined as: R i1 =R i -R1,i=2,3,...,M (2) The arrival time difference between the reference station and the observing station is expressed as: Where c is the signal propagation speed, n i1 represents the measurement noise of the arrival time difference between the i-th observation station and the reference station, and its known variance is σ n 2 ; Assumptions: Then we have: ΔR=R-R1+n (5) Assuming that the maximum likelihood method is used to estimate the unknown target, the maximum likelihood function is: Then the problem of solving the passive time difference positioning equation is transformed into finding the optimal Make L the largest, that is: So the objective function is set as:

3. The passive time difference positioning method based on the improved Kepler optimization algorithm according to claim 2, characterized in that: The specific implementation process of step 2 is as follows: Logistic chaos mapping is used to improve the initialization of the candidate solution position of the Kepler optimization algorithm. The improved candidate solution position initialization is as follows: Where X i j represents the j-th dimension variable of the i-th candidate solution in the search space, z i is the Logistic chaos factor, a is the control parameter, and takes values ​​in (0,4]. The larger a is, the higher the chaos is. When a=4, it is in a completely chaotic state. Finally, z i The value range of is still between [0,1]; N represents the number of candidate solutions in the search space; d represents the dimension of the optimization problem; and They represent the upper and lower limits of the j-th dimension variable of the i-th candidate solution; rand [0,1] is a random number between 0 and 1; Candidate solution eccentricity parameter e i The initialization definition is as follows: it is i =row [0,1] ,i=1,2,...,N (10) The cycle initialization is defined as: T i =|r1|,i=1,2,...,N (11) Where r1 is a random number generated according to normal distribution; The fitness is calculated using the initialized candidate solution positions according to formula (8), and the candidate solution with the current best fitness is selected as the optimal solution.

4. The passive time difference positioning method based on the improved Kepler optimization algorithm according to claim 3, characterized in that: The specific implementation process of step 3 is as follows: Optimal Solution X S and candidate solution X i The gravitational force between them is defined as follows in and Respectively represent X S and X i The quality of M S and m i The normalized value of ; ε is the minimum value, ζ is the gravitational constant, and r2 is a random number between 0 and 1; Yes X S and X i The Euclidean distance R i The normalized value of ; the quality of the optimal solution and candidate solution i at time t is calculated by the fitness evaluation as shown below: Among them, fit s (t) represents the fitness of the optimal solution, fit k (t) represents the fitness of the kth candidate solution, best(t) represents the current best fitness, worst(t) represents the current worst fitness, and r2 is a random number between 0 and 1. The above fitness are calculated according to formula (8).

5. The passive time difference positioning method based on the improved Kepler optimization algorithm according to claim 4, characterized in that: The specific process of calculating the speed of the candidate solution is as follows: μ(t) is a function that decreases exponentially with time t to control the search accuracy and is defined as follows where γ is a constant, μ0 is the initial value, t and T max They are the current number of iterations and the maximum number of iterations respectively; The speed of the candidate solution is calculated by formula (18): M＝(r3×(1-r4)+r4) (21) Among them, l, M, L, U2 are the parameters used in the calculation process, V i (t) represents the velocity of candidate solution i at time t, represents candidate solution i, represents the upper and lower bounds of the candidate solution i, r3 and r4 are random numbers between [0,1], and is a random vector between [0,1]; and represents a solution randomly selected from the population, M S and m i Represent the optimal solution X S and candidate solution X i The quality of R i (t) represents the optimal solution and candidate solution X at time t i The distance between them; F is used as a sign to change the search direction, a i (t) represents the semi-major axis parameter of candidate solution i at time t, which is defined as follows: R i-norm (t) represents the S and X i The Euclidean distance is normalized.

6. The passive time difference positioning method based on the improved Kepler optimization algorithm according to claim 5, characterized in that: The specific process of step 5 is as follows: Generate random numbers r and r1. When r≥r1, update the position of the candidate solution by the following formula: in, represents the optimal solution X S and candidate solution X i location; When r<r1, the candidate solution position is updated by the following formula: Where h is an adaptive factor that controls the distance between the optimal solution at time t and the current candidate solution, defined as: Where p is a number randomly generated according to a normal distribution, and e represents the natural logarithm; a2 is the loop control parameter. During the entire optimization process, a2 is The cycle gradually decreases from -1 to -2 and is defined as follows: Define η as nonlinearly decreasing as follows: The elite strategy in the Kepler optimization algorithm is summarized in the following formula: Where f represents the objective function defined in equation (8).

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