An improved cuckoo algorithm-based aircraft multilateration solution method

Abstract

The application discloses a kind of based on improved cuckoo algorithm's aircraft multilateration solution method, which includes obtaining multilateration mathematical model;Obtain the search area of cuckoo algorithm;Initialize population parameters;The fitness value of each bird nest is calculated, and the current optimal bird nest is recorded;The fitness value of updated bird nest is calculated and compared with the optimal bird nest fitness value;According to random preference walk, update bird nest position;Output optimal bird nest position, to obtain aircraft position and the like steps.The application utilizes Chan algorithm to process effective information, obtains initial estimated position of aircraft by twice least square estimation, reduces operation amount, and improves convergence rate of cuckoo algorithm;The initial population is chaotic in feasible region by using the thought of chaos algorithm, so that individual is relatively uniformly distributed in feasible region, and the diversity and ergodicity of initial population are strengthened;Self-adaptive discovery probability is introduced, which is conducive to balancing global exploration and local development capability of algorithm, accelerating convergence speed of algorithm, and increasing population diversity.

Description

Technical Field

[0001] This invention belongs to the field of multi-point positioning system technology, and specifically relates to an aircraft multi-point positioning solution method based on an improved cuckoo algorithm. Background Technology

[0002] my country's mainstream airport surface surveillance technologies mainly include multipoint positioning systems (MPS), primary and secondary surveillance radars, and Adaptive Digital Subsystem for Boundary (ADS-B) systems. Compared to ADS-B, MPS calculates the position of a target at the moment it sends a response signal, effectively avoiding the electronic deception issues present in ADS-B. However, MPS suffers from accuracy problems during operation, and the positioning algorithm is key to solving this issue. How to further improve target calculation accuracy to ensure the airport surface target monitoring accuracy required by international civil aviation has been a persistent and unresolved challenge since the advent of MPS.

[0003] Traditional multipoint localization (MRT) methods primarily employ line-of-sight (LAS) algorithms such as the Chan algorithm, Taylor series expansion, Friedlander algorithm, and Fang algorithm. In recent years, with the development of intelligent optimization algorithms, there has been extensive improvement to MRT methods, but most research focuses on combining traditional algorithms with intelligent optimization algorithms for optimized localization. The Chan algorithm, as a fundamental MRT algorithm, has been studied for many years due to its high positioning accuracy. However, no Chan localization algorithm based on an improved version of the Cuckoo algorithm has yet been discovered. Summary of the Invention

[0004] To address the aforementioned problems, the present invention aims to provide an aircraft multi-point positioning solution method based on an improved cuckoo algorithm, which can effectively improve the positioning accuracy of airport surface multi-point positioning systems.

[0005] To achieve the above objectives, the aircraft multi-point localization solution method based on the improved cuckoo algorithm provided by this invention includes the following steps performed in sequence:

[0006] Step 1: Based on the observation data from the base station, the likelihood function is obtained using the maximum likelihood method, and then a multi-point positioning mathematical model is obtained to solve the aircraft position.

[0007] Step 2: Use the Chan algorithm to solve the multi-point positioning mathematical model in Step 1 to obtain the initial estimated position of the aircraft, and then obtain the search area of ​​the Cuckoo algorithm;

[0008] Step 3: Initialize the population parameters using Circle chaotic mapping to ensure that the cuckoo population is evenly distributed throughout the search area.

[0009] Step 4: Set the fitness function of the cuckoo algorithm based on the likelihood function obtained in Step 1, and then use the fitness function to calculate the fitness value of each nest in the search area. The smaller the fitness value, the better the nest location, and record the current best nest.

[0010] Step 5: Update the position of each nest in Step 4 according to the Levy update formula of the Cuckoo Algorithm, then calculate the fitness value of the updated nest and compare it with the fitness value of the best nest in Step 4, and retain the nest with better fitness.

[0011] Step 6: Use adaptive discovery probability to filter the bird nests retained in Step 5, update the bird nest positions according to random preference walks, then calculate the fitness value of the updated bird nests and compare it with the fitness values ​​of the bird nests with better fitness in Step 5, and retain the bird nests with better fitness.

[0012] Step 7: Determine if the number of iterations is satisfied. If the number of iterations is satisfied, output the optimal bird's nest position among the bird's nests with good fitness in Step 6, thereby obtaining the aircraft position. Otherwise, return to Step 5.

[0013] In step 1, the method for obtaining the likelihood function based on the observation data from the base station using the maximum likelihood method, and then obtaining the multi-point positioning mathematical model for solving the aircraft position, is as follows:

[0014] In a multipoint positioning system, based on the TDOA values ​​of the aircraft's transponder signal reaching two base stations, it can be determined that the aircraft is on a hyperbola with those two base stations as its foci. A third base station can then be used to locate the aircraft at the intersection of the two hyperbolas. Given M base stations, (X... i ,Y i (x, y) represents the location of the i-th base station, (x, y) represents the aircraft's location, and r... i It is the distance from the aircraft to the i-th base station, r i,1 Let be the distance difference between the aircraft and the i-th base station and the aircraft and the 1st base station. If M base stations are observed simultaneously, then the matrix form of the observation equations is:

[0015] ΔR=R-R1+N (1)

[0016] Where ΔR is the distance difference r between the aircraft and the (2-M)th base station and the aircraft and the first base station. i,1 R1 is a (M-1)×1 matrix vector consisting of the distances r1 from the aircraft to the first base station, and R is a (M-1)×1 matrix vector consisting of the distances r1 from the aircraft to the second (M)th base station. i The matrix vector consisting of (M-1)×1 elements i = 2, ..., M, where N is the measurement noise ε from the (2-M)th base station to the 1st base station. i,1 A matrix vector consisting of (M-1)×1 elements, i = 2, ..., M;

[0017] Considering the case where M > 3 base stations, the aircraft position is estimated using the maximum likelihood method based on the above observation equations. It is assumed that the measurement noise ε from the i-th base station to the 1st base station... i,1 It has zero mean and variance δ 2 Given Gaussian white noise, and with all measurements independent of each other, the likelihood function is:

[0018]

[0019] Therefore, finding the coordinates that maximize the likelihood function is equivalent to solving the multi-point positioning mathematical model shown in equation (3) for determining the aircraft's position:

[0020] (x,y)=arg{min[(ΔR-R+R1) T (ΔR-R+R1)]} (3)

[0021] In step 2, the method of using the Chan algorithm to solve the multi-point positioning mathematical model in step 1 to obtain the initial estimated position of the aircraft, and then obtaining the search area of ​​the Cuckoo algorithm, is as follows:

[0022] First, the distance differences between each base station and the reference base station are calculated using the time difference between the point to be determined and each base station, and between the point to be determined and the reference base station. Then, using these distance differences, the initial position of the aircraft is approximated by applying the weighted least squares method twice, resulting in the two weighted least squares estimates of the aircraft's initial position.

[0023]

[0024] Where z′ is the initial solution (x0, y0) obtained from the second weighted least squares solution. This initial solution (x0, y0) is used as the center point for initialization of the Cuckoo Algorithm community, and the search area of ​​the Cuckoo Algorithm is obtained as follows:

[0025]

[0026] Wherein, C is a constant, which is taken as 0.25 in this invention.

[0027] In step 3, the method of initializing the population parameters using Circle chaotic mapping to ensure the cuckoo population is evenly distributed throughout the search area is as follows:

[0028] The population parameters are initialized using the Circle chaotic map. The expression for the Circle chaotic map is:

[0029]

[0030] Where α = 0.5, b = 0.2, xk It is the current random number, x k+1 It is the next generation of random numbers.

[0031] In step 4, the fitness function of the cuckoo algorithm is set according to the likelihood function obtained in step 1. Then, the fitness value of each nest in the search area is calculated using this fitness function. The smaller the fitness value, the better the nest location. The method for recording the current best nest is as follows:

[0032] Based on the likelihood function obtained in step 1, the fitness function of the Cuckoo algorithm is set as follows:

[0033]

[0034] Then, the fitness value of each nest in the search area is calculated using equation (7). The smaller the fitness value, the better the nest location. The current best nest is recorded.

[0035] In step 5, each nest position from step 4 is updated according to the Levy update formula of the cuckoo algorithm. Then, the fitness value of the updated nest is calculated and compared with the fitness value of the best nest in step 4. The method for retaining nests with better fitness is as follows:

[0036] The Levy update formula for the Cuckoo algorithm is as follows:

[0037]

[0038]

[0039]

[0040] Among them, X i t It is the location of the bird's nest in generation t; X i t+1 is the nest location of the (t+1)th generation; μ and v are random variables following a standard normal distribution; β is a control factor, which is taken as 1.5 in this invention;

[0041] Then, the fitness value of the updated bird nest is calculated according to equation (8) and compared with the fitness value of the best bird nest in step 4. The bird nest with better fitness is retained.

[0042] In step 6, the nests retained in step 5 are filtered using adaptive discovery probability. The nest locations are updated using a random preference walk. Then, the fitness value of the updated nests is calculated and compared with the fitness values ​​of the better-fitting nests from step 5. The method for retaining the better-fitting nests is as follows:

[0043] The formula for the adaptive discovery probability Pa is as follows:

[0044]

[0045] Where, r Pa =0.5, where T is the maximum number of iterations and t is the current iteration number;

[0046] A random number r distributed in [0,1] is generated using the rand function and compared with the adaptive discovery probability Pa. Nests with a probability greater than Pa undergo a preferred random walk, while nests with a probability less than Pa remain unchanged. The formula for the random preference walk is:

[0047] X i (t+1) =X i t +r(X j t -X k t (10)

[0048] Among them, X j t X k t Let t represent two random bird nests, and r be a random number that follows a normal distribution;

[0049] Then, the fitness value of the updated nest is calculated according to equation (10) and compared with the fitness value of the nest with better fitness in step 5. The nest with better fitness is retained.

[0050] The aircraft multi-point localization solution method based on the improved cuckoo algorithm provided in this invention has the following advantages compared with the prior art:

[0051] (1) This invention uses the Chan algorithm to process effective information and obtains the initial estimated position of the aircraft through two least squares estimations, thereby limiting the search area of ​​the Cuckoo algorithm, reducing the amount of computation, and improving the convergence rate of the Cuckoo algorithm.

[0052] (2) This invention introduces Circle chaotic mapping, which uses the idea of ​​chaotic algorithm to make the initial population chaotic in the feasible region, so that the individuals are relatively evenly distributed in the feasible region, enhance the diversity and ergodicity of the initial population, and reduce the risk of the algorithm getting trapped in local optima.

[0053] (3) The present invention introduces an adaptive discovery probability, which overcomes the drawbacks of taking a fixed discovery probability, and is conducive to balancing the global exploration and local development capabilities of the algorithm, accelerating the convergence speed of the algorithm, and increasing population diversity. Attached Figure Description

[0054] Figure 1The flowchart of the aircraft multi-point localization solution method based on the improved cuckoo algorithm provided by this invention is shown in the figure.

[0055] Figure 2 This is a schematic diagram of the multi-point positioning system used in this invention.

[0056] Figures 3(a) and (b) are the distribution diagram and histogram of the Circle chaotic mapping in this invention, respectively.

[0057] Figure 4 This is the curve showing the change in the adaptive discovery probability in this invention.

[0058] Figure 5 This is a schematic diagram of the cellular mesh base station distribution in this invention.

[0059] Figure 6 This is a comparison curve of the convergence curves of the Cuckoo Algorithm and the improved Cuckoo Algorithm in this invention.

[0060] Figure 7 This paper compares the MSE of the method of the present invention with those of Taylor's algorithm, Chan's algorithm, genetic algorithm and Cuckoo algorithm. Detailed Implementation

[0061] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0062] like Figure 1 As shown, the aircraft multi-point localization solution method based on the improved cuckoo algorithm provided by this invention includes the following steps performed in sequence:

[0063] Step 1: Based on the observation data from the base station, the likelihood function is obtained using the maximum likelihood method, and then a multi-point positioning mathematical model is obtained to solve the aircraft position.

[0064] In such Figure 2 In the multipoint positioning system shown, based on the TDOA values ​​of the aircraft's transponder signal reaching two base stations, it can be determined that the aircraft is on a hyperbola with those two base stations as its foci. A third base station can then be used to locate the aircraft at the intersection of the two hyperbolas. There are M base stations, (X... i ,Y i (x, y) represents the location of the i-th base station, (x, y) represents the aircraft's location, and r... i It is the distance from the aircraft to the i-th base station, r i,1 Let be the distance difference between the aircraft and the i-th base station and the aircraft and the 1st base station. If M base stations are observed simultaneously, then the matrix form of the observation equations is:

[0065] ΔR=R-R1+N (1)

[0066] Where ΔR is the distance difference r between the aircraft and the (2-M)th base station and the aircraft and the first base station.i,1 R1 is a (M-1)×1 matrix vector consisting of the distances r1 from the aircraft to the first base station, and R is a (M-1)×1 matrix vector consisting of the distances r1 from the aircraft to the second (M)th base station. i The matrix vector consisting of (M-1)×1 elements i = 2, ..., M, where N is the measurement noise ε from the (2-M)th base station to the 1st base station. i,1 A matrix vector consisting of (M-1)×1 elements, i = 2, ..., M;

[0067] Considering the case where M > 3 base stations, the aircraft position is estimated using the maximum likelihood method based on the above observation equations. It is assumed that the measurement noise ε from the i-th base station to the 1st base station... i,1 It has zero mean and variance δ 2 Given Gaussian white noise, and with all measurements independent of each other, the likelihood function is:

[0068]

[0069] Therefore, finding the coordinates that maximize the likelihood function is equivalent to solving the multi-point positioning mathematical model shown in equation (3) for determining the aircraft's position:

[0070] (x,y)=arg{min[(ΔR-R+R1) T (ΔR-R+R1)]} (3)

[0071] Step 2: Use the Chan algorithm to solve the multi-point positioning mathematical model in Step 1 to obtain the initial estimated position of the aircraft, and then obtain the search area of ​​the Cuckoo algorithm;

[0072] The Chan algorithm, a typical TDOA-based positioning algorithm, has advantages such as not requiring initial values ​​for calculation, high positioning accuracy in line-of-sight environments, and relatively simple computation. Due to its practicality, the Chan algorithm is also the most frequently chosen algorithm by researchers. Since the target location information in the actual scene of a multi-point positioning system is uncertain, this invention first uses the Chan algorithm to solve the multi-point positioning mathematical model in step 1 to obtain the initial estimated position of the aircraft. At the same time, it can limit the search area of ​​the Cuckoo algorithm and reduce the amount of computation.

[0073] First, the distance differences between each base station and the reference base station are calculated using the time difference between the point to be determined and each base station, and between the point to be determined and the reference base station. Then, using these distance differences, the initial position of the aircraft is approximated by applying the weighted least squares method twice, resulting in the two weighted least squares estimates of the aircraft's initial position.

[0074]

[0075] Where z′ is the initial solution (x0, y0) obtained from the second weighted least squares solution. This initial solution (x0, y0) is used as the center point for initialization of the Cuckoo Algorithm community, and the search area of ​​the Cuckoo Algorithm is obtained as follows:

[0076]

[0077] Wherein, C is a constant, which is taken as 0.25 in this invention.

[0078] Step 3: Initialize the population parameters using Circle chaotic mapping to ensure that the cuckoo population is evenly distributed throughout the search area.

[0079] The standard Cuckoo algorithm generates an initial random matrix using the rand function. However, the rand function has poor traversal of the value space, leading to poor population diversity and reduced global search capability. Chaotic mapping, characterized by uncertainty, non-repeatability, and unpredictability, uses pseudo-random numbers generated by chaotic operators to initialize population parameters, exhibiting strong diversity and robustness. Combining Figures 3(a) and (b), in chaotic mapping commonly used in swarm intelligence, the Circle chaotic mapping values ​​are between [0,1], showing a more uniform distribution within the non-edge range of the population, thus improving edge clustering. Therefore, this invention chooses the Circle chaotic mapping to initialize population parameters. The expression for the Circle chaotic mapping is:

[0080]

[0081] Where α = 0.5, b = 0.2, x k It is the current random number, x k+1 It is the next generation of random numbers.

[0082] Step 4: Set the fitness function of the cuckoo algorithm based on the likelihood function obtained in Step 1, and then use the fitness function to calculate the fitness value of each nest in the search area. The smaller the fitness value, the better the nest location, and record the current best nest.

[0083] Based on the likelihood function obtained in step 1, the fitness function of the Cuckoo algorithm is set as follows:

[0084]

[0085] Then, the fitness value of each nest in the search area is calculated using equation (7). The smaller the fitness value, the better the nest location. The current best nest is recorded.

[0086] Step 5: Update the position of each nest in Step 4 according to the Levy update formula of the Cuckoo Algorithm, then calculate the fitness value of the updated nest and compare it with the fitness value of the best nest in Step 4, and retain the nest with better fitness.

[0087] The Cuckoo algorithm assumes the following three idealized scenarios:

[0088] 1) Each nest contains only one egg, and the distribution of the eggs is random;

[0089] 2) During each evolution, retain the egg with the highest fitness value;

[0090] 3) The number of nests remains unchanged, and the parasitic birds discard cuckoo eggs based on the probability of discovery;

[0091] Based on the principles of the Cuckoo Algorithm and the above assumptions, the Levy update formula for the Cuckoo Algorithm is as follows:

[0092]

[0093]

[0094]

[0095] Among them, X i t It is the location of the bird's nest in generation t; X i t+1 is the nest location of the (t+1)th generation; μ and v are random variables following a standard normal distribution; β is a control factor, which is taken as 1.5 in this invention;

[0096] Then, the fitness value of the updated bird nest is calculated according to equation (8) and compared with the fitness value of the best bird nest in step 4. The bird nest with better fitness is retained.

[0097] Step 6: Use adaptive discovery probability to filter the bird nests retained in Step 5, update the bird nest positions according to random preference walks, then calculate the fitness value of the updated bird nests and compare it with the fitness values ​​of the bird nests with better fitness in Step 5, and retain the bird nests with better fitness.

[0098] The standard Cuckoo algorithm uses a constant discovery probability to balance global search and local exploitation capabilities. However, research has shown that a lower discovery probability in the early stages of global search can expand the search range and increase population diversity; a higher discovery probability in the later stages is beneficial for local exploitation, improving optimization accuracy and accelerating convergence. This requires the discovery probability to change dynamically with the number of iterations. Therefore, this invention proposes an adaptive discovery probability Pa, and the curve of the adaptive discovery probability Pa is shown below. Figure 4As shown in the figure, the improved adaptive discovery probability Pa changes dynamically with the iteration value. In the early stages of iteration, the adaptive discovery probability Pa is relatively small, indicating that global search is stronger than local exploration. The Cuckoo Algorithm can find a wide range of regions where the optimal value might exist. As the number of iterations increases, the Cuckoo Algorithm needs to perform local searches on the selected regions. At this point, the adaptive discovery probability Pa increases with the number of iterations, helping the Cuckoo Algorithm population to cluster locally, enabling it to find the best within a smaller area. By dynamically updating the adaptive discovery probability Pa, the Cuckoo Algorithm can simultaneously possess strong global search and local exploration capabilities.

[0099] The formula for the adaptive discovery probability Pa is as follows:

[0100]

[0101] Where, r Pa =0.5, where T is the maximum number of iterations and t is the current iteration number;

[0102] A random number r is generated in the [0,1] distribution using the rand function and compared with the adaptive discovery probability Pa. Nests with a probability greater than Pa are selected for a preferred random walk, while nests with a probability less than Pa remain unchanged. The formula for the random preference walk is as follows:

[0103] X i (t+1) =X i t +r(X j t -X k t (10)

[0104] Among them, X j t X k t Let t represent two random bird nests, and r be a random number that follows a normal distribution;

[0105] Then, the fitness value of the updated nest is calculated according to equation (10) and compared with the fitness value of the nest with better fitness in step 5. The nest with better fitness is retained.

[0106] Step 7: Determine if the number of iterations is satisfied. If the number of iterations is satisfied, output the optimal bird's nest position among the bird's nests with good fitness in Step 6, thereby obtaining the aircraft position. Otherwise, return to Step 5.

[0107] To verify the effectiveness of the method of the present invention, the inventors conducted the following experiments:

[0108] Adopting such Figure 5The cellular mesh base station distribution shown has a total of 5 base stations, with their location coordinates being (0,0), (0,-6) The unit is km, and the aircraft position is selected as (1.0, 2.5). The aircraft position is estimated 100 times, and the performance of the proposed method is evaluated using the mean position estimate (MV) and mean squared error (MSE). Simulations are performed to compare the proposed method with Taylor's algorithm, Chan's algorithm, genetic algorithm (GA), and Cuckoo algorithm (CS).

[0109] Figure 6 Comparing the convergence curves of the Cuckoo Algorithm and the Improved Cuckoo Algorithm, it can be seen that the Improved Cuckoo Algorithm has a better fitness value in the early stage of iteration because it introduces the Chan algorithm to limit the search area. The introduced Circle chaotic mapping and adaptive discovery probability Pa enable the Improved Cuckoo Algorithm to converge to the global optimum more quickly.

[0110] To further illustrate the superiority of this method, its positioning performance is compared with that of Taylor algorithm, Chan algorithm, genetic algorithm (GA), and Cuckoo algorithm (CS). The average estimated coordinates MV of the five algorithms obtained from simulation are shown in Table 1. The 10lg(cσ) in the first column of the table represents the noise level that may be encountered in a real cellular communication system.

[0111] Table 1. MV km for the 5 algorithms

[0112]

[0113] As can be seen from Table 1, when the noise variance is small, the average estimated coordinates MV of the five algorithms are very close to the actual coordinates of the target. As the noise variance increases, the gap between the average estimated coordinates MV of the five algorithms and the actual coordinates of the target also increases. It can be seen that the average estimated coordinates MV of the improved Cuckoo Algorithm (ICS) in this invention are closer to the actual coordinates of the target than the other four algorithms.

[0114] Figure 7A comparison of the MSE (Mean Sequence Size) of the five algorithms shows that when the noise variance is small, the localization performance of the five algorithms is similar. However, as the noise variance gradually increases, the Chan and Taylor algorithms deteriorate faster than the genetic algorithm. This is because the Chan algorithm uses two least squares calculations, introducing a quadratic term of noise. When the noise variance is small, this quadratic term can be ignored, and the algorithm's performance is close to optimal. As the noise variance increases, the noise quadratic term causes the Chan algorithm's performance to deteriorate more rapidly. The Taylor algorithm performs worse than the Chan algorithm in Gaussian environments, but the Taylor algorithm is affected by the initial value. When the initial value error is small, the Taylor algorithm has higher solution accuracy than the Chan algorithm. The genetic algorithm directly uses the maximum likelihood method to search for solutions and is not affected by the noise quadratic term, therefore its performance is better than the Chan and Taylor algorithms.

[0115] In intelligent optimization algorithms, the Cuckoo Algorithm is more versatile and robust than the Genetic Algorithm. The Levy flight algorithm, which alternates between large and small step sizes, enhances global search capabilities, thus outperforming the Genetic Algorithm. The improved Cuckoo Algorithm proposed in this invention introduces the Chan algorithm to narrow the search range, improving its convergence rate. It also incorporates Circle chaotic mapping during population parameter initialization, increasing population diversity and ensuring a relatively even distribution of individuals within the feasible region. Furthermore, it introduces an adaptive discovery probability Pa when updating the discovery probability for nest selection, giving the algorithm both strong global search and local exploration capabilities. Compared to the original Cuckoo Algorithm, this improves optimization accuracy and accelerates convergence. Therefore, the improved Cuckoo Algorithm proposed in this invention ensures superior positioning performance even under conditions of high noise variance.

Claims

1. A method for solving multi-point positioning of aircraft based on an improved cuckoo algorithm, characterized in that: The aircraft multi-point localization solution method based on the improved cuckoo algorithm includes the following steps performed in sequence: Step 1: Based on the observation data from the base station, the likelihood function is obtained using the maximum likelihood method, and then a multi-point positioning mathematical model is obtained to solve the aircraft position. Step 2: Use the Chan algorithm to solve the multi-point positioning mathematical model in Step 1 to obtain the initial estimated position of the aircraft, and then obtain the search area of ​​the Cuckoo algorithm; Step 3: Initialize the population parameters using Circle chaotic mapping to ensure that the cuckoo population is evenly distributed throughout the search area. Step 4: Set the fitness function of the cuckoo algorithm based on the likelihood function obtained in Step 1, and then use the fitness function to calculate the fitness value of each nest in the search area. The smaller the fitness value, the better the nest location, and record the current best nest. Step 5: According to the cuckoo algorithm The formula updates the position of each nest in step 4, then calculates the fitness value of the updated nest and compares it with the fitness value of the best nest in step 4, retaining the nests with better fitness. Step 6: Use adaptive discovery probability to filter the bird nests retained in Step 5, update the bird nest positions according to random preference walks, then calculate the fitness value of the updated bird nests and compare it with the fitness values ​​of the bird nests with better fitness in Step 5, and retain the bird nests with better fitness. Step 7: Determine if the number of iterations is satisfied. If the number of iterations is satisfied, output the optimal bird's nest position among the bird's nests with good fitness in Step 6, thereby obtaining the aircraft position; otherwise, return to Step 5. In step 2, the method of using the Chan algorithm to solve the multi-point positioning mathematical model in step 1 to obtain the initial estimated position of the aircraft, and then obtaining the search area of ​​the Cuckoo algorithm, is as follows: First, the distance differences between each base station and the reference base station are calculated using the time difference between the point to be determined and each base station, and between the point to be determined and the reference base station. Then, using these distance differences, the initial position of the aircraft is approximated by applying the weighted least squares method twice, resulting in the two weighted least squares estimates of the aircraft's initial position. (4)； in, The initial solution obtained from the second weighted least squares method. , this initial solution As the central point for the initialization of the Cuckoo Algorithm community, the search area for the Cuckoo Algorithm is obtained as follows: (5)； in, It is a constant, taken as 0.

25.

2. The aircraft multi-point positioning solution method based on the improved cuckoo algorithm according to claim 1, characterized in that: In step 1, the method for obtaining the likelihood function based on the observation data from the base station using the maximum likelihood method, and then obtaining the multi-point positioning mathematical model for solving the aircraft position, is as follows: In a multipoint positioning system, based on the TDOA values ​​of the aircraft's transponder signal reaching two base stations, it can be determined that the aircraft lies on a hyperbola with those two base stations as its foci. A third base station can then be used to locate the aircraft at the intersection of the two hyperbolas. One base station, It is the first The location of each base station. It's the aircraft's location. Is the plane to the The distance between base stations Is the plane to the The distance difference between each base station and the first base station, if observed simultaneously. If there are 1 base station, then the set of observation equations in matrix form is: (1)； in From the plane to the 2nd The distance difference between each base station and the first base station Composition Matrix vectors, It is the distance from the plane to the first base station. Composition Matrix vectors, From the plane to the 2nd Distance between base stations Composition Matrix vectors, It is the 2nd- Measurement noise from base station 1 to base station 2 Composition Matrix vectors; Consider base station Based on the above observation equations, the aircraft position is estimated using the maximum likelihood method. Let the first... Measurement noise from base station 1 to base station 2 It has zero mean and variance. Given Gaussian white noise, and with all measurements independent of each other, the likelihood function is: (2)； Therefore, finding the coordinates that maximize the likelihood function is equivalent to solving the multi-point positioning mathematical model shown in equation (3) for determining the aircraft's position: (3)。 3. The aircraft multi-point positioning solution method based on the improved cuckoo algorithm according to claim 1, characterized in that: In step 3, the method of initializing the population parameters using Circle chaotic mapping to ensure the cuckoo population is evenly distributed throughout the search area is as follows: The population parameters are initialized using the Circle chaotic map. The expression for the Circle chaotic map is: (6)； in, , It is the current random number. It is the next generation of random numbers.

4. The aircraft multi-point positioning solution method based on the improved cuckoo algorithm according to claim 2, characterized in that: In step 4, the fitness function of the cuckoo algorithm is set according to the likelihood function obtained in step 1. Then, the fitness value of each nest in the search area is calculated using this fitness function. The smaller the fitness value, the better the nest location. The method for recording the current best nest is as follows: Based on the likelihood function obtained in step 1, the fitness function of the Cuckoo algorithm is set as follows: (7)； Then, the fitness value of each nest in the search area is calculated using equation (7). The smaller the fitness value, the better the nest location. The current best nest is recorded.

5. The aircraft multi-point positioning solution method based on the improved cuckoo algorithm according to claim 1, characterized in that: In step 5, the method according to the cuckoo algorithm... The update formula updates each nest location in step 4, then calculates the fitness value of the updated nest and compares it with the fitness value of the best nest in step 4. The method for retaining nests with better fitness is as follows: The cuckoo algorithm The updated formula is: (8)； ； ； in, It is the first The location of the bird's nest in the era; It is the first The location of the bird's nest in the era; It is a random variable that follows a standard normal distribution; This is a control factor, set to 1.5; Then, the fitness value of the updated bird nest is calculated according to equation (8) and compared with the fitness value of the best bird nest in step 4. The bird nest with better fitness is retained.

6. The method for solving multi-point positioning of aircraft based on the improved cuckoo algorithm according to claim 1, characterized in that: In step 6, the nests retained in step 5 are filtered using adaptive discovery probability. The nest locations are updated using a random preference walk. Then, the fitness value of the updated nests is calculated and compared with the fitness values ​​of the better-fitting nests from step 5. The method for retaining the better-fitting nests is as follows: The adaptive discovery probability The formula is as follows: (9)； in, , It is the maximum number of iterations. This is the current iteration number; Generate random numbers distributed in the [0,1] range using the rand function. Compared with the above adaptive discovery probability The comparison shows that the probability of adaptive discovery is greater than the probability of discovery. The bird's nest is randomly visited with a preference, and the probability of discovery is less than the adaptive probability. If the bird's nest remains unchanged, the formula for random preference walk is: (10)； in, yes Replace two random bird nests, It is a random number that follows a normal distribution; Then, the fitness value of the updated nest is calculated according to equation (10) and compared with the fitness value of the nest with better fitness in step 5. The nest with better fitness is retained.

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