Fractional order chaotic circuit modeling method based on adaptive differential fusion ant colony optimization

Abstract

The invention provides a fractional order chaotic circuit system modeling method based on an adaptive differential fusion ant colony optimization algorithm, and belongs to the technical field of fractional order chaotic circuit system modeling. The technical problems that an existing method is high in dimensionality, high in nonlinearity and prone to falling into local optimum in fractional order chaotic circuit system parameter estimation are solved. Comprising the following steps: 1) constructing a multivariable optimization identification model of the fractional order chaotic circuit system; and step 2) providing a hybrid optimization algorithm fusing adaptive differential evolution SaDE and artificial ant colony ACO. The method has the advantages that the SaDE-ACO algorithm overcomes the premature convergence problem of a traditional ant colony algorithm through the random disturbance characteristic of differential evolution, and meanwhile optimal solution search is accelerated through an ant colony positive feedback mechanism.

Description

Technical Field

[0001] The present invention belongs to the technical field of fractional-order chaotic circuit system modeling, and specifically relates to a hybrid optimization algorithm (SaDE-ACO) based on an adaptive differential evolution algorithm (SaDE) and an artificial ant colony algorithm (ACO), which is used for parameter estimation and system identification of fractional-order chaotic circuit systems. Background Art

[0002] Currently, in the field of fractional-order chaotic system modeling and control, commonly used parameter identification techniques include least squares method (LSM), gradient descent (GD), particle swarm optimization (PSO), and adaptive differential evolution (SaDE), among others. Among these, constructing fractional-order chaotic system models using intelligent optimization algorithms can effectively address the strong coupling between system parameters and fractional orders, significantly improving the prediction accuracy of chaotic trajectories. Compared to traditional methods, swarm intelligence-based hybrid optimization techniques (such as ACO-DE) exhibit stronger global search capabilities and noise immunity when dealing with the historical dependencies of fractional-order systems. By combining pheromone guidance with an adaptive mutation mechanism, these techniques can overcome local extremum constraints and are therefore particularly suitable for high-dimensional parameter identification of fractional-order chaotic systems. The complex coupling of the memory effects of fractional-order calculus operators and the system's nonlinear terms is the core reason for the dimensionality explosion of parameter identification models. In order to improve modeling efficiency, constructing a hybrid optimization framework and combining the recursive calculation of fractional-order state equations with the parallel search of intelligent algorithms has become a key technical path to solve the modeling difficulties of fractional-order chaotic systems, and provides theoretical support for achieving chaotic synchronization and secure communication.

[0003] As an important research area in the field of nonlinear dynamics, the parameter estimation problem of chaotic systems is of key significance in both theoretical analysis and practical applications. Existing research has shown that the dynamic behavior of chaotic systems is highly dependent on system parameters. In particular, in fractional-order chaotic systems, the complexity of parameter identification increases significantly, becoming a core challenge in constraining system modeling and control. Currently, research on parameter estimation of chaotic systems involves a variety of methods. The paper "A delay-disturbance method to counteract the dynamical degradation of digital chaotic systems and its application" proposes a hybrid method combining parameter perturbation with delayed feedback control to improve the performance of chaotic systems by adjusting parameters. The paper "Intelligent parameter identification and prediction of variable-time fractional derivative and application in a symmetric chaotic financial system" integrates differential evolution algorithms with neural networks, achieving breakthroughs in parameter identification of financial chaos models. However, existing methods still face problems such as slow convergence, weak noise immunity, and low efficiency in processing high-dimensional data when dealing with fractional-order chaotic systems.

[0004] Traditional parameter estimation methods, such as least squares and gradient descent, are prone to falling into local optima when dealing with nonlinear, high-dimensional chaotic systems and are sensitive to initial parameters. The paper "A decomposition-based many-objective colony optimization algorithm with adaptive solution construction and selection approaches" points out that although the ant colony algorithm (ACO) demonstrates excellent global search capabilities by simulating the foraging behavior of ant colonies, its application to chaotic system parameter estimation has not yet been extensive, especially in fractional-order systems, where there is a lack of adaptive optimization mechanisms. On the other hand, while differential evolution (DE) and its adaptive improvements (such as SaDE) have demonstrated high efficiency in photovoltaic system parameter identification and synthetic aperture radar detection, they still suffer from premature convergence and strong parameter dependence when applied alone to chaotic systems. Summary of the Invention

[0005] The present invention provides a fractional-order chaotic system modeling method based on an adaptive differential evolution algorithm and an artificial ant colony algorithm. The loss function is established using the fractional order to be estimated and the system vector. In order to minimize this loss function, an artificial ant colony algorithm is applied for iterative optimization. In addition, a differential evolution algorithm and an adaptive differential evolution algorithm are introduced to improve the algorithm. Then, an ant colony algorithm method based on the adaptive differential evolution algorithm is derived. This method has a fast convergence speed and high identification accuracy.

[0006] The idea of ​​the present invention is that the fractional-order chaotic system reveals complex internal laws in the nonlinear dynamic system, but its parameter estimation has higher dimensionality and nonlinear characteristics challenges than the integer-order system. In response to the above problems, the present invention proposes a parameter identification method for the fractional-order chaotic system based on a fusion optimization strategy: first, the fractional-order chaotic system modeling is transformed into a multivariable optimization problem, and a mapping relationship between parameters and system dynamics is constructed; secondly, the mutation mechanism and parameter adaptation characteristics of the adaptive differential evolution algorithm are introduced to improve the pheromone update rule and path selection strategy of the artificial ant colony algorithm, forming a SaDE-ACO hybrid algorithm with global optimization capability. Among them, the dynamic scaling factor of the differential evolution algorithm enhances the ability of the algorithm to jump out of the local optimum, and the positive feedback mechanism of the ant colony algorithm ensures the efficiency of the search process. The simulation examples of the Lorenz chaotic circuit system and the Lu chaotic circuit system show that the SaDE-ACO algorithm proposed in the present invention has higher identification accuracy in the parameter estimation of the fractional-order chaotic circuit system, and significantly improves the identifiability of the complex chaotic system.

[0007] In order to achieve the above-mentioned purpose, the present invention adopts a technical solution specifically as follows: a fractional-order chaotic circuit modeling method based on adaptive differential fusion ant colony optimization, comprising the following steps:

[0008] Step 1) constructing a fractional-order chaotic system model of an ant colony algorithm based on an adaptive differential evolution algorithm to ultimately identify the unknown fractional order and coefficient vector of the chaotic system;

[0009] Step 1-1) constructing a fractional-order chaotic system model based on the integer-order chaotic system;

[0010] Step 1-2) Apply the formula to derive the numerical solution vector x(t i );

[0011] Steps 1-3) Define the estimation system and solve the numerical solution vector y(t i );

[0012] Steps 1-4) Construct the loss function of the fractional-order chaotic system

[0013] Step 2) constructing an ant colony algorithm based on differential evolution algorithm to optimize the fractional-order chaotic system model;

[0014] Step 2-1) Create the ant position coordinate vector and set the algorithm's parameter initialization;

[0015] Step 2-2) Calculate the fitness value Determine the optimal location

[0016] Step 2-3) Calculate P(i) to determine the search range of each ant;

[0017] Steps 2-4) Get the pre-update location And determine whether the ant position needs to be updated;

[0018] Step 2-5) Update the pheromone τ of each path i , and obtain the parameter vector

[0019] Step 2-6) Calculate the next iteration individual position vector

[0020] Step 2-7) Update pheromones Obtaining the best pheromones and the position vector at this time

[0021] Step 2-8) Obtain the coefficient vector of the estimated system and fractional order vectors

[0022] Step 2-9) Execute an iterative loop until the maximum number of iterations is reached, then stop and output the final estimated value.

[0023] The present invention provides a further optimization scheme for the fractional-order chaotic circuit modeling method based on adaptive differential fusion ant colony optimization, comprising the following steps:

[0024] (1-1) Construct an n-dimensional fractional-order chaotic system model:

[0025] D p x(t)=f(x(t),t,θ) (1)

[0026] Where t is the time variable, is the n-dimensional state vector of the chaotic system, and the initial condition of the system at time t0 is is the fractional order vector of the system, represents the coefficient vector of the system except the fractional order, and f(·) represents the unknown nonlinear function.

[0027] (1-2) Because p i >0 and Truncated p i The fractional derivative of GL is

[0028]

[0029] in, Indicates an integer, h is the time step. Fractional binomial coefficient Defined as

[0030]

[0031] Where Γ(·) is the Gamma function. Discretize the GL fractional derivative and convert In t k =kh(k=1,2, … ) is discretized to obtain

[0032]

[0033] Where h is the time step, is the Griinwald function, which can be obtained by recursion as follows

[0034]

[0035] Among them, the initial conditions According to formula (4), The numerical solution is

[0036]

[0037] (1-3) Similarly, the estimation system is defined as:

[0038]

[0039] in, To estimate the n-dimensional state variables of the system, the initial value of the system and are the fractional order vector and coefficient vector of the estimated system.

[0040] (1-4) Assume that the time series t0<t1< … <t l ,x(t i ) indicates that at t i (i=0,1, … ,l) is the real state variable sequence of the chaotic system at the moment, y(t i ) indicates that at t i (i=0,1, …,l) The estimated state variable sequence of the chaotic system at time ,and the unknown parameter vector is identified according to the optimization algorithm, and It is defined as the sum of squares of the errors between the true value and the estimated value of the state vector, that is:

[0041]

[0042] in, l is the length of the system state variable sequence, Indicates ι 2 -norm.

[0043] It can be seen that the fractional-order chaotic system model contains the parameter vector θ and the parameter vector of fractional order p. Therefore, it is necessary to use efficient and accurate algorithms for estimation.

[0044] (2-1) The ant colony algorithm based on adaptive differential evolution algorithm is used to solve the parameter identification problem of fractional-order chaotic system. Assume that the position coordinate vector of each ant is Where m is the number of ants, D is the vector dimension, the pheromone concentration corresponding to the i-th ant, that is, the fitness value is expressed as τ(i), and the optimal fitness of the i-th ant is recorded as τ best (i) The upper and lower limits of the ants’ activity range are and The initial position of the ant is randomly set to

[0045]

[0046] (2-2) Substitute the ant's position coordinates into formula (8) to calculate the fitness value Select the optimal position with the smallest fitness as the initial optimal position

[0047] (2-3) In the kth iteration, the i-th ant will decide the next direction of travel based on the pheromone concentration on the path. It uses the information stored at the current position to calculate the probability of reaching the next position, which is called the state transition probability P(i):

[0048]

[0049] Among them, τ best (i) is the optimal value of the fitness function corresponding to the i-th ant. For each ant, if P(i) < P0, it indicates that the fitness function has a higher probability of having an optimal solution near the point, and a local search is performed; if P(i) > P0, it indicates that the fitness function has a lower probability of having an optimal solution near the point, and a global search should be performed.

[0050] (2-4) After the search is completed, the pre-updated position of the i-th ant The calculation is as follows:

[0051]

[0052] in, Must be maintained If it overflows, boundary processing is performed

[0053]

[0054] Compare the fitness value of the new position vector with the fitness value of the original position vector. If the updated fitness value is better, the ant moves to the new position, otherwise the ant does not move.

[0055]

[0056] Then there is When all ant positions are updated, the pheromones on each path are updated, and their values ​​can be calculated by the following formula

[0057]

[0058] Perform k iterations to update the pheromone τ of each path i .

[0059] (2-5) In the mutation operation, in the kth iteration, the mutation operation is performed on individual i to generate a new vector The mutation operation mainly consists of differential vectors and basis vectors, and the basic form is

[0060]

[0061] in, represents the kth generation individual The mutation individual vector of represents the individual with the lowest fitness in the k-th generation population, that is, the optimal individual, r1, r2 represent the subscripts of individuals randomly selected from the population, and r1≠r2, r1, r2∈[1,K], K is the maximum number of iterations, are basis vectors, is the differential individual vector, and F is the learning factor, which is used to control the learning speed of the differential vector approaching the basis vector, and the range is between [0,1].

[0062] The crossover operation is used to increase the diversity of the mutation individual vector, the parent individual and mutation Cross-combination to form cross-individual In order to increase the diversity of the population, a crossover probability parameter CR is introduced in the crossover operation to control the selection probability of the two individual alleles. Commonly used crossover operations include binomial crossover and exponential crossover. The chaotic system parameters estimated by the present invention have no obvious relationship. Binomial crossover is used, and the form is as follows:

[0063]

[0064] jrand means [1,2, … ,D], j=jrand means at least one variable comes from the mutation individual vector, rand(0,1) means a random value in (0,1), represents the jth component in the i-th individual of the parent generation, is the jth component of the i-th individual in the mutant individual, where i = 1, 2, …, K, j = 1, 2, …, D, CR∈[0,1] is the crossover probability, which represents the proportion of mutant individuals in the ants.

[0065] The selection operation first calculates the fitness values ​​of all generated individuals, and then compares them with the fitness values ​​of the target vector one by one. In the minimum problem, the lower the fitness value, the better the vector, and the individual vector corresponding to the fitness value is retained to the next generation. Otherwise, the target vector is retained in the next generation of individuals and continues to iterate. The specific operation is

[0066]

[0067] Among them, J(·) represents the fitness value, represents individuals that successfully enter the k+1 generation.

[0068] (2-6) The differential evolution algorithm is prone to falling into local optimality and premature convergence as other evolutionary algorithms. This paper introduces the crossover probability constant μ to address the impact of the crossover probability CR of the differential vector on the algorithm and proposes an adaptive differential evolution algorithm. For the crossover probability CR,

[0069] CR=μ×(1+rand(0,1)) (18)

[0070] At the same time, the adaptive constant ε is introduced

[0071] ε=cos(1-(K / (K+1-k))) (19)

[0072] For the differential evolution coefficient of variation Introducing the initial variation coefficient F0 and adaptive constant ε of differential evolution

[0073]

[0074] Then the basic form of mutation operation becomes

[0075]

[0076] (2-7) According to the formula Solve Thus, the coefficient vector of the estimation system is obtained and fractional order vectors

[0077] (2-8) Perform an iterative loop until the maximum number of iterations is reached, stop and output the final result.

[0078] Compared with the prior art, the present invention has the following beneficial effects:

[0079] (1) This paper addresses the high complexity of parameter estimation in fractional-order chaotic systems by proposing an identification model based on multivariable optimization. By transforming the dynamic characteristics of fractional-order systems into a multidimensional parameter space optimization problem, this model effectively addresses the difficulty of traditional methods in dealing with fractional-order differential operators and nonlinear coupling. This model reduces the system dimensionality through mathematical reconstruction, significantly improving the theoretical feasibility and computational efficiency of parameter identification.

[0080] (2) This paper innovatively proposes an adaptive differential evolution-ant colony optimization (SaDE-ACO) hybrid algorithm, which realizes collaborative search of complex parameter spaces by integrating the global pheromone guidance mechanism of the ant colony optimization algorithm with the dynamic mutation strategy of the adaptive differential evolution algorithm. The introduction of adaptive mutation factors and crossover probabilities effectively avoids premature convergence. At the same time, the multi-innovation strategy expands the historical data dimension and enhances the robustness of the algorithm under noise interference. Simulation experiments show that the algorithm has good parameter identification for Lorenz and Chen fractional-order chaotic circuit systems, and still maintains stable convergence characteristics in high-dimensional and strongly nonlinear scenarios.

[0081] (3) The present invention proposes a parameter estimation method for fractional-order chaotic systems based on the adaptive differential evolution-ant colony optimization (SaDE-ACO) hybrid algorithm. This method combines the global search capability of ACO with the adaptive mutation mechanism of SaDE to effectively balance the exploration and development process. Specifically, the SaDE-ACO algorithm uses the pheromone positive feedback mechanism to optimize the parameter space search path, and at the same time enhances the local optimization efficiency by adaptively adjusting the mutation factor and crossover probability, thereby solving the problems of slow convergence speed and insufficient anti-interference ability in fractional-order chaotic systems that exist in traditional methods. Related experiments show that this method can still maintain high-precision parameter estimation in a noisy environment, providing reliable theoretical support for the engineering application of complex chaotic systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0082] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. They are used to explain the present invention together with the embodiments of the present invention and do not constitute a limitation of the present invention.

[0083] Figure 1 This is the Lorenz system simulation circuit diagram of the fractional-order chaotic circuit in the present invention.

[0084] Figure 2 This is the Chen system simulation circuit diagram of the fractional-order chaotic circuit in the present invention.

[0085] Figure 3 This is a three-dimensional diagram of the Lorenz system of the fractional-order chaotic circuit in the present invention.

[0086] Figure 4 This is a three-dimensional diagram of the Chen system of the fractional-order chaotic circuit in the present invention.

[0087] Figure 5 The present invention provides a flowchart of the operation steps of the fractional-order chaotic circuit modeling method based on adaptive differential fusion ant colony optimization.

[0088] Figure 6 This is an error curve diagram under the Chen system in Example 1 of the present invention.

[0089] Figure 7 This is a fitness curve diagram of the Chen system in Example 1 of the present invention.

[0090] Figure 8 This is an error curve diagram under the Lorenz system in Example 2 of the present invention.

[0091] Figure 9 This is a fitness curve diagram under the Lorenz system in Example 2 of the present invention. DETAILED DESCRIPTION

[0092] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. Of course, the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0093] See also Figures 6 to 9 The technical solution of the embodiment of the present invention is a fractional-order chaotic circuit modeling method based on adaptive differential fusion ant colony optimization SaDE-ACO. Figure 6The proposed method is applied to the identification of fractional-order Chen and Lorenz chaotic circuit systems. The parameter search space covers the nominal value range of the chaotic system's nonlinear terms, and the optimization objective is to minimize the mean square error of the output response. By integrating the adaptive mutation mechanism of differential evolution with the pheromone global search strategy of the ant colony algorithm, the system parameters and fractional orders are collaboratively optimized, ultimately generating a high-precision chaotic circuit model. The entire optimization process is implemented in Matlab, and the robustness of the algorithm is verified through multiple comparative experiments.

[0094] According to the attached Figure 6-9 The identification results show that the proposed method exhibits the following characteristics during the iterative process: the parameter estimation error decays exponentially with the number of iterations, and the fractional order estimates rapidly approach the true value. Compared with traditional optimization algorithms, its convergence curve enters a steady-state phase mid-iteration, and parameter oscillation is significantly suppressed. Experiments show that the output of the established chaotic circuit model closely matches the dynamic response of the real system, verifying the effectiveness and engineering applicability of this method for modeling complex nonlinear circuits.

[0095] Example 1

[0096] In Example 1, the parameter vector of the Chen circuit system is defined as:

[0097] θ=[a1,a2,a3,a4]=[35,3.5,28,-7]

[0098] The fractional order is defined as:

[0099] p=[b1,b2,b3] T =[0.9,0.9,0.9] T

[0100] In the simulation, the model sampling interval h = 0.005. The adaptive differential fusion ant colony optimization algorithm is used to identify the above Chen circuit system model. The basic parameters of the ant colony algorithm initialization stage are set as follows: the number of ants m = 10, the maximum number of iterations G max =50, pheromone evaporation coefficient ρ = 0.9, transfer probability constant P0 = 0.2, differential evolution initial variation coefficient F0 = 0.8.

[0101] Figure 6 and Figure 7 The ant colony algorithm parameter estimation error and fitness curve for the Chen circuit system are shown.

[0102] Example 2

[0103] In Example 2, the parameter vector of the Lorenz circuit system is defined as:

[0104] θ=[a1,a2,a3]=[10,28,8 / 3]

[0105] The fractional order is defined as:

[0106] p=[b1,b2,b3] T =[0.993,0.993,0.993] T

[0107] In the simulation, the model sampling interval h = 0.01. The adaptive differential fusion ant colony optimization algorithm was used to identify the Lorenz circuit system model. The basic parameters of the ant colony algorithm initialization phase were set as follows: number of ants m = 20, maximum number of iterations K = 50, pheromone evaporation coefficient ρ = 0.9, transition probability constant P0 = 0.2, and differential evolution initial variation coefficient F0 = 0.8.

[0108] Figure 8 and Figure 9 The parameter estimation error and fitness curve of the adaptive differential fusion ant colony optimization algorithm for the Lorenz circuit system are shown.

[0109] A fractional-order chaotic circuit modeling method based on adaptive differential fusion ant colony optimization in Examples 1 and 2 includes the following steps:

[0110] (1) Construct a fractional-order chaotic circuit system model based on adaptive differential fusion ant colony optimization. The specific steps are as follows:

[0111] Step 1: Construct an n-dimensional fractional-order chaotic system model:

[0112] D p x(t)=f(x(t),t,θ) (1)

[0113] Where t is the time variable, is the n-dimensional state vector of the chaotic system, and the initial condition of the system at time t0 is is the fractional order vector of the system, represents the coefficient vector of the system except the fractional order, and f(·) represents the unknown nonlinear function.

[0114] Step 2: Derivation of the formula to find the numerical solution of the state vector. The derivation steps are as follows:

[0115] Because p i >0 and Truncated p i The fractional derivative of GL is

[0116]

[0117] in, It represents an integer, and h is the time step.

[0118] Fractional binomial coefficient Defined as

[0119]

[0120] Where Γ(·) is the Gamma function. Discretize the GL fractional derivative and convert In t k =kh(k=1,2, … ) is discretized to obtain

[0121]

[0122] Where h is the time step, is the Griinwald function, which can be obtained by recursion as follows

[0123]

[0124] Among them, the initial conditions According to formula (4), The numerical solution is

[0125]

[0126] Step 3: Based on this model, the estimation system is defined as follows:

[0127]

[0128] in, To estimate the n-dimensional state variables of the system, the initial value of the system and are the fractional order vector and coefficient vector of the estimated system.

[0129] Step 4: Construct the error function between the true value and the estimated value of the state vector. The specific steps are as follows:

[0130] Assume that the time series t0<t1< … <t l ,x(t i ) indicates that at t i (i=0,1, … ,l) is the real state variable sequence of the chaotic system at the moment, y(t i ) indicates that at t i The estimated state variable sequence of the chaotic system at time (i=0,1,…,l) is used to identify the unknown parameter vector according to the optimization algorithm. and It is defined as the sum of squares of the errors between the true value and the estimated value of the state vector, that is:

[0131]

[0132] in, l is the length of the system state variable sequence, Indicates ι 2 -norm.

[0133] It can be seen that the fractional-order chaotic system model contains the parameter vector θ and the parameter vector of fractional order p. Therefore, it is necessary to use efficient and accurate algorithms for estimation.

[0134] (2) Construct an ant colony algorithm based on differential evolution algorithm to optimize the process of fractional-order chaotic circuit system model:

[0135] Step 1: Create the ant position coordinate vector and set the algorithm's parameter initialization;

[0136] Step 2: Calculate the fitness value Determine the optimal location

[0137] Step 3: Calculate P(i) to determine the search range of each ant;

[0138] Step 4: Get the pre-update location And determine whether the ant position needs to be updated;

[0139] Step 5: Update the pheromone τ of each path i , and obtain the parameter vector

[0140] Step 6: Calculate the individual position vector for the next iteration

[0141] Step 7: Update pheromones Obtaining the best pheromones and the position vector at this time

[0142] Step 8: Obtain the coefficient vector of the estimated system and fractional order vectors

[0143] Step 9: Execute the iterative loop until the maximum number of iterations is reached, stop and output the final estimate.

[0144] (3) According to the process of optimizing the fractional-order chaotic circuit system model using the ant colony algorithm based on the differential evolution algorithm, the optimization method of optimizing the fractional-order chaotic circuit system model using the ant colony algorithm based on the differential evolution algorithm is constructed as follows:

[0145]

[0146]

[0147] CR=μ×(1+rand(0,1)) (18)

[0148] ε=cos(1-(K / (K+1-k))) (19)

[0149]

[0150] See also Figure 5 , the specific steps of the above method are:

[0151] (1) Establish the ant position coordinate vector using formula (9) And initialize the parameters of the algorithm;

[0152] (2) Calculate the fitness value using formula (8) And determine the optimal position at this time

[0153] (3) Calculate P(i) using formula (10) to determine the search range of each ant;

[0154] (4) Obtain the pre-updated position from formula (11) And use formula (12) and formula (13) to determine whether the ant position needs to be updated at this time;

[0155] (5) Update the pheromone τ of each path through formula (14) i , and obtain the parameter vector

[0156] (6) Calculate the next iteration individual position vector by formula (15)

[0157] (7) Update the pheromone again according to formulas (14) and (15) Obtaining the best pheromones and the position vector at this time

[0158] (8) Obtain the coefficient vector of the estimated system and fractional order vectors

[0159] (9) Execute the iterative loop until the maximum number of iterations is reached, stop and output the final estimated value.

[0160] The definitions of the variables are as follows:

[0161] Define t as the time variable, is the n-dimensional state vector of the chaotic system, and the initial condition of the system at time t0 is is the fractional order vector of the system, represents the coefficient vector of the system except the fractional order, and f(·) represents a nonlinear function that is well defined but unknown in advance.

[0162] definition Indicates an integer, h is the time step, and definition is the fractional binomial coefficient, Γ(·) is the Gamma function, is the Griinwald function, the initial condition

[0163] definition To estimate the n-dimensional state variables of the system, the initial value of the system and are the fractional order vector and coefficient vector of the estimated system.

[0164] Define the time series t0<t1< … <t l ,x(t i ) indicates that at t i (i=0,1, … ,l) is the real state variable sequence of the chaotic system at the moment, y(t i ) indicates that at t i (i=0,1, … ,l) the estimated state variable sequence of the chaotic system at time, and Defined as the true value and estimated value of the state vector, l is the length of the system state variable sequence, Indicates ι 2 -norm.

[0165] Define the position coordinate vector of each ant as Where m is the number of ants, D is the vector dimension, the pheromone concentration corresponding to the i-th ant, that is, the fitness value is expressed as τ(i), and the optimal fitness of the i-th ant is recorded as τ best (i) The upper and lower limits of the ants’ activity range are and The initial optimal position is

[0166] Define P(i) as the state transition probability, and the pre-updated position of the i-th ant is definition represents the kth generation individual The mutation individual vector of represents the individual with the lowest fitness in the k-th generation population, that is, the optimal individual, r1, r2 represent the subscripts of individuals randomly selected from the population, and r1≠r2, r1, r2∈[1,K], K is the maximum number of iterations, are basis vectors, is the differential individual vector, and F is the learning factor, which is used to control the learning speed of the differential vector approaching the basis vector, and the range is between [0,1].

[0167] Define parent individuals and mutation Cross-combination to form cross-individual The crossover probability parameter CR is introduced in the crossover operation to control the selection probability of the two individual alleles. jrand represents [1,2, … ,D], j=jrand means at least one variable comes from the mutation individual vector, rand(0,1) means a random value in (0,1), represents the jth component in the i-th individual of the parent generation, is the jth component of the i-th individual in the mutation individual, where is the crossover probability, which represents the proportion of mutant individuals in ants.

[0168] Define J(·) to represent the fitness value, Indicates the individuals that successfully enter the k+1 generation, defines the crossover probability constant μ, the adaptive constant ε, and the differential evolution coefficient of variation The initial coefficient of variation F0 and adaptive constant ε of differential evolution.

[0169] Based on the attached Figure 6-9 The experimental results shown in the figure show that the SaDE-ACO algorithm proposed in this invention demonstrates significant technical advantages in the parameter identification of fractional-order chaotic circuit systems. Verification on two typical chaotic circuit systems, Chen and Lorenz, shows that: as the iterative process progresses, the system parameter estimation error shows a continuous decreasing trend and eventually approaches zero, and the parameter vector and fractional order estimation values ​​both converge stably to the true value neighborhood. Compared with traditional swarm optimization algorithms, this method shows a significant improvement in convergence speed, and the fitness curve completes the rapid decline stage in the early stage of iteration; in terms of anti-interference ability, its parameter estimation error band is significantly narrowed, and the characterization of the nonlinear characteristics of the system is more accurate. By integrating the adaptive perturbation mechanism of differential evolution with the parallel search characteristics of the ant colony algorithm, the present invention effectively avoids local extreme value traps in high-dimensional parameter space. Its hybrid optimization architecture significantly reduces the computational complexity while ensuring identification accuracy. Experimental data further show that the estimated output is highly consistent with the dynamic response of the real system, verifying the robustness and engineering practicality of this method in modeling complex chaotic circuit systems.

[0170] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A fractional-order chaotic circuit modeling method based on adaptive differential fusion ant colony optimization is characterized by: The steps include: Step 1) constructing a fractional-order chaotic circuit system model based on adaptive differential fusion ant colony optimization to ultimately identify the unknown fractional order and coefficient vector of the chaotic circuit system; Step 2) Construct an ant colony algorithm based on differential evolution algorithm to optimize the fractional-order chaotic circuit system model.

2. The fractional-order chaotic circuit system modeling method of the ant colony algorithm of the adaptive differential evolution algorithm according to claim 1 is characterized in that: The step 1) comprises the following steps: (1-1) Construct a fractional-order chaotic circuit system model based on the integer-order chaotic circuit system; (1-2) According to this model, the expression of the fractional-order chaotic circuit system model is as follows: D p x(t)=f(x(t),t,θ) (1) Where t is the time variable, is the n-dimensional state vector of the chaotic system, and the initial condition of the system at time t0 is is the fractional order vector of the system, represents the coefficient vector of the system except the fractional order, and f(·) represents the unknown nonlinear function; (1-3) According to the formula derivation, The numerical solution is Where h is the time step, is the Griinwald function, the initial condition t k =kh(k=1,2,…); (1-4) Based on the numerical solutions of the estimated system and the real system, the loss function is defined as follows: in, l is the length of the system state variable sequence, Indicates ι 2 -norm.

3. The method of claim 1, wherein the method comprises: The step 2) comprises the following steps: Step 2-1) Create the ant position coordinate vector and set the algorithm's parameter initialization; Step 2-2) Calculate the fitness value Determine the optimal location Step 2-3) Calculate P(i) to determine the search range of each ant; Steps 2-4) Get the pre-update location And determine whether the ant position needs to be updated; Step 2-5) Update the pheromone τ of each path i , and obtain the parameter vector Step 2-6) Calculate the next iteration individual position vector Step 2-7) Update pheromones Obtaining the best pheromones and the position vector at this time Step 2-8) Combine the differential evolution strategy to obtain the coefficient vector of the estimated system and fractional order vectors Step 2-9) Execute an iterative loop until the maximum number of iterations is reached, then stop and output the final estimated value.

4. The method of claim 1, wherein the method comprises: Introducing differential evolution operations into steps 2-8) of ant colony optimization includes the following steps: Step 2-8-1) In the kth iteration, perform mutation operation on individual i to generate a new vector Perform mutation operation, which consists of difference vector and basis vector. The learning factor is used to control the learning speed of the difference vector approaching the basis vector. Step 2-8-2) Perform a crossover operation to increase the diversity of the mutated individual vectors, the parent individual and mutation Cross-combination to form cross-individual The crossover probability parameter CR is introduced in the crossover operation to control the selection probability of the two individual alleles; Step 2-8-3) Perform the selection operation. First, calculate the fitness values ​​of all generated individuals, then compare them with the fitness value of the target vector one by one, and retain the individual vector corresponding to the fitness value to the next generation. Otherwise, retain the target vector to the next generation of individuals and continue iteration; Step 2-8-4) In view of the influence of the crossover probability CR of the differential vector on the algorithm, a crossover probability constant μ is introduced, and an adaptive differential evolution algorithm is proposed to improve the crossover probability and introduce an adaptive constant; Step 2-8-5) Solve Thus, the coefficient vector of the estimation system is obtained and fractional order vectors

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