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

Abstract

The application 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 application solves the technical problems of high dimension, strong nonlinearity and easy falling into local optimization in parameter estimation of the existing fractional order chaotic circuit system. The application comprises the following steps: step 1) constructing a multivariable optimization identification model of the fractional order chaotic circuit system; and step 2) proposing a hybrid optimization algorithm fusing adaptive differential evolution (SaDE) and artificial ant colony optimization (ACO). The application has the beneficial effect that the SaDE-ACO algorithm overcomes the premature convergence problem of the traditional ant colony algorithm through the random disturbance characteristics of differential evolution, and simultaneously accelerates the optimal solution search by using the positive feedback mechanism of the ant colony.

Description

Technical Field

[0001] This invention belongs to the field of fractional chaotic circuit system modeling technology, specifically involving a hybrid optimization algorithm (SaDE-ACO) based on adaptive differential evolution algorithm (SaDE) and artificial ant colony algorithm (ACO) for parameter estimation and system identification of fractional chaotic circuit systems. Background Technology

[0002] Currently, in the field of fractional-order chaotic system modeling and control, commonly used parameter identification techniques encompass a variety of methods, including least squares (LSM), gradient descent (GD), particle swarm optimization (PSO), and adaptive differential evolution (SaDE). Among these, using intelligent optimization algorithms to construct fractional-order chaotic system models can effectively address the strong coupling between system parameters and fractional-order terms, 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 resistance when dealing with the historical dependencies of fractional-order systems. Through the combination of pheromone guidance and adaptive mutation mechanisms, they can overcome local extrema limitations, making them particularly suitable for high-dimensional parameter identification of fractional-order chaotic systems. The memory effect of fractional-order calculus operators and the complex coupling of system nonlinear terms are the core reasons for the dimensionality explosion of parameter identification models. To improve modeling efficiency, a hybrid optimization framework has been constructed, which combines the recursive calculation of fractional-order state equations with the parallel search of intelligent algorithms. This has become a key technical approach to solving the modeling problem of fractional-order chaotic systems and provides theoretical support for achieving chaotic synchronization and secure communication.

[0003] Chaotic systems, as an important research direction in nonlinear dynamics, have crucial implications for parameter estimation in both theoretical analysis and practical applications. Existing research shows that the dynamic behavior of chaotic systems is highly dependent on system parameters, especially in fractional-order chaotic systems, where the complexity of parameter identification increases significantly, becoming a core challenge restricting system modeling and control. Currently, research on parameter estimation for chaotic systems involves various 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 and delay 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 and neural networks, achieving a breakthrough in parameter identification for financial chaotic models. However, existing methods still face problems such as slow convergence speed, weak noise resistance, 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 getting trapped in local optima when dealing with nonlinear, high-dimensional chaotic systems and are sensitive to initial parameters. The paper "A decomposition-based many-objectiveant colony optimization algorithm with adaptive solution construction and selection approaches" points out that although the Ant Colony Optimization (ACO) algorithm demonstrates excellent global search capabilities by simulating ant colony foraging behavior, its application in parameter estimation for chaotic systems is still limited, especially in fractional-order systems where it lacks adaptive optimization mechanisms. On the other hand, while differential evolution (DE) algorithms and their adaptive improvements (such as SaDE) have demonstrated high efficiency in areas such as photovoltaic system parameter identification and synthetic aperture radar detection, their application to chaotic systems alone still suffers from premature convergence and strong parameter dependence. Summary of the Invention

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

[0006] The core idea of ​​this invention is as follows: Fractional-order chaotic systems reveal complex intrinsic laws in nonlinear dynamic systems, but their parameter estimation presents higher dimensionality and nonlinearity challenges compared to integer-order systems. To address these challenges, this invention proposes a parameter identification method for fractional-order chaotic systems based on a fusion optimization strategy. First, the fractional-order chaotic system is modeled as a multivariate optimization problem, establishing a mapping relationship between parameters and system dynamics. Second, the mutation mechanism and parameter adaptive characteristics of the adaptive differential evolution algorithm are introduced, improving the pheromone update rules and path selection strategy of the artificial ant colony algorithm, forming a SaDE-ACO hybrid algorithm with global optimization capabilities. Specifically, the dynamic scaling factor of the differential evolution algorithm enhances the algorithm's ability to escape local optima, while the positive feedback mechanism of the ant colony algorithm ensures the efficiency of the search process. Simulation examples using Lorenz and Lu chaotic circuit systems demonstrate that the proposed SaDE-ACO algorithm achieves higher identification accuracy in parameter estimation of fractional-order chaotic circuit systems, significantly improving the identifiability of complex chaotic systems.

[0007] To achieve the aforementioned objectives, the present invention employs the following technical solution: a fractional-order chaotic circuit modeling method based on adaptive differential fusion ant colony optimization, comprising the following steps:

[0008] Step 1) Construct a fractional-order chaotic system model of the ant colony algorithm based on the adaptive differential evolution algorithm, so as to finally identify the unknown fractional order and coefficient vector of the chaotic system.

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

[0010] Step 1-2) Apply the formula to derive and solve for the numerical solution vector x(t) of the fractional-order chaotic system. i );

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

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

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

[0014] Step 2-1) Establish the ant's position coordinate vector and initialize the parameters of the algorithm;

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

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

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

[0018] Steps 2-5) Update the pheromone τ for each path i and obtain the parameter vector.

[0019] Steps 2-6) Calculate the individual position vector for the next iteration.

[0020] Steps 2-7) Update pheromones Obtain the optimal pheromone and the position vector at this time

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

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

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

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

[0025] D p x(t)=f(x(t),t,θ) (1)

[0026] Where t is the time variable, It is the n-dimensional state vector of the chaotic system, and the initial conditions of the system at time t0. It is the fractional order vector of the system. Let f(·) represent the coefficient vector of the system excluding fractional-order functions, and let f(·) represent an unknown nonlinear function.

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

[0028]

[0029] in, Indicates integer values, where h is the time step. Fractional binomial coefficients. Defined as

[0030]

[0031] Where Γ(·) is the Gamma function. The fractional derivative of GL is discretized, and... In t k =kh(k=1,2, … Discretizing at time 1000 yields the following results:

[0032]

[0033] Where h is the time step. It is the Grienwald function, which can be obtained recursively from the following formula.

[0034]

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

[0036]

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

[0038]

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

[0040] (1-4) Assume the time series t0 < t1 < … <t l ,x(t i ) indicates that at t i (i = 0, 1, … The sequence of real state variables of the chaotic system at time l), y(t) i ) indicates that at t i (i = 0, 1, …The estimated state variable sequence of the chaotic system at time l) is used to identify the unknown parameter vector based on the optimization algorithm. and Defined as the sum of squares of the errors between the true and estimated values ​​of the state vector, i.e.:

[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 includes a parameter vector θ and a parameter vector of fractional order p. Therefore, an efficient and accurate algorithm is needed for estimation.

[0044] (2-1) Solving the parameter identification problem of fractional-order chaotic systems using the ant colony algorithm with adaptive differential evolution. Let the position coordinate vector of each ant be... Where m is the number of ants, D is the vector dimension, the pheromone concentration of the i-th ant (i.e., its fitness value) is denoted as τ(i), and the optimal fitness of the i-th ant is denoted as τi. best (i) The upper and lower limits of the ant's activity range are respectively and The ants' initial positions are randomly set.

[0045]

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

[0047] (2-3) In the kth iteration, the i-th ant will decide its next walking direction based on the pheromone concentration along the path. Using the information stored at the current position, it calculates the probability of reaching the next position, which is called the state transition probability P(i):

[0048]

[0049] Where, τ best (i) represents 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 that point, and a local search should be performed; if P(i) > P0, it indicates that the fitness function has a lower probability of having an optimal solution near that point, and a global search should be performed.

[0050] (2-4) The pre-updated position of the i-th ant after the search ends. The calculation is as follows:

[0051]

[0052] in, Must remain If the value overflows within the specified range, boundary handling will be performed.

[0053]

[0054] The fitness value of the new position vector is compared 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 After all ant positions have been updated, the pheromones on each path are updated, and their values ​​can be calculated using the following formula.

[0057]

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

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

[0060]

[0061] in, Represents individuals of generation k The mutated individual vector, Let r1 and r2 represent the individuals with the lowest fitness in the k-th generation of the population, i.e., the optimal individuals. Let r1 and r2 represent the indices of individuals randomly selected from the population, and r1 ≠ r2, r1, r2 ∈ [1, K], where K is the maximum number of iterations. As basis vectors, F is the difference vector, and F is the learning factor, which controls the learning speed at which the difference vector approaches the basis vector, and its range is [0,1].

[0062] Crossover is used to increase the diversity of mutated individual vectors, parent individuals and mutation Cross-combination to form cross individuals This increases population diversity. A crossover probability parameter CR is introduced into the crossover operation to control the selection probability of alleles between two individuals. Commonly used crossover operations include binomial crossover and exponential crossover. Since the parameters of the chaotic system estimated in this invention do not have a clear relationship, binomial crossover is used, in the following form:

[0063]

[0064] jrand represents [1,2,...]. … Random values ​​within [D], j = jrand indicates that at least one variable comes from the mutated individual vector, and rand(0,1) represents random values ​​within (0,1). This represents the j-th component in the i-th individual of the parent generation. Let be the j-th component of the i-th individual among the mutant individuals, where i = 1, 2, ..., K, j = 1, 2, ..., D, CR ∈ [0, 1] is the crossover probability, representing the proportion of mutant individuals among the ants.

[0065] The selection operation first calculates the fitness value of all generated individuals, then compares it one by one with the fitness value of the target vector. In the minimization problem, the lower the fitness value, the better the vector, and the individual vector corresponding to that fitness value is retained in the next generation. Otherwise, the target vector is retained in the next generation of individuals for continued iteration. The specific operation is as follows:

[0066]

[0067] Where J(·) represents the fitness value, This indicates an individual that has successfully entered generation k+1.

[0068] (2-6) Like other evolutionary algorithms, the differential evolution algorithm is prone to getting trapped in local optima and exhibits premature convergence. This invention addresses the impact of the crossover probability CR of the difference vectors on the algorithm by introducing a crossover probability constant μ, and proposes an adaptive differential evolution algorithm. For the crossover probability CR, we have...

[0069] CR=μ×(1+rand(0,1)) (18)

[0070] Simultaneously, an adaptive constant ε is introduced.

[0071] ε=cos(1-(K / (K+1-k))) (19)

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

[0073]

[0074] The basic form of the mutation operation then becomes

[0075]

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

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

[0078] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0079] (1) This invention addresses the problem of high complexity in parameter estimation for 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, it effectively solves the difficulty of handling fractional-order differential operators and nonlinear coupling in traditional methods. This model reduces the system dimension through mathematical reconstruction, significantly improving the theoretical feasibility and computational efficiency of parameter identification.

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

[0081] (3) This invention proposes a parameter estimation method for fractional-order chaotic systems based on a hybrid algorithm of adaptive differential evolution-ant colony optimization (SaDE-ACO). This method combines the global search capability of ACO with the adaptive mutation mechanism of SaDE, effectively balancing the exploration and development process. Specifically, the SaDE-ACO algorithm utilizes a pheromone positive feedback mechanism to optimize the parameter space search path, while adaptively adjusting the mutation factor and crossover probability to enhance local optimization efficiency, thereby solving the problems of slow convergence speed and insufficient anti-interference capability of traditional methods in fractional-order chaotic systems. Related experiments show that this method can still maintain high-precision parameter estimation in noisy environments, providing reliable theoretical support for the engineering application of complex chaotic systems. Attached Figure Description

[0082] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with the embodiments of the invention to explain the invention and do not constitute a limitation thereof.

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

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

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

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

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

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

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

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

[0091] Figure 9 This is a fitness curve diagram of the Lorenz system in Embodiment 2 of the present invention. Detailed Implementation

[0092] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. Of course, the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0093] See Figures 6 to 9 The technical solution of this invention is a fractional-order chaotic circuit modeling method based on adaptive differential fusion ant colony optimization (SaDE-ACO). See also... Figure 6This invention applies the method to the identification of fractional-order Chen and Lorenz chaotic circuit systems. The parameter search space covers the nominal value range of the nonlinear terms of the chaotic system, 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 ant colony algorithm, the system parameters and fractional-order order are synergistically optimized, ultimately generating a high-precision chaotic circuit model. The entire optimization process is implemented in the Matlab platform, and the robustness of the algorithm is verified through multiple sets of comparative experiments.

[0094] According to the appendix Figure 6-9 Based on the identification results, the method of this invention exhibits the following characteristics during the iterative process: the parameter estimation error decreases exponentially with the number of iterations, and the fractional-order estimates rapidly approach the true values; compared with traditional optimization algorithms, its convergence curve enters the steady-state stage in the middle of the iteration and significantly suppresses parameter oscillations. Experiments show that the output of the established chaotic circuit model highly matches the dynamic response of the real system, verifying the effectiveness and engineering applicability of this method in complex nonlinear circuit modeling.

[0095] Example 1

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

[0097] θ=[a1,a2,a3,a4]=[35,3.5,28,-7]

[0098] 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 was used to identify the above Chen circuit system model. The basic parameters for the initialization phase of the ant colony algorithm were set as follows: number of ants m = 10, maximum number of iterations G. max =50, pheromone evaporation coefficient ρ = 0.9, transition probability constant P0 = 0.2, differential evolution initial variation coefficient F0 = 0.8.

[0101] Figure 6 and Figure 7 The parameter estimation error and fitness curves of the ant colony algorithm 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 follows:

[0104] θ = [a1, a2, a3] = [10, 28, 8 / 3]

[0105] 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 above Lorenz circuit system model was identified using the adaptive differential fusion ant colony optimization algorithm. The basic parameters for the initialization phase of the ant colony algorithm 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 coefficient of variation F0 = 0.8.

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

[0109] The 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 an adaptive differential fusion ant colony optimization fractional chaotic circuit system model. 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, It is the n-dimensional state vector of the chaotic system, and the initial conditions of the system at time t0. It is the fractional order vector of the system. Let f(·) represent the coefficient vector of the system excluding fractional-order functions, and let f(·) represent an unknown nonlinear function.

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

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

[0116]

[0117] in, This indicates taking integer values, where h is the time step.

[0118] Fractional binomial coefficients Defined as

[0119]

[0120] Where Γ(·) is the Gamma function. The fractional derivative of GL is discretized, and... In t k =kh(k=1,2, … Discretizing at time 1000 yields the following results:

[0121]

[0122] Where h is the time step. It is the Grienwald function, which can be obtained recursively from the following formula.

[0123]

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

[0125]

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

[0127]

[0128] in, To estimate the n-dimensional state variables of the system, the initial values ​​of the system are... and To estimate the fractional order vector and coefficient vector of the 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 the time series t0 < t1 < … <t l ,x(t i ) indicates that at t i (i = 0, 1, … The sequence of real state variables of the chaotic system at time l), 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 using an optimization algorithm. and Defined as the sum of squares of the errors between the true and estimated values ​​of the state vector, i.e.:

[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 includes a parameter vector θ and a parameter vector of fractional order p. Therefore, an efficient and accurate algorithm is needed for estimation.

[0134] (2) The process of constructing an ant colony algorithm based on differential evolution to optimize a fractional chaotic circuit system model:

[0135] Step 1: Establish the ant's position coordinate vector and initialize the parameters of the algorithm;

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

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

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

[0139] Step 5: Update the pheromone τ for 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 Obtain the optimal pheromone and the position vector at this time

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

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

[0144] (3) Based on the process of optimizing the fractional chaotic circuit system model using the ant colony algorithm based on the differential evolution algorithm, the optimization method for optimizing the fractional 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 Figure 5 The specific steps of the above method are as follows:

[0151] (1) Establish the ant position coordinate vector using formula (9) And perform parameter initialization for 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 for each ant;

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

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

[0156] (6) Calculate the individual position vector for the next iteration using formula (15).

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

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

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

[0160] The definitions of each variable are as follows:

[0161] Define t as a time variable. It is the n-dimensional state vector of the chaotic system, and the initial conditions of the system at time t0. , It is the fractional order vector of the system. Let f(·) represent the coefficient vector of the system excluding fractional-order functions, and let f(·) represent a well-defined but unknown nonlinear function.

[0162] definition Indicates taking integer values, where h is the time step, defined as follows: Let Γ(·) be the coefficient of the fractional binomial, and let Γ(·) be the Gamma function. It's the Grinwald function, initial conditions.

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

[0164] Define the time series t0 < t1 < … <t l ,x(t i ) indicates that at t i (i = 0, 1, … The sequence of real state variables of the chaotic system at time l), y(t) i ) indicates that at t i (i = 0, 1, … The estimated state variable sequence of the chaotic system at time l). 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 of the i-th ant (i.e., its fitness value) is denoted as τ(i), and the optimal fitness of the i-th ant is denoted as τi. best (i) The upper and lower limits of the ant's activity range are respectively and The initial optimal position is

[0166] Define P(i) as the state transition probability, and the pre-update position of the i-th ant is: definition Represents individuals of generation k The mutated individual vector, Let r1 and r2 represent the individuals with the lowest fitness in the k-th generation of the population, i.e., the optimal individuals. Let r1 and r2 represent the indices of individuals randomly selected from the population, and r1 ≠ r2, r1, r2 ∈ [1, K], where K is the maximum number of iterations. As basis vectors, F is the difference vector, and F is the learning factor, which controls the learning speed at which the difference vector approaches the basis vector, and its range is [0,1].

[0167] Define parent individual and mutation Cross-combination to form cross individuals In the crossover operation, the crossover probability parameter CR is introduced to control the selection probability of alleles between two individuals, where jrand represents [1,2, … Random values ​​within [D], j = jrand indicates that at least one variable comes from the mutated individual vector, and rand(0,1) represents random values ​​within (0,1). This represents the j-th component in the i-th individual of the parent generation. Let j be the j-th component of the i-th individual in the mutated individuals, where... The crossover probability represents the proportion of mutated individuals in the ant population.

[0168] Define J(·) as the fitness value. Let μ represent an individual that successfully enters generation k+1, and define the crossover probability constant, ε, and differential evolution mutation coefficient. The initial variation coefficient F0 and the adaptive constant ε of differential evolution.

[0169] Based on the appendix Figure 6-9 The experimental results show that the proposed SaDE-ACO algorithm demonstrates significant technical advantages in parameter identification of fractional-order chaotic circuit systems. Verification on two typical chaotic circuit systems, Chen and Lorenz, shows that as the iteration process progresses, the system parameter estimation error continuously decreases and eventually approaches zero; both the parameter vector and the fractional-order estimated value stably converge to the neighborhood of the true value. Compared to traditional swarm optimization algorithms, this method shows a significant improvement in convergence speed, with the fitness curve completing a rapid descent phase in the early stages of iteration. In terms of anti-interference capability, its parameter estimation error band is significantly narrowed, and it more accurately characterizes the nonlinear properties of the system. By integrating the adaptive perturbation mechanism of differential evolution with the parallel search characteristics of ant colony optimization, this invention effectively avoids local extremum traps in high-dimensional parameter spaces. Its hybrid optimization architecture significantly reduces computational complexity while maintaining identification accuracy. Experimental data further demonstrates that the estimated output highly matches the dynamic response of the real system, verifying the robustness and engineering applicability 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 within the protection scope of the present invention.

Claims

1. A fractional-order chaotic circuit modeling method based on adaptive differential fusion ant colony optimization, characterized in that, Includes the following steps: Step 1) Construct an adaptive differential fusion ant colony optimization fractional chaotic circuit system model to ultimately identify the unknown fractional order and coefficient vector of the chaotic circuit system; Step 1) includes the following steps: (1-1) Construct a fractional-order chaotic circuit system model based on an integer-order chaotic circuit system; (1-2) Based on this model, the expression for the fractional-order chaotic circuit system model is as follows: （1）； Where t is the time variable, It is a chaotic system A state vector, the system in Initial conditions at time , It is the fractional order vector of the system. This represents the coefficient vector of the system excluding fractional-order systems. Represents an unknown nonlinear function; (1-3) Based on the formula derivation, The numerical solution is （2）； in, For time step, It's the Grinwald function, initial conditions. , , ; (1-4) Based on the numerical solutions of the estimated system and the real system, the loss function is defined as follows: （3）； in, , , The length of the system state variable sequence. express Norm; Step 2) Construct an ant colony algorithm based on differential evolution to optimize the fractional-order chaotic circuit system model; Step 2) includes the following steps: Step 2-1) Establish the ant's position coordinate vector and initialize the parameters of the algorithm; Step 2-2) Calculate the fitness value Determine the optimal position ; Steps 2-3) Calculation Determine the search range for each ant; Steps 2-4) Obtain the pre-update location And determine whether the ant's position needs to be updated; Steps 2-5) Update the pheromones for each path and obtain the parameter vector. ; Steps 2-6) Calculate the individual position vector for the next iteration. ; Steps 2-7) Update pheromones To obtain the optimal pheromone and the position vector at this time ; Steps 2-8) Combine the differential evolution strategy to obtain the coefficient vector of the estimated system. and fractional order vectors ; Steps 2-9) Execute the iterative loop until the maximum number of iterations is reached, then stop and output the final estimated value.

2. The ant colony algorithm based on differential evolution algorithm for optimizing fractional-order chaotic circuit system models as described in claim 1, characterized in that, In steps 2-8) of ant colony optimization, differential evolution is introduced, including the following steps: Step 2-8-1) in the In the next iteration, for individuals Mutation operations generate new vectors The mutation operation is performed, consisting of difference vectors and basis vectors. The learning factor is used to control the learning speed at which the difference vectors approach the basis vectors. Step 2-8-2) Perform a crossover operation to increase the diversity of mutated individual vectors, parent individuals and mutation Cross-combination to form cross individuals Introducing a crossover probability parameter in crossover operations To control the probability of allele selection between two individuals; Step 2-8-3) Perform the selection operation. First, calculate the fitness value of all generated individuals, then compare it with the fitness value of the target vector one by one, and keep the individual vector corresponding to the fitness value in the next generation; otherwise, keep the target vector in the next generation of individuals to continue the iteration. Step 2-8-4) Crossover probability of the difference vector Impact on the algorithm, introducing the crossover probability constant An adaptive differential evolution algorithm was proposed, which improves the crossover probability and introduces an adaptive constant. Step 2-8-5) Solve for Thus, the coefficient vector of the estimated system can be obtained. and fractional order vectors .

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