A bouc-wen model parameter identification method based on multi-strategy improved balance optimization algorithm

Abstract

The application discloses a Bouc-Wen model parameter identification method based on a multi-strategy improved balance optimization algorithm; 1) obtaining experimental response data of a nonlinear hysteresis system under the action of external excitation; 2) establishing a Bouc-Wen nonlinear hysteresis model, taking the joint root mean square error between model output data and experimental response data as a target function; 3) embedding an improved balance pool construction strategy, a bidirectional guiding update strategy and a Gaussian-Cauchy hybrid mutation strategy into a standard balance optimization algorithm to form a multi-strategy improved balance optimization algorithm; 4) minimizing the target function by using the multi-strategy improved balance optimization algorithm to identify unknown parameters of the Bouc-Wen model, obtaining optimal parameters and forming a mathematical model; 5) fitting a hysteresis loop and predicting a system response according to the obtained optimal parameters and the model; the application has the characteristics of high search precision, good convergence performance, easy jumping out of local optimization and strong stability, and can effectively improve the Bouc-Wen model parameter identification precision.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear system modeling and intelligent optimization parameter identification technology, specifically involving a Bouc-Wen model parameter identification method based on a multi-strategy improved equilibrium optimization algorithm. Background Technology

[0002] Hysteresis nonlinearity is widely found in engineering fields such as piezoelectric actuators, magnetorheological dampers, shape memory alloys, magnetic materials, and structural vibration control systems. Unlike general nonlinear systems, the output of a hysteresis system is not only related to the current input but also closely related to the system's historical state and motion path, exhibiting a significant memory effect. Therefore, establishing a mathematical model that can accurately describe the hysteresis characteristics is of great importance in the modeling, analysis, and control of hysteresis nonlinear systems.

[0003] The Bouc-Wen model is a typical nonlinear hysteresis model. By introducing internal hysteresis variables to describe the memory effect in the system response, this model can effectively simulate smooth hysteresis loops of various shapes, and is therefore widely used in modeling nonlinear hysteresis systems. However, the Bouc-Wen model exhibits strong nonlinear characteristics, and there are coupling relationships between model parameters. The influence of different parameters on the shape, amplitude, and dynamic response of the hysteresis loop is complex, making it difficult to directly determine the model parameters analytically.

[0004] Currently, parameter identification methods for the Bouc-Wen model mainly include traditional numerical optimization methods and intelligent optimization algorithms. Traditional numerical optimization methods include least squares, gradient descent, and Kalman filtering. These methods can achieve certain results in some simple problems, but they are usually sensitive to initial values ​​and easily get trapped in local optima when the objective function is highly nonlinear or the parameters are significantly coupled, leading to insufficient identification accuracy. Intelligent optimization algorithms have strong global search capabilities and lower requirements for the continuity and differentiability of the objective function, making them suitable for handling complex nonlinear parameter identification problems. However, some classic intelligent optimization algorithms still suffer from slow convergence speed, insufficient search accuracy, poor result stability, and susceptibility to local optima in complex search spaces.

[0005] The equilibrium optimization algorithm is a metaheuristic optimization algorithm inspired by the idea of ​​mass balance in a controlled volume system. It has advantages such as simple structure, few parameters, and ease of implementation. However, when solving complex nonlinear optimization problems, the standard equilibrium optimization algorithm still suffers from drawbacks such as limited guidance information from the equilibrium pool, insufficient population diversity, limited local exploitation capabilities, and a tendency for premature convergence during the search process. Therefore, it is necessary to improve the standard equilibrium optimization algorithm and apply it to Bouc-Wen model parameter identification to improve the accuracy, convergence performance, and stability of model parameter identification. Summary of the Invention

[0006] To address the problems of low parameter identification accuracy, strong parameter coupling, reliance on initial values ​​in traditional numerical methods, and susceptibility to local optima in intelligent optimization algorithms in existing Bouc-Wen models, this invention proposes a parameter identification method for Bouc-Wen models based on a multi-strategy improved equilibrium optimization algorithm. This method introduces an improved equilibrium pool construction strategy, a bidirectional guided update strategy, and a Gaussian-Cauchy hybrid mutation strategy to the standard equilibrium optimization algorithm, forming a multi-strategy improved equilibrium optimization algorithm. This algorithm is then applied to the parameter identification of the Bouc-Wen nonlinear hysteresis model to obtain the optimal parameter combination. The Bouc-Wen model identified by this method can more accurately describe the dynamic response characteristics of the nonlinear hysteresis system and improve the fitting accuracy of the hysteresis loop.

[0007] The Bouc-Wen model parameter identification method based on a multi-strategy improved equilibrium optimization algorithm described in this invention includes the following steps:

[0008] 1) Obtain experimental response data of a nonlinear hysteresis system under external excitation, wherein the experimental response data includes displacement variable data and hysteresis internal variable data;

[0009] 2) Establish a Bouc-Wen nonlinear hysteresis model, calculate the model output response based on the model state equation, and use the error between the model output data and the experimental data as the objective function;

[0010] 3) The improved balance pool construction strategy, bidirectional guided update strategy, and Gauss-Cauchy hybrid mutation strategy are embedded into the standard balance optimization algorithm to form a multi-strategy improved balance optimization algorithm;

[0011] 4) The improved balance optimization algorithm based on the multi-strategy method is used to optimize the unknown parameters of the Bouc-Wen model in order to minimize the objective function and obtain the optimal parameter combination of the Bouc-Wen model.

[0012] 5) Establish a Bouc-Wen parameter identification model based on the obtained optimal parameters, and compare the model output with the experimental data to complete the hysteresis loop fitting and error analysis.

[0013] The Bouc-Wen model expression in the Bouc-Wen model parameter identification method based on a multi-strategy improved equilibrium optimization algorithm is as follows:

[0014]

[0015] In the Bouc-Wen nonlinear hysteresis model, x represents the system displacement variable. and Let z represent the system velocity and acceleration, u(t) represent the external excitation input, α represent the stiffness ratio, ξ represent the linear elastic viscous damping ratio, and ω represent the system velocity and acceleration, respectively; z represents the hysteresis internal variable, u(t) represents the external excitation input, α represents the stiffness ratio, ξ represents the linear elastic viscous damping ratio, and ω represents the linear elastic viscous damping ratio. n The pseudo-natural frequency of the system is represented by , A represents the hysteresis amplitude control parameter, and β, γ and n represent the hysteresis loop shape control parameters.

[0016] The external excitation input can be a periodic excitation function, as shown in the following expression:

[0017] u(t)=Bcos(ω·t) (2)

[0018] Where B represents the amplitude of the excitation signal, and ω represents the angular frequency of the excitation signal.

[0019] The parameter vector to be identified in the Bouc-Wen model is:

[0020] X=[α,β,γ,ξ,ωn,A,n] (3) To facilitate numerical solution, the Bouc-Wen model is rewritten as a spatial state equation. Let the state variables be:

[0021]

[0022] The state-space equation is then:

[0023]

[0024] For a given set of candidate parameters, the displacement variables and hysteresis internal variables of the model output are obtained by numerically solving the above state-space equations.

[0025] The objective function is the joint root mean square error of the model output and the experimental data, denoted as F. RMSE The formula for (X) is as follows:

[0026]

[0027] Where N represents the number of sample data, x 0,k and z 0，k Let x represent the experimental data respectively. BW，k and z BW，k These represent the model output data;

[0028] The process of finding the optimal parameters using the multi-strategy improved balance optimization algorithm in the Bouc-Wen model parameter identification method described above is as follows:

[0029] The algorithm begins the optimization process by initializing the population, where the initial concentration of particles is constructed by random initialization within the search space.

[0030] The initialization formula is as follows:

[0031]

[0032] in, Let C represent the initial concentration vector of the i-th particle. min and C max rand represents the lower and upper bounds of each dimension, respectively. i Let N be a random vector with values ​​in the range [0, 1], and let N be the population size.

[0033] The four particles in the equilibrium pool may contain more information about the optimal solution in their vicinity during the search process. Therefore, while retaining the four optimal particles, we calculate the maximum and minimum values ​​of the four optimal particles in each dimension to construct an elite search space, and generate a new candidate particle within this search space. We then reconstruct the equilibrium pool using this new candidate particle. The formula for constructing the new equilibrium pool is as follows:

[0034] C eq_max =max(C eq1 C eq2 C eq3 Ce q4) (8)

[0035] C eq_min =min(C eq1 C eq2 C eq3 C eq4 (9)

[0036] C eq_x =C eq_min +rand×(C eq_max -C eq_min (10)

[0037]

[0038] Among them, C eq_max and C eq_min These are vectors composed of the maximum and minimum values ​​of the four optimal particles in each dimension, respectively. `rand` is a random vector uniformly distributed within the interval [0, 1]. C eq _ x These are random particles generated within the elite search space, used to guide the search direction of a portion of the population. This is a newly constructed equilibrium pool. During the algorithm's iteration process, each particle is selected with equal probability from six candidate particles in the improved equilibrium pool, which helps in the generation of the global optimum.

[0039] The particle update formula in the standard equilibrium optimization algorithm is:

[0040]

[0041] Where C represents the current particle position, C eq Let F represent the candidate equilibrium particle selected from the equilibrium pool, G represent the generation rate, λ represent the turnover rate, and V represent the unit volume.

[0042] The exponential term F can be expressed as:

[0043] F = a1sign(r - 0.5)[e -λt -1] (13) where a1 is a parameter that controls the exploration capability of the algorithm, r and λ are random vectors in the interval [0, 1], and t is a time factor that changes with the number of iterations.

[0044] To enhance population information exchange capabilities, this invention introduces a bidirectional guided update strategy. This strategy simultaneously utilizes candidate particles in the balancing pool and the population center position to guide particle selection. The formula for calculating the population center position is:

[0045]

[0046] Among them, C m Indicates the location of the population center, N represents the population size, and C represents the population center location. i This represents the i-th individual in the population.

[0047] The bidirectional guided update formula is:

[0048] C i+1 =C r +θ1·(μ1·C eq -μ2·C i )+θ2·l·(μ1·C m -μ2·C i (15)

[0049] Among them, C i+1 Let C represent the updated position of the i-th individual. eq This represents the guide individual selected from the equilibrium pool, θ1 and θ2 are random perturbation coefficients, μ1 and μ2 are adjustment coefficients, and C r This refers to the reference particle selected from the current particle or candidate equilibrium particles based on the adjustment coefficient.

[0050] During the iteration process, the position update method is randomly selected to use either the standard balanced optimization algorithm or the bidirectional guided update method. When the random number is less than a preset threshold, the standard balanced optimization algorithm is used; otherwise, the bidirectional guided update strategy is used to update the particle position.

[0051] To further balance the algorithm's global search capability and local exploitation capability, this invention introduces a Gaussian-Cauchy hybrid mutation strategy. The Gaussian-Cauchy hybrid mutation formula is as follows:

[0052] Ci'=C i ·(1+w1·G+w2·L) (16)

[0053] Among them, C i C' represents the individual before the mutation. i Let G represent the mutated individual, G represent a Gaussian random variable following a standard normal distribution, L represent a Cauchy random variable following a standard Cauchy distribution, and w1 and w2 represent the adaptive weights of the Gaussian and Cauchy variants, respectively.

[0054] The weight calculation formula is as follows:

[0055]

[0056] Where Iter represents the current iteration number, and Max_iter represents the maximum iteration number. In the early stages of the algorithm iteration, the Cauchy mutation weight is relatively large, which is conducive to generating large step-size perturbations, expanding the search range, and improving the ability to escape local optima; in the later stages of the algorithm iteration, the Gaussian mutation weight gradually increases, which is conducive to performing fine search near the current dominant region, thereby improving the final convergence accuracy.

[0057] When using the above-mentioned multi-strategy improved equilibrium optimization algorithm for parameter identification of the Bouc-Wen model, the specific steps are as follows:

[0058] 1) Set the population size N, the maximum number of iterations Max_iter, the dimension of the parameter to be identified D, the lower boundary lb and the upper boundary ub of the search space;

[0059] 2) Randomly initialize population individuals within the search space, with each individual representing a set of candidate parameters for the Bouc-Wen model;

[0060] 3) Solve the Bouc-Wen model numerically based on the candidate parameters to obtain the displacement variables and hysteresis internal variables of the model output;

[0061] 4) Calculate the fitness value of each individual based on the joint root mean square error objective function;

[0062] 5) Select the four best individuals and their average individual based on their fitness values ​​to construct the initial balance pool;

[0063] 6) Based on the improved equilibrium pool construction strategy, generate new candidate equilibrium particles and construct the improved equilibrium pool;

[0064] 7) Update the position of individuals in the population by selecting either the standard balance optimization algorithm or the two-way guided update strategy based on random probability;

[0065] 8. Perform boundary processing on the updated individuals and recalculate their fitness values;

[0066] 9) Execute the Gaussian-Cauchy mixed mutation strategy according to the preset probability, and decide whether to replace the original individual based on the fitness values ​​of the individual before and after mutation;

[0067] 10) Update the balance pool and determine if the maximum number of iterations has been reached. If the maximum number of iterations has not been reached, return to step 6) to continue iterating; if the maximum number of iterations has been reached, output the optimal individual and its corresponding objective function value.

[0068] 11) The optimal individual output is used as the optimal parameter combination for the Bouc-Wen model, and this parameter combination is used for hysteresis loop fitting and system response prediction. Attached Figure Description

[0069] Figure 1 The accompanying drawings are a summary of the specification of this invention;

[0070] Figure 2 This is a mechanical equivalent structural diagram of the Bouc-Wen model described in this invention;

[0071] Figure 3 This is a flowchart of the multi-strategy improved equilibrium optimization algorithm described in this invention;

[0072] Figure 4 This is a graph showing the fitting effect of the Bouc-Wen model hysteresis loop described in this invention.

[0073] Figure 5 This is a comparison chart of parameter identification results using different methods of the present invention; Detailed Implementation

[0074] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. The described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0075] This invention provides a technical solution: a method for parameter identification of the Bouc-Wen model based on a multi-strategy improved equilibrium optimization algorithm, comprising the following steps:

[0076] 1) Obtain experimental response data of the nonlinear hysteresis system through the detection unit or data acquisition unit, including sampling time, external excitation input, system displacement response, and hysteresis internal variable response data;

[0077] 2) Establish a Bouc-Wen nonlinear hysteresis model, and use the joint root mean square error between the displacement response and the hysteresis internal variable response calculated by the model and the experimental response data as the objective function for optimization.

[0078] 3) The improved balance pool construction strategy, bidirectional guided update strategy, and Gauss-Cauchy hybrid mutation strategy are embedded into the standard balance optimization algorithm to form a multi-strategy improved balance optimization algorithm;

[0079] 4) Use a multi-strategy improved equilibrium optimization algorithm to minimize the objective function, identify the unknown parameters in the Bouc-Wen model, obtain the optimal parameter combination, and form the corresponding mathematical model;

[0080] 5) Based on the obtained optimal parameters of the model, the Bouc-Wen model is used to fit the response data of the nonlinear hysteresis system, and the hysteresis loop fitting results of the system are further obtained.

[0081] The Bouc-Wen model parameters identified by this method have high accuracy, can well reflect the relationship between input excitation and output response in nonlinear hysteresis systems, and can effectively describe the hysteresis memory characteristics of the system. This method has good application prospects in the modeling of nonlinear hysteresis systems such as piezoelectric actuators, magnetorheological dampers, shape memory alloys, and structural vibration control.

Claims

1. A parameter identification method for the Bouc-Wen model based on a multi-strategy improved equilibrium optimization algorithm. Its features include the following steps: 1) Obtain experimental response data of a nonlinear hysteresis system under external excitation, wherein the experimental response data includes system displacement data and hysteresis internal variable data; 2) Establish a Bouc-Wen nonlinear hysteresis model and determine the parameter vector to be identified. The parameters to be identified include stiffness ratio, linear elastic viscous damping ratio, system pseudo natural frequency, hysteresis amplitude control parameters, and hysteresis loop shape control parameters. 3) Construct an objective function based on the error between the Bouc-Wen model output data and the experimental response data; 4) The improved balance pool construction strategy, bidirectional guided update strategy, and Gauss-Cauchy hybrid mutation strategy are embedded into the standard balance optimization algorithm to form a multi-strategy improved balance optimization algorithm; 5) The objective function is minimized using the aforementioned multi-strategy improved equilibrium optimization algorithm to obtain the optimal parameter combination of the Bouc-Wen model; 6) Establish the Bouc-Wen nonlinear hysteresis model based on the obtained optimal parameter combination, and use it for response prediction or hysteresis loop fitting of nonlinear hysteresis systems.

2. The Bouc-Wen model parameter identification method based on a multi-strategy improved equilibrium optimization algorithm according to claim 1, characterized in that, The expression for the Bouc-Wen nonlinear hysteresis model is as follows: In the Bouc-Wen nonlinear hysteresis model, x represents the system displacement variable. and Let z represent the system velocity and acceleration respectively; z represent the hysteresis internal variable; u(t) represent the external excitation input; a represent the stiffness ratio; ξ represent the linear elastic viscous damping ratio; and ω represent the linear elastic viscous damping ratio. n The pseudo-natural frequency of the system is represented by , A represents the hysteresis amplitude control parameter, and β, γ and n represent the hysteresis loop shape control parameters.

3. The Bouc-Wen model parameter identification method based on a multi-strategy improved equilibrium optimization algorithm according to claim 1, characterized in that, The objective function is the joint root mean square error between the model output data and the experimental response data, and its calculation formula is as follows: Where N represents the number of sampling points, x 0,k and z 0，k Let x represent the experimental displacement data and the experimental hysteresis internal variable data at the k-th sampling point, respectively. BW，k and z BW，k F represents the displacement data and hysteresis internal variable data calculated by the Bouc-Wen model at the k-th sampling point, respectively. RMSE This represents the objective function value corresponding to the parameter to be identified; X = [α, β, γ, ξ, ω] n [A, n] are the seven parameters that need to be identified.

4. The Bouc-Wen model parameter identification method based on a multi-strategy improved equilibrium optimization algorithm according to claim 1, characterized in that, The execution steps of the multi-strategy improved equilibrium optimization algorithm are as follows: 1) Set the population size, maximum number of iterations T, problem dimension D, lower boundary lb and upper boundary ub of the search space for the parameters to be identified, and control parameters for the balanced optimization algorithm; 2) Randomly initialize population individuals within the search space of parameters to be identified, with each population individual representing a set of candidate parameters for the Bouc-Wen model; 3) Calculate the fitness value of each individual in the population according to the objective function, and select the four individuals with the best current fitness and their average value to construct the initial balance pool; 4) Regenerate the balance pool based on the improved balance pool construction strategy; 5) Update the individual positions in the population by selecting either the original balance optimization algorithm update method or the two-way guided update strategy based on random probability; 6) Perform boundary processing on the updated population individuals and recalculate their fitness values; 7) Perform Gaussian-Cauchy mixed mutation operation on individuals in the population according to the preset probability, and determine whether to replace the original individuals based on the fitness value; 8) Determine if the maximum number of iterations has been reached. If the maximum number of iterations has been reached, output the optimal individual and its objective function value; otherwise, return to step 4.

5. The Bouc-Wen model parameter identification method based on a multi-strategy improved equilibrium optimization algorithm according to claim 4, characterized in that, The improved balancing pool construction strategy includes the following steps: 1) Select the four best individuals in the current population based on their fitness values, and denote them as C. eq(1) C eq(2) C eq(3) C eq(4) ; 2) Calculate the average of the four optimal individuals to obtain the average equilibrium candidate individual C. eq(ave) ; 3) Calculate the maximum and minimum values ​​of the four optimal individuals in each dimension to obtain C. eq_max and C eq_min ; 4) In the case of C eq_max and C eq_min New candidate individuals C are randomly generated within the elite search space. eq_x The calculation formula is as follows: C eq_x =C eq_min +rand×(C eq_max -C eq_min ) (3) 5) By C eq(1) C eq(2) C eq(3) C eq(4) C eq(ave) and C eq_x Together they form a new equilibrium pool.

6. The Bouc-Wen model parameter identification method based on a multi-strategy improved equilibrium optimization algorithm according to claim 4, characterized in that, The bidirectional guided update strategy simultaneously incorporates candidate individuals from the balancing pool and population center location information to update individuals. The formula for calculating the population center location is as follows: Among them, C m Indicates the location of the population center, N represents the population size, and C represents the population center location. i This represents the i-th individual in the population; the individual update formula is: C i+1 =C r +θ1·(μ1·C eq -m2·C i )+θ2·l·(μ1·C m -m2·C i ) (5) Among them, C i+1 Let C represent the updated position of the i-th individual. eq This represents the guide individual selected from the equilibrium pool, θ1 and θ2 are random perturbation coefficients, μ1 and μ2 are adjustment coefficients, and C r This refers to the reference individual selected from the current individual or the candidate individuals in the balance pool based on the adjustment coefficient.

7. The Bouc-Wen model parameter identification method based on a multi-strategy improved equilibrium optimization algorithm according to claim 4, characterized in that, The mutation formula for the Gaussian-Cauchy mixed mutation strategy is as follows: C i '=C i ·(1+w1·G+w2·L) (6) Among them, C i C' represents the individual before the mutation. i Let G represent the mutated individual, L represent the Gaussian random variable following a standard normal distribution, and w1 and w2 represent the adaptive weights of the Gaussian and Cauchy variance terms, respectively. Their calculation formulas are as follows: Where Iter represents the current iteration number, and Max_iter represents the maximum iteration number.

8. The Bouc-Wen model parameter identification method based on a multi-strategy improved equilibrium optimization algorithm according to claim 1, characterized in that, The optimal parameter combination is used to reconstruct the hysteresis response curve of the nonlinear hysteresis system, minimizing the error between the displacement response and the hysteresis internal variable response output by the Bouc-Wen model and the experimental response data.

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