Model simplification and control method of nonlinear system

Abstract

The invention discloses a model simplification and control method of a nonlinear system. The method relates to the fields of computer systems based on specific calculation models, process control and the like. The method comprises the following steps: firstly, performing feedback linearization on an affine nonlinear model based on a Lie bracket and a Lie derivative of a differential geometry method, converting a nonlinear system into a linear system, and reducing the complexity of the model; secondly, on the basis of the genetic algorithm, by increasing the diversity of populations and the selection probability of high-quality individuals, the convergence speed and the global optimization capacity are improved, and it is ensured that better parameter configuration is obtained; compared with the prior art, the method has the advantages that a complex affine nonlinear system in the chemical process is simplified, the response speed and stability of a control system are improved, the production efficiency is improved, the global search capability of an optimization algorithm is enhanced, the dependency of parameter adjustment on manual adjustment is reduced, and the working efficiency is improved; the method has high technical innovation and application prospects in the chemical process of heat and mass transfer involving complex physical and chemical reactions.

Description

Technical Field

[0001] The present invention relates to a computer system and process control based on a specific calculation model, and in particular to a model simplification and control method for a nonlinear system. Background Art

[0002] In the industrial production processes of petroleum, chemical industry, light industry, etc., distillation, as a typical chemical unit operation, is currently the most widely used mass transfer and heat transfer process. The distillation process occupies an important position in petrochemical enterprises. It is used to separate crude oil into different fractions to provide raw materials for subsequent production, and can also directly separate industrial products that meet the requirements. However, in actual engineering, the safe, stable and efficient operation of the distillation process still faces many challenges.

[0003] The distillation tower is a typical multi-input, multi-output, multi-variable system with slow dynamic response, complex coupling relationships between variables, and significant differences in process flow and structure between different towers, which leads to extremely complex control requirements. In actual production, the control of distillation towers is highly dependent on mathematical models, but due to its inherent high-order nonlinear characteristics, establishing an accurate and controllable model has always been a core challenge in the engineering field.

[0004] The dimension of the state equation of the distillation process is usually directly related to the number of plates. The high order and nonlinear characteristics of complex distillation towers are particularly obvious, which brings difficulties to the practical application of the model. Although traditional model linearization and order reduction methods can improve computational efficiency, they often sacrifice accuracy and are difficult to fully adapt to the complex multi-condition characteristics of distillation towers. This limitation is particularly prominent in high-precision demand scenarios, posing a major challenge to achieving efficient control of the distillation process.

[0005] In view of the above-mentioned restrictive problems, the present invention proposes a model simplification and control method for nonlinear systems in chemical processes, which mainly decouples and linearizes affine nonlinear systems in chemical processes. First, the Lie bracket and Lie derivative of the differential geometry method are used to feedback linearize the affine nonlinear model, transform the nonlinear system into a linear system, and reduce the complexity of the model; secondly, based on the genetic algorithm, by increasing the diversity of the population and the probability of selecting high-quality individuals, the convergence speed and global optimization ability are improved, ensuring a better parameter configuration, improving the response speed and stability of the control system, improving production efficiency, and enhancing the global search ability of the optimization algorithm. Summary of the invention

[0006] The present invention designs a model simplification and control method for a nonlinear system.

[0007] The specific implementation steps are as follows:

[0008] Step 1: Construct the vector field And analyze whether it satisfies the full rank and involution conditions. In order to simplify the model of the affine nonlinear system, the Lie derivative operator in the differential geometry method is used to construct the vector field, and then it is necessary to analyze whether these vector fields satisfy the involution conditions and full rank conditions, and on this basis to explore the linearization relationship of the model.

[0009] Assume the affine nonlinear standard system ,

[0010] in, x = [ x 1 , x 2 , ⋯ , x n ] T ∈ D ( D ∈ R n ) . It is an open set The vector fields on are called dynamic vector field and control vector field respectively. It is defined in the open set A real-valued function on . The system is called an affine nonlinear system, which is characterized by the input Independent of the control vector field , and the right side of the output equation is the same as the input Not relevant.

[0011] Step 1.1: Construct the vector field

[0012] According to the definition of Lie function in differential geometry, we can calculate ,test , , The linear correlation between two functions is determined, thereby revealing their mutual dependence under specific conditions.

[0013] ,

[0014] ,

[0015] ∂ f ∂ x = [ ∂ f 1 ∂ x 1 ∂ f 1 ∂ x 2 ⋯ ∂ f 1 ∂ x n ∂ f 2 ∂ x 1 ∂ f 2 ∂ x 2 ⋯ ∂ f 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ f n ∂ x 1 ∂ f n ∂ x 2 ⋯ ∂ f n ∂ x n ] , ∂ g ∂ x = [ ∂ g 1 ∂ x 1 ∂ g 1 ∂ x 2 ⋯ ∂ g 1 ∂ x n ∂ g 2 ∂ x 1 ∂ g 2 ∂ x 2 ⋯ ∂ g 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ g n ∂ x 1 ∂ g n ∂ x 2 ⋯ ∂ g n ∂ x n ] ,

[0016] ad f g = ∂ g ∂ x f − ∂ f ∂ x g = [ ∂ g 1 ∂ x 1 ∂ g 1 ∂ x 2 ⋯ ∂ g 1 ∂ x n ∂ g 2 ∂ x 1 ∂ g 2 ∂ x 2 ⋯ ∂ g 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ g n ∂ x 1 ∂ g n ∂ x 2 ⋯ ∂ g n ∂ x n ] [ f 1 f 2 ⋮ f n ] − [ ∂ f 1 ∂ x 1 ∂ f 1 ∂ x 2 ⋯ ∂ f 1 ∂ x n ∂ f 2 ∂ x 1 ∂ f 2 ∂ x 2 ⋯ ∂ f 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ f n ∂ x 1 ∂ f n ∂ x 2 ⋯ ∂ f n ∂ x n ] [ g 1 g 2 ⋮ g n ] ,

[0017] in, is a first-order Lie bracket, is a second-order Lie bracket, calculate ,like , then the system model is linearly dependent.

[0018] Step 1.2: Inspection Whether the involution condition and full rank condition are satisfied. Before linearization, it is necessary to check whether the system satisfies the involution condition and full rank condition. The involution condition ensures that the distribution of the vector field is appropriate, while the full rank condition ensures the controllability of the system state. These conditions are necessary conditions for the system to be able to accurately linearize the state.

[0019] For an affine nonlinear system ,exist The necessary and sufficient conditions for accurate linearization of the state near the point are:

[0020] (1) Matrix [ g ( x 0 ) ad f g ( x 0 ) ⋯ ad f 2 g ( x 0 ) ] The rank is n;

[0021] (2) Distribution D = span [ g ad f g ⋯ ad f n − 2 g ] exist is an involution distribution in a neighborhood of , Lie bracket operation [ τ 1 , τ 2 ] ∈ D .

[0022] If the involution condition and full rank condition are satisfied, the system can be state-exactly linearized, otherwise it cannot.

[0023] Step 2: Get the output function The output function is a function that transforms the system state Function mapped to the output space. The choice of this function determines the form of the system after linearization and is crucial to the linearization of the system.

[0024] Calculate the output function based on the following conditions :

[0025] ∂ h ∂ x [ g ad f g ⋯ ad f n − 2 g ] = 0 ,

[0026] ,

[0027] in, is the output function, For the function Each variable in Find the partial derivative, Control vector, is a first-order Lie bracket, is a second-order Lie bracket, for brackets, for brackets;

[0028] Constructing state variables z ( x ) = [ h ( x ) L f h ( x ) ⋯ L f n − 1 h ( x ) ] T ,in The Lie derivative or Lie derivative operator is called the function Along the dynamic vector field The Lie derivative of for brackets, is a 2nd-order Lie bracket, for The order bracket is calculated as follows:

[0029] ,

[0030] ,

[0031] ,

[0032] Constructing state transitions z ( x ) = [ h ( x ) L f h ( x ) ⋯ L f n − 1 h ( x ) ] T , calculate the Jacobian matrix of this transformation:

[0033] ∇ Φ ( x ) = [ ∂ h ( x ) ∂ x 1 ∂ h ( x ) ∂ x 2 ⋯ ∂ h ( x ) ∂ x n ∂ L f h ( x ) ∂ x 1 ∂ L f h ( x ) ∂ x 2 ⋯ ∂ L f h ( x ) ∂ x 3 ⋮ ⋮ ⋱ ⋮ ∂ L f n − 1 h ( x ) ∂ x 1 ∂ L f n − 1 h ( x ) ∂ x 2 ⋯ ∂ L f n − 1 h ( x ) ∂ x n ] ,

[0034] calculate ,like , so the above nonlinear coordinate transformation is a differential homeomorphism.

[0035] Step 3: Accurately linearize the system using state feedback:

[0036] System Output For time Derivative, we have:

[0037] ,

[0038] like ,enter If it cannot appear explicitly, we continue to derive:

[0039] ,

[0040] like , then continue to derive until the input appears on the right side of the equation :

[0041] ,

[0042] When the number of derivatives is , the input appears on the right side of the equation , is the relative order of the system. , then the system can be accurately linearized, and it can be proved that is linearly independent and satisfies condition (1) in step 1.2, where is the system dimension, is a dynamic vector field, is the input variable, For the control vector Lie derivative or Lie derivative operator, is the number of system state variables, For along The rate of change of direction is Direction to find the rate of change.

[0043] Through the derivation process, the coordinate transformation formula of the differential homeomorphism can be obtained as follows:

[0044] z = [ z 1 z 2 ⋮ z n ] = Φ ( x ) = [ h ( x ) L f h ( x ) ⋮ L f n − 1 h ( x ) ] ,

[0045] Take the control input as:

[0046] ,

[0047] ,

[0048] ,

[0049] in , .

[0050] The original nonlinear system can be transformed into a controllable linearized system:

[0051] { z ˙ = [ 0 1 0 ⋯ 0 0 0 1 ⋯ 0 0 0 ⋯ 0 1 0 0 ⋯ 0 0 ] z + [ 0 ⋮ 0 1 ] u y = ( 1 0 ⋯ 0 ) z ,

[0052] Right now:

[0053] [ z ˙ 1 z ˙ 2 ⋮ z ˙ n − 1 z ˙ n ] = [ z 2 z 3 ⋮ z n v ] ,

[0054] ,

[0055] Step 4: Combine the Black Kite algorithm and the genetic algorithm to optimize the parameters of the simplified model. Use the Black Kite algorithm to initialize the population, and then apply the crossover and mutation operations of the genetic algorithm to explore the solution space, making full use of the advantages of the two algorithms to improve the efficiency of model parameter optimization.

[0056] Step 4.1: Initialize population parameters,

[0057] BK = [ BK 1 , 1 BK 1 , 2 ⋯ BK 1 , n BK 2 , 1 BK 2 , 2 ⋯ BK 2 , n ⋮ ⋮ ⋮ ⋮ BK m , 1 BK m , 2 ⋯ BK m , n ] ,

[0058] in, is the number of potential solutions, that is, the number of different combinations of operating parameters to be evaluated; n is the number of parameters to be optimized given the size of the problem dimension, It is a combination of parameters, and the parameter initialization rules are:

[0059] ,

[0060] in, is the value of the jth parameter in the ith potential solution, i is [ 1 , m ] Integers between and are the upper and lower bounds of the jth parameter in the ith potential solution, respectively, and rand is [ 0 , 1 ] A random number between .

[0061] During the initialization process, BKA selects the individual with the best fitness as the leader in the initial population. , which is considered the optimal position, taking the minimum value as an example The expression is as follows:

[0062] ,

[0063] ,

[0064] Step 4.2: Set the performance indicator function to evaluate the error size of the system in different time periods, reduce the steady-state error and overshoot of the system, and improve the response speed.

[0065] ,

[0066] ,

[0067] in, is the length of the time window, which can be adjusted dynamically according to the system performance or other external conditions, focusing only on errors within a period of time and ignoring the impact of errors at earlier time points on the current performance, thereby making the control system more agile, responsive to real-time changes, and better adapted to high-demand and fast-changing situations in industrial processes; is the adaptive time weight factor, It is the parameter of the time factor, which is used to dynamically adjust the time weight, so that the weight of the error in different time periods is different; Based on the error Adaptively adjust the weight function, is the error between the system output and the target value, The simulation ends at the time when the control strategy is dynamically adjusted according to the real-time performance of the system, while limiting the time window of integration, paying more attention to the recent performance of the system. This is suitable for actual industrial processes with high real-time requirements and frequent system changes.

[0068] Step 4.3: Select excellent individuals for selection, crossover and mutation operations to produce new offspring.

[0069] Step 4.3.1: Roulette wheel selection of excellent individuals

[0070] The performance index Convert to fitness value :

[0071] ,

[0072] Calculate the selection probability and assign the selection probability to each individual :

[0073] ,

[0074] Calculating Cumulative Probability , divide the roulette wheel into intervals:

[0075] ,

[0076] Select Satisfy , where the random number

[0077] The selected individuals are cross-fertilized to generate offspring.

[0078] Step 4.3.2: Black Kite Migration Optimization

[0079] The mathematical model of black kite migration behavior is introduced to further optimize the population and enhance the global search capability. The population is updated as follows:

[0080] ,

[0081] ,

[0082] in, The leading scorer of Black Winged Kite in the jth dimension at iteration t so far; and Respectively represent the i-th black-winged kite in the j-th dimension and The position in the iteration step, represents the Cauchy mutation, which enhances the global search capability of the algorithm. Is an individual The fitness value of is the fitness value of the best individual in the population.

[0083] Determine whether the conditions for ending the iteration are met, such as reaching the maximum number of iterations or fitness convergence. If the conditions are met, output the optimal parameter combination, otherwise return to update the population and continue iteration.

[0084] The present invention uses the differential geometry method to simplify the model of the affine nonlinear system of the chemical process, combines the black kite algorithm (BKA) and the genetic algorithm (GA) to optimize the parameters, so as to solve the problem of strong coupling between the variables of the multi-input and multi-output multivariable system of the chemical process. Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

[0085] 1. The Lie function and Lie derivative in differential geometry method are used to perform feedback linearization on the affine nonlinear system, which reduces the complexity of the model, improves the response speed of the control system, shortens the production cycle, reduces the transition time, and improves production efficiency.

[0086] 2. By combining the Black Kite algorithm with the genetic algorithm, the global search capability is enhanced, the dependence of parameter adjustment on manual adjustment is reduced, the control accuracy and product quality are improved, and the robustness and adaptability of the system are enhanced. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] Figure 1 It is an overall flow chart of an embodiment of the present invention. DETAILED DESCRIPTION

[0088] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0089] Attached Figure 1 is an overall flow chart of an embodiment of the present invention. This embodiment provides a model simplification and control method for an affine nonlinear system. Taking an ethanol-water binary distillation tower affine nonlinear system as an example, the method specifically includes the following steps: constructing a vector field And analyze whether it satisfies the full rank and involution conditions; obtain the output function ; Accurately linearize the system's state feedback; Blend the Black Kite algorithm and genetic algorithm to optimize the parameters of the simplified model.

[0090] The specific implementation instructions are as follows:

[0091] Step 1: Construct the vector field And analyze whether it satisfies the full rank and involution conditions. In order to simplify the model of the affine nonlinear system, the Lie derivative operator in the differential geometry method is used to construct the vector field, and then it is necessary to analyze whether these vector fields satisfy the involution conditions and full rank conditions, and on this basis to explore the linearization relationship of the model.

[0092] Assume the affine nonlinear standard system ,

[0093] in, x = [ x 1 , x 2 , ⋯ , x n ] T ∈ D ( D ∈ R n ) . It is an open set The vector fields on are called dynamic vector field and control vector field respectively. It is defined in the open set A real-valued function on . The system is called an affine nonlinear system, which is characterized by the input Independent of the control vector field , and the right side of the output equation is the same as the input Not relevant.

[0094] Step 1.1: Construct the vector field

[0095] According to the definition of Lie function in differential geometry, we can calculate ,test , , The linear correlation between two functions is determined, thereby revealing their mutual dependence under specific conditions.

[0096] ,

[0097] ,

[0098] ∂ f ∂ x = [ ∂ f 1 ∂ x 1 ∂ f 1 ∂ x 2 ⋯ ∂ f 1 ∂ x n ∂ f 2 ∂ x 1 ∂ f 2 ∂ x 2 ⋯ ∂ f 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ f n ∂ x 1 ∂ f n ∂ x 2 ⋯ ∂ f n ∂ x n ] , ∂ g ∂ x = [ ∂ g 1 ∂ x 1 ∂ g 1 ∂ x 2 ⋯ ∂ g 1 ∂ x n ∂ g 2 ∂ x 1 ∂ g 2 ∂ x 2 ⋯ ∂ g 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ g n ∂ x 1 ∂ g n ∂ x 2 ⋯ ∂ g n ∂ x n ] ,

[0099] ad f g = ∂ g ∂ x f − ∂ f ∂ x g = [ ∂ g 1 ∂ x 1 ∂ g 1 ∂ x 2 ⋯ ∂ g 1 ∂ x n ∂ g 2 ∂ x 1 ∂ g 2 ∂ x 2 ⋯ ∂ g 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ g n ∂ x 1 ∂ g n ∂ x 2 ⋯ ∂ g n ∂ x n ] [ f 1 f 2 ⋮ f n ] − [ ∂ f 1 ∂ x 1 ∂ f 1 ∂ x 2 ⋯ ∂ f 1 ∂ x n ∂ f 2 ∂ x 1 ∂ f 2 ∂ x 2 ⋯ ∂ f 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ f n ∂ x 1 ∂ f n ∂ x 2 ⋯ ∂ f n ∂ x n ] [ g 1 g 2 ⋮ g n ] ,

[0100] in, is a first-order Lie bracket, is a second-order Lie bracket, calculate ,like , then the system model is linearly dependent.

[0101] Step 1.2: Inspection Whether the involution condition and full rank condition are satisfied. Before linearization, it is necessary to check whether the system satisfies the involution condition and full rank condition. The involution condition ensures that the distribution of the vector field is appropriate, while the full rank condition ensures the controllability of the system state. These conditions are necessary conditions for the system to be able to accurately linearize the state.

[0102] For an affine nonlinear system ,exist The necessary and sufficient conditions for accurate linearization of the state near the point are:

[0103] (1) Matrix [ g ( x 0 ) ad f g ( x 0 ) ⋯ ad f 2 g ( x 0 ) ] The rank is n;

[0104] (2) Distribution D = span [ g ad f g ⋯ ad f n − 2 g ] exist is an involution distribution in a neighborhood of , Lie bracket operation [ τ 1 , τ 2 ] ∈ D .

[0105] If the involution condition and full rank condition are satisfied, the system can be state-exactly linearized, otherwise it cannot.

[0106] Step 2: Get the output function The output function is a function that transforms the system state Function mapped to the output space. The choice of this function determines the form of the system after linearization and is crucial to the linearization of the system.

[0107] Calculate the output function based on the following conditions :

[0108] ∂ h ∂ x [ g ad f g ⋯ ad f n − 2 g ] = 0 ,

[0109] ,

[0110] Compute state transitions z ( x ) = [ h ( x ) L f h ( x ) ⋯ L f n − 1 h ( x ) ] T

[0111] in is the Lie derivative or Lie derivative operator, called the function Along the vector field The Lie derivative of for The order brackets are calculated as follows:

[0112] ,

[0113] ,

[0114] ,

[0115] Constructing state transitions z ( x ) = [ h ( x ) L f h ( x ) ⋯ L f n − 1 h ( x ) ] T , calculate the Jacobian matrix of this transformation:

[0116] ∇ Φ ( x ) = [ ∂ h ( x ) ∂ x 1 ∂ h ( x ) ∂ x 2 ⋯ ∂ h ( x ) ∂ x n ∂ L f h ( x ) ∂ x 1 ∂ L f h ( x ) ∂ x 2 ⋯ ∂ L f h ( x ) ∂ x 3 ⋮ ⋮ ⋱ ⋮ ∂ L f n − 1 h ( x ) ∂ x 1 ∂ L f n − 1 h ( x ) ∂ x 2 ⋯ ∂ L f n − 1 h ( x ) ∂ x n ] ,

[0117] calculate ,like , so the above nonlinear coordinate transformation is a differential homeomorphism.

[0118] Step 3: Accurately linearize the system using state feedback:

[0119] System Output For time Derivative, we have:

[0120] ,

[0121] like ,enter If it cannot appear explicitly, we continue to derive:

[0122] ,

[0123] like , then continue to derive until the input appears on the right side of the equation :

[0124] ,

[0125] When the number of derivatives is , the input appears on the right side of the equation , is the relative order of the system. , then the system can be accurately linearized, and it can be proved that are linearly independent, satisfying condition (1) in step 1.2.

[0126] Through the derivation process, the coordinate transformation formula of the differential homeomorphism can be obtained as follows:

[0127] z = [ z 1 z 2 ⋮ z n ] = Φ ( x ) = [ h ( x ) L f h ( x ) ⋮ L f n − 1 h ( x ) ] ,

[0128] Take the control input as:

[0129] ,

[0130] ,

[0131] ,

[0132] in , .

[0133] The original nonlinear system can be transformed into a controllable linearized system:

[0134] { z ˙ = [ 0 1 0 ⋯ 0 0 0 1 ⋯ 0 0 0 ⋯ 0 1 0 0 ⋯ 0 0 ] z + [ 0 ⋮ 0 1 ] u y = ( 1 0 ⋯ 0 ) z ,

[0135] Right now:

[0136] [ z ˙ 1 z ˙ 2 ⋮ z ˙ n − 1 z ˙ n ] = [ z 2 z 3 ⋮ z n v ] ,

[0137] ,

[0138] Step 4: Combine the Black Kite algorithm and the genetic algorithm to optimize the parameters of the simplified model. Use the Black Kite algorithm to initialize the population, and then apply the crossover and mutation operations of the genetic algorithm to explore the solution space, making full use of the advantages of the two algorithms to improve the efficiency of model parameter optimization.

[0139] Step 4.1: Initialize population parameters,

[0140] BK = [ BK 1 , 1 BK 1 , 2 ⋯ BK 1 , n BK 2 , 1 BK 2 , 2 ⋯ BK 2 , n ⋮ ⋮ ⋮ ⋮ BK m , 1 BK m , 2 ⋯ BK m , n ] ,

[0141] in, is the number of potential solutions, that is, the number of different combinations of operating parameters to be evaluated; n is the number of parameters to be optimized given the size of the problem dimension, It is a combination of parameters, and the parameter initialization rules are:

[0142] ,

[0143] in, is the value of the jth parameter in the ith potential solution, i is [ 1 , m ] Integers between and are the upper and lower bounds of the jth parameter in the ith potential solution, respectively, and rand is [ 0 , 1 ] A random number between .

[0144] During the initialization process, BKA selects the individual with the best fitness as the leader in the initial population. , which is considered the optimal position, taking the minimum value as an example The expression is as follows:

[0145] ,

[0146] ,

[0147] Step 4.2: Set the performance indicator function to evaluate the error size of the system in different time periods, reduce the steady-state error and overshoot of the system, and improve the response speed.

[0148] ,

[0149] ,

[0150] in, is the length of the time window, which can be adjusted dynamically according to the system performance or other external conditions, focusing only on errors within a period of time and ignoring the impact of errors at earlier time points on the current performance, thereby making the control system more agile, responsive to real-time changes, and better adapted to high-demand and fast-changing situations in industrial processes; is the adaptive time weight factor, It is the parameter of the time factor, which is used to dynamically adjust the time weight, so that the weight of the error in different time periods is different; Based on the error Adaptively adjust the weight function, is the error between the system output and the target value, The simulation ends at the time when the control strategy is dynamically adjusted according to the real-time performance of the system, while limiting the time window of integration, paying more attention to the recent performance of the system. This is suitable for actual industrial processes with high real-time requirements and frequent system changes.

[0151] Step 4.3: Select excellent individuals for selection, crossover and mutation operations to produce new offspring.

[0152] Step 4.3.1: Roulette wheel selection of excellent individuals

[0153] The performance index Convert to fitness value :

[0154] ,

[0155] Calculate the selection probability and assign the selection probability to each individual :

[0156] ,

[0157] Calculating Cumulative Probability , divide the roulette wheel into intervals:

[0158] ,

[0159] Select Satisfy , where the random number

[0160] The selected individuals are cross-fertilized to generate offspring.

[0161] Step 4.3.2: Black Kite Migration Optimization

[0162] The mathematical model of black kite migration behavior is introduced to further optimize the population and enhance the global search capability. The population is updated as follows:

[0163] ,

[0164] ,

[0165] in, The leading scorer of Black Winged Kite in the jth dimension at iteration t so far; and Respectively represent the i-th black-winged kite in the j-th dimension and The position in the iteration step, represents the Cauchy mutation, which enhances the global search capability of the algorithm. Is an individual The fitness value of is the fitness value of the best individual in the population.

[0166] Determine whether the conditions for ending the iteration are met, such as reaching the maximum number of iterations or fitness convergence. If the conditions are met, output the optimal parameter combination, otherwise return to update the population and continue iteration.

Claims

1. A model simplification and control method for a nonlinear system, characterized in that: The following steps are involved: Step 1: Construct the vector field And analyze whether it satisfies the full rank and involution conditions; Step 2: Get the output function ; Specific implementation steps: Get the output function , the output function is a system state The function mapped to the output space. The choice of this function determines the form of the system after linearization and is crucial to the linearization of the system. ， ， in, is the output function, For the function Each variable in Find the partial derivative, Control vector, is a first-order Lie bracket, is a second-order Lie bracket, for brackets, for brackets; Constructing state variables ,in The Lie derivative or Lie derivative operator is called the function Along the dynamic vector field The Lie derivative of for brackets, is a 2nd-order Lie bracket, for The order bracket is calculated as follows: ， ， ， State Transition , calculate the Jacobian matrix of this transformation: ， calculate ,like , so the nonlinear coordinate transformation is a diffeomorphic mapping; Step 3: Accurately linearize the system state feedback; specifically, follow the following steps: System Output For time Derivative, we have: ， like ,enter If it cannot appear explicitly, we continue to derive: ， like , then continue to derive until the input appears on the right side of the equation : ， When the number of derivatives is , the input appears on the right side of the equation , is the relative order of the system. , then the system can be accurately linearized, and it can be proved that is linearly independent and satisfies condition (1) in step 1.2, where is the system dimension, is a dynamic vector field, is the input variable, For the control vector Lie derivative or Lie derivative operator, is the number of system state variables, For along The rate of change of direction is Direction to find the rate of change; Through the derivation process, the coordinate transformation formula of the differential homeomorphism can be obtained as follows: ， Take the control input as: ， ， ， in , , the original nonlinear system can be transformed into a controllable linearized system: ， Right now: ， ； Step 4: Combine the Black Kite algorithm and the genetic algorithm to optimize the parameters of the simplified model.

2. The model simplification and control method of a nonlinear system according to claim 1, characterized in that: The fusion black kite algorithm and genetic algorithm described in step 4 are used to optimize the parameters of the simplified model, which is specifically implemented in the following steps: Step 4: Combine the Black Kite algorithm and the genetic algorithm to optimize the parameters of the simplified model. Use the Black Kite algorithm to initialize the population, and then apply the crossover and mutation operations of the genetic algorithm to explore the solution space, making full use of the advantages of the two algorithms to improve the efficiency of model parameter optimization: Step 4.1: Initialize population parameters: ， in, is the number of potential solutions, that is, the number of different combinations of operating parameters to be evaluated; n is the number of parameters to be optimized given the size of the problem dimension, It is a combination of parameters, and the parameter initialization rules are: ， in, is the value of the jth parameter in the ith potential solution, i is Integers between and are the upper and lower bounds of the jth parameter in the ith potential solution, respectively, and rand is A random number between During the initialization process, BKA selects the individual with the best fitness as the leader in the initial population. , which is considered the optimal position, taking the minimum value as an example The expression is as follows: ， ， Step 4.2: Set the performance index function to evaluate the error size of the system in different time periods, reduce the steady-state error and overshoot of the system, and improve the response speed: ， ， in, is the length of the time window, which can be adjusted dynamically according to the system performance or other external conditions, focusing only on errors within a period of time and ignoring the impact of errors at earlier time points on the current performance, thus making the control system more agile, responsive to real-time changes, and better adapted to high-demand and fast-changing situations in industrial processes; is the adaptive time weight factor, It is the parameter of the time factor, which is used to dynamically adjust the time weight, so that the weight of the error in different time periods is different; Based on the error Adaptively adjust the weight function, is the error between the system output and the target value, To end the simulation, the control strategy is dynamically adjusted according to the real-time performance of the system, while limiting the time window of integration, paying more attention to the recent performance of the system. It is suitable for actual industrial processes with high real-time requirements and frequent system changes; Step 4.3: Select excellent individuals for selection, crossover and mutation operations to produce new offspring: Step 4.3.1: Roulette wheel selection of excellent individuals: The performance index Convert to fitness value : ， in, Is an individual The fitness value of Calculate the selection probability and assign the selection probability to each individual : ， Calculating Cumulative Probability , divide the roulette wheel into intervals: ， Select Satisfy , where the random number Crossover the selected individuals to generate offspring; Step 4.3.2: Black Kite Migration Optimization The mathematical model of black kite migration behavior is introduced to further optimize the population and enhance the global search capability. The population is updated as follows: ， ， in, The leading scorer of Black Winged Kite in the jth dimension at iteration t so far; and Respectively represent the i-th black-winged kite in the j-th dimension and The position in the iteration step, represents the Cauchy mutation, which enhances the global search capability of the algorithm. Is an individual The fitness value of is the fitness value of the best individual in the population; Determine whether the conditions for ending the iteration are met, such as reaching the maximum number of iterations or fitness convergence; if the conditions are met, output the optimal parameter combination, otherwise return to update the population and continue iteration.

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