阅读说明：本技术 一种双惯量伺服系统的自适应齿隙振荡抑制方法 (Self-adaptive backlash oscillation suppression method for double-inertia servo system ) 是由 尹忠刚 李通 白聪 罗培恩 于 2020-12-24 设计创作，主要内容包括：本发明公开了一种双惯量伺服系统的自适应齿隙振荡抑制方法,具体包括如下步骤：步骤1,建立永磁同步电机数学模型,得到电磁转矩表达式；步骤2,以转速和位置为状态变量建立双惯量伺服传动系统动力学数学模型；步骤3,转速环采用滑模控制方法,基于步骤2得到的双惯量系统动力学的数学模型和步骤1电磁转矩表达式,设计滑模控制器的滑模控制率；步骤4,建立可微非对称死区模型,采用差分进化算法对可微非对称死区模型参数进行辨识,所识别出的参数即为步骤3中滑模控制率的未知不可微齿隙轴转矩扰动项参数。本发明解决了现有双惯量伺服系统中由于齿隙引起振荡的问题。(The invention discloses a self-adaptive backlash oscillation suppression method for a double-inertia servo system, which specifically comprises the following steps: step 1, establishing a mathematical model of a permanent magnet synchronous motor to obtain an electromagnetic torque expression; step 2, establishing a dynamic mathematical model of the double-inertia servo transmission system by taking the rotating speed and the position as state variables; step 3, designing the sliding mode control rate of the sliding mode controller by adopting a sliding mode control method for the rotating speed ring based on the mathematical model of the double-inertia system dynamics obtained in the step 2 and the electromagnetic torque expression in the step 1; and 4, establishing a micro-asymmetric dead zone model, and identifying parameters of the micro-asymmetric dead zone model by adopting a differential evolution algorithm, wherein the identified parameters are unknown non-micro backlash shaft torque disturbance term parameters of the sliding mode control rate in the step 3. The invention solves the problem of oscillation caused by backlash in the conventional double-inertia servo system.)

1. A self-adaptive backlash oscillation suppression method of a dual-inertia servo system is characterized by comprising the following steps: the method specifically comprises the following steps:

step 1, establishing a mathematical model of a permanent magnet synchronous motor under a d-q coordinate system to obtain an electromagnetic torque expression;

step 2, establishing a dynamic mathematical model of the double-inertia servo transmission system by taking the rotating speed and the position as state variables;

step 3, designing the sliding mode control rate of the sliding mode controller by adopting a sliding mode control method for the rotating speed ring based on the mathematical model of the double-inertia system dynamics obtained in the step 2 and the electromagnetic torque expression in the step 1;

step 4, establishing a micro-asymmetric dead zone model, and adopting a differential evolution algorithm to carry out parameter k on the micro-asymmetric dead zone model_r、k_l、α_r、α_lAnd lambda is identified, and the identified parameters are unknown non-microminiature backlash shaft torque disturbance term parameters of the sliding mode control rate in the step 3.

2. The adaptive backlash oscillation suppression method for a dual inertia servo system according to claim 1, wherein: the specific process of the step 1 is as follows:

in d-q coordinatesSystem of electromagnetic torque T_eThe expression is shown in the following formula (1):

T_e＝1.5n_p(ψ_di_q-ψ_qi_d)＝1.5n_p[ψ_fi_q+(L_d-L_q)i_di_q] (1)；

wherein i_dIs the d-axis stator current component; i.e. i_qIs the q-axis stator current component; l is_dIs the d-axis stator inductance component; l is_qA stator inductance component that is the q-axis; psi_fIs a rotor permanent magnet flux linkage; n is_pThe number of pole pairs of the permanent magnet synchronous motor is; psi_dIs a d-axis flux linkage; psi_qIs a q-axis flux linkage.

3. The adaptive backlash oscillation suppression method for a dual inertia servo system according to claim 2, wherein: the mathematical model of the dynamics of the double-inertia servo transmission system in the step 2 is shown in the following formula (2):

wherein, J_mIs the driving side inertia; b is_mDrive side friction damping; j. the design is a square_lIs the driven side inertia; b is_lIs driven side friction damping; k_sThe elastic coefficient of the transmission shaft; b is_sFriction damping is adopted; omega_mA drive side rotational speed; omega_lThe rotation speed of the driven side; theta_mIs a driving side position; theta_lIs at the driven side position;is the drive side position differential;is the driven side position differential; t is_s[θ]Is the shaft torque; t is_lIs the load torque.

4. The adaptive backlash oscillation suppression method for a dual inertia servo system according to claim 3, wherein: the sliding mode control rate in the step 3 is shown in the following formula (3):

wherein the content of the first and second substances,s is a slip form surface; c is a constant greater than 0; epsilon is the sliding mode gain; k is a constant greater than 0;giving a differential for the rotation speed;setting for q-axis current;feedback for the driving side rotating speed; sign () is a function of the sign,is the differential of the shaft torque.

5. The adaptive backlash oscillation suppression method for a dual inertia servo system according to claim 4, wherein: the specific process of the step 4 is as follows:

step 4.1, establishing an asymmetric differentiable dead zone model shown in the following formula (4):

wherein θ is θ_m-θ_l(ii) a λ is a softness coefficient greater than 0; k is a radical of_rIs the slope of the dead zone of the positive half shaft；k_lIs the negative half-axis dead-zone slope; alpha is alpha_rIs a positive half shaft dead zone angle; alpha is alpha_lIs a negative half-shaft dead zone angle;

step 4.2, adopting a differential evolution algorithm to carry out the asymmetric microminiature area model parameter k_r、k_l、α_r、α_lAnd identifying the lambda parameter, specifically:

firstly, establishing a population and initializing the population, assuming NP is the size of the population scale, selecting n-dimensional vector parameters x as the number of variables to be identified_ij，i＝1,2,…,NP；j＝1,2,…,n，X_i＝[x_i1,x_i2,…,x_in]To obtain x_i,jThe expression of (a) is as follows:

x_i,j＝x_i,jmin+rand(0,1)*(x_i,jmax-x_i,jmin) (5)；

wherein x is_i,jmaxAnd x_i,jminRespectively an upper limit value and a lower limit value of the individual vector; x_iRepresents the ith "chromosome" or individual of the 0 th generation in the population; x is the number of_i,jThe ith "chromosome" representing the 0 th generation in the population or the jth "gene" of the individual; rand (0,1) represents random numbers uniformly distributed in the (0,1) interval;

4.3, generating a difference vector, and carrying out mutation operation to obtain a variant individualAs shown in the following equation (6):

wherein r1, r2, r3 are E {1,2, …, NP }, and r1, r2, r3 and i cannot be the same; f is a variation factor, and F belongs to [0,1 ]]；A j-th "genetic" variant individual representing the i-th "chromosome" of the k + 1-th generation of the variant individual;a j-th "gene" individual representing the r1 th "chromosome" in the k-th generation of individuals;a j-th "gene" individual representing the r2 th "chromosome" in the k-th generation of individuals;a j-th "gene" individual representing the r3 th "chromosome" in the k-th generation of individuals;

step 4.4, performing cross operation to obtain a test vectorThe specific expression of the crossover operation is as follows:

wherein eta is_jIs any number greater than 0 and less than 1; c_RIs a cross factor, C_RThe value range is [0,1 ]]；q_jE {1,2 …, n };a j-th "gene" test individual representing the i-th "chromosome" of the k + 1-th generation of the test individual;

in step 4.5, a selection operation is performed to generate k +1 generation individualsAs shown in the following equation (8):

where f is the objective function as follows:

wherein e is₁(t) is the position error at time t; t is_{posi_r-index}Is a position rise time index; t is_{speed_r-index}Is a rotating speed rising time index; delta N_indexIs a rotating speed error index; t is_re-indexThe rotating speed recovery time index after loading; t is_{posi_r}System position rise time; t is_{speed_r}The system rotation speed rise time; delta N is the system rotation speed error; t is_reThe system rotating speed recovery time after loading;

substituting the selected individual, namely the identified unknown parameter into the asymmetric microminiature area model to obtain the system corresponding position error e₁(T), rotation error Δ N, position rise time T_{posi_r}System speed rise time T_{speed_r}The responses are brought into an objective function f, the objective function f is minimized after a plurality of iterations, the individual at the moment is the optimal individual, and the identified parameter k_r、k_l、α_r、α_lAnd lambda is the unknown non-microminiature backlash shaft torque disturbance term parameter of the sliding mode control rate in the step 3.

Technical Field

The invention belongs to the technical field of high-precision alternating current servo control systems, and relates to a self-adaptive backlash oscillation suppression method for a double-inertia servo system.

Background

With the continuous development of servo driving technology, high performance servo systems have been widely applied in various fields, and related research and development have also received much attention. The servo system is firstly used for military and national defense, such as radar tracking and positioning, automatic aiming, missile launching and the like, is widely applied to industry later, and is suitable for numerical control machines, paper machines, mechanical arms, milling machines and the like. Direct drive systems with high stiffness drive components are used primarily for high performance industrial applications.

The servo drive control technology plays a significant role in intelligent manufacturing and industrial digitization. However, in actual conditions, some non-linear phenomena, such as non-linear friction, backlash and external interference, are common to mechanical systems. Among these non-linearities, the presence of backlash in the transmission significantly reduces the performance of the servo system. In order to ensure the normal operation of the gear transmission system, a certain gap is reserved on the premise of gear meshing, the larger the size of the gear is, the larger the corresponding gap is, and the transmission system is generally in multi-stage transmission, and the existence of the gap can make the mechanical resonance more severe and even make the control system divergent. When a part is processed by a full-digital numerical control machine tool, the processed surface is rough due to oscillation of gear transmission, and the estimated economic loss is caused; and in an industrial robot, when the mechanical arm is positioned, the tail end of the mechanical arm generates a buffeting phenomenon, so that stable stop and accurate stop cannot be realized, the adjusting time of the system is prolonged, the working efficiency is reduced, and some mechanical arms can even disperse the system to cause casualties. The sliding mode variable structure control is characterized in that the structure of the system is not fixed, and dynamic adjustment is performed according to the current state of the system, so that the system moves according to a preset sliding mode. In addition, the existence of the non-linearity of the backlash makes it difficult for the controller to accurately model unknown items of the system, and the defect directly influences the control performance of the dual-inertia servo system.

Disclosure of Invention

The invention aims to provide a self-adaptive backlash oscillation suppression method for a double-inertia servo system, which solves the problem of oscillation caused by backlash in the conventional double-inertia servo system.

The technical scheme adopted by the invention is that the self-adaptive backlash oscillation suppression method of the double-inertia servo system specifically comprises the following steps:

step 1, establishing a mathematical model of a permanent magnet synchronous motor under a d-q coordinate system to obtain an electromagnetic torque expression;

step 2, establishing a dynamic mathematical model of the double-inertia servo transmission system by taking the rotating speed and the position as state variables;

step 3, designing the sliding mode control rate of the sliding mode controller by adopting a sliding mode control method for the rotating speed ring based on the mathematical model of the double-inertia system dynamics obtained in the step 2 and the electromagnetic torque expression in the step 1;

step 4, establishing a micro-asymmetric dead zone model, and adopting a differential evolution algorithm to carry out parameter k on the micro-asymmetric dead zone model_r、k_l、α_r、α_lAnd lambda is identified, and the identified parameters are unknown non-microminiature backlash shaft torque disturbance term parameters of the sliding mode control rate in the step 3.

The present invention is also characterized in that,

the specific process of the step 1 is as follows:

electromagnetic torque T under d-q coordinate system_eThe expression is shown in the following formula (1):

T_e＝1.5n_p(ψ_di_q-ψ_qi_d)＝1.5n_p[ψ_fi_q+(L_d-L_q)i_di_q] (1)；

wherein i_dIs the d-axis stator current component; i.e. i_qIs the q-axis stator current component; l is_dIs the d-axis stator inductance component; l is_qA stator inductance component that is the q-axis; psi_fIs a rotor permanent magnet flux linkage; n is_pThe number of pole pairs of the permanent magnet synchronous motor is; psi_dIs a d-axis flux linkage; psi_qIs a q-axis flux linkage.

In the step 2, a dynamic mathematical model of the double-inertia servo transmission system is shown in the following formula (2):

wherein, J_mIs the driving side inertia; b is_mDrive side friction damping; j. the design is a square_lIs the driven side inertia; b is_lIs driven side friction damping; k_sThe elastic coefficient of the transmission shaft; b is_sFriction damping is adopted; omega_mA drive side rotational speed; omega_lThe rotation speed of the driven side; theta_mIs a driving side position; theta_lIs at the driven side position;is the drive side position differential;is the driven side position differential; t is_s[θ]Is the shaft torque; t is_lIs the load torque.

The sliding mode control rate in step 3 is shown in the following formula (3):

wherein the content of the first and second substances,s is a slip form surface; c is a constant greater than 0; epsilon is the sliding mode gain; k is a constant greater than 0;giving a differential for the rotation speed;setting for q-axis current;feedback for the driving side rotating speed; sign () is a function of the sign,is the differential of the shaft torque.

The specific process of the step 4 is as follows:

step 4.1, establishing an asymmetric differentiable dead zone model shown in the following formula (4):

wherein θ is θ_m-θ_l(ii) a λ is a softness coefficient greater than 0; k is a radical of_rIs the slope of the dead zone of the positive half shaft; k is a radical of_lIs the negative half-axis dead-zone slope; alpha is alpha_rIs a positive half shaft dead zone angle; alpha is alpha_lIs a negative half-shaft dead zone angle;

step 4.2, adopting a differential evolution algorithm to carry out the asymmetric microminiature area model parameter k_r、k_l、α_r、α_lAnd identifying the lambda parameter, specifically:

firstly, establishing a population and initializing the population, assuming NP is the size of the population scale, selecting n-dimensional vector parameters x as the number of variables to be identified_ij，i＝1,2,…,NP；j＝1,2,…,n，X_i＝[x_i1,x_i2,…,x_in]To obtain x_i,jThe expression of (a) is as follows:

x_i,j＝x_i,jmin+rand(0,1)*(x_i,jmax-x_i,jmin) (5)；

wherein x is_i,jmaxAnd x_i,jminRespectively an upper limit value and a lower limit value of the individual vector; x_iRepresents the ith "chromosome" or individual of the 0 th generation in the population; x is the number of_i,jThe ith "chromosome" representing the 0 th generation in the population or the jth "gene" of the individual; rand (0,1) represents random numbers uniformly distributed in the (0,1) interval;

4.3, generating a difference vector, and carrying out mutation operation to obtain a variant individualAs shown in the following equation (6):

wherein r1, r2, r3 are E {1,2, …, NP }, and r1, r2, r3 and i cannot be the same; f is a variation factor, and F belongs to [0,1 ]]；A j-th "genetic" variant individual representing the i-th "chromosome" of the k + 1-th generation of the variant individual;a j-th "gene" individual representing the r1 th "chromosome" in the k-th generation of individuals;a j-th "gene" individual representing the r2 th "chromosome" in the k-th generation of individuals;a j-th "gene" individual representing the r3 th "chromosome" in the k-th generation of individuals;

step 4.4, performing cross operation to obtain a test vectorThe specific expression of the crossover operation is as follows:

wherein eta is an arbitrary number greater than 0 and less than 1; c_RIs a cross factor, C_RThe value range is [0,1 ]]；q_jE {1,2 …, n }, as a random integer；A j-th "gene" test individual representing the i-th "chromosome" of the k + 1-th generation of the test individual;

in step 4.5, a selection operation is performed to generate k +1 generation individualsAs shown in the following equation (8):

where f is the objective function as follows:

wherein e is₁(t) is the position error at time t; t is_{posi_r-index}Is a position rise time index; t is_{speed_r-index}Is a rotating speed rising time index; delta N_indexIs a rotating speed error index; t is_re-indexThe rotating speed recovery time index after loading; t is_{posi_r}System position rise time; t is_{speed_r}The system rotation speed rise time; delta N is the system rotation speed error; t is_reThe system rotating speed recovery time after loading;

substituting the selected individual, namely the identified unknown parameter into the asymmetric microminiature area model to obtain the system corresponding position error e₁(T), rotation error Δ N, position rise time T_{posi_r}System speed rise time T_{speed_r}The responses are brought into an objective function f, the objective function f is minimized after a plurality of iterations, the individual at the moment is the optimal individual, and the identified parameter k_r、k_l、α_r、α_lAnd lambda is the unknown non-microminiature backlash shaft torque disturbance term parameter of the sliding mode control rate in the step 3.

The self-adaptive backlash oscillation suppression method for the double-inertia servo system has the beneficial effects that the self-adaptive backlash oscillation suppression method for the double-inertia servo system solves the problems of mechanical oscillation and impact oscillation caused by backlash; even when the backlash changes, the backlash model can be accurately identified, and shaft torque disturbance generated by the backlash is compensated in the controller in time, so that backlash oscillation is eliminated, and the stability, robustness and positioning accuracy of the system are improved.

Drawings

FIG. 1 is a block flow diagram of an adaptive backlash oscillation suppression method for a dual-inertia servo system according to the present invention;

FIG. 2 is a block flow diagram of an adaptive backlash oscillation suppression method for a dual-inertia servo system according to the present invention;

Detailed Description

The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

The invention discloses a self-adaptive backlash oscillation suppression method for a double-inertia servo system, and as shown in figure 1, a double-inertia servo control system with backlash comprises three-loop control of a position loop, a speed loop and a current loop. Phase current i_a、i_b、i_cObtaining a stator current component i under a two-phase static coordinate system through Clarke transformation after being measured by a current sensor_α、i_βThen obtaining a stator current component i under a two-phase rotating coordinate system through Park conversion_d、i_q. Position pulse given by theta^*Given by the upper computer, θ^*Position theta obtained from the driven encoder side_lThe difference is input to a position regulator, the output of which is used as the speed settingDrive side rotor speed ω_mMeasured by incremental encoders, omega_mWith speed givenThe difference is input into a sliding mode controller, and the output of the sliding mode controller is used as a quadrature axis current instruction value i_q ^*Direct axis current command value i_d ^*＝0，i_d、i_qAre respectively connected with i_d ^*And i_q ^*After making difference, output u of current regulator_d、u_qThen outputting u through Park inverse transformation_α、u_βFinally, six paths of PWM signals are output through the space vector pulse width modulation module to be supplied to an inverter to work, and the inverter enables the DC bus voltage U_dcThe PWM wave is applied to the permanent magnet synchronous motor to realize the positioning of the driven side.

The method specifically comprises the following steps:

step 1, establishing a mathematical model of a permanent magnet synchronous motor under a d-q coordinate system to obtain an electromagnetic torque expression; the specific process of the step 1 is as follows:

in a d-q coordinate system, a mathematical model of the permanent magnet synchronous motor is as follows:

wherein u is_d、u_qD-axis component of the stator voltage and q-axis component of the stator voltage respectively; psi_d、ψ_qRespectively a d-axis component of the stator flux linkage and a q-axis component of the stator flux linkage; i.e. i_dIs a d-axis stator current component, i_qIs the q-axis stator current component; r_sIs a stator resistor; theta_mIs the drive side position.

The flux linkage equation is:

the input power P of the permanent magnet synchronous motor is as follows:

the electromagnetic torque obtained by bringing the formula (1) and the formula (2) into the formula (3) does work as follows:

and (4) bringing the formula (2) into the formula (4), wherein the electromagnetic torque expression is as follows:

T_e＝1.5n_p[ψ_fi_q+(L_d-L_q)i_di_q] (5)；

wherein i_dIs the d-axis stator current component; i.e. i_qIs the q-axis stator current component; l is_dIs the d-axis stator inductance component; l is_qA stator inductance component of q; Ψ_fIs a rotor permanent magnet flux linkage; n is_pThe number of pole pairs of the permanent magnet synchronous motor is.

Step 2, establishing a dynamic mathematical model of the double-inertia servo transmission system by taking the rotating speed and the position as state variables;

the mathematical model of the dynamics of the double-inertia servo transmission system in the step 2 is shown in the following formula (6):

wherein, J_mIs the driving side inertia; b is_mDrive side friction damping; j. the design is a square_lIs the driven side inertia; b is_lIs driven side friction damping; k_sThe elastic coefficient of the transmission shaft; b is_sFriction damping is adopted; omega_mA drive side rotational speed; omega_lThe rotation speed of the driven side; theta_mIs a driving side position; theta_lIs at the driven side position;is the drive side position differential;is the driven side position differential; t is_s[θ]Is the shaft torque; t is_lIs the load torque.

Step 3, designing the sliding mode control rate of the sliding mode controller by adopting a sliding mode control method for the rotating speed ring based on the mathematical model of the double-inertia system dynamics obtained in the step 2 and the electromagnetic torque expression in the step 1;

establishing a slip form surface s is:

wherein the content of the first and second substances, setting the rotating speed of the driving side; c is a constant greater than 0.

The sliding mode controller adopts an exponential approximation rule as follows:

wherein epsilon is the sliding mode gain; k is a constant greater than 0; sign () is a sign function.

Neglecting the friction coefficient, the sliding mode controller output expression obtained by taking equation (7) into equation (8) to derive the combined equation (6) is as follows:

wherein the content of the first and second substances,s is a slip form surface; c is a constant greater than 0; epsilon is the sliding mode gain; k is a constant greater than 0;giving a differential for the rotation speed;setting for q-axis current;feedback for the driving side rotating speed; sign () is a function of the sign,is the differential of the shaft torque.

And 4, establishing an equivalent microminiature asymmetric dead zone model for the unknown non-microminiature backlash shaft torque disturbance term of the sliding mode control rate in the step 3, and accurately compensating unknown disturbance generated by backlash in a controller. Since the dead zone parameters of the system are unknown parts, model parameters need to be optimized. According to the obtained equivalent micro-asymmetric dead zone model, a differential evolution algorithm is adopted to carry out on the parameter k of the equivalent micro-asymmetric dead zone model_r、k_l、α_r、α_lAnd lambda is identified, and the equivalent asymmetric microminiature area model parameters are repeatedly optimized through a differential evolution algorithm according to the real-time response of the servo system, so that the objective function value is smaller and smaller until the static and dynamic performance of the system meets the set index. After finite algebra is optimized, the system finally reaches the index requirement, and k at the moment_r、k_l、α_r、α_lλ is the most suitable equivalent asymmetric differentiable zone model parameter, and the process is shown in fig. 2.

In step 4.1, aiming at the unknown ultramicro backlash shaft torque disturbance term of the sliding mode control rate in step 3, the invention provides an asymmetric microminiature area model, and because the parameter of the asymmetric microminiature area model is unknown, the invention adopts a differential evolution algorithm to carry out differential evolution on the parameter k of the proposed asymmetric microminiature area model_r、k_l、α_r、α_lAnd lambda is identified, and the asymmetric differentiable dead zone model has the following formula:

wherein θ is θ_m-θ_l(ii) a Lambda is the softness coefficient more than 0; k is a radical of_rIs the slope of the dead zone of the positive half shaft; k is a radical of_lIs the negative half-axis dead-zone slope; alpha is alpha_rIs a positive half shaft dead zone angle; alpha is alpha_lIs a negative half-axis dead-zone angle.

In step 4.2, the differential evolution algorithm is adopted to carry out asymmetric micro-deathRegion model parameter k_r、k_l、α_r、α_lFirstly establishing a population and initializing the population, supposing NP is the size of the population scale, selecting n-dimensional vector parameter x as the number of variables to be identified_ij(i＝1,2,…,NP；j＝1,2,…,n)，X_i＝[x_i1,x_i2,…,x_in]Then x can be calculated_i,jThe expression of (a) is as follows:

x_i,j＝x_i,jmin+rand(0,1)*(x_i,jmax-x_i,jmin) (11)；

wherein x is_i,jmaxAnd x_i,jminRespectively an upper limit value and a lower limit value of the individual vector; x_iThe ith "chromosome" (or individual) representing the 0 th generation in the population; x is the number of_i,jThe jth "gene" representing the ith "chromosome" (or individual) of the 0 th generation in the population; rand (0,1) represents random numbers uniformly distributed in the (0,1) interval.

In step 4.3, generating difference vector, performing mutation operation to obtain variant individualAs follows:

wherein r1, r2, r3 belongs to {1,2, …, NP }, and can not be the same as i; f is a variation factor, and F belongs to [0,1 ]]；A j-th "genetic" variant individual representing the i-th "chromosome" of the k + 1-th generation of the variant individual;a j-th "gene" individual representing the r1 th "chromosome" in the k-th generation of individuals;in representation of an individualThe j th "gene" individual of the r2 th "chromosome" of the k generation;represents the j 'gene' individual of r3 th chromosome of the k generation of individuals.

In step 4.4, cross operation is performed to obtain test vectorsThe specific expression of the crossover operation is as follows:

wherein eta is_jIs any number greater than 0 and less than 1; c_RIs a cross factor and has a value range of [0,1 ]]；q_jE {1,2 …, n };the j-th "gene" test individual representing the i-th "chromosome" of the k + 1-th generation in the test individual.

In step 4.5, a selection operation is performed to generate k +1 generation individualsAs follows:

where f is the objective function as follows:

wherein e is₁(t) is the position error at time t; t is_{posi_r-index}Is a position rise time index; t is_{speed_r-index}Is a rotating speed rising time index; delta N_indexIs error in rotation speedA difference index; t is_re-indexThe rotating speed recovery time index after loading; t is_{posi_r}System position rise time; t is_{speed_r}The system rotation speed rise time; delta N is the system rotation speed error; t is_reThe system rotating speed recovery time after loading.

Substituting the selected individual, namely the identified unknown parameter into the asymmetric microminiature area model to obtain the system corresponding position error e₁(T), rotation error Δ N, position rise time T_{posi_r}System speed rise time T_{speed_r}When the responses are brought into the objective function f, the objective function f is minimized after multiple iterations, the closer the model at the moment is to the system model, the more the individual is the most individual, namely k_r、k_l、α_r、α_lLambda optimum parameter.