CN110829921A - An Iterative Feedback Tuning Control of Permanent Magnet Synchronous Motor and Its Optimization Method - Google Patents

Abstract

The invention discloses an iterative feedback tuning control of a permanent magnet synchronous motor and an optimization method thereof, and relates to the field of servo control optimization. The closed-loop subspace identification method based on the proposed method, design the position loop double-loop iterative feedback tuning PI controller, and further analyze the convergence and convergence speed of the double-loop iterative feedback tuning algorithm, so as to effectively improve the positioning performance of the permanent magnet synchronous motor servo system. On the basis of and robustness, it satisfies the demand for the rapidity of the servo system.

Description

An Iterative Feedback Tuning Control of Permanent Magnet Synchronous Motor and Its Optimization Method

technical field

The invention relates to an iterative feedback tuning control of a permanent magnet synchronous motor and an optimization method thereof, belonging to the field of servo control optimization.

Background technique

Permanent magnet synchronous motor is a synchronous motor that generates a synchronous rotating magnetic field by permanent magnet excitation. In the fields of numerical control equipment, industrial robots and laser engraving, permanent magnetic synchronous Lightweight and other characteristics, it is widely used in position servo systems with high-speed, accurate tracking and positioning as the main goal. Aiming at the typical "three closed-loop" structure of the permanent magnet synchronous motor position servo system, scholars at home and abroad have done a lot of work in optimizing the traditional PID controller scheme in recent years. Among them, methods such as genetic algorithm, fuzzy control and neural network can effectively improve the positioning performance and robustness of permanent magnet synchronous motor servo system, but these methods may be difficult to meet the rapidity requirements of the system, or require sufficient modeling accuracy , all restrict the practical application of permanent magnet synchronous motor in industry to a certain extent.

The inherent nonlinearity and uncertainties of PMSM make it difficult to carry out accurate mathematical modeling. Model-based methods usually have high sensitivity to the accuracy of modeling. When searching for the controller parameters corresponding to the minimized performance criterion function, the traditional IFT (Iterative Feedback Tuning) algorithm usually uses an optimization algorithm like the Gauss-Newton method for parameter optimization. The required three experiments and the characteristics of linear approximation fundamentally limit the convergence speed of the algorithm, and it is difficult to meet the high-speed characteristics required by precision industries such as permanent magnet synchronous motor position loop control.

SUMMARY OF THE INVENTION

Aiming at the difficulty of traditional IFT algorithm to meet the rapidity requirement of parameter optimization of PMSM position loop PI controller, this application proposes an iterative feedback tuning control and optimization method for PMSM, which integrates a new IFT framework , the double-cycle IFT algorithm is applied to the parameter setting and control performance optimization of the position loop PI controller, in order to improve the fast and accurate tracking ability of the permanent magnet synchronous motor.

The technical scheme of the present invention is as follows:

An iterative feedback tuning control of a permanent magnet synchronous motor and an optimization method thereof include the following steps:

The first step is to construct the kinematic equation and vector control system of the permanent magnet synchronous motor;

The second step is to design the position loop PI controller of the permanent magnet synchronous motor servo system based on the double-loop iterative feedback tuning algorithm;

The third step is to analyze the convergence and convergence speed of the double-loop iterative feedback tuning algorithm;

The fourth step, the specific implementation of the dual-loop iterative feedback tuning control scheme of the position loop is given;

The first step is to construct the kinematic equation and vector control system of the permanent magnet synchronous motor:

Under the premise of ignoring harmonics, hysteresis and eddy current losses, the permanent magnet synchronous motor stator voltage equation in the synchronous rotating coordinate system d-q is shown in equation (1):

where _ud and u _q are the dq-axis components of the stator voltage vector, respectively, id and i _q are the _dq -axis components of the stator current vector, R is the stator resistance, and ψ _d and ψ _q are the dq of the stator flux vector, respectively. The shaft component, ω _c is the electrical angular velocity; the stator flux linkage equation is further obtained as:

where L _d and L _q are the dq-axis inductance components, respectively, and ψ _f represents the permanent magnet flux linkage; Equation (2) is substituted into Equation (1) to obtain the stator voltage equation:

According to the principle of electromechanical conversion, the electromagnetic torque T is the partial derivation of the magnetic field energy storage to the mechanical angular displacement, then the electromagnetic torque equation of the permanent magnet synchronous motor is:

where p _n is the pole pair number of the permanent magnet synchronous motor; the kinematic equation of the permanent magnet synchronous motor is further obtained as:

where ω _m is the mechanical angular velocity of the permanent magnet synchronous motor, J is the moment of inertia, B is the damping coefficient, and T _L is the load torque;

The vector control system of the permanent magnet synchronous motor is a "three closed-loop" structure composed of a position loop, a speed loop and a current loop. The PI controller of the position loop uses the double-loop iterative feedback tuning algorithm according to the input and output data and tracking error of the vector control system. Carry out parameter setting;

The second step is to design the position loop PI controller of the permanent magnet synchronous motor servo system based on the double-loop iterative feedback tuning algorithm:

The PI controller of the position loop is designed for the position loop of the vector control system of the permanent magnet synchronous motor, and the parameters are optimized by using the double-loop iterative feedback tuning algorithm. The PI controller C(z ^-1 ,ρ) of the position loop can be linearized as:

K_p、K_I为PI 控制器的比例及积分系数，T_s为采样时间；令S(z^-1,ρ)＝(P(z^-1)C(z^-1,ρ)+1)^-1， T(z^-1,ρ)＝P(z^-1)S(z^-1,ρ)，P(z^-1)为永磁同步电机的位置环模型；若r为系统输入，v 为均值为零的白噪声，可以得到位置环的跟踪误差e为：where the controller parameter ρ=[K _p ,K _I ] ^T , the controller linearization coefficient

K _p , K _I are the proportional and integral coefficients of the PI controller, and T _s is the sampling time; let S(z ^-1 ,ρ)=(P(z ^-1 )C(z ^-1 ,ρ)+1) ^{- 1} , T(z ^-1 ,ρ)=P(z ^-1 )S(z ^-1 ,ρ), P(z ^-1 ) is the position loop model of the permanent magnet synchronous motor; if r is the system input, v is For white noise with zero mean, the tracking error e of the position loop can be obtained as:

e=S(z ^-1 )(1-P(z ^-1 ))rS(z ^-1 )v (7)

The mathematical model of the speed loop of the PMSM can be considered as a typical second-order system. For a linear steady-state system with finite-time input, its initial state can be expressed as:

是由脉冲响应系数 h_s(s＝1,2,3…)组成的Toeplitz子空间矩阵，h_i于闭环状态下加入单位脉冲激励获 得；In formula (8), u and y are the input and output respectively, n is the number of sampling points,

is a Toeplitz subspace matrix composed of impulse response coefficients h _s (s=1, 2, 3...), and h _i is obtained by adding unit impulse excitation in a closed-loop state;

In order to improve the tracking effect of the position loop, it is advisable to define the performance criterion function J(ρ) as:

J(ρ)=eQe ^T (9)

N is the total number of sampling points, e is the tracking error matrix of the position loop; _Ly is the filter, usually _Ly = 1, Q is the identity matrix; the performance criterion function J(ρ) is minimized by the double-loop iterative feedback tuning algorithm It is used to obtain the optimal parameter ρ ^* of the PI controller of the position loop, so as to obtain the best control effect. Regarding the iterative way to obtain ρ ^* , the traditional iterative feedback tuning IFT algorithm usually uses the Gauss–Newton algorithm to calculate the next iteration Update value:

通常由给与矢量控制系统的三次参考输入 来进行无偏估计；where γ _i >0 represents the step size; R _i is a positive definite Hessian matrix representing the optimization search direction, is the partial derivative of J(ρ) with respect to the controller parameter ρ, R _i and

Unbiased estimation is usually performed by three reference inputs given to the vector control system;

和

若ρ_i+1＝ρ_i+Δρ_i+1，即已知 第i次迭代时控制器参数为ρ_i，Δρ_i+1为ρ_i到最佳控制器参数ρ^*的差值Δρ_i+1 ^*时， J(Δρ_i+1)的梯度为零，即：To simplify writing, denote Ci (z ^-1 ,ρ), Si (z ^-1 ,ρ) and Ti ₍ z ^-1 ,ρ ₎ for the _ith iteration as C _{i ,} Si and Ti _, Add P(z ^-1 ) and the corresponding Toeplitz matrix is

and

If ρ _i+1 =ρ _i +Δρ _i+1 , that is, the controller parameter at the i-th iteration is known to be ρ _i , and Δρ _i+1 is the difference between ρ _i and the optimal controller parameter ρ ^* Δρ _{i+ 1} ^* , the gradient of J(Δρ _i+1 ) is zero, that is:

From formula (7), the relationship between the error e i of the ith iteration and the error e _{i +1} of the i+1th iteration is:

Among them, P is the discrete function of the control object, e _j is the tracking error matrix under the jth iteration, and then we can get:

J(Δρ _i+1 )=e _i ^T Qe _i -2e _i ^T Qf(Δρ _i+1 )+f ^T (Δρ _i+1 )Qf(Δρ _i+1 ) (13)

in:

之 积之和：Under the i+1th iteration, α _j is the Toeplitz matrix corresponding to α _j (z ^-1 ), is the jth parameter Δρ _i+1,j ,

The sum of the products:

The gradient of J(Δρ _i+1 ) can be obtained from equation (11):

In order to obtain Δρ _i+1 ^* when the gradient of J(Δρ _i+1 ) is zero, a simple iterative method is used to define an iterative loop again, the number of iterations is represented by k, and formula (15) is substituted into (16) to get:

in:

则得到：like

then get:

Substituting equation (19) into equation (17) and setting it to zero yields:

并令：Obtained by S(z ^-1 ,ρ)=(P(z ^-1 )C(z ^-1 ,ρ)+1) ^-1

and order:

Then formula (20) can be transformed into:

为：definition

for:

You can further get:

进行计算，跟踪矩阵求逆原理可得：Next for

Carry out the calculation and follow the matrix inversion principle to obtain:

Substituting equation (25) into equation (23) can get:

根据式(12)得到：Calculate next

According to formula (12), we get:

的关系，进一步得到：And because the matrix M exists

relationship, further get:

In the end you can get:

令r＝0，u作为单位脉 冲输入，v为均值为零的白噪声，得到脉冲响应序列ζ_T和ζ_S来建立关于S_i和T_i的 Toeplitz矩阵

In the calculation, it needs to be obtained through an impulse response experiment

Let r=0, _u as the unit pulse input, v as white noise with zero mean, get the impulse response sequence _ζT and _ζS to establish the Toeplitz matrix about _Si and Ti

The third step is to analyze the convergence and convergence speed of the double-loop iterative feedback tuning algorithm:

The Gauss–Newton algorithm is usually used to calculate the update value for the next iteration:

且0<γ_i≤1时，ρ_i通常线性收敛，这种情况下存在：where R _i is a matrix and γ _i is a scalar, when

And 0<γ _i ≤ 1, ρ _i usually converges linearly, in this case:

||ρ _i+1 -ρ ^* ||≤||ρ _i -ρ ^* || (31)

是：known gradient

Yes:

Subtract (17) from equation (32) to get:

由式(21)做出定义，已知

是由

求得的，则式(33)可 以进一步得到：

Defined by equation (21), it is known that

By

can be obtained, then formula (33) can be further obtained:

且0<γ_i≤1的收敛条件，因此双循环迭代反馈整定算法是收 敛的，进一步考虑算法的收敛速度，根据式(22)内循环且

时，第一次 迭代下若

很小的时候，有下式：meets the

And the convergence condition of 0<γ _i ≤ 1, so the double-loop iterative feedback tuning algorithm is convergent, further considering the convergence speed of the algorithm, according to the inner loop of equation (22) and

, the first iteration if

When I was very young, there was the following formula:

where 0<β<1;

Theorem 2: Assuming that the number of iterations of the inner loop is m, when ||ρ _i -ρ ^* || is small, the algorithm converges to order m,

可以得到：Proof: By formula (35) and

You can get:

则最终可以得到：and

You can finally get:

||Δρ _i+1 -ρ ^* ||≤β ^m ||ρ _i -Δρ ^* || (37)

Theorem 2 shows that the double-loop iterative algorithm is convergent in order m, so it has a faster convergence rate than the traditional IFT algorithm;

The fourth step is to give the specific implementation of the double-loop iterative feedback tuning control scheme of the position loop:

The specific scheme of the dual-loop iterative feedback tuning trajectory tracking control method for the position loop of the permanent magnet synchronous motor servo system is as follows:

1) For the permanent magnet synchronous motor servo system, set the position signal to r=30°, the motor runs without load, and is affected by the white noise v with a mean value of zero during the working process, the sampling time T=1×10 ^-4 s, In order to further reduce overshoot in the simulation, e will be counted from the 50th sampling point;

2) Given the initial controller parameter ρ ₁ , establish the criterion function J(ρ _i ) according to formula (7), and set the outer loop iteration coefficient i=1;

3) Perform inner loop acquisition:

Step 1: Obtain Toeplitz Matrix by Impulse Response Experiment

e_i和ρ_i，设内循环系数k＝1，令

Step 2: Given

e _i and ρ _i , set the inner loop coefficient k=1, let

即

Step 3: Perform k iterations through equations (20), (21), (26) and (29) to obtain

which is

4) Calculate ρ _i+1 =ρ _i +Δρ _i+1 ^* ;

5) After obtaining ρ _i+1 , further calculate J(ρ _i+1 ) according to the criterion function, if the control requirement is met, go to step 6, otherwise go to step 3;

6) End.

The beneficial technical effects of the present invention are:

Taking the industrial equipment such as PMSM which is widely used in the industry as the research object, the controller parameters are optimized to achieve precise position control and further improve the rapidity of system positioning. The algorithm introduces the closed-loop subspace identification method based on the impulse response model, and uses the gradient of the minimization criterion function to find the optimal step size, which changes the limitation that the traditional IFT algorithm needs multiple experiments for each iteration and the convergence speed is generally slow; The cyclic IFT algorithm can realize online tuning during the operation process, and satisfies its robustness under different control input environments, so that the optimized PMSM can be further extended to practical engineering objects such as medical robots and high-precision numerical control equipment.

Description of drawings

FIG. 1 is a structural diagram of a vector control system of a permanent magnet synchronous motor disclosed in the present application.

Fig. 2 is a "three closed-loop" control structure diagram under the parameter setting of the double-cycle IFT algorithm disclosed in the present application.

FIG. 3 is a schematic diagram of an impulse response experiment of the dual-cycle IFT algorithm disclosed in the present application.

FIG. 4 is a graph showing the change of J(ρ) when K _p starts to iterate from 20 under the traditional IFT algorithm and the double-loop IFT algorithm, respectively.

FIG. 5 is a graph showing the change of J(ρ) when K _p starts to iterate from different starting points under the traditional IFT algorithm and the double-loop IFT algorithm, respectively.

FIG. 6 is a graph showing the changes of K _I , K _p and the criterion function J(ρ) in the two-dimensional case of the present application.

Fig. 7 is the graph of the position tracking situation of the permanent magnet synchronous motor under different algorithms in the present application and the enlarged view of the oscillation place.

Fig. 8 is the graph of the rotational speed variation of the permanent magnet synchronous motor under different algorithms in the present application and an enlarged view of the oscillation place.

_FIG . 9 is a graph showing the variation of stator current vectors id and i _q under different algorithms in the present application.

Detailed ways

The specific embodiments of the present invention will be further described below with reference to the accompanying drawings.

As shown in Fig. 1-Fig. 9, in this application, in order to realize the system concretely, a set of motor experiment platform built by Yaskawa permanent magnet synchronous motor, power circuit module and control circuit module is established, and the real-time data of the motor is collected by collecting the real-time data of the motor. The position and speed are transmitted to the control circuit module, and the closed-loop feedback control can be realized, which can reflect the effectiveness of the control strategy of the permanent magnet synchronous motor.

The power circuit module consists of a rectifier circuit, an inverter bridge and an isolated drive circuit. The rectifier circuit consists of a power-on protection circuit composed of a rectifier bridge module GBJ3510, a relay and a starting resistor. The inverter bridge uses 6 IGBTs and freewheeling diodes in parallel to form a three-phase full-bridge inverter circuit to convert the direct current into an equivalent three-phase sine alternating current. PC923 as the upper bridge arm driver chip and PC929 as the corresponding lower bridge arm driver chip together form the isolation drive circuit.

The control circuit module is built with DSP:TMS320F28335 as the core control chip, and specifically includes a peripheral interface circuit, a current detection circuit, a DC bus voltage detection circuit and a rotational speed detection circuit. The above circuits in this application are conventional circuits in the field. , so this application does not describe its circuit principle in detail. TMS320F28335 is a processing chip with high-speed floating-point computing capability, and its rich peripheral resources are very convenient for servo system control. The current detection circuit is composed of a current transformer and an instrument amplifier INA199. On the one hand, the motor current must be collected to realize the current loop control, and on the other hand, the cause of some faults can be determined according to the collected three-phase current. The DC bus voltage detection circuit uses the linear optical isolation chip HCNR201 as the core to obtain the accurate DC bus voltage value. The rotational speed detection circuit uses the incremental photoelectric encoder of 2500 lines to measure the rotational speed of the motor and send it to the DSP.

Based on the double-loop iterative feedback tuning algorithm, the position signal r=30° is set, the motor runs without load, and is affected by the white noise v with zero mean value during the working process. The sampling time is T=1×10 ^-4 s. In order to compare the traditional The performance of the IFT algorithm and the double-loop IFT algorithm is verified by setting up a set of simulation experiments in advance in this application. The system simulation parameters of this example are shown in Table 1 below:

Table 1

First, take ρ=[K _p 100] in the experiment, and the feedback controller C(z ^-1 ,ρ) is:

When the permanent magnet synchronous motor system is running, take the first N sampling points in the impulse response experiment, N=200. In the stable range with only one local optimum, the traditional IFT algorithm and the double-loop IFT algorithm are used respectively, and K _p is iterated from 20. The relationship between J(ρ) and K _p is shown in Figure 4. The specific , the curve of the change of K _p under the double-cycle IFT algorithm is divided into two sections, including curve 1 and curve 2. It can be seen that the traditional IFT algorithm linearly approximates the convergence interval from the starting point K _p1 =20, while under the double-loop IFT algorithm K _p can directly obtain the optimal step size, reducing the number of iterations to less than 3 times, but for The error brought by the identification of the speed loop and the limitation of the size of the sampling point N due to the consideration of computational complexity also make the double-loop IFT algorithm have a certain deviation in obtaining the optimal solution. Next, multiple sets of control experiments were set up, and each set of experiments K _p started to iterate from 25, 30, 35, and 40, respectively. The change trend of the two different algorithms is shown in Figure 5. It can be seen that in a one-dimensional space, the double loop Compared with the traditional IFT algorithm, the IFT algorithm has higher iterative efficiency. Take ρ=[K _p K _I ], K _p and K _I are proportional gain and integral gain respectively, according to the Ziegler-Nichols method of traditional PID parameter tuning method, ρ ₀ =[42.75 419] is established, and the feedback controller C(z ^-1 ,ρ) is:

After 20 iterations, the changes of K _I , K _p and the criterion function J(ρ) are shown in Figure 6. The double-loop algorithm can generally reach a local optimum within 3 iterations, which is linear compared to the traditional IFT algorithm. The method of approximation, K _I , K _p and criterion function J(ρ) under the double-loop algorithm have higher iterative efficiency. On the other hand, the double-loop algorithm also has some limitations. In addition to the error caused by the identification of the velocity loop, the multiple local optima may also cause the two algorithms to converge to a different optimal point ρ ^* . As shown in Figure 7, it shows the position tracking situation curve of the permanent magnet synchronous motor under different algorithms and its oscillating position magnification. Specifically, the curve 3 is the tracking situation curve of the given trajectory, and the curve 4 is the The tracking situation curve of the ZN algorithm, the curve 5 is the tracking situation curve of the double-sequential IFT algorithm, and the curve 6 is the tracking situation curve of the traditional IFT algorithm. 8 shows a graph of the speed change of the permanent magnet synchronous motor and an enlarged view of the oscillation place. Specifically, the curve 7 is the speed change curve of the traditional IFT algorithm, and the curve 8 is the speed change curve of the double-sequential IFT algorithm. Curve 9 is the speed change curve of the ZN algorithm. _FIG . 9 shows a graph showing the variation of the stator current vectors id and _iq .

The above descriptions are only preferred embodiments of the present application, and the present invention is not limited to the above embodiments. It can be understood that other improvements and changes directly derived or thought of by those skilled in the art without departing from the spirit and concept of the present invention should all be considered to be included within the protection scope of the present invention.

Claims (1)

1. A method for iterative feedback setting control and optimization of a permanent magnet synchronous motor is characterized by comprising the following steps:

firstly, constructing a kinematics equation and a vector control system of the permanent magnet synchronous motor;

secondly, designing a position loop PI controller of a permanent magnet synchronous motor servo system based on a double-loop iterative feedback setting algorithm;

thirdly, analyzing the convergence and the convergence speed of the double-loop iterative feedback setting algorithm;

step four, specific implementation of a double-loop iteration feedback setting control scheme of the position ring is given;

firstly, constructing a kinematics equation and a vector control system of the permanent magnet synchronous motor:

on the premise of ignoring harmonic, hysteresis and eddy current loss, a stator voltage equation of the permanent magnet synchronous motor under a synchronous rotation coordinate system d-q is shown as a formula (1):

in the formula u_d、u_qD-q axis components, i, of stator voltage vectors, respectively_d、i_qD-q axis components of stator current vector, R stator resistance,. psi_d、ψ_qD-q axis components, ω, of stator flux linkage vectors, respectively_cIs the electrical angular velocity; further obtaining a stator flux linkage equation as follows:

wherein L is_d、L_qRespectively d-q axis inductance component,. psi_fRepresents a permanent magnet flux linkage; the equation (2) is substituted for the equation (1) to obtain the stator voltage equation:

according to the electromechanical transformation principle, the electromagnetic torque T is the partial derivative of magnetic field energy storage to mechanical angular displacement, and the electromagnetic torque equation of the permanent magnet synchronous motor is as follows:

wherein p is_nThe number of pole pairs of the permanent magnet synchronous motor is shown; further obtaining a kinematic equation of the permanent magnet synchronous motor as follows:

wherein ω is_mIs the mechanical angular velocity of the permanent magnet synchronous motor, J is the rotational inertia, B is the damping coefficient, T_LIs the load torque;

the vector control system of the permanent magnet synchronous motor is a three-closed-loop structure formed by a position loop, a speed loop and a current loop, and a PI (proportional integral) controller of the position loop performs parameter setting by using the double-loop iterative feedback setting algorithm according to input and output data and a tracking error of the vector control system;

secondly, designing a position loop PI controller of a permanent magnet synchronous motor servo system based on a double-loop iterative feedback setting algorithm:

designing a PI controller of a position ring of a vector control system of the permanent magnet synchronous motor, and optimizing parameters by using the double-loop iterative feedback setting algorithm, wherein the PI controller C (z) of the position ring^-1ρ) can be linearized as:

wherein the controller parameter ρ ═ K_p,K_I]^TLinear coefficient of controller

K_p、K_IProportional and integral coefficients, T, of PI controllers_sIs the sampling time; let S (z)^-1,ρ)＝(P(z^-1)C(z^-1,ρ)+1)^-1，T(z^-1,ρ)＝P(z^-1)S(z^-1,ρ)，P(z^-1) A position ring model of the permanent magnet synchronous motor is obtained; if r is the system input and v is white noise with zero mean, the tracking error e of the position loop can be obtained as follows:

e＝S(z^-1)(1-P(z^-1))r-S(z^-1)v (7)

the speed loop mathematical model of the permanent magnet synchronous motor can be regarded as a typical second-order system, and for a finite time input linear constant system, the initial state can be expressed as follows:

in the formula (8), u and y are input and output respectively, n is the number of sampling points,

is formed by an impulse response coefficient h_s(s ═ 1,2,3 …) of Toeplitz subspace matrix, h_iAdding unit pulse excitation in a closed-loop state to obtain the product;

in order to improve the tracking effect of the position loop, the performance criterion function J (ρ) is not defined as:

J(ρ)＝eQe^T(9)

n is the total number of sampling points, and e is a tracking error matrix of the position ring; l is_yIs a filter, usually L_y1, Q is a unit array; minimizing the performance criteria function J (ρ) by the dual-loop iterative feedback tuning algorithm for finding an optimal parameter ρ of a PI controller of the position loop^*To thereby obtain an optimum control effect with respect to the acquisition of ρ^*In the conventional iterative feedback setting IFT algorithm, a Gauss-Newton algorithm is usually used to calculate an update value of the next iteration:

wherein gamma is_i>0 represents a step size; r_iTo determine the Hessian matrix representation to optimize the search direction,

is the partial derivative of J (p) with respect to the controller parameter p, R_iAnd

unbiased estimation is typically performed from a cubic reference input to the vector control system;

to simplify the writing, C of the ith iteration is_i(z^-1,ρ)、S_i(z^-1ρ) and T_i(z^-1ρ) is represented by C_i、S_iAnd T_iAddition of P (z)^-1) Corresponding to a Toeplitz matrix of

And

if ρ_i+1＝ρ_i+Δρ_i+1I.e. knowing the controller parameter at the i-th iteration as ρ_i，Δρ_i+1Is rho_iTo the optimal controller parameter p^*Difference Δ ρ of_i+1 ^*J (Δ ρ)_i+1) Is zero, i.e.:

the error e of the ith iteration is obtained from equation (7)_iError e from i +1 th iteration_i+1The relationship of (1) is:

where P is a discrete function of the control object, e_jIs the tracking error matrix at the jth iteration, then:

J(Δρ_i+1)＝e_i ^TQe_i-2e_i ^TQf(Δρ_i+1)+f^T(Δρ_i+1)Qf(Δρ_i+1) (13)

wherein:

at the (i + 1) th iteration,

is α_j(z^-1) The corresponding Toeplitz matrix is then used,

for the jth parameter Δ ρ_i+1,j、

Sum of products of:

J(Δρ_i+1) The gradient of (c) can then be derived from equation (11):

to obtain J (Δ ρ)_i+1) Δ ρ when the gradient of (b) is zero_i+1 ^*And (3) defining an iterative loop again by using a simple iteration method, wherein the iteration number is represented by k, and the formula (15) is substituted into the formula (16) to obtain:

wherein:

if it is

Then the following results are obtained:

formula (19) substitutes for formula (17) and makes it zero:

from S (z)^-1,ρ)＝(P(z^-1)C(z^-1,ρ)+1)^-1To obtain

And order:

then equation (20) can be:

definition of

Comprises the following steps:

can further obtain:

in the following toAnd calculating to obtain the following principle by a tracking matrix inversion principle:

by substituting formula (25) for formula (23):

then calculate

Obtained according to equation (12):

and because the matrix M exists

Further obtaining:

finally, the following is obtained:

the calculation needs to be acquired through one impulse response experiment

Let r be 0, u be the unit pulse input, v be white noise with mean zero, get the impulse response sequence ζ_TAnd ζ_STo establish a relation with S_iAnd T_iToeplitz matrix of

Thirdly, analyzing the convergence and the convergence speed of the double-loop iterative feedback setting algorithm:

the next iteration update value is typically calculated using the Gauss-Newton algorithm:

wherein R is_iIs a matrix, γ_iIs a scalar quantity when

And 0<γ_iAt most 1, rho_iGenerally linearly converging, in which case:

||ρ_i+1-ρ^*||≤||ρ_i-ρ^*|| (31)

gradient is known

The method comprises the following steps:

subtracting (17) from equation (32) yields:

is defined by the formula (21), known as

Is formed by

As obtained, equation (33) further yields:

conform to

And 0<γ_iA convergence condition of less than or equal to 1, so that the double-loop iterative feedback setting algorithm is converged, further considering the convergence speed of the algorithm, according to the inner loop of the formula (22) andthen, the first iteration isVery often, there is the following formula:

wherein 0< β < 1;

theorem 2: assuming that the iteration number of the internal loop is m, | | ρ_i-ρ^*When | | is smaller, the algorithm m-th order converges,

and (3) proving that: through formula (35) and

it is possible to obtain:

and is

Then the following results are obtained:

||Δρ_i+1-ρ^*||≤β^m||ρ_i-Δρ^*|| (37)

theorem 2 shows that the dual-loop iterative algorithm is m-order convergent, so that the convergence rate is higher than that of the traditional IFT algorithm;

step four, specific implementation of a double-loop iteration feedback setting control scheme of the position ring is given:

the specific scheme of the double-loop iteration feedback setting track tracking control method for the position ring of the permanent magnet synchronous motor servo system is as follows:

1) aiming at the permanent magnet synchronous motor servo system, setting a position signal to be r-30 degrees, wherein the motor runs in no-load mode and is subjected to white noise with zero mean value in the working processv, sample time T is 1 × 10^-4s, in order to further reduce overshoot in the simulation, e will count from the 50 th sampling point;

2) given initial controller parameter ρ₁Establishing a criterion function J (rho) according to equation (7)_i) Making an outer loop iteration coefficient i equal to 1;

3) carrying out internal circulation acquisition:

step 1: toeplitz matrix obtained by impulse response experiment

And 2, step 2: given a

e_iAnd ρ_iLet the internal circulation coefficient k equal to 1

And 3, step 3: k iterations through equations (20), (21), (26) and (29) are obtained

Namely, it is

4) Calculating rho_i+1＝ρ_i+Δρ_i+1 ^*；

5) To obtain rho_i+1Then, J (ρ) is further calculated according to a criterion function_i+1) If the control requirement is met, turning to step 6, otherwise, turning to step 3;

6) and (6) ending.

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