CN110829921B - Iterative feedback setting control and optimization method for permanent magnet synchronous motor - Google Patents

Iterative feedback setting control and optimization method for permanent magnet synchronous motor Download PDF

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CN110829921B
CN110829921B CN201911119936.5A CN201911119936A CN110829921B CN 110829921 B CN110829921 B CN 110829921B CN 201911119936 A CN201911119936 A CN 201911119936A CN 110829921 B CN110829921 B CN 110829921B
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loop
permanent magnet
synchronous motor
magnet synchronous
iteration
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CN110829921A (en
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陶洪峰
刘巍
张秀赟
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Suzhou Suli Energy Technology Co.,Ltd.
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Jiangnan University
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/0003—Control strategies in general, e.g. linear type, e.g. P, PI, PID, using robust control
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
    • H02P21/18—Estimation of position or speed
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/24—Vector control not involving the use of rotor position or rotor speed sensors
    • H02P21/28—Stator flux based control

Abstract

The invention discloses a permanent magnet synchronous motor iterative feedback setting control and an optimization method thereof, which relate to the field of servo control optimization.

Description

Iterative feedback setting control and optimization method for permanent magnet synchronous motor
Technical Field
The invention relates to an iterative feedback setting control and optimization method for a permanent magnet synchronous motor, and belongs to the field of servo control optimization.
Background
The Permanent magnet synchronous motor is a synchronous motor which generates a synchronous rotating magnetic field by Permanent magnet excitation, and is widely applied to a position servo system which mainly aims at high speed and accurate tracking and positioning due to the characteristics of high efficiency, excellent dynamic performance, light weight and the like of the Permanent Magnet Synchronous Motor (PMSM) in the fields of numerical control equipment, industrial robots, laser engraving and the like. In recent years, a large amount of work is done by domestic and foreign scholars on the scheme of optimizing the traditional PID controller aiming at the typical 'three closed loop' structure of the permanent magnet synchronous motor position servo system. The positioning performance and robustness of the permanent magnet synchronous motor servo system can be effectively improved by methods including genetic algorithm, fuzzy control, neural network and the like, but the methods either cannot meet the requirement of rapidity of the system or need enough modeling precision, and practical application of the permanent magnet synchronous motor in industry under the method is limited to a certain extent.
The inherent nonlinearity and uncertainty of the permanent magnet synchronous motor make it difficult to perform accurate mathematical modeling, and the model-based method generally has higher sensitivity to the modeling accuracy. When searching for controller parameters corresponding to a minimum performance criterion function, parameter optimization is usually performed by a traditional IFT (Iterative Feedback Tuning) algorithm through an optimization algorithm similar to a Gauss-Newton method, and the convergence speed of the algorithm is fundamentally limited by characteristics of three experiments and linear approximation required by each iteration of the method, so that the high-speed characteristics required by precision industries such as permanent magnet synchronous motor position loop control are difficult to meet.
Disclosure of Invention
Aiming at the problem that the traditional IFT algorithm is difficult to meet the requirement of the parameter optimization of the position ring PI controller of the permanent magnet synchronous motor, the application provides the iterative feedback setting control and the optimization method of the permanent magnet synchronous motor, which integrates a new IFT framework, popularizes and applies the dual-cycle IFT algorithm to the setting and control performance optimization of the position ring PI controller parameters, and aims to improve the rapid and accurate tracking capability of the permanent magnet synchronous motor.
The technical scheme of the invention is as follows:
a permanent magnet synchronous motor iteration feedback setting control and optimization method comprises 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):
Figure BDA0002275164650000021
in the formula ud、uqD-q axis components, i, of stator voltage vectors, respectivelyd、iqD-q axis components of stator current vector, R stator resistance,. psid、ψqD-q axis components, ω, of stator flux linkage vectors, respectivelycIs the electrical angular velocity; further obtaining a stator flux linkage equation as follows:
Figure BDA0002275164650000022
wherein L isd、LqRespectively d-q axis inductance component,. psifRepresents a permanent magnet flux linkage; the stator voltage equation obtained by substituting formula (2) into formula (1) is:
Figure BDA0002275164650000023
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:
Figure BDA0002275164650000024
wherein p isnThe 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:
Figure BDA0002275164650000025
wherein ω ismIs the mechanical angular velocity of the permanent magnet synchronous motor, J is the rotational inertia, B is the damping coefficient, TLIs 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 a 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:
a PI controller of a position ring is designed aiming at a position ring of a vector control system of a permanent magnet synchronous motor,PI controller C (z) using a two-cycle iterative feedback tuning algorithm to optimize parameters, position loop-1ρ) can be linearized as:
Figure BDA0002275164650000031
wherein the controller parameter ρ ═ Kp,KI]TLinear coefficient of controller
Figure BDA0002275164650000032
Kp、KIProportional and integral coefficients, T, of PI controllerssIs 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; 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 mathematical model of the speed loop of the permanent magnet synchronous motor can be considered as a typical second-order system, and for a finite-time input linear constant system, the initial state can be expressed as follows:
Figure BDA0002275164650000033
in the formula (8), u and y are input and output respectively, n is the number of sampling points,
Figure BDA0002275164650000034
is formed by an impulse response coefficient hs(s ═ 1,2,3 …) of Toeplitz subspace matrix, hiAdding unit pulse excitation in a closed-loop state to obtain the product;
to improve the tracking effect of the position loop, the performance criterion function J (ρ) is not defined as:
J(ρ)＝eQeT (9)
n is the total number of sampling points, and e is a tracking error matrix of the position loop; l isyIs a filter, usually L y1, Q is a unit array; minimizing a performance criterion function J (rho) by a dual-loop iterative feedback setting algorithm for finding an optimal parameter rho for a PI controller for a 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:
Figure BDA00022751646500000415
wherein gamma isi>0 represents a step size; riTo determine the Hessian matrix representation to optimize the search direction,
Figure BDA0002275164650000041
partial derivatives, R, of J (p) with respect to the controller parameter piAnd
Figure BDA0002275164650000042
unbiased estimation is usually done from a cubic reference input to the vector control system;
to simplify the writing, C of the ith iteration isi(z-1,ρ)、Si(z-1ρ) and Ti(z-1ρ) is represented by Ci、SiAnd TiAddition of P (z)-1) Corresponding to a Toeplitz matrix of
Figure BDA0002275164650000043
And
Figure BDA0002275164650000044
if ρi+1＝ρi+Δρi+1I.e. knowing the controller parameter at the i-th iteration as ρi，Δρi+1Is rhoiTo the optimal controller parameter p*Difference Δ ρ ofi+1 *J (Δ ρ)i+1) Has a gradient of zero, i.e.：
Figure BDA0002275164650000045
The error e of the ith iteration is obtained from equation (7)iError e from i +1 th iterationi+1The relationship of (1) is:
Figure BDA0002275164650000046
where P is a discrete function of the control object, ejIs the tracking error matrix at the jth iteration, then:
J(Δρi+1)＝ei TQei-2ei TQf(Δρi+1)+fT(Δρi+1)Qf(Δρi+1) (13)
wherein:
Figure RE-GDA0002302968960000047
at iteration i +1, αjIs alphaj(z-1) The corresponding Toeplitz matrix is then used,
Figure RE-GDA0002302968960000048
for the jth parameter Δ ρi+1,j、
Figure RE-GDA0002302968960000049
Sum of products of:
Figure BDA00022751646500000412
J(Δρi+1) The gradient of (c) can then be derived from equation (11):
Figure BDA00022751646500000413
to obtain J (Δ ρ)i+1) Δ ρ when the gradient of (b) is zeroi+1 *And (3) defining an iteration 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:
Figure BDA00022751646500000414
wherein:
Figure BDA0002275164650000051
if it is
Figure BDA0002275164650000052
Then the following results are obtained:
Figure BDA0002275164650000053
formula (19) substitutes for formula (17) and makes it zero:
Figure BDA0002275164650000054
from S (z)-1,ρ)＝(P(z-1)C(z-1,ρ)+1)-1To obtain
Figure BDA0002275164650000055
And order:
Figure BDA0002275164650000056
then equation (20) can be:
Figure BDA0002275164650000057
definition of
Figure BDA0002275164650000058
Comprises the following steps:
Figure BDA0002275164650000059
can further obtain:
Figure BDA00022751646500000510
in the following to
Figure BDA00022751646500000511
And calculating to obtain the following principle by a tracking matrix inversion principle:
Figure BDA00022751646500000512
by substituting formula (25) for formula (23):
Figure BDA00022751646500000513
then calculate
Figure BDA00022751646500000514
Obtained according to equation (12):
Figure BDA00022751646500000515
and because the matrix M exists
Figure BDA00022751646500000516
Further obtaining:
Figure BDA00022751646500000517
finally, the following can be obtained:
Figure BDA00022751646500000518
the calculation needs to be acquired through one impulse response experiment
Figure BDA00022751646500000519
Let r be 0, u be the unit pulse input, v be white noise with mean zero, get the impulse response sequence ζTAnd ζSTo establish a relation with SiAnd TiToeplitz matrix of
Figure BDA00022751646500000520
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:
Figure BDA0002275164650000061
wherein R isiIs a matrix, γiIs a scalar quantity when
Figure BDA0002275164650000062
And 0<γiAt most 1, rhoiGenerally linearly converge, in which case:
||ρi+1-ρ*||≤||ρi-ρ*|| (31)
gradient is known
Figure BDA0002275164650000063
The method comprises the following steps:
Figure BDA0002275164650000064
subtracting (17) from equation (32) yields:
Figure BDA0002275164650000065
Figure BDA0002275164650000066
is defined by the formula (21), known as
Figure BDA0002275164650000067
Is formed by
Figure BDA0002275164650000068
As obtained, equation (33) can further obtain:
Figure BDA0002275164650000069
conform to
Figure BDA00022751646500000610
And 0<γiA convergence condition of ≦ 1, so that the two-loop iterative feedback setting algorithm is convergent, further considering the convergence rate of the algorithm, according to equation (22) inner loop and
Figure BDA00022751646500000611
then, the first iteration is
Figure BDA00022751646500000612
Very often, there is the following formula:
Figure BDA00022751646500000613
wherein 0< β < 1;
theorem 2: assuming that the iteration number of the internal loop is m, | | ρi-ρ*When | | is smaller, the algorithm m-th order converges,
and (3) proving that:through formula (35) and
Figure BDA00022751646500000614
it is possible to obtain:
Figure BDA00022751646500000615
and is
Figure BDA00022751646500000616
Then finally it can be found:
||Δρi+1-ρ*||≤βm||ρi-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 double-loop iterative feedback setting track tracking control method for the position ring of the permanent magnet synchronous motor servo system has the following specific scheme:
1) aiming at a permanent magnet synchronous motor servo system, setting a position signal to be r-30 degrees, enabling the motor to run in a no-load mode, and influenced by white noise v with zero mean value in the working process, wherein the sampling time T is 1 multiplied by 10-4s, in order to further reduce overshoot in the simulation, e is counted 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
Figure BDA0002275164650000071
And 2, step 2: given a
Figure BDA0002275164650000072
eiAnd ρiLet the internal circulation coefficient k equal to 1
Figure BDA0002275164650000073
And 3, step 3: k iterations through equations (20), (21), (26) and (29) are obtained
Figure BDA0002275164650000074
Namely, it is
Figure BDA0002275164650000075
4) Calculating rhoi+1＝ρi+Δρi+1 *；
5) To obtain rhoi+1Then, J (ρ) is further calculated according to a criterion functioni+1) If the control requirement is met, turning to the step 6, otherwise, turning to the step 3;
6) and (6) ending.
The beneficial technical effects of the invention are as follows:
the method is characterized in that industrial equipment such as PMSM (permanent magnet synchronous motor) which is widely applied in the industry is taken as a research object, the parameters of a controller are optimized to realize accurate position control and further improve the rapidity of system positioning, a new IFT framework is integrated, namely, a closed-loop subspace identification method based on an impulse response model is introduced into a dual-cycle IFT algorithm, the optimal step length is obtained by using a minimum criterion function gradient, and the limitation that the traditional IFT algorithm needs multiple experiments for each iteration and the convergence speed is generally slow is changed; the dual-cycle IFT algorithm can realize online tuning in the operation process, and meets the robustness of the dual-cycle IFT algorithm in different control input environments, so that the optimized PMSM can be further popularized to practical engineering objects such as medical robots, high-precision numerical control equipment and the like.
Drawings
Fig. 1 is a block diagram of a vector control system of a permanent magnet synchronous motor disclosed in the present application.
FIG. 2 is a diagram of a "three closed loop" control architecture with parameter tuning by the dual-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 herein.
FIG. 4 shows K under the conventional IFT algorithm and the two-cycle IFT algorithm, respectivelypGraph of the variation of J (rho) at the beginning of the iteration from 20.
FIG. 5 shows K under the conventional IFT algorithm and the two-cycle IFT algorithm, respectivelypGraph of the variation of J (rho) when iteration is started from different starting points.
FIG. 6 is K in the two-dimensional case of the present applicationI、KpAnd a graph of the variation of the criterion function J (rho).
Fig. 7 is a graph of a position tracking situation of a permanent magnet synchronous motor under different algorithms in the application and an enlarged view of the oscillation position of the permanent magnet synchronous motor.
Fig. 8 is a graph of the variation of the rotation speed of the permanent magnet synchronous motor under different algorithms in the application and an enlarged view of the oscillation position of the permanent magnet synchronous motor.
FIG. 9 shows stator current vectors i under different algorithms in the present applicationd、iqGraph of the variation.
Detailed Description
The following further describes the embodiments of the present invention with reference to the drawings.
With reference to fig. 1 to 9, in the present application, in order to implement the system specifically, a set of motor experiment platform constructed by an ansha permanent magnet synchronous motor, a power circuit module and a control circuit module is established, and by collecting the real-time position and speed of the motor and transmitting the position and speed to the control circuit module, closed-loop feedback control can be implemented, so that the effectiveness of a control strategy of the permanent magnet synchronous motor can be embodied.
The power circuit module consists of a rectification circuit, an inverter bridge and an isolation driving circuit. The rectification circuit comprises a power-on protection circuit consisting of a rectifier bridge module GBJ3510, a relay and a starting resistor, and the inverter bridge adopts a three-phase full-bridge inverter circuit formed by connecting 6 IGBTs and a freewheeling diode in parallel to convert direct current into equivalent three-phase sinusoidal alternating current. The PC923 is used as an upper bridge arm driving chip, and the PC929 is used as a corresponding lower bridge arm driving chip to jointly form an isolation driving circuit.
The control circuit module is built by taking a DSP (digital signal processor) TMS320F28335 as a core control chip, and specifically comprises a peripheral interface circuit, a current detection circuit, a direct current bus voltage detection circuit and a rotating speed detection circuit, wherein the circuits in the application are conventional circuits in the field, so that the circuit principle of the control circuit is not described in detail in the application. TMS320F28335 is a processing chip with high-speed floating point arithmetic capability, and the abundant peripheral resources are very convenient for servo system control. The current detection circuit consists of a current transformer and an instrument amplifier INA199, so that the current loop control can be realized to acquire the motor current on one hand, and the reason of some faults can be determined according to the acquired three-phase current on the other hand. The direct current bus voltage detection circuit adopts the linear optical accident isolation chip HCNR201 as a core to acquire an accurate direct current bus voltage value. The rotating speed detection circuit measures the rotating speed of the motor by using a 2500-wire incremental photoelectric encoder and sends the rotating speed to the DSP.
Based on a double-loop iterative feedback setting algorithm, a position signal r is set to be 30 degrees, a motor runs in a no-load mode, the influence of white noise v with zero mean value is applied to the motor in the working process, and the sampling time T is 1 multiplied by 10-4s, in order to compare the performance of the conventional IFT algorithm and the dual-cycle IFT algorithm, a set of simulation experiments is first set for verification, and the system simulation parameters of the example are shown in the following table 1:
TABLE 1
Figure BDA0002275164650000091
First, in the experiment, rho is [ K ]p 100]Feedback controller C (z)-1ρ) is:
Figure BDA0002275164650000092
when the permanent magnet synchronous motor system runs, the first N sampling points in the pulse response experiment are taken, and N is 200. In a stable range with only one local optimum, respectively using the traditional IFT algorithm and the two-cycle IFT algorithm to make KpIterating from 20J (ρ) with respect to KpIn particular, K under the two-cycle IFT algorithmpThe curve of the variation of (2) is divided into two sections, including a curve 1 and a curve 2. It can be seen that the conventional IFT algorithm starts from the starting point K p120 linear approximation convergence interval, and K under dual-cycle IFT algorithmpThe optimal step length can be directly obtained, the iteration times are reduced to be within 3 times, and certain deviation exists when the optimal solution is obtained through the dual-cycle IFT algorithm due to errors brought by identification of the speed ring and the limitation of the size of the sampling point N in consideration of calculation complexity. Next, multiple control experiments were set up, each experiment KpThe iteration is started from 25, 30, 35 and 40 respectively, the change trend of the two different algorithms is shown in fig. 5, and it can be seen that the two-cycle IFT algorithm has higher iteration efficiency in one-dimensional space compared with the traditional IFT algorithm. Taking rho ═ Kp KI]，Kp、 KIRespectively proportional gain and integral gain, and determining rho according to the traditional PID parameter setting method Ziegler-Nichols method0＝[42.75 419]Feedback controller C (z)-1ρ) is:
Figure BDA0002275164650000093
after 20 iterations, KI、KpAnd the change condition of the criterion function J (rho) is shown in fig. 6, the dual-loop algorithm can reach a local optimum within 3 iterations, and compared with a traditional IFT algorithm linear approximation mode, the K under the dual-loop algorithmI、KpAnd the criterion function J (ρ) has a higher iteration efficiency. In another aspect, the dual-loop algorithm has some limitations, and besides the error caused by the identification of the speed loop, the local optima may cause the two algorithms to converge to a different optimal point ρ*. Referring to fig. 7, a graph of the position tracking of the pmsm under different algorithms and an enlarged view of the oscillation position are shown, specifically, curve 3 is a curve of the tracking of a given trajectory, curve 4 is a curve of the tracking of ZN algorithm, and curve 5 is a curve of the tracking of two cyclesThe trace situation curve of the sequence IFT algorithm, curve 6, is the trace situation curve of the conventional IFT algorithm. Fig. 8 is a graph showing a variation of the rotation speed of the permanent magnet synchronous motor and an enlarged view of the oscillation position thereof, and specifically, a curve 7 is a variation of the rotation speed of the conventional IFT algorithm, a curve 8 is a variation of the rotation speed of the dual sequential IFT algorithm, and a curve 9 is a variation of the rotation speed of the ZN algorithm. Fig. 9 shows the stator current vector id、iqGraph of the variation.
What has been described above is only a preferred embodiment of the present application, and the present invention is not limited to the above embodiment. It is to be understood that other modifications and variations directly derivable or suggested by those skilled in the art without departing from the spirit and concept of the present invention are to be considered as included within the 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;
the method comprises the following steps of firstly, constructing a kinematic equation and a vector control system of the permanent magnet synchronous motor, and specifically:
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):
Figure FDA0002812162470000011
in the formula ud、uqD-and q-axis components, i, of the stator voltage vector, respectivelyd、iqD-axis and q-axis components of the stator current vector, R is the stator resistance, #d、ψqD-axis and q-axis components, ω, of stator flux linkage vector, respectivelycIs the electrical angular velocity; further obtaining a stator flux linkage equation as follows:
Figure FDA0002812162470000012
wherein L isd、LqD-axis and q-axis inductance components, psifRepresents a permanent magnet flux linkage; the equation (2) is substituted for the equation (1) to obtain the stator voltage equation:
Figure FDA0002812162470000013
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:
Figure FDA0002812162470000014
wherein p isnThe 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:
Figure FDA0002812162470000021
wherein ω ismIs the mechanical angular velocity of the permanent magnet synchronous motor, J is the rotational inertia, B is the damping coefficient, TLIs 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, which specifically comprises the following steps:
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:
Figure FDA0002812162470000022
wherein the controller parameter ρ ═ Kp,KI]TLinear coefficient of controller
Figure FDA0002812162470000023
Kp、KIProportional and integral coefficients, T, of PI controllerssIs 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,ρ)(1-P(z-1))r-S(z-1,ρ)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:
Figure FDA0002812162470000024
in the formula (8), u and y are input and output respectively, n is the number of sampling points,
Figure FDA0002812162470000025
is formed by an impulse response coefficient hsA constituent Toeplitz subspace matrix, wherein s ═ 1,2,3 …, hsAdding unit pulse excitation in a closed-loop state to obtain the product;
to improve the tracking effect of the position loop, a performance criterion function J (ρ) is defined as:
J(ρ)＝eQeT (9)
n is the total number of sampling points, and e is a tracking error matrix of the position ring; l isyIs a filter, is Ly1, 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 iteration mode, the traditional iteration feedback setting IFT algorithm uses Gauss-Newton algorithm to calculate the next iteration update value:
Figure FDA0002812162470000031
wherein gamma isi>0 represents a step size; riTo determine the Hessian matrix representation to optimize the search direction,
Figure FDA0002812162470000032
is the partial derivative of J (p) with respect to the controller parameter p, RiAnd
Figure FDA0002812162470000033
unbiased estimation from a cubic reference input to the vector control system;
to simplify the writing, C of the ith iteration isi(z-1,ρ)、Si(z-1ρ) and Ti(z-1ρ) is represented by Ci、SiAnd TiAddition of P (z)-1) Corresponding to a Toeplitz matrix of
Figure FDA0002812162470000034
And
Figure FDA0002812162470000035
if ρi+1＝ρi+Δρi+1I.e. knowing the controller parameter at the i-th iteration as ρi，Δρi+1Is rhoiTo the optimal controller parameter p*Difference Δ ρ ofi+1 *J (Δ ρ)i+1) Is zero, i.e.:
Figure FDA0002812162470000036
the error e of the ith iteration is obtained from equation (7)iError e from i +1 th iterationi+1The relationship of (1) is:
Figure FDA0002812162470000037
where P is a discrete function of the control object, ejIs the tracking error matrix at the jth iteration, then:
J(Δρi+1)＝ei TQei-2ei TQf(Δρi+1)+fT(Δρi+1)Qf(Δρi+1) (13)
wherein:
Figure FDA0002812162470000038
at the (i + 1) th iteration,
Figure FDA0002812162470000039
is alphaj(z-1) The corresponding Toeplitz matrix is then used,
Figure FDA00028121624700000310
for the jth parameter Δ ρi+1,j、
Figure FDA00028121624700000311
Sum of products of:
Figure FDA0002812162470000041
J(Δρi+1) The gradient of (c) can then be derived from equation (11):
Figure FDA0002812162470000042
to obtain J (Δ ρ)i+1) Δ ρ when the gradient of (b) is zeroi+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:
Figure FDA0002812162470000043
wherein:
Figure FDA0002812162470000044
if it is
Figure FDA0002812162470000045
Then the following results are obtained:
Figure FDA0002812162470000046
formula (19) substitutes for formula (17) and makes it zero:
Figure FDA0002812162470000047
from S (z)-1,ρ)＝(P(z-1)C(z-1,ρ)+1)-1To obtain
Figure FDA0002812162470000048
And order:
Figure FDA0002812162470000049
then equation (20) can be:
Figure FDA00028121624700000410
definition of
Figure FDA00028121624700000411
Comprises the following steps:
Figure FDA00028121624700000412
can further obtain:
Figure FDA00028121624700000413
in the following to
Figure FDA00028121624700000414
And calculating to obtain the following principle by a tracking matrix inversion principle:
Figure FDA00028121624700000415
by substituting formula (25) for formula (23):
Figure FDA00028121624700000416
then calculate
Figure FDA00028121624700000417
Obtained according to equation (12):
Figure FDA0002812162470000051
and because the matrix M exists
Figure FDA0002812162470000052
Further obtaining:
Figure FDA0002812162470000053
finally, the following is obtained:
Figure FDA0002812162470000054
the calculation needs to be acquired through one impulse response experiment
Figure FDA0002812162470000055
Let r be 0, u be the unit pulse input, v be white noise with mean zero, get the impulse response sequence ζTAnd ζSTo establish a relation with SiAnd TiToeplitz matrix of
Figure FDA0002812162470000056
Thirdly, analyzing the convergence and the convergence speed of the double-loop iterative feedback setting algorithm, specifically:
the next iteration update value is calculated using the Gauss-Newton algorithm:
Figure FDA0002812162470000057
wherein gamma isi>0 represents a step size; riOptimizing the search direction for positive Hessian matrix representation
Figure FDA0002812162470000058
And 0<γiAt most 1, rhoiLinear convergence, in which case:
||ρi+1-ρ*||≤||ρi-ρ*|| (31)
gradient is known
Figure FDA0002812162470000059
The method comprises the following steps:
Figure FDA00028121624700000510
subtracting (17) from equation (32) yields:
Figure FDA00028121624700000511
Figure FDA00028121624700000512
is defined by the formula (21), known as
Figure FDA00028121624700000513
Is formed by
Figure FDA00028121624700000514
As obtained, equation (33) further yields:
Figure FDA00028121624700000515
conform to
Figure FDA0002812162470000061
And 0<γiA 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) and
Figure FDA0002812162470000062
then, the first iteration is
Figure FDA0002812162470000063
Very often, there is the following formula:
Figure FDA0002812162470000064
wherein 0< β < 1;
theorem 2: assuming that the iteration number of the internal loop is m, | | ρi-ρ*When | | is smaller, the algorithm m-th order converges,
and (3) proving that: through formula (35) and
Figure FDA0002812162470000065
it is possible to obtain:
Figure FDA0002812162470000066
and is
Figure FDA0002812162470000067
Then the following results are obtained:
||Δρi+1-ρ*||≤βm||ρi-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, providing the specific implementation of the double-loop iteration feedback setting control scheme of the position ring, which specifically comprises the following steps:
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, a position signal is set to be r-30 degrees, the motor runs in no-load, the influence of white noise v with zero mean value is received in the working process, and the sampling time T is 1 multiplied by 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
Figure FDA0002812162470000068
And 2, step 2: given a
Figure FDA0002812162470000069
eiAnd ρiLet the internal circulation coefficient k equal to 1
Figure FDA00028121624700000610
And 3, step 3: k iterations through equations (20), (21), (26) and (29) are obtained
Figure FDA00028121624700000611
Namely, it is
Figure FDA00028121624700000612
4) Calculating rhoi+1＝ρi+Δρi+1 *；
5) To obtain rhoi+1Then, J (ρ) is further calculated according to a criterion functioni+1) If the control requirement is met, turning to the step 6) to finish, otherwise, turning to the step 3) to execute the step of internal circulation acquisition again;
6) and (6) ending.
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