CN108599649A - PMSM positional servosystem High order Plant controller designs and parameter determination method - Google Patents

Abstract

The invention provides a PMSM position servo system high-order object controller design and parameter determination method. The position controller uses a PID regulator, and the PID regulator parameters are designed based on pole configuration; the current loop uses a PI regulator, which is corrected into an I-type system to determine the PI regulator. The current loop has a PWM inverter, and the characteristic equation of the closed-loop system JT _q s ⁴ +Js ³ +K _t K _d s ² +K _t K _p s+K _t K _i ＝0J is the moment of inertia, K _PWM is the proportional gain of the PWM inverter, R is the stator resistance of the motor, T _a is the electrical time constant of the motor, s is the controlled time variable, K _t is the torque constant, K _p , K _d , and K _i are the control parameters of the controller, _τi is the integral time constant of the PID regulator. It is of engineering significance to compare and analyze the different step responses of the high-order object control system when the pole parameters change, and to give the selection range of the relevant parameters.

Description

Design of High-order Object Controller and Parameter Determination Method for PMSM Position Servo System

technical field

The invention belongs to the field of servo system control, in particular to a PMSM position servo system high-order object controller design and parameter determination method based on pole configuration.

Background technique

In general, in the field of PMSM (permanent magnet synchronous motor, permanent magnet synchronous motor) position servo system control, the relevant design and parameter determination of the controller can establish a traditional second-order model through the mechanical motion equation of the permanent magnet synchronous motor, and then deduce And the corresponding simulation experiments, but the traditional second-order model cannot reflect the relevant physical characteristics of the actual system well, and cannot reflect the influence of current loop parameter changes on the system.

Therefore, it is an important point to optimize the servo system control to establish a high-order model of the servo system with higher precision by means of mathematical analysis.

Paper "Experimental Modeling and Analysis of High-Order Models of Velocity Loop of PMSM Servo System" (published in "Micro and Special Motors", 44(4): 52-55, 2016, published by Pan Haihong, Wang Ling, Chen Lin, Lin Xiaoci, He Yunda) based on the DSA (DigitalSignature Algorithm) algorithm PMSM servo system speed loop modeling experiment platform to establish the third-order, fourth-order and sixth-order high-order mathematical models of the PMSM speed loop; the paper "Permanent Magnet Synchronization Based on Feedforward Decoupling Motor Control System Research" (published in "Sichuan Electric Power Technology", 40(4): 74-78, 2017, published by Jing Shibo, Wang Weiqing, Wang Haiyun, Wu Xianyou, Jiang Zhongchuan) According to the connection between the controlled quantity and the PMSM mathematical model , constructed a system-wide simplified model based on Laplace transform to tune the PI parameters of the double closed-loop controller; and the paper "Optimization of PID parameters of position controller for high-order nonlinear systems" (published in "Motor Control and Application ", 44(9):84-87, 2017, published by Cao Wei, Xie Tianchi) conducted related research on the PID parameter optimization of position controllers for high-order nonlinear servo systems.

When performing pole configuration design on controller parameters of high-order objects, the influence of pole parameter changes on controller performance is also one of the issues that need to be further studied by those skilled in the art.

Contents of the invention

The invention provides a PMSM position servo system high-order object controller design and parameter determination method based on pole configuration, which is used to solve the above problems.

In order to achieve the above object, the present invention provides a PMSM position servo system high-order object controller design method, wherein, the controller adopts the PID closed-loop system controller, and the parameters of the controller are designed by the method of pole configuration, assuming that in the controller The current loop adopts a PI regulator, and corrects it into an I-type system to determine the parameters of the PI regulator. The current loop is provided with a PWM inverter. In the dq coordinate system, the characteristic equation of the PID closed-loop system is: JT _q s ⁴ +Js ³ +K _t K _d s ² +K _t K _p s+K _t K _i ＝0, where J is the moment of inertia, K _PWM is the proportional gain in PWM inverter, R is the stator resistance of the motor, T _a is the electrical time constant of the permanent magnet synchronous servo motor, s is the controlled time variable, K _t is the torque constant, K _p , K _d , and K _i are the three control parameters of the controller, in τ _i is the integral time constant of the PI regulator.

Preferably, in the dq coordinate system, assuming that the stator current vector of the permanent magnet synchronous servo motor is perpendicular to the d axis, that is, the current component id on the _dq axis =0, the transfer function of the permanent magnet synchronous servo motor

Where i _q is also the current component on the dq axis, u _q is the voltage component on the dq axis, E _f is the counter electromotive force, E _f = ω _m K _e , K _e is the counter electromotive force constant, ω _m is the mechanical angular velocity, L _q is Equivalent inductance on the dq axis.

Preferably, the transfer function of the current loop

As a preference, in order to improve the response speed of the current loop, the time constant τ _i of the PI regulator is set equal to the electrical time constant T _a of the permanent magnet synchronous servo motor, and the closed-loop transfer function of the current loop plus the PI regulator is: Among them T _PWM is the time constant of PWM inverter.

Preferably, the closed-loop transfer function of the second-order system of the closed-loop system of the controller where ω _n is the undamped oscillation frequency.

As a preference, according to the optimal tuning method of the second-order model, and the time constant T _PWM of the PWM inverter is regarded as 0, the high-order system transfer function of the current loop of the PMSM position servo system

Preferably, the closed-loop system of the controller is unit feedback, and the transfer function of the closed-loop system of the controller is

The present invention also provides a method for determining the parameters of the high-order object controller of the PMSM position servo system as described above, wherein the root of the characteristic equation is decomposed into

where ξ is the damping ratio, ω _n is the undamped oscillation frequency, k ₁ and k ₂ are the distances from the real axis pole to the origin, by determining the values of ξ, ω _n , k ₁ and k ₂ , the corresponding control parameter K is calculated Values of _p , K _i , K _d .

The present invention also provides a method for determining the parameters of the high-order object controller of the PMSM position servo system as described above, wherein the root of the characteristic equation is decomposed into

Where ξ ₁ and ξ ₂ are damping ratios, ω ₁ and ω ₂ are undamped oscillation frequencies, by determining the values of ξ ₁ , ξ ₂ , ω ₁ and ω ₂ , the corresponding control parameters K _p , K _i , K The value of _d .

The high-order object controller design and parameter determination method of the PMSM position servo system based on the pole configuration proposed by the present invention has the following advantages:

(1) Compared with the traditional second-order model established by the permanent magnet synchronous motor mechanical motion equation, the high-order object model of the position servo system proposed by the present invention can better reflect the relevant physical characteristics of the actual system, and can reflect the current The effect of ring parameter changes on the system.

(2) The present invention compares and analyzes the different step responses of the high-order object control system when the pole parameters change, and provides the relevant parameter selection range, which has certain practical engineering significance.

Description of drawings

Fig. 1 is the PMSM position servo system high-order object structural block diagram provided by the present invention;

Fig. 2 is the system block diagram of the high-order position loop PID controller provided by the present invention;

Fig. 3 is the actual Simulink simulation model of the permanent magnet synchronous motor position servo system provided by the present invention;

Fig. 4 is the same damping ratio and the pole configuration simulation result of real pole k ₁ provided by the present invention;

Fig. 5 is the same undamped oscillation frequency provided by the present invention and the pole configuration simulation result (0<ξ< ₁ ) of real pole k1;

Fig. 6 is the same undamped oscillation frequency provided by the present invention and the pole configuration simulation result of real pole k ₁ (ξ≥1);

Fig. 7 is the pole configuration simulation result of the same damping ratio and undamped oscillation frequency provided by the present invention;

Fig. 8 is the pole configuration simulation result that the characteristic root provided by the present invention is two pairs of conjugate poles;

Fig. 9 is the characteristic root that the present invention provides is a pair of conjugate poles, the pole configuration simulation result of a pair of real poles;

FIG. 10 shows the pole configuration simulation results where the characteristic roots provided by the present invention are two pairs of real poles.

Detailed ways

In order to make the above objects, features and advantages of the present invention more comprehensible, specific implementations of the present invention will be described in detail below in conjunction with the accompanying drawings.

The design and parameter determination method of the PMSM position servo system controller based on pole configuration provided by the present invention is mainly divided into three major steps: modeling of high-order objects of the position loop, design of the controller and parameter setting, and determination of PID through simulation analysis Controller parameters.

(1) Modeling of high-order objects in the position loop

Assuming that the saturation of the iron core is ignored, the eddy current and hysteresis loss are not calculated, and the induced electromotive force in the motor is a sine wave, then in the d-q coordinate system, the stator voltage equation of PMSM can be described as:

The stator flux equation is:

Then from (1) and (2), we can get:

Among them, u _d and u _q are the voltage components of the dq axis respectively; id and i _q are the current components on the dq axis respectively; L _d and _{L q are} the equivalent inductance on the dq axis; R is the stator resistance; ψ _d and ψ _q are the stator flux components on the dq axis; ω is the electrical angular velocity; ψ _f is the rotor flux corresponding to the permanent magnet.

Under the principle of constant power conversion, the output electromagnetic torque of the motor is obtained:

T _e ＝1.5p[ψ _f i _q +(L _d -L _q )i _d i _q ] (4)

where p is the number of pole pairs of the rotor.

If the reluctance torque is neglected, let L _d =L _q , and the torque equation becomes:

T _e ＝1.5pψ _f i _q ＝K _t i _q (5)

Where K _t is the torque constant. In addition, the mechanical equation of motion of the motor is:

Among them: J is the moment of inertia; ω _m is the mechanical angular velocity; T _L is the load torque; B is the damping coefficient.

In order to realize the decoupling of PMSM control parameters, the commonly used " _id = 0" control strategy is used, that is, the stator current vector is perpendicular to the d-axis, then when the ψ _f of the permanent magnet is a constant value, it is only necessary to adjust i _q , the direct control of the torque can be realized. The present invention establishes a servo system position servo system high-order object model based on the control strategy of " _id = 0", as shown in FIG. 1 .

First, the transfer function of the PWM inverter can be approximately equivalent to a first-order inertial link, written as:

Among them, T _PWM is the proportional gain, and T _PWM is the time constant of the inverter.

From (3), the transfer function of the motor is:

Where E _f is the back electromotive force, E _f = ω _m K _e , K _e is the back electromotive force constant, T _a is the electrical time constant of the permanent magnet synchronous servo motor,

In the case of ignoring the damping coefficient B, according to formulas (3), (5) and (6), after the current output is added to the load link, its transfer function is expressed as:

Where K _t =K _e =pψ _f , i _L is the load current.

The next step is the design of the current regulator. Considering the anti-disturbance recovery performance and follow-up performance of a typical I-type system, the current loop is designed as a typical I-type system, and a PI regulator is used. The transfer function is:

Where K _p is the proportional coefficient of the PI regulator, τ _i is the integral time constant of the regulator, Considering the delay effect of the inertia link on the system, in order to improve the response speed of the current loop, the time constant τ _i of the regulator is set equal to the electrical time constant T _a , then the open-loop transfer function of the current loop plus the PI regulator is:

in

Therefore, the closed-loop transfer function of the current loop is:

For a typical second-order system, its closed-loop transfer function can be expressed as:

According to the optimal tuning method of the second-order model, then:

ξ ² =0.5 (15)

From (13), (14) we know:

So from (15), (16), (17) we get

2kT _PWM = 1 (18)

In general, T _PWM is small, then (13) can be approximated as:

in

According to Figure 1, the high-order system transfer function of the current loop of the PMSM position servo system is obtained:

(2) Controller design and parameter tuning

The entire high-order position loop uses a PID controller, and the block diagram of the control system is shown in Figure 2.

Assuming that the closed-loop system is unit feedback, since _Gp (s) and G(s) are known, the transfer function of the closed-loop system can be obtained:

Get the characteristic equation of the system:

JT _q s ⁴ +Js ³ +K _t K _d s ² +K _t K _p s+K _t K _i ＝0 (23)

At this time, the root of the system characteristic equation can be decomposed into two forms, the first form:

where ξ is the damping ratio, _ωn is the undamped oscillation frequency, and _k1 and _k2 are the distances from the real axis poles to the origin. It can be seen that as long as the values of ξ, ω _n , k ₁ and k ₂ are determined, the corresponding K _p , K _i , K _d control parameters can be calculated according to the principle that the coefficients of formula (24) are equal.

In order (24) Equal to 0, get its characteristic root:

It can be seen that when 0<ξ<1, s _{1, 2} is a pair of conjugate complex roots; when ξ=1, s _{1, 2} is a pair of multiple roots; when ξ>1, s 1, ₂ is a pair The negative real roots are not equal, so according to the value of ξ, the characteristic roots can also be decomposed into the second form:

Among them, ξ ₁ and ξ ₂ are damping ratios, and ω ₁ and ω ₂ are undamped oscillation frequencies. Similarly, as long as the values of ξ ₁ , ξ ₂ , ω ₁ and ω ₂ are determined, the corresponding PID control parameters can also be calculated.

(3) Determine the parameters of the PID controller through simulation analysis

Since there are multiple variables in formulas (24) and (26), the method of controlling variables is used for experimental analysis. Figure 3 is the actual Simulink simulation model of the high-order model of the permanent magnet synchronous motor position servo system, in which the PI parameters of the current loop regulator can be obtained by (12), and the input angle θ of the simulation model is 1rad.

The simulation parameters selected for the simulation experiment are shown in Table 1:

Table 1 PMSM simulation parameters

The first form of the roots of the characteristic equation:

The first form is shown in formula (24). Since there are multiple variable parameters, the principle of controlling variables is adopted for simulation experiments:

a. Pole configuration strategy with the same damping ratio and real pole k ₁

It can be known from formula (25) that when 0<ξ<1, the system has a pair of conjugate complex roots and two real number poles, so take ξ=0.707, k ₁ =15, and ω _n (unit rad/s) take 10 , 20, 30, 40, 50, the PID controller parameters and system simulation results are shown in Table 2 and the figure:

Table 2 Same damping ratio and pole configuration parameters of real pole k ₁

ω _n k _p k _i k _d 10 0.3532 1.6943 0.0331 20 0.9284 6.7384 0.0490 30 1.7216 15.0742 0.0648 40 2.7290 26.6435 0.0806 50 3.9468 41.3880 0.0962

According to the simulation results shown in Figure 4, when the system damping ratio and a real-axis pole remain unchanged, as ω _n increases, the system adjustment time becomes shorter and the overshoot increases. When ω _n is greater than 30, the system begins to Oscillation occurs.

b. Pole configuration strategy with the same undamped oscillation frequency and real pole k ₁

When ω _n = 30, k ₁ = 15, ξ takes 0.4, 0.707, 0.9, 1, 2, 4 respectively, and the simulation experiment is divided into three cases: when 0<ξ<1, the system has a pair of conjugate complex root and two real number poles; when ξ=1, the system has a pair of negative heavy roots and two real number poles -k ₁ and -k ₂ on the real axis; when ξ>1, the system has two unequal negative real roots and two real poles -k ₁ and -k ₂ . The cases of ξ=1 and ξ>1 are analyzed uniformly, and the PID controller parameters and system simulation results are shown in Table 3 and Figures 5 and 6:

Table 3 The pole configuration parameters of the same undamped oscillation frequency and real pole k ₁

From the simulation results shown in Figure 5, it can be found that under the condition of constant undamped oscillation frequency, when 0<ξ<1, as the damping ratio increases, the system responds faster and the adjustment time becomes shorter, but slowly Oscillate slightly.

From the simulation results shown in Figure 6, it can be found that when ξ≥1, as the damping ratio increases, the system responds faster but the oscillation becomes larger, and the overshoot increases, which is not conducive to the performance index of the actual project.

c. Pole placement strategy with the same damping ratio and undamped oscillation frequency

Take ξ = 0.707, ω _n = 30, and k ₁ to be 1, 5, 10, 15, and 30 respectively for simulation experiments. The parameters of the PID controller and the system simulation results are shown in Table 4 and Figure 7:

Table 4 The pole configuration parameters of the same damping ratio and undamped oscillation frequency

k ₁ k _p k _i k _d 1 1.0588 1.0107 0.0492 5 1.2489 5.0453 0.0537 10 1.4858 10.0700 0.0593 30 2.4234 29.9633 0.0814

From the simulation results shown in Figure 7, it can be seen that with the increase of k ₁ , the system response becomes faster, but the overshoot becomes larger, and the system begins to oscillate; when k ₁ is small, the system overshoot is small, basically no Oscillation, good stability, but the response time is not very fast.

The first form of the roots of the characteristic equation:

The second form is shown in formula (26). According to the different value ranges of ξ ₁ and ξ ₂ , it is divided into three cases for discussion.

a. The characteristic root is the configuration strategy of two pairs of conjugate poles (0<ξ ₁ , ξ ₂ <1)

Given ξ ₁ = 0.707, ω _n = 30, respectively take ξ ₂ = 0.6, 0.7, 0.8, 0.9 for parameter configuration, the PID controller parameters and system simulation results are shown in Table 5 and Figure 8:

Table 5 The characteristic root is the pole configuration parameters of two pairs of conjugate poles

ξ ₂ k _p k _i k _d 0.6 82.3457 1725.6 1.9654 0.7 60.7672 1267.8 1.4567 0.8 46.7618 970.6660 1.1266 0.9 37.1598 766.9460 0.9002

It can be seen that the system does not respond at the beginning, and after about 0.45s, it is in a state of oscillation and divergence, and the expected target response cannot be achieved.

b. The case where the characteristic root is a pair of conjugate complex poles and a pair of real poles (0<ξ ₁ <1, ξ ₂ >1)

Take ξ ₁ =0.707, ω _n =30, respectively take ξ ₂ =3, 5, 15, and 30 for parameter configuration, the PID controller parameters and system simulation results are shown in Table 6 and Figure 9:

Table 6 The characteristic root is a pair of conjugate poles, and the pole configuration parameters of a pair of real poles

According to the simulation results in Figure 9, when _ξ1 and _ωn remain unchanged, as _ξ2 increases, the system response becomes slower, the oscillation and overshoot become smaller, and the system tends to be stable.

c. The case where the characteristic roots are two pairs of real poles (both the values of ξ ₁ and ξ ₂ are greater than 1)

According to the principle of control variables, given ξ ₁ = 2, ω _n = 30, respectively take ξ ₂ = 3, 5, 15, 30 to configure the PID controller parameters. The PID controller parameters and system simulation results are shown in Table 7 and Fig. 10 shows:

Table 7 The characteristic root is the pole configuration parameters of two pairs of real number poles

ξ ₂ k _p k _i k _d 3 9.6107 64.7360 0.2029 5 4.0865 23.3050 0.1569 15 1.3245 2.5894 0.1338 30 1.0655 0.6474 0.1317

From the simulation results, it can be seen that when the positions of one pair of real poles remain unchanged and the damping ratio of the other pair increases, the overshoot and oscillation of the system become smaller, the system tends to be stable, and the response becomes slower.

From the above analysis, taking into account the rapidity and stability of the system, we can roughly summarize the value range of the parameters as follows: the damping ratio ξ can generally be set between 0.6-0.8; ω _n (rad/s) can be between 20-40 value between; k ₁ can take a value between 10 and 20.

Obviously, those skilled in the art can make various changes and modifications to the invention without departing from the spirit and scope of the invention. If these modifications and variations of the present invention fall within the scope of the claims of the present invention and their technical equivalents, the present invention also intends to include these modifications and variations.

Claims (9)

1. a PMSM position servo system high-order object controller design method, it is characterized in that, described controller adopts PID closed-loop system controller, and with the parameter of the method design controller of pole configuration, assuming that the current loop in the controller adopts PI regulator, and correct it into an I-type system to determine the parameters of the PI regulator, the current loop is provided with a PWM inverter, and in the dq coordinate system, the characteristic equation of the PID closed-loop system is: where J is the moment of inertia, K _PWM is the proportional gain in PWM inverter, R is the stator resistance of the motor, T _a is the electrical time constant of the permanent magnet synchronous servo motor, s is the controlled time variable, K _t is the torque constant, K _p , K _d , and K _i are the three control parameters of the controller, in τ _i is the integral time constant of the PI regulator. 2. PMSM position servo system high-order object controller design method as claimed in claim 1, is characterized in that, in dq coordinate system, assumes that the stator current vector of described permanent magnet synchronous servo motor is perpendicular to d axis, that is The current component i _d on the dq axis =0, the transfer function of the permanent magnet synchronous servo motor Where i _q is also the current component on the dq axis, u _q is the voltage component on the dq axis, E _f is the counter electromotive force, E _f = ω _m K _e , K _e is the counter electromotive force constant, ω _m is the mechanical angular velocity, L _q is Equivalent inductance on the dq axis. 3. PMSM position servo system high-order object controller design method as claimed in claim 2, is characterized in that, the transfer function of described current loop 4. PMSM position servo system high-order object controller design method as claimed in claim 3 is characterized in that, in order to improve the response speed of current loop, make the time constant _τ of PI regulator equal to permanent magnet synchronous servo motor electrical time constant T _a , the closed-loop transfer function of the current loop plus the PI regulator is Among them T _PWM is the time constant of PWM inverter. 5. PMSM position servo system high-order object controller design method as claimed in claim 4, is characterized in that, the closed-loop transfer function of the second-order system of the closed-loop system of described controller where ω _n is the undamped oscillation frequency. 6. PMSM position servo system high-order object controller design method as claimed in claim 5 is characterized in that, according to second-order model optimal tuning method, and the time constant T _PWM of PWM inverter is regarded as 0, then Higher Order System Transfer Function of Current Loop in PMSM Position Servo System 7. PMSM position servo system high-order object controller design method as claimed in claim 6 is characterized in that, the closed-loop system of described controller is unit feedback, then the transfer function of the closed-loop system of described controller 8. a PMSM position servo system high-order object controller parameter determination method as claimed in claim 1, is characterized in that, the root of described characteristic equation is decomposed into where ξ is the damping ratio, ω _n is the undamped oscillation frequency, k ₁ and k ₂ are the distances from the real axis pole to the origin, by determining the values of ξ, ω _n , k ₁ and k ₂ , the corresponding control parameter K is calculated Values of _p , K _i , K _d . 9. a PMSM position servo system high-order object controller parameter determination method as claimed in claim 1, is characterized in that, the root of described characteristic equation is decomposed into Where ξ ₁ and ξ ₂ are damping ratios, ω ₁ and ω ₂ are undamped oscillation frequencies, by determining the values of ξ ₁ , ξ ₂ , ω ₁ and ω ₂ , the corresponding control parameters K _p , K _i , K The value of _d .