CN108599649A - PMSM positional servosystem High order Plant controller designs and parameter determination method - Google Patents
PMSM positional servosystem High order Plant controller designs and parameter determination method Download PDFInfo
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
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Abstract
The present invention provides PMSM positional servosystem High order Plant controller designs and parameter determination method, and positioner PID regulator designs PID regulator parameter based on POLE PLACEMENT USING；Electric current loop pi regulator, is corrected into the parameter that I type systems determine pi regulator, and electric current loop has PWM inverter, closed-loop system characteristic equation JTqs4+Js3+KtKds2+KtKps+KtKi=0J is rotary inertia,KPWMFor PWM inverter proportional gain,R is motor stator resistance, TaFor motor electrical time constant, s is controlled time variable, KtFor torque constant, Kp、Kd、KiDevice control parameter in order to control,τiFor the integration time constant of PID regulator, the different step response of High order Plant control system provides the selection range of relevant parameter, there is engineering significance come comparative analysis when for pole Parameters variation.
Description
Technical field
The invention belongs to servo system control field, more particularly to a kind of PMSM positional servosystems based on POLE PLACEMENT USING
High order Plant controller design and parameter determination method.
Background technology
Under normal circumstances, in PMSM (permanent magnet synchronous motor, permanent magnet synchronous motor) position
It sets in servo system control field, the relevant design of controller and parameter determination can pass through permanent magnet synchronous motor mechanical movement side
Then Cheng Jianli tradition second-order models carry out derivation and corresponding emulation experiment, but traditional second-order model can not be anti-well
The related physical feature for reflecting real system can not embody influence of the current ring parameter variation to system.
Therefore the high-order model for establishing the higher servo-drive system of precision by the method for mathematical analysis is optimization servo system
One emphasis of system control.
Paper " PMSM servo-drive systems speed ring high-order model Experimental modeling and analysis " (is published in《Small and special electric machine》, 44
(4)：52-55,2016, artificial Pan sea letter, Wang Ling, Chen Lin are delivered, woods knows word, He Yunda) it is based on DSA (Digital
Signature Algorithm) the PMSM servo-drive systems speed ring modeling experiment platform of algorithm establishes PMSM speed rings three
The high order mathematical model of rank, quadravalence and six ranks；Paper " the control system for permanent-magnet synchronous motor research based on Feedforward Decoupling " (is delivered
In《Sichuan Electric Power technology》, 40 (4)：74-78,2017, deliver artificial Jing Shibo, Wang Weiqing, Wang Haiyun, Wu Xianyou, Jiang Zhongchuan)
According to contacting between controlled volume and PMSM mathematical models, a kind of system-wide simplified model is constructed come to double based on Laplace transformation
The PI parameters of closed loop controller are adjusted；And paper " the positioner pid parameter of High Order Nonlinear System optimizes " (is delivered
In《Motor controls and application》, 44 (9):84-87,2017, deliver artificial Cao Wei, Xie Tianchi) then it is directed to high-order nonlinear servo
The pid parameter optimization problem of system position controller has carried out correlative study.
And when the controller parameter to High order Plant carries out POLE PLACEMENT USING design, the variation of pole parameter is to controller
The influence and those skilled in the art of energy need one of the problem of further investigation.
Invention content
The present invention provides a kind of PMSM positional servosystem High order Plant controller designs and parameter based on POLE PLACEMENT USING
Method is determined, for solving the above problems.
In order to achieve the above objectives, the present invention provides a kind of PMSM positional servosystems High order Plant controller design method,
Wherein, the controller uses PID closed-loop system controllers, and the method that POLE PLACEMENT USING is used in combination designs the parameter of controller, it is assumed that
Electric current loop in controller uses pi regulator, and is corrected into the parameter that I type systems determine pi regulator, is set in the electric current loop
It is equipped with PWM inverter, in d-q coordinate systems, the characteristic equation of the PID closed-loop systems is：JTqs4+Js3+KtKds2+KtKps
+KtKi=0, wherein J are rotary inertia,KPWMFor the proportional gain in PWM inverter,R is motor stator resistance, TaFor permanent magnet synchronous servo motor electrical time constant, s is the time variable being controlled, KtFor
Torque constant, Kp、Kd、KiThree control parameters of device in order to control, whereinτiIt is normal for the time of integration of pi regulator
Number.
Preferably, in d-q coordinate systems, it is assumed that the stator current vector of the permanent magnet synchronous servo motor hangs down with d axis
Current component i directly namely on d-q axisd=0, the transmission function of the permanent magnet synchronous servo motor
Wherein iqIt is also the current component on d-q axis, uqFor the electricity of d-q axis
Press component, EfFor counter electromotive force, Ef=ωmKe,KeFor back electromotive force constant, ωmFor mechanical angular speed, LqFor on d-q axis etc.
Imitate inductance.
Preferably, the transmission function of the electric current loop
Preferably, in order to improve the response speed of electric current loop, the timeconstantτ of pi regulator is enablediIt is equal to
Permanent magnet synchronous servo motor electrical time constant Ta, electric current loop add pi regulator after closed loop transfer function, beWherein TPWMFor 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
Wherein ωnFor undamped oscillation frequency.
Preferably, according to the best setting method of second-order model, and by the time constant T of PWM inverterPWMRegard 0 as,
The then high order system transmission function of PMSM positional servosystems electric current loop
Preferably, the closed-loop system of the controller is fed back for unit, then the transmission of the closed-loop system of the controller
Function
The present invention also provides a kind of PMSM positional servosystems High order Plant controller parameters as described above to determine method,
Wherein, the root of the characteristic equation is decomposed into
Wherein ξ is damping ratio, ωnFor undamped oscillation frequency, k1And k2It is real axis pole at a distance from origin, by true
Determine ξ, ωn, k1And k2Value, calculate corresponding control parameter Kp、Ki、KdValue.
The present invention also provides a kind of PMSM positional servosystems High order Plant controller parameters as described above to determine method,
Wherein, the root of the characteristic equation is decomposed into
Wherein ξ1、ξ2For damping ratio, ω1、ω2For undamped oscillation frequency, pass through determining ξ1、ξ2、ω1And ω2Value, meter
Calculate corresponding control parameter Kp、Ki、KdValue.
PMSM positional servosystem High order Plant controller designs and parameter proposed by the present invention based on POLE PLACEMENT USING are true
Determine method, has the following advantages that：
(1) present invention is compared with the traditional second-order model established by permanent magnet synchronous motor mechanical motion equation, the present invention
The positional servosystem High order Plant model of proposition can preferably reflect the related physical feature of real system, can embody
Current ring parameter changes the influence to system.
(2) the different step response of High order Plant control system has been carried out to score when the present invention is directed to pole Parameters variation
Analysis, and relevant parameter selection range is given, there is certain engineering significance.
Description of the drawings
Fig. 1 is PMSM positional servosystems High order Plant structure diagram provided by the invention；
Fig. 2 is the system block diagram of high-level position loop PID controller provided by the invention；
Fig. 3 is the practical Simulink simulation models of permanent magnetism synchronous electric machine position servo system provided by the invention；
Fig. 4 is same damping provided by the invention when real pole k1POLE PLACEMENT USING simulation result；
Fig. 5 is identical undamped oscillation frequency provided by the invention and real pole k1POLE PLACEMENT USING simulation result (0 ＜ ξ
＜ 1)；
Fig. 6 is identical undamped oscillation frequency provided by the invention and real pole k1POLE PLACEMENT USING simulation result (ξ >=
1)；
Fig. 7 is the identical POLE PLACEMENT USING simulation result for damping when undamped oscillation frequency provided by the invention；
Fig. 8 is the POLE PLACEMENT USING simulation result that characteristic root provided by the invention is two pairs of conjugate poles；
Fig. 9 is that characteristic root provided by the invention is a pair of of conjugate pole, the POLE PLACEMENT USING simulation result of a pair of of real pole；
Figure 10 is the POLE PLACEMENT USING simulation result that characteristic root provided by the invention is two pairs of real poles.
Specific implementation mode
In order to make the foregoing objectives, features and advantages of the present invention clearer and more comprehensible, below in conjunction with the accompanying drawings to the present invention
Specific implementation mode be described in detail.
The design and parameter determination method of PMSM positional servosystem controllers provided by the invention based on POLE PLACEMENT USING,
It is broadly divided into three big steps：The modeling of position ring High order Plant, the design of controller and parameter tuning are determined by simulation analysis
PID controller parameter.
(1) modeling of position ring High order Plant
Assuming that ignoring the saturation of iron core, vortex and magnetic hystersis loss are not calculated, the induced electromotive force in motor is sine wave, that
In d-q coordinate systems, the stator voltage equation of PMSM can be described as：
Stator magnetic linkage equation is：
So by (1) and (2), can obtain：
Wherein, udAnd uqThe respectively component of voltage of d-q axis；idAnd iqIt is the current component on d-q axis respectively；LdAnd LqFor
Equivalent inductance on d-q axis；R is stator resistance；ψdAnd ψqFor the stator magnetic linkage component on d-q axis；ω is angular rate；ψfIt is
The corresponding rotor flux of permanent magnet.
Under the principle of invariable power transformation, the output electromagnetic torque of motor is obtained：
Te=1.5p [ψfiq+(Ld-Lq)idiq] (4)
Wherein p is the number of magnetic pole pairs of rotor.
If ignoring reluctance torque, L is enabledd=Lq, torque equation becomes：
Te=1.5p ψfiq=Ktiq (5)
Wherein KtFor torque constant.In addition, the mechanical motion equation of motor is：
Wherein：J is rotary inertia；ωmFor mechanical angular speed；TLFor load torque；B is damped coefficient.
In order to realize the decoupling of PMSM control parameters, the more common " i of used=0 " control strategy, even stator current
Vector is vertical with d axis, then in the ψ of permanent magnetfIt is only to need by adjusting i in the case of definite valueq, so that it may to realize to torque
Directly control.The present invention is based on " id=0 " control strategy establishes the model of servo-drive system positional servosystem High order Plant
As shown in Figure 1.
First, the transmission function of PWM inverter can be write as with Approximate Equivalent for a first order inertial loop：
Wherein TPWMFor proportional gain, TPWMFor the time constant of inverter.
By (3), the transmission function of motor is：
Wherein EfFor counter electromotive force, Ef=ωmKe,KeFor back electromotive force constant, TaWhen electrical for permanent magnet synchronous servo motor
Between constant,
In the case where ignoring damped coefficient B, by formula (3), (5) and (6), after electric current exports and load link is added,
Transmission function is expressed as：
Wherein Kt=Ke=p ψf, iLIt is load current.
Next it is exactly the design of current regulator, it is contemplated that the disturbance rejection restorability and followability of typical I types system
Can be good, therefore by design of current ring at typical I types system, using pi regulator, transmission function is：
Wherein KpFor the proportionality coefficient of pi regulator, τiFor the integration time constant of adjuster,In view of inertia
Delayed-action of the link to system enables the timeconstantτ of adjuster to improve the response speed of electric current loopiEqual to it is electrical when
Between constant Ta, then electric current loop be plus the open-loop transfer function after pi regulator：
Wherein
So when electric current loop closed loop transfer function, be：
Typical second-order system, closed loop transfer function, can be expressed as：
According to the best setting method of second-order model, then：
ξ2=0.5 (15)
By (13), known to (14)：
Therefore by (15), (16), (17) obtain
2kTPWM=1 (18)
T under normal circumstancesPWMIt is smaller, then can be approximately by (13)：
Wherein
According to Fig. 1, the high order system transmission function of PMSM positional servosystem electric current loops is obtained：
(2) design and parameter tuning of controller
Entire high-level position loop uses PID controller, control system block diagram as shown in Figure 2.
If this closed-loop system is fed back for unit, due to Gp(s) and G (s) is it is known that the transmission letter of closed-loop system can be obtained
Number：
Obtain the characteristic equation of system：
JTqs4+Js3+KtKds2+KtKps+KtKi=0 (23)
At this moment two kinds of forms, the first form can be decomposed into for system features equation root：
Wherein ξ is damping ratio, ωnFor undamped oscillation frequency, k1And k2It is real axis pole at a distance from origin.Thus may be used
Know, as long as determining ξ, ωn, k1And k2Value, then corresponding K can be calculated according to the equal principle of formula (24) coefficientp、
Ki、KdControl parameter.
It enables in formula (24)Equal to 0, its characteristic root is obtained：
As can be seen that as 0 ＜ ξ ＜ 1, s1,2For a pair of of Conjugate complex roots；When ξ=1, s1,2For a pair of of repeated root；When ξ ＞ 1,
s1,2For a pair of negative real root such as not, therefore according to the value of ξ difference, characteristic root can also be decomposed into second of form：
Wherein ξ1、ξ2For damping ratio, ω1、ω2For undamped oscillation frequency.Similarly, as long as determining ξ1、ξ2、ω1And ω2's
Value, can also calculate corresponding pid control parameter.
(3) PID controller parameter is determined by simulation analysis
Due to formula (24), there are a variety of variables in (26), therefore the method for using control variable carries out experimental analysis.Fig. 3 is
The practical Simulink simulation models of permanent magnetism synchronous electric machine position servo system high-order model, wherein the PI ginsengs of electric current loop adjuster
Number can be found out by (12), and the input angle θ of simulation model is 1rad.
The simulation parameter that emulation experiment is chosen is as shown in table 1：
1 PMSM simulation parameters of table
The first form of the root of characteristic equation：
Shown in the first form such as formula (24), since there are multiple variable parameters, therefore the principle of control variable is taken to carry out
Emulation experiment：
A. same to damp when real pole k1Pole placement strategy
By as 0 ＜ ξ ＜ 1, system has a pair of of Conjugate complex roots and two real poles known to formula (25), thus take ξ=
0.707, k1=15, ωnWhen (unit rad/s) takes 10,20,30,40,50 respectively, PID controller parameter and system emulation result
As shown in table 2 and figure：
Table 2 is same to damp when real pole k1POLE PLACEMENT USING parameter
| ωn | kp | ki | kd |
| 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 |
Simulation result according to Fig.4, in the case where when a real axis pole is constant for system damping, with ωn's
Increase, the system adjustment time shortens, and overshoot increases, in ωnSystem starts to vibrate after more than 30.
B. identical undamped oscillation frequency and real pole k1Pole placement strategy
Take ωn=30, k1When=15, ξ takes 0.4,0.707,0.9,1,2,4 respectively, and emulation experiment is divided into three kinds of situations：When 0
When ＜ ξ ＜ 1, system has a pair of of Conjugate complex roots and two real poles；As ξ=1, system has a pair of of heavy burden root on real axis
And two real pole-k1With-k2；As ξ ＞ 1, there are two negative real root and two real pole-k such as or not system tool1With-k2。
The case where ξ=1 and ξ ＞ 1, is uniformly analyzed, PID controller parameter and system emulation result are as shown in table 3 and Fig. 5,6：
3 identical undamped oscillation frequency of table and real pole k1POLE PLACEMENT USING parameter
Simulation result is it can be found that in the case where undamped oscillation frequency is constant as shown in Figure 5, as 0 ＜ ξ ＜ 1, with
The increase of damping ratio, system response becomes faster, and regulating time shortens, but but small oscillations slowly occurs.
When simulation result can be found that ξ >=1 as shown in Figure 6, with the increase of damping ratio, system response becomes faster but vibrates change
Greatly, and overshoot increases, and is unfavorable for the performance indicator of Practical Project.
C. the identical pole placement strategy for damping when undamped oscillation frequency
Take ξ=0.707, ωn=30, k1Carry out emulation experiment when taking 1,5,10,15,30 respectively, PID controller parameter and
System emulation result is as shown in table 4 and Fig. 7：
The identical POLE PLACEMENT USING parameter for damping when undamped oscillation frequency of table 4
| k1 | kp | ki | kd |
| 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 |
Simulation result can be seen that as shown in Figure 7：With k1Increase, system response becomes faster, but overshoot becomes larger, system
Start to vibrate；And in k1When smaller, system overshoot is small, and basic dead-beat, stability is good, but the response time is not very
Rapidly.
The first form of the root of characteristic equation：
Shown in second of form such as formula (26), according to ξ1、ξ2Value range is different, is divided into three kinds of situation discussion.
A. characteristic root is configuration strategy (the 0 ＜ ξ of two pairs of conjugate poles1、ξ2＜ 1)
Given ξ1=0.707, ωn=30, ξ is taken respectively2=0.6,0.7,0.8,0.9 carries out parameter configuration, PID controller
Parameter and system emulation result are as shown in table 5 and Fig. 8：
5 characteristic root of table is the POLE PLACEMENT USING parameter of two pairs of conjugate poles
| ξ2 | kp | ki | kd |
| 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 system starts no response, it is in oscillation and divergence state after about 0.45s, is unable to reach the set goal sound
It answers.
B. the case where characteristic root is a pair of of complex-conjugate poles, a pair of of real pole (0 ＜ ξ1＜ 1, ξ2＞ 1)
Take ξ1=0.707, ωn=30, ξ is taken respectively2=3,5,15,30 carry out parameter configurations, PID controller parameter and are
Simulation result unite as shown in table 6 and Fig. 9：
6 characteristic root of table is a pair of of conjugate pole, the POLE PLACEMENT USING parameter of a pair of of real pole
According to Fig. 9 simulation results, in ξ1And ωnIn the case of constant, with ξ2Increase, system response it is slack-off, shake
It swings and overshoot becomes smaller, system tends towards stability.
C. the case where characteristic root is two pairs of real poles (ξ1、ξ21) value is all higher than
According to the principle of control variable, ξ is given1=2, ωn=30, ξ is taken respectively2=3,5,15,30 carry out PID controller
Parameter configuration, PID controller parameter and system emulation result are as shown in table 7 and Figure 10：
7 characteristic root of table is the POLE PLACEMENT USING parameter of two pairs of real poles
| ξ2 | kp | ki | kd |
| 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 |
Simulation result can be seen that：When the position of a pair of of real pole is constant, and the damping ratio of another pair increases, system overshoot
Amount and oscillation become smaller, and system tends towards stability, and response is slack-off.
By above analysis, the rapidity and stability of system are taken into account, we can be by parameter value range general summary
For：Damping ratio ξ generally can between 0.6-0.8 value；ωn(rad/s) can between 20-40 value；k1It can be between 10~20
Value.
Obviously, those skilled in the art can carry out invention spirit of the various modification and variations without departing from the present invention
And range.If these modifications and changes of the present invention is within the scope of the claims of the present invention and its equivalent technology, then
The present invention is also intended to including these modification and variations.
Claims (9)
1. a kind of PMSM positional servosystems High order Plant controller design method, which is characterized in that the controller uses PID
Closed-loop system controller, the method that POLE PLACEMENT USING is used in combination design the parameter of controller, it is assumed that the electric current loop in controller uses PI
Adjuster, and it is corrected into the parameter that I type systems determine pi regulator, it is provided with PWM inverter in the electric current loop, is sat in d-q
In mark system, the characteristic equation of the PID closed-loop systems is： Its
Middle J is rotary inertia,KPWMFor the proportional gain in PWM inverter,R is motor
Stator resistance, TaFor permanent magnet synchronous servo motor electrical time constant, s is the time variable being controlled, KtFor torque constant, Kp、
Kd、KiThree control parameters of device in order to control, whereinτiFor the integration time constant of pi regulator.
2. PMSM positional servosystems High order Plant controller design method as described in claim 1, which is characterized in that in d-
In q coordinate systems, it is assumed that the stator current vector of the permanent magnet synchronous servo motor is vertical with d axis namely d-q axis on electric current point
Measure id=0, the transmission function of the permanent magnet synchronous servo motor
Wherein iqIt is also the current component on d-q axis, uqFor the voltage point of d-q axis
Amount, EfFor counter electromotive force, Ef=ωmKe,KeFor back electromotive force constant, ωmFor mechanical angular speed, LqFor the equivalent electricity on d-q axis
Sense.
3. PMSM positional servosystems High order Plant controller design method as claimed in claim 2, which is characterized in that described
The transmission function of electric current loop
4. PMSM positional servosystems High order Plant controller design method as claimed in claim 3, which is characterized in that in order to
The response speed for improving electric current loop, enables the timeconstantτ of pi regulatoriEqual to permanent magnet synchronous servo motor electrical time constant Ta,
Electric current loop add pi regulator after closed loop transfer function, beWherein TPWM
For the time constant of PWM inverter.
5. PMSM positional servosystems High order Plant controller design method as claimed in claim 4, which is characterized in that described
The closed loop transfer function, of the second-order system of the closed-loop system of controllerWherein ωnFor undamped oscillation
Frequency.
6. PMSM positional servosystems High order Plant controller design method as claimed in claim 5, which is characterized in that according to
The best setting method of second-order model, and by the time constant T of PWM inverterPWMRegard 0 as, then PMSM positional servosystems electric current
The high order system transmission function of ring
7. PMSM positional servosystems High order Plant controller design method as claimed in claim 6, which is characterized in that described
The closed-loop system of controller is fed back for unit, then the transmission function of the closed-loop system of the controller
8. a kind of PMSM positional servosystems High order Plant controller parameter as described in claim 1 determines method, feature
It is, the root of the characteristic equation is decomposed into
Wherein ξ is damping ratio, ωnFor undamped oscillation frequency, k1And k2It is real axis pole at a distance from origin, by determining ξ,
ωn, k1And k2Value, calculate corresponding control parameter Kp、Ki、KdValue.
9. a kind of PMSM positional servosystems High order Plant controller parameter as described in claim 1 determines method, feature
It is, the root of the characteristic equation is decomposed into
Wherein ξ1、ξ2For damping ratio, ω1、ω2For undamped oscillation frequency, pass through determining ξ1、ξ2、ω1And ω2Value, calculate
Corresponding control parameter Kp、Ki、KdValue.
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| CN111082731A (en) * | 2020-01-07 | 2020-04-28 | 北京自动化控制设备研究所 | Pump speed-regulating motor parameter optimization method and pump speed-regulating motor |
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