CN106886152B - Magnetic suspension rotor odd harmonic current suppression method based on second-order odd repetitive controller - Google Patents
Magnetic suspension rotor odd harmonic current suppression method based on second-order odd repetitive controller Download PDFInfo
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Abstract
The invention discloses a magnetic suspension rotor odd harmonic current suppression method based on a second-order odd-order repetitive controller. The SOORC has a second-order inner membrane structure aiming at odd harmonic frequency suppression, and can improve the harmonic suppression capability under harmonic frequency variation or uncertain conditions, namely improve the control robustness of the system under the harmonic frequency variation or uncertain conditions. The invention also adopts phase lag-lead compensation to improve the steady state performance and the dynamic performance of the system, can realize the suppression of the odd harmonic current component generated by the magnetic bearing coil in the magnetic suspension rotor, and is suitable for the suppression of the odd harmonic current component of the magnetic suspension rotor system with rotor mass unbalance and sensor harmonic.
Description
Technical Field
The invention relates to the technical field of magnetic suspension rotor harmonic current suppression, in particular to a magnetic suspension rotor odd harmonic current suppression method based on a second-order odd-order repetitive controller, which is applied to harmonic current suppression in a magnetic suspension control moment gyroscope rotor system and provides technical support for application of a magnetic suspension control moment gyroscope on a super-silent satellite platform.
Background
The stable suspension of a rotor in a magnetic suspension Control Moment Gyro (CMG) is realized by adopting the electromagnetic force of a magnetic bearing, and the rotor and a stator of the magnetic bearing have no contact friction. Therefore, the magnetic bearing has many advantages compared with the traditional mechanical bearing: firstly, because the rotor and the stator of the magnetic bearing have no contact friction, the rotation speed of the CMG flywheel rotor can be greatly improved compared with that of a mechanical bearing, and the service life of a system is prolonged; secondly, the active control of the unbalanced vibration force of the magnetic suspension flywheel rotor is easy to realize, and the vibration intensity of the system in working is reduced; in addition, since the equivalent moment of inertia of the CMG frame is related to the rotor support stiffness, the bearing stiffness can be reduced by supporting the rotor based on the magnetic bearing, thereby increasing the equivalent moment of inertia of the CMG frame. Therefore, the magnetic bearing can improve the angular rate precision of the system frame and the CMG torque output precision under the condition of the same torque output, and finally improve the pointing precision and stability of the spacecraft. Magnetic bearings have been widely used in spacecraft high-precision long-life attitude control actuators. Therefore, a high-precision long-life CMG based on magnetic bearings is an ideal choice for spacecraft attitude control actuators.
Although the magnetic suspension control moment gyroscope has a plurality of advantages, high-frequency vibration in the system can be transmitted to the spacecraft through the magnetic bearing to indirectly influence the attitude control precision of the spacecraft, so that the pointing precision and the stability of the satellite platform are reduced. The high-frequency vibrations of the magnetically levitated control moment gyroscope CMG are mainly caused by rotor mass imbalances and sensor harmonics. The imbalance of the rotor mass is the main reason for generating vibration, and then the current components with the same frequency and frequency multiplication, namely sensor harmonic waves, can appear in the displacement sensor signals due to the roundness error of the detection surface of the sensor, the non-uniform electromagnetic property and the like, and the sensor harmonic waves can cause harmonic vibration.
Harmonic vibration suppression can be classified into three categories, zero current, zero displacement, and zero vibration. The zero-current vibration suppression has the advantages of small calculation amount and low power consumption. The current suppression technology mainly aims at the suppression of interference of single frequency, research work for harmonic current suppression is relatively few, and the mainstream method focuses on repeated control of an RC algorithm and a parallel multi-trap filter or multi-LMS filter algorithm. However, the parallel multi-trap algorithm cannot simultaneously suppress all vibrations, has a large calculation amount, needs to consider the problem of convergence speed among different filters, and is relatively complex to design; the repetitive control RC algorithm achieves simultaneous suppression of vibrations of different frequency components without the need for multiple filters in parallel. The repetitive control RC algorithm is a method for realizing the zero static error of the system according to the internal model principle, and the existing repetitive control algorithms applied to the magnetic suspension rotor system are all designed based on the inner membrane structure under the condition that the frequency of the harmonic current is accurately determined, and the problem of attenuation of the harmonic current suppression effect when the frequency of the harmonic current changes or is uncertain is not considered.
Disclosure of Invention
The purpose of the invention is as follows: the invention provides a magnetic suspension rotor odd harmonic current suppression method based on a second-order odd repetitive controller, which overcomes the defects of the prior art. The invention improves the odd harmonic current suppression precision of the magnetic suspension rotor system under the condition of harmonic current frequency change or uncertainty by introducing a second-order repetitive control algorithm.
The technical scheme adopted by the invention is as follows: a magnetic suspension rotor odd harmonic current suppression method based on a second-order odd repetitive controller comprises the following steps:
step (1) establishing a magnetic suspension rotor dynamic model containing rotor mass unbalance and sensor harmonic;
the radial two degrees of freedom of the magnetic suspension rotor are controlled by an active magnetic bearing, the other three degrees of freedom are passively and stably suspended by permanent magnet rings arranged on the rotor and the stator, Q is the geometric center of the stator of the magnetic bearing, O is the geometric center of the rotor, C is the center of mass of the rotor, an inertial coordinate system QXY is established by taking Q as the center, a rotating coordinate system O epsilon η is established by taking O as the center, and (x, y) are coordinate values of the geometric center O of the rotor under the inertial coordinate system.
For the X channel harmonic current, the modeling is as follows:
from newton's second law, the dynamic equation of the magnetic suspension rotor in the X direction is as follows:
where m is the rotor mass, fxThe bearing force of the magnetic bearing in the X direction, e is the deviation between the geometric center and the mass center of the rotor, omega is the rotating speed of the rotor, and phi is the initial phase of the unbalanced mass of the rotor.
The bearing force of the active and passive magnetic bearings is composed of the electromagnetic force of the active magnetic bearing and the permanent magnetic force of the passive magnetic bearing, and the bearing force f in the X channelxCan be written as:
fx＝fex+fpx
wherein f isexElectromagnetic force for X-channel active magnetic bearing, fpxIs the permanent magnetic force of the X-channel passive magnetic bearing. Wherein, permanent magnetic force and displacement are in a linear relation and are expressed as follows:
fpx＝Kprx
wherein, KprIs the displacement stiffness of the passive magnetic bearing;
when the rotor is suspended near the magnetic center, the active magnetic bearing electromagnetic force can be approximately linearized as:
fex≈Kerx+Kiix
wherein, Ker、KiRespectively, the displacement stiffness and the current stiffness of the active magnetic bearing ixOutputting current for the power amplifier;
for rotor systems containing mass unbalance, there are:
X(t)＝x(t)+Θx(t)
wherein X (t) is the displacement of the center of mass of the rotor, x (t) is the displacement of the geometric center of the rotor, and thetax(t) is the displacement disturbance caused by mass imbalance, and is recorded as:
Θx(t)＝lcos(Ωt+θ)
wherein l is the amplitude of mass unbalance, theta is the phase, and omega is the rotor speed;
considering the actual rotor system, the displacement x actually measured by the sensor is usually unavoidable due to the influence of mechanical processing precision, material unevenness and other factorss(t) can be expressed as:
xs(t)＝x(t)+xd(t)
wherein x isd(t) is the sensor harmonic, which can be rewritten as:
wherein, caIs the magnitude of the harmonic coefficient of the sensor, θaIs the phase of the sensor harmonic coefficient, w is the highest number of sensor harmonics;
will ix、X(t)、Θx(t)、xd(t) successively carrying out Laplace conversion into ix(s)、X(s)、Θx(s)、xd(s), the rotor dynamics equation is written as:
ms2X(s)＝(Ker+Kpr)(X(s)-Θx(s))+Kiix(s)
wherein the content of the first and second substances,
ix(s)＝-KsKiGc(s)Gw(s)(X(s)-Θx(s)+xd(s))
wherein, KsA gain element G of the displacement sensorc(s) is a controller element, Gw(s) is a power amplifier link;
from the above formula, it can be seen that due to the mass imbalance and the existence of sensor harmonic, the current component-K with the same frequency as the rotating speed exists in the coil currentsKiGc(s)Gw(s)(X(s)-Θx(s)) and a frequency-multiplied current component-KsKiGc(s)Gw(s)xd(s)。
In the controllable radial translation freedom degree X channel and the controllable radial translation freedom degree Y channel of the active magnetic bearing, the two channels are decoupled, so that a current model of the Y channel is similar to that of the X channel, and the specific analysis is as follows:
the rotor dynamics equation is:
ms2Y(s)＝(Ker+Kpr)(Y(s)-Θy(s))+Kiiy(s)
wherein Y(s) is the pull-type transformation of the displacement y (t) of the center of mass of the rotor, thetay(s) Displacement disturbance Θ caused by Mass imbalancey(t) pull transformation, iy(s) is the output current i of the Y-channel power amplifiery(t) pull transformation.
In the above formula, the first and second carbon atoms are,
iy(s)＝-KsKiGc(s)Gw(s)(Y(s)-Θy(s)+yd(s))
in the formula, yd(s) is the sensor harmonic yd(t) pull transformation.
From the above formula, it can be seen that due to the mass imbalance and the existence of sensor harmonic, the current component-K with the same frequency as the rotating speed exists in the coil currentsKiGc(s)Gw(s)(Y(s)-Θy(s)) and a frequency-multiplied current component-KsKiGc(s)Gw(s)yd(s)。
Designing a magnetic suspension rotor odd harmonic current suppression method based on a second-order odd repetitive controller;
the harmonic current is taken as a control target, and the algorithm controller is embedded into the original closed-loop system in an 'insertion' mode. Will harmonic current ixThe error signal is input to the plug-in repetitive controller module, and the output of the module is equivalently fed back to the power amplifier input end of the original control system. The design of the module mainly comprises the following two aspects:
①, by adopting an SOORC structure algorithm, carrying out spectrum analysis on harmonic current actually generated by a magnetic suspension rotor system at any rotating speed to obtain that main frequency components of the harmonic current are odd harmonic current, according to a general mode of SOORC structure design, designing an SOORC internal model link based on 2k +1(k 0.1.2.3.) second dominant frequency, obtaining the dominant harmonic component in the harmonic current by spectrum analysis, and completing the design by referring to the main harmonic component by the SOORC internal model structure.
② the phase lead-lag compensation link is composed of a phase lead-lag correction link and a first-order low-pass filter, which is determined according to the system function phase-frequency characteristic and the system stability condition.
Further, the harmonic current suppression algorithm in the step (2) is as follows:
① structural design of SOORC controller
The Repetitive Controller (RC) realizes error signal tracking based on an internal model principle and can eliminate error signal tracking by introducing infinite closed-loop poles
And frequency-multiplied harmonic components. The sorrc structure can be designed to contain an internal model of the 2k +1(k 0.1.2.3.) subharmonic frequency, i.e. an internal model link corresponding to the odd harmonic current frequency component is introduced, enabling accurate positioning and pole introduction of the odd harmonic frequency. Thus, it is possible to prevent the occurrence of,at the introduced frequency point, the system frequency response can obtain infinite gain.
SOORC structure transfer function GSOORC(z) can be expressed as:
wherein the content of the first and second substances,
krcis GSOORC(z) a controller gain, q (z) a low pass filter; n is a radical of2Discrete delay sampling points representing the SOORC;
by adopting an SOORC structure, the frequency spectrum analysis is carried out on the harmonic current actually generated by the magnetic suspension rotor system at any rotating speed, and the main frequency component of the harmonic current is odd harmonic current. According to the general approach of the sorc design, the sorc internal model element is designed based on 2k +1(k 0.1.2.3.) subharmonic frequencies as the dominant frequency.
As can be seen from the principle of the SOORC internal model, the frequency response of the odd harmonic components can be suppressed to almost zero. The SOORC provides some improvement in robustness of the control system when the harmonic current frequency is varied or uncertain compared to a conventional RC. At the same time, the controller gain krcThe system dynamic performance can be improved by proper adjustment;
② phase compensation function Kf(z) design of
In order to realize system stability, a system amplitude-frequency characteristic correction method based on serial connection of a plurality of leading links and lagging links is designed, namely: compensation function Kf(z) is designed as:
Kf(z)＝G1(z)G2 m(z)G3(z)q(z)(m＝0,1,2…)
wherein G is1(z) is a low-band compensation element, which is generally expressed in the form of: (Z (. cndot.) is a discretized notation)
The coefficient b is selected according to the system, so that the effective correction of the low frequency band of the system is realized, and meanwhile, the characteristic change of the medium and high frequency bands is small.
G2 m(z) is a middle frequency band lead compensation link, and the general expression form is as follows:
coefficient a, parameter TaAnd m is specifically selected according to the phase compensation requirement of the system, so that the intermediate frequency band of the system is effectively corrected.
G3(z) is the middle-low frequency band lag correction, and the general expression is as follows:
coefficient c, parameter TbAccording to G2 m(z) the look-ahead effect is designed so that the system is passing through G2 m(z) the intermediate frequency band after the advance correction meets the system stability condition.
q (z) is a cut-off frequency of ωcA low-pass filter of general expression:
wherein, ω iscThe system cutoff frequency.
By adopting the mode, the redundancy of the system stability design can be improved, and the dynamic performance and the steady-state performance of the system can be improved.
The basic principle of the invention is as follows: the conventional RC can realize effective suppression of harmonic current, but the conventional RC realizes effective suppression of harmonic current on the premise that the frequency of the harmonic current is accurately determined, and when the frequency of the harmonic current is changed or is uncertain, the harmonic current suppression effect of the conventional RC is greatly attenuated. The second-order RC has certain control robustness on frequency change or uncertainty in the harmonic current suppression process, and can improve the harmonic current suppression precision and convergence speed when the frequency of the harmonic current changes or is uncertain.
Compared with the prior art, the invention has the advantages that:
(1) the invention provides a magnetic suspension rotor odd harmonic current suppression method based on a second-order odd repetitive controller, aiming at effectively suppressing harmonic current in a magnetic suspension rotor system. The internal model link of the sorrc can realize accurate positioning and pole introduction for harmonic frequencies (including 2k +1 (k.: 0.1.2.3.) subharmonic frequencies), so as to realize effective suppression of system harmonic current. The SOORC structure can be realized by adjusting the parameter w of the internal mold link1And w2To improve the robustness of the system. The convergence rate of the SOORC on the harmonic current control is increased compared to the conventional RC, and the dynamic performance of the system is improved accordingly. The phase lag-lead compensation link is added, so that the system stability can be ensured on one hand, and the gain k of the system controller is widened on the other handrcThe upper limit of the value of (2) improves the dynamic performance of the system. The method is suitable for magnetic suspension rotor harmonic current suppression with mass unbalance and sensor harmonic.
(2) The invention combines the SOORC structure and the phase lag-lead link, improves the dynamic performance and the steady-state performance of the system, and optimizes the harmonic suppression effect of the system.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a schematic structural diagram of an active and passive magnetic suspension rotor system;
FIG. 3 is a schematic view of a static imbalance of a rotor;
FIG. 4 is a schematic diagram of sensor harmonics;
FIG. 5 is a block diagram of an X-channel magnetic bearing control system;
FIG. 6 is a block diagram of a Y-channel magnetic bearing control system;
FIG. 7 is a block diagram of the overall system for improving a plug-in repetitive controller for the X channel;
FIG. 8 is a block diagram of the overall system for improving a plug-in repetitive controller for the X channel;
fig. 9 is a block diagram of a specific structure of the plug-in repetitive controller.
Detailed Description
The invention is further described with reference to the following figures and specific examples.
As shown in fig. 1, an implementation process of a magnetic suspension rotor odd harmonic current suppression method based on a second-order odd-order repetitive controller is as follows: firstly, establishing a magnetic suspension rotor dynamic model containing rotor mass unbalance and sensor harmonic waves; then designing a magnetic suspension rotor odd harmonic current suppression method based on a second-order odd repetitive controller;
step (1) establishing a magnetic suspension rotor dynamic model containing mass unbalance and sensor harmonic
The structure schematic diagram of a magnetic suspension rotor system is shown in figure 2 and mainly comprises a permanent magnet (1), a driving magnetic bearing (2) and a rotor (3), wherein two radial degrees of freedom are controlled by the driving magnetic bearing, and the other three degrees of freedom are passively and stably suspended by permanent magnet rings arranged on the rotor and a stator, figure 3 is a static unbalance schematic diagram of the rotor, Q represents the geometric center of a magnetic bearing stator, O represents the geometric center of the rotor, C represents the mass center of the rotor, an inertial coordinate system QXY is established by taking Q as the center, a rotating coordinate system O epsilon η is established by taking O as the center, and X, y represent coordinate values of the geometric center O of the rotor under the inertial coordinate system, aiming at a radial translation degree of freedom X channel, the modeling is as follows:
according to Newton's second law, the dynamic equation of the magnetic suspension rotor in the X direction is as follows:
wherein m represents the rotor mass, fxThe bearing force of the magnetic bearing in the X direction is shown, e represents the deviation between the geometric center and the mass center of the rotor, omega represents the rotating speed of the rotor, and phi represents the initial phase of the unbalanced mass of the rotor.
The active and passive magnetic bearings are structurally composed of active magnetic bearings and passive magnetic bearings. The bearing force of the active and passive magnetic bearings is composed of the electromagnetic force of the active magnetic bearing and the permanent magnetic force of the passive magnetic bearing, so that the bearing force f in the X channelxCan be written as：
fx＝fex+fpx
Wherein f isexElectromagnetic force for X-channel active magnetic bearing, fpxIs the permanent magnetic force of the X-channel passive magnetic bearing. The permanent magnetic force is linearly related to the displacement and is expressed as:
fpx＝Kprx
wherein, KprIs the displacement stiffness of the passive magnetic bearing;
when the rotor is suspended near the magnetic center, the active magnetic bearing electromagnetic force can be approximately linearized as:
fex≈Kerx+Kiix
wherein, Ker、KiRespectively, the displacement stiffness and the current stiffness of the active magnetic bearing ixOutputting current for the power amplifier;
in an actual rotor system, due to the influence of factors such as magnetic bearing assembly error, rotor measurement surface roundness error, and electromagnetic unevenness in fig. 2, sensor harmonics such as those shown in fig. 4 are generated, where 4 denotes a sensor, 5 denotes a stator, and 6 denotes a rotor. Displacement x actually measured by the sensors(t) can be expressed as:
xs(t)＝x(t)+xd(t)
wherein x isd(t) is the sensor harmonic, which can be rewritten as:
wherein, caIs the magnitude of the harmonic coefficient of the sensor, θaIs the phase of the sensor harmonic coefficient, w is the highest number of sensor harmonics;
the magnetic bearing X-direction translation control system is shown in figure 5, wherein KsA gain element G of the displacement sensorc(s) is a controller element, Gw(s) is a power amplifier link, and P(s) is a transfer function of the rotor system; will ix、X(t)、Θx(t)、xd(t) successively carrying out Laplace conversion into ix(s)、X(s)、Θx(s)、xd(s), the rotor dynamics equation is written as:
ms2X(s)＝(Ker+Kpr)(X(s)-Θx(s))+Kiix(s) wherein (a) a,
ix(s)＝-KsKiGc(s)Gw(s)(X(s)-Θx(s)+xd(s))
wherein X (t) is the displacement of the center of mass of the rotor, and x (t) is the displacement of the geometric center of the rotor, thetax(t) displacement disturbances caused by mass imbalance.
As can be seen from the above formula, due to the mass unbalance and the existence of sensor harmonic wave, the current component-K with the same frequency as the rotating speed exists in the coil currentsKiGc(s)Gw(s)(X(s)-Θx(s)) and a frequency-multiplied current component-KsKiGc(s)Gw(s)xd(s)。
In the controllable radial translation freedom degree X channel and the controllable radial translation freedom degree Y channel of the active magnetic bearing, the two channels are decoupled, so that a current model of the Y channel is similar to that of the X channel, and the specific analysis is as follows:
the magnetic bearing Y-direction translation control system is shown in figure 6, wherein KsA gain element G of the displacement sensorc(s) is a controller element, Gw(s) is a power amplifier link, and P(s) is a transfer function of the rotor system.
The rotor dynamics equation is:
ms2Y(s)＝(Ker+Kpr)(Y(s)-Θy(s))+Kiiy(s)
wherein Y(s) is the pull-type transformation of the displacement y (t) of the center of mass of the rotor, thetay(s) Displacement disturbance Θ caused by Mass imbalancey(t) pull transformation, iy(s) is the output current i of the Y-channel power amplifiery(t) pull transformation.
In the above formula, the first and second carbon atoms are,
iy(s)＝-KsKiGc(s)Gw(s)(Y(s)-Θy(s)+yd(s))
in the formula, yd(s) is the sensor harmonic yd(t) pull transformation.
As can be seen from the above formula, due to the mass unbalance and the existence of sensor harmonic wave, the current component-K with the same frequency as the rotating speed exists in the coil currentsKiGc(s)Gw(s)(Y(s)-Θy(s)) and a frequency-multiplied current component-KsKiGc(s)Gw(s)yd(s)。
Harmonic vibration caused by harmonic current can be transmitted to the spacecraft through the magnetic bearing, so that the attitude control precision of the spacecraft is indirectly influenced. Therefore, effective suppression of harmonic currents is very essential;
step (2) magnetic suspension rotor odd harmonic current suppression method based on second-order odd repetitive controller
Aiming at the problem that harmonic current exists in the coil current in the step (1), the invention adopts a magnetic suspension rotor odd harmonic current suppression method based on a second-order odd-order repetitive controller to suppress the harmonic current.
For the X-channel harmonic current, the sorc is inserted on the basis of the original X-channel closed-loop system, as shown in fig. 7. Displacement deviations caused by unbalanced masses of the X-channel rotor and sensor harmonics as interference signals Rx(s) and Dx(s) through a controller Gc(s) and Power Amplifier GwAfter(s) a harmonic current I is formedx(s)。Ix(s) is fed back to the input via two paths, one path via rotor system Gp(s), the other path is through the "inserted" SOORC. FIG. 9 shows a specific block diagram of the SOORC in FIG. 7, where I (z) is the harmonic current I in the X channelx(s) discretized current sequence, i.e. tracking error, krcFor the gain of the SOORC controller, N is the period of the discrete current sequence I (z), and N ═ fs/f0，f0Is the fundamental frequency, f, of the harmonic current component of the X channelsThe system sampling frequency; n is a radical of2Represents discrete delayed sample points of the sorrc. Kf(z) represents the phase compensation function at the low and mid bands, which, in design,
represents the phase compensation function for the high frequency band, and q (z) is a low pass filter.
Since the X, Y channels are decoupled from each other, the harmonic current suppression mode of the X channel can be imitated for the harmonic current suppression of the Y channel. The specific implementation steps of the Y channel current suppression are as follows: the SOORC is inserted on the basis of the original Y-channel closed-loop system, as shown in FIG. 8, and displacement deviation caused by unbalanced mass of a Y-channel rotor and sensor harmonic are used as interference signals Ry(s) and Dy(s) through a controller Gc(s) and Power Amplifier GwAfter(s) a harmonic current I is formedy(s)，Iy(s) is fed back to the input via two paths, one path via rotor system Gp(s), the other path is through the "inserted" SOORC. FIG. 9 shows a block diagram of the specific structure of the SOORC controller in FIG. 8, where I (z) is the Y-channel harmonic current Iy(s) discretized current sequence, i.e. tracking error, krcFor the gain of the SOORC controller, N is the period of the discrete current sequence I (z), and N ═ fs/f0，f0Is the fundamental frequency, f, of the harmonic current component of the Y channelsThe system sampling frequency; n2 represents the number of discrete delayed sample points for the SOORC. Kf(z)，And Q (z) is consistent with the role and definition of X channel.
As can be seen from FIG. 9, the transfer function G of the SOORC structureSOORC(z) can be expressed as:
wherein the content of the first and second substances,
krcis GSOORC(z) the corresponding controller gain, and Q (z) is a low pass filter.
The SOORC controller design process is as follows:
the invention is applied toUnder the condition of high rotating speed, the magnetic suspension rotor system can know that the effective harmonic disturbance of the rotor is mainly expressed in the same frequency, the frequency tripling, the frequency quintupler, the frequency heptad and the frequency nonad according to the frequency spectrum analysis. Therefore, it can be known from the spectrum analysis result that the odd harmonic frequency components in the rotor system occupy the dominant position of the harmonic current component. For the magnetic levitation rotor system, a second-order odd harmonic frequency suppression Structure (SOORC) is designed as shown in FIG. 9, wherein the transfer function G of the inner mode structureSOORC(z) can be expressed as:
wherein, w1-w2＝1
Stability analysis and phase lag-lead compensation link design:
a) and stability analysis:
for the closed loop system as shown in fig. 7 and 8, the closed loop system is asymptotically stable if the following conditions are simultaneously satisfied:
condition 1:
if 0<w2<1, then
If-1<w2<0, then
Condition 2:
the following equation is satisfied when the condition 1 and the condition 2 are satisfied:
wherein the content of the first and second substances,
for the function of the system after the phase compensation,and
respectively representing the post-phase-compensation system function
The amplitude and phase angle of;for the low-band compensation function in the system,
and
respectively represent
The amplitude and phase angle of;
the function of the original system is represented,and
respectively representing system functions
The amplitude and phase angle of;
is a low-pass filter;a system high-frequency compensation link is adopted; t issSampling time for controlling a system; n is a radical of2The number of delay sampling points of the discrete system.
For the system shown in FIG. 9, krcIs given by the condition 1, N2The value range of (d) is given by the condition 2. In the case of a real system, for example,
and Tsω(ω≈ωc) Are known and, therefore, system parameters can be determined.
b) Designing a phase lag-lead compensation link:
phase compensation link KfThe general form of (z) is:
Kf(z)＝G1(z)G2 m(z)G3(z)q(z)(m＝0,1,2…)
wherein G is1(Z) is low band compensation, which is generally expressed in the form (Z (-) is a discretized notation):
the coefficient b is selected according to the system, so that the effective correction of the low-frequency band characteristic of the system is realized, and the change of the medium-high band characteristic is small.
G2 m(z) is the middle frequency band lead compensation, and the general expression is as follows:
middle frequency band phase compensation link G2 m(z) consists of m phase lead elements. Since each one isThe maximum lead angle provided by the phase lead element does not exceed 65 DEG, therefore G2(z) can provide a maximum lead angle of no more than m x 65 °. In order to ensure a certain phase redundancy and signal-to-noise ratio of the corrected system, the lead angle provided by each lead element is generally between 40 ° and 50 °. And determining the value m according to the angle required by the system to reach the stable state and the lead angle provided by each lead link. If the phase lowest point of the system bode diagram corresponds to a phase of
Selecting the lead angle provided by the lead link to be 45 degrees, and then:
G3(z) is the middle-low frequency band lag correction, and the general expression is as follows:
the system is passing through G2 mAfter the leading phase compensation of (z), the phase of the system in the medium and low frequency band may be changed greatly, thereby affecting the overall performance of the system. To weaken due to G2 m(z) to influence the low-frequency band in the system, a lag correction link G needs to be added3(z)。
Coefficient c, parameter TbAccording to G2 m(z) the look-ahead effect is designed so that the system is passing through G2 mAnd (z) the intermediate frequency phase after the advance correction meets the system stability condition.
q (z) is a cut-off frequency of ωcThe low-pass filter of (2), generally expressed in the form of:
wherein, ω iscFor system cut-off frequency, high-frequency compensation link can be weakened
The phase of the high frequency band of the system is influenced.
In summary, by introducing the phase compensation function and the gain coefficient, the stability of the system after the algorithm is added can be ensured.
Those skilled in the art will appreciate that the invention may be practiced without these specific details.
Claims (1)
1. A magnetic suspension rotor odd harmonic current suppression method based on a second-order odd repetitive controller is characterized by comprising the following steps: the method comprises the following steps:
step (1): establishing a magnetic suspension rotor dynamic model containing mass unbalance and sensor harmonic waves;
the two radial translational degrees of freedom of the magnetic suspension rotor are respectively controlled by an active magnetic bearing, and the two radial torsional degrees of freedom and the two axial translational degrees of freedom are respectively controlled by permanent magnet rings arranged on the rotor and the stator, namely, the passive magnetic bearing, so as to realize passive stable suspension control, wherein Q is the geometric center of the stator of the magnetic bearing, O is the geometric center of the rotor, C is the centroid of the rotor, an inertial coordinate system QXY is established by taking Q as the center, and a rotating coordinate system O epsilon η is established by taking O as the center, (x, y) represents the coordinate value of the geometric center O of the rotor in the inertial coordinate system;
for the X channel harmonic current, the modeling is as follows:
the dynamic equation of the magnetic suspension rotor in the X direction obtained from Newton's second law is as follows:
where m is the rotor mass, fxThe bearing force of the magnetic bearing in the X direction, e is the deviation between the geometric center and the mass center of the rotor, omega is the rotating speed of the rotor, and phi is the initial phase of the unbalanced mass of the rotor;
the active and passive magnetic bearings are composed of active and passive magnetic bearings, and the bearing force of the active and passive magnetic bearings is composed of the electromagnetic force of the active magnetic bearing and the electromagnetic force of the passive magnetic bearingPermanent magnetic force, so that the bearing force f in the X channelxCan be written as:
fx＝fex+fpx
wherein f isexElectromagnetic force for X-channel active magnetic bearing, fpxThe magnetic force is the permanent magnetic force of the X-channel passive magnetic bearing, wherein the permanent magnetic force of the passive magnetic bearing and the displacement are in a linear relation and are expressed as follows:
fpx＝Kprx
wherein, KprIs the displacement stiffness of the passive magnetic bearing; x is the displacement value of the rotor geometric center O in the X channel under the inertial coordinate system;
when the rotor is suspended near the magnetic center, the active magnetic bearing electromagnetic force can be approximately linearized as:
fex≈Kerx+Kiix
wherein, Ker、KiRespectively, the displacement stiffness and the current stiffness of the active magnetic bearing ixOutputting current for the power amplifier;
for rotor systems with mass unbalance there are:
X(t)＝x(t)+Θx(t)
wherein X (t) is the displacement of the center of mass of the rotor, x (t) is the displacement of the geometric center of the rotor, and thetax(t) is the displacement disturbance caused by mass imbalance, and is recorded as:
Θx(t)＝lcos(Ωt+θ)
wherein l is the amplitude of mass unbalance, theta is the phase, and omega is the rotor speed;
considering the influence of mechanical processing precision and material nonuniformity factors in an actual rotor system, sensor harmonic is usually unavoidable, so that the displacement x actually measured by the sensors(t) can be expressed as:
xs(t)＝x(t)+xd(t)
wherein x isd(t) is the sensor harmonic, which can be rewritten as:
wherein, caIs the magnitude of the harmonic coefficient of the sensor, θaIs the phase of the sensor harmonic coefficient, w is the highest number of sensor harmonics;
will ix、X(t)、Θx(t)、xd(t) successively carrying out Laplace conversion into ix(s)、X(s)、Θx(s)、xd(s), the rotor dynamics equation is written as:
ms2X(s)＝(Ker+Kpr)(X(s)-Θx(s))+Kiix(s)
wherein the content of the first and second substances,
ix(s)＝-KsKiGc(s)Gw(s)(X(s)-Θx(s)+xd(s))
wherein, KsA gain element G of the displacement sensorc(s) is a controller element, Gw(s) is a power amplifier link;
according to the formula, the current component-K with the same frequency as the rotating speed exists in the coil current due to the unbalanced rotor mass and the harmonic wave of the sensorsKiGc(s)Gw(s)(X(s)-Θx(s)) and a frequency-multiplied current component-KsKiGc(s)Gw(s)xd(s), and the same frequency current can be converted into frequency doubling current again under the nonlinear action of the magnetic bearing;
in the controllable radial translation freedom degree X channel and the controllable radial translation freedom degree Y channel of the active magnetic bearing, the two channels are decoupled, so that a current model of the Y channel is similar to that of the X channel, and the specific analysis is as follows:
the rotor dynamics equation is:
ms2Y(s)＝(Ker+Kpr)(Y(s)-Θy(s))+Kiiy(s)
wherein Y(s) is the pull-type transformation of the displacement y (t) of the center of mass of the rotor, thetay(s) Displacement disturbance Θ caused by Mass imbalancey(t) pull transformation, iy(s) is the output current i of the Y-channel power amplifiery(t) pull-type transformations;
in the above formula, the first and second carbon atoms are,
iy(s)＝-KsKiGc(s)Gw(s)(Y(s)-Θy(s)+yd(s))
in the formula, yd(s) is the sensor harmonic yd(t) pull-type transformations;
according to the formula, due to the existence of mass unbalance and sensor harmonic waves, a current component-K with the same frequency as the rotating speed exists in the coil currentsKiGc(s)Gw(s)(Y(s)-Θy(s)) and a frequency-multiplied current component-KsKiGc(s)Gw(s)yd(s)；
Step (2): magnetic suspension rotor odd harmonic current suppression method based on second-order odd repetitive controller
The second-order odd-order repetitive controller takes odd-order harmonic current suppression as a control target, the second-order odd-order repetitive controller is embedded into an original closed-loop system in an 'insertion' mode, and harmonic current ixThe error signal is input to the plug-in repetitive controller module, the output of the module is fed back to the power amplifier input end of the original control system, and the design of the module mainly comprises the following three steps:
① a second-order odd-order repetitive controller structure method, which comprises performing spectrum analysis on harmonic current generated by a magnetic suspension rotor system at any rotation speed to obtain that the frequency component of the harmonic current in the magnetic suspension rotor system is mainly odd harmonic, designing a second-order RC internal model link corresponding to the dominant frequency being odd harmonic frequency multiplication according to the general design mode of the second-order odd-order repetitive controller structure, and obtaining the dominant odd harmonic component in the harmonic current by spectrum analysis;
② design of weighting coefficient of second-order odd-order repetitive controller structure, and obtaining w in structure by stability analysis according to stability criterion for improved RC controller1And w2The relationship between the two weighting coefficients and the reference range can ensure that the second-order system has certain robustness in the control process when the frequency of the harmonic current is changed or uncertain by properly adjusting the weighting coefficients;
③ the phase lead-lag compensation link is composed of a phase lead-lag correction link and a first-order low-pass filter, which is determined according to the system function phase frequency characteristic and the system stability condition, the compensation link can improve the system stability, broaden the upper limit of the gain value of the controller, and increase the redundancy of the system stability design and improve the dynamic performance and the steady-state performance to a certain extent;
the current suppression method in the step (2) comprises the following steps:
① structural design of second-order odd-order repetitive controller
The repetitive controller RC realizes error signal tracking based on an internal model principle, can eliminate n frequency multiplication harmonic components by introducing infinite closed-loop poles,
although the traditional RC structure can realize the suppression of all frequency doubling harmonic components, when the traditional RC structure is adopted, the robustness of a control system to frequency change is poor, an internal model link for suppressing the frequency doubling harmonic components is introduced into the second-order odd-order repetitive controller structure, and the accurate positioning and the pole introduction can be realized aiming at the harmonic frequency components needing to be suppressed, so that the frequency response of the system is infinite gain at the introduced frequency points;
second order odd-order repetitive controller structure transfer function GSOORC(z) can be expressed as:
wherein the content of the first and second substances,
krcis GSOORC(z) a controller gain, q (z) a low pass filter; n is a radical of2The discrete delay sampling point number of the second-order odd-order repetitive controller is represented; n is the period of the discrete current sequence I (z) and has N ═ fs/f0，f0Is the fundamental frequency, f, of the harmonic current component of the X channelsThe system sampling frequency;
according to the frequency spectrum analysis of harmonic current generated by a magnetic suspension rotor system at any rotating speed, the main frequency component of the harmonic current is odd harmonic, and according to a general mode of designing a second-order odd repetitive controller, a second-order odd repetitive controller internal model link based on 2k +1 dominant frequency is designed, wherein k is 0.1.2.3;
according to the internal model principle of the second-order odd-order repetitive controller, the frequency response of odd harmonic components can be almost suppressed to zero, compared with the traditional RC, the second-order odd-order repetitive controller has the advantages that when the frequency of harmonic current is changed or uncertain, the robustness of a control system is improved to a certain extent, and meanwhile, the gain k of the controller is increasedrcThe system dynamic performance can be improved by proper adjustment;
② phase compensation function Kf(z) design of
In order to ensure the stability of the system, a system amplitude-frequency characteristic correction method based on the series connection of a plurality of leading links and lagging links is designed, namely: compensation function Kf(z) is designed as:
Kf(z)＝G1(z)G2 m(z)G3(z)q(z)，m＝0,1,2…
wherein G is1(z) is low band compensation, which is generally expressed as:
z (-) is a discretization mark, the coefficient b is specifically selected according to the system, effective correction of the low frequency band of the system is realized, and meanwhile, the characteristic change of the high frequency band of the system is small;
G2 m(z) is a medium-frequency phase compensation function of m leading phase compensation links connected in series, and the general expression form is as follows:
coefficient a, parameter TaM is specifically selected according to the phase compensation requirement of the system, so that the effective correction of the frequency band characteristic in the system is realized;
G3(z) is a medium or low frequencySegment lag correction, expressed generally as:
coefficient c, parameter TbAccording to G2 m(z) the look-ahead effect is designed so that the system is passing through G2 m(z) the intermediate frequency band after the advanced correction meets the system stability condition;
q (z) is a cut-off frequency of ωcA low-pass filter of general expression:
wherein, ω iscIs the system cutoff frequency;
by adopting the mode that the links are connected in series, the redundancy of the stability design of the system can be improved, and the dynamic performance and the steady-state performance of the system can be improved.
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| CN107844052B (en) * | 2017-10-14 | 2020-11-10 | 河南工业大学 | Method for suppressing harmonic current of magnetic bearing system based on parallel finite dimension repeated control |
| CN108106611B (en) * | 2017-12-21 | 2021-07-02 | 北京航空航天大学 | Magnetic suspension rotor harmonic current suppression method based on multiple phase shift quasi-resonance control |
| CN108490777B (en) * | 2018-03-19 | 2020-06-19 | 北京航空航天大学 | Magnetic suspension rotor harmonic vibration force suppression method based on improved odd-order repetitive control |
| CN108897218B (en) * | 2018-07-06 | 2021-02-02 | 北京航空航天大学 | Magnetic suspension rotor odd harmonic vibration suppression method based on mixed odd-order repetitive controller |
| CN111650975B (en) * | 2020-06-09 | 2021-05-25 | 北京航空航天大学 | Magnetic suspension rotor harmonic current suppression method based on multi-order repetitive controller |
| CN111752153B (en) * | 2020-06-24 | 2021-09-14 | 北京航空航天大学 | Harmonic current suppression method based on 1.5-order hybrid repetitive controller |
| CN112525179B (en) * | 2020-11-30 | 2023-03-14 | 北京航空航天大学 | Harmonic current suppression method based on virtual sampling second-order repetitive control |
| CN113741181A (en) * | 2021-01-18 | 2021-12-03 | 南京航空航天大学 | Rotating speed self-adaptive magnetic suspension rotor system odd harmonic current suppression method |
| CN112983989B (en) * | 2021-03-05 | 2022-06-07 | 广东美的暖通设备有限公司 | Magnetic suspension bearing and control method and system thereof |
| CN113188719B (en) * | 2021-04-21 | 2022-10-14 | 北京控制工程研究所 | Examine online dynamic balance system of accuse stator permanent-connection formula magnetic suspension rotary joint |
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