CN113114230A - Resonant frequency online identification and suppression method for servo system - Google Patents
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Abstract
The invention discloses a method for identifying and inhibiting resonant frequency of a servo system on line, which comprises the following steps: inputting a rotating speed deviation and setting an initial frequency; obtaining a transfer function according to the structure of the second-order generalized integrator, and calculating an output signal; calculating a frequency error signal, inputting the frequency error signal into a low-pass filter of a frequency locking ring, and multiplying the output of the low-pass filter by a proportional controller; multiplying the output of the proportional controller by a gain self-adjusting unit, inputting the output value into an integrator, summing the output frequency of the integrator with a set initial frequency to obtain an estimated resonant frequency, and judging whether convergence time is reached; when the convergence time is reached, the estimated resonance frequency is input to the notch filter, and the input is set as a notch frequency point of the notch filter, thereby realizing resonance suppression. The invention can accurately identify and inhibit the resonance of the servo system, particularly under the condition of higher rotating speed vibration frequency, thereby improving the performance of the servo system.
Description
Technical Field
The invention belongs to the technical field of servo drive control, and particularly relates to a resonant frequency online identification and suppression method for a servo system.
Background
The conventional mechanical resonant frequency identification methods are mainly classified into an off-line method and an on-line method. The off-line identification method is carried out when the servo system is in an abnormal working state, generally, an excitation signal is additionally applied, a bode diagram is drawn according to the rotating speed response of the system, and the characteristics of the resonant peak, such as frequency, amplitude and the like, are directly observed through the bode diagram. Although the method can accurately obtain the resonance characteristic of the system, the servo system is often not allowed to be in an abnormal working condition in some special occasions, so that the method has certain limitation. The on-line identification method is to directly identify and obtain the resonant frequency without affecting the normal operation of the servo system. The current mainstream online identification method is an FFT analysis method, when a system generates a resonance phenomenon, a rotating speed deviation signal of a motor is collected, FFT conversion is carried out on the signal to obtain a rotating speed deviation frequency spectrum, and a frequency point with the maximum energy in the frequency spectrum is the resonance frequency. The method is simple and easy to implement, but the sampling frequency of the signal and the selection of the number of points of FFT conversion both affect the accuracy and rapidity of the identification result, and simultaneously occupy a large amount of resources, so that the rapidity and accuracy of the identification result are difficult to ensure.
The second-order generalized integrator combined frequency-locked loop (SOGI-FLL) algorithm has been widely applied to the fields of grid synchronization, phase sequence detection, harmonic compensation algorithm, synchronous angular frequency estimation of the motor and the like. This is all due to the properties of the SOGI: the SOGI is capable of resonating at a specified frequency, producing two-phase quadrature output signals equal to the input signal resonant frequency, and therefore has a good filtering effect. And the resonant frequency is adjusted by combining with the FLL, so that the frequency self-adaptive function is realized. However, there are problems that the recognition accuracy is low after the digital realization and the recognition convergence time is affected by the characteristics of the frequency and amplitude of the input signal.
Disclosure of Invention
The invention mainly aims to overcome the defects of the prior art and provide an on-line identification and inhibition method for the resonant frequency of a servo system, which can accurately identify and inhibit the resonance of the servo system, and particularly improves the performance of the servo system under the condition of higher rotating speed and vibration frequency.
In order to achieve the purpose, the invention adopts the following technical scheme:
a resonant frequency online identification and suppression method for a servo system comprises the following steps:
s1, inputting the rotation speed deviation and setting an initial frequency, wherein the initial frequency is used for accelerating the convergence speed of the SOGI-FLL identification;
s2, obtaining a transfer function according to the structure of the second-order generalized integrator, and calculating an output signal of the second-order generalized integrator;
s3, calculating a frequency error signal, inputting the frequency error signal into a low-pass filter of a frequency lock loop, multiplying the output of the low-pass filter by a proportional controller, and calculating the convergence time of frequency identification;
s4, multiplying the output of the proportional controller by the gain self-adjusting unit, inputting the output value into the integrator, and summing the output frequency of the integrator with the set initial frequency to obtain the estimated resonance frequency;
s5, judging whether convergence time is reached; if the convergence time is not reached, feeding back the estimated resonance frequency to step S2, and calculating again;
if the convergence time is reached, the estimated resonance frequency is input into the notch filter and set as a notch frequency point of the notch filter, thereby realizing resonance suppression.
Further, the calculating the output signal of the second-order generalized integrator is specifically calculating the output signals u ', qu' and e of the second-order generalized integratoruThe method comprises the following steps:
obtaining a transfer function according to the structure of the second-order generalized integrator, wherein the transfer function comprises:
the transfer function d(s) from the second order generalized integrator input signal u to the noise-removed equiphase output signal u' is:
the transfer function q(s) from the input signal u to the quadrature phase output signal qu' is:
from the input signal u to the synchronization error signal euThe transfer function E(s) of (D) is:
wherein ω' is the estimated resonant frequency and k is the proportional controller of the SOGI;
multiplying the input rotation speed deviation u by corresponding transfer functions to obtain u ', qu' and euWhen the output signal is calculated for the first time, ω' takes the initial frequency ω0。
Further, the calculating the frequency error signal specifically includes:
frequency error signal ef＝eu*qu′；
When the frequency of the input signal u is lower than the estimated resonance frequency ω', the signal euThe phases of the sum qu' are in phase, and the frequency error signal e is generated according to the vector multiplication principlef＝euQu' and ef>0；
When the frequency of the input signal u is higher than the estimated resonance frequency ω', the signal euAnd qu' are in antiphase, when ef＜0；
Using frequency error signals efA frequency locking loop FLL is designed, which uses an integrator with negative gain and a self-adjusting unit of gain to realize a frequency adaptation function, so that the bandwidth center frequency ω' of the SOGI tracks to the resonance frequency ω of the input signal u.
Further, the multiplying the output of the low-pass filter by the proportional controller is specifically:
the output of the low-pass filter is multiplied by a proportional controller-gamma, gamma being a positive number, to determine the convergence time of the frequency identification.
Further, the step of calculating the convergence time of the frequency identification specifically comprises:
according to the structure of the second-order generalized integrator, the state space expression is specifically as follows:
the synchronization error signal and the frequency error signal are respectively expressed as:
ef＝qu'·eu (7)
when the SOGI tends towards steady state, i.e., ω' ≈ ω, equation (4) is written as:
wherein the content of the first and second substances,
is the state variable of the SOGI in steady state;
according to equation (8), rewriting equations (6) and (7) is:
when the input rotating speed deviation signal is
According to the structure of the second-order generalized integrator, the steady-state output value of the SOGI and the state variable have the following relationship:
(u′)2+(qu′)2＝R2 (12)
according to the structure of the SOGI-FLL, the frequency characteristic for the frequency locked loop FLL is expressed as:
wherein the content of the first and second substances,
a gain self-adjusting unit which is an input;
under steady state conditions, i.e.. omega '. apprxeq.omega, where omega'2-ω22(ω '- ω) ω', the average dynamic characteristic of the FLL is described as:
according to the formula (16), in the resonant frequency online identification and suppression method, the FLL simplified model with the frequency and gain self-adjustment performance is:
equation (17) is a first-order inertia element, and the frequency identification convergence time is specifically:
according to the formula (18), after the coefficient gamma of the first proportional controller of the FLL is determined, the convergence process of the online identification algorithm is not influenced by the vibration frequency and the amplitude of the input signal u, and the frequency identification convergence time is obtained according to the formula (18).
Further, the multiplying the output of the proportional controller by the gain self-adjusting unit is specifically as shown in equation (15), thereby obtaining a simplified model of the FLL under the steady-state condition as shown in equation (17);
according to the formula (17), the convergence of ω' is no longer affected by the amplitude and frequency of the rotational speed deviation signal, and is only related to the FLL gain element γ.
Further, the time for not reaching convergence specifically is:
if the convergence time is not reached, feeding back ω' to step S2, calculating the output signal of the second-order generalized integrator again, and continuing to execute the subsequent steps S2; when the subsequent steps adopt omega 'for calculation, the omega' calculated before feedback is adopted.
Further, the transfer function model of the notch filter is:
wherein the content of the first and second substances,
Bwand DpThe notch bandwidth and notch depth of the notch filter, respectively, and f is the notch frequency of the notch filter, i.e., ω ═ 2 π f.
Furthermore, when all integrators are dispersed, the integrators are dispersed by adopting an adaptive pre-distortion bilinear transformation method, and the numerical expression is as follows:
wherein, the input omega' of the discrete integrator is the center frequency of the SOGI, and Ts is the control period of the rotating speed loop of the servo system.
Compared with the prior art, the invention has the following advantages and beneficial effects:
1. the method is based on an improved second-order generalized integrator-frequency lock ring algorithm, has a good filtering function, and can generate output signals which are in phase and orthogonal with an input rotating speed deviation signal; the self-adaptive resonant frequency adjusting function is realized by utilizing the principle of vector multiplication and combining the phase locking function of the frequency locking ring.
2. On the basis of the traditional SOGI-FLL algorithm, the method introduces the pre-distortion bilinear transformation method to realize dynamic discretization on the integrator, so that the algorithm still keeps consistent with the frequency response characteristic in the original continuous domain after the digit is realized, the problem of large steady-state error after the discretization of the algorithm is well solved, and the accuracy of the identification result is improved.
3. According to the method, the gain self-adjusting unit is introduced into the frequency locking ring FLL, and the gain self-adjusting unit can dynamically adjust the frequency control gain according to the amplitude and the frequency of the vibration component in the rotating speed deviation signal, so that the frequency identification convergence time is not influenced by the input signal, and the stability and the rapidity of the identification process are improved.
4. According to the method, the frequency obtained by identifying the SOGI-FLL is used as the trap frequency of the trap, the trap filter is designed, and the trap filter is cut in between the rotating speed loop and the current loop controller, so that the resonance phenomenon of the system can be well eliminated to a certain extent.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a permanent magnet AC servo system for use with an embodiment of the present invention;
FIG. 3 is a block diagram of the SOGI-FLL according to the embodiment of the invention;
FIG. 4 is a bode plot of the transfer function of the output signal and the input signal of the SOGI-FLL of the present embodiment;
FIG. 5 is a comparison graph of the recognition performance of the method of the present invention and the conventional method.
Detailed Description
The present invention will be described in further detail with reference to examples and drawings, but the present invention is not limited thereto.
The invention aims to solve the problems of low accuracy, poor rapidity, more occupied resources and the like of mechanical resonant frequency of a servo system identified by adopting an FFT analysis method under the condition that an off-line resonant characteristic identification method is not suitable, provides an improved SOGI-FLL resonant frequency on-line identification method, and introduces a notch filter to realize resonance suppression; the method has the advantages of simple implementation process, high convergence speed, high identification precision and good control effect.
Examples
The invention discloses a resonant frequency online identification and suppression method for a servo system, which is implemented by a permanent magnet alternating current servo system with a flexible connection load, as shown in fig. 2. The system is a rotating speed and current double-closed-loop structure, the position information of the motor is collected through a high-precision encoder and converted into a rotating speed feedback signal through an M/T method, and the deviation between the given rotating speed and the rotating speed feedback is obtained through a speed controller ASR to obtain the given quantity i of the current inner loopqref. Sampling current i by using a traditional vector control modea、ib、icI under a rotating coordinate system is transformed through Clarke transformation and Park transformationd、iqAnd finally, a PWM control signal of the three-phase inverter is obtained through an SVPWM module, and double-loop control of the servo system is completed.
As shown in FIG. 3, isThe internal structure diagram of the SOGI-FLL is that the input signal of the second-order generalized integrator SOGI is u, and the output signal comprises u 'with the same phase as the original signal and an orthogonal signal qu'; synchronization error signal eu＝u-u′，euObtaining an error control signal ke after passing through a first proportional controlleru(ii) a The output of the first integrator is x1The output of the second integrator is x2The two integrators adopt frequency self-adaption to disperse in digital implementation; frequency error signal ef＝euQu' as input signal for frequency locked loop FLL, efObtaining a frequency compensation signal after passing through a first proportional controller and a gain self-adjusting unit, wherein the coefficient of the first proportional controller is-gamma, and gamma is a positive number; x is the number of3The bandwidth center frequency ω' ═ x of the SOGI, which is the output of the first integral controller3+ω0Wherein ω is0The frequency is initially set. In practical design, the rotation speed deviation signal is taken as the input u of the SOGI, and the bandwidth center frequency ω' is the mechanical resonance frequency of the servo system.
As shown in fig. 1, the present invention provides an online identification and suppression method for resonant frequency of a servo system, comprising the following steps:
s1, inputting the rotating speed deviation and setting the initial frequency omega0Setting an initial frequency for accelerating the convergence speed of the SOGI-FLL identification; the resonance frequency of the servo system is generally between 100Hz-2000Hz, and the initial frequency in this embodiment is 500 Hz.
S2, calculating output signals u ', qu' and e of second-order generalized integratoruThe method specifically comprises the following steps:
the transfer function d(s) from the input signal u to the noise-removed equiphase output signal u ', the transfer function q(s) from the input signal u to the quadrature-phase output signal qu', and the transfer function q(s) from the input signal u to the synchronization error signal e can be obtained from the block diagram of fig. 3uTransfer function E(s):
wherein ω' is the estimated resonant frequency, k is the proportional controller of the SOGI, which is taken as 1.414 in this embodiment, to set the damping of the system to 0.707, which can better ensure the dynamic performance and the steady-state performance;
multiplying the input rotation speed deviation u by corresponding transfer functions to obtain u ', qu' and eu(ii) a When the output signal is calculated for the first time, ω' may first take the initial frequency ω0。
S3, calculating a frequency error signal, inputting the frequency error signal into a low-pass filter of a frequency lock loop, multiplying the output of the low-pass filter by a proportional controller, and calculating the convergence time of frequency identification, wherein the calculation specifically comprises the following steps:
as can be seen from the block diagram of FIG. 3, ef＝eu*qu′；
FIG. 4 shows a bode plot of transfer functions Q(s) and E(s); in the following, taking ω 800Hz as an example, the frequency error signal is calculated:
when the frequency of the input signal u is lower than the estimated frequency ω' (ω)<ω'), signal euThe phases of the sum qu' are in phase, and the frequency error signal e is generated according to the vector multiplication principlef＝euQu' and ef＞0；
When the frequency of the input signal u is higher than the estimated frequency ω' (ω)>ω'), signal euAnd qu' are in antiphase, when ef＜0。
Using frequency error signals efThe frequency locking loop FLL is designed, and the frequency self-adaptive function is realized by adopting an integrator with a negative gain-gamma and a gain self-adjusting unit, so that the bandwidth center frequency omega' of the SOGI can accurately and quickly track the resonance frequency omega of the input signal u.
Inputting the error signal into a low-pass filter, and multiplying the output of the low-pass filter by a proportional controller, specifically:
will error signal efThe input low pass filter is added to the input end of the FLL, and the bandwidth is set to be 30Hz, considering that the resonant frequency of the servo system is generally high, and in order to reduce the burr in the steady state and improve the identification precision.
The output of the low pass filter is multiplied by a proportional controller-gamma, gamma being a positive number, gamma determining the convergence time of the frequency identification.
Calculating the convergence time of frequency identification, which comprises the following steps:
write the state space expression of FIG. 3:
the synchronization error signal and the frequency error signal are respectively expressed as:
ef＝qu'·eu (7)
when the SOGI tends to steady state, i.e. ω' ≈ ω
Equation (4) can be written as:
wherein the content of the first and second substances,
is a state variable at steady state.
According to equation (8), equations (6) and (7) are rewritten:
when the input rotating speed deviation signal is
As can be deduced from fig. 3, the steady state output value of the SOGI and the state variable have the following relationship:
(u′)2+(qu′)2＝R2 (12)
according to the structure of fig. 3, the frequency characteristic for the frequency locked loop FLL is represented as:
wherein the content of the first and second substances,
a gain self-adjusting unit for input.
Under steady state conditions ω '≈ ω, when ω'2-ω22(ω '- ω) ω', the average dynamic characteristic of the FLL can be described again as:
it can be seen from formula (16) that in the mechanical resonant frequency online identification algorithm of the present invention, the FLL simplified model with frequency and gain self-adjustment performance is:
equation (17) is a first-order inertia element, and the convergence time of frequency identification is as follows:
according to the formula (18), after the first proportional controller coefficient γ of the FLL is determined, the convergence process of the online identification algorithm is not affected by the vibration frequency and amplitude of the input signal u.
In this embodiment, taking the coefficient γ as 100, the convergence time of frequency identification is about 0.04s, which is obtained from the formula (18).
S4, multiplying the output of the proportional controller by the gain self-adjusting unit, inputting the output value into the integrator, and summing the output frequency of the integrator with the set initial frequency to obtain an estimated resonant frequency ω', specifically:
the multiplying the output of the proportional controller by the gain self-adjusting unit is specifically as shown in formula (15), so as to obtain a simplified model of the FLL under the steady-state condition as shown in formula (17);
according to the formula (17), the convergence of ω' is no longer affected by the amplitude and frequency of the rotational speed deviation signal, and is only related to the FLL gain element γ.
S5, determining whether the convergence time is reached, specifically:
if the convergence time is not reached, feeding back ω' to step S2, calculating the output signal of the second-order generalized integrator again, and continuing to execute the subsequent steps S2; when the subsequent steps adopt omega 'for calculation, the omega' calculated before feedback is adopted.
When the convergence time is reached, ω' is input to the notch filter and set as a notch frequency point of the notch filter, thereby realizing a function of resonance suppression.
In this embodiment, the specific transfer function model of the notch filter is as follows:
wherein the content of the first and second substances,
Bwand DpThe notch bandwidth and notch depth of the notch filter, respectively, and f is the notch frequency of the notch filter, i.e., ω ═ 2 π f.
In this embodiment, when all integrators are discretized, an adaptive pre-distortion bilinear transformation method is adopted to discretize the integrators, and the numerical expression is as follows:
in the formula (20), the input of the discrete integrator is the center frequency ω' of the SOGI, and Ts is the control period of the rotation speed loop of the servo system;
in the actual digital system implementation, even if the vibration frequency of the rotation speed deviation signal u is very high, the adaptive discretization method can still ensure that the SOGI generates two stable orthogonal signals, so that the bandwidth center frequency ω' can be accurately converged to the vibration frequency ω of u. Fig. 5 is a comparison graph of the recognition performance of the method of the present invention and the conventional digital implementation.
It should also be noted that in this specification, terms such as "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other identical elements in a process, method, article, or apparatus that comprises the element.
The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.
Claims (9)
1. A method for identifying and suppressing a resonant frequency of a servo system on line is characterized by comprising the following steps:
s1, inputting the rotation speed deviation and setting an initial frequency, wherein the initial frequency is used for accelerating the convergence speed of the SOGI-FLL identification;
s2, obtaining a transfer function according to the structure of the second-order generalized integrator, and calculating an output signal of the second-order generalized integrator;
s3, calculating a frequency error signal, inputting the frequency error signal into a low-pass filter of a frequency lock loop, multiplying the output of the low-pass filter by a proportional controller, and calculating the convergence time of frequency identification;
s4, multiplying the output of the proportional controller by the gain self-adjusting unit, inputting the output value into the integrator, and summing the output frequency of the integrator with the set initial frequency to obtain the estimated resonance frequency;
s5, judging whether convergence time is reached; if the convergence time is not reached, feeding back the estimated resonance frequency to step S2, and calculating again;
if the convergence time is reached, the estimated resonance frequency is input into the notch filter and set as a notch frequency point of the notch filter, thereby realizing resonance suppression.
2. The method as claimed in claim 1, wherein the calculating of the output signal of the second-order generalized integrator is specifically calculating the output signals u ', qu' and e of the second-order generalized integratoruThe method comprises the following steps:
obtaining a transfer function according to the structure of the second-order generalized integrator, wherein the transfer function comprises:
the transfer function d(s) from the second order generalized integrator input signal u to the noise-removed equiphase output signal u' is:
the transfer function q(s) from the input signal u to the quadrature phase output signal qu' is:
from the input signal u to the synchronization error signal euThe transfer function E(s) of (D) is:
wherein ω' is the estimated resonant frequency and k is the proportional controller of the SOGI;
multiplying the input rotation speed deviation u by corresponding transfer functions to obtain u ', qu' and euWhen the output signal is calculated for the first time, ω' takes the initial frequency ω0。
3. The on-line identification and suppression method for the resonant frequency of a servo system as claimed in claim 2, wherein said calculating the frequency error signal comprises:
frequency error signal ef＝eu*qu′；
When the frequency of the input signal u is lower than the estimated resonance frequency ω', the signal euThe phases of the sum qu' are in phase, and the frequency error signal e is generated according to the vector multiplication principlef＝euQu' and ef>0；
When the frequency of the input signal u is higher than the estimated resonance frequency ω', the signal euAnd qu' are in antiphase, when ef＜0；
Using frequency error signals efA frequency locking loop FLL is designed, which uses an integrator with negative gain and a self-adjusting unit of gain to realize a frequency adaptation function, so that the bandwidth center frequency ω' of the SOGI tracks to the resonance frequency ω of the input signal u.
4. The on-line identification and suppression method for the resonant frequency of a servo system as claimed in claim 3, wherein the multiplying the output of the low pass filter by the proportional controller is specifically:
the output of the low-pass filter is multiplied by a proportional controller-gamma, gamma being a positive number, to determine the convergence time of the frequency identification.
5. The method as claimed in claim 4, wherein the step of calculating the convergence time of the frequency identification comprises:
according to the structure of the second-order generalized integrator, the state space expression is specifically as follows:
the synchronization error signal and the frequency error signal are respectively expressed as:
ef＝qu'·eu (7)
when the SOGI tends towards steady state, i.e., ω' ≈ ω, equation (4) is written as:
wherein the content of the first and second substances,
is the state variable of the SOGI in steady state;
according to equation (8), rewriting equations (6) and (7) is:
when the input rotating speed deviation signal is
According to the structure of the second-order generalized integrator, the steady-state output value of the SOGI and the state variable have the following relationship:
(u′)2+(qu′)2＝R2 (12)
according to the structure of the SOGI-FLL, the frequency characteristic for the frequency locked loop FLL is expressed as:
wherein the content of the first and second substances,
a gain self-adjusting unit which is an input;
under steady state conditions, i.e.. omega '. apprxeq.omega, where omega'2-ω22(ω '- ω) ω', the average dynamic characteristic of the FLL is described as:
according to the formula (16), in the resonant frequency online identification and suppression method, the FLL simplified model with the frequency and gain self-adjustment performance is:
equation (17) is a first-order inertia element, and the frequency identification convergence time is specifically:
according to the formula (18), after the coefficient gamma of the first proportional controller of the FLL is determined, the convergence process of the online identification algorithm is not influenced by the vibration frequency and the amplitude of the input signal u, and the frequency identification convergence time is obtained according to the formula (18).
6. The on-line identification and suppression method for the resonant frequency of the servo system as claimed in claim 5, wherein the multiplying the output of the proportional controller by the gain self-adjusting unit is specifically as shown in equation (15), thereby obtaining a simplified model of the FLL under the steady state condition as shown in equation (17);
according to the formula (17), the convergence of ω' is no longer affected by the amplitude and frequency of the rotational speed deviation signal, and is only related to the FLL gain element γ.
7. The on-line identification and suppression method for resonant frequency of servo system as claimed in claim 6, wherein said time to not reach convergence is specifically:
if the convergence time is not reached, feeding back ω' to step S2, calculating the output signal of the second-order generalized integrator again, and continuing to execute the subsequent steps S2; when the subsequent steps adopt omega 'for calculation, the omega' calculated before feedback is adopted.
8. The method as claimed in claim 1, wherein the transfer function model of the notch filter is:
wherein the content of the first and second substances,
Bwand DpThe notch bandwidth and notch depth of the notch filter, respectively, and f is the notch frequency of the notch filter, i.e., ω ═ 2 π f.
9. The on-line identification and suppression method for the resonant frequency of the servo system as claimed in claim 1, wherein all integrators are discretized by adaptive predistortion bilinear transform method when discretized, and the numerical expression is:
wherein, the input omega' of the discrete integrator is the center frequency of the SOGI, and Ts is the control period of the rotating speed loop of the servo system.
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