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CN111082714A - Switched reluctance motor accurate modeling method based on small sample flux linkage characteristics - Google Patents

Switched reluctance motor accurate modeling method based on small sample flux linkage characteristics Download PDF

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CN111082714A
CN111082714A CN202010034404.8A CN202010034404A CN111082714A CN 111082714 A CN111082714 A CN 111082714A CN 202010034404 A CN202010034404 A CN 202010034404A CN 111082714 A CN111082714 A CN 111082714A
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flux linkage
fuzzy
membership function
rotor
switched reluctance
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李存贺
张存山
边敦新
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Shandong University of Technology
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    • HELECTRICITY
    • H02GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02PCONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P6/00Arrangements for controlling synchronous motors or other dynamo-electric motors using electronic commutation dependent on the rotor position; Electronic commutators therefor
    • H02P6/34Modelling or simulation for control purposes
    • HELECTRICITY
    • H02GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02PCONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P23/00Arrangements or methods for the control of AC motors characterised by a control method other than vector control
    • H02P23/0004Control strategies in general, e.g. linear type, e.g. P, PI, PID, using robust control
    • H02P23/0013Control strategies in general, e.g. linear type, e.g. P, PI, PID, using robust control using fuzzy control
    • HELECTRICITY
    • H02GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02PCONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P23/00Arrangements or methods for the control of AC motors characterised by a control method other than vector control
    • H02P23/14Estimation or adaptation of motor parameters, e.g. rotor time constant, flux, speed, current or voltage
    • HELECTRICITY
    • H02GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02PCONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P25/00Arrangements or methods for the control of AC motors characterised by the kind of AC motor or by structural details
    • H02P25/02Arrangements or methods for the control of AC motors characterised by the kind of AC motor or by structural details characterised by the kind of motor
    • H02P25/08Reluctance motors

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  • Automation & Control Theory (AREA)
  • Fuzzy Systems (AREA)
  • Control Of Electric Motors In General (AREA)

Abstract

本发明涉及开关磁阻电机磁链建模技术领域,具体涉及一种基于小样本磁链特性的开关磁阻电机精确建模方法,包括以下步骤:步骤1)、根据电机先验知识进行磁链模型输入和输出变量模糊空间划分;步骤2)、通过模糊逻辑系统提取模糊规则,建立模糊规则库;步骤3)、通过重心法解模糊求解任意电流和位置下的磁链值,完成磁链建模;该方法可以把SRM极对数和磁链随转子位置变化趋势等电机先验知识用于模糊隶属度函数的选取和模糊集划分,并且能够基于样本数据自动创建模糊规则库,实现了SRM小样本磁链特性下的精确建模。

Figure 202010034404

The invention relates to the technical field of switched reluctance motor flux linkage modeling, in particular to an accurate modeling method of switched reluctance motor based on small sample flux linkage characteristics, comprising the following steps: step 1), performing flux linkage according to the prior knowledge of the motor Model input and output variables are divided into fuzzy space; step 2), extract fuzzy rules through fuzzy logic system, and establish fuzzy rule base; step 3), solve the fuzzy value of flux linkage under any current and position by gravity center method, and complete the construction of flux linkage This method can use the prior knowledge of the motor, such as the SRM pole pair number and flux linkage with the rotor position change trend, for the selection of fuzzy membership function and fuzzy set division, and can automatically create a fuzzy rule library based on sample data, realizing SRM Accurate modeling of small sample flux linkage properties.

Figure 202010034404

Description

Switched reluctance motor accurate modeling method based on small sample flux linkage characteristics
Technical Field
The invention relates to the technical field of switched reluctance motor flux linkage modeling, in particular to a switched reluctance motor accurate modeling method based on small sample flux linkage characteristics.
Background
At present, the switched reluctance motor has wide application prospect in the fields of oil field pumping units, wind power generation, electric vehicles and the like due to the advantages of simple structure, large starting torque, wide speed regulation range, high reliability and efficiency and the like. Establishing an accurate mathematical model is critical to SRM performance assessment and implementing advanced control strategies. However, the doubly salient structure and the magnetic saturation characteristics of the SRM itself make it difficult to derive an accurate nonlinear mathematical model thereof through conventional electromagnetic and physical property derivation. The current nonlinear modeling method of the switched reluctance motor mainly comprises the following steps: interpolation iteration method, equivalent magnetic circuit method, function fitting method and neural network approximation method. The equivalent magnetic circuit in the equivalent magnetic circuit method is difficult to divide, the accuracy of magnetic resistance calculation depends on assumption and estimation, and the universality is poor; the function fitting method adopts an analytic expression to carry out nonlinear fitting on the flux linkage characteristic, and the precision of the function fitting method excessively depends on the form of the function analytic expression and the fitting precision of the analytic expression coefficient; both the interpolation iteration method and the neural network approximation method need a large amount of flux linkage sample data, so that the application range is not large.
Disclosure of Invention
In order to solve the deficiencies in the above technical problems, the present invention aims to: the accurate modeling method of the switched reluctance motor based on the small sample flux linkage characteristics is provided, the characteristics of less flux linkage sample data, high accuracy and good rapidity can be realized, and support is provided for performance evaluation and advanced control strategy implementation of the switched reluctance motor.
The technical scheme adopted by the invention for solving the technical problem is as follows:
the accurate modeling method of the switched reluctance motor based on the small sample flux linkage characteristic comprises the following steps:
step 1), carrying out fuzzy space division on input and output variables of a flux linkage model according to the prior knowledge of a motor;
step 2), extracting fuzzy rules through a fuzzy logic system, and establishing a fuzzy rule base;
and 3) solving the flux linkage value under any current and position by solving the fuzzy problem through a gravity center method to complete flux linkage modeling.
Preferably, the following method is specifically adopted in the step 1):obtaining enough sample data of flux linkage changing along with current through experimental measurement, comprehensively considering model complexity and precision, and dividing phase current into 21 intervals by adopting a triangular membership function to perform fuzzy set division; specifically, the jth phase current fuzzy set AjThe membership function of (d) can be described as:
Figure BDA0002365427080000011
wherein i is a current value, j is a phase current, imaxAt maximum current, a ═ imaxAnd/20 is the step size of the current membership function.
Preferably, in the step 1), the prior knowledge of the SRM is introduced into the membership function selection and fuzzy set division of the rotor position for supplement, and a curve of flux linkage changing along with the rotor position can be divided into three regions, namely a region I [ theta ] (theta)u1) Region II [ theta ]1hr) And region III [ theta ]hra],θaThe complete alignment position of the salient poles of the stator and the rotor is as follows: calculating the alignment position theta of the leading edge of the rotor pole and the leading edge of the stator pole1And rotor pole centerline and stator pole leading edge alignment position θhrThe following formula is adopted:
Figure BDA0002365427080000021
in the formula, βsAnd βrThe pole arc widths of the stator and rotor, respectively, and satisfy the following relationship:
Figure BDA0002365427080000022
wherein, in the formula, m and NrRespectively representing the number of stator phases and the number of rotor poles of the motor, and m is 3, N is used for a three-phase 12/8-pole SRM modelrWhen formula (3) is substituted for formula (2), 8 can be obtained:
Figure BDA0002365427080000023
preferably, the change of the flux linkage along with the position of the rotor can be approximated to a linear relation in a region II, the change of the flux linkage along with the position of the rotor can be approximated to a cosine relation in regions I and III, the flux linkage-position characteristic is approximated to a linear relation in a position interval [7.5 degrees, 15 degrees ], and the linear membership function is adopted for division; the interval [0 degrees, 7.5 degrees ] and [15 degrees, 22.5 degrees ] are approximately cosine characteristics, and are divided by cosine membership functions.
Preferably, in step 1), in order to implement accurate solution of the flux linkage characteristics, a more refined fuzzy partition is adopted for fuzzy space partition of input and output variables of the flux linkage model, specifically as follows: dividing flux linkage into 201 regions by using a triangular membership function, wherein the ith flux linkage fuzzy set ClThe membership function of (d) can be described as:
Figure BDA0002365427080000024
in the formula, #maxIs the maximum value of flux linkage, c ═ ψmaxAnd 200 is the step size of the flux linkage membership function.
Preferably, the following method is specifically adopted in the step 2): a method for designing a fuzzy inference system based on input and output data of the system is adopted, and fuzzy rules are automatically extracted from sample data.
Compared with the prior art, the invention has the following beneficial effects:
the invention adopts a fuzzy logic system to solve the problem of accurate modeling when the flux linkage sample data of the switched reluctance motor is insufficient. Firstly, fuzzy space division is carried out according to motor priori knowledge; secondly, extracting fuzzy rules from small sample flux linkage data only containing a plurality of special positions through a fuzzy logic system, and forming a fuzzy rule base; and finally, solving the ambiguity by adopting a gravity center method, solving the flux linkage characteristic under any current and any rotor position, and completing the nonlinear accurate modeling of the flux linkage characteristic. Specifically, a fuzzy logic system accurate modeling method suitable for SRM small sample flux linkage characteristics is provided. The method can use the prior knowledge of the motor such as the change trend of the SRM pole logarithm and flux linkage along with the rotor position and the like for the selection of the fuzzy membership function and the division of the fuzzy set, and can automatically create a fuzzy rule base based on sample data, thereby realizing the accurate modeling under the SRM small sample flux linkage characteristic. The fuzzy modeling method not only makes full use of sample data, but also well combines inherent prior knowledge of the motor, greatly improves the modeling precision of the SRM flux linkage under the small sample data, and can provide powerful support for the analysis of the SRM operating characteristics and the verification of advanced algorithms.
Drawings
FIG. 1 is a block diagram of the present invention;
FIG. 2 is a schematic view of flux linkage characteristics at four particular positions of a rotor according to the present invention;
FIG. 3 is a schematic diagram of the input division (current) of the fuzzy logic system of the present invention;
FIG. 4 shows the flux linkage of the present invention as a function of rotor position;
FIG. 5 is a schematic diagram of the fuzzy logic system input partitioning (rotor position) of the present invention;
FIG. 6 is a schematic diagram of the input division (current) of the fuzzy logic system of the present invention;
FIG. 7 is a schematic diagram of the flux linkage modeling results of the present invention.
Detailed Description
Embodiments of the invention are further described below with reference to the accompanying drawings:
example 1
As shown in fig. 1 to 7, the method for accurately modeling a switched reluctance motor based on small sample flux linkage characteristics according to the present invention includes the following steps:
step 1), carrying out fuzzy space division on input and output variables of a flux linkage model according to the prior knowledge of a motor;
step 2), extracting fuzzy rules through a fuzzy logic system, and establishing a fuzzy rule base;
and 3) solving the flux linkage value under any current and position by solving the fuzzy problem through a gravity center method to complete flux linkage modeling.
In the specific method, a torque balance position measurement method is adopted to obtain small sample flux linkage data of four special positions of the three-phase switched reluctance motor, namely 0 degree, 7.5 degrees, 15 degrees and 22.5 degrees, as shown in fig. 2.
The following method is specifically adopted in the step 1): obtaining enough sample data of flux linkage changing along with current through experimental measurement, comprehensively considering model complexity and precision, and dividing phase current into 21 intervals by adopting a triangular membership function to perform fuzzy set division; specifically, the jth phase current fuzzy set AjThe membership function of (d) can be described as:
Figure BDA0002365427080000041
wherein i is a current value, j is a phase current, imaxAt maximum current, a ═ imaxAnd/20 is the step size of the current membership function.
For the rotor position in the step 1), flux linkage information of four special positions can be measured and obtained only based on a torque balance method. Flux linkage data of limited positions brings difficulty to membership function selection and fuzzy set division of rotor positions. Therefore, the prior knowledge of the SRM is introduced into the membership function selection and fuzzy set division of the rotor position for supplement, and the approximate change trend is very similar although the numerical values of the flux linkage-position (psi-theta) characteristics have differences for any given SRM prototype. Fig. 4 shows the flux linkage as a function of rotor position at a particular current. The curve of flux linkage changing with the position of rotor can be divided into three regions, I [ theta ] respectivelyu1) Region II [ theta ]1hr) And region III [ theta ]hra],θaThe complete alignment position of the salient poles of the stator and the rotor is as follows: calculating the alignment position theta of the leading edge of the rotor pole and the leading edge of the stator pole1And rotor pole centerline and stator pole leading edge alignment position θhrThe following formula is adopted:
Figure BDA0002365427080000042
in the formula, βsAnd βrThe pole arc widths of the stator and rotor, respectively, and satisfy the following relationship:
Figure BDA0002365427080000043
wherein, in the formula, m and NrRespectively representing the number of stator phases and the number of rotor poles of the motor, and m is 3, N is used for a three-phase 12/8-pole SRM modelrWhen formula (3) is substituted for formula (2), 8 can be obtained:
Figure BDA0002365427080000044
the change of the magnetic linkage along with the position of the rotor can be approximated to a linear relation in a region II and approximated to a cosine relation in regions I and III by combining experimental data, the magnetic linkage-position characteristic is approximated to a linear relation in a position interval [7.5 degrees, 15 degrees ], and a linear membership function is adopted for division; the interval [0 degrees, 7.5 degrees ] and [15 degrees, 22.5 degrees ] are approximately cosine characteristics, and are divided by cosine membership functions. The fuzzy set partitioning for rotor position is shown in fig. 5.
Rotor position ambiguity set B in FIG. 51And B2The membership functions of (a) can be described as:
Figure BDA0002365427080000051
Figure BDA0002365427080000052
fuzzy set B3And B4Are respectively connected with B2And B1Regarding the middle position symmetry, the principle of its membership functions is the same.
In step 1), in order to realize accurate solution of flux linkage characteristics, more precise fuzzy division is adopted for flux linkage model input and output variable fuzzy space division, and the method specifically comprises the following steps: dividing flux linkage into 201 regions by using a triangular membership function, wherein the ith flux linkage fuzzy set ClThe membership function of (d) can be described as:
Figure BDA0002365427080000053
in the formula, #maxIs the maximum value of flux linkage, c ═ ψmaxAnd 200 is the step size of the flux linkage membership function.
The following method is specifically adopted in the step 2): a method for designing a fuzzy inference system based on input and output data of the system is adopted, and fuzzy rules are automatically extracted from sample data.
The fuzzy rule is composed of a front piece and a back piece and can be expressed as follows: "If x1is Ajand x2is Bkthen y isCl". For the SRM flux linkage fuzzy logic model, the fuzzy rule corresponding to each input/output data pair (i, theta; psi) can be described as "R(s):If i(s)is
Figure BDA0002365427080000054
andθ(s)i s
Figure BDA0002365427080000055
thenψ(s)is
Figure BDA0002365427080000056
". For each sample data, selecting the corresponding maximum membership function value, and calculating the corresponding fuzzy set as follows:
Figure BDA0002365427080000057
in the formula, q1,q2,q3The fuzzy set numbers of the front piece i, theta and the back piece psi, respectively. From the preceding fuzzy set partition, q1=4,q2=21,q3201. Extracted fuzzy rule R(s)Confidence of (D) (R)(s)) The following can be calculated:
Figure BDA0002365427080000061
it should be noted that some sample data have the same fuzzy front part, different fuzzy back parts will generate conflicting fuzzy rules, and the solution is to select the rule with the maximum confidence as the best fuzzy rule. Based on the measured special position flux linkage sample data, the fuzzy rule extraction is completed, as shown in table 1.
TABLE 1 fuzzy rule base extracted from experimental measurement data
Tab.1The final fuzzy rule base generated from the measured sampledata.
B1 B2 B3 B4
A1 C1 C1 C1 C1
A2 C3 C4 C8 C19
A3 C5 C8 C23 C37
A4 C8 C12 C35 C55
A5 C10 C15 C45 C71
A6 C12 C19 C61 C92
A7 C15 C23 C73 C109
A8 C17 C26 C80 C123
A9 C20 C30 C94 C137
A10 C22 C34 C105 C148
A11 C24 C38 C111 C157
A12 C27 C41 C120 C166
A13 C30 C45 C126 C172
A14 C32 C48 C132 C176
A15 C35 C52 C136 C181
A16 C37 C55 C141 C185
A17 C40 C58 C145 C188
A18 C43 C62 C149 C192
A19 C45 C65 C153 C195
A20 C48 C69 C157 C198
A21 C51 C72 C161 C200
Taking row 3, column 2 in table 1 as an example, the rules can be described as: "If phase current i is A3and rotor positionθis B2then flux-linkageψis C8”。
Solving the flux linkage value under any current and position by a gravity center method through ambiguity resolution to complete flux linkage modeling, which is concretely as follows:
flux linkage output of SRM under any current and position
Figure BDA0002365427080000071
The following can be calculated:
Figure BDA0002365427080000072
wherein
Figure BDA0002365427080000073
Figure BDA0002365427080000074
In the formula (I), the compound is shown in the specification,
Figure BDA0002365427080000075
representing fuzzy sets
Figure BDA0002365427080000076
The area of (a) is,
Figure BDA0002365427080000077
is that
Figure BDA0002365427080000078
The center of gravity of (a).
The SRM full period flux linkage values were calculated according to the proposed fuzzy logic system modeling method, as shown by the dotted line in fig. 7. In order to verify the modeling accuracy of the proposed method, the measured flux linkage value (shown as a solid line in fig. 7) is compared with the measured flux linkage value of the traditional rotor locking method, and the comparison result shows that the measured flux linkage value and the measured flux linkage value have better consistency.

Claims (6)

1.一种基于小样本磁链特性的开关磁阻电机精确建模方法,其特征在于,包括以下步骤:1. an accurate modeling method for switched reluctance motor based on small sample flux linkage characteristics, is characterized in that, comprises the following steps: 步骤1)、根据电机先验知识进行磁链模型输入和输出变量模糊空间划分;Step 1), carry out the fuzzy space division of the input and output variables of the flux linkage model according to the prior knowledge of the motor; 步骤2)、通过模糊逻辑系统提取模糊规则,建立模糊规则库;Step 2), extract fuzzy rules by fuzzy logic system, establish fuzzy rule base; 步骤3)、通过重心法解模糊求解任意电流和位置下的磁链值,完成磁链建模。Step 3), solve the fuzzy value of the flux linkage under any current and position by the centroid method, and complete the flux linkage modeling. 2.据权利要求1所述的基于小样本磁链特性的开关磁阻电机精确建模方法,其特征在于,步骤1)中具体采用以下方式:通过实验测量获得足够的磁链随电流变化的样本数据,综合考虑模型复杂度和精度,采用三角形隶属度函数对相电流进行模糊集划分,共划分为21个区间;具体表述为,第j个相电流模糊集Aj的隶属度函数可描述为:2. The method for accurate modeling of switched reluctance motors based on small sample flux linkage characteristics according to claim 1, characterized in that, in step 1), the following methods are specifically adopted: obtain sufficient flux linkages that vary with current through experimental measurements. For the sample data, considering the complexity and accuracy of the model, the triangular membership function is used to divide the phase current into a fuzzy set, which is divided into 21 intervals. Specifically, the membership function of the jth phase current fuzzy set A j can describe for:
Figure FDA0002365427070000011
Figure FDA0002365427070000011
式中,i为电流值,j为相电流,imax为电流最大值,a=imax/20为电流隶属度函数的步长。In the formula, i is the current value, j is the phase current, i max is the current maximum value, and a=i max /20 is the step size of the current membership function.
3.据权利要求1或2所述的基于小样本磁链特性的开关磁阻电机精确建模方法,其特征在于,步骤1)中将SRM的先验知识引入转子位置的隶属度函数选取和模糊集划分,进行补充,磁链随转子位置变化曲线可以划分为三个区域,分别是区域I[θu1),区域II[θ1hr)和区域III[θhra],θa是定转子凸极完全对齐位置,具体如下:计算转子极前沿和定子极前沿对齐位置θ1和转子极中心线和定子极前沿对齐位置θhr,采用如下公式:3. according to claim 1 and 2, it is characterized in that, in step 1), the a priori knowledge of SRM is introduced into the membership function of rotor position selection and The fuzzy set is divided and supplemented. The flux linkage curve with rotor position change can be divided into three regions, namely region I[θ u , θ 1 ), region II [θ 1 , θ hr ) and region III [θ hr , θ a ], θ a is the fully aligned position of the stator and rotor salient poles, as follows: to calculate the alignment position of the rotor pole leading edge and the stator pole leading edge θ 1 and the rotor pole centerline and the stator pole leading edge alignment position θ hr , use the following formula:
Figure FDA0002365427070000012
Figure FDA0002365427070000012
式中,βs和βr分别为定子和转子的极弧宽度,并且满足如下关系:where β s and β r are the pole arc widths of the stator and rotor, respectively, and satisfy the following relationship:
Figure FDA0002365427070000013
Figure FDA0002365427070000013
其中,式中,m和Nr分别代表电机定子相数和转子极数,对于三相12/8极SRM样机,m=3,Nr=8,故将式(3)代入式(2),可得:Among them, in the formula, m and N r represent the number of stator phases and rotor poles of the motor, respectively. For the three-phase 12/8-pole SRM prototype, m=3, N r =8, so the formula (3) is substituted into the formula (2) ,Available:
Figure FDA0002365427070000014
Figure FDA0002365427070000014
4.据权利要求1或2所述的基于小样本磁链特性的开关磁阻电机精确建模方法,其特征在于,结合实验数据得出,磁链随转子位置的变化在区域II可近似为线性关系,在区域I和III可近似为余弦关系,将磁链-位置特性在位置区间[7.5°,15°]上近似为线性关系,采用线性隶属度函数划分;在区间[0°,7.5°]和[15°,22.5°]上近似为余弦特性,采用余弦隶属度函数划分。4. The precise modeling method for switched reluctance motor based on small sample flux linkage characteristics according to claim 1 or 2, characterized in that, combined with experimental data, it is obtained that the flux linkage changes with rotor position in region II can be approximated as The linear relationship can be approximated as a cosine relationship in regions I and III, and the flux linkage-position characteristic is approximated as a linear relationship in the position interval [7.5°, 15°], and is divided by a linear membership function; in the interval [0°, 7.5° °] and [15°, 22.5°] are approximately cosine characteristics, and are divided by cosine membership function. 5.据权利要求1所述的基于小样本磁链特性的开关磁阻电机精确建模方法,其特征在于,步骤1)中为了实现磁链特性的精确求解,对磁链模型输入和输出变量模糊空间划分采用更加精细的模糊划分,具体如下:利用三角形隶属度函数将磁链划分为201个区域,第l个磁链模糊集Cl的隶属度函数可描述为:5. the accurate modeling method of switched reluctance motor based on small sample flux linkage characteristic according to claim 1, is characterized in that, in step 1), in order to realize the accurate solution of flux linkage characteristic, to flux linkage model input and output variable The fuzzy space division adopts a finer fuzzy division, as follows: The triangular membership function is used to divide the flux linkage into 201 regions, and the membership function of the lth flux linkage fuzzy set C l can be described as:
Figure FDA0002365427070000021
Figure FDA0002365427070000021
式中,ψmax为磁链最大值,c=ψmax/200为磁链隶属度函数的步长。In the formula, ψ max is the maximum value of the flux linkage, and c=ψ max /200 is the step size of the flux linkage membership function.
6.据权利要求1所述的基于小样本磁链特性的开关磁阻电机精确建模方法,其特征在于,步骤2)中具体采用以下方式:采用基于系统的输入输出数据设计模糊推理系统的方法,从样本数据中自动提取模糊规则。6. the precise modeling method of switched reluctance motor based on small sample flux linkage characteristic according to claim 1, is characterized in that, in step 2), adopt the following mode specifically: adopt the input and output data based on system to design fuzzy inference system. method to automatically extract fuzzy rules from sample data.
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Citations (2)

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