Stability analysis method of high-speed linear induction motor based on reduced-order differential equation

Abstract

The application provides a high-speed linear induction motor stability analysis method based on a reduced-order differential equation, and belongs to the technical field of high-speed linear induction motor dynamic stability analysis. The method comprises the following steps: establishing a motor body complex vector state equation set and a PI controller mathematical model; obtaining a reduced-order differential equation according to the motor body complex vector state equation set; obtaining a second-order homogeneous difference equation according to the reduced-order differential equation and the PI controller mathematical model; constructing a discrete system characteristic polynomial according to the second-order homogeneous difference equation, and obtaining an algebraic stability boundary equation according to a Jury stability criterion; and judging the stability of a high-speed linear induction motor current closed-loop system according to the algebraic stability boundary equation. The application overcomes the shortcomings of the traditional linearization method, such as dependence on a fixed steady-state operating point and incapability of adapting to parameter fast time variation, by means of discrete time domain modeling and explicit analytic analysis of the system stability boundary, and provides a more accurate theoretical basis for stable operation of the motor under high-speed working conditions.

Description

Technical Field

[0001] This invention belongs to the field of dynamic stability analysis technology for high-speed linear induction motors, specifically relating to a stability analysis method for high-speed linear induction motors based on reduced-order difference equations. Background Technology

[0002] High-speed linear induction motors are core equipment in electromagnetic propulsion ground-based ultra-high-speed test facilities, providing crucial ground testing methods for the development of high-end equipment such as new aerospace vehicles. To meet the demands of high-speed operation, high-speed linear induction motors typically employ a segmented stator topology. Under high-speed operating conditions exceeding 1000 km / h, the frequent movement of the rotor into and out of each stator segment causes the motor parameters to exhibit large-scale, nonlinear, and rapidly time-varying characteristics within a millisecond timescale. This can easily lead to instability in the current closed-loop system, becoming a core bottleneck restricting the improvement of motor operating speed.

[0003] Traditional linearization stability analysis methods have significant limitations when performing stability analysis on such systems. For example, time-domain linearization analysis based on the Lyapunov indirect method is theoretically applicable only to time-invariant systems and requires first-order linearization of the model using a fixed steady-state operating point. However, in the linear induction motor studied in this paper, the continuous movement of the mover into and out of the stator section causes the key system parameters to exhibit significant time-varying characteristics. Furthermore, as the mover speed increases, the coupling effect between the system's state variables continuously strengthens. Forcing a linearization expansion of the system model can easily overlook higher-order nonlinear terms that dominate the system's dynamic behavior. Frequency-domain linearization analysis based on transfer functions also assumes a linear time-invariant system and requires small-signal linearization using a fixed steady-state operating point. This assumption conflicts with the actual dynamic operating conditions of the motor and therefore fails to accurately reflect the system's instability characteristics. Summary of the Invention

[0004] The purpose of this invention is to provide a stability analysis method for high-speed linear induction motors based on reduced-order difference equations, which can be used to solve the current closed-loop instability problem caused by the large-range nonlinear and rapid time-varying parameters of high-speed linear induction motors.

[0005] To achieve the above objectives, this invention provides a stability analysis method for high-speed linear induction motors based on reduced-order difference equations. The method includes: Step S1, establishing a complex vector state equation set for the motor body and a mathematical model of the PI controller in a synchronously rotating dq coordinate system, with the stator current vector and mover flux linkage vector along the dq axis as state variables; Step S2, obtaining a reduced-order differential equation containing only stator current state variables based on the complex vector state equation set for the motor body; Step S3, obtaining a second-order homogeneous difference equation containing only current error increments based on the reduced-order differential equations and the PI controller mathematical model; Step S4, constructing a characteristic polynomial for the discrete system based on the second-order homogeneous difference equation containing only current error increments, and obtaining an algebraic stability boundary equation for judging the stability of the current closed-loop system based on the Jury stability criterion for the discrete system; Step S5, judging the stability of the current closed-loop system of the high-speed linear induction motor based on the algebraic stability boundary equation.

[0006] The beneficial effects of this invention are as follows:

[0007] By modeling and deriving in the discrete-time domain of digital control, the explicit analytical solution of the system stability boundary is approximately completed, overcoming the shortcomings of traditional linearization analysis methods in analyzing the stability problem of high-speed linear induction motors. It does not require linearization of the nonlinear motor model, nor does it rely on a fixed steady-state operating point. It can adapt to the rapidly changing operating characteristics of high-speed linear induction motor parameters, providing a more accurate parameter design basis for the current loop controller of linear induction motor under high-speed operating conditions, and can also be applied to the stability analysis of other types of time-varying motor systems. Attached Figure Description

[0008] The accompanying drawings are provided to further illustrate embodiments of the present invention and form part of the specification. They are used together with the following detailed description to explain the embodiments of the present invention, but do not constitute a limitation thereof. In the drawings:

[0009] Figure 1 A flowchart of the stability analysis method for high-speed linear induction motors based on reduced-order difference equations provided by this invention;

[0010] Figure 2 The different mover-stator coverage ratios a provided by this invention n With stator power supply frequency f s Critical proportional gain under operating conditions Contour map of stable boundary. Detailed Implementation

[0011] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other. To achieve the above objectives, this invention adopts the following technical solution.

[0012] Figure 1 The flowchart of the stability analysis method for high-speed linear induction motors based on reduced-order difference equations provided by this invention is as follows: Figure 1 As shown, the method includes:

[0013] Step S1: In the synchronously rotating dq coordinate system, establish the complex vector state equations of the motor body and the mathematical model of the PI controller with the stator current vector and the motor flux linkage vector of the dq axis as state variables;

[0014] Step S2: Based on the complex vector state equations of the motor body, obtain the reduced-order differential equations containing only stator current state variables;

[0015] Step S3: Based on the reduced-order differential equation and the mathematical model of the PI controller, a second-order homogeneous difference equation containing only the current error increment is obtained.

[0016] Step S4: Construct the characteristic polynomial of the discrete system based on the second-order homogeneous difference equation containing only the current error increment, and obtain the algebraic stability boundary equation for judging the stability of the current closed-loop system based on the Jury stability criterion of the discrete system.

[0017] Step S5: Determine the stability of the high-speed linear induction motor current closed-loop system based on the algebraic stability boundary equation.

[0018] Step S1 is to establish a high-speed linear induction motor control system model. Step S1 includes: establishing a complex vector state equation set of the motor body with the actual dq-axis stator current vector and dq-axis mover flux linkage vector as state variables in the synchronous rotating dq coordinate system. The complex vector state equation set of the motor body includes the stator current differential equation and the mover flux linkage differential equation; the current closed loop adopts a PI controller, and a complex vector form of the PI controller mathematical model is established.

[0019] The stator current differential equation is established based on the equivalent stator leakage inductance, differential operator, actual dq-axis stator current vector, mover coverage ratio, mover time constant, mover electric angular frequency, dq-axis mover flux linkage vector, stator equivalent resistance, supply angular frequency, and actual dq-axis stator voltage vector. The mover flux linkage differential equation is established based on the mover time constant, differential operator, dq-axis mover flux linkage vector, mutual inductance, actual dq-axis stator current vector, and slip angular frequency. The expression for the actual dq-axis stator voltage vector output by the complex vector form PI controller mathematical model is established based on the PI controller proportional coefficient, PI controller integral coefficient, given dq-axis stator current vector, and actual dq-axis stator current vector.

[0020] Specifically, in the synchronously rotating dq coordinate system, a stator current vector with the actual dq axis is established. With the dq axis moving part flux vector The complex vector state equation of the motor body with state variables is as follows (1):

[0021] (1)

[0022] Among them, the first equation The stator current differential equation, the second equation The equation is the differential equation for the motor flux linkage.

[0023] Among them, the mover time constant , For the sake of the child's own feelings, For the moving part resistance, The slip angular frequency is the equivalent stator leakage inductance. , For the stator self-inductance, a n The ratio of mover coverage to stator coverage. For mutual inductance, stator equivalent resistance , For stator resistance, The power supply angular frequency and the motor electric angular frequency are given. , denoted as the actual dq-axis stator voltage vector, p is the differential operator, and j is the imaginary unit.

[0024] In this embodiment, a PI controller is used for the current closed loop. A mathematical model of the PI controller in complex vector form is established, and its output is the actual dq-axis stator voltage vector. The expression is equation (2):

[0025] (2)

[0026] in, The proportional gain of the PI controller. The integral coefficient of the PI controller. Given the dq-axis stator current vector, t represents the actual dq-axis stator current vector, and t is the time variable.

[0027] Step S2 involves reducing the order of the mathematical model. Step S2 includes: establishing the mover flux linkage expression; substituting the mover flux linkage expression into the stator current differential equation in the complex vector state equation system of the motor body, eliminating the flux linkage variable, and obtaining a reduced-order differential equation containing only the stator current state variable.

[0028] The mover flux linkage expression is established based on the dq-axis mover flux linkage vector, mover time constant, mutual inductance, slip angular frequency, and given dq-axis stator current vector.

[0029] Specifically, due to the mover time constant Much larger than the stator time constant Therefore, the mover flux linkage is approximated as a slowly varying perturbation. The flux linkage differential equation is set to 0, and the expression for the mover flux linkage is obtained as follows (3):

[0030] (3)

[0031] Substituting the mover flux linkage expression into the stator current differential equation in the complex vector state equation system of the motor body, and eliminating the flux linkage variable, we obtain the reduced-order differential equation containing only the stator current state variable as shown in equation (4):

[0032] (4)

[0033] To simplify subsequent analysis and formula derivation, two core parameters are defined: The transient damping attenuation rate, The transient coupling angular frequency and transient damping attenuation rate are given. and transient coupling angular frequency The expression is shown in equation (5):

[0034] (5)

[0035] Step S3 is to derive the discrete difference equation of the system. Step S3 includes: discretizing the reduced-order differential equation containing only stator current state variables using the forward Euler method to obtain the discrete recursive equation of the current state variables; calculating the expression of the reduced-order stator current discrete difference equation based on the discrete recursive equation of the current state variables; calculating the output voltage increment of the PI controller based on the mathematical model of the PI controller; and obtaining the second-order homogeneous difference equation containing only the current error increment based on the expression of the reduced-order stator current discrete difference equation and the output voltage increment of the PI controller.

[0036] Step S3 further includes: calculating the transient damping attenuation rate based on the stator equivalent resistance, equivalent stator leakage inductance, mover coverage ratio of the stator, mutual inductance, mover time constant, mover electric angular frequency, and slip angular frequency; calculating the transient coupling angular frequency based on the power supply angular frequency, mover coverage ratio of the stator, mutual inductance, power supply angular frequency, equivalent stator leakage inductance, mover time constant, and slip angular frequency; and discretizing the reduced-order differential equation containing only stator current state variables using the forward Euler method based on the transient damping attenuation rate and the transient coupling angular frequency to obtain the discrete recursive equation for the current state variables.

[0037] In practical digital control systems, the sampling, calculation, and pulse width modulation (PWM) update process of the controller introduces an inherent delay of one control cycle, which is the voltage applied to the motor at the discrete sampling time k. The actual output of the PI controller at the discrete sampling time k-1 is calculated. Based on this, the forward Euler method is used to discretize the reduced-order differential equation containing only stator current state variables, and the resulting discrete recursive equation for the current state variables is as follows (6):

[0038] (6)

[0039] in, , For discrete damping factor, For discrete coupling deflection angle, This represents the value of the actual dq-axis stator current vector at the discrete sampling time k+1. T represents the value of the actual dq-axis stator voltage vector at the discrete sampling time k-1. s This represents the sampling time interval.

[0040] The output voltage increment of the PI controller can be simplified to equation (7):

[0041] (7)

[0042] in, This represents the value of the actual dq-axis stator voltage vector at the discrete sampling time k. This represents the value of the actual dq-axis stator current vector at the discrete sampling time k. This represents the value of the actual dq-axis stator current vector at the discrete sampling time k+1. This is the integral gain.

[0043] By shifting equation (6) forward by one period and subtracting it from equation (6), we can obtain the expression for the reduced-order stator current discrete difference equation. Substituting equation (7) into the expression for the reduced-order stator current discrete difference equation eliminates the voltage term and ignores the integral gain. The impact on high-frequency oscillations, and the definition of discrete proportional gain. The second-order homogeneous difference equation containing only the current error increment is obtained as shown in equation (8):

[0044] (8)

[0045] in, Let k be the stator current error increment at time k. Let be the stator current error increment at time k+1. This represents the stator current error increment at time k+2.

[0046] Step S4 is to derive the algebraic stability boundary equation. Step S4 includes: constructing the characteristic polynomial of the discrete system based on the second-order homogeneous difference equation containing only the current error increment; obtaining the characteristic root expression when the current closed-loop system is in a critical stable state based on the Jury stability criterion of the discrete system; and obtaining the algebraic stability boundary equation used to judge the stability of the current closed-loop system based on the characteristic polynomial and the characteristic root expression of the discrete system.

[0047] For the second-order homogeneous difference equation of formula (8) obtained in step S3, which contains only the current error increment, the characteristic polynomial of the discrete system is constructed as shown in formula (9):

[0048] (9)

[0049] In the formula Let be the eigenvalues ​​of the discrete system. According to the Jury stability criterion for discrete systems, the necessary and sufficient condition for a current-closed-loop system to remain stable is that all eigenvalues ​​of the characteristic polynomial lie within the unit circle in the Z-plane. When the current-closed-loop system is in a critically stable state, the eigenvalues ​​are... It will fall on the unit circle in the Z-plane, and the characteristic root expression is in the form of equation (10):

[0050] (10)

[0051] in, It is a unit complex exponent form modulo 1. Characteristic roots Argument in the Z-plane.

[0052] Based on equations (9) and (10), and combined with trigonometric identities Finally, the algebraic stable boundary equation in elliptic form is derived, as shown in equation (11):

[0053] (11)

[0054] The elliptic algebraic stable boundary equation is based on equation (11), and it can be seen from equation (5) that... and All are about For discrete damping factor, For discrete coupling deflection angle α n and The function iterates through the ratio a of the mover covering the stator. n ∈[0,1] and stator power supply frequency f s For the full operating range ∈ [0, 250Hz], then each group (a) can be solved. n ,f s The corresponding critical proportional gain .

[0055] Equation (11) gives the analytical expression of the stability boundary of the current closed-loop system of a high-speed linear induction motor in the discrete domain. If the discrete damping factor Discrete coupling deflection angle With discrete proportional gain K p The values ​​between these values ​​make the system stable when the left side of equation (11) is less than 1; critically stable when it is equal to 1; and unstable when it is greater than 1.

[0056] Figure 2 The different mover-stator coverage ratios a provided by this invention n With stator power supply frequency f s Critical proportional gain under operating conditions Contour map of stable boundary, with the moving part covering the stator at a scale of a. n The power supply frequency f for the horizontal axis and stator s Using the vertical axis as the ordinate, a contour map of the stable boundary is drawn, as shown below. Figure 2 As shown, based on the distribution characteristics of contour lines, the system operating conditions can be divided into three regions: a high stability margin region, a stability margin decay region, and an unstable region. The high stability margin region corresponds to low-speed operating conditions with low coverage ratios and low power supply frequencies, where the discrete coupling deflection angle... Smaller, critical proportional gain K p_max Larger values ​​result in a wider range of controller parameter tuning; the stability margin decay region corresponds to the ratio 'a' of the mover covering the stator. n With stator power supply frequency f s In synchronously increasing medium-to-high speed operating conditions, with the discrete coupling deflection angle... Significantly increase, critical proportional gain K p_max As the system is gradually compressed, its stability margin rapidly diminishes; the absence of an stable region corresponds to a high motor-to-stator ratio 'a'. n High stator power supply frequency fs In the extreme operating condition, the excessively large discrete coupling deflection angle... This results in the stability boundary equation having no positive real solutions, regardless of the proportional coefficient k of the PI controller. p No matter what value is chosen, the system cannot remain stable.

[0057] The optional embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the embodiments of the present invention are not limited to the specific details in the above embodiments. Within the scope of the technical concept of the embodiments of the present invention, various simple modifications can be made to the technical solutions of the embodiments of the present invention, and these simple modifications all fall within the protection scope of the embodiments of the present invention.

[0058] It should also be noted that the various specific technical features described in the above embodiments can be combined in any suitable manner without contradiction. To avoid unnecessary repetition, the embodiments of the present invention will not describe the various possible combinations separately.

[0059] Furthermore, various different implementations of the present invention can be combined arbitrarily, as long as they do not violate the spirit of the present invention, they should also be regarded as the content disclosed in the present invention.

Claims

1. A stability analysis method for high-speed linear induction motors based on reduced-order difference equations, characterized in that, The method includes: Step S1: In the synchronously rotating dq coordinate system, establish the complex vector state equations of the motor body and the mathematical model of the PI controller with the stator current vector and the motor flux linkage vector of the dq axis as state variables; Step S2: Based on the complex vector state equations of the motor body, obtain the reduced-order differential equations containing only stator current state variables; Step S3: Based on the reduced-order differential equation and the mathematical model of the PI controller, a second-order homogeneous difference equation containing only the current error increment is obtained. Step S4: Construct the characteristic polynomial of the discrete system based on the second-order homogeneous difference equation containing only the current error increment, and obtain the algebraic stability boundary equation for judging the stability of the current closed-loop system based on the Jury stability criterion of the discrete system. Step S5: Determine the stability of the high-speed linear induction motor current closed-loop system based on the algebraic stability boundary equation.

2. The stability analysis method for high-speed linear induction motors based on reduced-order difference equations according to claim 1, characterized in that, Step S1 includes: In the synchronously rotating dq coordinate system, a complex vector state equation set for the motor body is established with the actual dq-axis stator current vector and the dq-axis mover flux linkage vector as state variables. The complex vector state equation set for the motor body includes the stator current differential equation and the mover flux linkage differential equation. The current closed loop uses a PI controller, and a mathematical model of the PI controller in complex vector form is established.

3. The stability analysis method for high-speed linear induction motors based on reduced-order difference equations according to claim 2, characterized in that, The stator current differential equation is established based on the equivalent stator leakage inductance, differential operator, actual dq-axis stator current vector, mover coverage ratio of stator, mover time constant, mover electric angular frequency, dq-axis mover flux linkage vector, stator equivalent resistance, power supply angular frequency, and actual dq-axis stator voltage vector.

4. The stability analysis method for high-speed linear induction motors based on reduced-order difference equations according to claim 3, characterized in that, The differential equation of the mover flux linkage is established based on the mover time constant, differential operator, dq-axis mover flux linkage vector, mutual inductance, actual dq-axis stator current vector, and slip angular frequency.

5. The stability analysis method for high-speed linear induction motors based on reduced-order difference equations according to claim 4, characterized in that, The expression for the actual dq-axis stator voltage vector output by the mathematical model of the PI controller in complex vector form is established based on the proportional coefficient of the PI controller, the integral coefficient of the PI controller, the given dq-axis stator current vector, and the actual dq-axis stator current vector.

6. The stability analysis method for high-speed linear induction motors based on reduced-order difference equations according to claim 5, characterized in that, Step S2 includes: Establish the expression for the mover flux linkage; Substituting the mover flux linkage expression into the stator current differential equation in the complex vector state equations of the motor body, and eliminating the flux linkage variable, yields a reduced-order differential equation containing only the stator current state variable.

7. The stability analysis method for high-speed linear induction motors based on reduced-order difference equations according to claim 6, characterized in that, The mover flux linkage expression is established based on the dq-axis mover flux linkage vector, mover time constant, mutual inductance, slip angular frequency, and given dq-axis stator current vector.

8. The stability analysis method for high-speed linear induction motors based on reduced-order difference equations according to claim 7, characterized in that, Step S3 includes: The forward Euler method is used to discretize the reduced-order differential equation containing only stator current state variables, resulting in the discrete recursive equation of the current state variables. The expression for the reduced-order stator current discrete difference equation is calculated based on the discrete recursive equation of the current state variables. Calculate the output voltage increment of the PI controller based on the mathematical model of the PI controller; Based on the expression of the reduced-order stator current discrete difference equation and the output voltage increment of the PI controller, a second-order homogeneous difference equation containing only the current error increment is obtained.

9. The stability analysis method for high-speed linear induction motors based on reduced-order difference equations according to claim 8, characterized in that, Step S3 further includes: The transient damping attenuation rate is calculated based on the stator equivalent resistance, equivalent stator leakage inductance, mover coverage ratio, mutual inductance, mover time constant, mover electric angular frequency, and slip angular frequency. The transient coupling angular frequency is calculated based on the power supply angular frequency, the ratio of mover coverage to stator, mutual inductance, power supply angular frequency, equivalent stator leakage inductance, mover time constant, and slip angular frequency. Based on the transient damping attenuation rate and transient coupling angular frequency, the forward Euler method is used to discretize the reduced-order differential equation containing only stator current state variables, resulting in the discrete recursive equation of the current state variables.

10. The stability analysis method for high-speed linear induction motors based on reduced-order difference equations according to claim 9, characterized in that, Step S4 includes: Construct the characteristic polynomial of the discrete system based on the second-order homogeneous difference equation containing only the current error increment; Based on the Jury stability criterion for discrete systems, the characteristic root expression for the current closed-loop system when it is in a critical stable state is obtained; Based on the characteristic polynomial and characteristic root expression of the discrete system, the algebraic stability boundary equation for judging the stability of the current closed-loop system is obtained.

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