Method for analyzing bifurcation chaos stability of high-speed linear induction motor

Abstract

The invention discloses a bifurcation chaos stability analysis method for a high-speed linear induction motor. The bifurcation chaos stability analysis method comprises the following steps: S1, establishing a complex vector mathematical model of the high-speed linear induction motor under PI control; s2, exporting a state-space equation of the system based on the complex vector mathematical model; s3, solving the state-space equation to obtain a balance point of the system; s4, judging the local stability of the balance point obtained in the step S3; s5, identifying global bifurcation characteristics and chaotic dynamics behaviors of the system; s6, determining a stability domain boundary of the proportional-integral control parameters; and S7, analyzing the stability of the high-speed linear induction motor based on bifurcation chaos. According to the method, dynamic behaviors of the system are analyzed through numerical solution, the defect that a traditional transfer function method is insufficient in non-linear and time-varying parameter sensitivity problem analysis capacity is overcome, and a more accurate theoretical basis is provided for stable control of the motor under the ultra-high-speed working condition.

Description

Technical Field

[0001] This invention relates to the field of dynamic stability analysis of high-speed linear induction motors, and specifically to a bifurcation chaotic stability analysis method for high-speed linear induction motors. Background Technology

[0002] High-speed linear induction motors play an irreplaceable role in the testing of high-speed equipment in aviation, aerospace, and other fields due to their significant advantages of high thrust and high reliability. However, during high-speed operation, the motor's parameters exhibit highly complex nonlinear time-varying characteristics. This characteristic can easily lead to severe instability in the system, thus seriously restricting the improvement of system reliability.

[0003] Traditional frequency domain stability analysis methods have significant limitations when analyzing such systems: the transfer function method requires the system to meet the precondition of linear time invariance, which fundamentally conflicts with the strong nonlinearity and rapid time-varying characteristics of high-speed linear motor parameters, making this method difficult to apply; classic frequency domain tools such as Nyquist plots and Bode plots can only effectively analyze small disturbances near the equilibrium point, while high-speed linear motors are in a transient process throughout their operation and have no stable operating point. This characteristic severely limits the application of frequency domain analysis methods and makes it impossible to analyze stability problems involving global state changes. These limitations make it difficult to accurately analyze and guarantee the dynamic stability of high-speed linear induction motors under conditions of rapid nonlinear time-varying parameters. Summary of the Invention

[0004] To address the severe instability issues arising from the rapid time-varying nonlinear parameters of high-speed linear induction motors, this invention provides a bifurcation chaotic stability analysis method for such motors. This method analyzes the system's dynamic behavior through numerical solutions, thereby revealing the instability mechanism. This approach overcomes the limitations of traditional transfer function methods in analyzing nonlinear and time-varying parameter sensitivity issues, providing a more accurate theoretical basis for stable motor control under high-speed conditions.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A bifurcation chaotic stability analysis method for high-speed linear induction motors includes the following steps:

[0007] S1. Establish a complex vector mathematical model of a high-speed linear induction motor under PI control;

[0008] S2. Based on the complex vector mathematical model of the high-speed linear induction motor under PI control, the state-space equation of the system is derived.

[0009] S3. Solve the state-space equations to obtain the equilibrium point of the system;

[0010] S4. Determine the local stability of the equilibrium point obtained in S3;

[0011] S5. Identify the global bifurcation characteristics and chaotic dynamics of the system;

[0012] S6. Determine the stability domain boundary of the proportional-integral control parameters based on the global bifurcation characteristics and chaotic dynamics.

[0013] S7. Perform stability analysis on a bifurcation chaotic high-speed linear induction motor based on the stability domain boundary.

[0014] Further, S1 includes:

[0015] Establish the stator current vector along the dq axis With dq axis moving part flux vector The mathematical description of the complex vector mathematical model for state variables is as follows:

[0016] (1)

[0017] Among them, the rotor time constant , For the sake of the child's own feelings, For the moving part resistance, Slip angular frequency, equivalent stator leakage inductance , For the self-sensing of the stator, 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. , Let p be the dq-axis stator voltage vector, p be the differential operator, and j be the imaginary unit;

[0018] The expression for the motor thrust equation is as follows:

[0019] (2)

[0020] The current controller uses PI control, and the dq-axis stator voltage vector... The output expression of the PI controller is:

[0021] (3)

[0022] in, This is the proportionality coefficient. The integral coefficient is... Given the dq-axis stator current.

[0023] Further, S2 includes:

[0024] The nonlinear differential equations in S1 are rearranged as follows: The state-space equations of the form are as follows:

[0025] (4)

[0026] Where the state vector Input vector .

[0027] Further, S3 includes:

[0028] make And since the derivatives of all state variables in the state-space model are zero, the equilibrium point of the system can be obtained by solving the system of equations numerically. As shown in the following formula:

[0029] (5).

[0030] Further, S4 includes:

[0031] Equilibrium point obtained for S3 Linearization in its vicinity yields the Jacobian matrix J of the system at the equilibrium point:

[0032] (6)

[0033] In the formula:

[0034] .

[0035] Furthermore, let K p = 1.2, K i = 200, select different The values ​​are substituted into the Jacobian matrix J. By solving for the eigenvalues ​​of the Jacobian matrix, the stability of the equilibrium point is determined based on the sign of the real part of the eigenvalues: if the real part of all eigenvalues ​​is less than 0, the equilibrium point is locally stable; if there are eigenvalues ​​with a real part greater than or equal to 0, the equilibrium point is locally unstable, and the system may undergo complex behaviors such as bifurcation and chaos, thus further clarifying the critical conditions for stability.

[0036] Further, S5 includes:

[0037] Let K p = 1.2, K i = 200, select different Value, based on the seven-dimensional state equation of a high-speed linear induction motor Numerical simulation was used to continuously scan the selected parameters, and numerical integration was performed using the adaptive step-size Runge-Kutta method. Steady-state time series were extracted to generate a bifurcation diagram, and the Lyapunov exponent of the system was calculated based on the Wolf algorithm. The evolutionary characteristics of the bifurcation diagram and the maximum Lyapunov exponent diagram with parameters revealed the complete evolutionary path of the system's dynamic behavior: steady state → bifurcation point → periodic oscillation → chaotic state. Finally, the parameters inducing chaotic behavior were determined. Critical interval.

[0038] Further, S6 includes:

[0039] make =Critical parameter, set the proportional coefficient K p The integral coefficient K varies in steps of 0.01 within the range of [1.1, 1.3]. i Varying in steps of 10 within the range [50, 500], for each group (K) p , K i For parameter pairs, the maximum Lyapunov exponent is calculated as a chaos criterion, and a K-based system is constructed. p For the horizontal axis, K i The two-dimensional parameter plane with the vertical axis is colored and a heatmap is drawn based on the maximum Lyapunov exponent of each parameter point. Finally, the boundary of the stable parameter is obtained through the distribution characteristics of the exponent values ​​in the heatmap.

[0040] Further, S7 includes:

[0041] A seven-dimensional state-space equation system is established based on a complex vector mathematical model, and the equilibrium points of the state-space equation system are obtained. Expression; Let K p = 1.2, K i = 200, select different Value, calculate the eigenvalues ​​of the Jacobian matrix and determine Local stability; then, by using bifurcation diagrams and the maximum Lyapunov exponent, the global dynamic behavior of the system is identified, and the factors inducing bifurcation and chaotic behavior are determined. Critical interval; finally in Under critical conditions, construct K p For the horizontal axis, K i For each group (K), the two-dimensional parametric plane is defined as follows: p , K i The parameters are used to calculate the maximum Lyapunov exponent and color them, then a heatmap is plotted. The distribution characteristics of the exponent values ​​in the heatmap are used to determine the control parameter K that ensures the stable operation of the system. p K i Boundary conditions.

[0042] Beneficial effects:

[0043] 1. This invention effectively overcomes the shortcomings of traditional frequency domain analysis methods in analyzing the stability problem of high-speed linear induction motors;

[0044] 2. This invention can explain the problem of a system being locally stable but globally unstable;

[0045] 3. This invention can provide a more accurate theoretical basis for linear motors operating at high speeds;

[0046] 4. This invention is also applicable to the stability analysis of other rotating electric machines. Attached Figure Description

[0047] Figure 1 This is a flowchart of the bifurcation chaotic stability analysis method for the high-speed linear induction motor of the present invention.

[0048] Figure 2 This is the stator d-axis current bifurcation diagram for the state variables of this invention;

[0049] Figure 3 This is the maximum Lyapunov index diagram of the present invention;

[0050] Figure 4 The proportionality coefficients K of this invention p Integral coefficient K i The maximum Lyapunov index heatmap. Detailed Implementation

[0051] 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.

[0052] like Figure 1 As shown, the bifurcation chaotic stability analysis method for high-speed linear induction motors of the present invention includes the following steps:

[0053] S1. Establish a complex vector mathematical model of a high-speed linear induction motor under PI control;

[0054] Establish the stator current vector along the dq axis With dq axis moving part flux vector The mathematical description of the complex vector mathematical model for state variables is as follows:

[0055] (1)

[0056] Among them, the rotor 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 self-sensing of the stator, 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. , Let p be the dq-axis stator voltage vector, p be the differential operator, and j be the imaginary unit;

[0057] The expression for the electric motor thrust F is as follows:

[0058] (2)

[0059] The current controller uses PI control, and the dq-axis stator voltage vector... The output expression of the PI controller is:

[0060] (3)

[0061] in, This is the proportionality coefficient. The integral coefficient is... Given the dq-axis stator current;

[0062] t is a time variable.

[0063] S2. Based on the complex vector mathematical model, derive the state-space equations of the system;

[0064] The nonlinear differential equations in S1 are rearranged as follows: The state-space equations of the form are as follows:

[0065] (4)

[0066] Where the state vector Input vector ;

[0067] S3. Solve the state-space equations to obtain the equilibrium points of the system:

[0068] make Furthermore, since the derivatives of all state variables in the state-space model are zero, the equilibrium points of the system state variables can be obtained by solving the system of equations using numerical methods. As shown in the following formula, the superscript 'e' denotes the equilibrium point:

[0069] (5)

[0070] S4. Determine the local stability of the equilibrium point obtained in step S3:

[0071] Equilibrium point obtained for S3 Linearization is performed in its vicinity to obtain the Jacobian matrix J of the system at that point:

[0072] (6)

[0073] Among them, among them, , , , All are element matrices of the Jacobian matrix J.

[0074] In the formula:

[0075] ;

[0076] Let K p = 1.2, K i = 200, select different The values ​​are substituted into the Jacobian matrix J. By solving for the eigenvalues ​​of the Jacobian matrix, the stability of the equilibrium point is determined based on the sign of the real part of the eigenvalues: if the real part of all eigenvalues ​​is less than 0, the equilibrium point is locally stable; if there are eigenvalues ​​with a real part greater than 0, the equilibrium point is locally unstable, and the system may undergo complex behaviors such as bifurcation and chaos. Furthermore, the critical condition for stability can be clearly defined.

[0077] S5. Identifying the global bifurcation characteristics and chaotic dynamics of the system:

[0078] Let K p = 1.2, K i = 200, select different Value, based on the seven-dimensional state equation of a high-speed linear induction motor Numerical simulation was used to continuously scan the selected parameters, and numerical integration was performed using the adaptive step-size Runge-Kutta method to extract the steady-state time series (t>0.8t). end , t end The bifurcation graph is generated by representing the total time of numerical integration, and the Lyapunov exponents (LEs) of the system are calculated based on the Wolf algorithm. The results are as follows: Figure 2 and Figure 3 As shown, based on the bifurcation diagram and the maximum LEs diagram, the complete evolution path of the system's dynamic behavior can be revealed: steady state → bifurcation point → periodic oscillation → chaotic state, ultimately determining the parameters that induce chaotic behavior. The critical point is =1.

[0079] S6. Determine the stability domain boundary of the proportional-integral control parameters:

[0080] make =1, set the scaling factor K p The integral coefficient K varies in steps of 0.01 within the range of [1.1, 1.3]. i Varying in steps of 10 within the range [50, 500], for each group (K) p , K i For parameter pairs, the maximum Lyapunov exponent is calculated as a chaos criterion, and a K-based system is constructed. p For the horizontal axis, K i Using a two-dimensional parametric plane with the vertical axis as the coordinate axis, the parameters are colored according to the magnitude of the maximum Lyapunov exponent at each point, and a heatmap is plotted. The result is as follows: Figure 4 As shown, the boundary of the stable parameter is finally determined by the distribution characteristics of the exponential values ​​in the heatmap. p <1.2.

[0081] S7. Stability analysis of high-speed linear induction motors based on bifurcation chaos:

[0082] The stability analysis method for high-speed linear induction motors based on bifurcation chaos first establishes a seven-dimensional state-space equation based on a complex vector mathematical model, and then obtains the equilibrium point of the state-space equation system. Expression; then let K p = 1.2, K i =200, select different Value, calculate the eigenvalues ​​of the Jacobian matrix and determine Local stability; then, by using bifurcation diagrams and the maximum Lyapunov exponent, the global dynamic behavior of the system is identified, and the factors inducing bifurcation and chaotic behavior are determined. Critical interval. Finally in Under the critical condition that = 1, construct K p For the horizontal axis, K i For each group (K), the two-dimensional parametric plane is defined as follows: p , K i The parameters are used to calculate the maximum Lyapunov exponent and color them, then a heatmap is plotted. The distribution characteristics of the exponent values ​​in the heatmap are used to determine the control parameter K that ensures the stable operation of the system. p K i Boundary conditions.

[0083] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for bifurcation chaos stability analysis of high speed linear induction motor, characterized in that, The method comprises the following steps: S1, establishing a complex vector mathematical model of a high-speed linear induction motor under PI control; S2, deriving a state space equation of the system based on the complex vector mathematical model of the high-speed linear induction motor under PI control; S3, solving the state space equation to obtain an equilibrium point of the system; S4, judging the local stability of the equilibrium point obtained in S3; S5, identifying the global bifurcation characteristics and chaotic dynamic behavior of the system; S6, determining the stability domain boundary of the proportional integral control parameter according to the global bifurcation characteristics and chaotic dynamic behavior; S7, performing stability analysis on the bifurcation chaotic high-speed linear induction motor according to the stability domain boundary.

2. The bifurcation chaos stability analysis method of a high-speed linear induction motor according to claim 1, characterized by, The S1 comprises: establishing a dq-axis stator current vector and a dq-axis rotor flux vector is a complex vector mathematical model of state variables, which is mathematically described as follows: （1） wherein the rotor time constant , is the rotor self-inductance, is the rotor resistance, the slip angular frequency, the equivalent stator leakage inductance , is the stator self-inductance, is the rotor over-stator ratio, is the mutual inductance, the stator equivalent resistance , is the stator resistance, is the supply angular frequency, the rotor electrical angular frequency , is the dq-axis stator voltage vector, p is the differential operator, and j is the imaginary unit. The expression of the motor thrust equation is as follows: （2） The current controller adopts PI control, and the PI controller output expression of the dq-axis stator voltage vector is: The PI controller output expression of the dq-axis stator voltage vector is: （3） wherein, is a proportional coefficient, is an integral coefficient, is a given dq-axis stator current.

3. The bifurcation chaos stability analysis method of a high-speed linear induction motor according to claim 2, characterized by, The S2 comprises: The nonlinear differential equations in S1 are arranged into state space equations in the form as follows: （4） where the state vector , the input vector .

4. The bifurcation chaos stability analysis method of a high-speed linear induction motor according to claim 3, characterized by, The S3 comprises: Let and the derivatives of the individual state variables in the state space model are zero, the system equilibrium point is obtained by solving the equation set by numerical methods as shown in the following equation: （5）。 5. The bifurcation chaos stability analysis method of a high-speed linear induction motor according to claim 4, characterized by, The S4 comprises: The equilibrium point obtained for S3 Linearization around it gives the Jacobian matrix J of the system at the equilibrium point: （6） wherein , , , are element matrices of the Jacobian matrix J.

6. The bifurcation chaos stability analysis method of a high-speed linear induction motor according to claim 4, characterized by, In formula (6): 。 7. The bifurcation chaos stability analysis method of a high-speed linear induction motor according to claim 6, characterized by, Let K p = 1.2, K i = 200, select different values, substitute into the Jacobian matrix J, solve the eigenvalues of the Jacobian matrix, and judge the stability of the equilibrium point according to the sign of the real part of the eigenvalue: if the real part of all eigenvalues is less than 0, the equilibrium point is locally stable; if there is an eigenvalue with a real part greater than or equal to 0, the equilibrium point is locally unstable, and the system may have complex behaviors such as bifurcation, chaos, etc. Further clarify the stability critical condition.

8. The bifurcation chaos stability analysis method of a high-speed linear induction motor according to claim 7, characterized by, The S5 comprises: Let K p = 1.2, K i = 200, select different values, based on the seven-dimensional state equation of high-speed linear induction motor , the selected parameters are implemented by continuous scanning using numerical simulation method, combined with adaptive step Runge-Kutta method for numerical integration, the steady-state time sequence is intercepted to generate bifurcation diagram, and the Lyapunov exponent of the system is calculated based on Wolf algorithm, according to the evolution characteristics of bifurcation diagram and maximum Lyapunov exponent diagram with parameter evolution, the complete evolution path of system dynamics behavior is revealed: stable steady state→bifurcation point→periodic oscillation→chaotic state, and finally the parameters critical interval inducing chaotic behavior are determined.

9. The bifurcation chaos stability analysis method of a high-speed linear induction motor according to claim 8, characterized by, The S6 comprises: Let = critical parameter, set the scale factor K p In the range of [1.1, 1.3] with a step of 0.01, the integral coefficient K i In the range of [50, 500] with a step of 10, for each group of (K p , K i ) parameter pairs, calculate the maximum Lyapunov index as the chaos criterion, construct a two-dimensional parameter plane with K p as the horizontal axis and K i as the vertical axis, color and draw the heat map according to the maximum Lyapunov index of each parameter point, and finally obtain the boundary of the stable parameter through the distribution characteristics of the index value in the heat map.

10. The bifurcation chaos stability analysis method of a high-speed linear induction motor according to claim 9, characterized by, The S7 comprises: A seven-dimensional state space equation is established based on the complex vector mathematical model, and the state Equilibrium points of spatial system of equations Expression; let K p = 1.2, K i = 200, select different values, calculate the eigenvalue of Jacobian matrix Local stability; then through the bifurcation diagram and the maximum Lyapunov index, identify the global dynamic behavior of the system, determine the critical interval; finally, under the critical condition of , construct a two-dimensional parameter plane with K p as the horizontal axis and K i as the vertical axis, calculate the maximum Lyapunov index for each group of (K p , K i ) parameters, and color it, draw a heat map, and determine the boundary conditions of the control parameters K p , K i that ensure the stable operation of the system by the distribution characteristics of the index value in the heat map.

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