Axial magnetic suspension bearing system modeling and controller designing method based on different-element fractional order

Abstract

The invention belongs to the technical field of magnetic suspension, and discloses an axial magnetic suspension bearing system modeling and controller design method based on different-element fractional order, which comprises the following steps of: 1, deducing a fractional order axial magnetic suspension rotor mathematical model which considers an eddy current effect and has four denominators through a Maxwell differential equation set; 2, designing a fractional order expansion state observer according to the mathematical model so as to obtain accurate state estimation and system disturbance; and step 3, designing a fractional order sliding mode controller based on the mathematical model of known disturbance. According to the method, the axial fractional order magnetic suspension bearing with four denominators is established by adopting a method based on Maxwell's equations, the adaptive capacity of a system to parameter change and uncertainty is enhanced by combining a fractional order sliding mode control strategy (FSMC) of a fractional order extended state observer (FESO), and it is ensured that the system can still work normally even under extreme working conditions. And the system can maintain good dynamic response and stability.

Description

Technical Field

[0001] The present invention belongs to the technical field of magnetic suspension, and in particular relates to a modeling and controller design method of an axial magnetic suspension bearing system based on non-homologous fractional orders. Background Art

[0002] Active magnetic bearings are a new type of bearing that uses controllable electromagnetic force to stably suspend the magnetically suspended rotor in the air, with no mechanical contact between the stator and the rotor. Compared with traditional mechanical bearings, they have the characteristics of no friction, no need for lubrication, and long service life, and are suitable for special environments such as vacuum, ultra-clean, and high-speed.

[0003] Normally, the rotor of an axial magnetic bearing adopts a laminated structure, but due to structural strength and other reasons, it is changed to a solid structure. When a solid rotor rotates at high speed, strong eddy currents will be generated inside it, resulting in a large error between the analytical model of the electromagnetic bearing and the actual system. The effective magnetic resistance in the magnetic circuit plays an important role in modeling because it includes eddy current effects. When the frequency is large, a half-order term is introduced into the effective magnetic resistance, making the analytical model become fractional order. Therefore, it is necessary to establish a fractional-order axial magnetic bearing system to accurately describe the performance of the magnetic suspension system, thereby improving its accuracy. Summary of the invention

[0004] The object of the present invention is to provide a method for modeling and designing a controller of an axial magnetic bearing system based on non-homologous fractional orders, so as to solve the above-mentioned technical problems.

[0005] In order to solve the above technical problems, the specific technical solutions of the axial magnetic bearing system modeling and controller design method based on non-isotropic fractional order of the present invention are as follows: A method for modeling and controller design of an axial magnetic bearing system based on non-isotropic fractional order includes the following steps: Step 1: Derived the mathematical model of the fractional-order axial magnetic suspension rotor with four terms in the denominator considering the eddy current effect through the Maxwell differential equations; Step 2: Design a fractional-order extended state observer based on the mathematical model to obtain accurate state estimation and system disturbance; Step 3: Design a fractional-order sliding mode controller based on the mathematical model of the known disturbance.

[0006] Furthermore, the step 1 comprises the following steps: Step 1.1: Simplify Maxwell's differential equations to obtain the diffusion equation; Step 1.2: When the excitation is sinusoidal, a differential equation for the magnetic field intensity is obtained in the cylindrical coordinate system; Step 1.3: The differential equation obtained is consistent with the form of the Bessel equation, so the differential equation is solved by the Bessel function; Step 1.4: Express the Bessel function using its asymptotic expression.

[0007] Furthermore, the diffusion equation obtained by simplifying step 1.1 is: (1) In the formula, H is the magnetic field strength, is the vacuum permeability, is the relative magnetic permeability of the material, is the conductivity, is the Laplace operator, t For time, is the partial differential symbol.

[0008] Furthermore, the differential equation obtained in step 1.2 is: (2) In the formula , Hs is the external excitation magnetic field, is an imaginary number, j is an imaginary unit, is the frequency, r is the radius.

[0009] Furthermore, the general solution of the Bessel equation obtained by solving step 1.3 is: (3) After obtaining the general solution of magnetic field intensity, the two unknown coefficients A and B in the general solution are calculated and solved. The values ​​of A and B are obtained by setting boundary conditions. is the first kind Bessel function of an imaginary quantity, is the imaginary Bessel function of the second kind, , after solving, we can get the expression of magnetic field intensity as (4): (4) In the formula , are the first kind imaginary Bessel functions of order 0 and 1 respectively, are the second kind imaginary Bessel functions of order 0 and 1 respectively, r is the radius, is the radius from the axis to the outer edge of the inner ring, is the radius from the axis to the inner edge of the inner ring, is the external excitation magnetic field.

[0010] Furthermore, step 1.4 expresses the Bessel function using its asymptotic expression. The asymptotic expression of the imaginary Bessel function is I 0. I 1. K 0. K 1 is formula (5): (5) is the unknown parameter in the Bessel function, and the simplified expression of magnetic field intensity (6) is obtained: (6) In the formula r 0 and r 1 are the radius lengths of the inner and outer rings of the magnetic poles in the axial magnetic bearing respectively; Then the magnetic field strength B It is expressed as: (7) magnetic flux The relationship with magnetic induction intensity is: (8) Where A is the air gap magnetic pole area, then the axial magnetic bearing dynamic reluctance of the magnetic circuit in the time-varying magnetic field is R t Written as: (9) In the formula N is the number of coil turns, i is the bias current size; Open-loop transfer function of active axial magnetic bearing system , the current As input, the rotor displacement As output, it is represented as: (10) In the formula, is the effective magnetic resistance, R s is the static reluctance, s is the complex variable in the transfer function, static reluctance R s The expression is: (11) In the formula g is the air gap spacing, is the effective magnetic path length, M is the magnetic pole area of ​​the material; The concepts of effective magnetic resistance and effective magnetic permeability are introduced to analyze the system characteristics of the time-varying magnetic field. Therefore, by substituting equation (9) and the static magnetic resistance equation (11) into equation (10), the following fractional-order axial magnetic bearing model (12) is obtained: (12) In the formula are three coefficients, m is the mass of the axial magnetic bearing, is the current stiffness, is the displacement stiffness, There are three orders, among which , is the static magnetic resistance.

[0011] Furthermore, the step 2 includes the following specific steps: Step 2.1: Design the fractional-order extended state observer based on the mathematical model built in step 1. The method is as follows: Construct the state space equation based on the magnetic levitation system model: (13) In the formula, a , b , c Indicates the number of times, is the state variable, For system input, is the system output, is the system disturbance, is the system matrix, As the control gain, let , It's about the system status , , A nonlinear function of For external interference, the system disturbance is expanded into a new state quantity , and assuming it is differentiable, the state space equation can be rewritten as: (14) In the formula and Yes and Derivation; Step 2.2: Establish the extended state observer for the system described by equation (14): (15) in, System status The estimated value of is the estimation error of the system, is the observer gain coefficient, is the error function, Yes Derivative.

[0012] Furthermore, the step 3 includes the following specific steps: According to the mathematical model of known disturbance, the controller of the magnetic levitation system is designed as follows: Design a fractional-order integrated sliding surface to track the reference signal: (16) in a , b , c Indicates the number of times, t Indicates time, I represents the integral symbol, and are two weight factors in the sliding surface, which are used to adjust the influence of the first two items on the entire function; Taking the derivative of both sides of equation (16) we get: (17) Substituting equation (13) into equation (17), we obtain:

[0013] Therefore, from equation (18-20) we get: (twenty one) In the formula, is a state variable , is a state variable , that is , is a state variable , that is ;pass , and the equivalent control law is obtained , the equivalent control law and the law of approach Combine into new , that is, the adaptive control law ,in, are self-defined parameters, The function is a "smoother" that solves the jitter problem.

[0014] The modeling and controller design method based on non-homologous fractional-order axial magnetic bearing system of the present invention has the following advantages: the present invention adopts the method based on Maxwell equations to establish the axial fractional-order magnetic bearing with four terms as the denominator, and combines the fractional-order sliding mode control strategy (FSMC) of the fractional-order extended state observer (FESO). First, it can significantly improve the robustness of the system to unmodeled dynamics and external disturbances, and through the real-time estimation and compensation of FESO, it can effectively reduce the influence of these factors on the performance of the control system; secondly, the introduction of FESO can reduce the high-frequency chatter in the sliding mode control, improve the smoothness of the control signal, and improve the tracking performance of the system; finally, the combination of FSMC and FESO can enhance the system's adaptability to parameter changes and uncertainties, ensuring that the system can maintain good dynamic response and stability even under extreme working conditions. Therefore, this integrated fractional-order control method provides strong technical support for the precise control of complex systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 It is a flow chart for derivation of fractional-order transfer function of the axial magnetic bearing of the present invention; Figure 2 It is a schematic diagram of the overall structure of the controller system of the present invention. DETAILED DESCRIPTION

[0016] In order to better understand the purpose, structure and function of the present invention, the modeling and controller design method of the non-homologous fractional-order axial magnetic bearing system of the present invention will be further described in detail below in conjunction with the accompanying drawings.

[0017] The present invention provides a method for modeling and controller design of an axial magnetic bearing system based on non-homologous fractional order, comprising the following steps: Step 1: Figure 1 As shown, a mathematical model of a fractional-order axial magnetically suspended rotor with a denominator of four terms considering the eddy current effect is derived through the Maxwell differential equations; The magnetic levitation system is a complex nonlinear system that needs to take into account the influence of many nonlinear factors, and it is difficult to describe it with an accurate mathematical model. Although the integer-order model has a simple structure, it does not take into account the influence of nonlinear factors such as eddy current effect, so the accuracy of the integer-order model will be unsatisfactory. The fractional-order model established by the present invention is based on Maxwell's differential equations. The differential equations are solved using Bessel functions in a cylindrical coordinate system to obtain a fractional-order transfer function with a denominator of four terms to describe the performance of the axial magnetic levitation bearing. Through fractional-order modeling, the non-integer-order characteristics of the axial magnetic levitation bearing can be more accurately described, including the influence of eddy current effect. This modeling method can more flexibly adapt to different system dynamics and changes, and effectively improve the robustness of the magnetic bearing system.

[0018] Step 1.1: Simplify Maxwell's differential equations to obtain the diffusion equation. The simplified diffusion equation (1) is: (1) In the formula, H is the magnetic field strength, is the vacuum permeability, is the relative magnetic permeability of the material, is the conductivity, is the Laplace operator, t For time, is the partial differential symbol.

[0019] Step 1.2: When the excitation is sinusoidal, a differential equation (2) for the magnetic field intensity is obtained in the cylindrical coordinate system: (2) In the formula , Hs is the external excitation magnetic field, is an imaginary number, j is an imaginary unit, is the frequency, r is the radius.

[0020] Step 1.3: The differential equation obtained is consistent with the form of the Bessel equation, so the differential equation is solved by the Bessel function. The general solution of the Bessel equation obtained by solving (3) is: (3) After obtaining the general solution of magnetic field intensity, it is necessary to calculate and solve the two unknown coefficients A and B in the general solution. The values ​​of A and B can be quickly obtained by setting boundary conditions. is the first kind Bessel function of an imaginary quantity, is the imaginary Bessel function of the second kind, , after solving, we can get the expression of magnetic field intensity as (4): (4) In the formula , are the first kind imaginary Bessel functions of order 0 and 1 respectively, are the second kind imaginary Bessel functions of order 0 and 1 respectively, r is the radius, is the radius from the axis to the outer edge of the inner ring, is the radius from the axis to the inner edge of the inner ring, is the external excitation magnetic field.

[0021] Step 1.4: The expression of magnetic field intensity obtained by solution will contain Bessel function. In order to simplify, Bessel function is expressed by its asymptotic expression. The asymptotic expression of Bessel function of imaginary quantity isI 0. I 1. K 0. K 1 is formula (5).

[0022] (5) is the unknown parameter in the Bessel function, and the simplified expression of magnetic field intensity (6) is obtained: (6) In the formula r 0 and r 1 are the radius lengths of the inner and outer rings of the magnetic poles in the axial magnetic bearing respectively.

[0023] Then the magnetic field strength B It can be expressed as: (7) magnetic flux The relationship with magnetic induction intensity is: (8) Where A is the air gap magnetic pole area, then the axial magnetic bearing dynamic reluctance of the magnetic circuit in the time-varying magnetic field is R t can be written as: (9) In the formula N is the number of coil turns, i is the bias current size.

[0024] Open-loop transfer function of active axial magnetic bearing system , the current As input, the rotor displacement As output, it is represented as: (10) Static Reluctance R s The expression is: (11) In the formula g is the air gap spacing, is the effective magnetic path length, M is the magnetic pole area of ​​the material.

[0025] Since the axial magnetic bearing is a solid structure, the eddy current effect is more obvious, and the static magnetic resistance cannot accurately describe the magnetic field characteristics of the system. Therefore, it is necessary to introduce the concepts of effective magnetic resistance and effective magnetic permeability to analyze the system characteristics of the time-varying magnetic field. Therefore, by substituting equation (9) and static magnetic resistance equation (11) into equation (10), the following fractional-order axial magnetic bearing model (12) can be obtained: (12) In the formula are three coefficients, m is the mass of the axial magnetic bearing, is the current stiffness, is the displacement stiffness, There are three orders, among which , is the static magnetic resistance.

[0026] Existing magnetic bearings are mainly described by integer-order transfer functions, but magnetic bearings are affected by nonlinear factors such as eddy current effect, magnetic leakage, and magnetic saturation, resulting in unsatisfactory performance of magnetic bearings described by integer-order transfer functions, especially for non-laminated magnetic bearings. The error is large. In view of the above problems, this method introduces a fractional-order transfer function with a denominator of four terms to fully consider the influence of nonlinear factors such as eddy current effect. This transfer function can provide more complex dynamic characteristics than integer-order or fractional-order transfer functions with a denominator of three terms, which helps to improve the model's ability to predict system behavior.

[0027] Step 2: Design a fractional-order extended state observer based on the mathematical model to obtain accurate state estimation and system disturbance; like Figure 2 As shown, an axial magnetic bearing system based on non-homologous fractional order of the present invention comprises: Fractional-order sliding mode controller: used to calculate the control quantity of the magnetically suspended rotor system; PWM modulator: used to convert the control signal output by the controller into a switching signal recognizable by the power amplifier; Power amplifier: used to generate control current, thereby generating electromagnetic force to achieve stable suspension of the rotor; Fractional-order extended state observer: used to estimate the state variables and external disturbances of the system.

[0028] Step 2.1: Design the fractional-order extended state observer based on the mathematical model built in step 1. The method is as follows: Construct the state space equation based on the magnetic levitation system model: (13) In the formula, a , b , cIndicates the number of times, is the state variable, For system input, is the system output, is the system disturbance, is the system matrix. is the control gain. , It's about the system status , , A nonlinear function of is external interference. Expand the system disturbance into a new state quantity , and assuming it is differentiable, the state space equation can be rewritten as: (14) In the formula and Yes and Derivative.

[0029] Step 2.2: Establish the extended state observer for the system described by equation (14): (15) in, System status The estimated value of is the estimation error of the system, is the observer gain coefficient, is the error function, Yes Derivative.

[0030] Step 3: Design a fractional-order sliding mode controller based on the mathematical model of the known disturbance.

[0031] According to the mathematical model of the known disturbance, the controller of the magnetic levitation system is designed as follows: Design a fractional-order integrated sliding surface to track the reference signal: (16) in a , b , c Indicates the number of times, t Indicates time, I represents the integral symbol, and are two weight factors in the sliding surface, which are used to adjust the influence of the first two items on the entire function.

[0032] Taking the derivative of both sides of equation (16) we get: (17) Substituting equation (13) into equation (17), we obtain:

[0033] Therefore, from equation (18-20) we can get: (twenty one) In the formula, is a state variable , is the state variable , that is , is a state variable , that is ;pass , we can get the equivalent control law . The equivalent control law and the law of approach Combine into new , that is, the adaptive control law ,in, are self-defined parameters, The function is a "smoother" that solves the jitter problem.

[0034] It is to be understood that the present invention is described by some embodiments, and it is known to those skilled in the art that various changes or equivalent substitutions may be made to these features and embodiments without departing from the spirit and scope of the present invention. In addition, under the teachings of the present invention, these features and embodiments may be modified to adapt to specific circumstances and materials without departing from the spirit and scope of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are within the scope of protection of the present invention.

Claims

1. A method for modeling and controller design of an axial magnetic bearing system based on non-isotropic fractional order, characterized in that: The steps include: Step 1: Derived the mathematical model of the fractional-order axial magnetic suspension rotor with four terms in the denominator considering the eddy current effect through the Maxwell differential equations; Step 2: Design a fractional-order extended state observer based on the mathematical model to obtain accurate state estimation and system disturbance; Step 3: Design a fractional-order sliding mode controller based on the mathematical model of the known disturbance.

2. The method for modeling and controller design of an axial magnetic bearing system based on non-homologous fractional order according to claim 1 is characterized in that: The step 1 comprises the following steps: Step 1.1: Simplify Maxwell's differential equations to obtain the diffusion equation; Step 1.2: When the excitation is sinusoidal, a differential equation for the magnetic field intensity is obtained in the cylindrical coordinate system; Step 1.3: The differential equation obtained is consistent with the form of the Bessel equation, so the differential equation is solved by the Bessel function; Step 1.4: Express the Bessel function using its asymptotic expression.

3. The method for modeling and controller designing an axial magnetic bearing system based on non-homologous fractional order according to claim 2 is characterized in that: The diffusion equation obtained by simplifying step 1.1 is: （1） In the formula, H is the magnetic field strength, is the vacuum permeability, is the relative magnetic permeability of the material, is the conductivity, is the Laplace operator, t For time, is the partial differential symbol.

4. The method for modeling and controller designing an axial magnetic bearing system based on non-homologous fractional order according to claim 3 is characterized in that: The differential equation obtained in step 1.2 is: （2） In the formula , Hs is the external excitation magnetic field, is an imaginary number, j is an imaginary unit, is the frequency, r is the radius.

5. The method for modeling and controller designing of an axial magnetic bearing system based on non-homologous fractional order according to claim 4 is characterized in that: The general solution of the Bessel equation obtained by solving step 1.3 is: （3） After obtaining the general solution of magnetic field intensity, the two unknown coefficients A and B in the general solution are calculated and solved. The values ​​of A and B are obtained by setting boundary conditions. is the first kind Bessel function of an imaginary quantity, is the imaginary Bessel function of the second kind, , after solving, we can get the expression of magnetic field intensity as (4): （4） In the formula , are the first kind imaginary Bessel functions of order 0 and 1 respectively, are the second kind imaginary Bessel functions of order 0 and 1 respectively, r is the radius, is the radius from the axis to the outer edge of the inner ring, is the radius from the axis to the inner edge of the inner ring, is the external excitation magnetic field.

6. The method for modeling and controller design of an axial magnetic bearing system based on non-homologous fractional order according to claim 2 is characterized in that: Step 1.4 Express the Bessel function using its asymptotic expression. The asymptotic expression of the imaginary Bessel function is I 0. I 1. K 0. K 1 is formula (5): （5） is the unknown parameter in the Bessel function, and the simplified expression of magnetic field intensity (6) is obtained: （6） In the formula r 0 and r 1 are the radius lengths of the inner and outer rings of the magnetic poles in the axial magnetic bearing respectively; Then the magnetic field strength B It is expressed as: （7） magnetic flux The relationship with magnetic induction intensity is: （8） Where A is the air gap magnetic pole area, then the axial magnetic bearing dynamic reluctance of the magnetic circuit in the time-varying magnetic field is R t Written as: （9） In the formula N is the number of coil turns, i is the bias current size; Open-loop transfer function of active axial magnetic bearing system , the current As input, the rotor displacement As output, it is represented as: （10） In the formula, is the effective magnetic resistance, R s is the static magnetic resistance, s is the complex variable in the transfer function, and the static magnetic resistance R s The expression is: （11） In the formula g is the air gap spacing, is the effective magnetic path length, M is the magnetic pole area of ​​the material; The concepts of effective magnetic resistance and effective magnetic permeability are introduced to analyze the system characteristics of the time-varying magnetic field. Therefore, by substituting equation (9) and the static magnetic resistance equation (11) into equation (10), the following fractional-order axial magnetic bearing model (12) is obtained: （12） In the formula are three coefficients, m is the mass of the axial magnetic bearing, is the current stiffness, is the displacement stiffness, There are three orders, among which , is the static magnetic resistance.

7. The method for modeling and controller design of an axial magnetic bearing system based on non-homologous fractional order according to claim 1 is characterized in that: The step 2 comprises the following specific steps: Step 2.1: Design the fractional-order extended state observer based on the mathematical model built in step 1. The method is as follows: Construct the state space equation based on the magnetic levitation system model: （13） In the formula, a , b , c Indicates the number of times, is the state variable, For system input, is the system output, is the system disturbance, is the system matrix, As the control gain, let , It's about the system status , , A nonlinear function of For external interference, the system disturbance is expanded into a new state quantity , and assuming it is differentiable, the state space equation can be rewritten as: （14） In the formula and Yes and Derivation; Step 2.2: Establish the extended state observer for the system described by equation (14): （15） in, System status The estimated value of is the estimation error of the system, is the observer gain coefficient, is the error function, Yes Derivative.

8. The method for modeling and controller design of an axial magnetic bearing system based on non-homologous fractional order according to claim 1 is characterized in that: The step 3 comprises the following specific steps: According to the mathematical model of known disturbance, the controller of the magnetic levitation system is designed as follows: Design a fractional-order integrated sliding surface to track the reference signal: （16） in a , b , c Indicates the number of times, t Indicates time, I represents the integral symbol, and are two weight factors in the sliding surface, which are used to adjust the influence of the first two items on the entire function; Taking the derivative of both sides of equation (16) we get: （17） Substituting equation (13) into equation (17), we obtain: ， Therefore, from equation (18-20) we get: （21） In the formula, is a state variable , is a state variable , that is , is a state variable , that is ;pass , and the equivalent control law is obtained , the equivalent control law and the law of approach Combine into new , that is, the adaptive control law ,in, are self-defined parameters, The function is a "smoother" that solves the jitter problem.

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