Control method for solid thrust magnetic suspension bearing influenced by eddy current effect

Abstract

The invention provides a control method for a solid thrust magnetic suspension bearing influenced by an eddy current effect, and relates to the field of magnetic suspension bearing control. The method comprises the following steps: constructing a fractional order anti-interference control architecture of a controller in the solid thrust magnetic suspension bearing system, wherein the architecture comprises a pre-compensator, a low-pass filter and an expectation module; determining each transfer function of the pre-compensator, the low-pass filter, the expectation module and the controlled object; based on the pre-compensator, the low-pass filter, the expectation module and each transfer function of the controlled object, carrying out stability analysis on the solid thrust magnetic suspension bearing system, and determining a stability condition; adjusting the control parameters based on a stability condition; and substituting the adjusted control parameters into the transfer function of the pre-compensator and the transfer function of the low-pass filter to compensate the total disturbance of the system so as to realize the anti-interference control of the controlled object. According to the method, the control parameter adjusting process is simplified, and the stability and the anti-interference performance of the solid thrust magnetic suspension bearing system are improved.

Description

Technical Field

[0001] This application relates to the field of magnetic levitation bearing control, and particularly to a control method for a solid thrust magnetic levitation bearing affected by the eddy current effect. Background Art

[0002] Magnetic levitation bearing technology plays an important role in modern industry, especially in the field of high-speed rotating machinery, such as turbo molecular pumps, magnetic levitation blowers, and magnetic levitation centrifugal compressors. Active magnetic levitation bearings suspend the rotor through electromagnetic force to achieve non-contact support, with significant advantages such as no friction, no need for lubrication, and long service life.

[0003] However, due to its structural characteristics, a solid thrust magnetic levitation bearing is prone to being affected by the eddy current effect. The eddy current effect will lead to an enhanced nonlinear characteristic of the system, increasing the complexity of the control system and the difficulty of dynamic response. In related technologies, based on the fractional-order model of the magnetic levitation bearing, each parameter of the controller is designed according to the frequency response, which can enable the solid thrust magnetic levitation bearing control system to obtain better control performance. However, the parameter design process in this method is complex, with a large amount of calculation, and the anti-disturbance performance of the system is not studied. Summary of the Invention

[0004] Based on this, it is necessary to provide a control method for a solid thrust magnetic levitation bearing affected by the eddy current effect, which can simplify the control parameter adjustment process and improve the stability and anti-disturbance performance of the solid thrust magnetic levitation bearing system.

[0005] In a first aspect, this application provides a control method for a solid thrust magnetic levitation bearing affected by the eddy current effect. This method is applied to the controller of the solid thrust magnetic levitation bearing system, and the solid thrust magnetic levitation bearing system further includes a solid thrust magnetic levitation bearing, which is the controlled object. This method includes: Construct a fractional-order anti-disturbance control architecture for the controller in the solid thrust magnetic levitation bearing system. The architecture includes a pre-compensator, a low-pass filter, and a desired module; among them, the pre-compensator is a fractional-order PD controller; Determine the transfer function of the pre-compensator, the transfer function of the low-pass filter, the transfer function of the desired module, and the transfer function of the controlled object; Based on the transfer functions of the pre-compensator, the low-pass filter, the desired module, and the controlled object, conduct a system stability analysis on the solid thrust magnetic levitation bearing system to determine the stability conditions; Based on the stability conditions, adjust the control parameters in the fractional-order anti-disturbance control architecture; among them, the control parameters include the proportional coefficient of the differential term, the differential coefficient, and the fractional-order differential order of the pre-compensator, and the time constant of the low-pass filter; Substitute the adjusted control parameters into the transfer functions of the pre-compensator and the low-pass filter, and observe and compensate for the total disturbance of the solid thrust magnetic levitation bearing system to achieve disturbance rejection control of the controlled object.

[0006] In one embodiment, by taking the square root of the first parameter to convert it into the second parameter , the complex plane of consists of the imaginary axis and the real axis, the complex plane of includes a stable region and an unstable region, the complex plane of is divided into a right half-plane and a left half-plane by the imaginary axis, the unstable region is located in the right half-plane, the stable region is larger than the unstable region, and the stable region includes the left half-plane and a part of the right half-plane; the stability condition is that there exists a target constant greater than zero, when is greater than zero and less than the target constant: Condition a: The relative order of the transfer function of the controlled object after pre-compensation matches the transfer function of the desired module; Condition b: The zeros of the controlled object in the complex plane of

[0007] are located in the stable region; Condition c: The zeros of the pre-compensator in the complex plane of are located in the stable region. where, is used to represent the influence of the proportional coefficient of the differential term, the differential coefficient, the fractional-order differential order in the complex plane of and on the stability of the solid thrust magnetic levitation bearing system; Based on , by taking the square root of to convert it into , the transformation expression is obtained; By determining the distribution region of the roots where the characteristic polynomial equals zero when approaches zero, to determine Condition b and Condition c in the stability condition; where, the characteristic polynomial is related to the transformation expression .

[0008] In one embodiment, by determining when The distribution region of the roots when the characteristic polynomial is equal to zero to determine condition b and condition c in the stability condition, including: Equivalent the roots when the characteristic polynomial is equal to zero to the target expression Equal to zero roots; where the target expression Is the sum of two parts, one part is related to the proportional coefficient of the differential term and the differential coefficient, and the other part is related to Related; When Approaches zero, split the target expression Into the product of the first target expression And the second target expression Product; Based on The distribution region of the roots to determine condition b and condition c in the stability condition.

[0009] In one embodiment, based on The distribution region of the roots to determine condition b and condition c in the stability condition, including: Decompose the specific formula of Into the first expression and the second expression; where the parameters in the first expression are the same as those in the pre-compensator transfer function, and some of the parameters in the second expression are the same as some of the parameters in the controlled object expression; Determine condition c in the stability condition based on the root distribution region of the first expression, and determine condition b in the stability condition based on the root distribution region of the second expression.

[0010] In one embodiment, rewrite To integrate Containing two variables into the desired expression Containing one target variable; the desired expression Also includes the matching degree between the pre-compensated controlled object transfer function and the desired module transfer function; Among them, the stability condition also includes condition d: the roots when the desired expression Is equal to zero are negative real numbers.

[0011] In one embodiment, determine condition d in the stability condition, including: Assume that the first value is a root of the desired expression And the second value and the third value are two roots of the second target expression ;

[0012] According to Rouche's theorem, there exists Greater than zero and approaching zero, the square of the second value multiplied by Product, the square of the third value and The products are all equal to the first value; Through the square of the second value and the product, and the square of the third value and the product are all equal to the first value, it is obtained that when is greater than zero and approaches zero, the squares of the second value and the third value are both equal to the ratio of the first value to ; The squares of the second value and the third value are both equal to the ratio of the first value to ; it is obtained that in zero and the first value case, the second value and the third value are both pure imaginary numbers and are located in the stable region.

[0013] In one embodiment, based on the stability condition, the control parameters in the fractional-order disturbance rejection control architecture are adjusted, including: When approaches zero and the matching degree approaches zero, the transfer function of the controlled object after pre-compensation and the transfer function of the desired module have a matching degree higher than the matching threshold in the high-frequency band. By setting the differential coefficient of the pre-compensator, the proportional coefficient of the differential term of the pre-compensator is derived with respect to the transfer functions of the controlled object and the desired module and the differential coefficient . Based on this relationship, the proportional coefficient of the differential term is determined.

[0014] Determine the upper limit value of according to the stability condition; wherein, the upper limit value is the smallest value when the roots of the characteristic polynomial are located on the boundary line between the stable region and the unstable region, and the time constant is selected between zero and the upper limit value.

[0015] In one embodiment, is obtained by dynamic adjustment, and the dynamic adjustment specifically includes: Construct a simulation model based on the transfer function of the controlled object, and the simulation model includes a displacement outer loop and a current inner loop; When the simulation model is running, apply an impact disturbance in the step response, and observe the displacement drop value and the recovery time of the transfer function of the controlled object after pre-compensation after receiving the impact disturbance; Between zero and the upper limit value, gradually decrease until the displacement drop is less than the drop threshold and the recovery time is less than the recovery threshold.

[0016] In a second aspect, the present application also provides a solid thrust magnetic levitation bearing system affected by the eddy current effect. The system includes a controller and a solid thrust magnetic levitation bearing. The solid thrust magnetic levitation bearing is the controlled object, and the controller can execute the method for controlling a solid thrust magnetic levitation bearing affected by the eddy current effect in the first aspect.

[0017] The above method for controlling a solid thrust magnetic levitation bearing affected by the eddy current effect is applied to the controller of the solid thrust magnetic levitation bearing system, and specifically includes: constructing a fractional-order disturbance rejection control architecture of the controller, which specifically includes a pre-compensator, a low-pass filter, and an expectation module, where the pre-compensator is a fractional-order PD controller; determining the transfer functions of the pre-compensator, the low-pass filter, the expectation module, and the controlled object; further performing system stability analysis on the solid thrust magnetic levitation bearing system based on the transfer functions of the pre-compensator, the low-pass filter, the expectation module, and the controlled object to determine the stability conditions; adjusting the control parameters in the fractional-order disturbance rejection control architecture according to the stability conditions, including the proportional coefficient of the differential term, the differential coefficient, and the fractional-order differential order of the pre-compensator, and the time constant of the low-pass filter; substituting the adjusted control parameters into the transfer functions of the pre-compensator and the low-pass filter, and realizing disturbance rejection control of the controlled object by observing and compensating the total disturbance of the solid thrust magnetic levitation bearing system. This method can simplify the adjustment process of control parameters, improve the stability and disturbance rejection performance of the solid thrust magnetic levitation bearing system, and ensure that the solid thrust magnetic levitation bearing system can maintain stable operation when disturbed. Description of the Drawings

[0018] Figure 1 is a flowchart of the method for controlling a solid thrust magnetic levitation bearing affected by the eddy current effect in one embodiment; Figure 2 is a framework diagram of the solid thrust magnetic levitation bearing system in one embodiment; Figure 3 is a schematic diagram of the stable region and the unstable region in one embodiment; Figure 4 is a flowchart of determining the stability conditions in one embodiment; Figure 5 is a flowchart of determining condition b and condition c in the stability conditions in one embodiment; Figure 6 is a flowchart of obtaining the time constant in one embodiment; Figure 7 is a simulation result diagram in one embodiment. Detailed Embodiments

[0019] To make the objectives, technical solutions, and advantages of this application clearer, the following further elaborates on this application in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely used to explain this application and are not intended to limit this application.

[0020] In one embodiment, a control method for a solid thrust magnetic levitation bearing affected by the eddy current effect is provided. This method is applied to the controller of a solid thrust magnetic levitation bearing system, which also includes a solid thrust magnetic levitation bearing, and the solid thrust magnetic levitation bearing is the controlled object. As Figure 1 shown, this method includes the following steps: Step 101: Construct a fractional-order disturbance rejection control architecture for the controller in the solid thrust magnetic levitation bearing system. The architecture includes a pre-compensator, a low-pass filter, and a desired module; among them, the pre-compensator is a fractional-order PD controller.

[0021] Figure 2 is the framework diagram of the solid thrust magnetic levitation bearing system. Figure 2 It is shown in that the controller in the solid thrust magnetic levitation bearing system includes a pre-compensator, a low-pass filter, and a desired module, and the solid thrust magnetic levitation bearing is the controlled object. It should be noted that the pre-compensator is a fractional-order PD controller, which is used to pre-compensate the solid thrust magnetic levitation bearing system, adjust and optimize the solid thrust magnetic levitation bearing system in advance, and reduce the mismatch between the solid thrust magnetic levitation bearing system and the desired model; the low-pass filter can be used to reduce the influence of high-frequency noise on the disturbance observer, ensuring that the high-frequency noise from the outside will not significantly interfere with the normal operation of the solid thrust magnetic levitation bearing system during the disturbance rejection process; the desired module can express the desired dynamic response characteristics of the solid thrust magnetic levitation bearing system and provide a performance target for the controller. This performance target can describe the input-output relationship of the solid thrust magnetic levitation bearing system under ideal conditions, enabling the controller to adjust the actual response of the solid thrust magnetic levitation bearing system based on this performance target.

[0022] Step 102: Determine the transfer functions of the pre-compensator, the low-pass filter, the desired module, and the controlled object.

[0023] In the embodiments of this application, the expression of the pre-compensator can be the transfer function of the fractional-order PD controller, and the pre-compensator can be expressed as , and the transfer function of the pre-compensator is: ; where is the proportional coefficient of the differential term of the pre-compensator, is the differential coefficient of the pre-compensator, is the fractional-order differential term, and 1.5 is the fractional-order differential order. It should be noted that in the embodiments of the present application, the fractional-order differential order is set to 1.5, but this fractional-order differential order is not fixed. The selection of the fractional-order differential order can be adjusted according to specific system characteristics and control requirements, with the aim of optimizing the performance of the controller so that it can provide better control effects within a wider frequency range, thereby enhancing the stability and anti-interference ability of the solid thrust magnetic suspension bearing system.

[0024] The low-pass filter can be expressed as , and the transfer function of the low-pass filter is: ; where is the time constant of the low-pass filter, which is used to limit the influence of high-frequency noise on the solid thrust magnetic suspension bearing system, ensure the accuracy of disturbance observation, and does not affect the low-frequency response characteristics of the solid thrust magnetic suspension bearing system.

[0025] The transfer function of the desired module can be set according to the design requirements of the solid thrust magnetic suspension bearing system, and is usually a transfer function with desired dynamic characteristics for stability. The desired module can be expressed as , and the transfer function of the desired module is: = ; where is the time constant of the desired module, which can determine the response speed of the solid thrust magnetic suspension bearing system. The larger the time constant , the slower the response; the smaller the time constant , the faster the response.

[0026] The controlled object can be expressed as , and the transfer function of the controlled object is: ; where , and are all known quantities obtained through parameter identification of the solid thrust magnetic suspension bearing system.

[0027] It should be noted that is the Laplace operator.

[0028] Step 103: Based on the transfer functions of the pre-compensator, low-pass filter, desired module, and controlled object, perform system stability analysis on the solid thrust magnetic suspension bearing system to determine the stability conditions.

[0029] Exemplarily, the transfer function of the pre-compensator, the transfer function of the low-pass filter, the transfer function of the desired module, and the transfer function of the controlled object can be substituted into the expression of the closed-loop transfer function of the entire solid thrust magnetic levitation bearing system to construct the characteristic polynomial of the solid thrust magnetic levitation bearing system. Using stability criteria, such as Matignon's theorem, analyze the distribution of the roots of the characteristic polynomial in the complex plane (which can refer to the complex plane of , or the complex plane of other parameters). By ensuring that all roots of the characteristic polynomial are located in the stable region of the complex plane, the conditions for the stability of the solid thrust magnetic levitation bearing system are determined. For example, the relational expressions or ranges that the control parameters of the pre-compensator, low-pass filter, desired module, and controlled object need to satisfy, as well as the matching degree between the transfer function of the controlled object and the transfer function of the desired module, etc., so as to ensure that the entire solid thrust magnetic levitation bearing system can operate stably under various working conditions. of , or it can also be the complex plane of other parameters), and then ensure that the entire solid thrust magnetic levitation bearing system can operate stably under various working conditions.

[0030] Step 104: Adjust the control parameters in the fractional-order disturbance rejection control architecture based on the stability conditions; where the control parameters include the proportional coefficient of the differential term, the differential coefficient, and the fractional-order differential order of the pre-compensator, and the time constant of the low-pass filter .

[0031] Specifically, the control parameters include the proportional coefficient of the differential term of the pre-compensator , the differential coefficient and the fractional-order differential order, and the time constant of the low-pass filter . The purpose of adjusting these control parameters is to ensure that the entire solid thrust magnetic levitation bearing system meets the stability conditions and optimize the dynamic response performance of the solid thrust magnetic levitation bearing system.

[0032] According to the stability conditions, by adjusting the control parameters of the pre-compensator, low-pass filter, and desired module, the roots of the characteristic equation of the solid thrust magnetic levitation bearing system can be made to fall within the stable region, thereby ensuring the stable operation of the solid thrust magnetic levitation bearing system. Exemplarily, the adjustment methods may include step-by-step approximation, iterative optimization, or using specific control theory tools to find the optimal parameter combination that meets the stability conditions.

[0033] Step 105: Substitute the adjusted control parameters into the transfer function of the pre-compensator and the transfer function of the low-pass filter, and observe and compensate the total disturbance of the solid thrust magnetic levitation bearing system to achieve disturbance rejection control of the controlled object.

[0034] The updated pre-compensator inherits the new proportional coefficient of the differential term , the differential coefficient and the fractional-order differential order, and the low-pass filter applies the new time constant . Through the coordinated action of the desired module, pre-compensator, and low-pass filter, the solid thrust magnetic levitation bearing system can observe the total disturbance affecting the solid thrust magnetic levitation bearing in real time and compensate for it. This process effectively reduces the impact of the disturbance on the solid thrust magnetic levitation bearing system, ensures the stable operation of the solid thrust magnetic levitation bearing system, and successfully achieves the disturbance rejection control objective of the controlled object.

[0035] In this embodiment, the method is applied to the controller of the solid thrust magnetic levitation bearing system, and specifically includes: constructing a fractional-order disturbance rejection control architecture for the controller, which specifically includes a pre-compensator, a low-pass filter, and a desired module, where the pre-compensator is a fractional-order PD controller; determining the transfer functions of the pre-compensator, low-pass filter, desired module, and the controlled object; further performing system stability analysis on the solid thrust magnetic levitation bearing system based on the transfer functions of the pre-compensator, low-pass filter, desired module, and the controlled object to determine the stability conditions; adjusting the control parameters in the fractional-order disturbance rejection control architecture according to the stability conditions, including the proportional coefficient of the differential term, the differential coefficient, and the fractional-order differential order of the pre-compensator, and the time constant of the low-pass filter; substituting the adjusted control parameters into the pre-compensator and the low-pass filter, and realizing the disturbance rejection control of the controlled object by observing and compensating the total disturbance of the solid thrust magnetic levitation bearing system. This method can simplify the adjustment process of the control parameters, improve the stability and disturbance rejection performance of the solid thrust magnetic levitation bearing system, and ensure the stable operation of the solid thrust magnetic levitation bearing system when it is disturbed.

[0036] In one embodiment, by taking the square root of the first parameter to convert it into the second parameter , the complex plane of consists of the imaginary axis and the real axis, the complex plane of includes a stable region and an unstable region, , when is greater than zero and less than the target constant: Condition a: The relative order of the transfer function of the pre-compensated controlled object matches the desired transfer function; Condition b: In the complex plane of the zeros of the controlled object are located in the stable region; Condition c: In the complex plane of the zeros of the pre-compensator are located in the stable region.

[0037] Specifically, by taking the square root of the first parameter The square root is converted into the second parameter , that is . Among them, the first parameter is the Laplace operator parameter in the above steps. As Figure 3 shown , the complex plane of is composed of the real axis and the imaginary axis. The complex plane of is divided into the left half-plane and the right half-plane by the imaginary axis. The left half-plane (the shaded part in the figure) is the stable region, and the blank part in the right half-plane is the unstable region. And the area of the stable region is larger than that of the unstable region. The stable region includes the left half-plane and part of the right half-plane. There are two dividing lines between the stable region (the shaded part) and the unstable region (the blank part), which are located in the first quadrant and the fourth quadrant respectively. Among them, the angles between the two dividing lines and the real axis are both

[0038] . It should be noted that the angle between the dividing line and the real axis is not fixed, and the angle size can be adjusted according to the specific system characteristics . For example, if the relative order of the transfer function of the desired module is of the first order (such as ), and the relative order of the transfer function of the controlled object is 2.5 orders (such as ), then when the differential order of the pre-compensator is 1.5 (such as ), the relative order of

[0039] is 1, which matches the relative order of the transfer function of the desired module . Condition b is that the zeros of the controlled object in the complex plane of are located in the stable region. Among them, the zero refers to the zero of the transfer function of the controlled object, that is, the root of the numerator polynomial. The stable region refers to the left half-plane and part of the right half-plane of the complex plane of Figure 3 (the shaded part in

[0040] . Therefore, condition b requires that all zeros of the controlled object are located in the stable region. If the zeros of the controlled object are located in the stable region, the response characteristics of the solid thrust magnetic levitation bearing system are controllable, and there will be no undesired oscillations or unstable behaviors . Condition c is that the zeros of the pre-compensator in the complex plane of are located in the stable region. Similarly, the zeros of the pre-compensator should also be located in the left half-plane and part of the right half-plane of the complex plane of Figure 3(the shaded part). The role of the pre-compensator is to adjust the axial magnetic levitation bearing system to achieve the desired performance. If the zero point of the pre-compensator is located in the stable region, no new unstable factors will be introduced, and the stability of the magnetic levitation bearing system can be ensured.

[0041] In one embodiment, as Figure 4 shown, based on the transfer functions of the pre-compensator, low-pass filter, desired module, and controlled object, the system stability of the solid thrust magnetic levitation bearing system is analyzed to determine the stability conditions, including the following steps: Step 401: Derive the open-loop transfer function expression of the solid thrust magnetic levitation bearing system based on the transfer functions of the pre-compensator, low-pass filter, desired module, and controlled object ; Used to represent the proportional coefficient of the differential term, differential coefficient, and fractional differential order in the complex plane and the influence of on the stability of the bearing system.

[0042] It should be noted that the controlled object can represent the physical characteristics of the solid thrust magnetic levitation bearing, the pre-compensator can be used to adjust the dynamic characteristics of the solid thrust magnetic levitation bearing system, the low-pass filter can be used to reduce the influence of high-frequency noise, and the desired module can define the desired dynamic response of the solid thrust magnetic levitation bearing system. By combining these expressions, the open-loop transfer function of the solid thrust magnetic levitation bearing system can be obtained. The open-loop transfer function can describe the input-output relationship of the solid thrust magnetic levitation bearing system without feedback.

[0043] Exemplarily, assume , , and have the following transfer functions: , , , ; Based on , , and transfer functions, the open-loop transfer function expression of the solid thrust magnetic levitation bearing system can be derived as: .

[0044] Can be used to represent in The proportional coefficient of the differential term of the pre-compensator in the complex plane , the differential coefficient And the fractional-order differential order and the time constant of the low-pass filter On the stability of the bearing system.

[0045] Step 402: Based on By taking the Square root and converting it to , we get The transformation expression of .

[0046] Specifically, based on the open-loop transfer function expression of the solid thrust magnetic suspension bearing system , perform parameter transformation. The specific method is to take the Square root and convert it to , that is, let . Through parameter transformation, the original open-loop transfer function expression Can be converted to the transformation expression , and the transformation expression Is: .

[0047] Step 403: Determine the distribution region of the roots where the characteristic polynomial is equal to zero when the Of the low-pass filter, so as to determine condition b and condition c in the stability condition. Among them, the characteristic polynomial is related to the transformation expression .

[0048] Specifically, when the Of the low-pass filter approaches zero (i.e., ), the influence of the low-pass filter on the stability of the solid thrust magnetic suspension bearing system can be ignored. The characteristic polynomial is , and the roots where the characteristic polynomial is equal to zero are Of the roots.

[0049] By analyzing the distribution region of the roots of , the stability of the solid thrust magnetic suspension bearing system can be judged. If all the roots where the characteristic polynomial is equal to zero are located in the stable region of the Complex plane, then condition b and condition c are satisfied, that is, the zeros of the controlled object and the zeros of the pre-compensator are both located in the stable region, ensuring the stability of the solid thrust magnetic suspension bearing system.

[0050] In one embodiment, as Figure 5 Shown, The distribution region of the roots of the characteristic polynomial equal to zero when approaching zero is used to determine condition b and condition c in the stability condition, including the following steps: Step 501: Equivalent the roots of the characteristic polynomial equal to zero to the roots of the target expression equal to zero; the target expression is the sum of two parts, one part is related to the proportional coefficient and differential coefficient of the derivative term of the pre-compensator, and the other part is related to the time constant related.

[0051] Specifically, according to Matignon's theorem, the condition for the stability of the solid thrust magnetic levitation bearing system is that the roots of the characteristic polynomial equal to zero are all within Figure 3 the shown stable region, and the roots of the characteristic polynomial equal to zero are equivalent to the roots of the target expression equal to zero.

[0052] It should be noted that the expression of the roots of the characteristic polynomial equal to zero is: .

[0053] Furthermore, the expression of the roots of the equivalent target expression equal to zero is as follows: .

[0054] Among them, the target expression is the sum of two parts. The first half is related to the proportional coefficient of the derivative term of the pre-compensator, the differential coefficient of the pre-compensator, and the second half is related to the time constant of the low-pass filter.

[0055] Step 502: When approaches zero, split the target expression into the product of the first target expression and the second target expression .

[0056] Specifically, the target expression can be split into the first target expression and the second target expression . The specific formulas of the first target expression and the second target expression are as follows: .

[0057] Step 503: Determine condition b and condition c in the stability condition based on the distribution region of the roots.

[0058] Specifically, the stability condition of the solid thrust magnetic levitation bearing system is that the roots of the target expression are located in the stable region, that is, the roots of the first target expression and the second target expression are both located in the stable region.

[0059] When equals zero, and have the same roots. Let be the root of the first target expression , be the root of the target expression . There exists a minimum constant such that within the curve the first target expression has only one solution. The first target expression and are continuous in the curve , and there are and . It is obtained that there exists a constant such that when , holds. Let , . According to Rouche's theorem, it is obtained that the first target expression and the target expression have the same number of roots within the curve ; and at the same time , the limit , holds.

[0060] The roots of the first target expression include the zeros of the pre-compensator and the zeros of the controlled object. For the solid thrust magnetic levitation bearing system to be stable, the roots of the first target expression need to be located in the stable region. Therefore, the zeros of the pre-compensator and the zeros of the controlled object need to be both located in the stable region.

[0061] By analyzing the distribution of the roots of the first target expression in the complex plane, ensure that the roots of the first target expression are all located in the stable region, thereby ensuring the stability of the solid thrust magnetic levitation bearing system. Condition b requires that the zeros of the controlled object are located in the stable region, and condition c requires that the zeros of the pre-compensator are also located in the stable region. By verifying the root distribution, it can be ensured that these two conditions are met, thereby ensuring the stability of the entire solid thrust magnetic levitation bearing system.

[0062] In one embodiment, based on the distribution region of the roots of the first target expression conditions b and c in the stability condition are determined, including: Decompose the specific formula of the first target expression into a first expression and a second expression; among them, the parameters in the first expression are the same as the parameters in the pre-compensator transfer function, and some of the parameters in the second expression are the same as some of the parameters in the controlled object expression; Based on the root distribution region of the first expression, condition c in the stability condition is determined, and based on the root distribution region of the second expression, condition b in the stability condition is determined.

[0063] Specifically, the formula of the first target expression is as follows: .

[0064] Decompose the first target expression into a first expression and a second expression. The first expression is . Among them, the parameters in the first expression, such as the derivative term proportional coefficient , the derivative coefficient , are the same as the parameters in the expression of the pre-compensator; it should be noted that the first expression is the same as the expression of the pre-compensator. By analyzing the distribution of the roots of the first expression in the complex plane, condition c is determined, that is, whether the zeros of the pre-compensator are located in the stable region.

[0065] The second expression is . Among them, some of the parameters in the second expression, for example , are the same as the parameters in the expression of the controlled object. By analyzing the distribution of the roots of the second expression in the complex plane, condition b is determined, that is, whether the zeros of the controlled object are located in the stable region. By decomposing the first target expression and analyzing the distribution of the roots of each part in the complex plane respectively, the stability condition of the solid thrust magnetic levitation bearing system can be effectively determined.

[0066] In one embodiment, is rewritten, and the containing two variables is integrated into an expected expression containing one target variable; the expected expression also includes the matching degree between the pre-compensated controlled object transfer function and the expected module transfer function; The stability condition also includes condition d: the roots of the desired expression equal to zero are negative real numbers.

[0067] Specifically, by rewriting and letting , when the time constant approaches zero, the variable will approach infinity. This transformation can convert the second target expression containing two variables into a second target expression containing one variable .

[0068] The new second target expression is obtained by taking the limit when the time constant . In the limit case, is simplified to: ; where , is the matching degree between the transfer function of the controlled object after pre - compensation and the transfer function of the desired module.

[0069] In one embodiment, determining condition d in the stability condition includes: Assume that the first value is a root of the desired expression , and the second value and the third value are the two roots of the second target expression ; According to Rouche's theorem, there exists greater than zero and approaching zero, such that the product of the square of the second value and , and the product of the square of the third value and are both equal to the first value; From the fact that the product of the square of the second value and , and the product of the square of the third value and are both equal to the first value, it is obtained that when is greater than zero and approaching zero, the squares of the second value and the third value are both equal to the ratio of the first value to ; Since the squares of the second value and the third value are both equal to the ratio of the first value to , it is obtained that when is zero and the first value , the second value and the third value are both pure imaginary numbers and lie in the stable region.

[0070] Specifically, assume that the first value is a root of, the second value , the third value is the second target expression The two roots of. Then there exists a very small , such that , .

[0071] Let , . Expand to get: .

[0072] For a very small , , such that holds. According to Rouché's theorem, when , the two roots of the second value , the third value , that is, the limit , holds.

[0073] When the first value , the second value , the third value is in the stable region. At the same time, according to , the first value can be obtained. Therefore, is required.

[0074] holds. That is , holds, , the first value can obtain the second value and the third value are on the imaginary axis. That is, the second value and the third value are both pure imaginary numbers and are in the stable region shown in Figure 3 .

[0075] Furthermore, it can be obtained that when approaches zero and the system model of the solid thrust magnetic levitation bearing system has no zeros in the unstable region of the complex plane and the pre-compensator has no zeros in the unstable region of the complex plane, the roots of are all in the stable region shown in Figure 3 ; when , the roots of are negative real numbers, that is The roots of Figure 3 are all located in the stable region shown in

[0076] In one embodiment, based on the stability condition, the control parameters in the fractional-order disturbance rejection control architecture are adjusted, including: When tends to zero and the matching degree tends to zero, the transfer function of the controlled object after pre-compensation and the transfer function of the desired module have a matching degree higher than the matching threshold in the high-frequency band. By setting the differential coefficient of the pre-compensator, the proportional coefficient of the differential term of the pre-compensator is derived in relation to the transfer functions of the controlled object, the desired module, and the differential coefficient , and the proportional coefficient of the differential term is determined based on this relationship .

[0077] Determine the upper limit value of the time constant of the low-pass filter according to the stability condition; the upper limit value is the minimum value when the roots of the characteristic polynomial are located on the boundary line between the stable region and the unstable region. The time constant is selected between zero and the upper limit value.

[0078] Specifically, when and the matching degree , we get , which will further reduce the internal disturbance of the solid thrust magnetic levitation bearing system. The transfer functions of the controlled object and the desired module have a matching degree higher than the matching threshold in the high-frequency band. Among them, the pre-compensator can correct the phase and amplitude of the solid thrust magnetic levitation bearing system, making the transfer function closer to the desired dynamic characteristics in the high-frequency band, thereby improving the matching degree. By setting the differential coefficient of the pre-compensator, the proportional coefficient of the differential term of the pre-compensator is derived in relation to the transfer functions of the controlled object, the desired module, and the differential coefficient . Further, the relationship between the proportional coefficient of the differential term and the transfer functions of the controlled object, the desired module, and the differential coefficient can be derived as: where .

[0079] At this time, only the time constant has not been determined. As long as the time constant is selected small enough, the solid thrust magnetic levitation bearing system can be stable. The condition for the stability of the solid thrust magnetic levitation bearing system is that all roots of are located inFigure 3 The shown stable region, when the roots of or are such that the solid thrust magnetic levitation bearing system is in a critically stable state, i.e., the upper limit value of the time constant is the minimum value when the roots of the characteristic polynomial lie on the boundary line between the stable region and the unstable region. Where and are positive real numbers. Then the selected upper limit is: .

[0080] Therefore, is selected as a value as small as possible between to meet the performance requirements.

[0081] In one embodiment, as Figure 6 shown, the time constant is obtained by dynamic adjustment, and the dynamic adjustment specifically includes the following steps: Step 601: Based on the transfer function of the controlled object construct a simulation model, which includes a displacement outer loop and a current inner loop.

[0082] Specifically, based on the transfer function of the controlled object a simulation model can be constructed, and this simulation model can be used to simulate and analyze the dynamic behavior of the solid thrust magnetic levitation bearing system.

[0083] The simulation model includes a displacement outer loop and a current inner loop. The displacement outer loop is mainly responsible for controlling the position of the rotor to ensure its stable suspension within the desired position range; the current inner loop is used to precisely control the current in the electromagnet, so as to quickly respond to displacement changes and provide the corresponding electromagnetic force. Through the coordinated operation of this displacement outer loop and current inner loop, the solid thrust magnetic levitation bearing system can achieve precise control of the rotor position and ensure the stability and dynamic performance of the solid thrust magnetic levitation bearing system.

[0084] Step 602: When the simulation model is running, apply an impact disturbance in the step response, and observe the displacement drop value and recovery time of the pre-compensated transfer function of the controlled object after receiving the impact disturbance.

[0085] Apply a step response to the solid thrust magnetic levitation bearing system to make the rotor suspended at the central position. Further apply an impact disturbance to simulate sudden load addition or collision impact and other situations that may occur during actual operation. In this way, the disturbance situations that the solid thrust magnetic levitation bearing system may encounter under actual working conditions can be realistically simulated.

[0086] After applying an impact disturbance, the solid thrust magnetic levitation bearing system will respond to it, so that the dynamic response characteristics of the solid thrust magnetic levitation bearing system can be observed. The displacement drop value of the transfer function of the controlled object after receiving the impact disturbance can be observed, that is, the maximum deviation of the rotor position relative to the set point, which reflects the stability and anti-interference ability of the solid thrust magnetic levitation bearing system when it is disturbed. The time required for the solid thrust magnetic levitation bearing system to return to the stable state can also be recorded, that is, the recovery time, which reflects the dynamic response speed and regulation ability of the solid thrust magnetic levitation bearing system.

[0087] Step 603: Gradually decrease the time parameter between zero and the upper limit value Until the displacement drop is less than the drop threshold and the recovery time is less than the recovery threshold.

[0088] It should be noted that is selected as between. By continuously decreasing , the characteristics of the low-pass filter can be changed, thereby affecting the dynamic response of the solid thrust magnetic levitation bearing system. During the simulation process, observe the displacement drop value and recovery time of the solid thrust magnetic levitation bearing system after receiving an impact disturbance, and find the that meets the requirements. At this time, the solid thrust magnetic levitation bearing system exhibits good anti-disturbance performance.

[0089] In one embodiment, the transfer function expression of the solid thrust magnetic levitation bearing system after parameter identification is as follows: .

[0090] Select , then The value range of is: , select . The simulation process is to first give a step response to make the rotor levitate at the center position, and then apply an impact force with an amplitude of 250 N and a duration of 1 ms at 0.5 s. The simulation result of the displacement waveform of the solid thrust magnetic levitation bearing system is as Figure 7 shown.

[0091] As Figure 7 shown, the abscissa is time (s), and the ordinate is displacement (um), where the ordinate refers to the displacement of the system after receiving the impact force. After the solid thrust magnetic levitation bearing system receives the impact force, the displacement drops by about 10.3485 um, the peak-to-peak value of the oscillation after being impacted is about 14.90 um, and the time required for the displacement to return to within ±1 um again is 10.3 ms. The solid thrust magnetic levitation bearing system can operate stably and can return to the steady state after receiving the impact force, with a small displacement drop and adjustment time.

[0092] Based on the same concept, an embodiment of the present application further provides a solid thrust magnetic levitation bearing system affected by the eddy current effect. The solid thrust magnetic levitation bearing system includes a controller and a solid thrust magnetic levitation bearing. The solid thrust magnetic levitation bearing is the controlled object, and the controller can execute the above-mentioned control method for the solid thrust magnetic levitation bearing affected by the eddy current effect.

[0093] As the controlled object, the solid thrust magnetic levitation bearing is easily affected by the eddy current effect. The function of the controller is to execute the above-mentioned control method to achieve effective control of the solid thrust magnetic levitation bearing. By executing the above method, the controller can precisely control the solid thrust magnetic levitation bearing affected by the eddy current effect, thereby ensuring the stability and reliability of the system.

[0094] The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered as the scope described in this specification.

[0095] The above embodiments only represent several implementation manners of the present application. The description is relatively specific and detailed, but it should not be construed as a limitation on the patent scope of the present application. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several deformations and improvements can still be made, and these all belong to the protection scope of the present application. Therefore, the protection scope of the present application shall be subject to the appended claims.

Claims

1. A control method for a solid thrust magnetic levitation bearing affected by the eddy current effect, characterized in that, The method is applied to the controller of a solid thrust magnetic levitation bearing system, which also includes a solid thrust magnetic levitation bearing as the controlled object. The method comprises: Construct a fractional-order disturbance rejection control architecture for the controller in the solid thrust magnetic levitation bearing system, where the architecture includes a pre-compensator, a low-pass filter, and a desired module; among them, the pre-compensator is a fractional-order PD controller; Determine the transfer function of the pre-compensator, the transfer function of the low-pass filter, the transfer function of the desired module, and the transfer function of the controlled object; Based on the transfer functions of the pre-compensator, the low-pass filter, the desired module, and the controlled object, conduct a system stability analysis on the solid thrust magnetic levitation bearing system to determine the stability conditions; Adjust the control parameters in the fractional-order disturbance rejection control architecture based on the stability condition; wherein, the control parameters include the differential term proportional coefficient, differential coefficient, and fractional-order differential order of the pre-compensator, and the time constant of the low-pass filter ; Substitute the adjusted control parameters into the transfer functions of the pre-compensator and the low-pass filter, and observe and compensate for the total disturbance of the solid thrust magnetic levitation bearing system to achieve disturbance rejection control of the controlled object.

2. The control method for a solid thrust magnetic levitation bearing affected by the eddy current effect according to claim 1, wherein By taking the square root of the first parameter to convert it into a second parameter , the complex plane of the is composed of an imaginary axis and a real axis, and the complex plane of the includes a stable region and an unstable region. The complex plane of the is divided into a right half-plane and a left half-plane by the imaginary axis. The unstable region is located in the right half-plane, the stable region is larger than the unstable region, and the stable region includes the left half-plane and a part of the right half-plane; The stability condition is that there exists a target constant greater than zero, when the is greater than zero and less than the target constant: Condition a: The relative orders of the transfer function of the pre-compensated controlled object and the transfer function of the desired module match; Condition b: In the complex plane of the zeros of the controlled object are located in the stable region; Condition c: In the complex plane of the zeros of the pre-compensator are located in the stable region.

3. The control method for a solid thrust magnetic levitation bearing affected by the eddy current effect according to claim 2, wherein Based on the transfer functions of the pre-compensator, the low-pass filter, the desired module, and the controlled object, conduct a system stability analysis on the solid thrust magnetic levitation bearing system to determine the stability conditions, including: Derive the open-loop transfer function expression of the solid thrust magnetic levitation bearing system based on the transfer functions of the pre-compensator, the low-pass filter, the desired module, and the controlled object ; wherein, the is used to represent the proportional coefficient of the differential term, the differential coefficient, and the fractional-order differential order in the complex plane of the and the influence of the on the stability of the solid thrust magnetic levitation bearing system; Based on the above , by taking the square root of the above and converting it to the above , the transformation expression of the above is obtained ; By determining the distribution region of the roots for which the characteristic polynomial equals zero when approaches zero, to determine condition b and condition c in the stability condition; wherein, the characteristic polynomial is related to the transformation expression .​​​​ 4. The control method for a solid thrust magnetic levitation bearing affected by the eddy current effect according to claim 3, characterized in that By determining the distribution region of the roots for which the characteristic polynomial equals zero when the is satisfied, to determine condition b and condition c in the stability condition, including: Equate the roots of the characteristic polynomial to zero, which are equivalent to the target expression that are equal to zero; where the target expression is the sum of two parts, one part is related to the proportional coefficient of the differential term and the differential coefficient, and the other part is related to the related; When the approaches zero, split the target expression into the product of a first target expression and a second target expression ; Determine condition b and condition c in the stability condition based on the distribution area of the root.

5. The control method for a solid thrust magnetic levitation bearing affected by the eddy current effect according to claim 4, wherein Based on the distribution region of the root, determining condition b and condition c in the stability condition, including: Decompose the specific formula of into a first expression and a second expression; wherein, the parameters in the first expression are the same as the parameters in the pre-compensator transfer function, and some of the parameters in the second expression are the same as some of the parameters in the controlled object expression; Determine condition c in the stability conditions based on the root distribution region of the first expression, and determine condition b in the stability conditions based on the root distribution region of the second expression.

6. The control method for a solid thrust magnetic levitation bearing affected by the eddy current effect according to claim 4, characterized in that, The method further includes: taking the for rewriting, and integrating the including two variables into a desired expression including a target variable ; the desired expression further includes the matching degree between the pre-compensated transfer function of the controlled object and the transfer function of the desired module; Among them, the stability condition further includes condition d: the root of the desired expression equal to zero is a negative real number.

7. The control method for a solid thrust magnetic levitation bearing affected by the eddy current effect according to claim 6, characterized in that Determine condition d in the stability conditions, including: Assume that the first numerical value is a root of the said desired expression and the second and third numerical values are two roots of the second target expression ; According to Rouche's theorem, there exists the When it is greater than zero and approaches zero, the square of the second value multiplied by the product, and the square of the third value multiplied by the product are both equal to the first value; By the product of the square of the second value and the and the product of the square of the third value and the both being equal to the first value, it is obtained that when the is greater than zero and approaches zero, the squares of the second value and the third value are both equal to the ratio of the first value to the ; The square of the second value and the square of the third value are both equal to the ratio of the first value to the to obtain, in the is zero and the first value case, the second value and the third value are both pure imaginary numbers and are located in the stable region.

8. The control method for a solid thrust magnetic levitation bearing affected by the eddy current effect according to claim 2, characterized in that Based on the stability conditions, adjust the control parameters in the fractional-order disturbance rejection control architecture, including: When the approaches zero and the matching degree approaches zero, the transfer function of the controlled object after pre-compensation and the transfer function of the desired module have a matching degree higher than the matching threshold in the high-frequency band. By setting the differential coefficient of the pre-compensator, the proportional coefficient of the differential term of the pre-compensator is derived in relation to the transfer functions and differential coefficients of the controlled object and the desired module. Based on this relationship, the proportional coefficient of the differential term is determined; Determine the upper limit value of the according to the stability condition; wherein, the upper limit value is the smallest value when the roots of the characteristic polynomial are located on the boundary line between the stable region and the unstable region, and the time constant is selected between zero and the upper limit value.

9. The control method for a solid thrust magnetic levitation bearing affected by the eddy current effect according to claim 8, characterized in that, The said is obtained through dynamic adjustment. Dynamically adjusting the said specifically includes: Construct a simulation model based on the transfer function of the controlled object, where the simulation model includes a displacement outer loop and a current inner loop; When the simulation model is running, apply an impact disturbance in the step response, and observe the displacement drop value and recovery time of the transfer function of the pre-compensated controlled object after receiving the impact disturbance; Between zero and the upper limit value, gradually reduce the until the displacement drop is less than the drop threshold and the recovery time is less than the recovery threshold are satisfied.

10. A solid thrust magnetic levitation bearing system affected by the eddy current effect, characterized in that, The solid thrust magnetic levitation bearing system includes a controller and a solid thrust magnetic levitation bearing as the controlled object. The controller can execute the method for controlling a solid thrust magnetic levitation bearing affected by the eddy current effect according to any one of claims 1 to 9.

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