Design Method of Radial Decoupling and Axial PID Controller for AMBs–Rigid Rotor System

Abstract

The present invention discloses a design method for a radial decoupling and axial PID controller of an AMBs-rigid rotor system, constructs a radial decoupling PID controller, and gives a preliminary selection range for the parameters of the radial decoupling PID controller by using the requirements for the system stability and stiffness-damping characteristics; then, gives the tuning principles for each parameter of the radial decoupling PID controller, in particular, determines the tuning method for the proportional and derivative coefficients according to the variation relationship between the radial unbalanced vibration response and the unbalanced control current response of the system with respect to the proportional and derivative coefficients; finally, analyzes the dynamic characteristics of the axial motion and gives the parameter tuning basis and tuning method for the axial PID controller. The designed radial decoupling and axial PID controller of the present invention can not only quickly and stably levitate the electromagnetic bearing-rigid rotor system to the equilibrium position, but also enable the rotor system to have a smaller vibration response in the full speed range and a smaller control current when operating at the rated speed.

Description

Technical Field

[0001] The invention belongs to the technical field of electromagnetic bearing control, and in particular relates to a design method for radial decoupling and axial PID controller of an AMBs-rigid rotor system. Background Art

[0002] Active magnetic bearings (AMBs) have the advantages of no lubrication, no frequent maintenance, and active vibration control, and have been widely used in the field of high-speed rotating machinery. However, since the electromagnetic force provided by AMBs decreases with the increase of the air gap between the rotor and the magnetic pole surface, the AMBs-rigid rotor system is an open-loop unstable system with negative stiffness characteristics. In order to make the AMBs-rigid rotor system operate stably and have good dynamic performance, a suitable controller must be designed.

[0003] PID control has been widely used in AMBs-rigid rotor systems due to its advantages such as simple structure, convenient adjustment and strong applicability. How to adjust the parameters of the PID controller so that the system meets the requirements of stability, dynamic performance and other performance indicators is the key to the practical application of PID controllers. However, the current design methods of PID controllers are all used to adjust the decentralized PID control parameters, and the coupling of the radial translation and cone motion of the rotor is not considered. When the distances from the radial AMBs to the center of mass at both ends of the AMBs-rigid rotor system are quite different, the coupling of translation and cone motion is strong, and it is difficult to determine the changing relationship between each decentralized PID parameter and the motion characteristics of the translation and cone motion of the rotor system. At this time, the PID parameter tuning work is very difficult. In addition, in the actual application process, it is generally concerned whether the AMBs-rigid rotor system can safely and reliably pass through the rigid body critical speed under unbalanced excitation, and whether it can maintain low energy loss and heat generation during operation. In the existing PID controller design methods, the parameter tuning is often based on indicators such as stiffness, damping or stability margin, which is not intuitive and clear enough, and has limited guiding role in actual application occasions. Summary of the invention

[0004] The purpose of the present invention is to propose a design method for radial decoupling and axial PID controller of AMBs-rigid rotor system in view of the shortcomings of the prior art. The method is mainly used to solve the problem of setting parameters of PID controller of AMBs-rigid rotor system with strong coupling of translation and conical motion, so that the rotor system can operate safely, reliably and stably with smaller vibration response in the whole speed range while maintaining low energy loss and heat generation.

[0005] The object of the present invention is achieved by the following technical solutions: A design method for a radial decoupling and axial PID controller of an AMBs-rigid rotor system, the method comprising the following steps:

[0006] (1) Based on the requirements of the rotor system stability and stiffness-damping characteristics, construct a radial decoupling PID controller, associate the radial decoupling PID control parameters with the rotor system stability and the stiffness-damping characteristics of translation and coning respectively, and initially determine the value ranges of each PID control parameter;

[0007] (2) Based on the radial bias current of the rotor system and the proportional, differential and integral coefficients of the initially determined radial decoupling PID controller, further adjust the proportional and differential coefficients of the radial decoupling PID controller according to the variation relationships of the radial unbalance vibration response and unbalance control current response of the rotor system with the proportional and differential coefficients of the radial decoupling PID controller;

[0008] (3) Based on the requirements of the rotor system stability and stiffness-damping characteristics, associate the axial PID control parameters with the rotor system stability and stiffness-damping characteristics, and initially determine the value ranges of the axial PID control parameters;

[0009] (4) Based on the axial bias current of the rotor system and the initially determined axial PID control parameters, further adjust the proportional and differential coefficients of the axial PID controller according to the variation relationships of the axial vibration response and control current response of the rotor system to the axial excitation force with the proportional and differential coefficients of the axial PID controller.

[0010] Further, the specific process of the step (1) is as follows:

[0011] (1.1) The proportional coefficient matrix K Pr2 , differential coefficient matrix K Dr2 and integral coefficient matrix K Ir2 of the constructed radial decoupling PID controller are:

[0012]

[0013] Among them, K ri0 = diag([i 0a i 0b i 0a i 0b ), i 0a and i 0b are the bias currents at the A end and B end of the radial electromagnetic bearing AMB respectively; is the transformation matrix of the electromagnetic force, l a and l b are the axial distances from the rotor mass center to the A end and B end of the radial AMB respectively; kiri and k hri are parameters related to the number of turns of the coil and the cross-sectional area of the magnetic poles of the radial AMBs structure respectively. is the coordinate transformation matrix, l sa and l sb are the axial distances from the center of mass of the rotor to the radial sensor A and the radial sensor B respectively; K Pr3 = diag([k Prr k Prt k Prr k Prt ),k Prr and k Prt are the proportionality coefficients of the radial decoupling PID controller in the conical and translational systems respectively. K Ir3 = diag([k Irr k Irt k Irr k Irt ),k Irr and k Irt are the integral coefficients of the radial decoupling PID controller in the conical and translational systems respectively. K Dr3 = diag([k Drr k Drt k Drr k Drt ),k Drr and k Drt are the differential coefficients of the radial decoupling PID controller in the conical and translational systems respectively.

[0014] (1.2) According to the requirements of the rotor system for the stability, stiffness and damping characteristics of translational motion, determine the value range of the PID coefficients related to translation of the radial decoupling PID controller as:

[0015]

[0016] where m is the mass of the rotor.

[0017] According to the requirements of the rotor system for the stability, stiffness and damping characteristics of conical motion, determine the value range of the PID coefficients related to conical motion of the radial decoupling PID controller as:

[0018]

[0019] where J r is the equatorial moment of inertia of the rotor.

[0020] Furthermore, the specific process of the step (2) is as follows:

[0021] (2.1) The radial bias current i 0a and i 0bSetting

[0022] Set the radial static control current not to exceed 10% of the corresponding bias current to obtain the radial bias current i 0a and i 0b The value range is:

[0023]

[0024] where g is the acceleration due to gravity.

[0025] (2.2) Radial proportional and derivative coefficients k Prt 、k Drt 、k Prr and k Drr Setting

[0026] The radial displacement X sa (s) and X sb (s) and F εx1 (s) are related as follows:

[0027]

[0028]

[0029] where ε and ε z are the radial and axial offsets of the unbalanced mass of the rotor respectively, ω and φ are the angular velocity and angular displacement of the rotor rotating about the z-axis respectively, F εx1 (s) is the Laplace transform of the unit cosine function f εx1 (t) = -cosφ = -cos(ωt + φ0), and φ0 is the initial angle of the rotor.

[0030] The control currents I cxa (s) and I cxb (s) and F εx1 (s) are related as follows:

[0031]

[0032]

[0033] The specific parameter adjustment direction needs to be determined by combining the actual situations of the radial unbalance vibration response and unbalance control current response of the rotor system within the full speed range, as well as the specific relationships between the radial proportional and derivative coefficients and the unbalance vibration response and unbalance control current response at different speeds.

[0034] Furthermore, the specific process of step (3) is as follows:

[0035] According to the requirements of the rotor system for the stability of axial movement and the stiffness-damping characteristics, the value range of the axial PID control parameters is determined as follows:

[0036]

[0037] Further, the specific process of step (4) is as follows:

[0038] (4.1) Tuning of the axial bias current i 0z

[0039] It is set that when the rotor is axially subjected to an external force equal to the rotor gravity, the axial static control current does not exceed 40% of the corresponding bias current, and the value range of the axial bias current i 0z is obtained as follows:

[0040]

[0041] (4.2) Tuning of the axial proportional and derivative coefficients k Pz and k Dz

[0042] The relationship between the displacement Z(s) of the rotor in the axial direction and F dz1 (s) is:

[0043]

[0044] where z d is the amplitude of the axial synchronous excitation displacement signal received by the rotor, and F dz1 (s) is the Laplace transform of the unit sine function f dz1 (t) = sin(ωt + φ 0z ), and φ 0z is the initial angle of the excitation displacement signal.

[0045] The control current I cz (s) of the axial AMB and F dz1 (s) are related as follows:

[0046]

[0047] The specific parameter adjustment direction needs to be determined by combining the actual situations of the axial vibration response and the control current response of the rotor system to the axial excitation signal within the full speed range, as well as the specific relationships between k Pz and k Dz and the axial vibration response and the control current response at different speeds.

[0048] The advantages of the present invention compared with the prior art are as follows:

[0049] ​​(1) The design method of the radial decoupling PID controller proposed by the present invention can be applied to the translational-tilting strongly coupled AMBs-rigid rotor system, and provides an effective solution to the problem of difficult parameter tuning existing in the application of the currently widely used decentralized PID controller in the translational-tilting strongly coupled AMBs-rigid rotor system;

[0050] (2) Based on the form of the radial unbalanced force, the present invention specifically analyzes the relationship between the radial unbalanced vibration response and the unbalanced control current response and the proportional and differential coefficients of the radial decoupling PID controller, ensuring that the AMBs-rigid rotor system can safely and reliably cross the rigid body critical speed under the excitation of the radial unbalanced force and maintain low energy loss and heat generation during operation;

[0051] (3) Based on the form of the axial excitation force, the present invention specifically analyzes the relationship between the vibration response and the control current response of the rotor system to the axial excitation signal and the proportional and differential coefficients of the axial PID controller, ensuring that the AMBs-rigid rotor system can safely and reliably cross the critical speed under the action of the axial external excitation force with the same frequency as the rotational speed and maintain low energy loss and heat generation during operation;

[0052] (4) The design method of the radial decoupling and axial PID controller proposed by the present invention is simple, intuitive and purposeful, and can provide reasonable and correct guiding opinions for the parameter tuning process of the PID controller in actual situations. Description of the Drawings

[0053] Figure 1 Structural diagram of the translational-tilting strongly coupled AMBs-rigid rotor system adopted in the embodiment.

[0054] Figure 2 For different k Prt cases of X sa (s) / F εx1 (s) and X sb (s) / F εx1 (s) amplitude variation curves with rotational speed.

[0055] Figure 3 For different k Prr cases of X sa (s) / F εx1 (s) and X sb (s) / F εx1 (s) amplitude variation curves with rotational speed.

[0056] Figure 4 For different k Drt cases of X sa (s) / F εx1 (s) and Xsb (s) / F εx1 (s) amplitude vs. rotational speed curve.

[0057] Figure 5 For different k Drr cases of X sa (s) / F εx1 (s) and X sb (s) / F εx1 (s) amplitude vs. rotational speed curve.

[0058] Figure 6 For different k Prt cases of I cxa (s) / F εx1 (s) and I cxb (s) / F εx1 (s) amplitude vs. rotational speed curve.

[0059] Figure 7 For different k Prr cases of I cxa (s) / F εx1 (s) and I cxb (s) / F εx1 (s) amplitude vs. rotational speed curve.

[0060] Figure 8 For different k Drt cases of I cxa (s) / F εx1 (s) and I cxb (s) / F εx1 (s) amplitude vs. rotational speed curve.

[0061] Figure 9 For different k Drr cases of I cxa (s) / F εx1 (s) and I cxb (s) / F εx1 (s) amplitude vs. rotational speed curve.

[0062] Figure 10 For different k Pz and different k Dz cases of Z(s) / F dz1 (s) amplitude vs. rotational speed curve.

[0063] Figure 11 For different k Pz and different k Dz cases of I cz (s) / F dz1(s) amplitude variation curve with rotational speed.

[0064] Figure 12 It is the displacement response curve of the AMBs–rigid rotor system during the floating process after PID parameter tuning.

[0065] Figure 13 It is the vibration response curve of the AMBs–rigid rotor system during the acceleration process after PID parameter tuning.

[0066] Figure 14 It is the vibration waterfall plot of the AMBs–rigid rotor system during the acceleration process after PID parameter tuning.

[0067] Figure 15 It is the control current response curve of the AMBs–rigid rotor system during the acceleration process after PID parameter tuning. Specific implementation manner

[0068] The following further elaborates on the specific implementation manner of the present invention in conjunction with the accompanying drawings.

[0069] In this embodiment, the structure diagram of the translational-cone dynamic strongly coupled AMBs–rigid rotor system is as Figure 1 shown. A set of radial sensors and a set of radial AMBs are arranged on both sides of the rotor centroid, respectively used to monitor and control the movement of the rotor in four degrees of freedom in the radial direction. An axial sensor is arranged near the end face of the rotor close to the B end, and a pair of axial AMBs are arranged on both sides of the axial disk of the rotor, respectively used to monitor and control the movement of the rotor in the axial direction.

[0070] Taking the rotor centroid C as the origin, an xyz coordinate system is established, where the z direction is the axial direction of the rotor, and the x and y directions are located in the plane perpendicular to the rotor axial direction. They are orthogonal to each other and the included angle with the horizontal plane is 45°. Let the axial distances from the rotor centroid C to the radial AMBs -A and -B be l a and l b , and the axial distances to the radial sensors A and B be l sa and l sb .

[0071] According to the rotor dynamics theory, the motion differential equation of the AMBs–rigid rotor system can be obtained as

[0072]

[0073] In the formula, m is the rotor mass, z is the axial displacement of the rotor, f zAMB is the electromagnetic force of the axial AMB, f dzis the axial excitation force. M and G are respectively the mass matrix and gyro matrix of the radial motion, q is the generalized coordinate vector of the rotor's radial motion, L is the force transformation matrix, F rAMB is the electromagnetic force vector of the radial AMBs, F G is the gravity vector, F ε is the unbalance force vector, and their specific expressions are:

[0074]

[0075]

[0076] where, J r and J z are respectively the equatorial moment of inertia and polar moment of inertia of the rotor, θ x and θ y are respectively the angular displacements of the rotor rotating about the x and y axes, x and y are the displacements of the rotor's center of mass C in the x and y directions, ω and φ are respectively the angular velocity and angular displacement of the rotor rotating about the z axis, f xa 、f xb 、f ya and f yb are respectively the electromagnetic forces of AMB-A and -B in the x and y directions, ε and ε z are respectively the offsets of the rotor unbalance in the radial and axial directions.

[0077] After linearization, the radial electromagnetic force vector F rAMB and the axial electromagnetic force f zAMB can be expressed as:

[0078]

[0079] In the formula, K ir =diag([k ira k irb k ira k irb ) is the radial current stiffness matrix, k ira and k irb are respectively the current stiffness coefficients of radial AMB-A and -B; K hr =diag([k hra k hrb k hra k hrb ) is the radial displacement stiffness matrix, k hra and k hrb are respectively the displacement stiffness coefficients of radial AMB-A and -B; I cr =[i cxa i cxb i cya i cybT is the control current vector of the radial AMB, i cxa , i cxb , i cya and i cyb are the control currents of the radial AMBs - A and - B in the x and y directions respectively, q b = [x a x b y a y b T = L T q is the radial displacement vector of the rotor at the AMB, x a , x b , y a and y b are the radial displacements of the rotor at the AMBs - A and - B in the x and y directions respectively. k iz and k hz are the current stiffness coefficient and displacement stiffness coefficient of the axial AMB respectively, i cz is the control current of the axial AMB.

[0080] k ira and k irb , k hra and k hrb have the following relationships with the bias currents i 0a and i 0b of the radial AMBs - A and - B:

[0081]

[0082] wherein, k iri and k hri are parameters related to the structures of the radial AMBs.

[0083] k iz and k hz also have the following relationships with the bias current i 0z of the axial AMB:

[0084]

[0085] wherein, k izi and k hzi are parameters related to the structure of the axial AMB.

[0086] Combining equations (1) - (4), we can obtain:

[0087]

[0088] wherein, K ri0 = diag([i 0a i​​0b i 0a i 0b )。

[0089] Next, construct a radial decoupling PID controller, analyze the dynamic characteristics of the system's radial motion under decoupling PID control, and give the value ranges of the proportional, differential, and integral coefficients of the decoupling PID controller. The specific embodiments and their implementation processes are as follows:

[0090] (1) Construction of the radial decoupling PID controller

[0091] The constructed decoupling PID controller is:

[0092]

[0093] wherein, K Pr2 ∈R 4×4 , K Ir2 ∈R 4×4 and K Dr2 ∈R 4×4 are the proportional, integral, and differential coefficient matrices of the decoupling PID controller, respectively.

[0094] The differential equation of motion of the rotor's radial four degrees of freedom under decoupling PID control is

[0095]

[0096] wherein,

[0097] If the translational and conical motions of the system are to be decoupled, K D2 , K S2 and K I2 must all be diagonal matrices. For this purpose, K Pr2 , K Ir2 and K Dr2 can be constructed as:

[0098]

[0099] wherein, K Pr3 =diag([k Prr k Prt k Prr k Prt ), K Ir3 =diag([k Irr k Irt k Irr k Irt ), K Dr3 =diag([k Drr k Drt k Drr k Drt)。

[0100] Combining equations (7) and (8), we can obtain:

[0101]

[0102] Expanding equation (9) into the motion equations in four degrees of freedom, we can get:

[0103]

[0104] (2) Analysis of the translational dynamics characteristics of the system under decoupled PID control

[0105] As can be seen from equation (10), under the condition of neglecting external forces, the characteristic equations of the translational motion of the rotor in the x and y directions are:

[0106] ms 3 +k iri k Drt s 2 +k iri k Prt s + k iri k Irt =0 (11)

[0107] According to the Routh stability criterion, if the system is to be stable, the following conditions should be met:

[0108]

[0109] The integral coefficient k Irt mainly determines the speed at which the system eliminates the static error. Compared with k Prt and k Drt , its influence on the dynamic characteristics is very small. For the convenience of analysis, k Irt is neglected, and only the force components introduced by k Prt and k Drt are retained in the electromagnetic force of the AMBs, obtaining the characteristic equations of the translational motion of the rotor in the x and y directions:

[0110] ms 2 +k iri k Drt s + k iri k Prt =0 (13)

[0111] Regarding equation (13) as a single-degree-of-freedom vibration system, the damping ratio ζ t and the undamped natural frequency ω nt are:

[0112]

[0113] To ensure that the translational motion of the rotor has sufficient stiffness and damping, ζ t and ω nt should satisfy:

[0114]

[0115] Combined with Equation (12), the value ranges of k Prt , k Drt and k Irt are:

[0116]

[0117] (2) Analysis of the coning dynamics characteristics of the system under decoupled PID control

[0118] Let θ = θ x + jθ y , and combine the two rotor coning motion equations in Equation (10) into a motion differential equation with θ as the coordinate variable:

[0119]

[0120] Under the condition of neglecting external forces, the characteristic equation of the rotor's coning motion in the x and y directions is:

[0121] J r s 3 +(k iri k Drr - jJ z ω)s 2 + k iri k Prr s + k iri k Irr = 0 (18)

[0122] According to the complex coefficient Routh stability criterion, if the system is to be stable, the following conditions must be met:

[0123]

[0124] From Equation (19), it can be seen that under decoupled PID control, when the system runs at high speed, the value of J z ω introduced by the gyroscopic effect is relatively large, which will significantly weaken or even destroy the stability of the system. Therefore, in order to eliminate the adverse effects of the gyroscopic torque on the rotor dynamics characteristics, an additional gyroscopic compensation current vector I cg needs to be introduced:

[0125]

[0126] where,

[0127] After introducing the gyro compensation matrix, for the convenience of analysis, k is ignored Irr , Equation (18) is simplified to:

[0128] J r s 2 +k iri k Drr s + k iri k Prr =0 (21)

[0129] Similarly, the damping ratio ζ of the coning system can be determined r and the undamped natural frequency ω nr are

[0130]

[0131] To avoid the problem of excessive vibration caused by the rotor passing through both the translational and coning critical speeds within a short speed range, it is necessary to set k Prr and k Drr so that the coning modal frequency of the rotor is far from the translational modal frequency. At the same time, to ensure that the coning motion of the rotor has sufficient stiffness and damping, let ζ r and ω nr satisfy:

[0132]

[0133] Combined with Equation (19), the value ranges of k Prr , k Drr and k Irr are:

[0134]

[0135] The parameter tuning process of the radial decoupling PID controller is described below. The specific embodiments and their implementation processes are as follows:

[0136] (1) Tuning of the radial bias currents i 0a and i 0b The radial static control current vector I

[0137] used to compensate for the rotor gravity and the rotor gravity vector F c0 are related as follows: G k

[0138] k iri LK ri0 I c0 =-F G (25)

[0139] where I c0 =[icxa0 i cxb0 i cya0 i cyb0 T 。

[0140] Furthermore, it is obtained that:

[0141]

[0142] Let each element in I c0 be less than or equal to 10% of the corresponding radial bias current of the AMB, and the radial bias current i 0a and i 0b can be obtained with the value range of:

[0143]

[0144] (2) Radial ratio and differential coefficient k Prt , k Drt , k Prr and k Drr tuning

[0145] (2.1) Relationship between radial ratio and differential coefficient and unbalanced vibration response

[0146] Ignoring the integral term and not considering the rotor gravity and noise, the relationship between the translational displacement x(t) of the rotor in the x-direction and the unbalanced force f εx (t) it receives is as follows:

[0147]

[0148] In the formula,

[0149] The unbalanced force f εx (t) can be rewritten as:

[0150]

[0151] In the formula, f εx1 (t) = -cosφ = -cos(ωt + φ0), where φ0 is the initial angle.

[0152] Substitute Equation (29) into Equation (28) and perform Laplace transform, and it can be obtained that:

[0153]

[0154] Similarly, the relationship between the rotational angular displacement θ y (s) of the rotor about the y-axis in the complex frequency domain and F εx1 (s) is:

[0155] ​

[0156] Then, the radial displacements X sa (s) and X sb (s) of the rotor in the x-direction at the positions of sensors A and B have the following relationship with F εx1 (s):

[0157]

[0158]

[0159] Next, a set of coefficients is selected, and it is judged what kind of influence the change of a single radial ratio or differential coefficient value will have on the unbalanced vibration response of the system near the values of this set of coefficients. Let k Prt = 1000, k Drt = 2.5, k Prr = 20, k Drr = 0.02. Near these values, the values of k Prt , k Prr , k Drt and k Drr are adjusted respectively, and the curves of the variation of the amplitudes of X Prt (s) / F Prr (s) and X Drt (s) / F Drr (s) with the rotational speed in different cases of k sa (s) / F εx1 (s) and X sb (s) / F εx1 (s) with the rotational speed are shown respectively as Figures 2 to 5 shown.

[0160] It can be seen from Figure 2 that increasing k Prt will cause the increase of the vibration response peaks at the positions of the two sensors of the rotor system. It can be seen from Figure 3 that increasing k Prr will cause a slight increase in the vibration peak at sensor A of the rotor system and the vibration amplitude near the conical critical speed at sensor B, while the vibration amplitude near the translational critical speed at sensor B decreases. It can be seen from Figure 4 that increasing k Drt will cause a decrease in the vibration peak at sensor A of the rotor system and the vibration amplitude near the translational critical speed at sensor B, while the vibration amplitude near the conical critical speed at sensor B remains basically unchanged. It can be seen from Figure 5 that increasing k DrrIt will cause the vibration peak value of the rotor system at sensor A and the vibration amplitude near the conical critical speed at sensor B to decrease, while the vibration amplitude near the translational critical speed at sensor B remains basically unchanged. Therefore, in order to avoid large unbalanced vibration response peak values of the rotor system within the full speed range, the radial proportionality coefficient k Prt and k Prr should not be too large, and the radial differential coefficients k Drt and k Drr should not be too small.

[0161] During the actual tuning process of the proportional and differential coefficients, the vibration peak value of the rotor within the full speed range at sensor A and the vibration amplitudes near the translational critical speed and the conical critical speed at sensor B can be detected by sensors, and then it can be judged which of these values are too large and need to be further suppressed. Then, according to the relationship between the values of k Prt 、k Prr 、k Drt and k Drr analyzed above and these vibration amplitudes, the adjustment direction of the correlation coefficients can be judged.

[0162] (2.2) Relationship between radial proportional and differential coefficients and unbalanced control current response

[0163] As can be seen from Equation (8), the relationship between the radial control currents I cxa (s) and I cxb (s) in the x direction of AMB - A and - B and F εx1 (s) is:

[0164]

[0165]

[0166] When the values of the radial proportional and differential coefficients are near k Prt = 1000, k Drt = 2.5, k Prr = 20, k Drr = 0.02, respectively adjust the values of k Prt 、k Prr 、k Drt and k Drr to obtain the curves of the variation of the amplitudes of I Prt 、k Prr 、k Drt and k Drr in different radial cases with respect to I cxa (s) / F εx1 (s) and I cxb (s) / F εx1 (s) with the speed, respectively, as shown in Figures 6 to 9as shown

[0167] It is known that the rated speed of the rotor is 12,000 rpm. From Figure 6 it can be seen that increasing k Prt will cause the peak value of the unbalance control current response of the two ends of the AMBs in the full speed range and the amplitude of the unbalance control current response at the rated speed to increase. From Figure 7 it can be seen that increasing k Prr will cause the peak value of the unbalance control current response of the two ends of the AMBs in the full speed range and the amplitude of the unbalance control current response of AMB-B at the rated speed to increase, while the amplitude of the unbalance control current response of AMB-A at the rated speed decreases. From Figure 8 it can be seen that increasing k Drt will cause the peak value of the unbalance control current response of AMB-A in the full speed range and the amplitude of the unbalance control current response of AMB-B near the translational critical speed to decrease, while the amplitude of the unbalance control current response of AMB-B near the conical critical speed remains basically unchanged. From Figure 9 it can be seen that increasing k Drr will cause the peak value of the unbalance control current response of AMB-A in the full speed range to decrease slightly, the amplitude of the unbalance control current response of AMB-B near the conical critical speed to decrease, and the amplitude of the unbalance control current response near the translational critical speed to remain basically unchanged. In addition, excessive k Drt and k Drr will cause the amplitude of the unbalance control current response of the two ends of the AMBs at the rated speed to increase significantly. Therefore, in order to suppress the peak value of the unbalance control current response in the full speed range, the proportionality coefficients k Prt and k Prr should not be too large, and the differential coefficients k Drt and k Drr should not be too small; and in order to suppress the amplitude of the unbalance control current response at the rated speed, the differential coefficients k Drt and k Drr should not be too large either.

[0168] In order to reduce the power consumption of the AMBs and extend the service life of the equipment, on the basis of ensuring that the system can safely and reliably cross the critical speed with small vibrations, according to the peak value of the unbalance control current response of the two ends of the AMBs in the full speed range and the amplitude of the unbalance control current response at the rated speed obtained from the above analysis, as well as the variation relationship between the proportional and differential coefficients k Prt 、k Prr 、k Drt and k Drr between them, appropriately adjust their values so that the peak value of the unbalance control current response of the rotor in the full speed range and the unbalance control current response at the rated speed are both at a relatively low level.

[0169] (3) Radial integral coefficient k Irt and k Irr setting

[0170] Radial integral coefficient k Irt and k Irr The values of k Irt and k Irr mainly affect the speed at which the system eliminates the steady-state error and have a relatively small impact on the dynamic performance of the system. Therefore, after setting the proportional and differential coefficients to ensure that the system has sufficient stability and good dynamic performance, k Irt and k Irr can be set. To suppress the influence of the force components introduced by k Irt and k Irr on the system stability and dynamic performance, under the condition of ensuring that the system can eliminate the steady-state error in a short time, the values of k

[0171] should be taken as small as possible. The following analyzes the dynamic characteristics of the axial movement of the rotor system under PID control and gives the value ranges of the proportional, differential, and integral coefficients of the axial PID controller. The specific embodiments and their implementation processes are as follows:

[0172] The axial control current under PID control can be expressed as:

[0173]

[0174] Combined with Equation (5), we can obtain:

[0175]

[0176] Under the condition of ignoring the external excitation force, the characteristic equation of the system's movement in the z direction can be obtained:

[0177]

[0178] According to the Routh stability criterion, if the axial movement of the rotor system is to be stable, the following conditions should be met:

[0179]

[0180] For the convenience of analysis, ignoring k Iz , the characteristic equation of the rotor's axial movement is obtained as:

[0181]

[0182] Regarding Equation (40) as a single-degree-of-freedom vibration system, the damping ratio ζ z and the undamped natural frequency ω nz are respectively:

[0183]

[0184] To ensure that the axial movement of the rotor has sufficient stiffness and damping, ζ z and ω nz should satisfy:

[0185]

[0186] Combining equations (41) and (42), k Pz and k Dz should satisfy the following conditions:

[0187]

[0188] The following describes the parameter tuning process of the axial PID controller. The specific embodiments and their implementation processes are as follows:

[0189] (1) Tuning of the axial bias current i 0z

[0190] When an external force equal to the gravity of the rotor acts in the axial direction of the rotor, there is the following relationship between the axial static control current i cz0 and the gravity mg of the rotor:

[0191] k izi i 0z i cz0 = mg (44)

[0192] Furthermore, it can be obtained that:

[0193]

[0194] Let i cz0 be less than or equal to 40% of the axial AMBs bias current, and the value range of the axial bias current i 0z can be obtained as:

[0195]

[0196] The axial bias current i 0z should take a smaller value within the above range.

[0197] (2) Tuning of the axial proportional and differential coefficients k Pz and k Dz

[0198] (2.1) Relationship between the axial proportional and differential coefficients and the vibration response of the system to the axial excitation signal

[0199] The axial excitation displacement signal Δz can be expressed as: ​​

[0200] Δz = z d f dz1 (t) (47)

[0201] Where z d is the amplitude of the excitation displacement signal. For the convenience of analysis, it is considered a constant value. f dz1 (t) = sin(ωt + φ 0z ), where φ 0z is the initial angle of the excitation displacement signal.

[0202] Then, the axial excitation force f dz can be expressed as:

[0203]

[0204] Substitute Equation (48) into Equation (37), ignore the integral term, and perform Laplace transform on both sides of the equation simultaneously, the relationship between the axial displacement Z(s) of the rotor and the unit sine function F dz1 (s) is:

[0205]

[0206] When the values of the axial proportional and differential coefficients are around k Pz = 6000, k Dz = 6, adjust the values of k Pz and k Dz respectively, the curves of the amplitude of Z(s) / F Pz and different k Dz versus the rotational speed under different k dz1 (s) are as shown in Figure 10 Figure.

[0207] It can be seen from Figure 10 that increasing k Pz will cause the peak value of the axial vibration of the rotor system to increase, while increasing k Dz will cause the peak value of the axial vibration of the rotor system to decrease. Therefore, in order to avoid the peak value of the axial vibration of the rotor system from being too large, k Pz should not be taken too large, and k Dz should not be taken too small.

[0208] (2.2) Relationship between the axial proportional and differential coefficients of the rotor and the control current response of the system to the axial excitation signal. Combining Equation (36) and Equation (49), and ignoring the integral term, the relationship between the axial control current I cz (s) and F dz1 (s) is:

[0209]

[0210] When the axial ratio of the rotor and the values of the differential coefficient are also within k Pz = 6000, k Dz = 6, adjust k Pz and k Dz respectively. Different values of k Pz and different values of k Dz can result in the curves of the amplitude of I cz (s) / F dz1 (s) varying with the rotational speed as shown in Figure 11 .

[0211] As can be seen from Figure 11 , increasing k Pz will cause the control current response of the rotor system to the axial excitation signal to increase significantly in the rotational speed range from near the rotational speed where the axial vibration peak of the rotor system is located to the rated rotational speed of 12000 rpm. While increasing k Dz will cause the control current response of the system to the axial excitation signal to decrease near the rotational speed where the axial vibration peak of the rotor system is located, and increase near the rated rotational speed. Therefore, to avoid an overly large peak value of the axial control current response of the rotor system, k Pz should not be set too large, and k Dz should not be set too small; to avoid an overly large axial control current response of the rotor system at the rated rotational speed, k Dz should not be set too large either.

[0212] (3) Tuning of the axial integral coefficient k Iz The value of the axial integral coefficient k

[0213] mainly affects the speed at which the system eliminates the steady-state error in the axial direction and has a relatively small impact on the dynamic performance of the system. Therefore, after tuning the axial proportional coefficient k Iz and the differential coefficient k Pz to ensure that the axial motion of the system has sufficient stability and good dynamic performance, k Dz can be tuned. To suppress the influence of the force component introduced by k Iz on the stability and dynamic performance of the system, the value of k Iz should be taken as small as possible under the condition that the system can eliminate the steady-state error in a short time. Iz

[0214] Based on the above analysis process and combined with specific embodiments, the parameter tuning method of the radial decoupling and axial PID controller is summarized as follows:

[0215] 1) Determine the bias current value of the radial and axial AMBs as i according to the relative magnitudes of the radial and axial static control currents and the bias current​0a = 0.8 A, i 0b = 0.6 A, i 0z = 0.6 A;

[0216] 2) Based on the rotor system model parameters, the value ranges of the proportional, derivative, and integral coefficients of the radial decoupling and axial PID controllers are calculated as k Prt > 710, k Drt > 1.13, 0 < k Irt < 71316, k Prr > 8.5, k Drr > 0.009, 0 < k Irr < 1279, k Pz > 3432, k Dz > 2.3, 0 < k Iz < 145263;

[0217] 3) Select a set of initial parameter values within the value ranges of the radial and axial proportional, derivative, and integral coefficients in 2), specifically k Prt = 1000, k Drt = 2.5, k Irt = 500, k Prr = 20, k Drr = 0.02, k Irr = 10, i 0z = 0.6 A, k Pz = 6000, k Dz = 6, k Iz = 3000;

[0218] 4) To further reduce the peak values of the radial and axial vibrations at the positions of sensors A and B of the rotor system, while keeping the control current response of the system at a low level under the rated speed, appropriately reduce the values of k Prt 、k Prr and k Pz and appropriately increase the values of k Drt 、k Drr and k Dz Finally, their values are obtained as: k Prt = 900, k Drt = 4, k Prr = 18, k Drr = 0.032, k Pz = 5000, k Dz = 10.

[0219] 5) After determining the radial and axial proportional and derivative coefficients, to improve the speed of eliminating the steady-state error while ensuring the radial and axial motion stability of the rotor system, appropriately increase k Irt 、k Irrand k Iz The values are finally obtained as: k Irt = 1000, k Irr = 15, k Iz = 5000.

[0220] The displacement response curve of the translational-tilting strongly coupled AMBs-rigid rotor system during the floating process after the PID parameter tuning in this embodiment is as Figure 12 shown. It can be seen that the rotor can float to the given position with a small overshoot in a short time both in the radial and axial directions and can stably levitate at the given position.

[0221] The vibration response curve and vibration waterfall plot of the five-degree-of-freedom translational-tilting strongly coupled AMBs-rigid rotor system during the acceleration process after the PID parameter tuning in this embodiment are respectively as Figure 13 and Figure 14 shown. It can be seen that under the control of the decoupled PID controller and axial PID controller designed according to the parameter tuning method proposed by the present invention, the risk of the rotor colliding with the protection bearing is very small in the full speed range, and the system can safely and reliably cross the translational and tilting critical speeds and the axial rigid body critical speed.

[0222] The control current response curve of the translational-tilting strongly coupled AMBs-rigid rotor system during the acceleration process after the PID parameter tuning in this embodiment is as Figure 15 shown. Under the control of the decoupled PID controller and axial PID controller designed according to the parameter tuning method proposed by the present invention, the system can achieve a small control current response in the full speed range and at the rated speed under a small bias current condition, ensuring that the heat generation of the power amplifier and coil is at a low level, which is beneficial to the long-term stable operation of the system.

[0223] The above embodiments are used to explain the present invention rather than limit the present invention. Any modifications and changes made within the spirit and scope of the claims of the present invention fall within the protection scope of the present invention.

Claims

1. A design method for radial decoupling and axial PID controller of an AMBs-rigid rotor system, characterized in that: The method includes the following steps: (1) Based on the requirements of the rotor system stability and stiffness-damping characteristics, a radial decoupled PID controller is constructed. The proportional coefficient matrix \(K\) Pr2 of the constructed radial decoupled PID controller, the differential coefficient matrix \(K\) Dr2 and the integral coefficient matrix \(K\) Ir2 are as follows: ； Among them, , i 0a and i 0b are the bias currents at the A and B ends of the radial electromagnetic bearing AMB, respectively; is the transformation matrix of the electromagnetic force, l a and l b are the axial distances from the center of mass of the rotor to the A and B ends of the radial AMB, respectively; k iri and k hri are the parameters related to the number of turns of the coil and the cross-sectional area of the magnetic pole of the radial AMBs structure, respectively; is the coordinate transformation matrix, l sa and l sb are the axial distances from the center of mass of the rotor to the radial sensor A and the radial sensor B, respectively; K Pr3 = diag([k Prr k Prt k Prr k Prt ), k Prr and k Prt are the proportionality coefficients of the radial decoupling PID controller in the conical and translational systems, respectively; K Ir3 = diag([k Irr k Irt k Irr k Irt ), k Irr and k Irt are the integral coefficients of the radial decoupling PID controller in the conical and translational systems, respectively; K Dr3 = diag([k Drr k Drt k Drr k Drt ), k Drr and k Drt are the differential coefficients of the radial decoupling PID controller in the conical and translational systems, respectively; Associate the radial decoupling PID control parameters with the rotor system stability and the stiffness-damping characteristics of translational and conical motions respectively, and preliminarily determine the value ranges of the PID control parameters; (2) Based on the radial bias current of the rotor system and the preliminary determined proportional, differential and integral coefficients of the radial decoupling PID controller, further adjust the proportional and differential coefficients of the radial decoupling PID controller according to the variation relationships of the radial unbalance vibration response and the unbalance control current response of the rotor system with respect to the proportional and differential coefficients of the radial decoupling PID controller; (3) Based on the requirements of the rotor system stability and the stiffness-damping characteristics, associate the axial PID control parameters with the rotor system stability and the stiffness-damping characteristics, and preliminarily determine the value ranges of the axial PID control parameters; (4) Based on the axial bias current of the rotor system and the preliminary determined axial PID control parameters, further adjust the proportional and differential coefficients of the axial PID controller according to the variation relationships of the axial vibration response and the control current response of the rotor system to the axial excitation force with respect to the proportional and differential coefficients of the axial PID controller.

2. The design method of the radial decoupling and axial PID controller for the AMBs-rigid rotor system according to claim 1, characterized in that: In the step (1): According to the requirements of the rotor system for the stability and the stiffness-damping characteristics of translational motion, determine the value range of the PID coefficients related to translational motion of the radial decoupling PID controller as: ； where m is the mass of the rotor; According to the requirements of the rotor system for the stability and the stiffness-damping characteristics of conical motion, determine the value range of the PID coefficients related to conical motion of the radial decoupling PID controller as: ； Among them, J r is the equatorial moment of inertia of the rotor.

3. The design method of the AMBs-rigid rotor system radial decoupling and axial PID controller according to claim 2, characterized in that: The specific process of the step (2) is as follows: (2.1) Radial bias current i 0a and i 0b Setting Set the radial static control current not to exceed 10% of the corresponding bias current to obtain the radial bias current i 0a and i 0b The value range is: ； where g is the acceleration due to gravity; (2.2) Radial ratio and differential coefficient k Prt , k Drt , k Prr and k Drr setting The radial displacement X of the rotor in the x-direction at the positions of sensors A and B sa (s) and X sb (s) and F εx1 (s) has the relationship: ； where ε and ε z are the radial and axial offsets of the unbalanced mass of the rotor, respectively, and are the angular velocity and angular displacement of the rotor rotating about the z-axis, respectively. F εx1 (s) is the Laplace transform of the unit cosine function , and is the initial angle of the rotor; Control current I of radial AMB - A and - B in the x - direction cxa (s) and I cxb (s) and F εx1 (s) has the relationship: ； The specific parameter adjustment direction needs to be determined by combining the actual situations of the radial unbalance vibration response and the unbalance control current response of the rotor system within the full speed range, as well as the specific relationships between the radial proportional and differential coefficients and the unbalance vibration response and the unbalance control current response at different speeds.

4. The design method of the AMBs–rigid rotor system radial decoupling and axial PID controller according to claim 2, characterized in that: The specific process of the step (3) is as follows: According to the requirements of the rotor system for the stability and the stiffness-damping characteristics of axial motion, determine the value range of the axial PID control parameters as: 。 5. The design method of the AMBs–rigid rotor system radial decoupling and axial PID controller according to claim 2, characterized in that: The specific process of the step (4) is as follows: (4.1) Setting of the axial offset current i 0z ​ Set that when an external force equal to the gravity of the rotor acts axially on the rotor, the axial static control current does not exceed 40% of the corresponding bias current, and obtain the axial bias current i 0z The value range is: ； (4.2) Axial ratio and differential coefficient k Pz and k Dz setting The displacement Z(s) of the rotor in the axial direction and F dz1 (s) are related as follows: ； where z d is the amplitude of the axial synchronous excitation displacement signal received by the rotor, F dz1 (s) is the Laplace transform of the unit sine function , is the initial angle of the excitation displacement signal; Control current I of the axial AMB cz (s) and F dz1 (s) has the following relationship: ； The specific parameter adjustment direction needs to be determined by combining the actual conditions of the axial vibration response and the control current response of the rotor system to the axial excitation signal within the full speed range, as well as the specific relationships between k Pz and k Dz and the axial vibration response and the control current response at different speeds.

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