Active magnetic suspension bearing lumped disturbance suppression method based on adaptive finite time extended state observer

Abstract

The invention discloses an active magnetic suspension bearing lumped disturbance suppression method based on a self-adaptive finite time extended state observer. Establishing a mathematical model of the radial active magnetic suspension bearing containing lumped disturbance; constructing a self-adaptive nonsingular terminal sliding mode decoupling control algorithm according to the mathematical model of the radial active magnetic suspension bearing containing lumped disturbance; and analyzing reasons for system tracking errors caused by lumped disturbance, designing a self-adaptive finite time extended state observer, and carrying out decoupling of four radial degrees of freedom of the active magnetic suspension bearing and carrying out feed-forward compensation. According to the invention, the self-adaptive finite time extended state observer is adopted to accurately estimate the feedforward lumped disturbance, so that stable ultrahigh-speed operation of the magnetic suspension system is ensured; the finite time convergence theory in the sliding mode algorithm is introduced, it can be guaranteed that the system is rapidly converged and stably operates at an ideal working point, accurate estimation of various linear and nonlinear disturbances can be achieved, and the calculation amount is reduced for system model order reduction and decoupling processing.

Description

Technical Field

[0001] The present invention belongs to the technical field of radial active magnetic suspension bearing control, and in particular relates to a method for suppressing lumped disturbances of an active magnetic suspension bearing based on an adaptive finite-time extended state observer. Background Art

[0002] An active magnetic bearing (AMB) is a bearing system that uses electromagnetic force to support a rotating body. This force, either electromagnetic or permanent, suspends and supports rotating parts, avoiding the contact and friction issues found in traditional bearings. Active magnetic bearings are widely used in high-speed rotating equipment such as turbines, centrifuges, magnetic levitation trains, precision instruments, and some high-end motors and engines. There are two main types of active magnetic bearings. The first type uses electromagnetic force to generate a magnetic field, and the current used to adjust the magnetic field strength achieves rotor position control. These typically require a closed-loop control system to adjust the current in real time to maintain a stable suspension state. The second type is a permanent magnet bearing, which uses permanent magnets to generate a fixed magnetic field, suspending the rotor through magnetic force. These bearings do not require an external power source, but because the magnetic field cannot be adjusted, they are typically used in low-speed applications or where stability is less critical.

[0003] Although active magnetic bearings are frictionless, highly stable, and have a long service life, their control systems also face a series of challenges. (1) Dynamic stability issues. Active magnetic bearings require a precise control system to maintain stable suspension of the rotor. Since there is no physical contact between the rotor and the supporting structure, any slight deviation of the rotor will cause a change in the magnetic field, which may lead to system instability. Therefore, how to ensure the stability of the rotor at high speed is a very challenging task. Real-time feedback control, such as proportional integral differential control (PID), sliding mode control (SMC), and active disturbance rejection control (ADRC), is required to quickly correct the rotor position and suppress external disturbances. (2) High-frequency vibration and disturbance. When running at high speed, the rotor may be affected by uneven airflow or electromagnetic interference, resulting in high-frequency vibration. Precise sensors and control algorithms are required to sense and compensate for these disturbances in real time to avoid the accumulation of vibrations that affect the stability of the system. (3) Nonlinearity of magnetic force. The relationship between the magnetic force and current of the electromagnetic bearing is usually nonlinear, which makes the response of the control system complicated. It is usually difficult to obtain an accurate mathematical model of the active magnetic bearing. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for suppressing lumped disturbances in an active magnetic bearing based on an adaptive finite-time extended state observer, which solves the problems of coupling effects between the radial four degrees of freedom in the active magnetic bearing system and periodic disturbances caused by rotor mass imbalance.

[0005] The technical solution employed in this invention is a method for suppressing lumped disturbances in active magnetic bearings based on an adaptive finite-time extended state observer. Multiple disturbances exist in active magnetic bearing systems, which can be categorized as mechanical disturbances and control disturbances, referred to as lumped disturbances. Lumped disturbances make it difficult for the magnetically suspended rotor to maintain stable suspension at the operating point, potentially causing the rotor to fall at high speed and damage the magnetic suspension platform. This invention employs an adaptive finite-time extended state observer to accurately estimate the feedforward lumped disturbances, ensuring stable operation of the magnetic suspension system at ultra-high speeds. The specific steps are as follows:

[0006] Step 1, establishing a mathematical model of a radial active magnetic bearing containing lumped disturbances;

[0007] Step 2, constructing an adaptive non-singular terminal sliding mode decoupling control algorithm based on the mathematical model of the radial active magnetic bearing containing lumped disturbances in step 1;

[0008] Step 3: Analyze the causes of system tracking errors caused by lumped disturbances, design an adaptive finite-time extended state observer (AFTESO), decouple the radial four degrees of freedom of the active magnetic bearing, and perform feedforward compensation to offset the effects of the inaccurate system model and lumped disturbances on the system.

[0009] The present invention is also characterized in that:

[0010] Step 1 is as follows:

[0011] According to the force analysis of the five-degree-of-freedom active magnetic bearing system, the coordinate system of the active magnetic bearing rotor is established at the center of mass, C represents the center of mass position, where end a represents the left part of the bearing, end b represents the right part of the bearing, the x direction is defined as upward, the y direction is defined as outward, and the z direction is defined as rightward.

[0012] Consider the center distance between the two magnetic bearings. At this time, the distance between the two magnetic bearings is L, that is, L = l a +l b ; The displacement of the magnetic bearing at end a along the x-axis and y-axis is x a and x b , the displacement along the x-axis and y-axis at end b is x b and y b , then the displacement and rotation angle at the rotor mass center can be obtained as follows:

[0013]

[0014] In formula (1.1), x c and y c are the displacements of the center of mass C in the x and y directions, θ xand θ y are the angles of rotation of the rotor in the yz plane and the xz plane respectively;

[0015] According to Newton's law, the kinematic equation of the rotor of the radial active magnetic bearing with four radial degrees of freedom is derived as follows:

[0016]

[0017] In formula (1.2), m represents the mass of the rotor, F xa , F ya 、F xb , F yb represents the electromagnetic force of the electromagnetic bearing at ends a and b in the x-axis and y-axis directions, ω is the rotor speed, l a is the distance between the magnetic bearing end a and the center of mass, l b is the distance between the magnetic bearing at end b and the center of mass, is the perturbation in the x-axis direction, is the disturbance in the y-axis direction, J x 、J y 、J z are the moments of inertia of the magnetically suspended rotor in the x, y, and z axes respectively;

[0018] For a rotor with four radial degrees of freedom, the electromagnetic force in each direction is a function of the current stiffness coefficient and the displacement stiffness coefficient in formula (1.3). The magnitude of the electromagnetic force in each direction is expressed as:

[0019]

[0020] In formula (1.3), I xa , I xb They represent the control current of the active magnetic bearing at end a and end b in the x direction; I ya , I yb They represent the control current of the active magnetic bearing at end a and end b in the y direction; k x 、k i are the displacement stiffness coefficient and current stiffness coefficient of the magnetic bearing respectively;

[0021] To simplify the process of establishing the control model and designing the controller, substitute equations (1.1) and (1.3) into (1.2) and simplify to obtain:

[0022]

[0023] Wherein, L is the distance between the magnetic bearing at end a and the magnetic bearing at end b;

[0024] Furthermore, let the coefficients in equation (1.5) be expressed as follows:

[0025]

[0026] At this time, the four-degree-of-freedom mathematical model is as follows:

[0027]

[0028] The coupling B1x between the x-axis and the y-axis b +D1I xb 、B1y b +D1I yb , A2x a +C2I xa , A2y a +C2I ya and gyroscopic effect between the four degrees of freedom is regarded as an internal disturbance; together with the periodic disturbance d1 caused by the external rotor mass imbalance, the sum is defined as the lumped disturbance d, which is shown in the following formula:

[0029]

[0030] Among them, d xa , d ya , d xb , d yb are the lumped disturbances of the bearings at ends a and b in the x and y directions, respectively;

[0031] Furthermore, the decoupled four-degree-of-freedom mathematical model of the magnetic bearing system is derived:

[0032]

[0033] At this time, the decoupled four-degree-of-freedom mathematical model of the magnetic levitation bearing system lays a theoretical foundation for realizing decoupling control in four directions. This technical solution reduces the hardware requirements for the magnetic levitation controller, is easy to implement, and greatly reduces the amount of calculation.

[0034] Step 2 is as follows:

[0035] Step 2.1, verify the convergence of the non-singular terminal sliding surface;

[0036] The improved sliding surface, that is, the non-singular terminal sliding surface s, is used:

[0037]

[0038] In formula (2.1), γ is a designed positive number, x is the state variable of the active magnetic bearing system, q and p are positive odd numbers and satisfy

[0039] According to formula (2.1), we can deduce:

[0040]

[0041] Integrating both ends of equation (2.2) yields:

[0042]

[0043] The convergence time of the non-singular terminal sliding mode is calculated by formula (2.3):

[0044]

[0045] From the above formula, we can see that the non-singular terminal sliding surface can quickly converge to the equilibrium point in a finite time.

[0046] Step 2.2, derive the adaptive non-singular terminal sliding mode control algorithm for the radial x-axis of bearing a;

[0047] Let the reference position be x ref , define the rotor position error of the active magnetic bearing system as e1=x ref -x1, the displacement error change rate is Combined with the state equation of the active magnetic bearing system, the error state equation of the active magnetic bearing system is inferred to be:

[0048]

[0049] Among them, x1 and x2 are the rotor displacement and the rotor displacement derivative respectively;

[0050] The non-singular terminal sliding mode surface of the active magnetic bearing system is obtained:

[0051]

[0052] Taking the derivative of both ends of equation (2.6) we can get:

[0053]

[0054] Further launch:

[0055]

[0056] Where a0 is the redefined current stiffness coefficient, a0 = k i / m, b0 is the redefined displacement stiffness coefficient b0=k m / m, d is the lumped disturbance;

[0057] Furthermore, the non-singular terminal sliding mode controller of the active magnetic bearing system is derived as follows:

[0058]

[0059] ξ is the adaptive exponential reaching law coefficient, k is the exponential reaching rate coefficient, s is the non-singular terminal sliding surface, and r is the non-singular terminal sliding surface coefficient;

[0060] The adaptive law is designed as:

[0061]

[0062] In formula (2.10), η and ε are adjustable parameters of the adaptive law based on displacement error, ε>0.

[0063] Combined with the non-singular terminal sliding mode controller, the final adaptive non-singular terminal sliding mode decoupling control algorithm is as follows:

[0064]

[0065] In formula (2.11), we can see from the analysis that when it is far away from the sliding surface, e1 is large, then is also larger, which will accelerate the convergence speed; when it approaches the sliding surface, e1 is smaller, then The magnetic levitation rotor is also smaller, which further reduces the vibration. It also has a faster convergence speed and accuracy, ensuring that the magnetic levitation rotor can be more quickly and stably suspended at the ideal working point.

[0066] The specific process of step 3 is:

[0067] Step 3.1, analyze the reasons why the lumped disturbance causes the system tracking error;

[0068] The displacement tracking error e1 of the active magnetic bearing system is analyzed. The specific expression of the tracking error is shown in the following formula (3.1):

[0069]

[0070] In formula (3.1), τ is the control period, M1 is the upper bound of the lumped disturbance, m is the rotor mass, ξ is the sliding mode gain, and c is the sliding mode surface coefficient. When τ, m, and c in the system are determined, the only factor affecting the stability of the system is the upper bound of the lumped disturbance M1.

[0071] Step 3.2, design the adaptive finite-time extended state observer for the radial x-axis of bearing a as an example; the control of bearing b is the same as that of bearing a;

[0072] According to the mathematical model of the active magnetic bearing system, there is a lumped disturbance d in the mathematical model. If the above disturbance is expanded into a new state variable, the mathematical model of the active magnetic bearing system becomes:

[0073]

[0074] In formula (3.2), x3 is the newly expanded state variable, and h is a bounded function;

[0075] The specific expression derivation process of the adaptive finite-time extended state observer is as follows;

[0076] Adaptive variable coefficient integral sliding surface designed for the system:

[0077]

[0078] In formula (3.3), S1 is the designed adaptive integral sliding surface, K is the coefficient of the adaptive integral sliding surface, and n and N are adaptive coefficients;

[0079] Combining the decoupling mathematical model of the active magnetic bearing system, we can obtain:

[0080]

[0081] In formula (3.4), i c is the control current, e is the position error;

[0082] Define the following variables to simplify the equation:

[0083]

[0084] Substituting the defined variables into formula (1.6), we can obtain:

[0085]

[0086] At this time, the active magnetic bearing system is re-degraded and reorganized. Bringing it into the designed sliding surface will reduce the order of the system and converge to the ideal working point in a finite time. Combining the excellence of the extended state observer technology in estimating and compensating disturbances, a new state variable z1=S1 is redefined, and an extended state variable z2=W2 is defined, that is, Where g(t) is an unknown but bounded number; the reorganized system equations expand to the following form:

[0087]

[0088] set up For the observed value of the extended system state, the estimated error of the integral sliding surface z1 of the reorganized system equation is defined as:

[0089]

[0090] Design the following adaptive finite-time extended state observer:

[0091]

[0092] In formula (3.9), β1, β2, β3, α1, α2, and α3 are the parameters to be adjusted in the adaptive finite-time extended state observer.

[0093] In step 3.3, taking the radial x-axis of bearing A as an example, a finite-time extended state observer is used to estimate disturbances and decouple the four degrees of freedom. Based on the constructed adaptive finite-time extended state observer, the convergence of the estimation error of the finite-time extended state observer based on the adaptive integral sliding surface will be theoretically analyzed according to Lyapunov stability theory, and the range of tuning parameters will be finally derived.

[0094] Under ideal conditions, if the adaptive finite-time extended state observer is designed according to Equation (3.9) and appropriate parameters are selected to satisfy the constraints, the estimation error will converge to the origin within a finite time T2;

[0095] First, introduce an auxiliary state variable If the auxiliary state variables can converge to the origin within a finite time, the estimated errors ρ1, ρ2, and ρ3 will also converge to the origin within a finite time; where e1, e2, and e3 are the displacement tracking error, the first-order derivative of the displacement tracking error, and the second-order derivative of the displacement tracking error, respectively;

[0096] Auxiliary state variables for construction Taking the derivative with respect to time, we get:

[0097]

[0098] Among them, α1, α2, α3, ε, β1, β2, and β3 are the parameters to be tuned in the adaptive finite-time extended state observer. Then, the state correlation matrix is ​​given:

[0099]

[0100] in, i=1, 2, 3; I4 is the 4th-order identity matrix, 04 is the 4th-order zero matrix; (3.12)

[0102] Then the equation Can be expressed as:

[0103]

[0104] Finally, the value of the fast finite-time extended state observer constructed according to formula (3.9) will converge stably within a finite time; the expression of the convergence time T2 is as follows:

[0105]

[0106] The estimation error gradually decreases over time, and the area it finally enters is as follows:

[0107]

[0108] Where θ1 = (0, λ1) and θ2 = (0, λ2) are arbitrary constants. It is proved that the finite-time extended state observer based on the integral sliding surface can be stabilized within a finite time.

[0109] Furthermore, according to the above derivation process, the value range of the parameters of formula (3.9) can be obtained as follows: b i >0, and γ i is a positive definite diagonal matrix i=1,2,3;

[0110] The beneficial effects of the present invention are as follows: the method for suppressing lumped disturbances of active magnetic bearings based on an adaptive finite-time extended state observer provided by the present invention uses an adaptive non-singular terminal sliding mode controller to achieve independence of the controller from the system model, fast response speed and strong robustness, while also reducing the chattering problem of the traditional sliding mode controller; and for the coupling effect between the radial four degrees of freedom and external periodic disturbances in the radial active magnetic bearing system, a method based on an adaptive finite-time extended state observer is used to accurately estimate the internal and external disturbances of the system online and perform feedforward compensation to offset the influence of internal and external disturbances on the control performance of the system, and decouple the multi-input and multi-output strongly coupled system into four independent single-input and single-output systems, thereby improving the estimation accuracy and convergence speed of the external periodic disturbance term, enhancing the stability of the magnetic bearing system, and reducing the system calculation amount and hardware requirements. Ultimately, the present invention effectively solves the control problem caused by the gyroscopic effect when the magnetic bearing is running at high speed, and is suitable for a variety of practical application scenarios of high-speed rotating machinery. BRIEF DESCRIPTION OF THE DRAWINGS

[0111] Figure 1 This is a radial single-degree-of-freedom force analysis diagram of the magnetic suspension bearing of the present invention;

[0112] Figure 2 This is a force analysis diagram of the five-degree-of-freedom active magnetic bearing system of the present invention.

[0113] FIG3( a ) is a block diagram of the active magnetic suspension bearing lumped disturbance suppression based on the adaptive finite-time extended state observer of the present invention (a) a radial x-axis single degree of freedom model of the bearing.

[0114] FIG3( b ) is a schematic diagram of the structure of the adaptive finite-time extended state observer of the present invention.

[0115] Figure 4This is a block diagram of lumped disturbance suppression of an active magnetic suspension bearing based on an adaptive finite-time extended state observer (radial four-degree-of-freedom model) according to the present invention.

[0116] FIG5( a ) is a waveform diagram showing a comparison of the tracking step reference displacement according to the present invention;

[0117] FIG5( b ) is a comparative waveform diagram of the tracking step reference given current of the present invention;

[0118] FIG5( c ) is a waveform diagram showing a comparison of the sinusoidal reference displacement tracked by the present invention;

[0119] FIG5( d ) is a comparative waveform diagram of the present invention tracking a sinusoidal reference given current. DETAILED DESCRIPTION

[0120] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0121] Example 1

[0122] The present invention discloses a method for suppressing lumped disturbances in an active magnetic bearing based on an adaptive finite-time extended state observer. This method solves the coupling effect (especially gyroscopic effect coupling) between the four radial degrees of freedom and the external periodic disturbance (periodic disturbance caused by rotor mass imbalance) in the active magnetic bearing system, improves the estimation accuracy and convergence speed of the external periodic disturbance term, enhances the stability of the magnetic bearing system, and reduces the system computational complexity and hardware requirements. To achieve the above objectives, the present invention adopts a technical solution: lumped disturbance suppression in an active magnetic bearing based on an adaptive finite-time extended state observer, specifically comprising the following steps:

[0123] Step 1, establishing a mathematical model of a radial active magnetic bearing containing lumped disturbances;

[0124] Step 2, constructing an adaptive non-singular terminal sliding mode decoupling control algorithm based on the mathematical model of the radial active magnetic bearing containing lumped disturbances in step 1;

[0125] Step 3: Analyze the causes of system tracking errors caused by lumped disturbances and design an adaptive finite-time extended state observer (AFTESO) to decouple the radial four degrees of freedom of the active magnetic bearing and perform feedforward compensation to offset the effects of the inaccurate system model and lumped disturbances on the system.

[0126] Figure 1 The electromagnet in the figure refers to a simplified model of a magnetic bearing, which is a force analysis in one direction. i represents the control current and i0 is the bias current.

[0127] Example 2

[0128] Based on Example 1, the specific process of step 1 is as follows:

[0129] Step 1.1, construct a radial four-degree-of-freedom mathematical model of a radial active magnetic bearing with lumped disturbance;

[0130] like Figure 2 The figure shows a force analysis diagram for a five-degree-of-freedom active magnetic bearing system. The coordinate system of the active magnetic bearing rotor is established at the center of mass, where C represents the center of mass. End a is located on the left side of the bearing, and end b is located on the right side. The x-direction is defined as upward, the y-direction is defined as outward, and the z-direction is defined as rightward. When modeling a radial magnetic bearing system, some theoretical assumptions must be made, that is, under ideal conditions.

[0131] Assume that the mass of the rotor is m. The displacement sensor is responsible for detecting the displacement change of the rotor. Therefore, the displacement of the magnetic bearings on both sides can be used as the input. When establishing the overall length of the mathematical model, the center distance between the two magnetic bearings must be considered. At this time, the distance between the two magnetic bearings is L, that is, L = l a +l b The displacement of the magnetic bearing at end a along the x-axis and y-axis is x a and x b , the displacement along the x-axis and y-axis at end b is x b and y b , then the displacement and rotation angle at the rotor center of mass can be obtained by the following relationship.

[0132]

[0133] In formula (1.1), x c and y c are the displacements of the center of mass C in the x and y directions, θ x and θ y are the angles of rotation of the rotor in the yz plane and the xz plane respectively.

[0134] According to Newton's law, the kinematic equation of the rotor of the radial active magnetic bearing with four radial degrees of freedom can be derived as follows:

[0135]

[0136] In formula (1.2), m represents the mass of the rotor, F xa , F ya F xb , F yb represents the electromagnetic force of the electromagnetic bearing at ends a and b in the x-axis and y-axis directions respectively, ω is the rotor speed, l a is the distance between the magnetic bearing end a and the center of mass, lb is the distance between the magnetic bearing at end b and the center of mass, is the perturbation in the x-axis direction, is the disturbance in the y-axis direction, J x 、J y 、J z are the rotational inertia of the magnetic levitation rotor in the x, y, and z axes respectively.

[0137] For a rotor with four radial degrees of freedom, the electromagnetic force in each direction is a function of the current stiffness coefficient and the displacement stiffness coefficient in the formula. The magnitude of the electromagnetic force in each direction can be expressed as:

[0138]

[0139] In formula (1.3) I xa , I xb I represents the control current of the active magnetic bearing at end a and end b in the x direction respectively; ya , I yb They represent the control current magnitudes of the active magnetic bearing at end a and end b in the y direction; k x 、k i are the displacement stiffness coefficient and current stiffness coefficient of the magnetic bearing respectively.

[0140] In simplified models of active magnetic bearings, it is often assumed that the current stiffness and displacement stiffness coefficients are equal in all four directions. This assumption is made because the weight of the rotor itself is often neglected in these models, which simplifies the process of establishing the control model and designing the controller. Substituting equations (1.1) and (1.3) into equation (1.2) yields:

[0141]

[0142] Simplifying the above formula, we can get:

[0143]

[0144] Wherein, L is the distance between the magnetic bearing at end a and the magnetic bearing at end b.

[0145] Furthermore, let the coefficients in equation (1.5) be expressed as follows:

[0146]

[0147] At this time, the four-degree-of-freedom mathematical model is as follows:

[0148]

[0149] The coupling B1x between the x-axis and the y-axis b +D1Ixb 、B1y b +D1I yb , A2x a +C2I xa , A2y a +C2I ya and gyroscopic effect between the four degrees of freedom is regarded as an internal disturbance; together with the periodic disturbance d1 caused by the external rotor mass imbalance, the sum is defined as the lumped disturbance d, which is shown in the following formula:

[0150]

[0151] Among them, d xa , d ya , d xb , d yb are the lumped disturbances of bearings A and B in the x and y directions, respectively.

[0152] Furthermore, the decoupled four-degree-of-freedom mathematical model of the magnetic bearing system is derived:

[0153]

[0154] At this time, the decoupled four-degree-of-freedom mathematical model of the magnetic levitation bearing system lays a theoretical foundation for realizing decoupling control in four directions. This technical solution reduces the hardware requirements for the magnetic levitation controller, is easy to implement, and greatly reduces the amount of calculation.

[0155] Example 3

[0156] Based on Example 2, the specific process of step 2 is as follows:

[0157] Step 2.1, verify the convergence of the non-singular terminal sliding surface;

[0158] In view of the current control difficulties of active magnetic bearing systems, a control system with low dependence on system models, good robustness and fast response speed needs to be designed. The present invention adopts an improved sliding surface, namely a non-singular terminal sliding surface:

[0159]

[0160] In formula (2.1), γ is a designed positive number, x is the state variable of the active magnetic bearing system, q and p are positive odd numbers and satisfy

[0161] Formula (2.1) can be deduced:

[0162]

[0163] Integrating both ends of equation (2.2) yields:

[0164]

[0165] The convergence time of the non-singular terminal sliding mode can be calculated from formula (2.3):

[0166]

[0167] The above equation shows that a non-singular terminal sliding surface can converge quickly to an equilibrium point in a finite time. Furthermore, since it uses a nonlinear sliding surface, its convergence speed is comparable to that of a traditional linear sliding surface, significantly improving its convergence speed.

[0168] Step 2.2, derive the adaptive non-singular terminal sliding mode control algorithm based on the radial x-axis single degree of freedom model of bearing a;

[0169] Differential active magnetic bearing systems offer advantages such as no mechanical contact, high suspension stiffness, and ease of control. This "non-contact" approach significantly improves control accuracy and environmental friendliness. However, due to the inherent characteristics of active magnetic bearing systems, model uncertainty and external disturbances can affect system stability, making it crucial to consider system stability during design.

[0170] Let the reference position be x ref , define the rotor position error of the active magnetic bearing system as e1=x ref -x1, the displacement error change rate is Combined with the state equation of the active magnetic bearing system above, it can be inferred that the error state equation of the active magnetic bearing system is:

[0171]

[0172] Among them, x1 and x2 are the rotor displacement and the rotor displacement derivative respectively.

[0173] The non-singular terminal sliding mode surface of the active magnetic bearing system can be obtained:

[0174]

[0175] Taking the derivative of both ends of equation (2.6) we can get:

[0176]

[0177] It can be introduced:

[0178]

[0179] Where a0 is the redefined current stiffness coefficient, a0 = ki / m, b0 is the redefined displacement stiffness coefficient b0=k m / m, d is the lumped disturbance;

[0180] Example 4

[0181] Based on Example 3, the non-singular terminal sliding mode controller of the active magnetic bearing system is derived as follows:

[0182]

[0183] The adaptive law is designed as:

[0184] In formula (2.10), η and ε are adjustable parameters of the adaptive law based on displacement error, ε>0.

[0185] Furthermore, combined with the non-singular fast terminal sliding mode controller, the final adaptive non-singular fast terminal sliding mode control algorithm is as follows:

[0186]

[0187] In formula (2.11), we can see from the analysis that when it is far away from the sliding surface, e1 is large, then is also larger, which will accelerate the convergence speed; when it approaches the sliding surface, e1 is smaller, then The magnetic levitation rotor is also smaller, which will further reduce the "jitter". It will also have a faster convergence speed and accuracy, ensuring that the magnetic levitation rotor can be more quickly and stably suspended at the ideal working point.

[0188] Example 5

[0189] Based on Example 4, step 3 is as follows:

[0190] Step 3.1, analyze the reasons why the lumped disturbance causes the system tracking error;

[0191] The displacement tracking error e1 of the active magnetic bearing system is further analyzed. The specific expression of the tracking error is shown as follows:

[0192]

[0193] In equation (3.1), τ is the control period, M1 is the upper bound of the lumped disturbance, m is the rotor mass, ξ is the sliding mode gain, and c is the sliding mode surface coefficient. Once τ, m, and c are determined, the only factor affecting system stability is the upper bound of the lumped disturbance, M1. If some technical method can be used to effectively reduce the negative impact of the lumped disturbance, the displacement tracking error of the active magnetic bearing system can be further reduced, allowing the magnetic levitation platform to operate more stably and efficiently at the ideal operating point.

[0194] In addition, the mathematical model of the radial four-degree-of-freedom of the active magnetic bearing shows that, apart from the coupling between the radial and axial degrees of freedom, which is ignored, there is also a coupling effect between the x and y axes of the a and b bearings, and a gyroscopic effect also exists between the four radial degrees of freedom. According to its expression, it can be analyzed that as the speed increases, its impact on the active magnetic bearing system becomes greater, so this is also one of the important factors causing system tracking errors.

[0195] Therefore, the present invention adopts an adaptive finite-time extended state observer to decouple the four radial degrees of freedom, and at the same time performs feedforward compensation to the estimation of the lumped disturbance; the various coupling effects between the four radial degrees of freedom, the gyroscopic effect and the unknown disturbance of the unmodeled part can be regarded as the internal disturbance of the system, and the periodic vibration caused by the rotor mass imbalance can be regarded as the external disturbance, and the lumped disturbance is defined as the sum of the internal disturbance and the external disturbance.

[0196] Step 3.2, design an adaptive finite-time extended state observer taking the radial x-axis of bearing a as an example;

[0197] According to the mathematical model of the active magnetic bearing system, there is a lumped disturbance d in the mathematical model. If the above disturbance is expanded into a new state variable, the mathematical model of the active magnetic bearing system becomes:

[0198]

[0199] In formula (3.2), x3 is the newly expanded state variable, and h is a bounded function;

[0200] Furthermore, taking the radial x-axis of bearing A as an example, a finite-time extended state observer (FSTA) was designed to estimate disturbances and decouple the four degrees of freedom. Conventional methods design FSTAs based on nonsingular terminal sliding surfaces. However, the coefficients of the generalized uncertainty terms in these nonsingular terminal sliding surfaces are fractional powers of the velocity error. When using these generalized uncertainty terms in control law design, the estimated values ​​must be removed from the fractional powers of the velocity error. When the velocity error approaches zero, the fluctuations of the generalized uncertainty terms are amplified, and even singularities appear in the control law. To address this issue, a FSTA based on an adaptive variable constant coefficient integral sliding surface (hereinafter referred to as the adaptive FSTA) is proposed. The design concept is shown in Figure 3 below. In this case, after differentiating the adaptive variable constant coefficient integral sliding surface, the coefficients of the generalized uncertainty terms are adaptive variable constant coefficients. This means that the estimated values ​​can be used to replace the true values ​​in the control law design. This method effectively suppresses fluctuations in the generalized uncertainty terms, thereby avoiding singularities in the control law. Finally, the adaptive finite-time extended state observer of the present invention is introduced, and its specific expression derivation process is as follows:

[0201] Adaptive variable coefficient integral sliding surface designed for the system:

[0202]

[0203] In formula (3.3), S1 is the designed adaptive integral sliding surface, K is the coefficient of the adaptive integral sliding surface, and n and N are adaptive coefficients.

[0204] Combined with the single-freedom mathematical model of the active magnetic bearing system, we can obtain:

[0205]

[0206] In formula (3.4), i c is the control current, e is the position error;

[0207] To simplify the equation, the following variables can be defined:

[0208]

[0209] Substituting the defined variables into formula (3.4), we can obtain:

[0210]

[0211] At this point, the active magnetic bearing system is re-degraded and reorganized, and brought into the designed sliding surface, which will reduce the system order and converge to the ideal operating point within a finite time. Combining the advantages of the adaptive finite-time extended state observer technology in estimating and compensating disturbances, a new state variable z1 = S1 is redefined, and an extended state variable z2 = W2 is defined, that is, Where g(t) is an unknown but bounded number. The reorganized system equation can be expanded into the following form:

[0212]

[0213] set up For the observed value of the extended system state, the estimated error of the integral sliding surface z1 of the reorganized system equation is defined as:

[0214]

[0215] The following adaptive finite-time extended state observer can be designed:

[0216]

[0217] In formula (3.9), β1, β2, β3, α1, α2, and α3 are the parameters to be tuned in the adaptive finite-time extended state observer;

[0218] Based on the adaptive finite-time extended state observer constructed above, the following will theoretically analyze the convergence of the estimation error of the finite-time extended state observer based on the adaptive integral sliding surface according to the Lyapunov stability theory, and finally derive the parameter tuning range.

[0219] Under ideal conditions, such as Figure 4 , if the adaptive finite-time extended state observer is designed according to formula (3.9) and appropriate parameters are selected to meet the constraints, the estimation error will converge to the origin within the finite time T2.

[0220] First, introduce an auxiliary state variable If the auxiliary state variables can converge to the origin in a finite time, then the estimated errors ρ1, ρ2, and ρ3 will also converge to the origin in a finite time. Among them, e1, e2, and e3 are the displacement tracking error, the first-order derivative of the displacement tracking error, and the second-order derivative of the displacement tracking error, respectively;

[0221] Auxiliary state variables for construction Taking the derivative with respect to time, we get:

[0222]

[0223] In formula (3.10), α1, α2, α3, ε, β1, β2, and β3 are the parameters to be tuned in the adaptive finite-time extended state observer; then, the state correlation matrix is ​​given:

[0224]

[0225] in, Then the equation can be expressed as:

[0226]

[0227] Finally, the value of the fast finite-time extended state observer constructed according to Equation (3.9) will converge stably within a finite time. The expression of the convergence time T2 is as follows:

[0228]

[0229] The estimation error gradually decreases over time, and the area it finally enters is as follows:

[0230]

[0231] Where θ1 = (0, λ1) and θ2 = (0, λ2) are arbitrary constants. It can be proved that the above finite-time extended state observer based on the integral sliding surface can be stabilized in a finite time.

[0232] Furthermore, according to the above derivation process, the value range of the parameters of formula (3.9) can be obtained as follows: b i >0, and γ i is a positive definite diagonal matrix i=1,2,3; Combined with the definition of estimation error, the dynamic error of the above adaptive finite-time extended state observer is as follows:

[0233]

[0234] Finally, the present invention adopts a lumped disturbance suppression method based on an adaptive finite-time extended state observer, which effectively solves the problem of difficulty in accurately controlling the rotor displacement in the control system caused by the lack of accurate models (nonlinearity, multiple couplings, gyroscopic effects, etc.) and the presence of lumped disturbances (gyroscopic effects and external periodic disturbances caused by rotor mass imbalance) in the active magnetic bearing system.

[0235] By observation Figure 5(a)-Figure 5(d) The simulation results are summarized in Table 1 below;

[0236] Table 1 Comparison results between traditional method and the present invention

[0237]

[0238] As can be seen from Table 1, the algorithm of the present invention outperforms conventional algorithms in both dynamic response and robustness when tracking a step reference displacement. Similarly, when tracking a sinusoidal reference displacement, the response time of the algorithm of the present invention is 0.050s, which is still an improvement compared to the 0.065s of the conventional algorithm. In terms of control current, the algorithm of the present invention uses 0.8A, while the conventional algorithm uses 1.5A, confirming the theoretical analysis of suppressing system chattering.

[0239] Example 6

[0240] Figure 3(a)-Figure 3(b) The block diagram of the active magnetic bearing lumped disturbance suppression based on the adaptive finite-time state observer in the present invention is as follows:

[0241] 1. The dynamic model of the active magnetic bearing system is derived by analyzing the force on the radial single-degree-of-freedom model of the active magnetic bearing. Based on the dynamic model of the radial single-degree-of-freedom of the active magnetic bearing, the radial single-degree-of-freedom mathematical model and radial four-degree-of-freedom mathematical model of the active magnetic bearing system are further derived.

[0242] 2. Based on the mathematical model of the active magnetic bearing system, a non-singular terminal sliding mode controller is further constructed, whose input is the reference position x refThe error value e1 with the actual displacement x1 is output as the control current i of the active magnetic bearing. In addition, an adaptive sliding mode approach rate based on the position error is designed. Together, the two constitute an adaptive non-singular terminal sliding mode controller.

[0243] 3. However, due to the presence of lumped disturbance d in the active magnetic bearing system, which has a huge impact on the stability of the system, the present invention further designs an adaptive finite-time extended state observer to decouple the coupling between the four degrees of freedom in the system and perform estimated feedforward compensation to eliminate the system disturbance. Taking the radial x-axis single-degree-of-freedom model of bearing A as an example, its input is the control current i of the active magnetic bearing and the actual displacement x1 collected by the eddy current sensor, and the output is the estimated value of the lumped disturbance. Finally, the present invention provides a solution that can achieve good control effect on the active magnetic bearing system.

Claims

1. A method for suppressing lumped disturbances in active magnetic bearings based on an adaptive finite-time extended state observer, characterized in that: The specific steps are as follows: Step 1, establishing a mathematical model of a radial active magnetic bearing containing lumped disturbances; Step 2, constructing an adaptive non-singular terminal sliding mode decoupling control algorithm based on the mathematical model of the radial active magnetic bearing containing lumped disturbances in step 1; Step 3: Analyze the causes of system tracking errors caused by lumped disturbances, design an adaptive finite-time extended state observer (AFTESO), decouple the radial four degrees of freedom of the active magnetic bearing, and perform feedforward compensation to offset the effects of the inaccurate system model and lumped disturbances on the system.

2. The method for suppressing lumped disturbance of active magnetic bearing based on adaptive finite-time extended state observer according to claim 1 is characterized in that: Step 1 is as follows: According to the force analysis of the five-degree-of-freedom active magnetic bearing system, the coordinate system of the active magnetic bearing rotor is established at the center of mass, C represents the center of mass position, where end a represents the left magnetic bearing, end b represents the right magnetic bearing, the x direction is defined as upward, the y direction is defined as outward, and the z direction is defined as rightward. Consider the center distance between the two magnetic bearings. At this time, the distance between the two magnetic bearings is L, that is, L = l a +l b ; The displacement of the magnetic bearing at end a along the x-axis and y-axis is x a and x b , the displacement along the x-axis and y-axis at end b is x b and y b , then the displacement and rotation angle at the rotor mass center can be obtained as follows: In formula (1.1), x c and y c are the displacements of the center of mass C in the x and y directions, θ x and θ y are the angles of rotation of the rotor in the yz plane and the xz plane respectively; According to Newton's law, the kinematic equation of the rotor of the radial active magnetic bearing with four radial degrees of freedom is derived as follows: In formula (1.2), m represents the mass of the rotor, F xa , F ya 、F xb , F yb represents the electromagnetic force of the electromagnetic bearing at ends a and b in the x-axis and y-axis directions, ω is the rotor speed, l a is the distance between the magnetic bearing end a and the center of mass, l b is the distance between the magnetic bearing at end b and the center of mass, is the perturbation in the x-axis direction, is the disturbance in the y-axis direction, J x 、J y 、J z are the moments of inertia of the magnetically suspended rotor in the x, y, and z axes respectively; For a rotor with four radial degrees of freedom, the electromagnetic force in each direction is a function of the current stiffness coefficient and the displacement stiffness coefficient in formula (1.3). The magnitude of the electromagnetic force in each direction is expressed as: In formula (1.3), I xa , I xb They represent the control current of the active magnetic bearing at end a and end b in the x direction; I ya , I yb They represent the control current of the active magnetic bearing at end a and end b in the y direction; k x 、k i are the displacement stiffness coefficient and current stiffness coefficient of the magnetic bearing respectively; To simplify the process of establishing the control model and designing the controller, substitute formulas (1.1) and (1.3) into (1.2) to obtain: Simplifying the above formula, we can get: Furthermore, let the coefficients in equation (1.5) be expressed as follows: At this time, the four-degree-of-freedom mathematical model is as follows: The coupling B1x between the x-axis and the y-axis b +D1I xb 、B1y b +D1I yb , A2x a +C2I xa , A2y a +C2I ya and gyroscopic effect between the four degrees of freedom is regarded as an internal disturbance; together with the periodic disturbance d1 caused by the external rotor mass imbalance, the sum is defined as the lumped disturbance d, which is shown in the following formula: Among them, d xa , d ya , d xb , d yb are the lumped disturbances of the magnetic bearings at ends a and b in the x and y directions, respectively; Furthermore, the decoupled four-degree-of-freedom mathematical model of the magnetic bearing system is derived:

3. The method for suppressing lumped disturbance of active magnetic bearing based on adaptive finite-time extended state observer according to claim 2 is characterized in that: Step 2 is as follows: Step 2.1, verify the convergence of the non-singular terminal sliding surface; Step 2.2, derive the adaptive non-singular terminal sliding mode control algorithm for the radial x-axis of bearing a.

4. The method for suppressing lumped disturbance of active magnetic bearing based on adaptive finite-time extended state observer according to claim 3 is characterized in that: Step 2.1 is as follows: The improved sliding surface, that is, the non-singular terminal sliding surface s, is used: In formula (2.1), γ is a designed positive number, x is the state variable of the active magnetic bearing system, q and p are positive odd numbers and satisfy According to formula (2.1), we can deduce: Integrating both ends of equation (2.2) yields: The convergence time of the non-singular terminal sliding mode is calculated by formula (2.3): From the above formula, we can see that the non-singular terminal sliding surface can quickly converge to the equilibrium point in a finite time.

5. The method for suppressing lumped disturbance of active magnetic bearing based on adaptive finite-time extended state observer according to claim 4 is characterized in that: Step 2.2 is as follows: Let the reference position be x ref , define the rotor position error of the active magnetic bearing system as e1=x ref -x1, the displacement error change rate is Combined with the state equation of the active magnetic bearing system, the error state equation of the active magnetic bearing system is inferred to be: Among them, x1 and x2 are the rotor displacement and the rotor displacement derivative respectively; The non-singular terminal sliding mode surface of the active magnetic bearing system is obtained: Taking the derivative of both ends of equation (2.6) we can get: Further launch: Where a0 is the redefined current stiffness coefficient, a0 = k i / m, b0 is the redefined displacement stiffness coefficient b0=k m / m, d is the lumped disturbance; Furthermore, the non-singular terminal sliding mode controller of the active magnetic bearing system is derived as follows: ξ is the adaptive exponential reaching law coefficient, k is the exponential reaching rate coefficient, s is the non-singular terminal sliding surface, and r is the non-singular terminal sliding surface coefficient; The adaptive law is designed as: In formula (2.10), η and ε are adjustable parameters of the adaptive law based on displacement error, ε>0; Combined with the non-singular terminal sliding mode controller, the final adaptive non-singular terminal sliding mode decoupling control algorithm is as follows: In formula (2.11), we can see from the analysis that when it is far away from the sliding surface, e1 is large, then is also larger, thus accelerating the convergence speed; when approaching the sliding surface, e1 is smaller, then It is also smaller, which will further reduce vibration.

6. The method for suppressing lumped disturbance of active magnetic bearing based on adaptive finite-time extended state observer according to claim 5, characterized in that: The specific process of step 3 is: Step 3.1, analyze the reasons why the lumped disturbance causes the system tracking error; The displacement tracking error e1 of the active magnetic bearing system is analyzed. The specific expression of the tracking error is shown in the following formula (3.1): In formula (3.1), τ is the control period, M1 is the upper bound of the lumped disturbance, m is the rotor mass, ξ is the sliding mode gain, and c is the sliding mode surface coefficient. When τ, m, and c in the system are determined, the only factor affecting the stability of the system is the upper bound of the lumped disturbance M1. Step 3.2, design an adaptive finite-time extended state observer taking the radial x-axis of a bearing as an example; According to the mathematical model of the active magnetic bearing system, there is a lumped disturbance d in the mathematical model. If the above disturbance is expanded into a new state variable, the mathematical model of the active magnetic bearing system becomes: In formula (3.2), x3 is the newly expanded state variable, and h is a bounded function; In step 3.3, the radial x-axis of bearing a is designed as an example, and the finite-time extended state observer is used to estimate the disturbance and decouple the four degrees of freedom.

7. The method for suppressing lumped disturbance of active magnetic bearing based on adaptive finite-time extended state observer according to claim 6, characterized in that: The specific expression derivation process of the adaptive finite-time extended state observer is as follows; Adaptive variable coefficient integral sliding surface designed for the system: In formula (3.3), S1 is the designed adaptive integral sliding surface, K is the coefficient of the adaptive integral sliding surface, and n and N are adaptive coefficients; Combining the decoupling mathematical model of the active magnetic bearing system, we can obtain: In formula (3.4), i c is the control current, e is the position error; Define the following variables to simplify the equation: Substituting the defined variables into formula (1.6), we can obtain: At this time, the active magnetic bearing system is re-degraded and reorganized. Bringing it into the designed sliding surface will reduce the order of the system and converge to the ideal working point in a finite time. Combining the excellence of the extended state observer technology in estimating and compensating disturbances, a new state variable z1=S1 is redefined, and an extended state variable z2=W2 is defined, that is, Where g(t) is an unknown but bounded number; the reorganized system equations expand to the following form: set up For the observed value of the extended system state, the estimated error of the integral sliding surface z1 of the reorganized system equation is defined as: Design the following adaptive finite-time extended state observer: In formula (3.9), β1, β2, β3, α1, α2, and α3 are the parameters to be adjusted in the adaptive finite-time extended state observer.

8. The method for suppressing lumped disturbance of active magnetic bearing based on adaptive finite-time extended state observer according to claim 7, characterized in that: Step 3.3 is as follows: Based on the constructed adaptive finite-time extended state observer, the following will theoretically analyze the convergence of the estimation error of the finite-time extended state observer based on the adaptive integral sliding surface according to Lyapunov stability theory, and finally derive the range of the tuning parameters; Under ideal conditions, if the adaptive finite-time extended state observer is designed according to Equation (3.9) and appropriate parameters are selected to satisfy the constraints, the estimation error will converge to the origin within a finite time T2; First, introduce an auxiliary state variable If the auxiliary state variables can converge to the origin within a finite time, the estimated errors ρ1, ρ2, and ρ3 will also converge to the origin within a finite time; where e1, e2, and e3 are the displacement tracking error, the first-order derivative of the displacement tracking error, and the second-order derivative of the displacement tracking error, respectively; Auxiliary state variables for construction Taking the derivative with respect to time, we get: Among them, α1, α2, α3, ε, β1, β2, and β3 are the parameters to be tuned in the adaptive finite-time extended state observer. Then, the state correlation matrix is ​​given: in, I4 is the 4th-order identity matrix, 04 is the 4th-order zero matrix; Then the equation Can be expressed as: Finally, the value of the fast finite-time extended state observer constructed according to formula (3.9) will converge stably within a finite time; the expression of the convergence time T2 is as follows: The estimation error gradually decreases over time, and the area it finally enters is as follows: Where θ1 = (0, λ1) and θ2 = (0, λ2) are arbitrary constants. It is proved that the finite-time extended state observer based on the integral sliding surface can be stabilized within a finite time. Furthermore, according to the above derivation process, the value range of the parameters of formula (3.9) can be obtained as follows: And γ i is a positive definite diagonal matrix i=1,2,3;

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