CN115750592A - Decoupling and unbalanced vibration control method for active magnetic suspension bearing rotor - Google Patents

Abstract

The invention discloses a decoupling and unbalanced vibration control method of an active magnetic suspension bearing rotor, belongs to the field of magnetic suspension bearing control, and can solve the coupling problem of the magnetic suspension rotor and the vibration problem caused by the unbalance of the rotor and realize the stable control of a system. The invention comprises the following steps: firstly, establishing a system model of unbalanced vibration of a rotor according to a rotor dynamics equation, treating position coupling, gyroscopic effect coupling and unbalanced vibration coupling components existing in the system model as disturbance signals, and designing a state observer to estimate and compensate the disturbance, so that four position freedom degree subsystems are independent respectively; and a control method combining an adaptive notch filter and a nonlinear feedback control law is adopted to deduce a closed-loop feedback simplified structure of the subsystem, analyze the frequency characteristics of the subsystem, analyze the selection reason of the parameters of the controller from a BODE diagram and finally realize the stable control of the magnetic suspension rotor. The method can obviously decouple the system, inhibit the unbalanced vibration of the magnetic suspension bearing rotor and realize the stable suspension control of the suspension bearing rotor.

Description

Decoupling and unbalanced vibration control method for active magnetic suspension bearing rotor

Technical Field

The invention relates to the technical field of magnetic bearing control, in particular to a decoupling and unbalanced vibration control method for an active magnetic suspension bearing rotor.

Background

Even if the rotor of the rotating machinery has very high processing precision, the mass of the rotor is difficult to be uniformly and symmetrically distributed due to the factors of the uniformity of materials, assembly errors and the like, the vibration with the same frequency as the rotating speed is difficult to avoid due to the asymmetric mass in the working process, and the small eccentric force can generate larger centrifugal force at high speed to cause the unbalanced vibration of the rotor, which is more obvious in a rotor system taking a magnetic suspension bearing as a support.

In order to reduce or eliminate such unbalanced forces and to stabilize the operation of the magnetically levitated rotor, the rotor can be rotated about the bearing center by increasing the stiffness and damping of the magnetic bearings, but this increases the capacity and volume of the coils of the magnetic bearings. Other major solutions include: adding displacement stiffness force compensation on the basis of the self-adaptive notch filter; the generalized trap filter identifies and compensates the unbalanced same-frequency components of the rotor; a phase shift trap method for adaptively adjusting a phase shift angle at different rotation speeds. The above method for considering the problems mainly considers the problem of unbalanced vibration suppression of the rotor of the magnetic bearing under a single degree of freedom, and does not consider the problems of radial coupling, gyroscopic effect coupling and the like of the suspension rotor. According to rotor dynamics, nonlinear and strong coupling exists in four radial degrees of freedom of a magnetic suspension bearing rotor. Therefore, in the control process of the magnetic suspension rotor, the problems of unbalanced vibration of the rotor and radial position coupling and gyroscopic effect coupling of the rotor are fully considered. The method has high requirements on the mathematical model of the controlled object, and the control requirements are difficult to meet after the system control process contains a lot of nonlinearity and is interfered by external disturbance.

Disclosure of Invention

Aiming at the defects of the prior art, the invention aims to provide a decoupling and unbalanced vibration control method of a magnetic suspension bearing combined with an adaptive notch filter, a state observer and a nonlinear feedback controller, which is used for decoupling the position coupling and the gyro effect coupling of a system, and aiming at the vibration phenomenon of a rotor caused by mass imbalance, the state observer is used for taking the coupling and unbalanced components different from the series integral type as disturbance signals, effective estimation and control compensation are carried out, and meanwhile, the self-adaptive notch mode is used for inhibiting the vibration signals, so that the anti-interference capability of the system is enhanced, and the working stability of the system is improved.

1. In order to achieve the purpose, the technical scheme adopted by the invention is a decoupling and unbalanced vibration control method of an active magnetic bearing rotor, which comprises the following steps:

step 1: the model of the active magnetic suspension bearing system comprises a magnetic suspension bearing model, a driving amplification circuit and a displacement sensor; the transfer functions of the driving amplification circuit and the displacement sensor in the magnetic suspension bearing system model are described as follows:

wherein, f _uv Represents the stress of the rotor in the u-axis direction at the v end (v = a, b; u = x, y), x _j 、y _j Respectively, the rotor displacement at the j-th degree of freedom, i _uv Denotes the control current (v = a, b; u = x, y), K, of the rotor in the direction of the u-axis at the v-terminal _ir ，K _xr Current-force stiffness and displacement-force stiffness, respectively.

And 2, step: according to the geometric position relations of the position of a bearing coil, the position of a sensor, the suspension position of a rotor and the like of the active magnetic suspension bearing system, the relation of the static balance of the rotor and the dynamic unbalance of the rotor in a generalized coordinate system is sought, and a radial four-degree-of-freedom mathematical model of the magnetic suspension bearing system considering the unbalanced vibration problem of the rotor is established through a rotor dynamics method.

Where m is the rotor mass, ω is the rotational angular velocity of the rotor about the Z-axis, J _y And J _x The moment of inertia of the rotor about the Y-axis and the X-axis at the center of mass, respectively. ε and δ represent the amplitude and phase of the rotor static imbalance in a generalized coordinate system, σ and

indicating the amplitude and phase of the rotor at dynamic imbalance. l _a And l _b The distances from the centers of the two-end radial electromagnetic bearings a and b to the geometric center of the rotor.

And step 3: the mathematical model (2) is four second-order equations, terms which are different from state variables in each equation are taken as system disturbance terms, the coupling terms and the vibration problems are taken as system external disturbances, and then the system external disturbances are combined. Equation (2) can be written as:

wherein the content of the first and second substances,

m _x12 ＝m _x21 ＝(K _xr /m)-l _a l _b K _xr /J _y ，

n _x12 ＝n _x21 ＝(K _ir /m)-l _a l _b K _ir /J _y ，

m _y12 ＝m _y21 ＝(K _xr /m)-l _a l _b K _xr /J _x ，

n _y12 ＝n _y21 ＝(K _ir /m)-l _a l _b K _ir /J _x ，

w _x21 ＝l _b J _z ω/lJ _y ，w _x11 ＝-l _a J _z ω/lJ _y ，

w _y11 ＝l _a J _z ω/lJ _x ，w _y21 ＝-l _b J _z ω/lJ _x ，

and 4, step 4: the second-order nonlinear system with four degrees of freedom in the model equation (3) has a similar expression form, a state observer is designed for each equation in the equation set, so that the state observer can estimate the state variable and the disturbance sum of the system, and the designed observer needs to keep stable and track an observed value. Taking the first equation in the formula (3) as an example, the state observer is designed, and the other three equations of the system are designed in the same way. Let x ₁ ＝x _a ，

b ₀ ＝n _x11 ，u＝i _xa ，

Considering as a disturbance signal, the observer is designed as follows:

wherein z = [ z ] ₁₁ z ₁₂ z ₁₃ ] ^T Denotes x = [ x ] ₁ x ₂ x ₃ ] ^T Is determined by the estimated value of (c),

C＝[1 0 0]，B＝[β ₁ β ₂ β ₃ ] ^T to observe the gain.

Order to

The observation equation can be written as:

the pole of the observer is selected in the left half-plane-w ₀ At the handle

Value substitution, then there is:

to give out beta ₁ ＝3w ₀ ，

w ₀ Is the bandwidth of the observer. Bandwidth w ₀ When the observer is more than 0 and an appropriate value is selected, the observer is stable, and z can be ensured ₁₃ Can well track x ₃ The value is obtained.

And 5: and (4) designing a nonlinear feedback control law. First, a differential tracker is used for the transition, depending on the control target and the system endurance. Let ref be the given reference signal. Tracking signal v output by linear differential Tracker (TD) ₁ ＝ref，

Respectively with the observer estimate z ₁₁ ，z ₁₂ Forming a difference value, constructing a system error and error differential signal as follows:

control gain of nonlinear feedback control designed:

wherein

k _d ＝2w _c To control the gain, w _c Is the system bandwidth, u ₀ Is a control quantity before compensation.

The disturbance compensation procedure is as follows:

u＝(u ₀ -z ₁₃ )/b ₀ (9)

where u is the control quantity before compensation. After the disturbance is estimated and compensated, the system becomes a series integration subsystem, and the system is decoupled.

And 6: and designing an adaptive notch filter with the same frequency as the rotating speed in an output feedback loop of the decoupled subsystem, and combining the adaptive notch filter with the state observer. In the output of the wave trap, the same-frequency component with the same frequency and the same rotating speed in the input signal approaches to 0, and the same-frequency vibration is restrained.

Let y _s (t) is a concave feedback link N _f Input of (a) y _ω (t) is N _f The output of (c) then has:

can verify that y _s (t) and y _ω (t) satisfies the following differential equation:

wherein tau is a proportionality coefficient, then the concave feedback link N _f Has a transfer function of

Let s = j ω _r When τ ≠ 0, then there is,

wherein Δ ω is the wave trap N _f The bandwidth of (c). Therefore, when τ ≠ 0, at the wave trap N _f Of the outputs of(s), the input signal y _s And (t) the same-frequency component with the frequency omega approaches to 0, so that the rotating speed same-frequency vibration can be effectively inhibited.

And 7: and (5) deducing a closed loop feedback simplified structure of the subsystem for the unbalanced vibration subsystem of the magnetic suspension bearing rotor consisting of the wave trap, the observer and the nonlinear feedback controller, and analyzing frequency characteristics under different control parameters so as to guide parameter selection of the nonlinear feedback gain controller in the step 5.

The formula (4) is expanded and laplace transformed to derive z ₁₁ 、z ₁₂ 、z ₁₃ The functional expressions of (a) are respectively:

from the control gain equations (8) and (14) of the nonlinear feedback control, the expression of the control variable u of the system can be further derived from the principle of fig. 3:

and simultaneously substituting the formula (14) into the formula (15) to obtain:

simplifying the system structure form of the trapped and decoupled single-degree-of-freedom system, wherein the structure form of FIG. 3 is simplified into a structure diagram shown in FIG. 4, R(s) is a given reference signal, D(s) is a given reference signal _i And(s) is an external interference signal seen by the system, U(s) is an input signal, and Y(s) is an output signal. G _p (s) Single degree of freedom model after decoupling of control object, G _c (s) and H(s) are pending and are available:

U(s)＝[R(s)H(s)-Y(s)]G _c (s)＝R(s)H(s)G _c (s)-Y(s)G _c (s) (17)

laplace transform of equation (15) is:

the same Laplace transform is performed on the formula (14) and is substituted into the formula (18) to obtain

In the formula: c _n4 ＝-β ₃ -β ₂ k _d -β ₁ k _p -τk _p C _n3 ＝-β ₃ k _d -β ₂ k _p -β ₁ τk _p ，

C _d4 ＝1，C _d3 ＝β ₁ +k _d ，

H _n5 ＝k _p ，H _n4 ＝β ₁ k _p ，

H _d4 ＝C _n4 ，H _d3 ＝C _n3 ，H _d2 ＝C _n2 ，H _d1 ＝C _n1 ，H _d0 ＝C _n0

When model parameters are not determined, the closed-loop transfer function of the system is as follows:

the stability of the closed loop system is determined by the pole position of equation (21). Since the denominator term of H(s) can be known that the values of the characteristic roots are all negative numbers, and the Hurwitz condition is satisfied in the left half plane of the complex plane, the closed-loop system stability can be determined by equation (22):

according to equation (3), after removing the perturbation term, the system transfer function in a single degree of freedom can be expressed as:

wherein, b ₀ ＝n _x11 ，a＝m _x11 Thus, the characteristic polynomial for a closed loop system is:

1+G _p (s)G _c (s)＝s ⁷ +A ₆ s ⁶ +A ₅ s ⁵ +A ₄ s ⁴ +A ₃ s ³ +A ₂ s ² +A ₁ s+A ₀ (24)

in the formula: a. The ₆ ＝C _d3 ，A ₅ ＝C _d2 -a，A ₄ ＝C _d1 -aC _d3 +C _n4 ，A ₃ ＝C _d0 -aC _d2 +C _n3 ，A ₂ ＝-aC _d1 +C _n2 ，A ₁ ＝-aC _d0 +C _n1 ，A ₀ ＝C _n0

The open loop transfer function of the system is:

G _o (s)＝G _p (s)G _c (s) (25)

and analyzing the frequency characteristics of the system open loop transfer function under different control parameters. Analysis of omega ₀ ，ω _c Taking different values as the frequency characteristics of the system, obtaining a BODE graph of the system, observing cut-off frequency, phase margin and system gain when the parameters take different values, and finally determining omega according to the sensitivity degree of noise ₀ ，ω _c And (5) substituting the parameter values into the formula (8) to obtain the specific parameters of the nonlinear control law.

Adopt the produced beneficial effect of above-mentioned technical scheme to lie in: the invention provides a decoupling and unbalanced vibration control method of an active magnetic bearing rotor, which adopts a state observer to take coupling and unbalanced components different from a series integral type and the like as disturbance signals, effectively estimates and controls and compensates, enhances the anti-disturbance capability of a system, also decouples position coupling and gyroscopic effect coupling existing in the system, synchronously inhibits vibration signals caused by rotor imbalance by adopting a self-adaptive notch mode, and preferably controls the system by combining a nonlinear feedback controller to ensure that the rotor stably runs.

Drawings

FIG. 1 is a structure diagram of an active magnetic suspension bearing system with four radial degrees of freedom

FIG. 2 is a diagram of the relationship between a rotating coordinate system and a fixed coordinate system

FIG. 3 is a block diagram of single degree of freedom unbalanced vibration control for a suspension rotor

FIG. 4 is a simplified structure diagram of a single-degree-of-freedom subsystem of a suspension rotor

FIG. 5 four-degree-of-freedom magnetic bearing rotor decoupling and unbalanced vibration control block diagram

Detailed Description

The following describes the method of the present invention in detail with reference to the drawings and examples

Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the drawings are illustrative and intended to be illustrative of the invention and are not to be construed as limiting the invention.

As shown in fig. 1 to 5, the present invention provides a decoupling and unbalanced vibration control method based on an active magnetic suspension bearing rotor, comprising the following steps:

step 1: the model of the active magnetic suspension bearing system comprises a magnetic suspension bearing model, a driving amplification circuit and a displacement sensor; the transfer functions of the driving amplifying circuit and the displacement sensor in a magnetic suspension bearing system model are described as follows:

wherein f is _uv Represents the stress of the rotor in the u-axis direction at the v end (v = a, b; u = x, y), x _j 、y _j Respectively, the rotor displacement at the j-th degree of freedom, i _uv Denotes the control current (v = a, b; u = x, y), K, of the rotor in the direction of the u-axis at the v-terminal _ir ，K _xr Current-force stiffness and displacement-force stiffness, respectively.

Step 2: according to the geometric position relations of the position of a bearing coil, the position of a sensor, the suspension position of a rotor and the like of an active magnetic suspension bearing system, as shown in figure 1, the relation of the static balance of the rotor and the dynamic unbalance of the rotor in a generalized coordinate system is sought, as shown in figure 2, a radial four-degree-of-freedom mathematical model of the magnetic suspension bearing system considering the problem of the unbalanced vibration of the rotor is established through a rotor dynamics method.

Where m is the rotor mass, ω is the rotational angular velocity of the rotor about the Z-axis, J _y And J _x The moment of inertia of the rotor about the Y-axis and the X-axis at the center of mass, respectively. ε and δ represent the amplitude and phase of the rotor static imbalance in a generalized coordinate system, σ and

indicating the amplitude and phase of the rotor at dynamic imbalance. l _a And l _b The distance from the center of the radial electromagnetic bearings a and b to the geometric center of the rotor.

And step 3: the mathematical model (2) is four second-order equations, terms different from state variables in each equation are regarded as system disturbance terms, the coupling terms and the vibration problem are regarded as system external disturbance, and then the system external disturbance terms are combined. Equation (2) can be written as:

wherein the content of the first and second substances,

m _x12 ＝m _x21 ＝(K _xr /m)-l _a l _b K _xr /J _y ，

n _x12 ＝n _x21 ＝(K _ir /m)-l _a l _b K _ir /J _y ，

m _y12 ＝m _y21 ＝(K _xr /m)-l _a l _b K _xr /J _x ，

n _y12 ＝n _y21 ＝(K _ir /m)-l _a l _b K _ir /J _x ，

w _x21 ＝l _b J _z ω/lJ _y ，w _x11 ＝-l _a J _z ω/lJ _y ，

w _y21 ＝-l _b J _z ω/lJ _x ，

and 4, step 4: the second-order nonlinear system with four degrees of freedom in the model equation (3) has a similar expression form, a state observer is designed for each equation in the equation set, so that the state observer can estimate the state variable and the disturbance sum of the system, and the designed observer needs to keep stable and track an observed value. The state observer is designed by taking the first equation in the formula (3) as an example, and the other three equations of the system are designed in the same way. Let x ₁ ＝x _a ，

b ₀ ＝n _x11 ，u＝i _xa ，

Considering as a disturbance signal, the observer is designed as follows:

wherein z = [ z ] ₁₁ z ₁₂ z ₁₃ ] ^T Denotes x = [ x ] ₁ x ₂ x ₃ ] ^T Is determined by the estimated value of (c),

C＝[1 0 0]，B＝[β ₁ β ₂ β ₃ ] ^T to observe the gain.

Order to

The observation equation can be written as:

the pole of the observer is selected in the left half-plane-w ₀ At the handle

Value substitution, then there is:

to give out beta ₁ ＝3w ₀ ，

w ₀ Is the bandwidth of the observer. Bandwidth w ₀ When the observer is more than 0 and an appropriate value is selected, the observer is stable, and z can be ensured ₁₃ Can well track x ₃ The value is obtained.

And 5: and (4) designing a nonlinear feedback control law. First, a differential tracker is used for the transition, depending on the control target and the system endurance. Assume that the given reference signal is ref. Tracking signal v output by linear differential Tracker (TD) ₁ ＝ref，

Respectively with the observer estimate z ₁₁ ，z ₁₂ Forming a difference, constructing a system error and error differential signal as follows:

control gain of nonlinear feedback control designed:

wherein

k _d ＝2w _c To control the gain, w _c Is the system bandwidth, u ₀ Is a control quantity before compensation.

The disturbance compensation procedure is as follows:

u＝(u ₀ -z ₁₃ )/b ₀ (9)

where u is the control quantity before compensation. After the disturbance is estimated and compensated, the system becomes a series integration subsystem, and the system is decoupled.

Step 6: and designing an adaptive notch filter with the same frequency as the rotating speed in an output feedback loop of the decoupled subsystem, and combining the adaptive notch filter with the state observer. In the output of the wave trap, the same-frequency component with the same frequency and the same rotating speed in the input signal approaches to 0, and the same-frequency vibration is restrained.

Let y _s (t) is a concave feedback link N _f Input of (a) y _ω (t) is N _f The output of (c) then has:

can verify that y _s (t) and y _ω (t) satisfies the following differential equation:

wherein, tau is a proportionality coefficient, a concave feedback link N is formed _f Has a transfer function of

Let s = j ω _r When τ ≠ 0, then there is,

wherein, delta omega is a wave trap N _f The bandwidth of (c). Therefore, when τ ≠ 0, at the wave trap N _f Of the outputs of(s), the input signal y _s And (t) the same-frequency component with the frequency omega approaches to 0, so that the rotating speed same-frequency vibration can be effectively inhibited.

And 7: and (5) deducing a closed loop feedback simplified structure of the subsystem for the unbalanced vibration subsystem of the magnetic suspension bearing rotor consisting of the wave trap, the observer and the nonlinear feedback controller, and analyzing frequency characteristics under different control parameters so as to guide parameter selection of the nonlinear feedback gain controller in the step 5.

The formula (4) is expanded and laplace transformed to derive z ₁₁ 、z ₁₂ 、z ₁₃ The functional expressions of (a) are respectively:

from the control gain equations (8) and (14) of the nonlinear feedback control, the expression of the control variable u of the system can be further derived from the principle of fig. 3:

and simultaneously substituting the formula (14) into the formula (15) to obtain:

simplifying the system structure form of the trapped and decoupled single-degree-of-freedom system, wherein the structure form of FIG. 3 is simplified into a structure diagram shown in FIG. 4, R(s) is a given reference signal, D(s) is a given reference signal _i And(s) is an external interference signal seen by the system, U(s) is an input signal, and Y(s) is an output signal. G _p (s) Single degree of freedom model after decoupling of control object, G _c (s) and H(s) are pending and are available:

U(s)＝[R(s)H(s)-Y(s)]G _c (s)＝R(s)H(s)G _c (s)-Y(s)G _c (s) (17)

laplace transform of equation (15) is:

the same Laplace transform is performed on the formula (14) and is substituted into the formula (18) to obtain

Wherein: c _n4 ＝-β ₃ -β ₂ k _d -β ₁ k _p -τk _p C _n3 ＝-β ₃ k _d -β ₂ k _p -β ₁ τk _p ，

C _d4 ＝1，C _d3 ＝β ₁ +k _d ，

H _n5 ＝k _p ，H _n4 ＝β ₁ k _p ，

H _d4 ＝C _n4 ，H _d3 ＝C _n3 ，H _d2 ＝C _n2 ，H _d1 ＝C _n1 ，H _d0 ＝C _n0

When model parameters are not determined, the closed-loop transfer function of the system is as follows:

the stability of the closed loop system is determined by the pole position of equation (21). Since the denominator term of H(s) can be known that the values of the characteristic roots are all negative numbers, and the Hurwitz condition is satisfied in the left half plane of the complex plane, the closed-loop system stability can be determined by equation (22):

according to equation (3), after removing the perturbation term, the system transfer function in a single degree of freedom can be expressed as:

wherein, b ₀ ＝n _x11 ，a＝m _x11 Thus, the characteristic polynomial for a closed loop system is:

1+G _p (s)G _c (s)＝s ⁷ +A ₆ s ⁶ +A ₅ s ⁵ +A ₄ s ⁴ +A ₃ s ³ +A ₂ s ² +A ₁ s+A ₀ (24)

in the formula: a. The ₆ ＝C _d3 ，A ₅ ＝C _d2 -a，A ₄ ＝C _d1 -aC _d3 +C _n4 ，A ₃ ＝C _d0 -aC _d2 +C _n3 ，A ₂ ＝-aC _d1 +C _n2 ，A ₁ ＝-aC _d0 +C _n1 ，A ₀ ＝C _n0

The open loop transfer function of the system is:

G _o (s)＝G _p (s)G _c (s) (25)

and analyzing the frequency characteristics of the system open loop transfer function under different control parameters. Analysis of omega ₀ ，ω _c Taking different values as the frequency characteristics of the system, obtaining a BODE graph of the system, observing cut-off frequency, phase margin and system gain when the parameters take different values from the BODE graph, and finally determining omega according to the sensitivity degree of noise ₀ ，ω _c And (5) substituting the parameter values into the formula (8) to obtain the specific parameters of the nonlinear control law.

After the other three degrees of freedom in the system model (3) are subjected to the steps (4), (5), (6) and (7), decoupling and unbalanced vibration suppression are obtained, and a block diagram is shown in FIG. 5; meanwhile, the method can meet the stability of the control of the magnetic suspension rotor and has better anti-interference performance.

The above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it is apparent to those skilled in the art that various modifications and variations can be made in the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the claims of the present invention.

Claims (1)

1. A decoupling and unbalanced vibration control method for an active magnetic suspension bearing rotor is characterized by comprising the following steps:

step 1: the model of the active magnetic suspension bearing system comprises a magnetic suspension bearing model, a driving amplification circuit and a displacement sensor; the transfer functions of the driving amplifying circuit and the displacement sensor in a magnetic suspension bearing system model are described as follows:

wherein f is _uv Represents the stress of the rotor in the u-axis direction at the v end (v = a, b; u = x, y), x _j 、y _j Respectively the rotor displacement at the j-th degree of freedom, i _uv Denotes the control current (v = a, b; u = x, y), K, of the rotor in the direction of the u-axis at the v-terminal _ir ，K _xr Current-force stiffness and displacement-force stiffness, respectively.

Step 2: according to the geometric position relations of the position of a bearing coil, the position of a sensor, the suspension position of a rotor and the like of the active magnetic suspension bearing system, the relation of the static balance of the rotor and the dynamic unbalance of the rotor in a generalized coordinate system is sought, and a radial four-degree-of-freedom mathematical model of the magnetic suspension bearing system considering the unbalanced vibration problem of the rotor is established through a rotor dynamics method.

Where m is the rotor mass, ω is the angular speed of rotation of the rotor about the Z-axis, J _y And J _x The moment of inertia of the rotor about the Y-axis and the X-axis at the center of mass, respectively. ε and δ represent the amplitude and phase of the rotor static imbalance in a generalized coordinate system, σ and

indicating the amplitude and phase of the rotor at dynamic imbalance. l _a And l _b Are respectively two end diametersThe distance from the center of the electromagnetic bearings a and b to the geometric center of the rotor.

And step 3: the mathematical model (2) is four second-order equations, terms which are different from state variables in each equation are taken as system disturbance terms, the coupling terms and the vibration problems are taken as system external disturbances, and then the system external disturbances are combined. Equation (2) can be written as:

wherein the content of the first and second substances,

wherein the content of the first and second substances,

m _x12 ＝m _x21 ＝(K _xr /m)-l _a l _b K _xr /J _y ，

n _x12 ＝n _x21 ＝(K _ir /m)-l _a l _b K _ir /J _y ，

m _y12 ＝m _y21 ＝(K _xr /m)-l _a l _b K _xr /J _x ，

n _y12 ＝n _y21 ＝(K _ir /m)-l _a l _b K _ir /J _x ，

w _x21 ＝l _b J _z ω/lJ _y ，w _x11 ＝-l _a J _z ω/lJ _y ，

w _y11 ＝l _a J _z ω/lJ _x ，w _y21 ＝-l _b J _z ω/lJ _x ，

and 4, step 4: the second-order nonlinear system with four degrees of freedom in the model equation (3) has a similar expression form, a state observer is designed for each equation in the equation set, so that the state observer can estimate the state variable and the disturbance sum of the system, and the designed observer needs to keep stable and track an observed value. The state observer is designed by taking the first equation in the formula (3) as an example, and the other three equations of the system are designed in the same way. Let x ₁ ＝x _a ，

b ₀ ＝n _x11 ，u＝i _xa ，

Considering as a disturbance signal, the observer is designed as follows:

wherein z = [ z ] ₁₁ z ₁₂ z ₁₃ ] ^T Denotes x = [ x ] ₁ x ₂ x ₃ ] ^T Is determined by the estimated value of (c),

C＝[1 0 0]，B＝[β ₁ β ₂ β ₃ ] ^T to observe the gain.

Order to

The observation equation can be written as:

the pole of the observer is selected in the left half-plane-w ₀ At the handle

Value substitution, then there is:

to give out beta ₁ ＝3w ₀ ，

w ₀ Is the bandwidth of the observer. Bandwidth w ₀ When the observer is more than 0 and an appropriate value is selected, the observer is stable, and z can be ensured ₁₃ Can track x well ₃ The value is obtained.

And 5: and (4) designing a nonlinear feedback control law. First, a differential tracker is used for the transition, depending on the control target and the system endurance. Assume that the given reference signal is ref. Tracking signal v output by linear differential Tracker (TD) ₁ ＝ref，

Respectively with the observer estimate z ₁₁ ，z ₁₂ Forming a difference value, constructing a system error and error differential signal as follows:

control gain of nonlinear feedback control designed:

wherein

k _d ＝2w _c To control the gain, w _c Is the system bandwidth, u ₀ Is a control quantity before compensation.

The disturbance compensation procedure is as follows:

u＝(u ₀ -z ₁₃ )/b ₀ (9)

where u is the control quantity before compensation.

Step 6: and designing an adaptive notch filter with the same frequency as the rotating speed in an output feedback loop of the decoupled subsystem, and combining the adaptive notch filter with the state observer. In the output of the wave trap, the same-frequency component with the same frequency and the same rotating speed in the input signal approaches to 0, and the same-frequency vibration is restrained.

Let y _s (t) is a concave feedback link N _f Input of (a) y _ω (t) is N _f The output of (c) then has:

can verify that y _s (t) and y _ω (t) satisfies the following differential equation:

wherein tau is a proportionality coefficient, then the concave feedback link N _f Has a transfer function of

Let s = j ω _r When τ ≠ 0, then there is,

wherein, delta omega is a wave trap N _f Of the bandwidth of (c). Therefore, when τ ≠ 0, at the notch filter N _f Of the outputs of(s), the input signal y _s And (t) the same-frequency component with the frequency omega approaches to 0, so that the rotating speed same-frequency vibration can be effectively inhibited.

And 7: and (5) deducing a closed loop feedback simplified structure of the subsystem for the unbalanced vibration subsystem of the magnetic suspension bearing rotor consisting of the wave trap, the observer and the nonlinear feedback controller, and analyzing frequency characteristics under different control parameters so as to guide parameter selection of the nonlinear feedback gain controller in the step 5.

The formula (4) is expanded and laplace transformed to derive z ₁₁ 、z ₁₂ 、z ₁₃ The functional expressions of (a) are respectively:

from the control gain equations (8) and (14) of the nonlinear feedback control, the expression of the control variable u of the system can be further derived from the principle of fig. 3:

and simultaneously substituting the formula (14) into the formula (15) to obtain:

simplifying the system structure form of the single-degree-of-freedom system after completing trap and decoupling, setting R(s) as a given reference signal, and D _i And(s) is an external interference signal seen by the system, U(s) is an input signal, and Y(s) is an output signal. G _p (s) Single degree of freedom model after decoupling of control object, G _c (s) and H(s) are pending and are available:

U(s)＝[R(s)H(s)-Y(s)]G _c (s)＝R(s)H(s)G _c (s)-Y(s)G _c (s) (17)

laplace transform of equation (15) is:

the same Laplace transform is performed on the formula (14) and is substituted into the formula (18) to obtain

In the formula: c _n4 ＝-β ₃ -β ₂ k _d -β ₁ k _p -τk _p C _n3 ＝-β ₃ k _d -β ₂ k _p -β ₁ τk _p ，

C _d4 ＝1，C _d3 ＝β ₁ +k _d ，

H _n5 ＝k _p ，H _n4 ＝β ₁ k _p ，

H _d4 ＝C _n4 ，H _d3 ＝C _n3 ，H _d2 ＝C _n2 ，H _d1 ＝C _n1 ，H _d0 ＝C _n0

When model parameters are not determined, the closed loop transfer function of the system is as follows:

the stability of the closed loop system is determined by the pole position of equation (21). Since the denominator term of H(s) can be known that the values of the characteristic roots are all negative numbers, and the Hurwitz condition is satisfied in the left half plane of the complex plane, the closed-loop system stability can be determined by equation (22):

according to equation (3), after removing the perturbation term, the system transfer function in a single degree of freedom can be expressed as:

wherein, b ₀ ＝n _x11 ，a＝m _x11 Thus, the characteristic polynomial for a closed loop system is:

1+G _p (s)G _c (s)＝s ⁷ +A ₆ s ⁶ +A ₅ s ⁵ +A ₄ s ⁴ +A ₃ s ³ +A ₂ s ² +A ₁ s+A ₀ (24)

in the formula: a. The ₆ ＝C _d3 ，A ₅ ＝C _d2 -a，A ₄ ＝C _d1 -aC _d3 +C _n4 ，A ₃ ＝C _d0 -aC _d2 +C _n3 ，A ₂ ＝-aC _d1 +C _n2 ，A ₁ ＝-aC _d0 +C _n1 ，A ₀ ＝C _n0

The open loop transfer function of the system is:

G _o (s)＝G _p (s)G _c (s) (25)

and analyzing the frequency characteristics of the system open loop transfer function under different control parameters. Analysis of omega ₀ ，ω _c Taking different values as the frequency characteristics of the system, obtaining a BODE graph of the system, observing cut-off frequency, phase margin and system gain when the parameters take different values from the BODE graph, and finally determining omega according to the sensitivity degree of noise ₀ ，ω _c And (5) substituting the parameter values into the formula (8) to obtain the specific parameters of the nonlinear control law.

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