CN100587633C - Method for designing precession cross parameter of magnetic levitation high speed rotor - Google Patents

Abstract

The present invention provides a method for designing a precession cross parameter for a magnetic suspension high speed rotor system. A complex coefficients dynamics model for the magnetic suspensionhigh speed rotor system is established; a negative frequency Nyquist curve of complex coefficients loop transfer function at the highest speed is protracted; the low-pass cut-off frequency of the precession cross is calculated; and then the correction target at the highest speed is ensured according the phase angle margin requirement of the precession model; a designing frequency is searched; theprecession cross gain at the highest speed is ensured; and then the precession cross gains of whole rotate speed scope are ensured; so the phase angle stable margin of a rotor in the whole rotate speed scope is ensured and the robust stabilization design of a precession cross feedback is realized. The present invention provides the method for designing the precession cross parameter for the magnetic suspension high speed rotor system according to the design of the phase angle stable margin. Compared with the multivariable system of the prior art such as state space analytic technique and so on, the present invention is of very visual and is provided with good robustness. So the present invention is much more suitable for the practical magnetic suspension high speed rotor system such as a centrifugal machine, a high definition numerical control bathe, a turbine, a energy storage flywheel, a magnetic suspension flywheel and a magnetic suspension control force moment top and so on.

Description

A kind of method of designing precession cross parameter of magnetic levitation high speed rotor

Technical field

The present invention relates to a kind of method of designing precession cross parameter of magnetic levitation high speed rotor, can be used for the design of magnetic suspension rotor system precession cross parameter.

Background technology

With respect to traditional mechanical ball bearing, outstanding advantages such as that magnetic bearing has is contactless, rigidity and damping active controllable, thereby there is not a friction and wear, also need not to lubricate, allow rotor high-speed rotation, little, the supporting precision height of vibration, be particularly suitable for super-clean environment equipment, high rotating speed equipment and require low vibration, high precision, long-life space equipment.At present, the magnetic levitation high-speed rotor system is at hydro-extractor, high precision numerical control lathe, turbine, accumulated energy flywheel, and obtain increasingly extensive application in the civilian and space equipment such as magnetically levitated flywheel and magnetic suspension control torque gyroscope, and the trend that replaces mechanical bearing is arranged.

But, the flat high speed rotor of the especially big ratio of inertias of high speed rotor has strong gyroscopic effect, make the two-freedom of rotor decoupling zero when static suspension rotate generation coupling generation precession and nutating when rotating speed is arranged, and be tending towards unstable with rotating speed rising magnetic suspension rotor.The feedback of intersecting is a kind of inhibition gyroscopic effect, improve the effective ways of magnetic levitation high speed rotor precession and nutating stability, have remarkable result for the highest stabilizing rotating speed that promotes magnetic suspension rotor, key wherein is to design suitable cross parameter (mainly comprising filter cutoff frequency and cross-gain).Because the intrinsic undistinct shortcoming of mechanism of multivariable control theory, adopt the multivariate theory to carry out the cross parameter design and can not guarantee stability margin, be unfavorable for practical application, make the difficult point that is designed to of cross parameter, during practical application mainly by experiment method determine cross parameter, be theoretically unsound.

At present, double frequency Bode figure begins to be used for the stability analysis of magnetic levitation high speed rotor, and the double frequency Nyquist figure of amplification has been used for the design of nutating cross parameter, and obtains good design effect.But the design of precession cross parameter is significantly distinguished with the existence of nutating cross-over design: carry out in the positive frequency territory of the design s=j ω of (1) nutating cross parameter, and the precession design is to carry out in the negative frequency domain of s=-j ω; (2) nutation frequency with rotation speed change wide variation, and that precession frequency changes in the whole range of speeds is less; (3) directly selection unit's gain threshold frequency is as design frequency nutating cross parameter when design, but because LPF be differentiation element in negative frequency domain, directly selection unit's gain threshold frequency does not have to design to separate as design frequency usually.Therefore, the design procedure of magnetic levitation high speed rotor nutating cross parameter can not directly apply to the design of precession cross parameter.

Summary of the invention

Technology of the present invention is dealt with problems: at the singularity of precession, provide a kind of based on the negative precession cross parameter method for designing of Nyquist curve frequently, solve magnetic levitation high speed rotor precession cross parameter design problem, guaranteed the precession stability margin of magnetic levitation high-speed rotor system.

Technical solution of the present invention is: set up magnetic levitation closed loop rotor-support-foundation system complex coefficient kinetic model, draw and descend the negative frequency Nyquist curve of complex coefficient open-loop transfer function the most at a high speed and calculate the low-pass cut-off frequencies that precession intersects, require to determine correction target, search design frequency down according to the Phase margin of precession mode then and determine precession cross-gain under the high speed, and then determine the precession cross-gain that the whole range of speeds is interior, guarantee the phase angle stabilization, nargin of rotor precession in the whole range of speeds.

The present invention's advantage compared with prior art is: (1) is by being a complex coefficient single-variable system with original two variable system equivalences, thereby can continue to use classical single argument control theory and carry out stability analysis and design, thereby have intuitive and robustness, more be applicable to real system; (2) pass through at ω＞ω _P1Search makes in the scope

Phase angle just intersect the design frequency ω of correction angle for precession _P11, solve Nyquist figure and can not be directly used in the problem that the precession cross parameter designs, guaranteed the precession stability margin of magnetic levitation high-speed rotor system; (3) in the whole range of speeds, change features of smaller according to precession frequency, select the precession intersection of fixed cut-off frequency to use LPF, simplified the precession Switch Controller;

Description of drawings

Fig. 1 is magnetic levitation closed loop rotor-support-foundation system of the present invention and coordinate system definition;

Fig. 2 is a controller block diagram of the present invention;

The negative frequency Nyquist curve of complex coefficient open-loop transfer function (down) before and after Fig. 3 proofreaies and correct for precession of the present invention intersects.

Fig. 4 is a process flow diagram of the present invention;

Embodiment

With a kind of magnetic suspension rotor system is the embodiment of example explanation the inventive method, and design object is F _RmaxPrecession mode Phase margin reaches γ under the=400Hz _p=20 °.

Magnetic levitation closed loop rotor-support-foundation system and coordinate system definition are as shown in Figure 1, this system is made of displacement transducer, controller, power amplifier, electromagnet and rotor, oxyz is the rotor coordinate system, the o point is positioned at rotor centroid, x and y axle are along rotor radial and only follow the radially rotation of rotor and do not follow rotation, and the z axle is along rotor axial.A, β be rotor radial around x, the angular displacement that the y axle rotates, Ω is the rotation rotating speed of rotor.The electromagnet of rotor A, B two ends y direction that only draws among the figure, the x direction is similar with it.Each degree of freedom of magnetic suspension rotor system is all by the rotor displacement on this degree of freedom of displacement sensor, if rotor is not on given zero-bit, then error signal is by after the controller computing, by power amplifier output corresponding control current, drive the suitable magnetic attraction rotor of magnetic bearing electromagnet generation and get back on the given position.Set up the radially rotational motion dynamic differential equation model of magnetic suspension rotor system according to gyro technology equation:

$\left\{\begin{array}{c}{J}_y\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&beta;}}-H\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}-{2k}_{\mathrm{<figure-callout\; id="2"\; label="h\; l\; m"\; filenames="C2007101797220009H1.png"\; state="\{\{state\}\}">h</figure-callout>}}{l}_m^{}{}_2^{}\mathrm{\&beta;}=-{2l}_m{l}_s{k}_i{k}_s{g}_w(g_c\mathrm{\&beta;}-g_{\mathrm{cr}}\mathrm{\&alpha;})+p_{\mathrm{dy}}\\ J_x\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&alpha;}}+H\stackrel{\&CenterDot;}{\mathrm{\&beta;}}-{2k}_{\mathrm{<figure-callout\; id="2"\; label="h\; l\; m"\; filenames="C2007101797220009H1.png"\; state="\{\{state\}\}">h</figure-callout>}}{l}_m^{}{}_2^{}\mathrm{\&alpha;}=-{2l}_m{l}_s{k}_i{k}_s{g}_w(g_c\mathrm{\&alpha;}+g_{\mathrm{cr}}\mathrm{\&beta;})+p_{\mathrm{dx}}\end{array}\right.$

α, β represent the rotor radial relative stator around x in the following formula, the angular displacement that the y axle rotates, J _x=J _yAnd J _zBe respectively rotor radial and axial moment of inertia, H=J _zΩ is a rotor angular momentum, Ω=2 π F _rBe rotor speed, p _DxAnd p _DyBe the disturbing moment of rotor radial, k _iAnd k _hBe the displacement rigidity and the current stiffness of magnetic bearing, k _sBe the sensitivity of magnetic bearing displacement transducer, I _mAnd l _sBe respectively magnetic bearing and displacement transducer distance, g to rotor center _c, g _CrAnd g _wBe respectively the input-output transformation operator of PID controller, Switch Controller and power amplifier, promptly have $L\left[g_c\left(\frac{d}{\mathrm{dt}}\right)\right]=g_c\left(s\right),$ $L\left[g_{\mathrm{cr}}\left(\frac{d}{\mathrm{dt}}\right)\right]=g_{\mathrm{cr}}\left(s\right),$ $L\left[g_w\left(\frac{d}{\mathrm{dt}}\right)\right]=g_w\left(s\right),$ L represents Laplace transformation, and s is an operator, g _c(s), g _Cr(s) and g _w(s) be respectively the transport function of PID controller, precession Switch Controller and power amplifier.

The block diagram of the controller that adopts during the inventive method specific implementation comprises PID controller and precession Switch Controller two parts as shown in Figure 2, and wherein the precession Switch Controller is again by LPF link and gain link k _R11Polyphone constitutes.Behind the rotor displacement signal input controller, calculate a part of controlled quentity controlled variable through the PID controller on the one hand, on the other hand, the rotor displacement signal of x degree of freedom and y degree of freedom is input to the precession Switch Controller after asking difference separately, export summation or ask poor with y degree of freedom and x degree of freedom PID controller respectively after the calculating, be fed to power amplifier as total controller output signal.

The negative frequency Nyquist curve (down) of complex coefficient open-loop transfer function before and after Fig. 3 proofreaies and correct for the present invention adopts precession to intersect.This figure coordinate system transverse axis and the longitudinal axis are respectively the real axis and the imaginary axis, the corresponding g of the horizontal ordinate of each point and ordinate difference on the curve _OL(j ω) value real part and imaginary part, ω (0 ,+∞) between value.Curve before the representative of band point solid line is proofreaied and correct among the figure, the curve after the representative of band pecked line is proofreaied and correct, frequency range 2～700Hz, point of every Hz, arrow represent frequency rising direction.0 is the complex plane initial point, broken circle representation unit circle.Pass through a p ₁And p ₂Be respectively the intersection point of proofreading and correct front and back Nyquist curve and unit circle, p ₁₁For intersecting, precession proofreaies and correct the design frequency point.P before proofreading and correct ₁Frequency, phase place and the Phase margin of point are respectively f _P1=12Hz, φ _P1=176 °, γ _P1=-4 °, it is p that the precession intersection is proofreaied and correct design frequency ₁₁Dot frequency is f _P11=20Hz, the p after the correction ₂Dot frequency, phase place and Phase margin are respectively f _P2=20Hz, φ _P1=160 °, γ _P1=20 °.

The process flow diagram of precession cross parameter of the present invention design as shown in Figure 4, concrete steps are as follows:

(1) set up magnetic levitation closed loop rotor-support-foundation system complex coefficient kinetic model: set up the dynamic differential equation of the radially rotational motion of magnetic suspension rotor system according to gyro technology equation by Fig. 1 and Fig. 2:

$\left\{\begin{array}{c}{J}_y\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&beta;}}-H\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}-{2k}_{\mathrm{<figure-callout\; id="2"\; label="h\; l\; m"\; filenames="C2007101797220009H1.png"\; state="\{\{state\}\}">h</figure-callout>}}{l}_m^{}{}_2^{}\mathrm{\&beta;}=-{2l}_m{l}_s{k}_i{k}_s{g}_w(g_c\mathrm{\&beta;}-g_{\mathrm{cr}}\mathrm{\&alpha;})+p_{\mathrm{dy}}\\ J_x\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&alpha;}}+H\stackrel{\&CenterDot;}{\mathrm{\&beta;}}-{2k}_{\mathrm{<figure-callout\; id="2"\; label="h\; l\; m"\; filenames="C2007101797220009H1.png"\; state="\{\{state\}\}">h</figure-callout>}}{l}_m^{}{}_2^{}\mathrm{\&alpha;}=-{2l}_m{l}_s{k}_i{k}_s{g}_w(g_c\mathrm{\&alpha;}+g_{\mathrm{cr}}\mathrm{\&beta;})+p_{\mathrm{dx}}\end{array}\right.$

Use J _RrUnified expression J _x=J _y, and make φ=α+j β, p _d=p _Dx+ jp _Dy, wherein j is an imaginary unit, first formula of differential equation group be multiply by j be added to second formula again, does Laplace transformation again and obtains:

$J_{\mathrm{rr}}{s}^2-\mathrm{jHs}-{2k}_{\mathrm{<figure-callout\; id="2"\; label="h\; l\; m"\; filenames="C2007101797220009H1.png"\; state="\{\{state\}\}">h</figure-callout>}}{l}_m^{}{}_2^{}=-{2l}_m{l}_s{k}_i{k}_s{g}_w\left(s\right)[g_c\left(s\right)+g_{\mathrm{cr}}\left(s\right)]+p_d$

Order $g_{\mathrm{oeff}}\left(s\right)=\frac{1}{J_{\mathrm{rr}}{s}^2-\mathrm{jHs}-{2k}_{\mathrm{<figure-callout\; id="2"\; label="h\; l\; m"\; filenames="C2007101797220009H1.png"\; state="\{\{state\}\}">h</figure-callout>}}{l}_m^{}{}_2^{}},$ g _Ceff(s)=2l _ml _sk _ik _sg _w(s) [g _c(s)+g _Cr(s)], then the complex coefficient kinetic model of system (adopting open-loop transfer function here) is:

g _OL(s)＝g _oeff(s)g _ceff(s)

The negative frequency Nyquist curve of complex coefficient open-loop transfer function when (2) drawing high speed: the band pecked line that sees Fig. 3 for details.

(3) low-pass cut-off frequencies is chosen: the dominant pole frequency during static suspension is 2 π * 80rad/s, dominant pole frequency, i.e. ω when selecting precession intersection LPF cutoff frequency to equal static suspension _R1=2 π * 80rad/s.

(4) correction target under determining the most at a high speed: require γ according to design objective _p=20 °, then the phase angle of correction target is 180 °-γ _p=160 °, i.e. p among Fig. 3 ₂The point.

(5) search design frequency ω under the most at a high speed _P11(=2 π f _P11): can solve ω by following formula _P11

G wherein _CrLPFFor intersecting low pass LPF operator, p ₂Point is p ₁₁The correction target of some expectation, the intersection point of Nyquist curve and unit circle after just proofreading and correct.Find f in the negative frequency Nyquist curve (Fig. 3) before correction _P1=12Hz is at ω＞ω _P1Search p in=2 π * 12rad/s scope ₁₁Point obtains f _P11=20Hz, i.e. ω _P11=2 π * 20rad/s, corresponding correcting value

(6) the precession cross-gain under determining the most at a high speed: according to

$k_{\mathrm{rl}1d}=\left|\stackrel{\&RightArrow;}{p_{11}{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="C2007101797220009H1.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}\right|/\left[{2l}_m{l}_s{k}_i{k}_s\right|g_w\left({-\mathrm{j\&omega;}}_{p11}\right)g_{\mathrm{crLPF}}\left({-\mathrm{j\&omega;}}_{p11}\right)g_{\mathrm{oeff}}\left({-\mathrm{j\&omega;}}_{p11}\right)\left|\right]$

Precession cross-gain under determining the most at a high speed is k _Rl1d=0.3048.

(7) determine the cross-gain that the whole range of speeds is interior: at 0＜F _r＜F _RmaxIn the scope, the precession cross-gain is directly proportional with rotating speed, i.e. k _Rh1=k _Rh1dF _r/ F _Rmax=0.3048 * F _r/ 400, F wherein _rAnd F _RmaxBe respectively rotor speed and rotor maximum speed.

Among the present invention, magnetic bearing controller can be an analog or digital; The magnetic bearing power amplifier can be analog amplifier or digital power amplifier, can be Linear Power Amplifier or close power amplifier, close power amplifier can be again stagnate ring (Hysteresis) type, sampling maintenance (Sample/Hold) type, width modulation (PWM) type, minimum pulse width (MPW) type and three level types, and the collocation form of power amplifier output power pipe can be half-bridge or full-bridge; The magnetic bearing electromagnet can be to adopt electromagnetism biasing or permanent magnet bias mode; Displacement transducer can be current vortex type sensor or inductive type sensor.

Claims (8)

1, a kind of method of designing precession cross parameter of magnetic levitation high speed rotor, it is characterized in that: set up magnetic levitation closed loop rotor-support-foundation system complex coefficient kinetic model, draw and descend the negative frequency Nyquist curve of complex coefficient open-loop transfer function the most at a high speed and calculate the low-pass cut-off frequencies that precession intersects, correction target under determining the most at a high speed according to the Phase margin requirement of precession mode then, precession cross-gain under searching for design frequency and determining the most at a high speed, and then determine the precession cross-gain that the whole range of speeds is interior, guarantee the phase angle stabilization, nargin of rotor precession in the whole range of speeds.

2, the method for designing precession cross parameter of magnetic levitation high speed rotor according to claim 1 is characterized in that: the described step of setting up magnetic levitation closed loop rotor-support-foundation system complex coefficient kinetic model comprises:

(1) set up the radially rotational motion dynamic differential equation model of magnetic suspension rotor system according to gyro technology equation:

α, β represent the rotor radial relative stator around x in the following formula, the angular displacement that the y axle rotates, J _x=J _yAnd J _zBe respectively rotor radial and axial moment of inertia, H=J _zΩ is a rotor angular momentum, and Ω is the rotation rotating speed of rotor, Ω=2 π F _r, F _rBe rotor speed, p _DxAnd p _DyBe the disturbing moment of rotor radial, k _iAnd k _hBe the displacement rigidity and the current stiffness of magnetic bearing, k _sBe the sensitivity of magnetic bearing displacement transducer, l _mAnd l _sBe respectively magnetic bearing and displacement transducer distance, g to rotor center _c, g _CrAnd g _wBe respectively the input-output transformation operator of PID controller, precession Switch Controller and power amplifier, promptly have $L\left[g_c\left(\frac{d}{\mathrm{dt}}\right)\right]=g_c\left(s\right),$ $L\left[g_{\mathrm{cr}}\left(\frac{d}{\mathrm{dt}}\right)\right]=g_{\mathrm{cr}}\left(s\right),$ $L\left[g_w\left(\frac{d}{\mathrm{dt}}\right)\right]=g_w\left(s\right),$ L represents Laplace transformation, and s is an operator, g _c(s), g _Cr(s) and g _w(s) be respectively the transport function of PID controller, precession Switch Controller and power amplifier;

(2) adopt plural conversion that the two variable equations of the real coefficient in the step (1) are converted into complex coefficient single argument form, obtain complex coefficient single argument equivalent system, use J _RrUnified expression J _x=J _y, and make φ=α+j β, p _d=p _Dx+ jp _Dy, wherein j is an imaginary unit, first formula of Differential Equation Model be multiply by j be added to second formula again, does Laplace transformation again and obtains:

$J_{\mathrm{rr}}{s}^2-\mathrm{jHs}-2k_h{l}_m^2=-2l_m{l}_s{k}_i{k}_s{g}_w\left(s\right)[g_c\left(s\right)+g_{\mathrm{cr}}\left(s\right)]+p_d$

Order $g_{\mathrm{oeff}}\left(s\right)=\frac{1}{J_{\mathrm{rr}}{s}^2-\mathrm{jHs}-2k_h{l}_m^2},$ g _Ceff(s)=2l _ml _sk _ik _sg _w(s) [g _c(s)+g _Cr(s)], the complex coefficient open-loop transfer function g of system then _OL(s) be:

g _OL(s)＝g _oeff(s)g _ceff(s)。

3, the method for designing precession cross parameter of magnetic levitation high speed rotor according to claim 2, it is characterized in that: the negative frequency Nyquist curve of complex coefficient open-loop transfer function during described high speed, its coordinate system transverse axis and the longitudinal axis are respectively the real axis and the imaginary axis, the corresponding g of the horizontal ordinate of each point and ordinate difference on the curve _OL(j ω) value real part and imaginary part, ω (0 ,+∞) between value.

4, the method for designing precession cross parameter of magnetic levitation high speed rotor according to claim 1 is characterized in that: when described low-pass cut-off frequencies is chosen as static suspension 0.9～1 of dominant pole frequency times.

5, the method for designing precession cross parameter of magnetic levitation high speed rotor according to claim 2 is characterized in that: described correction target under the most at a high speed is the p on the complex plane unit circle ₂Point makes

Phase angle be 180 °-γ _p, wherein the o point is the complex plane initial point, γ _pFor proofreading and correct the stability margin requirement of precession mode under the maximum speed of back.

6, the method for designing precession cross parameter of magnetic levitation high speed rotor according to claim 5 is characterized in that: described the most following search design frequency is ω _P11, be to solve ω by following formula _P11

G wherein _CrLPFFor intersecting low pass LPF operator, p ₁₁Point is g before correspondence is proofreaied and correct on the negative Nyquist curve frequently _OL(j ω _P11) point of complex values.

7, the method for designing precession cross parameter of magnetic levitation high speed rotor according to claim 6 is characterized in that: the formula of described precession cross-gain under the most at a high speed is

$k_{\mathrm{rl}1d}=\left|\stackrel{\&RightArrow;}{p_{11}{p}_2}\right|/\left[2l_m{l}_s{k}_i{k}_s\right|g_w(-j{\mathrm{\&omega;}}_{p11})g_{\mathrm{crLPF}}(-j{\mathrm{\&omega;}}_{p11})g_{\mathrm{oeff}}(-j{\mathrm{\&omega;}}_{p11})\left|\right],$ k _Rl1dBe the precession cross-gain under the most at a high speed;

Be p ₁₁Point is to p ₂Distance; k _iBe the displacement rigidity of magnetic bearing, k _sBe the sensitivity of magnetic bearing displacement transducer, l _mAnd l _sBe respectively magnetic bearing and displacement transducer distance to rotor center.

8, the method for designing precession cross parameter of magnetic levitation high speed rotor according to claim 7 is characterized in that: the cross-gain k in the described whole range of speeds _Rh1At 0＜F _r＜F _RmaxDetermine by following formula in the scope:

k _rh1＝k _rh1dF _r/F _rmax

F wherein _rAnd F _RmaxBe respectively rotor speed and rotor maximum speed, k _Pl1dBe the precession cross-gain under the most at a high speed.

Images