CN102508433B - Method for compensating digital control delay of magnetic bearing switch power amplifier - Google Patents

Abstract

The invention discloses a method for compensating digital control delay of a magnetic bearing switch power amplifier, which is a method capable of compensating the digital control delay of a magnetic bearing switch power amplifier. The method comprises the steps of setting asymmetric factors of asymmetric sampling resistor networks according to a linear model predictive control theory on the basis of measuring the digital control delay of a switch power amplifier system and equivalent inductance and resistance of a magnetic bearing winding, and finally setting the asymmetric sampling resistance networks according to the asymmetric factors. The method for compensating the digital control delay of the magnetic bearing switch power amplifier belongs to the technical field of magnetic bearing control and can be applied to the fast response and high-stable control of a high-speed magnetic suspension rotor system.

Description

A kind of method that compensates magnetic bearing switch power amplifier digital control delay

Technical field

The present invention relates to a kind of method that compensates magnetic bearing switch power amplifier digital control delay, the fast-response and the high stable that are applicable to the high speed magnetic suspended rotor system are controlled, and belong to the magnetic bearings control technical field.

Background technology

Magnetic suspension bearing is because having without friction, low vibration, being easy to realize that the outstanding advantages such as high precision and long-life have broad application prospects in fields such as Aero-Space, commercial production, modern militaries.

In magnetic bearing control system, the effect of switch power amplifier is to provide corresponding electric current to produce needed electromagnetic force to the electromagnetic bearing coil.Switching Power Amplifier for Active Magnetic Bearing Applications mainly contains width modulation (PWM) type, sampling maintenance, stagnant chain rate than four kinds of forms such as type and minimum pulse width types.Wherein, the advantage of PWM close power amplifier is that switching frequency is fixed, and can limit the width of minimum turn-on and turn-off pulse, and the output waveform quality is good, stable state accuracy is high, reliability is high, in electromagnetic bearing, is widely applied.Electromagnetic bearing PWM close power amplifier mainly comprises controller, PWM generator, full-bridge main circuit and current sensor.

For PWM H full-bridge switch power amplifier, according to the difference that drives control mode, can be divided into again single-polarity PWM close power amplifier and bipolarity PWM close power amplifier.With bipolarity PWM close power amplifier, compare, the current ripples of single-polarity PWM close power amplifier does not increase along with the increase of DC bus-bar voltage, so the single-polarity PWM close power amplifier is applied even more extensively in the magnetic bearing switch power amplification system.

Traditional single-polarity PWM close power amplifier is to be realized by mimic channel, along with the development of digital signal processor (DSP), and the digital control magnetic bearing switch power amplification system that is applied to more and more widely.Control and compare with traditional simulation, digital control have many advantages, and such as moving more complicated control algolithm, environment and parameter are changed and have stronger robustness, but overprogram, and be convenient to fault diagnosis and fault-tolerant control etc.Yet, the digital control digital control delay that comprises A/D transfer delay, computation delay and PWM MDL modulation delay of inevitably introducing.These time delays will inevitably cause the phase place of power amplification system to lag behind, thereby the system dynamic responding speed of impact even affects the stability of system.Particularly, for the magnetic suspension rotor system with high rotating speed, large inductance and strong gyroscopic effect, too large phase place hysteresis has had a strong impact on the stability of system.

The phase place of Switching Power Amplifier for Active Magnetic Bearing Applications lags behind and mainly comprises digital control delay and controlled device time delay two parts.Wherein, digital control delay is a pure lag system, and its size is determined by digital control system itself, irrelevant with controlled device; And the object time delay is to be caused by inductive load, its size is also irrelevant with digital control system.In PWM H bridge unipolarity close power amplifier, for onesize inductive load, DC bus-bar voltage is larger, just less in the object time delay at same frequency place, but the control time delay of system does not change with the change of DC bus-bar voltage size.Therefore, improve the object time delay that DC bus-bar voltage can only compensate magnetic bearing, can not compensate its digital control delay.At present, the method for compensation magnetic bearing digital control delay mainly contains various Model Predictive Control, independent control etc.These control methods can realize accurate Current Control in theory, but make it very sensitive to system parameter variations because these methods depend on the accurate mathematical model of system.Especially, these advanced control methods often need more computational resource, cause the computation delay of system to increase, and this has further increased again the digital control delay of system.Therefore these methods also fail to be used widely in commercial production at present.Although TRAJECTORY CONTROL is relative simple with tracking control, it needs switching frequency higher and that change usually, and this has not only increased the power consumption of switching tube but also has strengthened the difficulty that output filter designs.

In order to overcome the deficiency of above control method, in recent years, the linear prediction control theory has obtained larger development, and is widely used in the commercial production such as rectifier, inverter.With model predictive control method, compare, it is a kind of simple and stronger method of robustness that linear prediction is controlled, and its main shortcoming is that steady-state tracking precision is poor and current noise is larger.

Summary of the invention

The objective of the invention is: overcome that linear prediction control method current noise is large, tracking accuracy is not high and the deficiency of needs increase extra computation resource, on the basis that does not increase any hardware circuit and software calculated amount, by asymmetric sampling resistor network, realize the effective compensation to magnetic bearing power amplification system digital control delay.

Technical solution of the present invention is: on the basis of measuring switch power amplification system digital control delay and magnetic bearing winding equivalent inductance, resistance, according to the linear prediction control theory, the dissymmetry factor of asymmetric sampling resistor network is set, finally according to dissymmetry factor, configures asymmetric sampling resistor network.Comprise the following steps:

(1) the digital control delay T of measuring switch power amplification system _d

T _d＝T _ad+T _cal+T _awa+0.5T _c

Wherein, T _AdMean power amplification system AD sampling time delay, T _calMean computation delay, T _awaMean that PWM waits for time delay, T _cThe servo period that means power amplification system;

(2) measure respectively magnetic bearing winding equivalent resistance R and inductance L with multimeter and electric inductance measuring-testing instrument;

(3) determine the big or small R of current sampling resistor _mWith the ratio k of current sample with divider resistance _Ab

(4), according to the linear prediction control theory, determine the degree of asymmetry factor of asymmetric sampling resistor network

T wherein _sFor the sampling period of power amplification system;

(5) take into account power consumption and the signal to noise ratio (S/N ratio) of sampling resistor network, according to the R of dissymmetry factor γ configuration sampling resistor network _a, R _b, R _cAnd R _d, R wherein _a=R _d,

Principle of the present invention is: utilize the linear prediction control principle, in conjunction with this concrete object of electromagnetic bearing switch power amplifier, provided the phase compensating method of the magnetic bearing switch power amplifier digital control delay based on asymmetric sampling resistor network.This phase compensating method, by the design of sampling resistor network degree of asymmetry, has been realized the compensation to the close power amplifier digital control delay when realizing current detecting.

Specifically, the digital control delay of magnetic bearing mainly comprises again AD sampling time delay T _Ad, computation delay T _cal, PWM waits for time delay T _awaWith PWM MDL modulation delay T _adjFour parts.Transform time delay T for AD _Ad, its big or small delay time with each passage is multiplied by the sampling channel number and can obtains, its transport function G _Ad(s) can be expressed as:

$G_{\mathrm{ad}}\left(s\right)=e^{-s\·T_{\mathrm{ad}}}---\left(1\right)$

Wherein s means the variable of the mathematical model-transport function of control system in complex field.In like manner, for same calculated amount, the computation delay of DSP depends on the processing speed of DSP, and its whole computation delay time can draw by the mode of measuring, and then obtains its transport function G _cal(s) be:

$G_{\mathrm{cal}}\left(s\right)=e^{-s\·T_{\mathrm{cal}}}---\left(2\right)$

For the digital control system of fixed switching frequency, because the dutycycle d calculated can only could upgrade in the beginning in each PWM cycle, it also inevitably brings a time delay, waits for time delay T _awa, this time delay can be expressed as:

$T_{\mathrm{awa}}=\mathrm{rem}\left(\frac{T_c-T_{\mathrm{cal}}}{T_p}\right)---\left(3\right)$

T wherein _pThe switch periods that means power amplification system, symbol " rem " means except rear complementation.Wait for time delay transport function G _awa(s) expression formula is:

$G_{\mathrm{awa}}\left(s\right)=e^{-s\·T_{\mathrm{awa}}}---\left(4\right)$

Due to each servo period T _cOnly upgrade the PWM dutycycle one time, so the PWM modulation is equivalent to sample-hold link, its transport function G _adj(s) can be expressed as:

$G_{\mathrm{adj}}\left(s\right)=\frac{1-e^{-s\·T_c}}{s}---\left(5\right)$

S=j ω (5) formula of bringing into is obtained,

$G_{\mathrm{adj}}\left(\mathrm{j\ω}\right)=\frac{1-e^{-\mathrm{j\ω}{T}_c}}{\mathrm{j\ω}}=\frac{2e^{-\mathrm{j\ω}{T}_c/2}(e^{{\mathrm{j\ωT}}_c/2}-e^{-\mathrm{j\ω}{T}_c/2})}{2\mathrm{j\ω}}=T_c\frac{\sin ({\mathrm{\ωT}}_c/2)}{{\mathrm{\ωT}}_c/2}{e}^{-\mathrm{j\ω}{T}_c/2}---\left(6\right)$

So, PWM MDL modulation delay T _adjFor

T _adj＝0.5T _c (7)

Therefore, whole digital control system time delay is:

T _d＝T _ad+T _cal+T _awa+T _adj (8)

Fig. 2 means the H bridge unipolarity magnetic bearing power amplification system theory diagram based on asymmetric sampling resistor network, its control signal and electric current loop feedback signal are done to export controlled quentity controlled variable through controller after difference, then carry out the PWM modulation and generate the PWM ripple, the PWM ripple drives the H full-bridge inverter to carry out the track reference electric current to produce corresponding control electric current, and last asymmetric sampling resistor network (as shown in empty frame in Fig. 2) is realized the detection of coil current and the compensation of magnetic bearing system phase place.R in figure _mIt is sampling resistor; R _a, R _b, R _cAnd R _dForm asymmetric resistor voltage divider network, asymmetric resistor voltage divider network and sampling resistor R _mForm asymmetric sampling resistor network; K is the operational amplifier gain of current sampling circuit; i _cIt is the electric current loop feedback factor; Q1, Q2, Q3, Q4 form four power switch pipes of H full-bridge, according to pwm signal, carry out turn-on and turn-off.

At first utilize State-space Averaging Principle, set up the small-signal model of H bridge unipolarity close power amplifier.According to the principle of H bridge single-polarity PWM close power amplifier, at a PWM, in the cycle, can obtain the equivalent electrical circuit topological structure of its charged state (0≤t<dT) and afterflow state (dT≤t<T) respectively as shown in Figure 3 and Figure 4.

According to Kirchhoff's current law (KCL), can proper 0≤t<during dT,

$\left\{\begin{array}{cc}\begin{array}{c}{i}_m\left(t\right)=i_L\left(t\right)+i_{\mathrm{cd}}\left(t\right)\\ U_d\left(t\right)=R_m{i}_m\left(t\right)+{\mathrm{Ri}}_L\left(t\right)+L\frac{{\mathrm{di}}_L\left(t\right)}{\mathrm{dt}}\\ (R_c+R_d)i_{\mathrm{cd}}\left(t\right)={\mathrm{Ri}}_L\left(t\right)+L\frac{{\mathrm{di}}_L\left(t\right)}{\mathrm{dt}}\\ u_s\left(t\right)={\mathrm{kk}}_{\mathrm{ab}}{U}_d-{\mathrm{kk}}_{\mathrm{cd}}({\mathrm{Ri}}_L\left(t\right)+L\frac{{\mathrm{di}}_L\left(T\right)}{\mathrm{dt}})\end{array}& \end{array}0\&le;t<\mathrm{dT}\right.---\left(9\right)$

Wherein, $k_{\mathrm{ab}}=\frac{R_b}{R_a+R_b},$ $k_{\mathrm{cd}}=\frac{R_d}{R_c+R_d}.$

Definition status variable x=i _L, input variable u=U _dWith output variable y=u _s.Correspondingly, t state variable, input variable and output variable constantly is designated as respectively x (t)=i arbitrarily _L(t), u (t)=U _dAnd y (t)=u (t) _s(t).So the state equation that can obtain system is:

$\left\{\begin{array}{c}\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=-\frac{a}{b}x\left(t\right)+\frac{1}{b}u\left(t\right)\\ y\left(t\right)=(\frac{{\mathrm{kk}}_{\mathrm{cd}}{d}_0}{c}-{\mathrm{kk}}_{\mathrm{cd}}R)x\left(t\right)+({\mathrm{kk}}_{\mathrm{ab}}-\frac{{\mathrm{kk}}_{\mathrm{cd}}}{c})u\left(t\right)\end{array}\right.---\left(10\right)$

Wherein $a=R_m+\frac{{\mathrm{RR}}_m}{R_c+R_d}+R,$ $b=L(1+\frac{R_m}{R_c+R_d}),$ $c=1+\frac{R_m}{R_c+R_d}$ And ${\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="HDA0000105859300000032.png"\; state="\{\{state\}\}">d</figure-callout>}}_0=R_m+R+\frac{{\mathrm{RR}}_m}{R_c+R_d}.$

Consider, in real system, R is arranged _c+ R _d>>R _mAnd R _c+ R _d>>RR _m, so a ≈ R _m+ R, b ≈ L, c ≈ 1 and d ₀≈ R _m+ R, can obtain its substitution (10),

$\left\{\begin{array}{c}\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=-\frac{R+R_m}{L}x\left(t\right)+\frac{1}{L}u\left(t\right)\\ y\left(t\right)={\mathrm{kk}}_{\mathrm{cd}}{R}_{m}x\left(t\right)+k(k_{\mathrm{ab}}-k_{\mathrm{cd}})u\left(t\right)\end{array}\right.---\left(11\right)$

In like manner utilize Kirchhoff's current law (KCL), when dT≤t<T, can obtain,

$\left\{\begin{array}{cc}\begin{array}{c}{i}_m\left(t\right)=i_L\left(t\right)+i_{\mathrm{cd}}\left(t\right)\\ 0=R_m{i}_m\left(t\right)+{\mathrm{Ri}}_L\left(t\right)+L\frac{{\mathrm{di}}_L\left(t\right)}{\mathrm{dt}}\\ (R_c+R_d)i_{\mathrm{cd}}\left(t\right)=-R_m{i}_m\left(t\right)\\ u_s\left(t\right)=-k{R}_c{i}_{\mathrm{cd}}\left(t\right)\end{array}& \mathrm{dT}\&le;t<T\end{array}\right.---\left(12\right)$

Wherein, i _Cd(t) mean that any t flows through divider resistance R constantly _cAnd R _dElectric current, i _m(t) mean that any t flows through sampling resistor R constantly _mElectric current.The state equation that similarly, can obtain system is:

$\left\{\begin{array}{c}\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=-\frac{R}{L}x\left(t\right)-\frac{R_m(R_c+R_d)}{L(R_c+R_d+R_m)}x\left(t\right)\\ y\left(t\right)=\frac{k{R}_c{R}_m}{R_c+R_d+R_m}x\left(t\right)\end{array}\right.---\left(13\right)$

Consider R in real system _c+ R _d>>R _m, so above formula can further be reduced to:

$\left\{\begin{array}{c}\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=-\frac{R+R_m}{L}x\left(t\right)\\ y\left(t\right)={\mathrm{kk}}_{\mathrm{cd}}{R}_{m}x\left(t\right)\end{array}\right.---\left(14\right)$

According to State-space Averaging Principle, (11) * d+ (14) * (1-d) can obtain at an average state space equation of PWM cycle, be:

$\left\{\begin{array}{c}\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=-\frac{R+R_m}{L}x\left(t\right)+\frac{d}{L}u\left(t\right)\\ y\left(t\right)={\mathrm{kk}}_{\mathrm{cd}}{R}_{m}x\left(t\right)+\mathrm{dk\&gamma;u}\left(t\right)\end{array}\right.---\left(15\right)$

Wherein, γ=k _Ab-k _Cd.

Equation (15) be one non-linear continuous time equation, can obtain its linear model by the small-signal disturbance, order

With

Wherein "～" represents small-signal, correspondingly,

With

The small-signal variable that means respectively any t state variable, input variable and output variable constantly.So can obtain steady-state equation and the condition of small signal equation of system is respectively:

$\left\{\begin{array}{c}\frac{\mathrm{dX}}{\mathrm{dt}}=-\frac{R+R_m}{L}X+\frac{D}{L}U\\ Y={\mathrm{kk}}_{\mathrm{cd}}{R}_{m}X+\mathrm{Dk\&gamma;U}\end{array}\right.---\left(16\right)$

$\left\{\begin{array}{c}\frac{d\tilde{x}\left(t\right)}{\mathrm{dt}}=-\frac{R+R_m}{L}\tilde{x}\left(t\right)+\frac{D}{L}\tilde{u}\left(t\right)+\frac{U}{L}\tilde{d}\left(t\right)\\ \tilde{y}\left(t\right)={\mathrm{kk}}_{\mathrm{cd}}{R}_m\tilde{x}\left(t\right)+\mathrm{Dk\&gamma;}\tilde{u}\left(t\right)+\mathrm{k\&gamma;U}\tilde{d}\left(t\right)\end{array},-,-,-,\left(17\right)\right.$

Like this, according to the Laplace conversion, the small-signal transport function that can obtain system is:

$G_i\left(s\right)=\frac{\tilde{x}\left(s\right)}{\tilde{u}\left(s\right)}=\frac{{\tilde{i}}_L\left(s\right)}{{\tilde{u}}_d\left(s\right)}=\frac{D}{R+R_m+\mathrm{Ls}}---\left(18\right)$

$G_y\left(s\right)=\frac{\tilde{y}\left(s\right)}{\tilde{u}\left(s\right)}=\frac{{\tilde{u}}_s\left(s\right)}{{\tilde{u}}_d\left(s\right)}=\mathrm{Dk}\frac{k_{\mathrm{ab}}{R}_m+\mathrm{\&gamma;R}+\mathrm{\&gamma;Ls}}{R+R_m+\mathrm{Ls}}---\left(19\right)$

Therefore, from

Arrive

Transport function can be expressed as:

$G_s\left(s\right)=\frac{{\tilde{u}}_s\left(s\right)}{{\tilde{i}}_L\left(s\right)}=k(k_{\mathrm{ab}}{R}_m+\mathrm{\&gamma;R}+\mathrm{\&gamma;Ls})---\left(20\right)$

Simultaneously, in H bridge single-polarity PWM close power amplifier,

$\frac{u_c}{U_{\mathrm{tri}}}=d---\left(21\right)$

U wherein _triIt is the amplitude of triangular carrier.

The tube voltage drop of ignoring MOSFET and diode, can obtain at a PWM in the cycle the internodal average terminal voltage u of a, b _AbWith the pass of DC bus-bar voltage, be:

u _ab＝dU _d (22)

Therefore, the transport function of whole PWM modulator can be expressed as following form:

$G_m\left(s\right)=\frac{u_{\mathrm{ab}}\left(s\right)}{u_c\left(s\right)}=\frac{U_d}{U_{\mathrm{tri}}}---\left(23\right)$

U wherein _c(s) be controller output u _cTransport function expression formula in frequency.In conjunction with (18) and (23), the transport function G of whole PWM modulator and H bridge power amplifier _p(s) be:

$G_p\left(s\right)=\frac{{\tilde{i}}_L\left(s\right)}{{\tilde{u}}_c\left(s\right)}=\frac{{\tilde{i}}_L\left(s\right)}{D{\tilde{u}}_d\left(s\right)}\frac{u_{\mathrm{ab}}\left(s\right)}{u_c\left(s\right)}=\frac{U_d}{U_{\mathrm{tri}}}\frac{1}{R+R_m+\mathrm{Ls}}---\left(24\right)$

Fig. 5 is Switching Power Amplifier for Active Magnetic Bearing Applications system equivalence closed loop controlling structure block diagram of the present invention, wherein G _p(s) mean the transport function of whole PWM modulator and H bridge power amplifier, G _d(s) representative digit is controlled the transport function of time delay, G _c(s) representative digit is controlled the transport function of time delay, G _s(s) mean the transport function of asymmetric sampling resistor network, i _cThe electric current loop feedback factor, i _refFor the given value of current value of close power amplifier system, i _cFor the output valve of magnetic bearing power amplification system controller, i _LFor winding current.

For traditional electric current detecting method, γ=0, by its substitution

In can obtain,

G _s(s)＝kk _abR _m (25)

Visible traditional electric current detecting method is a proportional component in essence.In like manner, if γ>0, the current detecting network has ratio-differential function.That is to say, compare with γ=0, introduced the current differential item in the feedback channel of power amplification system.

There is forecast function owing to differentiating, moreover traditional linear prediction control theory just is being based on the prediction of numerical differentiation computing realization to the state in future.Therefore, based on the linear prediction control theory, the parameter of utilizing the method for undetermined coefficients can adjust asymmetric sampling resistor network.

According to the linear prediction control theory, for compensating digits controls time delay T _d, PREDICTIVE CONTROL expression formula G _Pred(z) be:

$G_{\mathrm{pred}}\left(z\right)=1+\frac{T_d}{T_c}-\frac{T_d}{T_c}{z}^{-1}---\left(26\right)$

Wherein z means the variable of the mathematical model-transport function of discrete control system, T _cFor servo period.G _Pred(z) corresponding differential expressions i _{Pred, k}Can be expressed as:

$i_{\mathrm{pred},k}=(1+\frac{T_d}{T_c})\&CenterDot;i_{L,k}-\frac{T_d}{T_c}\&CenterDot;i_{L,k-1}---\left(27\right)$

I wherein _{L, k}The sample rate current that means current (k is constantly), i _{L, k-1}The sample rate current that means previous moment (k-1 constantly).

The method of application backward difference can show that the differential equation of (20) is:

$u_{s,k}=k(k_{\mathrm{ab}}{R}_m+\mathrm{\&gamma;R}+\frac{\mathrm{\&gamma;L}}{T_s})i_{L,k}-\frac{\mathrm{k\&gamma;L}}{T_s}{i}_{L,k-1}---\left(28\right)$

U wherein _{S, k}Mean current time (k constantly) output valve of asymmetric sampling resistor network after operational amplifier.

In conjunction with (27) and (28), make u _{S, k}=i _{Pred, k}, utilize the method for undetermined coefficients can show that the dissymmetry factor γ of asymmetric sampling resistor network and the gain k of operational amplifier are:

$\left\{\begin{array}{c}\mathrm{\&gamma;}=\frac{T_s{T}_d{k}_{\mathrm{ab}}{R}_m}{{\mathrm{LT}}_c-T_s{T}_{d}R}\\ k=\frac{{\mathrm{LT}}_c-T_s{T}_{d}R}{k_{\mathrm{ab}}{R}_m{\mathrm{LT}}_c}\end{array}\right.---\left(29\right)$

On this basis, the ratio k with divider resistance according to dissymmetry factor γ and current sample _AbCan further determine the R of asymmetric sampling resistor network _a, R _b, R _cAnd R _d.Usually choose R _a=R _d, by k _Ab, R _aSubstitution

In obtain R _b, then by R _a, R _b, R _dWith the γ substitution

In can obtain R _c.

The present invention is with the advantage that the linear prediction control method of existing compensation magnetic bearing power amplification system digital control delay is compared: the inventive method replaces the numerical differentiation computing of conventional linear forecast Control Algorithm by the analog differentiation function of utilizing asymmetric sampling resistor network, computation delay itself that not only avoided the Classical forecast control method to bring, and overcome that the factor word is differentiated and the problem of introducing excessive noise, improved the tracking accuracy of electric current.

The accompanying drawing explanation

The process flow diagram that Fig. 1 is the inventive method;

Fig. 2 is the magnetic bearing power amplification system structural drawing based on asymmetric sampling resistor network;

Fig. 3 is the equivalent topologies figure under asymmetric sampling resistor network charged state of the present invention;

Fig. 4 is the equivalent topologies figure under asymmetric sampling resistor network afterflow state of the present invention;

Fig. 5 is Switching Power Amplifier for Active Magnetic Bearing Applications system equivalence closed loop controlling structure block diagram of the present invention;

The contrast current-responsive figure that Fig. 6 is the inventive method and conventional digital control method;

The contrast current-responsive figure that Fig. 7 is the inventive method and conventional linear forecast Control Algorithm;

Embodiment

As shown in Figure 1, in specific implementation process, the specific embodiment of the invention step is as follows:

(1) measure the total digital control system time delay T of magnetic bearing switch power amplifier _d:

T _d＝T _ad+T _cal+T _awa+0.5T _c

Wherein, T _AdMean power amplification system AD sampling time delay, T _calMean computation delay, T _awaMean that PWM waits for time delay, T _cThe servo period of expression system.T _AdCan be multiplied by by the delay time of each sampling channel total sampling channel number just can draw; Computation delay can draw by testing directly; T _awaCan be according to formula

Calculate, wherein T _pThe switch periods that means power amplification system, symbol " rem " means except rear complementation.

(2) in radially load-bearing and dropping under the state of protection on bearing of magnetic suspension rotor, adopt respectively multimeter and secohmmeter can measure equivalent resistance R and the inductance L of bearing coil winding.

(3) determine the big or small R of current sampling resistor _mWith the ratio k of current sample with divider resistance _Ab, wherein

R _mAnd k _AbChoose and will be considered by the factors such as sample range of AD chip in conjunction with the scope of magnetic bearing winding current, size and the current sample in power amplifier voltage source.

(4), according to the linear prediction control theory, determine the dissymmetry factor of asymmetric sampling resistor network

T wherein _sFor the sampling period of power amplification system, γ is defined as

(5) take into account power consumption and the signal to noise ratio (S/N ratio) of sampling resistor network, according to the R of dissymmetry factor γ configuration sampling resistor network _a, R _b, R _cAnd R _d.

For the power consumption that reduces as much as possible system and improve signal to noise ratio (S/N ratio), 1000R should be arranged _m<R _a<10000R _mAnd 1000R<R _a<10000R for simplicity, chooses R simultaneously usually _a=R _d, according to

With Further obtain R _bAnd R _c.

Validity for checking the inventive method, the close power amplifier system of radially Ax passage of magnetic suspension control torque gyroscope magnetic bearing of take is verified as example, this system adopts 40-MIPS DSP TMS320C32 as control chip, and AD1671 is as sampling A/D chip, and DC bus-bar voltage is 28V.Current sample cycle and servo period all are set to 150 μ s, and switch periods is set to 50 μ s.At t=0.001s, the magnetic bearing reference current steps to 1A from 0.

Because switching time of each passage of AD1671 is 800ns, total sampling time of five passages of magnetic suspension control torque gyroscope electric current loop is 4 μ s so.The computation delay that can obtain after tested system is 38 μ s approximately, according to

The wait time delay that can draw PWM is 12 μ s, adds the MDL modulation delay of 75 μ s, draws the digital control delay T of system _d=129 μ s.Adopt respectively the equivalent resistance R=2.0 Ω of multimeter and secohmmeter test bearing coil winding, inductance L=21.3mH.The scope, the current sample that consider the magnetic bearing winding current are set to R by the sample range of AD chip and the signal to noise ratio (S/N ratio) sampling resistor of sampled signal _m=1.0 Ω, the ratio of divider resistance is set to k _Ab=1/6.The degree of asymmetry factor of calculating sampling resistor network

Choose R _a=R _d=7.500k Ω, according to

According to

Can obtain corresponding linear prediction control algolithm is $i_{\mathrm{pred},k}=(1+\frac{T_d}{T_c})\&CenterDot;i_{L,k}-\frac{T_d}{T_c}\&CenterDot;i_{L,k-1}=1.86i_{L,k}-0.86i_{L,k-1}.$

The contrast test waveform that the inventive method is compared with the conventional linear forecast Control Algorithm with traditional digital control method (without prediction) respectively as shown in Figure 6 and Figure 7, wherein thick line means to adopt the step response waveform of the inventive method, and fine rule means traditional digital control method or conventional linear forecast Control Algorithm.

In Fig. 6,7, horizontal ordinate means the time, and unit is s, and ordinate means the electric current of magnetic bearing radial passage, and unit is A.From Fig. 6, can obtain, with traditional digital control method (without prediction), compare, the inventive method does not almost have overshoot, 12% the overshoot and traditional digital control method is had an appointment, while adopting the inventive method, the rise time of system roughly remains unchanged simultaneously, so system bandwidth remains unchanged substantially.As can be seen from Figure 7, reach stable state after the electric current jerk value of (t>0.2s) the inventive method little and winding current waveform is Paint Gloss than the conventional linear forecast Control Algorithm.Therefore, with traditional linear prediction control method, compare, the inventive method has higher current tracking precision and less current noise.

The content be not described in detail in instructions of the present invention belongs to the known prior art of professional and technical personnel in the field.

Claims (1)

1. a method that compensates magnetic bearing switch power amplifier digital control delay, it is characterized in that: on the basis of measuring switch power amplification system digital control delay and magnetic bearing winding equivalent inductance, resistance, according to the linear prediction control theory, the dissymmetry factor of asymmetric sampling resistor network is set, finally according to dissymmetry factor, configure asymmetric sampling resistor network, comprise the following steps:

(1) the digital control delay T of measuring switch power amplification system _d

T _d＝T _ad+T _cal+T _awa+0.5T _c

Wherein, T _AdMean power amplification system AD sampling time delay, T _calMean computation delay, T _awaMean that PWM waits for time delay, T _cThe servo period of expression system; For AD sampling time delay T _Ad, its big or small delay time with each passage is multiplied by the sampling channel number and can obtains, its transport function G _Ad(s) can be expressed as:

$G_{\mathrm{ad}}\left(s\right)=e^{-s\&CenterDot;T_{\mathrm{ad}}}$

Wherein s means the variable of the mathematical model-transport function of control system in complex field; In like manner, for same calculated amount, the computation delay T of DSP _calThe processing speed that depends on DSP, its whole computation delay time can draw by the mode of measuring, and then obtains its transport function G _cal(s) be:

$G_{\mathrm{cal}}\left(s\right)=e^{-s\&CenterDot;T_{\mathrm{cal}}};$

For the digital control system of fixed switching frequency, because the dutycycle d calculated can only could upgrade in the beginning in each PWM cycle, it also inevitably brings a time delay, and PWM waits for time delay T _awa, this time delay can be expressed as:

$T_{\mathrm{awa}}=\mathrm{rem}\left(\frac{T_c-T_{\mathrm{cal}}}{T_p}\right)$

T wherein _pThe switch periods that means power amplification system, symbol " rem " means except rear complementation, waits for time delay transport function G _awa(s) expression formula is:

$G_{\mathrm{awa}}\left(s\right)=e^{-s\&CenterDot;T_{\mathrm{awa}}};$

(2) measure respectively magnetic bearing winding equivalent resistance R and inductance L with multimeter and electric inductance measuring-testing instrument;

(3) scope, current sample that considers the magnetic bearing winding current determined the big or small R of current sampling resistor by the signal to noise ratio (S/N ratio) of the sample range of AD chip and sampled signal _mWith the ratio k of current sample with divider resistance _Ab

(4) according to the linear prediction control theory, for compensating digits controls time delay T _d, PREDICTIVE CONTROL expression formula G _Pred(z) be:

$G_{\mathrm{pred}}\left(z\right)=1+\frac{T_d}{T_c}-\frac{T_d}{T_c}{z}^{-1}$

Wherein z means the variable of the mathematical model-transport function of discrete control system, G _Pred(z) corresponding differential expressions i _{Pred, k}Can be expressed as:

$i_{\mathrm{pred},k}=(1+\frac{T_d}{T_c})\&CenterDot;i_{L,k}-\frac{T_d}{T_c}\&CenterDot;i_{L,k-1}$

I wherein _L,kMean current i.e. k sample rate current constantly, i _{L, k-1}Mean the i.e. k-1 sample rate current constantly of previous moment;

According to Kirchhoff's current law (KCL), can proper 0≤t<during dT,

$\left\{\begin{array}{c}{i}_m\left(t\right)=i_L\left(t\right)+i_{\mathrm{cd}}\left(t\right)\\ U_d\left(t\right)=R_m{i}_m\left(t\right)+{\mathrm{Ri}}_L\left(t\right)+L\frac{{\mathrm{di}}_L\left(t\right)}{\mathrm{dt}}\\ (R_c+R_d)i_{\mathrm{cd}}\left(t\right)={\mathrm{Ri}}_L\left(t\right)+L\frac{{\mathrm{di}}_L\left(t\right)}{\mathrm{dt}}\\ u_s\left(t\right)={\mathrm{kk}}_{\mathrm{ab}}{U}_d-{\mathrm{kk}}_{\mathrm{cd}}({\mathrm{Ri}}_L\left(t\right)+L\frac{{\mathrm{di}}_L\left(t\right)}{\mathrm{dt}})\end{array}\right.0\&le;t<\mathrm{dT}---\left(9\right)$

Wherein,

The gain that k is operational amplifier, definition status variable x=i _L, input variable u=U _dWith output variable y=u _sCorrespondingly, t state variable, input variable and output variable constantly is designated as respectively x (t)=i arbitrarily _L(t), u (t)=U _dAnd y (t)=u (t) _s(t), so the state equation that can obtain system is:

$\left\{\begin{array}{c}\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=-\frac{a}{b}x\left(t\right)+\frac{1}{b}u\left(t\right)\\ y\left(t\right)=(\frac{{\mathrm{kk}}_{\mathrm{cd}}{d}_0}{c}-{\mathrm{kk}}_{\mathrm{cd}}R)x\left(t\right)+({\mathrm{kk}}_{\mathrm{ab}}-\frac{{\mathrm{kk}}_{\mathrm{cd}}}{c})u\left(t\right)\end{array}\right.---\left(10\right)$

Wherein $R_m+\frac{{\mathrm{RR}}_m}{R_c+R_d}+R,$ $b=L(1+\frac{R_m}{R_c+R_d}),$ $c=1+\frac{R_m}{R_c+R_d}$ And $d_0=R_m+R+\frac{{\mathrm{RR}}_m}{R_c+R_d},$

Consider, in real system, R is arranged _c+ R _d>>R _mAnd R _c+ R _d>>RR _m, so a ≈ R _m+ R, b ≈ L, c ≈ 1 and d ₀≈ R _m+ R, can obtain its substitution (10),

$\left\{\begin{array}{c}\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=-\frac{R+R_m}{L}x\left(t\right)+\frac{1}{L}u\left(t\right)\\ y\left(t\right)={\mathrm{kk}}_{\mathrm{cd}}{R}_{m}x\left(t\right)+k(k_{\mathrm{ab}}-k_{\mathrm{cd}})u\left(t\right)\end{array}\right.---\left(11\right)$

In like manner utilize Kirchhoff's current law (KCL), when dT≤t<T, can obtain,

$\left\{\begin{array}{c}{i}_m\left(t\right)=i_L\left(t\right)+i_{\mathrm{cd}}\left(t\right)\\ 0=R_m{i}_m\left(t\right)+{\mathrm{Ri}}_L\left(t\right)+L\frac{{\mathrm{di}}_L\left(t\right)}{\mathrm{dt}}\\ (R_c+R_d)i_{\mathrm{cd}}\left(t\right)=-R_m{i}_m\left(t\right)\\ u_s\left(t\right)=-{\mathrm{kR}}_c{i}_{\mathrm{cd}}\left(t\right)\end{array}\right.\mathrm{dT}\&le;t<T---\left(12\right)$

Wherein, i _Cd(t) mean that any t flows through divider resistance R constantly _cAnd R _dElectric current, i _m(t) mean that any t flows through sampling resistor R constantly _mElectric current; The state equation that similarly, can obtain system is:

$\left\{\begin{array}{c}\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=-\frac{R}{L}x\left(t\right)-\frac{R_m(R_c+R_d)}{L(R_c+R_d+R_m)}x\left(t\right)\\ y\left(t\right)=\frac{{\mathrm{kR}}_c{R}_m}{R_c+R_d+R_m}x\left(t\right)\end{array}\right.---\left(13\right)$

Consider R in real system _c+ R _d>>R _m, so above formula can further be reduced to:

$\left\{\begin{array}{c}\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=-\frac{R+R_m}{L}x\left(t\right)\\ y\left(t\right)={\mathrm{kk}}_{\mathrm{cd}}{R}_{m}x\left(t\right)\end{array}\right.---\left(14\right)$

According to State-space Averaging Principle, (11) * d+ (14) * (1-d) can obtain at an average state space equation of PWM cycle, be:

$\left\{\begin{array}{c}\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=-\frac{R+R_m}{L}x\left(t\right)+\frac{d}{L}u\left(t\right)\\ y\left(t\right)={\mathrm{kk}}_{\mathrm{cd}}{R}_{m}x\left(t\right)+\mathrm{dk\&gamma;u}\left(t\right)\end{array}\right.---\left(15\right)$

Wherein, γ=k _Ab-k _Cd, equation (15) be one non-linear continuous time equation, can obtain its linear model by the small-signal disturbance, order

With

Wherein "～" represents small-signal, correspondingly, and x

With Mean respectively the small-signal variable of any t state variable, input variable and output variable constantly, be respectively so can obtain steady-state equation and the condition of small signal equation of system:

$\left\{\begin{array}{c}\frac{\mathrm{dX}}{\mathrm{dt}}=-\frac{R+R_m}{L}X+\frac{D}{L}U\\ Y={\mathrm{kk}}_{\mathrm{cd}}{R}_{m}X+\mathrm{Dk\&gamma;U}\end{array}\right.---\left(16\right)$

$\left\{\begin{array}{c}\frac{d\tilde{x}\left(t\right)}{\mathrm{dt}}=-\frac{R+R_m}{L}\tilde{x}\left(t\right)+\frac{D}{L}\tilde{u}\left(t\right)+\frac{U}{L}\tilde{d}\left(t\right)\\ \tilde{y}\left(t\right)={\mathrm{kk}}_{\mathrm{cd}}{R}_m\tilde{x}\left(t\right)+\mathrm{Dk\&gamma;}\tilde{u}\left(t\right)+\mathrm{k\&gamma;U}\tilde{d}\left(t\right)\end{array}\right.---\left(17\right)$

Like this, according to the Laplace conversion, the small-signal transport function that can obtain system is:

$G_i\left(s\right)=\frac{\tilde{x}\left(s\right)}{\tilde{u}\left(s\right)}=\frac{{\tilde{i}}_L\left(s\right)}{{\tilde{u}}_d\left(s\right)}=\frac{D}{R+R_m+\mathrm{Ls}}---\left(18\right)$

$G_y\left(s\right)=\frac{\tilde{y}\left(s\right)}{\tilde{u}\left(s\right)}=\frac{{\tilde{u}}_s\left(s\right)}{{\tilde{u}}_d\left(s\right)}=\mathrm{Dk}\frac{k_{\mathrm{ab}}{R}_m+\mathrm{\&gamma;R}+\mathrm{\&gamma;Ls}}{R+R_m+\mathrm{Ls}}---\left(19\right)$

Therefore, from

Arrive Transport function can be expressed as:

$G_s\left(s\right)=\frac{{\tilde{u}}_s\left(s\right)}{{\tilde{i}}_L\left(s\right)}=k(k_{\mathrm{ab}}{R}_m+\mathrm{\&gamma;R}+\mathrm{\&gamma;Ls})---\left(20\right)$

Method to above formula application backward difference can draw the following differential equation:

$u_{s,k}=k(k_{\mathrm{ab}}{R}_m+\mathrm{\&gamma;R}+\frac{\mathrm{\&gamma;L}}{T_s})i_{L,k}-\frac{\mathrm{k\&gamma;L}}{T_s}{i}_{L,k-1}$

U wherein _s,kMean the i.e. output valve of k asymmetric sampling resistor network of the moment after operational amplifier of current time;

In conjunction with i _{Pred, k}And u _s,kExpression formula, make u _s,k=i _{Pred, k}, utilize the method for undetermined coefficients can show that the dissymmetry factor γ of asymmetric sampling resistor network and the gain k of operational amplifier are:

$\left\{\begin{array}{c}\mathrm{\&gamma;}=\frac{T_s{T}_d{k}_{\mathrm{ab}}{R}_m}{{\mathrm{LT}}_c-T_s{T}_{d}R}\\ k=\frac{{\mathrm{LT}}_c-T_s{T}_{d}R}{k_{\mathrm{ab}}{R}_m{\mathrm{LT}}_c}\end{array}\right.$

T wherein _sFor the sampling period of power amplification system;

(5) take into account power consumption and the signal to noise ratio (S/N ratio) of sampling resistor network, according to the R of dissymmetry factor γ configuration sampling resistor network _a, R _b, R _cAnd R _d, R wherein _a=R _d,

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