CN103944481A - AC asynchronous motor vector control system model parameter online modifying method - Google Patents

Abstract

The invention discloses an AC asynchronous motor vector control system model parameter online modifying method. The speed of a motor is controlled when the load is constant, and the speed of the motor is kept constant so that influences of rotor magnetic chain coupling terms on parameter dynamic online identification can be eliminated. According to the method, the requirements for high accuracy and instantaneity of parameter dynamic online identification of the motor can be met.

Description

The online modification method of a kind of AC Motor Vector Control System model parameter

Technical field

The present invention relates to variable-frequency control technique field, particularly the online modification method of a kind of AC Motor Vector Control System model parameter.

Background technology

The a lot of links of Vector Control System for Asynchronous Machine based on rotor field-oriented all relate to the use of the parameter of electric machine, as decoupling zero calculating of magnetic flux calculating, torque calculation, magnetic flux angle calculation and d axle excitation component and q shaft torque component etc.Therefore the order of accuarcy of parameter of electric machine identification has determined precision and the effect of vector control.Upgrading real-time and accurately AC Motor Vector Control System model parameter, is the prerequisite that realizes high-performance vector control frequency control technology.

Parameter of electric machine off-line identification method equal comparative maturity in theoretical and application in practice at present, and can reach higher parameter of electric machine identification precision.But the parameter of electric machine can be subject to the impact of the factors such as temperature rise and change in motor operation course, if cannot accomplish parameter real-time update in running, can cause very large impact to vector control effect, therefore in motor operation course, the parameter of electric machine be carried out to on-line identification and be necessary very much.

Current parameter of electric machine on-line identification algorithm is complicated if it were not for realizing, and can take a large amount of cpu resources, and real-time cannot ensure, it is exactly poor stability, affected by the factors such as noise jamming, or will ensure under good given initial condition, could obtain good identification precision.In the middle of numerous parameter of electric machine recognizers, RLS, because amount of calculation is little, does not need large matrix to invert, and calculating and convergence rate are very fast, in engineering reality, are widely adopted.But this algorithm is only successfully applied at present in parameter of electric machine off-line identification, and for on-line parameter identification, owing to cannot eliminating the rotating speed of motor conversion and the impact of magnetic linkage coupling, in the situation that motor mathematical model is not simplified, cannot applies recursive least-squares the parameter of electric machine is carried out to recursive operation.In existing on-line parameter discrimination method, also utilize RLS to carry out on-line identification calculating to the parameter of electric machine, but carry out under the complete prerequisite based on a kind of hypothesis, suppose that motor speed is constant, therefore also only limit to simulation study, cannot apply in practice.

Vector control is the main flow of current industrial system frequency-changing speed-regulating application, it is being analyzed on the Mathematical Modeling basis of AC induction motor, by coordinate transform and rotor field-oriented, realize the full decoupled of stator current torque component and excitation component, obtain the dynamic mathematical models of similar direct current machine, then analog DC motor is controlled, and can obtain good quiet, dynamic property.Its shortcoming is that control precision depends on stator resistance R _s, rotor resistance R _r, stator inductance L _s, inductor rotor L _r, mutual inductance L _metc. the accurate identification of multiple parameters of electric machine, especially the accuracy of detection of magnetic linkage closed-loop control system rotor magnetic flux is subject to rotor time constant τ _r(τ _r=L _r/ R _r) impact larger, due to rotor time constant τ _rit is larger that (being mainly rotor resistance) affected by the factors such as temperature, in motor whole service process, changes and can reach 150%, must detect by real-time online.Therefore the correct identification of the parameter of electric machine is the prerequisite that vector control system is realized the speed governing of high-performance motor.The method that the tradition off-line parameter of electric machine detects has unloaded and stall to test, but is often subject to the restriction of field condition and cannot implements.Some intelligent control algorithms have good precision for the offline inspection of the parameter of electric machine as particle cluster algorithm, ant group algorithm, genetic algorithm, neural net etc., but amount of calculation is very large, and convergence rate is not suitable for parameter slowly in line computation.Least square method of recursion is good to sensitivity to parameter, calculates and fast convergence rate, can synchronously disposablely pick out whole parameters, and highly versatile.But most research is in order to simplify computational process, is on actionless basis by Mathematical Models at motor, therefore can only be used for the static identification of parameter of electric machine off-line.Therefore how further to simplify the computation model of the dynamic on-line identification of the parameter of electric machine, improve computational speed and precision and enter practical application for accelerating the online detection algorithm of the parameter of electric machine, the performance that improves motor vector control is most important.

Summary of the invention

Technical problem to be solved by this invention is, for prior art deficiency, provides a kind of AC Motor Vector Control System model parameter online modification method, meets real-time and the pinpoint accuracy requirement of parameter of electric machine on-line identification.

For solving the problems of the technologies described above, the technical solution adopted in the present invention is: the online modification method of a kind of AC Motor Vector Control System model parameter, comprises the following steps:

1) obtain motor initial parameter accurately by the static discrimination method of the parameter of electric machine: stator resistance R _s, rotor resistance R _r, stator inductance L _s, inductor rotor L _rwith mutual inductance L _m(mutual inductance between stator and rotor);

2) parameter using above-mentioned motor initial parameter as vector control system model, for controlling driven by motor load running;

3) in load constant timing, above if rotating speed control fluctuation doubles on original basis, rotor flux is carried out to constant control;

4) in the time that rotor flux is constant, by torque reference electric current given closed loop switch to constant reference torque current , make to export electromagnetic torque and keep constant;

5) control motor speed, in the time that motor speed is constant, motor initial parameter is carried out to on-line identification calculating, obtain the up-to-date parameter of electric machine;

6) utilize the up-to-date parameter of electric machine to upgrade the motor initial parameter that participates in calculating in vector control system model, simultaneously by constant reference torque current switch to torque reference electric current , and cancel the constant control to rotor flux.

Described step 5) in, described step 5) in, to motor three-phase voltage u _a, u _b, u _cwith three-phase current i _a, i _b, i _csample, through coordinate transform, be separately converted to excitation and torque component u under two-phase rest frame _sd, u _sqand i _sd, i _sq, and by u _sd, u _sqand i _sd, i _sqas input, the parameter of electric machine is carried out to recursive least-squares calculating.

Compared with prior art, the beneficial effect that the present invention has is: method of the present invention is carried out in the time that motor load is constant, make it keep the constant impact of rotor flux coupling terms on dynamic state of parameters on-line identification of eliminating at load constant timing controlled motor speed, greatly improved accuracy and the real-time of the dynamic on-line identification of the parameter of electric machine.

Brief description of the drawings

Fig. 1 is one embodiment of the invention AC Motor Vector Control System model framework chart;

Fig. 2 is one embodiment of the invention flux estimator model framework chart;

Fig. 3 is the dynamic on-line identification rotor flux of one embodiment of the invention parameter of electric machine experimental waveform figure;

Fig. 4 is the dynamic on-line identification motor speed of one embodiment of the invention parameter of electric machine experimental waveform figure;

Fig. 5 is the dynamic on-line identification rotor resistance of one embodiment of the invention parameter of electric machine experimental waveform figure.

Embodiment

As shown in Figure 1, the dynamic on-line identification module of one embodiment of the invention parameter of electric machine Real-time Collection voltage, electric current and motor speed signal utilize least square method of recursion to carry out parameter of electric machine calculating, in system, comprise a flux ring, formed the closed loop constant control to rotor flux.

Fig. 2 is flux estimator model framework chart, estimates as long as utilize d shaft voltage and current signal can complete rotor flux.

Fig. 3 is the dynamic on-line identification rotor flux of parameter of electric machine experimental waveform, is R carrying out setting rotor resistance before Dynamic Identification with reference to initial value _s=1.405 Ω, and increase by 0.5 Ω while being allowed to condition at 6.15 seconds, saltus step is to R _s=1.905 Ω, and from this moment 2 paths of the K switch shown in Fig. 1 are cut off to K switch 3 closures, rotor flux is carried out to closed-loop control, make it be tending towards steady state, after dynamic on-line parameter identification finishes, cancel the constant control (disconnecting K3, closed K2) to rotor flux.

Fig. 4 is the dynamic on-line identification motor speed of parameter of electric machine experimental waveform, whole process, motor speed reference value is carried out to twice adjustment, carry to 800r/min from 400r/min for the first time, carry to 1200r/min from 800r/min again for the second time, its actual speed produced fluctuation between 6.15 seconds and 7.2 seconds, because there is larger Identification Errors at this time period rotor resistance, so affected the performance of this time period closed-loop vector control system, in addition during parameter of electric machine identification, owing to will keeping invariablenes turning speed, so by torque reference electric current as shown in Figure 1 switch to artificial given torque reference electric current , thereby rotating speed is within 7.0 seconds～8.0 second time period and have velocity deviation between reference rotation velocity.

Fig. 5 is the dynamic on-line identification rotor resistance of parameter of electric machine experimental waveform, because rotor flux and motor speed between 6.15 to 7.1 seconds are also unstable, therefore larger in this interval rotor resistance identification result fluctuation.But along with rotor flux and speed become constantly, rotor resistance identification result is restrained gradually close to reference value.After to rotor resistance, on-line identification completes, control system model parameter is upgraded, and in cutting speed, ring is controlled, as shown in Figure 4, in the time of motor speed-raising for the second time, because control system model parameter is accurate, velocity deviation and steady state ripple are eliminated completely, and system presents comparatively desirable speed closed loop control performance.

The dynamic on-line identification method of the parameter of electric machine of the present invention specifically comprises following steps:

Step 1: obtain motor initial parameter accurately by the static identification of the parameter of electric machine: stator resistance R _s, rotor resistance R _r, stator inductance L _s, inductor rotor L _rwith mutual inductance L _m;

Step 2: the parameter using the initial parameter of electric machine that obtains as vector control system model, for controlling the operation of motor bringing onto load;

Step 3: in load constant timing, if that rotating speed control fluctuation doubled on original basis was above (as being ± 1% originally, become now ± 2%), illustrate that significant variation has occurred the parameter of electric machine, cause control system model parameter inaccurate, thereby system control performance is declined, now cut rotor flux ψ _rdcontrol module, rotor flux is carried out to constant control, and (2 paths of the K switch shown in Fig. 1 are cut off, and K switch 3 closures, carry out closed-loop control to rotor flux, make it be tending towards steady state, can make ψ by pi regulator _rdapproach , and a given constant, rotor flux is stable);

Step 4: in the time that rotor flux is constant, by torque reference electric current given closed loop switch to artificial given constant reference torque current constant magnetic flux and torque current make to export electromagnetic torque and also keep constant, in load constant timing, it is constant that rotating speed also will keep, but now between rotating speed and set-point, there will be certain deviation, but the time very short (on-line parameter identification computing time once) that this rotating speed deviation exists, little to control system entire effect;

Step 5: in the time that control rotating speed makes it reach constant, starter motor dynamic state of parameters on-line identification module, carries out on-line identification calculating to the parameter of electric machine;

Step 6: after the dynamic on-line identification of the parameter of electric machine finishes, the parameter of electric machine of up-to-date acquisition is participated in to the parameter of electric machine calculating for upgrading the each module of vector control system, gain the given torque reference electric current of closed loop simultaneously , and cancel rotor flux ψ _rdconstant control.

Concrete principle is derived as follows:

The dynamic mathematical models of induction asynchronous machine are shown below:

$\begin{array}{c}\frac{d^2{i}_{\mathrm{sd}}}{{\mathrm{dt}}^2}=-(\mathrm{\ξ}+\frac{1}{{\mathrm{\τ}}_r})\frac{{\mathrm{di}}_{\mathrm{sd}}}{\mathrm{dt}}+\frac{L_m\mathrm{\η}-{\mathrm{\τ}}_r\mathrm{\ξ}}{{\mathrm{\τ}}_r^2}{i}_{\mathrm{sd}}+\frac{1}{\mathrm{\σ}{L}_s}\frac{{\mathrm{du}}_{\mathrm{sd}}}{\mathrm{dt}}+\frac{1}{\mathrm{\σ}{L}_s{\mathrm{\τ}}_r}{u}_{\mathrm{sd}}-\\ {\mathrm{\ω}}_r\frac{{\mathrm{di}}_{\mathrm{sq}}}{\mathrm{dt}}+\mathrm{\ωr}(\frac{\mathrm{\η}{L}_m}{{\mathrm{\τ}}_r}-\mathrm{\ξ})i_{\mathrm{sq}}+\frac{{\mathrm{\ω}}_r}{\mathrm{\σ}{L}_s}{u}_{\mathrm{sq}}+\mathrm{\η}\frac{d{\mathrm{\ω}}_r}{\mathrm{dt}}{\mathrm{\ψ}}_{\mathrm{rq}}\end{array}---\left(1\right)$

Wherein $\mathrm{\σ}=1-\frac{L_m^2}{L_s{L}_r},\mathrm{\η}=\frac{L_m}{\mathrm{\σ}{L}_s{L}_r},\mathrm{\ξ}=\frac{R_s}{\mathrm{\σ}{L}_s}+\frac{L_m^2}{\mathrm{\σ}{L}_s{L}_r{T}_r},$ I _sd, i _sqfor stator current vector d under two-phase rest frame, q axle component, u _sd, u _sqfor stator voltage vector d under two-phase rest frame, q axle component, ψ _rd, ψ _rqfor two-phase rest frame lower rotor part flux linkage vector d, q axle component, τ _rfor rotor time constant, L _s, L _rfor motor stator and rotor inductance, R _s, R _rfor motor stator and rotor resistance parameters, L _mfor motor mutual inductance, σ is motor flux leakage coefficient.

For formula (1), as motor speed ω _rwhen constant and non-vanishing, ω _rfor constant, have can be by formula, last relates to flux linkage calculation item to cancellation, thus the impact that magnetic linkage is coupled on parameter identifier eliminated.But in the time that the parameter of electric machine changes, keep motor speed constant, it is constant that prerequisite must keep load, and it is constant to control rotor flux.According to torque formula:

$T_{\mathrm{em}}=\frac{p{L}_m}{L_r}{i}_{\mathrm{sq}}{\mathrm{\ψ}}_{\mathrm{rd}}---\left(2\right)$

In formula, p is motor number of pole-pairs, as seen as rotor flux ψ _rdwhile keeping constant, electromagnetic torque is with torque current i _sqproportionally change, as long as holding torque current constant just can stable electrical magnetic torque be exported, in the situation that load is constant, motor speed also just can be stablized.In the time that motor speed is stablized, the magnetic linkage item ψ in formula (1) _rqitem just can be eliminated, thereby obtains:

$\begin{array}{c}\frac{d^2{i}_{\mathrm{sd}}}{{\mathrm{dt}}^2}=-(\mathrm{\ξ}+\frac{1}{{\mathrm{\τ}}_r})\frac{{\mathrm{di}}_{\mathrm{sd}}}{\mathrm{dt}}+\frac{L_m\mathrm{\η}-{\mathrm{\τ}}_r\mathrm{\ξ}}{{\mathrm{\τ}}_r^2}{i}_{\mathrm{sd}}+\frac{1}{\mathrm{\σ}{L}_s}\frac{{\mathrm{du}}_{\mathrm{sd}}}{\mathrm{dt}}+\\ \frac{1}{\mathrm{\σ}{L}_s{\mathrm{\τ}}_r}{u}_{\mathrm{sd}}-{\mathrm{\ω}}_r\frac{{\mathrm{di}}_{\mathrm{sq}}}{\mathrm{dt}}+{\mathrm{\ω}}_r(\frac{\mathrm{\η}{L}_m}{{\mathrm{\τ}}_r}-\mathrm{\ξ})i_{\mathrm{sq}}+\frac{{\mathrm{\ω}}_r}{\mathrm{\σ}{L}_s}{u}_{\mathrm{sq}}\end{array}---\left(3\right)$

Stablize rotor flux, first will estimate rotor flux, in rotor field-oriented control system, induction machine electric current and voltage computation model are:

$\left\{\begin{array}{c}{i}_{\mathrm{rd}}=\frac{1}{L_r}({\mathrm{\ψ}}_{\mathrm{rd}}-L_m{i}_{\mathrm{sd}})\\ u_{\mathrm{sd}}=R_s{i}_{\mathrm{sd}}+L_s\frac{{\mathrm{di}}_{\mathrm{sd}}}{\mathrm{dt}}+L_m\frac{{\mathrm{di}}_{\mathrm{rd}}}{\mathrm{dt}}\end{array}\right.---\left(4\right)$

First formula to formula (4) is carried out differential, and in second formula of substitution formula (4):

$\frac{{\mathrm{d\ψ}}_{\mathrm{rd}}}{\mathrm{dt}}=\frac{L_r}{L_m}[u_{\mathrm{sd}}-R_s{i}_{\mathrm{sd}}-(L_s-\frac{L_m^2}{L_r})\frac{{\mathrm{di}}_{\mathrm{sd}}}{\mathrm{dt}}]---\left(5\right)$

Again integration being carried out in formula (5) both sides obtains:

${\mathrm{\ψ}}_{\mathrm{rd}}=\frac{L_r}{L_m}[\∫(u_{\mathrm{sd}}-R_s{i}_{\mathrm{sd}})\mathrm{dt}-\mathrm{\σ}{L}_s{i}_{\mathrm{sd}}]---\left(6\right)$

Press the designed rotor flux computation model of formula (6) as shown in Figure 2.Only need to detect voltage and current signal, and computational process does not relate to this parameter of electric machine of rotor resistance of the easiest temperature influence.

In vector control system, add magnetic flux closed-loop control, for stablizing rotor flux, in the time of stabilization of speed, can utilize least square method of recursion to carry out dynamic state of parameters on-line identification by formula (3).First formula (3) is expressed as:

$\frac{d^2{i}_{\mathrm{sd}}}{{\mathrm{dt}}^2}={\mathrm{\λ}}_1\frac{{\mathrm{di}}_{\mathrm{sd}}}{\mathrm{dt}}+{\mathrm{\λ}}_2{i}_{\mathrm{sd}}+{\mathrm{\λ}}_3\frac{{\mathrm{du}}_{\mathrm{sd}}}{\mathrm{dt}}+{\mathrm{\λ}}_4{u}_{\mathrm{sd}}-{\mathrm{\λ}}_5\frac{{\mathrm{di}}_{\mathrm{sq}}}{\mathrm{dt}}+{\mathrm{\λ}}_6{i}_{\mathrm{sq}}+{\mathrm{\λ}}_7{u}_{\mathrm{sq}}---\left(7\right)$

λ in formula ₁～λ ₇for:

${\mathrm{\λ}}_1=\frac{{\mathrm{\δ}}_1+{\mathrm{\δ}}_2-1}{T_s^2}---\left(8\right)$

${\mathrm{\λ}}_2=\frac{{\mathrm{\δ}}_3+{\mathrm{\δ}}_4}{T_s^2}---\left(9\right)$

${\mathrm{\λ}}_3=\frac{{\mathrm{\δ}}_3}{T_s}---\left(10\right)$

${\mathrm{\λ}}_4=\frac{{\mathrm{\δ}}_1-2}{T_s}---\left(11\right)$

λ ₅＝ω _r (12)

${\mathrm{\λ}}_6={\mathrm{\ω}}_r(\frac{\mathrm{\η}{L}_m}{{\mathrm{\τ}}_r}-\mathrm{\ξ})---\left(13\right)$

${\mathrm{\λ}}_7=\frac{{\mathrm{\ω}}_r}{\mathrm{\σ}{L}_s}---\left(14\right)$

Wherein ω _rfor constant, formula (7) is carried out to discretization and obtains:

$\begin{array}{c}{i}_{\mathrm{sd}(k+2)}=({\mathrm{\λ}}_1{T}_s+2)i_{\mathrm{sd}(k+1)}+({\mathrm{\λ}}_2{T}_s^2-1)i_{\mathrm{sd}\left(k\right)}+{\mathrm{\λ}}_3{T}_s{u}_{\mathrm{sd}(k+1)}+\\ ({\mathrm{\λ}}_4{T}_s^2-{\mathrm{\λ}}_3{T}_s)u_{\mathrm{sd}\left(k\right)}-{\mathrm{\λ}}_5{T}_s{i}_{\mathrm{sq}(k+1)}+({\mathrm{\λ}}_6{T}_s^2+{\mathrm{\λ}}_5{T}_s)i_{\mathrm{sq}\left(k\right)}+{\mathrm{\λ}}_7{T}_s^2{u}_{\mathrm{sq}\left(k\right)}\end{array}---\left(15\right)$

Change formula (15) into least square form:

Y(k)＝Φ(k)θ(k)+δ(k) (16)

Wherein Y (k)=i _{sd (k+2)}, other is:

Φ(k)＝[i _sd(k+1) i _sd(k) u _sd(k+1) u _sd(k) i _sq(k+1) i _sq(k) u _sq(k)] (17)

$\begin{array}{c}\mathrm{\θ}\left(k\right)={\left[\begin{array}{ccccccc}{\mathrm{\λ}}_1{T}_s+2& {\mathrm{\λ}}_2{T}_s^2-1& {\mathrm{\λ}}_3{T}_s& {\mathrm{\λ}}_4{T}_s^2-{\mathrm{\λ}}_3{T}_s& -{\mathrm{\λ}}_5{T}_s& {\mathrm{\λ}}_6{T}_s^2+{\mathrm{\λ}}_5{T}_s& {\mathrm{\λ}}_7{T}_s^2\end{array}\right]}^T\\ ={\left[\begin{array}{ccccccc}{\mathrm{\δ}}_1& {\mathrm{\δ}}_2& {\mathrm{\δ}}_3& {\mathrm{\δ}}_4& {\mathrm{\δ}}_5& {\mathrm{\δ}}_5& {\mathrm{\δ}}_7\end{array}\right]}^T\end{array}---\left(18\right)$

In order to calculate δ ₁～δ ₇seven parameters, input and output need to be carried out 8 times with up-sampling, and carry out least square method of recursion calculating:

$\widehat{\mathrm{\θ}}(n+1)=\widehat{\mathrm{\θ}}\left(n\right)+G_{n+1}[Y(n+1)-{\mathrm{\Φ}}^T(n+1)\widehat{\mathrm{\θ}}\left(n\right)]---\left(19\right)$

$G_{n+1}=\frac{C_n\mathrm{\Φ}(n+1)}{\mathrm{\ϵ}+{\mathrm{\Φ}}^T(n+1)C_n\mathrm{\Φ}(n+1)}---\left(20\right)$

$C_{n+1}=\frac{C_n-G_{n+1}{\mathrm{\Φ}}^T(n+1)C_n}{\mathrm{\ϵ}}---\left(21\right)$

In formula: G _n+1for gain matrix, C _n+1for covariance matrix, ε is forgetting factor (0< η≤1).Its computational process is as follows:

(1) initialization: C ₀=10 ^ki, k gets larger positive integer,

$\widehat{\mathrm{\&theta;}}\left(0\right)={\left[\begin{array}{ccccccc}{10}^{-3}& {10}^{-3}& {10}^{-3}& 1^{-3}& {10}^{-3}& {10}^{-3}& 1^{-3}\end{array}\right]}^T,\mathrm{\&epsiv;}=1.$

(2) according to actual measurement output valve Φ (n+1) and C _nvalue is utilized formula (20) calculated gains matrix G _n+1.

(3) by G _n+1with substitution (19) formula is calculated one group of new parameter

(4) calculate next moment covariance matrix C according to formula (21) _n+1.

(5) forward (2) step to and carry out recursive operation next time.

Claims (2)

1. the online modification method of AC Motor Vector Control System model parameter, is characterized in that, comprises the following steps:

1) obtain motor initial parameter accurately by the static discrimination method of the parameter of electric machine: stator resistance R _s, rotor resistance R _r, stator inductance L _s, inductor rotor L _rwith mutual inductance L _m;

2) parameter using above-mentioned motor initial parameter as vector control system model, for controlling driven by motor load running;

3) in load constant timing, above if rotating speed control fluctuation doubles on original basis, rotor flux is carried out to constant control;

4) in the time that rotor flux is constant, by torque reference electric current given closed loop switch to constant reference torque current , make to export electromagnetic torque and keep constant;

5) control motor speed, in the time that motor speed is constant, motor initial parameter is carried out to on-line identification calculating, obtain the up-to-date parameter of electric machine;

6) utilize the up-to-date parameter of electric machine to upgrade the motor initial parameter that participates in calculating in vector control system model, simultaneously by constant reference torque current switch to torque reference electric current , and cancel the constant control to rotor flux.

2. the online modification method of AC Motor Vector Control System model parameter according to claim 1, is characterized in that described step 5) in, on-line identification is calculated and is referred to motor three-phase voltage u _a, u _b, u _cwith three-phase current i _a, i _b, i _csample, through coordinate transform, be separately converted to excitation and torque component u under two-phase rest frame _sd, u _sqand i _sd, i _sq, and by u _sd, u _sqand i _sd, i _sqas input, the parameter of electric machine is carried out to recursive least-squares calculating.