CN101832849B - Method for controlling soft start of vibrating meter based on three-parameter control - Google Patents

Abstract

The invention relates to a control method for removing the overshoot of a displacement generated by the soft start of a vibrating meter under three-parameter control. Based on the three-parameter control, the method provides an overshoot-correcting arithmetic of an off-line time domain: the method obtains an overshoot theory solution generated by the three-parameter control arithmetic by solving a three-parameter control differential equation in the time domain; and in order to directly remove a transient state component, and by taking trigonometric function and Fourier transform as tools, the method provides a correction method of the overshoot under any input. Compared with the conventional soft start in a windowing way, the method reduces the overshoot, and greatly reduces the on-line calculating amount, thereby improving the operating efficiency of a controller when removing the overshoot of the displacement of the vibrating meter.

Description

Soft start of vibrating control method based on tri-consult volume control

Technical field

The present invention relates to a kind of control method that is used for tri-consult volume control soft start of vibrating, belong to the experimental technique field.

Background technology

The tri-consult volume control algolithm proposed the seventies in 20th century, and mainly in order to compensate when having only displacement signal control in the earthquake simulation shaking table control, frequency band is not wide, can not realize the defective of acceleration, speed control.In the tri-consult volume control, low-frequency range by displacement control, Mid Frequency by speed control, high band by Acceleration Control, thereby reach the purpose of widening the control frequency band.Because advantages such as its control bandwidth, control mode are easy, tri-consult volume now is applied to the shaking table control in fields such as machinery, mining gradually.Tri-consult volume realizes by Fig. 1,2 modes that mainly input signal generates displacement, speed, three signals of acceleration by Fig. 1, then by Fig. 2 with three signal synthesis system driving command signals.The transport function that can generate signal (acceleration, speed, displacement) by Fig. 1 as shown in the formula:

$\left.\begin{array}{c}{e}_a=-\frac{R}{R_a}{G}_w{s}^2{u}_0\\ e_V=\frac{R}{R_a}{G}_w\frac{1}{R_1{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000084707300031.png"\; state="\{\{state\}\}">C</figure-callout>}}_1}{\mathrm{su}}_0\\ e_d=-\frac{R}{R_a}{G}_w\frac{1}{R_1{R}_2{C}_1{C}_2}{u}_0\end{array}\right\}$

In the formula

d _Max, v _Max, a _MaxBe respectively the displacement of shaking table design maximum, speed, acceleration, determine by the maximum function curve according to actual conditions.By adjusting D _w, n _w, R _a, R _V, R _dCan realize being equivalent to the control mode of different input quantities respectively etc. parameter: (1) works as n _w＜ω, D _w＜1 o'clock, input signal was suitable with the acceleration parameter, and drawing-in system control can realize Acceleration Control; (2) when getting

D _w＞1, R _V=R _aThe time input signal suitable with speed, drawing-in system control can realize speed control; (3) work as n _w＞ω, D _w＜1, R _d=R _aThe time input signal suitable with displacement, drawing-in system control can realize displacement control.

When the above-mentioned tri-consult volume of utilization is realized Acceleration Control, if the unloading phase the big amplitude signal of high frequency arranged, displacement over-control significantly can appear, for example middle-size and small-size earthquake simulation shaking table is when carrying out 5Hz sinusoidal acceleration input control, the displacement overshoot reaches 600% unloading phase of shaking table, general amplitude input all can be well beyond shaking table displacement threshold limit value, thereby the collision phenomenon of shaking table stage body and sidewall occurs, can make the breaking-up unloading phase of experiment of experiment test specimen when serious.The common method that solves overshoot at present is: input signal be multiply by one suc as formula x (t)=1-e ^-atWindowed function, to reduce a beginning section pumping signal amount,, be equivalent to add buffering to total system.The deficiency that this method exists is: in (1) 1～2s before entering the steady operation section, pumping signal is by excessive weakening.Tri-consult volume is more is used for experiment control system, can much often test and in several seconds, just finish, generally have only 3～6s such as the shaking-table test that reduced scale is bigger, peak accelerator often just in 1～2s, is difficult to accurately satisfy testing requirements like this; (2) in a single day selected windowed function at system, no matter incoming wave produces much overshoot, and the resize ratio of synchronization is all the same, can't specifically adjust according to actual incoming wave; (3) this mode all will be carried out the evaluation and the signal multiplication computing of exponential function in whole test in line process, has increased on-line calculation to a certain extent.Therefore, the algorithm of efficient, the high-precision correction tri-consult volume of research overshoot is significant to improving the shaking table experimental precision.

Summary of the invention

The present invention proposes a kind of off-line and eliminate the overshoot correction algorithm of tri-consult volume overshoot, particularly a kind of soft start of vibrating control method based on tri-consult volume control, this algorithm is converted to the differential equation with the tri-consult volume transport function, obtain the overshoot time domain analytic solution of tri-consult volume control algolithm under trigonometric function (sine or cosine) input by differential equation, overshoot under then being imported arbitrarily in conjunction with Fourier transform is similar to the time domain analytic solution, thus inverse modified tri-consult volume control algolithm displacement overshoot.Have characteristics such as on-line calculation is little, overshoot correction precision height, the influence of multiple input signals nothing itself, the unloading phase of shaking table, can well guarantee the fidelity of signal, can improve the shaking table experiment greatly, the control accuracy of particularly big reduced scale experiment.

To achieve these goals, the present invention adopts following technological means to realize:

By finding the solution the differential equation of tri-consult volume control algolithm, imported down the displacement overshoot (approximate analysis is separated) that tri-consult volume produces arbitrarily, at the displacement drive signal e of tri-consult volume generation _dDeduct the displacement overshoot on the basis, thereby eliminate the overshoot that produces in the tri-consult volume control algolithm, reach the purpose of soft start, may further comprise the steps:

Step 1, ask for tri-consult volume transport function under the Acceleration Control according to shaking table tri-consult volume design parameter;

Step 2, the tri-consult volume transport function is obtained the tri-consult volume control differential through inverse Laplace transform;

Step 3, find the solution trigonometric function when input tri-consult volume control differential, obtain the displacement overshoot of this moment;

Step 4, with arbitrary function form input by Fourier transform, will import arbitrarily and be transformed to limited trigonometric function input sum;

Step 5, based on step 3, obtain in the step 4 each trigonometric function input displacement overshoot of producing of tri-consult volume control algolithm down, and the displacement overshoot of summation under being imported arbitrarily;

Step 6, in the displacement drive signal that the tri-consult volume control algolithm produces, deduct the displacement overshoot that step 5 is tried to achieve, thereby eliminate the displacement overshoot that tri-consult volume produces at input signal.

Described step 1 is specially: according to Design of vibration table canonical parameter design maximum displacement d _Max, speed v _Max, acceleration a _Max, and relevant tri-consult volume design parameter R, R _d, R _v, R _aTry to achieve tri-consult volume control corresponding displacement down, speed, acceleration signal transport function, its formula is as follows:

$\left.\begin{array}{c}{e}_a=-\frac{R}{R_a}{G}_w{s}^2{u}_0\\ e_V=\frac{R}{R_a}{G}_w\frac{1}{R_1{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000084707300031.png"\; state="\{\{state\}\}">C</figure-callout>}}_1}{\mathrm{su}}_0\\ e_d=-\frac{R}{R_a}{G}_w\frac{1}{R_1{R}_2{C}_1{C}_2}{u}_0\end{array}\right\}$

In the formula

ω ₁, ω ₂Be shaking table maximum function curve respective frequencies lower limit, higher limit, C ₁, C ₂For adopting capacitance, D in the mimic channel _w, n _wBe respectively above frequency and the damping ratio that the tri-consult volume second order passes letter of trying to achieve, R amplifies resistance, R for the tri-consult volume generative circuit _d, R _v, R _aBe respectively tri-consult volume and produce displacement, speed, corresponding resistance, the u of amplifying of acceleration ₀Be the corresponding displacement of input acceleration, s is the laplace transform factor.

Described step 2 is specially: described transport function is obtained the corresponding differential equation of tri-consult volume control algolithm through inverse Laplace transform

In the formula, ω _n=n _w, ξ=D _w, f (t) is a reference-input signal, n _w, D _wBe respectively above frequency and the damping ratio that the tri-consult volume second order passes letter of trying to achieve.

Described step 3 is specially: when establishing the trigonometric function that is input as f (t)=Pcos (ω t+ ψ) (t is the time, and ψ, P are difference trigonometric function phase place and amplitude), obtain the displacement overshoot u that the tri-consult volume control algolithm produces by the differential equation in the solution procedure 2 _c(t), its formula is as follows:

$u_c\left(t\right)=e^{-\mathrm{\&zeta;}{\mathrm{\&omega;}}_{n}t}(A\cos {\mathrm{\&omega;}}_{d}t+B\sin {\mathrm{\&omega;}}_{d}t)$

In the formula: ${\mathrm{\&omega;}}_d={\mathrm{\&omega;}}_n\sqrt{1-{\mathrm{\&zeta;}}^2},\mathrm{ust}=P/{\mathrm{\&omega;}}_n^2.$

$\left.\begin{array}{c}A=-\sin \mathrm{\&psi;C}-\cos \mathrm{\&psi;D}\\ B=\frac{(\mathrm{\&omega;}\sin \mathrm{\&psi;}-\mathrm{\&zeta;}{\mathrm{\&omega;}}_n\cos \mathrm{\&psi;})D-(\mathrm{\&omega;}\cos \mathrm{\&psi;}-\mathrm{\&zeta;}{\mathrm{\&omega;}}_n\sin \mathrm{\&psi;})C}{{\mathrm{\&omega;}}_d}\end{array}\right\}$

$\left.\begin{array}{c}C=\mathrm{ust}\frac{2{\mathrm{\&zeta;\&omega;}/\mathrm{\&omega;}}_n}{{[1-{\left({\mathrm{\&omega;}/\mathrm{\&omega;}}_n\right)}^2]}^2+{\left(2{\mathrm{\&zeta;\&omega;}/\mathrm{\&omega;}}_n\right)}^2}\\ D=\mathrm{ust}\frac{1-{\left({\mathrm{\&omega;}/\mathrm{\&omega;}}_n\right)}^2}{{[1-{\left({\mathrm{\&omega;}/\mathrm{\&omega;}}_n\right)}^2]}^2+{\left(2{\mathrm{\&zeta;\&omega;}/\mathrm{\&omega;}}_n\right)}^2}\end{array}\right\}.$

Described step 4 is specially: the basic representation of Fourier transform is as follows:

$f\left(t\right)=\underset{j=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{c}_j{e}^{\mathrm{ijt}}=\underset{j=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{c}_j(\cos \mathrm{jt}+i\sin \mathrm{jt})$

C in the formula _jBe the Fourier's amplitude under the j point correspondence, t is the time, and i is an imaginary unit;

In cycle, is basic point sue for peace after with π at 2 π, following formula and following formula equivalence:

$f\left(t\right)=\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{p}_j\cos ({\mathrm{\&omega;}}_{j}t+{\mathrm{\&psi;}}_j)$

ω in the formula _i, ψ _i, p _iBe respectively frequency, phase place and the amplitude of j trigonometric function correspondence after the input record Fourier transform.Be Fourier transform with length be that any input of N is launched into n cosine function sum, times N is even number: n=N/2+1, and N is odd number: n=(N-1)/2+1.

Described step 5 is specially: promptly when input f (t) is the acceleration signal of arbitrary function, by Fourier transform it is transformed into limited trigonometric function input sum, then finds the solution and obtain the displacement overshoot u that the tri-consult volume control algolithm produces _Cr(t), its formula is as follows:

$u_{\mathrm{cr}}\left(t\right)=e^{{-\mathrm{\&zeta;\&omega;}}_{n}t}(\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{A}_j\cos {\mathrm{\&omega;}}_{d}t+\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{B}_j\sin {\mathrm{\&omega;}}_{d}t)$

In the formula,

$\left.\begin{array}{c}{A}_i=-\sin {\mathrm{\&psi;}}_i{C}_i-\cos {\mathrm{\&psi;}}_i{D}_i\\ B_i=\frac{({\mathrm{\&omega;}}_i\sin {\mathrm{\&psi;}}_i-\mathrm{\&zeta;}{\mathrm{\&omega;}}_n\cos {\mathrm{\&psi;}}_i)D_i-({\mathrm{\&omega;}}_i\cos {\mathrm{\&psi;}}_i-\mathrm{\&zeta;}{\mathrm{\&omega;}}_n\sin {\mathrm{\&psi;}}_i)C_i}{{\mathrm{\&omega;}}_d}\end{array}\right\}$

$\left.\begin{array}{c}{C}_i={\mathrm{ust}}_i\frac{{2\mathrm{\&zeta;\&omega;}}_i/{\mathrm{\&omega;}}_n}{{[1-{({\mathrm{\&omega;}}_i/{\mathrm{\&omega;}}_n)}^2]}^2+{\left(2{\mathrm{\&zeta;\&omega;}}_i/{\mathrm{\&omega;}}_n\right)}^2}\\ D_i={\mathrm{ust}}_i\frac{1-{({\mathrm{\&omega;}}_i/{\mathrm{\&omega;}}_n)}^2}{{[1-{({\mathrm{\&omega;}}_i/{\mathrm{\&omega;}}_n)}^2]}^2+{(2{\mathrm{\&zeta;\&omega;}}_i/{\mathrm{\&omega;}}_n)}^2}\end{array}\right\}$

ω _i, ψ _i, p _iBe respectively frequency, phase place and the amplitude of i trigonometric function correspondence after the input record Fourier transform.

Compared with prior art, advantage of the present invention is as follows:

1) at trigonometric function input and arbitrary input, directly obtained the analytic solution of overshoot signal, it is higher to revise precision;

2) safety of the present invention adopts calculated off-line, and online have only simple signal to superpose, and reduced on-line calculation greatly, improved the controller treatment effeciency.

Description of drawings

Fig. 1 tri-consult volume signal generation synoptic diagram;

Fig. 2 tri-consult volume signal synthesizes synoptic diagram;

Fig. 3 the inventive method realization flow figure;

The displacement signal that Fig. 4 uses the present invention front and back tri-consult volume control algolithm to produce.

Embodiment

Technical scheme of the present invention in conjunction with relevant drawings, is introduced implementation step of the present invention referring to shown in Figure 3 below in detail:

(1), according to Design of vibration table canonical parameter design maximum displacement d _Max, speed v _Max, acceleration a _Max, and be correlated with in the accompanying drawing 1 tri-consult volume design parameter R, R _d, R _v, R _aTry to achieve tri-consult volume control corresponding displacement down, speed, acceleration signal transport function, its formula is as follows:

$\left.\begin{array}{c}{e}_a=-\frac{R}{R_a}{G}_w{s}^2{u}_0\\ e_V=\frac{R}{R_a}{G}_w\frac{1}{R_1{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000084707300031.png"\; state="\{\{state\}\}">C</figure-callout>}}_1}{\mathrm{su}}_0\\ e_d=-\frac{R}{R_a}{G}_w\frac{1}{R_1{R}_2{C}_1{C}_2}{u}_0\end{array}\right\}$

In the formula

ω ₁, ω ₂Be shaking table maximum function curve respective frequencies lower limit, higher limit, C ₁, C ₂For adopting capacitance, D in the mimic channel _w, n _wBe respectively above frequency and the damping ratio that the tri-consult volume second order passes letter of trying to achieve, R amplifies resistance, R for the tri-consult volume generative circuit _d, R _v, R _aBe respectively tri-consult volume and produce displacement, speed, corresponding resistance, the u of amplifying of acceleration ₀Be the corresponding displacement of input acceleration, s is the laplace transform factor.

(2) the tri-consult volume transport function is obtained the tri-consult volume control differential through inverse Laplace transform: above transport function is obtained the corresponding differential equation of tri-consult volume control algolithm through inverse Laplace transform

In the formula, ω _n=n _w, ξ=D _w, f (t) is a reference-input signal;

Tri-consult volume control differential when (3) finding the solution trigonometric function (sine or cosine) input, obtain the displacement overshoot of this moment: (t is the time to establish the trigonometric function that is input as shape such as f (t)=Pcos (ω t+ ψ), ψ, P are difference trigonometric function phase place and amplitude), can obtain the displacement overshoot u that the tri-consult volume control algolithm produces by finding the solution the above differential equation _c(t), its formula is as follows:

$u_c\left(t\right)=e^{-\mathrm{\&zeta;}{\mathrm{\&omega;}}_{n}t}(A\cos {\mathrm{\&omega;}}_{d}t+B\sin {\mathrm{\&omega;}}_{d}t)$

In the formula:

T is the time,

$\left.\begin{array}{c}A=-\sin \mathrm{\&psi;C}-\cos \mathrm{\&psi;D}\\ B=\frac{(\mathrm{\&omega;}\sin \mathrm{\&psi;}-\mathrm{\&zeta;}{\mathrm{\&omega;}}_n\cos \mathrm{\&psi;})D-(\mathrm{\&omega;}\cos \mathrm{\&psi;}-\mathrm{\&zeta;}{\mathrm{\&omega;}}_n\sin \mathrm{\&psi;})C}{{\mathrm{\&omega;}}_d}\end{array}\right\}$

$\left.\begin{array}{c}C=\mathrm{ust}\frac{2{\mathrm{\&zeta;\&omega;}/\mathrm{\&omega;}}_n}{{[1-{\left({\mathrm{\&omega;}/\mathrm{\&omega;}}_n\right)}^2]}^2+{\left(2{\mathrm{\&zeta;\&omega;}/\mathrm{\&omega;}}_n\right)}^2}\\ D=\mathrm{ust}\frac{1-{\left({\mathrm{\&omega;}/\mathrm{\&omega;}}_n\right)}^2}{{[1-{\left({\mathrm{\&omega;}/\mathrm{\&omega;}}_n\right)}^2]}^2+{\left(2{\mathrm{\&zeta;\&omega;}/\mathrm{\&omega;}}_n\right)}^2}\end{array}\right\}$

(4) Fourier transform is passed through in the input of arbitrary function form, will be imported arbitrarily and be transformed to limited trigonometric function input sum:

$f\left(t\right)=\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{p}_j\cos ({\mathrm{\&omega;}}_{j}t+{\mathrm{\&psi;}}_j)$

ω in the formula _i, ψ _i, p _iBe respectively frequency, phase place and the amplitude of j trigonometric function correspondence after the input record Fourier transform.Be Fourier transform with length be that any input of N (N should enlarge corresponding multiple behind the seismologic record reduced scale) is launched into n (N is even number: n=N/2+1; N is odd number: n=(N-1) /+1) individual cosine function sum;

(5) based on step (3), obtain each trigonometric function input displacement overshoot of tri-consult volume control algolithm generation down in the step (4), and displacement overshoot is down imported in summation arbitrarily: promptly when importing f (t) and be the acceleration signal of arbitrary function, by Fourier transform it is transformed into limited trigonometric function input sum, then finds the solution and obtain the displacement overshoot u that the tri-consult volume control algolithm produces _Cr(t), its formula is as follows:

$u_{\mathrm{cr}}\left(t\right)=e^{{-\mathrm{\&zeta;\&omega;}}_{n}t}(\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{A}_j\cos {\mathrm{\&omega;}}_{d}t+\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{B}_j\sin {\mathrm{\&omega;}}_{d}t)$

In the formula,

$\left.\begin{array}{c}{A}_i=-\sin {\mathrm{\&psi;}}_i{C}_i-\cos {\mathrm{\&psi;}}_i{D}_i\\ B_i=\frac{({\mathrm{\&omega;}}_i\sin {\mathrm{\&psi;}}_i-\mathrm{\&zeta;}{\mathrm{\&omega;}}_n\cos {\mathrm{\&psi;}}_i)D_i-({\mathrm{\&omega;}}_i\cos {\mathrm{\&psi;}}_i-\mathrm{\&zeta;}{\mathrm{\&omega;}}_n\sin {\mathrm{\&psi;}}_i)C_i}{{\mathrm{\&omega;}}_d}\end{array}\right\}$

$\left.\begin{array}{c}{C}_i={\mathrm{ust}}_i\frac{{2\mathrm{\&zeta;\&omega;}}_i/{\mathrm{\&omega;}}_n}{{[1-{({\mathrm{\&omega;}}_i/{\mathrm{\&omega;}}_n)}^2]}^2+{\left(2{\mathrm{\&zeta;\&omega;}}_i/{\mathrm{\&omega;}}_n\right)}^2}\\ D_i={\mathrm{ust}}_i\frac{1-{({\mathrm{\&omega;}}_i/{\mathrm{\&omega;}}_n)}^2}{{[1-{({\mathrm{\&omega;}}_i/{\mathrm{\&omega;}}_n)}^2]}^2+{(2{\mathrm{\&zeta;\&omega;}}_i/{\mathrm{\&omega;}}_n)}^2}\end{array}\right\}$

ω _i, ψ _i, p _iBe respectively frequency, phase place and the amplitude of i trigonometric function correspondence after the input record Fourier transform;

(6) the displacement drive signal e that produces through the tri-consult volume control algolithm at input signal _dIn deduct the displacement overshoot u that step (5) is tried to achieve _CrThereby, eliminate the displacement overshoot that tri-consult volume produces; Promptly three signals that produce in the step (1) are superposeed, and deduct u by accompanying drawing 2 _CrAfter can get shaking table feedforward drive signal u ₀, its formula is as follows:

u ₀＝A _d(e _d-u _cr)+A _ve _v+A _ae _a

A in the formula _d, A _v, A _aBe respectively tri-consult volume control bottom offset, speed, feed forward of acceleration gain.

Design sketch of the present invention such as Fig. 4, adopt this method as can be seen after the overshoot of tri-consult volume control algolithm bottom offset obtained obvious improvement, when guaranteeing the shaking table safe operation, improved control accuracy.

It should be noted that at last: above summary of the invention only in order to the explanation the present invention, and and unrestricted technical scheme described in the invention; The present invention is not the control that only is applicable to shaking table, and the solution of the relevant overshoot problem that the tri-consult volume control algolithm relates to all should be encompassed in the middle of the claim scope of the present invention.

Claims (1)

1. soft start of vibrating control method based on tri-consult volume control is characterized in that: may further comprise the steps:

Step 1, ask for tri-consult volume transport function under the Acceleration Control, be specially: according to Design of vibration table canonical parameter design maximum displacement d according to shaking table tri-consult volume design parameter _Max, speed v _Max, acceleration a _Max, and relevant tri-consult volume design parameter R, R _d, R _v, R _aTry to achieve tri-consult volume control corresponding displacement down, speed, acceleration signal transport function, its formula is as follows:

$\left.\begin{array}{c}{e}_a=-\frac{R}{R_a}{G}_w{s}^2{u}_0\\ e_V=\frac{R}{R_a}{G}_w\frac{1}{R_1{C}_1}{\mathrm{su}}_0\\ e_d=-\frac{R}{R_a}{G}_w\frac{1}{R_1{R}_2{C}_1{C}_2}{u}_0\end{array}\right\}$

In the formula $G_w=\frac{1}{s^2+{2D}_w{n}_{w}s+n_w^2},$ $n_w^2=\frac{R}{R_d}\frac{1}{R_1{R}_2{C}_1{C}_2},$ $D_w=\frac{R}{R_V}\frac{1}{R_1{C}_1}\frac{1}{{2n}_w},$ $R_2=\frac{1}{{\mathrm{\&omega;}}_1{C}_2},$ $R_1=\frac{1}{{\mathrm{\&omega;}}_2{C}_1},$

ω ₁, ω ₂Be shaking table maximum function curve respective frequencies lower limit, higher limit, C ₁, C ₂For adopting capacitance, D in the mimic channel _w, n _wBe respectively above frequency and the damping ratio that the tri-consult volume second order passes letter of trying to achieve, R amplifies resistance, R for the tri-consult volume generative circuit _d, R _v, R _aBe respectively tri-consult volume and produce displacement, speed, corresponding resistance, the u of amplifying of acceleration ₀Be the corresponding displacement of input acceleration, s is the laplace transform factor;

Step 2, the tri-consult volume transport function is obtained the tri-consult volume control differential through inverse Laplace transform, be specially: described transport function is obtained the corresponding differential equation of tri-consult volume control algolithm through inverse Laplace transform

In the formula, ω _n=n _w, ζ=D _w, f (t) is a reference-input signal, n _w, D _wBe respectively above frequency and the damping ratio that the tri-consult volume second order passes letter of trying to achieve;

Step 3, find the solution trigonometric function when input tri-consult volume control differential, obtain the displacement overshoot of this moment, be specially: when establishing the trigonometric function that is input as f (t)=Pcos (ω t+ ψ), wherein t is the time, ψ, P are difference trigonometric function phase place and amplitude, obtain the displacement overshoot u that the tri-consult volume control algolithm produces by the differential equation in the solution procedure 2 _c(t), its formula is as follows:

$u_c\left(t\right)=e^{-\mathrm{\&zeta;}{\mathrm{\&omega;}}_{n}t}(A\cos {\mathrm{\&omega;}}_{d}t+B\sin {\mathrm{\&omega;}}_{d}t)$

In the formula: ${\mathrm{\&omega;}}_d={\mathrm{\&omega;}}_n\sqrt{1-{\mathrm{\&zeta;}}^2},\mathrm{ust}=P/{\mathrm{\&omega;}}_n^2,$

$\left.\begin{array}{c}A=-\sin \mathrm{\&psi;C}-\cos \mathrm{\&psi;D}\\ B=\frac{(\mathrm{\&omega;}\sin \mathrm{\&psi;}-\mathrm{\&zeta;}{\mathrm{\&omega;}}_n\cos \mathrm{\&psi;})D-(\mathrm{\&omega;}\cos \mathrm{\&psi;}-\mathrm{\&zeta;}{\mathrm{\&omega;}}_n\sin \mathrm{\&psi;})C}{{\mathrm{\&omega;}}_d}\end{array}\right\}$

$\left.\begin{array}{c}C=\mathrm{ust}\frac{2\mathrm{\&zeta;\&omega;}/{\mathrm{\&omega;}}_n}{{[1-{(\mathrm{\&omega;}/{\mathrm{\&omega;}}_n)}^2]}^2+{(2\mathrm{\&zeta;\&omega;}/{\mathrm{\&omega;}}_n)}^2}\\ D=\mathrm{ust}\frac{1-{(\mathrm{\&omega;}/{\mathrm{\&omega;}}_n)}^2}{{[1-{(\mathrm{\&omega;}/{\mathrm{\&omega;}}_n)}^2]}^2+{(2\mathrm{\&zeta;\&omega;}/{\mathrm{\&omega;}}_n)}^2}\end{array}\right\};$

Step 4, with arbitrary function form input by Fourier transform, will import arbitrarily and be transformed to limited trigonometric function input sum, be specially: the basic representation of Fourier transform is as follows:

$f\left(t\right)=\underset{j=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{c}_j{e}^{\mathrm{ijt}}=\underset{j=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{c}_j(c\cos \mathrm{jt}+i\sin \mathrm{jt})$

C in the formula _jBe the Fourier's amplitude under the j point correspondence, t is the time, and i is an imaginary unit;

In cycle, is basic point sue for peace after with π at 2 π, following formula and following formula equivalence:

$f\left(t\right)=\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{p}_j\cos ({\mathrm{\&omega;}}_{j}t+{\mathrm{\&psi;}}_j)$

ω in the formula _j, ψ _j, p _jBe respectively frequency, phase place and the amplitude of j trigonometric function correspondence after the input record Fourier transform, be Fourier transform with length be that any input of N is launched into n cosine function sum, times N is even number: n=N/2+1, and N is odd number: n=(N-1)/2+1;

Step 5, based on step 3, obtain each trigonometric function input displacement overshoot of tri-consult volume control algolithm generation down in the step 4, and the displacement overshoot of summation under being imported arbitrarily, be specially: promptly when input f (t) is the acceleration signal of arbitrary function, by Fourier transform it is transformed into limited trigonometric function input sum, then finds the solution and obtain the displacement overshoot u that the tri-consult volume control algolithm produces _Cr(t), its formula is as follows:

$u_{\mathrm{cr}}\left(t\right)=e^{-{\mathrm{\&zeta;\&omega;}}_{n}t}(\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{A}_j\cos {\mathrm{\&omega;}}_{d}t+\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{B}_j\sin {\mathrm{\&omega;}}_{d}t)$

In the formula, $\left.\begin{array}{c}{A}_j=-\sin {\mathrm{\&psi;}}_j{C}_j-\cos {\mathrm{\&psi;}}_j{D}_j\\ B_j=\frac{({\mathrm{\&omega;}}_j\sin {\mathrm{\&psi;}}_j-{\mathrm{\&zeta;\&omega;}}_n\cos {\mathrm{\&psi;}}_j)D_j-({\mathrm{\&omega;}}_j{\cos \mathrm{\&psi;}}_j-{\mathrm{\&zeta;\&omega;}}_n\sin {\mathrm{\&psi;}}_j)C_j}{{\mathrm{\&omega;}}_d}\end{array}\right\}$

$\left.\begin{array}{c}{C}_j=\mathrm{us}{t}_j\frac{2\mathrm{\&zeta;}{\mathrm{\&omega;}}_j/{\mathrm{\&omega;}}_n}{{[1-{({\mathrm{\&omega;}}_j/{\mathrm{\&omega;}}_n)}^2]}^2+{(2\mathrm{\&zeta;}{\mathrm{\&omega;}}_j/{\mathrm{\&omega;}}_n)}^2}\\ D_j=\mathrm{us}{t}_j\frac{1-{({\mathrm{\&omega;}}_j/{\mathrm{\&omega;}}_n)}^2}{{[1-{({\mathrm{\&omega;}}_j/{\mathrm{\&omega;}}_n)}^2]}^2+{(2\mathrm{\&zeta;}{\mathrm{\&omega;}}_j/{\mathrm{\&omega;}}_n)}^2}\end{array}\right\}$

${\mathrm{ust}}_j=p_j/{\mathrm{\&omega;}}_n^2;$

Step 6, in the displacement drive signal that the tri-consult volume control algolithm produces, deduct the displacement overshoot that step 5 is tried to achieve, thereby eliminate the displacement overshoot that tri-consult volume produces at input signal, three signals stacks that are about to produce in the step 1, and deduct u _CrAfter can get shaking table feedforward drive signal u ₀, its formula is as follows:

u ₀＝A _d(e _d-u _cr)+A _ve _v+A _ae _a

A in the formula _d, A _v, A _aBe respectively tri-consult volume control bottom offset, speed, feed forward of acceleration gain.

Images