SU1659981A2 - Device for optimal control of inertial objects with attached elastic members - Google Patents

Abstract

The invention relates to a technique for controlling the movement of dynamic objects, namely objects with attached elastic elements. The aim of the invention is to improve the accuracy and stability of the loading of the elastic element by using the terminal loading law. For this purpose, a device comprising an elastic element, a drive, a drive control unit, a deviation sensor, a differentiator, an acceleration measurement unit, a coordinate conversion unit, a phase trajectory coordinate generation unit, a control parameter selection unit, a key and a trigger unit are additionally equipped with a module former, a quadrant, two multipliers, an adder, two dividers, a constant voltage source, a null indicator and a key. 1 hp ff, 2 ill.

Description

(L

WITH

The invention relates to a technique for controlling the movement of dynamic objects, namely objects with attached elastic elements.

The purpose of the invention is to improve the accuracy and stability of the loading of the elastic element by using the terminal law loaded.

FIG. 1 shows a functional diagram of the proposed device; in fig. 2 -dynamic processes occurring during operation of the device.

The circuit contains an elastic structure element 1, a drive (load source) 2, a drive control unit 3, a deviation sensor 4, a differentiator 5, an acceleration measurement unit 6, a coordinate conversion unit 7, a phase trajectory coordinate generation unit 8, a control parameter selection unit 9, key 10, starting block 11, shaper 12 module, quad 1-3, first and second factors 14, 15, adder 16, first and second dividers 17. 18, constant voltage source 19, zero-indicator 20, key 21. Here, the drive control unit 3 consists of the integrator 22, ervogo and second keys 23, 24 and the actuator body 25.

The invention is based on the following. It is known, for example, that a flat angular motion of a rigid body with an attached weakly damped elastic element in the first approximation can be described by the following system of differential equations:

3iJ. (T);

.P,

where I is the moment of inertia of the object relative to the axis of rotation (dimension kg / cm2);

y (t) is the angle of rotation of the object;

y (t) is the angular acceleration, s

Os

cl

NQ

Yu

00

Yu

qi (t) are the generalized coordinates of oscillators with the dimension m (respectively, q i (t), m / s2), which characterize the dynamics of the main modes of oscillations of an elastic element;

gi are the coefficients of the mutual relationship between the solid and the corresponding vibration modes of the elastic element, kgf;

My (t) is the control moment (impact), kNm;

mi is the reduced mass of the main forms of oscillations of the elastic element. kgf / m; yy - the natural frequencies of the main forms of oscillations of the elastic element, p.

Allowing system (1) for y. obtainable

1 1 About

3-5:

Enter notation:, h

Mu (t)

.., no .-% a- /; e-T T

obtainable

eleven

W + W

where v (t) is the control acceleration, oa is the coefficient, m s. Substituting expression (3) into the equation of system (1), we get

) H “o; "; W-OrO ь

-

In view of the meaning

hi;

2 2, b V b --- r M

will get

W-wHlW + b- ltl-ZleCi) ,,

J VH (5)

The inertia of the control object or, which is the same, the inertia of the change in acceleration, communicated to the control object by the actuators (drive), is described by the expression

v-aUv (6)

where, 1 control parameter;

and ° 1 max. Islgn (L) - v. (t)) is a parameter characterizing the limited rate of change of the control acceleration;

U is the required steady-state value of the control acceleration.

The state of any of the oscillators of the system (5) with the minimum total energy (potential and kinetic) corresponding to the required steady-state value of the control acceleration U is determined by a point (/ g, 0) on the phase plane (qi, qi / od), where / 4 b, U / o /.

The task of optimal time control of the rate of change of the loading action is formulated as the need for optimal time transfer of the object (1) from the state q (0) {qi (0), i 1, n} T q0 to the state q (ti) {m 1 , n} T, q (ti) 0, nv (ti) U, using the limited control 6 0, 1.

The known device provides control of an inertial object with a simultaneous loaded attached elastic element along the main low-frequency form of oscillations (i 1) with a dynamic factor equal to unity.

/

(transfer to a point of dynamic equilibrium). In this case, the law of optimal control in the form of feedback, implemented by this device, can be represented as

(Q, .Q,

| (xi ,, W, X, + 4 jVsin / A1; 5u (xl | x2l a,) + s; n / A J. / 5U “4 -“ r / i / Хг./Ыо

sdgssoe

./ Х ver./ЦГ1 ok b

gh ", (b (Mo

hN) h) h

(7)

No - the required precision of the control acceleration U,

This law provides high-quality control not only for the basic low-frequency vibration of an elastic element, but also for the high-frequency components of the vibration spectrum. This is due to the fact that in the case of a linear law of variation of a loading effect with a loading process duration longer than the oscillation period of the oscillator, the energy of the residual oscillations of the loaded oscillator does not exceed 20% of the energy level with a relay (intermittent) change in the loading effect. However, in the case when the linear variation intervals of the loading effect turn out to be significantly shorter than the period of high-frequency oscillations, which can occur when gliding modes occur in the control system, occurrence of phenomena close to resonance may occur. The control law (7) allows the occurrence of sliding modes at the boundary of the switch line Gi (-) 0 in the presence of coordinate-parametric perturbations. In particular, this is possible if the real value of the product of the parameters a and bi turns out to be less than the calculated one, or when the frequency wi of the controlled low-frequency mode of oscillation turns out to be higher than the calculated one. In these cases, the error in the accuracy of the oscillation of the oscillator in the vicinity of the point of dynamic equilibrium may be less noticeable than the danger of the occurrence and consequence of high-frequency resonance.

In addition, a known device under certain conditions (when the actual acceleration is higher than the calculated or, what is the same, parameters a and bi will be greater, and U1 is smaller than the calculated one) may not provide the required accuracy in working out the loading program (error as v, so and qi) ..

These drawbacks can be avoided by using terminal (or finite) loading at the final stage of controlling the law. Such laws make it possible not only to strictly fulfill the specified boundary conditions, but also to ensure the required change in time of the control acceleration. In the general case, the problem of finite control of a linear stationary dynamic object is expressed as

X AH + We (8)

where X is the phase vector:

and is the vector of control parameters;

A, B - the object and control matrices. It is required to transfer the system (8) from some current state X (t) to the state Xm 0 for a limited time a T -1. where T is some fixed point in time. Office IF (t.). satisfies this requirement, is called finite and is

, (t) dt-e- «x (t), u

where eAG is the exponent matrix (the fundamental matrix).

I

The nature of the change in time of the vector if (t) can be imposed additional restrictions in the form

IFM-Sct. (YU)

where С {Cjl, j 1, m, I 1, k} is the coefficient matrix;

a col (l.a.o2o1 1) -vector

the so-called timer parameters.

In particular, when k 1 (t) const, and when k 2, it changes linearly from

Mt) C to IF (T) C i. Then,

if the boundaries of change are given at the ends of the interval t. T, then the coefficient matrix C corresponding to this can be determined. Substituting relation (10) into expression (9) and solving relatively unknown a and if. receive feedback control:

iF (X. Xt) F (X, XT); (X, XT).

where the second relation characterizes the predicted completion time of the control process.

In this case, the object is described by system (8) with

- “K“ A4 °.; L.Г01

s

(C)

where .XaW- tojfU ...

At the same time, g11, / ,, 1 i

Ai Soyil | -4fMSinCO, t |

p- I - - - - - - | - - - -. - I

W, 5iinCO, t) CoSCO.t j

The solution of this problem for the case of k 1 is known. Finite control is written as.

and ,, .. (12)

For case k 2, it is impossible to obtain a solution in closed form. However, such a solution is known for the case of control of a double integrato 2 (13)

The solution obtained from the case of k 1 for a double integrator differs only in the gain factor (1/2 replaced by 2/2), and the structure remains the same, i.e. we can assume that the finite control of the oscillator with k 2 also has the structure of formula (12). but with a modified composition of gain factors, i.e.

V)

and

f

,,

(14)

where a is a certain parameter, for example, given from considerations of continuity of the law of loading.

In addition, when crossing the phase point of the switch line G2 in the case of v (to) 4} 0, the loading effect, vyxe reaches U / 2, and the coordinates of the point C of the intersection of the Ga line are determined by

 ab,, with IT abt 2cog so5 la} coz

Suppose also that the drive efficiency (parameter a) is designed for a specific oscillation frequency w and maximum loading effect U so that the loading process of an elastic element with a dynamic factor equal to one is carried out strictly during the oscillation period T 2 / a). . Then a

U (0/2 l, and C: bi U bi U

Jtw

 .- Significantly

and

can be found from 2

o-.w-f.

X

M

Parameter setting

P-.-TtST.

those. we have about 3 / 2-4 / l. 1.1. Such a value of the coefficient a allows continuous (without control jumps) loading of the elastic element. Control jumps are possible only in the presence of parametric disturbances, but with a 30% spread, for example, in frequency, the magnitude of the jump will be in the order of 5% of the maximum value of the loading effect U. However, law (14) ensures that the boundary conditions are strictly fulfilled while simultaneously varying linearly. the law of magnitude of the loading effect, i.e., if we substitute (14) in (8) with allowance for (11), we have xi ha

, x; + - Ј - (, - t-about,) „-3

3

g

g

x; x, +

,

X1.b1u

B (15)

and integrating system (15) with initial conditions

biU X2 1FB2, and we get the condition

xi I ex U | if (:) - UI ei The control algorithm looks as follows.

At the initial moment, the control acceleration V (to) is applied to the control object, which begins to vary according to the linear law v (t) v (t0) + at, approaching the required value U. The image point of the phase plane (xi,

about

five

0

five

0

five

0

five

0

five

X2 / ohm) will move along the cycloid segment if I /. 2 l a I bi I nd, to point B, or cycloid, if

I // 1 I 2 l a (, to point / and further along the cycloid segment to point B. At point B, the intersection of the paths of the switching line G2 (-) 0 will occur. This will cause a switch in the control system, as a result of which the control value the acceleration will vary linearly, but will also be formed according to the law (14). Further movement of the imaging point (before the control is completed) will also occur in the polycycloid until the terminal conditions are met.The control acceleration value will increase from Urp / 2 at point B to Morning at the end of control.

The difference of this algorithm from the algorithm by which the prototype operates is that, when moving in the vicinity of the line G (-) 0, the possibility of short-term transitions from the controlled loading mode to the uncontrolled and back, i.e. from the mode of change v (t) according to the linear law to the mode of motion with v (t) const U.

The proposed algorithm is implemented by a device whose functional scheme is shown in FIG. 1. At the same time, the shaper 12 module forms I xil, the quad 13 - ha, the output of the multiplier 14

we get aol I xi I, at the output of the adder 16 - g (x2 / xi + aafi I xi I), .as

the output of the multiplier 15 is removed the value of the required finite acceleration if (1)

TJ (xi 14 - “I am I xi I) sign (v - UTp).

Zero-indicator 20 controls I v - UTp I for smallness, constant voltage source 19 forms bi value so that from the output of divider 18 the value of the required control IF (1) IF (g) /.

The device works in the same way as the prototype (the work starts on the Start signal from block 11) with the only exception that after crossing the switching point G2 (-) 0 with the dot, when block 9 generates a zero signal at the output, the block chain 12-21, forming the required finite control, varying linearly. The generated control is fed to the second information input of the unit 3 and through the key 23 is fed to the input of the actuator 25. The operation of the device is terminated by the inhibit signal from the null indicator 20 when the condition le -Uipl fiu is met. where is some small parameter.

The proposed device in comparison with the prototype has significantly greater roughness with respect to the coordinate-parametric disturbances in the process of control, due to the use of the terminal control law in the final part of the loading process.

This allows, on the one hand, at the final stage of control, to eliminate sliding modes, which are accompanied by a large number of control switchings and can largely be the cause of the oscillation of the elastic element at frequencies that are multiple to the switching frequency, i.e. the cause of resonance and near-resonance effects. The occurrence of vibrations in the structure causes the quality of the target application of the control object to deteriorate. The device allows to provide guaranteed quality of loading with a coefficient of dynamicity Co not less than 1.2 (linear loading provides a deterioration in the quality of high-frequency components by no more than 20% compared with the optimal loading). A prototype, providing loading on a low-frequency harmonic with close to unity, on high-frequency components, can not always provide even KD 2. In this connection, the overall dynamic coefficient in a whole number of experiments exceeds even 1.8-2.5, which requires additional time or energy costs to ensure a given quality control of the movement of an elastic object.

On the other hand, this device has higher accuracy properties (reduction in speed, which does not exceed 5-10% of the optimal loading duration).

Claims (2)

Claim 1. Device for optimal control of the inertial object with attached elastic element according to ed. St. Mg 1381445, characterized in that, in order to increase accuracy and stability, the device is additionally equipped with a constant voltage source, a null indicator, a module driver, a quad, a first and second divider and a sum, a first and second multiplier, a second key, the input of the modulator module is connected with the output of the deviation sensor, and the output is connected with the first inputs of the first multiplier and the first divider, the input of the quad is connected to the output of the differentiator. and the output of the quadrant is connected to the second input of the first divider, the output of which is connected to the first input of the adder, the second input of which is connected to the output of the first multiplier, the second input the first multiplier is connected to the fourth output of the coordinate conversion unit, the first output of which is associated with the first input of the second multiplier, the second input of which is connected to the output of the adder, output the second multiplier is connected to the information input of the second key, the control input of which is connected to the output of the null indicator, the input of the null indicator is connected to the third output of the conversion unit coordinates, the output of the second key is connected to the first input of the second divider, the second input of which is connected to the output of a constant voltage source, and the output to the second information input of the drive control unit, the first control input of which is connected to the output of the trigger unit, and the second control input is with the output of the control parameter selection block. 2. The device of claim 1, wherein the drive control unit consists of an integrator, a first and second keys and an executive body, the first information input of the drive control unit connected to the information integrator input, the output of which is connected to the information input of the first key, the second information input of the control unit is connected to the information input of the second key, the first control input of the drive control unit is connected to the integrator's control input, the second control input of the control unit drive - with the control the inputs of the first and second keys, the input of the executive body is connected to the outputs of the first and second keys, the output of the executive body is connected to the output of the drive control unit. & V