Научная статья на тему 'Stability and accuracy of the robust system for stabilizing the rotor flux-linkage of an asynchronous electric drive at random variations of the uncertain parameters within the specified boundaries'

Stability and accuracy of the robust system for stabilizing the rotor flux-linkage of an asynchronous electric drive at random variations of the uncertain parameters within the specified boundaries Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

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Ключевые слова
electric drive / vector control / flux-linkage channel / stabilizing robust system / stability / accuracy / электропривод / векторное управление / канал потокосцепления / робастная система стабилизации / устойчивость / точность

Аннотация научной статьи по электротехнике, электронной технике, информационным технологиям, автор научной работы — I. N. Khlopenko, S. A. Rozhkov, N. J. Khlopenko

Purpose. The aim is to investigate the stability and the accuracy of a robust system for stabilizing the rotor flux-linkage of an asynchronous electric drive at random variations of the uncertain parameters of the object and the regulator within the specified boundaries. Methodology. To make the research, the mathematical model of the rotor flux-linkage channel of the vector control system of an asynchronous electric drive with parametric uncertainty was applied. The transfer function of the Н∞-suboptimal regulator was calculated using the mixed sensitivity method. This transfer function was used to construct the regulator structural scheme in the form of a connection of proportional and integrating links and several adders. Analytical dependences of the coefficients of the regulator's transfer function on the parameters of links of such a connection are determined. These dependences served to researching the influence of uncertain parameters of the regulator links and the object on the stability of the robust system and the accuracy of flux-linkage stabilization. Results. Investigations of the robust system stability and the accuracy of flux-linkage stabilization in the Robust Control Toolbox are done. The curves of the fluxlinkage transient processes and the Bode diagram for the open system at random variations of the indeterminate parameters of the object and the regulator links within the specified boundaries are constructed. A choice of variable parameters was carried out by the Monte Carlo method. By the scatter of the obtained curves of the transient processes, the accuracy of flux-linkage stabilization was determined, and according to the Bode diagram, stability reserves in the amplitude and the phase of the robust system were determined. A high accuracy of flux-linkage stabilization (deviation less than 1 %) in fairly wide ranges of changing the uncertain parameters of the object and the regulator, while maintaining the stability of the system with permissible reserves in amplitude and phase, is established. Originality. For the first time, analytical dependences of the coefficients of the transfer function of the Н∞-suboptimal regulator on the parameters of its structural scheme, which represented in the form of a connection of proportional and integrating links, are obtained. The method for calculating the stability of a robust flux-linkage control system and the accuracy of its stabilization at random variations of the uncertain parameters of the object and the regulator links within the specified boundaries is developed. Practical value. The use of the proposed method allows, during the design of the regulator, to ensure the selection of its elements from standard series. References 10, figures 3.

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Цель. Целью работы является исследование устойчивости и точности робастной системы стабилизации потокосцепления ротора асинхронного электропривода при случайных вариациях неопределенных параметров объекта и регулятора в заданных границах. Методология. Для проведения исследований применялась математическая модель канала потокосцепления ротора системы векторного управления асинхронного электропривода с параметрической неопределенностью. Рассчитывалась передаточная функция Н∞субоптимального регулятора по методу смешанной чувствительности. Эта передаточная функция использовалась для построения структурной схемы регулятора в виде соединения пропорциональных и интегрирующих звеньев и нескольких сумматоров. Определялись аналитические зависимости коэффициентов передаточной функции регулятора от параметров звеньев такого соединения. Эти зависимости служили для исследования влияния неопределенных параметров звеньев регулятора и объекта на устойчивость робастной системы и точность стабилизации потокосцепления. Результаты. Проведены исследования устойчивости робастной системы и точности стабилизации потокосцепления в пакете Robust Control Toolbox. Построены кривые переходных процессов потокосцепления и диаграмма Боде для разомкнутой системы при случайных вариациях неопределенных параметров объекта и звеньев регулятора в заданных границах. Выбор варьируемых параметров осуществлялся по методу МонтеКарло. По разбросу полученных кривых переходных процессов определялась точность стабилизации потокосцепления, а по диаграмме Боде – запасы устойчивости по амплитуде и фазе робастной системы. Установлена высокая точность стабилизации потокосцепления (отклонение менее 1 %) в достаточно широких диапазонах изменения неопределенных параметров объекта и регулятора при сохранении устойчивости системы с допустимыми запасами по амплитуде и фазе. Новизна. Впервые получены аналитические зависимости коэффициентов передаточной функции Н∞-субоптимального регулятора от параметров его структурной схемы, представленной в виде соединения пропорциональных и интегрирующих звеньев. Построена методика расчета устойчивости системы робастного управления потокосцепления и точности его стабилизации при случайных вариациях неопределенных параметров объекта и звеньев регулятора в заданных границах. Практическое значение. Использование предложенной методики позволяет в процессе конструирования регулятора обеспечить выбор его элементов из стандартных рядов. Библ. 10, рис. 3.

Текст научной работы на тему «Stability and accuracy of the robust system for stabilizing the rotor flux-linkage of an asynchronous electric drive at random variations of the uncertain parameters within the specified boundaries»

UDC 621.3.07

doi: 10.20998/2074-272X.2018.4.06

I.N. Khlopenko, S.A. Rozhkov, N.J. Khlopenko

STABILITY AND ACCURACY OF THE ROBUST SYSTEM FOR STABILIZING THE ROTOR FLUX-LINKAGE OF AN ASYNCHRONOUS ELECTRIC DRIVE AT RANDOM VARIATIONS OF THE UNCERTAIN PARAMETERS WITHIN THE SPECIFIED BOUNDARIES

Purpose. The aim is to investigate the stability and the accuracy of a robust system for stabilizing the rotor flux-linkage of an asynchronous electric drive at random variations of the uncertain parameters of the object and the regulator within the specified boundaries. Methodology. To make the research, the mathematical model of the rotor flux-linkage channel of the vector control system of an asynchronous electric drive with parametric uncertainty was applied. The transfer function of the Hm-suboptimal regulator was calculated using the mixed sensitivity method. This transfer function was used to construct the regulator structural scheme in the form of a connection of proportional and integrating links and several adders. Analytical dependences of the coefficients of the regulator's transfer function on the parameters of links of such a connection are determined. These dependences served to researching the influence of uncertain parameters of the regulator links and the object on the stability of the robust system and the accuracy of flux-linkage stabilization. Results. Investigations of the robust system stability and the accuracy offlux-linkage stabilization in the Robust Control Toolbox are done. The curves of the flux-linkage transient processes and the Bode diagram for the open system at random variations of the indeterminate parameters of the object and the regulator links within the specified boundaries are constructed. A choice of variable parameters was carried out by the Monte Carlo method. By the scatter of the obtained curves of the transient processes, the accuracy of flux-linkage stabilization was determined, and according to the Bode diagram, stability reserves in the amplitude and the phase of the robust system were determined. A high accuracy of flux-linkage stabilization (deviation less than 1 %) in fairly wide ranges of changing the uncertain parameters of the object and the regulator, while maintaining the stability of the system with permissible reserves in amplitude and phase, is established. Originality. For the first time, analytical dependences of the coefficients of the transfer function of the Hm-suboptimal regulator on the parameters of its structural scheme, which represented in the form of a connection of proportional and integrating links, are obtained. The method for calculating the stability of a robust flux-linkage control system and the accuracy of its stabilization at random variations of the uncertain parameters of the object and the regulator links within the specified boundaries is developed. Practical value. The use of the proposed method allows, during the design of the regulator, to ensure the selection of its elements from standard series. References 10, figures 3.

Key words: electric drive, vector control, flux-linkage channel, stabilizing robust system, stability, accuracy.

Цель. Целью работы является исследование устойчивости и точности робастной системы стабилизации потокосцепления ротора асинхронного электропривода при случайных вариациях неопределенных параметров объекта и регулятора в заданных границах. Методология. Для проведения исследований применялась математическая модель канала потокосцепления ротора системы векторного управления асинхронного электропривода с параметрической неопределенностью. Рассчитывалась передаточная функция Нт-субоптимального регулятора по методу смешанной чувствительности. Эта передаточная функция использовалась для построения структурной схемы регулятора в виде соединения пропорциональных и интегрирующих звеньев и нескольких сумматоров. Определялись аналитические зависимости коэффициентов передаточной функции регулятора от параметров звеньев такого соединения. Эти зависимости служили для исследования влияния неопределенных параметров звеньев регулятора и объекта на устойчивость робастной системы и точность стабилизации потокосцепления. Результаты Проведены исследования устойчивости робастной системы и точности стабилизации потокосцепления в пакете Robust Control Toolbox. Построены кривые переходных процессов потокосцепления и диаграмма Боде для разомкнутой системы при случайных вариациях неопределенных параметров объекта и звеньев регулятора в заданных границах. Выбор варьируемых параметров осуществлялся по методу Монте-Карло. По разбросу полученных кривых переходных процессов определялась точность стабилизации потокосцепления, а по диаграмме Боде - запасы устойчивости по амплитуде и фазе робастной системы Установлена высокая точность стабилизации потокосцепления (отклонение менее 1 %) в достаточно широких диапазонах изменения неопределенных параметров объекта и регулятора при сохранении устойчивости системы с допустимыми запасами по амплитуде и фазе. Новизна. Впервые получены аналитические зависимости коэффициентов передаточной функции Нт-субоптимального регулятора от параметров его структурной схемы, представленной в виде соединения пропорциональных и интегрирующих звеньев. Построена методика расчета устойчивости системы робастного управления потокосцепления и точности его стабилизации при случайных вариациях неопределенных параметров объекта и звеньев регулятора в заданных границах. Практическое значение. Использование предложенной методики позволяет в процессе конструирования регулятора обеспечить выбор его элементов из стандартных рядов. Библ. 10, рис. 3.

Ключевые слова: электропривод, векторное управление, канал потокосцепления, робастная система стабилизации, устойчивость, точность.

Introduction. In [1] the method of structural synthesis is constructed and the structure of the stabilizing robust ^„-suboptimal regulator is obtained in the form of

a connection of proportional and integrating links for the flux-linkage channel of the vector control system of an

© I.N. Khlopenko, S.A. Rozhkov, N.J. Khlopenko

asynchronous electric drive with parametric uncertainty of the control object. However, when designing such a regulator from analog devices (operational amplifiers and RC-circuit) there are rounding errors of its gain factors and time constants due to the selection of elements (resistors, capacitors) of these devices from standard series. The consideration of such rounding errors in the calculation model of the regulator with parametric uncertainty of the object is of fundamental importance for ensuring the stability of the robust system and the necessary accuracy of flux-linkage stabilization.

Robust systems of stabilizing the parameters of asynchronous electric drives are engaged in a number of domestic and foreign scientists [2-9]. They solved many problems both in the development of mathematical methods of research, and in studying the stability, accuracy of regulation, and the speed of systems with a given uncertainty of the object. However, the problem of the influence of the parametric uncertainty of the robust regulator on the stability and accuracy of the flux-linkage stabilization system was not considered. In this connection, the problem of studying the stability of a robust system and the accuracy of stabilizing the rotor flux-linking with the parametric uncertainty of the object and the regulator within given boundaries seems to be actual.

The goal of the work is study of the stability and accuracy of a robust system for stabilizing the rotor flux-linkage of an asynchronous electric drive at random variations of the indeterminate parameters of the object and the regulator within given boundaries.

Methods and results of research. The paper [1] contains a system of equations of state of an object consisting of a frequency converter and stator and rotor windings in the normal operator form:

1

Pxi = _ ~xi + T

L12Jn

Px2 =-

T

-Xo +

E„

1eq

2 ^--x3 ■

R T J ■iMeqi 1eq-' n

(1)

1 KfcUn

px3 =--x3 +— u

T T E

where

w

E

U

u = ■

xl W ; xl j ; x3 e ' U 1 n 1 n ^n w n

p is the Laplace operator; E is the frequency converter's EMF; U is the control action; J is the current in the rotor flux-linkage channel; W is the rotor flux-linkage vector's module; Tfc is the time constant of the frequency converter; T1eq=L1eq/R1eq is the electromagnetic time constant of the stator winding, where R1eq=R1+(^r)2R2 and L1eq=aL1 are its equivalent resistance and the leakage inductance; R1, R2 are the active resistances of the stator and rotor windings; T2=L2/R2 is the electromagnetic time constant of the rotor winding; L1, L2 are the inductances of the stator and rotor windings; L12 is the mutual

inductance of the stator and rotor windings; o=1 - (L12)2/(L1 L2) is the coefficient of magnetic field scattering; &r=Li2/L2.

In this paper, this system of equations, together with the undefined parameters Kfc, R1eq, R2, L1, L2 and Li2 of the object, is used to construct a mathematical model for the stability and accuracy of a robust system for stabilizing the rotor flux-linkage of an asynchronous electric drive at random variations of uncertain parameters within given boundaries.

To construct such a model, the system of equations (1) is reduced to the canonical form [1]: px = Ax + Bjw + B2U ; z = Cjx + Dnw + D12U ; y = C2x + D21W + D22U ,

(2)

where

A =

2n

L2n

R

L 0

0

R2n 0

L2n

R ■'Heqn R ^cqn

L1eqn L1eq n

1

0

_ Tf"

B, =

0 0 0 Pr2 L2n PL12 L2n _ pl2 Pr2 L2n

0 PL1eq PR1eq L1eqn 0 0 0 0

PKfc 0 0 0 0 0 0

c, =

0 0

0 0 0

_ R2n L2n R2n

0

0

r

r

1eqn -fl1eqn

L1eqn R1eqn R2n

R2n R2n

L2n

0

l

1eq n

0

0 0

0 0

D11 =

BT =

00

Tf

fc

C2 =[1 0 0];

"0 0 0 0 0 0 0

PR1eq L1eqn

0 PL1eq 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 Pr2 0 0 0

0 0 0 Pr2 pl\2 _ Ph2 Pr2

L2n L2n L2n

0 0 0 0 0 0 0

; DjT2 = [1 0 0 0 0 0 0];

D21 =[0 0 0 0 0 0 0]; D22 =[0];

T

fc

X

2

1

J

1

x = (x1, x2, x3) is the phase vector; y is the one-dimensional output vector which closes the feedback; z = (z1, z2,..., z7)T, w = (wu w2,..., w7)T are, respectively, the input and output uncertainty vectors interconnected by the matrix expression w(p)=A(p)-z(p) in which the uncertainty matrix A(p) has a diagonal form.

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The written canonical form of equations (2) together with the weight functions [10] for control the quality of the robust stabilization system, allows in the Robust Control Toolbox to calculate the transfer function of the ^„-suboptimal regulator for the nominal object. This transfer function can be represented in the form

a 2 = b2 + k

K ( p) = k

p + bip + b2

3 2

p + ai p + a2 p + a3

(3)

where k, b1, b2 are the regulator parameters.

We assume that the transfer function of the regulator (3) retains its form for random variations of the parameters k, a1, a2, a3, b1, b2.

Then, expanding (3) into a continued fraction by the Euclidean algorithm, we obtain the block diagram of the regulator shown in Fig. 1. It contains the undefined parameters k, k1, k2, k3, T1, T2, caused, as already noted earlier, by rounding errors that occur when designing the regulator.

k2+(k3 "k2)

T T

a3 =

kik 2 k3 T1T2

Fig. 1. Structural scheme of H„-suboptimal stabilizing robust controller

We replace the structural scheme shown in Fig. 1 by equivalent circuit under the rules of transformation of structural schemes. Then we obtain the following expressions for the coefficients of the polynomials of the numerator and denominator of the transfer function of the robust regulator (3):

b1 = -1 (k2 -k1 (k3 -k2); a1 = k1 + b1;

T1 T2

b2 =

■[^3 (k2 - k1 ) + k1k 2

(4)

12

The system of equations (1) together with (3) and expressions (4) serves to investigate the stability and accuracy of the rotor flux-linkage stabilization system at random variations of the indeterminate parameters of the object and regulator within given boundaries.

The accuracy of flux-linkage stabilization is determined by the spread of the curves of its modulus of transient processes, and the stability reserves by amplitude and phase - according to the Bode diagram at various random variations of the undefined parameters of the regulator k, k1, k2, k3, T1, T2 and the object Kfc, R1eq, R2, L1, L2, L12 within specified boundaries. In this case, the Monte Carlo method is used for random choice of parameters [10]. Calculations are performed in MATLAB and terminated when, in the steady state mode of the system, transients do not exceed the boundaries of a one-percent «tube».

The procedure for calculating the stability and accuracy of the rotor flux-linkage stabilization system at random variations of undefined parameters within given boundaries is reduced to the following sequence of actions:

1. The transfer function (3) of the regulator for the nominal object is calculated.

2. The decomposition of the found transfer function into a continued fraction is performed.

3. The block diagram of the regulator is formed (see Fig. 1) corresponding to a continued fraction, and its nominal parameters kn, k1n, k2n, k3n, T1n, T2n are calculated.

4. The transfer function of the object (1) is programmatically determined.

5. The system is formed by the command of flux-linkage stabilization from the series-connected transfer functions of the regulator (3) and the object (1) covered by a single feedback.

6. The curves of the flux-linkage transients and the Bode diagram for the open system are calculated with random variations of the parameters of the object Kfc, R1eq, R2, L1, L2, L12 and regulator k, k1, k2, k3, T1, T2 within the specified boundaries.

7. The accuracy of the flux-linkage stabilization is determined from the scales of the curves of the transients, and the stability reserves in the amplitude and phase are determined from the Bode diagram.

The numerical solution was carried out at the following values of the initial data: Tfc = 0.001 s; R1n = 2.65 Q; R2n = 2.0 Q; L1n = 0.186 H; L2n = 0.189 H; L12n = 0.179 H; a = 0.0996 corresponding the asynchronous electric drive with motor MDXMA100-32.

The nominal parameters of the regulator calculated from these data turned out to be equal to: ki=5.016T05; k1n=1.436-104; k2n=1.752-104; k3n=3.473-103; T1n=19.70 s; T2n=1.256-103 s.

1

Undefined parameters of the object varied in the ranges ±90 %, and the parameters of the regulator ku k2 -in the ranges ±3 %, k -in the range ±15 %, and k3, Tu T2 in the ranges ±20 % of their nominal values.

Fig. 2 shows 20 curves of transients of the rotor flux-linkage corresponding to random variations of the undefined parameters of the object and the regulator selected within the prescribed boundaries by the Monte Carlo method. They are obtained in the packages of the MATLAB application for a single step change in the control action.

As can be seen, the curves of the transients shown in Fig. 2 do not exceed the limits of 1 % of the tube.

*F

Fig. 2. Transients of the rotor flux-linkage

Fig. 3 shows the Bode diagram with 20 generated curves of amplitude L(m) and 20 phase ^(m) frequency characteristics with the same uncertain parameters as in the previous case.

L(a>), dB 0 -50 -LOO

Fig. 3. The Bode diagram of the open system

From the amplitude L(m) and phase ^(m) characteristics presented in this diagram, it is seen that the system is stable, since the amplitude characteristic crosses the abscissa axis before the phase characteristic, finally decaying, goes over the value of the angle -180°. In this case, the calculated value of

the stability reserve in amplitude is 19.9 dB, and in phase - 47.9° for nominal values of the object and regulator parameters for variance of random curves not exceeding 4 dB for amplitude and 15° for phase frequency characteristics.

Thus, the results of the calculations confirm the expediency of using the proposed method for constructing robust ^„-suboptimal regulators from elementary links. Conclusions.

1. For the first time, analytical dependences of the coefficients of the transfer function of the ^„-suboptimal regulator from the parameters of its structural scheme represented as a combination of proportional and integrating links are obtained.

2. A method is developed for calculating the stability and accuracy of a robust system for stabilizing the rotor flux-linkage of an asynchronous electric drive at random variations of the indeterminate parameters of the object and the regulator within given boundaries.

3. The results of the calculations show a high accuracy of flux-linkage stabilization (deviation less than 1 %) and a low sensitivity of the robust stabilization system to random variations of uncertain parameters within given wide enough boundaries.

REFERENCES

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Received 23.04.2018

I.N. Khlopenko1, Master of Science,

S.A. Rozhkov , Doctor of Technical Sciences, Professor,

N.J. Khlopenko?, Doctor of Technical Sciences, Professor,

1 Kherson State Marine Academy,

20, Ushakov Ave., Kherson, 73009, Ukraine, e-mail: rozhkov_ser@meta.ua

2 Admiral Makarov National University of Shipbuilding, 3, Central Ave., Nikolaev, 54021, Ukraine,

e-mail: khlopenko.n@gmail.com

How to cite this article:

Khlopenko I.N., Rozhkov S.A., Khlopenko N.J. Stability and accuracy of the robust system for stabilizing the rotor flux-linkage of an asynchronous electric drive at random variations of the uncertain parameters within the specified boundaries. Electrical engineering & electromechanics, 2018, no.4, pp. 35-39. doi: 10.20998/2074-272X.2018.4.06.

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