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15
ITERATIVE ALGORITHM FOR ADJUSTING THE STABILIZING PARAMETER OF THE COUPLING SCHEME IN THE DECOMPOSED
RADIO CIRCUIT
DOI 10.24411/2072-8735-2018-10251
Andrey V. Sklyar,
Southern Federal University" (SFU), Taganrog, Russia, [email protected]
Keywords: HIL-simulation, iterative algorithm, radio circuit, decomposition, coupling scheme, stabilizing parameter, coupling quadripole, Schur complement, electric equilibrium equations, node-voltage analysis, Y-parameters.
The subject of the researching is a radio system, divided (or decomposed) into several fragments using "matching devices" between them which called "coupling interface" (another names are coupling scheme and coupling quadripole). Some fragments of the decomposed system can be presented in hardware, and others fragments can be presented in software. This method is called HIL-simu-lation and it allows us to identify disadvantages of the radio system in the early stages of its design. The main problem in system decomposition is the correct adjustment of the stabilizing parameters in a coupling interface. Their correct adjustment allows to achieve convergence of the parameters (such as simultaneous voltage between nodes or current in branche if radio system is considered as a radio circuit) of the decomposed system to according parameters of the initial system in fewer "time steps" (in this paper it is called "iterations"). The adjustment of the stabilizing parameters can be performed in two ways: using the Schur complement before simulation starting of the decomposed radio circuit or using the iterative adjustment algorithm of the stabilizing parameters (IAASP) during the simulation. Unfortunately, in a system, which decomposed into two parts, the adjustment of both stabilizing parameters is not always realizable in practice. The purpose of this paper is to prove that for matching fragments of a decomposed (into two parts) system, it is sufficient to correctly adjust only one of the two stabilizing parameters according to the IAASP. For clarity, the simplest radio system (radio circuit), which consisting of current sources and conductivities, acts as a simple example. Two stabilizing elements are presented as conductances in this case, but only one of them will adjust correctly. On the basis of the obtained results, conclusions were drawn about the admissibility of the practical application of such method in HIL-simulation of radio systems.
Information about author:
Andrey V. Sklyar, Postgraduate student of "Southern Federal University" (SFU), Taganrog, Russia Для цитирования:
Скляр А.В. Итерационный алгоритм подстройки стабилизирующего параметра схемы сшивания при декомпозиции радиотехнической цепи // T-Comm: Телекоммуникации и транспорт. 2019. Том 13. №3. С. 64-70.
For citation:
Sklyar A.V. (2019). Iterative algorithm for adjusting the stabilizing parameter of the coupling scheme in the decomposed radio circuit. T-Comm, vol. 13, no.3, pр. 64-70.
T-Comm 1ом 13. #3-2019
A creation of radio system includes the following steps: simulation, analyzing of simulation results and creating a prototype in hardware lorm. Such approach makes it possible to identify problems in the creating radio system ai early stages of projection. If the radio system consists of a set of blocks (cascades), then it is advisable to carry out a simulation, which includes both computational (on a PC) and hardware parts of the created radio system. Such approach of "system's divison" into the parts is called system decomposition [!]. In scientific papers, this method of simulation has name "H[L modeling" [2 - 9|. For example, let the radio system consists of 4 stages, as shown in Fig. 1.
A radio system
Cascade 1 —► Cascade 2 —► Cascade 3 — Cascade 4
Fig. i. An initial muiti-cascadc radio system
Initially, all four cascades are simulated numerically on a PC. For it can be used such tools as N1 Multisim, N1 LabVlEW and other tools. In case of getting positive results of such simulation it is possible to start assembling the first cascade in a hardware form, and then to make a HIL-simulation including the first cascade in the hardware form and other cascades in a numerical (software) form. Structural scheme of radio system simulation at this stage is presented in Fig. 2.
A radio system
Haidwíü jMrt
Cascade J
Coupling interface
Nwwifcalpaif
Cascade 2
Cascade 3
Cascade 4
Fig. 2. HIL-simulation of the initial multi-cascade radio system
As shown in Fig. 2, a coupling interface is using for matching hardware and numerical parts. The coupling interface includes signal amplifiers, analog-to-digital and digital-to-analog converters (ADC and DAC), and a block of implementation of an interface algorithm (BflA), whose task is to ensure stability and convergence of parameters of the decomposed radio system to parameters of the initial radio system during the simulation |2, 10]. Unfortunately, the coupling interface has delay in data exchange between hardware and numerical parts. Thus, the following requirements are placed for the coupling interface: fast data exchange between the parts of the radio system (minimal delays) and a minimal data distortion (e.g. a minimal distortion of instantaneous values of currents and voltages which can be transmitted from one part of the decomposed system to another part). In case of getting positive results at this stage of the simulation it is possible to create the second cascade in the hardware form and to make one more HIL-simulation where the hardware part will include first and second cascades, and the numerical part will include third and fourth cascades. Thus, there is a gradual "build-up" of the hardware and reduction of the numerical part. At final stage, the radio system must be fully implemented in the hardware from [11], This is an essence of HIL-simulation.
In this paper for simplification of calculations and simplicity understanding of simulation processes, the radio system will be presented as a replacement scheme which can be divided into the parts, thus the replacement scheme of the coupling interface will perform a role of interface algorithm.
An initial system in a form of an arbitrary electronic circuit is pictured in Fig. 3. This system consists of two parts A and B, and one of these parts is a hardware, and the other part is a software. These parts of the initial system are presented as equivalent circuits with equivalent current sources Jj(t) and Ji(t), aslo Y/ and 7; are output and input equivalent conductances for parts A and B, respectively [7, 8. 10. 11, 12, 13].
J KD I Yi
Yi
0
B
Ji(t)
Fig. 3. Parts A and B of an initial system
In Fig. 4, the initial system is presented in a decomposed form and in an operator form. The parts of the decomposed system are connected using a coupling scheme D.
Ji(p)\Y-
i'<p) r » 1
J
» hip)
hlple*
ii(p)e*hi=l/yi I Ui<p)e*
-e-
t i
4
B
Iii J:fp>
Fig. 4. A decomposed system with a coupling scheme
The coupling scheme consists of two parts, containing idealized current sources {I i(p)e'" and I ¡(pie) and idealized voltages sources (U,(p)epl and Uifpje1"), as well as stabilizing elements Zi~J/yi and z?=7/v?.
Consequently, the coupling schcmc is a quadripole, that matches the hardware part with the numerical part and ensures data exchange between fhe parts of the decomposed system using Gauss-Jacobi method, according to which the solutions of electric equilibrium equations are performed parallel to each other for both parts of the decomposed system. After it, the data exchange will be performed between two parts of the decomposed system, and then equations will be solved for each part of the decomposed system again, etc. [9, 12] A time interval from the n to n+1 data exchanges between two parts of the decomposed system is called an integration step [12]. One integration step, as a rule, includes several iterations of the solution of electric equilibrium equations for the decomposed system.
In order to research a communication of the coupling scheme (D) with the parts (A and B) of the decomposed system, it is necessary to use methods of radio circuits analyzing [10, 12].
Y parameters of the coupling scheme D can be presented as follows:
Y =
I I
,-2 pi
(z, z.
1 -e':i"
—+—2e"3p' 2, z,
(1)
The coupling scheme can perform a matching task, if at least one of these following conditions can he met [13]:
zr-l/Yi (2)
zrl/Y,
The values zt and zj in the article [13] can he calculated according to Schur complement. This method of adjusting the stabilizing elements allows us to adjust the values of the stabilizing parameters z/ and z2 before starling the simulation. However, the adjustment of these values can he done during the simulation. The purpose of ihis article is to prove a possibility of adjusting the stabilizing parameters during the simulation, provided that at least one of conditions (2) [ 131 is fulfilled, and it is not necessary thai both conditions from (2) 110, 11, 14,15] are fulfilled simultaneously,
A behavior of the parts A and B in the decomposed system is described by matrices in general form, as given below [10, 11, 14, 15]:
For the decomposed system, which is shown in Fig. 4, we can find Y-paramelers and present them as a system of equations composed by node-voltage analysis:
I !
—+—-e ' z,
1 1
2 7
1-e'2''
z, z, J 1-e"2'"
\-e'lpt
—+—2e"î,w z^-
1-e ^
~VV rJ\
(6)
Further, it is necessary to proceed from a continuous form of system representation (6) to a discrete form and perform the replacement:
c'pr = z~!
Xu rA12~ A
Y .42] Y U2 .J2.
'y 1 BW Ym m V,"
Y J B21 P, /*_
and
and multiply both parts of the system by ( I -z " ):
(3)
A behavior of the initial system (radio circuit), not divided into parts, is described by a system of equations using node-voltage analysis [12, 13, 14, 15]:
(4)
where is U_,=U}.
Then the decomposed system will be described by the following system of equations [13]:
y m YAl2 0 " 3 A
7*21 yAI 2+ ' m i Y ' B]2 U2 - J2 +J
0 Y ' B21 Y ' B22. P< J4
—+- «ip(~2pr) + 7-\ 7i " I 2pr)
(z+ z
\~tm-2pr) 0
exp(-pr)
Y„u +
t^exp(- 2pr)
-- + —exp(-2/)r} Z, Z,
1 -e\f(-2pr)
r„,
Ymt ^itll
u, A J2
u'ï =
J,
Jt
UK
If coefficients of matrices 1a and I'd are unknown, then the decomposed system can be presented in form: [13]:
A J, o
—+-Iu:'-i:
Y«, yM 0 0 0 0 0 0
)',!, Km 1 0 0 t> 0 0
u.
0 l 0 -1 0 0 0 0
1 I,
0 0 -1 « 0 0 0
z.
0 0 0 0 1 ■ 1 0 0
Z;
0 0 0 0 -i 0 1 0
0 Ü 0 0 0 l rm> y 1 fliz U,
0 0 0 0 0 0
0
&
J;
(5)
where l„ and Ih are currents which are flow into D from the parts A and B.
z, z,
-(-UV zi z;
11-2
z
ZI h. 1 1
-{-+—Jz"1
Z, 7.2
, 1 1. z, z,
z, z.
/ 1 1 ^ -I
Z, Z;
,11 Y2(l—z)+—+—z'
Z, z,
Then we need to proceed to difference equations:
Z, 2, z, z.
z, z,
z, z.
J,(l-z-2)" J2( 1-2-)
J, -J--
(7)
In the above equations, indices -I, -2 were used for characterize values of currents and voltages obtained at previous iterations.
If the values of current sources are constant during the integration step, then the right-hand side of equality (7) will be have zero values, since J\-J{2 - 0 and J?Ji = 0.
The values Y\ and from the expression (7) can be written in the following form:
1 1 1 ,, I
t-i
{-+—jt/;1--^-—u;2
Zj Z-, Z|
(U, ~u-3)
z, z, z, z, ~
(U2 -U?)
Substituting expressions (2) into expressions (8), the follow ing expressions Will be obtained:
Vi-v?)
T-Comm Tom 13. #3-2019
Based on the expressions above, the execution of iterative algorithm for adjusting the stabilizing parameters (IAASP) will include the following steps.
1) Prior to the simulation, the stabilizing elements Zj and Zj arc given by arbitrary values.
2) Simulation begins and a full integration step, which including three iterations, is performed.
3) Before performing the first iteration of the next integration step, at least one of the stabilizing parameters is adjusted according to the corresponding expression from (9).
4) The iterations are continue, after which the parameters in the decomposed system converge to the corresponding parameters in the initial system.
For an explanation of calculations, the simplest initial radio circuit, which consist of current sources and conductances, as shown in Fig. 5, will be considered as a simple example.
(V Ya^Q)
Mp)
® Q
Yai
Ya3
0
§
¿81 (3) B
Hp)
□ ®
Ya =
0.021 -0.02 -0.02 0.145
>« =
-0.2 0.21
In this case, a system, according to (4), will take the form: 0.021 -0.02 0 -0.02 0.145 + 0.2 -0.2 0 -0.2 0.21
For a system which based on the above system using modified node-voltage analysis, according to (5):
Fig. 5. Ati initial system (radio circuit)
A Iter decomposition of the same circuit with the addition of a coupling scheme is shown in Fig. 6.
Conductances are given by the following values: ^, = 0.001 S; Yja ~ 0.02 S; ^j = 0.125S; Ym = 0,2 S; yb2 = 0.01 S. A current from source jt is a unipolar "square wave" with an amplitude of 2 A and with a pulse duration which equal to ten integration steps. A current from source j2 is varies according to the law: 5sin (8 x jr * n / N) A , where n is the current number of the integration step, and N is the total number of integration steps. In this experiment, N = 100. Then, conduction matrices }'A and according to (3), can be written in the following form:
" 0.2 -0.2
"iV
Vt ii 0
A
0.021 -0.02 -0.02 O.Mi
0 0 — -I
I 0
0.2 -0,2 -0.2 OJI
■MH^
o
o j,
It should be noted that voltage U„ is the voltage between the nth and zero nodes in the above systems. The main task in this experiment is a calculation with a subsequent comparison of solutions of these two systems under the condition that the stabilizing parameters will be calculated during the simulation. An experiment will consist of two parts. First, the voltage between the zero and first nodes will be calculated for both the initial system (Ul) and the decomposed system (UU1). The latter is must correspond to the voltage Ul according to theoretical calculations. Then the voltage value between zero and third nodes in the initial system (U3) and the voltage between zero and eighth nodes (UU8) in the decomposed system will be calculated. The latter is must correspond to the voltage U3 according to theoretical calculations.
The calculations will be performed in the MATLAB.
The first part of this experiment, as mentioned above, includes the researching of the voltages Ul in the initial system and UU1 in the decomposed system.
If we select the values of the stabilizing parameters, as shown below:
zl = 75 ft;
z2 = 1000 Q,
then after three iterations on the first integration step, the stabilizing parameter z2 will be adjusted according to expression (6) and will become equal to 8 Cl. The results are presented in Fig. 7.
Fig. 6. A radio circuit after decomposition
In Fig. 11 - 14, the solid line is indicate the voltage U3 between the nodes 3 and 0 in the initial system depending on the integration step number n, and the "circles" are indicate the voltage UU8 between the nodes 8 and 0 in the decomposed system depending on the integration step number n.
If the value of the stabilizing parameter Zj is equal to 10k£l, and the stabilizing parameter z2 is equal to 8 Mii, then after three iterations on the first integration step, the stabilizing parameter z2 will be adjusted according to expression (6) and will become equal to 8 il The results are presented in Fig. 12.
Fig. 12. Voltage between the nodes 3 and 0 (U3) in the initial and voltage between the nodes 8 and 0 (UU8) in decomposed systems while Z) = 10 kii
Suppose, that the value of the stabilizing parameter z2 is equal to 75 ii. and the stabilizing parameter Z| is 50 Q. After three iterations on the first integration step, the stabilizing parameter Z| will be adjusted according to expression (6) and will become equal to 105 £2. The results are presented in Fig, 13.
Fig. 13. Voltage between the nodes 3 and 0 (U3) in the initial and voltage between the nodes 8 and 0 (UU8) in the decomposed systems while zj = 75 ii
Suppose, that the value of the stabilizing parameter z2 is equal to 10 k£2, and the stabilizing parameter Z\ is 200 £1. After three iterations on the first integration step, the stabilizing parameter z, will be adjusted according to expression (6) and will become equal to 105 ii. The results are presented in Fig. 14.
Based on the above results, we can conclude that the voltage between the nodes 0 and 8 in the decomposed system (UU8) nearly coincides with the voltage between the nodes 0 and 3 in the initial system (U3), if at least one of the stabilizing parameters (zi or z2) was adjusted correctly.
Fig. 14. Voltage between the nodes 3 and 0 (U3) in the initial
and voltage between the nodes 8 and 0 (UU8) iti the decomposed systems while z2 = 10 kii
Similarly, the voltage between the nodes 0 and 1 in the decomposed system (UU1) nearly coincides with the voltage between the nodes 0 and I in die initial system (Ul). Therefore, we can conclude, that if only one of the stabilizing parameters is adjusted correctly based on the results obtained in previous three iterations on the first step, the parameters of the decomposed system will converge to the parameters of the initial system on the second integration step. This algorithm of adjustment is implemented during the simulation process and allows us to not use the Schur complement, which can be used to calculate the stabilizing parameters before the starting of the simulation. Therefore, there is no necessary to adjust both stabilizing parameters (by the way, it is not a simplest task) and a process of PHIL-simulation will be significantly simplified. Then it is sufficient to set the value or the stabilizing parameter in the hardware variously, as well as to adjust the stabilizing parameter in the software part, and therefore we can obtain convergence of parameters in the initial system to parameters in the decomposed system.
References
1. Kopysov S.P. (2006). Decomposition methods and parallel distributed technologies for adaptive versions of the finite element method, PhD thesis: 05.13.18. Izhevsk,
2. Ren W. (2007). Accuracy Evaluation of Hardware-in-the-Loop (PHIL) Simulation. PhD thesis. Florida State University, Tallahassee.
3. Avras A., Roscoe AJ„ Burt G.M. (2014). Scalable Real-Time Controller Hardware-ln-the-Loop Testing for Multiple Interconnected Converters. The UPEC 2014 conference, IEEE. URL: www.dx.doi.oi-g/I0.H09/UPEC 2014.6934620 (appeal date: 02/27/2019).
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8. Merezhin N.I. (2009). Stand for analog-to-digital modeling using adaptive Coupling schemes. Proceedings of the international scientific conference "Methods and algorithms for making effective decisions". Taganrog: SFU. Vol. 2, pp. 47-51.
9. Maksimov M.N., Merezhin N.I., Sklyar A.V., Merezhin D.N. (2015). Using the Poincare-Steklov operator to ensure the sustainabiliiy of PHIL modeling. Cooperation of the BRICS countries for sustainable development: proceedings of the International Scientific and Practical Conference of Young Scientists of the BRICS Countries, Rostov-oil-Don: SFU. Vol.2, pp. 81-82.
10. Maksimov M., Merezhin N., Lyashev V., Sinyutin S. (2017). Poincare-Steklov filter in hardware-in-the-loop modeling. 20! 7 International Siberian Conference on Control and Communications (S1BCON 2017). URL: www.doi.org/10.M09/SIBCON.20l7.799853i (appeal date: 02/27/2019).
ИТЕРАЦИОННЫЙ АЛГОРИТМ ПОДСТРОЙКИ СТАБИЛИЗИРУЮЩЕГО ПАРАМЕТРА СХЕМЫ СШИВАНИЯ ПРИ ДЕКОМПОЗИЦИИ РАДИОТЕХНИЧЕСКОЙ ЦЕПИ
Скляр Андрей Вадимович, Южный федеральный университет (ЮФУ), г. Таганрог, Россия, [email protected]
Аннотация
Предметом исследования является радиотехническая система, разделённая (декомпозированная) на несколько фрагментов с использованием между ними "согласующих устройств", именуемых интерфейсом (схемой, или четырёхполюсником) сшивания. Одни фрагменты декомпозированной системы могут быть представлены в аппаратном виде, а другие - в виде программной модели. Такой метод, называемый полунатурным моделированием (HIL-simulation), позволяет выявить недостатки системы на ранних этапах её проектирования. Основной проблемой при декомпозиции системы является подстройка стабилизирующих параметров согласующих элементов интерфейса сшивания. Их правильная подстройка позволяет достичь сходимости параметров декомпозированной и исходной систем за меньшее количество "временных шагов" (итераций). Подстройку стабилизирующих параметров можно выполнить двумя способами: до начала моделирования декомпозированной радиотехнической цепи с использованием дополнения Шура, либо во время моделирования с использованием итерационного алгоритма подстройки. В декомпозированной на две части системе подстройка обоих стабилизирующих параметров не всегда реализуема на практике. Целью статьи является доказательство того, что для согласования фрагментов декомпозированной на две части системы достаточно корректно подстроить только один из двух стабилизирующих параметров согласно итерационному алгоритму подстройки (IAASP). Для наглядности в качестве исследуемого образца выступает простейшая радиотехническая система, состоящая из источников тока и проводимостей, а стабилизирующие элементы представлены в виде проводимостей. На основе полученных данных были сделаны выводы о допустимости применения на практике такого способа в полунатурном моделировании радиотехнических систем.
Ключевые слова: HIL-моделирование, итерационный алгоритм, радиотехническая цепь, декомпозиция, схема сшивания, стабилизирующий параметр, сшивающий четырехполюсник, дополнение Шура, уравнения электрического равновесия, метод узловых напряжений, Y-параметры.
Литература
1. Копысов С.П. Методы декомпозиции и параллельные распределённые технологии для адаптивных версий метода конечных элементов // Дис. док. физ.-мат. наук: 05.13.18. Ижевск. 2006.
2. Ren W. Accuracy Evaluation of Hardware-in-the-Loop (PHIL) Simulation // PhD thesis. Florida State University, Tallahassee. 2007.
3. Avras A., A.J. Roscoe and G.M. Burt. Scalable Real-Time Controller Hardware-In-the-Loop Testing for Multiple Interconnected Converters // The UPEC 2014 conference, IEEE. URL: www.dx.doi.org/l0.ll09/UPEC 2014.6934620 (дата обращения: 02/27/2019).
4. Скляр А.В. Полунатурное моделирование многофункциональных модульных систем. Условия выбора модели // Инженерный вестник Дона. 2017. № 2. URL: ivdon.ru/ru/magazine/archive/N2y20l7/4236 (дата обращения: 02/27/2019).
5. Шайкин А.С., Шайкина Е.В. Применение комплекса полунатурного моделирования в процессе проектирования информационно-измерительных и управляющих систем // Инженерный вестник Дона. 2014. № 1. URL: ivdon.ru/ru/magazine/archive/nly20l4/2248 (дата обращения: 02/27/2019).
6. Скляр А.В., Мережин Н.И. Полунатурное моделирование комплексных систем // Компьютерные и информационные технологии в науке, инженерии и управлении "КомТех-20!7": материалы Всероссийской научно-технической конференции с международным участием. Таганрог: ЮФУ. 20l7. С. 2l-24.
7. Попов В.П., Максимов М.Н., Мережин Н.И. Об устойчивости и сходимости моделирования по частям // Вестник Южного научного центра РАН. T.l. № 3. 2005. С. ll-2l.
8. Мережин Н.И. Стенд для аналого-цифрового моделирования с использованием адаптивных схем сшивания // Материалы международной научной конференции "Методы и алгоритмы принятия эффективных решений". Таганрог: ТТИ ЮФУ. 2009. Ч.2. С. 47-5l.
9. Максимов М.Н., Мережин Н.И., Скляр А.В., Мережин Д.Н. Использование оператора Пуанкаре-Стеклова для обеспечения устойчивости PHIL моделирования // Сотрудничество стран БРИКС для устойчивого развития: материалы Международной научно-практической конференции молодых учёных стран БРИКС. Ростов-на-Дону: ЮФУ. 20l5. Т.2. С. 8l-82.
10. Maksimov M., Merezhin N., Lyashev V., Sinyutin S. Poincare-Steklov filter in hardware-in-the-loop modeling. 2017 International Siberian Conference on Control and Communications (SIBCON 20l7). URL: www.doi.org/l0.ll09/SIBC0N.20l7.799853l. (дата обращения: 02/27/20l9).
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Информация об авторе: Скляр Андрей Вадимович, аспирант Южного федерального университета (ЮФУ), г. Таганрог, Россия
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