THE METHOD OF SYNTHESIS OF THE DIGITAL CONTROLLER FOR A SOLAR ENERGY CONVERSION CHANNEL OF THE SOLAR BATTERY IN THE POWER SUPPLY SYSTEM OF A SPACECRAFT

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Introduction. One of the most important onboard sys- tems of automatic spacecraft (SC) is the power supply system (PSS), which is a combination of primary and secondary current sources, energy conversion equipment and output voltage stabilization with the necessary control and automation [1; 2]. When developing a PSS of a SC, it is necessary to take into account many disturbing influences that affect the quality of the output voltage, the static and dynamic characteristics of the PSS adversely. Such impacts include changes in load resistance and power supply sources, degradation of primary energy sources, etc., which, in turn, leads to the need in building an automatic control system that ensures stable PSS operation with a given accuracy over the entire range of changes in these distur- bances. At present, control systems (CS) on an analog element base [3-7], which are often called analog control systems for short, are widely used in the PSS of a SC. When solv- ing problems of improving the reliability of the PSS and at the same time improving its overall mass and func- tional characteristics, it becomes obvious that analog control systems fit the physical limit of the possibility of improving their functional parameters and overall mass characteristics [8]. The next stage in the development of the PSS of a SC is the transition to control systems built on a digital ele- ment base, also called digital control systems (DCS). Digital control systems, compared to analogue CSs, can provide [9] higher reliability, reduced weight and dimen- sions of the power supply system (including the cable network), and electronic anticountermeasures of transmis- sion of the control signal. In addition, digital control will allow to expand the functionality of the CS, simplify the creation of their modifications and embeddability of various devices in the structure. Digital control systems for their intended pur- pose allow to exchange data with the on-board control complex over a high-speed data interface, and this func- tion is already widely used. However, the systems of direct digital control of energy-converting equipment, including pulse converters of electrical energy parame- ters, despite these obvious advantages, are practically not used at the moment, and analog control systems are widely used. This situation in the field of direct digital control systems application is connected with the fact that digital systems contain analog-digital converters, digital blocks for implementing signal modulators and digital communication interfaces for closing negative feedback circuits. Аs it is well known digital nodes and blocks cause delays in signals that are similar in nature to breaks of transport delays, which complicates the synthesis of correcting sections greatly. Purpose. The purpose of this work is to develop a method of synthesis of the digital controller for a solar energy conversion channel of a solar battery in a power supply system of a spacecraft. Method of synthesis of the digital controller. The method of synthesis of the digital controller will be considered in the following sequence: 1. Develop a mathematical model of a pulse con- verter (control object) in the form of a block diagram constructed in the basis of discontinuous switching functions according to a given functional diagram. 2. Select nonlinear links and linearize them at a given operating point on the block diagram obtained in the basis of discontinuous switching functions, presenting each nonlinear link by a combination of linearized links, in order to obtain a linearized block diagram of the control object. 3. Determine the necessary transfer functions of the system, including the linearized transfer function of the open-ended control object based on the obtained lin- earized block diagram. 4. Compare the frequency characteristics of the lin- earized open unadjusted control object, obtained from its linearized transmission links, with the experimental “low- signal” frequency characteristics of the open unadjusted control object, taken on its simulation model built in the Simulink package of the Matlab design environment; if the compared frequency characteristics coincide with a given quantitative accuracy, the linearized model can be considered adequate. 5. Build a family of frequency characteristics of an open unadjusted control object using its linearized trans- fer functions at the “extreme” operating points of the sys- tem, meaning the “extreme” working points of the steady state point of the controlled object at the minimum and maximum values of all disturbing factors. 6. Synthesize the necessary type of analog correction link and its parameters assuming that the linearized model adequately describes the open control object and using the classical methods of synthesizing linear control systems for this reason. 7. Build a simulation model (for example, in the Simulink package of the Matlab design environment) of a closed-loop control system with the parameters of the corrective element obtained in step 6, and check the static and dynamic characteristics of the synthesized analogue controller. 8. Make a z-transformation of the transfer function of the discrete circuit and synthesize a digital correction link using the known transfer function of the analog correction link and a bilinear transition. If necessary, check the dynamic characteristics of the automatic control system (ACS) with a digital correction link on the simulation model in the Matlab environment, taking into account the delay. The control object for the direct digital control system is the voltage stabilization module (VSM) of the energy- conversion complex, which includes three types of energy conversion channels: - “SlB conversion channel” intended to convert the energy of on-board solar batteries into a constant stabi- lized output bus voltage to which the load is connected; - “memory conversion channel” intended for charging onboard batteries with a stabilized current, the value of which can be changed by an external signal; - “StB conversion channel” intended to convert the energy of the on-board storage batteries into a constant stabilized voltage of the output bus to which the load is connected; Each of the energy conversion channels is a power pulse converter of electrical energy with or without feed- back, providing a given law of operation. It can be made on the basis of parallel connection of several pulse con- verters. The number of channels, the number of parallel- connected converters in the channel and the order of their inclusion is determined by the load power, the number of solar and rechargeable batteries, and the accepted opera- tion algorithm. Let us consider the PSS of a SC when operating a SlB conversion channel, made on the basis of a shunt con- verter (fig. 1), on the topology of a boost converter (L1, VT1, VD1, C3) with negative feedback implemented by a digital control system (DCS). Since the solar battery op- erates on the current branch of its current-voltage charac- teristic, it was represented in this diagram as a current source ISlB. The feedback signal Uf from the voltage sen- sor VS, connected to the output of the SlB channel, is converted by the ADC into digital for, subtracted from the signal of the setting voltage u and then through the cor- recting element CS, the digital pulse-width modulator PWM and driver D are fed to the power switch VT1 of the shunt converter. Fig. 1. Functional diagram of the shunt converter with a digital control system Рис. 1. Функциональная схема шунтового преобразователя с цифровой системой управления The scheme uses a pulse-width modulator of the first kind (PWM1) with a delay of one sweep signal. It is PWM1 that is implemented in digital control systems based on microcontrollers. The current source IL is de- In this diagram, the nonlinear signal transmission links are the pulse-width modulator PWM (its analog model) and the multipliers X1, X2, which are described by the relations signed to simulate the disturbing effect of the load current on the stabilization system. The input filter, assembled on iC (t ) = ëé1- FK (t )ûù × iL1 (t ), (1) elements С1, С2, R1, is intended [10] for smoothing the voltage ripples on the solar battery (current source ISlB). The output filter С3 is designed to smooth the ripple on the load, indicated by the equivalent resistance RL. To implement the proposed method for the synthesis of a corrective element, in accordance with step 1, it is necessary to present a mathematical model of a shunt en- ergy converter in the form of a structural diagram built in the basis of discontinuous switching functions. The digital control system, at this stage of the study, is considered ideal, assuming that the ADC performs the conversion instantaneously with infinitely high frequency, and the adder S1 and the corrective section of the short circuit are equivalent under these assumptions to the ideal analog link. Such a structural scheme, built according to the method described in [11], is presented in fig. 2. This block diagram is obtained by representing the inductances L and the capacities C of the shunt converter by ideal in- tegrators with the corresponding coefficients 1/L and 1/C. The resistors in the power circuit of the shunt converter are replaced by instantaneous links with transfer coeffi- cients R, if it is a current-to-voltage link, or 1/R, if it is a voltage-to-current link. The summation of voltages and currents in the respective voltage circuits and current nodes reflect the adders S1 - S6 of the block diagram. The key part of the shunt converter, consisting of the power transistor VT1 and diode VD1, is represented by two uni- directional controlled transmission links made in the form uL (t ) = éë1- FK (t )ûù × uC 3 (t ), where iC(t), uL(t) - are output signals of multipliers X1, X2; iL1(t), uC3(t) - are input signals of the multipliers (state variables of the described dynamic system), which are fed to the "power" input of the corresponding multi- plier; [1-FK(t)] is a control signal that is generated by a PWM block and is fed to the control inputs of both mul- tipliers. Assuming that the continuous part of the SlB trans- form channel (control object) has the properties of a low- pass filter, which is naturally always performed with high quality requirements for the output voltage, the impulse switching function [1-FK(t)] can be reduced to a continu- ous function. Indeed, in the steady state operation of the system, a continuous signal x(t) at the input of AM PWM can be represented as the sum of the constant component of the signal Х0, which characterizes the operating point of the steady state mode, and the small deviation dx of this signal from X0. In this case, the output of AM PWM sets the switching function FK0, the relative duration of which will correspond to the signal X0, if the signal x(t) is normalized and its maximum value is equal to one. Thus, under the condition of signal normalization x(t), expres- sions (1), describing the operation of multipliers, can be rewritten in the form (2), where the time dependence of the values used is not indicated to simplify the expression. i = (1- x)× i , of multipliers marked with X. And one of the links X1 plays the role of the current transfer link iL1 of the input choke L1 to the output capacitor C3, and the second X2 C L1 uL = (1- x)× iC 3 . (2) takes the role of the transmission link of the output volt- age UC3 in the input circuit of the choke L1. Both links of the multipliers X1 and X2 are controlled by a common signal (1-FK(t)), which is fed to their control inputs. As it was mentioned above, an ideal digital control system is Since now expressions (2) describe the product of continuous functions, they can be linearized by the for- mula for finding the differential of the product, which allows us to obtain analog and consists of an instantaneous voltage sensor with a transfer coefficient Cs; source of voltage u(t); ad- der S7; analog model of the correcting section (AMCS), diC = (1- X 0 )× diL1 - IL10 × dx, duL = (1- X 0 )× duC 3 -UC 30 × dx, (3) with an unknown transfer function and analog model of a pulse-width modulator (AM PWM), which in its turn converts a continuous signal x(t) acting on the output of the AMCS in a pulse signal FK(t), which is a switching function of the shunt converter, and accepts values “0” or “1”. The analog models of the corrective link of the AMCS and the pulse-width modulator AM PWM will be understood as some decisive blocks of an analog com- puter that implement the corresponding signal conversion functions. This condition will allow us to present block diagrams in the form of temporary functions. The block diagram of fig. 2 which is built in the basis of switching discontinuous functions, clearly shows all the internal feedbacks of the shunt converter and the role that these connections play in the system, and also allows one to select non-linear transmission links. where diC, diL1, duL, duC3, dx - are small deviations of the corresponding signals from their steady-state values IC0, IL0, UL0, UC30, X0. According to expressions (3), each of the multipliers and analog models of PWM, which are nonlinear links, are replaced by two linear instantaneous links [12] with transfer coefficients (1-X0), IL10 and (1-X0), UC30, respectively, as well as adders (subtractors ) S8 and S9, as shown in fig. 3. Since the rest sections of the block diagram in fig. 2 are linear, then the whole sys- tem becomes linearized. However, it should be remem- bered that in a linearized system, the variables are small deviations of signals from their steady-state value, there- fore these signals are marked with a “tilde” icon. Further simplification of the linearized block diagram and its reduction to the form shown in fig. 4 is a cumber- some task rather than a complex one. Fig. 2. Block diagram of the shunt converter with input filter and control system Рис. 2. Структурная схема шунтового преобразователя с входным фильтром и системой управления Fig. 3. Linearized block diagram of the shunt converter with input filter and control system Рис. 3. Линеаризованная структурная схема шунтового преобразователя с входным фильтром и системой управления Fig. 4. Simplified linearized mathematical model of the shunt converter with input filter and control system Рис. 4. Упрощенная линеаризованная математическая модель шунтового преобразователя с входным фильтром и системой управления Formulas 4-6 allow us to determine the transfer func- tions of the links W1(p) - W3(p) shown in fig. 4 given in [13]. This technique means to make a simulation model of a pulsed shunt converter with open-loop feed- back in the software design environment Matlab. W1 ( p) = % ( p) = I C X% ( p) Simulink; to take down logarithmic amplitude and phase-frequency characteristics on this model using the means provided by the software environment during é U × 1 × 1 × (1- X ) ù the experiment; to compare the characteristics obtained ê C 30 ê = ê RL1 æ L1 ç R è L1 p +1ö ÷ ø 0 ú ú - I ú ; (4) experimentally on the model, with the characteristics calculated by the small-signal model (linearized block × × + + ê 1 1 æ 1 R1 ö L0 ú ê R æ L1 ö ç C1p R1C2 p +1 ÷ 1 ú diagram) with a broken feedback loop. Feedback loop in fig. 3 is broken at the output of the analog model of the ê L1 ëê ç R è L1 p +1÷ ø è ø ú úû I% ( p) Подпись: 2 correcting section AMCS, and the delay link, shown con- ventionally by a dotted line, is not yet taken into account. This means that in the synthesis of the transfer function of the correcting element, the influence of the delay % W2 ( p) = I ( p) = is neglected. The test signal is sent to the system input SlB 1 1 æ 1 R1 ö x(t) = X0 + (dx)·sinwt, while X0 determines the operating R × æ L1 ö ×ç C1p + R1C2 p +1 ÷ × (1- X 0 ) point of the steady state, the “small” amplitude dx of the R L1 ç = è L1 p +1÷ è ø ø ; (5) sine determines the small deviation from the operating point, and the frequency w changes in the desired range. 1 × 1 ×æ 1 + R1 ö +1 The transfer function of an open object with respect to a RL1 æ L1 R ç ÷ è ø è L1 p +1ö ç C1p R1C2 p +1 ÷ ø control action, the adequacy of which is further proved, will have the form (7), with ĨL = 0 и ĨSlB = 0. W ( p) = U%С 3 ( p) = W ( p) ×W ( p) . (7) W ( p) = U%C 3 ( p) = у X% ( p) 1 2 3 3 I% ( p) The results of such a test are presented in fig. 5. RL é 1 × 1 ×æ 1 + R1 ö + ù Dependencies shown in fig. 5 confirm the adequacy p +1 è ø ú R C3 p +1 ê R æ L1 L ê L1 ç ö ç C1p R1C2 p +1 ÷ 1ú ÷ of the linearized block diagram in fig. 3 and the simplified scheme in fig. 4 in the considered frequency range. = êë è RL1 ø úû . (6) RL × 1 × 1 × (1- X )2 + To determine the parameters of the correcting section, in accordance with paragraph 6 of the proposed method- R C3 p +1 R æ L1 ö 0 L L1 ç R è ø è L1 p +1÷ ø ology, it is necessary to build a family of Bode diagrams of an open loop uncorrected system at the “extreme” op- + 1 × 1 ×æ 1 + R1 ö +1 erating points. The definition of “extreme” operating RL1 é L1 êë RL1 p +1ù úû ç C1p R1C2 p +1 ÷ points becomes clear if we pay attention to the fact that in expressions 4-6, the electrical parameters of the circuit characterizing its mode of operation are used as transfer Checking the adequacy of the obtained linearized structural diagram in fig. 3 and the simplified block dia- gram in fig. 4, which can also be called small-signal mod- els of the system, is carried out according to the method coefficients of the links, for example, UC30, X0, IL0, which, in turn, depend on current ISlB and load resistance RL. Fig. 6, a shows the families of characteristics when the current of the solar battery ISlB is changed and the load resistance RL is in the range from their minimum value to the maximum, which explains the use of the term “ex- treme” operating points. Based on this family of characteristics, obtained according to a linearized block diagram and presented in fig. 6, a, in accordance with paragraph 6 of the pro- posed methodology, the type of correcting section (7) and its coefficients were synthesized: of the correcting section parameters and requires their experimental verification. The results of the simulation modeling of the SC PSS during SlB channel operation with closed feedback and a synthesized corrective link are shown in fig. 8. Analysis of the simulation modeling results shows that when the In load current varies in the range from 3 to 7 A, the steady-state voltage Un on the power bus of the power X% ( p) T p + 1 supply load does not change, which indicates a zero static WCS ( p) = = Kkr 1 , (8) control error. From the given time diagrams it can be seen g% ( p) p(T2 p + 1) that with sharp changes in the load current In, the voltage spikes Un do not go beyond the permissible range. The kr 1 2 where K = 2,4·104 с-1; T = 0,005 с; T = 6,6·10-6. overshoot of voltage during load dropping is 0.8 %, the The Bode diagrams of the corrected system, corre- sponding to the same “extreme” points of the variation range of the parameters, are presented in fig. 6, b, from which it is possible to determine the obtained stability reserves (16 dB, 35 degrees) with the worst combinations of the considered parameters. To test the dynamic characteristics of the SlB chan- nel, in the Simulink package of the Matlab design envi- ronment, its simulation model was built (fig. 7, a) with closed feedback and a synthesized analog correcting section (8). The model parameters are presented in table, voltage fall during load rise is 0.94 %, the durations of transient processes during loading current dropping and rise are 3.3 ms and 4.3 ms respectively. The obtained transfer functions and the achieved dy- namic characteristics correspond to the analog correcting section. To determine the parameters of the transfer func- tion of the corresponding discrete correcting section, it is necessary to implement the bilinear form of the transition from the Laplace transform to the z-transform, by replac- ing the Laplace operator p [15] in the transfer function Wcs ( p) (analogue function of the prototype): the internal structure of the PWM1 unit is shown in fig. 7, b, and the scheme of the T1 trigger subsystem is shown in fig. 7, c. If it is necessary, the proposed Wcs (z) = Wcs ( p) p , when p = 2 ( z -1) , T ( z +1) (9) simulation model can take into account and verify the effect of the delay section, which, according to the authors of [14], does not allow accurate calculation where T is the sampling period of the digital correcting section. After that, it is necessary to reduce the transfer function to the canonical form (10). Fig. 5 Bode diagrams in the nominal mode of operation constructed on the small-signal model of the expression (7) (solid lines) and taken on the simulation model (marked with a “+” sign) Рис. 5. ЛАЧХ и ЛФЧХ в номинальном режиме работы, построенные по малосигнальной модели выражение (7) (сплошные линии) и снятые на имитационной модели (помечены знаком «+») а A1, F1: Х0 = 0.9324, ISlB = 7.4А, RL = 200Ω A2, F2: Х0 = 0.3243, ISlB = 7.4А, RL = 20Ω A3, F3: Х0 = 0.8936, ISlB = 4.7А, RL = 200Ω A4, F4: Х0 = 0.2908, ISlB = 4.7А, RL = 30Ω b Fig. 6. Bode diagrams: а - open-loop uncorrected system; b - open-loop corrected system Рис. 6. Логарифмические частотные характеристики: а - разомкнутой нескорректированной системы; b - разомкнутой скорректированной системы а b c Fig. 7. Simulation model: а - energy conversion channel SlB; b - subsystem model PWM1; c - subsystem model Т1 Рис. 7. Имитационная модель: а - канала преобразования энергии БС; b - модель подсистемы ШИМ I; c - модель подсистемы Т1 The parameters of the simulation model elements of the shunt converter Symbol Value Symbol Value С1 1 µF RL Varies in range 13.5 ... 135 Ohm С2 1 µF ISlB 7.4 А R1 10 Ohm IL Impulse: from 0 to 4 А. L1 170 µF Diode1, Switch 1 Ideal RL1 33 mOhm F 100000Hrz C3 1200 µF Uз Setpoint voltage: 1V Fig. 8. Timing diagrams of the load current (In), on-load voltage (Un), the control signal at the input of the PWM modulator of the SlB channel (Gamma) Рис. 8. Временные диаграммы тока нагрузки (In), напряжения на нагрузке (Un), сигнала управления на входе ШИМ-модулятора канала БС (Gamma) For the transfer function of the correcting section with the parameters specified in (8), the coefficients of expres- sion (10) have the following numerical values: b0 = 51.7759; b1 = 0.1034; b2 = -51.6724; a1 = 1.1380; a2 = -0.1380; T = 10-5 с. (b + b z-1 + b z-2 ) The diagrams shown in fig. 10 are obtained on a simu- lation model of a closed SAR (fig. 7) with an analog cor- recting section, the parameters of which are given on the model, and with a digital correcting section (10) under the same conditions as the SAR with an analog correcting section. W (z) = 0 1 2 , (10) Comparison of timing diagrams in fig. 10 with timing cs (1- a z-1 - a z-2 ) 1 2 diagrams in fig. 8 shows that they differ quantitatively Expression (10) makes it easy to go to the recurrence relation (11) and the difference equation (12) by no more than 0.01-0.02 V, that is, they practically coincide. m n x%[i] = åb × g%[i - j] +å a × x%[i - j], (11) Conclusion. The method of synthesizing a digital con- where j =0 j = 0,1...m j j j =1 - is the number of coefficients in the troller for a channel of converting the solar battery energy in a power supply system of a spacecraft based on a block diagram of a converter and on the method of switching expression (10), i - is the current reading of the input g(i) and output x(i) signals of the correcting section x(k) = b0 × g(k) + b1 × g(k -1) + b2 × g(k - 2) - a1 ´ ´ x(k -1) - a2 × x(k - 2). (12) Differential equation (12) allows us to calculate the output signal of the digital correcting section from the readings of the input signal, thus realizing the digital implementation of the obtained analog correcting section, in accordance with paragraph 9 of the proposed method. The loading diagram of the synthesized correcting section, which is also a numerical simulation model of a digital controller, is shown in fig. 9. This scheme is constructed according to expression (12), and the z-1 link presented in the diagram is a delay link for one sampling rate period. Fig. 10 shows the timing diagrams of the output voltage, obtained by abruptly changing the load current, with analog and digital correcting sections. discontinuous functions has been developed. On the basis of the proposed methodology, the type and coefficients of the correcting section were selected, providing the necessary stability stocks. The parameters of the correcting section are chosen so that the characteristics of the system are close to the set- tings of the technical optimum. It was shown on the simulation model built in the Simulink package of the Matlab design environment that with the selected type of correcting section and the se- lected numerical values of its coefficients, the system ensures stable operation throughout the claimed range of disturbance changes. This suggests that, despite the pres- ence of delays in the digital information processing chan- nel and in the PWM channel, the proposed method allows us to synthesize a digital control system of a power supply device for a spacecraft that meets the specified require- ments for accuracy and speed. Fig. 9. Simulation model of the digital controller Рис. 9. Имитационная модель цифрового регулятора Fig. 10. Timing diagrams of the output voltage with an abrupt change in the load current, obtained in a system with analog and digital correcting sections Рис. 10. Временные диаграммы выходного напряжения при скачкообразном изменении тока нагрузки, полученные в системе с аналоговым и цифровым корректирующими звеньями

About the authors

V. N. Shkolnyi

JSC “Academician M. F. Reshetnev “Information Satellite Systems”

52, Lenin St., Zheleznogorsk, Krasnoyarsk region, 662972, Russian Federation

V. D. Semenov

Tomsk State University of Control Systems and Radioelectronics, Research Institute of Space Technologies

40, Lenina Av., Tomsk, 634050 Russian Federation

V. A. Kabirov

Tomsk State University of Control Systems and Radioelectronics, Research Institute of Space Technologies

40, Lenina Av., Tomsk, 634050 Russian Federation

M. P. Sukhorukov

Tomsk State University of Control Systems and Radioelectronics, Research Institute of Space Technologies

Email: max_sukhorukov@mail.ru
40, Lenina Av., Tomsk, 634050 Russian Federation

D. S. Torgaeva

Tomsk State University of Control Systems and Radioelectronics, Research Institute of Space Technologies

40, Lenina Av., Tomsk, 634050 Russian Federation

References

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