Learning and Adaptation for Optimization and Control of Complex Renewable Energy Systems
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Cheng Guo, Linzhen Zhong, Jun Zhao, Guanbin Gao, Yingbo Huang, "FirstOrder and HighOrder Repetitive Control for SinglePhase GridConnected Inverter", Complexity, vol. 2020, Article ID 1094386, 10 pages, 2020. https://doi.org/10.1155/2020/1094386
FirstOrder and HighOrder Repetitive Control for SinglePhase GridConnected Inverter
Abstract
With the increasing demand of users for power sources and quality, how to provide highquality renewable clean energy has become a key issue of power electronics. The main idea of this paper is to develop a composite control including a PI control and repetitive control for a singlephase gridconnected inverter to eliminate the effects of harmonics, which can obtain better steadystate and dynamic responses of the singlephase inverter system and reduce the net current harmonics. The modelling of a singlephase inverter is first introduced; then a firstorder repetitive control is developed for the proposed gridconnected inverter. Moreover, a highorder repetitive controller is adopted to further improve the robustness against the uncertainties in the period of signals. The stability and performance analysis are given for the firstorder repetitive control and highorder repetitive control. Finally, comparative simulations are conducted in a circuitlevel inverter model, which show the effectiveness of the proposed method.
1. Introduction
The development of traditional fuelbased energy is becoming stringent due to its serious pollution problem and other shortcomings, while with the rapid development of human economy and society, the investigation of clean and highefficiency energy has been gradually expanded [1, 2]. In fact, the exploration of renewable energy and alternative energy has been gradually becoming a key research area in various countries and industries. In the fields of power generation, thermal power is now being gradually replaced by renewable energy such as hydropower, photovoltaic and wind power, nuclear energy, and solar energy [3, 4].
However, there are some critical issues to be addressed in the renewable energy powered gridconnected systems. For instance, photovoltaic power generation is greatly affected by many factors such as weather and region, and thus the power supply is not stable [3, 5]. In addition, due to the characteristics of large investment and wide land occupation, the centralized photovoltaic gridconnected system has limited application, though the distributed photovoltaic gridconnected system has more potential applications, that is, smallscale photovoltaic gridconnected system. However, it remains as an open problem to ensure that a large number of distributed systems can be successfully connected to the grid through power electronic methods, that is, gridconnected inverter [6–9], which can retain the stability of public grid while ensuring the efficiency of the grid connection and avoiding the interference of largescale grid connection to public grid. This is of great significance and necessity not only for the photovoltaic power generation system but also for other renewable energy powered systems [5]. Among these gridconnected systems, the gridconnected inverter is the core part [10–12]. As an interface device, the gridconnected inverter used to transform the unstable DC power received by photovoltaic cells into a more stable AC power supply and feed it into the public grid plays an important role, which has also attracted significant attentions in the power electronics industry [13–15].
The gridconnected inverter is to convert the DC power output extracted from the photovoltaic array into the AC power with the same frequency and phase as the grid voltage, while the output current harmonic should be small and the sinusoidal AC is incorporated into the grid. Hence, proper control strategies are essential for this purpose. There are two basic control strategies of the gridconnected current, that is, indirect current control and direct current control [16–19]. Indirect current control uses the vector relationship of the output voltage, so that the gridconnected current can achieve the expected amplitude and phase, without the introduction of AC current feedback. Direct current control applies the AC input current command and AC current feedback, so as to make the input current tracking command change through the regulator’s tracking control; it has good dynamic performance. In this line, there are many control methods developed for gridconnected inverter, such as PI control, hysteresis control, deadbeat control, robust control, repetitive control, and adaptive control [20–26]. PI control is one of the commonly used control methods since this algorithm is simple and reliable, but it cannot realize the current adjustment without static difference. It may contain oscillations, so that the current quality of the network side is limited. Hysteresis control is designed to control the error of the inverter output current to track a sinusoidal reference current within the hysteresis width, which has the advantages of realtime control and fast response. However, this method suffers the problems of switching loss and tracking accuracy. Deadbeat control is a method based on the precise mathematical model of the controlled plant. The basic principle is to calculate the duty cycle of the next switching cycle of the inverter power device according to the state equation and the output feedback signal and the required reference signal at the next time. Deadbeat control has the advantages of fast response and low harmonic content. However, it has specific requirements for the calculation accuracy and speed of the controller.
Based on the above discussions, we will develop a repetitive control for a gridconnected inverter. Repetitive control is a control method developed based on the internal model principle, where the basic principle is that the harmonic distortion of the previous cycle will be eliminated in the next cycle. The controller is used to add control signals in the next cycle for correction and compensation, so as to eliminate the periodic interference. To this end, we first introduce the modelling of a singlephase inverter. Then, a firstorder repetitive control is developed for the proposed gridconnected inverter. Moreover, a highorder repetitive controller is also adopted to further improve the robustness against the uncertainties in the period of signals. The stability and performance analysis are all given for the proposed two control strategies. Finally, comparative simulations are conducted in a circuitlevel inverter model to show that the highorder repetitive control can obtain better steadystate and dynamic features of the singlephase inverter system and reduce the net current harmonics.
The major contributions of this paper include the following:(1)The repetitive control is used to eliminate the total harmonics in the current of the gridconnected inverter(2)A highorder repetitive control is proposed to address the variation in the period of grid, which can guarantee the control system robustness of the gridconnected inverter(3)The stability and performance analysis for the proposed firstorder repetitive control and highorder repetitive control are all given
This paper is organized as follows: In Section 2, we introduce the singlephase inverter type and modelling. In Section 3, a firstorder repetitive control and highorder repetitive control are introduced based on the proposed gridconnected inverter to suppress the total harmonics in the current. The stability and performance analysis are also given. Comparative simulations are conducted in Section 4, and some conclusions are stated in Section 5.
2. Preliminaries and Problem Formulation
2.1. SinglePhase Inverter Type
Gridconnected inverter is the poststage structure of photovoltaic grid, which is a device to convert DC into AC. Inverters are widely used in the electric, traffic, military, and other fields. There are three main topologies of singlephase inverter: pushpull inverter, halfbridge inverter, and fullbridge inverter [27–31].
2.1.1. PushPull Inverter
The pushpull inverter has a simple structure, which consists of two common negative power switches and a stepup transformer with a central tap on its original side. The main disadvantage is that the output transformer is easy to saturate. The utilization ratio of transformer is relatively low, and it is difficult to drive the inductive load, which is suitable for the occasion of low DC bus voltage. The topology of pushpull inverter is given in Figure 1.
2.1.2. HalfBridge Inverter
The halfbridge inverter circuit is composed of two capacitors in series, a pair of controllable devices, and a bridge arm for antiparallel diodes. When the capacity of the two voltage dividing capacitors is large enough, the capacitor voltage of the power switch device keeps when being in switching on and off state, which has a strong ability to resist voltage output imbalance. However, its disadvantage is that the AC output voltage amplitude of the main circuit is half of the input voltage, and the DC side still needs the voltage balance of two capacitors, the DC side voltage utilization is low, and the harmonic of the grid current is large. Halfbridge inverter has been widely used in the low power level inverter. The topology of halfbridge inverter is shown in Figure 2.
2.1.3. FullBridge Inverter
The fullbridge inverter circuit has two bridge arms, which can be composed of two halfbridge circuits. The topology is given in Figure 3. One pair of fullcontrol devices is and , and the other pair is and . The same pair of fullcontrol devices is on at the same time, and the two pairs of devices are complementary.
Under the same DC input voltage, the maximum output voltage of the fullbridge inverter is twice of that of the halfbridge inverter. When the power is the same, the output current and the current through the switch element are half of the halfbridge inverter circuit. Fullbridge inverter circuit is simple in terms of structure and easy to control; therefore, it has been widely used in the highpower occasions.
At the same time, the inverter is divided into the active inverter and passive inverter according to whether there is a power supply or not. Active and passive refer to whether the inverter is connected to the power supply. In this project, the photovoltaic gridconnected inverter needs to be connected to the grid, which belongs to the gridconnected inverter, so that the active inverter circuit is selected.
From Figure 3, for the inverter with load, its filter circuit is usually composed of an LC filter circuit with inductance and capacitance. For the gridconnected inverter, the output terminal needs to be connected to the grid. The filter capacitor C will be clamped by the power grid and cannot play the role of filtering, and the parallel capacitor can be removed equivalently, which can be seen in Figure 4.
This paper mainly focusses on the later stage of the gridconnected inverter system, which consists of a DC power supply provided by the former DCDC part, four power switch devices (IGBT) inverter bridge, and filter inductor. The inverter outputs sinusoidal current with the same frequency and phase the same as the grid voltage and enters the grid.
2.2. Modelling of SinglePhase Inverter
The main circuit of the singlephase gridconnected inverter is shown in Figure 5, which is a voltage type fullbridge circuit composed of four IGBTs and continuous current diodes in reverse parallel. is the IGBT, is the continuous current diode, is the filter inductance on one side of the power grid, and is the parasitic resistance of the filter inductance. and are the corresponding AC power supply and DC power supply, respectively.
According to Figure 5, we can obtain the flowing equation:
Then, we have
For the DC side, we get
Then, we know that the singlephase gridconnected inverter adopts the double loop control strategy of voltage and current loop, and the outer loop is the DC side voltage loop, whose function is to keep the DC side voltage stable. The inner loop is a current loop, the output current of the inverter is in the same phase as the grid voltage, and the current amplitude is determined by the output of the voltage loop regulator. This paper only studies the inverter function of the later stage of the gridconnected system; therefore, the front end of the inverter bridge can be regarded as a constant voltage DC power supply, while the voltage loop is not considered.
3. Repetitive Control Design of SinglePhase GridConnected Inverter
Repetitive control [32] is derived from the internal model principle (IMP). The merit of IMP is that, in a stable closedloop control system, the internal control includes the generator of external controlled signals, so as to realize the adjustment without steadystate error and suppress the periodic interference. When the steadystate error of a system is zero, thus the input of the controller is zero. At this time, the reference signal and the feedback signal still exist, the periodic disturbance still changes according to the original dynamic characteristics, and the controller still outputs the corresponding control signal to keep the system adjusted without static error. Therefore, the essence of realizing no steadystate error is that the controller is a structural model, which can reflect and process external signals.
The proposed controller should be designed to suppress the current harmonics to reduce the distortion of sine wave of current feed into the network. However, the jamming signal in the power electronic system is periodic; then a particular control structure shown in Figure 6 will be used.
Remark 1. This control system in Figure 6 consists of an RC controller and PI controller. The PI controller is designed to stabilize the closedloop system, whose parameters can be tuned as [33]. This paper will focus on the design of repetitive control (e.g., FORC and HORC) and their comparisons.
3.1. FirstOrder Repetitive Control (FORC)
3.1.1. FirstOrder Repetitive Control Structure
As shown in Figure 7, repetitive control can be designed based on an internal model, which can learn a signal of the period T and duplicate it, even if the input of this model is set to zero. Consider the frequencydomain response; this internal model introduces infinite gains (without filter ) at the specific frequencies rad/s . Then, according to the wellknown IMP, zeroerror tracking or rejection of period signals at these frequencies can be guaranteed if the closedloop system is stable.
A lowpass filter is usually used to improve the robustness of controlled systems, although it may reduce the gains in these specific frequencies. is set as a secondorder Butterworth filter in this paper by designing its cutoff frequency. The compensator is utilized to retain the system stability when the internal model is added to the closedloop system. Thus, the transfer function of RC is described by
It is found that the use of lowpass filter will lead to an unavoidable phase lag, which will shift the frequencies at which the maximum gains are obtained. Specifically, the frequency shift is mainly related to its phase. However, a proper compensation can be further incorporated to address this issue as [34].
3.1.2. Stability and Performance Analysis
In this part, we will give a proposition regarding the system stability. The following conditions should be fulfilled to guarantee the stability of the closedloop system.
Proposition 1. The closedloop system in Figure 6 with FORC in Figure 7 is stable if the following conditions are fulfilled [35, 36]:(1)The closedloop system without RC is stable; that is, is stable(2); that is, the filter should have a gain close to 1 within the suitable bandwidth(3)
Proof. The proof is similar to that given in [36] and thus is not presented here.
According to these three conditions, we have different design methods to satisfy the conditions. The PI control is designed to fulfill condition (1), a lowpass filter is used to fulfill condition (2), and can be designed to satisfy condition (3). For any minimumphase plant , a constructive selection of is given aswhere is a constant.
By substituting (5) into condition (3) of Proposition 1, we have the following.
Corollary 1. The closedloop control system shown in Figure 6 with a stable minimumphase and compensator (5) is stable if the following conditions are fulfilled:(1) is a stable system, (2) is a constant fulfilling,
Remark 1. The compensator of (5) is valid for stable minimumphase plants, because there are no zeropole cancellations in the righthalf plane in this case. For nonminimumphase plants, an alternative approach is to cancel the minimumphase zeroes and poles and to compensate for the phase of the nonminimumphase ones as [37].
3.2. HighOrder Repetitive Control (HORC)
3.2.1. HighOrder Repetitive Control Structure
Although FORC in Figure 7 is easy to implement, its performance may degrade when the period of the reference or disturbance has uncertainties; that is, it is sensitive to the variation in the period of signals. In order to tackle these disadvantages, highorder repetitive control (HORC) was proposed in [38–40] because of its strong robustness, which is given in Figure 8.
In Figure 8, is a weighted sum function of repetitive loops. Thus, the transfer function of HORC can be written as
It is shown that HORC uses a multiple loop internal model, which replaces the delay by a weighted sum function of several delays given as follows:where is the number of delays, that is, the order of RC loop.
Similar to FORC, the gains of HORC are infinite at the multiples of the input signal frequency. In particular, the gain of (6) will tend to infinity if . As shown in [38], we substitute into (7) and have
It is noted that, for and , (8) is reduced to FORC. For the case , we need to determine the weight parameters . Inspired by the idea of [38], we can make the firstorder derivative of with respect to zero at the specific frequency. This imposes the condition
According to (9), the following equation is true:
As shown in [38], to further decrease the sensitivity for periodtime variations by using the weight parameters of HORC, we can make th derivatives equal zero, so that
The weight parameters can be calculated based on (7)–(11). For example, when and 3, we can set and ; , , and , respectively.
3.2.2. Stability and Performance Analysis
The stability analysis of HORC system is similar to that given in Proposition 1, and thus we have the following.
Proposition 2. The closedloop system in Figure 6 with HORC in Figure 8 is stable if the following conditions hold:(1)The closedloop system without the HORC is stable; that is, is stable(2)(3)
Proof. The proof is similar to that of Proposition 1 by replacing with .
For any minimumphase , we design asBy substituting (12) into the third condition in Proposition 2, we can get. . Furthermore, condition (2) in Corollary 2 can be derived as . Hence, we have the following.
Corollary 2. For stable, minimumphase , the closedloop control system shown in Figure 8 with HORC is stable if the following conditions are true:(1)(2) is a constant fulfilling
4. Simulation Results and Analysis
In this section, we validate the effectiveness of the proposed control algorithms by using Simpower systems Toolbox in Matlab/Simulink. The simulation model of a singlephase gridconnected inverter is built to compare the current harmonic suppression of FORC and HORC. The FFT analysis tool in Simulink is used to get the total harmonic distortion (THD). Through the block diagram of current loop control system in Figure 6 and the topology structure in Figure 5 of the singlephase gridconnected inverter, the simulation diagram of the singlephase inverter can be obtained as Figure 9. In addition, the main circuit parameters are shown in Table 1.

The main circuit simulation module is composed of a DC power supply, an IGBT inverter bridge, a filter inductor, an AC source of public power grid, and an output current detection module. The control part consists of a control circuit, a PWM generator module, and a PLL module. The current loop is used in the control loop. The control module includes a PI control, a firstorder repetitive control, or a highorder repetitive control, respectively. In this paper, PI control, PI + FORC, and PI + HORC are all simulated. In order to fully analyze the characteristics of these three control methods, the steadystate waveform, harmonic suppression, and dynamic tracking are all given.
4.1. The SteadyState Waveform
Figures 10–12 show the voltage and current of AC side when the PI control, PI + FORC, and PI + HORC reach the steady state, respectively. Figures 13–15 are the corresponding harmonic analysis results. It can be seen from Figures 10–12 that the three methods can achieve the same phase of voltage and current in the steady state. However, the current harmonic is relatively large and the THD reaches 2.10% in Figure 12. In Figure 11, the harmonics are correspondingly less and the THD is reduced to 1.98% when PI + FORC is adopted. Finally, PI + HORC is used to suppress the current harmonics more restrictively, where the THD is reduced to 1.84% in Figure 15.
4.2. Harmonic Suppression
In order to further verify the harmonic suppression of the three control methods, highorder harmonics are added to the system manually. The frequency of the grid is set as 50 Hz; then the 7th and 9th harmonics are injected. It is shown in Figure 16 that PI control has poor performance to suppress harmonics and it leads to great distortions in the current. It can be seen from Figure 17 that, after the introduction of FORC, the highorder harmonics are suppressed and the output current retains a better sinusoidal waveform. After using HORC, we can get a smoother sine curve, as shown in Figure 18.
4.3. Dynamic Tracking
Through the above analysis, we can find that the HORC has a better harmonic suppression performance. In order to further verify the tracking performance of HORC, the reference current is suddenly reduced by 30% in 0.26s to simulate the sudden drop of current. It is shown in Figure 19 that the current can be recovered in 0.28s. Therefore, when the reference current changes suddenly, the current can respond quickly, and the reference current can be tracked after a short transient; that is, the system has fast dynamic response.
5. Conclusion
In this paper, a novel control method combining PI control and repetitive control is proposed for a singlephase gridconnected inverter. After introducing the singlephase inverter type and modelling, a firstorder repetitive control and a highorder repetitive control are developed for the gridconnected inverter, respectively. The stability and performance analysis are all given for the proposed two repetitive controls. Finally, comparative simulations are conducted based on the proposed singlephase gridconnected inverter to show that the highorder repetitive control can obtain better steadystate and dynamic features and reduce the net current harmonics. Future work will focus on the reactive power compensation control and its practical application.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by China Southern Power Grid Corporation (no. YNKJXM20180366) and the Scientific Research Fund of Yunnan Education Department under Grant 2020J0067.
References
 C. Zhang, Y. Ye, and H. Chen, “Photovoltaic grid connected inverter based on output current control,” Journal of Electrical Technology, vol. 22, no. 8, pp. 41–45, 2007. View at: Google Scholar
 F. Wang, S. Yu, and J. Su, “Research on solar photovoltaic grid connected power generation system,” Journal of Electrical Technology, vol. 20, no. 5, pp. 72–74, 2005. View at: Google Scholar
 Z. Yao, F. Yu, and Q. Zhao, “Simulation of large photovoltaic grid connected system based on modular multilevel converter,” Chinese Journal of Electrical Engineering, vol. 33, no. 36, pp. 27–33, 2013. View at: Google Scholar
 T. Cheng, M. Chen, Y. Wang et al., “Adaptive robust method for dynamic economic emission dispatch incorporating renewable energy and energy storage,” Complexity, vol. 2018, Article ID 2517987, 13 pages, 2018. View at: Publisher Site  Google Scholar
 X. Guo, X. Zhang, Z. Lu, B. Wang, H. Qi, and X. Sun, “Power/current quality coordinated control strategy of photovoltaic grid connected inverter under unbalanced grid voltage,” Chinese Journal of Electrical Engineering, vol. 34, no. 3, pp. 346–353, 2014. View at: Google Scholar
 Z. Yao and L. Xiao, “Control of singlephase gridconnected inverters with nonlinear loads,” IEEE Transactions on Industrial Electronics, vol. 60, no. 4, pp. 1384–1389, 2013. View at: Publisher Site  Google Scholar
 Q. Yan, X. Wu, X. Yuan, and Y. Geng, “An improved gridvoltage feedforward strategy for highpower threephase gridconnected inverters based on the simplified repetitive predictor,” IEEE Transactions on Power Electronics, vol. 31, no. 5, pp. 3880–3897, 2016. View at: Publisher Site  Google Scholar
 Y. He, H. S.H. Chung, C.T. Lai, X. Zhang, and W. Wu, “Active cancelation of equivalent grid impedance for improving stability and injected power quality of gridconnected inverter under variable grid condition,” IEEE Transactions on Power Electronics, vol. 33, no. 11, pp. 9387–9398, 2018. View at: Publisher Site  Google Scholar
 C. Zhao, J. Liu, Z. Xie, and F. Zhou, “Coordinate control of power/current for gridconnected inverter based on PCI controller under unbalanced grid conditions,” Complexity, vol. 2019, Article ID 5968984, 14 pages, 2019. View at: Publisher Site  Google Scholar
 Q. L. Zhao, X. Q. Guo, and W. Y. Wu, “Research on grid connected control technology of singlephase inverter,” Chinese Journal of Electrical Engineering, vol. 27, no. 16, pp. 60–64, 2007. View at: Google Scholar
 M. Dong and A. Luo, “Design and control method of inverter in photovoltaic grid connected power generation system,” Power System Automation, vol. 30, no. 20, pp. 97–102, 2006. View at: Google Scholar
 X. B. Ruan and Y. G. Yan, “Control strategy of four leg threephase inverter,” Transactions of China Electrotechnical Society, vol. 15, no. 1, pp. 61–64, 2000. View at: Google Scholar
 B. Sahan, A. N. Vergara, N. Henze, A. Engler, and P. Zacharias, “A singlestage PV module integrated converter based on a lowpower currentsource inverter,” IEEE Transactions on Industrial Electronics, vol. 55, no. 7, pp. 2602–2609, 2008. View at: Publisher Site  Google Scholar
 J. Selvaraj and N. A. Rahim, “Multilevel inverter for gridconnected PV system employing digital PI controller,” IEEE Transactions on Industrial Electronics, vol. 56, no. 1, pp. 149–158, 2009. View at: Publisher Site  Google Scholar
 S. Wang, L. Tao, Q. Chen, J. Na, and X. Ren, “USDEbased sliding mode control for servo mechanisms with unknown system dynamics,” IEEE/ASME Transactions on Mechatronics, vol. 25, no. 2, pp. 1056–1066, 2020. View at: Publisher Site  Google Scholar
 J. Fei, T. Li, F. Wang, and W. Juan, “A novel sliding mode control technique for indirect current controlled active power filter,” Mathematical Problems in Engineering, vol. 2012, Article ID 549782, 18 pages, 2012. View at: Publisher Site  Google Scholar
 Y.P. Liu, K.Z. Liu, and X. Yang, “Nonlinear current control for reluctance actuator with hysteresis compensation,” Journal of Control Science and Engineering, vol. 2014, Article ID 150345, 7 pages, 2014. View at: Publisher Site  Google Scholar
 G. Drexlin, V. Hannen, S. Mertens, and C. Weinheimer, “Current direct neutrino mass experiments,” Advances in High Energy Physics, vol. 2013, Article ID 293986, 39 pages, 2013. View at: Publisher Site  Google Scholar
 M. RuizRuigomez, J. Badiola, S. M. SchmidtMalan et al., “Direct electrical current reduces bacterial and yeast biofilm formation,” International Journal of Bacteriology, vol. 2016, Article ID 9727810, 6 pages, 2016. View at: Publisher Site  Google Scholar
 R. Grino, R. Cardoner, R. CostaCastello, and E. Fossas, “Digital repetitive control of a threephase fourwire shunt active filter,” IEEE Transactions on Industrial Electronics, vol. 54, no. 3, pp. 1495–1503, 2007. View at: Publisher Site  Google Scholar
 R. CostaCastello, R. Grino, and E. Fossas, “Oddharmonic digital repetitive control of a singlephase current active filter,” IEEE Transactions on Power Electronics, vol. 19, no. 4, pp. 1060–1068, 2004. View at: Publisher Site  Google Scholar
 G. Weiss, Q.C. Zhong, T. C. Green, and J. Liang, “Repetitive control of DCAC converters in microgrids,” IEEE Transactions on Power Electronics, vol. 19, no. 1, pp. 219–230, 2004. View at: Publisher Site  Google Scholar
 K. Zhou and D. Wang, “Digital repetitive learning controller for threephase CVCF PWM inverter,” IEEE Transactions on Industrial Electronics, vol. 48, pp. 820–830, 2001. View at: Google Scholar
 S. Wang, J. Na, and Y. Xing, “Adaptive optimal parameter estimation and control of servo mechanisms: theory and experiments,” IEEE Transactions on Industrial Electronics, vol. 99, p. 1, 2020. View at: Publisher Site  Google Scholar
 S. Wang and J. Na, “Parameter estimation and adaptive control for servo mechanisms with friction compensation,” IEEE Transactions on Industrial Informatics, vol. 99, p. 1, 2020. View at: Publisher Site  Google Scholar
 J. Zhao, J. Na, and G. Gao, “Adaptive dynamic programming based robust control of nonlinear systems with unmatched uncertainties,” Neurocomputing, vol. 395, pp. 56–65, 2020. View at: Publisher Site  Google Scholar
 P. Giri, A. Das, and A. Bhattacharya, “Pushpull inverter based wireless power transfer system,” in Proceedings of the 7th International Conference on Signal Processing and Integrated Networks (SPIN), pp. 489–493, Noida, India, February 2020. View at: Google Scholar
 Z. Kaczmarczyk and W. Jurczak, “A pushpull classE inverter with improved efficiency,” IEEE Transactions on Industrial Electronics, vol. 55, no. 4, pp. 1871–1874, 2008. View at: Publisher Site  Google Scholar
 E. Babaei, E. Shokati Asl, and M. Hasan Babayi, “Steadystate and smallsignal analysis of highvoltage gain halfbridge switched boost inverter,” IEEE Transactions on Industrial Electronics, vol. 63, no. 6, pp. 3546–3553, 2016. View at: Publisher Site  Google Scholar
 S.H. Ryu, D.G. Woo, M.K. Kim, and B.K. Lee, “Analysis and design of modified halfbridge seriesresonant inverter with DClink neutralpointclamped cell,” IEEE Transactions on Power Electronics, vol. 31, no. 3, pp. 2282–2295, 2016. View at: Publisher Site  Google Scholar
 L. Zhang, K. Sun, Y. Xing, and J. Zhao, “A family of fivelevel dualbuck fullbridge inverters for gridtied applications,” IEEE Transactions on Power Electronics, vol. 31, pp. 7029–7042, 2016. View at: Google Scholar
 X. Liu, G. Ma, P. R. Pagilla, and S. S. Ge, “Stateestimatorbased asynchronous repetitive control of discretetime markovian switching systems,” Complexity, vol. 2020, Article ID 6195162, 13 pages, 2020. View at: Publisher Site  Google Scholar
 R. Yi and C. Boshi, Electric Drive Automatic Control SystemMotion Control System, China Machine Press, Beijing, China, 2009.
 J. Na, X. Ren, R. CostaCastelló, and Y. Guo, “Repetitive control of servo systems with time delays,” Robotics and Autonomous Systems, vol. 62, no. 3, pp. 319–329, 2014. View at: Publisher Site  Google Scholar
 R. Griñó and R. CostaCastelló, “Digital repetitive plugin controller for oddharmonic periodic references and disturbances,” Automatica, vol. 41, no. 1, pp. 153–157, 2005. View at: Publisher Site  Google Scholar
 R. CostaCastello, J. Nebot, and R. Grino, “Demonstration of the internal model principle by digital repetitive control of an educational laboratory plant,” IEEE Transactions on Education, vol. 48, no. 1, pp. 73–80, 2005. View at: Publisher Site  Google Scholar
 W. S. Chang, I. H. Suh, and T. W. Kim, “Analysis and design of two types of digital repetitive control systems,” Automatica, vol. 31, no. 5, pp. 741–746, 1995. View at: Publisher Site  Google Scholar
 M. Steinbuch, “Repetitive control for systems with uncertain periodtime,” Automatica, vol. 38, no. 12, pp. 2103–2109, 2002. View at: Publisher Site  Google Scholar
 T. Inoue, “Practical repetitive control system design,” in Proceedings of the 29th IEEE Conference on Decision and Control, pp. 1673–1678, Honolulu, HI, USA, December 1990. View at: Google Scholar
 R. CostaCastello, G. A. Ramos, J. M. Olm, and M. Steinbuch, “Secondorder oddharmonic repetitive control and its application to active filter control,” in Proceedings of the IEEE Conference on Decision & Control, Atlanta, GA, USA, December 2011. View at: Google Scholar
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