Synchronous Generator Rectiﬁcation System Based on Double Closed-Loop Control of Backstepping and Sliding Mode Variable Structure

: Low-voltage and high-current direct current (DC) power supplies are essential for aerospace and shipping. However, its robustness and dynamic response need to be optimized further on some special occasions. In this paper, a novel rectiﬁcation system platform is built with the low-voltage and high-current permanent magnet synchronous generator (PMSG), in which the DC voltage double closed-loop control system is constructed with the backstepping control method and the sliding mode variable structure (SMVS). In the active component control structure of this system, reasonable virtual control variables are set to obtain the overall structural control variable which satisﬁed the stability requirements of Lyapunov stability theory. Thus, the fast-tracking and the global adjustment of the system are realized and the robustness is improved. Since the reactive component control structure is simple and no subsystem has to be constructed, the SMVS is used to stabilize the system power factor. By building a simulation model and experimental platform of the 5 V/300 A rectiﬁcation module based on the PMSG, it is veriﬁed that the power factor of the system can reach about 98.5%. When the load mutation occurs, the DC output achieves stability again within 0.02 s, and the system ﬂuctuation rate does not exceed 2%. control structure is compared by using two methods, the backstepping control method and the SMVS. The simulation results demonstrate that the system robustness is better and the dynamic performance is greatly improved when the SMVS is applied to the reactive component control structure in the rectiﬁcation control system. The backstepping control method and the SMVS constitute the double closed-loop control system, whose stability is not easily disturbed by large signals, and the power factor can quickly reach a steady state. The experimental results are consistent with the simulation results which prove that the control strategy combining the backstepping control method and the SMVS is an effective and reliable control scheme. This study is only a preliminary discussion of the design and performance of the three-phase voltage rectiﬁcation double closed-loop control system. Furthermore, study the situation of the rectiﬁcation system with nonlinear load, and make it design together with the whole system of the post-nonlinear load, optimize the matching, and get the best performance and reliability.


Introduction
In the fields of shipping, electrolysis, and electroplating, direct current (DC) power supply is required to operate in low-voltage and high-current modes. For example, in the all-electric propulsion system, the generator has an obligation to have lower voltage and higher power output characteristics due to the limitation of output voltage. However, the process of commutating the rectification current leads to the distortion of the alternating current (AC) voltage waveform, which leads to harmonic loss. Therefore, realizing the high performance and high-power DC power supply system is a problem that has received considerable attention in recent years. Since the setting of the proportional-integral (PI) parameters in conventional PI control is sensitive to the rectification itself and includes the hysteresis of the PI control itself, it is difficult to maintain excellent dynamic performance and robustness in four-quadrant operation. Meanwhile, due to the time-varying of the switching power supply system, the control method needs to be optimized to keep the output of the converter in a stable state. The application of the modern nonlinear control method has become one of the research hotspots in power electronic control [1,2]. Commonly, there are internal model control, direct power control, passive theoretical control, and exact feedback linearization control. The purpose of control is to maintain the DC output voltage at the expected value [3]. Direct power control method indirectly controls the output voltage by directly controlling the active and reactive power. In [4], a (1) The asymptotically stable nonlinear controller is designed using the backstepping control method for the single-input single-output (SISO) model of the active component control structure. The reactive component control structure is uncomplicated and it does not reflect the advantages of the backstepping control method, so this paper uses the sliding mode variable structure (SMVS) to improve it. (2) The rectification system can realize low-voltage and high-current DC power generation and high-power output; Electronics 2021, 10, 1832 3 of 18 (3) The rectification system achieves better dynamic and static performance, improves responsiveness and robustness, and maintains the high power factor operation after load mutation.
The main structure of this paper is as follows, and the second part introduces the controller design combining the backstepping control method and the SMVS. The third part describes the circuit simulation design of the system based on Simulink. Experimental results are provided in the fourth part, and the fifth part concludes the paper. Figure 1 shows the outline of the double closed-loop control system. The control system is composed of the active component control structure controlled by the backstepping control method and the reactive component control structure controlled by SMVS.

Methods
Electronics 2021, 10, x FOR PEER REVIEW 3 of 19 (2) The rectification system can realize low-voltage and high-current DC power generation and high-power output; (3) The rectification system achieves better dynamic and static performance, improves responsiveness and robustness, and maintains the high power factor operation after load mutation.
The main structure of this paper is as follows, and the second part introduces the controller design combining the backstepping control method and the SMVS. The third part describes the circuit simulation design of the system based on Simulink. Experimental results are provided in the fourth part, and the fifth part concludes the paper.

Active Component Control Structure
The mathematical model of voltage and current under the a-b-c three-phase stationary axis is converted to the d-q dynamic axis, and the mathematical model of the threephase rectification system under the d-q axis can be obtained as [19]

Active Component Control Structure
The mathematical model of voltage and current under the a-b-c three-phase stationary axis is converted to the d-q dynamic axis, and the mathematical model of the three-phase rectification system under the d-q axis can be obtained as [19] where i d and i q are the active and reactive current components of the AC input current in the d-q coordinate system; e d and e q are the active and reactive voltage components of the AC input voltage in the d-q dynamic system; S d and S q are voltage modulation ratios on d and q axis; u gd and u gq are the voltage modulation control variables on d and q axis; u dc is the DC output voltage; i dc is the DC output current; i L is the load current; R L is the load; ω is the angular velocity of AC current; L is the equivalent inductor in the three-phase permanent magnet synchronous generator (PMSG); R is the equivalent resistance in the three-phase PMSG; C is the DC energy storage capacitance. This paper adopts the quasi-linearization processing used in [19], which is defined as When using the equivalent coordinate transformation, the AC and DC sides keep the input and output power conserved. The AC side active power and DC side power can be expressed as Multiplying both ends of Equation (2) by u dc and put them into Equation (4). Then the improved quasi-linearized model can be derived by The improved Equation (6) is formally a linear system, but considering the variables substitution described above, Equation (2) is still essentially a nonlinear system. Therefore Equation (6) is known as a quasilinear model [20]. The variables substitution is formally linearized and does not change the nature of the system as a nonlinear system. The purpose of using variables substitution here is intended to facilitate the subsequent design of the controller.
When the three-phase AC input voltage is symmetrical and the d-axis of the d-q dynamic system coincides with the three-phase AC input voltage vector, it follows that where E m is the amplitude of AC input voltage. After bringing Equations (4) and (7) into Equations (1) and (6), the SISO system with the active component structure can be represented by In this paper, the backstepping control method is used to control the active component control structure, which is to achieve the goal of maintaining the stable DC output voltage and to ensure that the rectification system can operate normally.
The backstepping control method is built with a subsystem and radiates out another subsystem. Because of this recursive structure, it is possible to start the design process on a known stable system and 'step back', outputting a new controller and gradually stabilizing each external subsystem. The process terminates when the final external control is reached after several steps. Therefore, this process is known as backstepping control [21].
The mathematical model of a SISO nonlinear system is x n = ϕ n (x 1 , · · · , x n )µ + f n (x 1 , · · · , x n ) Electronics 2021, 10, 1832 where µ ∈ R is the state variables of the system, which can be considered as the virtual control variable of the subsystem; f i is the uncertainty of the system; ϕ i is a smooth function and ϕ i (x 1 , · · · x i ) = 0, and the control objective is to drive the state variables x 1 toward its expected value x 1 * . The design concept of the backstepping control method is to take x i+1 (i = 1, 2, · · · , n − 1) as the virtual control variables, then construct a Lyapunov function V i and design a virtual feedback control ϕ i (x 1 , · · · x i ) so that the error has asymptotic characteristics. The backstepping control method ensures the convergence of the error to zero [22]. Finally, construct a Lyapunov function and control variable µ for the whole system so as to achieve asymptotic stability of the whole system [20].
In this paper, the control structure is divided into two parts. The first part is the outer voltage loop subsystem and the second part is the inner current loop subsystem. The output i d of this subsystem can be obtained by using the squared error between the actual DC output voltage value and the expected DC output voltage value as the input of the outer voltage loop subsystem. Then the i d is set as a virtual control variable, and the second part of the subsystem can be designed by tracking the error between i d and the d-axis component of the AC input current, which is the input into the inner current loop subsystem. After that, the control function of the active component control structure is obtained by satisfying the Lyapunov stability theory. Convert the SISO system of active component structure into the mathematical model of the backstepping control method, namely, transform Equation (8) into the form of Equation (9). If [x 1 x 2 ] = [u 2 dc i d ] T is set, we can obtain the following: The step of applying the backstepping control method to design the active component control structure is as follows: Step 01: Define the tracking error as input in the outer voltage loop subsystem, in which u dcref is the expected value of the DC output voltage.
The derivative of θ 1 is .
Step 02: Since i d is not the control input, in the backstepping design, i d is called the virtual control variable, denoted as ϕ 1 (θ 1 ). PI control is used to eliminate the static deviations of the input of the outer voltage loop subsystem [20], so that ϕ 1 (θ 1 ) is expressed as where k 1 is the proportional gain, k 2 is the integral gain. Bringing Equation (16) into Equation (15), the following is given: .
Electronics 2021, 10, 1832 6 of 18 Step 03: Define the Lyapunov function V 1 and its derivative for the outer voltage loop subsystem of .
Step 04: Define the new tracking error as input to the inner current loop subsystem.
The derivative of θ 2 is .
Select control variable: Bringing Equation (23) into Equation (21), we can obtain Step 05: Define the Lyapunov function V 2 and its derivative for the overall active component control structure .
From Equation (26), the reaching condition can be derived by Then the derivative of the Lyapunov function V 2 is negative definite and the active component control structure is global asymptotic stability.

Reactive Component Control Structure
The control objective of the reactive component control structure is to enable the AC input current to quickly track the voltage and to present a nearly complete smooth sine wave so that the power factor of the rectification system can reach 1.
The i q is selected as the input of the reactive component control structure to control the power factor, and the SISO system of the reactive component structure is obtained The reactive component control structure is controlled using the SMVS which is a nonlinear control method that switches from one continuous structure to another according to the current position in the state space. Therefore, SMVS is a variable structure control method that will slide along the boundary of the control structure. The motion of the system as it slides along these boundaries is referred to as the sliding mode, and the geometric trajectory formed by these boundaries is called the sliding surface [23].
An exponential convergence function is utilized to design the control variables of the reactive component control structure. The main advantage is that the process of approaching the sliding mode surface converges with the changed velocity. It starts with the fast velocity and then gradually decreases, and the velocity is close to zero when the sliding mode surface is about to be reached. This not only shortens the convergence time, but also narrows the velocity of the motion point when it reaches the switching surface. The chatter is weakened while the dynamic characteristics of the control structure are improved.
Here, an exponential convergence function is used, making where ε > 0, k > 0, . s = −ks is the exponential convergence term, which solution is s = s(0)e −kt ; . s = −εsgn(s) is the isokinetic convergence term. In which The exponential convergence term in Equation (29) can only guarantee that the motion point approaches the switching surface infinitely, and does not guarantee that it can be reached in finite time. Thus, it is an important condition for the control of the SMVS that the motion point can reach the switching surface in finite time. Therefore, it is necessary to add the isokinetic convergence term to ensure that the convergence velocity tends to be zero, so that the condition of reaching in finite time can be satisfied. To satisfy the relevant performance indicators of the control structure and play a fast convergence while weakening the chattering effect, the two parameters of the exponential convergence term should be as large as possible while the value should be as small as possible.
Taking the error between the expected value and the actual value as the new state variable, Equation (28) can be rewritten as The purpose of the control is to obtain the unit power factor, which is to control i q to 0. Following Equation (30) and the theory of SMVS, it can be inferred that the choice of sliding mode surface can ensure the robustness of the closed-loop system, which can be defined as Therefore, by combining Equations (28), (31), and (32), the control variable of reactive component structure can be calculated as Once u gq is obtained, the pulse can be generated with the space vector pulse width modulation.
The purpose of the control variable of reactive component structure is to ensure that the structure state can reach the switching surface quickly and remain near the sliding surface. To avoid a series of problems such as discretization of continuous variables, the u gq is converted to a voltage signal in the a-b-c stationary axis as the input signal, and the control pulses of the metallic oxide semiconductor field effect transistors (MOSFETs) in the three-phase rectification bridge are obtained directly by the modulation with fixed switching frequency.

Results
For the convenience of the study, a single three-phase rectification module is selected which consists of the three-phase AC input voltage section, the rectification and filtering section, the phase locked loop section, and the double closed-loop control section. Figure 2 shows the simulation model of DC voltage control system, in which the reactive component control structure of the double closed-loop control system will be analyzed by using the backstepping control method and the SMVS, respectively. Table 1 shows the simulation parameters of the three-phase rectification module.    Figure 3 shows the simulation model of the active component control structure. The input of the outer voltage loop subsystem is the squared error between the actual voltage value and the expected voltage value on the DC side. The id is selected as the virtual control variable to construct the state expression of the outer voltage loop subsystem, and the stability of the subsystem is judged by the Lyapunov stability theory, then the reasonable expression can be obtained. According to the definition of the backstepping control method, it is known that the virtual control variable constructed by the outer voltage loop subsystem is used to control the inner current loop subsystem. The φ1(θ1) module is constructed in the outer voltage loop subsystem so that the error between φ1(θ1) and virtual control variable of the outer voltage loop is used as the input of the inner current loop subsystem. The δ2 module is constructed to derive the ugd, which is the output of the active component control structure, and the output satisfies Lyapunov stability theory.   variable to construct the state expression of the outer voltage loop subsystem, and the stability of the subsystem is judged by the Lyapunov stability theory, then the reasonable expression can be obtained. According to the definition of the backstepping control method, it is known that the virtual control variable constructed by the outer voltage loop subsystem is used to control the inner current loop subsystem. The ϕ 1 (θ 1 ) module is constructed in the outer voltage loop subsystem so that the error between ϕ 1 (θ 1 ) and virtual control variable of the outer voltage loop is used as the input of the inner current loop subsystem. The δ 2 module is constructed to derive the u gd , which is the output of the active component control structure, and the output satisfies Lyapunov stability theory.  Figure 4 shows the simulation model of the reactive component control structure using SMVS. The reactive component control structure is employed to make the input current fast track the voltage. Thus, the iq, which is the current of the input reactive component axis, is selected and its expected value is set to zero to achieve the objective of controlling the reactive power nearly to zero. Thence, the error between iq and its expected value is used as input to obtain the ugq that is the output of the reactive component control structure.

Reactive Component Control Structure
s 

Phase Locked Loop
In the phase locked loop of three-phase rectification system as shown in Figure 5, the voltage of q axis is controlled to be equal to 0, which is combined with the feedback output voltage to generate an error voltage [24], and then PI control is used to adjust the voltage without static error. The output of the PI controller is superimposed with the actual input rated frequency to obtain the output frequency, and in turn, the angle and sine values can be obtained.  Figure 4 shows the simulation model of the reactive component control structure using SMVS. The reactive component control structure is employed to make the input current fast track the voltage. Thus, the i q , which is the current of the input reactive component axis, is selected and its expected value is set to zero to achieve the objective of controlling the reactive power nearly to zero. Thence, the error between i q and its expected value is used as input to obtain the u gq that is the output of the reactive component control structure.  Figure 4 shows the simulation model of the reactive component control structure using SMVS. The reactive component control structure is employed to make the input current fast track the voltage. Thus, the iq, which is the current of the input reactive component axis, is selected and its expected value is set to zero to achieve the objective of controlling the reactive power nearly to zero. Thence, the error between iq and its expected value is used as input to obtain the ugq that is the output of the reactive component control structure.

Reactive Component Control Structure
s 

Phase Locked Loop
In the phase locked loop of three-phase rectification system as shown in Figure 5, the voltage of q axis is controlled to be equal to 0, which is combined with the feedback output voltage to generate an error voltage [24], and then PI control is used to adjust the voltage without static error. The output of the PI controller is superimposed with the actual input rated frequency to obtain the output frequency, and in turn, the angle and sine values can be obtained.

Phase Locked Loop
In the phase locked loop of three-phase rectification system as shown in Figure 5, the voltage of q axis is controlled to be equal to 0, which is combined with the feedback output voltage to generate an error voltage [24], and then PI control is used to adjust the voltage without static error. The output of the PI controller is superimposed with the actual input rated frequency to obtain the output frequency, and in turn, the angle and sine values can be obtained.

Simulation Waveform Comparison and Analysis
3.4.1. Startup Characteristics Figure 6 exhibits the startup response of DC output voltage and current in the rectification system at steady state. It can be observed that the rectification system starts at 0 s and quickly enters the steady state without overshoot. At the time of 0.07 s, the DC output voltage reaches a stable value of 5 V, and the DC voltage starts with excellent response characteristics. The simulation results indicate that the double closed-loop control system operates stably and achieves the expected object.

Load Mutation
The load mutation is that the load increases and decreases suddenly at 0.5 s. Figure  7 demonstrates the waveform of DC output voltage and current when the load suddenly decreases from 32 Ω to 16 Ω at 0.5 s, respectively. The analysis demonstrates that after the load decreasing, the voltage value returns to stability after reaching about 4.9 V. Meanwhile, the current stabilizes at 300 A, and the whole response process lasts for 15 ms.

Simulation Waveform Comparison and Analysis
3.4.1. Startup Characteristics Figure 6 exhibits the startup response of DC output voltage and current in the rectification system at steady state. It can be observed that the rectification system starts at 0 s and quickly enters the steady state without overshoot. At the time of 0.07 s, the DC output voltage reaches a stable value of 5 V, and the DC voltage starts with excellent response characteristics. The simulation results indicate that the double closed-loop control system operates stably and achieves the expected object.

Load Mutation
The load mutation is that the load increases and decreases suddenly at 0.5 s. Figure  7 demonstrates the waveform of DC output voltage and current when the load suddenly decreases from 32 Ω to 16 Ω at 0.5 s, respectively. The analysis demonstrates that after the load decreasing, the voltage value returns to stability after reaching about 4.9 V. Meanwhile, the current stabilizes at 300 A, and the whole response process lasts for 15 ms.

Load Mutation
The load mutation is that the load increases and decreases suddenly at 0.5 s. Figure 7 demonstrates the waveform of DC output voltage and current when the load suddenly decreases from 32 Ω to 16 Ω at 0.5 s, respectively. The analysis demonstrates that after the load decreasing, the voltage value returns to stability after reaching about 4.9 V. Meanwhile, the current stabilizes at 300 A, and the whole response process lasts for 15 ms. Figure 8 shows the waveform of DC output voltage and current when the load suddenly increases. It can be noticed that the DC voltage value starts to approach the steady state value after reaching about 5.1 V, and the current is stable at 75 A. The time of the entire response process is about 16 ms.

Comparative Simulation Analysis
The recursive structure of backstepping control method is to start the design process with a known stable system and 'back' to a new structure for the output, which is to gradually stabilize each external subsystem. However, for the reactive component control structure, there is only a single structure without a gradual stabilization process, so the SMVS is used for its control. The SMVS has been verified to be an efficient and pretty popular robust control technique since it owns some well-known and strong properties in disturbance rejection and insensitivity to model uncertainties and parameter variations occurring in dynamical systems [25,26] which can regulate reactive power and reduce harmonic distortion to reach better system requirements. Therefore, for comparison during load mutation, the simulation is concentrated on the reactive component control structure with backstepping control method and SMVS, respectively.

Comparative Simulation Analysis
The recursive structure of backstepping control method is to start the design process with a known stable system and 'back' to a new structure for the output, which is to gradually stabilize each external subsystem. However, for the reactive component control structure, there is only a single structure without a gradual stabilization process, so the SMVS is used for its control. The SMVS has been verified to be an efficient and pretty popular robust control technique since it owns some well-known and strong properties in disturbance rejection and insensitivity to model uncertainties and parameter variations occurring in dynamical systems [25,26] which can regulate reactive power and reduce harmonic distortion to reach better system requirements. Therefore, for comparison during load mutation, the simulation is concentrated on the reactive component control structure with backstepping control method and SMVS, respectively.
To verify the performance difference between SMVS and backstepping control method, the simulation models of the reactive component control structure using these

Comparative Simulation Analysis
The recursive structure of backstepping control method is to start the design process with a known stable system and 'back' to a new structure for the output, which is to gradually stabilize each external subsystem. However, for the reactive component control structure, there is only a single structure without a gradual stabilization process, so the SMVS is used for its control. The SMVS has been verified to be an efficient and pretty popular robust control technique since it owns some well-known and strong properties in disturbance rejection and insensitivity to model uncertainties and parameter variations occurring in dynamical systems [25,26] which can regulate reactive power and reduce harmonic distortion to reach better system requirements. Therefore, for comparison during load mutation, the simulation is concentrated on the reactive component control structure with backstepping control method and SMVS, respectively.
To verify the performance difference between SMVS and backstepping control method, the simulation models of the reactive component control structure using these two kinds of control methods respectively are presented in Figure 9. To verify the performance difference between SMVS and backstepping control method, the simulation models of the reactive component control structure using these two kinds of control methods respectively are presented in Figure 9. Figure 10 shows the Fast Fourier Transform comparison between backstepping control method and SMVS. As a standard sine wave power supply, the synchronous generator contains only odd harmonics. However, the non-linearity of PWM rectifier components, the dead zone of the control signal and the lag of the phase-locked loop all lead to the existence of even harmonics, but they are still far lower than the odd harmonics. For the reactive component control structure controlled by backstepping control method, as shown in Figure 10a, the fundamental voltage is 5 V and the total harmonic distortion of voltage is 0.45%; for the reactive component control structure controlled by SMVS, as shown in Figure 10b, the fundamental voltage is 5 V and the total harmonic distortion of voltage is 0.13%. It is generated that the SMVS is better than the backstepping control method for voltage harmonic suppression.  Figure 10 shows the Fast Fourier Transform comparison between backstepping control method and SMVS. As a standard sine wave power supply, the synchronous generator contains only odd harmonics. However, the non-linearity of PWM rectifier components, the dead zone of the control signal and the lag of the phase-locked loop all lead to the existence of even harmonics, but they are still far lower than the odd harmonics. For the reactive component control structure controlled by backstepping control method, as shown in Figure 10a, the fundamental voltage is 5 V and the total harmonic distortion of voltage is 0.45%; for the reactive component control structure controlled by SMVS, as shown in Figure 10b, the fundamental voltage is 5 V and the total harmonic distortion of voltage is 0.13%. It is generated that the SMVS is better than the backstepping control method for voltage harmonic suppression.  Figure 11 displays the comparison of A-phase input current between backstepping control method and SMVS after halving the load. The analysis demonstrates that the AC input current value quickly doubles to the stable value of 360 A. From Figure 11a, it can be noticed that A-phase input current which is controlled by the backstepping control method has the large overshoot during the two cycles of AC input current, and over after two cycles, the A-phase current reaches the stable value with the response time of 0.04 s. After the load mutation, the state point of the system deviates from the switching surface, but under the control of the SMVS, the state point tends to move towards the switching surface. From Figure 11b, it can be seen that the AC input current soon returns to the new   Figure 10 shows the Fast Fourier Transform comparison between backstepping control method and SMVS. As a standard sine wave power supply, the synchronous generator contains only odd harmonics. However, the non-linearity of PWM rectifier components, the dead zone of the control signal and the lag of the phase-locked loop all lead to the existence of even harmonics, but they are still far lower than the odd harmonics. For the reactive component control structure controlled by backstepping control method, as shown in Figure 10a, the fundamental voltage is 5 V and the total harmonic distortion of voltage is 0.45%; for the reactive component control structure controlled by SMVS, as shown in Figure 10b, the fundamental voltage is 5 V and the total harmonic distortion of voltage is 0.13%. It is generated that the SMVS is better than the backstepping control method for voltage harmonic suppression.  Figure 11 displays the comparison of A-phase input current between backstepping control method and SMVS after halving the load. The analysis demonstrates that the AC input current value quickly doubles to the stable value of 360 A. From Figure 11a, it can be noticed that A-phase input current which is controlled by the backstepping control method has the large overshoot during the two cycles of AC input current, and over after two cycles, the A-phase current reaches the stable value with the response time of 0.04 s. After the load mutation, the state point of the system deviates from the switching surface, but under the control of the SMVS, the state point tends to move towards the switching surface. From Figure 11b, it can be seen that the AC input current soon returns to the new  Figure 11 displays the comparison of A-phase input current between backstepping control method and SMVS after halving the load. The analysis demonstrates that the AC input current value quickly doubles to the stable value of 360 A. From Figure 11a, it can be noticed that A-phase input current which is controlled by the backstepping control method has the large overshoot during the two cycles of AC input current, and over after two cycles, the A-phase current reaches the stable value with the response time of 0.04 s. After the load mutation, the state point of the system deviates from the switching surface, but under the control of the SMVS, the state point tends to move towards the switching surface. From Figure 11b, it can be seen that the AC input current soon returns to the new steady state value of 360 A. The current overshoot is obviously reduced, and the whole process time of adjustment is within 0.03 s. Figure 12 displays the analytical comparison of AC input power factor between backstepping control method and SMVS after halving the load. The AC input current has a fast response, so it has the good tracking effect on the input voltage. Analysis shows that compared with the backstepping control method, the SMVS control the reactive component control structure can accelerate the responsiveness of the AC input current and enter a new steady state faster after load mutation. Furthermore, the power factor reaches stability in 30 ms. steady state value of 360 A. The current overshoot is obviously reduced, and the whole process time of adjustment is within 0.03 s.
A-phase current/A A-phase current/A t (100 ms/div) t (100 ms/div) ( a) (b) Figure 11. A-phase current waveform when the load suddenly decreases: (a) Controlled by backstepping control method; (b) Controlled by SMVS. Figure 12 displays the analytical comparison of AC input power factor between backstepping control method and SMVS after halving the load. The AC input current has a fast response, so it has the good tracking effect on the input voltage. Analysis shows that compared with the backstepping control method, the SMVS control the reactive component control structure can accelerate the responsiveness of the AC input current and enter a new steady state faster after load mutation. Furthermore, the power factor reaches stability in 30 ms.  Figure 13 shows the comparison of A-phase input current between backstepping control method and SMVS when the load suddenly increases from 32 Ω to 64 Ω. The analysis shows that the value of AC input current soon reaches a stable value of 90 A. From Figure  13a, it can be seen that the reactive component control structure controlled by the backstepping control method has the response time of 0.04 s and reach a steady state value after two cycles of AC input current; for the reactive component control structure controlled by the SMVS, as shown in Figure 13b, the AC input current reaches a new steady state value of 90 A within 0.03 s. A-phase current/A A-phase current/A t (100 ms/div) t (100 ms/div) ( a) (b) Figure 11. A-phase current waveform when the load suddenly decreases: (a) Controlled by backstepping control method; (b) Controlled by SMVS. Figure 12 displays the analytical comparison of AC input power factor between backstepping control method and SMVS after halving the load. The AC input current has a fast response, so it has the good tracking effect on the input voltage. Analysis shows that compared with the backstepping control method, the SMVS control the reactive component control structure can accelerate the responsiveness of the AC input current and enter a new steady state faster after load mutation. Furthermore, the power factor reaches stability in 30 ms.  Figure 13 shows the comparison of A-phase input current between backstepping control method and SMVS when the load suddenly increases from 32 Ω to 64 Ω. The analysis shows that the value of AC input current soon reaches a stable value of 90 A. From Figure  13a, it can be seen that the reactive component control structure controlled by the backstepping control method has the response time of 0.04 s and reach a steady state value after two cycles of AC input current; for the reactive component control structure controlled by the SMVS, as shown in Figure 13b, the AC input current reaches a new steady state value of 90 A within 0.03 s.  Figure 13 shows the comparison of A-phase input current between backstepping control method and SMVS when the load suddenly increases from 32 Ω to 64 Ω. The analysis shows that the value of AC input current soon reaches a stable value of 90 A. From Figure 13a, it can be seen that the reactive component control structure controlled by the backstepping control method has the response time of 0.04 s and reach a steady state value after two cycles of AC input current; for the reactive component control structure controlled by the SMVS, as shown in Figure 13b, the AC input current reaches a new steady state value of 90 A within 0.03 s. Figure 14 shows the analytical comparison of AC input power factor between backstepping control method and SMVS when the load suddenly increases from 32 Ω to 64 Ω. Compared to the backstepping control method, the SMVS makes the reactive component control structure enter the new steady state faster after the load mutation and the power factor reaches stability in 30 ms.
According to the above simulation results, the active component control structure controlled by the backstepping control method can enhance the robustness of DC output voltage, and the squared error between the value of DC output voltage and the predicted value of DC output voltage makes the DC output voltage respond quickly. Meanwhile, the reactive component control structure controlled by the SMVS can effectively enhance the robustness of the system, which has low current distortion. It can be observed that compared with the global backstepping control method, the rectification system with the combination of backstepping control method and SMVS has better performance. The DC output voltage is more stable, the current distortion is smaller, and the AC input current track AC input voltage faster in real time. Under the load mutation, both DC output voltage and AC input current can reach the steady state quickly.  Figure 14 shows the analytical comparison of AC input power factor between backstepping control method and SMVS when the load suddenly increases from 32 Ω to 64 Ω. Compared to the backstepping control method, the SMVS makes the reactive component control structure enter the new steady state faster after the load mutation and the power factor reaches stability in 30 ms. According to the above simulation results, the active component control structure controlled by the backstepping control method can enhance the robustness of DC output voltage, and the squared error between the value of DC output voltage and the predicted value of DC output voltage makes the DC output voltage respond quickly. Meanwhile, the reactive component control structure controlled by the SMVS can effectively enhance the robustness of the system, which has low current distortion. It can be observed that compared with the global backstepping control method, the rectification system with the combination of backstepping control method and SMVS has better performance. The DC output voltage is more stable, the current distortion is smaller, and the AC input current track AC input voltage faster in real time. Under the load mutation, both DC output voltage and AC input current can reach the steady state quickly.

Verification of the Experimental Result
According to the experimental principles, an experimental test platform for low-voltage and high-current DC power generation system with double closed-loop control based  Figure 14 shows the analytical comparison of AC input power factor between backstepping control method and SMVS when the load suddenly increases from 32 Ω to 64 Ω. Compared to the backstepping control method, the SMVS makes the reactive component control structure enter the new steady state faster after the load mutation and the power factor reaches stability in 30 ms. According to the above simulation results, the active component control structure controlled by the backstepping control method can enhance the robustness of DC output voltage, and the squared error between the value of DC output voltage and the predicted value of DC output voltage makes the DC output voltage respond quickly. Meanwhile, the reactive component control structure controlled by the SMVS can effectively enhance the robustness of the system, which has low current distortion. It can be observed that compared with the global backstepping control method, the rectification system with the combination of backstepping control method and SMVS has better performance. The DC output voltage is more stable, the current distortion is smaller, and the AC input current track AC input voltage faster in real time. Under the load mutation, both DC output voltage and AC input current can reach the steady state quickly.

Verification of the Experimental Result
According to the experimental principles, an experimental test platform for low-voltage and high-current DC power generation system with double closed-loop control based

Verification of the Experimental Result
According to the experimental principles, an experimental test platform for lowvoltage and high-current DC power generation system with double closed-loop control based on DSP28335 was built to verify the effectiveness of backstepping control method and SMVS, as shown in Figure 15. The platform can directly output DC power to supply the load after the three-phase PMSG generates AC power and rectifies it by the confluence plate. The confluence plate is composed of upper and lower legs of the bridge and in order to improve the conduction capability, each leg is connected in parallel with five MOSFETs which are fixed on the heat sink. The experiments of load mutation are carried out on the experimental platform, on which the phase voltage on the AC side of the three-phase PMSG is 3.8 V and the output voltage on the DC side is 5 V. The overall structure of the rectification double closed-loop control system consisting of active and reactive component control structure is shown in Figure 13.
The DC output voltage of the rectification system is shown in Figure 16. From the figure, it can be seen that the output waveform is smooth, with the voltage value of 5 V and the current value of 150 A, which is consistent with the DC characteristics. The experimental results are basically consistent with the dynamic simulation results and satisfy the DC output requirements. on DSP28335 was built to verify the effectiveness of backstepping control method and SMVS, as shown in Figure 15. The platform can directly output DC power to supply the load after the three-phase PMSG generates AC power and rectifies it by the confluence plate. The confluence plate is composed of upper and lower legs of the bridge and in order to improve the conduction capability, each leg is connected in parallel with five MOSFETs which are fixed on the heat sink. The experiments of load mutation are carried out on the experimental platform, on which the phase voltage on the AC side of the three-phase PMSG is 3.8 V and the output voltage on the DC side is 5 V. The overall structure of the rectification double closed-loop control system consisting of active and reactive component control structure is shown in Figure 13. The DC output voltage of the rectification system is shown in Figure 16. From the figure, it can be seen that the output waveform is smooth, with the voltage value of 5 V and the current value of 150 A, which is consistent with the DC characteristics. The experimental results are basically consistent with the dynamic simulation results and satisfy the DC output requirements. The experimental waveform is shown in Figure 17 which shows the DC output voltage and current waveform when the load suddenly increases and decreases, respectively. Owing to the presence of internal resistance, the external characteristics of the rectification experimental platform, on which the phase voltage on the AC side of the three-phase PMSG is 3.8 V and the output voltage on the DC side is 5 V. The overall structure of the rectification double closed-loop control system consisting of active and reactive component control structure is shown in Figure 13. The DC output voltage of the rectification system is shown in Figure 16. From the figure, it can be seen that the output waveform is smooth, with the voltage value of 5 V and the current value of 150 A, which is consistent with the DC characteristics. The experimental results are basically consistent with the dynamic simulation results and satisfy the DC output requirements. The experimental waveform is shown in Figure 17 which shows the DC output voltage and current waveform when the load suddenly increases and decreases, respectively. Owing to the presence of internal resistance, the external characteristics of the rectification The experimental waveform is shown in Figure 17 which shows the DC output voltage and current waveform when the load suddenly increases and decreases, respectively. Owing to the presence of internal resistance, the external characteristics of the rectification system change when the load changes under the condition of the same duty cycle. The DC output voltage in both Figure 17a,c were able to quickly return to a stable state with a brief fluctuation and the DC output voltage remains at 5 V. Meanwhile, the DC output current in Figure 17b is able to quickly reach 300 A in 20 ms; the DC output current in Figure 17d is also able to reach 75 A in 30 ms, returning to a new steady state. The experimental results are basically consistent with the dynamic simulation results, which prove that the dynamic control performance of the system is better after the combination of backstepping control method and SMVS.
The waveform of A-phase input current is shown in Figure 18 when the load suddenly increases and decreases under the combination of backstepping control method and SMVS. It can be observed that A-phase input current rises and falls more smoothly without overshoot and reaches stability after 30 ms. The experimental results are basically consistent with the dynamic simulation results. Figure 17a and Figure 17c were able to quickly return to a stable state with a brief fluctuation and the DC output voltage remains at 5 V. Meanwhile, the DC output current in Figure 17b is able to quickly reach 300 A in 20 ms; the DC output current in Figure 17d is also able to reach 75 A in 30 ms, returning to a new steady state. The experimental results are basically consistent with the dynamic simulation results, which prove that the dynamic control performance of the system is better after the combination of backstepping control method and SMVS. The waveform of A-phase input current is shown in Figure 18 when the load suddenly increases and decreases under the combination of backstepping control method and SMVS. It can be observed that A-phase input current rises and falls more smoothly without overshoot and reaches stability after 30 ms. The experimental results are basically consistent with the dynamic simulation results.

Conclusions
In this study, a rectification control system by using of backstepping control method and SMVS in a multi-three-phase PMSG is proposed with the aim of stabilizing the DC output at low-voltage and high-current. In the active component control structure, the backstepping control method is used for optimization, which can achieve fast tracking and global adjustment of the system by setting intermediate virtual control variables and satisfy the stability requirement of Lyapunov stability theory so that a reasonable control variable is finally designed. The reactive component control is stabilized by the SMVS to stabilize the power factor. In this paper, the performance of the reactive component control structure is compared by using two methods, the backstepping control method and the SMVS. The simulation results demonstrate that the system robustness is better and the dynamic performance is greatly improved when the SMVS is applied to the reactive component control structure in the rectification control system. The backstepping control

Conclusions
In this study, a rectification control system by using of backstepping control method and SMVS in a multi-three-phase PMSG is proposed with the aim of stabilizing the DC output at low-voltage and high-current. In the active component control structure, the backstepping control method is used for optimization, which can achieve fast tracking and global adjustment of the system by setting intermediate virtual control variables and satisfy the stability requirement of Lyapunov stability theory so that a reasonable control variable is finally designed. The reactive component control is stabilized by the SMVS to stabilize the power factor. In this paper, the performance of the reactive component control structure is compared by using two methods, the backstepping control method and the SMVS. The simulation results demonstrate that the system robustness is better and the dynamic performance is greatly improved when the SMVS is applied to the reactive component control structure in the rectification control system. The backstepping control method and the SMVS constitute the double closed-loop control system, whose stability is not easily disturbed by large signals, and the power factor can quickly reach a steady state. The experimental results are consistent with the simulation results which prove that the control strategy combining the backstepping control method and the SMVS is an effective and reliable control scheme. This study is only a preliminary discussion of the design and performance of the three-phase voltage rectification double closed-loop control system. Furthermore, study the situation of the rectification system with nonlinear load, and make it design together with the whole system of the post-nonlinear load, optimize the matching, and get the best performance and reliability.

Conflicts of Interest:
The authors declare no conflict of interest.