Dynamics and Model Predictive Control of Current-Fed Dickson Voltage Multiplier: TS Fuzzy Approach

This study presents a new approach to modeling and control the current-fed Dickson voltage multiplier (CF-DVM). The capacitor voltage relation and the input current are obtained. As all switching intervals are considered in detail, a highly accurate dynamic model is obtained, which can be easily extended for a CF-DVM with an arbitrary number of stages. Using the precise extracted model, the Takagi-Sugeno fuzzy model (TSFM) of the CF-DVM is provided, which is an exact equivalent representation of the CF-DVM nonlinear model. Then, a highly accurate and responsive model predictive controller (MPC) is designed based on an obtained TSFM of the CF-DVM to control the output voltage in an optimal and constrained manner. To obtain the control signal, the suggested optimization problem is converted to a quadratic programming (QP)-based problem which has a low online computational burden. Moreover, the performance of the proposed MPC is compared with the PI controller and the linear MPC. Finally, the simulation and experimental results demonstrate the promising merits of the proposed model and control approaches.


I. INTRODUCTION
Utilizing small-size renewable resources have been rapidly increased in recent years. One of the most significant applications of the power electronic (PE) converters is integrating the power generation of the micro-sources [1]. Because the output voltage of renewable resources is generally low, high step-up DC/DC converters are required to boost the output voltage [2]. These converters can be categorized as isolated and non-isolated converters [3].
The voltage gain can be adjusted by changing the winding turn ratio in the isolated structures. However, achieving a very high voltage gain requires a high turn ratio, which increases the converter's weight, volume, cost [3] and often necessitates employing snubber circuits due to the high voltage spikes across the semiconductors caused by the leakage inductance [4]. Compared to the isolated converters, non-isolated DC/DC converters offer simplicity, compact size, and low cost [3]. However, the classical non-isolated boost converters are unable to achieve a high voltage gain at extremely-high duty cycles ( ) due to excessive conduction losses [3], minimal efficiency [4], poor transient response [2], and protection issues.
The SC-based converters are inductor-less circuits that step up/down the input voltage [9]. The absence of magnetic elements in these converters leads to a high-power density [10], [11]. However, their output voltage regulation is limited due to the discrete nature of stages and voltage gain. Moreover, the capacitor charging and discharging rapidly in voltage-fed (VF) structures generates input inrush current [12]. In response to these issues, an inductor or current source can be integrated into the SC converters, also leading to soft switching of semiconductor switches and enhanced efficiency [1], [2], [13]. The SC converters applications include high side MOSFET gate drive circuitry [2], battery charge stabilization [14], high-efficiency DC/DC converters [11], photovoltaic (PV) interface systems [15], power supply processors, flash memory [13], fuel cell (FC) [16], energy extraction [17], and IoT systems [18].
The most well-known SC-VMs are Cockcroft-Walton (CWVM), Fibonacci (FVM), and Dickson (DVM) voltage multipliers. Compared to other structures, the DVM can provide a high voltage gain with a low output impedance anis independent of parasitic capacitors, particularly for many stages [19]. Thus, the DVM is appropriate for high step-up applications [2]. Nevertheless, to achieve a high voltage gain with SC converters, some limitations should be considered, such as input peak current, the effect of charging and discharging capacitors, voltage level, number of stages, the impact of the parasitic elements on the circuit's performance [20]- [22].
The DVM switching is accomplished by pulse-widthmodulation (PWM); hence, throughout employing a control system, the output voltage can be regulated around an operating point even during input voltage and output load disturbances. The controller design process requires mathematical equations and analysis of the converter. A dynamic model of the DVM was derived in [23], [24]. However, one operation mode is ignored for simplicity, reducing the derived model's accuracy. Furthermore, a bilinear model for controller design is extracted in [25]. However, the model contains only two operating modes, while CS converters have several operating modes. Another dynamic model is represented in [26], with a phase-shift switching strategy, where time intervals are identified. However, for the non-phase-shift method, other time intervals must be calculated.
Other studies also considered the output voltage regulation [27]- [29]. A sliding mode controller (SMC) was implemented based on the linearized model of the SC converter in [27]. Also, a dynamic modeling and controller design for a high step-up DC/DC converter is provided in [28] using a reduced-order model. Then, the control design is provided based on the linear systems theorem by obtaining the linear model of the converter.
Model predictive control (MPC) is a very effective control method that has been employed to regulate the output voltage of the converters [30], [31]. In the MPC strategy, a dynamic model of the system is used to predict the system's future behavior [32]. The control signal is calculated based on an optimization problem. The accuracy of the adopted model in the MPC scheme directly affects the controller's performance. In [30], a linear model and MPC method were proposed to control a constant switching frequency DC/DC converter. Also, a maximum power point tracking (MPPT) approach was designed for PV systems based on the given MPC. In [29], a fuzzy-based MPC was proposed based on the linear matrix inequality (LMI) to regulate the output voltage of a DC/DC boost converter. To do this, the Takagi-Sugeno fuzzy model (TSFM) of the converter was obtained, and the control signal was calculated based on the LMIbased online optimization problem. However, the proposed LMI-based optimization problem is computational heavy. A computationally efficient MPC for a DC/DC boost converter was designed in [33], based on the linear model and continuous time.
This study proposes a highly accurate dynamic model by considering all operating intervals with no simplification compared to [23], [34], and also an MPC controller design based on an obtained TSFM for a two-stage CF-DVM to regulate the output voltage. The proposed method can easily be extended for any desired stages. The fuzzy-based MPC is designed based on the quadratic optimization problem. Additionally, the proposed TSFM is presented based on the sector nonlinearity approach, which represents an exact approximation of the nonlinear dynamic model of the CF-DVM. Then, the optimization problem of the nonlinear MPC is converted to quadratic programming (QP)-based MPC compared to [29], [35], which has the benefit of a very low online computational burden compared to [36].
This paper is organized as follows: In section II, the dynamic modeling of the CF-DVM is presented. In section III, the fuzzy-based MPC of the CF-DVM is designed. Then, simulations and experiments in section IV validate the accuracy and merits of the proposed model and control approach. Finally, the paper is concluded in section V .

II. Dynamic modeling of the CF-DVM
In this section, the converter topology and operating states are explained. Then the dynamic model of the CF-DVM is extracted using the state-space averaging method.

A. Extracting large-signal model of the CF-DVM
The structure of the two-stage CF-DVM is shown in Figure  1. The CCM operation includes three operating modes, as discussed in the following. Equivalent circuits with the current flow path in each state and the main waveforms with associated timing are illustrated in Figure 2 and Figure 3, respectively. It is hypothesized that the converter operates in steady-state CCM, S1-S4 have no deadtime, and all components are ideal except for the diodes with an internal resistor rd.
Mode 1 ( 1 ): In the time interval of 1 , 2 and 4 are conducting. The inductor current ( ) passes through 1 − 2 − 2 current flow path as shown in Figure 2(a). As a result, the voltage across the capacitor 1 ( 1 ) decreases, and the voltage across the capacitor 2 ( 2 ) increases. This condition lasts until the voltages of the capacitors 1 and 2 become equal. Furthermore, the capacitor 3 supplies the load current and gets discharged. The state equations are given by: This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and Mode 2 (( − 1) ): In this mode, all switches maintain their previous state, keeps its current flow path through the diode 0 and increases linearly by the input voltage source ( ). Diodes D1, D2, D3 are reversed-biased. Therefore, there is no current flowing through capacitors 1 and 2 . Capacitor 3 discharge current supplies the load. The equivalent circuit of this state is shown in Figure 2 Figure 2(c). We use the diode current equations to obtain the inductor's voltage and the capacitors' current equations. For this purpose, diode current equations are extracted: The averaged state-space model of the converter can be obtained by applying the averaging state-space technique to (1), (2), and (5) as follows: where the state space variables are: The parameter 1 can also be obtained in terms of , where is the duty cycle of 2 and 4 . In the time interval of 1 , the charged energy 1 stored in 1 is divided equally between 1 and 2 . During the interval of (1 − ) , capacitor 1 receives the 1 2 ⁄ from the input source ( ) through 1 . Capacitor 2 also transfers the 1 2 ⁄ energy received from the previous step to the output load. As both capacitors are in the inductor current path, the charge energy 1 is drawn from the input source during this time interval. Since other parameters such as inductor size, diode

FIGURE 3. Converter's main waveforms with associated timing in CCM
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3186370 resistance ( ) and capacitances are constant, the (1 − ) interval lasts twice as long as the 1 interval. Therefore, this relationship can be expressed by the number of current paths through the diodes in each mode of operation. This theory is accurate for the DVM with any number of stages.
where is the ratio time of the Mode 1 to the Mode 3, 1 ℎ and (1 − ) ℎ are the number of inductor current paths through the diodes in 1 and (1 − ) time intervals, respectively. Substituting (12) into (7), The averaged statespace model of the converter obtained:

B. STEADY-STATE ANALYSIS 1) STEADY-STATE EQUATION AND VOLTAGE GAIN
The voltage gain for the converter can be obtained by using the volt-second balance on inductor voltage [16].
The can be achieved by applying capacitor charge balance for capacitors current: By substituting (19) in (18), the steady-state output voltage is equal to: Furthermore, the voltage gain for number of capacitors can be obtained as:

2) LOSSES AND EFFICIENCY
The existence of non-idealities in power elements leads to unavoidable losses and decreased efficiency. The converter losses are categorized into conduction losses and switching losses. This section represents the converter's non-ideal voltage gain and efficiency. The parasitic elements considered for conduction losses are the inductor's

C. COMPONENT SELECTION
By choosing proper values for inductors and capacitors, we can achieve a compact design with an acceptable ripple size and fulfill control demands such as zeros and poles movement on the RPH axis, improving the phase margin, quality, and damping coefficient. The inductor current and capacitor voltage ripples in the ratio form are expressed as: For this kind of converter, the ratio is designed to be 10 − 20% for inductor current and 1 − 2% for capacitor voltages [25].

1) CAPACITOR VOLTAGE AND INDUCTOR CURRENT RIPPLE
The inductor current and capacitor voltage ripples in CCM conditions according to (1), (2), and (5) are given by:

2) CALCULATION OF SMALL-SIGNAL MODEL AND INPUT-TO-OUTPUT TRANSFER FUNCTION
Small-signal equations by using the small-perturbation technique are obtained as follows. The input variables and small-signal variables are given by (32) and (33), respectively.

̇= +
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. [̂̂] (32) The input-to-output voltage ( ) and output voltage to duty cycle ( ) transfer functions are achieved based on the values used for simulations in Table I

3) STABILITY CRITERIA FOR PASSIVE COMPONENTS
The pole-zero map of the transfer function in (35) is plotted in Figure 5. Accordingly, with the presence of a zero on the Right Half Plane (RHP), the converter is Non-Minimum Phase (NMP), and the closed-loop operation without a controller is unstable and has a non-regulated output, especially in high gain applications. Also, approaching the zero along the positive real axis toward the origin will increase control constraints, and the converter cannot achieve the desired fast response [2].
The NMP systems behave inversely to the input step changes because their open/close loop transfer functions can have the same zeros in the RHP. This means in case of an increase in the input voltage, the output performs an undershoot before the response approaches its final value, which makes the controller design challenging.
As shown in Figure 6, the zero and the origin pole remain unchanged by increasing the inductance value (L). Also, the imaginary axis left side conjugate poles approach the real axis, which reduces the response oscillations. But on the other hand, approaching the RHP zero to the origin causes response oscillation. Thus, the RHP zero moves away from the origin with a lower inductance value and reduces the response oscillation. Nevertheless, according to (29), the converter's switching frequency must increase to maintain the inductor current ripple.
The effect of the capacitance value on zeros and poles is illustrated in Figure 7. Accordingly, the dominant conjugate poles approach the real axis as the capacitances decrease. Conversely, as the capacitances increase, the conjugate poles move away from the real axis and then toward the origin. Nevertheless, the value of the capacitances does not affect the imaginary axis right side zero. Moreover, by reducing the  value of the capacitances to achieve a better dynamic response, the converter's switching frequency must increase to maintain the desired voltage ripple, with respect to (27) and (28).

III. DESIGN OF THE FUZZY-BASED MPC OF CF-DVM A. TS FUZZY REPRESENTATION OF CURRENT-FED DICKSON VOLTAGE MULTIPLIER
This subsection obtains the TSFM of the CF-DVM. The nonlinear dynamic model represented in (6) has some nonlinear terms. The sector nonlinearity approach can calculate the TSFM of this system. The nonlinear model is rephrased as follows: where ( ) is the duty cycle and, According to the maximum and minimum values of each state based on the physical limitations, the premise variables are defined as follows: Moreover, the weighted average of each variable can be considered as follows: where ̅ and are the maximum and minimum values of the variables. Due to the orthogonality of the membership functions , and , each has: The state-space representation of each local linear subsystem of TSFM can be obtained as follows: , ℬ 8 = Based on the derived model, the fuzzy-based MPC of the CF-DVM is designed in the following subsection.

B. TSFM-BASED MODEL PREDICTIVE CONTROL OF CF-DVM
MPC requires a dynamical model of the controlled system to predict the system states' future behavior. The main advantages of the MPC are optimal performance and guaranteed constrained performance of the system [37]. In order to design the TSFM-based MPC for the CF-DVM, the obtained TSFM in (41) is employed. Then, the optimization problem is converted to a QP optimization that is solved in each time step. The suitable value of the control signal is obtained based on the minimization of the QP optimization. The fuzzy MPC scheme based on the discrete-time model of the TSFM is defined as follows: are the desired values of the system states and control input. Also, is the prediction horizon, > 0, > 0, and ℜ > 0 are proper weighting coefficient matrices. The sets and are the polyhedrons that specify the constraints on the system states and the control input, respectively.
To implement the fuzzy-based MPC, the vector form of the optimization problem (43) must be obtained. Therefore, considering the predetermined prediction horizon , the system's output can be calculated as follows: According to (45), does not affect the optimization problem. Consequently, the cost function is similar to a QP optimization problem and has less computational complexity than the online optimization problem based on the linear matrix inequality [38]. The block diagram of the control system is illustrated in Figure 8.

IV. SIMULATION RESULTS
This section presents simulations and experimental results to evaluate the proposed modeling and control methods. The converter with the parameters shown in TABLE 1 is simulated to validate the proposed model.

A. OPEN-LOOP OPERATION
First, to demonstrate the accuracy of the proposed modeling approach, the transients of the output voltage ( 3 ( )) and input inductor current ( ( )) are plotted in Figure 9 and Figure10, respectively. Accordingly, the obtained results confirm consistency and accuracy between modeling and simulation responses of the CF-DVM.   This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Moreover, to demonstrate the feasibility of the theoretical concepts, a laboratory prototype of the proposed converter was implemented and tested. The experimental test setup is shown in Figure 11. The converter's output voltage and input current transients are shown in Figure 12, which shows consistency with simulations. The converter's output voltage measured to be = 70.5 and the input current at = 10.7 that validate the calculated values in (22) and (23), respectively. The output power of the converter with the input voltage of = 10 , was measured to be = 100 . Figure 13 shows the output voltage ripple of the capacitors alongside the switches 2 and 4 gating pulse. Figure 14 demonstrates the inductor voltage and current ripple. Finally, the obtained waveforms from the experimental results confirm the accuracy of the modeling and simulation under open-loop operation. It should be noted that the minor differences between the theoretical and experimental results are due to the influence of parasitic elements.  Output Voltage (20V/div) Input Current (2.5A/div) 5ms/div

B. CLOSED-LOOP OPERATION
In this scenario, the output voltage of the CF-DVM is controlled by the TSFM-based MPC. To demonstrate the merits of the proposed MPC, simulations are used to compare the converter's output voltage with the proposed MPC, the LMPC [33], and the PI-controller. As shown in Figure 15, the proposed MPC's performance is more precise and faster than the PI-controller and the LMPC. Moreover, the LMPC's output voltage tracking error is higher due to the linearization of the nonlinear model in the design process. Meanwhile, the TSFM of the CF-DVM represents the exact model of the nonlinear systems. Therefore, the predicted output is more accurate and improves the controller's performance. Moreover, the laboratory prototype was tested under closed-loop operation to illustrate the advantages of theoretical concepts and the proposed control method. The converter's output voltage and input current under the TSFM-based MPC are shown in Figure 16. Accordingly, the performance of the proposed MPC in the experimental test is promising. Finally, the obtained waveforms from the experimental results confirm the accuracy of the modeling and controller design under closed-loop operation. It should be noted that the minor differences between the theoretical and experimental results are due to the influence of parasitic elements.

Inductor
L Voltage (10V/div) This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3186370 Output Voltage (20V/div) Input Current (2.5A/div) 5ms/div