A Novel LQI Control Technique for Interleaved-Boost Converters

Abstract

Hybrid electric vehicles (HEVs) and fuel cell electric vehicles (FCEVs) utilize boost converters to gain a higher voltage than the battery. Interleaved boost converters are suitable for low input voltage, large input current, miniaturization, and high-efficiency applications. This paper proposes a novel linear quadratic integral (LQI) control for the interleaved boost converters. First, the small-signal model of the interleaved-boost converter is derived. In the proposed method, an output voltage and a current signal error between two-phase input currents are selected to control not only the output voltage but also a balance between two-phase input currents. Furthermore, steady-state characteristics in terms of the output voltage and the input current are demonstrated by experiments and simulations using an experimental apparatus with a rated power of 700 W. The validity of the proposed method’s tracking performance and load response is demonstrated by comparing it with that of the conventional PI control. The tracking performance of the LQI control for the 40 V step response has a ten times faster response than that of the PI control. Also, the experimental results demonstrate that the proposed method maintains a constant output voltage for a 300 W load step while the PI control varies by 10 V during 70 ms. Additionally, the proposed method has an excellent disturbance rejection.

1. Introduction

CO₂ reduction is the biggest challenge in the world, and automobiles play an important role [1]. DC link voltage variable systems with DC/DC converters are used in hybrid electric vehicles (HEVs) and fuel cell electric vehicles (FCEVs) [2]. Interleaved-boost converters can increase power capability and overall switching frequency by using interleaved switching, a well-known concept of sequential switching. Therefore, interleaved-boost converters have been widely studied for automobiles and industry applications such as grid systems. For example, reference [3] reports a boost converter with a coupled inductor and a common clamping capacitor capable of high gain, high efficiency, and miniaturization for integrating photovoltaic power into a standalone DC microgrid. Reference [4] proposes an averaged small-signal model of a dual-interleaved buck converter using sampler decomposition to include the phase interaction effects that arise from the interleaved sampling of the phase currents. In order to improve stationary characteristics, stability, and transient response, reference [5] proposes a method of modeling and implementation for an interleaved tri-state buck-boost converter. Reference [6] presents a generalized state-space average model for the multi-phase interleaved buck, boost, and buck-boost converters. Using the model, the converter can regulate the system to achieve its maximum efficiency.

Generally, the boost converter system requires specifications such as miniaturization, large power density, high efficiency, a high step-up ratio, fast-tracking performance, and robust stability. With respect to circuit topology, several methods exist to obtain a high step-up ratio and large power density, such as an interleaved boost converter with coupled inductors and a double-boost converter topology [7,8,9]. To downsize the system, there are methods to reduce the ripple of input current by applying a multiphase-interleaving method and a multidevice interleaved boost converter [10]. In reference [10], conventional PI controllers are used, and an improvement in the transient response of the output voltage for a load step is reported by applying the interleaving method. However, there is a problem with the responses of the input current and input voltage. As for the control strategy to obtain a fast-tracking performance and robust stability, there are several methods, such as an LMI approach [11], a modified linear quadratic regulator (LQR) [12], and a sliding mode control [13]. For high efficiency, there are methods such as two-phase boost converters with an Electric Double-Layer Capacitor [14,15] and an LQR with a converter loss function [16].

Reference [17] reports a method for downsizing the output filter capacitor by suppressing the output voltage variation with a load current feedforward control for a conventional boost converter. This means that the fast-tracking performance and the disturbance rejection are effective for the miniaturization of a system. From another point of view, the controller needs to handle many variables, such as the output voltage, the input current, and so on, when the converter is connected to the voltage-source three-phase inverters in the HEVs. Generally, the LQR is suitable for multi-input and multi-output systems and is well-known for its fast-tracking performance and complexity for implementation. In reference [18], LQR methods are applied to the grid system. We believe that the LQR is a superior method for establishing a high-tracking performance and downsizing the passive components in power converters. Reference [18], however, requires adding another controller to control the output variables of the converter in addition to the LQR. That leads to more complexity of the control system.

In this paper, in order to improve the complexity of LQR implementation, we propose a novel linear quadratic integral (LQI) control for the interleaved boost converter suitable for HEVs and FCEVs [19]. The proposed method selects the output variables as the output voltage and a current error signal between the two-phase currents, and it can control the output voltage with only the LQI control. To demonstrate the validity of the proposed method, we carry out a comparison verification between the proposed LQI control and conventional PI control because there are a few references for interleaved boost converters with LQI control. For this purpose, steady-state characteristics in the output voltage and the input current are presented. Experimental results demonstrate that the proposed method has a better step response that is ten times faster than that of the PI controller. This article’s novelty is applying a novel LQI control to the interleaved boost converter, which can control the output voltage with only the LQI control. The proposed LQI has a simpler control system than conventional LQR and a better tracking performance than conventional PI control. Consequently, this paper contributes not only to an amelioration in the tracking performance and the complexity of implementation but also to the miniaturization of the motor drive system in HEVs and so on. The organization of the paper is as follows. Section 2 describes switching modes using the interleaving method; Section 3 describes the derivation of the small-signal model and the control method of the proposed LQI control; Section 4 presents the steady-state characteristics and transient response of the proposed method. We present a response comparison of the output voltage tracking performance between the proposed LQI and conventional PI controls. The load response comparison is also carried out. Simulation and experimental results demonstrate the superiority of the proposed LQI control. Section 5 describes the conclusions.

2. Interleaved Boost Converter

In this chapter, the basic principle of the interleaved boost converter and its switching mode analysis are introduced [19]. It is assumed that a three-phase voltage-source inverter is combined with the load of the boost converter. However, the inverter is replaced with a resistor because this paper focuses on the characteristics of the output voltage and input current ripple with the proposed LQI control.

2.1. Circuit Configuration

Figure 1 shows the interleaved boost converter, in which the duty ratios are defined as D_p and D_n given to the switches (S₁ and S₂), respectively. Figure 2 shows the principle of the interleaving method where D is the duty cycle/ratio when D_p and D_n are the same and D > 0.5. In Figure 2, the gate signals for the two switches are generated by comparing the duty ratio with two carriers of f_c1 and f_c2 that have a 180-phase shift. Here, carrier1 f_c1 and duty ratio D_p are for S₁, and carrier2 f_c2 and duty ratio D_n are for S₂, respectively. As a consequence of the interleaving method, the input current amplitude is reduced compared to the phase input current, and the converter produces an input current ripple that has twice the carrier frequency.

2.2. Switching Modes

Figure 3 shows four modes of the boost converter, which depend on the duty cycle of D. MODE1 is a mode in which both S₁ and S₂ are on, MODE2 is a mode in which both S₁ and S₂ are off, and MODE3 is a mode in which S₁ is off, and S₂ is on. In MODE4, S₁ is on, and S₂ is off. In case D > 0.5, in Figure 2, both reactors store the magnetic energy with MODE 1 in Figure 3a. Each reactor boosts the output voltage alternately with MODE 3 in Figure 3c and MODE 4 in Figure 3d.

3. Control Methods

This chapter describes a theory of the proposed novel linear quadratic integral (LQI) control for the interleaved boost converter. The LQI control controls the output voltage and the balance between two-phase input currents. A small-signal model is demanded for the design of the LQI control. Therefore, the state-space averaging equations are derived. Note that this paper assumes a current continuous mode in the analysis.

3.1. Derivation of State Averaging Equations

Generally, the state space averaging equation and the output equation are expressed as follows:

$$\dot{\mathit{x}}\left(t\right)=\mathit{A}\mathit{x}\left(t\right)+\mathit{B}\mathit{u}\left(t\right)$$

(1)

$$\mathit{y}\left(t\right)=\mathit{C}\mathit{x}\left(t\right)$$

(2)

State-space variables are selected as the phase input currents and the output voltage in this paper. Next, state-space equations for the switching modes in Figure 2, in which the duty ratio is D > 0.5, are derived. In mathematical modeling, ideal switching elements are assumed for simplifying, but the resistance of a DC reactor is considered. The state-space equation in Mode1 is expressed as the following:

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{L1}\\ i_{L2}\\ v_{dc}\end{array}\right]=\left[\begin{array}{ccc}-{\displaystyle \frac{r_1}{L_1}}& 0& 0\\ 0& -{\displaystyle \frac{r_2}{L_2}}& 0\\ 0& 0& -{\displaystyle \frac{1}{CR}}\end{array}\right]\left[\begin{array}{c}{i}_{L1}\\ i_{L2}\\ v_{dc}\end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{1}{L_1}}\\ {\displaystyle \frac{1}{L_2}}\\ 0\end{array}\right]V_i,$$

$$\dot{\mathit{x}}\left(t\right)={\mathit{A}}_1\mathit{x}\left(t\right)+{\mathit{B}}_1{V}_i$$

(3)

The state-space equation in Mode3 is expressed as the following:

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{L1}\\ i_{L2}\\ v_{dc}\end{array}\right]=\left[\begin{array}{ccc}-{\displaystyle \frac{r_1}{L_1}}& 0& -{\displaystyle \frac{1}{L_1}}\\ 0& -{\displaystyle \frac{r_2}{L_2}}& 0\\ {\displaystyle \frac{1}{C}}& 0& -{\displaystyle \frac{1}{CR}}\end{array}\right]\left[\begin{array}{c}{i}_{L1}\\ i_{L2}\\ v_{dc}\end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{1}{L_1}}\\ {\displaystyle \frac{1}{L_2}}\\ 0\end{array}\right]V_i,$$

$$\dot{\mathit{x}}\left(t\right)={\mathit{A}}_3\mathit{x}\left(t\right)+{\mathit{B}}_3{V}_i$$

(4)

The state-space equation in Mode4 is expressed as the following:

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{L1}\\ i_{L2}\\ v_{dc}\end{array}\right]=\left[\begin{array}{ccc}-{\displaystyle \frac{r_1}{L_1}}& 0& 0\\ 0& -{\displaystyle \frac{r_2}{L_2}}& -{\displaystyle \frac{1}{L_2}}\\ 0& {\displaystyle \frac{1}{C}}& -{\displaystyle \frac{1}{CR}}\end{array}\right]\left[\begin{array}{c}{i}_{L1}\\ i_{L2}\\ v_{dc}\end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{1}{L_1}}\\ {\displaystyle \frac{1}{L_2}}\\ 0\end{array}\right]V_i,$$

$$\dot{\mathit{x}}\left(t\right)={\mathit{A}}_4\mathit{x}\left(t\right)+{\mathit{B}}_4{V}_i$$

(5)

Coefficient matrices A and B for the state-space equation in Equation (1) are derived by applying the state-space averaging method with A₁, A₃, A₄, B₁, B₃, and B₄, according to Figure 2 [19].

$$\mathit{A}=\left[\begin{array}{ccc}-{\displaystyle \frac{r_1}{L_1}}& 0& -{\displaystyle \frac{\overline{D_p}}{L_1}}\\ 0& -{\displaystyle \frac{r_2}{L_2}}& -{\displaystyle \frac{\overline{D_n}}{L_2}}\\ {\displaystyle \frac{\overline{D_p}}{C}}& {\displaystyle \frac{\overline{D_n}}{C}}& -{\displaystyle \frac{1}{CR}}\end{array}\right], \mathit{B}=\left[\begin{array}{c}{\displaystyle \frac{1}{L_1}}\\ {\displaystyle \frac{1}{L_2}}\\ 0\end{array}\right]$$

(6)

where $i_{L1} \mathrm{a}\mathrm{n}\mathrm{d} i_{L2}$ are the phase input currents,$v_{dc}$ is the output voltage, $V_i$ is the input voltage, $L_1 {\mathrm{a}\mathrm{n}\mathrm{d} L}_2$ are the inductances of the DC reactors, $r_1 {\mathrm{a}\mathrm{n}\mathrm{d} r}_2$ are the resistances of the DC reactors, C is the capacitance of the output capacitor, and $\overline{D_p}=1-D_p, \overline{D_n}=1-D_n$.

3.2. Derivation of Small-Signal Model

In this section, the linearization of the state-space Equation is carried out since Equation (1) is a nonlinear system. Equation (1) is represented by Equation (6) as the following:

$$\dot{\mathit{x}}\left(t\right)=\mathit{f}\left(\mathit{x}\left(t\right),\mathit{u}\left(t\right)\right)=\left[\begin{array}{c}-{\displaystyle \frac{r_1}{L_1}}{i}_{L1}-{\displaystyle \frac{\overline{D_p}}{L_1}}{v}_{dc}+{\displaystyle \frac{1}{L_1}}{V}_i\\ -{\displaystyle \frac{r_2}{L_2}}{i}_{L2}-{\displaystyle \frac{\overline{D_n}}{L_2}}{v}_{dc}+{\displaystyle \frac{1}{L_2}}{V}_i\\ {\displaystyle \frac{\overline{D_p}}{C}}{i}_{L1}+{\displaystyle \frac{\overline{D_n}}{C}}{i}_{L2}-{\displaystyle \frac{1}{CR}}{v}_{dc}\end{array}\right]$$

(7)

For this purpose, the linearization process with the Jacobian matrix is used in this paper [20]. The state-space equation is linearized around the equilibrium point. The equilibrium point is set as x(t) = $\mathit{X}$, u(t) = $\mathit{U}$. A small-signal model is derived by assuming the fluctuation in the equilibrium point is satisfyingly slower than the carrier period.

$${\displaystyle \frac{d\Delta \mathit{x}\left(t\right)}{dt}}=\Delta \mathit{A}\Delta \mathit{x}\left(t\right)+\Delta \mathit{B}\Delta \mathit{u}\left(t\right)$$

(8)

$$\Delta \mathit{A}={\left[{\displaystyle \frac{\partial \mathit{f}\left(\mathit{X},\mathit{U}\right)}{\partial \mathit{x}\left(t\right)}}\right]}^T=\left[\begin{array}{ccc}-{\displaystyle \frac{r_1}{L_1}}& 0& -{\displaystyle \frac{\overline{D_p}}{L_1}}\\ 0& -{\displaystyle \frac{r_2}{L_2}}& -{\displaystyle \frac{\overline{D_n}}{L_2}}\\ {\displaystyle \frac{\overline{D_p}}{C}}& {\displaystyle \frac{\overline{D_n}}{C}}& -{\displaystyle \frac{1}{CR}}\end{array}\right]$$

(9)

$$\Delta \mathit{B}={\left[{\displaystyle \frac{\partial \mathit{f}\left(\mathit{X},\mathit{U}\right)}{\partial \mathit{u}\left(t\right)}}\right]}^T=\left[\begin{array}{cc}-{\displaystyle \frac{V_{dc}}{L_1}}& 0\\ 0& -{\displaystyle \frac{V_{dc}}{L_2}}\\ {\displaystyle \frac{I_{L1}}{C}}& {\displaystyle \frac{I_{L2}}{C}}\end{array}\right]$$

(10)

$$\mathit{x}\left(t\right)≔\mathit{X}+\Delta \mathit{x}\left(t\right),\Delta \mathit{x}\left(t\right)≔[{\Delta i}_{L1}{\Delta i}_{L2}\Delta v_{dc}{]}^T$$

$$\mathit{u}(t)≔\mathit{U}+\Delta \mathit{u}\left(t\right),\Delta \mathit{u}(t)≔{\left[\Delta \overline{D_p},\Delta \overline{D_n}\right]}^T$$

$$\mathit{X}≔[I_{L1}{I}_{L2}{V}_{dc}{]}^T,\mathit{U}≔{\left[\overline{D_p},\overline{D_n}\right]}^T$$

where $\overline{D_p}=1-D_p, \overline{D_n}=1-D_n.$ The values of the equilibrium point when $D_p=D_n$ are derived as follows:

$$V_{dc}={\displaystyle \frac{V_i}{\overline{D}}},I_{L1}=I_{L2}={\displaystyle \frac{V_i}{2R{\overline{D}}^2}}$$

(11)

where $\overline{D}=1-D$.

3.3. Derivation of Small-Signal-Model for Servo System

In this section, Equation (8) is extended to be developed as the servo controller so that the output voltage v_dc is capable of following the step response, and the phase input currents $i_{L1}$ and $i_{L2}$ are balanced. For this purpose, the output equation, the output variables, and its coefficient matrix are determined as follows:

$$\Delta \mathit{y}\left(t\right)=\Delta \mathit{C}\Delta \mathit{x}\left(t\right)$$

(12)

$$\Delta \mathit{y}\left(t\right)≔\left[\begin{array}{c}{\Delta v}_{dc}\\ \Delta i_L\end{array}\right],\Delta \mathit{C}=\left[\begin{array}{ccc}0& 0& 1\\ 1& -1& 0\end{array}\right]$$

Two expanded state-space variables are introduced to develop the servo system. Additional state-space variables ω₁(t) and ω₂(t) are defined as follows.

$${\omega }_1\left(t\right)≔{\int }_0^t{e}_1\left(\tau \right)d\tau , {\omega }_2\left(t\right)≔{\int }_0^t{e}_2\left(\tau \right)d\tau$$

$$e_1\left(\tau \right)≔v_{dc\_ref}\left(\tau \right)-v_{dc}\left(\tau \right) , e_2\left(\tau \right)≔{\Delta i}_{L\_ref}\left(\tau \right)-{\Delta i}_L\left(\tau \right)$$

The small-signal model for the servo system can be expressed as follows:

$${\displaystyle \frac{d{\mathit{x}}_{\mathit{e}}\left(t\right)}{dt}}={\Delta \mathit{A}}_{\mathit{e}}{\mathit{x}}_{\mathit{e}}\left(t\right)+{\Delta \mathit{B}}_{\mathit{e}}{\mathit{u}}_{\mathit{e}}\left(t\right)$$

(13)

$${\mathit{x}}_{\mathit{e}}\left(t\right)≔[\Delta \mathit{x}\left(t\right)\mathit{\omega }\left(t\right){]}^T=[{\Delta i}_{L1}\Delta i_{L2}{\Delta v}_{dc}{\omega }_1{\omega }_2{]}^T$$

(14)

$${\Delta \mathit{A}}_e=\left[\begin{array}{cc}\Delta \mathit{A}& \mathit{O}\\ -\Delta \mathit{C}& \mathit{O}\end{array}\right],{\Delta \mathit{B}}_e=\left[\begin{array}{c}\Delta \mathit{B}\\ \mathit{O}\end{array}\right]$$

(15)

The system input can be given as the following equation:

$${\mathit{u}}_{\mathit{e}}\left(t\right)=\Delta \overline{\mathit{D}}=\mathit{F}{\mathit{x}}_{\mathit{e}}\left(t\right)$$

(16)

where, ${\mathit{u}}_{\mathit{e}}\left(t\right)=[u_{e1}\left(t\right)u_{e2}\left(t\right){]}^T$,

$$\mathit{F}=\left[\mathit{K} \mathit{G}\right], \mathit{K}=\left[\begin{array}{ccc}{K}_1& K_2& K_3\\ K_4& K_5& K_6\end{array}\right], \mathit{G}=\left[\begin{array}{cc}{G}_1& G_2\\ G_3& G_4\end{array}\right]$$

(17)

As shown in Equation (17), feedback gain F consists of the gains K and G, in which K is defined as the gain for the state-space variables, and G is for the expanded state-space variables. Therefore, the system input can be represented as the following:

$${\mathit{u}}_{\mathit{e}}\left(t\right)=\mathit{K}\Delta \mathit{x}\left(t\right)+\mathit{G}\mathit{\omega }\left(t\right)$$

(18)

where, $\mathit{\omega }\left(t\right)=[{\omega }_1\left(t\right){\omega }_2\left(t\right){]}^T$.Considering Equations (13)–(18), the configuration of the proposed LQI control is shown in Figure 4. In Figure 4, $K_1,K_2,K_3,G_1,\mathrm{a}\mathrm{n}\mathrm{d} G_2$ are the feedback gains for input $u_{e1}$, and $K_4,K_5,K_6,G_3,\mathrm{a}\mathrm{n}\mathrm{d} G_4$ are the feedback gains for input $u_{e2}$. These feedback gains are obtained by solving the Riccati algebraic equation as explained in Equation (20). In the theory of optimal control, the performance index so that the system is stable is described as the following equations:

$$J={\int }_0^{\infty }\left\{{\mathit{x}}_{\mathit{e}}{\left(t\right)}^{⊺}\mathit{Q}{\mathit{x}}_{\mathit{e}}\left(t\right)+{\mathit{u}}_{\mathit{e}}{\left(t\right)}^{⊺}\mathit{R}{\mathit{u}}_{\mathit{e}}\left(t\right)\right\}dt$$

(19)

where, Q is a non-negative definite symmetric matrix, which is the weight coefficient matrix for each state-space variable, and R is a positive definite matrix, the weight coefficient matrix for the control input. The weight coefficient matrices Q and R of the LQI control are set as Q = diag [q₁, q₂, q₃, q₄, q₅] and R = diag [r₁, r₂], in which diag denotes a diagonal matrix. Each coefficient of the weight coefficient matrices Q and R corresponds to q₁:i_L1, q₂:i_L2, q₃:v_dc, q₄:ω₁, q₅:ω₂, r₁:$\overline{D_p}$, and r₂:$\overline{D_n}$. The feedback gain F can be given as the following:

$$\mathit{F}=-{\mathit{R}}^{-1}{\Delta \mathit{B}}_{\mathit{e}}^{⊺}\mathit{P}$$

(20)

where, the matrix P is the positive definite symmetric matrix that satisfies the Riccati algebraic equation as the following.

$$\mathit{P}{\Delta \mathit{A}}_{\mathit{e}}+\Delta {\mathit{A}}_{\mathit{e}}^{⊺}\mathit{P}-\mathit{P}\Delta {\mathit{B}}_{\mathit{e}}{\mathit{R}}^{-1}\Delta {\mathit{B}}_{\mathit{e}}^{⊺}\mathit{P}+\mathit{Q}=0$$

(21)

3.4. PI Controller

This section investigates the configuration of the PI controller for comparison with the proposed method. Figure 5 shows the configuration of the PI control system. The PI control system consists of a major loop of voltage control and a minor loop of current control, which has two PI controllers so as to control each phase’s input current [19].

4. Verification

Figure 6 shows the experimental setup. The experiment was carried out with a proto-type converter with a rated power of 700 W. For controllers, the PE-Expert4 is used, which consists of a DSP board (TMS320C6657), an FPGA board (MWPE4-IPFPGA-24), an AD board (AD7357, 14-bit), and a DA board (AD5547,16-bit).

Table 1. Circuit conditions.

Input voltage Vi	100 V
Carrier frequency fc1 and fc2	20 kHz
IGBT	FGH40T120SMD
rated voltage	1200 V
rated current	40 A
Diode	FEP16DT
rated voltage	600 V
rated current	16 A
Load (Resistance) R	100 Ω
DC reactor	-
inductance L1 and L2	1.8 mH
rated current	9 A
core	silicon steel plate
resistance r1 and r2	68.6 mΩ
Capacitance C	750 µF
Converter	-
Power rating	700 W
Output voltage Vdc	250 V
Duty cycle	0.6
Input current	7 A
Output current	2.8 A

shows the parameters of the main circuit. The prototype has snubber circuits with a resistance of 1 kΩ and a capacitance of 1nF for compensating the effect of parasitic inductances. In this paper, the weight coefficients in Equation (19) are designed through simulations [19]. The “Arimoto-Potter method,” used to solve Riccati algebraic equations and gains using the Hamiltonian matrix, is employed for gain calculation.

Table 2. The parameters of LQR for Q = diag [1, 10, 0, 100,000, 100,000] and R = diag [1, 1].

Feedback Gain
K1	2.2138	K6	1.9795
K2	1.1652	G1	−497.60
K3	2.0918	G2	−97.733
K4	1.1604	G3	−478.74
K5	2.1158	G4	106.94

shows the gains designed. On the other hand, the PI control parameters were determined using Bode diagrams and simulations. The simulation of the PI controller was carried out to ensure stable control of the input current. As a result, a control bandwidth for the current control system was designed at 1000 rad/s. The control bandwidth of the voltage control was designed at 1/10 of that of the current control.

Table 3. Parameters of PI controllers.

Parameter	Value
Kpv	0.15
Tiv	0.02
Kpi	4.0
Tii	0.002

shows the parameters of the PI control. The software used for simulation is the PSIM 64-bit ver.11.1.7. In the simulation, ideal switching models are used for IGBTs and Diodes, and the parasitic inductance is not considered for simplification.

4.1. Steady-State Characteristics

Figure 7 shows the steady-state characteristics of the output voltage, the input current, and the phase input current when the reference voltage is 150 V. Figure 7a,b are the simulated and experimental results, respectively. It can be seen that the output voltage is controlled to its reference value, the input current ripple is reduced, and its frequency is twice the carrier frequency, 40 kHz. Figure 8 shows the steady-state characteristics when the reference voltage is 200 V and the duty ratio is 0.5. In this condition, the input current ripple is approximately zero. Similarly, Figure 9 shows the characteristics when the reference voltage is 250 V. The ripples of input current and phase input current for simulation are 0.53 A and 1.53 A, respectively. On the other hand, those values for the experiment are 0.6 A and 1.57 A, respectively. Figure 7, Figure 8 and Figure 9 show that the simulated results agree well with the experimental results.

4.2. Tracking Performance and Load Response

Figure 10 shows the simulated results of the tracking performance of the output voltage, in which the reference voltage changes in steps from 150 V to 190 V. In the experiment, the step voltage was chosen so that the parasitic inductance between the input voltage and the load does not affect the output voltage response. Because the value of the parasitic inductance is very small with $\mathsf{\mu }\mathrm{H},$ and its time constant is sufficiently smaller than that of output voltage, its effect is very small. The higher the output voltage, the output voltage response of the converter has a large influence. However, the experiment was carried out on the condition that the parasitic inductance does not affect the output voltage response. Figure 10a,b are the results of the proposed method and the PI control, respectively. In case Figure 10a, the output voltage reaches its reference value in about 10 ms, while that of the PI control reaches about 80 ms. Figure 11 shows the experimental results in the same condition as Figure 10. The tracking performance is 10 ms, which is the same as the simulation, while that of the PI control is about 100 ms, which is a little longer than that of the simulation due to an overshoot. The proposed method has a ten times faster tracking performance than the PI control. Figure 12 shows the load response when the load is changed from 200 W to 500 W. Figure 12a,b are the results of the proposed method and the PI control, respectively. In Figure 12b, the output voltage drops by a maximum of about 10 V and fluctuates for 70 ms. On the other hand, in Figure 12a, the output voltage keeps a constant value without fluctuation. Figure 13 shows the response when the load changes from 500 W to 200 W. In Figure 13b, the output voltage rises about 10 V and varies over 70 ms. Figure 13a shows that the output voltage keeps a constant value without fluctuation. The experimental results demonstrate that the proposed method has an excellent disturbance rejection for a load step response compared to the PI control. In the experiment, the buttery and the capacitor have slight voltage variations because we assume the converter has a relatively stable voltage source. The variation of the voltage source affects a disturbance in the control system. If the voltage source is unstable, the converter system will likely have a serious influence, such as input current ripple distortion, an imbalance between two-phase input currents, output voltage variation, and delayed output voltage response.

5. Conclusions

This paper proposed a novel linear quadratic integral (LQI) control for the interleaved boost converters suitable for HEVs and FCEVs. The conclusions obtained in this paper are as follows:

• The small-signal model for the servo system of the interleaved-boost converter was derived. The LQI control was proposed using the small-signal model.
• In the proposed method, an output voltage and a current signal error between two-phase input currents are selected as output variables to control the output voltage and balance between two-phase input currents. The proposed method has a simple system and can control both output variables without additional controllers.
• Steady-state characteristics in terms of the output voltage and the input current were demonstrated by experiments and simulations using an experimental apparatus with a rated power of 700 W.
• The validity of tracking performance and load response of the proposed method was demonstrated by comparing it with that of the conventional PI control. The tracking performance of the LQI control for the 40 V step response has a ten times faster response than that of the PI control. Also, the experimental results demonstrated that the proposed method can keep the output voltage constant for a load step of 300 W while that of the PI control varies by 10 V during 70 ms and has an excellent disturbance rejection.
• Therefore, the proposed LQI control improves the tracking performance and the load response and miniaturizes the motor-drive systems in HEVs and FCEVs.

For future research, current sharing control in the interleaved boost converters using the proposed LQI control is considered for a multi-buttery management system combined with an electric double-layer capacitor and fuel-cell battery.