Sliding Mode Control of Buck DC–DC Converter with LC Input Filter

Abstract

The employment of input filters in modern DC–DC converters is mandatory in order to ensure EMC (Electro-Magnetic Compatibility), provide power electronics with decent voltage and minimize the converter influence on the power grid. The LC type input filters bring also a possibility of voltage and current oscillation that may occur in the system. This oscillation may rise to a certain level that affects system stability. A wide range of methods are employed in order to attenuate these oscillations and allow appropriate system transient response. In this paper, the sliding mode control (SMC) strategy is proposed to reduce the input LC filter voltage and current oscillation and allow DC (Direct Current) output voltage control for both resistive and constant power load. The proposed control algorithm is intended for use in railway, tram and trolleybus DC/DC converters. For the proposed scheme of control simulation, a model is developed using Simulink software. Furthermore, laboratory stand experiments are carried out to verify simulation results.

1. Introduction

The purpose of using input filters in power electronic converter systems is to reduce the negative impact of the converters on the network, ensuring electromagnetic compatibility and appropriate quality of voltage directly supplying power electronic components. The most common solution is an LC type filter. It is a practically lossless second-order system, which tends to oscillate under certain circumstances. Oscillations are particularly likely to occur in systems where size minimization is sought at the expense of capacitance and inductance values. This fact is the most important for control systems with a load that has constant power character. Load systems characterized by constant power drawn from the power source are an increasingly common type in lighting and drive control systems. Their nature of impact on the converter from which they are supplied makes them perceived as a load with negative dynamic resistance, which means that they tend to oscillate in the LC input filters. From our point of view, research on the issue of ensuring voltage stability in these filters by means of control methods allows reducing the capacitance and inductance of the filters. This problem becomes particularly important in high-power systems, where reducing the capacitance value allows for a significant reduction in the size and weight of the device. So, the problem is still valid, and looking for methods which can positively affect the stability of the DC–DC converter by using appropriate control algorithms is worth consideration.

Modern methods of suppressing current and voltage oscillations in input LC filters of power electronics systems [1,2,3,4,5,6,7,8,9,10,11] can be divided into two groups: passive and active methods. Passive methods usually are realized by connecting an additional parallel RC branch to the input path of the power electronic converter, which is responsible for damping.

Another frequently used method is a significant oversizing of the input filter, so that its limit parameters are far away from the frequencies that can cause oscillations. When using passive suppression methods, the control algorithm most often does not take into account the presence of the LC filter, and the control method is adapted to the second-order object (which is the step-down, step-up, step-up-down converter in the basic configuration), and not the fourth-order object (the converter including the filter). Important works on passive methods of damping oscillations in power electronic systems are included in [1,2,3,4,5,6]. Active methods are based on appropriately selected control strategies, taking into account selected state variables responsible not only for the desired output values, but also those related to the input LC filter. The most frequently active methods of oscillation damping rely on taking into account the signal compensating the filter properties in the choke current control loop or the output capacitor voltage of the DC–DC converter. Such methods can be also called indirect active methods. Direct active methods include algorithms in which the compensating signal flows directly into the PWM modulator or directly affects the switching times. Interesting papers dealing with active damping of oscillations in input LC filters of power electronic converters include [8,9,11,12]. A classified table with research on suppressing oscillations in input LC filters is presented below (

Table 1. Classification of input LC filter oscillation suppression.

Passive Suppression	Active Suppression
Parallel to input LC filter capacitance RC series branch	In classic cascade regulators, the compensating signal addition occurs in: Current regulator loop, Voltage regulator loop, Directly to switches.
Parallel to input LC filter inductance RL series branch	Full state feedback controller
Series connection of parallel RL branch upon input LC filter	AI methods (neural networks, etc.)
Oversizing of input filter parameters	SMC

).

In article [9], the authors focused on the active damping of oscillations of the input LC filter of the step-down converter. It should be noted that the presented state-space model has five variables (choke currents, capacitor voltages, and filtered input capacitor voltage). In addition to the four directly related to the converter itself, the model has one additional variable that is responsible for the value of the input capacitor voltage filtered using a first-order low-pass filter. In the next chapter, the authors propose a modified version of the PI controller from the formula

$$u=(U_{zad}-U_{C2})k_p+{\displaystyle \int (U_{zad}-U_{C2})}{k}_{i}dt+U_{C1*}{k}_s$$

(1)

where U_zad is the set value of the output voltage, U_C2 is the output capacitor voltage, U_C1* means the input capacitor voltage after filtering with a low-pass filter, and k_p, k_i and k_s are appropriately selected constants.

Paper [10] discusses the sliding control of a DC–DC converter with a step-down LC filter in the input path. The load in the analyzed case has a resistive character. Initially, the authors propose a sliding variable in the form of

$$s(x)=\alpha x_1+\beta x_2+\gamma x_3+\delta x_4-k,$$

(2)

where x_1…4 mean the normalized currents flowing in the chokes and the voltages appearing on the capacitors of the system, while k, α, β, γ and δ mean appropriately selected constants. On the basis of this, they derive the equivalent control and obtain a description of the dynamics, which is unfortunately non-linear. Then, the reduced sliding variable is analyzed in the form:

$$s(x)=x_3+\delta x_4-k.$$

(3)

This is a function of the voltage on the capacitor (x₃) of the input LC filter and the voltage on the capacitor (x₄) of the DC–DC converter’s output stage. The description of the dynamics model during the sliding motion is non-linear. After its linearization, the conditions of asymptotic stability of the system can be obtained. For the discussed sliding variable, simulation tests were carried out for the DC–DC converter with the input LC filter in dynamic states, i.e., start and step load change. On the other hand, article [8] is devoted to methods of active suppression of input LC filters in power electronic converters, supplied from DC microgrids, whose loads have a constant power character. The authors, at the beginning of their work, note that converters and power electronic converters working with a constant power load can lead to instability of the input LC filter, and thus the entire control structure. In the case of DC microgrids, the methods of active stabilization consist mainly of adding an appropriate compensating structure in the control system of the power electronic converter or inverter. In this work, the authors propose including a compensating loop in the control structure of a converter or inverter supplying a DC microgrid. The authors present a new method for the active damping of oscillations in the LC filters of converters and power electronic inverters, intended for networks with one or many receivers. The stability of control systems is achieved by using the method of pole placement and also obtaining the assumed dynamic properties of the objects. As an example, they illustrate the application of the control method adopted by them for a step-down DC–DC converter that lowers the voltage supplying the network, whose receiver or receivers are devices of a constant power nature. The proposed control law takes into account the voltage of the converter output capacitor and the current in the inductance of the LC filter. Theoretical considerations are illustrated by numerous simulation results and tests of real systems.

Many works on the sliding control of various types of converters have been created [8,13,14,15,16,17,18,19,20,21,22,23,24,25]. However, most consider the case of a resistive/constant power load alone without the presence of an input LC filter [12,25,26,27]. Only a few authors consider the presence of an LC input filter in the creation of a sliding control strategy and, at the same time, a constant power load [8].

The properties of the sliding control make it ideal for the implementation of control structures in power electronic systems. They ensure immunity to a wide class of disturbances and enable effective control of state variables. The wide range of sliding control techniques [28,29] and a relatively small computational effort required to create control signals gives real opportunities to use SMC algorithms in practical applications. A very important feature of these algorithms is the flexible choice of the sliding variable and the control law. In the case of a DC–DC converter, this gives a chance to simultaneously control the output voltage and the current and voltage values in the input LC filter. In a later part of the work, an analysis of the sliding control structures is carried out in terms of their use to regulate the output voltage of the converter and reduce the oscillations generated in the input LC circuits.

2. Methodology

The general structure of the system under consideration is shown in the Figure 1.

The system dynamics can be described using state equations, provided that the load of the system is known. From an analytical point of view, the simplest case is a resistive load. The equations describing the dynamics of this type of system are shown by the Formula (4). Unfortunately, in modern devices, a purely resistive load is definitively rare. It is much more common to encounter systems that, thanks to an internal control structure that ensures the stabilization of many different voltages, behave as a load with constant power from the point of view of power supply terminals. Systems of this type cause the most problems with stability, because they have negative dynamic resistance properties. A description of the dynamics of DC–DC converter with constant power load is provided in Equation (5).

$$\begin{array}{l}{\dot{i}}_{L1}=\frac{1}{L_1}\left(U_w-U_{C1}\right)\\ {\dot{U}}_{C1}=\left\{\begin{array}{cc}\frac{1}{C_1}\left(i_{L1}-i_{L2}\right)& \begin{array}{ccc}& \mathrm{for}& \end{array}{T}_{ON}\\ \frac{1}{C_1}{i}_{L1}& \begin{array}{ccc}& \mathrm{for}& \end{array}{T}_{OFF}\end{array}\right.\\ {\dot{i}}_{L2}=\left\{\begin{array}{cc}\frac{1}{L_2}\left(U_{C1}-U_{C2}\right)& \begin{array}{ccc}& \mathrm{for}& \end{array}{T}_{ON}\\ -\frac{1}{L_2}{U}_{C2}& \begin{array}{ccc}& \mathrm{for}& \end{array}{T}_{OFF}\end{array}\right.\\ {\dot{U}}_{C2}=\frac{1}{C_2}\left(i_{L2}-i_r\right)=\frac{1}{C_2}\left(i_{L2}-\frac{U_{C2}}{R}\right)\end{array}$$

(4)

$$\begin{array}{l}{\dot{i}}_{L1}=\frac{1}{L_1}\left(U_w-U_{C1}\right)\\ {\dot{U}}_{C1}=\left\{\begin{array}{cc}\frac{1}{C_1}\left(i_{L1}-i_{L2}\right)& \begin{array}{ccc}& \mathrm{for}& \end{array}{T}_{ON}\\ \frac{1}{C_1}{i}_{L1}& \begin{array}{ccc}& \mathrm{for}& \end{array}{T}_{OFF}\end{array}\right.\\ {\dot{i}}_{L2}=\left\{\begin{array}{cc}\frac{1}{L_2}\left(U_{C1}-U_{C2}\right)& \begin{array}{ccc}& \mathrm{for}& \end{array}{T}_{ON}\\ -\frac{1}{L_2}{U}_{C2}& \begin{array}{ccc}& \mathrm{for}& \end{array}{T}_{OFF}\end{array}\right.\\ {\dot{U}}_{C2}=\frac{1}{C_2}\left(i_{L2}-i_o\right)=\frac{1}{C_2}\left(i_{L2}-\frac{P}{U_{C2}}\right)\end{array}$$

(5)

where i_L1 and i_L2 and U_C1 and U_C2 are, respectively, the currents and voltages in the circuit, U_w is the input voltage, R and P, respectively, are the resistance and power of the load. T_ON refers to the time interval in which transistor T₁ is on and transistor T₂ is off, while T_OFF refers to the time interval in which transistor T₂ is on and transistor T₁ is off.

2.1. Sliding Mode Control Concept

The analysis of the system showed that the suppression of the resulting oscillations is possible through the appropriate control of the transistors T₁ and T₂. When the voltage on the capacitor C₁ is greater than the input voltage, the transistor T₁ should be switched on and T₂ should be switched off. On the other hand, the increase in the oscillation is caused by switching off of the transistor T₁ and the complementary switching on of T₂ when the voltage across the capacitor C₁ is lower than the input voltage. The above considerations and the analysis of the system without the input LC filter led to the selection of the following sliding variable:

$$s=\left(U_{zad}-U_{C2}\right)+c_2\left(-{\dot{U}}_{C2}\right)+c_3\left(U_{C1}-U_w\right)$$

(6)

Making the assumption that the sliding variable and its derivative are equal to 0 during the sliding motion, we obtain the following equivalent control described by Equations (7) and (8).

$$\begin{array}{l}\\ \delta =\frac{C_1{C}_2{L}_2}{c_3{C}_2{L}_2{i}_{L2}+U_{C1}{c}_2{C}_1}\left\{i_{L1}\frac{1}{C_1}{c}_3+U_{C2}\left[\frac{1}{R{C}_2}\left(1-\frac{c_2}{R{C}_2}\right)+\frac{c_2}{L_2{C}_2}\right]+i_{L2}\frac{1}{C_2}\left(\frac{c_2}{R{C}_2}-1\right)\right\}\\ \end{array}$$

(7)

$$\begin{array}{l}\\ \delta =\frac{C_1{C}_2{L}_2}{c_3{C}_2{L}_2{i}_{L2}+U_{C1}{c}_2{C}_1}\left\{i_{L1}\frac{c_3}{C_1}+i_{L2}\left(-\frac{1}{C_2}-\frac{c_{2}P}{C_2^2{U}_{C2}^2}\right)+\frac{c_2{U}_{C2}}{L_2{C}_2}+\frac{P}{C_2{U}_{C2}}+\frac{c_2{P}^2}{C_2^2{U}_{C2}^3}\right\}\end{array}$$

(8)

From Equations (7) and (8), it follows that the obtained equivalent controls, marked as δ, are expressed by non-linear dependencies. Substitution of both the dependencies (7) and (8) to the Equation (4) or (5) describing the dynamics of the object leads to a non-linear dependence describing the closed control system. At this stage of the work, the authors made the linearization of the control system to prove the stability of the proposed solution. The state matrix of the linearized system has the form shown by the dependencies (9) and (10):

$$A_{lin}=\left[\begin{array}{cccc}0& -\frac{1}{L_1}& 0& 0\\ \frac{\partial U_{C1}}{\partial i_{L1}}& \frac{\partial U_{C1}}{\partial U_{C1}}& \frac{\partial U_{C1}}{\partial i_{L2}}& \frac{\partial U_{C1}}{\partial U_{C2}}\\ \frac{\partial i_{L2}}{\partial i_{L1}}& \frac{\partial i_{L2}}{\partial U_{C1}}& \frac{\partial i_{L2}}{\partial i_{L2}}& \frac{\partial i_{L2}}{\partial U_{C2}}\\ 0& 0& \frac{1}{C_2}& -\frac{1}{R{C}_2}\end{array}\right]$$

(9)

$$A_{lin}=\left[\begin{array}{cccc}0& -\frac{1}{L_1}& 0& 0\\ \frac{\partial U_{C1}}{\partial i_{L1}}& \frac{\partial U_{C1}}{\partial U_{C1}}& \frac{\partial U_{C1}}{\partial i_{L2}}& \frac{\partial U_{C1}}{\partial U_{C2}}\\ \frac{\partial i_{L2}}{\partial i_{L1}}& \frac{\partial i_{L2}}{\partial U_{C1}}& \frac{\partial i_{L2}}{\partial i_{L2}}& \frac{\partial i_{L2}}{\partial U_{C2}}\\ 0& 0& \frac{1}{C_2}& \frac{P}{C_2{U}_{C2}^2}\end{array}\right]$$

(10)

where the second and third rows of the matrix are the appropriate partial derivatives resulting from the substitution of the equivalent control into the system of equations describing the control object.

To calculate the range of parameters c₂ and c₃, for which the linearized control system is stable, the values of the inverter elements shown in the

Table 2. DC–DC converter parameters.

Name of the Parameter	Value
L1	100 µH
C1	600 µF
L1	990 µH
C1	1 mF
Uzad	24 V
Uw	48 V
R	2–12 Ω
P	50–300 W

were used.

To present a graphical interpretation of the influence of the selection of c₂ and c₃ parameters on the stability of the system, a special script was created in the program Matlab that calculates values of the dominant transmittance modes.

The analysis of the sets of points shown in Figure 2 and Figure 3 allows the determination of the influence of the load resistance or load power of the DC–DC converter on the stability of the system for given values of the coefficients c₂ and c₃. The graphs in Figure 2 and Figure 3 show that the range of parameters ensuring stability decreases with increasing load (decrease in resistance or increase in load power). At the same time, the parameters that guarantee stability for the largest analyzed load also ensure stability for smaller loads.

One final property that needs to be demonstrated is that the representative point will converge to the sliding hyperplane. This can be carried out using the Lyapunov method, as follows. We propose the function

$$V=\frac{1}{2}{s}^2$$

(11)

and we will show that its time derivative is negative when $V\ne 0$. We note that $\dot{V}=s\dot{s},$ and in the following analysis we will consider positve and negative $s$ separately. For positive s, the transistor is switched on, and the value of s can be obtained using (5) and (6) as

$$\begin{gathered} \dot{s}=\left(U_{zad}-\frac{1}{C_2}\left(i_{L2}-\frac{U_{C2}}{R}\right)\right)+c_2\left(-\frac{1}{C_2}\left(i_{L2}-\frac{1}{R{C}_2}\left(i_{L2}-\frac{U_{C2}}{R}\right)\right)\right) \\ +c_3\left(\frac{1}{C_1}\left(i_{L1}-i_{L2}\right)-U_w\right) \end{gathered}$$

(12)

On the other hand, for negative $s$, the transistor is switched on, and we obtain

$$\dot{s}=\left(U_{zad}-\frac{1}{C_2}\left(i_{L2}-\frac{U_{C2}}{R}\right)\right)+c_2\left(-\frac{1}{C_2}\left(i_{L2}-\frac{1}{R{C}_2}\left(i_{L2}-\frac{U_{C2}}{R}\right)\right)\right)+c_3\left(\frac{1}{C_1}{i}_{L1}-U_w\right)$$

(13)

Thus, the convergence of the representative point to the hyperplane (and, therefore, the stability of the system) is ensured in the subspace of the state space in which either $s>0$ and the right hand side of (12) is negative, or $s<0$ and the right hand side of (13) is positive.

The next sections of the work are dedicated to the description of the simulation model and the laboratory stand for experimental tests.

2.2. Simualtion Model of DC–DC Converter with Input LC Filter

In order to test the previously proposed control algorithms, a model of an electronic power converter with an input LC filter (Figure 4) was created in the Matlab/Simulink environment using the SimPower library.

To solve the problem of the so-called chattering, as well as to be able to use the PWM modulator, a well-known and often-used [30,31] control signal with saturation function was selected.

$$u=sat(s)=\left\{\begin{array}{ccc}-1& \mathrm{for}& s<-e\\ \frac{1}{e}s& \mathrm{for}& \left|s\right|\le e\\ 1& \mathrm{for}& s>e\end{array}\right.$$

(14)

Unfortunately, in this case, a steady-state error occurs in the control system, resulting from the introduction of a continuous control law. It can be eliminated by adding an integral term to the sliding variable. Sliding variable with a damping term taking into account the voltage on the capacitor C₁ takes the form

$$s=\left(U_{zad}-U_{C2}\right)+c_2\left(-{\dot{U}}_{C2}\right)+c_3\left(U_{C1}-U_w\right)+\frac{1}{T_i}{\displaystyle \int (U_{zad}-U_{C2})}$$

(15)

Assuming that the constant T_i of the integral term is limited and that the term itself affects only slow-changing signals significantly deviating from the frequencies important from the point of view of system stability, it can be assumed that the range of coefficients for which the closed control system is stable will not change. Parameter values calculated for sliding variables and plants from chapter 2.1 will also be valid for the discussed cases. The structure of the regulator is shown in the Figure 5.

2.3. Laboratory Stand Setup

An illustrative diagram of the laboratory stand and its view are shown in the Figure 6 and Figure 7, respectively.

In Figure 6, several basic elements of the system are presented:

• dSPACE DS1104 card, which, together with a PC computer with Matlab/Simulink and ControlDesk software installed, constitutes the acquisition and control part;
• DC power supply with capacitance, which is the power source of the system;
• DC–DC step-down converter with input LC filter, made in the form of a circuit placed on a printed circuit board;
• Measurement card cooperating with LEM current and voltage converters, adjusting the voltage level for the dSPACE card;
• 2SC0108T2A0 controllers responsible for controlling the transistors of the converter itself as well as the configurable resistive load, optional step-down DC–DC converter simulating a constant power load, four-channel oscilloscope for measuring voltages and currents of energy storage components.

Laboratory stand is illustrated in Figure 7.

3. Results

3.1. Simulation Results

The first load for which simulation tests were carried out was the resistive load. Data on the values of the system elements are taken from Table 2. The converter starts with a load of 8 Ω, and then, after 0.03 s from the start of the simulation, a step change in the load is made to the value of 4 Ω.

Figure 8 shows the waveforms of currents and voltages on the energy storage elements, the trajectory of the system, and the waveform of the control signal when the sliding variable includes the term containing the voltage on the capacitor C₁, responsible for damping the oscillations of the LC input filter.

The second type of load for which simulation tests were carried out was a constant power load. The parameters of the system elements are taken from Table 2. The converter started with a load of 100 W, and then, after 0.03 s from the start of the simulation, a step load change was made to the value of 200 W.

Figure 9 shows the waveforms of currents and voltages on the energy storage elements, the trajectory of the system, and the waveform of the control signal when the sliding variable includes the term containing the voltage on the capacitor C₁, responsible for the damping of the oscillations of the LC input filter.

3.2. Experimental Results

In order to demonstrate the usefulness and correctness of the analytical considerations in this chapter, laboratory tests were carried out on a step-down DC–DC converter with an input LC filter for a resistive load and a constant power load. A number of tests were carried out; each time data acquisition was performed. The following tests were carried out: an algorithm with a sliding variable not taking into account the damping term, and an algorithm with a sliding variable taking into account the voltage on the capacitor C1. The following tests of the dynamic states of the DC–DC converter with the input LC filter were carried out:

• Start of the converter with open output terminals;
• Start of the converter with resistive load;
• A step change in the load from 8 Ω to 4 Ω;
• A step change in the load from 4 Ω to 8 Ω;
• Start of the converter with open output terminals;
• A step change in the load from 72 W to 144 W;
• A step change in the load from 144 W to 72 W.

The acquisition of measurement data and the control process itself were carried out using the dSPACE card. The selected signals of the system were also recorded in parallel using the MS07034A oscilloscope. Some of the research results are presented below.

Figure 10 shows the waveforms obtained for starting the converter with a load of 8 Ω where the sliding variable includes a damping term containing the voltage value on the capacitor C₁. The converter is started 5 ms after the start of data acquisition.

Figure 11 shows the waveforms obtained for a step change in the converter load from 72 W to 144 W when the sliding variable includes the damping term with the voltage on the capacitor C₁. A step change in load takes place 5 ms after the start of data acquisition.

4. Discussion

The aim of the study was to perform simulation and laboratory tests of sliding mode control algorithms dedicated to a DC–DC buck converter with input LC filter for both resistive and constant power loads. The experiments revealed the possibility of current and voltage input LC filter oscillation attenuation and appropriate system response at the same time by novel sliding mode variable usage. As part of the research, a novel sliding variable for a DC–DC buck converter with an input LC filter was proposed. Next, the stability of the proposed method was proven by means of equivalent control and linearization for wide range of system loads (both resistive and constant power). The next step was to develop a simulation model for the system, including the proposed control strategy. The final stage of the work was to implement the control structures in the MATLAB/Simulink environment and run them on a dSpace card controlling DC–DC buck converter with the input LC filter. The results of laboratory tests confirmed the effectiveness of the proposed method. The presented algorithm, through appropriate control, allows the reduction in the size of the filters, and the simplicity of the implementation allows it to be incorporated in industrial railway, tram, and trolleybus DC–DC converters.