Numerical Validation of a Boost Converter Controlled by a Quasi-Sliding Mode Control Technique with Bifurcation Diagrams

Abstract

A boost converter is an electronic circuit that generates a higher voltage in the output than in the input. The most common method to regulate the DC/DC converter is pulse-width modulation (PWM), and some techniques such as sliding mode control help perform a switching frequency to determine the duty cycle. However, some instabilities at different operating points have been detected with the controllers that have not yet been studied. Therefore, this paper presents a numerical validation of the boost converter with bifurcation diagrams. The pulse-width modulation is controlled by using a quasi-sliding mode control technique, such as the zero average dynamics, because it allows for the reduction of some phenomena such as chattering, ripple, and distortions. The results show that $N-T$ periodic orbits are detected with this technique from an initial operating point and they present a qualitative symmetry in both voltage and current variables. This technique is helpful to study a whole range of instability problems resulting from the different power converters and the controllers.

1. Introduction

A boost converter is a device that helps to step up voltage and regulate the output signal. Boost converters are frequently used to condition renewable energy sources such as solar panels [1]. Other important applications of this power converter are described in [2]. This device can be controlled with pulse-width modulation (PWM), a method to vary the duty cycle of a signal (defined as the time the switch is in the ON position), to control the output voltage with an equivalent fixed frequency. However, one of the main issues with the converters is the system stability under different disturbances.

Some control techniques help calculate the duty cycle related to the time the switch remains open and closed. In the 1980s, the sliding mode control was presented as a technique to perform this task [3]. In [4], the authors presented a control system for power conditioning based on sliding mode control (SMC) theory, showing some advantages of using this technique. However, the design generated chattering, which is not desirable because it can produce ripple and distortion at the output of the circuit.

One solution to improve chattering was presented in 2000 by [5] by considering zero average dynamics (ZAD). This technique helps create a switching surface that can be used as a reference for the system. One advantage of the technique is that it allows a fixed switching frequency [6]. This controller has been used in several applications. For example, the authors of [7] used the ZAD to control the boost converter for power factor correction. In [8], the authors used ZAD and SMC to design and implement a multi-phase step-down converter. Other authors have used this technique to control DC motors with a buck converter [9]. The experimental implementation of a buck converter with quasi-SMC and a loss estimator function was presented [10].

However, bifurcations in DC/DC converters have been reported in the literature because of the nonlinear switching actions [11,12,13,14]. In [12], the authors studied one- and two-periodic orbits, and besides, an attractor was detected. In [11], the authors studied the symmetrical dynamic behaviors of the two converters by using time-domain waveforms and phase portraits. In [14], the authors studied the behavior of a buck converter when the error amplifier coefficient changes, and employed sine voltage compensation to stabilize the system and expand the stability boundary. In [13], the authors proposed a novel discrete-time nonlinear map for converter configuration changes during modulation; they also derive analytical criteria for periodic orbits and flip bifurcations.

Furthermore, chaos has been presented as reported by [15,16]. In [17], the authors confirmed the presence of chaos in a buck converter experimentally. Stability analysis, paths to chaos analysis, and converter analysis in discontinuous driving mode can also be used to investigate nonlinear phenomena [18,19,20]. Hence, in [21], the authors investigated the behavior of a two-dimensional system defined by a boost converter controlled by ZAD. They discovered saturated periodic orbits and a period addition event. In [22], the authors use a boost converter to study the period addition phenomena in a model with parasitic resistances, similar to the results obtained from the experimental model. Additionally, the authors of [23] presented a switching surface that incorporates the current in the capacitor and a fixed-point induction control (FPIC) technique to respond to chaos.

The literature shows that an analysis of the behavior of a boost converter with control parameter variations applying bifurcation diagrams has not been made, nor applying a quasi-sliding mode control that considers time delays. In addition, the Jacobian matrix of the Poincaré map has not been used to evaluate periodic orbits in boost converters. Therefore, this paper presents a numerical validation of the boost converter using bifurcation diagrams obtained with the Poincaré technique, which help characterize the stability behavior of the system. The paper includes simulation tests of the boost converter with the ZAD technique and variation of parameters associated with the switching surface.

The novelties of the paper are related to: (a) the determination of the stability of a piecewise linear system with a quasi-sliding mode control that considers time delays to represent real systems close to physical phenomenon; (b) the inclusion of time delays generated by electronic devices, close-loop control, sampling, and processing times; and (c) the use of spectral radius to determine the stability of periodic orbits. Moreover, the main contributions in this paper are as follows: (1) the system is discretized through Poincaré application to reduce the samples of the system variables and the results demonstrate that with few samples, the control is efficient; (2) the system is discretized for simple and easy implementation in real-time digital processors, which are usually limited in sampling speed, quantization errors, and processing speed; (3) a sampling of current and voltage variables is synchronized at the beginning of each period of the PWM signal to facilitate the implementation in real-time applications; and (4) the method allows determining 1T-orbits and a flip bifurcation was found when the control parameter is changed.

Three additional sections are included in the paper. Section 2 outlines the materials and methods utilized to formulate the problem mathematically. Next, in Section 3, the findings and analysis are presented. Finally, Section 4 summarizes the findings and presents the conclusion.

2. Materials and Methods

The boost converter model controlled by ZAD is presented in this section. In addition, the mathematical equations for calculating the duty cycle and the switching surface, $s\left(x\right(t\left)\right)$, used to characterize the dynamics of the boost system, are included. The switching surface is represented by a piecewise linear function to achieve an explicit representation of the duty cycle.

2.1. Boost Converter

Equations (1) and (2) present the differential equation of a system defined as the two-state model.

$$\frac{dv}{dt}=-\frac{1}{RC}v+\frac{1}{C}i(1-u),$$

(1)

$$\frac{di}{dt}=-\frac{1}{L}v(1-u)+\frac{V_{in}}{L}.$$

(2)

The implementation of the boost converter is relatively simple. The basic circuit to represent a boost converter is shown in Figure 1. This diagram depicts a series source with an impedance of L, a shunt switch S, a diode D, a capacitor C, and a load represented by R. The input of the source is presented as $V_{in}$, the current in the inductor is defined as i, and v is the output voltage.

The boost converter works according to the switch. First, when the switch is on, the inductor stores the energy from the source, and the load is fed by the capacitor. Second, when the switch changes from on to off, the current flows through the diode and feeds the capacitor and the load.

The state-space expressions are obtained as presented in Equations (3)–(5). Equation (3) represents the variable $x_1$ that considers changes between the output voltage v and the input voltage $V_{in}$.

$$x_1=\frac{v}{V_{in}}.$$

(3)

Equation (4) represents the variable $x_2$, which considers changes between the current i and the input voltage $V_{in}$. This equation also considers the constant values of the inductance L and the capacitance C.

$$x_2=\sqrt{\frac{L}{C}}\frac{i}{V_{in}}.$$

(4)

Finally, in Equation (5), the variable S is considered with the time t. This equation considers the root square of the constants defined as the inductance L and the capacitance C.

$$S=\sqrt{LC}t.$$

(5)

Additionally, the system considers the parameter $\gamma =\sqrt{\frac{L}{R^{2}C}}$ in Equation (6).

$$\left(\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right)=\left(\begin{array}{cc}-\gamma & 1-u\\ u-1& 0\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)+\left(\begin{array}{c}0\\ 1\end{array}\right).$$

(6)

The boost converter is built to obtain an output voltage value higher than the input voltage ($v>V_{in}$). According to Equation (3), the resulting value must be $x_1>1$. In addition, the differential equations are solved when u takes values from the following set of $\{0,1\}$. These two values are used to represent two operating topologies, as defined in Equations (7) and (8).

$$\dot{X}\left(t\right)=A_{1}X\left(t\right)+B,\mathrm{if}u=1,$$

(7)

$$\dot{X}\left(t\right)=A_{2}X\left(t\right)+B,\mathrm{if}u=0.$$

(8)

The parameters $A_1$, $A_2$, and B correspond to the matrices obtained from Equation (6).

$$A_1=\left(\begin{array}{cc}-\gamma & 0\\ 0& 0\end{array}\right),$$

(9)

$$A_2=\left(\begin{array}{cc}-\gamma & 1\\ -1& 0\end{array}\right),$$

(10)

$$B=\left(\begin{array}{c}0\\ 1\end{array}\right).$$

(11)

These two differential equations, defined previously as (7) and (8), can be rewritten as in Equation (12):

$$\dot{X}=A_{1}X+B+(A_2-A_1)Xu.$$

(12)

This last system represents the continuous conduction mode and will be considered in the simulation. However, the discontinuous conduction mode is not considered to simplify the numerical simulation.

Now, the solution that is obtained after solving for each topology proposed in Equations (7) and (8) and presented as follows [24]:

$$X_i\left(t\right)={\varphi }_i(t-t_0)X\left(t_0\right)+{\psi }_i(t-t_0),$$

(13)

$${\varphi }_i(t-t_0)=e^{A_i(t-t_0)},$$

(14)

and

$${\psi }_i(t-t_0)={\displaystyle {\int }_{t_0}^t{e}^{A_i(t-\tau )}Bd\tau }.$$

(15)

The following step calculates the exponential matrices as presented in Equations (16) and (17).

$${\psi }_1(t-t_0)=B(t-t_0),$$

(16)

$${\psi }_2(t-t_0)=A_2^{-1}\left(e^{A_2(t-t_0)}-I_2\right)B.$$

(17)

where $I_2$ is a two-by-two identity matrix.

2.2. Duty Cycle

A duty cycle in a PWM controller is the amount of time the output is in a high state in a given period. The duty cycle in a PWM controller is calculated by dividing the periods by the number of cycles. This means that it will always be equal to 1 if there are equal periods and cycles, but it can be less than 1 if there are more periods or cycles. The duty cycle can be used to regulate the voltage supplied to the load.

The ZAD technique can be used to generate a duty cycle according to the behavior of the system. Thus, the duty cycle obtained with this technique is represented in Equation (18). This equation can obtain $d<0$ and $d>T$ values.

$$d={\displaystyle \frac{2s\left(\mathrm{x}\left(nT\right)\right)+T{\dot{s}}_2\left(\mathrm{x}\left(nT\right)\right)}{{\dot{s}}_2\left(\mathrm{x}\left(nT\right)\right)-{\dot{s}}_1\left(\mathrm{x}\left(nT\right)\right)}}.$$

(18)

Electronic devices managed with a pulse-width modulator must consider a delay period. Next, the duty cycle must consider this period as $t=(n-1)T$ and not at $t=nT$. Consequently, the duty cycle with the delay period consider the following mathematical expression:

$$d_c={\displaystyle \frac{2s\left(\mathrm{x}\left((n-1)T\right)\right)+T{\dot{s}}_2\left(\mathrm{x}\left((n-1)T\right)\right)}{{\dot{s}}_2\left(\mathrm{x}\left((n-1)T\right)\right)-{\dot{s}}_1\left(\mathrm{x}\left((n-1)T\right)\right)}}.$$

(19)

The analytical study of the system stability that governs the boost converter, including the duty cycle with a delay period (Equation (19)), is relatively complex. For example, the stability of the $1T$-periodic orbit can be obtained by calculating the Floquet exponents. After incorporating a delay period into the system, the term t must be changed to $t-T$. The term T is defined as the switching period. The equation that defines the boost converter is expressed as follows:

$$\begin{array}{cc}\hfill \left(\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right)& =\left(\begin{array}{cc}-\gamma & 0\\ 0& 0\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)+\left(\begin{array}{c}0\\ 1\end{array}\right)+\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)·\theta \left(t-\left(T+\frac{d}{2}\right)\right)\hfill \\ \hfill & +\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)·\theta \left(t-\left(2T-\frac{d}{2}\right)\right).\hfill \end{array}$$

(20)

Note that the lag period has already been added to the equations of the system, and the duty cycle that appears in Equation (20) is that of Equation (19). The analysis of the stability of the $1T$-periodic orbit using the calculation of the Floquet exponents associated with the solution of Equation (20) will be left as a future task.

2.3. Discretization of the System

It is known that any continuous dynamic system can be discretized by choosing a suitable hyperplane in which the trajectories cross it, forming an angle other than zero (transversality condition) [25,26], as shown in Figure 2.

The Poincare application is defined as:

$$\begin{array}{c}P:\mathsf{\Sigma }\to \mathsf{\Sigma }\hfill \\ x\mapsto P\left(x\right).\hfill \end{array}$$

(21)

where $P\left(x\right)$ is the intersection of the orbit passing through x with the hyperplane $\mathsf{\Sigma }$.

The dynamics of a continuous system can be characterized entirely by applying the Poincaré technique. Note that if $L_0$ is a limit cycle, the orbit passing through $x_0$ always intersects $\mathsf{\Sigma }$ at $x_0$. Thus, a $1T$-periodic orbit corresponds to a single point in $\mathsf{\Sigma }$; a $2T$-periodic orbit corresponds to two points in $\mathsf{\Sigma }$, and so on. The above is because the system is T-periodically forced.

To build the application of Poincaré, we will take a system sampling each period, concatenating the solutions in each of the intervals $[nT,nT+\frac{d_n}{2}]$, $[nT+\frac{d_n}{2},(n+1)T-\frac{d_n}{2}]$, and $[(n+1)T-\frac{d_n}{2},(n+1)T]$, following the methodology presented in [27].

Equation (13) is used if $x\left(nT\right)$ is the status of the system in $NT$. Next, at the end of the first stretch $[nT,nT+\frac{d_n}{2}]$, the following expression is obtained:

$${\mathbf{x}}_1\left(nT+\frac{d_n}{2}\right)={\varphi }_1\left(\frac{d_n}{2}\right)\mathbf{x}\left(nT\right)+{\psi }_1\left(\frac{d_n}{2}\right).$$

(22)

Now, at the end of the second stretch $[nT+\frac{d_n}{2},(n+1)T-\frac{d_n}{2}]$, the following expression is obtained:

$${\mathbf{x}}_2\left((n+1)T-\frac{d_n}{2}\right)={\varphi }_2\left(T-d_n\right){\mathbf{x}}_1\left(nT+\frac{d_n}{2}\right)+{\psi }_2\left(T-d_n\right).$$

(23)

Moreover, replacing $x_1$ defined in Equation (22) in this last expression, the following equality is obtained:

$$\begin{array}{cc}{\mathbf{x}}_2\left((n+1)T-\frac{d_n}{2}\right)\hfill & ={\varphi }_2\left(T-d_n\right){\varphi }_1\left(\frac{d_n}{2}\right)\mathbf{x}\left(nT\right)\hfill \\ \hfill & +{\varphi }_2\left(T-d_n\right){\psi }_1\left(\frac{d_n}{2}\right)+{\psi }_2\left(T-d_n\right).\hfill \end{array}$$

(24)

Now, at the end of the third stretch $[(n+1)T-\frac{d_n}{2},(n+1)T]$, the following equivalence is obtained:

$${\mathbf{x}}_3\left((n+1)T\right)={\varphi }_1\left(\frac{d_n}{2}\right){\mathbf{x}}_2\left((n+1)T-\frac{d_n}{2}\right)+{\psi }_1\left(\frac{d_n}{2}\right).$$

(25)

Consequently, at the end of the third stretch, the following expression is obtained:

$$\begin{array}{cc}\hfill {\mathbf{x}}_3\left((n+1)T\right)& ={\varphi }_1\left(\frac{d_n}{2}\right){\varphi }_2\left(T-d_n\right){\varphi }_1\left(\frac{d_n}{2}\right)\mathbf{x}\left(nT\right)\hfill \\ \hfill & +{\varphi }_1\left(\frac{d_n}{2}\right){\varphi }_2\left(T-d_n\right){\psi }_1\left(\frac{d_n}{2}\right)\hfill \\ \hfill & +{\varphi }_1\left(\frac{d_n}{2}\right){\psi }_2\left(T-d_n\right)+{\psi }_1\left(\frac{d_n}{2}\right).\hfill \end{array}$$

(26)

The application of Poincaré P that we will use is the one corresponding to $P\left(\mathbf{x}\left(nT\right)\right)={\mathbf{x}}_3\left((n+1)T\right)$, where ${\mathbf{x}}_3\left((n+1)T\right)$ is given by Equation (26). Note that, in this case, we are assuming that there is no saturation of the duty cycle, that is, $d_n\in \left(0,T\right)$. Further, the saturated duty cycle can be represented as follows:

1.

If $d_n=0$, the Poincaré map corresponds to:

$$P\left(\mathbf{x}\left(nT\right)\right)={\varphi }_2\left(T\right)\mathbf{x}\left(nT\right)+{\psi }_2\left(T\right);$$

(27)

2.

If $d_n=T$, the Poincaré map corresponds to:

$$P\left(\mathbf{x}\left(nT\right)\right)={\varphi }_1\left(T\right)\mathbf{x}\left(nT\right)+{\psi }_1\left(T\right).$$

(28)

The Poincaré application is presented graphically in Figure 3 [27], where $d_n=d$ is used to simplify the terms. Note that the Poincaré P is a function of the system status at $NT$ and the duty cycle in the $\left[nT,(n+1)T\right]$, that is, $P=P\left(\mathbf{x}\left(nT\right),d_n\right)$. This last expression is considered when the stability of periodic orbits is analyzed.

3. Results

This section presents the simulation results performed on the boost converter with the ZAD technique. In this case, some changes in the parameters associated with the switching surface are carried out. In addition, some bifurcation diagrams are obtained with the Poincaré technique to characterize the behavior of the system.

Bifurcations

Figure 4 presents the variation of the duty cycle in terms of $k_1$. In addition, the term $k_2=0.5$ and the parameters $\gamma =0.35$ and $T=0.18$. Moreover, the initial condition is selected at the point ${\left(2.52.1875\right)}^T$. In this figure, the cyan curve corresponds to the $1T$-periodic orbit, which is unstable and the result shows that when $k_1=-0.26$ (marked in the figure), the system loses stability.

Figure 5 presents the variation of the voltage magnitude in terms of $k_1$. This figure is obtained considering that $k_2$ = 0.5. In addition, the parameters $\gamma =0.35$ and $T=0.18$ are selected as constants and the initial condition is selected at the point ${\left(2.52.1875\right)}^T$. The result shows that the voltage has the same behavior of the duty cycle, starting form the stable zone and reaching stability for values higher than $k_1=-0.26$.

Similar to the two previous results, Figure 6 presents the variation of the current magnitude in terms of $k_1$. Likewise, the parameters $\gamma =0.35$ and $T=0.18$ are selected as constants and the initial condition is selected at the point ${\left(2.52.1875\right)}^T$. The figure shows a complete behavior of the current when $k_1$ changes from −0.4 to 0.5. The behavior is similar to the two previous figures, where the system is stable before reaching the maximum value at $k_1=-0.26$ and, after this, the system is unstable.

Figure 7 shows the resulting values obtained with the Jacobian matrix of the Poincaré map. The initial values are inside a unit circle, which shows the stability of the $1T$-periodic orbit. By increasing $k_1$, the value goes away from the circle ($-1$), reaching approximately $k_1=-0.26$, which means it is a flip bifurcation [25]. The bifurcation related to the $1T$-periodic orbit is unstable and a new $2T$-periodic orbit is born where the bend occurs.

Figure 8 shows the variation in the spectral radius in terms of $k_1$. In this case, the parameters $\gamma =0.35$, $T=0.18$, and $k_2=-0.5$ were considered as constant values, and an initial point ${\left(2.52.1875\right)}^T$ was selected. The result shows that $k_1=0.26$ is the value where the spectral radius is equal to 1. For values of $k_1>0.26$, the spectral radius is less than 1, and for values of $k_1<0.26$, the spectral radius is greater than 1.

Figure 9 shows the variation of the duty cycle depending on the parameter $k_1$. In addition, the parameters $\gamma =0.35$, $T=0.18$, and $k_2=-0.5$ were considered as constant values, and an initial point ${\left(2.52.1875\right)}^T$ was selected.

Figure 10 shows the variation of the voltage magnitude according to the changes of $k_1$. Similar to the previous studies, the parameters $\gamma =0.35$, $T=0.18$, and $k_2=-0.5$ were considered as constant values, and an initial point ${\left(2.52.1875\right)}^T$ was selected.

Figure 11 shows the variation of the current magnitude depending on the parameter $k_1$. In addition, the parameters $\gamma =0.35$, $T=0.18$, and $k_2=-0.5$ and an initial point ${\left(2.52.1875\right)}^T$ were selected.

Figure 12 shows the evolution of the eigenvalues according to the changes of $k_1$. In addition, the parameters $\gamma =0.35$, $T=0.18$, and $k_2=-0.5$ were treated as constants, and an initial point ${\left(2.52.1875\right)}^T$ was selected.

In these figures, the instability of the $1T$-periodic orbit is observed (cyan) for $k_1$ with values less than $0.26$. After this value, the $1T$-periodic orbit is stabilized. Figure 12 shows that there are values under $-1$ and that they enter into the unit circle for $-1$ just when $k_1=0.26$.

The result shows that there is a flip bifurcation. The analytical study of the transition to chaos is relatively complex because the range to determine unstable periodic orbits is too small. After calculating the spectral radius with the Jacobian matrix of the Poincaré evaluated in $1T$-, $2T$-, and $3T$-periodic orbits, the Matlab code is not able to differentiate when the spectral radius of the Jacobian matrix becomes greater than 1 at each of the orbits mentioned above.

This behavior is observed in Figure 13 and Figure 14. For example, in Figure 13, the blue curve corresponds to the variation of the spectral radius of the Jacobian matrix of the Poincaré map evaluated in $1T$-periodic orbits, in terms of $K_1$. Furthermore, the cyan curve corresponds to the variation of the spectral radius of the Jacobian matrix of the Poincaré map evaluated in the $2T$-periodic orbits, in terms of $k_1$.

In Figure 14, the variation of the spectral radius is observed for several orbits and it shows symmetry around the point −0.26. Starting from the $1T$-periodic orbit (cyan) to the $6T$-periodic orbit (in yellow), the spectral radius becomes less than 1 approximately at the same point. This phenomenon has been observed in a buck converter [28]. Apparently, this type of behavior does not depend on the type of converter, but rather the ZAD technique.

Figure 15, Figure 16 and Figure 17 show the bifurcation diagrams of the duty cycle, voltage, and current when $k_1$ changes and a delay period is presented.

The $\gamma =0.35$, $T=0.18$, $k_2=0.5$, and initial condition ${\left(2.52.1875\right)}^T$ were selected. Comparing the bifurcation diagrams of Figure 4, Figure 5 and Figure 6, it is observed that the $1T$-periodic orbit loses stability in certain ranges in which it was previously stable.

To sum up, the results showed that the system presents different period orbits, loses stability, and has a flip bifurcation under variation of the parameter $k_1$. The bifurcation diagrams showed that the system becomes more unstable when a time delay is included in the boost converter.

4. Conclusions

This paper presented a numerical validation of the boost converter controlled by a quasi-sliding mode control technique with bifurcation diagrams. The research considered simulation tests of the boost converter with the ZAD technique and variations of the parameters associated with the switching surface. Voltage and current behaviors were studied by changing the control parameter $k_1$ to determine ranges where the $1T$-periodic orbits lose stability. In addition, $NT$-periodic orbits are detected from an initial operating point and the results showed qualitative symmetry in both voltage and current variables. Synchronization of the system variables with a periodic Poincaré trajectory, in a suitable hyperplane, allowed for the sampling of voltages and currents periodically at a very low speed, without losing system information. Furthermore, the inclusion of a time delay, as in real control systems, helped find that the control technique continued working at a permitted level in voltage regulation. Moreover, a large delay close to the steady-state natural frequency of the system variables, as usually visualized in the ripple of the output voltage signals of power converters, has been considered and the results showed that the ZAD regulates the signal correctly.