Zero–Average Dynamics Technique Applied to the Buck–Boost Converter: Results on Periodicity, Bifurcations, and Chaotic Behavior

Abstract

This study addresses chaos control in a Buck–Boost converter using ZAD technique and FPIC. The system analysis identified 1-periodic orbits and observed the occurrence of flip bifurcations, indicating chaotic behavior characterized by sensitivity to initial conditions. To mitigate these instabilities, FPIC was successfully applied, stabilizing periodic orbits and significantly reducing chaos in the system. Numerical simulations verified the presence of chaos, confirmed by positive Lyapunov exponents, and demonstrated the effectiveness of the applied control methods. Steady-state and transient responses of the open-loop model and experimental system were evaluated, showing a strong correlation between them. Under varying load conditions, the numerical model accurately predicted the converter’s real dynamics, validating the proposed approach. Additionally, closed-loop control with ZAD exhibited robust performance, maintaining stable inductor current even during abrupt load changes, thus achieving effective control in non-minimum phase systems. This work contributes to the design of robust control strategies for power converters, optimizing their stability and dynamic response in applications that require precise management of power under variable conditions. Finally, a comparison was made between the performance of the Buck–Boost converter controlled with ZAD and the one controlled by PID. It was observed that both controllers effectively regulate the current, with a steady-state error of less than 1%. However, the system controlled with ZAD maintains a fixed switching frequency, whereas the PID-controlled system lacks a fixed switching frequency and operates with a very high PWM frequency. This high frequency in the PID-controlled system presents a disadvantage, as it leads to issues such as chattering, duty cycle saturation, and consequently, overheating of the MOSFET.

1. Introduction

The Buck–Boost converter is a pivotal component in the realm of power electronics, serving as a versatile solution for voltage regulation by either stepping up (Boosting) or stepping down (Bucking) voltage levels. This dual functionality is particularly advantageous in applications where the input voltage may fluctuate above or below the desired output voltage. Recent advancements in control strategies, particularly the implementation of Zero Average Dynamics (ZAD) and Fixed Point Induced Control (FPIC) [1], have opened new avenues for enhancing the stability and performance of these converters. The current study investigates the dynamics of a Buck–Boost converter under these control techniques, revealing the existence of 1-periodic orbits, the occurrence of Flip bifurcations, and the manifestation of chaotic behavior, as evidenced by the calculation of Lyapunov exponents. The FPIC technique is employed to stabilize the 1-periodic orbit, and numerical simulations are conducted to validate the analytical findings.

The state of the art in Buck–Boost converter technology has seen significant developments, particularly in control methodologies aimed at improving performance metrics such as efficiency, stability, and transient response. The recent literature highlights various innovative designs and control strategies that address the inherent challenges associated with Buck–Boost converters. For instance, Narasimha and Salkuti [2] propose a closed-loop triple-deck Buck–Boost converter that emphasizes high gain and soft switching, showcasing the importance of control in enhancing converter performance. Similarly, Kurniawan et al. [3] introduce a novel architecture aimed at improving transient response and output voltage ripple, indicating a growing focus on optimizing converter dynamics. The work of Mutta and Goswami [4] further emphasizes the role of Buck–Boost converters in power-electronic systems, underscoring their capability to manage voltage swings effectively.

Despite these advancements, the Buck–Boost converter remains susceptible to instability, particularly during transitions between Buck and Boost modes. Abdelsalam et al. [5] discuss the complexities involved in controlling these transitions, highlighting the need for robust control strategies to maintain stable operation. The challenges of chaotic behavior in power converters are also well-documented, with studies such as that by Trujillo et al. [6] exploring the chaotic dynamics associated with Buck–Boost converters controlled using ZAD techniques. The existence of chaotic behavior, characterized by sensitive dependence on initial conditions, necessitates the implementation of effective control strategies to ensure reliable operation.

Regarding the control of instabilities that lead to chaotic behavior, several methodologies have been proposed, each with its own advantages and disadvantages. For example, in Ref. [7], a nonlinear feedback controller is proposed, given by a piecewise quadratic function, with the advantage of adjusting the feedback gain in the nonlinear feedback controller. In Ref. [8], an iterative map is built for the Buck–Boost converter under current-mode control. Bifurcations associated with variations in several model parameters are shown, particularly the input voltage, reference current, and load resistance. In Ref. [9], a control method based on the energy balance of a cycle is proposed. The underlying idea is to control a cycle by maintaining the energy balance of the system during the cycle. The advantage of this method is that the poles of the transfer function are adjusted, increasing the range of variability of the parameters where the system remains stable, meaning that system instabilities disappear, simultaneously eliminating bifurcation phenomena. In Ref. [10], the system’s instabilities are controlled using the weak periodic perturbation (WPP) method, when operating in current mode, demonstrating that the system can be driven from a chaotic region to a periodic one. In Ref. [11], chaos control is used to design a controller that eliminates chaotic behavior. The control is based on delayed feedback, controlled in peak current mode, operating in continuous conduction mode, demonstrating the system’s robustness. All the above techniques naturally require the measurement of one or more state variables to function and be implemented. However, in this paper, the FPIC technique is implemented in the Buck–Boost converter. This technique only requires knowledge of the steady-state control signal or the unstable fixed point, and in this way, the control forces the system to evolve towards the fixed point, so that by varying a certain control parameter, all the eigenvalues of the Jacobian matrix of the controlled system fall within the unit circle, stabilizing the unstable orbit.

The Buck–Boost converter has been widely studied and utilized in various applications due to its ability to efficiently handle voltage conversion in systems with variable inputs and specific output requirements. However, despite advances in control techniques, the ZAD control method has never been applied to a Buck–Boost converter. This control strategy has been successfully implemented in other converters, such as the Buck converter, where it has proven effective in regulating output voltage and controlling the system’s dynamic behavior. The ZAD strategy ensures a fixed switching frequency, allowing for greater control over the system and reducing the effects of nonlinear phenomena such as chaos and unwanted oscillations [12,13]. ZAD control applied to Buck converters has shown promising results in reducing steady-state errors and improving robustness against load disturbances [14]. However, no previous studies have addressed the implementation of this technique in a Buck–Boost converter, representing a significant research opportunity. This gap in the scientific literature is partly due to the inherent complexities of the Buck–Boost mathematical model, as well as the practical challenges associated with implementing ZAD control in converters with more complex topologies [15].

Moreover, the ZAD technique [16] has demonstrated its effectiveness in signal regulation with low steady-state error in Buck converters when combined with FPIC [17]. The use of these combined techniques could provide additional benefits to the Buck–Boost system, such as enhanced stability during switching, especially in applications where maintaining a fixed switching frequency is crucial to avoid instability issues and improve overall system performance [15]. The development of this paper was primarily motivated by the study of the dynamics of Buck and Boost converters when controlled using Zero Average Dynamics (ZAD). Significant theoretical and applied references for this work include the studies presented in [16,18], with particular emphasis on [19,20,21]. In these works, the dynamics of converters such as Buck, Boost, Sepic, among others, as well as converters with the addition of parasitic resistances, have been studied.

These devices can be controlled by PWM, a method that allows varying the duty cycle of a signal to adjust the output voltage with an equivalent fixed frequency. However, Buck–Boost DC–DC converters are nonlinear systems due to their switching device. In cases of low loads, the converter may enter discontinuous conduction mode, which occurs when the inductor current crosses zero. Bifurcation and chaotic events have been demonstrated in DC–DC converters due to the nonlinear switching action and the different types of controllers [22,23,24,25,26]. Consequently, these devices exhibit different behaviors when parameters and operating conditions change or when the system experiences perturbations [27].

Several authors have studied the behavior of power converters and established the conditions under which the system presents changes in stability or periodicity (periodic, quasi-periodic, or chaotic orbits) [13,28,29]. The implementation of various control techniques has enabled testing modern controllers, such as nonlinear control techniques [30]. Recently, techniques such as ZAD have been used to control DC–DC converters, as shown in [30,31,32], for both center-pulse and side-pulse control. From the reviewed literature, it is observed that there is no analytical or numerical analysis that evaluates the behavior of the Buck–Boost converter controlled with ZAD as presented here.

Recently, Ref. [33] presented an innovative proposal for a four-switch Buck–Boost power converter (FSBBC) utilizing current-mode control based on PI controllers. This system enabled smooth transitions between different operating modes (Buck, Boost, and Buck–Boost) and ensured robust voltage and current control, resulting in rapid response to voltage fluctuations and load changes. The proposed methodology facilitated precise tuning of the converter’s duty cycle, optimizing its performance and efficiency, particularly in renewable energy and energy storage applications.

On the other hand, the main contribution of Ref. [33] was the implementation of current-based control, which reduced switching losses and improved converter stability through rapid response to voltage and load demand changes. Simulation results performed in MATLAB demonstrated that the converter maintained stable performance across an input voltage range of 5 to 75 V and output currents of up to 6 A. This research highlighted the flexibility and efficiency of the FSBBC in various applications, such as photovoltaic and energy storage systems, making it an attractive option for high-frequency, low-cost applications.

The primary objective of this research is to explore the dynamics of a Buck–Boost converter under the influence of ZAD and FPIC control techniques, aiming to achieve enhanced stability and performance. By investigating the existence of 1-periodic orbits and Flip bifurcations, this study seeks to contribute to the understanding of nonlinear dynamics in power converters. Additionally, the calculation of Lyapunov exponents will provide insights into the chaotic behavior exhibited by the converter, further informing the design of control strategies that can mitigate instability. The validation of analytical results through numerical simulations will serve to reinforce the findings and provide a comprehensive understanding of the system’s behavior.

The contributions of this research extend beyond theoretical insights, offering practical implications for the design and control of Buck–Boost converters in various applications. By demonstrating the effectiveness of the FPIC technique in stabilizing the 1-periodic orbit, this study provides a foundation for future research aimed at developing more robust control methodologies. Furthermore, the exploration of chaotic dynamics in power converters opens new avenues for understanding and managing instability, ultimately leading to more reliable and efficient power electronic systems.

Therefore, this paper provides an analysis and control of chaos in the Buck–Boost converter using ZAD and FPIC. The contributions of this paper are as follows:

• A detailed construction of the piecewise approximation of the switching surface.
• Existence of chaos is demonstrated numerically for positive Lyapunov exponents.
• Chaos in the Buck–Boost converter controlled with ZAD and FPIC is minimized, increasing the parameter $\gamma$ of the FPIC.
• Bifurcation diagrams are used to visualize the chaos is reduced when the value of $\gamma$ is increased from $0.1$ to 1.
• The equations for a non-ideal Buck–Boost converter in continuous conduction mode (CCM) are presented, considering parasitic resistances of components and the mathematical model is experimentally validated, accounting for the effects of parasitic resistances. ZAD control is then applied to the model, including these parasitic effects, to effectively regulate the inductor current $i_L$.
• A comparison of ZAD and PID control shows that both regulate current with less than 1% steady-state error.
• The ZAD control maintains a balanced duty cycle without saturation, avoiding chattering, while the PID control exhibits saturation in the duty cycles, which can cause stability issues.
• This study presents the first application of ZAD control to a Buck–Boost converter with parasitic resistances, creating a more realistic model. Unlike previous works on Boost converters, this approach addresses the non-minimum phase behavior and stability challenges of Buck–Boost converters, enhancing control reliability.

The rest of the paper was divided into three more sections. Section 2 includes a detailed mathematical formulation of the Buck–Boost converter, the duty cycle for two control techniques ZAD and FPIC, and a definition of chaos. Subsequently, Section 3 presents the results and analysis, where simulations of chaos in the Buck–Boost converter demonstrate the presence and control of chaos in the Buck–Boost converter. Finally, in Section 4 the conclusions are presented.

2. Materials and Methods

In this section, the materials and methods are focused on presenting the different mathematical models of the Buck–Boost converter with the ZAD and FPIC.

2.1. Buck–Boost Converter

The studied system is a Buck–Boost converter, the basic diagram of which is represented in Figure 1. The operation of this circuit is as follows: when the transistor switches to the ON state (conducting state), the diode is reverse-biased. During this period, the current is generated from the voltage source E and passes through the inductor L. While the diode remains reverse-biased, the circuit is said to be operating in the “charging period”. On the other hand, when the transistor switches to the OFF state, the diode becomes forward-biased. This period is known as the “discharging period” because the energy stored in the inductor L is completely transferred to the load R of the system. The fundamental difference between this type of converter and the Buck and Boost converters is that the output voltage is of the opposite sign to that of the constant voltage source E [34].

Taking into account Figure 1 and using Kirchoff’s laws of voltage and current, the following system of differential equations is obtained, which models the dynamics of the system:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_C}{dt}}& =-{\displaystyle \frac{v_C}{RC}}-(1-u){\displaystyle \frac{i_L}{C}},\hfill \\ \hfill {\displaystyle \frac{d{i}_L}{dt}}& =(1-u){\displaystyle \frac{v_C}{L}}+{\displaystyle \frac{E}{L}}u.\hfill \end{array}$$

(1)

2.2. Dimensionless System

In order to obtain a topologically equivalent system (1), we perform the change of variables

$$x_1={\displaystyle \frac{1}{E}}{v}_C,x_2={\displaystyle \frac{1}{E}}\sqrt{{\displaystyle \frac{L}{C}}}{i}_L,\tau ={\displaystyle \frac{t}{\sqrt{LC}}},$$

(2)

which reduces the number of parameters and, in turn, allows us to have a dimensionless system. As is well known, this significantly simplifies the analysis of the system’s dynamics.

Differentiating each equation in (2) with respect to $\tau$, combining these derivatives with the system described in (1), and setting $Q=\sqrt{{\displaystyle \frac{R^{2}C}{L}}}$ the following system is obtained, which is topologically equivalent to system (1):

$$\begin{array}{cc}\hfill & {\dot{x}}_1=-Q^{-1}{x}_1-(1-u)x_2,\hfill \\ \hfill & {\dot{x}}_2=(1-u)x_1+u.\hfill \end{array}$$

(3)

Note that the system (3) depends only on the parameter Q and can be written in matrix form as

$${\displaystyle \frac{d\mathbf{x}}{dt}}=f(\mathbf{x},u)=\left\{\begin{array}{cc}{\mathbf{A}}_{\left[1\right]}\mathbf{x}+{\mathbf{B}}_{\left[1\right]}\hfill & , u=1\hfill \\ {\mathbf{A}}_{\left[0\right]}\mathbf{x}\hfill & , u=0.\hfill \end{array}\right.$$

(4)

where ${\mathbf{A}}_{\left[1\right]}$, ${\mathbf{B}}_{\left[1\right]}$ and ${\mathbf{A}}_{\left[0\right]}$ are the matrices given by (notice that ${\mathbf{B}}_{\left[0\right]}=\mathbf{0}$ and therefore it is not necessary to include it in (4))

$${\mathbf{A}}_{\left[1\right]}=\left[\begin{array}{cc}-Q^{-1}& 0\\ 0& 0\end{array}\right],{\mathbf{B}}_{\left[1\right]}=\left[\begin{array}{c}0\hfill \\ 1\hfill \end{array}\right],{\mathbf{A}}_{\left[0\right]}=\left[\begin{array}{cc}-Q^{-1}& -1\\ 1& 0\end{array}\right].$$

(5)

Note that we have now used t instead of $\tau$, in order to use the traditional notation for the time variable. This does not alter the analysis of the system, as now, in Equation (6), the variable $\tau$ acts as an integration variable that is only used to complete the mathematical structure. Denoting by ${\mathbf{A}}_{\left[u\right]}$ as the state matrix and ${\mathbf{B}}_{\left[u\right]}$ as the input vector, we can compactly write Equation (4) for $u\in \{0,1\}$ as

$$\dot{\mathbf{x}}\left(t\right)={\mathbf{A}}_{\left[u\right]}\mathbf{x}\left(t\right)+{\mathbf{B}}_{\left[u\right]},u\in \{0,1\}.$$

(6)

Based on the above, the solution for each topology (u in the set $\{0,1\}$) can establish a solution in time by integrating both sides of Equation (6) between $t_0$ and t. It follows that

$$\mathbf{x}\left(t\right)={\mathbf{\Phi }}_{\left[u\right]}(t-t_0)\mathbf{x}\left(t_0\right)+{\mathbf{\Psi }}_{\left[u\right]}(t-t_0),$$

(7)

where

$${\mathbf{\Phi }}_{\left[u\right]}\left(t\right)=exp\left[{\mathbf{A}}_{\left[u\right]}\left(t\right)\right],{\mathbf{\Psi }}_{\left[u\right]}(t-t_0)={\int }_{t_0}^t{\mathbf{\Phi }}_{\left[u\right]}(t-s){\mathbf{B}}_{\left[u\right]}ds.$$

(8)

Explicit forms of the solution $\mathbf{x}\left(t\right)$ can be found in terms of t, the parameter Q and the initial state. For example, it is easy to see that the closed form of ${\mathbf{\Psi }}_{\left[1\right]}\left(t-t_0\right)$ is ${\left[0\mid t-t_0\right]}^{\top }$. However, the closed forms of the transition matrices ${\mathbf{\Phi }}_{\left[u\right]}(t-t_0)$ will be treated numerically.

2.3. Switching Surface

The switching surface (9) used in this paper is a linear combination of the current error and the voltage error, where greater weight is assigned to the control parameter $k_2$ to primarily regulate the inductor current $i_L$, which corresponds to $x_2$. Meanwhile, the value of $k_1$ is kept lower to avoid controlling the capacitor voltage $v_C$, represented as $x_1$. This choice is due to the inherent difficulty in controlling $v_C$ in this converter, as the Buck–Boost topology operates as a non-minimum phase system. The switching surface is presented below:

$$\begin{array}{cc}\hfill s\left(\mathbf{x}\right(t\left)\right)& =\mathbf{K}·(\mathbf{x}\left(t\right)-{\mathbf{x}}_{\mathrm{ref}})=k_1(x_1\left(t\right)-x_{1\mathrm{ref}})+k_2(x_2\left(t\right)-x_{2\mathrm{ref}}),\hfill \end{array}$$

(9)

where

• $\mathbf{x}\left(t\right)={\left[\begin{array}{cc}{x}_1\left(t\right)& x_2\left(t\right)\end{array}\right]}^{\top }$ is the state vector, with $x_1\left(t\right)$ as the voltage state and $x_2\left(t\right)$ as the current state,
• ${\mathbf{x}}_{\mathrm{ref}}\left(t\right)={\left[\begin{array}{cc}{x}_{1\mathrm{ref}}& x_{2\mathrm{ref}}\end{array}\right]}^{\top }$ is the reference vector, whose components are the voltage and current reference signals. This vector corresponds to the state to which we want the system to evolve.
• $\mathbf{K}=\left[\begin{array}{cc}{k}_1& k_2\end{array}\right]$ is the parameter vector of the switching surface, which contains the constant components $k_1$ and $k_2$ associated with the error between the output signal and the reference signal.
• The difference $\mathbf{x}\left(t\right)-{\mathbf{x}}_{\mathrm{ref}}$ represents the error in voltage and current, respectively.

ZAD condition [15,35,36]. Given a fixed period T and $n\in \mathbb{N}$, the system is required to switch a finite number of times in the interval $[nT,(n+1\left)T\right]$, on a switching surface $s\left(\mathbf{x}\right(t\left)\right)$, which is equivalent to

$${\int }_{nT}^{(n+1)T}s\left(\mathbf{x}\left(t\right)\right)dt=0.$$

(10)

2.4. Duty Cycle

PWM is a widely used control technique in power electronics to regulate the output voltage and current of switching converters by adjusting the duty cycle of the switching device [37]. In this method, the switch alternates between ON and OFF states within a fixed switching period, and the ratio of ON time to the total period, known as the duty cycle (d), directly determines the average output voltage. The duty cycle is dynamically modulated in response to input voltage fluctuations, load variations, or reference signals from a feedback control system, enabling efficient power conversion with minimal losses. PWM schemes can be classified into fixed-frequency and variable-frequency types, with further distinctions such as center-aligned or edge-aligned modulation, each affecting spectral performance and harmonic content. In modern applications, PWM signals are generated by digital controllers like microcontrollers (MCUs), digital signal processors (DSPs), or field-programmable gate arrays (FPGAs), allowing precise real-time control. When combined with advanced control strategies such as ZAD and FPIC, PWM enhances stability, mitigates chaotic behavior, and ensures reliable operation in DC–DC power converters under varying load and input conditions. For a LPWM or a $\{1,0\}$-scheme, in a time period T, a switching is performed such that the time interval $[nT,(n+1\left)T\right]$ is subdivided into the intervals $[nT,nT+d_n]$ and $(nT+d_n,(n+1)T]$, where the system evolves according to the topologies $u=1$ and $u=0$, respectively, (see Figure 2). It should be noted that the duty cycle is not necessarily the same in each switching period.

Referring to Figure 2, the control variable u can be written as

$$\begin{array}{cc}\hfill u\left(t\right)& =\left\{\begin{array}{cc}1,\hfill & nT\le t\le nT+d_n\hfill \\ 0,\hfill & nT+d_n<t\le (n+1)T.\hfill \end{array}\right.\hfill \end{array}$$

(11)

Next, we approximate the switching surface $s\left(\mathbf{x}\right(t\left)\right)$ under the following assumptions:

(i)

The dynamics of the error or switching surface behave as piecewise linear segments.

(ii)

The slopes of the error dynamics in the intervals $[nT,nT+d_n]$ and $[nT+d_n,(n+1)T]$ are determined by ${\dot{s}}_{\left[1\right]}$ and ${\dot{s}}_{\left[0\right]}$, calculated at the moment of switching. Both slopes correspond to the time derivative of the switching surface for $u=1$ and $u=0$, respectively, evaluated at $\mathbf{x}\left(nT\right)$.

Under these assumptions, the function $\hat{s}\left(\mathbf{x}\left(t\right)\right)$ approximating $s\left(\mathbf{x}\right(t\left)\right)$ is defined as follows:

$$\hat{s}\left(\mathbf{x}\left(t\right)\right)=\left\{\begin{array}{cc}{\dot{s}}_{\left[1\right]}(t-nT)+s\hfill & ,nT\le t\le nT+d_n\hfill \\ {\dot{s}}_{\left[0\right]}\left(t-nT-d_n\right)+d_n{\dot{s}}_{\left[1\right]}+s\hfill & ,nT+d_n<t<(n+1)T.\hfill \end{array}\right.$$

(12)

Figure 3 summarizes clearly the process that was carried out to obtain Equation (12).

In Figure 3, note that both s, ${\dot{s}}_{\left[1\right]}$, and ${\dot{s}}_{\left[0\right]}$ depend implicitly on $\mathbf{x}\left(nT\right)$:

$$s=s\left(\mathbf{x}\left(nT\right)\right),{\dot{s}}_{\left[1\right]}={\dot{s}}_{\left[1\right]}\left(\mathbf{x}\left(nT\right)\right),{\dot{s}}_{\left[0\right]}={\dot{s}}_{\left[0\right]}\left(\mathbf{x}\left(nT\right)\right).$$

(13)

Now, using the piecewise linear approximation given in (12), the zero-average condition (10) is applied, from which the following quadratic equation in the variable $d_n$ is obtained:

$$d_n^2-2T{d}_n+{\displaystyle \frac{T^2{\dot{s}}_{\left[0\right]}+2Ts}{{\dot{s}}_{\left[0\right]}-{\dot{s}}_{\left[1\right]}}}=0.$$

(14)

Therefore, $d_n$ can be written as

$$d_n=\left(1-\sqrt{g\left({\mathbf{x}}_n\right)}\right)T,n=0,1,2,\dots$$

(15)

where $g\left({\mathbf{x}}_n\right)$ is defined as

$$g\left({\mathbf{x}}_n\right)={\displaystyle \frac{{\dot{s}}_{\left[1\right]}\left({\mathbf{x}}_n\right)+2T^{-1}s\left({\mathbf{x}}_n\right)}{{\dot{s}}_{\left[1\right]}\left({\mathbf{x}}_n\right)-{\dot{s}}_{\left[0\right]}\left({\mathbf{x}}_n\right)}}.$$

(16)

Since $g_n:=g\left({\mathbf{x}}_n\right)$ appears as a sub-radical quantity in Equation (15), it becomes necessary to discriminate between the cases where $g_n\le 0,0<g_n<1$, and $g_n\ge 1$. This will allow us to define the duty cycle correctly for each switching period:

• Topology $u=1$. Suppose $g_n\le 0$, then $d_n$ takes complex values. To resolve this issue, $d_n$ is chosen as the value minimizing $p\left(d_n\right)=d_n^2-2T{d}_n+T^2{g}_n$ [19]. By differentiation, the minimum of this polynomial is attained at $d_n=T$. Therefore, $d_n$ is chosen as $d_n=T$.
• Topologies $u=0$ and $u=1$. Suppose $0<g_n<1$, then $0<\sqrt{g_n}<1$, implying $0<(1-\sqrt{g_n})T<T$. Thus, it follows that $0<d_n<T$. When $d_n\in (0,T)$, we say the duty cycle is unsaturated.
• Topology $u=0$. Suppose $g_n\ge 1$, then $(1-\sqrt{g_n})T\le 0$, and therefore $d_n\le 0$. Since $d_n$ is strictly greater than zero, we choose $d_n=0$.

We can summarize the previous analysis with the following decision rule:

$$d_n=\left\{\begin{array}{cc}T,\hfill & \mathrm{if}{g}_n\le 0\hfill \\ \left(1-\sqrt{g_n}\right)T,\hfill & \mathrm{if}0<g_n<1\hfill \\ 0,\hfill & \mathrm{if}{g}_n\ge 1.\hfill \end{array}\right.$$

(17)

2.5. Poincaré Map

Based on Figure 4, we consider the system state at time instant $t=nT$ as $\mathbf{x}\left(nT\right)$. Next, we allow the system to evolve during the time interval $[nT,nT+d_n]$ with topology $u=1$. Then, in the interval $[nT+d_n,(n+1)T]$, we let the system evolve with topology $u=0$, taking as initial condition the system state at the end of the interval $[nT,nT+d_n]$, that is, $\mathbf{x}(nT+d_n)$. In this way, we obtain the final state $\mathbf{x}\left(\right(n+1\left)T\right)$ at time $t=(n+1)T$.

Once the system evolution according to the switching scheme is completely understood, Equation (7) is used to concatenate the states $\mathbf{x}\left(nT\right)$, $\mathbf{x}(nT+d_n)$, and $\mathbf{x}\left(\right(n+1\left)T\right)$. With these considerations, the Poincaré map $\mathbf{P}\left({\mathbf{x}}_n\right)$ for the $\{1,0\}$ scheme transforms the state ${\mathbf{x}}_n:=\mathbf{x}\left(nT\right)$ into the state ${\mathbf{x}}_{n+1}:=\mathbf{x}\left((n+1)T\right)$ through the following recurrence relation:

$${\mathbf{x}}_{n+1}=\mathbf{P}\left({\mathbf{x}}_n\right)=\left\{\begin{array}{cc}{\mathbf{\Phi }}_{\left[0\right]}\left(T\right){\mathbf{x}}_n\hfill & ,d_n\le 0\hfill \\ {\mathbf{\Phi }}_{\left[0\right]}(T-d_n)[{\mathbf{\Phi }}_{\left[1\right]}\left(d_n\right){\mathbf{x}}_n+{\mathbf{\Psi }}_{\left[1\right]}\left(d_n\right)]\hfill & ,d_n\in (0,T)\hfill \\ {\mathbf{\Phi }}_{\left[1\right]}\left(T\right){\mathbf{x}}_n+{\mathbf{\Psi }}_{\left[1\right]}\left(T\right)\hfill & ,d_n\ge T\hfill \end{array}\right.$$

(18)

being

$$\begin{array}{ccc}{\mathbf{\Phi }}_{\left[0\right]}\left(T-d_n\right)=exp\left[{\mathbf{A}}_{\left[0\right]}\left(T-d_n\right)\right],\hfill & {\mathbf{\Phi }}_{\left[1\right]}\left(d_n\right)=exp\left[{\mathbf{A}}_{\left[1\right]}{d}_n\right],\hfill & {\mathbf{\Psi }}_{\left[1\right]}\left(d_n\right)={\mathbf{B}}_{\left[1\right]}{d}_n,\\ {\mathbf{\Phi }}_{\left[0\right]}\left(T\right)=exp\left[{\mathbf{A}}_{\left[0\right]}T\right],\hfill & {\mathbf{\Phi }}_{\left[1\right]}\left(T\right)=exp\left[{\mathbf{A}}_{\left[1\right]}T\right],\hfill & {\mathbf{\Psi }}_{\left[1\right]}\left(T\right)={\mathbf{B}}_{\left[1\right]}T.\end{array}$$

(19)

As mentioned earlier, $\mathbf{P}\left({\mathbf{x}}_n\right)$ allows us to obtain the state ${\mathbf{x}}_{n+1}$ from the state ${\mathbf{x}}_n$ and the duty cycle $d_n$, without the need to explicitly go through the state of the system at time $t=nT+d_n$. In summary, $\mathbf{P}\left({\mathbf{x}}_n\right)$ generates a discrete sequence of successive states starting from the initial condition ${\mathbf{x}}_0$, namely:

$$\mathcal{O}\left({\mathbf{x}}_0\right)=\{{\mathbf{x}}_0,{\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_n,{\mathbf{x}}_{n+1},\dots \}$$

(20)

which is precisely the orbit $\mathcal{O}$ of the system, as a function of the initial condition ${\mathbf{x}}_0$ (see Figure 5).

2.6. Stability of Periodic Orbits

In this subsection, the stability of the $1T$-periodic orbits of the system will be analyzed. For this purpose, the Jacobian matrix of the Poincaré map will be obtained and subsequently evaluated at its fixed points. If the eigenvalues of this matrix lie inside the unit circle (the unit circle, denoted by $S^1$, is defined as the set of all points $z\in \mathbb{C}$ such that $\left|z\right|=1$), then the $1T$ periodic orbit is stable. If there is an eigenvalue outside the unit circle, then the $1T$-periodic orbit is unstable.

Next, we will explicitly calculate the Jacobian matrix of the Poincaré map for both the unsaturated case and the saturated cases. Given that the Poincare map is a function of ${\mathbf{x}}_n$, and ${\mathbf{x}}_n$ is itself a function of $d_n$, we can differentiate (18) with respect to ${\mathbf{x}}_n$. From the chain rule [38], we obtain:

$$\mathbf{JP}({\mathbf{x}}_n;d_n)={\displaystyle \frac{\partial \mathbf{P}}{\partial {\mathbf{x}}_n}}+{\displaystyle \frac{\partial \mathbf{P}}{\partial d_n}}·{\displaystyle \frac{\partial d_n}{\partial {\mathbf{x}}_n}}.$$

(21)

When the duty cycle is not saturated, Equation (21) is given by

$$\begin{array}{cc}\hfill \mathbf{JP}\left({\mathbf{x}}_n\right)& ={\mathbf{\Phi }}_{\left[0\right]}(T-d_n){\mathbf{\Phi }}_{\left[1\right]}\left(d_n\right)+\cdots \hfill \\ \hfill & + {\mathbf{\Phi }}_{\left[0\right]}(T-d_n)[({\mathbf{A}}_{\left[1\right]}-{\mathbf{A}}_{\left[0\right]}){\mathbf{\Phi }}_{\left[1\right]}\left(d_n\right){\mathbf{x}}_n+({\mathbf{I}}_2-d_n{\mathbf{A}}_{\left[0\right]}){\mathbf{B}}_{\left[1\right]}]\hfill \\ \hfill & •\left[\begin{array}{cc}{\displaystyle \frac{B_q\left({\mathbf{x}}_n\right)q_{n,1}-A_p\left({\mathbf{x}}_n\right)p_{n,1}}{2T^{-1}{A}_p^2\left({\mathbf{x}}_n\right)\sqrt{g\left({\mathbf{x}}_n\right)}}}& {\displaystyle \frac{B_q\left({\mathbf{x}}_n\right)q_{n,2}-A_p\left({\mathbf{x}}_n\right)p_{n,2}}{2T^{-1}{A}_p^2\left({\mathbf{x}}_n\right)\sqrt{g\left({\mathbf{x}}_n\right)}}}\end{array}\right].\hfill \end{array}$$

(22)

where $A_p\left({\mathbf{x}}_n\right)$, $B_q\left({\mathbf{x}}_n\right)$ and $p_{n,i}$, $q_{n,i}$$i\in \{1,2\}$ are scalar quantities given by

$$\begin{array}{cc}\hfill A_p\left({\mathbf{x}}_n\right)& ={\dot{s}}_{\left[1\right]}\left({\mathbf{x}}_n\right)-{\dot{s}}_{\left[0\right]}\left({\mathbf{x}}_n\right)\hfill \\ \hfill B_q\left({\mathbf{x}}_n\right)& ={\dot{s}}_{\left[1\right]}\left({\mathbf{x}}_n\right)+2T^{-1}s\left({\mathbf{x}}_n\right)\hfill \end{array},\begin{array}{cc}\hfill p_{n,1}& =(2T^{-1}-Q^{-1})k_1\hfill \\ \hfill q_{n,1}& =-k_2\hfill \end{array},\begin{array}{cc}\hfill p_{n,2}& =2k_2{T}^{-1}\hfill \\ \hfill q_{n,2}& =k_1\hfill \end{array}.$$

(23)

and finally, when the duty cycle is saturated, in either case, we have ${\displaystyle \frac{\partial d_n}{\partial {\mathbf{x}}_n}}=0$, and therefore $\mathbf{JP}={\displaystyle \frac{\partial \mathbf{P}}{\partial {\mathbf{x}}_n}}$, thus

• For $d_n=0$, we have

  $$\mathbf{JP}({\mathbf{x}}_n;0)={\mathbf{\Phi }}_{\left[0\right]}\left(T\right){\displaystyle \frac{\partial \mathrm{x}\left(nT\right)}{\partial {\mathrm{x}}_n}}={\mathbf{\Phi }}_{\left[0\right]}\left(T\right).$$

  (24)
• And for $d_n=T$, we have

  $$\mathbf{JP}({\mathbf{x}}_n;T)={\mathbf{\Phi }}_{\left[1\right]}\left(T\right){\displaystyle \frac{\partial \mathrm{x}\left(nT\right)}{\partial {\mathrm{x}}_n}}={\mathbf{\Phi }}_{\left[1\right]}\left(T\right).$$

  (25)

In order to determine the stability of $1T$-periodic orbits, we need to solve the equation $\mathbf{P}\left({\mathbf{x}}_n\right)={\mathbf{x}}_n$ for ${\mathbf{x}}_n$, which means, when the duty cycle is not saturated, that

$${\mathbf{x}}_n={\mathbf{\Phi }}_{\left[0\right]}(T-d_n){\mathbf{\Phi }}_{\left[1\right]}\left(d_n\right){\mathbf{x}}_n+{\mathbf{\Phi }}_{\left[0\right]}(T-d_n){\mathbf{\Psi }}_{\left[1\right]}\left(d_n\right).$$

(26)

Solving this equation for ${\mathbf{x}}_n$, we have

$${\mathbf{x}}_n={[{\mathbf{I}}_2-{\mathbf{\Phi }}_{\left[0\right]}(T-d_n){\mathbf{\Phi }}_{\left[1\right]}\left(d_n\right)]}^{-1}{\mathbf{\Phi }}_{\left[0\right]}(T-d_n){\mathbf{\Psi }}_{\left[1\right]}\left(d_n\right).$$

(27)

According to Neumann’s lemma [39], the existence of periodic orbits $1T$ reduces to proving that

$${\mathbf{I}}_2-{\mathbf{\Phi }}_{\left[0\right]}(T-d_n){\mathbf{\Phi }}_{\left[1\right]}\left(d_n\right)$$

(28)

is a non-singular matrix. This is guaranteed if the spectral radius (the spectral radius of a matrix $\mathbf{A}$, denoted by $\rho \left(\mathbf{A}\right)$, is defined as $\rho \left(\mathbf{A}\right)=max\left\{\right|\lambda |:\lambda \in \sigma (\mathbf{A}\left)\right\}$, where $\sigma \left(\mathbf{A}\right)=\{\lambda \in \mathbb{C}:det(\mathbf{A}-\lambda \mathbf{I})=0\}$ represents the spectrum of $\mathbf{A}$) of ${\mathbf{\Phi }}_{\left[0\right]}(T-d_n){\mathbf{\Phi }}_{\left[1\right]}\left(d_n\right)$ is less than 1, for some R and all integer n such that $n\ge R$. That is to say:

$$\rho \left[{\mathbf{\Phi }}_{\left[0\right]}(T-d_n){\mathbf{\Phi }}_{\left[1\right]}\left(d_n\right)\right]<1,\forall n\ge R.$$

(29)

On the other hand, when there is saturation of the duty cycle, Equation (27) still applies, and thus the condition (29) holds. The cases $d_n=0$ and $d_n=T$ are discriminated according to the conditions established in Inequality (17). In the results Section 3.2 (Figures 18 and 19, the condition given in Inequality (29) will be analyzed numerically using various approaches.

Therefore, if the eigenvalues of $\mathbf{JP}({\mathbf{x}}_n;d_n)$, evaluated at these fixed points, all have a modulus less than one, the stability of this type of orbit is guaranteed.

2.7. Chaos Existence in the Buck–Boost Converter

This subsection analyzes the existence of chaos in the Buck–Boost converter. First, the concept of chaos that we will adopt will be defined, and then its existence will be demonstrated numerically by calculating the Lyapunov exponents. We will consider the following definitions [40].

Definition 1.

A dynamical system exhibits sensitivity to initial conditions when small variations in initial conditions lead to significant changes in the separation between trajectories that initially start close together, causing them to diverge exponentially.

Definition 2.

With $\mathbf{JP}\left(x\right)$ as the Jacobian matrix of the Poincare application and $q_i\left(\mathbf{JP}\left(\mathbf{x}\right)\right)$ the i-eigenvalue of $\mathbf{JP}\left(x\right)$. The Lyapunov exponent ${\lambda }_i$ for each eigenvalue is given by

$${\lambda }_i=\underset{n\to \infty }{lim}\left({\displaystyle \frac{1}{n}}\sum _{k=0}^{n}log\left|q_i\left(\mathbf{JP}\left({\mathbf{x}}_k\right)\right)\right|\right)$$

(30)

We will adopt Equation (30) to numerically calculate the Lyapunov exponents. To illustrate the chaotic behavior of the system, we will adopt the following definition [41].

Definition 3.

A system is considered chaotic if it exhibits long-term aperiodic behavior in a deterministic framework with sensitive dependence on initial conditions or, equivalently, if it has positive Lyapunov exponents.

In order to demonstrate that our system exhibits chaotic behavior, we first observe from the solutions of system (18) that it is deterministic. Moreover, numerical simulations (as observed in the Figure 6, Figure 7 and Figure 8), show that the system exhibits aperiodic behavior, which is also reflected in the fact that the matrix (28) is singular.

2.8. Chaos Control with FPIC

One possible cause of chaotic behavior in a discrete dynamical system is the instability of one of its periodic orbits, as occurs in the case of the logistic map. If a mechanism is implemented to expand the range of parameters in which the $1T$-periodic orbit is unstable, then the region of the parameter space where this instability occurs is reduced. Consequently, if chaos in the system arises due to the instability of the $1T$-periodic orbit, it would be controlled. This is the central idea of the FPIC technique, which we present below:

Theorem 1

(FPIC [18]). Given a discrete dynamical system defined by the following difference equation

$${\mathbf{x}}_{k+1}=f({\mathbf{x}}_k,u\left({\mathbf{x}}_k\right)),where{\mathbf{x}}_k\in {\mathbb{R}}^n,u:{\mathbb{R}}^n\to \mathbb{R},f:{\mathbb{R}}^{n+1}\to {\mathbb{R}}^n.$$

(31)

Suppose that the system has a fixed point given by

$$({\mathbf{x}}^{*},u\left({\mathbf{x}}^{*}\right)):=({\mathbf{x}}^{*},u^{*}).$$

(32)

Let $\mathbf{J}={\mathbf{J}}_{\mathbf{x}}+{\mathbf{J}}_u$ be the Jacobian of the system evaluated at this fixed point, that is,

$${\mathbf{J}}_{\mathbf{x}}=\left({\displaystyle \frac{\partial f}{\partial \mathbf{x}}}\right){|}_{\left({\mathbf{x}}^{*},u^{*}\right)}and{\mathbf{J}}_u=\left({\displaystyle \frac{\partial f}{\partial u}}{\displaystyle \frac{\partial u}{\partial \mathbf{x}}}\right){|}_{\left({\mathbf{x}}^{*},u^{*}\right)}.$$

(33)

Then, if the spectral radius of ${\mathbf{J}}_{\mathbf{x}}$ is less than one, that is, $\rho \left({\mathbf{J}}_{\mathbf{x}}\right)<1$, there exists a control signal $\widehat{u}$ such that

$${\widehat{u}}_k={\displaystyle \frac{u\left({\mathbf{x}}_k\right)+\gamma u^{*}}{\gamma +1}},$$

(34)

which guarantees the stability of the fixed point $({\mathbf{x}}^{*},u^{*})$ for some positive real γ.

In our context, we have

$${\mathbf{x}}_{k+1}=\mathbf{P}({\mathbf{x}}_k;d_k),\mathrm{with}\mathbf{JP}={\displaystyle \frac{\partial \mathbf{P}}{\partial {\mathbf{x}}_k}}+{\displaystyle \frac{\partial \mathbf{P}}{\partial d_k}}{\displaystyle \frac{\partial d_k}{\partial {\mathbf{x}}_k}}.$$

(35)

Let us denote by ${\mathbf{y}}^{*}=({\mathbf{x}}^{*},d_{\mathrm{ref}})$ a fixed point of the Poincaré map such that

$$\rho \left({\displaystyle \frac{\partial \mathbf{P}}{\partial {\mathbf{x}}_k}}{|}_{{\mathbf{y}}^{*}}\right)<1.$$

(36)

By the FPIC theorem, there exists a control signal ${\hat{d}}_k$ given by

$${\hat{d}}_k={\displaystyle \frac{d_k+\gamma d_{\mathrm{ref}}}{\gamma +1}},$$

(37)

such that the system has ${\mathbf{y}}^{*}$ as a stable fixed point.

In Equation (37), the quantity $d_{\mathrm{ref}}$ corresponds to the steady-state duty cycle, which can be obtained by setting ${\mathbf{x}}_k={\mathbf{x}}_{\mathrm{ref}}$ in Equation (15), that is:

$$d_{\mathrm{ref}}=d_k\left({\mathbf{x}}_{\mathrm{ref}}\right)=\left(1-\sqrt{{\displaystyle \frac{{\dot{s}}_{\left[1\right]}\left({\mathbf{x}}_{\mathrm{ref}}\right)+2T^{-1}s\left({\mathbf{x}}_{\mathrm{ref}}\right)}{{\dot{s}}_{\left[1\right]}\left({\mathbf{x}}_{\mathrm{ref}}\right)-{\dot{s}}_{\left[0\right]}\left({\mathbf{x}}_{\mathrm{ref}}\right)}}}\right)T.$$

(38)

2.9. Model with Parasitic Resistors

This section presents the equations for a non-ideal Buck–Boost DC–DC converter in continuous conduction mode (CCM). The basic system is illustrated in Figure 9, with elements such as switch, diode, inductor, capacitor and load resistor. For an accurate model, the parasitic resistances [42] of each component are considered. The variables $V_{in}$, $v_o$, $v_C$ and $v_L$ represent the input, output, capacitor and inductor voltages, respectively; $i_L$ and $i_C$ denotes the inductor and capacitor currents, respectively.

Let u in $\{0,1\}$ denote the states ON $(u=1)$ and OFF $(u=0)$, respectively. We set $x_1\left(t\right)=v_C\left(t\right)$ and $x_2\left(t\right)=i_L\left(t\right)$.

When $u=1$ (see Figure 10), employing Kirchhoff’s voltage and current laws, the following system of equations is obtained:

$$\begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)& =a{x}_1\left(t\right).\hfill \\ \hfill {\dot{x}}_2\left(t\right)& =b{x}_2\left(t\right)+{\displaystyle \frac{1}{L_f}}{V}_{in},\hfill \end{array}$$

(39)

where a and b are parameters given by

$$a=-{\displaystyle \frac{1}{(r_{Cf}+R_L)C}},b=-{\displaystyle \frac{(r_{in}+r_{sw}+r_{Lf})}{L_f}}.$$

(40)

Now, when $u=0$ (see Figure 11) applying Kirchhoff’s voltage and current laws and setting again $x_1\left(t\right)=v_C\left(t\right)$ and $x_2\left(t\right)=i_L\left(t\right)$, we obtain the system

$$\begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)=& f{x}_1\left(t\right)+g{x}_2\left(t\right),\hfill \\ \hfill {\dot{x}}_2\left(t\right)=& p{x}_1\left(t\right)+q{x}_2\left(t\right)-{\displaystyle \frac{1}{L_f}}{v}_{fd},\hfill \end{array}$$

(41)

where f, g, p and q are parameters given by

$$\begin{array}{ccc}f=-{\displaystyle \frac{1}{(R_L+r_{Cf})C_f}},\hfill & & g={\displaystyle \frac{R_L}{(R_L+r_{Cf})C_f}},\hfill \\ p={\displaystyle \frac{1}{L_f}}\left({\displaystyle \frac{r_{Cf}}{R_L+r_{Cf}}}-1\right),\hfill & & q=-{\displaystyle \frac{1}{L_f}}\left(r_{Lf}+r_d+{\displaystyle \frac{R_L{r}_{Cf}}{R_L+r_{Cf}}}\right).\hfill \end{array}$$

(42)

From a theoretical perspective, the analysis of the Buck–Boost converter is conducted exclusively under the assumption of continuous conduction mode (CCM). This assumption is justified as it provides a more predictable system behavior, facilitating the controller design. In this context, the ZAD technique, combined with FPIC, is employed to stabilize the system and enhance its dynamic response. Implementing this control method in CCM ensures that the eigenvalues of the system’s Jacobian matrix remain within the unit circle, stabilizing the previously unstable orbit and guaranteeing robust converter performance.

However, in the simulation performed in MATLAB R2018a with Simulink, discontinuous conduction mode (DCM) is also considered, as it can occur when the converter’s load varies significantly. This behavior is observed in the paper’s results, where abrupt load changes lead to transitions between the two operating modes. The ability to analyze these variations in simulation allows for validating the controller’s effectiveness, which was designed for CCM, under adverse operating conditions. It is crucial to highlight that although the controller is specifically designed for CCM, the inclusion of DCM scenarios in simulation enables the evaluation of its robustness against load disturbances and other factors that may induce bifurcations or chaotic behavior. This is essential to ensure that the converter maintains stable performance in real-world applications, where load conditions are often dynamic rather than constant.

Next, we define the switching surface as in Equation (9), more specifically

$$S\left(\mathbf{x}\left(t\right)\right)=k_1(x_1\left(t\right)-x_{1\mathrm{ref}})+k_2(x_2\left(t\right)-x_{2\mathrm{ref}}).$$

(43)

In order to carry out the numerical simulations, we will work with the CPWM scheme. The selection of the CPWM scheme for simulations and experimental validations is justified by its ability to reduce noise in voltage and current measurements. By sensing at the midpoint of the pulse, the influence of transients caused by switching is minimized, resulting in more stable and accurate measurements. In power electronics systems, abrupt changes in current and voltage during switching can introduce interference and complicate data interpretation. CPWM positions the switching edges symmetrically, ensuring that at the midpoint of the switching cycle, signals are less susceptible to disturbances. This strategy enhances the correlation between simulations and experimental measurements, enabling a more reliable analysis of system behavior.

For this scheme (CPWM), it is known [6] that the duty cycle is given by the expression

$$d_{ZAD}={\displaystyle \frac{2S\left(\mathbf{x}\left(t\right)\right)+T{S}^{-}\left(\mathbf{x}\left(t\right)\right)}{S^{-}\left(\mathbf{x}\left(t\right)\right)-S^{+}\left(\mathbf{x}\left(t\right)\right)}},$$

(44)

where $S^{-}$ and $S^{+}$ correspond to the derivatives of the switching surface when $u=1$ and $u=0$, that is

$$S^{+}\left(\mathbf{x}\left(t\right)\right)=\dot{S}\left(\mathbf{x}\left(t\right)\right){|}_{u=1}\mathrm{and}{S}^{-}\left(\mathbf{x}\left(t\right)\right)=\dot{S}\left(\mathbf{x}\left(t\right)\right){|}_{u=0}.$$

(45)

In our case $S^{+}$ and $S^{-}$ are given by

$$\begin{array}{cc}\hfill S^{+}\left(\mathbf{x}\left(t\right)\right)& =k_1\left[a{x}_1\left(t\right)\right]+k_2\left[b{x}_2\left(t\right)+{\displaystyle \frac{1}{L_f}}{V}_{in}\right],\hfill \\ \hfill S^{-}\left(\mathbf{x}\left(t\right)\right)& =k_1[f{x}_1\left(t\right)+g{x}_2\left(t\right)]+k_2\left[p{x}_1\left(t\right)+q{x}_2\left(t\right)-{\displaystyle \frac{1}{L_f}}{v}_{fd}\right].\hfill \end{array}$$

(46)

The use of linear approximations for determining the duty cycle in the ZAD strategy is justified in this study due to its effectiveness in control design for switched systems and its ability to provide a computationally efficient solution. While it is acknowledged that accuracy may be affected in highly nonlinear regions, the chosen approximation adequately captures the system dynamics in the most relevant operating conditions, particularly within the typical operating ranges of the Buck–Boost converter.

3. Results

In this section, we present the numerical simulations of the model. We begin by discussing the performance of the ZAD strategy, followed by an analysis of the bifurcations observed in the converter. Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22 illustrate the numerical simulations without FPIC control. Next, we examine the existence and stability of $1T$-periodic orbits, addressing the onset of chaos within the converter. Finally, we apply the FPIC technique to control the instabilities, with Figure 23a, Figure 24 and Figure 25b showcasing the results under FPIC control. Additionally, we validate the mathematical model experimentally, including the effects of parasitic resistances, and apply the ZAD control to the model with parasitic resistances to regulate the inductor current $i_L$. A comparison with the PID control is also presented, highlighting the performance differences, where the PID system shows higher sensitivity to switching frequency variations, leading to issues such as chattering and MOSFET overheating.

3.1. Performance of the ZAD Strategy

Figure 12 and Figure 13 present the results obtained regarding the performance of the ZAD technique in terms of regulation when the switching surface is approximated using linear segments given by Equation (12). The parameters used were $T=0.17$, $Q=0.62$, $x_{1\mathrm{ref}}=-1.2$, $k_1=-6$, and $k_2=1.35$. The initial state of the system is represented by a green circle, while the final state is marked in red.

Figure 12. Poincare map associated with input parameters $T=0.17$, $Q=0.5$, $x_{1\mathrm{ref}}=-1.5$, $k_1=0.5$, and $k_2=1.35$.

Figure 12. Poincare map associated with input parameters $T=0.17$, $Q=0.5$, $x_{1\mathrm{ref}}=-1.5$, $k_1=0.5$, and $k_2=1.35$.

It is observed that, with this choice of parameters, the system exhibits good voltage regulation. The absolute error is approximately $|-1.2000-(-1.2226\left)\right|=0.0226$, corresponding to a relative error of $1.88\%$. However, the current evolution over time is not optimal, as the absolute error is $|4.2581-3.8902|=0.3679$, resulting in a relative error of 8.64%.

Figure 13. Comparison of voltage and current evolution based on the reference value, with input parameters $T=0.17$, $Q=0.5$, $x_{1\mathrm{ref}}=-1.5$, $k_1=0.5$, and $k_2=1.35$. The upper plot shows voltage fluctuations around the reference value, while the lower plot shows current fluctuations, both with the red line indicating the target value. These plots illustrate the system’s performance relative to the reference.

Figure 13. Comparison of voltage and current evolution based on the reference value, with input parameters $T=0.17$, $Q=0.5$, $x_{1\mathrm{ref}}=-1.5$, $k_1=0.5$, and $k_2=1.35$. The upper plot shows voltage fluctuations around the reference value, while the lower plot shows current fluctuations, both with the red line indicating the target value. These plots illustrate the system’s performance relative to the reference.

On the other hand, Figure 14 shows the evolution of the duty cycle during each switching period. It is observed that the duty cycle stabilizes quickly at a value of $d=0.0906$.

Figure 14. Evolution of the duty cycle.

Figure 14. Evolution of the duty cycle.

3.2. Bifurcations in the Buck–Boost Converter

For the simulation, the following parameters have been chosen: $T=0.17$, $Q=0.5$, $x_{1\mathrm{ref}}=-1.1$, and $k_2=-1.5$. The range of variation for the parameter $k_1$ has also been kept in the interval $[-2,0.5]$. This “wide” interval has been intentionally chosen to clearly observe the phenomenon of the $2T$-period bifurcation near $k_1=-0.5$ and its evolution towards chaos as $k_1$ approaches the branch value. The bifurcation diagrams for the voltage $\left(x_1\right)$ and current $\left(x_2\right)$ states are presented in Figure 15 and Figure 16. On the other hand, Figure 17 shows the bifurcation diagram of the duty cycle as a function of $k_1$.

Figure 15. Voltage bifurcation diagram as a function of $k_1$ when $k_1\in [-2,0.5]$ and $k_2=-1.5$.

Figure 15. Voltage bifurcation diagram as a function of $k_1$ when $k_1\in [-2,0.5]$ and $k_2=-1.5$.

Figure 16. Current bifurcation as a function of $k_1$ when $k_1\in [-2,0.5]$ and $k_2=-1.5$.

Figure 16. Current bifurcation as a function of $k_1$ when $k_1\in [-2,0.5]$ and $k_2=-1.5$.

Figure 17. Duty cycle bifurcation diagram as a function of $k_1$ when $k_1\in [-2,0.5]$ and $k_2=-1.5$.

Figure 17. Duty cycle bifurcation diagram as a function of $k_1$ when $k_1\in [-2,0.5]$ and $k_2=-1.5$.

3.3. Existence and Stability of $1T$-Periodic Orbits

The condition given in Equation (29) will be studied numerically according to the following approaches:

(i)

The initial condition ${\mathbf{x}}_0$ was kept fixed, equal to or close to the reference values.

(ii)

The parameter vector $\mathbf{K}$ of the switching surface was fixed.

(iii)

The system parameter Q was varied over an appropriate range of values.

To test this approach, a simulation was carried out to study the evolution of the spectral radius of the product matrix

$${\mathbf{\Phi }}_{\left[0\right]}\left(T-d_n\right){\mathbf{\Phi }}_{\left[1\right]}\left(d_n\right),$$

(47)

using $T=0.17$, $Q\in (0.2,3)$, $k_1=0.17$, $k_2=0.17$, and $x_{1\mathrm{ref}}=-1.5$. The result of this simulation is presented in Figure 18. As can be seen, for this range of parameters, the spectral radius of the matrix is always less than one.

Figure 18. Evolution of the spectral radius as a function of the parameter Q. This plot shows the evolution of the spectral radius of the product matrix given in Equation (47), using the parameters $T=0.17$, $Q\in (0.2,3)$, $k_1=0.17$, $k_2=0.17$, and $x_{1\mathrm{ref}}=-1.5$. The black curve represents how the spectral radius varies with Q, showing a pattern where the spectral radius decreases to a minimum and then increases. The red horizontal line marks the reference value of 1. For this range of parameters, the spectral radius remains below one, indicating the existence of $1T$-periodic orbits, but it does not provide information on their stability.

Figure 18. Evolution of the spectral radius as a function of the parameter Q. This plot shows the evolution of the spectral radius of the product matrix given in Equation (47), using the parameters $T=0.17$, $Q\in (0.2,3)$, $k_1=0.17$, $k_2=0.17$, and $x_{1\mathrm{ref}}=-1.5$. The black curve represents how the spectral radius varies with Q, showing a pattern where the spectral radius decreases to a minimum and then increases. The red horizontal line marks the reference value of 1. For this range of parameters, the spectral radius remains below one, indicating the existence of $1T$-periodic orbits, but it does not provide information on their stability.

Figure 19 shows the behavior of the eigenvalues of the product matrix (47) when $Q\in (0.2,3)$.

Figure 19. Behavior of the eigenvalues of the product matrix (47), when $Q\in [0.2,3]$. The plot shows the real and imaginary parts of the eigenvalues in the complex plane, with the blue circles representing the eigenvalues for different values of Q. The fact that all eigenvalues lie within the unit circle implies that the spectral radius is less than 1, which is consistent with the existence of periodic orbits, though it does not provide direct information on their stability.

Figure 19. Behavior of the eigenvalues of the product matrix (47), when $Q\in [0.2,3]$. The plot shows the real and imaginary parts of the eigenvalues in the complex plane, with the blue circles representing the eigenvalues for different values of Q. The fact that all eigenvalues lie within the unit circle implies that the spectral radius is less than 1, which is consistent with the existence of periodic orbits, though it does not provide direct information on their stability.

In Figure 15 and Figure 16, the presence of $1T$-periodic orbits for certain values of $k_1$ can be observed. To determine the stability limit of these orbits, we will use the eigenvalues of the Jacobian of the Poincaré map (22), evaluated at the fixed points, which are shown in Figure 20.

Figure 20. Behavior of the eigenvalues of the Jacobian of the Poincaré application, with $k_1$ variable and $k_2=-1.5$. The plot shows the real and imaginary parts of the eigenvalues of the Jacobian evaluated at the fixed points of the system. The blue circles represent the eigenvalues for different values of Q, and the unit circle is shown for reference. The arrows indicate the movement of the eigenvalues as changes.

Figure 20. Behavior of the eigenvalues of the Jacobian of the Poincaré application, with $k_1$ variable and $k_2=-1.5$. The plot shows the real and imaginary parts of the eigenvalues of the Jacobian evaluated at the fixed points of the system. The blue circles represent the eigenvalues for different values of Q, and the unit circle is shown for reference. The arrows indicate the movement of the eigenvalues as changes.

It can also be seen in Figure 20 that initially, the eigenvalues lie within the unit circle, which ensures the stability of the $1T$ orbit. However, as the value of $k_1$ increases, some of the eigenvalues exit the circle through the point $-1$, moving from right to left. This indicates that the $1T$ orbits lose their stability. In fact, a bifurcation of orbits occurs when $k_1$ reaches approximately the value of $-0.486622$ (Figure 21). This bifurcation is of the flip type [40]. In this type of bifurcation, a stable orbit of a dynamical system loses its stability and gives rise to a new orbit with twice the period of the original one.

Figure 21. Variation of the spectral radius of the Jacobian as a function of $k_1$, with $k_2=-1.5$. The black curve shows how the spectral radius evolves as $k_1$ increases. The red horizontal line represents the critical value $\rho =1$, indicating the stability limit.

Figure 21. Variation of the spectral radius of the Jacobian as a function of $k_1$, with $k_2=-1.5$. The black curve shows how the spectral radius evolves as $k_1$ increases. The red horizontal line represents the critical value $\rho =1$, indicating the stability limit.

3.4. Chaos in the Buck–Boost Converter

To calculate the Lyapunov exponents, we must consider the system’s final states during each switching period $[kT,(k+1\left)T\right]$. For this purpose, we consider the Poincaré map written in the following form:

$${\mathbf{x}}_{k+1}=\mathbf{P}\left({\mathbf{x}}_k;d_k\right)={\mathbf{\Phi }}_{\left[0\right]}\left(T-d_k\right)\left[{\mathbf{\Phi }}_{\left[1\right]}\left(d_k\right){\mathbf{x}}_k+{\mathbf{\Psi }}_{\left[1\right]}\left(d_k\right)\right]$$

where ${\mathbf{x}}_k:=\mathbf{x}\left(kT\right)$ and ${\mathbf{x}}_{k+1}:=\mathbf{x}\left((k+1)T\right)$, respectively. The Jacobian matrix of the Poincaré map is given by Equation (22). We will denote by ${\nu }_i\left(\mathbf{JP}\left({\mathbf{x}}_k\right)\right)$ the i-th eigenvalue of $\mathbf{JP}\left({\mathbf{x}}_k\right)$ at the switching instant $t=kT$. Then, the Lyapunov exponent ${\lambda }_i$ associated with each eigenvalue will be calculated using Equation (30).

To determine the transition limit to chaos, we will use the Lyapunov exponents. We take $k_1$ in the interval $[-2,0.5]$, with $k_2=-1.5$, period $T=0.17,x_{1\mathrm{ref}}=-1.1$, and parameter $Q=0.5$. The results of the evolution of the exponents are shown in Figure 22, where the red line corresponds to the Lyapunov exponents related to voltage, and the blue line corresponds to the Lyapunov exponents related to current. Simulation confirms the existence of chaos in the Buck–Boost converter for a range of the parameter $k_1$, due to the presence of positive Lyapunov exponents. Figure 22 shows that the stability is lost when the parameter takes the approximate value of $k_1\approx 0.0568$.

Figure 22. Lyapunov exponents as a function of $k_1$ when $k_1\in [-2,0.5]$ and $k_2=-1.5$. The plot shows the evolution of the Lyapunov exponents for both voltage (red line) and current (blue line) in the BuckBoost converter system. The red curve represents the Lyapunov exponents associated with voltage, while the blue curve corresponds to the Lyapunov exponents related to current.

Figure 22. Lyapunov exponents as a function of $k_1$ when $k_1\in [-2,0.5]$ and $k_2=-1.5$. The plot shows the evolution of the Lyapunov exponents for both voltage (red line) and current (blue line) in the BuckBoost converter system. The red curve represents the Lyapunov exponents associated with voltage, while the blue curve corresponds to the Lyapunov exponents related to current.

3.5. Instabilities Control with FPIC

Referring to the FPIC technique, we must find the spectral radius of the matrix

$${\displaystyle \frac{\partial \mathbf{P}}{\partial {\mathbf{x}}_n}}={\mathbf{\Phi }}_{\left[0\right]}(T-d_n){\mathbf{\Phi }}_{\left[1\right]}\left(d_n\right),$$

(48)

evaluated at the fixed point ${\mathbf{y}}^{*}$ of the Poincaré map (27) and see if it is less than 1. Recall that the fixed points of the map are given by the expression (27). It is numerically verified that $\rho \left(\partial \mathbf{P}/\partial {\mathbf{x}}_n\right)<1$, indicating that the technique is applicable. Next, we will proceed to apply the FPIC technique with a value of $\gamma =0.1$. Figure 23 and Figure 25a confirm that the region in which the system presents chaotic behavior has decreased. In addition, the $1T$-periodic orbit is stable over a larger range of the parameter $k_1$. By approaching values of $\gamma \to 1$, it is observed that the stability limit increases, and the chaotic region becomes compressed. For instance, for $\gamma =1$, the results shown in Figure 24 and Figure 25b are obtained.

Figure 23. Bifurcation diagrams as functions of $k_1$ when FPIC is applied with $\gamma =0.1$.

Figure 23. Bifurcation diagrams as functions of $k_1$ when FPIC is applied with $\gamma =0.1$.

Figure 24. Bifurcation diagrams as functions of $k_1$ when FPIC is applied with $\gamma =1$.

Figure 24. Bifurcation diagrams as functions of $k_1$ when FPIC is applied with $\gamma =1$.

Figure 25. Duty cycle bifurcation diagrams as functions of $k_1$ when FPIC is applied with different $\gamma$ values.

Figure 25. Duty cycle bifurcation diagrams as functions of $k_1$ when FPIC is applied with different $\gamma$ values.

3.6. Validation of the Buck–Boost Converter Model with Parasitic Resistances: Experimental and Numerical Analysis

For the laboratory experiment, we used the equipment shown in Figure 26, including a 10 VDC power supply (model MCH-305B), which is manufactured by Shenzhen Meichuang Instrument Co., Ltd., located in Shenzhen, Guangdong Province, China. Additionally, we used a Tektronix AFG1062 arbitrary function generator, manufactured by Tektronix, Inc., a leading American company specializing in test and measurement equipment. Tektronix is headquartered in Beaverton, OR, USA. This signal generator has a 60 MHz bandwidth and is capable of producing a variable PWM square wave signal ranging from 0 to 5 volts at a frequency of 20 kHz.

We conducted experimental testing with a duty cycle $d=0.8$. The current signals in the inductor, $i_L$, were measured using a 0.5-ohm shunt resistor, and both this current and the capacitor voltage, $v_c$, were directly monitored on a Tektronix MSO2024B Mixed Signal Oscilloscope, manufactured by Tektronix, Inc., headquartered in Beaverton, OR, USA. Additionally, the DC input voltage was measured using a Fluke 179 True RMS Digital Multimeter, which is manufactured by Fluke Corporation, headquartered in Everett, WA, USA. The device is assembled in the USA.

For constructing the converter, we used a ferrite-core inductor with an inductance of $L_f=0.784\mathrm{mH}$, a load composed of several resistors in series and parallel, each rated at 10 $\Omega$ and 20 watts, which together form a total resistance of 59 $\Omega$ and can dissipate up to 480 W. We also used two capacitors, each with a commercial value of 220 $\mathsf{\mu }\mathrm{F}$, which, when connected in series, form an equivalent total capacitance of 105.2 $\mathsf{\mu }\mathrm{F}$.

A Vishay BYV28-200 fast-switching diode was used, manufactured by Vishay Intertechnology, Inc., located in Malvern, PA, USA, along with an IRFZ44N transistor from Infineon Technologies AG, located in Neubiberg, Germany, for switching purposes. The actual internal resistance of the inductor is 0.2 $\Omega$, but by adding a 0.5 $\Omega$ shunt resistor ($r_{measure}$), the total resistance is considered to be $r_{Lf}=0.7\Omega$. This value is necessary to ensure that the dynamics are not affected and to match the model with the experimental setup. All experimental parameters and their respective values are detailed in

Table 1. Parameter values used in the Buck–Boost converter circuit.

Parameters	Value
Input voltage $\left(V_{\mathrm{in}}\right)$	10 V
Source resistance $\left(r_{\mathrm{in}}\right)$	$0.3\Omega$
Ferrite core inductor $(L_f/r_{Lf})$	$0.784$ mH$/0.7\Omega$
Electrolytic capacitor $(C_f/r_{cf})$	$105.2\mathsf{\mu }\mathrm{F}/0.2\Omega$
Diode (BYV28-200) forward drop $\left(V_{fd}\right)$	0.457 V
Diode resistance $\left(r_d\right)$	$0.03\Omega$
Switch (IRFZ44N) resistance $\left(r_{\mathrm{sw}}\right)$	$0.0175\Omega$
Switching frequency $\left(f_{\mathrm{sw}}\right)$	20 kHz
Load resistance $\left(R_L\right)$	$59\Omega$

. The impact of parasitic resistances on the converter’s performance has been carefully considered in the mathematical model, incorporating their effects into the system equations to provide a more accurate representation of real-world operation. The inclusion of these resistances affects key performance metrics such as efficiency, output voltage regulation, and transient response. Specifically, parasitic resistances contribute to power losses, influencing overall efficiency, and modifying the dynamic behavior of the system, impacting its stability and transient characteristics. The mathematical model and experimental validation demonstrate that the converter’s response, including steady-state and transient behavior, aligns well with real conditions, confirming the relevance of including parasitic effects.

For the ideal converter, the output is given by $v_c=-\left({\displaystyle \frac{d}{1-d}}\right)·V_{in}$. Figure 27 shows the experimental behavior of the converter. In Figure 27a, the steady-state behavior for a duty cycle $d=0.3$ is displayed, where the ideal output voltage should be $v_c=-\left({\displaystyle \frac{0.3}{1-0.3}}\right)·10=-4.28$ volts. However, due to system losses, the actual output voltage is approximately $-3.86$ V, as indicated by the yellow line. All yellow lines represent the voltage across the capacitor ($v_c$), while the green signals in the experimental result figures represent the current through the inductor ($i_L$), as initially shown in Figure 27a. This demonstrates that for duty cycles below $0.5$, the converter operates as a step-down converter. Figure 27b shows the start-up process with the initial condition of the power source being off, illustrating the transient response in open-loop mode. Here, the evolution of the current in the inductor is shown in the upper part, and the capacitor voltage in the lower part. Figure 27c presents the steady-state behavior for a duty cycle of $d=0.5$, where the ideal output voltage should be $v_c=-\left({\displaystyle \frac{0.5}{1-0.5}}\right)·10=-10$ volts. Experimentally, the output voltage was approximately $-8.95$ V, which can be attributed to losses or parasitic resistances in the experimental model. Finally, Figure 27d illustrates the response when the DC source is initially off and then turned on, showing the damped progression of the inductor current to the steady-state condition, reaching both steady-state voltage and current levels as described in Figure 27c.

Across all these results, noise in the current measurement is observed, caused by its low magnitude and electromagnetic interference from the converter’s switching operations. This is a typical effect in switched converters. The voltage is less affected by this noise due to its higher magnitude; however, when zoomed in, some electromagnetic interference in the voltage signal can also be detected.

In Section 2.9, the Buck–Boost converter model with parasitic resistances was presented, including the parasitic resistances described in Table 1. To demonstrate that the model used in this paper is accurately calculated and aligns with real-world behavior, experimentally obtained data is shown in Figure 28. This figure presents both the experimental and numerical evolution of the system using the model with parasitic resistances described in Section 2.9, under both steady-state and transient conditions for a duty cycle of $d=0.8$. For an ideal converter, the expected output voltage is $v_c=-\left({\displaystyle \frac{0.8}{1-0.8}}\right)·10=-40$ V. However, in the experimental setup, a voltage level of $v_c=-27.6$ V was achieved, as shown in Figure 28b, represented by the yellow signal.

Figure 28a,b illustrates the system’s behavior in both dynamic and steady-state conditions. The inductor current, $i_L$, represented in green, reaches a value close to 2.73 A. This current level is also observed in the model that includes parasitic resistances, as shown in Figure 28b,c, where the current in the model reaches approximately 2.34 A. Finally, Figure 28e,f shows the dynamic and steady-state voltage behavior in the simulation when parasitic resistances are included. For the experimental case shown in Figure 28b, the yellow voltage signal reached $v_c=-27.6$ V for a duty cycle of $d=0.8$, while in the simulation (Figure 28e,f), the voltage reached an optimized value of $v_c=-27.8$ V.

In the transient state of the current, the experimental results, shown in green in Figure 28a, indicate that the current reaches a transient maximum value of approximately 6.4 A, with a settling time of about 10 ms. In the simulation, as shown in Figure 28c, the inductor current $i_L$ reaches a transient maximum of approximately 5.8 A. The settling time for the simulation is approximately 9 ms. For the voltage signals, the experimental results, shown in Figure 28a, indicate a maximum value of $v_c=-27.6$ V, with a settling time of 10 ms. In the simulation, as shown in Figure 28e, the maximum voltage is $v_c=-28.29$ V, with a settling time of 9 ms.

Overall, the converter’s behavior in steady-state and transient conditions shows a strong correlation between the numerical and experimental results, confirming that the Buck–Boost converter model was accurately developed. This validation assures that the results presented in this paper reflect real-world performance and support the precision of the model employed in this study.

3.7. Closed-Loop Current Control of a Buck–Boost Converter Using the ZAD Technique

In Figure 29, the closed-loop control of the Buck–Boost converter using the ZAD control technique is presented. Since the converter exhibits a non-minimum phase behavior, its voltage response to control inputs can initially counter the desired outcome, making direct control of the output voltage challenging within certain operating ranges. Therefore, in this study, we focus on controlling the inductor current $i_L$ of the Buck–Boost converter. ZAD control is particularly effective for non-minimum phase systems like the Buck–Boost converter, as it allows for a smoother switching response and improved closed-loop stability. In Figure 29a, the behavior of the converter is shown when the current reference $i_{L\mathrm{ref}}\left(t\right)$ is set to 2 A. The reference $i_{L\mathrm{ref}}\left(t\right)$ is depicted in blue, while the converter’s response, $i_L\left(t\right)$, controlled by ZAD, is shown in black. Additionally, in this same setup, a load change is introduced: initially, the load $R_L$ is set to 73 $\Omega$, but at $t=0.04$ s, it is abruptly reduced to 7.3 $\Omega$ to assess the robustness of the current control against load variations.

Figure 29c illustrates the output voltage $v_c$, which is not directly controlled. When the load change occurs, $v_c$ drops abruptly to a lower value, owing to the increase in output power as $R_L$ is reduced to 7.3 $\Omega$. Since the voltage is not controlled, it directly reflects the load variations. A similar behavior is observed when the current reference $i_{L\mathrm{ref}}\left(t\right)$ is set to a constant value of 4 A. Upon the load change at $t=0.04$ s, the converter current $i_L$ closely tracks the reference signal $i_{L\mathrm{ref}}\left(t\right)$ without being disrupted by the disturbance. This result demonstrates the effectiveness of ZAD control in maintaining a regulated current in the Buck–Boost converter. Figure 29d further shows that, as the load voltage is not controlled, it varies in response to changes in the load. In conclusion, the ZAD control technique proves effective for controlling the inductor current in the Buck–Boost converter, even in the presence of load changes or disturbances.

3.8. Comparison of ZAD and PID Control for Buck–Boost System

In this section, a comparison is made between the behavior of the Buck–Boost system controlled by ZAD and PID, with the aim of evaluating the performance and effectiveness of each controller. The controlled variable in this case is the inductor current $i_L$. All results were analyzed up to a total time of $t=0.004\mathrm{s}$. A change in the load $R_L$ was applied as follows: from $t=0\mathrm{s}$ to $t=0.002\mathrm{s}$, the load was $R_L=5.9\Omega$, and from $t=0.002\mathrm{s}$ to $t=0.004\mathrm{s}$, the load was $R_L=0.059\Omega$. This extremely low resistance load was chosen to subject both controllers to extreme operating conditions. The other parameters used are detailed in Table 1.

The control parameters for ZAD control in this section are as follows: $k_2=0.05$ and $k_1=0.0000005$, with $k_1$ being very small to focus primarily on the switching surface $s=k_1(x_1-x_{1,\mathrm{ref}})+k_2(x_2-x_{2,\mathrm{ref}})$, especially in controlling the current $i_L=x_2$. The switching frequency is $f_{\mathrm{sw}}=40\mathrm{kHz}$. For PID control, the parameters were calculated following the procedure described in [43], resulting in the following values: $K_p=1500$, $K_i=100$, and $K_d=0.05$. The Buck–Boost converter parameters are the same for both PID and ZAD control and are described in Table 1. The only parameter that changes is the load resistance $R_L$, as a load disturbance is intentionally applied to test the robustness of both controllers under the same system conditions. The results are presented in Figure 30, Figure 31 and Figure 32, which are explained in detail below.

In Figure 30a, the behavior of the current controlled by both techniques, ZAD and PID, is shown. Both have a settling time of $0.8$ ms with no overshoot. In general, it can be observed that both controllers follow the reference signal ($i_{L\mathrm{ref}}\left(t\right)=4$ A) in a similar manner. In Figure 30c, a zoom of Figure 30a is presented, which allows for a detailed view. Here, it can be seen that the ZAD control causes a larger ripple in the current. This is because the switching frequency in ZAD is constant at 40 kHz, which is reflected in the oscillations of the current $i_L$, which occur at the same frequency of 40 kHz. The blue current corresponds to the PID-controlled current, which exhibits a smaller ripple, as PID control, due to its lack of a fixed switching frequency, operates at a higher switching speed, resulting in a smaller ripple in the current signal. However, the current controlled with ZAD remains balanced around the reference signal $i_{L\mathrm{ref}}\left(t\right)=4$ A symmetrically, while the current controlled by PID stays below the reference signal until $t=2\mathrm{ms}$, and then exceeds it. Figure 30b shows the global error for both controllers, and Figure 30d presents a zoom of the errors, where it can be seen that for both controllers, the error is less than 1%. This indicates that both controllers are performing well. The larger error in the current controlled by ZAD is due to the lower switching frequency in the ZAD-controlled system. However, this is one of the advantages of ZAD control, as it operates with a fixed switching frequency, which will be explained in more detail in Figure 31.

In Figure 31a, it can be observed that the duty cycle under ZAD control does not saturate at 1 but remains at an intermediate level between 0 and 1. This is one of the main advantages of ZAD control. Before $t=2$ ms, the duty cycle is $d=0.6$, and after $t=2$ ms, it decreases to $d=0.36$. However, these values do not exhibit saturation or chaotic behavior, ensuring a fixed switching frequency of 40 kHz. As shown in Figure 31c, only the pulse width varies due to the change in duty cycle at $t=2$ ms, while the switching frequency remains unchanged. This characteristic is further confirmed in Figure 31a,c. Specifically, Figure 31a illustrates the system’s fixed switching frequency, which is also reflected in Figure 31c, where the control signal u is displayed. In this case, u corresponds to a PWM signal with a fixed frequency of 40 kHz, as programmed and predicted in the controller design.

In contrast, Figure 31b,d shows the duty cycle for PID control. In Figure 31b, it can be seen that the duty cycle with PID is constantly saturating at 1 and 0, which leads to chattering problems and undesired switching, as the PWM signals do not have a fixed frequency and instead vary at high speed. This poses a significant drawback, as is well known both in control theory and experimental practice. High switching frequencies are associated with several issues, including increased electromagnetic noise, excessive heating, component malfunction, and potential chaotic behavior, among others. These effects can degrade system performance, reduce efficiency, and compromise reliability, making frequency selection a critical aspect in power electronics and control applications.

Finally, it can also be observed that the switching frequency in Figure 31c is lower than that in Figure 31d. Additionally, it is important to note that the PWM signal frequency in PID control is not fixed; instead, it exhibits multiple values and operates at a very high speed.

In Figure 31d, a zoomed-in view is provided to illustrate the high frequency and variability of the PWM signal in PID control. These results emphasize the advantages of ZAD control, which, if experimentally implemented in the future, could offer a significant improvement over PID control. This advantage aligns with the findings reported in previous studies [13,43,44], where alternative control strategies have demonstrated enhanced performance in terms of stability, noise reduction, and efficiency.

Finally, in Figure 32, the voltage across the capacitor $v_c$ and the currents in the load are shown for both ZAD and PID control. In Figure 32a, it can be seen that the voltage for both controllers is very similar, with noticeable abrupt changes when the load is altered. This is expected, as the voltage signals $v_c$ are not actively controlled, and therefore this behavior is normal. In Figure 32b, the current $i_{R_L},$ for the system controlled with ZAD is shown, where the same oscillations present in the inductor current $i_L$ are reflected here in $i_{R_L}$, caused by the 40 kHz switching frequency. In contrast, Figure 32c shows the behavior of the current $i_{R_L}$ for the system controlled with PID. While the shape and magnitudes of the current are similar to those shown in Figure 31b, the oscillation frequency is higher, as it originates from a PWM with a non-fixed, high-speed frequency, characteristic of PID control. In general, the behavior of both currents is quite similar.

4. Conclusions

The primary objective of this paper was to study the nonlinear dynamics of the Buck–Boost converter when controlled via LPWM combined with ZAD strategy. To analyze the system, the scheme $\{1,0\}$ was used. The main results obtained are as follows: A variable change was performed, allowing the states of the Buck–Boost converter to be non-dimensionalized, which facilitated the analysis of its dynamics. The explicit expression for the duty cycle was obtained when the system was controlled by LPWM and ZAD, using a linear approximation of the switching surface. The Poincaré map associated with the system’s $\{1,0\}$ scheme was explicitly constructed, supported by graphs. A detailed explanation was provided on how the system evolves during the n-th switching period, starting from a given initial condition. An analytical expression was found for a sufficient condition for the existence of $1T$-periodic orbits. To validate and support these results, numerical simulations were carried out in MATLAB R2018a with Simulink, following the conditions established by the analytical expressions, effectively verifying the existence of the expected periodic orbits. The analytical expression for the Jacobian matrix of the Poincaré map was obtained.

The results from the bifurcation diagrams of the duty cycle d as a function of $k_1$ show that the duty cycle does not fully saturate, meaning it neither reaches zero nor one. This indicates that the system maintains a fixed switching frequency, which is a significant contribution and a key objective of this research. It is demonstrated that the ZAD technique ensures a fixed switching frequency, achieving the desired control in the Buck–Boost converter.

Both the experimental and numerical responses of the open-loop system exhibit a sawtooth pattern in the current due to the switching effect. This phenomenon, observed in both representations, confirms that the mathematical model effectively captures the converter’s switching dynamics in open-loop operation. The experimental validation was performed under identical conditions, ensuring high-precision measurements with state-of-the-art instrumentation. This rigorous approach yielded a strong correlation between the experimental data and the mathematical model, representing a significant contribution given the limited availability of such detailed validations in the literature. A thorough comparison of transient and steady-state responses demonstrated a remarkable consistency between both models. While incorporating noise into the mathematical framework would be highly complex, measurement errors are minimal due to the use of high-accuracy equipment. Future work will focus on implementing closed-loop experimental control, as demonstrated in simulations, where the control system actively compensates for errors and disturbances. This underscores the key advantage of advanced control strategies: continuously correcting deviations to enhance system stability and performance.

The results demonstrate that the ZAD control keeps the inductor current $i_L$ in the converter aligned with the set reference, even in the presence of load disturbances, while also reducing the transient response time. This highlights the effectiveness of ZAD control in stabilizing the inductor current and its potential for applications requiring both current stability and rapid adaptation to load changes.

The comparison between ZAD and PID control for the Buck–Boost converter demonstrates the significant advantages of ZAD control in maintaining system stability and efficiency. ZAD control ensures a fixed switching frequency, leading to a balanced duty cycle with minimal risk of chattering and undesirable switching effects. This stability prevents issues such as duty cycle saturation and excessive MOSFET heating, which are common drawbacks of high-frequency PID control. In contrast, PID control exhibits saturated duty cycles at 0 and 1, generating PWM signals with variable and high switching frequencies. This variability increases the risk of instability, electromagnetic interference, and thermal stress on switching components, which could compromise system performance, particularly in experimental implementations. These findings reaffirm the robustness of ZAD control in power electronics applications, where precise and stable switching is essential. Future work should focus on experimental validation to further assess its benefits and explore its applicability to other converter topologies.