Research on Nonlinear Dynamic Characteristics of Fractional Order Resonant DC-DC Converter Based on Sigmoid Function

Abstract

Resonant DC-DC converters are a class of strongly nonlinear systems with rich nonlinear phenomena. In order to describe the dynamic behavior of resonant DC-DC converters more accurately, the nonlinear dynamic behavior of fractional order (FO) resonant DC-DC converters is studied deeply, based on the fractional order nature of inductance and capacitance. Firstly, a Sigmoid function state model of the fractional order resonant converter is established and integrated with phase shift control. A discrete model of the converter is established by using an estimation correction algorithm. Secondly, the mathematical and equivalent circuit models of the fractional order converter are constructed in MATLAB. The circuit simulations and the experimental results verified the correctness of the Sigmoid function model. Thirdly, the effect of circuit parameters on the converter’s nonlinear dynamics is analyzed using bifurcation diagrams, time-domain waveforms, and phase diagrams. Finally, an experimental platform is established to validate the theoretical analysis. The results demonstrate that increasing the proportional coefficient and load resistance destabilizes the system, leading to rich nonlinear phenomena such as bifurcation and chaos. Compared to integer order converters, fractional order converters offer a broader stable operating range. Fractional order models can more accurately reflect the nonlinear dynamic characteristics of resonant DC-DC converters.

1. Introduction

Compared with traditional DC-DC converters, DC-DC resonant converters have the advantages of a higher efficiency and smaller size, and are frequently and widely used for distributed energy generation, electric vehicles, data centers, and aerospace [1,2,3].

The DC-DC resonant converter is a highly nonlinear system characterized by complex nonlinear behaviors, including bifurcation and chaotic dynamics, which manifest under specific circuit parameters and operational conditions [4,5,6,7]. Studies on the nonlinear dynamic characteristics of DC-DC converters have been extensively conducted since the 1980s [8,9,10]. The latest research results indicate that the inductors and capacitors are in fact fractional order elements and not integer order components [11,12]. As a result, the integer order models used to describe the characteristics of inductance and capacitance are not accurate; they cannot accurately reflect the physical nature of a fractional order. Inductance and capacitance are universally recognized as fundamental components for energy storage in converters. Therefore, the converter is also fractional order. But most existing work on resonant DC-DC converters is still based on integer order models, which are not accurate enough to describe the true dynamic characteristics of the converter [13]. Therefore, it is necessary to study the nonlinear dynamics of fractional order resonant DC-DC converters; not only does this help us to understand and comprehend the complex behavior, but it also facilitates the optimization of converter design and further improves converter performance.

One study [14] modeled the simulation model of the FO Buck converter and studied the chaotic behavior with a reference voltage proportional amplification factor as the chaotic control variable, but did not establish a discrete model of the FO converter. In [15], a fractional order model of a magnetically coupled Boost converter was proposed and a bifurcation diagram of the inductor current was obtained. In addition, a discrete model for the converter was provided. In [16,17], the discrete model of the FO Buck converter and Z-source converter was established by using the estimation correction method and bifurcation diagrams with fractional order as a bifurcation parameter were obtained, but the influence of other circuit parameters on the performance of FO model was not analyzed and only the continuous conduction mode (CCM) converter was discussed. The nonlinear behavior of the CCM and discontinuous conduction mode DCM FO Buck-Boost converters are analyzed in [18] by using the estimation correction method. The mathematical model of the FO Luo converter is established in [19], and the influence of the order on the nonlinear dynamic characteristics of the system is studied. Compared with traditional integer order models, the FO models have higher accuracy, which helps to better reveal and analyze the nonlinear behavior of DC-DC converters.

At present, most research on the nonlinear behavior of the FO DC-DC converter has focused on the basic converter, such as Buck, Boost, and Buck-Boost converters, with less research focusing on FO resonant DC-DC converters. And most research objectives of resonant DC-DC converters are based on the IO models, which are inconsistent with the fractional order nature of converters. Therefore, it is necessary to research the nonlinear behavior of an FO resonant DC-DC converter to improve the accuracy of modeling and dynamic performance. Consequently, this study investigates the bifurcation and chaotic behaviors in fractional order (FO) resonant DC-DC converters through the integration of nonlinear dynamic system theory and fractional calculus. The Sigmoid function and predictor–corrector method of fractional calculus are adopted to obtain the FO discrete model for the FO resonant DC-DC converter. Compared with standard modeling methods, the Sigmoid function is characterized by its property of global differentiability, which renders it particularly efficacious for the modeling of discontinuous transitions and square wave signals within dynamic systems. Its smooth, continuous nature enables the precise representation of abrupt system changes, making it an essential tool for analyzing and simulating systems exhibiting such nonlinear behaviors. Therefore, the FO model of the FO converter would be more accurate. The nonlinear dynamic characteristics are analyzed based on this model.

This paper is organized as follows: The Sigmoid function is introduced in Section 2. In Section 3, the Sigmoid function model of FO resonant DC-DC converter is established based on the state-space average method. In Section 4, the bifurcation diagrams of the converter with proportional coefficient, load resistance and fractional order as bifurcation parameters are obtained, and the V-I phase diagrams and time domain waveform diagrams in different parameters are drawn. The impact of the system parameters on the stability is analyzed. The simulation model of the FO resonant DC-DC converter is built in MATLAB R2022a/SIMULINK in Section 5. A FO resonant DC-DC converter experimental platform is built to verify the correctness of theoretical analysis, and the stability is discussed. Final conclusions are offered in Section 6.

2. Sigmoid Function

The Sigmoid function is known for its continuity, smoothness, boundedness, and differentiability, and it can describe the modal transformation of circuits [20]. The Tan Sigmoid function (tanh(x)) is a common Sigmoid function, which has the property of global differentiability. It has a domain of any real number R and a range of (−1, 1). The function is expressed as follows:

$$\mathit{tanh}\left(\mathrm{x}\right)={\displaystyle \frac{{\mathrm{e}}^{\mathrm{x}}-{\mathrm{e}}^{-\mathrm{x}}}{{\mathrm{e}}^{\mathrm{x}}\mp {\mathrm{e}}^{-\mathrm{x}}}}={\displaystyle \frac{2}{1+{\mathrm{e}}^{-2\mathrm{x}}}}-1$$

(1)

By introducing a steepness factor b into the tanh(x) function, the slope of the tanh(x) curve can be adjusted. The tanh(bx) function curves of different b values are shown in Figure 1. As observed in Figure 1, the larger b, the closer the T-function curve is to the sign function curve. For this reason, a hyperbolic tangent function with a steep factor b is used to continuously process discontinuous transition points in the system in this paper. Figure 2 illustrates the function y = tanh[bsin(ωt)] curves at different b values. As shown in Figure 2, as b increases, the waveform becomes closer to an ideal square wave. Therefore, the tanh-based Sigmoid function with a steepness factor b is particularly effective in modeling discontinuous transitions and square wave signals in dynamic systems.

3. Sigmoid Function Model of FO Resonant DC-DC Converter

3.1. Mathematical Modeling

Fractional calculus is primarily defined by three seminal definitions: the Riemann–Liouville definition, the Grünwald–Letnikov definition, and the Caputo definition. Each of these definitions requires different initial conditions, leading to distinct analytical results depending on the selected definition. This paper does not analyze the discrepancies arising from these fractional calculus frameworks, as its core objective focuses on investigating the nonlinear dynamics of fractional order series resonant converters.

The topology of the FO resonant DC-DC converter circuit is shown in Figure 3, in which V_in is the input voltage, R_o is the output load, S₁–S₄ are ideal switching transistors, D₁–D₄ are also ideal diodes, Lr^α is α-order inductor, Cr^β and Co^γ are, respectively, β- and γ-order capacitors, and 0 < α, β, γ < 1. k_p is the controller’s proportional coefficient, k_i is the integral coefficient, and V_ref is the output reference voltage.

The FO inductor voltage v_L and FO capacitor current i_c can be expressed as in [21].

$$\left\{\begin{array}{c}{v}_L(t)=L{\displaystyle \frac{d^{\alpha }{i}_L}{d{t}^{\alpha }}}\\ i_C(t)=C{\displaystyle \frac{d^{\beta }{v}_C}{d{t}^{\beta }}}\end{array}\right.$$

(2)

where α ∈ (0, 1), β ∈ (0, 1).

The operating mode of a resonant DC-DC converter is determined by the switching frequency f_s and the resonant frequency f_r. When f_s < f_r/2, the system operates in discontinuous conduction mode (DCM). When f_s > f_r/2, the system operates in continuous conduction mode (CCM). In this paper, the switching frequency f_s is greater than the resonant frequency f_r. FO resonant DC-DC converter contains four operating states in CCM, as shown in Figure 4.

State 1 (t₀ < t < t₀ + d₁T): S₁, S₄, D₁, and D₄ are on, and S₂, S₃, D₂ and D₃ are off, as illustrated in Figure 4a. The state-space equations are as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{i}_{Lr}}{d{t}^{\alpha }}}={\displaystyle \frac{V_{in}-v_o-v_{Cr}}{L_r}}\\ {\displaystyle \frac{d^{\beta }{v}_{Cr}}{d{t}^{\beta }}}={\displaystyle \frac{i_{Lr}}{C_r}}\\ {\displaystyle \frac{d^{\gamma }{v}_o}{d{t}^{\gamma }}}={\displaystyle \frac{i_{Lr}}{C_o}}-{\displaystyle \frac{v_o}{R{C}_o}}\end{array}\right.$$

(3)

State 2 (t₀ + d₁T < t < t₀ + d₁T + d₂T): S₁, S₄, D₂ and D₃ are off, and S₂, S₃, D₁ and D₄ are on, as illustrated in Figure 4b. The state-space equations are as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{i}_{Lr}}{d{t}^{\alpha }}}={\displaystyle \frac{{-V}_{in}-v_o-v_{Cr}}{L_r}}\\ {\displaystyle \frac{d^{\beta }{v}_{Cr}}{d{t}^{\beta }}}={\displaystyle \frac{i_{Lr}}{C_r}}\\ {\displaystyle \frac{d^{\gamma }{v}_o}{d{t}^{\gamma }}}={\displaystyle \frac{i_{Lr}}{C_o}}-{\displaystyle \frac{v_o}{R{C}_o}}\end{array}\right.$$

(4)

State 3 (t₀ + d₁T + d₂T < t < t₀ + d₁T + d₂T + d₃T): S₁, S₄, D₁ and D₄ are off, and S₂, S₃, D₂ and D₃ are on, as illustrated in Figure 4c. The state-space equations are as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{i}_{Lr}}{d{t}^{\alpha }}}={\displaystyle \frac{-V_{in}+v_o-v_{Cr}}{L_r}}\\ {\displaystyle \frac{d^{\beta }{v}_{Cr}}{d{t}^{\beta }}}={\displaystyle \frac{i_{Lr}}{C_r}}\\ {\displaystyle \frac{d^{\gamma }{v}_o}{d{t}^{\gamma }}}={\displaystyle \frac{{-i}_{Lr}}{C_o}}-{\displaystyle \frac{v_o}{R{C}_o}}\end{array}\right.$$

(5)

State 4 (t₀ + d₁T + d₂T + d₃T < t < t₀ + T): S₂, S₃, D₁ and D₄ are off, and S₁, S₄, D₂ and D₃ are on, as illustrated in Figure 4d. The state-space equations are as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{i}_{Lr}}{d{t}^{\alpha }}}={\displaystyle \frac{V_{in}+v_o-v_{Cr}}{L_r}}\\ {\displaystyle \frac{d^{\beta }{v}_{Cr}}{d{t}^{\beta }}}={\displaystyle \frac{i_{Lr}}{C_r}}\\ {\displaystyle \frac{d^{\gamma }{v}_o}{d{t}^{\gamma }}}={\displaystyle \frac{-i_{Lr}}{C_o}}-{\displaystyle \frac{v_o}{R{C}_o}}\end{array}\right.$$

(6)

By applying the state-space averaging method, the state-space model of the FO resonant DC-DC converter is derived from Equations (3)–(6) as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{i}_{Lr}}{d{t}^{\alpha }}}={\displaystyle \frac{v_a-v_{o}sgn\left(i_{Lr}\right)-v_{Cr}}{L_r}}\\ {\displaystyle \frac{d^{\beta }{v}_{Cr}}{d{t}^{\beta }}}={\displaystyle \frac{i_{Lr}}{C_r}}\\ {\displaystyle \frac{d^{\gamma }{v}_o}{d{t}^{\gamma }}}={\displaystyle \frac{i_{Lr}sgn\left(i_{Lr}\right)}{C_o}}-{\displaystyle \frac{v_o}{R{C}_o}}\end{array}\right.$$

(7)

where sgn(x) represents the sign function, and the AC square wave voltage v_a(t) is as follows:

$$sgn\left(t\right)=\left\{\begin{array}{c}1 t>0\\ 0 t=0\\ -1 t<0\end{array}\right.$$

(8)

$$v_a\left(t\right)=\left\{\begin{array}{l}{V}_{in} 0\ll t\ll {\displaystyle \frac{T}{2}}\\ {-V}_{in} {\displaystyle \frac{T}{2}}\ll t\ll T\end{array}\right.$$

(9)

When α = β = γ = 1, Equation (6) represents the state-space model of the integer order (IO) resonant DC-DC converter.

Constant frequency phase-shifting control with PI control is used to regulate the output voltage. In this way, the input voltage transitions from an AC square wave voltage v_a to a step voltage v_ab. As a result, the new state-space model is as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{i}_{Lr}}{d{t}^{\alpha }}}={\displaystyle \frac{v_{ab}-v_{o}sgn\left(i_{Lr}\right)-v_{Cr}}{L_r}}\\ {\displaystyle \frac{d^{\beta }{v}_{Cr}}{d{t}^{\beta }}}={\displaystyle \frac{i_{Lr}}{C_r}}\\ {\displaystyle \frac{d^{\gamma }{v}_o}{d{t}^{\gamma }}}={\displaystyle \frac{i_{Lr}sgn\left(i_{Lr}\right)}{C_o}}-{\displaystyle \frac{v_o}{R{C}_o}}\end{array}\right.$$

(10)

where v_ab is the output step voltage of the inverter, which can also be approximated by the Sigmoid function as follows:

$$v_{ab}\approx 0.5V_{in}[\mathit{tanh}\left(bsin\left(\omega t\right)\right)-\mathit{tanh}\left(bsin\left(\omega t-\theta \right)\right)]$$

(11)

where θ represents the control phase shift angle and ω represents the system angular frequency.

$$\theta =\pi -\omega (k_p{V}_{ref}-k_p{v}_o+k_i\rho )$$

(12)

$$\rho =\int (V_{ref}-V_o)dt$$

(13)

When θ = 90°, the step voltage v_ab can be approximated by tanh functions with different steepness coefficients b, as shown in Figure 5. It can be seen from Figure 5 that the larger the steepness coefficient b, the better the tanh function can represent the switching action of the converter.

The symbol function sgn is equivalently replaced by the Sigmoid function as follows:

$$sgn\left(i_{Lr}\right)=tanh\left(b{i}_{Lr}\right)$$

(14)

Combining Equations (10), (11), and (14), the state-space model of a FO resonant DC-DC converter with phase-shift control can be obtained by using the Sigmoid function:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{i}_{Lr}}{d{t}^{\alpha }}}={\displaystyle \frac{0.5V_{in}\left[\tanh \left(bsin\left(\omega t\right)\right)-\tanh \left(bsin\left(\omega t-\theta \right)\right)\right]-v_o\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(b{i}_{Lr}\right)-v_{Cr}}{L_r}}\\ {\displaystyle \frac{d^{\beta }{v}_{Cr}}{d{t}^{\beta }}}={\displaystyle \frac{i_{Lr}}{C_r}}\\ {\displaystyle \frac{d^{\gamma }{v}_o}{d{t}^{\gamma }}}={\displaystyle \frac{i_{Lr}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(b{i}_{Lr}\right)}{C_o}}-{\displaystyle \frac{v_o}{R{C}_o}}\end{array}\right.$$

(15)

3.2. Simulation

The circuit parameters of Figure 3 are listed in

Table 1. Resonant converter parameters.

Parameters	Values
Resonant inductor (Lr)	1.19 × 10−4
Resonant capacitor (Cr)	8.49 × 10−6
Input voltage (Vin)	30 V
Switching frequency (f)	10,000 Hz
Load (R)	10 Ω
Reference voltage (Vref)	18 V
Proportional coefficient (kp)	2.5 × 10−5
Integral coefficient (ki)	2 × 10−3
Inductance order (α)	0.9
Capacitance order (β)	0.9

.

Based on the relationship of inductance current and capacitance voltage in (15), the mathematic model of the FO resonant DC-DC converter is established, as shown in Figure 6a. The FO resonant DC-DC converter modeled by the Oustaloup’s filter-based approximation method [22] and the simulation model in MATLAB/Simulink are shown in Figure 6b.

The simulation waveforms of the mathematical model and circuit model are illustrated in Figure 7. The blue solid line and the red dotted line in the figure are the curves obtained by the circuit model and mathematical model, respectively. As shown in Figure 7, the waveform of the mathematical model and circuit model are in good agreement, confirming the accuracy of the Sigmoid function mathematical model of the Sigmoid function established in this article.

4. Dynamics Behavior and Stability Analysis

The resonant converter is a quintessential, highly nonlinear system that exhibits diverse complex dynamic behaviors. The circuit parameters have a significant impact on the stability of the system. If the circuit parameters are not appropriate, bifurcation and chaos would occur in the system which could even result in instability. By using the FO estimation correction method, a discrete mathematical model based on the Sigmoid function for phase-shift-controlled resonant DC-DC converters is developed. The detailed analysis of the dynamic behavior of the system and the impact of key parameters on system stability are carried out through bifurcation diagrams and V-I phase diagrams.

4.1. FO Estimation Correction Algorithm

The fractional calculus prediction correction algorithm was developed by K. Diethelm et al. in 2002 [22]; it extends the classic one-step estimation correction method for first-order differential equations to the FO field. As a time-domain approximation technique for fractional differential equations, this method demonstrates superior modeling accuracy and computational efficiency compared to conventional algorithms. The FO estimation correction algorithm is outlined below.

For any FO differential equation with initial values, it can be expressed as follows:

$$\begin{array}{l}{D}_{*}^{q}y\left(x\right)=f\left(x,y\left(x\right)\right)\\ y^{\left(k\right)}\left(0\right)=y_0^{\left(k\right)},k=\mathrm{0,1},\dots ,m-1\end{array}$$

(16)

where $D_{*}^q$ is the q-order differential operator defined by Caputo and $y_0^{\left(k\right)}$ is the known initial values.

Equation (16) can be equivalent to the Volterra integral equation as follows:

$$y\left(x\right)=\sum _{k=0}^{\left[q\right]-1}{y}_0^{\left(k\right)}{\displaystyle \frac{x^k}{k!}}+{\displaystyle \frac{1}{\Gamma \left(q\right)}}{\int }_0^x{\left(x-t\right)}^{q-1}f\left(t,y\left(t\right)\right)dt$$

(17)

Take h = T/N, t_n = n_h, n = 0, 1, …, N, with N being an integer. The integrals on the right-hand side of Equation (13) are approximated by using the FO estimation correction algorithm. The estimated correction model for fractional differential equations can be obtained as follows:

$$y_h\left(t_{n+1}\right)=\sum _{k=0}^{\left[q\right]-1}{\displaystyle \frac{t_{n+1}^k}{k!}}{y}_0^{\left(k\right)}+{\displaystyle \frac{h^q}{\Gamma \left(q+2\right)}}f\left(t_{n+1},y_h^p\left(t_{n+1}\right)\right)+{\displaystyle \frac{h^q}{\Gamma \left(q+2\right)}}\sum _{j=0}^n{a}_{j,n+1}f(t_j,y_h\left(t_j\right))$$

(18)

where a_j,n+1 are correction coefficients; it can be expressed as follows:

$$a_{j,n+1}=\left\{\begin{array}{l}{n}^{q+1}-\left(n-q\right){\left(n+1\right)}^q, j=0\\ {(n-j+2)}^q+{(n-j)}^{q+1}-2{\left(n-j+1\right)}^{q+1}, 1\ll j\ll n\\ 1, j=n+1\end{array}\right.$$

(19)

The initial estimated approximation of the state variable y_h^p(t_n+1) can be expressed as follows:

$$y_h^p\left(t_{n+1}\right)=\sum _{k=0}^{\left[q\right]-1}{\displaystyle \frac{t_{n+1}^k}{k!}}{y}_0^{\left(k\right)}+{\displaystyle \frac{1}{\Gamma \left(q\right)}}\sum _{j=0}^n{b}_{j,n+1}f(t_j,y_h\left(t_j\right))$$

(20)

where b_j,n+1 is estimated coefficients; it can be expressed as follows:

$$b_{j,n+1}={\displaystyle \frac{h^q}{q}}({\left(n+1-j\right)}^q-{\left(n-j\right)}^q)$$

(21)

4.2. Discrete Mathematical Model Based on Sigmoid Function Mathematical Model

The estimation correction method can solve any fractional order differential equation with initial conditions in the time domain, and the Sigmoid function model can also be solved by using the estimation correction method. First, the state equations of the FO resonant DC-DC converter are discretized. The discretizing inductor current and the output voltage are defined as i_L = i_h(t_n+1), v_c = v_h(t_n+1), respectively. The initial values of the inductor current and output voltage are zero, and the initial estimated values are i_hp(t_n+1) = v_p,n+1.

$$\left\{\begin{array}{l}{i}_{Lr n+1}=i_{Lr 0}+{\displaystyle \frac{h^{\alpha }}{\Gamma \left(q+2\right)}}\left({\displaystyle \frac{v_a+v_{o p,n+1}\tanh \left(b{i}_{Lr p,n+1}\right)-v_{Cr p,n+1}}{L_r}}\right)+{\displaystyle \sum _{j=0}^n}{a}_{j,n+1}\left({\displaystyle \frac{v_a+v_{o j}\tanh \left(b{i}_{Lr}\right)-v_{Crj}}{L_r}}\right)\\ v_{Cr n+1}=v_{Cr 0}+{\displaystyle \frac{h^{\beta }}{\Gamma \left(q+2\right)}}\left({\displaystyle \frac{i_{Lr p,n+1}}{C_r}}\right)+{\displaystyle \sum _{j=0}^n}{\beta }_{j,n+1}\left({\displaystyle \frac{i_{Lr j}}{C_r}}\right)\\ v_{o n+1}=v_{o 0}+{\displaystyle \frac{h^{\gamma }}{\Gamma \left(q+2\right)}}\left({\displaystyle \frac{i_{Lr p,n+1}\tanh \left(b{i}_{Lr p,n+1}\right)}{C_o}}-{\displaystyle \frac{v_{o p,n+1}}{R{C}_o}}\right)+{\displaystyle \sum _{j=0}^n}{\gamma }_{j,n+1}\left({\displaystyle \frac{i_{Lr j}\tanh \left(b{i}_{Lr j}\right)}{C_o}}-{\displaystyle \frac{v_{o j}}{R{C}_o}}\right)\end{array}\right.$$

(22)

where the expression of correction coefficients α_{j, n+1}, β_{j, n+1}, γ_{j, n+1} are as follows:

$$a_{j,n+1}=\left\{\begin{array}{l}{n}^{\alpha +1}-\left(n-\alpha \right){\left(n+1\right)}^{\alpha }, j=0\\ {(n-j+2)}^{\alpha }+{(n-j)}^{\alpha +1}-2{\left(n-j+1\right)}^{\alpha +1}, 1\ll j\ll n\\ 1, j=n+1\end{array}\right.$$

(23)

$${\beta }_{j,n+1}=\left\{\begin{array}{l}{n}^{\beta +1}-\left(n-\beta \right){\left(n+1\right)}^{\beta }, j=0\\ {(n-j+2)}^{\beta }+{(n-j)}^{\beta +1}-2{\left(n-j+1\right)}^{\beta +1}, 1\ll j\ll n\\ 1, j=n+1\end{array}\right.$$

(24)

$${\gamma }_{j,n+1}=\left\{\begin{array}{l}{n}^{\gamma +1}-\left(n-\gamma \right){\left(n+1\right)}^{\gamma }, j=0\\ {(n-j+2)}^{\gamma }+{(n-j)}^{\gamma +1}-2{\left(n-j+1\right)}^{\gamma +1}, 1\ll j\ll n\\ 1, j=n+1\end{array}\right.$$

(25)

The initial estimated values of the resonant current, resonant voltage, and output voltage in Equation (18) can be expressed as follows:

$$\left\{\begin{array}{l}{i}_{Lr p,n+1}=i_{Lr 0}+{\displaystyle \frac{1}{\Gamma (\alpha )}}{\displaystyle \sum _{i=0}^n}{b}_{i,n+1,\alpha }({\displaystyle \frac{v_a+v_{o }\tanh \left(b{i}_{Lr}\right)-v_{Cr }}{L_r}})\\ v_{Cr n+1}=v_{Cr 0}+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\displaystyle \sum _{i=0}^n}{b}_{i,n+1,\beta }({\displaystyle \frac{i_{Lr}}{C_r}})\\ v_{o n+1}=v_{o 0}+{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\displaystyle \sum _{i=0}^n}{b}_{i,n+1,\gamma }({\displaystyle \frac{i_{Lr}\tanh \left(b{i}_{Lr}\right)}{C_o}}-{\displaystyle \frac{v_o}{R{C}_o}})\end{array}\right.$$

(26)

The estimated coefficients for the resonant current, resonant voltage, and output voltage can be expressed as follows:

$$\left\{\begin{array}{c}{b}_{i,n+1,\alpha }={\displaystyle \frac{h^{\alpha }}{\alpha }}({\left(n-i+1\right)}^{\alpha }-{\left(n-i\right)}^{\alpha })\\ b_{i,n+1,\beta }={\displaystyle \frac{h^{\beta }}{\beta }}({\left(n-i+1\right)}^{\beta }-{\left(n-i\right)}^{\beta })\\ b_{i,n+1,\gamma }={\displaystyle \frac{h^{\gamma }}{\gamma }}({\left(n-i+1\right)}^{\gamma }-{\left(n-i\right)}^{\gamma })\end{array}\right.$$

(27)

where h = T_n/N₀, T_n is the period of the input clock signal, N₀ is the number of points, and n is the number of calculation periods. In this paper, N₀ = 500, n > 100.

For analysis purposes, we calculate 500 points per cycle and run more than 100 calculation cycles. To simplify this, the order of inductance and capacitance is made equal, α = β = γ = 0.9, and the circuit parameters are the same as Table 1.

4.3. Proportional Coefficient k_p as Bifurcation Parameter

The circuit parameters are the same as Table 1. The bifurcation diagram of the FO resonant DC-DC converter with proportional coefficient k_p as the bifurcation parameter is shown in Figure 8, in which the horizontal direction is the proportional coefficient k_p between 1 × 10⁻⁵ and 9 × 10⁻⁵, and the vertical direction is the output voltage v_o which ranges from 11 V to 19 V. Enlarge Figure 8a to obtain Figure 8b. As seen in Figure 8, the system undergoes a bifurcation at k_p = 4.5 × 10⁻⁵. When k_p < 4.5 × 10⁻⁵, the output voltage is equal to the reference value and the system is in stable operation. When k_p is near the bifurcation point, even small changes in k_p can lead to fluctuations between stability and instability. This indicates that the system is in a critical state. With the increase in k_p, the output voltage v_o oscillates significantly; the system becomes chaotic.

When α = β = γ = 1, the system is an integer order one. The bifurcation diagram of the IO converter with k_p as the bifurcation parameter is shown in Figure 9. Enlarge Figure 9a to obtain Figure 9b. It can be seen from Figure 9 that when k_p < 3.1 × 10⁻⁵, the system is stable. The bifurcation occurs at k_p = 3.1 × 10⁻⁵. As the k_p is continuously increases, the converter eventually exhibits chaos. From Figure 8 and Figure 9, the bifurcations of the FO and IO converter occur at different k_p values. The bifurcation points of the FO converter are larger than those of the IO converters, so the FO converter system exhibits a broader range of stable operational parameters compared to the IO converter.

The V-I phase diagram and time-domain waveforms for the FO converter are plotted at k_p = 2.5 × 10⁻⁵ and k_p = 7.5 × 10⁻⁵, as shown in Figure 10 and Figure 11, respectively. From Figure 10, it can be seen that the phase diagram displays a smooth closed limit cycle at k_p = 2.5 × 10⁻⁵, and the system is stable. However, the limit cycle becomes concave and the trajectory overlaps at k_p = 7.5 × 10⁻⁵, so the system becomes chaotic. Figure 10a,b show that when k_p = 2.5 × 10⁻⁵, the output voltage closely matches the reference voltage, and when k_p = 7.5 × 10⁻⁵, the output voltage waveform oscillates.

4.4. Load Resistance R as Bifurcation Parameter

The bifurcation diagrams of the FO resonant DC-DC converter with the load resistance R as the bifurcation parameter are shown in Figure 12. Enlarge Figure 12a to obtain Figure 12b. From Figure 12, it can be observed that the FO converter undergoes a transition at the bifurcation point R = 48 Ω. The system is stable when R < 48 Ω, while the system is chaotic when R > 48 Ω.

Figure 13 shows the bifurcation diagram of the IO resonant DC-DC converter. Enlarge Figure 13a to obtain Figure 13b. When R < 35 Ω, the converter works in a stable state. When R = 35 Ω, the bifurcation takes place. When R > 35 Ω, the converter operates in a chaotic state. From Figure 12 and Figure 13, it can be observed that the value of the load R of the FO converter is larger than that of the IO converters when the bifurcation takes place.

Figure 12 shows that the system is stable when R < 48 Ω, while the system is chaotic when R > 48 Ω. The V-I phase diagram and time-domain waveform of the FO converter are drawn at R = 10 Ω, as shown in Figure 10a and Figure 11. The V-I phase diagram and time-domain waveform of the FO converter are drawn at R = 80 Ω, as shown in Figure 14 and Figure 15, respectively. As shown in Figure 14, the phase diagram indicates that limit cycle trajectories overlap at R = 80 Ω, hence, the system is chaotic. Figure 15 shows that the output voltage waveform oscillates; therefore, the system is operating in a chaotic state.

4.5. Fractional Order of Inductance and Capacitance as Bifurcation Parameter

Setting the proportional coefficient as k_p = 2.5 × 10⁻⁵ and the load resistance as R = 40 Ω, the complete circuit parameters are listed in Table 1. Bifurcation diagrams using the fractional order of inductance and capacitance as the bifurcation parameter are illustrated in Figure 16. Enlarge Figure 16a to obtain Figure 16b.

As shown in Figure 16, when the fractional order α = β = γ < 0.85, the converter operates stably. When the fractional order α = β = γ > 0.85, the output voltage begins to oscillate. By further increasing the fractional order α, β, and γ, the converter finally goes into a chaotic state. By observing the variation in the output voltage with fractional order, the change in the operating state of the converter can be clearly seen; that is, as the fractional order increases, the system progresses sequentially from stable operation to bifurcation and ultimately enters a chaotic regime.

5. Experiments

In order to validate the correctness of theoretical analysis, experiments have been set up. In the experimental platform, the switch tube is designated as the IRF640 model, the resistor is a metal film resistor, the inductor is an iron–silicon–aluminum magnetic ring inductor, and the capacitor is an aluminum electrolytic capacitor. The gate drive circuit employs a pulse-width modulated (PWM) signal generator connected to the input of a gate driver IC, which subsequently controls the MOSFET’s switching operation through isolated voltage amplification. Currently, fractional order components are typically implemented by using approximation methods. The improved Oustaloup approximation method is the most widely used due to its broader fitting bandwidth and higher accuracy [23]. Hence, an improved Oustaloup approximation method is adopted to fit FO inductance and capacitance, as shown in Figure 17. The parameter values of the power circuit fractional capacitor and inductor can be seen in

Table 2. The parameter values of fractional component.

	L = 1.19 × 10−4 H		C = 8.49 × 10−6 F
	α = 0.9		β = 0.9
i	RLi (Ω)	Li (H)	RCi (Ω)	Ci (F)
1	229.920 k	26.18 m	190.091
2	96.564	10.834 m	169.8 m	40.073 m
3	190.4 m	21.363 m	130.058	52.332 m
4	775.330	8.701 m	100.326 k	67.835 m
5	1.528	17.112 m	14.594	46.619 m
6	6.624 k	7.433 m	10.475 k	64.957 m
7	12.207	13.428 m	1.613	42.204 m
8	21.960 m	23.8 m	1.162 k	58.309 m
9			7456.597 k	9.127 m

, in which m represents 1 × 10⁻³, k represents 1 × 10³.

The circuit parameters of FO resonant DC-DC converter are listed in Table 1. The FO resonant DC-DC converter experimental platform is shown in Figure 18.

5.1. The Effect of Proportional Coefficient kp on System Dynamics Behavior

The time-domain waveform and V-I phase diagram of the FO resonant DC-DC converter at k_p = 2.5 × 10⁻⁵ and k_p = 7.5 × 10⁻⁵ are shown in Figure 19 and Figure 20, respectively. From Figure 19, it can be clearly seen that when k_p = 2.5 × 10⁻⁵, the output voltage v_o = 18 V and the phase diagram is a smooth closed limit cycle. Therefore, the system is stable. According to Figure 20, it can be observed that when k_p = 7.5 × 10⁻⁵, the limit cycle becomes unsmooth and the trajectory overlaps, and the output voltage waveform oscillates. Hence, the system operates in a chaotic state. This is consistent with the simulation results in Section IV. Compared with the simulation results, experimental results exhibit more pronounced fluctuations in current/voltage waveforms and less defined limit cycle boundaries in phase portraits compared with the simulation data. This discrepancy can be attributed to the fact that the fractional order inductors and capacitors in the circuit model are implemented through approximate equivalence methods, rather than truly equivalent fractional order components. In addition, the parasitic resistance of diodes, switches, and other circuit components also leads to errors in output voltage measurement.

5.2. The Effect of Load R on System Dynamics Behavior

The time-domain waveforms and V-I phase diagram of the FO resonant DC-DC converter at k_p = 2.5 × 10⁻⁵ and R = 80 Ω are shown in Figure 21. As can be observed from Figure 21, the output voltage waveform fluctuates, and the V-I phase diagram exhibits irregular rings, which indicates that the system is unstable. Compared with the simulation results, the output voltage and the resonant capacitor voltage are slightly lower and the area of the overlapping limit cycle trajectories in the phase diagram diminishes. This is because the simulation environment and electronic components are ideal, without considering actual situations such as switch losses and heat losses. In summary, the experimental results are consistent with the simulation results of the theoretical analysis, verifying the accuracy of the theoretical analysis.

6. Conclusions

Firstly, based on the fractional order theory, a Sigmoid function state model of the FO resonant converter is established and integrated with phase shift control. A discrete model of the converter is established by using an estimation correction algorithm. Secondly, the mathematical and equivalent circuit models of the FO converter are constructed in MATLAB. The circuit simulations are carried out, and the experimental results were able to verify the correctness of the established Sigmoid function model. Thirdly, the impact of circuit parameters on the nonlinear dynamic characteristics of the converter is investigated in detail by bifurcation diagrams, time-domain waveforms, and phase diagrams. Finally, an experimental platform is established to validate the theoretical analysis. The research results show that with the increase in the proportional coefficient and load resistance, the system transitions from a stable state to an unstable state, which exhibits rich nonlinear phenomena such as bifurcation and chaos. Compared to integer order converters, FO converters offer a broader stable operating range. FO models can more accurately reflect the nonlinear dynamic characteristics of resonant DC-DC converters.