Modeling and virtual simulation of the boost chopper by DCM using the optimal PIDF control

Abstract

This work introduces an improved method for modeling and simulating the Boost Converter utilizing Duty Cycle Modulation (DCM) regulated by an optimum PIDF (Proportional-Integral-Derivative with Filter) regulator. We optimized the characteristic parameters of the PIDF regulator for a second-order system generated from its transfer function by using a mix of theoretical study and simulation using the Matlab/LQR tool. The conventional PID parameters in the time domain were converted into their corresponding LQR (Linear Quadratic Regulator) counterparts, allowing for the solution of the Riccati problem and the creation of an optimum state trajectory model. The results of analog virtual simulations done in a Multisim environment indicate that the system has improved dependability. It maintains a high level of accuracy in a stable condition, with no static error and a reaction time of 1.5 ms, without any overshooting. The effectiveness of the optimum PIDF control in regulating the DCM Boost Converter is highlighted by the system's strong ability to handle changes in load during transient states within a time frame of 300 ms. This study represents a substantial enhancement compared to conventional PID-based approaches, providing valuable knowledge about the possible uses in power electronics and control systems.

1. Introduction

1.1. Background of study

The use of DC-DC converters, specifically Boost choppers, in renewable energy applications such as solar systems, has required the creation of sophisticated control algorithms to maximize the energy output in terms of voltage and current. Duty Cycle Modulation (DCM) has become an attractive substitute for Pulse Width Modulation (PWM) because of its simplicity and efficacy in decreasing the number of components and improving system implementation.

1.2. Literature review

Multiple studies have investigated diverse control systems for Boost choppers. Nonlinear Composite Control has been utilized to regulate input voltage and load fluctuations [1]. This study presents a nonlinear Composite Control method to handle changes in input voltage and load in DC-DC boost converters; while some researchers have concentrated on the small-signal modeling in DCM [2]. In addition, the use of hybrid feedforward and sliding-mode control techniques has effectively tackled specific obstacles such as system nonlinearity and voltage balancing [3,4]. Recent advancements in the field have introduced the application of strong PID controllers developed using Quantitative Feedback Theory (QFT) [5]. [6] presents a tracking strategy that uses energy shaping along with a self-tuning mechanism and a disturbance observer. However, it fails to consider the impact of input saturation on the performance of the system. The authors [7]propose a dual-loop control system for three-phase floating interleaved boost converters in order to ensure a stable output voltage.

The application of Linear Quadratic Regulator (LQR) methods in Ref. [8] shows enhanced regulation of output voltage for parallel-connected boost DC-DC converters. However, the susceptibility to parameter fluctuations and input saturation still poses a hurdle [9]. presents a new design structure for a PID controller that includes an Elman neural network. This design has important implications for improving system performance.

Saleem et al. [10] present a strong and efficient control method that utilizes a Fractional-order Proportional-Integral (FOPI) controller to improve the accuracy of output-voltage tracking and control. Furthermore [11], examines the utilization of Quantitative Feedback Theory (QFT) in the development of a resilient PID regulator for voltage control in DC-DC boost converters.

The study presented in Ref. [12] introduces a sophisticated order PI controller that demonstrates improved performance when compared to conventional controllers for the purpose of controlling DC-DC converters. In addition [13], investigates the optimization of Interval Type-2 Fuzzy Logic PID controllers to achieve optimal performance in boost DC-DC converters.

The incorporation of FTIλDN controllers, as described in Ref. [14], presents a new method for enhancing the performance of Automatic Generation Control (AGC) in interconnected power systems. In addition [15], presents a Fractional Order Proportional Tilt Integral Derivative Plus One (FOPTID+1) controller for Automatic Generation Control (AGC) design. The controller is optimized using the global neighborhood algorithm (GNA) to regulate frequency during load disturbances.

Ultimately, the implementation of duty cycle modulation (DCM) as a streamlined control scheme, as described in Ref. [16], highlights continuous endeavors to streamline control strategies for industrial systems, as illustrated in the following Fig. 1.

Fig. 1.

a) PWM circuit diagram; b) DCM circuit diagram.

Table 1 provides a comparative study of some of the weaknesses identified in the PWM, which are remedied by the DCM circuit, a significant competitor.

Table 1.

Comparison between PWM and DCM.

Some weaknesses of PWM	DCM advantages
Frequency spectrum too rich in harmonics to be filtered, complicating the downstream signal processing with a power filter	Offers interesting properties in terms of robustness and modulation spectrum control
Open-loop operating structure results in low robustness	The self-oscillating architecture is equipped with two loops (+ and -)
Noise injection in non-linear control systems	Exact knowledge of analytical characteristics
Complexity of implementation due to the use of a large number of components	Simplicity in cost and execution
Fixed frequency	self-adjusting frequency

The effectiveness of this new, innovative approach has been demonstrated in numerous scientific research projects in a variety of fields, including: ECG signal processing [17] Here, we demonstrate the feasibility of using DCM adapted to this type of signal by building an experimental prototype that attests to the high performance offered by this new system for transmission via coaxial cable and optical fiber, with very good quality of the reconstructed signal, with RMSE < 0.06 and SNR = 30 dB. Quality factors (SNR = −2.60 dB, SINAD = − 2.72 dB, THD = −6.77 dB, SFDR = 0.47 dB) of the reconstructed ECG signal are very close to the parameters of the reference (noise-free) ECG signal to be transmitted. In Ref. [18], we prototyped on an FPGA target the design of electrocardiographic (ECG), respiratory and photoplethysmography (PPG) biological signals. In Refs. [19,20] [5], DCM was optimized for A/D and D/A conversion of low-frequency signals using an over-sampling architecture based on the DCM principle, resulting in an overall resolution of the analog-to-digital converter (ADC) of around 12 bits, an SNR of 70.45 dB and an SFDR of 58.76 dB. In power electronics, DCM has proved its worth in the field of inverters: in Refs. [[21], [22], [23]] the design of the very first single-phase H-bridge solar inverter with sinusoidal duty cycle modulation (SDCM) was set up and experimentally tested on a test bench against single-phase PWM inverters, 23 dB, SINAD is 41.61 dB, SFDR 50.33 dB and SNR is 41.86 dB. However, this open-loop system is not stable and lacks robustness. The design and implementation of digital and active filters by DCM in Refs. [24,25] significantly reduced the THD to 2.82 %, compared with 3.12 % for control using the hysteresis approach. As regards the Buck chopper class [26,27], established the basis for optimizing control using the PID/LQR controller. The comparative study carried out between the analog and digital domains by discretizing the system's closed-loop transfer functions resulted in a perfect juxtaposition of the system's responses in these different domains. In Ref. [28]the DCM principle is established on the Boost chopper class, controlled using a conventional PID close control the simulation results obtained are encouraging. In Ref. [28], the discretization of the new DCM Boost chopper is carried out using PIDF control. The simulations carried out, taking into account scenarios involving disturbances of −2 V and +2 V respectively, introduced at instants between 0 ≤ t ≤ 7.5 ms, validate the stability and robustness criteria specific to DCM Boost chopper control. 1.3. Research Gap and Motivation.

This work identifies a critical research gap in the optimal control of the Boost chopper in DCM using PIDF controllers. While the literature addresses various aspects of Boost chopper control, there is limited exploration of PIDF control in DCM, particularly in the face of parameter uncertainties and system non-linearities. The need for a control strategy that not only enhances performance but also ensures robustness and stability under dynamic conditions motivates this research.

1.3. Challenges

The main challenges in controlling Boost choppers include maintaining stability and performance across a wide range of operating conditions and handling the inherent non-linearities of DCM. These challenges are compounded by the need for controllers that can adapt to sudden load variations and input uncertainties without compromising on the quality of the output.

1.4. Contribution

The main objective of this article is to optimize the performance of the PID-F controller by equating the parameters of the standard PID simulated in Refs. [28,29] to those of the robust LQR control, which will offer more satisfactory results than those reported in previous work.in particular, the quality of the response (output voltage) obtained must be stable despite the many fluctuations that may be observed during operation of the proposed system.

• •The performance of the optimal PIDF controller has been developed for output voltage control of the DCM Boost chopper model using the LQR command.
• •The performance of the PIDF controller has been tested by varying the desired output voltage values and also changing the load resistance value of the DCM Boost chopper circuit as a parameter uncertainty problem.

1.5. Paper organization

The subsequent sections of this paper are organized in the following manner: Section 2 provides a comprehensive explanation of the techniques and instruments used to achieve the most efficient control of the DCM Boost Chopper. Section 3 showcases the outcomes of virtual simulations that illustrate the efficacy of the suggested PIDF controller. Furthermore, the document includes discussions of the consequences and significance of these findings. The paper finishes in Section 4, where it provides a summary of the findings and presents future research directions.

2. Methods and tools

This part will utilize analytical and mathematical models to operate the DCM Boost chopper. Virtual simulations will be conducted using the Multisim software.

2.1. Virtual modeling of the boost to DCM converter

The virtual open-loop simulation of the DCM Boost chopper, with a base frequency of 1 KHz and an amplitude of 15V as depicted in Fig. 2, exhibits inadequate resilience and a suboptimal response with insufficient cushioning, as demonstrated in Fig. 3.

Fig. 2.

Boost chopper controlled by the DCM circuit.

Fig. 3.

Response graph at Echelon E0 = 6 V.

In order to systematically analyze the values obtained by graphical representation, we begin by examining the Overrun, which is determined through the application of Equation (1):

$$d=\frac{\left(36,7-29,5\right)}{29,5}=0,2241$$ (1)

Next, we analyze the weather parameter, denoted as $T_m$, which has a value of 6.2 ms, as stated in Equation (2).

$$T_m=6,2\text{ms}$$ (2)

Additionally, the static gain, represented as $K_s$, can be calculated by dividing $Y_m\left(\infty \right)$ by $E_0$, as shown in Equation (3).

$$K_s=\frac{Y_m\left(\infty \right)}{E_0}=\frac{V_S}{E_0}=\frac{29,5}{6}=4,92$$ (3)

To elaborate on these principles, we comprehend that the transfer function of the Boost chopper with DCM in an open loop can be expressed by Equation (4) [16].

$$G\left(p\right)=\frac{Y\left(p\right)}{U\left(p\right)}=\frac{K_s{w}_n^2}{p^2+2\xi w_{n}p+w_n^2}$$ (4)

Where the damping coefficient, ξ, is determined Equation (5).

$$\xi =\frac{\log \left(d\right)}{\sqrt{\left({\log \left(d\right)}^2\right)+{\pi }^2}}=0,4095$$ (5)

Lastly, the natural pulse, $w_n$, is established by the following Equation (6):

$$w_n=\frac{\pi }{\pi \sqrt{1-{\xi }^2}}=\frac{\log \left(\frac{100}{r}\right)}{\xi T_r}$$ (6)

By determining these parameters, we know the nature of the system, which is under-damped because ξ < 1 and the numerical value _(p) of the transfer function of the open-loop system transfer function.

2.2. Principle of the DCM

The DCM circuit shown in Fig. 4 is an astable oscillator, i.e., controlled with negative resistance, whose operating principle is based on the charging and discharging of a capacitor. The u_C1 voltage obtained is compared with a sinusoidal voltage (modulating signal) to obtain a control signal needed to drive the switch of the DCM DC-DC converters [18].

Fig. 4.

DCM circuit.

The DCM consists of an operational amplifier supplied with a symmetrical voltage ± E, a capacitor C₁ and resistors R₁, R₂ and R₃. Knowing that the period of the DCM is given by Equation (7) [19]:

$$T_m\left(x\right)=T_{ON}\left(x\right)+T_{OFF}\left(x\right)$$ (7)

The duty cycle of the DCM reported in the literature [16,22] is given by the following.

Equation (8):

$$R_m\left(x\right)=\frac{T_{ON}\left(x\right)}{T_m\left(x\right)}$$ (8)

This implies:

$$R_m\left(x\right)=\frac{\ln \left(\frac{{\alpha }_{2}x-\left(1+{\alpha }_1\right)E}{{\alpha }_{2}x+\left({\alpha }_1-1\right)E}\right)}{\ln \frac{{\left({\alpha }_{2}x\right)}^2-{\left(\left(1+{\alpha }_1\right)E\right)}^2}{{\left({\alpha }_{2}x\right)}^2-{\left(\left({\alpha }_1-1\right)E\right)}^2}}$$ (9)

Where the coefficients ${\alpha }_1$ and ${\alpha }_2$ in Equation (9) are given by Equation (10):

$${\alpha }_1=\frac{R_1}{\left(R_1+R_2\right)}and{\alpha }_2=1-\frac{R_1}{\left(R_1+R_2\right)}$$ (10)

These works [9,22] shown that linearization of Rm (x) at points 0 and $\frac{1}{2}$ provides an approximation of the duty cycle of the DCM given by Equation (11):

$$R_m\left(x\right)=\frac{1}{2}+P_m\left(x\right)with{P}_m\left(x\right)=\left(\frac{\frac{{\alpha }_1{\alpha }_2}{E\left(1-{{\alpha }_1}^2\right)}}{\log \left(\frac{1+{\alpha }_1}{1-{\alpha }_1}\right)}\right)$$ (11)

Knowing that the base period and the base frequency expressed in Equation (12) are obtained for the value of $x=0$ in other words

$$T_m\left(0\right)=2\tau \mathit{log}\left(1+\frac{R_1}{R_2}\right)ie,F_m\left(0\right)=\frac{1}{2\tau \mathit{log}\left(1+\frac{R_1}{R_2}\right)}with\tau =R_3{C}_1$$ (12)

2.3. Optimal PID-F controller implementation equivalent to LQR

The conventional PID (Proportional-Integral-Derivative) controller is widely acknowledged for its efficacy in many applications within control systems. The system consists of three separate elements: the proportional, integral, and derivative components, each contributing to the overall control action in unique ways. In the PIDF (Proportional-Integral-Derivative with Filter) controller, the derivative term is enhanced by including a low-pass filter. This change plays a crucial role by reducing the impact of high-frequency noise on the derivative calculation. The filter improves the stability and performance of the controller by reducing fluctuations, making the PIDF arrangement highly beneficial in applications that demand precision and stability in noisy settings.

Initially we'll determine the characteristic parameters of the standard PID then optimize them by equating to the LQR parameters.

2.3.1. Determination of the characteristic parameters of the PID standard

The transfer function of the PID standard is given by Equation (13) [11]:

$$D\left(p\right)=\frac{U\left(p\right)}{E\left(p\right)}=K_p+\frac{K_i}{p}+K_{d}p=K_p\left(1+\frac{1}{T_{i}p}+T_d\right)$$ (13)

With.

• ⁃K_P: Proportional gain
• ⁃$T_i$ = $\frac{K_p}{K_i}\text{:}$ Integral time constant
• ⁃$T_d=\frac{K_d}{K_p}$: Differential time constant

In order to optimize the performance of the characteristic parameters of this controller, which offers fairly satisfactory results. We are interested in reformulating and solving the state - space PID controller problem as that of an LQR (linear quadratic regulator), which consists in finding a vector K(t) that minimizes or maximizes, depending on the case, a functional criterion is given by Equation (14):

$$J=\frac{1}{2}{\int }_0^{\infty }\left(Z^T\left(t\right)QZ\left(t\right)+u^T\left(t\right)\left)d\left(t\right)\right)\right)$$ (14)

With R >, 0 Q ≥ 0 (Qx = N^TQN) which are symmetric matrices.

Note that the optimal control obtained is written as state feedback given by Equation (15)

$$u=-K\left(t\right)x+RefwithK=-R^{-1}{B}^{T}P$$ (15)

We propose to equip the Boost Chopper in DCM with an LQR controller with parameters $K_1$, $K_2$ and $k_3$ which were initially equipped with a conventional PID corrector. The 2nd order differential equation obtained by inverse Laplace transformation is established respectively by (16), (17) [26]:

$$\frac{d^{2}y\left(t\right)}{dt}=-2\mathit{\xi }{w}_n\frac{dy}{dt}‐w_n^{2}y+K_s{w}_n^{2}u\left(t\right)$$ (16)

In other words:

$$\frac{d^{2}e\left(t\right)}{dt}=-2\xi w_n\frac{dy}{dt}-w_n^2\mathrm{y}+K_s{w}_n^{2}u\left(t\right)$$ (17)

The state variables $x_1$, $x_2$ and $x_3$ indicated in Fig. 5 are defined in the time domain by Equation (18):

$$x_1=e=Ref-y,x_2=\int edtand{x}_3=\frac{de}{dt}$$ (18)

Fig. 5.

Results of optimal control by optimal PID controller.

At this stage the relationship between Equation (11) and Equation (13) can be expressed as:

$$\{\begin{array}{c}\frac{d{x}_1}{dt}=\frac{de}{dt}=x_3\\ \frac{d{x}_2}{dt}=e=x_1\\ \frac{d{x}_3}{dt}=\frac{d^{2}e}{d{t}^2}=-2\xi w_n{x}_3-(Ref-x_1{)w}_n^2+K_S{w}_n^{2}u\left(t\right)\end{array}$$ (19)

Equation (19) corresponds to the following state model which can be expressed by Equation (20) and Equation (21)

$$\left[\begin{array}{c}\frac{d{x}_1\left(t\right)}{dt}\\ \frac{d{x}_2\left(t\right)}{dt}\\ \frac{d{x}_3\left(t\right)}{dt}\end{array}\right]=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ -w_n^2& 0& -2\xi w_n\end{array}\right]\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ K_S{w}_n^2\end{array}\right]\mathrm{u}\left(\mathrm{t}\right)+\left[\begin{array}{c}0\\ 0\\ w_n^2\end{array}\right]$$ (20)

$$\left[\begin{array}{c}\frac{d{x}_1\left(t\right)}{dt}\\ \frac{d{x}_2\left(t\right)}{dt}\\ \frac{d{x}_3\left(t\right)}{dt}\end{array}\right]=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ -w_n^2& 0& -2\xi w_n\end{array}\right]\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\\ K_S{w}_n^2& w_n^2\end{array}\right]\left[\begin{array}{c}u\left(t\right)\\ Ref\end{array}\right]$$ (21)

Assuming that matrices A, B, C and D have the components shown in (22), (23):

$$\mathrm{A}=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ -w_n^2& 0& -2\xi w_n\end{array}\right]\text{and}\mathrm{B}=\left[\begin{array}{c}0\\ 0\\ w_n^2(1+K_S\end{array}\right]$$ (22)

$$\mathrm{C}=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]and\mathrm{D}=$$ (23)

Practical constraints on the implementation of this state feedback control system given by Equation (24) can be verified more quickly by executing the Matlab command Mc = rank (ctrb (A, B)), which demonstrates that the pair (A, B) is controllable with matrix A of dimension n*n and matrix B of dimension n*1.

$$\mathrm{R}=\mathrm{r}\text{and}Q=\left[\begin{array}{ccc}{q}_1& 0& 0\\ 0& q_2& 0\\ 0& 0& q3\end{array}\right]$$ (24)

The solution of the infinite time horizon LQR problem associated with (13) is given by (25a), (25b)):

$$u\left(t\right)=-Kx\left(t\right)+RefwithK=-R^{-1}{B}^{T}P$$ (25a)

or P is the solution of the Riccati equation expressed as:

$$PA+A^{T}P-PB{R}^{-1}{B}^{T}P+Q=0$$ (25b)

Equation (15) can be expressed as Equation (26):

$$P=\left[\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right]$$ (26)

So the expression of $u\left(t\right)$ for the gain K defined in Equation (15) becomes Equation (27) subsequently we arrive at an expression of u(t) given by Equation (28).

$$u\left(t\right)=-\frac{1}{r}\left[\begin{array}{ccc}0& 0& \left(K_s+1\right)w_n^2\end{array}\right]\left[\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right]x\left(t\right)$$ (27)

$$u\left(t\right)=\left[\begin{array}{ccc}\frac{\left(K_S+1\right)w_n^2}{r}{P}_{31}& \frac{\left(K_S+1\right)w_n^2}{r}{P}_{32}& \frac{\left(K_S+1\right)w_n^2}{r}\end{array}\right]x\left(t\right)$$ (28)

2.3.2. PID/LQR equivalence principle

It can be seen that the PID controller defined in Equation (13) and the LQR controller given by Equation (27) are structurally equivalent, in other words the relationship governing the principle of equivalence between the PID and LQR controllers results from the equality in the time domain of expressions (13) and (27) leads to the expression (29):

$$u\left(t\right)=\underset{LQR}{\underbrace{-\left[\begin{array}{ccc}{K}_1& K_2& K_3\end{array}\right]\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\end{array}\right]}}=-\underset{PID}{\underbrace{\left[\begin{array}{ccc}{K}_p& K_i& K_d\end{array}\right]\left[\begin{array}{c}e\left(t\right)\\ \int e\left(t\right)dt\\ \frac{de\left(t\right)}{dt}\end{array}\right]}}$$ (29)

In terms of earnings, the equivalence is as follows by expression (30):

$$[\begin{array}{c}\frac{\left(K_S+1\right)w_n^2}{r}{P}_{31}=K_1=K_p\\ \frac{\left(K_S+1\right)w_n^2}{r}{P}_{32}=K_2=K_i\\ \frac{\left(K_S+1\right)w_n^2}{r}{P}_{33}=K_3=K_d\end{array}$$ (30)

For a reference Ref = 6 V from the initial conditions set, we obtain the trajectories of the optimal PID controller control system equivalent to the following LQR. The results of optimal control by the optimal PID controller and the optimal trajectories of the states x₁, x₂ and x₃. Fig. 5, Fig. 6 show respectively the results of the optimal PID controller and the optimal state trajectories.

Fig. 6.

Optimal state trajectories.

2.3.3. PIDF optimal controller design

Given that the transfer function of the PID controller is [28]. The transfer function of the system regulated by the standard closed-loop PID controller is given by Equation (31).

$$Dp=K_p\left(1+\frac{1}{T_{i}p}+T_{d}p\right)F_{1}p=\frac{V_S\left(p\right)}{V_{ref}\left(p\right)}=\frac{G\left(p\right)D_1\left(p\right)}{1+G\left(p\right)D_1\left(p\right)}=\frac{K_S{w}_n^2\left(K_d{p}^2+K_{p}p+K_i\right)}{p^3+\left(2\xi w_n+K_s{K}_d{w}_n^2\right)p^2+\left(1+K_s{K}_p\right)p{w}_n^2+K_s{K}_i{w}_n^2}$$ (31)

F₁(p) is the transfer function of the Boost chopper regulated by the conventional PID controller.

The transfer function of the optimal PIDF controller is given by the expression expression (32) following:

$$D_1\left(p\right)=K_p+\frac{K_i}{p}+\frac{K_{d}p}{1+T_{f}p}=K_p(1+\frac{1}{T_{i}p}+\frac{T_{d}p}{1+T_{f}p})$$ (32)

Since the parameters $K_p$, $K_i$ and $K_d$ are known thanks to relation (30), we can easily deduce T_d, and T_i invoked in Equation (13). The final parameter to be determined is the time constancy T_f of the first order low pass filter. Knowing that (33) is the closed loop system transfer function regulated by the optimal PIDF controller so the expression is as follows:

$$F_2\left(p\right)=\frac{V_S\left(p\right)}{V_{Ref}\left(p\right)}=\frac{G\left(p\right)D_2\left(p\right)}{1+G\left(p\right)D_2\left(p\right)}$$ (33)

By replacing respectively in the Ti, $K_p$ , T_d and fixing T_f=0.25s by their values in the expression (32) we obtain the following closed-loop transfer function given by (34) after development and simplification.

$$F_2\left(p\right)=\frac{K_s{w}_n^2\left[2,7568995p^2+0,93944p+1\right]}{0,17236p^4+\left[0,3447\xi w_n+0,68944\right]p^3+\left[w_n^2\left(0,25+2,7568995\right)\right]p^2+\left[w_n^2\left(0,93944\right)\right]p+w_n^2{K}_S}$$ (34)

Fig. 7 shows the prototyping the DCM Boost chopper controlled by optimal PIDF has been implemented in Multisim software. This electronic schematic presented numbered as follows: the analog PIDF control divided into 3 parts: a) (difference circuit with output with output.

Fig. 7.

Overall diagram of the analog adjustment of the Boost to DCM converter by optimal PIDF.

$e=V_S-Ref)$ , b) (PIDF input circuit) and c) (output inverter). additional relevant parts are numbered also: d) (DCM) driver, e) (Boost chopper), f) (virtual multichannel oscilloscope), g) (the power supply) and the load variation. Table 2 shows the technical information and data for the prototyping control system.

Table 2.

Characteristic elements of the functioning of the overall system.

REF	PID/LQR control	Optimal PIDF circuit	Error amplifier	DCM Circuit	Boost Chopper Circuit
6V	KP = 6, 8944	RS = 75KΩ	R7 = 100KΩ	U1 = TL082CD	E = 15 V
	Ki = 10	Ra = 120KΩ	R8 = 100KΩ	R1 = 1,2 KΩ	R0 = 6Ω
	Kd = 2, 2700	Ca = 0,725uF	R9 = 100KΩ	R2 = 10 KΩ	L = 7.5 mH
	r = 20	Cb = 0.53 nF	R11 = 100KΩ	R3 = 1,2KΩ	C1 = 4,167 mF
	q1 = 0,08	Rb = 1,65MΩ	R12 = 100KΩ	C1 = 3,3 nF	Q = IRLZ14
	q2 = 2000	U (2,3,4) = TL082CD	R14 = 100KΩ	R6 = 1 KΩ	D = HF04TB60
	$q_3=0,1e^{-7}$	–	–	–	–

2.4. system modeling and simulation in the frequency domain

However, certain performance criteria that we suggest can be understood in terms of frequency analysis. For instance, the THD (total harmonic distortion), SINAD (signal to noise distortion), and the FFT (Fast Fourier Transform) can provide information about the SFDR (spurious free dynamic range) of the optimal PIDF controlled DCM Boost chopper. (11), (12) are derived from the global Fourier equation of DCM, which stands for Discrete Cosine Transform.

$$x_m\left(t\right)=\left(2R_m\left(x\left(t\right)\right)-1\right)E+\sum _{n=1}^{\infty }\left(\left(\frac{4E}{\pi }\right)\frac{\mathit{sin}\left(n\pi R_m\left(x\left(t\right)\right)\right)}{n}\right)\mathit{cos}\left(2\pi n\frac{t}{T_m\left(x\left(t\right)\right)}\right)$$ (35)

By breaking down this equation, we derive the equations ${\beta }_1$ and ${\beta }_2$, which respectively represent a low frequency and a high frequency in a repeated manner. The respective expressions of ${\beta }_1$ and ${\beta }_2$ are represented in (36) [30].

$$\{\begin{array}{c}{\beta }_1=\left(2R_m\left(x\left(t\right)-1\right)\right)E\\ {\beta }_2=\sum _{n=1}^{\infty }\left(\left(\frac{4E}{\pi }\right)\frac{\mathit{sin}\left(n\pi R_m\left(x\left(t\right)\right)\right)}{n}\right)\mathit{cos}\left(2\pi n\frac{t}{T_m\left(x\left(t\right)\right)}\right)\end{array}$$ (36)

Equation (37) gives the static gain $K_f$ of a low-pass filter needed to extract x(t) from the nearly linear range of (10) can be obtained by utilizing the first-order Taylor series of (10). Consequently, the value of $K_f$ is provided in the following manner:

$$K_f=\frac{1}{2P_{m}E}=\frac{\left(1-{\alpha }_1^2\right)}{2{\alpha }_1{\alpha }_2}\log \left(\frac{1+{\alpha }_1}{1-{\alpha }_1}\right)$$ (37)

The Fourier series expressed in equation (35) depicts a modulated wave $x_m\left(t\right)$. It is applicable only when the maximum frequency of x(t) is significantly lower than the fundamental modulation frequency, denoted as $F_m\left(0\right)=\frac{1}{T_m\left(0\right)}$ and derived from the following relationship given by the expression (38):

$$\begin{array}{c}{T}_m\left(x\left(t\right)\right)=\frac{T_{ON}\left(x\left(t\right)\right)}{T_{Off}\left(x\left(t\right)\right)}=\tau \mathit{log}\left(\frac{{\left({\alpha }_{2}x\left(t\right)\right)}^2-{\left(\left(1+{\alpha }_1\right)E\right)}^2}{{\left({\alpha }_{2}x\left(t\right)\right)}^2-{\left(\left({\alpha }_1-1\right)E\right)}^2}\right)\\ with\tau =R_3{C}_1\end{array}$$ (38)

In the initial functional diagram proposed in Fig. 7, it would be necessary to adjust it by taking into account The processing block for analog filter: $R_6=15700\Omega$, $R_4=15700\Omega$, $C_3=4,7nF$ , $C_4=10nF$ proposed in Ref. [30]. (Fig. 8(j)) is ideally a linear second order (or more if necessary) low-pass filter with unit static gain. It owes its linear filtering nature to the fact that the modulated wave (2), consists of a sum of a single low frequency part given by, ${\beta }_1$ and of high frequency ${\beta }_2$. with the following time-varying amplitude and frequency given by Equation (39) [30]

$$F_{m}x\left(t\right)=\frac{n}{T_{m}x\left(t\right)}\text{with}\mathrm{n}=1,2$$ (39)

Fig. 8.

System response in the frequency domain.

Fig. 9 shows the optimal response of the Boost chopper with DCM controlled by the PIDF controller.

Fig. 9.

Closed-loop analog simulation of the complete system for R₀ = 6 Ω.

3. Results and discussions

The evaluation of the response obtained at the output of the DCM Boost chopper controlled by the PIDF optimal controller is carried out by taking into account the values of 3 distinct loads and by considering 2 scenarios, the steady state and the transient state.

3.1. Case 1: steady-state system response

Fig. 9 shows the modulating signal U_m and the output voltage, which tends to become stable at time t = 125 ms, for an initially fixed load of R₀ = 6 Ω at the system output. Fig. 10 shows the output voltage Vs.

Fig. 10.

Closed-loop voltage Vs for (R₀ = 6Ω) at t = 250 ms.

For a load greater than the initial load, assuming R₀ = 12Ω, we have at time = 125 ms, the output voltage Vs tends to stabilize. Fig. 11 shows the shape of the modulating signal U_m and the voltage Vs at the output of the DCM boost chopper, while Fig. 12 shows the shape of the output voltage Vs for such a load.

Fig. 11.

Closed-loop system response for (R₀ = 12Ω).

Fig. 12.

Vs signal at t = 250 ms for R₀ = 12Ω.

For a load lower than the initial load at we set R₀ = 3Ω, Fig. 13 shows the modulating signal U_m and the output voltage Vs at time t = 85 ms. Fig. 14 shows the shape of the output voltage Vs.

Fig. 13.

Closed-loop system response for a load R₀ = 3Ω.

Fig. 14.

Vs signal profile for R₀ = 3Ω.

3.2. Case 2: transient system response

In this case, the system is subjected to a different load variation thanks to an integrated relay. The aim here is to observe whether the output voltage Vs remains stable. We will take the cases where the different values R₀ = 6Ω and $R_{01}=\frac{R_0}{2}$ respectively.

Fig. 15 shows the closed-loop response of the system for load R₀ = 6Ω with variation at t = 300 ms.

Fig. 15.

Closed-loop system response for (R₀ = 6Ω) with variation t = 300 ms.

Fig. 16 shows the closed-loop system response for load R₀ = 12 Ω and with load variation performed at time t = 300 ms and R₀₁ = 6Ω.

Fig. 16.

Vs signal after a load variationR₀ = 12Ω and R₀₁ = 6 Ω at time t = 300 ms.

Fig. 17 shows the closed-loop system response for load R₀ = 3 Ω and with load variation performed at time t = 300 ms and R₀₁ = 1,5Ω.

Fig. 17.

Vs signal after a load variation R₀ = 3 Ω and R₀₁ = 1,5 Ω at time t = 300 ms.

The response of the system is carried out for a stable amplitude a = 15V, for the different load values, we observe a response without overshoot, zero static error, response time Tr = 1.5 ms these characteristic parameters sufficiently testify to the robustness offered by the optimization of the PIDF controller compared to those of the open-loop response. Figs. 10, 12 and 14, show the shape of the Vs signal respectively for these different loads. Overall, it can be seen that Vs becomes constant and stable from the instant t = 125 ms for Fig. 9, t = 135 ms for Fig. 11 and t = 85 ms for Fig. 13 after simulations carried out respectively with the initial load R₀ = 6Ω and for a load higher than it and another lower than it, the same shape of the Vs signal is observed. However, the purpose of the load variation carried out at time t = 300 ms is to see the steady-state behavior of the system following an unexpected disturbance that may occur during operation of the device. It can be seen that the Vs signal stabilizes again after a certain time, which is respectively t = 350 ms with Vs = 26V for load R₀ = 6Ω, t = 385 ms with Vs = 27 V where R₀ = 12Ω and at the end of t = 335 ms with Vs = 24V for R₀ = 3Ω. This return of the system to its stable position testifies to the robustness of the PIDF optimal controller. For the various loads used in this paper report $\frac{V_S}{E}$ > 1 shows that it is indeed a step-up chopper. Fig. 15, Fig. 16, Fig. 17 give a fairly accurate picture of the system's overall behavior at the various critical instants for the different load values used to perform the virtual simulation of the Boost chopper with DCM controlled by the PIDF optimal controller. The voltage V_S is kept constant, which testifies to the performance of the PIDF optimal controller on the effect of load variation on the output voltage Vs.

3.3. Case 3: system response in the frequency domain

Fig. 18 presents the computed standard performances of Optimal PIDF controlled DCM Boost chopper models within the modulating bandwidth. These performances include the total harmonic distortion (THD) and the signal-to-noise distortion ratio (SINAD).

Fig. 18.

Processing of Optimal PIDF controlled DCM Boost chopper modulating waveforms.

The spectrum analysis conducted in Ref. [30] using Fast Fourier Transform (FFT) with N = 16384 points reveals the spurious free dynamic range (SFDR) of the best PIDF DCM Boost chopper, as shown in Fig. 19. Table 3 presents a concise overview of the findings from a comparative analysis of THD and SIGNAD conducted within the scope of this research.

Fig. 19.

spectral analysis of optimal PIDF DCM Boost chopper.

Table 3.

Transient response parameters of Boost system with different controllers.

PARAMETERS	PI [10]	PID [11]	FoPI [10]	N-FoPI [10]	COPI [11]	FOPID [11]	PID [28]	PIDF [29]	OPTIMAL PIDF CONTROL DCM BOOST (present study)
Overshoot (%)	12,89	21	9,92	7,68	7,4	8,2	0	3	5
Rise time (ms)	149,8	0,5	89,3	53	0,5	0,5	2,5	0,8	0,3
Settling time (ms)	183	69	116	53,5	50	67	–	–	85
Static error	–	–	–	–	–	–	0	0	0
Setting controller	–	–	–	–	–	–	Successive	Successive	Direct procurement
Robustness	–	–	–	–	–	–	Acceptable	Good	Great under 50 % load variation

All the previous research on enhancing the parameters of the PID controller in the Boost chopper class has exclusively utilized a PWM signal. Meanwhile, we can compare the standard PID controller with enhanced versions proposed in Refs. [10,11] and The pioneering work that has demonstrate the feasibility of using the DCM for Boost choppers that of [28,29] at this level it would be judicious to compared this work with the new approach developed to optimize the characteristics of the PIDF controller using the LQR proposed in this article. Table 3 provides a comparison of the Boost Converter Response using different controllers.

This comparative Table 3 showcases the performance metrics of different control methodologies applied to a specified system. The proposed methodology, exhibits superior performance with minimal overshoot at 5 %, the lowest rise time at 0.3 ms, and an excellent settling time of 85 ms. Remarkably, this method achieves zero static error, underscoring its precision and efficiency. The robustness of the proposed system is also noteworthy, although the exact robustness measure isn't specified. These results suggest that our methodology not only meets but surpasses the performance of traditional PID and other controller settings, making it a compelling choice for systems where precision and speed are paramount.

4. Conclusion

The study's findings unequivocally demonstrated the best PIDF regulator's ability to control a Boost chopper's functioning using duty cycle modulation (DCM). With its capacity to provide both accuracy and stability in steady-state operation and a strong reaction during transient load fluctuations, the PIDF regulator performs much better than typical PID controllers, particularly in situations where load fluctuates quickly.

The study has a broad range of real-world applications, especially in power electronics where Boost Converters are crucial for applications including industrial automation, electric cars, and renewable energy systems. These systems may operate more efficiently and last longer if the ideal PIDF regulator can reduce output variations and enhance power supply dependability.

The durability of the ideal PIDF regulator designed for the controlled DCM Boost chopper is confirmed by the results achieved in the frequency domain, particularly a much lower SINAD. Similarly, the self-adjusting frequency of the DCM or even the extended DCM, namely the NDCM mentioned in Ref. [30], might be an attractive way to lower the rate of harmonic distortion.

In the future, we will concentrate our research efforts on a few major topics. In order to confirm our simulation findings and further optimize the PIDF control settings to better suit real-world scenarios, we initially want to carry out hardware implementation experiments. This is an essential step in turning theoretical discoveries into workable solutions. In order to automatically optimize control settings and adjust to changing environmental circumstances without requiring human recalibration, we also want to include machine learning methods.

Additionally, more research will look at how well this control strategy works with other kinds of converters and if it can be scaled for higher-power applications. We want to contribute to the creation of more robust and effective energy systems that are better able to satisfy the changing needs of contemporary technology applications by expanding the focus of our research into these fields.

Funding

This research received no external funding.

Data availability

We did not use any data in this study.

CRediT authorship contribution statement

Ella NKOUNA. Paul Lionnel: Writing – original draft, Visualization, Software, Methodology, Formal analysis, Conceptualization. Arnaud Obono Biyobo: Conceptualization, Formal analysis, Investigation. Paul OWOUNDI. Etouke: Conceptualization, Software, Writing – original draft. Yves Paulin Dangwe Sounsoumou: Conceptualization, Resources, Software, Writing – original draft. Reagan Jean Jacques Molu: Methodology, Writing – review & editing. Serge Raoul Dzonde Naoussi: Supervision, Validation, Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.