Geostatistical modelling of soil contamination with arsenic, cadmium, lead, and nickel: the Silesian voivodeship, Poland case study

1.   Introduction

Power converters are increasingly employed in numerous applications, such as industrial electronics, portable devices, energy storage systems, and consumer electronics. In recent applications, such as artificial intelligence, networks related to the Internet of Things, and wireless sensors, power converters are critical components. Switch-mode DC-DC converters transfer the DC voltage from one level to another. High-level DC voltage is reduced to a low-level DC voltage by a DC-DC buck converter, which is widely used in modern-era applications such as photovoltaic systems, electric vehicles etc.

Regulating the output voltage of DC-DC converters in the presence of external noise and system parameter variations has proven to be challenging. Rigorous investigations are still ongoing in this subject, and different control algorithms have been proposed for controlling the output voltage of the converter. Chebyshev's neural network-based adaptive controller was designed to control the output voltage of a buck converter ^[1]. In another work, a new control algorithm was proposed to track the output voltage of a buck converter with overcurrent protection to hardware elements ^[2]. In Miao (2019) and Wang (2019), current-constrained controllers were proposed for the buck converter ^[3,4]. A linear matrix inequality-based H-infinity controller has been demonstrated to regulate the output voltage of buck and boost converters ^[5,6]. An artificial neural network (ANN)-based control algorithm was proposed for a DC-DC buck converter, and an optimal controller was designed using approximate dynamic programming ^[7]. Several observer-based control algorithms have been proposed, both for buck and boost converters ^[8,9,10,11]. A data-driven control algorithm was suggested for the interleaved DC-DC boost converter based on online integral reinforcement learning ^[12]. In 2022, the non-minimum-phase model of a boost converter was modified to a minimum-phase model for the purpose of controller design ^[13].

The proportional-integral-derivative (PID) controller and its modified versions are widely used in different applications. A sigmoid function-based data-driven proportional-integral (PI) controller was proposed to control the angular speed of a DC motor that is energized by a DC-DC converter ^[14]. Here, a sigmoid function represents the limits of the parameters of the PI controller. In ^[15], a neuroendocrine PID controller was designed for a multivariable crane system where the hormone discharge rate of neuroendocrine-PID can be regulated as per the error change. The output voltage of the DC-DC boost converter has been tracked using a brain emotional learning-based intelligent controller (BELBIC), where parameters are determined using the particle swarm optimization technique ^[16]. The performance of this controller outperformed the classical PID controller. Fractional-order proportional-integral (FOPI) and fractional-order proportional-integral-derivative (FOPID) controllers are very popular in industrial applications ^[17,18,19]. The FOPID controller is designed to stabilize the output power of the buck-boost converter for proton exchange membrane fuel cells ^[20]; an optimal FOPI controller was proposed for a solar photovoltaic system with a DC-DC boost converter ^[21]. The FOPID controller was also implemented in a real-time system, like a DC-DC boost converter and a magnetic levitation system ^[22,23]. A modified version of the FOPID controller, namely the tilt integral derivative (TID) and hybrid proportional integral derivative (PID) controllers, was proposed for load frequency control in multi-area interconnected power systems and second-order time delay systems, respectively ^[24,25].

Though different control techniques have been proposed for tracking the output voltage of the DC-DC converter, the Kalman filter-based control algorithm has not been addressed before. The measurement of the output voltage of a DC-DC converter is essential for control purposes. However, measured outputs may be affected by the sensor noise. To deal with model uncertainty, noise must be considered. As a result, this research work designed a Kalman filter (KF) along with a controller to reduce the effects of sensor and process noises on the converter output. Modified versions of PID controllers have been employed for DC-DC converters, but some complications limit their widespread use. Instead, fractional-order PID controllers and variants have shown superior control abilities. Fractional-order proportional-integral (FOPID) controllers suffer from tuning complexity, and TID controllers are sensitive to parameter variations. To avoid these problems, a KF-based new controller, named blended proportional-integral (BPI) controller, is proposed to track the output voltage of the DC-DC buck converter. The BPI controller is a combination of FOPI and integer-order integral (I) controllers.

For parameter determination of the PID and FOPID controllers, optimization techniques are essential; for that purpose, metaheuristic algorithms like particle swarm optimization (PSO), artificial bee colony (ABC), genetic algorithm (GA), or firefly algorithm (FA) play a critical role ^[26]. The PID, FOPID, and FOPI parameters have been determined via the PSO technique ^[27], and a modified version of the sine-cosine algorithm was employed to estimate the PID controller parameters ^[28]. In ^[29], six parameters of a complex controller, which was a combination of the PID controller with an N-filter and a PD controller, were predicted using the golden eagle optimization technique. A new metaheuristic optimization technique, which is a combination of pattern search with opposition-based snake optimizer, was introduced and utilized to calculate the parameters of two cascaded first-order low-pass filters with the PID (PID-FF) controller ^[30]. Due to fast convergence rate, ease of implementation, and simplicity, the firefly algorithm (FA) was applied to estimate the parameters of the proposed BPI controller, which was introduced by Xin-She Yangin, 2008 ^[31]. This FA was then modified by different researchers and has been used in different applications for optimization purposes ^{[32,33,34,35]}.

The contributions of the proposed work are presented below.

●  A new Kalman filter-based controller is designed for the DC-DC buck converter. To control the output voltage of the converter, a blended proportional-integral (BPI) controller is proposed, which is the fractional order proportional-integral (FOPI) plus integer integral (I) controller.

●  The parameters of the suggested controller are estimated using four metaheuristic optimization techniques: firefly algorithm (FA), artificial bee colony (ABC), particle swarm optimization (PSO), and Harris Hawks optimization (HHO). The FA-based controller outperforms the other three metaheuristic algorithms.

●  The Kalman filter minimizes the effect of sensor and process noises on the output of the system, while the Kalman filter-based BPI controller tracks the output voltage of the converter with excellent transient response specifications in the presence of sensor and process noises. When the performance of the suggested BPI controller is compared with the PI, PID, FOPI, and FOPID controllers, BPI outperforms the others. The output of the closed-loop system is also compared with the artificial neural network-based controller presented by Dong et al., 2021 ^[7]; the proposed controller outperforms this controller.

2.   Materials and methods

2.1.   Dynamics of DC-DC buck converter

Figure 1 represents the circuit diagram of the buck converter. The circuit elements include the controlled switch MOSFET (${S}_{Bu})$, diode (${D}_{Bu})$, inductor (${L}_{Bu}$) with internal resistance ${R}_{L}$, and capacitor (${C}_{Bu}$) with internal resistance ${R}_{C}$, load resistance $R$, and DC input voltage ${V}_{in}$. The converter circuits with controlled switch (${S}_{Bu}$) during on and off conditions are also shown. The parameter values for the buck converter circuit are given in Table 1 and are based on ^[7].

Figure 1.  Buck converter circuit (a); on-mode circuit (b); off-mode circuit (c).

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Table 1.  Values of different parameters of the buck converter ^[7].

Parameter	Value
Input voltage, ${V}_{in}$ (𝑉)	42
Load resistance, $R$ ($\mathrm{\Omega }$)	10
Internal resistance of inductor, ${R}_{L}$ ($\mathrm{\Omega }$)	0.3
Internal resistance of capacitor, ${R}_{C}$ ($\mathrm{\Omega }$)	0.02
Inductance, ${L}_{Bu}$(m𝐻)	5.63
Capacitance, ${C}_{Bu}$ ($\mu F$)	5

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The voltage current of the converter during the on state is represented by the following equations:

where ${i}_{L}$ and ${v}_{c}$ are the inductor current and capacitor voltage, respectively, and ${v}_{o}$ is the output voltage.

The equations below characterize the converter during the off state:

Thus, the average model of the converter is given by the following equations:

Here, $D$represents the duty ratio (control input), which is provided by the pulse width modulator (PWM), and$D\in [0, 1]$.

The state-space model of the converter is illustrated below:

2.2.   Kalman filter-based fractional-order controller

The Kalman filter-based controller is used to regulate the output voltage of the DC-DC buck converter. The parameters of the proposed BPI controller are determined by applying the metaheuristic firefly optimization technique. The block diagram of the buck converter with the Kalman filter-based controller is shown in Figure 2. The Kalman filter eliminates the effects of sensor and process noise on the output voltage, which is used for comparison with the reference voltage. The control input $u\left(t\right)$ is applied to the buck converter through the PWM.

Figure 2.  Buck converter with the Kalman filter-based BPI controller.

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2.2.1.   Kalman filter

The Kalman filter identifies the state variables of the system from the input and output data. This estimated output is used for control purposes. R. E. Kalman proposed the idea of the Kalman filter in 1960 ^[36]; since then, the Kalman filter and its modified versions have been used extensively in real-time systems for state estimation ^{[37,38,39,40,41]}.

In discrete time domain, the state-space model of a linear time invariant is represented as

Here, $k$ represents the time instant; $x\in {\mathbb{R}}^{{n}_{x}}$, $u\in {\mathbb{R}}^{{n}_{u}}$, and $y\in {\mathbb{R}}^{{n}_{y}}$ are the state, input, and measurement vectors, respectively; ${y}_{t}$ is the output without sensor noise; ${w}_{k}$ and ${n}_{k}$ are the process and sensor noise, respectively.

It is assumed that process (${w}_{k})$ and sensor (${n}_{k}$) noises are Gaussian white noises with zero-mean and are mutually uncorrelated signals. The process and sensor noise covariance matrices are $E\left({w}_{k}{w}_{k}^{T}\right) = Q$ and $E\left({n}_{k}{n}_{k}^{T}\right) = R$, respectively. The algorithm of the Kalman filter is given in Table 2. The Kalman filter estimates the state variables of the system from the information of input and noisy output of the system. It estimates the state variables by minimizing the mean squared error $E\left[\left({x}_{k}-{\widehat{x}}_{k}\right){\left({x}_{k}-{\widehat{x}}_{k}\right)}^{T}\right] = {P}_{K}$ where ${\widehat{x}}_{k}$ is the prior estimated state vector and ${P}_{K}$ is the error covariance matrix at $k$. These estimated states are free from sensor and process noise and are used for control purposes. Accordingly, the Kalman filter minimizes the effects of noise on the output of the system. The algorithm of the Kalman filter is given in Table 2 ^[37].

Table 2.  Kalman filter (KF) algorithm.

Step 1: Initialization (k = 0)	Mean value of states: ${\widehat{x}}_{0}=E\left({x}_{0}\right)={x}_{in}$ Error covariance matrix: ${P}_{0}=E\left[\left({x}_{0}-{\widehat{x}}_{0}\right){\left({x}_{0}-{\widehat{x}}_{0}\right)}^{T}\right]={P}_{in}$
Step 2: Prediction (time update)	Predicted state vector: ${\widehat{x}}_{k+1\mid k}=A{\widehat{x}}_{k\mid k}+B{u}_{k}$ A priori covariance matrix: ${P}_{k+1\mid k}=A{P}_{k\mid k}{A}^{T}+{Q}_{w}$
Step 3: Correction (measurement update)	The updated state vector: ${\widehat{x}}_{k+1\mid k+1}={\widehat{x}}_{k+1\mid k}+{K}_{k+1}\left[{y}_{k+1\mid k}-C{\widehat{x}}_{k+1\mid k}\right]$ Posteriori covariance matrix: ${P}_{k+1\mid k+1}=(I-{K}_{k+1}C){P}_{k+1\mid k}$ The Kalman filter gain: ${K}_{k+1}={P}_{k+1\mid k}{C}^{T}{({Q}_{v}+C{P}_{k+1\mid k}{C}^{T})}^{-1}$ The algorithm returns to step 2.

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It is challenging to select the process noise covariance matrix $Q$ as it depends on model uncertainty. For this, $Q$ is chosen by the trial-and-error method as $diag(0.00001, 0.00001)$. The sensor noise covariance ($R$) is chosen as 0.1, which depends on the accuracy of the voltage sensor. The Kalman filter is implemented using MATLAB software, and the state estimation error covariance matrix becomes $\left[\begin{array}{cc}2.728\times {10}^{-7} & 2.725\times \times {10}^{-8}\\ 2.725\times {10}^{-8} & 2.975\times {10}^{-9}\end{array}\right]$. The resulting Kalman filter estimates the state variables ${i}_{L}$ and ${v}_{c}$ accurately, and accuracy is computed in terms of root mean square error (RMSE) given by $RMSE = {\left(\frac{\sum _{k = 1}^{{T}_{s}}{\left({x}_{1}-{\widehat{x}}_{1}\right)}^{2}}{{T}_{s}}\right)}^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}$, where ${x}_{1}$ and ${\widehat{x}}_{1}$ are the actual and estimated state variables, respectively. The RMSE values of state variables ${i}_{L}$ and ${v}_{c}$ are 0.1261 and 0.2167, respectively. The covariance of the measured error (error between ${y}_{t}$ and measured output, $y$) and the estimated error (error between ${y}_{t}$ and estimated output, $\widehat{y}$) are 0.0994 and 0.0465, respectively. From these covariances, it can be concluded that KF minimizes the effect of noise on the output of the system.

2.2.2.   Blended proportional-integral (BPI) controller

Linear controllers, as the integer and fractional-order proportional-integral (PI) and proportional-integral-derivative (PID) controllers, are used extensively for the fulfillment of the transient as well as steady-state response specifications of the system. The transfer function of PID and fractional-order PID (FOPID) controllers is as follows ^[17,18].

where ${k}_{p}$, ${k}_{d}$, and ${k}_{i}$ are the gain of the proportional, derivative, and integral controllers, respectively; $\lambda$ and $\mu$ are real fractional numbers ($0\le (\lambda, \mu)\le 1$).

The FOPID has more control on the system than PID, since the former has five control parameters, as opposed to the three from PID. Still, the issue lies in optimizing these five parameters in FOPID instead of the three in PID.

For this study, a new fractional-order controller is proposed, which is a combination of integer and fractional-order PI controllers and is named the blended proportional-integral (BPI) controller. The structure of the controller is shown in Figure 3, and the transfer function of the controller is given by

where ${k}_{p}$ is the gain of the proportional controller, ${k}_{i}$ is the gain of the integer-order integral controller, ${k}_{f}$ is the gain of the fractional-order integral controller, and the exponent $\lambda$ is a real number ($0\le \lambda \le 1$).

Figure 3.  Structure of BPI controller.

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The number of parameters that need to be optimized for the BPI controller is four, which is less than the FOPID (five parameters). The merit of the BPI controller over other fractional-order controllers like FOPID and TID is that it has no derivative term. Actually, the derivative term is required to damp out oscillations, but it can strengthen noise if parameter estimations are incorrect. The results section proves that BPI presents robust performance when facing parameter variations and noise.

2.2.3.   Firefly algorithm for the design of BPI controller

The firefly algorithm (FA) is an extremely popular metaheuristic optimization technique that takes inspiration from nature and is extensively used for solving engineering problems. In this algorithm, the objective function changes proportionally with the brightness of the firefly; this means that when brightness is reduced, the firefly will travel or search for a brighter firefly. If there is no brighter firefly, then the firefly will move aimlessly.

Consider there are n fireflies where $f\left(d\right)$ represents the fitness function or objective function. The brightness of a firefly $\left({I}_{i}\right)$ determines the current position ${d}_{i}$. The distance between two fireflies is given ^[31,32] by

where ${d}_{i}$ and ${d}_{j}$ represent the position of i^th and j^th fireflies, respectively.

The relationship between the attractiveness function $\beta$ and distance $r$ is $\beta = {\beta }_{0}{e}^{-\gamma {r}^{2}}$, where ${\beta }_{0}$ signifies the attractiveness at $r = 0$ and the light absorption coefficient is denoted by $\gamma$.

Suppose that in a t-th iteration, the firefly $i$ at position ${d}_{i}$ is moved to a brighter firefly $j$. Now, the new position of firefly $i$ in $(t+1)$-th iteration is expressed by the following equation ^[33,34]:

where ${d}_{i}\left(t\right)$ represents the old position of firefly i; ${ϵ}_{i}$ is a random number that can be determined from Gaussian, uniform, or any other distribution; $\alpha$ is called the randomization parameter $\left(0\le \alpha \le 1\right)$.

The pseudocode of FA is illustrated in Table 3. Optimization problems like extremely nonlinear and multi-modal can be handled efficiently using FA; its speed of convergence is also extraordinary.

Table 3.  Algorithm of the firefly optimization technique.

Pseudocode of firefly algorithm (FA)

• Input: Number of fireflies (n), no. of maximum iterations (MaxIter), attractiveness factor (${\beta }_{0})$, randomization parameter ($\alpha$), light absorption coefficient ($\gamma$). • Output: Best result originated by the FA. 1.       Initialization of n, MaxIter, ${\beta }_{0}$, $\alpha$, $\gamma$. 2.       Determine initial population of fireflies ${d}_{i}$, where $i=\mathrm{1, 2}, \dots, n$. 3.       Define objective function $f\left(d\right)$ where $d={[{d}_{1}, \dots, {d}_{n}]}^{T}$. 4.       Calculate light intensity ${I}_{i}$ for firefly at ${d}_{i}$ using $f\left({d}_{i}\right)$ for all fireflies Main Loop 5. While ($t < MaxIter$)             For $i=1\;\mathrm{t}\mathrm{o}\;n$             For $j=1\;\mathrm{t}\mathrm{o}\;n$             if $({I}_{j} > {I}_{i})$             Travel firefly i towards firefly j according to equation (9)                                 end if                                 Evaluate new $f\left({d}_{i}\right)$ and light intensity ${I}_{i}$                             end for $j$             end for $i$ Rank the fireflies and determine the best global solution                 end while

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For the optimization, the fitness function or cost function is considered as integral time absolute error (ITAE), which computes the integral of the product of the absolute value of difference between reference input and actual output and time. The ITAE is given by

where ${v}_{r}$ and ${\widehat{v}}_{o}$ are the reference voltage and estimated output voltage by Kalman filter respectively.

One of the key objectives of this research work is to minimize the time response specifications, like maximum peak overshoot, settling time, and steady-state error. The cost function ITAE fulfils these requirements better than other cost functions, like integral square error (ISE) and integral absolute error (IAE). The design procedure of the BPI controller using the FA algorithm is presented in Figure 4.

Figure 4.  Design procedure of the BPI controller using the FA algorithm.

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3.   Results

The values of different parameters used for the buck converter are presented in Section 2.1 ^[7]. The transfer function of the considered DC-DC buck converter is$G\left(s\right) = \frac{(148.9s+1.489\times {10}^{9})}{({S}^{2}+2.002\times {10}^{4}s+3.652\times {10}^{7})}$.

For the BPI controller, four parameters, ${k}_{p}$, ${k}_{i}$, ${k}_{f}$, and $\lambda$ need to be optimized using the optimization technique FA. The lower limit (LL) and upper limit (UL) of these four parameters are chosen from experience as follows:

Applying FA, the gains of the controller BPI are optimized considering the cost function ITAE, ISE, and IAE. The best cost function versus the number of iteration curves is shown in Figure 5a for 25 iterations.

Figure 5.  Fitness function vs. number of iteration curve (a) for different cost functions and (b) for different controllers.

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In this case, PI, PID, FOPI, and FOPID controllers are also designed for the buck converter using FA to compare the potential of BPI. Table 4 represents the values of the parameters of the different controllers. The objective function ITAE vs. the number of iterations (25 iterations) of different controllers is shown in Figure 5b; it shows that, among all controllers, BPI provides the minimum value of ITAE. The fractional-order controller blocks are obtained using the FOMCON toolbox. Figure 6 represents the closed-loop output voltage and inductor current responses of the buck converter with Kalman filter-based controllers for an 18 V reference voltage. Performance measures of different controllers are given in Table 5.

Table 4.  Parameters of different controllers obtained using different metaheuristic algorithms.

Optimization technique	Control algorithm	Value of the parameters
		${k}_{p}$	${k}_{i}$	${k}_{d}$	${k}_{f}$	$\lambda$	$\mu$
	PI	0.001	9.8729	-	-	-	-
	PID	0.0998	10	0.0362	-	-	-
	FOPI	0.0012	9.9803	-	-	0.999	-
FA	FOPID	0.001	9.9317	0.0157		0.999	0.001
	BPI	0.0017	9.99	-	5.9858	0.999	-
ABC	BPI	0.0112	10	-	6	0.999	-
PSO	BPI	0.001	10	-	6	0.999	-
HHO	BPI	0.09	10	-	6	0.999	-

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Figure 6.  Output voltage (a) and inductor current (b) of the buck converter with different controllers.

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Table 5.  Performance measures of different controllers using different metaheuristic algorithms.

	Time response specifications	Settling-time ($\mathit{s})$	Maximum overshoot	Under shoot	Integral time absolute error (ITAE)	Integral square error (ISE)	Integral absolute error (IAE)
FA-based	PI	0.0095	0	0	9.108×10-5	0.4736	0.04471
	PID	0.001	4.54%	0	0.0007329	0.05703	0.03065
	FOPI	0.0095	0	0	9.125×10-5	0.4636	0.04412
	FOPID	0.02	0	0	0.0001613	0.3343	0.04433
	BPI	0.0055	0.06%	0	3.101×10-5	0.3177	0.02766
ABC-based	BPI	0.01	0%	0	4.672×10-5	0.2337	0.02756
PSO-based	BPI	0.0051	0.236%	0	3.044×10-5	0.3261	0.02774
HHO-based	BPI	0.028	0%	0	0.0001776	0.08423	0.02754

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To test the performance of the Kalman filter, outputs are collected for a reference voltage of 18V with a sensor noise power of 0.1 and process noise power of 0.000001. The output voltage and inductor current of the buck converter with a KF-based BPI controller and without KF (only with BPI controller) are shown in Figure 7.

Figure 7.  Output voltage of the closed-loop system with and without the Kalman filter.

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3.1.   Comparison with other metaheuristic algorithms

Besides the firefly algorithm (FA), three other metaheuristic algorithms [artificial bee colony (ABC), particle swarm optimization (PSO), and Harris Hawks optimization (HHO)] were used to estimate parameters (${k}_{p}$, ${k}_{i}$, ${k}_{f}$, and $\lambda$) of the BPI controller. The algorithms of these optimization techniques are given in the literature ^[42]. The estimated parameters of the BPI controller using these four optimization techniques are presented in Table 4. Performance measures of different optimization techniques for the KF-based BPI controllers are given in Table 5 and shown in Figure 8.

Figure 8.  Performance measures of different algorithm-based BPI controllers with KF.

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The statistical analysis of the fitness function (ITAE) with different optimization algorithms is presented in Table 6. As the metaheuristic algorithms FA, PSO, ABC, and HHO are stochastic in nature, the best value, standard deviation, and mean value of ITAE are calculated for 15 individual trials. The p-value of Wilcoxon's rank-sum test is also calculated and provided in Table 6. Wilcoxon's test, a nonparametric test, is used to compare two paired groups. A p-value lower than 0.05 means there is no variance between the two compared groups.

Table 6.  Statistical analysis of fitness function (ITAE) for different optimization algorithms.

Algorithm	Best	Standard deviation	Mean	Wilcoxon's rank-sum test p-value FA (BPI) vs
FA (BPI)	1.75× 10-5	1.81548× 10-6	1.88× 10-5
ABC (BPI)	1.77× 10-5	6.36946× 10-6	2.54× 10-5	6.0111× 10-4
HHO (BPI)	1.78× 10-5	4.55625× 10-6	2.34× 10-5	0.0012
PSO (BPI)	1.77× 10-5	5.10655× 10-6	2.30× 10-5	0.0047

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3.2.   Comparison with other control techniques

For comparison with the proposed KF-based BPI controller, an artificial neural network (ANN) -based controller, suggested by Dong (2021), was considered. The same system is considered, and simulation results are compared in both cases. The KF-based BPI controller is tested with the same load and parameter variations as mentioned in ^[7]. Responses under various conditions are illustrated in Figure 9, and performance measures are given in Table 7.

Figure 9.  Output voltage and inductor current responses when (a) load changes from 7.33 to 11 Ω; (b) reference voltage varies from 18 to 24 V; (c) input voltage ${V}_{in}$ changes from 42 to 47 V; (d) ${L}_{Bu}$ decreases by 50% with the same load change as in (a); (e)${L}_{Bu}$ decreases by 85% with the same load change as in (a).

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Table 7.  Performance comparison with the ANN-based controller.

Changed condition	% Overshoot		Settling time (s)
	ANN controller	Proposed controller	ANN controller	Proposed controller
Load resistance $R$ changes from 7.33 to 11 Ω	38.89%	1.39%	0.002	0.002
Reference voltage ${v}_{r}$ varies from 18 to 24 V	0%	0%	0.007	0.005
Input voltage ${V}_{in}$ is changed from 42 to 47 V	4.44%	17.2%	0.006	0.0035
Inductance ${L}_{Bu}$ decreases by 50% and load resistance $R$ changes from 7.33 to 11 Ω	38.89%	1.39%	0.005	0.003
Inductance ${L}_{Bu}$ decreases by 85% and load resistance $R$ changes from 7.33 to 11 Ω	44.44%	1.66%	0.0017	0.0018

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Table 6 shows that the proposed BPI controller performs better than the ANN-based controller in terms of overshoot and settling time. Some ringing (oscillations) can be seen in both voltage and current waveforms for the ANN-based controller, which is not present at all in the proposed BPI controller.

4.   Discussion

The best cost function values for IAE, ISE, and ITAE were 0.011533, 0.010004, and 1.75 × 10^-5, respectively. From the convergence curve (Figure 5a) and these cost values, it is evident that ITAE provides better results than the other cost functions.

From Table 5, it is clear that the proposed BPI controller demonstrates better performance in terms of settling time, ITAE, and IAE than the other designed controllers. Figure 6 shows that the output voltage with the KF-based BPI controller accurately tracks the reference voltage of 18 V, with a negligible amount of overshoot in both the output voltage and inductor current waveforms. Hence, the BPI controller is considered for the regulation of the output voltage of the buck converter.

From Figure 7, it is clear that the converter outputs with the KF-based BPI controller are not influenced by the sensor and process noise; Additionally, the controller provides overcurrent protection to the hardware components of the buck converter. This indicates that the KF eliminates the influence of sensor and process noise on the system output.

Table 5 shows that the ABC and HHO-based BPI controllers are capable of producing zero overshoot, but the settling time are higher than with the FA and PSO-based controllers. Also, the maximum peak overshoot of the PSO-based controller is greater than that of the FA-based controller. By comparing these performance measures, it is clear that the FA-based BPI controller outperforms the other three metaheuristic algorithm-based controllers. From Table 6, it is clear that the FA-based BPI controller provides the lowest best value, standard deviation, and mean value compared to the other techniques.

Figure 9 shows the responses of the KF-based BPI controller under various parameter variations. A performance comparison between the proposed controller and the ANN-based controller ^[7] is presented in Table 7. The BPI controller improves maximum overshoot by 96.42% compared to the ANN controller under load resistance variations.

5.   Conclusions

This study describes the design of a new Kalman filter-based blended proportional-integral (BPI) controller and demonstrates the regulation of the output voltage of a DC-DC buck converter. The responses with and without the Kalman filter reveal that it eliminates the effect of sensor and process noises on the output voltage. The proposed fractional-order KF-based BPI controller gives better time response with respect to settling time, maximum overshoot, and ITAE than other KF-based controllers like PI, PID, FOPI, and FOPID. Statistical analysis shows that the performance of the proposed FA-based controller is superior to that of other metaheuristic algorithms. A comparative study also shows that the KF-based BPI controller outperforms other control techniques like ANN: the peak value of the voltage and transient time is much lower for the BPI controller than the ANN controller under different load conditions and parameter variations. The BPI controller decreases the maximum overshoot by 96.26% when the inductance of the buck converter is reduced by 85% under a load change from 7.33 Ω to 11 Ω. Hence, the proposed controller is capable to provide excellent transient responses and robust performance. In the future, the proposed control algorithm may be experimentally validated with a real-time system.

Author contributions

Debarchita Mishra: Conceptualization, simulation, writing – original draft; Sharmistha Mandal: Analysis, drafting and editing of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

On behalf of all authors, the corresponding author declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.