Fuzzy Modelling Algorithms and Parallel Distributed Compensation for Coupled Electromechanical Systems

Abstract

Modelling and controlling an electrical Power Generation System (PGS), which consists of an Internal Combustion Engine (ICE) linked to an electric generator, poses a significant challenge due to various factors. These include the non-linear characteristics of the system’s components, thermal effects, mechanical vibrations, electrical noise, and the dynamic and transient impacts of electrical loads. In this study, we introduce a fuzzy modelling identification approach utilizing the Takagi–Sugeno (T–S) structure, wherein model and control parameters are optimized. This methodology circumvents the need for deriving a mathematical model through energy balance considerations involving thermodynamics and the non-linear representation of the electric generator. Initially, a non-linear mathematical model for the electrical power system is obtained through the fuzzy c-means algorithm, which handles both premises and consequents in state space, utilizing input–output experimental data. Subsequently, the Particle Swarm Algorithm (PSO) is employed for optimizing the fuzzy parameter m of the c-means algorithm during the modelling phase. Additionally, in the design of the Parallel Distributed Compensation Controller (PDC), the optimization of parameters pertaining to the poles of the closed-loop response is conducted also by using the PSO method. Ultimately, numerical simulations are conducted, adjusting the power consumption of an inductive load.

1. Introduction

Electric Power Generation Systems (PGSs) based on Internal Combustion Engines (ICEs) in combination with an electric generator are widely used in industrial and commercial applications due to their low maintenance and ease of operation under several environmental conditions. However, the low energy efficiency of ICEs reduces the performance of the whole power generation system. In recent years, many researches have proposed new strategies to optimize the results of this type of power generation systems. Such research includes the development of new fuels and explosion chamber systems with new materials for ICE operation, electrical generators with new architectures to maximize fuel efficiency, and voltage and power regulation systems with control algorithms using hybrid automatic identification and control techniques [1].

In order to optimize these systems, different strategies have been addressed, such as the use of innovative control techniques [2,3] or the study of the compartmentalization of electric power generation systems with different topologies of artificial neural networks to predict and be efficient in the complicated characteristics of various types of engine with different fuels [4].

The authors in [5] reviewed typical applications of machine learning (ML) and data-driven control (DDC) at the level of monitoring, control, optimization, and fault detection of power generation systems, with a particular focus on finding out how these methods can be used to evaluate, counteract, or withstand the effects of the associated uncertainties. The benefits of ML and DDC techniques are accordingly interpreted in terms of visibility, manoeuvrability, flexibility, profitability, and safety. The authors in [6] provide an overview of artificial-intelligence (AI) applications for power electronic systems. The applications of four categories of AI are discussed, which are expert system, fuzzy logic, metaheuristic method (like PSO), and machine learning. In [7], the authors discussed the design choices and parameters that affect the system performance, the closed-loop stability, and the controller robustness. Moreover, they presented the solvers and control platforms that can be employed for the real-time implementation of Model Predictive Control (MPC) algorithms. Furthermore, in [8], a comprehensive review is conducted on artificial-intelligence applications in regards to optimization system configuration and energy control strategy, along with the applicability of different energy storage technologies.

Most of the control techniques used for voltage regulation require a mathematical model of the ICE coupled to the electric generator. However, obtaining a complete mathematical model is a challenging task due to the interrelation between all the system components and the corresponding determination of its parameters. In order to deal with this issue, the authors in  [9] proposed the use of a non-linear mathematical model, as well as the combination of the first principle of thermodynamics, physical laws, and steady state data of the engine mass air flow, obtained empirically. This type of approach typically results in inaccurate models due to unmodelled system dynamics and, in addition, the resulting complex mathematical models are not suitable to be used to derive voltage regulation control laws.

One of the major potential areas of application of this type of power generator system is autonomous vehicles. For instance, the authors in [10] developed an unmanned military action vehicle, which uses a power generator system composed of a gasoline engine together with an electric generator and a lithium iron phosphate cell. Similarly, the American company Top Flight created the Airborg H8 10k [11], a hybrid powered UAV with eight rotors and an estimated flight time of two hours with a 4 kg payload or one hour of flight with a 10 kg payload, with a maximum speed of 40 mph. In addition, Sullivan UV is one of the leading companies that develop power generator systems in the expanding market of unmanned aerial vehicles [12].

1.1. Related Work

Techniques based on machine learning have proven to be effective approaches for modelling and controlling ICEs, due to their robustness, which allows compensating variations in the electrical load and whose approach guarantees stability [3]. Nevertheless, artificial-intelligence techniques require significant amounts of data to achieve a good model; in addition, they are merely black boxes, in the sense of interpreting the system’s parameters and its mathematical structure. This contrasts with fuzzy models and controllers, such as Mamdani linguistic systems, where an expert in the physical system dictates the basis of operation rules, which are not optimal, but rather based on human experience. On the other hand, the fuzzy modelling and control technique proposed by Takagi–Sugeno (T–S) can be complemented with optimal parameter identification algorithms, as well as with different control techniques. In the Mamdani fuzzy model, both the antecedent and the consequent are fuzzy propositions, which are chosen by experts on the system to be modelled or by an expert in fuzzy modelling, while in the T–S fuzzy modelling, the antecedent is a fuzzy proposition and the consequent is a mathematical function, which can have a polynomial state-space architecture [13]. This represents a strength for the modelling of complex systems and control based on submodels represented in state space. For instance, the authors in [14] presented a fuzzy logic-based control and power management strategy for unmanned aerial vehicles, in which they developed a set of fuzzy rules for controlling and effectively dividing the power between the two power sources: electric generator and ICE. Similarly, the authors in [15] used a fuzzy logic controller to improve the performance of the traditional perturb and observe algorithm for power generation using a photovoltaic system. The proposed fuzzy logic controller with a rule base collection of 25 rules outperformed the controller using the traditional perturb and observe algorithm due to its adaptive step size, enabling the fuzzy logic controller to adapt the photovoltaic system faster to changing environment conditions.

However, the selection of the fuzzy parameter m of the fuzzy c-means algorithm is a crucial aspect in the implementation of this type of techniques. The authors in [16] found that the value of m must be greater than $n/(n-2)$ (where n is the total number of data objects in the data set) viewed from the perspective of the algorithm convergence. More recently, the authors in [17] proposed, based in their experience, a range of $1.1\le m\le 5$, whereas [18] proposed a new guideline for m selection based on robust fuzzy c-means analysis. They suggested to define m in the range [1.5–4]. The authors in [19] proposed a method for the selection of the optimal m value using four cluster validity indices, obtaining an optimal m value in the interval [2.5–3]. The authors in [20] analyzed the use of soft computing techniques based on the c-means fuzzy classification algorithm and the T–S type fuzzy rule structure in research developed in the field of ICE for the generation of electric power. In [21], the authors proposed a novel semi-supervised fuzzy c-means clustering algorithm using multiple fuzzification coefficients instead of an auxiliary matrix to adjust the membership grade of the data elements, aiming to make the proposed fuzzy c-means algorithm a semi-supervised machine-learning method. The authors demonstrated the convergence of the algorithm and validated the efficiency of the method through a numerical example and derived a supervised approach to determine the parameter m of the c-means algorithm.

In our previous work [22], we used the fuzzy T–S technique to estimate the remaining battery energy in an autonomous vehicle. We used the fuzzy c-means algorithm for the assumptions and the consequents were obtained using linear polynomials, while the m parameter used in the fuzzy c-means algorithm was found by implementing the Particle Swarm Optimization (PSO) algorithm. The PSO metaheuristic algorithm has been previously used in several research works in combination with fuzzy logic techniques; for example, the authors in [23] developed a time-delay network control model to predict the output of the control system. In order to deal with the problem of falling into local optimality, they proposed a non-linear fitting formula to obtain the PSO algorithm parameters based on the number of iterations and using the T–S algorithm to obtain a global optimal solution. Similarly, the authors in [24] employed a fuzzy logic control structure for load frequency control of a dual-area power system, in addition to the PSO method to improve its performance.

In the present research work, the technique proposed by Takagi–Sugeno and Tanaka is used to identify a power generation system based on an internal combustion engine–generator scheme controlled using a fuzzy strategy that allows the regulation of the voltage generated by the system. With this strategy, it is sought to avoid the need to derive a complex mathematical model. The contribution of the proposed strategies is not in the development of a new fuzzy control law, but rather in the development of system identification and control algorithms for a specific application to a power generation system.

1.2. Contributions

The main contributions of this research work are summarized in the development of the following two algorithms:

A1.

An algorithm that uses input–output data corresponding to the control signal sent to the Internal Combustion Engine (ICE) and the voltage signal acquired from the electric generator. This algorithm aims to identify offline a T–S type fuzzy model of the power generation system, where the consequent consists of four state-space fuzzy rules. The model consists of four linear submodels, which represent the input–output operation within the learning space. In order to derive the fuzzy model, the consequents are established using the fuzzy data clustering technique, namely the fuzzy c-means algorithm. This method enables an optimal search for the premises of the fuzzy rules. In this technique, an open problem is to determine the fuzzy parameter m, which allows classification of the data into groups with fuzzy affinity. In the present paper, we propose to employ the Particle Swarm Optimization (PSO) method to identify the optimal value of the fuzzy parameter m from within the search space.

A2.

An algorithm that uses Parallel Distributed Compensation (PDC) to regulate the output voltage, whose gains are determined through the combination of pole placement and PSO methods. This control technique, rooted in the T–S fuzzy model, leverages the premises of the rules and adjusts the consequents using state feedback.

The remainder of the article is organized as follows: In Section 2, the description of the power generation system is presented. In Section 3, the T–S fuzzy modelling algorithm with the assumptions in the form of clustering and linear consequents, as well as the construction of the fuzzy system, applied to the power generation system, are derived. The PDC controller for the voltage regulation of the system is shown in Section 4. Finally, the conclusions and future directions of this research work are presented in Section 5.

2. Simulation Platform

To obtain the required experimental data, the system is considered an open-loop Power Generation System (PGS) in series configuration with an electric generator powered by an Internal Combustion Engine (ICE) [25]. In this configuration, the generated voltage is measured in open loop to first identify a fuzzy model of the power generation system and then design a Parallel Distributed Compensation Controller (PDC) that regulates the generated voltage in closed loop.

The PGS is composed of a myRIO-1900 device from National Instruments used for data acquisition and processing. Both the generator/electric motor and the electric motor used as an electrical load are U12 II (120 KV) brushless motors, while the Electronic Speed Controller (ESC) is a 100 Amp Flame HV, all from T-Motor. The servomotor is a DS3230MG, which is 30 kg/cm. The three-phase diode rectifier is 100 amps and 1600 volts. The ICE used is the Stihl-2MIX model, which is a 1.1 HP/0.8 kW and 22.2 cm³ single-cylinder engine, whose operation is two-stroke. We used a resistive voltage sensor to measure the output voltage of the PGS.

A connection diagram of the proposed power generation system is shown in Figure 1, where $r\left(k\right)$ is the desired voltage, $k_r$ is a precompensation gain, $e\left(k\right)$ is the voltage error, $u\left(k\right)$ is the obtained control signal, and $VG\left(k\right)$ represents the generated voltage, and the variable load block corresponds to an inductive load used to produce a richer system behaviour in open loop in order to improve the off-line identification process.

3. T–S Fuzzy Modelling

Two main topics of research and application for the development of fuzzy systems, known as Mamdani and Takagi–Sugeno, are highlighted in the bibliography [26]. The If–Then fuzzy rules contain two fundamental parts, which are the premises and the consequents. In this work, we demonstrate a fuzzy algorithm with the T–S fuzzy system, where the premises are extracted with the clustering technique, which consists of grouping the learning data into subsets based on their similarity to each other. Since the fuzzy c-means algorithm was developed for pattern recognition by [16], it has been of great technical value for the fuzzy modelling and control of non-linear dynamic systems, as presented in this work. We use the c-means clustering algorithm to extract the premises or firing degrees of the fuzzy T–S rules, because it is effective and feasible in the search for the best prototypes of the membership values; Equation (1).

The iterative fuzzy c-means algorithm is applied to find the clustering centres, $c_i$, which are the prototypes of the membership functions and the membership values, ${\mu }_{ik}\in [0,1]$, in the learning phase [27].

$$J(\mathbf{Z};\mathbf{U},\mathbf{C})=\sum _{i=1}^c\sum _{k=1}^N{\left({\mu }_{ik}\right)}^m{∥z_k-c_i∥}_I^2.$$

(1)

where N is the pairs of input–output data used in order to perform learning, c is the number of clusters proposed by the expert designer using a certain heuristic rule (in this paper, we set $c=3$ to illustrate the example of the fuzzy identification, modelling, and control algorithm), $\mathbf{Z}=\left\{z_1,z_2,\dots ,z_N\right\}$ is the learning matrix, which can be a time series of input–output measurements, $\mathbf{U}=\left[{\mu }_{ik}\right]$ is the fuzzy partition matrix, with $i=1,2,\dots ,c$ and $k=1,2,\dots ,N$, and $\mathbf{C}=[c_1,c_2,\dots ,c_c]$ is the vector of centres to identify. Normally, the parameter fuzzy $m>1$ defines the fuzziness, adjusted heuristically or with some genetic algorithm, and has an important relevance for the representation of the premises in the fuzzy rules. Here, we propose the PSO algorithm to find the best prototypes of the membership functions within the learning space.

Several classification and modelling problems in hard sciences and in different topics have been solved efficiently using the fuzzy c-means algorithm and its different variants. In [27], the authors detail the steps to follow in order to implement the c-means algorithm; the Euclidean distances of each sample of the data to be classified from the centres is as follows:

$$D_{ikI}^2={∥z_k-c_i∥}_I^2={(z_k-c_i)}^{T}I(z_k-c_i),$$

(2)

where I is the identity matrix. In the learning phase, the solution of Equation (1) for the membership functions of each cluster is given by Equations (3) and (4) and is, iteratively:

$${\mu }_{ik}={\displaystyle \frac{1}{{\sum }_{j=1}^c{\left(\frac{D_{ikI}}{D_{jkI}}\right)}^{(2/m-1)}}}$$

(3)

$$c_i={\displaystyle \frac{{\sum }_{k=1}^N{\left({\mu }_{ik}\right)}^m{z}_k}{{\sum }_{k=1}^N{\left({\mu }_{ik}\right)}^m}},$$

(4)

where ${\mu }_{ik}$ is the firing value of the i-th cluster for the k-th sample, with $1\le i\le c$, $1\le k\le N$.

The construction of the T–S fuzzy model from experimental data was solved in two steps: (1) the structure identification and (2) the estimation of parameters [28]. In step (1), the structure of the model is proposed, using expert knowledge and observations of the response of the system to be modelled.

Based on the model structure already defined, the learning matrix is defined for the learning matrix for parameter estimation, using time sequences of the input–output $X\in {\mathbb{R}}^{N\times n}$, where n is the number of state-space variables, as follows,

$$X=\left[\begin{array}{cccc}{x}_{11}& x_{12}& \cdots & x_{1n}\\ x_{21}& x_{22}& \cdots & x_{2n}\\ ⋮& ⋮& \cdots & ⋮\\ x_{N1}& x_{N2}& \cdots & x_{Nn}\end{array}\right]$$

and the output vector $Y\in {\mathbb{R}}^{N\times 1}$, $Y={\left[y_1,\cdots ,y_N,\right]}^T$; using Equation (5), the solution of the weighted average least squares, the identification of the linear consequent parameters $\widehat{{\theta }_i}$ for each i-th rule,

$$\widehat{{\theta }_i}={\left[X^T{\Gamma }_{i}X\right]}^{-1}{X}^T{\Gamma }_{i}Y,$$

(5)

is defined by the matrix ${\Gamma }_i=$ diag $\left({\mu }_{ik}\right)$$\in {\mathbb{R}}^{N\times N}$, which contains in its main diagonal the firing value of the i-th fuzzy rule, which gives it a weighting due to the membership values of each rule $(i=1,\dots ,c)$ and each instant $(k=1,\dots ,N)$. A fuzzy T–S model is proposed (6), based on 4 dynamic linguistic response labels within the discourse universe: low, zero, medium, and high, $R_i$, $i=1,2,3,$ 4 rules. The premises are represented by clustering centres; a discrete-time of the state-space model is proposed [29]:

$${\mathrm{R}}_i:\mathrm{If}{z}_k\mathrm{is}{c}_i\mathrm{then}\left\{\begin{array}{c}{x}^i(k+1)=A_{i}x\left(k\right)+B_{i}u\left(k\right)\\ y^i\left(k\right)=C_{i}x\left(k\right)\end{array}\right.$$

(6)

where $x^i(k+1)$ is the state vector at time $(k+1)$, $A_i\in {\mathbb{R}}^{n\times n}$ is the state matrix of the system, $x\left(k\right)\in {\mathbb{R}}^n$ is the state vector at time $\left(k\right)$, $B_i\left(k\right)\in {\mathbb{R}}^{n\times m}$ is the input matrix of the system, $C_i\left(k\right)\in {\mathbb{R}}^{q\times n}$ is the output matrix, $u\left(k\right)\in {\mathbb{R}}^m$ is the input vector of the system, and $y^i\left(k\right)\in {\mathbb{R}}^q$ is the output vector. The overall output of the fuzzy rule base (defuzzification) is the weighted sum of the firing values of the fuzzy rules [28], given as:

$$x(k+1)={\displaystyle \frac{{\sum }_{i=1}^c{\mu }_{ik}\left\{A_{i}x\left(k\right)+B_{i}u\left(k\right)\right\}}{{\sum }_{i=1}^c{\mu }_{ik}}}.$$

(7)

3.1. Fuzzy Identification of a Second-Order System

It is well known that any real-time system identification technique is commonly affected by uncertainties and identification errors. In fact, if multiple runs of an identification process applied to exactly the same model were performed, we would end up obtaining similar but slightly different results [30]. For this reason, we decided to use fuzzy logic to determine a model that was sufficiently close to the physical system and that would allow us to perform the desired voltage regulation despite the inherent errors and uncertainties.

Since the central axis of the modelling with fuzzy logic lies in the knowledge base of the system to be modelled, we rely on the observation of the operation of the power generation system of this research work, and the experimental measurements of the input–output, to propose four fuzzy rules.

Thus, the grouping of the power generator operation dynamics was divided into four clusters, using the learning data $Z=\left[VG\right(k-1);VG(k);u(k\left)\right]$. These clusters correspond to the time series of the voltage measurements of the electrical generator and the excitation of the system, which are respectively in variables of state $x_1$ and $x_2$ and the input signal $u\left(k\right)$.

The centroids of the prototype membership functions were determined using the c-means algorithm.

Table 1. Centres of the prototype membership functions.

Centroids	${\mathit{x}}_1$	${\mathit{x}}_2$	u
1	45.5960	45.5858	0.7158
2	38.8837	38.8710	0.6258
3	21.2521	21.2416	0.3369
4	25.3004	23.9919	0.4860

presents the 4 coordinates of the centroids, which were obtained in Matlab^® using the fcm() function inside the search loop for the best parameter m with Algorithm 1. These values correspond to the time series of the measurements of the voltage of the electric generator and the PWM excitation, which are, respectively, in the identification of the states $x_1$, $x_2$, and $u\left(k\right)$ control signal, as shown in Figure 1.

Algorithm 1 Fuzzy system identification and m parameter optimization.

$VG,u$← Load open_loop_experimental_data.mat $Z\leftarrow \left[VG\right(k-1),VG(k),u(k\left)\right]$ $c\leftarrow 4$                     ▹ Number of rules or clusters $N\leftarrow$ Total number of data function fuzzy c-means($Z,c$)     Calculate the centroids $\mathbf{c}$ end function return $\mathbf{c}$ for i ← 1 to c do                  ▹ Obtaining of consequents     $A_i,B_i\leftarrow$ Equation (5)                      ▹ Least squares end for return $A_i,B_i$ function PSO Algorithm                  ▹ Open Loop System     for m← 1.1 to 3 do                     ▹ Fuzzy Parameter m         Optimize m using Equation (10)            ▹ Objective function     end for end function return m

Four clusters and their corresponding fuzzy rules, which map fuzzy linear subspaces, were proposed. Each clustering operation is defined by locally mapping the system of equations into state-space variables; Equation (6). The global model is the defuzzification of the rules, which infers the non-linear response. The non-linearities of the ICE driver are represented within the ensemble of the four T–S fuzzy rules; Equation (7).

Figure 2 shows the dynamics of operation, due to the excitation input and the response in the state variables, as well as the clustering centres.

The proposed state-space structure, Equation (8), for each of the four rules locally does not represent all the effects acting on the system, but globally the rule base integrates the non-linearity of the identified system.

$$\left[\begin{array}{c}{x}_1^i\left(k+1\right)\\ x_2^i\left(k+1\right)\end{array}\right]=\underset{A_i}{\underbrace{\left[\begin{array}{cc}0& 1\\ a_2^i& a_1^i\end{array}\right]}}\left[\begin{array}{c}{x}_1\left(k\right)\\ x_2\left(k\right)\end{array}\right]+\underset{B_i}{\underbrace{\left[\begin{array}{c}0\\ b^i\end{array}\right]}}u\left(k\right)$$

(8)

where $a_1^i$, $a_2^i$, and $b^i$ are the linear parameters that are identified with the weighted least squares algorithm.

Table 2. Resulting values for $A_i$ and $B_i$.

$A_1$		$B_1$	$A_2$		$B_2$
0	1	0	0	1	0
0.8465	$-0.0828$	14.9186	1.0667	$-0.1268$	3.7456
$A_3$		$B_3$	$A_4$		$B_4$
0	1	0	0	1	0
0.9635	$-0.07516$	7.0516	0.9589	$-0.1102$	9.4053

shows the obtained values of $A_i$ and $B_i$ for the four identified discrete systems.

In order to obtain the required data to conduct the system identification, we conducted open-loop experiments under different scenarios. Figure 3 depicts the open-loop system response in contrast to the output obtained from the identified fuzzy system (identified voltage) with a multiple step input. The identified voltage is smooth with respect to the actual signal from the electrical generator (generated voltage), mainly due to the linear nature of the fuzzy rule consequents, as shown in Equation (6).

Figure 4 depicts the open-loop system response in contrast to the output obtained from the identified fuzzy system with a constant input and an inductive load used to produce a richer system behaviour in open loop in order to improve the off-line identification process.

To validate the discordance of the model, the Quality of Fit (QoF) was calculated by dividing the Mean Squared Error (MSE) of the identified model by the Mean Squared Output (MSO) (9), which must be much smaller than 1 [31], resulting in a QoF $=6.1264\times {10}^{-4}$

$${\displaystyle \frac{MSE}{MSO}}={\displaystyle \frac{\frac{1}{N}\left|\right|\theta X-{Y\left|\right|}^2}{\frac{1}{N}{\left|\right|Y\left|\right|}^2}}$$

(9)

3.2. Optimization of the Fuzzy Parameter m

To improve the approximation of the T–S fuzzy rule base with the input–output learning data, it is proposed to use the PSO algorithm for the optimal search of the m parameter of the c-means fuzzy algorithm. In addition, the PSO algorithm is also used for the optimal selection of the poles of the closed-loop system of the fuzzy rules defined in Equation (6), with a state feedback control structure for each of the fuzzy rules. In this search for the best parameters within the learning space with the input–output measurements, the PSO algorithm was used, as shown in [32,33].

The proposed cost functions for the optimization problem should be defined according to the characteristics of the system one intends to work with. In our experiment, the cost function of Algorithm 1 is proposed in order to minimise the error of the system identification. On the other hand, the corresponding number of clusters in Algorithm 1 was defined based on the open-loop response of the PGS. Since it presented four operational set points, as depicted in Figure 3, we decided to use four clusters in order to capture most of the system behaviour. This behaviour in the PGS is due to the hysteresis phenomena introduced by the servomotor used to control the ICE’s accelerator.

In Algorithm 2, the cost function was defined based on the desired transient response. In this sense, the system poles of the considered electromechanical system were defined in the positive real axis inside the unite circle, since the desired system response should be similar to the response of a first-order system. For a different system, the location of the desired closed-loop poles and the number of clusters/rules will depend on the characteristics of the system, as well as on the desired system response.

Algorithm 2 Pole placement optimization and PDC control.

function PSO Algorithm                  ▹ Close Loop System     for $p_i\leftarrow 0.01$ to $0.9$ do                        ▹ Poles $p_i$         Optimize $p_i$ using Equation (12)              ▹ Objective function     end for end function return $p_i$ Calculate $K_i$ for each $c_i$ using pole assignment function PDC Control($A_i,B_i,\mathbf{c},c,K_i,m,x_1\left(0\right),x_2\left(0\right)$)     for i ← 1 to N do         Calculate Euclidean distances using (2)         Calculate the membership degrees using (3)         Calculate the strategy control using (14)         Perform defuzzification using (7)     end for end function

The cost function employed in the PSO to optimize the parameter m, shown in Equation (3), in Algorithm 1 is based on the RMSE between the measurement obtained from the PGS and the output of the identified fuzzy model, and is given as follows:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{{\sum }_{k=1}^N{\left(VG\left(k\right)-x_2\left(k\right)\right)}^2}{N}}}$$

(10)

where $VG\left(k\right)$ is the value of the physical system output and $x_2\left(k\right)$ is the output (state variable) of the system identified through fuzzy modelling.

Table 3. Parameters used in the PSO algorithm and obtained value for the parameter m.

Optimal Value m	Bounds	# of Particles	Inertia Coefficient $\mathit{\omega }$	${\mathit{\varsigma }}_1,{\mathit{\varsigma }}_2$	# of Iterations
2.97	$[1.1,3]$	300	0.0917	0.5959	100

shows the values used in the PSO algorithm, as well as the search values assigned to the m parameter and the obtained optimal value of m.

To obtain the value of the constants ${\varsigma }_1$ and ${\varsigma }_2$, parameter $\xi$ is calculated as:

$$\xi ={\displaystyle \frac{2\kappa }{\left|2-\varphi -\sqrt{{\varphi }^2-4\varphi }\right|}}$$

(11)

where $\varphi ={\varphi }_1+{\varphi }_2$. The constants ${\varphi }_1$ and ${\varphi }_2$ are bounded as $\varphi \ge 4$ and $\le \kappa \le 1$, whose values are defined in [34] for the PSO algorithm. Thus, ${\varsigma }_1$ and ${\varsigma }_2$ are obtained as ${\varsigma }_1=\xi {\varphi }_1$ and ${\varsigma }_2=\xi {\varphi }_2$.

Algorithm 1 shows the procedure used to obtain the parameter m as well as to obtain the identified system based on the number of defined clusters/rules.

3.3. Closed-Loop System Poles Optimization

For the optimization process of the poles of the closed-loop system, we used the following cost function:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{{\sum }_{k=1}^N{\left(r\left(k\right)-x_2\left(k\right)\right)}^2}{N}}}$$

(12)

where $r\left(k\right)$ is the desired voltage reference and $x_2\left(k\right)$ is the state variable, which approximates in a fuzzy way the voltage generated in closed loop. As explained before, in order to obtain the best poles that minimises the system response guaranteeing that the poles remain on the real axis inside the unit circle, the search limits were proposed in the interval $[0.01,0.9]$.

Table 4. Parameters used in the PSO algorithm and obtained values for the poles.

Optimal Poles	Bounds	# of Particles	Inertia Coefficient $\mathit{\omega }$	${\mathit{\varsigma }}_1,{\mathit{\varsigma }}_2$	# of Iterations
$0.3$, $0.8$	$[0.01,0.9]$	300	$0.3820$	$0.9549$	100

shows the values used in the PSO algorithm, as well as the values obtained for the closed-loop poles.

In order to highlight the advantages in the implementation of the proposed PSO algorithms, we used the index errors based on the Integral of the Squared Error (ISE), the Integral of the Absolute Value of the Error (IAE), and the Integral of the Absolute Value of the Error multiplied by Time (ITAE).

Table 5. Error indexes using the algorithm PSO and without the algorithm PSO.

	ISE	IAE	ITAE
	${\sum }_{k=0}^N{e}^2\left(k\right)T_s$	${\sum }_{k=0}^N\left|e\left(k\right)\right|T_s$	${\sum }_{k=0}^{N}k\left|e\left(k\right)\right|T_s$
With PSO	2.6677$\times {10}^3$	51.6501	8.02$\times {10}^5$
Without PSO	6.9970$\times {10}^6$	2.6452$\times {10}^3$	2.1061$\times {10}^9$

Where $N=301$ is the total number of samples, $T_s=1$ s is the sampling time, and $e\left(k\right)$ is the system output tracking error.

depicts the obtained values from the three indexes applied to the simulation scenarios when the PSO algorithm was used and when it was not employed.

From Table 5, we can see that the error indexes are bigger when the PSO algorithm is not used. This is due the fact that the value of the fuzzy parameter has to be proposed based on the designer expertise at the time of conducting the fuzzy modelling by trial and error. Furthermore, since the optimal location of the system poles is obtained with the PSO algorithm, the system performance is improved

4. PDC Fuzzy Control

To achieve the regulation of the T–S system, the control technique based on the distributed parallel compensator was used. In this control strategy, the consequent of each fuzzy rule of the model (6) is replaced by a control law with the form of linear state feedback ([29,35]). In the PDC approach with pole assignment, the eigenvalues of $(A_i-B_i{K}_i)$ are proposed. The consequent for each PDC fuzzy rule is given by:

$$u_i\left(k\right)=-K_{i}x\left(k\right)$$

(13)

The global output of the PDC, in other words, the fuzzy aggregation of all control rules, has the following form:

$$u\left(k\right)={\displaystyle \frac{{\sum }_{i=1}^c{\mu }_{ik}{u}_i\left(k\right)}{{\sum }_{i=1}^c{\mu }_{ik}}}$$

(14)

where $K_i$ is the vector of gains of the i-th rule, as shown in

Table 6. Defined gain values for each submodel.

Gains	Values
$K_1$	$0.0448,-0.0559$
$K_2$	$0.2130,-0.2118$
$K_3$	$0.0529,-0.0632$
$K_4$	$0.0720,-0.0809$

, and $k_r=1.38$ is a precompensation gain, as shown in Figure 1. For obtaining the PDC gains, the method by pole assignment was used, as obtained by the PSO algorithm.

Figure 5 shows the simulation results of the voltage regulation at different reference values, in which it can be seen that the PDC controller regulates the generated voltage, defined by reference steps of 30, 40, and 25 volts. These reference values are typical voltage values in applications that employs LiPo batteries. In such applications, 25, 40, and 30 volts are nominal voltages of LiPo batteries with different numbers of cells.

Furthermore, Figure 5 shows the control signal applied to the PGS, which is bounded to values in a range of $[0,1]$. This variable regulates the amount of gasoline in the explosion chamber by manipulating the throttle unit (carburettor) with a servomotor. It is seen that the PDC fuzzy controller regulates the voltage in the simulation test to the reference voltage value.

Algorithm 2 describes the required steps for the optimization of the controller’s gains as well as the implementation of the PDC controller. The fuzzy model tracking of the closed-loop power generation system achieves a QoF $=3.4763\times {10}^{-4}$, given by Equation (9). The QoF is much less than 1, even though the PGS is subjected to noise in its generated voltage, natural dead zones of the control actuator in the throttle body (carburettor servo motor), and non-linear dynamically varying electrical disturbances (load).

Comparison of Implementing a PDC vs. PD Controller

In order to highlight the advantages of using the proposed strategies, we simulated a classic PD controller and compared its performance with the one obtained with the PDC strategy. We can see from Figure 6 that the PD controller is not able to reach the desired voltages at different step inputs due to the fact that its gains can only be tuned for a specific region of operation. When the desired voltage is defined at a different region of operation, the proposed PDC controller uses the obtained firing value ${\mu }_{ik}$ from the fuzzy strategy to modify the corresponding controller’s gains. It is worth mentioning that this type of electromechanical systems has different regions of operation due to its intrinsic characteristics. These characteristics are given by the non-linear nature and complexity of the system due to its different components, such as dead zones stemming from the actuator utilized in the internal combustion engine’s accelerator, voltage fluctuations, and non-linearities within the electric load and its associated electronic speed controller.

5. Conclusions

The proposed algorithms based on the fuzzy c-means technique, along with the PDC controller and the PSO metaheuristic optimization method, demonstrated effectiveness in modelling, identifying, and controlling a complex system. This capability was verified through numerical tests conducted on the obtained power generator system model.

Taking advantage of experimental data from the open-loop plant, the optimization strategy utilizing the PSO algorithm enabled the determination of the optimal fuzzy parameter m for the c-means algorithm. It was demonstrated that, by optimizing the value of m, the learning data pertaining to the premises of fuzzy rules are classified with greater similarity. This leads to an enhanced interpretation and reduced approximation error of the fuzzy model. Additionally, it facilitated the selection of poles within the unit circle for each closed-loop rule’s consequent, aiding in the fine-tuning of fuzzy controllers via state feedback.

By utilizing input and output data, the process of identifying a fuzzy model for the power generation system bypasses the necessity to derive and identify a mathematical model through conventional techniques. This is particularly advantageous given the significant challenge posed by the complexity of the entire system, which encompasses both an internal combustion engine and a mechanically coupled electric generator.

The comparison of the proposed control strategies with a classical control law demonstrated that classical controllers are not able to reach the desired voltages at different step inputs due to the controller’s gains only being tuned for a certain region of operation. However, when the desired voltage is required to be defined at a different region of operation, the proposed PDC controller modifies the corresponding controller’s gains thorough the obtained firing value, ${\mu }_{ik}$, from the fuzzy logic strategy.

In our upcoming research work, we aim to deploy the developed algorithms onto a physical power generation system. This step is crucial to validate their efficacy in addressing the inherent complexities characteristic of such systems. These complexities include dead zones stemming from the actuator utilized in the internal combustion engine’s accelerator, voltage fluctuations, and non-linearities within the electric load and its associated electronic speed controller.

The obtained quality-of-fit indexes between the regulated voltage and the identified voltage are $QoF=3.4763\times {10}^{-4}$ and $QoF=6.1264\times {10}^{-4}$, respectively. For this index, values much smaller than 1 are considered a good performance.