A multi-objective indirect neural adaptive processes control design for minimization of energy consumption: An experimental validation on a transesterification reactor

Abstract

Nowadays, energy management environment is a very important issue that technologies have focused on in order to save costs and minimize energy waste. This objective can be achieved by means of an energy resource management approach through an appropriate optimization technique. However, energy savings can conflict with other objective functions, to solve the problem a multi-objective optimization that considers the minimization of the control energy can be adopted. In this paper, we propose to use a multi-objective indirect neural adaptive control concept for nonlinear systems with unknown dynamics. The control scheme consists of an adaptive instantaneous neural emulator and neural controller built on fully connected real-time recurrent learning networks (RTRL). The multi-objective particle swarm optimization (MOPSO) algorithm is used as a mechanism to find NE and NC adaptive learning rates that optimize the proposed objective functions: control energy minimization and closed-loop preservation. Comparative studies and experimental validation on a semi-batch reactor, used to generate cleaner biofuels, are performed to confirm the effectiveness of the proposed strategy.

Keywords

indirect adaptive neural control MOPSO algorithm nonlinear systems control energy

1. Introduction

In previous years, the monitoring of industrial processes was principally based on keeping a stable operation at all desired operating conditions and minimizing the effect of external disturbances. To reach these goals, classical controllers have been successfully employed. But nowadays, with the globalization of markets, the evolution of customers: needs and demands and the increasing social attention to environmental issues linked to production processes, industries are also obliged to develop control strategies to stay both competitive and profitable. It is then needed to consider an optimal control with a multi-objective criterion.

The field of multi-objective optimization control has attracted considerable interest in scientists motivated by real-world engineering problems (Hong et al., 2021). It involves optimizing two or more conflicting objectives at the same time, under certain constraints.

Based on the non-probabilistic multi-objective optimization, Yang and Xia (2022) introduced a two-step strategy to select the optimum displacement sensor configurations in structural health monitoring. The non-dominated sorting genetic algorithm is used first to get the preliminary Pareto front. Then, to determine the final sensor configuration, a new interval time series model is created using the ratio of reduced and full intervals. In Civera et al. (2021) and through optimal sensor placement (OSP) techniques, a multi-objective optimization approach and genetic algorithms (GAs) are used for damage detection after a potentially catastrophic event. The proposed approach ensures a higher level of robustness than standard procedures, where potential damage-induced structural changes are completely ignored in the OSP problem. In Ma et al. (2022b), authors applied a global multi-objective optimization, genetic algorithm, to the problem of energy saving and nitrogen oxides emission reduction of part-load gas turbines. The level of energy economy is evaluated by determining the cost of fuel. In Al Shidhani et al. (2020), a multi-objective optimization model integrating four objective functions is proposed to obtain the optimal combination of power generation expansion of a country by integrating sustainability indicators. It evolves minimizing total discounted costs, carbon emissions, land use, and social opposition. For the optimization of an air-conditioning system based on indoor thermal comfort, Haniff et al. (2019) proposed a new multi-objective particle swarm optimization (MOPSO) algorithm able to find better optimal solutions faster than its original version, the standard particle swarm optimization (PSO) algorithm. The indoor air temperature is optimized in such a way that the air-conditioning electrical power consumption is minimized while keeping the predicted mean vote (PMV) comfort index values around the PMV values that are targeted in the schedule.

Yang et al. (2023) also put forward a cuckoo optimization algorithm (COA) used to solve energy production cost minimization in a combined heat and power (CHP) generation system. The obtained results confirm the applicability of the COA and reveal the efficient and potential of the suggested framework for solving non-convex, non-smooth, and nonlinear real-world challenges. Abdelkader et al. (2018) used a genetic algorithm to optimize the sizing of an autonomous renewable energy in which two objective functions are used: the loss of power supply probability (LPSP) and the levelized cost of energy (LCE). The genetic algorithm is used to find the most appropriate configuration of the different subsystems: battery capacity, photovoltaic (PV) array capacity, and upper water tank capacity (Ghanjati and Tnani, 2022).

Atig et al. (2022) introduce a multi-objective indirect neural adaptive control concept for nonlinear multi-variable systems having unknown behaviors. A multi-objective criterion, which takes the minimization of the control energy into account, is considered by optimizing the neural controller adaptive rate using Lyapunov stability analysis and tracking error dynamics. The performance of the suggested multi-objective approach in terms of regulation and control energy minimization is tested both in simulation and in real time on a perturbed thermal process.

This paper proposes an indirect multi-objective neural adaptive control approach taking into account the minimization of the control energy and the tracking error. The control scheme consists of an adaptive instantaneous neural emulator and a neural controller with fully connected real-time recurrent learning networks (RTRL). The proposed NE and NC adaptive learning rates are developed to ensure that closed-loop performance targets are both economical and satisfactory. For this, MOPSO algorithm has been employed, where the designer is capable of choosing appropriate NE and NC whose parameters are determined by the trade-off between the different objectives: control energy and tracking error.

The main motivations of this paper are as follows: (1) to define a multi-objective criterion to design new NE and NC adaptive learning rates, ensuring satisfactory performance in terms of regulation, tracking, and minimization of control energy, (2) present a comparative study with existing methods, and (3) apply the multi-objective neural adaptive control approach in real time to a transesterification reactor.

This paper is planned as follows: Section 2 defines the indirect multi-objective neural adaptive control. The proposed algorithm MOPSO is developed in Section 3. Section 4 presents the results of a numerical simulation. The multi-objective neural adaptive control applied to a transesterification reactor is shown in Section 5. Section 6 concludes the paper.

2. Indirect multi-objective neural adaptive control

The basic indirect neural network control scheme consists of recurrent NE and NC based on fully connected real-time recurrent learning (RTRL) networks (Farhat et al., 2022). NE and NC parameters change with different dynamics in order to minimize the emulation and control instantaneous errors. Using an online adjustment of the adaptive learning rates of the NE and NC, the indirect neural adaptive structure helps the plant output to asymptotically approximate the desired signal and reduce the convergence time (Sardahi, 2016). In this paper, the adaptive rates are updated by the MOPSO algorithm to improve the control design by considering a multi objective criterion: control energy minimization and closed-loop performances preservation (Figure 1).

Figure 1.

Structure of an indirect neural adaptive control based on MOPSO algorithms.

2.1. Neural emulator

The NE is utilized to provide an emulation of the instantaneous outputs of the process. The adaptive algorithm defined in this study does not store the system dynamics. The NE parameters are used to adapt the controller in the tracking phase (Hong et al., 2019). Therefore, the size of such a network is only dependent on the number of inputs and outputs, and such a structure is small in comparison to other neural networks that can be found in the literature (Farhat et al., 2021).

The adaptation algorithm is based on the RTRL algorithm and the major benefit of such a method is that it does not require any prior knowledge of the dynamics. In continuous time, the dynamics of the NE neurons, shown in Figure (2), are defined by equation (1), for i = 1, …, 2N, in continuous time:

\frac{1}{|τ_{e} (t)|} \frac{d (s_{i} (t))}{d t} + s_{i} (t) = \tanh (\sum_{j = 1}^{N_{e}} w_{i j} (t) s_{j} (t) + u_{i} (t))

(1)

Figure 2.

Fully connected structure for the NE.

where s_i(t) is the i^th neuron state of the emulator and x_i(t) presents the input.

$x_{i} (t) = u_{i} (t) i f i \in \{1, \dots, N\}, x_{i} (t) = 0 i f i \in \{N + 1, \dots, 2 N\}$ , and w_ij(t) are the emulator weights and $1 / |τ_{e} (t)|$ presents the emulator time parameter. The outputs of the NE are described as follows: ${\hat{y}}_{i - N} (t) = s_{i} (t) i f i \in \{N + 1, \dots, 2 N\}$ .Our goal is to minimize the emulation error e_e(t) given by the following equation:

e_{e} (t) = \frac{1}{2} E_{e}^{T} (t) E_{e} (t)

(2)

where

E_{e} (t) = e_{e l} (t) \in R^{N \times 1}

is the error column vector for the emulation:

e_{e l} (t) = {\hat{y}}_{l} (t) - y_{l} (t)

. The adjustment of the NE weight is based on the gradient of the instantaneous error equation (3):

\frac{d w_{i j} (t)}{d t} = - |η_{e} (t)| \sum_{l = 1}^{N} ({\hat{y}}_{l} (t) - y_{l} (t)) P_{lij} (t)

(3)

where $η_{e} (t)$ is the emulator adaptive rate and $P_{lij} (t) = \partial {\hat{y}}_{l} (t) / \partial w_{i j} (t)$ are the NE sensitivity functions defined by equation (4):

\begin{gathered} \frac{1}{|τ_{e} (t)|} \frac{d P_{lij} (t)}{d t} = \frac{d \tanh}{d t} (\sum_{m = 1}^{N_{e}} w_{l m} (t) s_{m} (t) + x_{l} (t)) \\ \times (δ_{i}^{l} {\hat{y}}_{j} (t) + \sum_{m = 1}^{N_{e}} w_{l m} (t) P_{mij} (t)) - P_{lij} (t) \end{gathered}

(4)

with

δ_{i}^{l}

is the Kronecker symbol (Zerkaoui et al., 2009; Atig et al., 2010b).

2.2. Neural controller

A neural controller (NC) with a fully connected recurrent structure that resembles the NE structure is also considered (Figure (3)). This structure is composed of 2N neurons (Chen et al., 2022). The inputs of the NC are the N plant desired outputs y_ci and the N output error functions (y_ci − y_i) and the NC outputs are the control inputs u_i (Atig et al., 2010a).

The dynamic activation of neurons of the NC, in continuous time, is given by the following equation:

\frac{1}{|τ_{c} (t)|} \frac{d (o_{i} (t))}{d t} + o_{i} (t) = \tanh (\sum_{i = 1}^{2 N} ϕ_{i j} (t) o_{j} (t) + z_{i} (t))

(5)

with o_i(t) as the i^th neuron state of the NC, u_i(t) = o_i(t). If

i \in \{1, \dots, N\}

, ϕ_ij(t) is the controller weight from neuron j to neuron i and

1 / |τ_{c} (t)|

is the controller time parameter. z_i(t) = y_ci(t) − y_i(t) if

i \in \{1, \dots, N\}

z_{i} (t) = y_{c_{(i - N)}} (t)

i \in \{N + 1, \dots, 2 N\}

y_{c_{i}} (t)

is the desired output and y_i(t) is the system output.

For the NC, our objective is to minimize the tracking error e_c(t), given by equation (6):

e_{c} (t) = \frac{1}{2} E_{c}^{T} (t) E_{c} (t)

(6)

where

E_{c} (t) = (e_{c l} (t)) \in R^{N \times 1}

is the column vector of the following error:

e_{c l} (t) = y_{c_{l}} (t) - y_{l} (t)

Using equation (7), the NC weights are updated by the tracking equation:

\frac{d ϕ_{i j} (t)}{d t} = |η_{c} (t)| \sum_{l = 1}^{N} ((y_{c l} (t) - y_{l} (t)) \sum_{b = 1}^{N_{c}} Q_{bij} (t) J_{l b} (t))

(7)

Q_bij and J_lb present the NC sensitivity functions presented by relations (8) and (9), respectively.

Q_{bij} (t) = \frac{\partial o_{b} (t)}{\partial ϕ_{i j} (t)}

(8)

J_{l b} (t) = \frac{\partial {\hat{y}}_{l} (t)}{\partial o_{b} (t)}

(9)

The sensitivity function Q_bij(t) is given by the following equation:

\begin{matrix} \frac{1}{|τ_{e} (t)|} \frac{d Q_{bij} (t)}{d t} + Q_{bij} (t) = \frac{d \tanh}{d t} (\sum_{h = 1}^{N_{e}} ϕ_{b h} (t) o_{h} (t) + z_{b} (t)) \\ ({δ_{i}}^{b} o_{j} (t) + \sum_{h = 1}^{N_{e}} ϕ_{b h} (t) Q_{bij} (t) + \frac{\partial z_{b} (t)}{\partial ϕ_{i j} (t)}) \end{matrix}

(10)

J_lb(t) is computed as follows:

\begin{matrix} \frac{1}{|τ_{e} (t)|} \frac{d J_{l b} (t)}{d t} + J_{l b} (t) = \frac{d \tanh}{d t} (\sum_{h = 1}^{N_{e}} w_{l j} (t) s_{h} (t) + x_{l} (t)) \\ ({δ_{b}}^{l} + \sum_{h = 1}^{N_{e}} w_{l j} (t) J_{h b} (t)) \end{matrix}

(11)

Figure 3.

Fully connected structure for the NC.

2.3. Basic concept of MOPSO

The objectives to be optimized in multi-objective problems are generally in conflict with one another. As a result, in these problems, no unique solution can be reached. However, a set of Pareto-optimal that would represent the best possible trade-offs between the objectives can be provided for the optimization problem.

A standard problem with multiple objectives can be formulated as follows (Panichella et al., 2015):

Minimize: $\vec{f} (\vec{x}) = [{\vec{f}}_{1} (\vec{x}), {\vec{f}}_{2} (\vec{x}), \dots, {\vec{f}}_{i} (\vec{x})]$

subject to the constraint functions.

Minimize: $g_{i} (\vec{x}) ⩽ 0$ if: i = 1, 2, …, m

Minimize: $h_{i} (\vec{x}) = 0$ if: i = 1, 2, …, p

Here, $\vec{x} = [x_{1}, x_{2}, \dots, x_{n}]$ is the vector of the decision variables, $f_{i} (\vec{x})$ is the objective function, and $g_{i} (\vec{x}), h_{i} (\vec{x})$ are the constraint functions.

For two vectors x₁, x₂, in the feasible domain, if $\vec{x_{1}} ⩽ \vec{x_{2}}$ , then ${\vec{x}}_{1}$ dominates $\vec{x_{2}}$ . A vector of decision variables $\vec{x_{u}}$ is called non-dominated or Pareto-optimal with regard to a feasible region, if it does not exist, further $\vec{x_{v}}$ results $\vec{f} (\vec{x_{u}}) ≺ \vec{f} (\vec{x_{v}})$ . The Pareto front is constituted by the set of Pareto-optimal solutions.

2.4. General algorithm for PSO

The MOPSO algorithm is developed from single objective PSO algorithm. The standard PSO algorithm operates by updating the position of particles from a swarm to search the optimal solution (Kennedy and Eberhart, 1997). Each particle adapts its flying trajectory by continuously updating its position and velocity (Das et al., 2014; Ma et al., 2022a). Computed by the following relations:

v_{i d} (k + 1) = w v_{i d} (k) + C_{1} r_{1} (p_{i d} (k) - x_{i d} (k)) + C_{2} r_{2} (p_{g d} (k) - x_{i d} (k))

(12)

x_{i d} (k + 1) = x_{i d} (k) + v_{i d} (k + 1) 1 ⩽ d ⩽ D

(13)

where v_id is the velocity of the particle i in the D-dimensional search space, and w is the inertia weight that determines the influence of the velocity of the previous iteration. p_id is the personal best position of the i^th particle, p_g represents the global best position among all particles, C₁ and C₂ represent the learning factor, and r₁ and r₂ are two similar random numbers in [0, 1].

In each generation, the inertia factor w is reduced during a run from 1 to near 0 and it is determined as follows:

w = w_{\max} - \frac{w_{\max} - w_{\min}}{i t e r_{\max}} \times i t e r

(14)

where iter_max is the maximum number of iterations, and iter is the current number of iteration.

2.5. MOPSO algorithm

In the MOPSO algorithm, the particles are analyzed and the positions of the particles that are classified as non-dominated are saved in the archive. In addition, the search area which has been explored so far is partitioned into hypercubes and all particles are positioned in a hypercube according to the particle’s position in the objective area.

The steps of MOPSO algorithm are given in this table.

1: Create and initialize a swarm

2: while stopping condition has not been reached do

3: for each particle in the swarm, do:

4: Calculate new velocity

5: Calculate new position

6: Manage boundary constraint violations

7: Update archive

8: Update the particle’s allocation hypercubes

9: for each particle in the swarm, do:

10: Update pbest

3. The proposed algorithm

In the past years, optimal design methods for energy savings have used the single optimization objective. Although single-objective models can give decision makers with an overview, they generally cannot provide a set of alternative solutions that balance different objectives (Altiparmak et al., 2006). There is an urgent need to develop a multi-objective optimal design method for these systems. It is a complex multi-objective optimization problem with multiple variables and multiple constraints. Actually, the principal methods for solving this type of multi-objective optimization problems are multi-objective evolutionary algorithms. The main idea of multi-objective evolutionary algorithms based on the dominant relationship is to search all non-dominant individuals in the current population through the use of a Pareto-based fitness allocation theory. In this work, the proposed approach is developed to get sufficient conditions for the adaptive learning rates of the NE and NC. The adaptive rates need, at each time, to be updated by the MOPSO algorithm so as to optimize the proposed multi-objective criterion: control energy and tracking error minimization. Generally for the control of nonlinear systems, the energy gain is correlated with a considerable performance degradation in closed-loop performances. In this work, our objective is to define cost functions taking into account environmental and safety criteria and allow to minimize the control energy consumption with no significative degradation of the closed-loop system performances. The problem to be solved consists in computing the adaptive learning rates to meet objective functions in the best manner.

The proposed method for the online adjustment of the adaptive learning rates for the NE and NC comprises the following steps:Step1: Initialization of the NE (w_ij, P_lij), the NC (ϕ_ij, Q_bij), and the MOPSO parameters (v_bd, x_bd, w_max, w_min, iter_max).Step2: Compute the emulation e_e(k) and the tracking e_c(k) errors.Step3: Search optimal NE and NC adaptive learning rates that minimize simultaneously the control energy and the tracking error.Step4: Update the NE and NC parameters and calculate the control action, then stop, else go to step 2.

The different steps of the proposed real-time approach implementation can be also presented in the following Figure 4:

Figure 4.

Flowchart of the real-time implementation.

4. Simulation example

Energy consumption will remain a major concern in the process industries. In this section, the main goal is to evaluate the proposed strategy to improve the performance of the process from an energetic point of view. For this, consider a nonlinear system described by the following equation. The proposed system has a strong nonlinearity in its operating space (Farhat et al., 2022).

\begin{gathered} y (k) = 0.4 u (k - 1) + 0.12 y (k - 1) \\ + \frac{0.4 u (k - 1) y (k - 2) (u (k - 1) + 2.5)}{1 + u {(k - 1)}^{2} + y {(k - 2)}^{2}} \end{gathered}

(15)

The retained computing step ΔT is equal to 0.1 s (t = k ∗ ΔT). An external perturbation is applied in the regulation phase from 3000 s to 4000 s. Using the MOPSO, the NE and NC adaptive rates are updated in ways to minimize the proposed multi-objective functions: control energy and tracking errors.

Figure (5) gives the evolution of the desired and system outputs with the MOPSO algorithm. We remark that the proposed algorithm guarantees a faster output convergence. Also, in the presence of disturbance, we can note that the suggested control method can swiftly and exactly change the output of the system to the reference signal y_c(t).

The evolution of the control signal applied to the nonlinear system is displayed in Figure (6).

Figure 5.

Evolution of the desired and real outputs with the MOPSO algorithm.

Figure 6.

Evolution of the control signal.

The NC adapting rate η_c is displayed in figure (7). To evaluate the performance of the proposed algorithm, comparative studies with the method proposed in Yassin et al. (2023) are elaborated. In the proposed method, the PSO algorithm is considered. A single objective function (the tracking error) is used for optimizing NE and NC adaptation rates. As shown in figure (8), the system output cannot closely follow the reference signal throughout the entire operating space.

Figure (9) shows the evolution of the control signal nonlinear system with PSO.

The evolution of the PSO adapting rate η_c is shown in figure (10).

To evaluate the performance obtained with the proposed multi-objective approach, the instantaneous normalized mean square error (NMSE), control energy (CE), and energy gain (EG) relations are considered:

N M S E = \frac{\sum_{k = 1}^{N} {(y_{c} (k) - y (k))}^{2}}{\sum_{k = 1}^{N} y_{c} {(k)}^{2}}

(16)

E G = \frac{\sum_{k = 1}^{N} {(u_{a})}^{2} - \sum_{k = 1}^{N} {(u_{p})}^{2}}{\sum {(u_{a})}^{2}} \times 100

(17)

C E = \sum_{k = 1}^{N} \frac{u^{2} (k + 1)}{N}

(18)

u_p: the control value obtained with the MOPSO algorithm.u_a: the control value obtained the PSO algorithm.The NMSE and CE indices, calculated for different strategies, are summarized in Table 1.

Figure 7.

Evolution of η_c with MOPSO.

Figure 8.

Evolution of the desired and real outputs with the PSO algorithm.

Figure 9.

Evolution of the control signal.

Figure 10.

Evolution of η_c.

Table 1.

NMSE and CE with MOPSO and PSO algorithms.

	MOPSO	PSO
NMSE	0.0048	0.0061
CE	0.0081	0.0306

Compared to the PSO algorithm, the optimization achieved through MOPSO exhibited a decrease in NMSE and significant reduction in control energy. An important energy gain (EG = 73%) is achieved with no significant performance degradation in term of tracking.

5. A practical validation on a transesterification reactor

Chemical reactors are the most often used industrial processes in the field of pharmaceutical and chemical applications. Their flexibility and versatility have made them interesting for various applications in research. Without doubt, process nonlinearity and the energy control are the most relevant factors in characterizing process control problems. Batch reactors can be used to extract oils. Vegetable oils are mostly extracted and undergo reactions to generate fuels. In fact, the oils require little modifications like the transesterification reaction to generate cleaner biofuels. To speed up the transformation, a catalyst like a strong acid or base is required.

The chemical process in Figure (11) is a stirred tank equipped with a jacket where the heat exchange takes place between a cooling fluid and the reaction mixture. The flow rate of the heating and cooling reactor fluid is kept constant. The temperature of the fluid inside the jacket is monitored by an external control system comprising a plate heat exchanger equipped with electric resistors. The heating of the fluid is performed by electric resistances, while the cooling fluid is obtained by a plate heat exchanger.

Figure 11.

Transesterification reactor used for the experiments.

The fluid temperature within the jacket is regulated with an external servo system including a plate heat exchange with electric resistors.

The heating of the fluid is insured by electric resistors, whereas the cooling fluid which is also named tap water is provided by a plate exchanger. Many temperature sensors are employed to collect the temperature of chemical processes and the inlet and the outlet jacket temperature. The chemical reactor operates in batch mode in which the biodiesel is utilized. The transesterification reaction proceeds in several steps. Vegetable oils which are considered fatty materials (FM) are mixed with alcohol to give ester and glycerol.

The described reaction is represented as following equation:

F M + A l c o h o l ⇌ E s t e r s + G l y c e r o l

(19)

Biodiesels are recognized as an attractive alternative to bio-based fuel, which can compensate fossil fuel resources that will be expended in the nearest future.

As they are environmentally friendly, biodiesels have received more attention and have been developed for further application as a renewable energy source. Indeed, biofuels help to decrease global warming by reducing emissions of particulate matter, sulfur, and carbon dioxide. The studied system can be regarded as a single-input single-output one. The heating power Q and reactor temperature T_r are the input and the output of the reactor, respectively. The desired reference trajectory T_c(°C) is composed of three basic phases: heating, reaction, and cooling.

It is desired that the outlet temperature of the reactor T_r(°C) tracks the reference trajectory, minimizing power consumption. The energy consuming element is the electrical resistance. The experimental results are presented in Figure (12). This figure illustrates the evolution of the reactor output and shows the efficiency of the proposed MOPSO algorithm in determining appropriate NE and NC adaptive learning rates.

Figure 12.

Evolution of the desired and system outputs with MOPSO.

The reactor output can achieve the desired performance in the three considered phases (heating, reaction, and cooling). The evolution of the heating power Q (W) is presented in Figure (13). The evolution of the adaptive rate η_c(t) is illustrated in Figure (14).

Figure 13.

Evolution of the control signal Q(W) with MOPSO.

Figure 14.

Evolution of the controller parameter η_c(t) with MOPSO.

Figure (17) presents the results when the PSO algorithm with a single optimization function (the tracking error) is used as a mechanism to optimize the adaptive learning rates of the NE and NC. In this situation, the reactor outlet can follow the desired signal in all operating modes (heating, reaction, and cooling) as presented in Figure (15). Figure (16) illustrates the obtained control signal.

Figure 15.

Evolution of the desired and system outputs with PSO.

Figure 16.

Evolution of the control signal Q(W) with PSO.

Figure 17.

Evolution of the controller parameter η_c(t) with PSO.

Table 2 presents a comparative analysis of the two types of control. As can be seen, the NMSE for the control based exclusively on the MOPSO algorithm is practically equal to the NMSE obtained by using the PSO algorithm. But, for the case of multi-objective optimization, the CE is reduced significantly. An energy saving of about 28% was achieved. It can be concluded that the proposed approach guarantees energy savings while maintaining the same tracking performance.

Table 2.

NMSE and CE with MOPSO and PSO algorithms.

	MOPSO	PSO
NMSE	0.1335	0.1386
CE	3.4099 × 10¹¹	4.8 × 10¹¹

6. Conclusion

In this paper, an indirect multi-objective adaptive neural control scheme is proposed. The NE and NC adaptive learning rates used here are adjusted in ways to optimize the proposed multi-objective functions: control energy minimization and closed-loop preservation. The closed-loop performance of the developed method is compared to that obtained where the standard PSO algorithm is proposed as an optimization mechanism. A real application on a transesterification reactor has shown that the proposed method ensures both economical objectives and satisfactory closed-loop performances. Future work should focus on extending the proposed approach to multivariable systems. A comparative study with other control methods will also be carried out.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Ministry of Higher Education and Scientific Research, Tunisia.

ORCID iDs

Amel Bentaher

Zribi Ali

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