Furnace Temperature Model Predictive Control Based on Particle Swarm Rolling Optimization for Municipal Solid Waste Incineration

Abstract

Precise control of furnace temperature (FT) is crucial for the stable, efficient operation and pollution control of the municipal solid waste incineration (MSWI) process. To address the inherent nonlinearity and uncertainty of the incineration process, a FT control strategy is proposed. Firstly, by analyzing the process characteristics of the MSWI process in terms of FT control, the secondary air flow is selected as the manipulated variable to control the FT. Secondly, an FT prediction model based on the Interval Type-2 Fuzzy Broad Learning System (IT2FBLS) is developed, incorporating online parameter learning and structural learning algorithms to enhance prediction accuracy. Next, particle swarm rolling optimization (PSRO) is used to solve the optimal control law sequence to ensure optimization efficiency. Finally, the stability of the proposed method is validated using Lyapunov theory, confirming the controller’s reliability in practical applications. Experiments based on actual operational data confirm the method’s effectiveness.

1. Introduction

Rapid urbanization and increasing consumption levels have led to a significant accumulation of municipal solid waste (MSW) [1], posing serious threats to environmental protection and human health. Improper management of MSW, which contains toxic substances, flammable materials, and large amounts of organic matter, can result in soil, water, and air pollution, endangering ecological balance and safety [2]. In densely populated developing countries like China, the challenge of MSW treatment and disposal is even more pronounced [3]. MSW Incineration (MSWI) technology is crucial for advancing renewable energy recycling, sustainable development, and environmental protection in cities worldwide. It has been widely adopted by developed countries such as in the US, Europe, and Japan [4,5]. However, due to the variability in MSW composition in developing countries like China, which introduces uncertainty into the incineration process, on-site control technology in MSWI plants still largely relies on manual control by field experts, despite the long-term industrial application and localization of automatic combustion control systems imported from developed countries [3,6]. Therefore, there is a current research focus on combining practical experience to develop large-scale, integrated, and intelligent MSWI control systems with localized MSW characteristics. The goal is to optimize the incineration process using efficient intelligent control technology to achieve sustainable, low-carbon, and harmless development [3].

The effective control of furnace temperature (FT) is critical for the MSWI process. Proper FT control ensures complete combustion, improves energy efficiency, and reduces harmful gas emissions and waste residuals, minimizing environmental impact [7]. Current research on intelligent FT control primarily focuses on fuzzy neural network (FNN) controllers. For instance, Tian et al. [8] designed a self-organizing FNN controller based on a growth-deletion-merging mechanism. Ding et al. [9] designed a self-organizing FNN controller based on multi-task learning. Ding et al. [10,11,12] and He et al. [13] developed various event-triggered mechanisms for FNN controllers to enhance the efficiency of controller updates. Furthermore, to address the strong uncertainty in the MSWI process, Tang et al. [14] designed an Interval Type-2 FNN (IT2FNN) controller, which improves uncertainty handling through re-fuzzification of membership functions, and its effectiveness was demonstrated through simulations with real industrial data. While fuzzy neural controllers integrate the benefits of fuzzy logic and neural networks to manage system nonlinearity and uncertainty, they still require accurate system models and often rely on trial and error to determine control rules, which can lead to incomplete or inaccurate rules and necessitates further performance improvement.

Model predictive control (MPC), a sophisticated control algorithm, predicts future dynamic changes in complex industrial processes by constructing a predictive model and optimizing future control actions based on this model. MPC accounts for both dynamic characteristics and constraints, providing optimal control actions within a finite time domain [15,16]. By considering both current and future states, MPC responds effectively to system changes, offering advantages in control performance and accuracy. MPC has been extensively studied for complex industrial processes. For example, Han et al. [17] proposed an MPC based on a self-organizing Radial Basis Function Neural Network (RBFNN) for controlling the dissolved oxygen concentration in municipal wastewater treatment processes, predicting future dynamic changes through the RBFNN. Xie et al. [18] proposed an adaptive MPC based on neuron adaptive splitting and merging RBFNN and applied it to the control of the iron removal process in a wet zinc smelting plant. For the MSWI process, Qiao et al. [19] proposed an event-triggered adaptive MPC strategy for flue gas oxygen content control using a time-based gradient descent method to update model parameters online to address uncertainty. Sun et al. [20] used a self-organizing dual long short-term memory neural network to construct a compact model of flue gas oxygen content, developing an MPC strategy. Meng et al. [21] constructed a self-organizing FNN based on a self-organizing addition–deletion mechanism and the LM algorithm, obtaining a predictive model with a simplified structure and high prediction accuracy. Tang et al. [22] designed an IT2FNN predictive model with online parameter learning and self-organizing mechanisms and developed an MPC control strategy for FT based on this model. These applications demonstrate MPC’s effectiveness in managing process dynamics and complexity.

MPC solves control laws through rolling optimization, adjusting system behavior in each control cycle. While gradient descent methods have achieved good control results, they may encounter local optima or slow convergence in highly nonlinear systems [23]. The particle swarm optimization (PSO) algorithm, known for its global optimization capability and fast convergence, has been applied to MPC rolling optimization. For example, Yan et al. [24] proposed a nonlinear NOx emission prediction model with an improved random configuration network and used PSO for constraint backstepping optimization. Feng et al. [25] developed a parameter-adaptive PSO algorithm for optimizing control laws in rolling mills. These studies show that PSO-based rolling optimization outperforms gradient-based methods. However, research on applying PSO to FT control in MSWI is limited.

Given that MPC’s effectiveness relies on the predictive model’s accuracy and robustness [26], neural networks are often used for this purpose but are constrained by their depth [27,28]. Shallow networks may underperform in complex process modeling, while deep networks may become too complex, affecting real-time performance. To address this, Chen et al. [29] designed a broad learning system (BLS) based on the random vector functional link neural network, training parameters using a ridge regression pseudo-inverse algorithm [30]. This approach saves computational resources and shortens model training time while improving model performance. BLS has been widely applied in data-driven modeling, text classification, and computer vision [31,32,33]. To further enhance network generalization and feature extraction capabilities, Feng et al. [34] combined BLS and FNN [35] to design a fuzzy BLS (FBLS), replacing the feature nodes of BLS with FNN subsystems. However, the deterministic membership degree of FBLS makes it difficult to achieve good predictive performance under strong uncertainties. To address this, Han et al. [36] proposed IT2FBLS, which showed enhanced identification and uncertainty handling capabilities for nonlinear systems in experimental results. Liu et al. [37] designed an IT2FBLS-based sludge self-healing controller, identifying system fault conditions to accurately extract fault features and designing adaptive expansion strategies to improve the fitting accuracy and robustness of IT2FBLS. Experimental results validated its effectiveness.

In summary, this article presents an MPC strategy for managing FT in the MSWI process, utilizing particle swarm rolling optimization (PSRO). The strategy involves the following steps: first, describe the MSWI process technology for FT control and examine its control characteristics. Next, develop an FT prediction model using IT2FBLS to forecast future FTs. Then, apply PSRO to determine the optimal sequence of control actions to enhance accuracy. Finally, assess the convergence of the PSRO process using the Lyapunov method. Simulations using actual MSWI process data confirmed the effectiveness of the proposed strategy.

Through a literature review, it was found that MPC has been applied to FT control in the MSWI process. However, in existing FT control research, the application of meta-heuristic algorithms to MPC has not been reported. Therefore, the purpose of this study was to investigate an intelligent control method suitable for FT control in the MSWI field. The main innovations of this method are as follows: (1) IT2FBLS is used to construct a prediction model for FT control to improve accuracy and overcome process uncertainty; (2) the PSO algorithm is applied to the rolling optimization process of MPC to design a control-oriented PSRO strategy; (3) the convergence of the proposed control strategy is analyzed to ensure the theoretical foundation of the algorithm.

2. Problem Formulation

2.1. Overview and Characteristic Analysis of the MSWI Process in Terms of FT Control

In the grate furnace used in the MSWI process, MSW is fed into the incinerator by a grab and undergoes drying, combustion, and burnout stages. These processes are supported by combustion air, high-temperature radiation, and heating. During incineration, the organic matter in the MSW is gasified and pyrolyzed, releasing heat and destroying pathogenic organisms such as viruses and bacteria. The entire process is divided into six subsystems: solid waste fermentation, solid waste combustion, waste heat exchange, steam power generation, flue gas cleaning, and flue gas emission, as shown in Figure 1.

As shown in Figure 1, the MSWI process flow is as follows:

MSW is delivered to the plant by compactor collection vehicles, weighed at a weighbridge, and unloaded into the deposit pool from the platform. In the pool, the MSW is thoroughly crushed, mixed, and stacked using a crane grab, allowing for microorganisms to ferment and naturally dehydrate the waste [38]. This fermentation process, which usually takes 5–7 days, increases the calorific value of the solid portion by about 30%. The grab then lifts the fermented MSW and transfers it to the hopper [39], where it slides into the chute and is fed into the incinerator. The MSW is dried by the furnace’s heat radiation and the preheated primary air before entering the combustion stage. During combustion, air is added to provide the necessary oxygen, and in some cases, other media may assist. After several hours of high-temperature combustion, the combustible components are fully burned, generating heat, while the non-combustible ash is pushed out of the furnace by the burnout grate. The high-temperature flue gas produced from MSW combustion passes through various boiler heating surfaces, where it is absorbed and cooled. Toxic substances and heavy metals in the flue gas are treated through denitrification [40], desulfurization, dust removal, and ash collection, converting them into non-toxic, harmless gases that meet environmental standards and are released into the atmosphere through a chimney by an induced draft fan [41]. Meanwhile, deionized water in the waste heat boiler absorbs the heat generated by incineration, converting it into high-temperature steam. This steam expands to generate power, driving the turbine and generator to produce electricity.

The primary objective of the MSWI process is to safely treat MSW, with power generation or heat production serving as secondary goals. Typically, the power output of the steam turbine generator in an MSWI plant is synchronized with the incinerator’s operation, and the external power grid does not impose restrictions on power dispatching. Consequently, the main goal of the MSWI process’s automatic control system is to ensure stable MSW combustion, maintain consistent boiler steam production, minimize the loss on ignition of slag, and reduce pollutant emissions as much as possible.

FT is directly influenced by the solid waste combustion process. During this process, MSW is mechanically fed onto the grate, where it undergoes drying, combustion, and burnout as it moves along the grate, eventually transforming into ash and high-temperature flue gas. Primary air is introduced from beneath the grate to supply oxygen for combustion, while secondary air is injected above the flame to provide oxygen for redox reactions in the flue gas and to enhance turbulence within the furnace. This makes the MSWI process an “air and material distribution” system, with key manipulated variables being primary air flow, secondary air flow, and grate speed.

Through analysis and insights from operational engineers at the MSWI plant [42], five key variables were identified as critical to influencing FT: primary air flow, secondary air flow, the average speed of the feeder, the average speed of the drying grate, and urea injection flow—the latter being used to lower pollutant concentrations within the incinerator. These five inputs formed the basis of the controlled object model for FT, from which the manipulated variable (MV) was chosen. Pearson correlation coefficient (PCC) values between these variables and FT were calculated using specific operational data from the plant, with detailed results available in [14].

Among the variables considered, aqueous ammonia showed the highest PCC value with FT, followed by secondary air flow. However, the reliance on aqueous ammonia stems from differences in control technologies between Chinese MSWI plants and those in developed countries, leading to its excessive use in some plants to meet emission standards. As a result, aqueous ammonia was not selected as the manipulated variable for FT. Instead, since stable combustion is the central goal of MSWI control through the “air and material distribution” process, secondary air flow was chosen as the manipulated variable for FT in this study.

This study specifically examined data from a single day of operation, where FT ranged between 880 °C and 988 °C. Future research will explore FT fluctuations across different ranges and conduct multi-condition studies over extended periods.

2.2. Model Predictive Control

MPC is a type of multi-objective discrete control method that repeatedly solves optimization problems in each control iteration to obtain the optimal control law within a finite time horizon at each moment. The implementation of optimization algorithms is typically based on accurate predictions of the controlled variables by the predictive model, and its objective function is usually set as follows:

$$\begin{array}{ll}\widehat{J}(t)& ={\displaystyle \sum _{i_{\mathrm{p}}=1}^{H_{\mathrm{p}}}e{\left(t+i_{\mathrm{p}}\right)}^{\mathrm{T}}{w}_{i_{\mathrm{p}}}^{y}e\left(t+i_{\mathrm{p}}\right)}\\ & +{\displaystyle \sum _{j_{\mathrm{u}}=1}^{H_{\mathrm{u}}}\Delta u{(t+j_{\mathrm{u}}-1)}^{\mathrm{T}}{w}_{j_{\mathrm{u}}}^u\Delta u(t+j_{\mathrm{u}}-1)}\end{array}$$

(1)

where $H_p$ and $H_{\mathrm{u}}$ represent the prediction horizon and control horizon of the system (i.e., the step size of the predicted values and control law set in each iteration, satisfying $H_p>H_{\mathrm{u}}$ [43]); $e\left(t\right)$ is the deviation between the predicted value and the reference trajectory at time t; $\Delta u\left(t\right)$ is the change in the control law at time t; $w_{i_{\mathrm{p}}}^y$ and $w_{j_{\mathrm{u}}}^u$ are the weighting parameters of the objective function; and $\widehat{\mathit{J}}$ is the objective function for the system’s online optimization control.

The data-driven FT prediction model constructed in this article is described as a nonlinear autoregressive exogenous (NARX) system, with inputs $X\left(t\right)=\left[u(t-1),\cdots ,u(t-n_u),y(t-1),\cdots ,y(t-n_y)\right]$, and outputs for future steps $y_p\left(t\right)$, where $y$ and $\mathit{u}$ are the system’s output and input, and ${\mathit{n}}_y$ and ${\mathit{n}}_u$ are the maximum lags of the system’s output and input.

2.3. Particle Swarm Optimization (PSO)

PSO is a widely used computational method for solving optimization problems by mimicking the social behavior of birds [44]. Its simplicity and effectiveness make it a popular choice for finding optimal solutions in complex scenarios.

In PSO, a swarm of particles represents potential solutions to the optimization problem [45]. These particles traverse the solution space, adjusting their positions based on their own experiences and those of neighboring particles. Each particle is characterized by a position and velocity, which are updated using following formulas [46]:

$$\left\{\begin{array}{l}{v}_n\left(t+1\right)=a\cdot v_d^n\left(t\right)+b_1\cdot r_1\cdot (p_{\mathrm{best}}^n\left(t\right)-x_n\left(t\right))+b_2\cdot r_2\cdot (g_{\mathrm{best}}^{ }\left(t\right)-x_n\left(t\right))\\ x_n\left(t+1\right)=x_n\left(t\right)+v_n\left(t+1\right)\end{array}\right.$$

(2)

where $n\in \left[1,N\right]$, $N$ is the number of particle; $d\in \left[1,D\right]$, $D$ is the dimension of particle; $\mathit{a}$ is the inertia coefficient; $v_d^n\left(t\right)$ is the d-dimensional velocity of the nth particle at the current time; ${\mathit{b}}_1$ and ${\mathit{b}}_2$ are acceleration coefficients; ${\mathit{r}}_1$ and ${\mathit{r}}_2$ are uniformly distributed random numbers in the interval [0, 1]; $p_{\mathrm{best}}^n\left(t\right)$ is the individual optimal position of the nth particle at the current time; $g_{\mathrm{best}}^{ }\left(t\right)$ is the global optimal position of the population; and $x_n\left(t\right)$ is the particle position.

3. Materials and Methods

3.1. Materials

Some detailed information about an MSW plant in Beijing is as follows. The capacity of MSW is 628.8 t/d. The length of grate is 11 m, and the width is 12.9 m. The primary air flow is 67,500 m³/h. The temperature of air is 200 °C. Primary air enters the bed from four independent parts of grate below. The flow rate of each part accounts for 24.31%, 43.35%, 19.27%, and 13.07% of primary air, respectively.

The data were collected through an edge verification platform of an MSWI power plant equipped with a safety isolation acquisition device. The device facilitates one-way data transmission, mitigating interference issues during the data acquisition process, as shown in Figure 2.

Figure 2 illustrates the edge verification platform, which includes a safety isolation acquisition device, an OPC server, and a data acquisition and storage platform. The safety isolation device ensures unidirectional data flow, safeguarding the system from external signal interference and malicious attacks. Using hardware isolation technology, data are securely transmitted from the plant’s DCS net to the OPC server via high-speed Ethernet and TCP/IP protocols and stored according to predefined intervals or trigger conditions. The OPC server uniformly manages and distributes various types of data and communicates with both upper-level computers and the lower-level data storage platform. The data acquisition and storage platform collects, processes, and stores data received from the OPC server.

The experimental data were obtained from the operational records of an actual MSWI plant in Beijing for a specific day from 8:00 to 24:00. After preprocessing, the dataset consisted of 857 samples with 6 features, including 1 manipulated variable (secondary air flow), 4 disturbance variables (primary air flow, feeder speed, dryer grate speed, ammonia injection rate), and 1 controlled variable (FT).

3.2. Methods

The proposed strategy is shown in Figure 3.

Figure 3 shows that the proposed PSRO-MPC control strategy includes a Particle Swarm Rolling Optimization Module (PSROM) and an Interval Type-2 Fuzzy Broad Prediction Module (IT2FBPM). The functions of each module are as follows.

(1) Interval Type-2 Fuzzy Broad Prediction Module (IT2FBPM). The inputs are the historical MV and FT, defined as $X\left(t\right)=\left[u\left(t-1\right),\cdots ,u\left(t-n_u\right),y\left(t-1\right),\cdots ,y\left(t-n_y\right)\right]$, and the output is the predicted output $y_p\left(t\right)$ for the next 1 to $H_p$ moments. It includes four submodules, among which the IT2FBLS Prediction Submodule is used to calculate the model predicted output for the next 1 to $H_p$ moments. The Parameter Learning Submodule and Structural Learning Submodule are used for online updating of IT2FBLS parameters and the number of enhancement nodes to improve prediction accuracy and reduce model redundancy. The Feedback Correction Submodule corrects $\widehat{y}\left(t\right)$ through the prediction error $\epsilon \left(t\right)$ to obtain the predicted output $y_p\left(t\right)$.

(2) Particle Swarm Rolling Optimization Module (PSROM). The input is the system error, and its output is the corrected secondary air flow. The input of the interval type-2 fuzzy broad prediction module consists of the current and historical MV and FT, and its output is the FT prediction.

The PSRO-MPC control process operates as follows: first, the particle swarm rolling optimization module adjusts the secondary air flow based on system error. Next, the IT2FBLS prediction submodule calculates the model’s predicted output using current and historical MV and FT as inputs. Concurrently, the parameter learning submodule and structural learning submodule update the IT2FBLS parameters and the number of enhancement nodes in real time to enhance prediction accuracy and reduce model redundancy. Finally, the feedback correction submodule corrects the predicted output based on the prediction error, producing the final predicted output.

Taking time t as an example, the calculation process of the control system is described as follows. First, we calculate the system error between the FT reference trajectory $y_r\left(t\right)$ and the predicted value $y_p\left(t\right)$ at this moment, which serves as the input of the PSROM. After optimization using the PSO algorithm, the optimal control sequence of MV $u^{\ast }\left(t\right)=\left[u\left(t\right),\dots ,u\left(t+H_u\right)\right]$ is calculated. This sequence is then used as the input of the controlled object model to calculate the FT value at the current moment. Next, $u^{\ast }\left(t\right)$ and $y\left(t\right)$ are used as inputs of the IT2FBPM, and the IT2FBLS prediction model sub-module calculates the model output $\widehat{y}\left(t\right)=\left[\widehat{y}\left(t\right),\dots ,\widehat{y}\left(t+H_p\right)\right]$. The difference in this output $y\left(t\right)$ is taken to calculate the prediction error $\epsilon \left(t\right)$. Then, parameter learning and structure learning are performed on the IT2FBLS prediction model at the current moment to reduce $\epsilon \left(t\right)$. Finally, $\epsilon \left(t\right)$ is used to perform feedback correction on $\widehat{y}\left(t\right)$, obtaining the FT prediction value of the system $y_p\left(t\right)$, thus completing the calculation of the control system at time t.

3.2.1. Interval Type-2 Fuzzy Broad Prediction Module (IT2FBPM)

IT2FBLS Prediction Submodule

The IT2FBLS prediction submodule includes an input layer, IT2FNN layer, enhancement layer, and output layer. Taking time t as an example, the calculation process of the k-th subsystem of the prediction model is obtained as follows:

$$z^k=\varphi \left(X\left(t\right)\right)$$

(3)

where $\varphi \left(\cdot \right)$ is a nonlinear function, and the calculation process can refer to reference [36].

The output of the IT2FNN layer composed of K subsystems is denoted as

$$\mathit{z}=[z^1,\cdots ,z^K]$$

(4)

The enhancement layer receives the output of the IT2FNN layer as its input and performs nonlinear transformations, consisting of $L$ enhancement nodes. Taking the l-th enhancement node as an example ($l=1,\cdots ,L$), its output is

$$h_l=\tanh \left(\mathit{z}{\mathit{w}}_l^{f,e}+{\beta }_l^{f,e}\right)$$

(5)

where ${\mathit{w}}_l^{f,e}={\left[w_{1l}^{f,e},\cdots ,w_{Kl}^{f,e}\right]}^{\mathrm{T}}$ and ${\beta }_l^{f,e}$ are the connection weight vector and bias term coefficient between the IT2FNN layer and the l-th enhancement layer, respectively.

(1)

The output layer linearly combines the outputs of the IT2FNN layer and the enhancement layer to obtain the final model output, as follows:

$$\begin{array}{ll}\widehat{y}\left(t\right)& =\mathit{z}\left(t\right){\mathit{w}}_f\left(t\right)+\mathit{h}\left(t\right){\mathit{w}}_e\left(t\right)\\ & ={\displaystyle \sum _{k=1}^K{z}^k\left(t\right)w_{fk}\left(t\right)}+{\displaystyle \sum _{l=1}^L{h}_l\left(t\right)w_{el}\left(t\right)}\end{array}$$

(6)

where ${\mathit{w}}_f\left(t\right)={\left[w_{f1}\left(t\right),\cdots ,w_{fK}\left(t\right)\right]}^{\mathrm{T}}$ is the weight vector between the IT2FNN layer output and the output layer, and ${\mathit{w}}_e\left(t\right)={\left[w_{e1}\left(t\right),\cdots ,w_{eL}\left(t\right)\right]}^{\mathrm{T}}$ is the weight vector between the enhancement layer output and the output layer.

The $\widehat{y}\left(t\right)$ obtained from Equation (6) is used as the input of the IT2FBLS model at the next time step, repeating Equations (3)–(6), to calculate the output at future times $t+1,\dots ,t+H_p$, where $H_p$ is the prediction horizon.

As a broad learning model, IT2FBLS does not rely on the backpropagation algorithm of sequential sample inputs for pre-training. Instead, it solves the weights between the IT2FNN layer, enhancement layer, and output layer by simultaneously inputting large-scale sample data in one batch. The initial values of parameters ${\mathit{w}}_f$ and ${\mathit{w}}_e$ are solved using the ridge regression approximation algorithm, as follows:

$${\mathit{W}}_{f,e}\left(t\right)={\left[{\overline{A}}^{\mathrm{T}}\left(t\right)\overline{A}\left(t\right)+\lambda I\right]}^{-1}{\overline{A}}^{\mathrm{T}}\left(t\right)\times \overline{Y}\left(t\right)$$

(7)

where ${\mathit{W}}_{f,e}\left(t\right)={\mathit{w}}_f^{\mathrm{T}}\left(t\right)\left|{\mathit{w}}_e^{\mathrm{T}}\left(t\right)\right.$, $\overline{A}\left(t\right)=z\left(t\right)\left|h\left(t\right)\right.$, $I$ is the identity matrix, $\lambda$ is the regularization coefficient, and $\overline{Y}\left(t\right)$ is the matrix of the actual output values of all samples.

Parameter Learning Submodule

To enhance the predictive model’s adaptability to real scenarios, online parameter updates based on the prediction error $\epsilon \left(t\right)$ are performed during actual industrial control processes, using gradient descent to update parameters $w_f^{ }\left(t\right)$ and $w_e^{ }\left(t\right)$.

First, define the performance function as follows:

$$E(t)={\displaystyle \frac{1}{2}}{\epsilon }^2(t)={\displaystyle \frac{1}{2}}{(y(t)-y_p(t))}^2$$

(8)

Next, compute the partial derivatives of the above parameters, specifically as follows:

$${\displaystyle \frac{\partial E\left(t\right)}{\partial w_{fk}}}={\displaystyle \frac{\partial E\left(t\right)}{\partial e\left(t\right)}}{\displaystyle \frac{\partial e\left(t\right)}{\partial y\left(t\right)}}{\displaystyle \frac{\partial y\left(t\right)}{\partial w_{fi}}}=-e\left(t\right)z^k$$

(9)

$${\displaystyle \frac{\partial E\left(t\right)}{\partial w_{el}}}={\displaystyle \frac{\partial E\left(t\right)}{\partial e\left(t\right)}}{\displaystyle \frac{\partial e\left(t\right)}{\partial y\left(t\right)}}{\displaystyle \frac{\partial y\left(t\right)}{\partial w_{el}}}=-e\left(t\right)h_l$$

(10)

Finally, update the parameters based on the gradient descent method, as follows:

$$w_{fk}\left(t+1\right)=w_{fk}\left(t\right)-\eta {\displaystyle \frac{\partial E\left(t\right)}{\partial w_{fk}}}$$

(11)

$$w_{el}\left(t+1\right)=w_{el}\left(t\right)-\eta {\displaystyle \frac{\partial E\left(t\right)}{\partial w_{el}}}$$

(12)

where $\eta \in \left(0,1\right)$ is the learning rate.

Structural Learning Submodule

For the MSWI process, the initial structure of IT2FBLS is inadequate to cope with the system’s high uncertainty and dynamic changes. Therefore, an adaptive structural optimization strategy is designed to extend or reduce the number of enhancement nodes in the enhancement layer during control to improve the model’s robustness and adaptability.

(1) Enhancement Node Growth: The prediction error $\epsilon \left(t\right)$ between the actual system output and the predictive model output is used as the criterion for the enhancement node growth. When the square of $\epsilon \left(t\right)$ exceeds a certain threshold, a set of enhancement nodes is added, denoted as $J+1$. It follows the following specific rules:

$$\left\{\begin{array}{l}\mathrm{IF}\begin{array}{c} \end{array}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c} \end{array}\end{array}\end{array}\end{array}:{\displaystyle \frac{1}{2}}{\epsilon }_{ }^2\left(t\right)\ge G_{\mathrm{th}}\\ \mathrm{THEN}:y_p\left(t\right)=\mathit{z}(t){\mathit{w}}_z(t)+{\mathit{h}}_{J+1}\left(t\right){\mathit{w}}_{e,J+1}(t)\end{array}\right.$$

(13)

where $G_{\mathrm{th}}\in \left[0,1\right]$ is the threshold for structural growth determination; ${\mathit{h}}_{J+1}$ and ${\mathit{w}}_{e,J+1}$ are the output vector and weight vector of the expanded enhancement layer, respectively, which are as follows:

$${\mathit{h}}_{J+1}=\left[h_1\left(t\right),h_2\left(t\right),\cdots ,h_J\left(t\right),h_{J+1}\left(t\right)\right]$$

(14)

$${\mathit{w}}_{e,J+1}={\left[w_{e,1},w_{e,2},\dots ,w_{e,J},w_{e,J+1}\right]}^{\mathrm{T}}$$

(15)

where $h_{J+1}\left(t\right)=\tanh \left(\mathit{z}\left(t\right){\mathit{w}}_{J+1}\left(t\right)+{\beta }_{J+1}\left(t\right)\right)$; ${\mathit{w}}_{J+1}$ and ${\beta }_{J+1}$ are the connection weights and bias terms after structural expansion, with initial values set as follows:

$$\left\{\begin{array}{l}{\mathit{w}}_{J+1}\left(t\right)=\mathrm{rand}\left(0,1\right)\\ {\beta }_{J+1}\left(t\right)=\mathrm{rand}\left(0,1\right)\\ w_{e,J+1}=0\end{array}\right.$$

(16)

where $\mathrm{rand}\left(0,1\right)$ represents a random number uniformly distributed between 0 and 1.

(2) Enhancement Node Reduction: To prevent structural redundancy due to excessive growth of enhancement nodes, the useless rate is used as the criterion for structural reduction. The useless rate is calculated by the accumulated number of times each set of enhancement nodes plays an important role within the assessment period.

Taking the j-th set of enhancement nodes as an example, its useless rate is as follows:

$$Rate\_useles{s}_j={\displaystyle \frac{R_j}{T_d}}$$

(17)

where $R_j$ is the number of times the j-th set of enhancement nodes did not play an important role, and $T_d$ is the assessment period for structural reduction. The calculation for $R_j$ is as follows:

$$R_j=\left\{\begin{array}{l}{R}_j+1,h_j\left(t\right)<R_{th}\\ R_j,h_j\left(t\right)>R_{th}\end{array}\right.$$

(18)

where $R_j$ is initially 0, $R_{th}$ is the threshold for determining the importance of enhancement nodes, and $h_j\left(t\right)$ is the output value of the j-th set of enhancement nodes at time t ($t=1,2,\dots ,T_d$).

When the useless rate exceeds a certain threshold, the set of nodes is deleted according to the following specific rules:

$$\left\{\begin{array}{l}\mathrm{IF}\begin{array}{c} \end{array}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c} \end{array}\end{array}\end{array}\end{array}:Rate\_useles{s}_j\ge D_{\mathrm{th}}\\ \mathrm{THEN}:\widehat{y}\left(t\right)=\mathit{z}\left(t\right){\mathit{w}}_z\left(t\right)+{\mathit{h}}_{J-1}\left(t\right){\mathit{w}}_{e,J-1}\left(t\right)\end{array}\right.$$

(19)

where $D_{\mathrm{th}}$ is the threshold for structural reduction determination; ${\mathit{h}}_{J-1}$ and ${\mathit{w}}_{e,J-1}$ are the reduced output vector and weight vector, respectively.

To ensure the stability of network output and avoid the impact of structural reduction on the network, network parameters need to be compensated. Assuming the j-th set of enhancement nodes has the smallest Euclidean distance to the deleted nodes (the j-th set of enhancement nodes), the parameters are compensated as follows:

$$\left\{\begin{array}{l}{\overline{\mathit{w}}}_{\tilde{j}}\left(t\right)={\mathit{w}}_{\tilde{j}}\left(t\right)\\ {\overline{\beta }}_{\tilde{j}}\left(t\right)={\beta }_{\tilde{j}}\left(t\right)\\ {\overline{w}}_{e,\tilde{j}}\left(t\right)=w_{e,\tilde{j}}\left(t\right)+\left(h_j\left(t\right)/h_{\tilde{j}}\left(t\right)\right)w_{e,\tilde{j}}\left(t\right)\end{array}\right.$$

(20)

where ${\mathit{w}}_{\tilde{j}}$, ${\beta }_{\tilde{j}}$, and $w_{e,\tilde{j}}$ are the corresponding parameters of the $\tilde{\mathit{j}}$-th set of enhancement nodes before compensation; ${\overline{w}}_{\tilde{j}}$, ${\overline{\beta }}_{\tilde{\mathit{j}}}$, and ${\overline{\mathit{w}}}_{e,\tilde{j}}$ are the corresponding parameters of the $\tilde{j}$-th set of enhancement nodes after compensation.

Feedback Correction Submodule

As a complex industrial process, the inherent high uncertainty of the MSWI process makes it difficult for the IT2FBLS prediction model to accurately reflect the true dynamic changes in FT, resulting in a deviation between the predicted output value $\widehat{y}(t)$ and the actual value $y(t)$. In this control strategy, the purpose of error compensation is to reduce this deviation. By using the prediction error $\epsilon (t)$ at the current time, a series of subsequent predictive model outputs within the prediction horizon are compensated for. The compensated output is the FT prediction output of the system, as follows:

$$y_p(t+i)=\widehat{y}(t+i)+\epsilon (t)$$

(21)

where $i=1,\dots ,H_p$.

3.2.2. Particle Swarm Rolling Optimization Module (PSROM)

The primary purpose of online optimization control is to make the objective function of the MPC system gradually approach its minimum within a given prediction and control horizon, thereby solving for the optimal control variables in a fixed horizon to achieve predictive control of the controlled variables. For ease of calculation, the objective function in Equation (1) is rewritten as follows:

$$\widehat{J}(t)={\rho }_1{\left[\mathit{r}(t)-\widehat{\mathit{y}}(t)\right]}^{\mathrm{T}}\left[\mathit{r}(t)-\widehat{\mathit{y}}(t)\right]$$

(22)

where $\mathit{r}(t)$ is the tracking value vector; and ${\mathit{y}}_{\mathrm{p}}(t)$ is the prediction value vector. They are denoted as follows:

$$\mathit{r}(t)={\left[r(t+1),r(t+2),\dots ,r(t+H_{\mathrm{p}})\right]}^{\mathrm{T}}$$

(23)

$${\mathit{y}}_{\mathrm{p}}(t)={\left[y_{\mathrm{p}}(t+1),y_{\mathrm{p}}(t+2),\dots ,y_{\mathrm{p}}(t+H_{\mathrm{p}})\right]}^{\mathrm{T}}$$

(24)

To optimize control performance and improve FT control accuracy, this study used the PSO algorithm to achieve the rolling optimization process of MPC.

The rolling optimization process for the nth particle can be rewritten according to Equation (2) as follows:

$$v_d^n\left(t+1\right)=a{v}_d^n\left(t\right)+b_1{r}_1\left(p_{\mathrm{best}}^n\left(t\right)-u\left(t\right)\right)+b_2{r}_2\left(g_{\mathrm{best}}^{ }\left(t\right)-u\left(t\right)\right)$$

(25)

$$u_d^n\left(t+1\right)=u_d^n\left(t\right)+v_d^n\left(t+1\right)$$

(26)

where $d=1,\dots ,H_u$, $H_u$ is the control horizons, i.e., particle dimensions; $u\left(t\right)$ is the MV, i.e., the current position of the particle.

By using random numbers $r_1$ and $r_2$, randomness is introduced into the exploration of the search space, enhancing the particle’s tendency to explore unknown spaces, but also slowing down the convergence speed. Considering the real-time requirements of the control process, this article adopts a deterministic PSO [47], which sets the random numbers to their expected values, i.e., $r_1=r_2=0.5$.

Remark: Given the dynamic unknowns, strong disturbances, and uncertainties in the FT control of the MSWI process, applying intelligent optimization algorithms for MPC must prioritize safety. As this is the first application of the PSO algorithm for FT control, starting with a simple PSO aids in convergence analysis and establishes a theoretical foundation for understanding the algorithm’s convergence. This meets the safety and performance requirements for FT control. In future work, we will explore improved PSO algorithms to further enhance control performance while maintaining safety.

3.2.3. Convergence Analysis

The PSO algorithm defines each particle as a potential optimal solution to a multidimensional space problem. In addition to the velocity component, it stores the historical individual optimum and global optimum. During each iteration, the particle calculates a new position by adjusting the velocity along each dimension. Since each dimension is updated independently of others, and the only link between the dimensions in the search space is the objective function, this article analyzes the convergence of the PSO algorithm in one dimension while ensuring generality.

For ease of explanation in the following text, the symbols in Equation (25) are simplified as follows:

$$b={\displaystyle \frac{b_1+b_2}{2}}$$

(27)

$$p={\displaystyle \frac{b_1}{b_1+b_2}}{p}_{\mathrm{best}}^n\left(t\right)+{\displaystyle \frac{b_2}{b_1+b_2}}{g}_{\mathrm{best}}^{ }\left(t\right)$$

(28)

Substituting Equations (27) and (28) into Equations (25) and (26), the deterministic PSO algorithm can be described in the following state equation form:

$$\left\{\begin{array}{l}y\left(t+1\right)=Ay\left(t\right)+Bp\\ \begin{array}{cc} & y\left(t+1\right)=\left[\begin{array}{c}u\left(t\right)\\ v_{\left(t\right)}\end{array}\right]\end{array},\\ \begin{array}{cc} & A=\left[\begin{array}{cc}1-b& a\\ -b& a\end{array}\right]\end{array}\\ \begin{array}{c} \\ & B=\left[\begin{array}{c}b\\ b\end{array}\right]\end{array}\end{array}\right.$$

(29)

In the context of dynamic systems theory, $y\left(t\right)$ is the particle state composed of its current position and velocity, $A$ is the dynamic matrix, whose properties determine the temporal behavior of the particle, $p$ is the external input used to drive the particle to a specific position, and $B$ is the input matrix representing the influence of the external input on the particle state.

Typically, the initial particle state is not in equilibrium. Determining whether the particle will eventually stabilize at an equilibrium state, i.e., whether the PSO algorithm converges and how the particle moves within the state space, is crucial for the algorithm’s engineering application. Based on Lyapunov’s dynamic system theory, it is shown that the convergence of the particle depends on the eigenvalues of the dynamic matrix $A$, with the solution process as follows:

$$\left|\lambda I-A\right|={\lambda }^2-\left(1+a-b\right)\lambda +a=0$$

(30)

The necessary and sufficient condition for the existence of an equilibrium point in the particle swarm dynamic system described by Equation (29) is that the magnitudes of the two eigenvalues of $A$ are both less than 1. In this case, the particle will eventually stabilize at the equilibrium point as shown in Equation (31), and the PSO algorithm will converge.

$$\begin{array}{l}{y}^{\mathrm{eq}}={\left[\begin{array}{c}p,0\end{array}\right]}^{\mathrm{T}},\begin{array}{cc} & \mathrm{that} \mathrm{is}\end{array}\\ \begin{array}{cc} & \end{array}\begin{array}{ccc}{x}^{\mathrm{eq}}=p& and& v^{\mathrm{eq}}=0\end{array}\end{array}$$

(31)

where $y^{\mathrm{eq}}$ is the equilibrium point, $x^{\mathrm{eq}}$ and $v^{\mathrm{eq}}$ are the position and velocity at the equilibrium point, and $p=p_{\mathrm{best}}^n=g_{\mathrm{best}}^{ }$.

From Equation (30), it can be seen that to satisfy the equilibrium point condition, i.e., the magnitudes of the eigenvalues are both less than 1, the parameters of the state equation must meet the following conditions:

$$a<1,\hspace{1em}b>0,\hspace{1em}2a-b+2>0$$

(32)

According to Lyapunov’s first method, when the PSO algorithm parameters satisfy Equation (32), the particle will converge to the equilibrium point, and the PSO algorithm will be convergent.

4. Results and Discussion

4.1. Performance Metrics

The experimental section describes the building of predictive models and tracking control experiments. To accurately and effectively analyze the performance of the predictive models, this study uses Root Mean Square Error (RMSE) and Coefficient of Determination (R-squared, R²) as evaluation metrics, defined as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{N_s}}{\displaystyle {\sum }_{n=1}^{N_s}{(y_n-y_{p,}{ }_n)}^2}}$$

(33)

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_n{(y_n-y_{p,}{ }_n)}^2}}{{\displaystyle {\sum }_n{(y_n-\overline{y})}^2}}}$$

(34)

where $y_n$ is the actual value of the sample, ${\widehat{y}}_n$ is the predicted output of the sample, $\overline{y}$ is the mean of all sample data, and $N_s$ is the total number of samples.

For the analysis of the proposed control strategy’s performance, Integral of Squared Error (ISE), Integral of Absolute Error (IAE), and Maximal Deviation from setpoint (Dev^max) are chosen as control metrics. These metrics define the evaluation criteria for the proposed PSRO-MPC control strategy as follows: [48,49]

$$\mathrm{ISE}={\displaystyle \frac{1}{t_f-t_0}}{\displaystyle {\sum }_{t=t_0}^{t_f}{e}_{ }^2\left(t\right)}$$

(35)

$$\mathrm{IAE}={\displaystyle \frac{1}{t_f-t_0}}{\displaystyle {\sum }_{t=t_0}^{t_f}\left|e\left(t\right)\right|}$$

(36)

$${\mathrm{Dev}}^{\max }=\max \left(e\left(t_0\right),\cdots ,e\left(t_f\right)\right)$$

(37)

where ${\mathit{t}}_0$ and ${\mathit{t}}_f$ are the initial time and number of iterations of the controller.

4.2. Prediction Model Experimental Results

In the prediction experiment, the experimental dataset was divided into training, validation, and testing sets in a ratio of 2:1:1. The maximum lag for input and output was set to 1, with an initial number of 20 IT2FNN subsystems, each with three fuzzy rules. The initial number of enhancement layer nodes was 9, with 1 neuron in the prediction output layer, and a regularization parameter of 2⁻³. Backpropagation NN (BPNN) and FBLS were used for comparative experiments. The hyperparameters for BPNN were set as follows: 8 neurons in the hidden layer, 1 neuron in the output layer, a maximum of 3000 iterations, and a learning rate of 0.005. The hyperparameters for FBLS were set as follows: 9 fuzzy subsystems, 3 fuzzy rules, 10 enhancement nodes, and a regularization parameter of 2⁻³.

The prediction results of FT by the IT2FBLS model are shown in Figure 4. The performance metrics of each prediction model across the three datasets are shown in

Table 1. Performance indicators of predictive models.

Model	Dataset	RMSE	R2
IT2FBLS	Training	3.2470 × 100	9.7098 × 10−1
	Validation	3.2974 × 100	9.7013 × 10−1
	Testing	3.4554 × 100	9.6730 × 10−1
FBLS	Training	3.3715 × 100	9.6871 × 10−1
	Validation	3.3774 × 100	9.6867 × 10−1
	Testing	3.6378 × 100	9.6376 × 10−1
BPNN	Training	4.0779 × 100	9.5422 × 10−1
	Validation	3.9428 × 100	9.5729 × 10−1
	Testing	4.0666 × 100	9.5472 × 10−1

.

From Figure 4 and Table 1, it can be seen that the IT2FBLS prediction model can effectively predict FT, and compared to other models, IT2FBLS has the best performance.

4.3. Tracking Control Experiment Results

Two sets of simulation experiments were designed: constant value tracking with a setpoint of 930 °C and a maximum of 150 iterations; and variable tracking with a setpoint of 930–935 °C, a maximum of 200 iterations, and change setpoints at the 50th, 100th, and 150th iterations. A 60dBW Gaussian white noise was applied to each disturbance. To evaluate the control effectiveness of the proposed method, comparative experiments were conducted using gradient descent-based MPC (GD-MPC), an IT2FBLS controller, and an incremental PID controller. The controller hyperparameters were set as follows: PSRO-MPC, Hu = 1, Hp = 3, 5 particles, 10 optimization iterations, a = 0.9, b₁ and b₂ both 2; GD-MPC, Hu =8, Hu = 1, and parameter learning rate 0.7; IT2FBLS, 3 IT2FNN subsystems, 6 fuzzy rules, 9 enhancement nodes, and regularization parameter 2⁻³⁰; PID, Proportional parameter 0.05, integral parameter 0.005, and differential parameter 0.005.

4.3.1. Constant Setpoint Value Experimental Results

The experimental results are shown in Figure 5, Figure 6, Figure 7 and Figure 8 and

Table 2. Comparison of constant setpoint value tracking performance indicators.

	Performance Index
	ISE	IAE	DEVmax
PSRO−MPC	1.165 × 10−1	1.173 × 10−1	2.2429 × 100
GD−MPC	4.6300 × 10−2	1.3970 × 10−1	1.6269 × 100
IT2FBLS	5.8105 × 10−2	1.3652 × 10−1	1.6915 × 100
PID	6.7040 × 10−1	7.0620 × 10−1	1.7428 × 100

.

From Figure 5, Figure 6, Figure 7 and Figure 8 and Table 2, it can be seen that PSRP-MPC can effectively control FT, making it quickly converge to the setpoint tracking value, with tracking errors consistently at a low level, within ±0.1 °C. After stabilization, the tracking error is much lower than that of other controllers, demonstrating better stability and robustness, indicating the effectiveness of the PSRO algorithm compared to gradient-based algorithms. Due to initial value settings, its convergence speed is relatively slow, but it has smaller errors after convergence, reflected in performance metrics as IAE being superior to other algorithms. The objective function can converge to a lower level, indicating the effectiveness of the PSRO algorithm. As the control error gradually decreases, the prediction model reduces its enhancement nodes through structural learning algorithms, maintaining performance while alleviating computational resource consumption.

4.3.2. Experimental Results of Variable Setpoint Values

The results are shown in Figure 9, Figure 10, Figure 11 and Figure 12 and

Table 3. Comparison of variable setpoint value tracking performance indicators.

	Performance Index
	ISE	IAE	DEVmax
PSRO−MPC	2.6051 × 100	2.0791 × 10−1	3.2747 × 100
GD−MPC	4.2040 × 10−1	3.7190 × 10−1	3.3937 × 100
IT2FBLS	4.8357 × 10−1	3.3440 × 10−1	5.1309 × 100
PID	3.2857 × 100	1.5120 × 100	4.3929 × 100

.

From Figure 9, Figure 10, Figure 11 and Figure 12 and Table 3, it can be seen that PSRP-MPC can quickly re-track the setpoint value after a change, demonstrating good rapidity. The stabilized tracking error consistently remains at a low level, within ±0.1 °C. Compared to other controllers, the proposed method has the best IAE and DEVmax, and the second-best ISE. Notably, due to the characteristics of the PSO algorithm, a non-tracking phenomenon occurred after stable tracking, affecting performance metrics, reflected in some peaks in the objective function curve, while the number of enhancement nodes increased to reduce control error impact. Additionally, due to the influence of the particle’s initial values, the proposed method has a narrower variable setpoint value control range.

4.4. Experimental Results of Hyperparameter Analysis

This section describes a sensitivity analysis on the hyperparameters that affect the performance of the PSRO-MPC. The hyperparameters include weight coefficients $a$, $b_1$, and $b_2$ in Equation (25) and the number of particles $N$ of PSRO. In addition, there is also the hyperparameters $H_p$ and $H_u$ of MPC. The corresponding range of values is shown in

Table 4. Range of controller hyperparameter values.

Hyperparameter	Symbol	Range	Step
Prediction horizon	$H_p$	[1, 20]	2
Control horizon	$H_u$	[1, 20]	2
Weight coefficient	$a$	[0.0001, 1]	0.1
Weight coefficient	$b_1$	[0.0001, 4]	0.4
Weight coefficient	$b_2$	[0.0001, 4]	0.4
The number of particles	$N$	[3, 30]	3

.

The experiment used a constant tracking setpoint value for analysis, the evaluation metric was the ISE, and a single factor analysis strategy was employed. The experimental results are as follows.

From Figure 13, it can be seen that the larger the number of particles $N$, the smaller the ISE, approaching zero, but computation time and resource consumption also increase. Once $N$ increases beyond a certain value, the ISE does not significantly decrease. Therefore, it is advisable to select a value within the range of [6, 10]. At the same time, the control system’s ISE is negatively correlated with $\mathit{a}$, and increasing its value will significantly reduce system error. Therefore, it is advisable to choose a larger value. ${\mathit{b}}_1$ and ${\mathit{b}}_2$ have similar effects on the system ISE. When their values are in the range of [2, 2.4], the ISE is minimized. It is advisable to select values within this range, but the constraints of Equation (32) must also be considered. When $H_p$ increases, the ISE shows a decreasing trend, but if it exceeds a certain range, the ISE increases, and computational resource consumption also rises. Therefore, $H_p$ should not be too large and can be selected within the range of [1, 10]. The value of $H_u$ is negatively correlated with the ISE, and when it is in the range of [9, 13], the ISE is minimized. However, its value is constrained and cannot exceed the value of $H_p$.

4.5. Comprehensive Analysis

The method proposed in this article has several limitations at each stage, as outlined below.

(1)

In the PSRO phase, the use of the PSO algorithm for online control law optimization results in high computational costs due to the large number of particles and dimensions. This challenges the real-time performance of the control process. Future work will explore the use of a surrogate model to reduce computational resource consumption.

(2)

During variable setpoint value tracking, the PSRO algorithm may struggle with significant setpoint changes due to limitations of initial particle values, resulting in slow convergence of the control error. Previous studies using gradient descent for rolling optimization have shown the possibility of compensating for the aforementioned disadvantages. Future work will focus on integrating gradient descent with the PSO algorithm to enhance performance.

(3)

The proposed control strategy mainly achieves the optimization of control laws through particle exploration and exploitation. However, unconstrained exploration may cause the FT to deviate from the setpoint value range, leading to untraceable results, which is a clear limitation of existing methods. Therefore, in the future, we will achieve safe tracking of setpoint values by imposing constraints on control laws.

5. Conclusions

To address the nonlinearity and uncertainty inherent in the MSWI process, this article proposes an MPC strategy based on PSRO for managing FT. The main contributions include the following: (1) designing an IT2FBLS-based prediction model and developing online parameter and structural learning algorithms to enhance prediction accuracy. (2) Using PSRO to determine the optimal control law sequence for efficient optimization. (3) Analyzing the convergence of the PSRO process to ensure controller reliability in practical applications. Experimental validation with real MSWI process data confirmed the effectiveness of the proposed approach.

It is worth noting that the proposed IT2FBLS prediction model significantly advances the control of FT in the MSWI process. By effectively modeling the complex, nonlinear dynamics of FT, the IT2FBLS model enhances the predictive accuracy of MPC strategies. This advancement ensures more stable and efficient FT operations, leading to reduced emissions and improved energy recovery. The model’s adaptability to the intricate data in MSWI processes underscores its relevance and potential impact, making a valuable contribution to the broader field of sustainable waste management technologies.

We developed an IT2FBLS prediction model using operational data from an MSWI plant in Beijing, China. The model’s adequacy and adaptability were rigorously evaluated by comparing its predictions to actual FT values and benchmarking its performance against alternative methods. The results confirmed the model’s superior accuracy in capturing the complex dynamics of the MSWI process and its robustness under varying operational conditions. Moreover, the IT2FBLS model was successfully integrated into the MPC framework for FT control, leading to improved stability and efficiency in temperature control. This outcome underscores the model’s practical applicability and its potential to enhance control strategies in the MSWI field.

Future research will focus on the following: (1) optimizing initial value settings in the PSO algorithm to enhance the range of FT control. (2) Implementing safety constraints to prevent deviations during tracking. (3) Evaluating the robustness of the controller under varying operating conditions. (4) Investigating the potential applications and challenges of the proposed method in real industrial systems.

The advantages of this study include the following: (1) the novel application of the PSO algorithm in MPC rolling optimization for the MSWI process, specifically for FT control. (2) A high tracking accuracy achieved with the PSRO-MPC strategy. (3) Effective handling of process uncertainty through the IT2FBLS-based FT prediction model. (4) An online parameter update and structural optimization strategy that enhances prediction accuracy and reduces model redundancy. (5) The research is grounded in real MSWI plant data, providing significant practical insights.

The limitations of this study include the following: (1) the strategy’s dependence on initial values, which can restrict the control range during variable setpoint tracking. To address this, future work will integrate gradient descent algorithms to optimize initial values and develop a hybrid optimization strategy. (2) The broad constraints applied to the control law may result in FT deviations from the setpoint. Future research will focus on dynamically applying safety constraints to ensure stable FT tracking.