A Review of Model Predictive Control for the Municipal Solid Waste Incineration Process

Abstract

Municipal solid waste incineration (MSWI) is essential for tackling urban environmental challenges and facilitating renewable energy recycling. The MSWI process has characteristics of multiple variables, strong coupling, and complex nonlinearity, requiring advanced process control (APC) technology. Although there have been several reviews on the modeling and control of the MSWI process, there is a lack of focus on model predictive control (MPC), a widely used APC technology. This article aims to comprehensively review MPC strategies in the MSWI process. First, it describes MSWI process technology in detail, examining control issues and objectives to highlight the complexity and challenges in controller design while providing an overview of MPC methods and their benefits. Second, it reviews incinerator modeling for control, including traditional modeling techniques and machine learning technologies such as fuzzy neural networks. Third, it reviews the controllers used for MSWI process, emphasizing the advantages of MPC over existing control methods. Fourth, it discusses the current status of MPC design and online updates, covering the need for an accurate dynamic predictive model and objective function and the online updates components such as predictive modeling, rolling optimization, and feedback correction. Finally, the study concludes with a summary of the findings.

1. Introduction

Pollution control and prevention is crucial for the sustainable development of the global economy and environment and for protecting public health, making it a global priority [1,2,3,4]. Municipal solid waste (MSW) is growing annually from 8% to 10% worldwide [5,6]. In developing countries like China, more cities are at risk of being overwhelmed by MSW, causing significant environmental issues [7,8]. As part of its national strategy, the developing country of China has prioritized “pollution prevention and control” [9] and aims to protect “blue skies, clear waters, and clean land” by 2035, emphasizing the need for continued efforts [5].

Municipal solid waste incineration (MSWI) is a complex system that converts waste into energy (WTE) [10,11,12,13]. It plays a key role in addressing urban environmental challenges and supporting renewable energy recycling [14,15]. With the push for “net-zero” goals [16], developing countries like China have promoted MSWI for over 30 years. By January 2024, there were 948 operational MSWI plants, with 137 new plants in 2023. Although MSWI is a scientifically validated waste treatment method, emissions from these plants have made them significant pollution sources [17,18,19], often facing opposition due to the “not in my backyard” (NIMBY) effect [20,21].

The combustion stage in the MSWI process is the main source of environmental indicators (EIs) like NOx, CO, HCl, and SO₂. It also generates trace dioxins, which are difficult to measure [5,22,23]. The flue gas cleaning stage then reduces these pollution emissions by transferring them to fly ash through adsorption and catalysis. Thus, controlling the combustion stage is a key research focus of MSWI process [5]. The “3T + E” principle, which stands for temperature, time, turbulence, and excess air, is critical for effective combustion in MSWI processes. Experience from grate-type MSWI plants in developed countries shows that following the “3T + E” principle—maintaining furnace temperatures above 850 °C [24,25], ensuring flue gas residence time over 2 s, promoting sufficient turbulence, and using an appropriate excess air coefficient—can effectively decompose and combust harmful substances [26,27].

Researchers have identified the main manipulated variables (MVs) in this stage as feed rate, grate speed, and primary/secondary air flow, with key-controlled variables (CVs) including combustion line length, furnace temperature, flue gas oxygen content, and steam flow. Variables like flue gas residence time, turbulence intensity, excess air coefficient, and MSW layer thickness are auxiliary variables (AVs). These insights led to the development of the automatic combustion control (ACC) system [28,29], incorporating advanced proportional-integral-derivative (PID) controllers [30], combustion characteristic curves [31], and expert rule bases [5,32]. This system ensures the stable, long-term operation of MSWI processes, provided that MSW composition and calorific value remain consistent. According to World Bank statistics and data from Gu and Yamada et al. [33], MSW in developed countries is sorted for a long time, resulting in stable calorific value fluctuations. In contrast, developing countries like China are still improving MSW classification policies and management systems step by step. Consequently, collected MSW in these regions is often unpredictable in composition, with low and fluctuating calorific values [5]. Additionally, high-precision instruments needed for detecting key controlled variables like combustion line length and material layer thickness are prone to damage and are costly. This makes it challenging to directly apply ACC systems from developed countries to MSWI operations in developing countries like China [34]. Currently, MSWI plants primarily rely on manual control, often requiring interventions from domain experts [35]. This control model operates as a “perception–prediction–cognition–execution” system, driven by the experts’ accumulated experience. However, this dependence on domain engineers is increasingly inadequate for the advanced demands of intelligent incineration in modern MSWI plants [36]. Consequently, many plants struggle to meet national regulatory standards for pollution emissions. As artificial intelligence (AI) technology becomes essential for optimizing complex industrial processes [37], developing advanced intelligent control systems for MSWI that can match or surpass expert engineers’ capabilities is crucial for the industry’s sustainable development [38].

Model predictive control (MPC), a key advanced process control (APC) technology [39,40], has garnered significant academic attention for its predictability and robustness [41]. It is widely applied in industrial processes [42,43]. MPC works by predicting future dynamic changes in complex processes, constructing an approximate model, and solving optimization problems to determine optimal control actions within a set time frame. Recently, MPC has been increasingly used in the MSWI process.

Several review articles have been published on related topics. Ding et al. [44] describes the typical MSWI process and its control requirements, classifying and summarizing existing methods like PID, fuzzy, neural network, and hybrid controls while highlighting the complex nature of MSWI control. However, it does not address MPC in depth. Tang et al. [5] discusses intelligent optimization control technology for MSWI, analyzing process flow and control characteristics, emphasizing its environmental importance, and reviewing intelligent control aspects like modeling and control strategies. It offers a multidimensional outlook on future research but does not detail existing MPC methods or algorithms. Wang et al. [45] examines AI technology’s role in optimizing MSWI control and categorizing AI applications in model building, control, optimization, and maintenance. However, it also lacks an in-depth exploration of MPC algorithms.

Existing reviews on intelligent control for the MSWI process have not thoroughly addressed MPC. This article seeks to fill this gap by systematically reviewing MPC research for MSWI, examining aspects like process complexities, MPC advantages, incinerator modeling, controller design, and online updates. The databases reviewed include Web of Science (WoS), IEEE Xplore, ScienceDirect, SpringerLink, and CNKI, using keywords such as (“Waste-to-energy” OR “WTE” OR “MSWI” OR “municipal solid waste incineration” OR “MSWC” OR “municipal solid waste combustion”) AND (“Model Predictive Control” OR “Predictive Control” OR “MPC”).

The remainder of this article is organized as follows. Section 2 introduces the MSWI process and control complexities as well as MPC methods and advantages. Section 3 analyzes the current status of incinerator modeling for control. Section 4 reviews the references on controllers used in MSWI. Section 5 discusses the current status of MPC design and online updates in detail. Finally, Section 6 concludes this article.

2. Description of Municipal Solid Waste Incineration (MSWI) Process and the Model Predictive Control (MPC) Method

2.1. Typical Process Flow Description

In the MSWI process using a grate furnace, MSW is fed into the incinerator by a grab and sequentially undergoes drying [46], combustion, and burnout stages. This occurs with the help of combustion-supporting air, high-temperature radiation, and heating [47]. The organic matter in the MSW is gasified and pyrolyzed at high temperatures, releasing heat and destroying pathogenic organisms like viruses and bacteria through high-temperature incineration [48]. The entire process involves six subsystems [49]: solid waste fermentation, solid waste combustion, waste heat exchange, steam power generation, flue gas cleaning, and flue gas emission, as illustrated in Figure 1.

The MSWI process flow is as follows [5,45]:

MSW is transported to the plant by dedicated compactor collection vehicles, weighed at a weighbridge, and then dumped into the MSW deposit pool from the unloading platform. In the pool, the MSW is thoroughly crushed, mixed, and stacked using a crane grab, facilitating natural fermentation and dehydration by microorganisms [50]. This fermentation process, which typically takes 5–7 days, increases the calorific value of the solid portion by about 30%, thereby enhancing its efficiency and stability during combustion. The grab lifts the fermented MSW and sends it to the hopper, where it slides into the chute and is pushed into the incinerator by the feeder. The MSW is dried under the heat radiation of the furnace wall and the blowing and baking of 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 combustion). After several hours of high-temperature combustion, the combustible components are fully burned, generating heat, and the non-combustible ash is pushed out of the furnace by the burnout grate. As the high-temperature flue gas from MSW combustion passes through various boiler heating surfaces, it undergoes absorption and cooling. Toxic substances and heavy metals are meticulously treated through processes like denitrification, desulfurization, dust removal, and ash collection. These processes transform harmful emissions into non-toxic gases that meet stringent environmental standards, which are then safely discharged into the atmosphere through a chimney with the help of an induced draft fan [51]. 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 the generator to produce electricity [52].

The primary goal of the MSWI process is the harmless treatment of MSW, with power generation or heat production as secondary objectives. Typically, the power output of the steam turbine generator in an MSWI plant aligns with the state of the incinerator, and the external power grid does not limit generator power dispatching [53]. Therefore, the primary goal of the MSWI process’s automatic control system is to achieve stable combustion of MSW, stabilize boiler steam production, minimize the slag’s loss on ignition rate, and reduce pollutant emissions as much as possible.

The detailed descriptions of each subsystem are as follows.

2.1.1. Solid Waste Fermentation Subsystem

The solid waste fermentation subsystem in an MSWI plant is responsible for receiving and storing MSW. Typically, MSW is transported by vehicles, weighed, recorded by a weighing system, and then dumped into the deposit pool through the unloading platform and port [50]. Large MSW requires coarse crushing before entering the pool. The purpose of storing MSW in the deposit pool is for dewatering and fermentation. The grab stirs the MSW to evenly distribute its components and remove some mud and sand. The MSW deposit pool is usually designed to store enough MSW for 5–7 days of incineration.

This subsystem includes the MSW deposit pool, grab, crusher, feed hopper, and fault removal and monitoring devices [54]. The MSW deposit pool provides a space for storage, mixing, and removal of large MSW. A large MSWI plant typically has one deposit pool to supply 3–4 incinerators, each with multiple feed hoppers, usually served by 1–2 cranes and grabs above the pit. Operators monitor screens or visually check the speed at which MSW slides into the furnace from the feed hopper to determine feed frequency. If large objects block the feed port, the fault removal device in the feed hopper can push the objects back into the deposit pool [55]. Operators can also use the grab to pick up large items and send them to the crusher above the deposit pool for crushing [56].

2.1.2. Solid Waste Combustion Subsystem

The solid waste combustion subsystem is the key of an MSWI plant, determining the overall process flow and equipment configuration [57]. This subsystem typically includes an incinerator, feeder, combustion air supply equipment, auxiliary fuel supply and combustion equipment, reagent supply equipment, and slag discharge and treatment devices [58]. Key components within the incinerator are the grate and combustion chamber [59]. The grate is usually a mechanically movable surface where MSW is flipped and burned. Above the grate, the combustion chamber allows several seconds of residence time for combustion gases. Primary air is sprayed from below the grate to mix with the MSW layer, while secondary air from above enhances the mixing of combustion gases.

The combustion process of MSW involves the following steps: ① evaporation of moisture on the solid surface; ② evaporation of moisture inside the solid; ③ ignition and combustion of volatile components in the solid; ④ surface combustion of solid carbon; ⑤ completion of combustion. These stages can be grouped into two main processes: drying (steps 1 and 2) and combustion (steps 3 to 5). Combustion is further categorized into primary and secondary stages, where the primary stage initiates combustion, and the secondary stage completes it. MSW combustion primarily relies on decomposition combustion, and primary air alone is insufficient for complete combustion [60]. Its role is to burn volatile components and decompose macromolecular elements. During primary combustion, CO₂ may sometimes be reduced, significantly affected by temperature. In secondary combustion, flammable gases and particulate carbon produced during primary combustion undergo homogeneous gaseous combustion [61]. The efficiency of this process, indicated by CO concentration, is crucial not only for energy recovery but also for suppressing the formation of harmful dioxins, thus ensuring environmental safety. Therefore, the incineration process in a grate furnace must be designed based on these combustion mechanisms and characteristics. In MSWI plants, incinerators can be divided into different zones: the solid-phase MSW combustion zone on the grate, the gas-phase component combustion zone, the selective non-catalytic reduction (SNCR) [62] denitrification zone, the waste heat boiler heat exchange zone, and the flue gas cooling zone [63].

The primary objectives of an MSWI power plant are the reduction, harmless treatment, and resource utilization of MSW. Specific process parameters in the incinerator include ① combustion temperature: above 850 °C (900 °C is optimal); ② flue gas residence time: more than 2 s; ③ CO concentration: below 100 mg/m³ (1-h average); ④ stable combustion: avoid instantaneous CO concentrations above 100 × 10⁻⁶ g/m³ as much as possible; ⑤ daily management: set up thermometers, continuous CO analyzers, and continuous O₂ analyzers to monitor and control combustion process parameters in real time [64].

2.1.3. Waste Heat Exchange Subsystem

After cooling, the high-temperature flue gas from the incinerator can be treated through waste heat recovery and water spray cooling [65]. Waste heat boilers recover heat in three ways, namely, producing steam for power generation, combining heat and power, and providing hot water. Waste heat boilers in MSWI plants are either duct-type or integrated. Duct-type boilers are similar to conventional boilers, where fully combusted MSW in the incinerator allows flue gas to undergo heat exchange, reducing its temperature and generating steam or hot water [66]. Integrated boilers combine waste heat boilers and incinerators, often using water-cooled walls as combustion chambers [67].

2.1.4. Steam Power Generation Subsystem

In the power generation process, waste heat boilers transfer heat from the flue gas to water, transforming it into superheated steam with a precise pressure and temperature, which is critical for efficient energy conversion. This steam drives a turbo generator, converting thermal energy into electrical energy [68]. Practice shows significant heat loss during conversion, influenced by waste calorific value, boiler thermal efficiency, and turbo generator efficiency [69]. Combined heat and power can improve efficiency, as the turbine and generator efficiency during steam power generation is high (62–67%), whereas direct heating efficiency is relatively higher.

2.1.5. Flue Gas Cleaning Subsystem

Flue gas from the incinerator is the primary source of pollution in the MSWI process [70], containing a wide range of particulate and gaseous pollutants. A flue gas treatment system is required to remove these pollutants and achieve compliant emission standards. Particulate pollutants are removed by gravity settling, centrifugal separation, electrostatic precipitation, and bag filtration. Gaseous pollutants like SO₂, NO₂, HCL, and organic gases are purified through absorption, adsorption, and redox technologies [71,72]. The main equipment includes settling chambers, which remove larger particles by gravity; cyclone dust collectors, which use centrifugal force to capture finer particles; electrostatic precipitators, which apply electric charges to remove particulates; scrubbers, which neutralize gaseous pollutants through chemical reactions; and bag filters, which trap particulate matter in fabric bags.

2.1.6. Flue Gas Emission Subsystem

After dust removal, flue gas is discharged into the atmosphere by an induced draft fan, monitored by a continuous emission monitoring system (CEMS). CEMS continuously monitors flue gas emissions, detecting concentrations of key pollutants such as particulate matter (PM), CO, SO₂, NOx, HCl, and O₂, along with operational parameters including flow rate, temperature, and pressure. CO₂ is also monitored, primarily for greenhouse gas reporting. It displays, prints, and transmits data to the MSWI process control system and regulatory authorities [5]. CEMS includes a gas pollutant content analysis system, dust concentration monitoring, flow detection system, and data collection and analysis system. The gas pollutant content analysis system comprises sampling probes, pipelines, pre-treatment systems, and analyzers. Common analysis methods are dilution, extraction, and direct measurement, with the extraction method being prevalent in MSWI plants.

2.2. Control Problem Description and Control Target Analysis

The MSWI process is a complex system of multiple industrial equipment, converting MSW into electricity, non-toxic flue gas, and non-combustible ash. To optimize operational efficiency, combustion quality, material consumption, and pollutant emission concentrations, multiple subsystems must be coordinated and controlled for whole-process optimization of this complex industrial process [73].

As shown in Figure 1, the MSWI process’s six subsystems can be summarized into four main stages: solid waste fermentation, solid waste combustion, waste heat exchange, and flue gas cleaning. During the solid waste fermentation stage, MSW undergoes biological fermentation and dehydration in storage tanks for 3 to 7 days, then is fed into hoppers by mechanical grabs for incineration. The main variable at this stage is the MSW calorific value, a key factor in optimizing the MSWI process. In the solid waste combustion stage, fermented MSW is fed into the incinerator, dried, burned, and fully combusted using push feeders and grates, eventually converting into ash and high-temperature flue gas. Key MVs at this stage are MSW feed rate, grate speed, and furnace air intake, while main CVs include furnace temperature, flue gas oxygen content, steam flow, and combustion line. In the waste heat exchange stage, high-temperature flue gas transfers heat through water-cooled walls and equipment, generating high-temperature and high-pressure steam that drives the turbo generator. Some steam is used for heating. Cooling rate control is crucial, with the main MV being boiler feed water quantity and the main CV being boiler steam flow. In the flue gas cleaning stage, flue gas undergoes purification processes such as SNCR denitrification, semi-dry desulfurization, activated carbon adsorption, and bag filtration. Finally, flue gas that meets emission standards is discharged into the atmosphere through a chimney. Key MVs are the consumption amounts of urea, activated carbon, and lime, while the main EI is pollutant emission concentration.

Based on the above analysis, the key to combustion process control is ensuring stable combustion through “air and material distribution” operations. Actuators of key MVs include primary and secondary air’s damper blades; grate equipment for drying, burning, and burnout speed; and their hydraulic power units. Key CVs include furnace temperature (FT), flue gas oxygen content (FGOC), steam flow (SF), and combustion line. The detailed description of these CVs is as follows.

Various physical and chemical reactions, heat and mass transfer, and multiphase fluid flow in the MSW combustion process, coupled with high temperature, high pressure, and multiphase interactions, make stable control of furnace temperature particularly important. The furnace temperature is typically measured by thermocouples, an important parameter characterizing combustion stability and directly related to pollutant emissions. Its precise control is necessary. The “3T + E” principle states that the furnace’s operating temperature should be maintained between 850 and 950 °C. This range facilitates the complete combustion of MSW, enhances burnout degree, achieves desired volume and mass reduction rates, and reduces pollutant concentration in the emitted flue gas.

Flue gas oxygen content is closely related to incineration efficiency and pollutant emissions. In practical operations, sufficient oxygen is needed for complete combustion, desulfurization, and denitrification in the incinerator. However, excessive oxygen increases flue gas loss, fan power consumption, and exhaust gas treatment scale, carrying away much heat and dust, as well as increasing fuel-derived NOx pollutant emissions. Conversely, a low excess air coefficient leads to increased incomplete combustion heat loss; reduced incineration efficiency; and production of large amounts of toxic gases such as dioxins, CO, and SO₂. Therefore, controlling flue gas oxygen content within a reasonable range is crucial for improving incineration efficiency, reducing pollutant emissions, and enhancing economic benefits.

During the MSWI process, waste burns on the grate with oxygen supplied to both the solid waste layer and the gas phase through the grate and furnace walls. The resulting flue gas, containing fly ash particles, transfers heat to the steam system, which is then converted into thermal and/or electrical energy. Economically, maximizing waste treatment and energy output are mutually supportive, as more material means more energy output. However, considerations of equipment lifespan impose constraints on process variables, particularly those affecting corrosion, such as furnace temperature and boiler steam flow. There is a direct conflict between economic goals, which demand high furnace temperatures and steam flow, and equipment lifespan, which can be compromised by these conditions. Therefore, stabilizing boiler steam flow is essential for maintaining both operational stability and equipment longevity [74].

Taking the key CVs of the MSWI process described above as an example, the multivariable intelligent optimization control problem can be described as follows:

$$\begin{array}{l}\left\{Y_{\mathrm{Tem}}^{*},Y_{\mathrm{Oxy}}^{*},Y_{\mathrm{Ste}}^{*}\right\}=\arg \left(\min \left\{\Vert R-R^{*}\Vert \right\}\right)\\ \mathrm{s}.\mathrm{t}.\{\begin{array}{l}\mathrm{Equality} \mathrm{constraints}:\{\begin{array}{l}\widehat{R}=f_R(R,\left\{Y_{\mathrm{Tem}},Y_{\mathrm{Oxy}},Y_{\mathrm{Ste}}\right\},D)\\ \left\{{\widehat{Y}}_{\mathrm{Tem}},{\widehat{Y}}_{\mathrm{Oxy}},{\widehat{Y}}_{\mathrm{Ste}}\right\}=f_Y(\left\{Y_{\mathrm{Tem}},Y_{\mathrm{Oxy}},Y_{\mathrm{Ste}}\right\},\left\{U_{\mathrm{Tem}},U_{\mathrm{Oxy}},U_{\mathrm{Ste}}\right\},D)\\ \left\{{\widehat{U}}_{\mathrm{Tem}},{\widehat{U}}_{\mathrm{Oxy}},{\widehat{U}}_{\mathrm{Ste}}\right\}=f_U((Y_{\mathrm{Tem}}^{*}-Y_{\mathrm{Tem}}),(Y_{\mathrm{Oxy}}^{*}-Y_{\mathrm{Oxy}}),(Y_{\mathrm{Ste}}^{*}-Y_{\mathrm{Ste}}),D,{\theta }_{\mathrm{Tem}},{\theta }_{\mathrm{Oxy}},{\theta }_{\mathrm{Ste}})\end{array}\\ \mathrm{Inequality} \mathrm{constraints}:\{\begin{array}{l}{R}^{\min }\le R\le R^{\max }\\ Y_{\mathrm{Tem}}^{\min }\le Y_{\mathrm{Tem}}\le Y_{\mathrm{Tem}}^{\max }\\ Y_{\mathrm{Oxy}}^{\min }\le Y_{\mathrm{Oxy}}\le Y_{\mathrm{Oxy}}^{\max }\\ Y_{\mathrm{Ste}}^{\min }\le Y_{\mathrm{Ste}}\le Y_{\mathrm{Ste}}^{\max }\\ U_{\mathrm{Tem}}^{\min }\le U_{\mathrm{Tem}}\le U_{\mathrm{Tem}}^{\max }\\ U_{\mathrm{Oxy}}^{\min }\le U_{\mathrm{Oxy}}\le U_{\mathrm{Oxy}}^{\max }\\ U_{\mathrm{Ste}}^{\min }\le U_{\mathrm{Ste}}\le U_{\mathrm{Ste}}^{\max }\end{array}\end{array}\end{array}$$

(1)

where $\left\{Y_{\mathrm{Tem}},Y_{\mathrm{Oxy}},Y_{\mathrm{Ste}}\right\}$, $\left\{Y_{\mathrm{Tem}}^{*},Y_{\mathrm{Oxy}}^{*},Y_{\mathrm{Ste}}^{*}\right\}$, and $\left\{{\widehat{Y}}_{\mathrm{Tem}},{\widehat{Y}}_{\mathrm{Oxy}},{\widehat{Y}}_{\mathrm{Ste}}\right\}$ are the actual, desired, and dynamically modeled estimated values of the furnace temperature, flue gas oxygen content, and boiler steam flow, respectively; $R$, $R^{*}$, and $\widehat{R}$ are the actual, desired, and dynamically modeled estimated values of the operating indicators, respectively; $U$ is the manipulated variable of the basic control loop; $U_{\mathrm{Tem}}$ and ${\widehat{U}}_{\mathrm{Tem}}$ are the actual output value and dynamically modeled estimated value of the furnace temperature controller, respectively; $U_{\mathrm{Oxy}}$ and ${\widehat{U}}_{\mathrm{Oxy}}$ are the actual output value and dynamically modeled estimated value of the flue gas oxygen content controller, respectively; $U_{\mathrm{Ste}}$ and ${\widehat{U}}_{\mathrm{Ste}}$ are the actual output value and dynamically modeled estimated value of the boiler steam flow controller, respectively; $f_R(\cdot )$, $f_Y(\cdot )$, and $f_U(\cdot )$ are the unknown nonlinear dynamic functions of the operating indicators, controlled objects, and controllers, respectively; $D$ is external disturbances including MSW composition fluctuations, operating condition drifts, equipment wear, and maintenance; ${\theta }_{\mathrm{Tem}},{\theta }_{\mathrm{Oxy}}$ and ${\theta }_{\mathrm{Ste}}$ are the parameters of the furnace temperature, flue gas oxygen content, and boiler steam flow controllers, respectively; and superscripts min and max denote the minimum and maximum values of the variables.

In summary, the various control processes within an MSWI plant are interdependent and must be coordinated to maintain the incineration process within specified ranges [75]. To meet environmental regulations and optimize operation, the control objectives are as follows: (1) a furnace temperature above 850 °C with minimal fluctuations; (2) consistent thermal output, or uniform steam flow; (3) stable oxygen content in flue gas at the boiler outlet. These targets ensure effective operation [5]. However, due to the uncertain composition and calorific value of MSW, the combustion process often operates under dynamic and variable conditions. Effective controllers are necessary to achieve operational goals, guiding the design of MSWI process controllers.

2.3. Control Difficulties and Complexities Analysis

The stable operation of an MSWI plant relies on coordinating six subsystems: solid waste fermentation, combustion, waste heat exchange, steam power generation, flue gas cleaning, and emission. Each subsystem is distinct, influenced by external disturbances, and interconnected through material and energy flows (MSW, flue gas, steam, fly ash). This leads to multivariable coupling, nonlinearity, and time-varying complexities, detailed as follows.

(1)

Online real-time detection of key EIs

From the perspective of control science research, the normal operation of intelligent optimization control, intelligent optimization decision making, and integrated intelligent optimization decision making and control systems depend on real-time detection of key operating indicators [76]. However, in the actual MSWI process, due to limitations in detection technology and the characteristics of key EIs, some are difficult to detect in real-time online. For example, the dioxins (DXN) emitted by MSWI require high specificity, selectivity, and sensitivity in detection technology [5]. Currently, commonly used DXN detection methods are offline direct detection methods, such as high-resolution gas chromatography/high-resolution mass spectrometry (HRGC/HRMS), bioassay, and immunoassay [77]. The main means of monitoring DXN emissions in industrial sites and environmental protection departments are still the above-mentioned offline direct detection methods, characterized by high complexity, large time lag scales (monthly/weekly), laboratory analysis, and high cost. In this context, to reduce DXN emission concentrations, most MSWI power plants adopt cost-insensitive adsorption strategies, which not only increase operating costs but also increase fly ash treatment costs.

(2)

Multivariable coupling characteristics

From the previous analysis, it is clear that the process of MSW from entering the incinerator feed hopper to burning in the furnace to final burnout into slag involves major operating objects, including primary air volume, secondary air volume, drying grates, combustion grates, burnout grates, and their hydraulic power units. The main controlled variables include furnace temperature, flue gas oxygen content, main steam flow, furnace negative pressure, material layer thickness, and combustion line. The MSWI process system consists of multiple operating objects that need to closely coordinate and work together to complete the whole process, exhibiting significant multivariable characteristics. Additionally, there is a strong coupling between multiple controlled variables and manipulated variables across several loops. Strong coupling exists between inputs like primary and secondary air flows and outputs such as furnace temperature and flue gas oxygen content. Additionally, there is a strong coupling between the system’s inputs such as primary and secondary air volumes and the system’s outputs such as furnace temperature and flue gas oxygen content outputs, as well as between changes in feeder and grate speed and primary air flow.

(3)

Nonlinear and time-varying characteristics

Typically, the designed annual operating time of the MSWI process is not less than 8000 h, making the operating process prone to changes in dynamic characteristics due to condition drift caused by equipment wear and MSW components. Additionally, seasonal and climatic changes lead to changes in the composition and characteristics of MSW fed into the furnace, also causing changes in the characteristics of the MSWI process. Clearly, the unknown time-varying characteristics of numerous disturbance factors can significantly impact and disrupt the operation of the MSWI process. Due to the complex and variable composition of MSW, condition drift in operating conditions, and factors such as equipment failure and maintenance, there are unknown nonlinear dynamic relationships between various process parameters, making it difficult to qualitatively and quantitatively describe them using mathematical and linear relationships. For example, there are obvious nonlinear dynamic relationships between primary air flow and flue gas oxygen content, secondary air flow and furnace temperature, primary and secondary air flows and flue gas oxygen content, and grate speed and boiler steam flow. Additionally, other influencing factors, such as leachate recirculation, turbine power generation, and boiler feedwater, will also change with the actual operating conditions, further exacerbating the nonlinear and time-varying characteristics affecting the MSWI process.

(4)

Time-lag characteristics

The MSWI system is characterized by slow processing and large time lag system. Various process equipment and subsystems in the incineration process, such as waste grabs, grate propulsion speed, slag extractors, boiler drums, flue gas turbulence, and slag conveying processes, all exhibit typical time lag characteristics. Additionally, the basic loop control of air and material distribution also has a lag, causing a significant delay in the process from MSW incineration initiation to the final production of ash and flue gas and obtaining flue gas emission detection values.

(5)

Multi-source disturbed and uncertain dynamic characteristics

The actual MSWI process has numerous disturbing factors and significant uncertain dynamic characteristics, primarily related to the composition and calorific value of MSW. Significant fluctuations in MSW composition are reflected in changes in the proportion of combustibles, causing the combustion line on the grate to move forward or backward accordingly. Accordingly, operating experts need to predict MSW composition changes based on changes in furnace flames and process data and then intervene in grate operating speed control. Additionally, if the calorific value of MSW is high enough, the incineration process can maintain furnace temperature using the flames produced by its own combustion; otherwise, auxiliary fuel is needed for combustion, introducing corresponding uncertain disturbances.

2.4. Model Predictive Control (MPC) Method and Its Advantages

Research indicates that the stable operation of the MSWI process is crucially dependent on the incinerator. The combustion process is characterized by complex physical and chemical reactions, making it a multiple input multiple output (MIMO) system with multiple loops, strong coupling, and nonlinearity, common in process industries [78]. Effective control and stability rely on coordinating various manipulated and controlled variables, while also addressing MSW variability, equipment maintenance, and operational management. Therefore, APC technologies [79,80,81] are essential for managing the complex and nonlinear nature of the combustion process.

Model predictive control (MPC) [82,83] is one of the most widely used APC technologies [84] and has been extensively researched and applied in fields such as urban wastewater treatment, cement kilns, and blast furnace ironmaking. MPC is a multi-objective [85] discrete control method that solves an optimization problem at each iteration to determine the optimal control law within a finite time horizon and specified constraints [86]. This iterative optimization process minimizes the objective function to derive the best control sequence, of which only the first control action is applied to the process [87].

The classic control structure of MPC is shown in Figure 2, where $u$ is the control input (i.e., MV), $y_r$ is the setpoint reference trajectory, $y$ is the control output (i.e., CV), $e$ is the system error, $d$ is the external disturbance, $\widehat{y}$ is the predicted model output value, $\epsilon$ is the prediction error, and $y_p$ is the feedback-corrected output value.

As shown in Figure 2, MPC mainly consists of three parts: predictive model, rolling optimization, and feedback correction. The functions of each part are as follows.

(1)

Predictive model: It is used to predict the future behavior of the system. By establishing a mathematical model of the system, the predictive model can estimate future outputs based on current and past data for use in subsequent rolling optimizations. Its inputs are the historical and current state, and its output is the predicted future CV. For data-driven controlled variable predictive models, it is typically described as a nonlinear autoregressive exogenous (NARX) system:

$$\begin{array}{c}\widehat{y}(t)=f[\widehat{y}(t-1),\widehat{y}(t-2),\cdots ,\widehat{y}(t-n_y),\\ \mathit{u}(t-1),\mathit{u}(t-2),\cdots ,\mathit{u}(t-n_u)]\end{array}$$

(2)

where $f(\cdot )$ is the unknown nonlinear multiple-input multiple-output model used to construct the predictive model; $\widehat{y}$ and $\mathit{u}$ are the system’s output value and input vector, respectively; and $n_y$ and $n_u$ are the maximum lags of the system output and input, respectively.

(2)

Rolling optimization: It is used to minimize a predefined objective function by optimizing the future control input sequence. Rolling optimization is performed at each sampling instant, and only the first optimized control input is executed, with the remaining parts being re-optimized at the next instant. The implementation of the optimization algorithm is based on the accurate prediction of controlled variables by the predictive model and the design of the optimization algorithm. As a necessary condition for MPC, the objective function describes various error metrics and energy usage as a quadratic function. In classic MPC, the objective function is usually set as follows:

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

(3)

where $H_p$ and $H_{\mathrm{u}}$ are the prediction and control horizons set by the system, respectively (i.e., the steps of the predicted values and control law sets at each system iteration, satisfying $H_p>H_{\mathrm{u}}$ [42]); $e\left(t\right)=y_r\left(t\right)-y_p\left(t\right)$ is the system error vector at time $t$; $\Delta u\left(t\right)$ is the change in control law at time $t$; $w_{i_{\mathrm{p}}}^y$ and $w_{j_{\mathrm{u}}}^u$ are the weight parameters of the objective function; and $J$ is the online optimization control objective function of the system.

(3)

Feedback correction: It is used to adjusts the predictive model output $\widehat{y}$ at each control instant to address model mismatches and external disturbances. Through real-time measured feedback, MPC can correct the predictive model’s error, thereby maintaining system stability and performance. The general form of feedback correction can be expressed as

$$y_p\left(t\right)=\widehat{y}\left(t\right)+K\left(\widehat{y}\left(t\right)-y\left(t\right)\right)$$

(4)

where $K>0$ is the feedback gain.

The main advantages of MPC include [41]: (1) applicability to uncertain MIMO systems with constraints, significant delays, nonlinearity, and randomness; (2) rolling optimization and feedback correction mechanisms that compensate for uncertainties due to model mismatches and external disturbances; (3) prediction of future system behavior using a predictive model, allowing potential issues to be anticipated and addressed, which ensures accurate control strategies and system stability; (4) optimization of the control input sequence at each step to improve system performance indicators (e.g., energy consumption, efficiency, quality), enhancing overall system performance and economic benefits. Consequently, MPC is widely recognized in complex industrial process control.

3. Current Status of Incinerator Modeling for Control

Combustion is a natural process that converts the chemical energy (potential energy) of materials into thermal and radiant energy (kinetic energy) through rapid heat release. As a low-quality fuel, municipal solid waste (MSW) contains high moisture, various non-combustibles, and pollutants, leading to compositional heterogeneity and size non-uniformity. Thus, the combustion complexity of MSW exceeds that of conventional fuels like coal and wood. The production of toxic gases (DXN, NOx, SO₂) during the MSWI process raises environmental concerns [89]. Effectively controlling and optimizing MSWI requires a precise understanding of the physical and chemical reactions involved [90]. The unique properties of MSW pose significant challenges. These challenges make it difficult to develop an accurate mathematical model for the combustion and heat exchange processes [91]. Therefore, understanding MSW combustion characteristics is crucial.

Complex industrial processes often involve mechanisms that are hard to describe. These processes also face high uncertainty in disturbances and fluctuating operating conditions. Therefore, constructing accurate data-driven models, guided by expert knowledge, is crucial for intelligent control research [92,93]. In the MSWI process, from a control perspective, the combustion and heat exchange processes involve manipulated variables such as feed rate, air flow rate, and grate speed; controlled variables like furnace temperature, steam flow, flue gas oxygen content, and combustion line position; and auxiliary variables (AVs) or disturbance variables (DVs) including MSW composition and calorific value, layer thickness, combustion state, equipment wear and maintenance status, and ash and slag buildup in the furnace.

Table 1 shows various modeling techniques for controlled objects for MSWI process control.

Table 1. Modeling technology of controlled objects for MSWI process control.

Method	MSWI Parameters	Comments and Summary	Author/Year	Reference
System identification and dynamic models based on first principles	· MVs: MSW inlet flow rate, grate speed, primary air flow rate, secondary air flow rate · CVs: Steam flow rate, flue gas oxygen content	· Model verification through system identification ensures the success of the model in practical applications; dynamic models contribute to optimizing process operational performance	Leskens et al., 2002	[94]
Auto-regressive with extra inputs (ARX) models	· MVs: MSW inlet flow rate, grate speed, primary air flow rate, secondary air flow rate · CVs: Steam flow rate, flue gas oxygen content	· Model reduction is applied to each transfer function of the MIMO ARX model separately, which simplifies the model while maintaining its essential dynamics	Leskens et al., 2002	[95]
System identification, state–space models	· MVs: Input mass flow rate of Ca (OH)2 · CVs/EIs: Outlet concentration of HCl, pressure drop across the baghouse filter	· Verified through system identification techniques, reliable and suitable for control, capable of reducing reagent consumption and optimizing control performance	Riccardo et al., 2022	[96]
Dynamic process and disturbance models obtained experimentally	· MVs: MSW inlet flow rate, grate speed, primary air flow rate, secondary air flow rate · CVs: Steam flow rate, flue gas oxygen content	· Capable of handling multivariable interactive processes, multiple conflicting objectives, and constraint conditions	Leskens et al., 2005	[97]
Neural networks models	· MVs: MSW inlet flow rate, grate speed, primary air flow rate, secondary air flow rate · CVs: Steam flow rate, flue gas oxygen content	· Data-driven neural network models can capture and describe the unique dynamic behavior of each controlled object	Qin et al., 2008	[98]
Multi-zone, zero-dimensional modeling approach	· MVs: Air flow rate, MSW supply rate, steam valve · CVs: Oxygen mole fraction in exhaust gas, steam generation rate, steam pressure	· Capable of simulating real plant operation data, accurately predicting system behavior, with relatively low computational resource requirements	Cho et al., 2017	[99]
Ensemble decision tree	· MVs: Air flow rate, MSW feed rate · CVs: Furnace temperature, flue gas oxygen concentration, steam flow	· Combines random forest and gradient boosting decision tree to construct a MIMO controlled object model with historical data	Wang et al., 2021	[100]
Cascaded transfer tunction, particle swarm optimization (PSO)	· MVs: Primary air flow rate · CVs: Furnace temperature, flue gas oxygen content, steam flow	· The predictive values of the established system model are consistent with the actual situation, capable of reflecting the dynamic characteristics of the system under actual working conditions	Chen et al., 2023	[101]
Random forest (RF) and gradient boosting decision tree (GBDT)	· MVs: Air flow rate of the left 1 drying grate, drying grate, right 1 air flow rate, total 37 variables · CVs: Furnace temperature, flue gas oxygen concentration, and steam flow	· Developed for actual industrial processes, provides a reliable engineering verification environment, and considers the MIMO characteristics of the MSWI process	Wang et al., 2023	[102]
Linear regression decision tree (LRDT)	· MVs: Primary air flow, secondary air flow, drying grate average speed, feeder average speed · CVs: Furnace temperature	· Furnace temperature model based on LRDT with leaf predictions based on a single path as the final output, and it has high interpretability	Xia et al., 2023	[103]
LRDT	· MVs: Primary air flow, secondary air flow, average drying grate speed, average feeder speed · CVs: Furnace temperature, flue gas oxygen content, and boiler steam flow rate	· Realize multivariable coupled control based on off-site data simulation	Wang et al., 2023	[104]
Mechanistic modeling based on material and energy balance	· MVs: MSW grab quantity, MSW push quantity, drying grate speed, combustion grate 1 speed, combustion grate 2 speed, ash grate speed, primary air flow and its branch flows, secondary air flow, induced draft fan flow · CVs: Furnace temperature, flue gas oxygen content, and steam flow	· Provides a mechanistic modeling based on material and energy balance, enabling independent analysis of the internal characteristics, and achieving dynamic simulation of the whole process by calculating the inputs and outputs of each subprocess	Ding et al., 2023	[105]
T-S FNN	· MVs: Primary air flow, secondary air pressure, primary air heating temperature · CVs: Furnace temperature	· Capable of reflecting the trend of furnace temperature changes with high fitting accuracy and realize furnace temperature control based on off-site data simulation	He et al., 2023	[106]
XGBoost-based serial-parallel ensemble model	· MVs: Feeder average speed, drying grate average speed, ammonia water injection amount, activated carbon injection amount, lime injection amount · CVs/EIs: Furnace temperature, boiler steam flow rate, flue gas oxygen content of G1 and G3, NOx, CO, CO2	· Provides a whole-process MSWI model in terms of MVs as inputs	Wang et al., 2024	[107]

Table 1. Modeling technology of controlled objects for MSWI process control.

Method	MSWI Parameters	Comments and Summary	Author/Year	Reference
System identification and dynamic models based on first principles	· MVs: MSW inlet flow rate, grate speed, primary air flow rate, secondary air flow rate · CVs: Steam flow rate, flue gas oxygen content	· Model verification through system identification ensures the success of the model in practical applications; dynamic models contribute to optimizing process operational performance	Leskens et al., 2002	[94]
Auto-regressive with extra inputs (ARX) models	· MVs: MSW inlet flow rate, grate speed, primary air flow rate, secondary air flow rate · CVs: Steam flow rate, flue gas oxygen content	· Model reduction is applied to each transfer function of the MIMO ARX model separately, which simplifies the model while maintaining its essential dynamics	Leskens et al., 2002	[95]
System identification, state–space models	· MVs: Input mass flow rate of Ca (OH)2 · CVs/EIs: Outlet concentration of HCl, pressure drop across the baghouse filter	· Verified through system identification techniques, reliable and suitable for control, capable of reducing reagent consumption and optimizing control performance	Riccardo et al., 2022	[96]
Dynamic process and disturbance models obtained experimentally	· MVs: MSW inlet flow rate, grate speed, primary air flow rate, secondary air flow rate · CVs: Steam flow rate, flue gas oxygen content	· Capable of handling multivariable interactive processes, multiple conflicting objectives, and constraint conditions	Leskens et al., 2005	[97]
Neural networks models	· MVs: MSW inlet flow rate, grate speed, primary air flow rate, secondary air flow rate · CVs: Steam flow rate, flue gas oxygen content	· Data-driven neural network models can capture and describe the unique dynamic behavior of each controlled object	Qin et al., 2008	[98]
Multi-zone, zero-dimensional modeling approach	· MVs: Air flow rate, MSW supply rate, steam valve · CVs: Oxygen mole fraction in exhaust gas, steam generation rate, steam pressure	· Capable of simulating real plant operation data, accurately predicting system behavior, with relatively low computational resource requirements	Cho et al., 2017	[99]
Ensemble decision tree	· MVs: Air flow rate, MSW feed rate · CVs: Furnace temperature, flue gas oxygen concentration, steam flow	· Combines random forest and gradient boosting decision tree to construct a MIMO controlled object model with historical data	Wang et al., 2021	[100]
Cascaded transfer tunction, particle swarm optimization (PSO)	· MVs: Primary air flow rate · CVs: Furnace temperature, flue gas oxygen content, steam flow	· The predictive values of the established system model are consistent with the actual situation, capable of reflecting the dynamic characteristics of the system under actual working conditions	Chen et al., 2023	[101]
Random forest (RF) and gradient boosting decision tree (GBDT)	· MVs: Air flow rate of the left 1 drying grate, drying grate, right 1 air flow rate, total 37 variables · CVs: Furnace temperature, flue gas oxygen concentration, and steam flow	· Developed for actual industrial processes, provides a reliable engineering verification environment, and considers the MIMO characteristics of the MSWI process	Wang et al., 2023	[102]
Linear regression decision tree (LRDT)	· MVs: Primary air flow, secondary air flow, drying grate average speed, feeder average speed · CVs: Furnace temperature	· Furnace temperature model based on LRDT with leaf predictions based on a single path as the final output, and it has high interpretability	Xia et al., 2023	[103]
LRDT	· MVs: Primary air flow, secondary air flow, average drying grate speed, average feeder speed · CVs: Furnace temperature, flue gas oxygen content, and boiler steam flow rate	· Realize multivariable coupled control based on off-site data simulation	Wang et al., 2023	[104]
Mechanistic modeling based on material and energy balance	· MVs: MSW grab quantity, MSW push quantity, drying grate speed, combustion grate 1 speed, combustion grate 2 speed, ash grate speed, primary air flow and its branch flows, secondary air flow, induced draft fan flow · CVs: Furnace temperature, flue gas oxygen content, and steam flow	· Provides a mechanistic modeling based on material and energy balance, enabling independent analysis of the internal characteristics, and achieving dynamic simulation of the whole process by calculating the inputs and outputs of each subprocess	Ding et al., 2023	[105]
T-S FNN	· MVs: Primary air flow, secondary air pressure, primary air heating temperature · CVs: Furnace temperature	· Capable of reflecting the trend of furnace temperature changes with high fitting accuracy and realize furnace temperature control based on off-site data simulation	He et al., 2023	[106]
XGBoost-based serial-parallel ensemble model	· MVs: Feeder average speed, drying grate average speed, ammonia water injection amount, activated carbon injection amount, lime injection amount · CVs/EIs: Furnace temperature, boiler steam flow rate, flue gas oxygen content of G1 and G3, NOx, CO, CO2	· Provides a whole-process MSWI model in terms of MVs as inputs	Wang et al., 2024	[107]

4. Current Status of Intelligent Controller Design for the MSWI Process

Effectively controlling the combustion process with numerous process parameters, strong coupling, and high nonlinearity has been a major research focus in both the industry and academia fields [108,109]. With advancements in industrial AI technology, developing intelligent control technologies for MSWI plants that match or exceed expert capabilities is essential for sustainable development.

Current reports highlight controllers aimed at achieving stable operation of key CVs in the MSWI process, including expert systems, human-simulated intelligent control, fuzzy logic control, machine learning control, and self-tuning PID control. However, these systems require significant manual adjustments and rely on precise knowledge of the MSWI process. MPC is anticipated to become the preferred method for future MSWI process control. MPC uses predictive models to forecast the system’s future behavior, including disturbances, ensuring effective control. The following sections and Table 2 review controllers used in the MSWI process.

4.1. Expert Systems and Human-Simulated Intelligent Control

Early MSWI process control relied on domain experts manually managing the incineration process based on experience, which often resulted in inaccuracies and slow response times. Expert systems simulate the IF-THEN rules of human reasoning [110], organizing domain experts’ knowledge systematically to enable automatic adjustment of control strategies. These systems assist in real-time process control decisions within complex systems [111]. They solve problems that require expert knowledge by using data, knowledge bases, and control mechanisms, incorporating expert reasoning methods through AI techniques [112]. However, expert systems struggle with dynamic, unstructured problems in the MSWI process, as they require detailed rules and knowledge bases that are difficult to update in real time and lack reasoning and approximation capabilities under uncertainty [113].

Human-simulated intelligent control (HSIC) offers a knowledge-driven architecture to address precise control issues in complex systems [114]. HSIC leverages existing expert experience and theoretical insights to generalize human control skills and reasoning logics, proving more effective in dynamic environments compared to expert systems [115]. HSIC integrates multiple knowledge sources, identifies process dynamic features online, uses heuristic and intuitive reasoning, and combines open-loop and closed-loop control with qualitative and quantitative decision making [116]. Recent studies in the MSWI process include Li et al. [116], who addressed pollution and energy waste with a human-simulated intelligent control algorithm; Ju et al. [117], who proposed an optimization strategy for secondary flue gas pollution based on HSIC, showing rapid process stabilization; Ni et al. [118] and Xiao et al. [119], who developed furnace temperature control strategies simulating expert cognitive mechanisms; and Wu et al. [120], who improved an HSIC temperature control algorithm with a PSO algorithm for furnace control.

Due to the complex dynamics of the MSWI process and the need for precise and rapid control responses, HSIC faces limitations in these environments. More mature intelligent control technologies, such as fuzzy logic control, neural network control, and MPC, are often used to handle nonlinear behavior, environmental changes, and model uncertainties in complex industrial processes [121,122].

4.2. Fuzzy Logic Control

Given the limitations of expert systems in managing uncertainties in the MSWI process, fuzzy logic control (FLC), based on fuzzy set theory [123], provides an effective approach to address uncertainty and disturbance noise through fuzzification, fuzzy inference, and defuzzification [124]. Unlike expert systems with explicit rules, FLC uses fuzzy rules derived from expert knowledge or experience [125], allowing it to manage fuzzy concepts and uncertain information effectively, which is advantageous in complex systems [126].

Implementing a fuzzy logic controller for the MSWI process involves the following steps.

First, precise system errors are used as inputs, mapped to fuzzy numbers between 0 and 1 using predefined membership functions [127], to manage process uncertainty.

Next, the fuzzy inference process uses fuzzified inputs and a fuzzy rule base to generate fuzzy outputs [128]. Fuzzy rules, typically derived from expert experience or historical data, are expressed as “IF-THEN” statements and are categorized into two types [129]: Mamdani [130] and Takagi–Sugeno–Kang (TSK) [131], as shown in (5)–(6), respectively:

$$\begin{array}{l}{\tilde{R}}^j: \mathrm{IF} x_1 \mathrm{is} {\tilde{X}}_1^j, x_2 \mathrm{is} {\tilde{X}}_2^j, \cdots , \mathrm{and} x_n \mathrm{is} {\tilde{X}}_n^j,\\ \mathrm{THEN} {\delta }_j={\tilde{Y}}_j \end{array}$$

(5)

$$\begin{array}{l}{\tilde{R}}^j: \mathrm{IF} x_1 \mathrm{is} {\tilde{X}}_1^j, x_2 \mathrm{is} {\tilde{X}}_2^j, \cdots , \mathrm{and} x_n \mathrm{is} {\tilde{X}}_n^j,\\ \mathrm{THEN} {\delta }_j=w_0^j+w_1^j{x}_1+\cdots +w_n^j{x}_n\end{array}$$

(6)

where ${\tilde{R}}^j$ is the jth fuzzy rule $\left(j=1,\cdots ,J\right)$, J is the total number of fuzzy rules, ${\tilde{X}}_n^j$ is the fuzzy set corresponding to the jth rule for the nth input of the FLS, ${\delta }_j$ is the consequent output corresponding to the jth rule, ${\tilde{Y}}_j$ is the fuzzy set, and ${\mathit{w}}^j=[w_0^j,\cdots ,w_n^j]$ is the weight of the consequent connection. This process enables the controller to simulate expert decision making by comprehensively considering relevant fuzzy rules to manage fuzzy information and determine the corresponding control actions.

Finally, the fuzzy inference results are converted into precise control actions through defuzzification. These actions can be directly applied to the controlled object or system [132]. This process achieves the desired control effect.

To implement these steps, the design of a fuzzy logic controller involves selecting appropriate membership functions and fuzzy rules. Membership functions typically include Gaussian, elliptical, and piecewise linear types [133], while fuzzy rules mainly consist of Mamdani type and TSK type rules. The choice of different membership functions and rules directly affects the controller’s performance, so they should be selected based on the specific needs of the problem. References [129,134] provide a detailed discussion of these issues from both theoretical and practical perspectives.

Research on FLC-based MSWI intelligent control was primarily concentrated in its early stages. Qian et al. [135] addressed the compensation problem of the constructed MSW moisture content estimation model by designing a pusher control strategy based on fuzzy rules. Shen et al. [136,137] proposed a fuzzy rule controller for the furnace temperature control problem. Chang et al. [138] summarized domain engineer experience into fuzzy control rules applied to an MSWI power plant in Shenzhen and designed a fuzzy rule controller. Notably, studies focusing solely on designing control strategies using fuzzy logic for the MSWI process have become rare. This is because the performance of a fuzzy logic controller heavily depends on its structural design, making it challenging to find the optimal fuzzy rules and membership functions for complex control systems like the MSWI process. Furthermore, once fuzzy rules and membership functions are set, they remain fixed, making it difficult to adapt to complex dynamic changes in the process, highlighting a lack of adaptability in the control system. Consequently, studies have introduced adaptive mechanisms to improve the performance of fuzzy logic controllers, such as integrating fuzzy logic with machine learning algorithms [139] or intelligent optimization algorithms [140].

4.3. Machine Learning Control

Machine-learning-based controllers [141,142] interact with the control system through trial and error, adaptively searching for the optimal control law $u^{\ast }$. In MSWI process intelligent control, machine learning controllers are mainly applied in controller design, self-tuning PID parameters, and constructing predictive models for MPC. Self-tuning PID parameters and predictive model construction for MPC will be discussed in detail in the following section. This section primarily introduces research on machine learning algorithms in controller design. Regarding its application in MSWI process intelligent control, it can be divided into neural network control and fuzzy neural control from the perspective of uncertainty handling [143].

4.3.1. Neural Network Control

Neural network control [144,145] leverages the nonlinear mapping ability and self-learning capability of artificial neural networks to achieve complex system control. It learns from data samples to simulate the system’s dynamic behavior and automatically adjusts the controller’s internal parameters based on system errors to achieve the desired control effect. Through learning, a neural network control system can continuously optimize via online learning, adapt to environmental changes and system parameter variations, and possesses nonlinear processing and adaptive learning capabilities. For the control problem of lime slurry flow at the desulfurization tower inlet of an MSWI power plant, You et al. [146] designed a neural controller using a BPNN and trained the artificial neural network with a large amount of process data accumulated under manual control mode to replace manual control. Luo et al. [147] incorporated neural network algorithms into the original traditional PID controller, effectively overcoming the problems of traditional PID control.

The adaptive capability of neural networks has led to their application in MSWI process control. However, the “black box” nature [148,149] of neural networks makes their decision-making process difficult to interpret, and they face limitations in handling uncertainty. Consequently, studies focusing solely on using neural networks for intelligent control are currently rare, with most existing research combining neural networks with other intelligent control techniques in a hybrid strategy.

4.3.2. Fuzzy Neural Network Control

Given the limitations of FLC in adaptability and the self-learning and self-adaptive capabilities of neural networks, Lee et al. [150] first introduced the fuzzy concept into neurons, proposing the fuzzy neural network (FNN). Subsequently, Strefezza et al. [151] used it to construct an FNN controller, employing the backpropagation algorithm to adaptively update the controller’s membership functions, achieving position tracking of a direct current motor. Since then, FNN has been widely applied in the intelligent control of industrial processes [152,153].

The FNN structure can be described as a connectionist model implementing FLC based on neural networks. It integrates traditional FLC functions into a network structure through distributed learning capabilities [154], translating control language into definite control laws while retaining the qualitative states and linguistic interpretations of fuzzy rules. Simultaneously, fuzzy rules and membership functions are adaptively optimized through backpropagation or other learning algorithms, achieving online updates of the network.

FNN combines the reasoning ability of FLC with the learning capability of NN, making it widely applicable in the MSWI process. For single-loop control of furnace temperature, Tian et al. [155] designed a furnace temperature controller based on the gradient descent method and a self-organizing algorithm of add–delete–reorganize mode; He et al. [156] designed a self-organizing FNN controller for furnace temperature and developed an event-triggered mechanism based on fixed thresholds to improve the update efficiency of the controller; Ding et al. [157] proposed an event-triggered online learning fuzzy neural robust controller, established a TS-FNN-based controlled object model and an online learning fuzzy neural controller for furnace temperature, and designed an event-triggered mechanism to reduce equipment wear while maintaining control performance. To further address uncertainty issues in furnace temperature control, Tang et al. [158] designed an IT2FNN controller to enhance interpretability and robustness against uncertainty, validating its effectiveness through simulation based on actual industrial data. For the simultaneous control of furnace temperature and flue gas oxygen content, Ding et al. [159] proposed a self-organizing FNN controller. However, the above studies apply to single operating conditions, and their generalizability requires enhancement. To reduce mechanical wear and energy consumption during the control process, Ding et al. [160] introduced an event-triggered mechanism in the controller, with experimental results showing that this control strategy reduced energy consumption by 66.23% and improved multivariable tracking control accuracy. Furthermore, to address strong coupling and time-varying dynamics in the control process, Ding et al. [161] proposed a collaborative event-triggered self-organizing FNN multivariable controller with multi-task learning to achieve adaptive multivariable control of the MSWI process, establishing an FNN multivariable controller for synchronous control of furnace temperature and flue gas oxygen content.

4.4. Self-Tuning PID Control

Due to their simple structure and ease of design, PID controllers account for about 95% of use in engineering automatic control systems [162,163] and have significant influence [164]. In actual incineration processes, the nonlinearity, time-variability, and large lag characteristics of the controlled object, combined with environmental noise and other external interference, make it difficult to establish an accurate mathematical model and determine optimal controller parameters. Under these conditions, traditional PID control methods have limitations in control accuracy, self-learning ability, and adaptability. As previously mentioned, intelligent algorithms, including fuzzy logic, neural networks, and metaheuristic algorithms, are often used for the adaptive selection of controller parameters [165]. Therefore, exploring PID control with online adaptive parameter tuning capabilities based on these intelligent algorithms has become a key research direction in the application of intelligent control in industrial processes [166].

Currently, PID controller parameter tuning methods are primarily divided into model-free tuning and tuning based on the controlled object model. In the MSWI process, given the difficulty in obtaining an accurate furnace temperature-controlled object model, existing research primarily focuses on model-free tuning based on intelligent algorithms. For furnace temperature control, He et al. [167] used a radial basis function neural network to tune the PID controller; Dai et al. [168] established a rule base suitable for the combustion process based on expert experience and designed a fuzzy PID controller; for multi-loop control of furnace temperature, flue gas oxygen content, and main steam flow, Wang et al. [104] designed a single-neuron adaptive PID controller; Ding et al. [169] constructed an adaptive PID controller based on a quasi-diagonal recurrent NN. Additionally, existing studies have reported research on self-tuning PID controllers based on metaheuristic algorithms [147], but these do not directly apply optimization algorithms to PID parameter tuning; instead, they adjust the PID controller by optimizing neural network parameters.

The various controller design techniques for the MSWI process mentioned above are shown in Table 2.

Table 2. Design techniques of various intelligent controllers for the MSWI process.

Controller Type	Process Model	MSWI Parameter	Inference	Author/Year	Reference
Expert systems and human-simulated intelligent control	Transfer function	· MVs/AVs: Furnace temperature, mixing degree of MSW, flue gas residence time, excess air ratio · CVs/EIs: Dioxin-like organic chloride, particulate matter, NOx, and CO, etc.	· Integrated control strategy, based on HSIC and integrated with expert knowledge · With high control accuracy, good dynamic and steady-state quality, and strong robustness	Ni et al., 2014	[118]
	Second-order lag transfer function	· MVs/AVs: MSW quantity, air flow volume · CVs: Furnace temperature	· Demonstrated strong anti-interference capability, good control quality, fast response, and small steady-state error	Xiao et al., 2015	[119]
	Transfer function	· MVs/AVs: Oxygen concentration, etc. · CVs: Furnace temperature, furnace pressure	· HSIC combined with PSO algorithm · The improved PSO algorithm enhances the ability to resist external pulse interference, showing strong robustness even when process parameters change significantly	Wu Qian et al., 2018	[120]
	Transfer function	· MVs: (Not explicitly mentioned) · CVs: Furnace temperature	· Strong robustness, small overshoot and oscillation, control quality superior to traditional PID control	Li et al., 2019	[116]
	(Not explicitly mentioned)	· MVs: (Not explicitly mentioned) · CVs: Furnace temperature	· HSIC combined with an improved PSO algorithm · Intelligent optimization control during the incineration process is achieved, improving control accuracy and the system’s dynamic stability	Ju et al., 2022	[117]
Fuzzy logic control	Neural network	· MVs/AVs: Pile material speed, air flow volume · CVs: Steam flow, furnace temperature, flue gas oxygen content	· Improves control performance, solves the delay problem of moisture content estimation in traditional control methods	Qian et al., 1993	[135]
	Transfer function	· MVs/AVs: MSW feeding quantity · CVs: Furnace temperature	· Fuzzy adaptive control for furnace temperature · Designed a basic fuzzy controller and an intelligent synchronous modification weight factor module	Chang et al., 2004	[138]
	Shenzhen 150 t/d MSWI power plant	· MVs/AVs: MSW feeding quantity (feeding time and stopping time) · CVs: Furnace temperature	· Fuzzy adaptive control strategy and parameter correction module to ensure the effectiveness of the adaptive factor	Shen et al., 2004	[136]
	Shenzhen 150 t/d MSWI power plant	· MVs/AVs: Waste feeding amount (feeding time and stopping time) · CVs: Furnace temperature	· Adaptive fuzzy control strategy and combining genetic algorithms and neural network technology to optimize control parameters	Shen et al., 2005	[137]
Neural network control	Second-order inertia transfer function and lag delay transfer function	· MVs/AVs: (Not explicitly mentioned) · CVs: Furnace temperature	· Combining BPNN with PID for adjusting PID parameters · Optimizing the BPNN through PSO to improve the global optimality of control parameters	Luo et al., 2023	[147]
	BPNN	· MVs: Lime slurry flow · CVs/EIs: Emission concentration of acidic gases (SO2 and HCl)	· Using a BPNN genetic algorithm strategy, replacing manual control by training an artificial neural network · The control output is smoother than the PID mode, can achieve the effect of manual control, and can reduce workload at the same time	You et al., 2024	[146]
Fuzzy neural control	LRDT	· MVs: Primary air flow, secondary air flow, average feeder speed, average drying grate speed, and ammonia water injection amount · CVs: Furnace temperature	· Adaptive TS-FNN controller · Good decoupling performance, robustness, and anti-interference capabilities	Tian et al., 2023	[155]
	TS-FNN	· MVs: Secondary air pressure, primary air heating temperature · CVs: Furnace temperature	· Event-triggered self-organizing FNN control	He et al., 2023	[156]
	TS-FNN	· MVs: Primary air flow, drying grate speed · CVs: Furnace temperature	· Event-triggering mechanism · Controller convergence based on Lyapunov’s second law	Ding et al., 2023	[157]
	MISO-LRDT	· MVs: Primary air flow, secondary air flow, feeder average speed, drying grate average speed, ammonia water injection amount · CVs: Furnace temperature	· Interval type-2 FNN controller · Stability analysis	Tang et al., 2023	[158]
	MIMO-TSFNN	· MVs: Primary air flow, drying grate speed, secondary air flow · CVs: Furnace temperature, flue gas oxygen content	· Data-driven modeling and self-organizing control method · Effectiveness of the model and controller was verified	Ding et al., 2023	[159]
	MIMO-TSFNN	· MVs: Primary air flow, drying grate speed · CVs: Furnace temperature, flue gas oxygen content	· Event-triggered FNN multivariable controller	Ding et al., 2023	[160]
	MIMO-TSFNN	· MVs: Primary air flow, drying grate speed · CVs: Furnace temperature, flue gas oxygen content	· Dynamic self-organizing mechanism	Ding et al., 2023	[161]
Self-tuning PID control	Transfer function	· MVs: MSW feeding quantity (feeding time and stopping time), primary air flow · CVs: Furnace temperature	· Fuzzy control method, rule base established based on operator experience · Compare fuzzy control with PID control algorithms	Dai et al., 2019	[168]
	TS-FNN	· MVs: Primary air flow · CVs: Furnace temperature	· Event-triggered RBF–PID controller · Online updating of network parameters through gradient descent algorithm and recursive least squares algorithm	He et al., 2022	[167]
	MIMO-TSFNN	· MVs: Primary air flow, secondary air flow, drying grate speed · CVs: Flue gas oxygen content, furnace temperature, steam flow	· Multi-loop PID controller based on quasi-diagonal recurrent neural network · Adaptively adjusting PID parameters	Ding et al., 2022	[169]
	Tikhonov-LRDT	· MVs: Primary air flow, secondary air flow, feeder average speed, drying grate average speed · CVs: Furnace temperature, steam flow, flue gas oxygen content	· Multi-input multi-output control method based on single-neuron adaptive PID controller · Online continuous adjustment of neuron weight coefficients through supervised Hebb learning algorithm to achieve set-point tracking control	Wang et al., 2023	[104]

Table 2. Design techniques of various intelligent controllers for the MSWI process.

Controller Type	Process Model	MSWI Parameter	Inference	Author/Year	Reference
Expert systems and human-simulated intelligent control	Transfer function	· MVs/AVs: Furnace temperature, mixing degree of MSW, flue gas residence time, excess air ratio · CVs/EIs: Dioxin-like organic chloride, particulate matter, NOx, and CO, etc.	· Integrated control strategy, based on HSIC and integrated with expert knowledge · With high control accuracy, good dynamic and steady-state quality, and strong robustness	Ni et al., 2014	[118]
	Second-order lag transfer function	· MVs/AVs: MSW quantity, air flow volume · CVs: Furnace temperature	· Demonstrated strong anti-interference capability, good control quality, fast response, and small steady-state error	Xiao et al., 2015	[119]
	Transfer function	· MVs/AVs: Oxygen concentration, etc. · CVs: Furnace temperature, furnace pressure	· HSIC combined with PSO algorithm · The improved PSO algorithm enhances the ability to resist external pulse interference, showing strong robustness even when process parameters change significantly	Wu Qian et al., 2018	[120]
	Transfer function	· MVs: (Not explicitly mentioned) · CVs: Furnace temperature	· Strong robustness, small overshoot and oscillation, control quality superior to traditional PID control	Li et al., 2019	[116]
	(Not explicitly mentioned)	· MVs: (Not explicitly mentioned) · CVs: Furnace temperature	· HSIC combined with an improved PSO algorithm · Intelligent optimization control during the incineration process is achieved, improving control accuracy and the system’s dynamic stability	Ju et al., 2022	[117]
Fuzzy logic control	Neural network	· MVs/AVs: Pile material speed, air flow volume · CVs: Steam flow, furnace temperature, flue gas oxygen content	· Improves control performance, solves the delay problem of moisture content estimation in traditional control methods	Qian et al., 1993	[135]
	Transfer function	· MVs/AVs: MSW feeding quantity · CVs: Furnace temperature	· Fuzzy adaptive control for furnace temperature · Designed a basic fuzzy controller and an intelligent synchronous modification weight factor module	Chang et al., 2004	[138]
	Shenzhen 150 t/d MSWI power plant	· MVs/AVs: MSW feeding quantity (feeding time and stopping time) · CVs: Furnace temperature	· Fuzzy adaptive control strategy and parameter correction module to ensure the effectiveness of the adaptive factor	Shen et al., 2004	[136]
	Shenzhen 150 t/d MSWI power plant	· MVs/AVs: Waste feeding amount (feeding time and stopping time) · CVs: Furnace temperature	· Adaptive fuzzy control strategy and combining genetic algorithms and neural network technology to optimize control parameters	Shen et al., 2005	[137]
Neural network control	Second-order inertia transfer function and lag delay transfer function	· MVs/AVs: (Not explicitly mentioned) · CVs: Furnace temperature	· Combining BPNN with PID for adjusting PID parameters · Optimizing the BPNN through PSO to improve the global optimality of control parameters	Luo et al., 2023	[147]
	BPNN	· MVs: Lime slurry flow · CVs/EIs: Emission concentration of acidic gases (SO2 and HCl)	· Using a BPNN genetic algorithm strategy, replacing manual control by training an artificial neural network · The control output is smoother than the PID mode, can achieve the effect of manual control, and can reduce workload at the same time	You et al., 2024	[146]
Fuzzy neural control	LRDT	· MVs: Primary air flow, secondary air flow, average feeder speed, average drying grate speed, and ammonia water injection amount · CVs: Furnace temperature	· Adaptive TS-FNN controller · Good decoupling performance, robustness, and anti-interference capabilities	Tian et al., 2023	[155]
	TS-FNN	· MVs: Secondary air pressure, primary air heating temperature · CVs: Furnace temperature	· Event-triggered self-organizing FNN control	He et al., 2023	[156]
	TS-FNN	· MVs: Primary air flow, drying grate speed · CVs: Furnace temperature	· Event-triggering mechanism · Controller convergence based on Lyapunov’s second law	Ding et al., 2023	[157]
	MISO-LRDT	· MVs: Primary air flow, secondary air flow, feeder average speed, drying grate average speed, ammonia water injection amount · CVs: Furnace temperature	· Interval type-2 FNN controller · Stability analysis	Tang et al., 2023	[158]
	MIMO-TSFNN	· MVs: Primary air flow, drying grate speed, secondary air flow · CVs: Furnace temperature, flue gas oxygen content	· Data-driven modeling and self-organizing control method · Effectiveness of the model and controller was verified	Ding et al., 2023	[159]
	MIMO-TSFNN	· MVs: Primary air flow, drying grate speed · CVs: Furnace temperature, flue gas oxygen content	· Event-triggered FNN multivariable controller	Ding et al., 2023	[160]
	MIMO-TSFNN	· MVs: Primary air flow, drying grate speed · CVs: Furnace temperature, flue gas oxygen content	· Dynamic self-organizing mechanism	Ding et al., 2023	[161]
Self-tuning PID control	Transfer function	· MVs: MSW feeding quantity (feeding time and stopping time), primary air flow · CVs: Furnace temperature	· Fuzzy control method, rule base established based on operator experience · Compare fuzzy control with PID control algorithms	Dai et al., 2019	[168]
	TS-FNN	· MVs: Primary air flow · CVs: Furnace temperature	· Event-triggered RBF–PID controller · Online updating of network parameters through gradient descent algorithm and recursive least squares algorithm	He et al., 2022	[167]
	MIMO-TSFNN	· MVs: Primary air flow, secondary air flow, drying grate speed · CVs: Flue gas oxygen content, furnace temperature, steam flow	· Multi-loop PID controller based on quasi-diagonal recurrent neural network · Adaptively adjusting PID parameters	Ding et al., 2022	[169]
	Tikhonov-LRDT	· MVs: Primary air flow, secondary air flow, feeder average speed, drying grate average speed · CVs: Furnace temperature, steam flow, flue gas oxygen content	· Multi-input multi-output control method based on single-neuron adaptive PID controller · Online continuous adjustment of neuron weight coefficients through supervised Hebb learning algorithm to achieve set-point tracking control	Wang et al., 2023	[104]

While intelligent control algorithms, with their self-learning and adaptive capabilities, are effective at handling unknown dynamic changes in the MSWI process, they still face certain shortcomings. The main challenge lies in their limitations in systematically addressing the following characteristics of MSWI control problems.

(1)

Multi-dimensional and conflicting objectives: In the MSWI process, economic, environmental, and production indicators must be considered simultaneously, often leading to conflicting objectives. For instance, in furnace temperature control, increasing the combustion temperature can enhance energy recovery efficiency but may also damage the incineration equipment and increase emissions of harmful gases. This multi-objective optimization problem makes it challenging for a single control algorithm to find the optimal balance among all objectives.

(2)

Multivariable and interaction characteristics: The MSWI process involves multiple interacting process variables, and the complex coupling relationships between these variables make multi-loop control difficult to achieve. Existing intelligent controllers have limitations in handling multivariable interactions and must rely on advanced MIMO methods to effectively manage these complex relationships [117].

(3)

Constraints on process variables: Most process variables have strict operating ranges, and exceeding these ranges can lead to equipment damage or safety issues. Therefore, control algorithms must consider these constraints during execution to ensure that all MVs remain within safe limits. This requires control algorithms suitable for the MSWI process to have strong constraint-handling capabilities [44].

As analyzed in Section 2.4, the control strategy that can provide such a systematic approach is MPC [97], which is reviewed and detailed as follows.

5. Current Status of Model Predictive Controller for the MSWI Process

5.1. MPC Controller Design

MPC is an advanced control method that uses process models to predict the future behavior of the controlled object. By solving a constrained optimization problem, MPC implicitly determines the control law through rolling optimization, thereby shifting the challenge of controller design to modeling the controlled object [88].

The main advantage of MPC is its intuitive handling of multivariable situations, system constraints, and nonlinearity. Therefore, MPC has been successfully applied in various industrial fields, including blast furnace ironmaking [170], wastewater treatment [171,172], and refining processes [173]. These successful applications in other process industries suggest that MPC can effectively manage the dynamic and complex nature of processes, predicting future behavior and optimizing control signals to improve the efficiency and stability of the MSWI process while reducing pollutant emissions.

Early research on the MSWI process often used mechanistic models to construct predictive models for MPC. For example, Leskens et al. [97] proposed a linear MPC (LMPC) strategy for the simultaneous control of steam flow and flue gas oxygen content, using state–space equations to construct the model of the controlled object. To address the challenge posed by strong nonlinearity, which makes it difficult for LMPC to achieve optimal control effects, Leskens et al. [174,175] proposed a nonlinear MPC (NMPC) strategy by estimating the parameters of nonlinear state equations through a rolling horizon state estimator.

It is worth noting that the complex nonlinearity and dynamic characteristics of the MSWI process make it difficult to obtain precise mechanistic expressions for its predictive model, rendering conventional MPC methods based on dynamic matrix control and quadratic dynamic matrix control unsuitable [176]. Specifically, MPC methods that use linear state–space forms with observers or describe nonlinear quadratic matrix control forms cannot accurately predict future behavior and optimize control signals [177,178]. With the advent of AI technologies and industrial big data, extracting the essential characteristics of system dynamics through advanced machine learning techniques and constructing predictive models driven by large-scale, high-dimensional, and dynamically changing data has become a key focus in MSWI process research and application. Designing and constructing a typical MPC controller involves building a predictive model and selecting an objective function. By establishing a mathematical model that predicts the future behavior of the system, the controller can optimize current and future control inputs to achieve the best control effect. The objective function is typically quadratic, incorporating input terms, output terms (error terms), and special terms [84]. The input and output terms of the objective function evaluate the deviation of MV and CV from their reference trajectories to optimize tracking performance and limit energy usage. Special terms are additional components added to the objective function to address issues related to strict constraints, often using equal constraints and severe penalties to replace inequality constraints. Table 3 provides an analysis of predictive model construction methods and objective function selection in MPC, based on mechanistic and data-driven approaches reported in the reference.

Table 3. Statistical results of constructing prediction models and selecting objective functions in MSWI process MPC.

Prediction Model Type	Prediction Model Design	Objective Function Form	Special Term	Reference
Mechanism driven	Rolling time-domain state estimator, nonlinear state–space equations	Output terms	\	[174]
	Linear state–space equations	Input and output terms	\	[97,175,179]
	Subspace identification, partial differential equations	Input and output terms	\	[180]
	Linear state–space equations	Input and output terms	Terminal constraint terms	[181]
Data driven	IT2FNN	Output terms	\	[158]
	SOFNN	Input and output terms	\	[182]
	Improved long short-term memory (LSTM)	Input and output terms	\	[183]
	Improved LSTM	Input and output terms	\	[184]
	Improved random weight neural network (RWNN)	Input and output terms	\	[185]
	RWNN	Input and output terms	\	[186]
	LSTM	Input and output terms	\	[187]
	RBFNN	Input and output terms	\	[188]
	BPNN	Output terms	\	[189]

Table 3. Statistical results of constructing prediction models and selecting objective functions in MSWI process MPC.

Prediction Model Type	Prediction Model Design	Objective Function Form	Special Term	Reference
Mechanism driven	Rolling time-domain state estimator, nonlinear state–space equations	Output terms	\	[174]
	Linear state–space equations	Input and output terms	\	[97,175,179]
	Subspace identification, partial differential equations	Input and output terms	\	[180]
	Linear state–space equations	Input and output terms	Terminal constraint terms	[181]
Data driven	IT2FNN	Output terms	\	[158]
	SOFNN	Input and output terms	\	[182]
	Improved long short-term memory (LSTM)	Input and output terms	\	[183]
	Improved LSTM	Input and output terms	\	[184]
	Improved random weight neural network (RWNN)	Input and output terms	\	[185]
	RWNN	Input and output terms	\	[186]
	LSTM	Input and output terms	\	[187]
	RBFNN	Input and output terms	\	[188]
	BPNN	Output terms	\	[189]

As shown in Table 3, the choice of the MPC objective function is largely independent of the predictive model type. Whether the predictive model is constructed using mechanistic or data-driven methods, the standard form of the objective function, composed of input and output terms, is predominantly used. Only Riccardo et al. [181] consider terminal constraints within the objective function. In early MSWI process research, challenges in data acquisition resulted in a scarcity of industrial process data, leading to the use of mechanistic models for predictive modeling. While these models can accurately represent physical and chemical processes, their high complexity, limited adaptability, and strong reliance on experimental data restrict their practical application. Recently, the advancement of industrial big data and machine learning technologies has led to an increased use of data-driven methods for predictive modeling in the MSWI process, primarily utilizing various neural network models. These models are highly valued for their ability to adapt to changing conditions through online updates and their rapid construction and application in industrial processes.

The MPC controllers reported for the MSWI process incorporate online updates of the predictive model, rolling optimization, and feedback correction. Rolling optimization applies the optimization results only for the current moment in each control cycle and updates the optimization window continuously to respond to real-time changes in the system state. Feedback correction involves comparing actual measurements with predicted values to adjust the controller’s behavior. This process corrects model errors, preventing system deviations from the desired trajectory and ensuring stable operation. Online updating of the predictive model is a fundamental requirement for MPC in the MSWI process. Designing an MPC controller typically requires an accurate predictive model, but inevitable model mismatches in the actual MSWI process necessitate real-time adjustments to model parameters.

5.2. MPC Controller Online Update

This section presents the content of online updates for MPC, including predictive models, rolling optimization, and feedback correction [84]. We have modified it in terms of the MSWI process, which is shown in Figure 3.

Optimization technology is primarily used to design MPC online update algorithms and evaluate improvement through these updates [84]. Initially, process data describing incineration dynamics is collected from industrial data acquisition platforms in MSWI power plants. Next, correlation analysis identifies MV and CV, and a model of the controlled object is constructed, using other less relevant parameters as external disturbance variables. Then, models describing input–output relationships and time-domain dynamics are then built using data-driven modeling techniques and machine learning algorithms. The selected model is used to implement the MPC controller. The objective function describes standard controller performance indicators, including input, output, and special terms, which also serve as constraints to prevent instability and untunable parameters. Optimization algorithms are also used for online MPC updates, which can be done sequentially or in parallel, with formulated constraint conditions. This allows for the selection of parameters for optimization according to industrial demands, such as output accuracy and energy savings. The performance of the online-updated MPC replaces the default MPC controller based on historical data. Finally, performance evaluation based on different operational indicators is conducted, and industrial application in the MSWI process is realized.

This section surveys various MPC online update algorithms, determining their feasibility in MSWI process applications and identifying challenges related to MPC updates in disturbance scenarios.

5.2.1. Online Update of Prediction Model

Current research constructs predictive models using machine learning algorithms, updating parameters with gradient methods. To fully utilize historical sample data containing prior knowledge and expert experience from MSWI plants, researchers have improved learning algorithms to enhance feature extraction and learning capabilities. Qiao et al. [183] proposed an event-triggered adaptive MPC strategy for flue gas oxygen content control, using time-based gradient descent (GD) for online model parameter updates to handle uncertainties and employing the PSO algorithm to optimize hyperparameters. Sun et al. [184] proposed an MPC strategy based on dual LSTM neural networks, employing self-organizing LSTM to construct a compact model for flue gas oxygen content and using time-based GD for online parameter updates. Wang et al. [185] developed a nonlinear predictive model for NOx emissions using an improved RWNN, employing recursive least squares for online updating of the RWNN hidden layer’s output weights. Hu et al. [186] proposed a nonlinear MPC method for furnace temperature, using RWNN to establish an offline static nonlinear predictive model and employing recursive least squares to update the RWNN hidden layer neurons’ output weights, thus creating a dynamic nonlinear predictive model. To adapt models to frequent fluctuations in MSWI process conditions, researchers have introduced structural update strategies into the online update process. Sun et al. [187] proposed a data-driven method for predicting and controlling flue gas oxygen content in the MSW incineration process, designing a self-organizing LSTM-based predictive model that dynamically adjusts the hidden layer structure based on neuron activity and significance, improving prediction accuracy and using GD to solve control laws. Meng et al. [181] developed a self-organizing FNN using self-organizing add–delete neuron mechanisms and the Levenberg–Marquardt (LM) algorithm, resulting in a predictive model with a simplified structure and high accuracy, and they designed an MPC method for MSWI furnace temperature based on this model. Tang et al. [158] developed an IT2FNN predictive model with online parameter learning and self-organizing mechanisms and designed an MPC control strategy for furnace temperature.

Table 4. Statistical analysis of prediction models and methods used for online model updating in MSWI process MPC.

Prediction Model	Parameter Update	Structure Update	Performance Improvement	Author/Year	Reference
IT2FNN	GD	√	· Dynamic optimization of model parameters and adjustment of network structure · Designed to improve prediction effects, enhance anti-interference capabilities, and ensure the closed-loop stability of the control system	Tang et al., 2023	[158]
SOFNN	LM algorithm	√	· Obtained a prediction model with streamlined structure and higher prediction accuracy · Enhancing control precision and robustness	Meng et al., 2023	[181]
Improve LSTM	Back propagation algorithm	√	· Self-organization of network structure based on neuron activity and importance	Sun et al., 2023	[184]
Improved RWNN	Recursive least squares	√	· Online update of RWNN hidden layer output weights using recursive least squares to adapt to the dynamic characteristics of the MSWI process · Selection of the hidden layer neurons number through a supervisory mechanism to improve model generalization	Wang et al., 2023	[185]
RWNN	Recursive least squares	\	· Improved the accuracy of the furnace temperature prediction model, adapting to the time-varying characteristics	Hu et al., 2023	[186]
LSTM	GD	√	· Used adaptive fuzzy C-means algorithm to assist in determining RBF network parameters and online updating	Sun et al., 2023	[187]
RBFNN	GD	\	· Improved the accuracy of the prediction model, enhancing the adaptability and stability of the control method	Sun et al., 2023	[188]
BPNN	GD	\	· Real-time adjustment of BPNN weight parameters to improve tracking performance and interference capability	Wang et al., 2023	[189]
Improved LSTM	Backpropagation through time algorithm	√	· Dynamic parameter adjustment based on prediction error and model parameter changes · Optimization of network structure through self-organizing methods and algorithms to enhance the adaptability of complex industrial dynamics	Qiao et al., 2024	[183]

analyzes the online update methods used for predictive models and models requiring online updates.

5.2.2. Online Update of Rolling Optimization

MPC dynamically solves the control law through rolling optimization, enabling system behavior prediction and online updates in each control cycle. Traditionally, this relies on mechanistic numerical methods, requiring an in-depth understanding and precise mathematical description of the system’s physical or chemical processes. Due to the MSWI process’s complexity and uncertainty, its predictive model is constructed using “black box” machine learning models, making conventional mechanistic numerical methods challenging to apply directly. Therefore, control strategies now rely on optimization algorithms. Among these, GD is widely used for its simplicity and applicability. Qiao et al. [183], Sun et al. [184,187,188], Tang et al. [158], and Wang et al. [189] all utilize GD to solve control laws in rolling optimization. To further expand the search scope and achieve optimal performance, metaheuristic algorithms are also applied to the rolling optimization process. Yan et al. [185] designed a PSO method to calculate the optimal control law for each control sequence in constraint-backward layer optimization. Hu et al. [186] combined an improved seagull optimization algorithm with a set-point evaluation and learning model, resulting in an enhanced rolling optimization strategy to improve NMPC solution performance.

Table 5. Statistical results of online update methods used for MPC rolling optimization in the MSWI process.

Manipulated Variable	Rolling Optimization Method	Prediction Horizon	Control Horizon	Performance	Author/Year	Reference
MSW feed rate, grate speed, primary air flow, secondary air flow	Quadratic programming (QP)	\	\	Improved performance compared to conventional MSWI control systems, especially in handling large temporary disturbances	Leskens et al., 2005	[97]
MSW feed rate, grate speed, primary air flow, secondary air flow	QP, state estimation	5	5	Demonstrated good closed-loop performance	Leskens et al., 2005	[174]
MSW feed rate, grate speed, primary air flow, secondary air flow	Gradient method, sequential quadratic programming (SQP)	\	\	Demonstrated better disturbance rejection capabilities compared to multivariable PID and linear MPC	Leskens et al., 2008	[175]
Throttle valve position, pump speed, flu gas speed, air flow speed	System identification, QP	10	10	Achieved good transient and steady-state responses, as well as decoupling performance and satisfactory disturbance rejection	Zhang et al., 2013	[179]
MSW feed rate, feedwater, primary air flow, secondary air flow	QP, feedforward Control	\	\	MPC with feedforward provided superior performance, maintaining stable temperatures throughout the process	Nathan et al., 2018	[180]
Drying grate speed, combustion grate 1 speed, combustion grate 2 speed	Improved seagull optimization algorithm	3	2	Achieved smooth control of furnace temperature in the MSWI process	Hu et al., 2022	[186]
Secondary air flow	GD	8	1	Demonstrated effective control of furnace temperature, ensuring safe and stable operation and reducing pollutant emissions in flue gas	Tang et al., 2023	[158]
\	GD	\	\	Validated by actual industrial data, the proposed method is effective and excellent in furnace temperature prediction modeling and control	Meng et al., 2023	[181]
Drying grate primary air flow, combustion grate 1 primary air flow, secondary air flow	Adaptive mutation PSO, GD	1–5	1–5	Achieved promising tracking control performance for flue gas oxygen content and improved operational performance	Sun et al., 2023	[184]
Drying grate 1 primary air flow, combustion grate 1 primary air flow, secondary air flow	PSO	5	1	Exhibited small control errors and good tracking performance	Wang et al., 2023	[185]
Primary air flow and combustion grate 1 section primary flow	GD	5	1	Demonstrated better control performance in flue gas oxygen content control, with fast response and stable performance	Sun et al., 2023	[187]
Drying grate primary air flow, combustion grate 1 primary air flow, secondary air flow	GD	5	1	Demonstrated better adaptability and disturbance rejection capabilities, achieving more precise control of flue gas oxygen content	Sun et al., 2023	[188]
Secondary air flow	GD	\	\	Demonstrated the effectiveness of the method, showing enhanced tracking and anti-interference capabilities	Wang et al., 2023	[189]
Drying grate, primary air flow, combustion grate 1 primary air flow	Event-triggered GD	5	1	Achieved satisfactory tracking control performance with fewer triggering events	Qiao et al., 2024	[183]

analyzes the online update methods used in rolling optimization.

As shown in Table 5, for the MSWI process, existing MPC methods mainly use GD and its variants for parameter updates and rolling optimization, with limited application of metaheuristic algorithms [183,185]. However, this does not necessarily mean that gradient-based algorithms are optimal, as their performance depends on the actual application.

5.2.3. Online Update of Feedback Correction

System disturbances or model mismatches inevitably cause deviations between predicted and actual outputs. To reduce these deviations, the difference between predicted and actual outputs at the last moment is typically used for error feedback compensation [190]. References [185,186,187,188] employ error compensation-based feedback correction methods. However, these methods have significant lag and lack adaptability to complex nonlinear systems [191]. To address this, only Sun et al. [184] proposed a data-driven optimal control scheme with a hierarchical control structure, using the superior time series analysis capability of LSTM to establish another LSTM for predicting and correcting compensation errors.

Table 6. Statistical analysis of online update methods used for feedback correction in MPC for the MSWI process.

Whether Feedback Correction	Methods	Formula	Author/Year	Reference
Not explicitly mentioned	\	\	Leskens, et al., 2005	[97]
Not explicitly mentioned	\	\	Leskens, et al., 2005	[174]
Not explicitly mentioned	\	\	Leskens et al., 2008	[175]
Not explicitly mentioned	\	\	Zhang et al., 2013	[179]
Not explicitly mentioned	\	\	Nathan, et al., 2018	[180]
√	Error Feedback	$\begin{array}{l}e(k)=T(k)-T_m(k)\\ T_p(k+j)=T_m(k+j)+\eta e(k)\end{array}$	Hu et al., 2022	[186]
√	Error Feedback	$\begin{array}{l}\widehat{\epsilon }\left(t\right)=y(t)-\widehat{y}(t)\\ \widehat{y}(t+i)=\widehat{y}(t+i)+\widehat{\epsilon }\left(t\right)\end{array}$	Tang, et al., 2023	[158]
Not explicitly mentioned	\	\	Meng et al., 2023	[181]
√	Error Feedback	$\begin{array}{l}\overline{e}(k+1)=\overline{f}(y(k),u(k),e(k),e(k-1),\dots ,e(k-kd))\\ \widehat{y}(k+i)=y_p(k+i)+\overline{e}(k+1)\end{array}$	Sun et al., 2023	[184]
√	Error Feedback	$\begin{array}{l}e(k)=y(k)-y_p(k)\\ y_m(k+j)=y_m(k+j)+\lambda e(k)\end{array}$	Wang et al., 2023	[185]
√	Error Feedback	$\begin{array}{l}err(t-1)=y_d(t-1)-y_p(t-1)\\ \widehat{y}(t)=y_p(t)+err(t-1)\end{array}$	Sun, et al., 2023	[187]
√	Error Feedback	$\begin{array}{l}err(t-1)=y(t-1)-{\widehat{y}}_p(t-1)\\ \widehat{y}(t+1)={\widehat{y}}_p(t+j)+err(t)\end{array}$	Sun, et al., 2023	[188]
√	Error Feedback	$\begin{array}{l}e(t)=y(t)-{\widehat{y}}_p(t)\\ \widehat{y}(t+j)={\widehat{y}}_p(t+j)+e(t)\end{array}$	Wang et al., 2023	[189]
√	LSTM, PSO	Refer to Part IV of the reference for details	Qiao, et al., 2024	[183]

analyzes the online update methods used for feedback correction in the reference.

As shown in Table 6, feedback correction of the predictive model compensates for deviations between actual and predicted values. Most existing research uses simple cumulative methods for compensation. Most of the studies in Table 6 are based on error compensation; only [184] designed an adaptive compensation strategy using LSTM. Thus, more research should be done in this area.

Abbreviations and their meanings are as Abbreviations in the back matter.

6. Conclusions

Developed and developing countries differ in their focus on MSWI research. Developed countries resolved MSWI process control problems under stable MSW calorific value 20 years ago based on their MSW characteristics, successfully applying MPC strategies. However, MPC control research is just beginning in developing countries due to the highly uncertain composition of MSW. Given its importance, this article systematically reviews existing MPC research on the MSWI process globally. The main findings include: (1) Through the analysis of the MSWI process, selecting appropriate MV, AV, and CV is an important foundation for achieving its optimized control. (2) Against the backdrop of the rapid development of industrial big data and artificial intelligence, data-driven predictive modeling is the current main trend in MSWI process MPC. (3) The application of meta-heuristic algorithms in the rolling optimization and feedback correction of MSWI process MPC is still rarely reported. As an effective optimization strategy, its application in MPC can be a key research direction in the future.