The distributed parameter system (DPS) has complex spatiotemporal dynamics, with their energy or mass constantly changing in time and space (Li and Qi in J Process Control 20:891–901, 2024). This type of system widely involves a large number of physical processes in various fields such as engineering, physics, chemistry, etc., such as charging and discharging of the lithium battery, particle diffusion, chemical reaction, heat conduction processes, fluid flow, elastic vibration (Wang in IEEE Trans Cybern 49:3041–3051, 2019; Xu et al. in Int J Heat Mass Transf 209, 2023a), requiring additional consideration of the spatial distribution compared with the centralized parameter systems (CPSs). Generally, distributed parameter systems are typically described by partial differential equations (PDEs), which can be summarized as three categories, namely parabolic, hyperbolic, and elliptical forms, corresponding to different types of physical processes such as heat conduction, wave motion, and steady-state problems. Besides, some more complex processes may also involve integral equations or differential integral equations. These equations form the mathematical and theoretical foundation of the distributed parameter system, reflecting the continuous variations of different physical quantities such as temperature, pressure, concentration in space and time, further affecting the production and manufacturing efficiency as well as the final product quality. However, these processes have obvious nonlinear, time-varying, and coupling effect, as well as infinite spatial dimensions. In addition, the complex internal and external energy exchange processes make the system face complex boundary and initial conditions. Besides, in practical applications, these thermal processes often exhibit complex and unknown nonlinear characteristics in space and time due to operating conditions or usage scenarios. Therefore, aiming for these issues, many modeling and control frameworks or methods have been proposed for different types of DPSs, and evolved with the development of algorithms such as machine learning, and deep learning, pursuing performance in terms of accuracy, adaptability, and computational efficiency from different perspectives (Ai et al. in Symmetry 13:453, (2021), Bombard et al. in Chem Eng Sci 65:962–975, (2010), Bonis et al. in AIChE J 801–811, (2012), Lei et al. in Nonlinear Anal-Model Control 27:234–253, (2022c), Qazani et al. in IEEE Trans Syst Man Cybern Syst 51(10):6096–6110, (2021a), Qazani et al. in IEEE Syst J 15(1):445–453, (2021b), Qing et al. in AIChE J 67:17246, (2021), Rapoport and Pleshivtseva in Optoelectron InstrumData Process 57:345–355, (2021), Schaft and Maschke in JGeom Phys 42(1–2):166–194, (2002), Wang et al. in Int J Fuzzy Syst 18:792–805, (2016), Wang et al. in IEEE Trans Fuzzy Syst 26:2967–2980, (2018), Xie et al. in J Process Control 35:50–58, (2015), Xu et al. in IEEE Trans Industr Electron 65:9767–9776, (2018), Xu and Lu in Nonlinear Dyn 108. (2022b), Zhang et al. in Ind Eng Chem Res 21:10510–10516, (2010), and Zhou et al. in IEEE Trans Transport Electrif 7:2260–2268, (2021)). In this chapter, a systematic overview and classification is first presented on modeling and control of different types of DPSs. Limitations and advantages of various approaches are also discussed.

Introduction

章节
发表日期：

2026年

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Bowen Xu,
Xinjiang Lu

摘要

The distributed parameter system (DPS) has complex spatiotemporal dynamics, with their energy or mass constantly changing in time and space (Li and Qi in J Process Control 20:891–901, 2024). This type of system widely involves a large number of physical processes in various fields such as engineering, physics, chemistry, etc., such as charging and discharging of the lithium battery, particle diffusion, chemical reaction, heat conduction processes, fluid flow, elastic vibration (Wang in IEEE Trans Cybern 49:3041–3051, 2019; Xu et al. in Int J Heat Mass Transf 209, 2023a), requiring additional consideration of the spatial distribution compared with the centralized parameter systems (CPSs). Generally, distributed parameter systems are typically described by partial differential equations (PDEs), which can be summarized as three categories, namely parabolic, hyperbolic, and elliptical forms, corresponding to different types of physical processes such as heat conduction, wave motion, and steady-state problems. Besides, some more complex processes may also involve integral equations or differential integral equations. These equations form the mathematical and theoretical foundation of the distributed parameter system, reflecting the continuous variations of different physical quantities such as temperature, pressure, concentration in space and time, further affecting the production and manufacturing efficiency as well as the final product quality. However, these processes have obvious nonlinear, time-varying, and coupling effect, as well as infinite spatial dimensions. In addition, the complex internal and external energy exchange processes make the system face complex boundary and initial conditions. Besides, in practical applications, these thermal processes often exhibit complex and unknown nonlinear characteristics in space and time due to operating conditions or usage scenarios. Therefore, aiming for these issues, many modeling and control frameworks or methods have been proposed for different types of DPSs, and evolved with the development of algorithms such as machine learning, and deep learning, pursuing performance in terms of accuracy, adaptability, and computational efficiency from different perspectives (Ai et al. in Symmetry 13:453, (2021), Bombard et al. in Chem Eng Sci 65:962–975, (2010), Bonis et al. in AIChE J 801–811, (2012), Lei et al. in Nonlinear Anal-Model Control 27:234–253, (2022c), Qazani et al. in IEEE Trans Syst Man Cybern Syst 51(10):6096–6110, (2021a), Qazani et al. in IEEE Syst J 15(1):445–453, (2021b), Qing et al. in AIChE J 67:17246, (2021), Rapoport and Pleshivtseva in Optoelectron InstrumData Process 57:345–355, (2021), Schaft and Maschke in JGeom Phys 42(1–2):166–194, (2002), Wang et al. in Int J Fuzzy Syst 18:792–805, (2016), Wang et al. in IEEE Trans Fuzzy Syst 26:2967–2980, (2018), Xie et al. in J Process Control 35:50–58, (2015), Xu et al. in IEEE Trans Industr Electron 65:9767–9776, (2018), Xu and Lu in Nonlinear Dyn 108. (2022b), Zhang et al. in Ind Eng Chem Res 21:10510–10516, (2010), and Zhou et al. in IEEE Trans Transport Electrif 7:2260–2268, (2021)). In this chapter, a systematic overview and classification is first presented on modeling and control of different types of DPSs. Limitations and advantages of various approaches are also discussed.