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Many distributed parameter systems (DPSs) have strongly nonlinear spatiotemporal dynamics, unknown parameters and complex boundary conditions, which make it difficult to obtain accurate prediction and control in actual practice. In this paper, a data-driven spatiotemporal model predictive control (MPC) strategy is proposed for nonlinear DPSs. It first develops a low-order nonlinear spatiotemporal model by using kernel principal component analysis to reconstruct the nonlinear spatial dynamics, so that the spatial nonlinearity is better reserved in contrast with the traditional data-driven DPS modeling methods. On this basis, a spatiotemporal MPC is proposed for nonlinear DPSs. In this control strategy, a new objective function is constructed with consideration of errors on not only time but also space, which overcomes the shortcoming of the traditional MPC due to the ignorance of nonlinear spatial dynamics. The stability and effectiveness of the proposed spatiotemporal control strategy are demonstrated by mathematical stability and comparative case studies.

Many advanced industrial processes are a class of time-varying distributed parameter systems (DPSs). It is not an easy task for traditional spatiotemporal modeling methods to approximate these systems because of the inherent time-varying and strong nonlinear characteristics. To address this problem, a novel online spatiotemporal modeling method using Kernel-based Multilayer Extreme Learning Machine is proposed to model the time-varying DPSs. First, the Kernel-based Multilayer Extreme Learning Machine is designed to create a deep network through stacking multiple Kernel-based Extreme Learning Machine Autoencoders and one original Extreme Learning Machine Autoencoder. In this step, the spatiotemporal output of time-varying DPSs is transformed into low-dimensional time coefficients directly. Then Online Sequential Regularized Extreme Learning Machine is developed to predict temporal dynamics of time-varying DPSs. Finally, based on the temporal dynamics model, Kernel-based Extreme Learning Machine is applied to reconstruct the spatiotemporal dynamics. Simulations on the thermal processes of a lithium-ion battery and a snap curing oven are presented to validate the performance and effectiveness of the proposed modeling method.

In actual industrial systems, numerous processes are characterized as distributed parameter systems (DPSs) that exhibit strong nonlinearity, complex boundary conditions, and unknown dynamics. Achieving accurate control of such processes poses a significant challenge. In light of this, a data-driven spatiotemporal least squares support vector machine (LS-SVM) inverse control method has been developed specifically for complex DPSs. First, a spatiotemporal LS-SVM model is proposed to capture the dynamics of DPSs by leveraging available data. Subsequently, by employing Taylor expansion on this spatiotemporal model, a state model is derived to explicitly establish the relationship between the control input and output variables. Building upon this foundation, an explicit control input is obtained through inversion and spatial fuzzy strategy, allowing for effective tracking of spatiotemporal dynamics. This control approach takes into account the influence of each input on all spatial points, thereby ensuring a favorable control effect for DPSs. Theoretical analysis and stability proofs affirm the stability of the proposed control approach for nonlinear DPSs. Furthermore, the efficacy of this controller is demonstrated through two actual experiments. First, the proposed approach exhibits superior control performance, as evidenced by a nearly threefold improvement in tracking accuracy on several sensors compared to the fuzzy controller. Second, the proposed controller maintains a smaller error during the tracking process, with deviations bounded within 1 °C even in the presence of disturbances.

The present work extends known finite dimensional Luenberger observer and Kalman filter designs to the realm of linear transport-reaction systems frequently present in chemical engineering practice. A unified modelling framework for distributed parameter systems (DPS) which does not account for any type of spatial approximation or order reduction is developed. The Cayley–Tustin transformation of continuous linear distributed parameter system yields structure and properties preserving discrete distributed parameter models, amenable to observer and filter design developments. Designs presented here explore well known state reconstruction methodologies starting from least square estimation, continuous and discrete Luenberger observers and one-step predictor Kalman filter realization. Simple implementation and realization account for the appealing nature of the discrete system observers and filter designs for linear transport-reaction systems. The simulation scenarios cover the majority of representative examples found in the common engineering process control practice.

At present, there are many studies on \"computer immunology\" in the world, and there are many applications of immune genetic algorithm in engineering.The article discusses the immune genetic algorithm in the design of PID intelligent controller.Most objects in the practical industrial process fall in the distributed parameter system (DPS), where the proportional-integral-derivative (PID) control method is used to control the field. In this paper, the PID controller optimization method based on the immune genetic algorithm (IGA) (referred to as the PODM method) is applied to a class of distributed parameter objects to design the optimal controller, which is compared with several controllers based on the conventional tuning formulas. The simulation results suggest that the PID controller designed based on the PODM method can obtain relatively superior control effects in overshoot, tuning time, time integrated time absolute error (ITAE), and other indexes with relatively less control energy. The application of the PODM method to DPS can improve the control level of the PID controller in the industry at present.

This paper concerns the problem of boundary time-varying feedback controller for fixed-time stabilization of a linear parabolic distributed parameter system with spatially and temporally varying reactivity. By utilizing the continuous backstepping approach, the invertible Volterra integral transformation with the time-dependent gain kernel is introduced to convert the closed-loop system into a target system with a time-dependent coefficient. Meanwhile, the convergence of the target system within the prescribed time is guaranteed via the Lyapunov method. The well-posedness of the resulting kernel partial differential equations is also proven by exploiting the method of successive approximation. In addition, the growth-in-time of the kernel functions is estimated by applying the generalized Laguerre polynomials and the modified Bessel functions. Subsequently, the fixed-time stability of the closed-loop system under state feedback control within the prescribed time is proven by using the fixed-time stability of the target system and the time-varying kernel functions. Finally, a numerical example is provided to illustrate the effectiveness of the proposed control method.

Distributed parameter systems (DPSs) frequently appear in industrial manufacturing processes, with complex characteristics such as time–space coupling, nonlinearity, infinite dimension, uncertainty and so on, which is full of challenges to the modeling of the system. At present, most DPS modeling methods are offline. When the internal parameters or external environment of DPS change, the offline model is incapable of accurately representing the dynamic attributes of the real system. Establishing an online model for DPS that accurately reflects the real-time dynamics of the system is very important. In this paper, the idea of reinforcement learning is creatively integrated into the three-dimensional (3D) fuzzy model and a reinforcement learning-based 3D fuzzy modeling method is proposed. The agent improves the strategy by continuously interacting with the environment, so that the 3D fuzzy model can adaptively establish the online model from scratch. Specifically, this paper combines the deterministic strategy gradient reinforcement learning algorithm based on an actor critic framework with a 3D fuzzy system. The actor function and critic function are represented by two 3D fuzzy systems and the critic function and actor function are updated alternately. The critic function uses a TD (0) target and is updated via the semi-gradient method; the actor function is updated by using the chain derivation rule on the behavior value function and the actor function is the established DPS online model. Since DPS modeling is a continuous problem, this paper proposes a TD (0) target based on average reward, which can effectively realize online modeling. The suggested methodology is implemented on a three-zone rapid thermal chemical vapor deposition reactor system and the simulation results demonstrate the efficacy of the methodology.

This study addresses the standardization of Singular Distributed Parameter Systems (SDPSs). It focuses on classifying and simplifying first- and second-order linear SDPSs using characteristic matrix theory. First, the study classifies first-order linear SDPSs into three canonical forms based on characteristic curve theory, with an example illustrating the standardization process for parabolic SDPSs. Second, under regular conditions, first-order SDPSs can be decomposed into fast and slow subsystems, where the fast subsystem reduces to an Ordinary Differential Equation (ODE) system, while the slow subsystem retains the spatiotemporal characteristics of the original system. Third, the standardization and classification of second-order SDPSs is proposed using three reversible transformations that achieve structural equivalence. Finally, an illustrative example of a building temperature control is built with SDPSs. The simulation results show the importance of system standardization in real-world applications. This research provides a theoretical foundation for SDPS standardization and offers insights into the practical implementation of distributed temperature systems.

The problem of controllability for problems of optimal control and optimization of distributed parameter systems governed by partial differential equations is considered. The concept of controllability understood as Tikhonov correctness for solving optimization problems is introduced. A theorem formulating controllability conditions for directly solving optimization problems (direct minimization of the objective functional) is presented. A test example of the numerical solution of the optimization problem for a nonlinear hyperbolic system describing the unsteady flow of water in an open channel is considered. The analysis of controllability is demonstrated that ensures the correctness of the problem solution and high accuracy of optimization of the distributed friction coefficient in the flow equations.

Modeling distributed parameter systems (DPSs) is challenging due to their strong nonlinearity and spatiotemporal coupling. In this study, a three-dimensional fuzzy modeling method combining genetic algorithm (GA)-based automatic clustering and hierarchical extreme learning machine (HELM) is proposed for DPS modeling. The method utilizes GA-based automatic clustering to learn the premise part of 3D fuzzy rules, while HELM is employed to learn spatial basis functions and construct a complete fuzzy rule base. This approach effectively captures the spatiotemporal coupling characteristics of the system and mitigates the information loss commonly observed in dimensionality reduction in traditional fuzzy modeling methods. Through experimental verification, the proposed method is successfully applied to a rapid thermal chemical vapor deposition system. The experimental results demonstrate that the method can accurately predict temperature distribution and maintain good robustness under noise and disturbances.