Distributed Parameter Systems Research Articles

Online Three-Dimensional Fuzzy Reinforcement Learning Modeling for Nonlinear Distributed Parameter Systems

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.

Nonlinear Frequency Response Analysis of Distributed Parameter Systems with One Spatial Coordinate

AbstractThe objective of this paper is to investigate possible strategies for applying nonlinear frequency response (NFR) analysis based on the concept of higher‐order frequency response functions to distributed parameter systems. Three approaches are presented and compared: one based on applying the existing procedure directly to the partial differential model, and two based on approximation of the distributed parameter system with a series of lumped parameter segments. One of them treats the complete series of segments integrally, while the other treats it segment by segment, so it uses only the model of a single segment. A simple example, an isothermal plug‐flow reactor with a simple reaction mechanism, is used as a case study. Pros and cons for all three approaches are given.

Adaptive fault estimation and accommodation for distributed parameter systems with coupled parabolic partial differential equations

This paper presents a novel model-based approach for fault detection, estimation and accommodation in linear distributed parameter systems (DPS) governed by parabolic partial differential equations (PDEs), prone to sensor and actuator faults. Unlike traditional methods, our approach utilises a filter-based observer directly employing the PDE representation and relies solely on boundary measurements, eliminating the need for extensive sensor arrays. By generating fault detection residuals from a comparison of measured and observer outputs, our method offers efficient fault detection. Innovative parameter update laws facilitate accurate estimation of fault parameters, enabling precise adjustment of the nominal controller to mitigate fault effects. Rigorous Lyapunov analysis ensures the robustness of our approach. Additionally, we derive a formula for estimating time-to-accommodation (TTA), providing valuable insights into fault resolution timelines. Simulations on a linearised diffusion process validate the effectiveness of our scheme, highlighting its superiority over existing approaches.

Globalization of distributed parameter self‐optimizing control

AbstractNumerous nonlinear distributed parameter systems (DPSs) operate within an extensive range due to process uncertainties. Their spatial distribution characteristic, combined with nonlinearity and uncertainty, poses challenges in optimal operation under two‐step real‐time optimization (RTO) and economic model predictive control (EMPC). Both methods necessitate substantial computational power for prompt online reoptimization. Recent local distributed parameter self‐optimizing control (DPSOC) achieves optimality without repetitive optimization. However, its effectiveness is confined to a narrow range around a nominal operation. Here, globalized DPSOC is introduced to surmount the limitation of the local DPSOC. A global loss functional concerning controlled variables (CVs) is formulated using linear operators and Fubini's theorem. Minimizing the loss with a numerical optimization procedure yields CVs exhibiting global optimality. Maintaining these CVs at constants ensures such a process has a minimal average loss in a large operating space. The effectiveness of the proposed method is substantiated through a transport reaction simulation.

Computation-Efficient Fault Detection Framework for Partially Known Nonlinear Distributed Parameter Systems.

Fault detection for distributed parameter systems (DPSs) generally requires the complete model information to be known so far. However, for numerous industrial applications, it is common that accurate first-principles physical models are extremely difficult to obtain. Hence, the applicability of traditional model-based methods is being restricted. To pave the way, an adaptive neural network (AdNN) is constructed to simultaneously estimate the state variable and the unknown nonlinearity for a class of partially known nonlinear DPSs. Moreover, considering that full-state measurement is unrealistic in applications, the proposed adaptive neural observer is based on a reduced-order model, which also increases the computation efficiency. Then, the residual generation and evaluation are conducted using the output estimation error of the proposed adaptive neural observer. Bearing the effects of the neglected fast dynamics in mind, a data-driven threshold generation scheme is proposed. Extensive experimental results are presented and analyzed to validate the effectiveness of the proposed method.

Exponential Boundary Control for 2-D Spatial Distributed Parameter Systems Under Boundary Collocated and Planar Linear Measurements.

For a class of 2-D spatial distributed parameter systems (DPSs) with space-dependent diffusivity, this article aims to achieve exponential realization of their desired profiles. To reduce the number of required sensors and actuators, a planar output feedback boundary control strategy is proposed with combining two nonfull-domain measurement methods, boundary collocated measurement and planar linear measurement, in which only two boundaries of the considered 2-D spatial DPSs are controlled and a little output information is measured. Moreover, by employing the Poincaré-Wirtinger inequality and variable substitution dexterously, the final exponential convergence criteria of the error system can be obtained with method of "Diverse treatment for same term." Finally, we provide a general numerical example and an application example in 2-D heat conduction systems to illustrate the effectiveness and practicability of the proposed measurement and control schemes.

A dual scale spatio‐temporal control for complex distributed parameter systems

AbstractIt is very difficult to control the distributed parameter system (DPS) for consistent spatial performance. In this paper, a dual scale spatio‐temporal control is designed for the DPS to achieve a consistent performance across the entire workspace under unknown exogenous disturbances. First, the generalized “spatial observer” should be designed under spectral decomposition/synthesis, to act as the nominal model of the DPS, through which the spatial mismatch between the model and the process could be estimated. Then a distributed disturbance compensator can be constructed to suppress all the undesirable spatial disturbance through the inner loop, and the compensated system will become closer to the nominal model and easier to control. Third, a dual controller will be designed in the outer loop for the final consistent spatial performance. Since the spatial performance is mainly affected by the dominant dynamics on the slow scale, a convex optimization algorithm can be effectively established in terms of nonlinear matrix inequalities for the dual controller. In this way, the nonlinear optimization problem is solved by converting it into the problem of a linear matrix inequalities with constraints. Besides, the spatio‐temporal state variable of the controlled plant is demonstrated to converge in Hilbert space. The feasibility and effectiveness of the proposed control method are verified by temperature control of a catalytic rod, a benchmark in chemical reactors.

Knacks of marine predator heuristics for distributed energy source-based power systems harmonics estimation

The power system incorporates renewable energy resources into the main utility grid, which possesses low or no inertia, and these systems generate harmonics due to the utilization of power electronic equipment. The precise and effective assessment of harmonic characteristics is necessary for maintaining power quality in distributed power systems. In this paper, the Marine Predator Algorithm (MPA) that mimics the hunting behavior of predators is exploited for harmonics estimation. The MPA utilizes the concepts of Levy and Brownian motions to replicate the movement of predators as they search for prey. The identification model for parameter estimation of harmonics is presented, and an objective function is developed that minimizes the difference between the real and predicted harmonic signals. The efficacy of the MPA is assessed for different levels of noise, population sizes, and iterations. Further, the comparison of the MPA is conducted with a recent metaheuristic of the Reptile Search Algorithm (RSA). The statistical analyses through sufficient autonomous executions established the accurate, stable, reliable and robust behavior of MPA for all variations. The substantial enhancement in estimation accuracy indicates that MPA holds great potential as a strategy for estimating harmonic parameters in distributed power systems.

Spatiotemporal Transformation-Based Neural Network With Interpretable Structure for Modeling Distributed Parameter Systems.

Many industrial processes can be described by distributed parameter systems (DPSs) governed by partial differential equations (PDEs). In this research, a spatiotemporal network is proposed for DPS modeling without any process knowledge. Since traditional linear modeling methods may not work well for nonlinear DPSs, the proposed method considers the nonlinear space-time separation, which is transformed into a Lagrange dual optimization problem under the orthogonal constraint. The optimization problem can be solved by the proposed neural network with good structural interpretability. The spatial construction method is employed to derive the continuous spatial basis functions (SBFs) based on the discrete spatial features. The nonlinear temporal model is derived by the Gaussian process regression (GPR). Benefiting from spatial construction and GPR, the proposed method enables spatially continuous modeling and provides a reliable output range under the given confidence level. Experiments on a catalytic reaction process and a battery thermal process demonstrate the effectiveness and superiority of the proposed method.

Consensus control and vibration suppression for multiple flexible nonlinear Timoshenko manipulators under undirected communication topology

In this paper, consensus control and vibration suppression problems are considered for the flexible nonlinear Timoshenko manipulator multi-agent system. The multi-agent system comprises multiple identical flexible Timoshenko manipulators, which can realize cooperation utilizing local information exchange between various agents. The bending deformation and shear deformation generated by the practical robotic manipulators during motion both affect the control effect, so the flexible Timoshenko manipulator is used to describe the dynamic characteristics. The flexible Timoshenko manipulator is a typical distributed parameter system, the deformation states are not only related to time but also to position, and the partial differential equations (PDEs) models are studied in this paper. Based on the PDEs models, the consensus control scheme is developed under the undirected communication topology structure, which can realize joint angle consensus tracking and vibration suppression of all the flexible Timoshenko manipulator agents simultaneously. Finally, the control effect is verified by the numerical simulations.

Physical Informed Sparse Learning for Robust Modeling of Distributed Parameter System and Its Industrial Applications

Physical Informed Sparse Learning for Robust Modeling of Distributed Parameter System and Its Industrial Applications

A fast data-driven fault detection and location method for unknown distributed thermal processes

The distributed thermal processes are widely present in various industrial operations, but the presence of time delay effect and unknown model information make achieving fast and timely fault detection challenging. Furthermore, the involvement of spatiotemporal coupled data in measurement processes further exacerbates the situation. To solve these problems, this paper proposes a data-driven fault detection and location method for thermal processes described by distributed parameter systems (DPS). Firstly, a Time/Space separation is employed to decoupling the spatio-temporal data into spatial basis functions and temporal coefficients. Subsequently, dynamic partial least squares (DPLS) is employed to obtain the temporal information of the model with fault-free data. Monitoring statistics in both the temporal and spatial domains are then constructed, and their thresholds are estimated by kernel density function. Finally, an online strategy for fault detection and location is presented. The method was validated on the catalytic rod, and an experimental snap curing oven.

Linear, Nonlinear, and Distributed-Parameter Observers Used for (Renewable) Energy Processes and Systems—An Overview

Full- and reduced-order observers have been used in many engineering applications, particularly for energy systems. Applications of observers to energy systems are twofold: (1) the use of observed variables of dynamic systems for the purpose of feedback control and (2) the use of observers in their own right to observe (estimate) state variables of particular energy processes and systems. In addition to the classical Luenberger-type observers, we will review some papers on functional, fractional, and disturbance observers, as well as sliding-mode observers used for energy systems. Observers have been applied to energy systems in both continuous and discrete time domains and in both deterministic and stochastic problem formulations to observe (estimate) state variables over either finite or infinite time (steady-state) intervals. This overview paper will provide a detailed overview of observers used for linear and linearized mathematical models of energy systems and review the most important and most recent papers on the use of observers for nonlinear lumped (concentrated)-parameter systems. The emphasis will be on applications of observers to renewable energy systems, such as fuel cells, batteries, solar cells, and wind turbines. In addition, we will present recent research results on the use of observers for distributed-parameter systems and comment on their actual and potential applications in energy processes and systems. Due to the large number of papers that have been published on this topic, we will concentrate our attention mostly on papers published in high-quality journals in recent years, mostly in the past decade.

Fuzzy Boundary Sampled-Data Control for Nonlinear Parabolic DPSs.

For a nonlinear parabolic distributed parameter system (DPS), a fuzzy boundary sampled-data (SD) control method is introduced in this article, where distributed SD measurement and boundary SD measurement are respected. Initially, this nonlinear parabolic DPS is represented precisely by a Takagi-Sugeno (T-S) fuzzy parabolic partial differential equation (PDE) model. Subsequently, under distributed SD measurement and boundary SD measurement, a fuzzy boundary SD control design is obtained via linear matrix inequalities (LMIs) on the basis of the T-S fuzzy parabolic PDE model to guarantee exponential stability for closed-loop parabolic DPS by using inequality techniques and a acrlong LF. Furthermore, respecting the property of membership functions, we present some LMI-based fuzzy boundary SD control design conditions. Finally, the effectiveness of the designed fuzzy boundary SD controller is demonstrated via two simulation examples.

Controllability of Distributed Parameter Systems

Controllability of Distributed Parameter Systems

Multidirectional cascade of square MIMO LTI subsystems: Transfer function representation

This paper considers continuous transfer function representation for a cascade composed of N mutually interconnected square multiple-input multiple-output (MIMO) linear time-invariant (LTI) subsystems with general multidirectional input–output configurations. Such cascaded structure can arise, e.g., from the spatial discretization of one-dimensional distributed parameter systems (DPSs) described by partial differential equations (PDEs) with boundary conditions representing different configurations of boundary input signals. As shown in the paper, the transfer function matrix of the resulting cascade can be calculated using the Redheffer star product of the subsystems’ transfer function matrices, which simplifies into the usual matrix product for the unidirectional cascade. For the particular case of 2 × 2 cascade composed of rational transfer function subsystems, the recursive formulas for its boundary and distributed transfer functions have been derived for both uni- and multidirectional configurations. The well-posedness and the stability criteria are also analyzed, proving to be more restrictive for the multidirectional cascade than for the unidirectional one due to the internal positive feedback in the first case. As shown in the attached example, the presented results can be used to approximate one-dimensional distributed parameter systems using rational transfer function subsystems representing the spatial sections of the original DPS.

Concurrent Learning Robust Adaptive Fault Tolerant Boundary Regulation of Hyperbolic Distributed Parameter Systems.

This article develops a robust adaptive boundary output regulation approach for a class of complex anticollocated hyperbolic partial differential equations subjected to multiplicative unknown faults in both the boundary sensor and actuator. The regulator design is based on the internal model principle, which amounts to stabilize a coupled cascade system, which consists of a finite-dimensional internal model driven by a hyperbolic distributed parameter system (DPS). To this end, a systematic sliding mode equipped with a backstepping approach is developed such that the robust state feedback control can be realized. Moreover, since the available information is a faulty boundary measurement at the right side point, state estimation is required. However, due to the presence of boundary unknown faults, we need to solve an issue of joint fault-state estimation. Restrictive persistent excitation conditions are usually required to guarantee the exact estimation of faults but are unrealistic in practice. To this end, a novel concurrent learning (CL) adaptive observer is proposed so that exponential convergence is obtained. It is the first time that the spirit of CL is introduced to the field of DPSs. Consequently, the observer-based adaptive boundary fault tolerant control scheme is developed, and rigorous theoretical analysis is given such that the exponential output regulation can be achieved. Finally, the effectiveness of the proposed methodology is demonstrated via comparative simulations.

Event-triggered iterative learning control for discrete parabolic distributed parameter systems

Iterative learning control (ILC) is an intelligent control strategy which can achieve the desired trajectory tracking by continuously renewing the control signal based on the previous experience. In this article, an event-triggered condition that can save system resources by reducing the number of iterative updates is proposed. Moreover, an event-triggered ILC algorithm is designed for linear discrete parabolic distributed parameter systems, the system input is updated with the learning law only when the derived triggered condition is met. Through rigorous mathematical analysis, the tracking error can converge to zero when the iterative batch approaches infinity under the given sufficient condition. Then, we also extended the above ideas to nonlinear case. By simulation, the effectiveness of the designed algorithm are illustrated.

Event-triggered adaptive vibration control for a flexible satellite with time-varying actuator faults

In this study, vibration control is studied for the flexible satellite through wireless communication, and given the fact that the actuators are prone to unknown faults, the input signal event-triggering and time-varying actuator faults are considered for the flexible satellite simultaneously. The satellite system is composed of two symmetrical flexible panels attached to a centrebody, which is a distributed parameter system and the established model is represented by partial differential equations (PDEs). Based on the PDEs model of the flexible satellite, the input event-triggered mechanism using the relative threshold strategy is designed firstly to lower the communication burden, and then the event-triggered-based adaptive fault-tolerant control laws are developed with the aid of the designed auxiliary signals for the system with time-varying actuator faults. With the proposed control scheme, the vibration of the system is controlled to the small neighborhood of zero, and the angle of the panels is regulated to the desired position. Finally, numerical simulation results are given to further show the effectiveness of the designed control scheme.

Development of a continuous reheating furnace state-space model based on the finite volume method

This study developed a modeling approach of a continuous steel slab reheating furnace process as a particular case of spatially distributed parameter systems involving radiative heat transfer. The aim of the resulting mathematical model, which is both detailed and computationally tractable, is to serve in prospective advanced process control (APC) and model-based optimization. The two-dimensional state-space model is introduced to accurately simulate the temperature distribution and dynamics, using the finite volume method (FVM) to incorporate essential heat transfer phenomena, including radiation, conduction, convection, advection, and simple combustion. The study presents a novel furnace measurement model that interprets temperature sensor readings (useful for state estimation), a benefit of the FVM treatment of radiative heat transfer. Strategies for linearization and model order reduction, such as balanced truncation, are proposed to facilitate real-time control. The simulation case study demonstrates the targeted capabilities of the model. The accuracy of the model is verified through comparisons with more complex computational fluid dynamics (CFD) software models. The study prioritizes theoretical modeling over empirical validation of a specific furnace unit, omitting experimental validation at this stage.