Complex Distributed Parameter Systems Research Articles

Adaptive spatial-model-based predictive control for complex distributed parameter systems

Many industrial thermal processes can be classified as complex distributed parameter systems (DPS). The traditional control method has become ill-suited for complex DPS since it ignores the time-varying dynamics and spatially distributed behavior. Moreover, the controller incorporating a nonlinear model imposes substantial computational demands, hindering the timely acquisition of control strategies in practical scenarios. In this study, an adaptive spatial-model-based predictive controller with real-time linearization for complex DPS is proposed. Initially, an adaptive spatiotemporal modeling method is developed to characterize the time-varying dynamics of DPS more accurately. Based on this developed spatiotemporal model, a spatial model predictive control strategy with real-time linearization is further designed to reduce the computational burden of the controller. In addition, a weighted terminal cost is embedded to replace terminal constraints in the controller, ensuring stability while improving calculation efficiency. Theoretical analysis confirms the stability of the proposed controller. Finally, furnace simulation and oven experiments are performed to comprehensively evaluate the performance of the proposed controller. It shows that the proposed method significantly improves tracking performance and robustness while reducing computational load.

A spatiotemporal fuzzy control approach for complex distributed parameter system with application to thermal processes

Many industrial processes belong to distributed parameter systems (DPSs) with unknown spatiotemporal dynamics and complex boundary conditions. However, attentions to both temporal and spatial dynamics in the control law are less paid in the existing control methods, which often makes them challenging to achieve accurate control. Here, a spatiotemporal fuzzy control approach with consideration of both temporal and spatial dynamics is proposed for unknown and complex DPSs. First, a spatiotemporal fuzzy model is developed to reconstruct dynamics of nonlinear DPS. Since it constructs a two-layer fuzzy rules (i.e., temporal fuzzy rule and spatial fuzzy rule) to consider both the nonlinear dynamics on time and the nonlinear energy transfer on space, it can effectively achieve the model of DPSs. On this basis, a spatiotemporal fuzzy controller is developed according to the developed two-layer fuzzy rules, which means that this derived control law considers both temporal and spatial dynamics. Then, a two-step solution method is designed for the spatiotemporal control law. Stability conditions of the control system are rigorously deduced by transforming the solving process to a series of linear matrix inequalities (LMIs). The stability and effect of the proposed spatiotemporal control strategy are demonstrated by mathematical proof and case studies. Comparison results demonstrate that the proposed method has better control performance (relative errors and mean absolute percentage errors are bounded in 4%).

Deep Extreme Learning Machines Based Two-Phase Spatiotemporal Modeling for Distributed Parameter Systems

Accurate and robust modeling of complex distributed parameter systems (DPSs) is a challenge for three reasons: 1) they have infinite-dimensional characteristics; 2) they are time/space coupled; and 3) there are model uncertainties. In this article, a two-phase spatiotemporal (S/T) modeling framework based on deep extreme learning machine (DELM) is proposed for DPSs. The modeling process consists of two S/T models in two phases: Phase I: a DELM model and Phase II: a Karhunen–Loève (KL) based ELM (KL-ELM) model. In phase I, the DELM model is constructed by combing the multilayer ELM (ML-ELM), ELM, and kernel-based ELM (K-ELM) to approximate the dominant S/T dynamics of DPSs. Since DPSs have an infinite-dimensional characteristic that can hardly be handled directly, ML-ELM is first employed to transform the infinite-dimensional systems into finite-dimensional systems. Then, the ELM model is adopted to further approximate the finite-dimensional systems to ensure the model can predict future dynamic behavior. Finally, the K-ELM is used to reconstruct the infinite-dimensional systems, which can be considered as the inverse process of ML-ELM. Thus, the final DELM model can be used for prediction in both space and time directions. In phase II, a KL-ELM model is constructed to compensate for modeling errors caused by reconstruction error or unknown nonlinear dynamics. By integrating the obtained DELM and KL-ELM models, the proposed two-phase S/T model can be constructed. Experiments on a typical industrial thermal process verified that the proposed method may work better in complex DPSs.

Approximate Inertial Manifold-Based Model Reduction and Vibration Suppression for Rigid-Flexible Mechanical Arms

The dynamic characteristics of the mechanical arm with a rigid-flexible structure are very complex. The reason is that it is a complex DPS (distributed parameter system) with infinite dimension and nonlinearity in essence due to the rigid-flexible coupling. So, accurately positioning and controlling the rigid-flexible mechanical arms could be difficult. Therefore, a model reduction method of rigid-flexible mechanical arms based on the approximate inertial manifold is put forward. To repress the residual vibration of the end of the mechanical arm, a feedforward control strategy is designed. The high-dimensional solution of the vibration equation of the rigid-flexible mechanical arms is projected into the complete space composed of orthogonal decomposition modes. By using Galerkin’s method, the system is simplified and the approximate solution is obtained through the interaction between high-order and low-order modes. The truncated finite mode is also used to construct a lowest-order dynamic model on the basis of approximate inertia manifold. Given the reduced-order rigid-flexible mechanical arms dynamic model, dynamic response analysis is conducted to optimize the target position error and end residual vibration. A limited number of sinusoidal signals approximately combine the input signal, by using the particle swarm optimization algorithm to optimize the input signal, and the amplitude of the sinusoidal signal is corrected. The simulation results depict the superiority of the proposed method, which greatly suppresses the end residual vibration of the mechanical arm and realizes the accurate positioning of the end of the mechanical arm. In addition, the hardware experimental device of the rigid-flexible mechanical arms is constructed, and the experimental verification of the above method is put into effect. The simulation results of angular displacement and end vibration of the reduced model are accordant which is shown by the experimental results of the hardware platform.

Dissimilarity Analysis-Based Multimode Modeling for Complex Distributed Parameter Systems

For complex distributed parameter systems (DPSs) with strong nonlinearities and time-varying dynamics, the conventional spatiotemporal modeling methods become ill-suited since the elementary assumption that the process data follow a unimodal Gaussian distribution usually becomes invalid. In this paper, a multimode method is proposed for modeling of such systems. First, the original operating space is partitioned along the time dimension into several subspaces via modified dissimilarity analysis. Each subspace represents the local spatiotemporal characteristics of the original system. Second, the Karhunen–Loeve decomposition (KLD)-based spatiotemporal modeling approach is applied to approximate the local dynamics of each subspace. Finally, an ensemble model is obtained using the soft weighting sum of the local ones, where the corresponding weights are calculated by principal component regression. By properly decomposing the original space into several local parts, the ensemble model is capable of handling the strong nonlinearities and time-varying dynamics of the system. The validity and efficiency of the proposed method are verified on two representative applications: 1) a one-dimensional parabolic catalytic rod and 2) a two-dimensional curing thermal process. The experimental results show that the proposed method provides a superior performance regarding modeling accuracy compared to several baselines.

Modelling and boundary optimal control design of hybrid column flotation

AbstractA three‐phase continuous hybrid flotation column that seeks to obtain the benefits of both mechanical cells and flotation columns is modelled as the interconnection of a CSTR representing the well‐mixed zone and two plug‐flow reactors (PFR) representing pulp and froth zones. The plant model accounts for the micro‐scale processes such as bubble‐particle collision and attachment and the appearance and breakage of bubbles. This complex distributed parameter system (DPS) is described by sets of nonlinear coupled conservation counter‐current hyperbolic partial differential equations (PDEs) and one set of ordinary differential equations (ODEs). The dynamic conservation law‐based model for the continuous hybrid flotation column including well‐stirred, pulp (bubbly), and froth zones, is utilized in an optimal model‐based controller design. This linear quadratic regulator (LQR)‐based controller accounts for optimality, stability, and performance. The controller design utilizes a linear model obtained by linearization at operating steady states of interest. A full‐state optimal feedback control law is designed and controller performance has been demonstrated through a numerical simulation of physically meaningful and relevant plant operating conditions. The LQR‐based optimal controller outperforms proportional‐integral (PI)‐based control by more than an order of magnitude in terms of a return to steady state after a perturbation in the initial condition.

Locally Weighted Principal Component Analysis-Based Multimode Modeling for Complex Distributed Parameter Systems.

Global principal component analysis (PCA) has been successfully introduced for modeling distributed parameter systems (DPSs). In spite of the merits, this method is not feasible due to parameter variations and multiple operating domains. A novel multimode spatiotemporal modeling method based on the locally weighted PCA (LW-PCA) method is developed for large-scale highly nonlinear DPSs with parameter variations, by separating the original dataset into tractable subsets. This method implements the decomposition by making full use of the dependence among subset densities. First, the spatiotemporal snapshots are divided into multiple different Gaussian components by using a finite Gaussian mixture model (FGMM). Once the components are derived, a Bayesian inference strategy is then applied to calculate the posterior probabilities of each spatiotemporal snapshot belonging to each component, which will be regarded as the local weights of the LW-PCA method. Second, LW-PCA is adopted to calculate each locally weighted snapshot matrix, and the corresponding local spatial basis functions (SBFs) can be generated by the PCA method. Third, all the local temporal models are estimated using the extreme learning machine (ELM). Thus, the local spatiotemporal models can be produced with local SBFs and corresponding temporal model. Finally, the original system can be approximated using the sum form of each local spatiotemporal model. Unlike global PCA, which uses global SBFs to construct a global spatiotemporal model, LW-PCA approximates the original system by multiple local reduced SBFs. Numerical simulations verify the effectiveness of the developed multimode spatiotemporal model.

A Novel Spatiotemporal Fuzzy Method for Modeling of Complex Distributed Parameter Processes

Fuzzy modeling has been widely used to model lumped parameter systems. However, it cannot be used to model complex distributed parameter systems (DPS) due to its inability to handle spatial dynamics. In this paper, we propose a novel spatiotemporal fuzzy method for the modeling of complex nonlinear DPSs. A spatial fuzzy model is first constructed to represent the nonlinear spatial dynamics. This process ensures that the space information is inherently considered in the spatiotemporal fuzzy model. A fuzzy model is then used to represent the nonlinear temporal dynamics. These two fuzzy models are further integrated to construct a spatiotemporal fuzzy model, which allows for the reconstruction of the DPS. Additionally, it can improve the modeling robustness even in the presence of noise due to the robust ability of fuzzy modeling. Performance analyses and experimental validations further show that the proposed method can effectively model complex nonlinear DPSs and has the better modeling ability than several commonly used methods.

Parallel Control of Distributed Parameter Systems.

In this paper, we study the control problems of distributed parameter systems, and discuss the limitations of traditional control methods. In recent years, social factors have gradually become an essential parameter of system modeling. For complex distributed parameter systems, the accurate modeling becomes difficult. With the rapid development of the network and the technology of big data and cloud computing, based on the advanced control theory of large-scale computing, we introduce the idea of parallel control to the control of distributed parameter systems. Parallel control is a method to accomplish tasks through the interaction of virtual and actual. Its core is to model the complex distributed parameter system on artificial society or artificial system, then analyze and evaluate it by computational experiment, and finally control and manage the distributed parameter system by parallel execution. Data-driven control and computational control are used in this method, which is a control idea that adapts to the rapid development of society.

Optimization of ASP flooding based on dynamic scale IDP with mixed-integer

The alkali/surfactant/polymer (ASP) flooding is a complex distributed parameter system (DPS). In this paper, an optimization model of ASP flooding is developed, which takes net present value (NPV) as the performance index, oil/water seepage continuity equations and adsorption diffusion equations of displacing agents as the governing equations, physicochemical algebraic equations and boundary conditions of displacing agents as the constraint equations. To solve the injection concentration and size of each slug of the model and terminal flooding time, a dynamic scale iterative dynamic programming with mixed-integer (DSMI-IDP) is proposed. The essence of slug size is time, it can only be integer. In DSMI-IDP, the integer truncation is carried out by a proportion method after time normalization which can convert the free time terminal problem to a fixed time terminal problem. A dynamic contraction factor and a principle of adjustment factors are introduced to realize the dynamic scale. To test the algorithm, three examples are solved by DSMI-IDP. At last, the DSMI-IDP is applied to optimize an optimization model of ASP flooding. The solving effect is shown by the comparison with IGA, MIDP and trial and error solutions.

Spatial-Temporal Separation Based on the Dynamic Recurrent Wavelet Neural Network Modelling for ASP Flooding

In this paper, a three-dimensional spatial-temporal decomposition modelling method is proposed to build the alkali-surfactant-polymer (ASP) flooding model, in which a new dynamic recurrent wavelet neural network (DRWNN) is presented to identify the temporal coefficients. At first, the detailed mathematical model of ASP flooding is described which is a complex distributed parameter system. Then a three-dimensional spatial-temporal modelling method is inferred based on Karhunen-Loeve (K-L) decomposition to decompose the water saturation of reservoir into a series of spatial basis functions and corresponding temporal coefficients. Furthermore, the recurrent wavelet neural network is used to acquire the identification model, in which the injection concentrations of ASP flooding and temporal coefficients are taken as the input and output information. In order to improve the capability of dynamic modelling, DRWNN is proposed through adding feedback layers and setting the different weights with time to achieve dynamic memory of the past information. Considering the gradient descent method for the neural networks training easily leads to local minimum and slow convergence speed, the spectral conjugate gradient method is introduced to optimize the weights of DRWNN. At last, DRWNN is used to build the relation between the moisture content of production wells and the water saturation of the corresponding grids. Thus, the final approximate model of ASP flooding is finished. The accuracy is proved by model with four injection wells and nine production wells through data from the mechanism model.

Distributed parameter models for vehicle–suspension systems

Recent advances in the modelling of complex distributed parameter systems incorporating discrete dynamic components (such as a continuously elastic chassis attached to lumped parameter suspension and control components) and the active control of such devices have appeared in the literature. These techniques are extremely efficient and make possible the notion of real time control of suspension kinematics and forces, as well as of chassis motion. In the present paper, the dynamic analysis of a class of systems which includes multiple–point connected discrete–distributed structures is accomplished using Green's function methods in conjunction with a newly developed hierarchical algorithm for eigenvalue extraction. The modal analysis of such systems, wherein a general rigid body possessing translational and rotational degrees of freedom is connected to a distributed substructure at multiple points, is addressed in a compact and efficient way.

Substructure system identification from coupled system test data

System identification in the context of linear structural dynamics and mechanical vibrations is addressed. More specifically, the focus is on large systems built up by components and the identification of its subsystems using test data of the complete system. A method based on reconstruction of the interface forces acting between the unknown subsystem and its neighbour is presented. Theoretical aspects regarding frequency domain load identification are illuminated. The causes and consequences of an ill-conditioned matrix inversion are emphasized. It is shown that the load identification and thus also the subsystem identification is sensitive to the existence of general anti-resonances in the frequency domain of interest. Different methods for validation of an identified subsystem are discussed. Two numerical examples with artificial test data are given to illustrate the procedure: a simple lumped parameter system and a slightly more complex distributed parameter system. The technique is also successfully applied to a physical test setup consisting of a plane frame structure built up from two components.

Transfer Functions of One-Dimensional Distributed Parameter Systems by Wave Approach

An alternative analysis technique, which does not require eigensolutions as a priori, for the dynamic response solutions, in terms of the transfer function, of one-dimensional distributed parameter systems with arbitrary supporting conditions, is presented. The technique is based on the fact that the dynamic displacement of any point in a waveguide can be determined by superimposing the amplitudes of the wave components traveling along the waveguide, where the wave numbers of the constituent waves are defined in the Laplace domain instead of the frequency domain. The spatial amplitude variations of individual waves are represented by the field transfer matrix and the distortions of the wave amplitudes at point discontinuities due to constraints or boundaries are described by the wave reflection and transmission matrices. Combining these matrices in a progressive manner along the waveguide using the concepts of generalized wave reflection and transmission matrices leads to the exact transfer function of a complex distributed parameter system subjected to an externally applied force. The transient response solution can be obtained through the Laplace inversion using the fixed Talbot method. The exact frequency response solution, which includes infinite normal modes of the system, can be obtained in terms of the complex frequency response function from the system’s transfer function. This wave-based analysis technique is applicable to any one-dimensional viscoelastic structure (strings, axial rods, torsional bar, and beams), in particular systems with multiple point discontinuities such as viscoelastic supports, attached mass, and geometric/material property changes. In this paper, the proposed approach is applied to the flexural vibration analysis of a classical Euler–Bernoulli beam with multiple spans to demonstrate its systematic and recursive formulation technique.

Dynamic Optimization of Complex Distributed Process Systems

The dynamic optimization (DO) of complex distributed parameter systems (DPSs), like e.g., reaction–diffusion processes, is a challenging task. Most of the existing numerical approaches imply a large computational effort, therefore precluding its application to demanding applications like real time DO or model predictive control. This work, based on the control vector parameterization (CVP) approach, describes two ways to enhance the efficiency of the resulting nonlinear programming (NLP) problem solution. On the one hand, the convergence properties of the NLP solver are enhanced through the use of exact gradients and projected Hessians (H.p). On the other hand, simulation efficiency is improved through the use of reduced order descriptions of the DPSs. The capabilities and possibilities of these two enhancements are illustrated with a number of complex distributed case studies.

Formulation of Reduced-Order Models for the Dynamic and Stability Analyses of Autothermal Radial Flow Reactors

Autothermal radial flow reactors typically consist of a reactor setup of multiple catalyst-beds with internal heat exchange. These reactors are widely used because of their high efficiency due to the internal heat exchange, and radial flow arrangements are preferred due to their low pressure drops. Although an efficient multi-functional reactor arrangement, this setup has shown to provide for an additional destabilizing mechanism via the heat feedback. Thus, additional stability considerations are necessary when operating autothermal or non-adiabatic reactors at high conversions. This work proposes the formulation of a simplified model to investigate the effect of the heat transfer feedback on the stability of autothermal radial flow reactors. The present work focuses on a lumping approach to reduce the order of a complex distributed parameter system. The model is complex enough so as to preserve the intricacies of this reactor arrangement, but still yield a tractable dynamic formulation. The industrial ammonia synthesis process has been chosen as a case study to illustrate the proposed methodology. The lumped model predictions are qualitatively compared against numerical simulations of a detailed mathematical model.

Subspace identification based inferential control applied to a continuous pulp digester

Abstract The idea of constructing a data-driven stochastic system model through subspace identification for the purpose of inferential control is investigated. Various available methods for designing an inferential controller are discussed and their limitations are brought out, particularly in applications involving multi-variable processes. Practical issues that arise in identifying a system model geared toward inferential control using a subspace method are discussed. They include: handling of nonstationary disturbances, handling of multi-rate measurements/missing data, and secondary measurement selection. With the identified stochastic system model, a multi-rate Kalman filter can be designed and coupled with a model predictive controller. The method is applied to a continuous pulp digester, which is a complex distributed parameter system involving heterogeneous reactions. The application study indicates much potential for the data-based approach.

Closed-Form Transient Response of Distributed Damped Systems, Part I: Modal Analysis and Green’s Function Formula

An analytical method is developed for closed-form estimation of the transient response of complex distributed parameter systems that are nonproportionally damped, and subject to arbitrary external, initial, and boundary excitations. A new modal analysis leads to the Green’s function formula for the distributed system and an eigenfunction expansion of the system Green’s function. The legitimacy of the modal expansion is also shown.

Distributed Transfer Function Analysis of Complex Distributed Parameter Systems

This paper presents a new analytical and numerical method for modeling and synthesis of complex distributed parameter systems that are multiple continua combined with lumped parameter systems. In the analysis, the complex distributed parameter system is first divided into a number of subsystems; the distributed transfer functions of each subsystem are determined in exact and closed form by a state space technique. The complex distributed parameter system is then assembled by imposing displacement compatibility and force balance at the nodes where the subsystems are interconnected. With the distributed transfer functions and the transfer functions of the constraints and lumped parameter systems, exact, closed-form formulation is obtained for various dynamics and vibration problems. The method does not require a knowledge of system eigensolutions, and is valid for non-self-adjoint systems with inhomogeneous boundary conditions. In addition, the proposed method is convenient in computer coding and suitable for computerized symbolic manipulation.

Dynamic behaviour of coupled CSTRs operating under different conditions

The present paper analyzes the consequences of coupling two independent CSTRs operating under different stability conditions. A variety of behavioural patterns of the coupled system have been shown to exist for different types of reactor combinations, viz. stable-stable, oscillatory-stable and oscillatory-oscillatory. The reactor output is in general sensitive to the exchange coefficient, a proper choice of which can be of use in practice. The results obtained also provide an explanation for the occurrence of aperiodic behaviour. More importantly, the present simple model, viz. the linear coupling of the lumped parameter systems, offers guidelines for the analysis of more complex distributed parameter systems such as catalyst pellet or the fixed bed.