Handling sensor malfunctions in control of particulate processes

This work focuses on feedback control of particulate processes in the presence of sensor data losses. Two typical particulate process examples, a continuous crystallizer and a batch protein crystallizer, modeled by population balance models (PBMs), are considered. In the case of the continuous crystallizer, a Lyapunov-based nonlinear output feedback controller is first designed on the basis of an approximate moment model and is shown to stabilize an open-loop unstable steady-state of the PBM in the presence of input constraints. Then, the problem of modeling sensor data losses is investigated and the robustness of the nonlinear controller with respect to data losses is extensively investigated through simulations. In the case of the batch crystallizer, a predictive controller is first designed to obtain a desired crystal size distribution at the end of the batch while satisfying state and input constraints. Subsequently, we point out how the constraints in the predictive controller can be modified as a means of achieving constraint satisfaction in the closed-loop system in the presence of sensor data losses.     

Predictive control of transport-reaction processes

This work focuses on the development of computationally efficient predictive control algorithms for nonlinear parabolic and hyperbolic PDEs with state and control constraints arising in the context of transport-reaction processes. We first consider a diffusion-reaction process described by a nonlinear parabolic PDE and address the problem of stabilization of an unstable steady-state subject to input and state constraints. Galerkin’s method is used to derive finite-dimensional systems that capture the dominant dynamics of the parabolic PDE, which are subsequently used for controller design. Various model predictive control (MPC) formulations are constructed on the basis of the finite dimensional approximations and are demonstrated, through simulation, to achieve the control objectives. We then consider a convection-reaction process example described by a set of hyperbolic PDEs and address the problem of stabilization of the desired steady-state subject to input and state constraints, in the presence of disturbances. An easily implementable predictive controller based on a finite dimensional approximation of the PDE obtained by the finite difference method is derived and demonstrated, via simulation, to achieve the control objective.     

Robust model predictive control of nonlinear process systems: Handling rate constraints

This work considers the problem of stabilization of control affine nonlinear process systems subject to constraints on the rate of change and magnitude of control inputs in the presence of uncertainty. We first handle rate constraints within a soft constraints framework. A new robust predictive controller formulation that minimizes rate constraint violation while guaranteeing stabilization and input constraint satisfaction from an explicitly characterized stability region is designed. We then derive conditions that allow for guaranteed satisfaction of hard rate constraints. Subsequently, a predictive controller is designed that ensures rate constraints satisfaction when the required conditions are satisfied, relaxing them otherwise to preserve feasibility and robust stability. The implementation of the proposed predictive controllers is illustrated via a chemical reactor example.     

Predictive control of parabolic PDEs with boundary control actuation

This work focuses on predictive control of linear parabolic partial differential equations (PDEs) with boundary control actuation subject to input and state constraints. Under the assumption that measurements of the PDE state are available, various finite-dimensional and infinite-dimensional predictive control formulations are presented and their ability to enforce stability and constraint satisfaction in the infinite-dimensional closed-loop system is analyzed. A numerical example of a linear parabolic PDE with unstable steady state and flux boundary control subject to state and control constraints is used to demonstrate the implementation and effectiveness of the predictive controllers.     

Model-based control of particulate processes

In this work, we present an overview of recently developed methods for model-based control of particulate processes. We primarily discuss methods developed in the context of our previous research work and use examples of crystallization, aerosol and thermal spray processes to motivate the development of these methods and illustrate their application. Specifically, we initially discuss control methods for particulate processes which utilize suitable approximations of population balance models to design nonlinear, robust and predictive control systems and demonstrate their application to crystallization and aerosol processes. Finally, we discuss the issues of control problem formulation and controller design for high-velocity oxygen-fuel (HVOF) thermal spray processes and close with few thoughts on unresolved research challenges on control of particulate processes.     

A unified framework for detection, isolation and compensation of actuator faults in uncertain particulate processes

This paper presents a methodology for the robust detection, isolation and compensation of control actuator faults in particulate processes described by population balance models with control constraints and time-varying uncertain variables. The main idea is to shape the fault-free closed-loop process response via robust feedback control in a way that enables the derivation of performance-based fault detection and isolation (FDI) rules that are less sensitive to the uncertainty. Initially, an approximate finite-dimensional system that captures the dominant process dynamics is derived and decomposed into interconnected subsystems with each subsystem directly influenced by a single manipulated input. The decomposition is facilitated by the specific structure of the process input operator. A robustly stabilizing bounded feedback controller is then designed for each subsystem to enforce an arbitrary degree of asymptotic attenuation of the effect of the uncertainty in the absence of faults. The synthesis leads to (1) an explicit characterization of the fault-free behavior of each subsystem in terms of a time-varying bound on an appropriate Lyapunov function and (2) an explicit characterization of the robust stability region in terms of the control constraints and the size of the uncertainty. Using the fault-free Lyapunov dissipation bounds as thresholds for FDI in each subsystem, the detection and isolation of faults in a given actuator is accomplished by monitoring the evolution of the system within the stability region and declaring a fault if the threshold is breached. The thresholds are linked to the achievable degree of asymptotic uncertainty attenuation and can therefore be properly tuned by proper tuning of the controllers, thus making the FDI criteria less sensitive to the uncertainty. The robust FDI scheme is integrated with a robust stability-based controller reconfiguration strategy that preserves closed-loop stability following FDI. Finally, the implementation of the fault-tolerant control architecture on the particulate process is discussed and the proposed methodology is applied to the problem of robust fault-tolerant control of a continuous crystallizer with a fines trap.     

Integral approximation—an approach to reduced models for particulate processes

In this contribution, a model reduction technique for population balance systems describing particulate processes is presented. This technique is based on integral approximation and allows the derivation of highly accurate moment models. In contrast to other model reduction methods which can be found in literature, this integral approximation technique can be applied for arbitrarily complex phenomena specifications. The applicability of the presented method will be demonstrated for different example processes by comparing the dynamic behavior of the original population balance models with those of the derived reduced models of moments.     

Measurement based modeling and control of bimodal particle size distribution in batch emulsion polymerization

In this article, a novel modeling approach is proposed for bimodal Particle Size Distribution (PSD) control in batch emulsion polymerization. The modeling approach is based on a behavioral model structure that captures the dynamics of PSD. The parameters of the resulting model can be easily identified using a limited number of experiments. The resulting model can then be incorporated in a simple learning scheme to produce a desired bimodal PSD while compensating for model mismatch and/or physical parameters variations using very simple updating rules. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Application of Model Predictive Control to a Continuous Multiple‐Effect Crystallizer

An infinite horizon model predictive controller (IHMPC) with zone control is applied to a continuous five‐effect evaporative sodium chloride crystallizer. Firstly, a phenomenological dynamic model of the process is developed considering mass, energy, and moment balances coupled to crystallization kinetics. The developed model plays the role of the real system in order to study the proposed optimization/control strategy. The proposed approach is compared to a classical proportional integral derivative (PID) control system. The control strategy based on the prediction of the future state of the plant provides a faster response, a better stability to the process, and a reduction in energy consumption.     

A Laplace transformation based technique for reconstructing crystal size distributions regarding size independent growth

This article introduces a technique for reconstructing crystal size distributions (CSDs) described by well-established batch crystallization models. The method requires the knowledge of the initial CSD which can also be used to calculate the initial moments and initial liquid mass. The solution of the reduced four-moment system of ordinary differential equations (ODEs) coupled with an algebraic equation for the mass gives us moments and mass at the discrete points of the given computational time domain. This information can be used to get the discrete values of size independent growth and nucleation rates. The discrete values of growth and nucleation rates along with the initial distribution are sufficient to reconstruct the final CSD. In the derivation of current technique the Laplace transformation of the population balance equation (PBE) plays an important role. The proposed technique has dual purposes. Firstly, it can be used as a numerical technique to solve the given population balance model (PBM) for batch crystallization. Secondly, it can be used to reconstruct the final CSD from the initial one and also vice versa. The method is very efficient, accurate and easy to implement. Several numerical test problems of batch crystallization processes are considered here. For validation, the results of the proposed technique are compared with those from the high resolution finite volume scheme which solves the given PBM directly.     

Part I: Dynamic evolution of the particle size distribution in particulate processes undergoing combined particle growth and aggregation

The present study provides a comprehensive investigation on the numerical problems arising in the solution of dynamic population balance equations (PBEs) for particulate processes undergoing simultaneous particle growth and aggregation. The general PBE was numerically solved in both the continuous and its equivalent discrete form using the orthogonal collocation on finite elements (OCFE) and the discretized PBE method (DPBE), respectively. A detailed investigation on the effect of different particle growth rate functions on the calculated PSD was carried out over a wide range of variation of dimensionless aggregation and growth times. The performance (i.e., accuracy and stability) of the employed numerical methods was assessed by a direct comparison of predicted PSDs or/and their respective moments to available analytical solutions. It was found that the OCFE method was in general more accurate than the discretized PBE method but was susceptible to numerical instabilities. On the other hand, for growth dominated systems, the discretized PBE method was very robust but suffered from poor accuracy. For both methods, discretization of the volume domain was found to affect significantly the performance of the numerical solution. The optimal discretization of the volume domain was closely related with the satisfactory resolution of the time-varying PSD. Finally, it was shown that, in specific cases, further improvement of the numerical results could be obtained with the addition of an artificial diffusion term or the use of a moment-weighting method to correct the calculated PSD.     

A generalized population balance model for the prediction of particle size distribution in suspension polymerization reactors

In the present study, a comprehensive population balance model is developed to predict the dynamic evolution of the particle size distribution in high hold-up (e.g., 40%) non-reactive liquid-liquid dispersions and reactive liquid(solid)-liquid suspension polymerization systems. Semiempirical and phenomenological expressions are employed to describe the breakage and coalescence rates of dispersed monomer droplets in terms of the type and concentration of suspending agent, quality of agitation, and evolution of the physical, thermodynamic and transport properties of the polymerization system. The fixed pivot (FPT) numerical method is applied for solving the population balance equation. The predictive capabilities of the present model are demonstrated by a direct comparison of model predictions with experimental data on average mean diameter and droplet/particle size distributions for both non-reactive liquid-liquid dispersions and the free-radical suspension polymerization of styrene and VCM monomers.     

Inferential closed-loop control of particle size distribution for styrene emulsion polymerization

In this work, a new control strategy for controlling the particle size distribution (PSD) in emulsion polymerization has been proposed. It is shown that the desired PSD can be achieved by controlling the free surfactant concentration which in turn can be done by manipulating the surfactant feed rate. Simulation results show that the closed-loop control of free surfactant concentration results in a better control of PSD compared to open-loop control strategy, in presence of model mismatch and disturbances. Since the on-line measuring of ionic free surfactant concentration is difficult, conductivity which is related to it is measured instead and used for control purposes. The closed-loop control of conductivity also results in a better control of PSD compared to open-loop control strategy, but its performance is not as good as controlling free surfactant concentration in presence of model mismatch.     

Part II: Dynamic evolution of the particle size distribution in particulate processes undergoing simultaneous particle nucleation, growth and aggregation

The present study provides a comprehensive investigation on the solution of the dynamic population balance equation (PBE) for particulate processes undergoing simultaneous particle nucleation, growth and aggregation. The general PBE was numerically solved in both the continuous and its equivalent discrete form using the orthogonal collocation on finite elements and the discretized PBE method, respectively. A detailed investigation on the effect of particle nucleation rate on the calculated particle size distribution (PSD) was carried out over a wide range of variation of dimensionless aggregation, nucleation and growth times. The performance (i.e., accuracy and stability) of the two numerical methods was assessed by a direct comparison of predicted PSDs and/or their respective moments to available analytical solutions. For combined aggregation and nucleation problems, the numerical error scaled with the product of the dimensionless aggregation and nucleation times. On the other hand, for combined growth and nucleation problems, the numerical error scaled only with the dimensionless growth time. For particulate systems with minimal particle growth, constant particle nucleation rate and Brownian aggregation, the total particle number approached a “steady-state” value characterized by the equilibrium of particle aggregation and nucleation rates. When the particle nucleation rate followed a pulse-like function, the PSD converged to a self-similar distribution after the end of particle nucleation. Moreover, for particulate systems exhibiting a constant particle nucleation rate and a Brownian-type particle aggregation kernel, an increase in the particle growth rate resulted in a decrease in the final total number of particles. On the other hand, for a constant particle nucleation rate and an electrostatically stabilized Brownian aggregation kernel, an increase in the particle growth rate can lead to an increase in the final total number of particles.     

Control of particulate processes: Recent results and future challenges

This paper provides a discussion of the existing results on control of particulate processes using population balance models and presents an overview of future research directions in this field in the context of chemical, materials and biological process systems.     

Feedback controllability assessment and control of particle size distribution in emulsion polymerisation

In this article, the importance of particle size distribution (PSD) control as a means for the inferential control of the rheology of emulsion polymers is illustrated. A controllability assessment is presented to illustrate the attainability or otherwise of bimodal PSD using feedback control through a consideration of the process mechanisms—measurement limitations and process constraints that prevent the implementation of feedback corrections. The suitability of a batch-to-batch iterative feedback PSD control is demonstrated, which could act in addition to any in-batch feedback control, the latter being less feasible in certain cases, as argued in this article. A detailed population balance model is used for the batch-to-batch control, which simplifies model update and feedback correction.     

Part IV: Dynamic evolution of the particle size distribution in particulate processes. A comparative study between Monte Carlo and the generalized method of moments

The present work provides a comparative study on the numerical solution of the dynamic population balance equation (PBE) for batch particulate processes undergoing simultaneous particle aggregation, growth and nucleation. The general PBE was numerically solved using three different techniques namely, the Galerkin on finite elements method (GFEM), the generalized method of moments (GMOM) and the stochastic Monte Carlo (MC) method. Numerical simulations were carried out over a wide range of variation of particle aggregation and growth rate models. The performance of the selected techniques was assessed in terms of their numerical accuracy and computational requirements. The numerical results revealed that, in general, the GFEM provides more accurate predictions of the particle size distribution (PSD) than the other two methods, however, at the expense of more computational effort and time. On the other hand, the GMOM yields very accurate predictions of selected moments of the distribution and has minimal computational requirements. However, its main disadvantage is related to its inherent difficulty in reconstructing the original distribution using a finite set of calculated moments. Finally, stochastic MC simulations can provide very accurate predictions of both PSD and its corresponding moments while the MC computational requirements are, in general, lower than those required for the GFEM.     

A control oriented study on the numerical solution of the population balance equation for crystallization processes

The population balance equation provides a well established mathematical framework for dynamic modeling of numerous particulate processes. Numerical solution of the population balance equation is often complicated due to the occurrence of steep moving fronts and/or sharp discontinuities. This study aims to give a comprehensive analysis of the most widely used population balance solution methods, namely the method of characteristics, the finite volume methods and the finite element methods, in terms of the performance requirements essential for on-line control applications. The numerical techniques are used to solve the dynamic population balance equation of various test problems as well as industrial crystallization processes undergoing simultaneous nucleation and growth. The time-varying supersaturation profiles in the latter real-life case studies provide more realistic scenarios to identify the advantages and pitfalls of a particular numerical technique.The simulation results demonstrate that the method of characteristics gives the most accurate numerical predictions, whereas high computational burden limits its use for complex real crystallization processes. It is shown that the high order finite volume methods in combination with flux limiting functions are well capable of capturing sharp discontinuities and steep moving fronts at a reasonable computational cost, which facilitates their use for on-line control applications. The finite element methods, namely the orthogonal collocation and the Galerkin's techniques, on the other hand may severely suffer from numerical problems. This shortcoming, in addition to their complex implementation and low computational efficiency, makes the finite element methods less appealing for the intended application.     

Predictive control of particlesize distribution of crystallization process using deep learning based image analysis

The challenges to regulate the particle-size distribution (PSD) stem from on-line measurement of the full distribution and the distributed nature of crystallization process. In this article, a novel nonlinear model predictive control method of PSD for crystallization process is proposed. Radial basis function neural network is adopted to approximate the PSD such that the population balance model with distributed nature can be transformed into the ordinary differential equation (ODE) models. Data driven nonlinear prediction model of the crystallization process is then constructed from the input and output data and further be used in the proposed nonlinear model predictive control algorithm. A deep learning based image analysis technology is developed for online measurement of the PSD. The proposed PSD control method is experimentally implemented on a jacketed batch crystallizer. The results of crystallization experiments demonstrate the effectiveness of the proposed control method.