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.     

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.     

Optimal boundary control of a diffusion–convection-reaction PDE model with time-dependent spatial domain: Czochralski crystal growth process

In this paper the optimal boundary control problem for diffusion–convection-reaction processes modeled by partial differential equations (PDEs) defined on time-dependent spatial domains is considered. The model of the transport system with time-varying domain arises in the context of high energy consuming Czochralski crystal growth process in which the crystal temperature regulation must successfully account for the change in the crystal spatial domain due to the crystal growth process realized by the pulling crystal out of melt. Starting from the first principles of continuum mechanics and transport theorem the time-varying parabolic PDE describing temperature evolution is derived and represented as a nonautonomous parabolic evolution system on an appropriately defined function space which is exactly transformed in the infinite-dimensional boundary control problem for which a boundary linear quadratic regulator is proposed. Properties of the solution of the time-varying parabolic PDEs given by the two-parameter evolutionary system are utilized in the synthesis of the optimal boundary regulator, and the control law is applied to the model given by a two-dimensional partial differential equation in the cylindrical coordinates representing the Czochralski crystal growth process with one-dimensional growth direction. Finally, numerical results demonstrate optimal stabilization of the two-dimensional temperature distribution in the crystal.     

Modelling of a clinker rotary kiln using operating functions concept

Modelling and parameter identification of complex dynamic systems/processes is one of the main challenging problems in control engineering. An example of such a process is clinker rotary kiln (CRK) in cement industry. In the prevailing models independently of which structure is used to describe the kiln's dynamics and the identification algorithm, parameters are assumed to be centralised and constant while the CRK is well known as a distributed parameter system with a strongly varying dynamic through time. In this work, the kiln's dynamic is described in the form of a state‐space representation with three state variables using a system of partial differential equations (PDE). The structure is chosen so that it can easily be embedded in classical state‐space control algorithms. The parameters of the PDE system are called operating functions since their numerical values vary with respect to different operating conditions of the kiln, to their position in the kiln, and through time. A phenomenological approach is also proposed in this paper to identify the operating functions for a given steady‐state operation of the kiln. The model is then used to perform a semi‐dynamic simulation of the process through manipulating main process variables.     

Dynamic optimization of dissipative PDE systems using nonlinear order reduction

This article presents computationally efficient methods for the solution of dynamic constraint optimization problems arising in the context of spatially distributed processes governed by highly dissipative nonlinear partial differential equations (PDEs). The methods are based on spatial discretization using the method of weighted residuals with spatially global basis functions (i.e., functions that cover the entire domain of definition of the process and satisfy the boundary conditions). More specifically, we perform spatial discretization of the optimization problems using the method of weighted residuals with analytical or empirical (obtained via Karhunen-Loève expansion) eigenfunctions as basis functions, and combination of the method of weighted residuals with approximate inertial manifolds. The proposed methods account for the fact that the dominant dynamics of highly dissipative PDE systems are low dimensional in nature and lead to approximate optimization problems that are of significantly lower order compared to the ones obtained from spatial discretization using finite-difference and finite-element techniques, and thus, they can be solved with significantly smaller computational demand. The resulting dynamic nonlinear programs include equality constraints that constitute a low-order system of coupled ordinary differential equations and algebraic equations, which can then be solved with combination of standard temporal discretization and nonlinear programming techniques. We employ backward finite differences (implicit Euler) to perform temporal discretization and solve the nonlinear programs resulting from temporal and spatial discretization using reduced gradient techniques (MINOS). We use two representative examples of dissipative PDEs, a diffusion-reaction process with constant and spatially varying coefficients, and the Kuramoto-Sivashinsky equation, a model that describes incipient instabilities in a variety of physical and chemical systems, to demonstrate the implementation and evaluate the effectiveness of the proposed optimization algorithms.     

On the Evaluation of the Exit Boundary Condition for the Axial Dispersion Bioreactor System

Employment of the axial dispersion theory to model chemical or biochemical processes often results in coupled parabolic partial differential equations (PDEs). The classical Danckwerts boundary conditions have been widely used to solve these PDEs despite the fact that artificial suppression of the exit concentration gradient to zero may be physically unrealistic and may cause numerical instability. In this study, a recently developed exit boundary condition is shown to be inapplicable to model processes which demonstrate significant differences in the dynamics of components – typically found in biochemical processes. Using an activated sludge process and a pilot-scale subsurface flow (SSF) constructed wetland as case studies, we demonstrated in this study that a time-dependent exit boundary condition is more appropriate for use with the Danckwerts inlet boundary condition and the axial dispersion theory to model a biological system operating at near-plug flow conditions. Instead of using steady-state results we found that evaluation of alternative exit boundary condition using dynamic simulation results is more realistic.     

Feedback control of hyperbolic distributed parameter systems

Hyperbolic distributed parameter systems (DPS) represent a large number of industrial processes with spatially nonuniform operating variable profiles. Research has been conducted to develop high-performance control strategies for these systems by exploiting their high-fidelity models. In this paper, a feedback control method that yields improved performance is proposed for DPS modelled by first-order hyperbolic partial differential equations (PDEs) using the method of characteristics. Simulation results show that this method can provide effective control for the systems modelled by a scalar PDE as well as a system of PDEs. Further, it can efficiently compensate the effect of model-plant mismatch and effectively reject the disturbances.     

Grey-box modelling and control of chemical processes

The success of model based control of chemical processes is dependent on good process models. Many of these processes exhibit strong nonlinearity and time varying parameters and are often difficult to model accurately. The ‘grey-box’ model which combines partial knowledge of the process, with a neural network to capture the remaining dynamics, is a promising modelling tool for nonlinear processes. This modelling methodology maximizes the use of a priori process knowledge. This, in turn, reduces the size of the neural network required to capture the remaining dynamics, hence, less data for training and faster convergence can be achieved. The grey-box model is combined with a generic model control structure and applied to a number of simulations as well as a real-time process.     

Dynamic output feedback covariance control of stochastic dissipative partial differential equations

In this work, we develop a method for dynamic output feedback covariance control of the state covariance of linear dissipative stochastic partial differential equations (PDEs) using spatially distributed control actuation and sensing with noise. Such stochastic PDEs arise naturally in the modeling of surface height profile evolution in thin film growth and sputtering processes. We begin with the formulation of the stochastic PDE into a system of infinite stochastic ordinary differential equations (ODEs) by using modal decomposition. A finite-dimensional approximation is then obtained to capture the dominant mode contribution to the surface roughness profile (i.e., the covariance of the surface height profile). Subsequently, a state feedback controller and a Kalman-Bucy filter are designed on the basis of the finite-dimensional approximation. The dynamic output feedback covariance controller is subsequently obtained by combining the state feedback controller and the state estimator. The steady-state expected surface covariance under the dynamic output feedback controller is then estimated on the basis of the closed-loop finite-dimensional system. An analysis is performed to obtain a theoretical estimate of the expected surface covariance of the closed-loop infinite-dimensional system. Applications of the linear dynamic output feedback controller to both the linearized and the nonlinear stochastic Kuramoto-Sivashinsky equations (KSEs) are presented. Finally, nonlinear state feedback controller and nonlinear output feedback controller designs are also presented and applied to the nonlinear stochastic KSE.     

Approximate ODE models for population balance systems

We propose an approximate polynomial method of moments for a class of first-order linear PDEs (partial differential equations) of hyperbolic type, involving a filtering term with applications to population balance systems with fines removal terms. The resulting closed system of ODEs (ordinary differential equations) represents an extension to a recently published method of moments which utilizes least-square approximations of factors of the PDE over orthogonal polynomial bases. An extensive numerical analysis has been carried out for proof-of-concept purposes. The proposed modeling scheme is generally of interest for control and optimization of processes with distributed parameters.     

Feedback control of dissipative PDE systems using adaptive model reduction

The problem of feedback control of spatially distributed processes described by highly dissipative partial differential equations (PDEs) is considered. Typically, this problem is addressed through model reduction, where finite dimensional approximations to the original infinite dimensional PDE system are derived and used for controller design. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional ordinary differential equation (ODE) models using the method of weighted residuals. A common approach to this task is the Karhunen‐Loève expansion combined with the method of snapshots. To circumvent the issue of a priori availability of a sufficiently large ensemble of PDE solution data, the focus is on the recursive computation of eigenfunctions as additional data from the process becomes available. Initially, an ensemble of eigenfunctions is constructed based on a relatively small number of snapshots, and the covariance matrix is computed. The dominant eigenspace of this matrix is then utilized to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is recomputed with the addition of each snapshot with possible increase or decrease in its dimensionality; due to its small dimensionality the computational burden is relatively small. The proposed approach is applied to representative examples of dissipative PDEs, with both linear and nonlinear spatial differential operators, to demonstrate its effectiveness of the proposed methodology. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

High-resolution method for numerically solving PDEs in process engineering

Abrupt phenomena in modelling real-world systems indicate the importance of investigating systems with steep gradients. However, it is difficult to solve such systems either analytically or numerically. In 1993, Koren developed a high-resolution numerical computing scheme to deal with compressible fluid dynamics with Dirichlet boundary condition. Recently, Qamar adapted this scheme to numerically solve population balance equations without diffusion terms. This paper extends Koren’s scheme for partial differential equations (PDEs) that describe both nonlinear propagation and diffusive effects, and for PDEs with Cauchy or Neumann boundary condition. Accurate and convergent numerical solutions to the test problems have been obtained. The new results are also compared to those obtained by wavelet-based methods. It is shown that the method developed in this paper is more efficient.     

The Complexity and Multiplicity of the Specific cAMP Phosphodiesterase Family: PDE4, Open New Adapted Therapeutic Approaches

Cyclic nucleotides (cAMP, cGMP) play a major role in normal and pathologic signaling. Beyond receptors, cyclic nucleotide phosphodiesterases; (PDEs) rapidly convert the cyclic nucleotide in its respective 5′-nucleotide to control intracellular cAMP and/or cGMP levels to maintain a normal physiological state. However, in many pathologies, dysregulations of various PDEs (PDE1-PDE11) contribute mainly to organs and tissue failures related to uncontrolled phosphorylation cascade. Among these, PDE4 represents the greatest family, since it is constituted by 4 genes with multiple variants differently distributed at tissue, cellular and subcellular levels, allowing different fine-tuned regulations. Since the 1980s, pharmaceutical companies have developed PDE4 inhibitors (PDE4-I) to overcome cardiovascular diseases. Since, they have encountered many undesired problems, (emesis), they focused their research on other PDEs. Today, increases in the knowledge of complex PDE4 regulations in various tissues and pathologies, and the evolution in drug design, resulted in a renewal of PDE4-I development. The present review describes the recent PDE4-I development targeting cardiovascular diseases, obesity, diabetes, ulcerative colitis, and Crohn’s disease, malignancies, fatty liver disease, osteoporosis, depression, as well as COVID-19. Today, the direct therapeutic approach of PDE4 is extended by developing allosteric inhibitors and protein/protein interactions allowing to act on the PDE interactome.     

Temperature distribution reconstruction in Czochralski crystal growth process

A mechanical geometric crystal growth model is developed to describe the crystal length and radius evolution. The crystal radius regulation is achieved by feedback linearization and accounts for parametric uncertainty in the crystal growth rate. The associated parabolic partial differential equation (PDE) model of heat conduction is considered over the time‐varying crystal domain and coupled with crystal growth dynamics. An appropriately defined infinite‐dimensional representation of the thermal evolution is derived considering slow time‐varying process effects. The computational framework of the Galerkin's method is used for parabolic PDE order reduction and observer synthesis for temperature distribution reconstruction over the entire crystal domain. It is shown that the proposed observer can be utilized to reconstruct temperature distribution from boundary temperature measurements. The developed observer is implemented on the finite‐element model of the process and demonstrates that despite parametric and geometric uncertainties present in the model, the temperature distribution is reconstructed with the high accuracy. © 2014 American Institute of Chemical Engineers AIChE J, 60: 2839–2852, 2014     

Optimal modelling and experimentation for the improved sustainability of microfluidic chemical technology design

Optimization of the dynamics and control of chemical processes holds the promise of improved sustainability for chemical technology by minimizing resource wastage. Anecdotally, chemical plant may be substantially over designed, say by 35–50%, due to designers taking account of uncertainties by providing greater flexibility. Once the plant is commissioned, techniques of nonlinear dynamics analysis can be used by process systems engineers to recoup some of this overdesign by optimization of the plant operation through tighter control. At the design stage, coupling the experimentation with data assimilation into the model, whilst using the partially informed, semi-empirical model to predict from parametric sensitivity studies which experiments to run should optimally improve the model. This approach has been demonstrated for optimal experimentation, but limited to a differential algebraic model of the process. Typically, such models for online monitoring have been limited to low dimensions.Recently it has been demonstrated that inverse methods such as data assimilation can be applied to PDE systems with algebraic constraints, a substantially more complicated parameter estimation using finite element multiphysics modelling. Parametric sensitivity can be used from such semi-empirical models to predict the optimum placement of sensors to be used to collect data that optimally informs the model for a microfluidic sensor system. This coupled optimum modelling and experiment procedure is ambitious in the scale of the modelling problem, as well as in the scale of the application – a microfluidic device. In general, microfluidic devices are sufficiently easy to fabricate, control, and monitor that they form an ideal platform for developing high dimensional spatio-temporal models for simultaneously coupling with experimentation.As chemical microreactors already promise low raw materials wastage through tight control of reagent contacting, improved design techniques should be able to augment optimal control systems to achieve very low resource wastage. In this paper, we discuss how the paradigm for optimal modelling and experimentation should be developed and foreshadow the exploitation of this methodology for the development of chemical microreactors and microfluidic sensors for online monitoring of chemical processes. Improvement in both of these areas bodes to improve the sustainability of chemical processes through innovative technology.     

Generalized Diffusion Tensor Imaging (GDTI): A Method for Characterizing and Imaging Diffusion Anisotropy Caused by Non-Gaussian Diffusion

For non-Gaussian distributed random displacement, which is common in restricted diffusion, a second-order diffusion tensor is incapable of fully characterizing the diffusion process. The insufficiency of a second-order tensor is evident in the limited capability of diffusion tensor imaging (DTI) in resolving multiple fiber orientations within one voxel of human white matter. A generalized diffusion tensor imaging (GDTI) method was recently proposed to solve this problem by generalizing Fick's law to a higher-order partial differential equation (PDE). The relationship between the higher-order tensor coefficients of the PDE and the higher-order cumulants of the random displacement can be derived. The statistical property of the diffusion process was fully characterized via the higher-order tensor coefficients by reconstructing the probability density function (PDF) of the molecular random displacement. Those higher-order tensor coefficients can be measured using conventional diffusion-weighted imaging or spectroscopy techniques. Simulations demonstrated that this method was capable of quantitatively characterizing non-Gaussian diffusion and accurately resolving multiple fiber orientations. It can be shown that this method is consistent with the q-space approach. The second-order approximation of GDTI was shown to be DTI.     

Guaranteed cost distributed fuzzy observer‐based control for a class of nonlinear spatially distributed processes

The guaranteed cost distributed fuzzy (GCDF) observer‐based control design is proposed for a class of nonlinear spatially distributed processes described by first‐order hyperbolic partial differential equations (PDEs). Initially, a T–S fuzzy hyperbolic PDE model is proposed to accurately represent the nonlinear PDE system. Then, based on the fuzzy PDE model, the GCDF observer‐based control design is developed in terms of a set of space‐dependent linear matrix inequalities. In the proposed control scheme, a distributed fuzzy observer is used to estimate the state of the PDE system. The designed fuzzy controller can not only ensure the exponential stability of the closed‐loop PDE system but also provide an upper bound of quadratic cost function. Moreover, a suboptimal fuzzy control design is addressed in the sense of minimizing an upper bound of the cost function. The finite difference method in space and the existing linear matrix inequality optimization techniques are used to approximately solve the suboptimal control design problem. Finally, the proposed design method is applied to the control of a nonisothermal plug‐flow reactor. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2366–2378, 2013     

Invariant manifolds and the calculation of the long-term asymptotic response of nonlinear processes using singular PDEs

The present research work proposes a new systematic approach to the problem of finding invariant manifolds and computing the long-term asymptotic behavior of nonlinear dynamical systems. In particular, nonlinear processes are considered whose dynamics is driven by an external time-varying ‘forcing’ or input/disturbance term, or by a set of time-varying process parameters, or by the autonomous dynamics of an upstream process. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear PDEs, and a rather general set of conditions for solvability is derived. In particular, within the class of analytic solutions the aforementioned set of conditions guarantees the existence and uniqueness of a locally analytic solution. The solution of the system of singular PDEs is then proven to be a locally analytic invariant manifold of the nonlinear dynamical system. The local analyticity property of the invariant manifold enables the development of a series solution method, which can be easily implemented with the aid of a symbolic software package such as maple. Under a certain set of conditions, it is shown that the invariant manifold computed exponentially attracts all system trajectories, and therefore, the long-term asymptotic process response is described and calculated through the restriction of the process dynamics on the invariant manifold. Finally, in order to illustrate the proposed approach and method, a representative enzymatic bioreactor example is considered.