Output feedback control of dissipative PDE systems with partial sensor information based on adaptive model reduction

We address the problem of control of spatially distributed processes in the presence of measurement constraints. Specifically, we assume the availability of sensors that measure part of the state spatial profile. The measurements are utilized for the derivation and on‐demand update of reduced order models (ROM) based on an extension of the adaptive proper orthogonal decomposition (APOD) method using a snapshot reconstruction technique. The proposed Gappy‐APOD methodology constructs locally accurate low‐dimensional ROM thus resulting in a computationally efficient alternative to using a large‐dimensional ROM with global validity. Based on the low‐dimensional ROM and continuous measurements available from point sensors a Lyapunov‐based static output feedback controller is subsequently designed. The proposed controller design method is illustrated on an unstable process modeled by the Kuramoto‐Sivashinsky equation, when the designed controller successfully stabilizes the process even in the presence of model uncertainty. © 2012 American Institute of Chemical Engineers AIChE J, 59: 747–760, 2013     

Order‐reduction of parabolic PDEs with time‐varying domain using empirical eigenfunctions

A novel methodology for the order‐reduction of parabolic partial differential equation (PDE) systems with time‐varying domain is explored. In this method, a mapping functional is obtained, which relates the time‐evolution of the solution of a parabolic PDE with time‐varying domain to a fixed reference domain, while preserving space invariant properties of the initial solution ensemble. Subsequently, the Karhunen–Loève decomposition is applied to the solution ensemble on fixed spatial domain resulting in a set of optimal eigenfunctions. Further, the low dimensional set of empirical eigenfunctions is mapped on the original time‐varying domain by an appropriate mapping, resulting in the basis for the construction of the reduced‐order model of the parabolic PDE system with time‐varying domain. This methodology is used in three representative cases, one‐ and two‐dimensional (1‐D and 2‐D) models of nonlinear reaction‐diffusion systems with analytically defined domain evolutions, and the 2‐D model of the Czochralski crystal growth process with nontrivial geometry. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4142–4150, 2013     

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     

Low‐order optimal regulation of parabolic PDEs with time‐dependent domain

Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two‐dimensional (2‐D) parabolic partial differential equation with time‐varying domain is realized in this work. The transformation of boundary actuation to distributed control setting allows to represent the system's model in a standard evolutionary form. By exploring dynamical model evolution and generating data, a set of time‐varying empirical eigenfunctions that capture the dominant dynamics of the distributed system is found. This basis is used in Galerkin's method to accurately represent the distributed system as a finite‐dimensional plant in terms of a linear time‐varying system. This reduced‐order model enables synthesis of a linear optimal output tracking controller, as well as design of a state observer. Finally, numerical results are prepared for the optimal output tracking of a 2‐D model of the temperature distribution in Czochralski crystal growth process which has nontrivial geometry. © 2014 American Institute of Chemical Engineers AIChE J, 61: 494–502, 2015     

Delay‐range‐dependent guaranteed cost control for batch processes with state delay

A guaranteed cost control scheme is proposed for batch processes described by a two‐dimensional (2‐D) system with uncertainties and interval time‐varying delay. First, a 2‐D controller, which includes a robust feedback control to ensure performances over time and an iterative learning control to improve the tracking performance from cycle to cycle, is formulated. The guaranteed cost law concept of the proposed 2‐D controller is then introduced. Subsequently, by introducing the Lyapunov–Krasovskii function and adding a differential inequality to the Lyapunov function for the 2‐D system, sufficient conditions for the existence of the robust guaranteed cost controller are derived in terms of matrix inequalities. A design procedure for the controller is also presented. Furthermore, a convex optimization problem with linear matrix inequality (LMI) constraints is formulated to design the optimal guaranteed cost controller that minimizes the upper bound of the closed‐loop system cost. The proposed control law can stabilize the closed‐loop system as well as guarantee H_∞ performance level and a cost function with upper bounds for all admissible uncertainties. The results can be easily extended to the constant delay case. Finally, an illustrative example is given to demonstrate the effectiveness and advantages of the proposed 2‐D design approach. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2033–2045, 2013     

Modification to adaptive model reduction for regulation of distributed parameter systems with fast transients

We focus on output feedback control of distributed processes whose infinite dimensional representation in appropriate Hilbert subspaces can be decomposed to finite dimensional slow and infinite dimensional fast subsystems. The controller synthesis issue is addressed using a refined adaptive proper orthogonal decomposition (APOD) approach to recursively construct accurate low dimensional reduced order models (ROMs) based on which we subsequently construct and couple almost globally valid dynamic observers with robust controllers. The novelty lies in modifying the data ensemble revision approach within APOD to enlarge the ROM region of attraction. The proposed control approach is successfully used to regulate the Kuramoto‐Sivashinsky equation at a desired steady state profile in the absence and presence of uncertainty when the unforced process exhibits nonlinear behavior with fast transients. The original and the modified APOD approaches are compared in different conditions and the advantages of the modified approach are presented. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4595–4611, 2013     

Economic model predictive control of nonlinear time‐delay systems: Closed‐loop stability and delay compensation

Closed‐loop stability of nonlinear time‐delay systems under Lyapunov‐based economic model predictive control (LEMPC) is considered. LEMPC is initially formulated with an ordinary differential equation model and is designed on the basis of an explicit stabilizing control law. To address closed‐loop stability under LEMPC, first, we consider the stability properties of the sampled‐data system resulting from the nonlinear continuous‐time delay system with state and input delay under a sample‐and‐hold implementation of the explicit controller. The steady‐state of this sampled‐data closed‐loop system is shown to be practically stable. Second, conditions such that closed‐loop stability, in the sense of boundedness of the closed‐loop state, under LEMPC are derived. A chemical process example is used to demonstrate that indeed closed‐loop stability is maintained under LEMPC for sufficiently small time‐delays. To cope with performance degradation owing to the effect of input delay, a predictor feedback LEMPC methodology is also proposed. The predictor feedback LEMPC design employs a predictor to compute a prediction of the state after the input delay period and an LEMPC scheme that is formulated with a differential difference equation (DDE) model, which describes the time‐delay system, initialized with the predicted state. The predictor feedback LEMPC is also applied to the chemical process example and yields improved closed‐loop stability and economic performance properties. © 2015 American Institute of Chemical Engineers AIChE J, 61: 4152–4165, 2015     

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     

A thermal‐flywheel approach to distributed temperature control in microchannel reactors

Microchannel reactors are a promising route for monetizing distributed natural gas resources. However, intensification and miniaturization represent a significant challenge for reactor control. Focusing on autothermal methane‐steam reforming reactors, a novel microchannel reactor temperature control strategy based on confining a layer of phase‐change material (PCM) between the reactor plates is introduced. Melting‐solidification cycles, which occur with latent heat exchange at constant temperature, allow the PCM layer to act as an energy storage buffer—a “thermal flywheel”—constituting a distributed controller that mitigates temperature excursions caused by fluctuations in feedstock quality. A novel stochastic optimization algorithm for selecting the PCM layer thickness (i.e., distributed controller “tuning”) is introduced. Furthermore, a hierarchical control structure, whereby the PCM layer is complemented by a supervisory controller that addresses persistent disturbances, is proposed. The proposed concepts are illustrated in a comprehensive case study using a detailed two‐dimensional reactor model. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2051–2061, 2013     

Sequential and iterative architectures for distributed model predictive control of nonlinear process systems

In this work, we focus on distributed model predictive control of large scale nonlinear process systems in which several distinct sets of manipulated inputs are used to regulate the process. For each set of manipulated inputs, a different model predictive controller is used to compute the control actions, which is able to communicate with the rest of the controllers in making its decisions. Under the assumption that feedback of the state of the process is available to all the distributed controllers at each sampling time and a model of the plant is available, we propose two different distributed model predictive control architectures. In the first architecture, the distributed controllers use a one‐directional communication strategy, are evaluated in sequence and each controller is evaluated only once at each sampling time; in the second architecture, the distributed controllers utilize a bi‐directional communication strategy, are evaluated in parallel and iterate to improve closed‐loop performance. In the design of the distributed model predictive controllers, Lyapunov‐based model predictive control techniques are used. To ensure the stability of the closed‐loop system, each model predictive controller in both architectures incorporates a stability constraint which is based on a suitable Lyapunov‐based controller. We prove that the proposed distributed model predictive control architectures enforce practical stability in the closed‐loop system and optimal performance. The theoretical results are illustrated through a catalytic alkylation of benzene process example. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Optimization‐Based Nonlinear Centralized Controller Tuning of Liquid‐Liquid Extraction Processes

Abstract

The control problem of an agitated contactor is considered in this work. A Scheibel extraction column is modeled using the non‐equilibrium backflow mixing cell model. Model dynamic analysis shows that this process is highly nonlinear, thus the control problem solution of such a system needs to tackle the process nonlinearity efficiently. The control problem of this process is solved by developing a multivariable nonlinear control system implemented in MATLAB?. In this control methodology, a new controller tuning method is adopted, in which the time‐domain control parameter‐tuning problem is solved as a constrained optimization problem. A MIMO (multi‐input multi‐output) PI controller structure is used in this strategy. The centralized controller uses a 2×2 transfer function and accounts for loops interaction. The controller parameters are tuned using an optimization‐based algorithm with constraints imposed on the process variables reference trajectories. Incremental tuning procedure is performed until the extractor output variables transient response satisfies a preset uncertainty which bounds around the reference trajectory. A decentralized model‐based IMC (internal model control) control strategy is compared with the newly developed centralized MIMO PI control one. Stability and robustness tests are applied to the two algorithms. The performance of the MIMO PI controller is found to be superior to that of the conventional IMC controller in terms of stability, robustness, loops interaction handling, and step‐change tracking characteristics.     

Evaluation of wall boundary condition parameters for gas–solids fluidized bed simulations

Wall boundary conditions for the solids phase have significant effects on numerical predictions of various gas–solids fluidized beds. Several models for the granular flow wall boundary condition are available in the open literature for numerical modeling of gas–solids flow. A model for specularity coefficient used in Johnson and Jackson boundary conditions by Li and Benyahia (Li and Benyahia, AIChE J. 2012;58:2058–2068) is implemented in the open‐source CFD code‐MFIX. The variable specularity coefficient model provides a physical way to calculate the specularity coefficient needed by the partial‐slip boundary conditions for the solids phase. Through a series of two‐dimensional numerical simulations of bubbling fluidized bed and circulating fluidized bed riser, the model predicts qualitatively consistent trends to the previous studies. Furthermore, a quantitative comparison is conducted between numerical results of variable and constant specularity coefficients to investigate the effect of spatial and temporal variations in specularity coefficient. Published 2013 American Institute of Chemical Engineers AIChE J, 59: 3624–3632, 2013     

Dynamic modelling and predictive control for the sequential collaborative reactors of cobalt removal process under time‐varying conditions

The accuracy of the process model directly affects the performance of the model‐based controller. In zinc hydrometallurgy, the overall dynamics of the cobalt removal process can hardly be described by a fixed model since there are a series of interconnected reactors working together under time‐varying inlet and reaction conditions. In this study, an interacting continuously stirred tank reactors (ICSTR) model is developed to describe the cooperative relationship of these cascaded reactors. Considering the time‐varying inlet and reaction conditions, the reaction surface conversion coefficient is defined and incorporated into the ICSTR model, and the kernel partial least squares (KPLS) is employed to update the dynamic model online. The effectiveness of the time‐varying ICSTR model is validated using industrial data. Based on the proposed time‐varying ICSTR model, a predictive controller is designed to realize the optimal operation of the cobalt removal process. Simulation results indicate that compared with conventional predictive control and manual manipulation, the time‐varying ICSTR model‐based predictive control method can not only maintain the outlet cobalt ion concentration but also reduce the zinc dust dosage.     

Error‐triggered on‐line model identification for model‐based feedback control

In industry, it may be difficult in many applications to obtain a first‐principles model of the process, in which case a linear empirical model constructed using process data may be used in the design of a feedback controller. However, linear empirical models may not capture the nonlinear dynamics over a wide region of state‐space and may also perform poorly when significant plant variations and disturbances occur. In the present work, an error‐triggered on‐line model identification approach is introduced for closed‐loop systems under model‐based feedback control strategies. The linear models are re‐identified on‐line when significant prediction errors occur. A moving horizon error detector is used to quantify the model accuracy and to trigger the model re‐identification on‐line when necessary. The proposed approach is demonstrated through two chemical process examples using a model‐based feedback control strategy termed Lyapunov‐based economic model predictive control (LEMPC). The chemical process examples illustrate that the proposed error‐triggered on‐line model identification strategy can be used to obtain more accurate state predictions to improve process economics while maintaining closed‐loop stability of the process under LEMPC. © 2016 American Institute of Chemical Engineers AIChE J, 63: 949–966, 2017     

Robust continuous velocity control of convective spatially distributed systems

In this work a robust nonlinear scheme is proposed to control spatially distributed convective systems described by first-order hyperbolic partial differential equations by manipulating the flow velocity. The proposed scheme is designed after the method of characteristics is used to establish key structural properties of the system dynamics. The resulting feedback control, which can be seen as a proportional integral controller with variable integration time, does not require measurements for several axial points nor infinite dimensional state estimations. The proposed controller is applied successfully to two heat exchange simulation examples and a nonisothermical plug flow reactor. It is shown that it is robust in the face of uncertain parameters and load disturbances. Finally, the performance of the robust controller is compared to other control applications.     

Thermal analysis of a cylindrical lithium-ion battery

This work investigates the heat generation characteristics of a cylindrical lithium-ion battery. The battery consists of the graphite, LiPF₆ of the propylene carbonate/ethylene carbonate/dimethyl carbonate (PC/EC/DMC) solution, and spinal as anode, electrolyte and cathode, respectively. The coupled electrochemical–thermal model is developed with full consideration of electrolyte transport properties as functions of temperature and Li ion concentration. A truly conservative finite volume numerical method is employed for the spatial discretization of the model equations. Three types of heat generation sources including the ohmic heat, the active polarization heat and the reaction heat are quantitatively analyzed for the battery discharge process. The ohmic heat is found to be the largest contribution with around 54% in the total heat generation. About 30% of the total heat generation in average is ascribed to the electrochemical reaction. The active polarization contributes the least comparing to the ohmic heat and reactions heat. The results also show that the Li ion concentration and its gradient in electrolyte are the main factors giving the effect on the heat generations of active polarization and electrolyte electric resistance. The raised temperature in the battery discharge is positive related with the thickness of both separator and electrodes.     

Real‐time economic model predictive control of nonlinear process systems

Closed‐loop stability of nonlinear systems under real‐time Lyapunov‐based economic model predictive control (LEMPC) with potentially unknown and time‐varying computational delay is considered. To address guaranteed closed‐loop stability (in the sense of boundedness of the closed‐loop state in a compact state‐space set), an implementation strategy is proposed which features a triggered evaluation of the LEMPC optimization problem to compute an input trajectory over a finite‐time prediction horizon in advance. At each sampling period, stability conditions must be satisfied for the precomputed LEMPC control action to be applied to the closed‐loop system. If the stability conditions are not satisfied, a backup explicit stabilizing controller is applied over the sampling period. Closed‐loop stability under the real‐time LEMPC strategy is analyzed and specific stability conditions are derived. The real‐time LEMPC scheme is applied to a chemical process network example to demonstrate closed‐loop stability and closed‐loop economic performance improvement over that achieved for operation at the economically optimal steady state. © 2014 American Institute of Chemical Engineers AIChE J, 61: 555–571, 2015     

Parameter estimation of linear MIMO systems using sequential relay feedback test

The interactions between inputs and outputs in the multi input multi output (MIMO) systems yield responses from closed‐loop off‐diagonal elements. To characterize these responses through modeling, a systematic approach is followed to derive analytical expressions for relay feedback responses on multivariable systems. Time domain analytical expressions are helpful in deriving boundary conditions to estimate the parameters of MIMO systems. Unknown system parameters are identified using limit cycle data of off‐diagonal closed‐loop relay response curves obtained from single relay feedback test on these MIMO systems. Developed models are found to be in close agreement with real time system, and the parametric identification procedure is also efficient. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1672–1681, 2014     

Economic model predictive control with time‐varying objective function for nonlinear process systems

Economic model predictive control (EMPC) is a control scheme that combines real‐time dynamic economic process optimization with the feedback properties of model predictive control (MPC) by replacing the quadratic cost function with a general economic cost function. Almost all the recent work on EMPC involves cost functions that are time invariant (do not explicitly account for time‐varying process economics). In the present work, we focus on the development of a Lyapunov‐based EMPC (LEMPC) scheme that is formulated with an explicitly time‐varying economic cost function. First, the formulation of the proposed two‐mode LEMPC is given. Second, closed‐loop stability is proven through a theoretical treatment. Last, we demonstrate through extensive closed‐loop simulations of a chemical process that the proposed LEMPC can achieve stability with time‐varying economic cost as well as improve economic performance of the process over a conventional MPC scheme. © 2013 American Institute of Chemical Engineers AIChE J 60: 507–519, 2014     

Feedback control design for an anaerobic digestion process

An approach for the design of linear feedback controllers for anaerobic digestion systems is presented. The effluent chemical oxigen demand (COD) concentration and the dilution rate are taken respectively as the regulated and the manipulated variables. The control design is based on simple step‐response models of the process endowed with an input delay to account for dead‐times induced by measurement devices. The resulting feedback controller has a traditional proportional‐integral (PI) control structure, so it can be easily implemented with conventional control technologies. Since the concentration of volatile fatty acids can be easily and quickly measured as compared with COD concentration, it is used as a secondary measurement that is incorporated into the feedback loop scheme to enhance the robustness of the control scheme with respect of influent disturbances. The performance of the proposed control scheme is illustrated via numerical simulations and experimental work. © 2002 Society of Chemical Industry