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     

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     

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     

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     

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.     

Linear model predictive control for transport‐reaction processes

The article deals with systematic development of linear model predictive control algorithms for linear transport‐reaction models emerging from chemical engineering practice. The finite‐horizon constrained optimal control problems are addressed for the systems varying from the convection dominated models described by hyperbolic partial differential equations (PDEs) to the diffusion models described by parabolic PDEs. The novelty of the design procedure lies in the fact that spatial discretization and/or any other type of spatial approximation of the process model plant is not considered and the system is completely captured with the proposed Cayley‐Tustin transformation, which maps a plant model from a continuous to a discrete state space setting. The issues of optimality and constrained stabilization are addressed within the controller design setting leading to the finite constrained quadratic regulator problem, which is easily realized and is no more computationally intensive than the existing algorithms. The methodology is demonstrated for examples of hyperbolic/parabolic PDEs. © 2017 American Institute of Chemical Engineers AIChE J, 63: 2644–2659, 2017     

A linear model predictive control algorithm for nonlinear large‐scale distributed parameter systems

This work provides a framework for linear model predictive control (MPC) of nonlinear distributed parameter systems (DPS), allowing the direct utilization of existing large‐scale simulators. The proposed scheme is adaptive and it is based on successive local linearizations of the nonlinear model of the system at hand around the current state and on the use of the resulting local linear models for MPC. At every timestep, not only the future control moves are updated but also the model of the system itself. A model reduction technique is integrated within this methodology to reduce the computational cost of this procedure. It follows the equation‐free approach (see Kevrekidis et al., Commun Math Sci. 2003;1:715–762; Theodoropoulos et al., Proc Natl Acad Sci USA. 2000;97:9840‐9843), according to which the equations of the model (and consequently of the simulator) need not be given explicitly to the controller. The latter forms a “wrapper” around an existing simulator using it in an input/output fashion. This algorithm is designed for dissipative DPS, dissipativity being a prerequisite for model reduction. The equation‐free approach renders the proposed algorithm appropriate for multiscale systems and enables it to handle large‐scale systems. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

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     

State‐Space Digital PI Controller Design for Linear Stochastic Multivariable Systems with Input Delay

In this paper, a centralized digital PI control scheme is proposed for linear stochastic multivariable systems with input delay. The discrete linear quadratic regulator (LQR) approach with pole placement is used to achieve satisfactory set‐point tracking with guaranteed closed‐loop stability. In addition, the innovation form of Kalman gain is employed for state estimation with no prior knowledge of noise properties. Compared with existing designs, the proposed scheme provides an optimal closed‐loop design via the digitally implementable PI controller for linear stochastic multivariable systems with input delay. Its effectiveness will be demonstrated by the simulation study on examples from both industrial process control and aircraft control.     

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     

Generalized hybrid control synthesis for affine systems using sequential adaptive networks

BACKGROUND: A generalized methodology for the synthesis of a hybrid controller for affine systems using sequential adaptive networks (SAN) is presented. SAN consists of an assembly of neural networks that are ordered in a chronological sequence, with one network assigned to each sampling interval. Using a suitable process model based on oxygen metabolism and an a priori objective function, a hybrid control law is derived that can use online measurements and the states predicted by SAN for computing the desired control action. RESULTS: The performance of the SAN–hybrid controller is tested for simulated fed‐batch production of methionine for three different process conditions. Simulations assume that online measurements of dissolved oxygen (DO) concentration are available. The performance of the SAN–hybrid controller gave an NRMSE of ～10^?4 in the absence of noise, ～10^?3 and ～10^?2 for ± 5% and ± 10% noise in the DO measurement and ～10^?2 for parameter uncertainty when compared with the ideal model prediction. CONCLUSIONS: The observed performance for unmeasured state prediction and control implementation shows that the proposed SAN–hybrid controller can efficiently compute the manipulated variable required to maintain methionine production along the optimized trajectory for different conditions. The test results show that the SAN–hybrid controller can be used for online real‐time implementation in fed‐batch bioprocesses. Copyright © 2009 Society of Chemical Industry     

On‐line multivariate optimization of injection molding

An auxiliary process controller was designed, implemented, and validated for on‐line process and quality optimization. The objective function included terms related to the process variation, model uncertainty, and control energy. The controller architecture relied on characterized models including both process transfer functions and principal components analysis to perform on‐line optimization in parallel with the physical molding process. New process and quality observations were input to the controller to update the models and provided new settings for the machine controller. Experimentation included characterization with a D‐optimal design of experiments followed by a validation to measure the controller's performance with respect to controller stability, extrinsic material variation, cycle time reduction, and other common manufacturing goals. In every case, the controller was able to reduce the value of the objective function while also improving the part dimensions relative to tight tolerance specifications. While characterization experiments could be costly, the use of the resulting process models greatly speeds convergence and facilitates the consideration of various cost and quality terms in the objective function. POLYM. ENG. SCI., 55:2743–2750, 2015. © 2015 Society of Plastics Engineers     

Nonlinear model predictive control for distributed parameter systems using data driven artificial neural network models

In this work the radial basis function neural network architecture is used to model the dynamics of Distributed Parameter Systems (DPSs). Two pure data driving schemes which do not require knowledge of the governing equations are described and compared. In the first method, the neural network methodology generates the full model of the system that is able to predict the process outputs at any spatial point. Past values of the process inputs and the coordinates of the specific location provide the input information to the model. The second method uses empirical basis functions produced by the Singular Value Decomposition (SVD) on the snapshot matrix to describe the spatial behavior of the system, while the neural network model is used to estimate only the temporal coefficients. The models produced by both methods are then implemented in Model Predictive Control (MPC) configurations, suitable for constrained DPSs. The accuracies of the modeling methodologies and the efficiencies of the proposed MPC formulations are tested in a tubular reactor and produce encouraging results.     

Economic model predictive control of stochastic nonlinear systems

This work focuses on the design of stochastic Lyapunov‐based economic model predictive control (SLEMPC) systems for a broad class of stochastic nonlinear systems with input constraints. Under the assumption of stabilizability of the origin of the stochastic nonlinear system via a stochastic Lyapunov‐based control law, an economic model predictive controller is proposed that utilizes suitable constraints based on the stochastic Lyapunov‐based controller to ensure economic optimality, feasibility and stability in probability in a well‐characterized region of the state‐space surrounding the origin. A chemical process example is used to illustrate the application of the approach and demonstrate its economic benefits with respect to an EMPC scheme that treats the disturbances in a deterministic, bounded manner. © 2018 American Institute of Chemical Engineers AIChE J, 64: 3312–3322, 2018     

Temporal clustering for order reduction of nonlinear parabolic PDE systems with time‐dependent spatial domains: Application to a hydraulic fracturing process

A temporally‐local model order‐reduction technique for nonlinear parabolic partial differential equation (PDE) systems with time‐dependent spatial domains is presented. In lieu of approximating the solution of interest using global (with respect to the time domain) empirical eigenfunctions, low‐dimensional models are derived by constructing appropriate temporally‐local eigenfunctions. Within this context, first of all, the time domain is partitioned into multiple clusters (i.e., subdomains) by using the framework known as global optimum search. This approach, a variant of Generalized Benders Decomposition, formulates clustering as a Mixed‐Integer Nonlinear Programming problem and involves the iterative solution of a Linear Programming problem (primal problem) and a Mixed‐Integer Linear Programming problem (master problem). Following the cluster generation, local (with respect to time) eigenfunctions are constructed by applying the proper orthogonal decomposition method to the snapshots contained within each cluster. Then, the Galerkin's projection method is employed to derive low‐dimensional ordinary differential equation (ODE) systems for each cluster. The local ODE systems are subsequently used to compute approximate solutions to the original PDE system. The proposed local model order‐reduction technique is applied to a hydraulic fracturing process described by a nonlinear parabolic PDE system with the time‐dependent spatial domain. It is shown to be more accurate and computationally efficient in approximating the original nonlinear system with fewer eigenfunctions, compared to the model order‐reduction technique with temporally‐global eigenfunctions. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3818–3831, 2017     

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     

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     

Integrating data‐based modeling and nonlinear control tools for batch process control

A data‐based multimodel approach is developed in this work for modeling batch systems in which multiple local linear models are identified using latent variable regression and combined using an appropriate weighting function that arises from fuzzy c‐means clustering. The resulting model is used to generate empirical reverse‐time reachability regions (RTRRs) (defined as the set of states from where the data‐based model can be driven inside a desired end‐point neighborhood of the system), which are subsequently incorporated in a predictive control design. Simulation results of a fed‐batch reactor system under proportional‐integral (PI) control and the proposed RTRR‐based design demonstrate the superior performance of the RTRR‐based design in both a fault‐free and faulty environment. The data‐based modeling methodology is then applied on a nylon‐6,6 batch polymerization process to design a trajectory tracking predictive controller. Closed‐loop simulation results illustrate the superior tracking performance of the proposed predictive controller over PI control. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

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