This paper studies the application of proper orthogonal decomposition (POD) to reduce the order of distributed reactor models with axial and radial diffusion and the implementation of model predictive control (MPC) based on discretetime linear time invariant (LTI) reducedorder models. In this paper, the control objective is to keep the operation of the reactor at a desired operating condition in spite of the disturbances in the feed flow. This operating condition is determined by means of an optimization algorithm that provides the optimal temperature and concentration profiles for the system. Around these optimal profiles, the nonlinear partial differential equations (PDEs), that model the reactor are linearized, and afterwards the linear PDEs are discretized in space giving as a result a highorder linear model. POD and Galerkin projection are used to derive the loworder linear model that captures the dominant dynamics of the PDEs, which are subsequently used for controller design. An MPC formulation is constructed on the basis of the loworder linear model. The proposed approach is tested through simulation, and it is shown that the results are good with regard to keep the operation of the reactor.
Advances in power computing have propelled the development of process models increasingly detailed and precise, which are then used in the design, optimization, monitoring, and diagnosis of faults, among other tasks. Usually distributed chemical reactors are described by partial differential equations (PDEs) that show the spacetime evolution of some variables of interest. In order to simulate PDEs, the spatial domain is discretized, obtaining a large number of ordinary differential equations (ODEs). However, a fine discretization leads to an increase in model complexity. To reduce the complexity of the models, a technique based on orthogonal decomposition of a set of measurements of physical quantities (such as temperature and concentration) is used to represent these quantities along the space and time. This technique, proper orthogonal decomposition (POD), has been used to reduce the order of a large number of systems. This method is based on orthonormal basis functions generated from process data (snapshot matrix) which are obtained by simulation or experimentation on the process; These data are taken by excitation of the process through manipulated variables, external inputs and disturbances of the process. The main idea is to consider POD basis functions which capture the spatial dynamics of our system. The advantage of working with these basis functions is that it is possible to reduce the model order from hundreds or thousands to a few tens. This reduction, resulting in ease of simulation, assimilation and optimization, enables that such models can be applied in real time applications.
There are several methods available for reducing the dimension of a system. The most immediate is a heuristic approach that consists of proposing a priori solution to the equations of motion on the grounds of symmetry and boundary conditions. These solutions usually take the form of a truncated series in terms of general sets of orthogonal functions, such as Fourier modes or spherical harmonics. Antoulas [
Model reduction framework.
Proper orthogonal decomposition (POD) is a powerful method for data analysis aimed at obtaining lowdimensional approximate descriptions of a highdimensional process. POD provides a basis for the modal decomposition of an ensemble of functions, such as data obtained in the course of experiments or numerical simulations. The basis functions are commonly called empirical eigenfunctions, empirical basis functions, empirical orthogonal functions, proper orthogonal modes, or basis vectors. The most striking feature of the POD is its optimality: it provides the most efficient way of capturing the dominant components of an infinitedimensional process with only a finite number of modes, and often surprisingly few modes. In general, there are two different interpretations for the POD. The first interpretation regards the POD as the KarhunenLoeve decomposition (KLD), and the second one considers that the POD consists of three methods: the KLD, the principal component analysis (PCA), and the singular value decomposition (SVD). In recent years, there have been many reported applications of the POD methods in engineering fields such as in studies of turbulence [
Let
In POD, the orthonormal basis vectors are calculated in such a way that the reconstruction of the snapshots using the first
The aim of the previous constraints is to ensure the basis vector orthogonality. The orthonormal basis vectors that solve (
The derivation of the dynamical model for the POD coefficients can be done in two ways, by using the Galerkin projection or by means of other system identification techniques. The Galerkin projection is the most common way of deriving the dynamical model for the POD coefficients, and it will be the method used in this work.
For explaining the ideas, we suppose that the dynamical behaviour of the highdimensional system from which we want to find a reducedorder model is described by the following nonlinear model in state space form:
Let us define a residual function
Finally, the reduced order model of (
In the following subsection we will describe in detail the kind of tubular reactor for which we will design and implement PODbased model predictive control (MPC) control strategies.
The system to be controlled is a nonisothermal tubular reactor with three phenomena: axial and radial diffusion, reaction and convection, where a reversible, secondorder, exothermic, catalyzed reaction takes place (
Tubular chemical reactor with 3 cooling/heating jackets.
It is assumed that radial and axial variations in concentration and temperature are present; also we consider laminar flow regime. In this study we are neglecting the heat transfer effects between the jackets fluids and the reactor wall. Under the previous assumptions, the mass balance of specie
Cylindrical shell of thickness
with the following boundary conditions:
radial:
at
at
at
axial:
at the outlet of the reactor
Here,
The parameter values of the reactor model are taken from [
Values of the reactor parameters with axial and radial diffusion, reaction, and convection.
Parameter  Value 



































The temperature of the jacket sections
The operating profiles (steady state concentration and temperature profiles) of the reactor are derived by means of an optimization algorithm, which minimizes an objective function subject to the steady state equations of the reactor described by (
with
with
subject to
The algorithm was executed in MATLAB with the following parameters:
The algorithm was executed using different initial conditions. Along the experiments, one local minima was found. The operating point was given by
Steady state concentration and temperature profiles.
Steady state concentration and temperature profiles at
The linear model of the tubular chemical reactor is obtained by linearizing (
radial:
at
at
at
axial:
at the outlet of the reactor
Here,
Since the spatial domain of the reactor is divided into
The derivation of a reduced order model of (
We have created a snapshot matrix from the system response (
The POD basis vectors are obtained by computing the SVD of the snapshot matrix
The
Logarithmic plot of
The Galerkin projection is the most common way of deriving the dynamical model for the POD coefficients, and it will be the method used in this work. Let us define a residual function
For validating the reduced order model of the reactor, we applied constant input signals
Temperature and concentration profiles and
Temperature and concentration profiles at
Temperature and concentration profiles at
Average of the absolute error between the fullorder model (
The discretetime version of (
Model predictive control (MPC), also referred to as receding horizon control (RHC) or moving horizon control, is a control strategy where a finite or infinite horizon openloop optimal control problem is solved online at each sampling time using the current state of the plant as the initial state, in order to get a sequence of future control actions from which only the first one is applied to the plant. The fact of solving an optimization problem online where common plant constraints are included makes MPC different from conventional optimal control which uses a precomputed control law [
A modelling approach frequently adopted in model predictive controller (MPC) considers a discretetime statespace model in incremental form [
It is clear that the model defined in (
MPC is usually based on a discretetime statespace model as shown in (
Finally, the control optimization problem of the infinite horizon MPC can be formulated as
For large changes on
To produce an infinite horizon MPC, which is implementable in practice, the objective function of infinite horizon MPC is redefined as follows:
Finally, the control optimization problem of the extended infinite horizon MPC can be formulated as follows:
For a stable system, if in the control objective defined in (
The proof is provided in [
The control objective is to reject the disturbances that affect the reactor, that is, the changes in the temperature and concentration of the feed flow. In addition, the control actions must satisfy the input constraints of the process (
In this formulation
The control horizon
In order to evaluate the performance of the control system, the following tests were carried out.
These disturbances have a big impact on the temperature profile of the reactor.
The simulation results for Test 1 are presented in Figures
Steady state temperature and concentration profiles for Test 1 at
Steady state temperature and concentration profiles for Test 1 at
Steady state error (temperature and concentration) for Test 1.
Control actions (jackets temperatures) of the MPC controller for Test 1.
Steady state temperature and concentration profiles for Test 2 at
Steady state temperature and concentration profiles for Test 2 at
Steady state error (temperature and concentration) for Test 2.
Control actions (jackets temperatures) of the MPC controller for Test 2.
Furthermore, some quantities of interest are given in Table
Performance parameters of the control systems.
Quantities  Test 1  Test 2 


332.3218  323.6579 

−1.0531  0.8732 
In general, the control schemes showed a good behavior for rejecting the disturbances (typical magnitudes:
In this work, it is shown how POD and Galerkin projections can be used for deriving reduced order model of systems with reaction, diffusion, and convection in two dimensions. The method proposed here is illustrated with a nonisothermal reactor and based on the proposed reduced model, a state observer and a predictive controller are designed and tested.
The algorithm proposed in [
The POD method is characterized for its capability to describe the spatial distribution of the relevant physical variables in terms of a set of orthonormal basis functions. These basis functions are selected from observed data and are optimal in a welldefined sense. In the nonisothermal tubular reactor model, the spatial domain is discretized into a high number of grid cells, while in POD models, the spatial distributions are described by the first few and most relevant POD basis functions. The timedependent characteristics of the variables are given by the time varying coefficients of the POD basis functions. The model of the time varying coefficients is denominated by the reduced order model and is obtained by projecting the POD basis functions onto the original governing equations. Throughout the results presented in this work, it is shown that with very few POD basis functions (less than 3% of the number of grid cells), the temporal and spatial dynamics of the nonisothermal tubular reactor with diffusion, reaction and, convection have an acceptable approximation.
In the application of POD technique, the data matrix (
In Section
The authors acknowledge the European 7th framework STREP project Hierarchical and Distributed Model Predictive Control (HDMPC). Contract no. INFSOICT223854 for funding this work.