Reducción de modelos dinámicos usando POD para sistemas con

Transcription

Reducción de modelos dinámicos usando POD para
sistemas con reacción, convección y difusión radial y
axial con aplicaciones en simulación y control
Alejandro Marquez Ruiz
Universidad Nacional de Colombia
Facultad de Minas, Escuela de Procesos y Energı́a
Medellı́n, Colombia
2011
Reducción de modelos dinámicos usando POD para
sistemas con reacción,convección y difusión radial y
axial con aplicaciones en simulación y control
Alejandro Marquez Ruiz
Thesis Work to Obtain the Degree of:
Magister en Ingenierı́a - Ingenierı́a Quı́mica
Advisor:
Jairo José Espinosa Oviedo
Lı́nea de Investigación:
Lı́nea de Matemáticas Avanzadas para el Control y los Sistemas Dinámicos
Grupo de Investigación:
GAUNAL
Universidad Nacional de Colombia
Facultad de Minas, Escuela de Procesos y Energı́a
Medellı́n, Colombia
2011
Todo lo que aprendas, procura aprenderlo con
la máxima profundidad posible. Los estudios
superficiales producen con harta frecuencia
hombres mediocres y presuntuosos.
Silvio Pellico
Acknowledgements
I am heartily thankful to my supervisor, Jairo José Espinosa Oviedo, whose encouragement,
guidance and support from the initial to the final level enabled me to develop an understanding of the subject.
I owe my deepest gratitude to Hernan Dario Alvarez for his guidance in developing this thesis.
I am grateful with the ”Grupo de Automatica de la Universidad Nacional (GAUNAL)” specially with the boys of the ”Lı́nea de Matemáticas Avanzadas para el Control y los Sistemas
Dinámicos” , Jose Fernando Garcia, Julian Patiño, Pablo Andres Deossa, Felipe Valencia,
Richard Rios, Juan Esteban Castaño and Jose David López who supported me in any aspect
during the completion of this thesis.
I would like to show my gratitude to Jarol Esneider Molina and Juan Carlos Carmona for
their contributions and views.
I would like to thank my family for their support.
v
Resumen
En esta tesis se presenta el resultado de aplicar POD (Proper Orthogonal Decomposition)
e IHMPC (Infinite Horizon Model Predictive Control) para el control de un reactor tubular
no isotérmico con tres fenómenos: reacción, difusión y convección. El objetivo de control
es mantener el reactor en un perfil deseado de operación a pesar de las perturbaciones en
el flujo de alimentación. El perfil de operación deseado es determinado por medio de un
algoritmo de optimización que proporciona el perfil óptimo de temperatura y concentración
para el sistema. Alrededor de este perfil se linealizan las ecuaciones diferenciales parciales
que rigen el comportamiento del reactor para luego realizar una discretización espacial de
dichas ecuaciones, dando como resultado un modelo lineal de alto orden. POD y proyecciones de Galerkin son usados para encontrar un modelo lineal de orden reducido que capture
las dinámicas dominantes del sistema no lineal. El modelo de orden reducido es usado para
diseñar un sistema de control para el reactor. Una formulacion de IHMPC es construida y
se demuestra a partir de simulación que se puede alcanzar el objetivo de control.
Palabras clave: POD, Reducción de Modelos, Reacción,Difusión, Convección, Control predictivo de Horizonte infinito.
Abstract
This thesis presents the result of applying POD (Proper Orthogonal Decomposition) and
IHMPC (Infinite Horizon Model Predictive Control) to the control of a non-isothermal tubular reactor with three phenomena: diffusion, reaction and convection. In this thesis 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 non-linear PDEs(Partial Differential Equations) that model the reactor are linearized and afterwards the linear PDEs are discretized
in space giving as result a high order linear model. POD and Galerkin projection are used to
derive the low order linear model that capture the dominant dynamics of the PDEs, which
are subsequently used for controller design. One IHMPC formulation is constructed on the
basis of the low order linear model and is demonstrated, through simulation, to achieve the
control objectives.
Keywords:POD, Model reduction, reaction, diffusion, convection, Infinite Horizon
Model Predictive Control
Contents
Acknowledgements
iv
Abstract
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1 Introduction
1.1 Model reduction of the reaction - convection - diffusion processes . . .
1.1.1 Model reduction Techniques . . . . . . . . . . . . . . . . . . . .
1.1.2 Reduced-order models in reaction-convection-diffusion processes
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Chapter by chapter overview . . . . . . . . . . . . . . . . . . . . . . . .
2 Proper Orthogonal Decomposition (POD)
2.1 Proper Orthogonal Decomposition . . .
2.1.1 General procedure . . . . . . .
2.1.2 Galerkin Projection . . . . . . .
2.2 Example . . . . . . . . . . . . . . . . .
2.2.1 Operating point . . . . . . . . .
2.2.2 Linear Model . . . . . . . . . .
2.2.3 Model Reduction Using POD .
and Galerkin
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3 Tubular chemical reactor with diffusion, reaction and convection
3.1 Non-isothermal tubular reactor model . . . . . . . . . . . . . . . . . . . . . .
3.2 Approximation techniques to solve diffusion, reaction, convection equation .
3.2.1 Approximation techniques . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Courant-Friedrichs-Lewy condition: Space and time step . . . . . . .
3.2.3 Simulation of non-isothermal tubular reactor model with axial and
radial diffusion, reaction and convection . . . . . . . . . . . . . . . .
3.3 Operating profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Model reduction using POD for a non-isothermal tubular reactor with diffusion, reaction and convection . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
4 Infinite Horizon Model Predictive Control (IHMPC)
4.1 Infinite Horizon Model Predictive Control . . . . . . . . .
4.2 Extended Infinite Horizon Model Predictive Control . . . .
4.3 IHMPC and POD applied to Control of a Tubular Reactor
4.3.1 Simulation Results . . . . . . . . . . . . . . . . . .
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5 Conclusions
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Bibliografı́a
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List of Tables
2-1 Values of the reactor parameters with convection and reaction . . . . . . . .
12
3-1 Values of the reactor parameters with axial and radial diffusion, reaction and
convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-2 Values of the reactor parameters with axial diffusion, reaction and convection
23
25
4-1 Performance parameters of the control systems . . . . . . . . . . . . . . . . .
52
List of Figures
2-1 Tubular Chemical Reactor with 3 cooling/heating jackets. . . . . . . . . . .
2-2 Steady-state concentration and temperature profiles (Operating Profiles) whit
TJ1 = 374.6 K, TJ2 = 310.1 K and TJ3 = 325.2 K. . . . . . . . . . . . . . .
2-3 Temperature and concentration deviation profiles at t = 8s and t = 15s. Solid
line - full order Model. Dashed line - reduced order model. . . . . . . . . . .
Tubular Chemical Reactor with 3 cooling/heating jackets. . . . . . . . . . . . . .
Cylindrical shell of thickness ∆r, lenght ∆z, and volume 2πr∆r∆z. . . . . . . . .
Steady state temperature and concentration profile for Pe = 105 . . . . . . . . . .
Steady state temperature and concentration profile for Pe = 106 . . . . . . . . . .
Temperature at the reactor output for Pe = 106 . . . . . . . . . . . . . . . . . . .
Temperature at the reactor output for Pe = 103 . . . . . . . . . . . . . . . . . . .
Simulation of non-isothermal tubular reactor model with axial and radial diffusion,
reaction and convection in Matlab and COMSOL Multiphysics . . . . . . . . . .
3-8 Steady-state concentration and temperature profiles. . . . . . . . . . . . . . . .
3-9 Steady-state concentration and temperature profiles at r = 0 m and z = 10m. . .
3-10 Logarithmic plot of Pn and 1 − Pn for determining the truncation degree of the
POD basis vectors in the reactor case . . . . . . . . . . . . . . . . . . . . . . .
3-11 Temperature and concentration profiles at t = 100 s and t = 1000 s at r = 0 m.
Solid line - full order Model. Dashed line - reduced order model. . . . . . . . . .
3-12 Temperature and concentration profiles at t = 1500 s and t = 8000 s at z = 10 m.
Solid line - full order Model. Dashed line - reduced order model. . . . . . . . . .
3-13 Temperature and Concentration profiles at t = 10000 s: full order Model and reduced order model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-14 Average of the absolute error between the full order model (3-1) and the reduced
order model (3-22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-1
3-2
3-3
3-4
3-5
3-6
3-7
4-1 Steady state Temperature and concentration profiles for Test 1 at z = 10 m. Solid
line-IHMPC. Dashed line-Nominal profile (reference). . . . . . . . . . . . . . . .
4-2 Steady state Temperature and concentration profiles for Test 1 at r = 0 m. Solid
4-3 Steady state error (Temperature and concentration) for Test 1 . . . . . . . . . .
4-4 Control actions (jackets temperatures) of the MPC controller for Test 1 . . . . . .
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List of Figures
4-5 Steady state Temperature and concentration profiles for Test 2 at z = 10 m. Solid
line- IHMPC. Dashed line-Nominal profile (reference). . . . . . . . . . . . . . . .
4-6 Steady state Temperature and concentration profiles for Test 2 at r = 0 m. Solid
4-7 Steady state error (Temperature and concentration) for Test 2 . . . . . . . . . .
4-8 Control actions (jackets temperatures) of the MPC controller for Test 2 . . . . . .
1
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1 Introduction
1.1 Model reduction of the reaction - convection diffusion processes
1.1.1 Model reduction Techniques
There are various methods available for reducing the dimension of a system. The most
immediate approach is probably heuristic. It consists of proposing a priori solutions 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 [6] proposes a classification into
two main groups, namely, Krylov and singular value decomposition (SVD) methods. Krylov
methods make use of iterations for finding approximations to large-scale dynamical systems.
The SVD methods are based on the decomposition of the state vector into a set of vectors that
can be ordered in a certain sense. These methods include balanced truncation and Hankel
approximations for linear systems, and the proper orthogonal decomposition (POD) methods
and empirical grammians for nonlinear systems. In this thesis we are concerned with the
POD and its combination with Galerkin projection to produce reduced dynamical versions
of the original large-scale system. The interested reader is referred to [6] for a more complete
account of the available reduction methods. The POD [26, 33] is a statistical technique to
extract features from a given dataset by searching for patterns that optimally represent the
dataset with respect to quantities such as variance or energy. The output of the POD is a
set of time-independent orthogonal functions called empirical orthogonal functions (EOFs).
Each EOF is associated with a certain amount of variance or energy. The first suggestion
to use the POD in the analysis of dynamical systems was due to Lumley [35]. One of the
first phenomena to be studied by means of reduced models arising from the use of the POD
was turbulence [11]. Perhaps for nonlinear systems the most studied method is the POD in
conjunction with Galerkin projections [6]. However, this is not the only available method
and some other ideas have been proposed, such as a decomposition into principal interaction
patterns (PIPs), which explicitly incorporate information about the systems dynamics within
the determination of the patterns [29, 30]. There are other techniques that might be suitable
for developing low-order models such as the independent component analysis (ICA) [14]. The
ICA aims at answering questions such as, what signal comes from what source? This problem
1.2 Motivation
3
is known as the cocktail-party problem. The solution relies on the assumption of statistical
independence of the sources and, therefore, of the signals. The outcome is a set of modes
and a mixing matrix that sets the relationship between the modes in order to reconstruct
the original mixed signal. The ICA modes are statistically independent but not necessarily
orthogonal and there is no specific order. Furthermore, they are not clearly related to any
physical quantity that can help to decide what modes to retain and what to rule out. These
features may constitute a disadvantage when trying to construct low-order models since a
truncation using these modes would lack an appropriate parameter to measure the expected
accuracy of the resulting model. Original algorithms assume linearity although some efforts
have been made to extend the formalism to nonlinear systems. So far these ideas have not
yet been further investigated in the context of dynamical systems.
1.1.2 Reduced-order models in reaction-convection-diffusion processes
In 1996 Panagiotis [40] used model reduction to design a control system for a non-isothermal
packed bed reactor, taking into account reaction, convection and diffusion in one dimension.
The method for finding the reduced order model is similar to POD but differs in the fact
that the basis functions are analytic.
In 2000 Bendersky [10] used POD for optimization in reaction, convection processes, the
illustrative example used in this work was a steady-state catalytic reactor with diffusion in
one dimension. In 2003 Rowley [47] applied POD and Galerkin projection for compressible
fluids. In 2004 Astrid [7] presented the application of POD for computational fluid dynamics
models, focused on control system design and optimization techniques. In 2005 Huisman [24]
used POD for estimation and control of glass melt temperatures in industrial glass melting
tanks and glass melt feeders. In 2007 Li and Panagiotis [32] designed a controller for reaction,
convection and diffusion processes in a single dimension using orthogonal collocation. In
2009 Agudelo [1] applied proper orthogonal decomposition to the design of model predictive
control schemes for tubular chemical reactors with diffusion and convection in one dimension.
1.2 Motivation
The intensive use of models in engineering is more than evident. Advances in computing
power have catapulted the development of process models increasingly detailed and precise,
which are then used in design, optimization, monitoring and diagnosis of faults, among other
tasks. Usually Chemical processes are described by partial differential equations (PDE) that
show the space-time evolution of some variable of interest. Representative examples of this
are the deposition of semiconductor materials, phenomena of mass transfer, momentum and
others. 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
4
1 Introduction
orthogonal decomposition of a set of measurements of physical quantities (such as temperature, concentration) in a position and time is used. This technique, POD (Proper Orthogonal
Decomposition) has been used to reduce the order of a large number of systems. This method
is based on orthonormal basis functions generated by the data process (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 thousands or hundreds to a few tens. This reduction,
resulting in ease of simulation, assimilation and optimization, enables that such models can
be operated in real time. POD method is currently a research topic of several researchers
([7, 24, 1]), because there some numerical issues that increase the complexity of the problem.
Such issues motivate the research presented in this thesis.
1.3 Objectives
General Objective
To extend the methodology of Proper Orthogonal Decomposition (POD) to systems that
include reaction, convection and 2D diffusion and study the impact on the control system.
Specific Objectives
• To apply POD to a tubular reactor that includes diffusion, convection and reaction.
• To formulate predictive controllers that impose operating restrictions that comply with
the expected functions from a reduced order model.
1.4 Chapter by chapter overview
This thesis is organized in 5 chapters. A brief description of each chapter is given as follows.
• Chapter 2: This chapter describes the model reduction method that is used. First
the fundamentals of Proper Orthogonal Decomposition are presented. Subsequently,
the chapter shows how POD and Galerkin projection are used for deriving reduced
order models from high-dimensional systems. These techniques are used together to
find a reduced order model for the non-isothermal tubular reactor.
• Chapter 3: This chapter describes in detail the non-isothermal tubular reactor model
with three phenomena: diffusion, reaction and convection. Subsequently an optimization algorithm for deriving the operating profile in steady state of the reactor is
1.4 Chapter by chapter overview
5
introduced. In this chapter a linear model is obtained around the optimal operation
profile and a reduced order model using POD.
• Chapter 4: This chapter addresses the control of a non-isothermal tubular reactor
using a reduced model and an Infinite Horizon Model Predictive Controller (IHMPC).
• Chapter 5: In this chapter the most important conclusions that can be drawn from
this thesis are presented.
2 Proper Orthogonal Decomposition
(POD) and Galerkin projection
Proper orthogonal decomposition (POD) is a powerful method for data analysis aimed at
obtaining low-dimensional approximate descriptions of a high-dimensional 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 it’s optimality: it provides the most efficient way of capturing the dominant components
of an infinite-dimensional 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 Karhunen-Loeve 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 [17, 41, 49, 50, 52], vibration analysis [5, 20, 22, 4, 18], process identification
[28, 21, 27, 12] and control in chemical engineering [1, 31, 23, 36, 8, 39, 55, 43, 44, 54],
etc. In general POD is a methodology that first identifies the most energetic modes in a
time-dependent system, and subsequently provides a means of obtaining a low-dimensional
description of the system’s dynamics where the low-dimensional system is obtained directly
from the Galerkin projection of the governing equations on the empirical basis set (the POD
modes).
2.1 Proper Orthogonal Decomposition
2.1.1 General procedure
Let x(t) ∈ <N = [x1 (t), x2 (t), ..., xN (t)]T be the state vector of a given dynamical system,
and let X ∈ <N ×N d with Nd ≥ N be the so-called snapshot matrix that contain a finite
number of samples or snapshots of the evolution of x(t) at t = t1 , t2 , ..., tNd . In POD, we
start by observing that each snapshot can be written as a linear combination of a set of
ordered orthonormal basis vectors (POD basis vectors) ϕj ∈ <N , ∀j = 1, 2, ..., N :
x(ti ) =
N
X
aj (ti )ϕj , ∀i = 1, 2, ..., Nd
7
(2-1)
j=1
aj (ti ) = hx(ti ), ϕj i, ∀j = 1, 2, ..., N
where aj (ti ) is the coordinate of x(ti ) with respect to the basis vector ϕj (it is also called
time-varying coefficient or POD coefficient) and h·, ·i denotes the Euclidean inner product.
Since the first n most relevant basis vectors capture most of the energy in the data collected,
we can construct an n − th order approximation of the snapshots by means of the following
truncated sequence
xn (ti ) =
n
X
aj (ti )ϕj , ∀i = 1, 2, ..., Nd , n N
(2-2)
j=1
In POD, the orthonormal basis vectors are calculated in such a way that the reconstruction
of the snapshots using the first n most relevant basis vectors is optimal in the sense that the
Sum-Squared-Error (SSE) between x(ti ) and xN (ti ), ∀i = 1, ..., Nd ,
En =
Nd
X
kx(ti ) − xn (ti )k22
(2-3)
1=1
is minimized. Herein k · k2 denotes the L2 -norm or Euclidean Norm. In other words, the
POD basis vectors are the ones that solve the following constrained optimization problem:
2
Nd n
X
X
hx(ti ), ϕj iϕj min
x(ti ) −
ϕ1 ,...,ϕn
i=1
j=1
(2-4)
2
subject to
ϕTi ϕj
=
1
0
if i = j
otherwise
The constraint in 2-4 imposes the orthonormality condition of the basis vectors. The orthonormal basis vectors that solve 2-4 can be found by calculating the singular value decomposition of the snapshot matrix (Xsnap ).
Proof.
8
2 Proper Orthogonal Decomposition (POD) and Galerkin projection
• Given: x1 (t), x2 (t)...., xn (t) ∈ <m ; set ν = span{x1 (t), x2 (t)...., xn (t)} ⊂ <m
• Goal: find l ≤ dim(ν) orthonormal vectors {ϕi }li=1 in <m minimizing
2
n l
X
X
J(ϕ1 , ..., ϕl ) =
hx(ti ), ϕj iϕj x(ti ) −
i=1
j=1
2
• Constrained optimization:
min J(ϕ1 , ..., ϕl ) subject to
ϕTi ϕj
=
1
0
if i = j
otherwise
• Lagrange functional:
L(ϕ1 , ..., ϕl , λ11 , ....., λll ) = J(ϕ1 , ..., ϕl ) +
l X
l
X
λij (ϕTi ϕj − δij )
j=1 i=1
with the Kronecker symbol δij =
1
0
if i = j
otherwise
• Optimality conditions:
∂L
(ϕ1 , ..., ϕl , λ11 , ....., λll ) = 0 ∈ <m f or i = 1, ..., l
∂ϕi
∂L
(ϕ1 , ..., ϕl , λ11 , ....., λll ) = 0 ∈ <m f or i, j = 1, ..., l
∂λij
• Necessary optimality conditions:
n
X
∂L
=0⇔
xj (xTj ϕi ) = λii ϕi and λij = 0 f or i 6= j
∂ϕi
j=1
∂L
= 0 ⇔ ϕTi ϕj = δij
∂λij
9
Setting λi = λij and X = [x1 (t), ...., xn (t)] ∈ <m×n we have
XX T ϕi = λi ϕi f or i = 1, ..., l
i.e., necessary optimality conditions are given by a symmetric m×m eigenvalue problem
• Solution by SVD: d = rank(X), σ1 ≥ ... ≥ σd > 0, U = [u1 , ..., um ] ∈ <m×m and
V = [v1 , ..., vm ] ∈ <n×n orthogonal with:
T
U XV =
D 0
0 0
= Σ ∈ <m×n
where D = diag(σ1 , ...., σd ) ∈ <d×d . Moreover, for 1 ≤ i ≤ d
Xvi = σi ui , X T ui = σi vi , XX T ui = σi2 ui , X T Xvi = σi2 vi
• POD basis: ϕi = ui and λi = σi2 > 0 for i = 1, ..., ≤ d = dim(ν)
• Data ensemble: ν = span{x1 (t), ...., xn (t)} ⊂ <m and d = dim(ν), POD basis of rank
l: ϕi = ui and λi = σi2 > 0 for i = 1, ..., ≤ d
The singular values of Xsnap are positive real numbers that are ordered in a decreasing way,
σ1 ≥ σ2 ... ≥ σn·m ≥ 0. These values quantify the importance of the basis vectors in capturing
the information present in the data. Therefore, the first POD basis vector is the most relevant
one and last POD basis vector is the least important element. For the application of POD
to practical problems, the choice of the n most relevant basis vectors is certainly of central
importance. A criterion commonly used for choosing n based on heuristic considerations is
the so called energy criterion [7]. In this criterion we check the ratio between the modelled
energy and the total energy contained in Xsnap ,
Pn =
n
P
j=1
N
P
j=1
σj2
, n = 1, ..., (N)
σj2
(2-5)
10
The ratio Pn is used to determine the truncation degree of the selected POD basis vectors.
The number of POD basis elements should be chosen such that the fraction of the first
singular values in (2-5) is large enough to capture most of the information in the data [7].
An ad-hoc rule frequently applied is that n has to be determined for Pn = 0.99. The closer
Pn to 1, or similarly the closer 1 − Pn to 0, the better the approximation of X will be.
2.1.2 Galerkin Projection
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 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 thesis.
For explaining the ideas, we are going to suppose that the dynamical behaviour of the highdimensional system from which we want to find a reduced order model, is described by the
following non-linear model in state space form,
ẋ(t) = f (x(t), u(t))
y(t) = g(x(t), u(t))
(2-6)
Let us define a residual function R(ẋ, x) for equation (2-6) as follows:
R(x) = ẋ(t) − f (x(t), u(t))
(2-7)
and let R(ẋn , xn ) be the residual when the state vector x(t) is approximated by its n − th
order approximation
xn (t) =
n
X
aj (t)ϕj = ΦN a(t), n N
j=1
where Φn = [ϕ1 , ϕ2 , ..., ϕn ] and a(t) = [a1 (t), a2 (t), ..., an (t)]T . In the Galerkin projection,
the projection of the residual R(ẋ, x) on the space spanned by the basis vectors Φn vanishes,
that is
hR(ẋ, x), ϕj i = 0; ∀j = 1, ...n
(2-8)
where h·, ·i denotes the Euclidean inner product. Replacing x(t) by its n − th order approximation xn (t) = Φn a(t) in equation (2-6),
2.2 Example
11
Φn ȧ(t) = f (Φn a(t), u(t))
and then we apply the inner product criterion (2-8) as follows,
hΦn ȧ(t), ϕj i = hf (Φn a(t), u(t)), ϕj i, ∀j = 1, 2, ..., n
ΦTn Φn ȧ(t) = ΦTn f (Φn a(t), u(t))
and given that ΦTn Φn = In because of the orthonormality of the basis vectors, we have that
the model for the POD coefficients reduces to
ȧ(t) = ΦTn f (Φn a(t), u(t))
Finally, the reduced order model of (2-8) with only n states has the following form,
ȧ(t) = ΦTn f (Φn a(t), u(t))
y(t) = g(Φn a(t), u(t))
(2-9)
2.2 Model Reduction for a non-isothermal tubular reactor
with convection and reaction
The system to be reduced is a non-isothermal tubular reactor where a single, first order,
irreversible, exothermic reaction takes place (A→B). The reactor is surrounded by 3 cooling/heating jackets as it is shown in Figure 2-1. It is assumed that the reacting mixture
flows as a plug through the reactor body in the axial direction. In this dynamics only three
phenomena are taken into account, namely, convection, reaction and heat transfer (between
the reactor and its jackets). In this study we are not considering the diffusion/dispersion
phenomena and we are neglecting the effects of the reactor wall. Under the previous assumptions, the mathematical model of the tubular chemical reactor consists of the following
coupled nonlinear PDEs:
E
∂C
∂C
= −v
− k0 Ce− RT .
∂t
∂z
E
∂T
∂T
= −v
− Gr Ce− RT + Hr (Tw − T )
∂t
∂z
∆Hk0
4h
Gr = −
, Hr = −
ρCp
2rs ρCp
(2-10)
12
Figure 2-1: Tubular Chemical Reactor with 3 cooling/heating jackets.
Table 2-1: Values of the reactor parameters with convection and reaction
Parameter value
v
0.1 m · s−1
L
1m
k0
106 s−1
E
11250 cal · mol−1
R
1.986 cal · mol−1 · K −1
Cin
0.02 mol · l−1
Tin
340 K
Gr
4.25 · 109 l · K · mol−1 · s−1
Hr
0.2 s−1
with the following boundary conditions:
C = Cin at z = 0 and T = Tin at z = 0.
Here C(z, t) is the reactant concentration in [mol/l], T (z, t) is the reactant temperature in
[K] and Tw (z, t) is the reactor wall temperature in [K] defined as follows (see Figure 2-1),

 Tj1 , 0 ≤ z < Za
Tw =
T , Za ≤ z < Zb
 j2
Tj3 , Zb ≤ z < L
(2-11)
The parameters of the reactor are presented in Table 2-1.
The temperature of the jacket sections (Tj1 , Tj2 and Tj3 ) must be between 280 K and 400 K
(input constraints), in addition, the temperature inside the reactor must be smaller than 400
K (state constraint) in order to avoid the formation of side products.The kind of disturbances
that affect the reactor are the variations in the temperature and concentration of the feed
2.2 Example
13
flow. In this system, only the temperature of the feed flow is measured directly. In addition,
the reactor has a temperature sensor at the output and 4 temperature sensors (s1 ,s2 ,s3 and
s4 ) distributed in its interior as it is shown in Figure 2-1.
2.2.1 Operating point
The desired operating profiles (steady-state concentration and temperature profiles) of the
reactor are derived by means of an optimization algorithm, which minimizes a cost function
subject to the steady-state equations of the reactor described by 2-10, and the input and
state constraints defined previously. The steady-state model of the reactor is given by the
following Ordinary Differential Equations (ODEs):
E
k0
dC
= − Ce− RT
dz
v
Gr − E
Hr
dT
=
Ce RT +
(Tw − T )
dz
v
v
(2-12)
with T = Tin at z = 0 and C = Cin at z = 0, and the discrete version of 2-12 can be found
by replacing the spatial derivatives by forward difference approximations as follows:
− E
k0 ∆z
C i+1 = C i −
C i e RTf T i
v
− E
Hr ∆z
Gr ∆zCf
Hr ∆z
RTf T i
+
+
T i+1 = T i 1 −
C ie
T w,i
v
vTf
v
Tin
Cin
, for i = 1, 2, ....., N = 300
T0 =
, C0 =
Tf
Cf
(2-13)
where C i = Ci /Cf and T i = Ti /Tf are the normalized concentration and temperature of the
i − th increment, T w,i = Tw,i /Tf is the normalized temperature of the reactor wall of the
i − th increment, N is the number of increments in which the reactor is divided, Tf and Cf
are normalization factors, and ∆z is the length of the space increment. The variables are
normalized in order to avoid possible numerical problems. The minimization problem that
is solved by the optimization algorithm is defined as:
N
1 X
min ω(C r − C N ) + (1 − ω)
(T r,i − T i )2
N i=1
T j1 ,T j2 ,T j3
2
(2-14)
14
0.02
390
Temperature [K]
Concentration [mol/l]
380
0.015
0.01
370
360
0.005
350
0
0
0.2
0.4
0.6
0.8
Length z[m]
1
340
0
0.2
0.4
0.6
0.8
1
Length z[m]
Figure 2-2: Steady-state concentration and temperature profiles (Operating Profiles) whit
TJ1 = 374.6 K, TJ2 = 310.1 K and TJ3 = 325.2 K.
subject to
steady-state model given by 2-13
Tjmax
Tjmin
≤ T j1 , T j2 , T j3 ≤
Tf
Tf
Tmax
, for i = 1, 2, ...., N = 300
Ti ≤
Tf
where C r is the desired concentration (normalized) at the reactor output, T r,i is the desired
temperature (normalized) inside the reactor of the i − th increment, C N is the concentration
(normalized) at the reactor output, ω is a trade-off coefficient, Tjmin and Tjmax are the lower
and upper temperature values of the fluids in the jackets, and Tmax is the maximum allowed
temperature inside the tubular reactor. In this problem C r was set to 0, T r,i was selected
equal to the normalized temperature of the feeding flow ( T in = Tin /Tf ) for i = 1, 2, ..., N.
The trade-off parameter ω can take values from 0 to 1. To solve the optimization problem
described by 2-14 a sort of Sequential Quadratic Programming was proposed by [2].
The algorithm was executed using different initial conditions. Along the experiments, three
local minima were found. The selection of the optimal temperature and concentration profiles
was done by checking the value of the cost function and the deviation of the temperature at
the reactor output with respect to the temperature of the feed flow.
From the three local minima it was adopted the operating point with TJ1 = 374.6 K,
TJ2 = 310.1 K and TJ3 = 325.2 K, since it has the smallest cost function value and a small
temperature deviation at the reactor output. The optimal concentration and temperature
profiles can be observed in Figure 2-2.
2.2 Example
15
2.2.2 Linear Model
The linear model of the tubular chemical reactor is obtained by linearizing 2-10 around the
jacket’s temperatures and the operating profiles presented in Figure 2-2. This linear model
is given by,
∆
v ∆
Tf ∆
dC i
∆
∆
=−
C i − C i−1 − αA,i C i − αB,i T i
dt
∆z
Cf
∆
v
Cf ∆
dT i
∆
∆
∆
∆
=−
T i − T i−1 − αC,i C i − αD,i T i + Hr T w,i
dt
∆z
Tf
(2-15)
where
E − E∗
e RT
RT ∗2
E
E − E∗
αC (z) = −Gr e− RT ∗ , αD (z) = −Gr C ∗
e RT + Hr
RT ∗2
for i = 1, 2, ....., N = 300
E
αA (z) = k0 e− RT ∗ , αB (z) = k0 C ∗
whit
∆
∆
∆
∆
C 0 = C in , T 0 = T in
where αA,i = αA (zi ), αB,i = αB (zi ), αC,i = αC (zi ), αD,i = αD (zi ), zi = i∆z, Ci , Ti , Tw,i
are the concentration, temperature and reactor wall temperature of the i − th increment,
∗
Ci∗ ,Ti∗ , Tw,i
are the steady state concentration, temperature and reactor wall temperature
∗
∗
corresponding to zi , Cin
and Tin
are the the steady state concentration and temperature
∆
∆
∆
∗
∗
of the feed flow, C i = (Ci − Ci )/Cf , T i = (Ti − Ti∗ )/Tf , T w,i = (Tw,i − Tw,i
)/Tf are the
normalized deviations from steady state of the concentration, temperature and reactor wall
∆
∆
∗
∗
temperature of the space increment, C in = (Cin − Cin
)/Cf , T in = (Tin − Tin
)/Tf are the
normalized deviations from steady state of the concentration and temperature of the feed
flow, N is the number of space increments in which the reactor is divided, and ∆z is the
length of each increment.
The linear system 2-15 can be written as follows:
ẋ(t) = Ax(t) + Bu(t) + Fd(t)
(2-16)
Where A,B and F are the matrices describing the system, x(t) is the state vector,u(t) is the
vector of the inputs and d(t) is the vector of the disturbances.
16
Since the spatial domain of the reactor is divided into N = 300 sections, the number of
states of 2-16 is equal to 600. Given that such large number of states makes the design and
implementation of feedback controllers for the reactor difficult, in the next section a reduced
order model will be derived using POD and Galerkin projection.
2.2.3 Model Reduction Using POD
Let x(t) ∈ <2N = [x1 (t), x2 (t), ..., x2N ]T be the state vector of a given dynamical system,
and let X ∈ <2N ×N d with Nd ≥ 2N be the so-called snapshot matrix that contain a finite
number of samples or snapshots of the evolution of x(t) at t = t1 , t2 , ..., tNd . In POD, we
start by observing that each snapshot can be written as a linear combination of a set of
ordered orthonormal basis vectors (POD basis vectors) ϕj ∈ <N , ∀j = 1, 2, ..., N :
x(ti ) =
2N
X
aj (ti )ϕj , ∀i = 1, 2, ..., Nd
(2-17)
j=1
where aj (ti ) is the coordinate of x(ti ) with respect to the basis vector ϕj (it is also called timevarying coefficient or POD coefficient). Since the first n most relevant basis vectors capture
most of the energy in the data collected, we can construct an nth order approximation of
the snapshots by means of the following truncated sequence
x(ti ) =
n
X
aj (ti )ϕj , ∀i = 1, 2, ..., Nd , n 2N
(2-18)
j=1
This is the essence of model reduction by POD.
The POD basis Functions are determined from simulation or experimental data (Snapshot
matrix) of the process. The dynamic model for the first n time varying coefficients can be
found by means of the Galerkin projection.
The derivation of a reduced order model of 2-16 was done in 5 steps. These steps are
described as follows:
A. Generation of the Snapshot Matrix. We have created a snapshot matrix Xsnap ∈
<600×1500 from the system response where independent step changes were made in the
input u(t) and perturbation d(t) signals of the nonlinear model 2-10
Xsnap = [x(t = ∆t), x(t = 2∆t), ..., x(t = 1500∆t)]
B. Derivation of the POD basis vectors. The POD basis vectors are obtained by
computing the SVD of the snapshot matrix Xsnap ,
2.2 Example
17
Xsnap = ΦΣΨT
where Φ ∈ <600×600 and Ψ ∈ <1500×1500 are unitary matrices, and Σ ∈ <600×1500 is
a matrix that contains the singular values of Xsnap in a decreasing order on its main
diagonal. The left singular vectors, i.e. the columns of Φ are the POD basis vectors.
C. Selection of the most relevant POD basis vectors.It was done by checking
the singular values of Xsnap . The larger the singular value the more relevant the basis
function is. The first 20 basis functions associated to the first 20 largest singular values
were selected. The 20th order approximation of x(t) is given by
x(ti ) =
20
X
aj (ti )ϕj = Φn a(t)
(2-19)
j=1
D. Construction of the model for the first n = 20 POD coefficients. 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 paper. Let us define a residual
function R(x) for equation 2-16 as follows:
R(x) = ẋ(t) − Ax(t) − Bu(t) − Fd(t),
(2-20)
and we replace x(t) by its nth order approximation xn (t) = Φn a(t) in equation 2-20,
the Galerkin projection states that the projection of R(xn ) on the space spanned by
the basis functions Φn vanishes. That is,
hR(xn ), ϕj (zd )i = 0; j = 1, ...n
(2-21)
Replacing x(t) by its nth order approximation xn (t) = Φn a(t) in equation 2-16, and
applying the inner product criterion (Galerkin projection) to the resulting equation we
have:
ȧ(t) = ΦTn AΦn a(t) + ΦTn Bu(t) + ΦTn Fd(t)
(2-22)
xn (t) = Φn a(t)
and we obtain the model for the first n POD coefficients.
Profiles at t = 8sec
0.04
0.02
0
−0.02
0
0.5
1
Concentration [mole/l]
Concentration [mole/l]
18
Profiles at t = 15sec
0.04
0.02
0
−0.02
0
450
400
350
300
0
0.5
Length z [m]
0.5
1
Length z [m]
Temperature [K]
Temperature [K]
Length z [m]
1
450
400
350
300
0
0.5
1
Length z [m]
Figure 2-3: Temperature and concentration deviation profiles at t = 8s and t = 15s. Solid
line - full order Model. Dashed line - reduced order model.
E. Validation of the reduced order model. For validating the reduced order model
∆
of the reactor, we applied constant input signals (cooling/heating jackets) T J1 (t) =
∆
∆
∆
∆
∆
(t) = −20 K), T J2 (t) = 0.25 (TJ2
(t) =
0.125 (TJ1
(t) = 10 K), T J2 (t) = −0.25 (TJ2
∆
∆
−3
20 K), constant perturbation signals C in (t) = 0.05 (Cin (t) = 10
mole/l) and
∆
∆
T in (t) = 0.0625 (Tin (t) = 5 K) to both the full order model 2-10 and the reduced
order model 2-22, and afterwards we compared their responses. Figure 2-3 show
the temperature and concentration deviation profiles of the reactor at different time
instants for each model. From the previous results we can conclude that the reduced
order model with only 20 states provides an acceptable approximation of the full order
model (600 states).
3 Tubular chemical reactor with axial
and radial diffusion, reaction and
convection
A chemical reactor is basically a vessel where chemical reactions take place. A reactor is
usually the heart of an overall chemical o biochemical process. For modelling the behaviour
of most chemical reactors there are three main basic models that are commonly used, namely,
the batch reactor model (batch), the Continuous Stirred-Tank Reactor (CSTR) model and
the Plug Flow Reactor (PFR) model. Nowadays Plug flow reactors also, called Continuous
Tubular Reactors (CTRs) or simply tubular reactors, are widespread in chemical industry.
Typically, they are operated under steady-state conditions which leads to the production
of large amounts of products with constant and high quality. One big advantage of this
kind of reactors is the possibility of large-scale and low-cost production related to their
continuous operation since there are not down times as there are in the batch processes.
Furthermore, they are suitable for advanced, automated process control and optimization
techniques, and they deliver constant and high product quality due to tight monitoring and
control of the reaction environment. However, they have some disadvantages, the investment
costs are larger than in the other kinds of reactors and they are not suitable to produce
a variety of products in small amounts since switching between products can lead to a
considerable amount of off-spec production [34]. For the sake of the generality of the results
and conceptual contributions, in this dissertation i will focus our attention on an elementary
reaction in an ideal plug-flow reactor model, instead of reactions in specific complex industrial
reactors. In the following subsection i will describe in detail the kind of tubular reactor
for which i will design and implement POD-based MPC(Model Predictive Control) control
strategies.
3.1 Non-isothermal tubular reactor model with axial and
radial diffusion, reaction and convection
The system to be controlled is a non-isothermal tubular reactor with three phenomena: axial
and radial diffusion, reaction and convection, where a reversible, second order, exothermic,
catalyzed reaction takes place (A+B 2C). The reactor is surrounded by 3 cooling/heating
20
Figure 3-1: Tubular Chemical Reactor with 3 cooling/heating jackets.
Figure 3-2: Cylindrical shell of thickness ∆r, lenght ∆z, and volume 2πr∆r∆z.
jackets as it is shown in Figure 3-1. The temperature of the jackets fluids (TJ1 , TJ2 and TJ3 )
can be manipulated independently in order to control the concentration and temperature
profiles in the reactor.
It is assumed that radial and axial variations in concentration and temperature are present
also we consider laminar flow regime. In this study i am neglecting the heat transfer effects
between the jackets fluids and the reactor wall. Under the previous assumptions we are
going to carry out differential mole and energy balances on the differential cylinder show in
Figure 3-2.
In order to derive the governing equations, Fogler in [19] defines a couple of terms. The
first is the molar flux of species i, Wi (mol/m2 s). The molar flux has two components, the
radial component Wir , and the axial component Wiz . The molar flow consists of a diffusional
component and a convective flow component. The second term is the energy flux vector, e
(J/m2 s) that includes conduction and convection of energy. The equations for these terms
are,
3.1 Non-isothermal tubular reactor model
Wiz = −Dez
∂Ci
+ Uz Ci
∂z
Wir = −Der
∂Ci
+ Ur Ci
∂z
21
e = q + ΣWi Hi
where Dez and Der are the effective diffusivity in [m2 /s], Uz and Ur are the axial and radial
velocity in [m/s] respectively and q is given by Fourier’s law. For this work i neglect the
velocity in the radial directions and i suppose that Dez = Dez = De . A mole and energy
balance on specie A on a cylindrical system volume of lenght ∆z and thickness ∆r gives:
dC(2πr∆r∆z)
=WAr 2πr∆z |r −WAr 2πr∆z |r+∆r
dt
+WAz 2πr∆r |z −WAr 2πr∆r |z+∆z +rA 2πr∆r∆z
dT (2πr∆r∆zρCp )
=qr 2πr∆z |r −qr 2πr∆z |r+∆r +qz 2πr∆r |z −qz 2πr∆r |z+∆z
dt
+Uz 2πr∆rρCp |z −Uz 2πr∆rρCp |z+∆z +rA ∆Hrx 2πr∆r∆z
Dividing by 2πr∆r∆z and taking the limit as ∆r, ∆z → 0 i obtain:
∂C
∂ 2 C De ∂C
∂2C
∂C
= De 2 +
+ De 2 − Uz
+ rA
∂t
∂r
r ∂r
∂z
∂z
(3-1)
2
2
Ke ∂ T
Ke 1 ∂T
Ke ∂ T
∂T
∆Hrx
∂T
=
+
+
−
U
+
rA
z
∂t
ρCp ∂r 2
ρCp r ∂r
ρCp ∂z 2
∂z
ρCp
where,
E
− RT
rA = −k0 e
XA =
ρcat
C2
CCB − C
Keq
1
− T1 )]
[− ∆HRrx ( 303
, Keq = Keq0 e
CA0 − C
, CB = CB0 − CA0 XA , CC = 2CA0 XA ,
CA0
with the following boundary conditions,
, Uz = 2U0 1 −
r 2 R
22
• Radial:
1. At r = 0, we have symmetry:
∂T
∂r
= 0 and
∂C
∂r
=0
2. At r = R, the temperature flux to the wall on the reaction side equals the convective flux out of the reactor into shell side of the heat exchanger.
−Ke
∂T
|R = Hw (T (R, z) − Tw )
∂r
3. At r = R there is no mass flow through the tube walls
∂C
∂r
=0
• Axial:
1. T = Tin and C = Cin at z = 0,
2. At the outlet of the reactor z = L:
∂T
∂z
= 0 and
∂C
∂z
=0
Here C(r, z, t) is the reactant concentration in [mol · m−3 ], T (r, z, t) is the reactant temperature in [K], De is the diffusivity of all species in [m2 · s−1 ], Ke is the thermal conductivity
of the reaction mixture in [J · m−1 · s−1 · K −1 ], Keq0 is the equilibrium constant at 300K,
Uz is the fluid velocity in [m · s−1 ], (−∆Hrx ) is the heat of the reaction in [J · mol−1 ], ρ
and Cp are the density in [kg · m−3 ] and the specific heat in [J · kg −1 · K −1 ] of the mix
respectively, k0 is the rate constant, in [m6 · mol−1 · s−1 · kg −1], E is the activation energy in
[J · mol−1 ], R is the ideal gas constant in [J · mol−1 · K −1 ], Hw is the heat transfer coefficient
in [J · m−2 · s−1 · K −1 ], L is the reactor length in [m], Cin and Tin are the concentration in
[mol · m−3 ] and the temperature in [K] of the feed flow, z is the axial coordinate in [m], r is
the radial coordinate in [m], t is the time in [s] and Tw (z, t) is the reactor wall temperature
in [K] defined as follows

 TJ1 , 0 ≤ z < Za
Tw =
T , Za ≤ z < Zb
 J2
TJ3 , Zb ≤ z < L
The parameter values of the reactor model are taken from [19]. These values are presented
in Table 3-1.
The temperature of the jacket sections TJ1 , TJ2 and TJ3 must be between 280 K and 330 K.
In addition, the temperature inside the reactor must be below 400 K in order to avoid the
formation of side products. The kind of disturbances that affects the reactor are principally
variations in temperature and concentration of the feed flow. Typically, such variations are in
the range of ±10 K for the temperature and ±5% of the nominal value for the concentration.
In this system, only the temperature of the feed flow is measured directly.
3.2 Approximation techniques to solve diffusion, reaction, convection equation
23
Table 3-1: Values of the reactor parameters with axial and radial diffusion, reaction and
convection
Parameter value
De
10−9 m2 · s−1
v0
10−5 m3 · s−1
L
10 m
R
0.05 m
ρcat
1500 kg · m−3
ρ
1000 kg · m−3
Cp
4180 J · kg −1 · K −1
∆Hrx
−83680 J · mol−1
Keq0
1000
Ke
0.559 J · m−1 · s−1 · K −1
k0
1.1 × 108 m6 · mol−1 · s−1 · kg −1
E
95238 J · mol−1
R
8.314 J · mol−1 · K −1
Cin
500 mol · m−3
Tin
320 K
Hw
1300 J · m−2 · s−1 · K −1
Hr
0.2 s−1
3.2 Approximation techniques to solve diffusion, reaction,
convection equation
3.2.1 Approximation techniques
Systems of non-linear PDEs are not easily amenable to approximation techniques as they
depend on both the nature of the problem under investigation and on the properties of
the numerical techniques themselves. It is well known that convection-dominated PDEs
present serious numerical difficulties due to the moving steep fronts present in the solutions
of convection-diffusion transport PDEs or shock discontinuities in the solutions of pure convection PDEs [3]. Because of the extensive research carried out in these areas, it is impossible
to describe adequately all developed methods (finite difference and finite elements) for this
type of systems in this thesis.
The simulation of time-dependent convection-diffusion-reaction equations is required in various applications. A typical example is the simulation of processes which involve a chemical
reaction in a flow field.Such reactions are modelled by a non-linear system of time-dependent
convection-diffusion-reaction equations for the concentrations of the reactants and the products. These equations are strongly coupled such that inaccuracies in one concentration directly affect all other concentrations. Typically, the size of the diffusion is smaller by several
24
orders of magnitude compared to the size of the flow field, that means, the convectiondiffusion-reaction equations are convection-dominated, often, there is also a strong chemical
reaction such that the equations become reaction-dominated, too. A characteristic feature
of solutions of convection- and reaction-dominated equations is the presence of sharp layers [25]. The accurate simulation of such processes requires numerical methods which are,
on the one hand, able to compute sharp layers and which prevent, on the other hand, the
occurrence of spurious oscillations. There are many works about the numerical solution of
the reaction-diffusion-convection equation ([53, 13, 25, 51, 48]...),however in the following
example i show the solution of this equation by means of finite differences and recommend
a method.
The system to be simulated is a non-isothermal tubular reactor where a single, first order, irreversible, exothermic reaction takes place (A→B). The reactor is surrounded by 3
cooling/heating jackets. It is assumed that the reacting mixture flows as a plug through the
reactor body in the axial direction. In this dynamics four phenomena are taken into account,
namely, axial diffusion, convection, reaction and heat transfer (between the reactor and its
jackets). Under the previous assumptions, the mathematical model of the tubular chemical
reactor consists of the following coupled nonlinear PDEs:
E
∂C
∂2C
∂C
= DC 2 − v
− k0 Ce− RT .
∂t
∂z
∂z
E
∂T
∂2C
∂T
= DT 2 − v
− Gr Ce− RT + Hr (Tw − T )
∂t
∂z
∂z
Gr = −
(3-2)
∆Hk0
4h
, Hr = −
ρCp
2rs ρCp
whit,

 Tj1 , 0 ≤ z < Za
Tw =
T , Za ≤ z < Zb
 j2
Tj3 , Zb ≤ z < L
where DC and DT are the mass and energy dispersion coefficients in [m2 ·s−1 ]. Note, however,
that in practice the dimensionless mass and energy Peclet numbers, i.e., PeC = vL/DC ,
and PeT = vL/DT , are mostly used for indicating the level of dispersion. The boundary
conditions of the previous PDEs are the classical Danckwerts boundary conditions [9] given
by:
25
Table 3-2: Values of the reactor parameters with axial diffusion, reaction and convection
Parameter value
v
0.1 m · s−1
L
1m
k0
106 s−1
E
11250 cal · mol−1
R
1.986 cal · mol−1 · K −1
Cin
0.02 mol · l−1
Tin
340 K
Gr
4.25 · 109 l · K · mol−1 · s−1
Hr
0.2 s−1
∂C
∂z
∂T
DT
∂z
∂C
∂z
∂T
∂z
DC
= v(C − Cin )
at z = 0
= v(T − Tin )
at z = 0
=0
at z = L
=0
at z = L
The parameters of the reactor are presented in Table 3-2.
For carrying out the simulations of the dispersive plug flow reactor model, the non-linear
model 3-2 was discretized in space by using two schemes:
• Central difference scheme(CDS):
un − uni−1
∂u
≈ i+1
∂x
2∆x
• Upwind difference scheme(UDS): Upwind schemes use an adaptive or solution-sensitive
finite difference stencil to numerically simulate more properly the direction of propagation of information in a flow field. The upwind schemes attempt to discretize
hyperbolic partial differential equations by using differencing biased in the direction
determined by the sign of the characteristic speeds. To illustrate the method, consider
the following one-dimensional linear wave equation
∂u
∂u
+a
=0
∂t
∂x
26
It describes a wave propagating in the x-direction with a velocity a. The preceding
equation is also a mathematical model for one-dimensional linear convection. Consider
a typical grid point i in the domain. In a one dimensional domain, there are only two
directions associated with point i- left and right. If a is positive the left side is called
upwind side and right side is the downwind side. Similarly, if a is negative the left
side is called downwind side and right side is the upwind side. The simplest upwind
scheme possible is the first-order upwind scheme. It is given by:
un − uni−1
un+1
− uni
i
+a i
= 0 for a > 0
∆t
∆x
un − uni
un+1
− uni
i
+ a i+1
= 0 for a < 0
∆t
∆x
Figures 3-3, 3-4 show the steady state profile of the reactor for different Peclet number, in
these Figures can be seen that central difference scheme (CDS) has a small oscillation when
the Peclet number is bigger while upwind difference scheme (UDS) has well performance in
both cases. However in order to show this fact the Figures 3-5, 3-6 show the temperature
at the reactor outlet, in these Figures it is easy to see that when the Peclet number is bigger
than 103 the CDS has a big oscillation regardless the number of nodes increases while the
UDS has well performance in all cases. Due of this the recommended method for simulate
convection-diffusion-reaction equations by means of finite differences is Upwind difference
scheme.
3.2.2 Courant-Friedrichs-Lewy condition: Space and time step
In order to select the space step the Courant-Friedrichs-Lewy condition (CFL) is used [42, 37].
The the Courant-Friedrichs-Lewy condition (CFL condition) is a necessary condition for
convergence while solving certain partial differential equations (usually hyperbolic PDEs)
numerically. (It is not in general a sufficient condition.) It arises when explicit time-marching
schemes are used for the numerical solution. As a consequence, the time step must be
less than a certain time in many explicit time-marching computer simulations, otherwise
the simulation will produce wildly incorrect results. The condition is named after Richard
Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper [16, 15]. For
one-dimensional case, the CFL condition is given by:
ν=
U · ∆t
≤C
∆x
In the two-dimensional case this becomes,
(3-3)
5
Pe = 10
0.02
0.018
0.016
0.014
0.012
CDS
UDS
1
2
3
4
5
6
Length z [m]
7
8
9
10
4
5
6
Length z [m]
7
8
9
10
Temperature [K]
340
335
330
325
320
CDS
UDS
1
2
3
Figure 3-3: Steady state temperature and concentration profile for Pe = 105 .
P = 106
e
0.02
0.018
0.016
0.014
0.012
CDS
UDS
1
2
3
4
5
6
Length z [m]
7
8
9
10
Temperature [K]
340
335
330
325
320
CDS
UDS
1
2
3
4
5
6
Length z [m]
7
8
9
10
Figure 3-4: Steady state temperature and concentration profile for Pe = 106 .
27
28
Pe = 106
n = 10
CDS
UDS
339.5
339
338.5
n = 50
340.1
Temperature [K]
Temperature [K]
340
0
5
10
15
time [s]
20
25
CDS
UDS
340
339.9
339.8
30
0
5
10
n = 150
25
30
20
25
30
340.04
CDS
UDS
340.02
Temperature [K]
340.02
Temperature [K]
20
n = 300
340.04
340
339.98
339.96
339.94
339.92
15
time [s]
CDS
UDS
340
339.98
339.96
339.94
0
5
10
15
time [s]
20
25
339.92
30
0
5
10
15
time [s]
Figure 3-5: Temperature at the reactor output for Pe = 106 .
Pe = 103
n = 10
n = 50
340
CDS
UDS
339.5
Temperature [K]
Temperature [K]
340
339
338.5
338
0
5
10
15
time [s]
20
25
339.8
339.7
339.6
30
CDS
UDS
339.9
0
5
10
n = 150
25
30
25
30
340
339.95
Temperature [K]
Temperature [K]
20
n = 300
340
CDS
UDS
339.9
339.85
339.8
15
time [s]
0
5
10
15
time [s]
20
25
30
CDS
UDS
339.95
339.9
339.85
339.8
0
5
10
15
time [s]
20
Figure 3-6: Temperature at the reactor output for Pe = 103 .
Ux · ∆t Uy · ∆t
+
≤C
∆x
∆y
29
(3-4)
where U is the velocity, ∆t is the time step, ∆x is the length interval and the constant C
depends on the particular equation to be solved and not on ∆t and ∆x. The number ν is
called the Courant number. For upwind difference scheme for one-dimensional case the CFL
condition states that:
U · ∆t ∆x ≤ 1
(3-5)
3.2.3 Simulation of non-isothermal tubular reactor model with axial
and radial diffusion, reaction and convection
In order to simulate the non-isothermal tubular reactor model with axial and radial diffusion,
reaction and convection two facts were taken into account:
• The spatial derivative (radial and axial) in 3-1 was replaced by upwind difference
scheme to obtain 2 · n × m ordinary differential equations.
• The number of sections in axial and radial directions in which the reactor is divided
was selected agree with CFL condition as follows:
Uz · ∆t Ur · ∆t
+
≤1
∆z
∆r
(3-6)
Like Ur = 0 and ∆z = L/m i obtain that:
∆t ≤
L
m · Uz
(3-7)
According to the above the simulation of the non-isothermal tubular reactor model
5
was made in Matlab with n = m = 30 with ∆t ≤ ∆tmax = 103 . This simulation
was validated with a model of COMSOL Multiphysics with 26000 elements. The
Figure 3.2.3 show a comparison between the simulation made in Matlab and COMSOL
Multiphysics.
30
Matlab-Number of elements : 1800
z[m]
Comsol-Number of elements : 13000
r[m]
Figure 3-7: Simulation of non-isothermal tubular reactor model with axial and radial diffusion,
reaction and convection in Matlab and COMSOL Multiphysics .
3.3 Operating profile
The operating profiles (steady-state concentration and temperature profiles) of the reactor
are derived by means of an optimization algorithm, which minimizes a cost function subject
to the steady-state equations of the reactor described by (3-1), and the input and state
constraints defined previously. The steady-state model of the reactor is given by the following
Partial Differential Equations (PDEs):
De
∂ 2 C De ∂C
∂2C
∂C
+
+
D
− Uz
+ rA = 0
e
2
2
∂r
r ∂r
∂z
∂z
(3-8)
2
2
Ke 1 ∂T
Ke ∂ T
∂T
∆Hrx
Ke ∂ T
rA = 0
+
+
− Uz
+
2
2
ρCp ∂r
ρCp r ∂r
ρCp ∂z
∂z
ρCp
With T (r, 0) = Tin , C(r, 0) = Cin , C0,j = C1,j , Cn+1,j = Cn,j , Ci,m+1 = Ci,m , T0,j = T1,j ,
Tn+1,j = Tn,j and Tm+1,j = f (Tm,j , Tw ), where f (Tm,j , Tw ) is a function that depends of the
boundary conditions when r = R. The discrete version of (3-8) can be found by replacing
the spatial derivatives by upwind difference approximations as follows:
31
θ1 C i+1,j + θ2 C i,j+1 + θ3 C i,j + θ4 C i−1,j + θ5 C i,j−1 −
1
rA = 0
Cf
(3-9)
ϑ1 T i+1,j + ϑ2 T i,j+1 + ϑ3 T i,j + ϑ4 T i−1,j + ϑ5 T i,j−1 −
1
ϑ6 r A = 0
Tf
for i = 1, 2, ....., n = 30 and j = 1, 2, ....., m = 30
with
T wm,j

TJ1


, ∀j = 1, ....., za
T J1 =


Tf



TJ2
, ∀j = za + 1, ....., zb
T J2 =
=
Tf





T

 T J3 = J3 , ∀j = zb + 1, ....., n
Tf

θ1
θ2
θ3
θ4
θ5


(De /∆r 2 )
(De /∆z 2 )
(−2De /∆r 2 (1/i − 2) − 2De /∆z 2 − Uz /∆z)
De /∆r 2 (1 − 1/i)
De /∆z 2 + Uz /∆z

ϑ1
ϑ2
ϑ3
ϑ4
ϑ5
ϑ6


Ke /(ρCp ∆r 2 )
Ke /(ρCp ∆z 2 )
(−Uz /∆z − (2Ke /(ρ ∗ Cp ))(1/∆z 2 + 1/∆r 2 ) + Ke /(iρCp ∆r 2 ))
Ke /(ρCp ∆r 2 ) − Ke /(iρCp ∆r 2 )
Ke /(ρCp ∆z 2 ) + Uz /∆z
∆H/(ρCp )














 
 
 
=
 
 
 
 
 
 
=
 
 
 
−
r A = −k0 e
E
RTf T i,j
















1
1
rx
−
− ∆H
CC2
R
303
Tf T i,j
ρcat Ci,j CB −
, K eq = Keq0 e
Keq
where n and m are the number of sections in axial and radial directions in which the reactor is
divided, za and zb are the reactor sections defining the ending of the first and second jacket
respectively, Tf and Cf are normalization factors, C i,j = Ci,j /Cf and T i,j = Ti,j /Tf are
the normalized concentration and temperature of the i, j-th section of the reactor, T wm,j =
32
Twm,j /Tf is the normalized reactor wall temperature of the i, j-th section, and ∆z and ∆r
are the length and thickness of each section respectively. The variables are normalized in
order to avoid possible numerical problems. The optimization problem that is solved for
deriving the operating profiles is defined as:
m
min
T J 1 ,T J 2 ,T J 3
m
n
ω X
(1 − ω) X X
(C ri,n − C i,n )2 +
(T ri,j − T i,j )2
m i=1
n · m i=1 j=1
(3-10)
subject to
θ1 C i+1,j + θ2 C i,j+1 + θ3 C i,j + θ4 C i−1,j + θ5 C i,j−1 −
TJ min
Tf
≤ T J1 , T J2 , T J3 ≤
T i,j ≤
1
r
Cf A
1
Tf
=0
ϑ6 r A = 0
TJ max
Tf
Tmax
,
Tf
f or i = 1, 2, ...., m = 30 and f or j = 1, 2, ...., n = 30
where Cri,n is the desired concentration (normalized) at the reactor output, T ri,j is the desired
temperature (normalized) inside the reactor of the i, j-th increment, C i,n is the concentration
(normalized) at the reactor output, ω is a trade-off coefficient, TJmin and TJmax are the lower
and upper temperature values of the fluids of the jackets, and Tmax is the maximum allowed
temperature inside the tubular reactor. The first term of the cost function corresponds to
the squared error of the normalized concentration at the reactor output (terminal cost), and
the second term is related to the mean squared error of the normalized temperature along
the reactor (integral cost). In this problem Cri,n was set to 0, T ri,j was selected equal to the
normalized temperature of the feed flow (T in = Tin /Tf ) for i = 1, 2, ..., m and j = 1, 2, ..., n
. The trade-off parameter ω can take values between 0 and 1. When ω goes to 1, the
reduction of the reactant concentration at the reactor output becomes more important than
the temperature deviations. On the other hand when ω goes to 0, the temperature deviations
become more important than the concentration at the reactor output and the risk of the
formation of hot spots is reduced. To solve the optimization problem described by (3-10)
the following algorithm (a sort of Sequential Quadratic Programming - SQP) was proposed
by [2]:
i
h ∗
∗
∗
∗ T
1. Choose the initial values of the jackets temperatures T J = T J1 , T J2 , T J3 in such a
way that the constraints are satisfied.
33
∗
∗
2. Using T J , simulate (3-9) in order to obtain the temperature (T ∈ <n·m ) and concen∗
tration (C ∈ <n·m )profiles of the reactor steady state.
∗
∗
∗
3. Linearize the nonlinear model given by (3-9) around T , C ,and T J by means of Taylor
series. The resulting linear model would have the following structure:
∆
∆
∆
∆
∆
∆
θ1 C i+1,j + θ2 C i,j+1 + θ3 C i,j + θ4 C i−1,j + θ5 C i,j−1 − αa C i,j −
∆
∆
∆
∆
∆
Tf
∆
αb T i,j = 0
Cf
Cf
∆
∆
ϑ6 αa C i,j − ϑ6 αb T i,j = 0
Tf
(3-11)
for i = 1, 2, ....., n = 30 and j = 1, 2, ....., m = 30
with
∗ Cin − Ci,j
i,j ρ
cat 2Ci,j − 8
Keq
− RTE∗
αa = k 0 e
− RTE∗
αb = k 0 e
i,j
ρcat
∗ 2
(Cin − Ci,j
) ∆H
4
∗2
Ke RTi,j
!
Ek0 − RTEi,j
CC∗2
∗
∗
∗
+
e
ρcat Ci,j CB −
∗2
RTi,j
Keq
The system (3-11) can be written as:
∆
Ass X + Bss T wn,j = 0
(3-12)
with
iT
h ∆
∆
∆
∆
∆
X = C 1,1 , C 1,2 , ....., C m,n , T 1,1 , ........T m,n
∗
where Ass (i) and Bss (i) are the matrices describing the system in steady-state, C i,j ,
∗
∗
T i,j and T wm,j are the normalized operating points of the concentration, tempera∆
∆
∆
ture and reactor wall temperature of the i, j-th increment, C i,j , T i,j and T wm,j are the
34
normalized deviation variables of the concentration, temperature and reactor wall temperature respectively.
4. Solve the following Quadratic optimization Problem (QP):
m
∆
min
∆
∆
T J 1 ,T J 2 ,T J 3
m
n
ω X
(1 − ω) X X
(C ri,n − C i,n )2 +
(T ri,j − T i,j )2
m i=1
n · m i=1 j=1
(3-13)
subject to
∆
AssX + Bss T wn,j = 0
TJ min
Tf
∆
−Tjmax
Tf
≤ T J1 , T J2 , T J3 ≤
∆
∆
∆
≤ T J1 , T J2 , T J3 ≤
T i,j ≤
TJ max
Tf
∆
Tjmax
Tf
Tmax
,
Tf
f or i = 1, 2, ...., m = 30 and f or j = 1, 2, ...., n = 30
∆
∆
∆
where T J1 , T J2 , T J3 are the normalized deviation variables of the jackets temperatures,
and Tjmax are local input constraints which limits the range of the jackets temperatures in such a way that the linear model (3-12) is still a good approximation of the
nonlinear model (3-9). If this is not the case, then i would have convergence problems.
op
5. Calculate the new jackets temperatures T j ∈ <3 as follows:
op
∆,op
TJ = TJ
∗
+ TJ
h ∆,op ∆,op ∆,op iT
∆,op
= T j1 , T j2 , T j3
is the solution of the QP problem stated in the
where T J
previous step.
35
TA
Temperature[k]
325
330
320
320
315
310
310
300
305
290
0
0
0.02
5
300
295
0.04
10
0.06
z[m]
r[m]
Concentration [mol/L]
CA
0.2
0.4
0.2
0.15
0
0.1
2
4
6
8
z[m]
10
0.05
0.04
0.03
0.02
0.01
0.05
r[m]
Figure 3-8: Steady-state concentration and temperature profiles.
op
6. Using T J , simulate (3-9) in order to obtain the new temperature (T op ∈ <n·m ) and
op
concentration (C ∈ <n·m ) profiles of the reactor in steady state.
op
∗
∗
op
∗
op
∗
7. If max(|T J − T J |) ≤ T ol then stop, else make T J = T J , C = C , T = T
go to step 3.
op
and
The algorithm proposed by [2] was executed in MATLAB with the following parameters:
n = m = 30, Tf = 320 K, Cf = 500 mole/m3 , TJmin = 280 K, TJmax = 340 K,
∆
Tmax = 335 K, T ol = 10−3 , ω = 0.7 and Tjmax
= 20 K.
The maximum allowed temperature (Tmax ) inside the reactor was chosen 5 degrees below
the actual limit (340 K) in order to give to the feedback controller enough room of maneuverability. The trade-off coefficient ω was found by trial and error and the local input
∆
constraint TJmax
was selected in such a way that the differences between the non-linear and
linear model are small.
The algorithm was executed using different initial conditions. Along the experiments, one
local minima was found. The operating point was given by TJ1 = 291.1705 K, TJ2 =
293.6938 K and TJ3 = 294.8280 K. The optimal concentration and temperature profiles
can be observed in Figure 3-8 and 3-9.
36
TA at r=0
330
400
325
300
320
A
T (K)
CA (mol/m3)
CA at r=0
500
200
100
0
315
310
0
2
4
6
8
305
10
0
2
4
z(m)
6
8
10
0.03
0.04
0.05
z(m)
CA at z=10
TA at z=10
45
308
306
40
302
A
T (K)
CA (mol/m3)
304
35
30
300
298
25
20
296
0
0.01
0.02
0.03
0.04
0.05
294
0
0.01
r(m)
0.02
r(m)
Figure 3-9: Steady-state concentration and temperature profiles at r = 0 m and z = 10m.
3.4 Linear model
The linear model of the tubular chemical reactor is obtained by linearizing (3-1) around the
jackets temperatures and the operating profiles presented in Figure 3-8. This linear model
is given by,
∂ 2 C ∆ De ∂C ∆
∂2C ∆
∂C ∆
∂C ∆
= De
+
+
D
−
U
− αa C ∆ − αb T ∆
e
z
∂t
∂r 2
r ∂r
∂z 2
∂z
∂T ∆
∆Hrx
Ke ∂ 2 T ∆
Ke 1 ∂T ∆
Ke ∂ 2 T ∆
∂T ∆ ∆Hrx
αa C ∆ −
αb T ∆
=
+
+
−
U
−
z
2
2
∂t
ρCp ∂r
ρCp r ∂r
ρCp ∂z
∂z
ρCp
ρCp
(3-14)
with the following boundary conditions,
• Radial:
1. At r = 0, we have symmetry:
∂T ∆
∂r
= 0 and
∂C ∆
∂r
=0
3.4 Linear model
37
2. At r = R,
−Ke
∂T ∆
|R = Hw (T ∆ (R, z) − Tw∆ )
∂r
3. At r = R:
∂C ∆
∂r
=0
• Axial:
∆
∆
1. T ∆ = Tin
and C ∆ = Cin
at z = 0,
2. At the outlet of the reactor z = L:
∂T ∆
∂z
= 0 and
∂C ∆
∂z
=0
Here C ∆ = C − C ∗ , T ∆ = T − T ∗ and Tw∆ = Tw − Tw∗ are the deviations from steady state
of the concentration, temperature and reactor wall temperature; C ∗ , T ∗ and Tw∗ are the
steady state profiles (operating profiles) of the concentration, temperature and reactor wall
temperature respectively. In order to reduce the infinite dimensionality of 3-14, the partial
derivatives with respect to space are replaced by backward difference approximations leading
to the following system of ODEs:
∆
De
dC
De
De
=
(C i+1,j − C i,j + C i−1,j ) +
(C i,j − C i−1,j ) +
(C i,j+1 − C i,j + C i,j−1)
2
2
dt
∆r
i∆r
∆z 2
−
Tf
Uz
∆
∆
αb T i,j
(C i,j − C i,j−1) − αa C i,j −
∆z
Cf
∆
dT
Ke
Ke
Ke
=
(T i+1,j − T i,j + T i−1,j ) +
(T i,j − T i−1,j ) +
(T i,j+1 − T i,j + T i,j−1 )
2
2
dt
ρCp ∆r
iρCp ∆r
ρCp ∆z 2
−
∆Hrx
Uz
Cf ∆Hrx
∆
∆
αa C i,j −
αb T i,j
(T i,j − T i,j−1 ) −
∆z
Tf ρCp
ρCp
(3-15)
for i = 1, 2, ....., n = 30 and j = 1, 2, ....., m = 30
Where Ci,j , Ti,j , Twm,j are the concentration, temperature and reactor wall temperature
∗
∗
of the i, j-th section, Tf and Cf are normalization factors, Ci,j
, Ti,j
, Tw∗m,j are the steady
∆
state concentration, temperature and reactor wall temperature of the i, j-th section, C i,j =
∆
∆
∗
∗
)/Tf , T wm,j = (Twm,j − Tw∗m,j )/Cf are the normalized
(Ci,j − Ci,j
)/Cf , T i,j = (Ti,j − Ti,j
deviations from steady state of the concentration, temperature and reactor wall temperature
of the i, j-th section.
38
If i define the following vectors,
h ∆
iT
∆
∆
∆
∆
x(t) = C 1,1 , C 1,2 , ....., C m,n , T 1,1 , ........T m,n
h ∆ ∆ iT
d(t) = C in , T in
h ∆ ∆ ∆ iT
u(t) = T J1 , T J2 , T J3
Then (3-15) can be cast as follows:
ẋ(t) = Ax(t) + Bu(t) + Fd(t)
(3-16)
where A ∈ <m·n×m·n , B ∈ <m·n×3 , F ∈ <m·n×2 are the matrices describing the system,
x(t) ∈ <m·n is the state vector, u(t) ∈ <3 is the vector of the inputs and d(t) ∈ <2 is the
vector of the disturbances.
Since the spatial domain of the reactor is divided into n × m = 900 sections, the number of
states of (3-16) is equal to 1800. Given that such large number of states makes the design
and implementation of feedback controllers for the reactor difficult, in the next section a
reduced order model will be derived using POD and Galerkin projection.
3.5 Model reduction using POD for a non-isothermal
tubular reactor with diffusion, reaction and convection
The derivation of a reduced order model of (3-16) was done in 5 steps. These steps are
described in the following subsections.
1. Generation of the Snapshot Matrix. We have created a snapshot matrix from
the system response (Xsnap ∈ <1800×10000 ) when independent step changes were made
in the input u(t) and perturbation d(t) signals on the non-linear model (3-1)
Xsnap = [x(t = ∆t), x(t = 2∆t), ..., x(t = 1000∆t)]
(3-17)
Along the simulations 10000 samples were collected using a sampling time ∆t of 1 s.
The amplitude of the step changes was chosen in such a way as to produce changes
of similar magnitude in the temperature and concentration profiles. This avoids a
possible bias in the resulting model.
3.5 Model reduction using POD for a non-isothermal tubular reactor with diffusion,
reaction and convection
39
2. Derivation of the POD basis vectors. The POD basis vectors are obtained by
computing the SVD of the snapshot matrix Xsnap ,
Xsnap = ΦΣΨT
where Φ ∈ <1800×1800 and Ψ ∈ <10000×10000 are unitary matrices, and Σ ∈ <1800×10000 is
a matrix that contains the singular values of Xsnap in a decreasing order on its main
diagonal. The left singular vectors, i.e. the columns of ,Φ
Φ = [ϕ1 , ϕ2 , ..., ϕ1800 ]
are the POD basis vectors.
3. Selection of the most relevant POD basis vectors.
The n most relevant POD basis vectors are chosen using the energy criterion presented
in section 2.1. The plot of 1 − PN (see equation 2-5) for the first 100 basis vectors is
shown in Figure 3-10. In this problem, i chose the first n = 50 POD basis vectors
based on their truncation degree 1 − Pn = 3.3 × 10−4 (Pn = 0.9996). The 50th order
approximation of x(t) is given by the following truncated sequence:
xn (t) =
50
X
aj (t)ϕj = ΦN a(t)
(3-18)
j=1
where Φn = [ϕ1 , ϕ2 , ..., ϕ50 ] and a(t) = [a1 (t), a2 (t), ..., a50 (t)]T
4. Construction of the model for the first n = 50 POD coefficients. 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 thesis. Let us define a residual
function R(x) for equation (3-16) as follows:
R(x) = ẋ(t) − Ax(t) − Bu(t) − Fd(t),
(3-19)
and we replace x(t) by its n-th order approximation xn (t) = Φn a(t) in equation (3-19),
the Galerkin projection states that the projection of R(xn ) on the space spanned by
the basis functions Φn vanishes. That is,
40
1
0.95
Pn(i)
0.9
0.85
0.8
0.75
0
10
20
30
40
50
grado i
60
70
80
90
100
60
70
80
90
100
1−Pn
0
10
−5
1−Pn
10
−10
10
−15
10
−20
10
0
10
20
30
40
50
grado i
Figure 3-10: Logarithmic plot of Pn and 1 − Pn for determining the truncation degree of the POD
basis vectors in the reactor case
hR(xn ), ϕj )i = 0; j = 1, ...n
(3-20)
where h·, ·i denotes inner product. Replacing x(t) by its n-th order approximation
xn (t) = Φn a(t) in equation (3-16), and applying the inner product criterion (Galerkin
projection) to the resulting equation we have:
hΦn ȧ(t), ϕj i = hAΦn a(t) + Bu(t) + Fd(t), ϕj i
(3-21)
by evaluating the inner product in (3-21),
ȧ(t) = ΦTn AΦn a(t) + ΦTn Bu(t) + ΦTn Fd(t)
and we obtain the model for the first n POD coefficients. The reduced order model of
the reactor with only 50 states is then given by
41
ȧ(t) = Ar a(t) + Br u(t) + Fr d(t)
(3-22)
xn (t) = Φn a(t)
where Ar = ΦTn AΦn ,Br = ΦTn B and Fr = ΦTn F. The initial condition for a(t) reads as
a(0) = 0.
For validating the reduced order model of the reactor, i applied constant input sig∆
∆
∆
∆
∆
nals T J1 (t) = 0.0312 (TJ1
(t) = 10 K), T J2 (t) = −0.0312 (TJ2
(t) = 10 K), T J3 (t) =
∆
∆
∆
(t) =
0.0625 (TJ3
(t) = 20 K) and constant perturbation signals C in (t) = 2 × 10−4 (Cin
∆
∆
0.1 mole/l) and T in (t) = 0.0156 (Tin (t) = 5 K) to both the full order model (3-1) and
the reduced order model (3-22), and afterwards we compared their responses. Figures
3-11, 3-12 and 3-13 show the temperature and concentration profiles of the reactor at
different time instants and coordinate for each model. In order to measure the quality
of the reduced order model the averages of the absolute error for the temperature (ET )
and concentration (EC ) were calculated by means of the following formulas:
ET =
EC =
1
Ns
1
Ns
Ns
P
|T (k∆t) − Tn (k∆t)|
k=1
Ns
P
|C(k∆t) − Cn (k∆t)|
k=1
where Ns = 10000 is the number of time steps and ∆t = 1 s. The plots of ET and
EC are shown in Figure 3-14. For the temperature profile, the maximum value for
the error is ET is 0.7542 K. For the concentration, the maximum peak for the error
EC is 3.3 × 10−3 mol/l. From the previous results we can conclude that the reduced
order model with only 50 states provides an acceptable approximation of the full order
model.
The discrete-time version of (3-22) that is used for designing the digital controller, was
obtained using the discretization method known as zero-order hold (ZOH) with a sampling
time of 0.2 s,
a(k + 1) = Ãa(k) + B̃u(k) + F̃d(k)
(3-23)
xn (k) = Φn a(k)
42
Profiles at t=100 s and r=0 m
Profiles at t=1000 s and r=0 m
0.4
Concentration[mol/L]
0.4
0.3
0.2
0.1
0
0
2
4
6
8
0.3
0.2
0.1
0
10
0
2
4
z[m]
330
10
6
8
10
330
Temperature[K]
Temperature[K]
8
335
325
320
315
310
305
6
z[m]
325
320
315
310
0
2
4
6
8
305
10
0
2
z[m]
4
z[m]
Figure 3-11: Temperature and concentration profiles at t = 100 s and t = 1000 s at r = 0 m.
Solid line - full order Model. Dashed line - reduced order model.
Profiles at t=8000 s and z=10 m
0.05
0.04
0.045
Profiles at t=1500 s and z=10 m
0.045
0.035
0.03
0.04
0.035
0.025
0.02
0.03
0
0.01
0.02
0.03
0.04
0.05
0.025
0
0.01
0.02
r[m]
0.03
0.04
0.05
0.03
0.04
0.05
r[m]
316
316
315
314
Temperature[K]
Temperature[K]
314
312
310
313
312
311
310
308
309
306
0
0.01
0.02
0.03
r[m]
0.04
0.05
308
0
0.01
0.02
r[m]
Figure 3-12: Temperature and concentration profiles at t = 1500 s and t = 8000 s at z = 10 m.
Solid line - full order Model. Dashed line - reduced order model.
C Full order model
C Reduced order model
A
A
0.12
0.12
0.2
0.1
0.15
0.1
0.08
0.05
0.06
0
0.05
0.03
0.02
0.01
r[m]
10
8
6
2
4
0.2
0.1
0.1
0.08
0.06
0
0.05
0.04
0.03
0.02
0.01
r[m]
0.04
0.04
0.02
z[m]
0.04
10
4
6
8
2
0.02
z[m]
T Full order model
T Reduced order model
A
A
330
330
320
Temperature[k]
Temperature[k]
320
350
310
300
300
250
350
310
300
300
250
10
10
0.01
0.01
8
0.02
6
0.03
2
8
0.02
290
4
0.04
r[m] 0.05
43
6
0.03
0.04
r[m] 0.05
z[m]
290
4
2
z[m]
Figure 3-13: Temperature and Concentration profiles at t = 10000 s: full order Model and reduced
order model.
−3
ET
EC
x 10
0.7
3
−3
0.6
1.5
1
0.5
4
Temperature[k]
2
x 10
2.5
2
0
10
8
1
0.5
0.5
0.4
0
10
0.3
0.2
8
0.05
6
0.04
4
z[m]
0.05
6
2
0.02
0.01
r[m]
0.04
z[m]
0.1
0.03
4
0.03
2
0.02
0.01
r[m]
Figure 3-14: Average of the absolute error between the full order model (3-1) and the reduced
order model (3-22).
44
where Ã, B̃ and F̃ are the matrices describing the new system. A modeling approach frequently adopted in model predictive controller (MPC) considers a discrete-time state-space
model in the incremental form [45], hence (3-23) can be represented in the following form:
xs (k + 1)
xd (k + 1)
y(k) =
C
s
=
C
d
Iny 0
0 P
xs (k)
xd (k)
xs (k)
xd (k)
+
Ds
Dd
∆u(k) +
Fs
Fd
∆d(k)
(3-24)
(3-25)
where xd (k) = V1 a(k), xs (k) = V2 a(k − 1), ∆uk = uk − uk−1 is the input increment
∆dk = dk − dk−1 is the disturbance increment and V1 , V2 are transformation matrices. In the
state equation defined in (3-24), the state component xs corresponds to the integrating poles
produced by the incremental form of the model, and xd (k) = a(k) corresponds to the system
modes. For stable systems, it is easy to show that when the system approaches steady state,
component xd tends to zero. P is a diagonal matrix with components corresponding to the
poles of the system.
4 Infinite Horizon Model Predictive
Control (IHMPC)
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 open-loop
optimal control problem is solved on-line 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 on-line an optimization problem where commonly plant constraints are included, makes MPC different from conventional
control which uses a pre-computed control law [38]. MPC has been widely adopted by the
industrial process control community and implemented successfully in many applications.
First of all, the MPC algorithms can handle in a very natural way constraints on both process inputs (manipulated variables or control actions) and process outputs values (controlled
variables), which often have a significant impact on the quality, effectiveness and safety of
the production. Additionally, the MPC controllers can take into account the internal interactions within the process, thanks to the multivariable models on which they are typically
based. This make the MPC algorithms a quite suitable option for multivariable process
control. Another reason of the success of MPC is the fact that the principle of operation is
comprehensible and relatively easy to explain to process operators and engineers. This is an
important aspect at the moment of introducing new techniques into industrial practice.
4.1 Infinite Horizon Model Predictive Control
A modelling approach frequently adopted in model predictive controller (MPC) considers a
discrete-time state -space model in incremental form [46],
x(k + 1) = Ax(k) + B∆u(k)
y(k) = Cx(k)
The model in incremental form can be represented in the following way:
(4-1)
46
xs (k + 1)
xd (k + 1)
=
Iny 0
0 P
y(k) =
xs (k)
xd (k)
Cs Cd
+
Ds
Dd
s
x (k)
xd (k)
∆u(k) +
Fs
Fd
∆d(k)
(4-2)
where xd (k) = V1 x(k), xs (k) = V2 x(k − 1), ∆uk = uk − uk−1 is the input increment
∆dk = dk − dk−1 is the disturbance increment and V1 , V2 are transformation matrices. In the
state equation defined in (4-2), the state component xs corresponds to the integrating poles
produced by the incremental form of the model, and xd (k) = x(k) corresponds to the system
modes. For stable systems, it is easy to show that when the system approaches steady state,
component xd tends to zero. P is a diagonal matrix with components corresponding to the
poles of the system.
MPC is usually based on a discrete state-space model as shown in (4-1). In the outputtracking problem, the IHMPC cost can be defined as follows:
Jk,∞ =
∞
P
j=1
eTk+j Qek+j +
m−1
P
j=1
∆uTk+j R∆uk+j
(4-3)
where ek+j = y(k + j) − r(j); y(k + j) is the output prediction at time instant k + j made
at time k; r is the desired output reference, m is the control horizon, Q ∈ <ny ×ny and
R ∈ <nu ×nu are positive definite weighting matrices. The controller that is based on the
minimization of the above cost function corresponds to the IHMPC [46] for the outputtracking case. Most of the infinite horizon controllers reduce to finite horizon controllers by
defining a terminal state penalty Q. For the cost defined in (4-3) such a terminal penalty is
computed by the following Lyapunov equation :
T
Q − P T QP = P T C d QC d P
(4-4)
Since an infinite horizon is used and the model defined in (4-2) has integrating modes,
terminal constraints must be added to prevent the cost from becoming unbounded. Hence
constraints can be written as follows:
s
s
C xs (k) − ysp + C D̃ s ∆uk = 0
where
(4-5)
4.1 Infinite Horizon Model Predictive Control
D̃ s =
s
D s ...D s
47
C = diag [C s ...C s ]
with the terminal penalty, the cost defined in (4-3) reduces to
Jk,∞ =
m−1
P
j=1
T
eTk+j Qek+j + xdk+m Qxdk+m +
m−1
P
j=1
∆uTk+j R∆uk+j
(4-6)
Finally, the control optimization problem of the infinite horizon MPC can be formulated as:
min Jk,∞ =
∆uk
m
P
j=1
eTk+j Qek+j + eTk+m Qek+m +
m−1
P
j=1
∆uTk+j R∆uk+j
(4-7)
subject to
xs (k + 1)
xd (k + 1)
=
Iny 0
0 P
y(k) =
s
xs (k)
xd (k)
s
d
C
C
+
Ds
Dd
xs (k)
xd (k)
∆u(k) +
Fs
Fd
∆d(k)
s
C xs (k) − ysp + C D̃ s ∆uk = 0
(4-8)
−∆umax ≤ ∆uk+j ≤ ∆umax
∆uk+j = 0 ; j ≥ m
umin ≤ uk−1 +
j
P
∆uk+i ≤ umax ; j = 0, 1, ..., m − 1
i=0
For large changes on xs (k) or ysp or if ysp corresponds to an unreachable steady state, then the
optimization problem defined through (4-7)-(4-8) may become infeasible because of a conflict
between constraints. Consequently, the IHMPC as defined above cannot be implemented in
practice.
48
4.2 Extended Infinite Horizon Model Predictive Control
To produce an infinite horizon MPC, which is implementable in practice, the objective
function of infinite horizon MPC is re-defined as follows:
Jk,∞ =
∞
P
(ek+j − δk )T Q (ek+j − δk ) +
j=1
m−1
P
j=1
∆uTk+j R∆uk+j + δkT Sδk
(4-9)
Where δk ∈ <ny is a vector of slack variables and S ∈ <ny ×ny is assumed positive definite.
Observe that each slack variable refers to a given controlled output. Weight matrix S should
be selected such that the controller tends to zero the slacks or at least minimize them
depending on the number of inputs, which are not constrained.
Analogously to the IHMPC, the extended infinite horizon controllers reduce to finite horizon
controllers by defining a terminal state penalty Q that is obtained by solving (4-4) and
terminal constraints must be added to prevent the cost from becoming unbounded, this
constraint can be written as follows:
s
s
C xs (k) − ysp + C D̃ s ∆uk − δk = 0
(4-10)
where
D̃ s =
s
D s ...D s
C = diag [C s ...C s ]
Hence, the control objective defined in (4-9) becomes:
Jk,∞ =
m
P
j=1
(ek+j − δk )T Q (ek+j − δk ) + eTk+m Qek+m +
m−1
P
j=1
Finally, the control optimization problem of the extended infinite horizon MPC (IHMPC)
can be formulated as:
min Jk,∞ =
∆uk ,δk
subject to
m
P
j=1
T
(ek+j − δk )T Q (ek+j − δk ) + xdk+j Qxdk+j
m−1
P
j=1
(4-11)
xs (k + 1)
xd (k + 1)
=
Iny 0
0 P
y(k) =
s
xs (k)
xd (k)
s
d
C
C
+
Ds
Dd
xs (k)
xd (k)
∆u(k) +
Fs
Fd
49
∆d(k)
s
C xs (k) − ysp + C D̃ s ∆uk − δk = 0
(4-12)
−∆umax ≤ ∆uk+j ≤ ∆umax
∆uk+j = 0 ; j ≥ m
umin ≤ uk−1 +
j
P
∆uk+i ≤ umax ; j = 0, 1, ..., m − 1
i=0
4.3 IHMPC and POD applied to Control of a Tubular
Reactor
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 (280K ≤ TJ1 (t), TJ2 (t), TJ3 (t) ≤ 335K), and the
control system should keep the temperature inside the reactor below 335K. In this scheme,
the control of the temperature and concentration profiles is achieved indirectly by controlling
the POD coefficients. The references (aref ) of these POD coefficients can be calculated by
aref = ΦTn xref
where xref is the reference of the vector x(t) and is equal to 0 (the model of the MPC is a
linear model) since the control system has to keep the reactor operating around the profiles
shown in Figure 3-8. The IHMPC controller, which uses model (4-2) to predict the future
behaviour of the reactor, is formulated as (4-11) and (4-12).
In this formulation C d = I50×50 V1 , C s = 050×50 V2 . Since the state vector a(k) is unknown
∆
and the changes in the concentration of the feed flow (d1(k) = Cin
(k)) are not measured
directly, they are estimated by means of an observer (in this case a Kalman filter) with the
following formulation:
50
"
â(k + 1)
dˆ1 (k + 1)
#
=
"
Ã F̃C
0 1
#"
â(k)
dˆ1 (k)
La
Ld
+
#
+
"
B̃
0
#
u(k) +
(y(k) − ŷ(k))
"
F̃T
0
#
d2 (k)
(4-13)
y(k) = Csel x̂n (k) = Csel Φn â(k)
∆
where â is the estimated vector of the POD coefficients, dˆ1 (k) is the estimation C in , d2 (k)
∆
is the normalized temperature deviation of the feed flow T in (k), y(k) ∈ <10 is a vector
containing ten temperature measurements (normalized deviations) along the reactor, ŷ(k) is
the estimate of y(k), La and Ld are the sub-matrices of the observer gain (Kalman gain), FC
and FT are the column vectors of F̃ = [F̃C , F̃T ] and Csel is a selection matrix which selects
the measured temperatures from the vector xn (k).
The control horizon m was set to 10 samples umin and umax were selected according to
the input constraints of the process and the operating temperatures of the jackets, and the
weighting matrices in this way: Q = 1 · I50×50 , R = 10 · I3×3 , S = 1 · I50×50 . The Kalman
gain matrix was computed from the following covariance matrices: Rw = 1 · I51×51 ,Rv =
10−3 · I10×10 .
4.3.1 Simulation Results
In order to perform the closed-loop simulations of the control systems described in the
previous sections, the non-linear model of the process given in 3-1 was discretized by replacing
the partial derivatives with respect to space by backward difference approximations using
an ”upwind” scheme [9]. The use of low order approximations for the spatial derivatives
is known to produce excessive smoothing of the profiles due to numerical diffusion, and on
the other hand, high-order approximations lead to excessive non-physical oscillations due to
numerical dispersion [34]. Notice that both numerical diffusion and dispersion are two kinds
of computational errors that occur as a result of the discretization process, and therefore
they should not be confused with their physical counterparts. One way of decreasing these
undesirable effects is again by increasing the grid density (finer grid), but this measure leads
to an increment of the computational burden. So, a trade-off between computational time
and accuracy must be found. At the beginning, we divided the reactor into n = m = 10, n =
m = 30, and n = m = 50, and we found that a partition of 30 sections provides a good
trade-off.
In order to evaluate the performance of the control system, the following tests were carried
out:
51
• Test1: The temperature of the feed flow is increased 10K at the 5000s and the concentration of the feed flow is decreased 25 × 10−3 mole/L at the 5s.
• Test2: The temperature of the feed flow is decreased 10K at the 5000s and the concentration of the feed flow is increased 25 × 10−3mole/L at the 5s.
These disturbances have a big impact on the temperature profile of the reactor.
The simulation results for the Test1 are presented in Figure 4-1, 4-2, 4-3, 4-4 and the
simulation results for the Test2 are presented in Figure 4-5, 4-6, 4-7, 4-8.
Profiles at z = 10 m
0.05
0.04
0.03
0.02
0.01
0
0.01
0.02
0.03
0.04
0.05
0.03
0.04
0.05
r[m]
Temperature[K]
310
300
290
280
0
0.01
0.02
r[m]
Figure 4-1: Steady state Temperature and concentration profiles for Test 1 at z = 10 m. Solid
line-IHMPC. Dashed line-Nominal profile (reference).
Furthermore, some quantities of interest are given in Table 4-1. In this table, Tmax is the
maximum temperature reached inside the reactor during the test. ∆Cout is the percentage
of the change of the mean steady state concentration at the reactor outlet with respect to
its nominal value. That is,
∆Cout % =
CN − CN∗
CN∗
(4-14)
where CN∗ is the mean nominal value (0.0339 mol/l) and CN is the mean concentration at
the reactor outlet in steady state after the test.
In general, the control schemes showed a good behavior for rejecting the disturbances (typical
magnitudes: Cin = ±25 × 10−3 mol/l and Tin = ±10 K) and both presented a similar
performance.
52
Profiles at r = 0 m
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
6
8
10
z[m]
Temperature[K]
340
330
320
310
300
0
2
4
z[m]
Figure 4-2: Steady state Temperature and concentration profiles for Test 1 at r = 0 m. Solid
Table 4-1: Performance parameters of the control systems
Quantities
Test 1
Test 2
Tmax [k] 332.3218 323.6579
∆Cout [%] -1.0531
0.8732
53
Steady state profile
0.01
0
0.05
−0.01
0
−0.05
10
8
6
4
z[m]
0.05
0.04
0.03
0.02
0.01
r[m]
2
−0.02
−0.03
4
Temperature[K]
2
10
0
0
−10
10
−2
0.05
0.04
8
−4
0.03
6
0.02
4
0.01
2
z[m]
−6
r[m]
Figure 4-3: Steady state error (Temperature and concentration) for Test 1
TJ1
TJ2
292
296
291.5
295
TJ3
298
296
291
294
289.5
289
Temperature[K]
290
292
291
292
290
290
288.5
288
289
288
288
287.5
287
294
293
Temperature[K]
Temperature[K]
290.5
286
0
1
t[s]
287
2
4
x 10
0
1
t[s]
2
0
4
x 10
1
t[s]
2
4
x 10
Figure 4-4: Control actions (jackets temperatures) of the MPC controller for Test 1
54
Profiles at z = 10 m
0.05
0.04
0.03
0.02
0
0.01
0.02
0.03
0.04
0.05
0.03
0.04
0.05
r[m]
Temperature[K]
310
305
300
295
290
0
0.01
0.02
r[m]
Figure 4-5: Steady state Temperature and concentration profiles for Test 2 at z = 10 m. Solid
line- IHMPC. Dashed line-Nominal profile (reference).
Profiles at r = 0 m
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
6
8
10
z[m]
Temperature[K]
330
325
320
315
310
305
0
2
4
z[m]
Figure 4-6: Steady state Temperature and concentration profiles for Test 2 at r = 0 m. Solid
55
Steady state profile
0.03
0.02
0.05
0.01
0
2
−0.05
0.05
0
4
0.04
6
0.03
−0.01
8
0.02
0.01
10
z[m]
r[m]
6
Temperature[K]
4
2
10
0
0
−10
0.05
2
−2
4
0.04
6
0.03
−4
8
0.02
0.01
z[m]
10
r[m]
Figure 4-7: Steady state error (Temperature and concentration) for Test 2
TJ1
TJ2
295
TJ3
299
294.5
303
302
298
301
294
297
300
293
292.5
292
Temperature[K]
Temperature[K]
Temperature[K]
293.5
296
295
294
291.5
298
297
296
295
293
291
294
292
290.5
290
299
0
1
t[s]
291
2
4
x 10
293
0
1
t[s]
292
2
4
x 10
0
1
t[s]
2
4
x 10
Figure 4-8: Control actions (jackets temperatures) of the MPC controller for Test 2
5 Conclusions
Concluding Remarks
This thesis considers a major research topic: The use of POD and the Galerkin projection
as a model reduction technique for a non-isothermal tubular reactor with diffusion, reaction
and convection phenomena. With this research topic, the following concluding remarks are
highlighted:
General conclusion
In this thesis was shown how POD and Galerkin projections can be used for deriving reduced
order model of systems with reaction, diffusion and convection in two dimensions. In this
particular case a non-isothermal tubular reactor was used and Control systems and observers
were designed from the reduced order model.
Particular conclusions
• The algorithm proposed in [1] to find the operating profiles was extended for the reactor
with diffusion, reaction and convection in two dimensions, and it was shown that it
works properly complying with the reactor operating constraints.
• 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 well-defined
sense. In the non-isothermal 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 time-dependent
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 the reduced
order model, and is obtained by projecting the POD basis functions onto the original
governing equations. Throughout the results presented in this thesis, 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 non-isothermal tubular reactor with diffusion,
reaction and convection have an acceptable approximation.
57
• In the application of POD technique the data matrix (Xsnap ) was taken from the nonlinear system unlike in [1]. The reason why this was done is that the linear model did
not capture the reaction kinetics (irreversible reaction) in a desired way. The use of the
non-linear model data gives a more realistic sense to the results of this work because
usually the data would be taken from the process. However, using non-linear model
data increases the number of basis functions.
• The reduced order model could be used as a base model for controller design. In Chapter 4 an Infinite Horizon MPC controller has been designed for the non-isothermal
tubular reactor model with diffusion, reaction and convection on the basis of the reduced order model. Given that the original model is a non-linear one and, although
there is a linear model, it is very difficult to solve the optimization problem due to the
high order model . The control and optimization problem becomes very tractable if
the model can be reduced based on a small number of POD basis functions inferred
from the open loop data. It is shown in Chapter 4 that the desired temperature and
concentration distribution can be controlled using the reduced order model as the basemodel for the controller (Figure 4-1 to 4-7); In this case, the control of the reactor
profiles is achieved indirectly by controlling the POD coefficients which have no physical meaning. This makes the tuning of the controller less intuitive and the definition
of the control goals less flexible.
Finally in spite of the spatial discretization of the non-linear PDE’s describing the
reactor, the linearization and the dramatic reduction of the order by means of POD, the
controller has an acceptable performance. However, if larger disturbances are applied
to the tubular chemical reactor, the behaviour of the IHMPC controllers would not be
as good as it has been thus far. This is due to the differences between the non-linear
model and linear model and consequently the reduced order model.
Future Research
• To take into account the non-linear nature of the reactor in the design of the control
schemes.
• Derivation of POD models from closed-loop experiments.
• Model reduction of uncertain systems.
• Model reduction of differential-algebraic (DAE) systems.
• Development/validation of control algorithms based on reduced models
Bibliography
[1] Agudelo, O.M.: The application of Proper Orthogonal Decomposition to the control
of tubular reactors. En: PhD thesis, Katholieke Universiteit Leuven (2009)
[2] Agudelo, O.M. ; Espinosa, J.J. ; De Moor, B.: Control of a Tubular Chemical
Reactor by means of POD and Predictive Control Techniques. En: in Proceedings of
the European Control Conference (ECC 2007) 20 (2007), p. 1046–1053
[3] Alhumaizi, Khalid ; Henda, Redhouane ; Soliman, Mostafa: Numerical analysis
of a reaction-diffusion-convection system. En: Computers & Chemical Engineering 27
(2003), Nr. 4, p. 579 – 594. – ISSN 0098–1354
[4] Amabili, M. ; Sarkar, A. ; Paı̈doussis, M. P.: Reduced-order models for nonlinear
vibrations of cylindrical shells via the proper orthogonal decomposition method. En:
Journal of Fluids and Structures 18 (2003), Nr. 2, p. 227 – 250. – Axial and Internal
Flow Fluid-Structure Interactions. – ISSN 0889–9746
[5] Amabili, M. ; Sarkar, A. ; Paı̈doussis, M.P.: Chaotic vibrations of circular cylindrical shells: Galerkin versus reduced-order models via the proper orthogonal decomposition method. En: Journal of Sound and Vibration 290 (2006), Nr. 3-5, p. 736 –
762. – ISSN 0022–460X
[6] Antoulas, A.C.: Approximation of large-scale dynamical systems. En: in Advances
in design and control. SIAM Number 6 (2005)
[7] Astrid, P.: Reduction of Process Simulation Models: a proper orthogonal decomposition approach. En: PhD thesis, , Technishche Universiteit Eindhoven (2004)
[8] Atwell, J. A. ; King, B. B.: Proper orthogonal decomposition for reduced basis
feedback controllers for parabolic equations. En: Mathematical and Computer Modelling
33 (2001), Nr. 1-3, p. 1 – 19. – Computation and control VI proceedings of the sixth
Bozeman conference. – ISSN 0895–7177
[9] Beers, K.: Numerical Methods for Chemical Engineering. Massachusetts, United
States : Cambridge University Press, 2007
[10] Bendersky, E. ; Panagiotis, D.: Optimization of transport reaction processes using
nonlinear model reduction. En: Chemical Engineering Science 55 (2000), p. 4349–4366
Bibliography
59
[11] Berkooz, G. ; Holmes, P. ; Lumley, J.L.: The proper orthogonal decomposition in
the analysis of turbulent flows. En: Fluid Mech 25 (1993), p. 539–575
[12] Chelidze, David ; Zhou, Wenliang: Smooth orthogonal decomposition-based vibration mode identification. En: Journal of Sound and Vibration 292 (2006), Nr. 3-5, p.
461 – 473. – ISSN 0022–460X
[13] Cherniha, Roman: New Q-conditional symmetries and exact solutions of some
reaction-diffusion-convection equations arising in mathematical biology. En: Journal
of Mathematical Analysis and Applications 326 (2007), Nr. 2, p. 783 – 799. – ISSN
0022–247X
[14] Comon, P.: Independent component analysis: a new concept. En: Signal processing
36 (1994), p. 287–314
[15] Courant, R. ; Isaacson, E. ; Rees, M.: On the solution of non-linear hyperbolic
differential equations by finite differences. En: Communications on Pure and Applied
Mathematics 5 (1952), Nr. 243
[16] Courant, R.and K.0. F. ; Lewy, H.: Uber die partiellen differenzengleichungen der
mathematischen physik. En: Mathematische Annalen 100 (1928), Nr. 32
[17] Druault, Philippe ; Delville, Joël ; Bonnet, Jean-Paul: Proper Orthogonal Decomposition of the mixing layer flow into coherent structures and turbulent Gaussian
fluctuations. En: Comptes Rendus Mécanique 333 (2005), Nr. 11, p. 824 – 829. – ISSN
1631–0721
[18] Feeny, B. F. ; Liang, Y.: Interpreting proper orthogonal modes of randomly excited
vibration systems. En: Journal of Sound and Vibration 265 (2003), Nr. 5, p. 953 – 966.
– ISSN 0022–460X
[19] Fogler, S.: Elements of Chemical Reaction Engineering. Fourth. Massachusetts,
United States : Prentice Hall, 2008
[20] Galvanetto, Ugo ; Violaris, George: Numerical investigation of a new damage
detection method based on proper orthogonal decomposition. En: Mechanical Systems
and Signal Processing 21 (2007), Nr. 3, p. 1346 – 1361. – ISSN 0888–3270
[21] Gilliam, Xiaoning ; Dunyak, James P. ; Smith, Douglas A. ; Wu, Fuqiang: Using
projection pursuit and proper orthogonal decomposition to identify independent flow
mechanisms. En: Journal of Wind Engineering and Industrial Aerodynamics 92 (2004),
Nr. 1, p. 53 – 69. – ISSN 0167–6105
60
Bibliography
[22] Gonçalves, P.B. ; Silva, F.M.A. ; Prado, Z.J.G.N. D.: Low-dimensional models for
the nonlinear vibration analysis of cylindrical shells based on a perturbation procedure
and proper orthogonal decomposition. En: Journal of Sound and Vibration 315 (2008),
Nr. 3, p. 641 – 663. – EUROMECH colloquium 483, Geometrically non-linear vibrations
of structures. – ISSN 0022–460X
[23] Hömberg, D. ; Volkwein, S.: Control of laser surface hardening by a reduced-order
approach using proper orthogonal decomposition. En: Mathematical and Computer
Modelling 38 (2003), Nr. 10, p. 1003 – 1028. – ISSN 0895–7177
[24] Huisman, L.: Control of Glass Melting Processes Based on Reduced CFD models. En:
PhD thesis, , Technishche Universiteit Eindhoven (2005)
[25] John, Volker ; Schmeyer, Ellen:
Finite element methods for time-dependent
convection-diffusion-reaction equations with small diffusion. En: Computer Methods
in Applied Mechanics and Engineering 198 (2008), Nr. 3-4, p. 475 – 494. – ISSN
0045–7825
[26] Karhunen, K.: Zur spektral Theorie stochastischer Prozesse.
Sci.Fenicae (1946)
En: Ann. Acad.
[27] Katayama, Tohru ; Kawauchi, Hidetoshi ; Picci, Giorgio: Subspace identification
of closed loop systems by the orthogonal decomposition method. En: Automatica 41
(2005), Nr. 5, p. 863 – 872. – ISSN 0005–1098
[28] Khalil, Mohammad ; Adhikari, Sondipon ; Sarkar, Abhijit: Linear system identification using proper orthogonal decomposition. En: Mechanical Systems and Signal
Processing 21 (2007), Nr. 8, p. 3123 – 3145. – ISSN 0888–3270
[29] Kwasniok, F.: The reduction of complex dynamical systems using principal interaction
patterns. En: Physica D 92 (1996), p. 28–60
[30] Kwasniok, F.: Optimal Galerkin approximations of partial differential equations using
principal interaction patterns. En: Physica D (1997), p. 5365–5375
[31] Leibfritz, F. ; Volkwein, S.: Reduced order output feedback control design for
PDE systems using proper orthogonal decomposition and nonlinear semidefinite programming. En: Linear Algebra and its Applications 415 (2006), Nr. 2-3, p. 542 – 575.
– Special Issue on Order Reduction of Large-Scale Systems. – ISSN 0024–3795
[32] Li, M. ; Panagiotis, D.: Optimal Transition Control of diffusion-Convection- reaction
processes. En: 8th International IFAC Symposium on Dynamics and Control of Process
System (2007)
Bibliography
61
[33] Loeve, M.: Fonctions aleatoire de second ordre. En: C. R. Acad. Sci. Paris (1945)
[34] Logist, F.: Model based optimization and control of chemical reactors with distributed
parameters. En: PhD thesis, Katholieke Universiteit Leuven (2008)
[35] Lumley, J.L.: Stochastic tools in turbulence. En: Applied mathematics and mechanics
12 (1970)
[36] Ly, Hung V. ; Tran, Hien T.: Modeling and control of physical processes using proper
orthogonal decomposition. En: Mathematical and Computer Modelling 33 (2001), Nr.
1-3, p. 223 – 236. – Computation and control VI proceedings of the sixth Bozeman
conference. – ISSN 0895–7177
[37] Mahgerefteh, Haroun ; Rykov, Yuri ; Denton, Garfield: Courant, Friedrichs and
Lewy (CFL) impact on numerical convergence of highly transient flows. En: Chemical
Engineering Science 64 (2009), Nr. 23, p. 4969 – 4975. – ISSN 0009–2509
[38] Mayne, D. Q. ; Rawlings, J. B. ; Rao, C.V. ; M.Scokaert., P. O.: Constrained
model predictive control: Stability and optimality. En: Automatica 36 (2000), p.
789–814
[39] Padhi, Radhakant ; Balakrishnan, S. N.: Proper orthogonal decomposition based
optimal neurocontrol synthesis of a chemical reactor process using approximate dynamic
programming. En: Neural Networks 16 (2003), Nr. 5-6, p. 719 – 728. – Advances in
Neural Networks Research: IJCNN ’03. – ISSN 0893–6080
[40] Panagiotis, D. ; Prodromus, D.: Nonlinear Control of Diffusion-ConvectionReaction Processes. En: Computers Chemical 20 (1996), p. S1071–S1076
[41] Rahal, S. ; Cerisier, P. ; Azuma, H.: Application of the proper orthogonal decomposition to turbulent convective flows in a simulated Czochralski system. En: International Journal of Heat and Mass Transfer 51 (2008), Nr. 17-18, p. 4216 – 4227. – ISSN
0017–9310
[42] Ramos, J.I.: Numerical methods for nonlinear second-order hyperbolic partial differential equations. I. Time-linearized finite difference methods for 1-D problems. En: Applied
Mathematics and Computation 190 (2007), Nr. 1, p. 722 – 756. – ISSN 0096–3003
[43] Ravindran, S. S.: Control of flow separation over a forward-facing step by model
reduction. En: Computer Methods in Applied Mechanics and Engineering 191 (2002),
Nr. 41-42, p. 4599 – 4617. – ISSN 0045–7825
[44] Ravindran, S.S.: Optimal boundary feedback flow stabilization by model reduction.
En: Computer Methods in Applied Mechanics and Engineering 196 (2007), Nr. 25-28,
p. 2555 – 2569. – ISSN 0045–7825
62
Bibliography
[45] Rodrigues, M.A. ; Odloak, D.: MPC for stable linear systems with model uncertaintly. En: Automatica 39 (2003), p. 569–583
[46] Rodrigues, M.A. ; Odloak, D.: MPC for stable linear systems with model uncertainty. En: Automatica 39 (2003), p. 569–583
[47] Rowley, C. ; Colonius, T. ; Murray, R.: Model Reduction for compressible flows
using POD and Galerkin projection. En: Physica 189 (2003), p. 115–129
[48] Sinha, Rajen K. ; Geiser, Jürgen: Error estimates for finite volume element methods
for convection-diffusion-reaction equations. En: Applied Numerical Mathematics 57
(2007), Nr. 1, p. 59 – 72. – ISSN 0168–9274
[49] Solari, G. ; Tubino, F.: A turbulence model based on principal components. En:
Probabilistic Engineering Mechanics 17 (2002), Nr. 4, p. 327 – 335. – ISSN 0266–8920
[50] Tabib, Mandar V. ; Joshi, Jyeshtharaj B.: Analysis of dominant flow structures and
their flow dynamics in chemical process equipment using snapshot proper orthogonal
decomposition technique. En: Chemical Engineering Science 63 (2008), Nr. 14, p. 3695
– 3715. – ISSN 0009–2509
[51] Tang, Shimin ; Wu, Jianghang ; Cui, Maochang: The nonlinear convection-reactiondiffusion equation for modelling El Niño events. En: Communications in Nonlinear
Science and Numerical Simulation 1 (1996), Nr. 1, p. 27 – 31. – ISSN 1007–5704
[52] Utturkar, Yogen ; Zhang, Baoning ; Shyy, Wei: Reduced-order description of fluid
flow with moving boundaries by proper orthogonal decomposition. En: International
Journal of Heat and Fluid Flow 26 (2005), Nr. 2, p. 276 – 288. – ISSN 0142–727X
[53] Wang, Yuan-Ming ; Lan, Xiao-Lin: Higher-order monotone iterative methods for finite
difference systems of nonlinear reaction-diffusion-convection equations. En: Applied
Numerical Mathematics 59 (2009), Nr. 10, p. 2677 – 2693. – ISSN 0168–9274
[54] Xie, Weiguo ; Bonis, Ioannis ; Theodoropoulos, Constantinos: Off-line Model
Reduction for On-line Linear MPC of Nonlinear Large- Scale Distributed Systems. En:
Computers & Chemical Engineering In Press, Accepted Manuscript (2011), p. –. – ISSN
0098–1354
[55] Xu, Chao ; Ou, Yongsheng ; Schuster, Eugenio: Sequential linear quadratic control
of bilinear parabolic PDEs based on POD model reduction. En: Automatica 47 (2011),
Nr. 2, p. 418 – 426. – ISSN 0005–1098

Reducción de modelos dinámicos usando POD para sistemas con

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