Tuning the parameters of the Model Predictive Control (MPC) of an industrial Crude Distillation Unit (CDU) is considered here. A realistic scenario is depicted where the inputs of the CDU system have optimizing targets, which are provided by the Real Time Optimization layer of the control structure. It is considered the nominal case, in which both the CDU model and the MPC model are the same. The process outputs are controlled inside zones instead of at fixed set points. Then, the tuning procedure has to define the weights that penalize the output error with respect to the control zone, the weights that penalize the deviation of the inputs from their targets, as well as the weights that penalize the input moves. A tuning approach based on multi-objective optimization is proposed and applied to the MPC of the CDU system. The performance of the controller tuned with the proposed approach is compared through simulation with the results of an existing approach also based on multi-objective optimization. The simulation results are similar, but the proposed approach has a computational load significantly lower than the existing method. The tuning effort is also much lower than in the conventional practical approaches that are usually based on ad-hoc procedures.

1. Research Article
Tuning the Model Predictive Control of a Crude Distillation Unit
André Shigueo Yamashita a,n
, Antonio Carlos Zanin b
, Darci Odloak a
a
Department of Chemical Engineering, University of São Paulo, Av. Prof. Luciano Gualberto, trv 3 380, 05424-970 São Paulo, Brazil
b
Petrobras S.A., Center of Excellence for Technology Application in Industrial Automation, São Paulo, SP, Brazil
a r t i c l e i n f o
Article history:
Received 23 March 2015
Received in revised form
11 September 2015
Accepted 19 October 2015
Available online 6 November 2015
Keywords:
MPC tuning algorithm
MPC with input targets and zone control
Crude Distillation Unit
a b s t r a c t
Tuning the parameters of the Model Predictive Control (MPC) of an industrial Crude Distillation Unit
(CDU) is considered here. A realistic scenario is depicted where the inputs of the CDU system have
optimizing targets, which are provided by the Real Time Optimization layer of the control structure. It is
considered the nominal case, in which both the CDU model and the MPC model are the same. The
process outputs are controlled inside zones instead of at fixed set points. Then, the tuning procedure has
to define the weights that penalize the output error with respect to the control zone, the weights that
penalize the deviation of the inputs from their targets, as well as the weights that penalize the input
moves. A tuning approach based on multi-objective optimization is proposed and applied to the MPC of
the CDU system. The performance of the controller tuned with the proposed approach is compared
through simulation with the results of an existing approach also based on multi-objective optimization.
The simulation results are similar, but the proposed approach has a computational load significantly
lower than the existing method. The tuning effort is also much lower than in the conventional practical
approaches that are usually based on ad-hoc procedures.
& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
The petrochemical and oil processing industries usually take
advantage of advanced control platforms to improve the product
quality and yield, as well as to optimize the operation and to
reduce unnecessary costs. The first MPC application was reported
back in the late 1970's, and its formulation has become more
complex, allowing it to account for output zone control, input
optimizing targets [1], nominal stability and robustness to model
uncertainty [2] since then. However, MPC tuning guidelines,
aiming at the trade-off between performance and robustness, are
still scarce. The literature usually separates the tuning techniques
in two major classes. The first one considers the techniques based
on system approximations, heuristic definition of certain variables
and sensitivity functions to calculate analytical equations for the
tuning parameters. For example, [3] calculated analytical expres-
sions for the weights on the control efforts assuming, in the tuning
step, that the system is approximated by first order plus dead time
transfer functions and that the conditioning number of the
dynamic matrix of the control problem is 500; in [4], the Small
Gain theorem was applied to draw general guidelines for the
tuning parameters of a MPC in model mismatch scenario; [5]
developed an analytical expression for the input moves weighting
matrix, R, based on the performance observation of a DMC in
closed-loop with a first-order process.
The second class contains the tuning techniques in which a
multi-objective optimization problem is defined by the tuning
goals. Some examples are: the controller matching strategy [6], to
use different parameterizations of the multi-objective problem
and to calculate sets of Pareto solutions [7], to solve the multi-
objective tuning problem using evolutionary algorithms such as
particle swarm optimization [8] or a genetic algorithm [9]. In
[10,11], the tuning problem was posed as an optimization problem,
in which the coefficients of a polynomial that represents the
controller are obtained through the minimization of the distance
between the adopted polynomial and a characteristic polynomial
obtained through the specification of a desired output response. In
[10], it was considered a similar goal definition procedure, but it
was adopted a goal attainment oriented multi-objective optimi-
zation algorithm to obtain the optimum values of the weights of
the input moves (R) and output errors (Qy) for a conventional MPC
controller. It was proposed in [13] a frequency domain based
approach to define the tuning problem as the minimization of a
mixed sensitivity function to obtain the optimum R. A similar
approach was proposed in [12], in which the effects of R was
studied in scenarios where uncertainties affect the system inputs
and outputs. In [7] the MPC tuning problem was defined as a
multi-objective optimization problem, which can be solved
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2015.10.017
0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author. Tel.: þ55 1130912261.
E-mail addresses: andre.yamashita@usp.br (A.S. Yamashita),
zanin@petrobras.com.br (A.C. Zanin), odloak@usp.br (D. Odloak).
ISA Transactions 60 (2016) 178–190

2. considering different techniques to obtain a discretized set of
Pareto optimum solutions, from which a final solution is selected.
To the best of our knowledge, the literature has not addressed
the tuning of a MPC with output zone control and input targets.
Moreover, it is sought a practical and straightforward method,
capable of taking into account tuning goals defined in terms of
reference trajectories. The survey reported in [13] showed that the
majority of the MPC suppliers and its customers in industry regard
the required manpower for the controller tuning as the most
important cost factor; moreover, 30% of the survey responders
mentioned that MPC commissioning and tuning is among the top
three most relevant cost factors. The particular development of a
tuning framework for a Crude Distillation Unit is welcome because
such process units play a major role in oil refineries [14]. Moreover,
the CDU is a system with several inputs and outputs, and coupled
variables, which makes it a challenging control problem.
In this work, we try to fill a gap in the tuning literature by
providing a case study in which a compromise optimization based
technique is proposed to tune the MPC of an industrial Crude
Distillation Unit. Section 2 presents the MPC formulation tuned
here. Section 3 defines the proposed multi-objective optimization
based tuning framework. Section 4 introduces the CDU system
from Petrobras, Brazil, in which the controller is implemented. In
Section 5, it is discussed the strategy of definition of the tuning
goals used here, it summarizes the tuning results and presents
simulations of the CDU system for the different operational con-
ditions that were considered to validate the method. The paper
closes with some conclusions in Section 6.
2. The MPC with input targets and zone control problem
The MPC studied here is based on the state-space model
description given below:
x kþ1ð Þ ¼ Ax kð ÞþBΔu kð Þ
y kð Þ ¼ Cx kð Þ ð1Þ
where x kð ÞAℜnx
is the state vector at time instant k,y kð ÞAℜny
and
Δu kð ÞAℜnu
Δu kð Þ ¼ uðkÞÀuðkÀ1Þ
À Á
are the system outputs and
input increments at time instant k; matrices A AℜnxÂnx
; BA
ℜnxÂnu
; C AℜnyÂnx
are the model matrices, defined according to
the formulation presented in [15].
The MPC cost function takes into account the deviation
between the predicted system outputs, calculated using (1) over
the prediction horizon, and the output set points; the penalization
of the input increments and the deviation between the system
inputs and the input targets, both over the control horizon. Then,
the MPC cost function can be written as follows:
V1;k ¼
Xp
i ¼ 1
‖yðkþi k
ÞÀysp;k‖2
Qy þ
XmÀ 1
i ¼ 0
Δu kþi k
Þ 2
R
À
þ
XmÀ1
i ¼ 0
‖uðkþi k
ÞÀudes;k‖2
Qu
ð2Þ
where p is the prediction horizon, y kþijkð ÞAℜny
, Δu kþi k
Þ
À
Aℜnu
, and u kþijkð ÞAℜnu
are, respectively, the system outputs,
input increments and inputs at time instant kþi, calculated using
information available up to time instant k and the model defined
in (1). The parameter udes;k Aℜnu
is an input target, assumed to be
defined by an upper layer of the control structure and ysp;k Aℜny
is
the output set point that is an additional decision variable of the
control problem; assumed to be restricted within a control zone.
Qy AℜnyÂny
, Qu AℜnuÂnu
and RAℜnuÂnu
are positive definite dia-
gonal weighting matrices.
The MPC considered here is defined through the solution to the
following problem:
Problem 1
min
Δuk;ysp
V1;k ð3Þ
subject to (1) and
ÀΔumax rΔuðkþj kÞrΔumax; j ¼ 0; 1; :::; mÀ1
ð4Þ
umin ruðkþj kÞrumax; j ¼ 0; 1; :::; mÀ1
ð5Þ
ymin rysp;k rymax ð6Þ
where Δumax Aℜnu
, umin Aℜnu
, umax Aℜnu
, ymin Aℜny
and ymax Aℜny
are the input increment, input and output bounds, respectively.
The MPC approach assumes the rolling horizon strategy [16], in
which Problem 1 is solved at a time instant k, and the input incre-
ment corresponding to the first time step of the control horizon
value is fed to the real system represented in (1). The new system
inputs and outputs are obtained and the procedure is repeated at
time instant kþ1. The MPC formulation used here is a simplified,
finite horizon version of the one proposed in [15], for open-loop
stable systems.
In order to reduce the computational time required to tune the
MPC defined in Problem 1, the analytical solution to Problem 1 was
employed in the tuning algorithm developed here, disregarding
constraints (4)–(6) and adopting ysp as a fixed parameter instead of
a decision variable.
The effects of the control and prediction horizons on the per-
formance of the closed-loop system with MPC have been studied
to some extent and we can find in the literature satisfactory tuning
guidelines for parameters m and p [17,18]. In this work, we set p
and m to pre-defined values and focus on the selection of opti-
mum values for the weighting matrices Qy, Qu and R. In this
fashion, a tuning framework based on multi-objective optimiza-
tion is proposed, in which the tuning objectives are selected with
respect to the deviation between the closed-loop input and output
responses and reference trajectories. Finally, the decision variables
of the optimization problem are the matrices Qy, Qu and R,
assumed to be time-invariant and diagonal.
3. Multi-objective optimization
Before posing the tuning framework developed for the MPC
with input targets and zone control, brief considerations about
general multi-objective optimization problems are presented.
These methods solve problems with competing goals. There are
two main alternatives to deal with the trade-off between diverging
objectives: properly weighting the objectives prior to the problem
solution or choosing an optimum solution based on subjective
criteria, from a set of Pareto, or non-dominated optimum solu-
tions. A general multi-objective problem can be posed as follows:
Problem 2
min
x
F xð Þ ¼ F1 xð Þ F2 xð Þ ⋯ Fw xð Þ
 ÃT
ð7Þ
subject to
gj xð Þr0; j ¼ 1; …; z ð8Þ
hl xð Þ ¼ 0; l ¼ 1; …; e ð9Þ
where F xð Þ is a vector of w objectives Fi xð Þ. Functions gj xð Þ and hl xð Þ
are related with the inequality and equality constraints, respec-
tively, xAℜndec
is the vector of decision variables and ndec is the
number of decision variables. The feasible design space is defined
as X ¼ fxAℜndec
gj xð Þr0; j ¼ 1; …; z and hl xð Þ ¼ 0; l ¼ 1; …; eg
, and
the feasible criterion space is defined as Z ¼ zAℜw
jz
È
¼ F xð Þ; xAXg. The objectives Fi xð Þ are defined in terms of
A.S. Yamashita et al. / ISA Transactions 60 (2016) 178–190 179

3. preferences, imposed by the decision-maker. The following state-
ments characterize the optimum solutions in the multi-objective
optimization problem.
Definition 1. A point xÃ
AX is a Pareto optimum, or a non-
dominated solution, iff there does not exist another point xAX,
such that F xð ÞrF xÃ
ð Þ, and Fi xð ÞoFi xÃ
ð Þ for at least one i.
Definition 2. A point F1
xð ÞAZ is an Utopia point iff for each i¼1,…,
w, Fi1 xð Þ ¼ min
x
Fi xð ÞjxAΧ
È É
.
The previous statements formally define a multi-objective
optimization problem. It has been proposed in the literature sev-
eral alternatives to solve such problem, and compromise optimi-
zation is an attractive one because it is relatively simple and does
not require the selection of a single optimum from a set of non-
dominated solutions.
3.1. Multi-objective tuning technique based on compromise
optimization
The MPC tuning method proposed here is based on a com-
promise multi-objective optimization approach. The goals are
defined as the difference between the closed-loop responses with
the proposed controller and previously defined reference profiles.
The closed-loop response is calculated solving the unconstrained
MPC. The reference trajectories are specified either as the step
response approximations or linear profiles. The following subsec-
tions provide the steps of the proposed Compromise Tuning
Technique (CTT). The authors recommend to normalize the system
inputs and outputs, as well as the model transfer function gains to
avoid numerical problems.
3.1.1. Tuning goals definition
The MPC of the CDU system considered here, must yield good
performance at the usual operating scenarios, which consider that
there are input targets that must be tracked in order to optimize
the process operation. At the same time, the controller must be
tuned so that the process outputs shall be kept inside their control
zones. To cope with these requirements, a set of goals or objectives
is formulated aiming to emulate the typical operating require-
ments of the controller.
3.1.2. Output goals
To define the goals associated with the tracking of targets by the
process outputs, first, we assume that an output priority list is defined
in terms of their relative importance. This is reasonable because in
most real operations, some outputs are considered more important
than others, based on economic, operational or other criteria. Next, an
input variable is assigned to each output; the repetition of inputs is
not prohibited. Finally, first order transfer functions are selected to
approximate the real transfer functions for each pair defined pre-
viously. Their time constants are multiplied by response factors that
take into account the output priority. Response factors lower than one
means a reference trajectory faster than the approximated open-loop
response. Observe that the above tuning goal definition assumes that
the importance of a system output is directly related to a fast closed-
loop dynamics, however, it is reasonable to expect a slow response as
goal instead. The output oriented goal definition is defined as follows:
Fi xð Þ ¼
Xθt
k ¼ 1
yref
i kð ÞÀyi kð Þ
2
i ¼ 1; …; w ð10Þ
θt is the tuning horizon, yref
i kð Þ and yi kð Þ are respectively the
reference output trajectory and the closed-loop output trajectory
for output i at time step k.
3.1.3. Input goals
Considering that the Real Time Optimization is performed in an
upper layer of the control structure, economic tuning goals can be
defined in terms of input reference trajectories as well. In this case,
two possibilities need to be considered: the presence of input
optimizing targets and the need to compensate the effects of
unmeasured disturbances that tend to force the controlled outputs
outside their control zones. In the first case, the dynamics of the
desired trajectory is defined in terms of the value of Δumax, while
the ideal trajectory in the latter case is a constant value or set point
for the input. Associated with the input tracking, we need to
consider the possibility of an output bound being reached if the
input target is unreachable. Then, the input oriented tuning goals
are defined according to (11).
Fi xð Þ ¼
Xθt
k ¼ 1
ysp
i Àyi kð Þ
À Á2
þ
Xθt
k ¼ 1
uref
i kð ÞÀui kð Þ
2
i ¼ 1; …; w ð11Þ
where θt is the tuning horizon, uref
i kð Þ and ui kð Þ are respectively the
input reference value and the input closed-loop value, at time
instant k and ysp
i is a fixed set point representing an activated
bound. Observe that if Qy ¼ diagðqy;1; :::; qy;nyÞ, Qu ¼ diag qu;1; :::;
À
qu;nuÞ and R ¼ diag r1; :::; rnuð Þ, then the vector of decision variables
of the tuning problem is x ¼ qy;1; :::; qy;ny; qu;1;
:::; qu;nu; r1; :::; rnuÞ.
Also, since yi kð Þ and ui kð Þ are the output and input trajectories of
the system in closed loop, obviously they can be considered as
explicit functions of x.
3.1.4. Solving the multi-objective optimization problem
The compromise optimization is based on finding the closest
feasible solution, in terms of the Euclidian distance, to the Utopia
solution of a multi-objective optimization problem. Fig. 1 illus-
trates the methodology for a problem with two objectives.
First, the Utopia solution is obtained as the solution to the
Problem 3a) defined as follows:
Problem 3a)
Fi1 xð Þ ¼ min
x
Fi xð Þ; i ¼ 1; …; w ð12Þ
subject to (10), (11) and
LBrxrUB ð13Þ
where xAℜny þ2nu
is the vector of decision variables and vectors LB
and UB are its upper and lower bounds. The Utopia solution vector
is defined as F1 ¼ F1 1ð Þ ⋯ F1 wð Þ
 Ã
, F1Aℜw
, which is unfeasible
unless all the goals share the same solution.
Observe that, according to Definition 2, a total of w optimiza-
tion problems are defined, and each problem takes into account
Fig. 1. Geometric representation of the compromise optimization method con-
sidering two objectives.
A.S. Yamashita et al. / ISA Transactions 60 (2016) 178–190180

4. one objective at a time, optimizing all the decision variables
simultaneously.
Once F1
is available, we proceed by calculating the compromise
solution F xð Þ, xAX that is the closest feasible point to the Utopia
solution, in terms of the Euclidian distance. The following problem
is solved:
Problem 3b)
min
x
‖F1
ÀF xð Þ‖2
ð14Þ
subject to (10) or (11) and (13).
Before applying the method proposed here to tune the MPC in
closed-loop with the Crude Distillation Unit, the system is briefly
described in the next session. The identified operating character-
istics allow for a comprehensive definition of the tuning goals. A
later section will illustrate the solution of the tuning problem.
4. The Crude Distillation Unit
The CDUs are arguably one of the most fundamental, complex
and energy-intensive processes in oil refineries [19,20]. It is also
one of the most common separation process units in the chemical
industry [14]. Moreover, all the crude oil must initially go through
a CDU before being further processed in subsequent refining units.
Even though chemical reactions do not take place along the dis-
tillation process, the amount of mass and energy involved, the
number of equipment, the environmental and operational con-
straints, and the conflicting financial goals are sufficient to make
the CDU system a challenging control problem.
The system studied here is a process unit at the RECAP refinery
of Petrobras, located in São Paulo, Brazil that processes about
50,000 bbl/day of light crude oil. Fig. 2 shows a schematic repre-
sentation of the process system, including the relevant process
equipment, streams and regulatory control loops. The crude pre-
heating system is composed of one preheating train with two
desalters operating in series. The pre-flash column N-507 sepa-
rates light naphtha and light diesel from the crude. Its bottom
product is preheated in two parallel trains of heat exchangers and
is partially vaporized in the furnace L-506, before being fed to the
atmospheric fractionator N-506, which produces: heavy naphtha,
kerosene, heavy diesel and the atmospheric residue. Kerosene and
heavy diesel are blended to produce the commercial diesel pro-
duct. The bottom product is sent directly to a Residue Catalytic
Cracking Unit, not considered here.
The economic objectives of the CDU, in order of priority, are:
(i) Maximize the diesel production by minimizing the flow rates of
heavy naphtha and atmospheric residue while enforcing that the
main product specifications are satisfied (minimum bound on the
naphtha flash point and a maximum bound on the diesel ASTM D-
86 distillation curve 95%). (ii) Supply the required amount of light
naphtha that feeds the solvent unit. (iii) Minimize the flow rate of
stripping steam injected into the system. (iv) Minimize the con-
sumption of fuel oil at the preheating furnace.
Based on the economic objectives defined above, related to the
profit optimization, the unit operational goals are listed as follows:
(1) The crude oil flow rate (u1) should be maximized in all the
scenarios. (2) The flow rate of the stripping steam to the pre-flash
column (u3) should be driven to its target that is calculated by the
RTO layer. (3) The light naphtha outlet stream flow rate (y9) and
the ASTM D-86 end point (y7) are important controlled variables
to ensure the appropriate quantity and quality of the light naphtha
sent to the solvent production unit. (4) The pre-flash overhead
temperature set-point (u2) is a manipulated variable with direct
influence on the light naphtha quality, but it is constrained by the
top reflux flow rate (y8). (5) Two properties of the diesel must be
enforced as constraints, its flash point (y6) and its ASTM D86 95%
(y5). The former is important in the minimization of the fractio-
nator top temperature (u4). This minimization is constrained by
the minimum bound on the heavy naphtha flow rate (y2) and by
the maximum bound on the reflux flow rate (y3). The diesel ASTM
D86 95% constraint tends to become active when the outlet tem-
perature of the crude furnace (u8) is maximized, which is also
constrained by the maximum bound on the furnace heat duty
(y10). (6) The pumparound flow rate (u6) should be maximized in
order to save fuel in the furnace. It is constrained by the ratio
between kerosene and heavy diesel flow rates that can be repre-
sented by the kerosene withdraw temperature (y4). (7) The diesel
pumpdown reflux flow rate (u5) should be manipulated to
improve the fractionation between the heavy diesel and the
atmospheric residue.
The controlled and manipulated variables are illustrated in
Fig. 2, and are described in Tables 1 and 2, respectively, along with
their operational bounds and usual values. The output set points,
or control zones, are enforced directly by the MPC framework,
while the input targets are calculated at the RTO layer. The reader
can find the system transfer functions, obtained from fitting the
operating data of the CDU system by up to seventh order transfer
functions, in Appendix A.
Fig. 2. Schematic representation of the Crude Distillation Unit [21].
A.S. Yamashita et al. / ISA Transactions 60 (2016) 178–190 181

5. 5. Tuning the MPC of the CDU system
In the previous section, it was listed all the important inputs
that should be driven to their targets and the outputs that should
be kept inside their control zones. The MPC tuning analysis pre-
sented in this work will consider two different scenarios. The first
one will focus only at the output tracking performance of the
controller. This means that the weighting matrices Qy, and R are
selected solely based on the performance of the output responses.
The second scenario will consider output and input tracking
simultaneously, and tuning will focus on Qy, Qu, and R.
The CTT method proposed here is compared with a MPC tuning
technique from the literature that is also based on multi-objective
optimization. Hereafter this technique is addressed as the Normal
Boundary Intersection (NBI) approach. It is based on a posteriori
choice of the optimum solution from a set of non-dominated
solutions. The set of solutions is obtained through a grid search
over an evenly spaced parameterized unitary segment sið Þ for each
objective. The searching points lie on the (quasi-) normal direction
ÀλΦe
À Á
to a plane defined by the individual optimum solutions,
indicated by Φs. The following optimization problem is solved:
Problem 4
min
x AX;λ
Àλ ð15Þ
Subject to
ΦsÀλΦ1ny;1 ¼ F xð ÞÀF1
ð16Þ
si Z0;
Pσ
i ¼ 1
si ¼ 1. The i-th column of the pay-off matrix Φ is
defined as F xi1ð ÞÀF1.
The reader is referred to [7,22] and the references therein for
more information. The NBI method was implemented using an
increment of 0.5 to define the vector s. The decision regarding the
best Pareto solution was done following the weighted sum method
[23], in which a performance index named WSM is calculated
according to (17). The non-dominated solution that yields the
largest index value is selected as the best solution.
WSM ¼
Xw
j ¼ 1
aijf j; i ¼ 1; …; M ð17Þ
where w is the number of objectives, M is the number of non-
dominated solutions, fj is the j-th objective weight and aij is the
value of the cost function of the j-th objective calculated using the
i-th Pareto solution.
The objective weights of the output tuning approach are the
inverse of the response factors, defined in Section 5.1, i.e.
f output ¼ 1
f res
, and the input tuning approach weighting vector was
chosen arbitrarily as f input ¼ 1 2 3 4 5 6 7 8
 Ã
.
The two tuning approaches considered here consider the following
lower and upper bounds for the tuning parameters, which correspond
to Qy, Qu, and R: LB ¼ 1nyÂ1 1nuÂ1 Â 10À 3
1nuÂ1 Â 10À2
h i
,
UB ¼ 1nyÂ1 Â 20 1nuÂ1 Â 0:1 1nuÂ1 Â 102
h i
; and the tuning hor-
izon is θt ¼ 300 min.
The parameters originally implemented in the MPC of the real
CDU system are Qy ¼ diag 5 2 1 3 10 5 5 1 10 20
 Ã
,
Qu ¼ diag 1 1 1 1 1 1 1 1
 Ã
,
R ¼ diag 0:1 6 2:93 5 0:15 4 1 20
 Ã
.
The performance of the controller with this tuning set, which
was obtained by trial and error, based on economic factors and the
system knowledge of control engineers and operators, is addres-
sed here as ‘Existing’, is also compared to the other tuning sets
obtained here.
In both scenarios, the transfer function gains, inputs and out-
puts are normalized considering the input and output operating
ranges. The prediction horizon is chosen as p¼40 and the control
horizon as m¼5. These parameters were selected following the
literature guidelines and also based on the analysis of the simu-
lation results. The MPC control problem was solved analytically,
disregarding the constraints (Problem 1b). The CTT multi-objective
optimization problem (Problem 3) and the NBI problems (Problem
4) were solved using fmincon (default settings), MATLABs
2013, on
an Intels
Core i5, 3.20 Ghz, 4 Gb RAM computer.
5.1. Tuning the MPC for the output set point tracking scenario
The selection of the input–output pairs for the definition of the
reference trajectories is performed based on the input–output
relationship matrix represented in Table 4 and the RGA approach
[24]. Since the closed loop transfer functions relate the outputs
with their set points, all the gains are unitary, and the time con-
stants are obtained by multiplying the time constants of the
approximate transfer functions by the response factors defined
below:
f res ¼ 0:25 0:30 0:45 0:50 0:55 0:70 0:85 0:90 1:0 1:0
 Ã
:
Table 4 summarizes the transfer functions that define the
reference trajectory for each output. In the tuning scenario cor-
responding to the output tracking, matrix Qu is not included as a
decision variable of the tuning problem. Therefore, the objective
function of the MPC considered here does not include the term
corresponding to the input target. Also, ysp is considered as a pre-
defined parameter, instead of a decision variable of the MPC pro-
blem, because the main focus of this tuning step is to obtain a set
of parameters such that the MPC will be capable of adequately
driving the outputs back to their zones when a disturbance affects
the CDU system. To compute the values of function (10) that
characterizes the performance of the controller, the system starts
Table 1
CDU controlled outputs.
Tag Variable name Unit Bounds y0
y1 N-507 bottom stripping steam to reduced
crude ratio
kg/m3
1 4 1.03
y2 Heavy naphtha flow rate m3
/d 40 380 220
y3 N-506 top reflux flow rate m3
/d 800 1850 1670.41
y4 Kerosene withdraw temperature °C 180 190 184.38
y5 Diesel ASTM D-86 95% °C 368 370 368.6
y6 Diesel flash point °C 35 65 39.7
y7 Light naphtha ASTM D-86 end point °C 160 180 173.68
y8 N-507 top reflux flow rate m3
/d 600 1290 1050
y9 Light naphtha flow rate m3
/d 600 1342 911.62
y10 L-506 heat duty Gcal/h 15 21 17.7
Table 2
CDU manipulated inputs.
Tag Variable name Unit Bounds u0 Δumax udes
u1 Crude feed flow rate m3
/d 8000 8500 8000 20 8100
u2 N-507 top tempera-
ture set-point
°C 128 131 129.95 0.1 129.9
u3 N-507 bottom strip-
ping steam flow rate
t/h 0.3 1 0.3 0.03 0.5
u4 N-506 top tempera-
ture set-point
°C 105 110 108.54 0.1 109
u5 Heavy diesel pump-
down reflux flow rate
m3
/d 1000 1670 1380 10 1365
u6 Diesel pumparound
reflux flow rate
m3
/d 5800 6400 6400 8 6400
u7 Kerosene outlet flow
rate
m3
/d 900 1050 1050 6 1049.5
u8 L-506 outlet
temperature
°C 365 372 368.98 0.07 368.5
A.S. Yamashita et al. / ISA Transactions 60 (2016) 178–190182

6. from a steady state that corresponds to 50% of the nominal output
values and according input values, in which the outputs are all
inside of their control zones and the output set points are moved
to a value equal to 75% of their nominal value at time instant
1 min. At time instant 100 min, they are moved to values equal to
50% of the nominal values and finally, at time 200 min, the set
points are fixed at 25% of the nominal values.
The tuning parameters resulting from the application of the
proposed CTT method to this first scenario are the following:
Qy ¼ diag 1 2:07 1 1 1 2:93 8:3 20 10:2 1:34
 Ã
;
R ¼ diag 0:65 0:011 0:01 0:01 0:01 0:01 0:01 0:01
 Ã
and the tuning parameters obtained with the application of the
NBI method are the following:
Qy ¼ diag 1 20 1 1 1 1 1 20 1:54 4:41
 Ã
;
R ¼ diag 0:01 3:49 0:11 0:01 0:012 0:01 0:015 0:235
 Ã
We observe that there are significant differences between some
of the tuning parameters produced by the two methods. We also
observe that the existing tuning parameters of the CDU controller
are completely different from the parameters obtained here.
The performances of these three sets of parameters are com-
pared in a simulated scenario in which five outputs are driven to
different set points, following the pattern proposed in the tuning
section, while the remaining outputs are assumed to remain
within their control zones. Fig. 3 shows the closed-loop responses
of the selected outputs. It is also shown the responses of the
controller with the parameters corresponding to the utopia
solution.
From Fig. 3, we observe that the responses of the MPC with the
tuning parameters defined through the CTT method are closer to
the Utopia responses than the MPC tuned through the NBI
method. We also observe that the existing tuning parameters are
quite conservative and lead to more sluggish responses than the
tuning parameters obtained here. The controllers resulting from
the two tuning techniques analyzed here successfully tracked all
the output set points, but the existing controller failed to track the
set point moves of y3 within the simulation horizon considered
simulated example. The previous results are consistent with the
data shown in Table 5. The CTT set of tuning parameters yield the
overall lowest Sum of Squared Errors (SSE) for all the considered
outputs.
5.2. Tuning for the input target tracking scenario
As can be seen in Table 3, each input of the CDU system is
related to several outputs through non-zero transfer functions.
This multivariable character may result in a conflict between the
input targets and the output zones. This means that some of the
outputs may tend to be driven to outside their zones when the
MPC forces the inputs toward their targets. The tuning parameters
of the MPC should be selected such that this conflict is minimized
in the practical case. This means that if the input targets are
unreachable, the priority should be to keep the outputs inside the
control zones and allow for offset in the inputs. The following
paragraphs present some specific scenarios where an input is
driven to a target while one or more associated outputs tend to be
driven to the border of their control zones, becoming active con-
straints. The reference trajectories for the inputs are defined as
linear functions with angular coefficient corresponding to Δumax.
In the input optimization, the output constraints are included for
the outputs strongly correlated with the input. Observe that, here
the definition of the input tuning objective is more complex, and it
is heavily dependent on a practical knowledge of the process
system and its operating peculiarities. It is assumed that the out-
put initial value lies in the middle of its operating range and at
time instant 1 min, their set points are changed to 75% of their
nominal values. The inputs initial conditions and targets, in the
units shown in Tables 1 and 2 are
u0 ¼ 8000 128 1 110 1271:5 6400 1050 372
 ÃT
and udes ¼ 8125 128:75 0:825 108:75 1439:1
Â
6250 1012:5
370:25ŠT
:
Each of the cases described below considers one input with its
target. The outputs more heavily connected to this input and that
can reach a constraint are also considered one at a time, even
though, in practice, the input may be constrained by more than
one output simultaneously. In the definition of the tuning goal
related with each case, it is assumed that the output should be
driven to a set point, while the input should follow a trajectory
towards its target.
5.2.1. Objective I – flow rate of crude to the CDU
The target to the flow rate of crude oil (u1) is defined by the
RTO layer that tries to force the maximization of the flow rate of
diesel that is produced by the CDU. The outputs that may constrain
the input optimization are the light naphtha flow rate (y9) and the
furnace heat duty (y10). Only the latter is considered in the defi-
nition of this tuning objective.
5.2.2. Objective II – flow rate of diesel pumparound
The diesel pumparound flow rate (u6) is set by the RTO to
maximize the heat that is recovered in the crude oil preheating
trains and to save fuel oil/gas in the oil heating furnace. This input
is mainly constrained by the top reflux flow rate of the main
fractionator (y3).
5.2.3. Objective III – flow rate of stripping steam
The RTO layer computes an optimum target to the flow rate of
the stripping to the bottom of the pre-flash column (u3) in order to
maximize the flow rate of light naphtha that is produced by the
CDU. The related output constraints are the pre-flash reflux flow
Table 3
CDU input–output relationship matrix.
u1 u2 u3 u4 u5 u6 u7 u8
y1 x
y2 x x
y3 x x x x x
y4 x x x x
y5 x x x x
y6 x x x
y7 x x
y8 x x x
y9 x x x
y10 x x x x
Table 4
Reference trajectories transfer function time constants.
Output Input Time constant
y1 u3 1.1080
y2 u4 12.6710
y3 u6 1.0684
y4 u7 10.6011
y5 u5 8.3741
y6 u4 2.7598
y7 u2 8.1778
y8 u3 7.3582
y9 u1 32.0422
y10 u8 3.8556
A.S. Yamashita et al. / ISA Transactions 60 (2016) 178–190 183

7. rate (y8) and the ratio between the flow rate of the stripping steam
and the flow rate of reduced crude (y1). The former is chosen in
the definition of this objective.
5.2.4. Objective IV – temperature at the top of the pre-flash column
The set point to the controller of the temperature at the top of
the pre-flash column (u2) is set by the RTO to maximize the pro-
duction of light naphtha. The constraints associated with this
target are the bounds of the light naphtha flow rate (y9) and the
Fig. 3. Output tracking tuning analysis, CTT (solid line), NBI (dashed line), existing controller (dotted line), Utopia (dot-dashed line).
Table 5
Sum of Squared Errors (SSE).
Output tuning
y1 y3 Â 10À5
y5 y6 y7
CTT 2.38 7.59 1.56 536.97 208.96
NBI 3.24 7.22 2.05 838.90 631.74
Existing 4.57 128.77 1.30 1642.10 583.85
A.S. Yamashita et al. / ISA Transactions 60 (2016) 178–190184

8. maximum D-86 end point of the light naphtha (y7). Here, the
objective is formulated in terms of y9 only.
5.2.5. Objective V – temperature at the top of the atmospheric
column
The RTO sets a target to the set point of the temperature con-
troller at the top of the fractionator (u4) in order to maximize the
production of heavy naphtha. The constraints associated with this
input are the bounds on the heavy naphtha flow rate (y2) and the
bounds on the diesel flash point (y6). The objective pair is selected
in terms of y2
5.2.6. Objective VI – flow rate of heavy diesel pumpdown reflux
The target to the flow rate of the heavy diesel pumpdown
reflux in the fractionator (u5) is set by the RTO to maximize the
production of diesel. This target may be constrained by the diesel
ASTM D-86 95% (y5).
5.2.7. Objective VII – kerosene outlet flow rate
The kerosene outlet flow rate (u7) is also set by the RTO when
the diesel production is maximized. This target may be con-
strained by the temperature of the tray where the kerosene is
drawn from the atmospheric column (y4).
Fig. 4. Input tracking tuning analysis, CTT II-A (dashed line), CTT II-B (solid line), existing controller (dotted line), reference trajectory (dot-dashed red line).
A.S. Yamashita et al. / ISA Transactions 60 (2016) 178–190 185

9. 5.2.8. Objective VIII – furnace temperature
The temperature at the outlet of the crude heating furnace (u8)
is also set by the RTO. This target may be constrained by the diesel
ASTM D-86 95% specification (y5).
Two different approaches are proposed: in Scenario II-A, the
optimum values of Qy and R obtained in Section 5.1 are inherited
and used as fixed parameters instead of decision variables. In
Scenario II-B, all elements of matrices Qy, Qu and R are considered
as decision variables of the tuning problem.
The input tracking analysis compares the closed-loop responses
with the controller tuned through the CTT method using approa-
ches II-A and II-B to the controller with the existing parameters.
The following pairs were chosen to illustrate the closed-loop
performance of the MPC for the input tracking scenario: y3 Àu6,
y8 Àu3, and y10 Àu1, which represent the objectives II, III, and I,
respectively. The remaining outputs are assumed to be kept within
their control zones and other inputs do not have active targets. The
input references are also included in the figures.
From Fig. 4, we observe that the output responses of the con-
trollers tuned with the two CTT methods are fast and not too
different from each other, but the CTT II-A tends to produce an
oscillatory behavior, which is not desirable in practice. The per-
formance of the MPC based on the CTT II-B method is slightly
more sluggish but not oscillatory. It is observed in Table 6 that the
CTT II-A method in fact yields the smallest sum of square errors for
both outputs and inputs. Regarding the input reference trajec-
tories, none of the proposed tuning approaches yielded an out-
standing tracking. This shows that the CTT approaches prioritize
the control of the system outputs while the optimizing input tar-
gets will be tracked only after the outputs have been driven to
their set points. Comparing the values of the tuning parameters
obtained with CTT approach and the existing tuning parameters
that have been used in the real plant, in the existing controller all
the values of Qu are set equal to 1, which is quite large compared
to the values of Qu obtained here. The effect that is observed in
Fig. 4 is that with the existing controller the process response is
slower than in the controller tuned with CTT. The result will be a
slower output response if the input target is changed because the
RTO has computed a new optimum operating point. The con-
sequence is that we may lose the product specification for a sig-
nificant period of time and the expected economic benefit will be
jeopardized.
5.3. Additional remarks and simulations
Regarding the NBI method, when it is implemented in the
output tuning case, there are 10 objectives (one objective per
output), which require a total of 55 Pareto solutions. In the input
tuning case, the NBI method requires 8 additional objectives (one
per input), which lead to 36 Pareto solutions. Therefore, the
number of solutions to be considered in the NBI method is much
larger than the number of solutions that are needed to apply the
CTT method. In the NBI method, the increment interval of para-
meter s affects the distribution and spread of the Pareto solutions
over the Pareto curve. The adopted value of s in the imple-
mentation of the method was rather large as a smaller value
would require too many solutions and an unacceptable com-
puter time.
Table 6 shows the optimum tuning parameters obtained with
the CTT and NBI methods, whereas Table 7 summarizes the com-
putational time and the resulting minimum value of the tuning
cost function for each method. It is clear that the proposed method
(CTT) is much faster than the existing method of the MPC tuning
literature (NBI). It is also clear that the solution obtained with the
CTT method is better than the solution obtained with NBI, in terms
of the cost function value. In the input tracking scenario, the
adopted upper bound on Qu is 10 times lower than the lower
bound of Qy. With these constraints, we expect that the input
targets will be tracked only if the related outputs are lying in
their zones.
To wrap up the performance assessment of the tuning techni-
que developed here, we include a simulation, considering the
constrained MPC with input targets and zone control (Problem 1),
tuned according to the techniques CTT II-B and NBI II-B, which
yielded the best results in terms of the tuning cost function values
shown in Table 7, and the MPC with the existing tuning para-
meters. The system initial condition, output zone values and input
targets are the same given in Tables 1 and 2 and correspond to a
typical operating condition. It is assumed that the system starts
from a condition where the input target and all the computed
output set points lie in their operating ranges and, consequently,
the inputs are not constrained by the output zones. In this case,
the MPC can drive the system inputs to their targets without off-
set. This scenario can be observed in Figs. 4 and 5, from time
instant 0 until time instant 1000 min. In this case, all the inputs
reach their targets and the outputs stabilize in the control zones.
At time instant 1000 min, the target of input u5 is changed from
udes;5 ¼ 1365 m3
=d to udes;5 ¼ 1250 m3
=d, which is unreachable
because it corresponds to a steady state where y5 would lie above
its maximum bound. From Fig. 4, we observe that the output
responses of the closed loop system with the MPC tuned with the
proposed method are quite similar to the responses of MPC tuned
with the NBI method and the MPC with the existing parameters.
The three controllers try to drive u5 to its new target, which cannot
be reached because y5 is constrained by its maximum bound. We
observe that all the controllers with the different sets of tuning
parameters behave adequately in the sense that the maximum
bound on y5 is not surpassed, but inputs u4, u5, u7 and u8 that have
non-zero transfer functions relating them to y5, tend to show
offsets with respect to their targets. Fig. 5 shows that the three
controllers stabilize the system at a new steady state where the
Table 6
Sum of Squared Errors (SSE).
Input Tuning
y3 Â 10À5
y8 Â 10À4
y10 u1 Â 10À4
u3 u6 Â 10À5
II-A 1.56 4.44 3.98 7.2 0.03 2.87
II-B 2.92 6.72 15.83 12.63 0.06 6.2
Existing 5.81 5.52 12.08 17.05 0.13 2.16
Table 7
Tuning results for Scenario II-A and II-B.
CTT NBI
II-A II-B II-A II-B
Qu Qy Qu R Qu Qy Qu R
0.1 1 0.1 0.035 0.001 1 0.099 0.082
0.1 2.359 0.1 0.03 0.1 2.067 0.001 0.01
0.1 1 0.1 0.142 0.001 1.746 0.001 0.01
0.1 1 0.1 0.044 0.1 1.6 0.1 1.497
0.001 1 0.001 0.01 0.1 1.006 0.1 0.202
0.1 1 0.1 0.01 0.1 3.857 0.001 0.075
0.1 1.924 0.1 9.435 0.1 1 0.018 0.028
0.061 3.012 0.018 5.048 0.1 1 0.076 2.499
1 1
1 4.616
A.S. Yamashita et al. / ISA Transactions 60 (2016) 178–190186

10. offsets in the inputs are distributed differently by each controller.
We also observe that the responses of inputs u6 and u7 show some
large excursions for the controller tuned with the NBI method.
At time instant 2000 min, the target of input u1 is changed from
udes;1 ¼ 8100 m3
=d to udes;1 ¼ 8470 m3
=d. With this new target, the
controller tends to be constrained by the upper bound on output
y10. Again the three controllers try to drive the CDU system to an
optimum point that is unreachable, and the offset is distributed
between inputs u1, u2, u3 and u8, which are the inputs that are
related to y10. The MPC tuned with the method proposed here
behaves similarly to the existing MPC tuned with ad-hoc proce-
dures based on trial and error, which has an inferior performance
with respect to the response of y10 that tends to slightly surpass its
maximum bound. However, the former shows large excursions
upon the second input target change for u6 and u8, which is
attributed to the low value of the upper bounds on Qu. Again, the
MPC tuned with the NBI method shows unjustified large excur-
sions of inputs u2 and u3.
Table 8 summarizes the computational time elapsed to the
completion of the tuning techniques and the final cost function
values. It is observed that the proposed method always reaches a
lower cost function value than the NBI method. More importantly,
the CTT method is at least one order of magnitude faster than the
NBI method in all observed instances.
To further compare the performance of the proposed tuning
strategy with the method from the literature and the existing set
of tuning parameters, mathematical indexes were also included. In
this simulation, since the outputs remain within their ranges all
the time, it is not helpful to calculate the output sum of squared
errors. Then, the performance indexes employed in this analysis
are the sum of squared errors between inputs and input targets,
and the sum of the absolute values of control actions throughout
the simulation (SACA). The former index is useful to identify which
set of tuning parameters yielded the most economic MPC. The
indexes are presented in Tables 9 and 10. It is observed that the for
the NBI, the SSE for inputs u2, u6 and u7 is 3 orders of magnitude
higher than the respective values for the CTT. The existing set of
tuning parameters yields good performance in tracking the target
of u6. Regarding the SACA index, the values yielded by CTT, NBI and
the existing sets of tuning parameters are within the same order of
magnitude, except for inputs u2 and u7 for the NBI method, in
which the large excursions can be observed in Fig. 6.
6. Conclusions
The tuning problem of the MPC of a Crude Distillation Unit was
addressed. A tuning framework was proposed here, based on the
compromise solution of a multi-objective optimization problem
was developed to overcome the shortcomings of the current
approaches of the literature. The controller considered here is a
Fig. 5. Outputs of the CDU in closed loop simulation with MPC tuned with CTT II-B
(solid line), NBI II-B (dashed line), existing controller (dotted line) and bounds
(dashed-dotted lines).
Table 8
Tuning strategies comparison.
Method Elapsed time (h) Cost fuction value
Scenario I-outputs only
CTT 2.54 9.34
NBI 68.37 12.52
Scenario IIA-inputs only
CTT 0.69 237
NBI 24.5 385.7
Scenario IIB-outputs and inputs simultaneously
CTT 7.34 180.5
NBI 34.5 241.76
Table 9
Simulation Sum of Squared Errors (SSE).
u1 Â 10À6
u2 u3 u4 u5 Â 10À7
u6 u7 u8 Â 10À3
CTT 2.17 0.21 0.13 0.81 1.11 78.35 84.32 1.55
NBI 3.02 784.07 59.19 5.39 1.04 14200 6098 302.85
Existing 2.79 1.24 0.18 3.93 1 8.94 271.23 472.13
Table 10
Simulation sum of the absolute values of control actions (SACA).
u1 u2 u3 u4 u5 u6 u7 u8
CTT 475.31 0.18 0.23 0.62 98.97 12.16 3.15 1.94
NBI 477.15 3.75 0.89 0.65 89.68 17.12 22.82 1.96
Existing 475.68 0.25 0.22 0.70 108.24 3.41 8 1.99
A.S. Yamashita et al. / ISA Transactions 60 (2016) 178–190 187

11. Fig. 6. Inputs of the CDU in closed loop simulation with MPC tuned with CTT II-B (solid line), NBI II-B (dashed line), existing controller (dotted line) and targets (dashed-
dotted lines).
A.S. Yamashita et al. / ISA Transactions 60 (2016) 178–190188

12. finite horizon MPC with output zone control and input targets. The
method was compared to a similar multi-objective tuning
approach, based on an a posteriori solution. The latter was, in
average, 22 times more expensive in terms of computational effort
than the proposed method, which might be a remarkable differ-
ence for large scale systems, even though both techniques are off-
line. Two different tuning strategies were assessed: in the first one
(Scenario II-A), matrices Qy and R are tuned in a first step, con-
sidering the output tracking goals and in the second step, matrix
Qu is tuned, assuming that the optimum values of Qy and R are
inherited from the previous step and the input tracking goals are
addressed. The second tuning strategy (Scenario II-B) considers
the case in which Qy; Qu and R are tuned simultaneously, con-
sidering the input tracking goals. In both strategies, typical oper-
ating scenarios are defined, in which an output is assumed to act
as an active constraint to an input reference trajectory tracking.
The results showed that the methodology defined in Scenario II-A
might lead to faster but oscillatory responses. A simulation study
was performed, considering the output zones and input targets
defined in a real operating scenario of the CDU system. A quali-
tative analysis of the results showed that the CTT method yielded
similar responses to the existing set of tuning parameters that
were obtained by trial and error. The input tracking capability of
the latter was more efficient, but the former yielded better results
in the output tracking scenario analysis. The quantitative analysis
of mathematical indexes such as the sum of square errors between
inputs and their targets and the sum of absolute value of control
actions was in agreement with the previously mentioned obser-
vations, and large control action costs were obtained for the tun-
ing method from the literature. Nonetheless, the trial and error
approach, used to obtain the existing tuning parameters, is cum-
bersome and time consuming and therefore, should be used as a
supplementary tuning method, instead of in the early stages of the
MPC tuning procedure.
Acknowledgments
The authors would like to thank the financial support provided
by CNPq under Grant 140677/2011-9 and FUNDESPA under Grant
15015..
Appendix A
The omitted entries in Table A1 represent null transfer
functions.
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Table A1
CDU transfer functions
u1
y8 7:31Â10À 4
s5
À 1:82Â10À 3
s4
þ 1:49Â10 À 2
s3
þ 2:67Â10 À 3
s2
þ1:32Â10 À 3
sþ1:71Â10 À 4
s6 þ 2:18s5 þ 5:83s4 þ 2:14s3 þ 0:56s2 þ 0:03sþ 9:07Â10 À 4
y9 À 2:63Â10À 2
s6
À 0:17s5
þ 0:36s4
þ0:15s3
þ0:12s2
þ5:55Â10À 3
sþ8:86Â10 À 4
s7 þ 8:16s6 þ 29:45s5 þ 19:87s4 þ 7:98s3 þ 1:11s2 þ 0:10sþ 3:20Â10À 3
y10 À 1:88Â10À 4
s5
þ 1:11Â10 À 4
s4
þ1:53Â10 À 4
s3
þ8:67Â10À 5
s2
À 1:76Â10À 5
sþ 3:90Â10À 5
s6 þ3:83s5 þ 6:67s4 þ 2:84s3 þ 0:68s2 þ 0:12sþ 4:88Â10À 3
u2
y3 ðÀ 1:29s2
À 1:37sÀ2:13Þ eÀ 14s
s3 þ1:99s2 þ 0:99sþ0:19
y7 4:60Â10 À 3
s2 þ 0:06sþ 4:11Â10 À 3
s3 þ 0:39s2 þ 0:07sþ 2:93Â10À 3
y8 11:31s6
þ 102:90s5
À 58:29s4
À 171:20s3
À 362:10s2
À 159:70sÀ 56:27
s7 þ 8:84s6 þ 38:89s5 þ 71:07s4 þ 83:25s3 þ 53:24s2 þ 19:11sþ 3:16
y9 À 8:80s5 þ13:69s4 þ8:60s3 þ5:95s2 þ0:89sþ0:18
s6 þ 4:13s5 þ 4:94s4 þ 2:19s3 þ 0:65s2 þ 0:11sþ 7:57Â10 À 3
y10 À4:67Â10 À 3
s2 À5:59Â10À 4
sÀ6:76Â10À 4
s3 þ0:32s2 þ0:06sþ 5:19Â10 À 3
u3
y1 0:46s3
À1:15s2
þ7:40sþ0:91
s4 þ2:57s3 þ11:90s2 þ3:96sþ 0:32
Table A1 (continued )
u3
y7 À 0:20s3
À1:08s2
þ3:67sþ 3:13
s4 þ 9:85s3 þ 35:88s2 þ 12:34sþ 1:62
y8 À 0:83s2 þ 15:19sþ 4:86
s3 þ 1:85s2 þ 0:46sþ 0:06
y9 0:56s2
þ 14:74sþ 5:08
s3 þ 1:96s2 þ 0:52sþ 0:05
y10 1:82Â10 À 3
s3 þ0:20s2 À0:41sÀ 0:78
s4 þ 3:55s3 þ 14:10s2 þ 9:24sþ2:22
u4
y2 0:24s6 À 0:14s5 þ 1:71s4 þ3:63s3 þ3:72s2 þ0:66sþ 0:05
s7 þ5:58s6 þ 19:65s5 þ 39:65s4 þ 11:93s3 þ 3:85s2 þ 0:20sþ4:13Â10À 3
y3 0:14s5 þ 3:82s4 À 7:56s3 À 12:02s2 À 0:30sÀ0:39
s6 þ5:22s5 þ 18:97s4 þ 17:63s3 þ 3:14s2 þ 0:32sþ0:01
y4 À 0:13s2
þ 0:26sþ 0:29
s3 þ4:16s2 þ 1:90sþ0:32
y5 À 6:42Â10 À 4
s6 À 0:02s5 þ0:04s4 þ0:05s3 þ0:04s2 þ0:03sþ6:327Â10 À 3
s7 þ4:27s6 þ16:04s5 þ10:95s4 þ6:98s3 þ 1:72s2 þ 0:51sþ0:02
y6 0:03s3
À 0:12s2
þ 0:60sþ0:42
s4 þ3:26s3 þ 13:41s2 þ 7:94sþ1:64
u5
y5 À 6:499s2
Â10 À 4
À 6:382sÂ10 À 4
À 1:923Â10À 4
s3 þ0:58s2 þ 0:09sþ5:371Â10 À 3
u6
y3 À 0:05s3 þ 0:52s2 À 1:92sÀ 2:32
s4 þ 4:56s3 þ 17:19s2 þ 17:01sþ5:58
y4 0:01s3
þ 0:03s2
þ 0:05sÀ 0:44
s4 þ 7:28s3 þ 29:63s2 þ 53:21sþ33:54
y6 9:46Â10À 4
s2 À 1:39Â10 À 5
sþ 1:72Â10 À 4
s3 þ 0:46s2 þ 0:12sþ 0:01
u7
y2 0:01s5 À 0:02s4 À0:02s3 À0:01s2 À1:84Â10 À 4
sÀ1:89Â10 À 4
s6 þ 3:39s5 þ 4:97s4 þ2:45s3 þ0:47s2 þ0:03sþ 9:58Â10 À 4
y3 À1:84Â10À 3
s2 À 6:35Â10À 4
sÀ 1:07Â10À 4
s3 þ0:26s2 þ0:02sþ 3:83Â10 À 4
y4 À6:67s4 Â10 À 4
þ 1:26Â10À 3
s3 þ 1:71Â10À 3
s2 þ 1:68Â10 À 4
sþ 6:58Â10 À 5
s5 þ2:48s4 þ3:34s3 þ 0:66s2 þ 0:08sþ4:09Â10À 3
y5 À8:51Â10À 4
s5
À 1:52Â10À 4
s4
À 1:48Â10À 4
s3
À 1:01Â10 À 5
s2
À1:32Â10 À 6
sÀ 6:Â10 À 8
s6 þ 0:31s5 þ 0:21s4 þ 0:04s3 þ6:14s2 Â10À 3
þ2:93Â10À 4
sþ5:26Â10 À 6
y6 À8:431Â10 À 4
s3
þ5:02Â10À 5
s2
À 0:02sÀ3:67Â10À 3
s4 þ 3:34s3 þ 13:05s2 þ 3:09sþ 0:25
u8
y3 0:5407s6
þ 1:312s5
þ 0:5546s4
þ 0:3327s3
þ 0:06388s2
þ0:005414sþ 0:0003131
s7 þ 0:4485s6 þ0:599s5 þ0:1516s4 þ0:06437s3 þ0:009542s2 þ 0:0005524sþ1:289eÀ 05
y4 0:008947s3 À0:3574s2 þ1:39sþ 0:9825
s4 þ 4:132s3 þ 15:16s2 þ 7:629sþ 1:396
y5 À 0:02s5
À 0:26s4
þ 0:74s3
þ 0:21s2
þ 0:11sþ 0:04
s6 þ 5:85s5 þ 20:46s4 þ 13:84s3 þ 4:03s2 þ 0:67sþ 0:04
y10 0:03s3 À 0:10s2 þ 0:51sþ 0:11
s4 þ 2:96s3 þ 12:66s2 þ 5:60sþ0:70
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