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Steady-state target calculation for constrained predictive control systems based on goal programming.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2008; 3: 648–655
Published online 17 October 2008 in Wiley InterScience
(www.interscience.wiley.com) DOI:10.1002/apj.200
Special Theme Research Article
Steady-state target calculation for constrained predictive
control systems based on goal programming
Shaoyuan Li,* Yi Zheng and Baiping Wang
Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
Received 23 July 2008; Accepted 23 July 2008
ABSTRACT: A new method of steady-state target calculation for constrained model predictive control (MPC) using
goal programming has been developed to solve the problem that the result is not satisfactory when the optimization
problem is infeasible or the feasible region is away from the desired working point owing to system dynamics. In
this model, soft constraints adjustment and target relaxation have been adopted simultaneously to coordinate with
the result. The goal priority factors are introduced to describe the priority of constraints and targets; thereby, the
steady-state target calculation is transformed into a goal-programming problem with a standard linear programming
form and is solved with some calculation in real time. Simulation is processed with the example of the Shell heavy
oil fractionators’ benchmark problem, and the result shows the validity of the proposed algorithm.  2008 Curtin
University of Technology and John Wiley & Sons, Ltd.
KEYWORDS: predictive control; constraint priority; target priority; goal programming; soft constraint
INTRODUCTION
Currently, the optimization and control of complex systems have become very important issues as good optimization results can bring tremendous economic benefits and good dynamic behavior.[1 – 3] In the modern
process industry, simple control methods cannot satisfy
the control requirements of large enterprises along with
the scale of commercial enterprises expanding continuously; therefore, many large enterprises use a control
system, which is structured hierarchically into several
layers, and each layer operates on a different timescale.
Typically, layers include global steady-state calculation
(everyday or week), local steady-state calculation (every
30 min to 6 h), model predictive control (MPC) (every
1–2 min) and basic dynamic control (every second). As
one part of a multilevel hierarchy of control functions,
MPC plays an important role in the complex control
industry. It can move the plant from one constrained
steady state to another while minimizing constraint violations along the way.[4] Recently, the separation of the
MPC algorithm into a steady-state and dynamic calculation has been a common part of industrial MPC
technology.[4 – 7] Here, the goal of the steady-state target calculation is to recalculate the targets from the local
optimizer every time the MPC controller executes. This
*Correspondence to: Shaoyuan Li, Department of Automation,
Shanghai Jiao Tong University, Shanghai 200240, China.
E-mail: syli@sjtu.edu.cn
 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
must be done because disturbances entering the system
or new input information from the operator may change
the location of the optimal steady state.[4]
MPC has become an industry standard to solve complex constrained multivariable control problems in the
process industry with its wide usage in refinery and
petrochemical processes.[8] Currently, constraint MPC
is being successfully used in the control of various dynamic systems, and its use is becoming more
widespread.[9 – 11] However, chemical plants in modern
industries have become increasingly complex, which
leads to excessive targets and constraints. The study
of constraint MPC is also a major problem.[12 – 14] In
Olaru and Dumur,[13] the methods to avoid constraints
redundancy in MPC are discussed. In Wang et al .,[12] a
hybrid approach using a mixed logical dynamical framework to handle infeasibility and constraint prioritization
issues in MPC based on dynamic model is introduced.
As an important part of a two-stage constraint MPC
algorithm, steady-state target calculation faces many
problems and challenges. There are many manipulated
variables and controlled variables in the process of MPC
dynamic optimization. The steady-state target must satisfy the requirements of system safety, energy, and technical conditions. In addition, these requirements have
different priority requests according to their extent of
importance. In essence, these problems are multiobjective and multidegree of the freedom proposed in
Ref. [15] In the classical constraint programming, the
constraint cannot be changed. Once the intersection of
Asia-Pacific Journal of Chemical Engineering
these ‘hand constraints’ does not exist, or constraints
are violated due to disturbances, operator intervention,
model mismatch, and plant failure, it might result in
infeasibility of the MPC optimal program. However,
in satisfactory optimization control, soft constraints can
be used to reduce the probability of infeasibility.[16,17]
Therefore, how to use the adjustable character of system constraints to get satisfying results has become one
of the urgent problems in MPC steady-state target calculation.
MPC steady-state target calculation comes down to
constrained multiobjective multidegree of freedom optimization problem (CMMO) of complex industrial process under a dynamic uncertainty environment.[15] Reference [5] discussed the case in which the optimization
problem was feasible, but it did not consider how to
handle the problem in which the optimization problem was infeasible. Though Ref. [4] used soft constraints to deal with the infeasibility of the steady-state
target optimization problem, it treated all constraints
equally and did not consider the effects of soft constraints adjustment on control targets and the problem
of multiobjectivity and priority. In Xi and Gu,[18] the
problem of feasibility of CMMO is discussed. As there
is an inherent relationship between the solution and the
feasible sets of the optimization problem, it is necessary to consider the desired target while adjusting
the constraints.[19] Reference [19] investigated the relationship between feasibility and objective coordination
of the CMMO problem under flexible constraint conditions and used the mixed logic method to describe
the priority of the constraints and targets. Then, the
optimization problem is transformed into a mixed integer quadratic programming problem. However, a mixed
integer quadratic programming problem is complex and
the solution is time consuming, and it is not only used
in the local steady-state calculation (every 30 min to
6 h) layer but also in the MPC (every 1–2 min) layer.
Compared with the restriction of the single goal and
the single optimum solution in linear programming, goal
programming[20,21] permits to deal with different layers
of conflicted goals. As the goal-programming model,
whose form is a standard linear programming model,
the goal-programming problem can be solved by the
simplex method of the linear programming problem,
which is fast and involves little calculation.
In this study, on the basis of the structure of the
MPC steady-state target calculation and dynamic control, we studied the feasibility and target coordination
problems of CMMO systematically and put soft constraints adjustment and target coordination into MPC
steady-state target calculation. By adopting the goal priority factor, the priority strategy is introduced into the
description of system target and constraints. The priority
level of soft constraints adjustment and target coordination are determined according to the requirement of system conditions. Then, the steady-state target calculation
 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
STEADY-STATE TARGET USING GOAL PROGRAMMING
is transformed into a goal programming problem with a
standard linear programming form and is solved with a
little calculation in real time. Through this method, the
problem that the result is not satisfactory when the optimization problem is infeasible or the feasible region is
away from the desired working point owing to system
dynamic is systematically solved in steady-state target
calculation of constrained MPC.
FEASIBILITY AND TARGET COORDINATION IN
STEADY-STATE TARGET OPTIMIZATION
Feasibility of CMMO
For the linear time-invariant multivariable system, we
use the transfer function model of the plant
y(s) = G(s)u(s)
(1)
where y ∈ R m is the output vector and u ∈ R n is the
input vector, G(s) = [gij (s)]m×n , i = 1, 2, · · · , m; j =
1, 2, · · · , n. Suppose the steady-state gain matrix exists,
the steady-state process can be described by
ys = Gu s
(2)
subject to
umin ≤ us ≤ umax ,
ymin ≤ ys ≤ ymax
(3)
where ys ∈ R m is the steady-state value of output, us ∈
R n the steady-state value of input, and G ∈ R m×n the
steady-state gain matrix. If the pair (us , ys ) exists, which
satisfies the condition (2) and (3), the CMMO problem
is feasible.
When CMMO problem is infeasible, some inequality
constraints can be softened to guarantee the feasibility
of the optimization problem. Then, the system constraints can be described by Eqn (4)

y = Gu s

 s
umin − umin ≤ us ≤ umax + umax

 ymin − ymin ≤ ys ≤ ymax + ymax T
= [umin , umax , ymin , ymax ] ≥ 0
(4)
Hard constraints correspond to i = 0 and soft
constraints correspond to i ≥ 0. x1 , x2 , x3 , and x4
are auxiliary variables denoting the adjustment of the
constraints, defined as

x

 1
x2

 x3
x4
= us − umin + umin
= umax − us + umax
= Gu s − ymin + ymin
= ymax − Gu s + ymax
(5)
Asia-Pac. J. Chem. Eng. 2008; 3: 648–655
DOI: 10.1002/apj
649
650
S. LI, Y. ZHENG AND B. WANG
Asia-Pacific Journal of Chemical Engineering
(ys∗ , us∗ ) is the steady-state
us∗ = ud . However, (ys∗ , us∗ )
solution. Ideally, ys∗ = yd ,
may be away from (ud , yd )
because of the existence of constraints. Reference [15]
introduces objective coordination algorithm based on
sensitivity analysis.
Moreover, define
z = [x1T , x2T , x3T , x4T , T ]T
umax − umin b = ymax − Gu min Gu
max − ymin
I I 0 0 −I
= G 0 0 I −G
0 G I 0
0
−I
0
−G
0
0
−I
0
−I
0
Relationship between feasibility and target
coordination of CMMO
Then, the system constraints can be rewritten as
z = b
(6)
z ≥0
The feasibility of CMMO can be judged by the
following linear programming
min W = c T , c T = [1, · · · , 1]
s.t. (6)
(7)
If W = 0, CMMO is feasible, or, if W > 0, CMMO
is infeasible. Reference [18] determined the soft constraints and its adjustable range based on the
human–machine interface and adjusted the weight factor c on the basis of the requirement of the operators,
then the optimization problem in (8) is solved
min W = c T , c T = [c1 , · · · , c2(m+l ) ]
s.t.
(6)
(8)
However, in the process control, a case usually exists
that a constraint is required to be softened first when the
constraint region is infeasible. If the relaxation results
cannot make the optimization feasible, then consider
the next possibility. Obviously, it is difficult to obtain
a feasible optimum by the human–machine interface.
Target coordination of CMMO problem
The target coordination problem in CMMO is to find an
appropriate pair (us , ys ) under equality and inequality
constraints. The mathematical description of the optimization problem can be formulated as
min J = (us − ud 2R + ys − yd 2Q )
(9)
s.t. (2), (3)
where ud ∈ U ⊂ R n is the desired value of MVs and
yd ∈ Y ⊂ R m is the desired value of CVs. (ud , yd )
comes from the local optimizer.
In CMMO, the objective of the target calculation
is to find the feasible pair (us , ys ), such that (us , ys )
is as close as possible to (ud , yd ). When CMMO
problem is feasible, there exist ys and us , which satisfy
the equality and inequality constraints. The optimum
 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
As can be seen from Eqn (5), feasibility is a necessary condition for the system optimization problem.
However, it is crucial to consider a satisfactory target
while designing a feasible region in system optimization. Now, we discuss two aspects of the relationship
between feasibility and target coordination of CMMO.
First, if the feasible region is nonempty, we can get
an optimum that satisfies certain priority conditions.
However, the function index J becomes large when
the feasible region is away from the desired working
point, which means that we might not get a satisfying
result, though there is a large feasible region. In this
case, feasibility is not the only necessary condition for
CMMO. On the one hand, if there is surplus freedom
for the steady-state target, this freedom can be utilized
to adjust the feasible region so that it comes close to the
desired working point; on the other hand, targets may
be relaxed while softening the constraints.
Second, when the feasible region is empty, it must
make the optimization problem feasible before coordinating the targets, which can be done by soft constraints
adjustment. Generally, the polyhedral and anomalistic
for the feasible region are composed of linear constraint conditions in U , Y .[19] The feasible region reconstructed by constraints adjustment may not be close to
the desired target, thus leading to an unsatisfying result.
Therefore, for CMMO, attention should also be given
to targets along with the optimization feasibility region.
In summary, it is necessary to use the adjustable
character of system freedom to acquire a satisfying
result whenever the optimization problem is unfeasible
or the feasible region is away from the desired working
point.
CONSTRAINT ADJUSTMENT AND TARGET
COORDINATION BASED ON GOAL
PROGRAMMING
In this article, target coordination is taken into account
along with constraint adjustment. We use goal programming to deal with the constraint adjustment and target
coordination problem in steady-state target calculation.
Moreover, the goal priority factor is introduced to combine the priority of the constraints and targets.
Asia-Pac. J. Chem. Eng. 2008; 3: 648–655
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
STEADY-STATE TARGET USING GOAL PROGRAMMING
By introducing positive and negative error variables,
Eqn (12) is transformed into equality constraints as
Eqn (13)
Soft constraints adjustment in steady-state
target calculation
Reference [18] stated the mathematical description of
the soft constraints adjustment in steady state and,
in this description, both the inputs and outputs were
considered as adjustable variables, whereas, in practice,
constraints of manipulated variables are generally hard
and cannot be violated at all.
According to the principle of goal programming, the
constraints in steady-state target calculation are divided
into three types by introducing positive and negative
error variables.
Hard constraints composed by equality and
input constraints
Constraints of manipulated variables are usually derived
from the physical property and equipment limitation,
such as valve, which has limited active scope. Thereby,
constraints of manipulated variables are hard
ys = Gu s ,
umin ≤ us ≤ umax
(10)
Soft constraints
Constraints of controlled variables can be usually violated if safety is not a concern. Therefore, steady-state
output constraints can be considered as soft constraints.
By introducing positive and negative error variables,
the steady-state output constraints in Eqn (3) can be
rewritten as
ysi + d1i− − d1i+ = ymax
ysi + d2i− − d2i+ = ymin
i = 1, 2, · · · , m
(11)
where, for ‘≤’ constraints, each positive error variable d1i+ (i = 1, 2, · · · , m) is minimized and, for ‘≥’
constraints, each negative error variable d2i− (i =
1, 2, · · · , m) is minimized.
Goal constraints
The difference between performance and constraints
is diminishing as far as the optimization problem
in the industry is concerned, and can converge into
each other under some conditions. The minimization
of the performance index may be comprehended as
‘soft’ constraints, whose values are zero, while various
constraints can be regarded as targets inversely.[19]
In this way, the quadratic performance index in
Eqn (5), whose objective is to minimize the differences
between us and ud and the differences between ys and
yd , can be transformed into inequality constraints as in
Eqn (12)
|us − ud | ≤ 0
(12)
|ys − yd | ≤ 0
 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
usi + d3i− − d3i+ = ud
ysj +
d4j−
−
d4j+
= yd
i = 1, 2, · · · , n
j = 1, 2, · · · , m
(13)
where only if all the positive and negative error
variables d3i+ , d3i− (i = 1, 2, · · · , n) are equal to zero,
then us is equal to ud . Minimizing d3i+ , d3i− (i =
1, 2, · · · , n) at the same time implies that us is as close
as possible to the desired ud . Similarly, only if all
the positive and negative error variables d4j+ , d4j− (j =
1, 2, · · · , m) are equal to zero, then ys is equal to yd .
Minimizing d4j+ , d4j− (j = 1, 2, · · · , m) at the same time
implies that ys is as close as possible to the desired yd .
Priority of constraint adjustment and target
coordination
The core in CMMO is based on the human–machine
interface to determine the requirement of control, to
emphasize on people’s request, and to make the result
satisfy people’s requirement.[19] The priority in soft
constraint adjustment in the industrial process was
mentioned in Ref. [18]. The idea can be stated as
follows: when there is no feasible region or the solution
of the optimization problem is unsatisfying, some
constraints should be softened. If the relaxation results
cannot make the optimization feasible or the optimum
of the problem is still not satisfactory, the next should
be considered.
In the same way, soft constraint adjustment in steadystate target calculation also has a priority request.
According to the operator’s desire, soft constraints
are set to different priorities. The highest priority is
designated to the constraint, which is preferred to be
violated first, then the second and so on. At the same
time, when coordinating the targets, we may want
to reach some targets first and then another. So the
target coordination in CMMO has additional priority
requests. Targets with higher priorities are met first, and
then those with lower priorities are satisfied as much
as possible. Generally, priorities of output targets are
higher than those of input targets.
In this article, we focus on soft constraint adjustment
and target coordination in steady-state target calculation
when the optimization problem is infeasible or the
feasible region is away from the desired working
point. According to the system requirement, we can
determine the priority of each target and constraint,
use the goal priority factor to describe the priority
of the constraints and targets and transform the target
coordination problem into a goal constraint satisfactory
problem. Finally, steady-state target calculation can be
realized by solving a goal-programming problem.
Asia-Pac. J. Chem. Eng. 2008; 3: 648–655
DOI: 10.1002/apj
651
652
S. LI, Y. ZHENG AND B. WANG
Asia-Pacific Journal of Chemical Engineering
Mathematical description of steady-state
target calculation
The goal function in goal programming is constituted by goal priority factors: positive and negative
error variables of each goal. When each target is
determined, the objective of the optimization problem
is as close as possible to the desired target. Priority is determined according real requirement. If there
are s priorities, then goal priority factors Pi (i =
1, 2, · · · , s) satisfy P1 P2 · · · Ps , where ‘’
denotes priority.
The priority of goal constraints is higher than the priority of output constraints and the priority of output constraints is higher than the priority of input constraints.
In practice, we can set the priority of constraints and
targets according to the system requirement correspondingly. Assuming that d + , d − denote all the positive and
negative error variables, the steady-state target calculation problem is achieved by the goal-programming
problem as follows:
min P1
m
− −
+ +
(ω4j
d4j + ω4j
d4j )
+ P2
− −
+ +
(ω3i
d3i + ω3i
d3i ) + P3
i =1
+
m
m
+ +
ω1i
d1i
i =1
− −
ω2i
d2i
(14)
i =1
s.t.
usi + d3i− − d3i+ = ud
ysj + d4j− − d4j+ = yd
ysi +
ysi +
d1i−
d2i−
−
−
d1i+
d2i+
Operators can determine the priority and judge whether
the optimization solution is satisfying or not according
to their desire, and the adjusting process would not finish until the optimization result is satisfactory. If all the
adjustable constraints change into hard constraints, the
original optimization problem is infeasible. This indicates that there is not enough freedom of the system
and it cannot satisfy the control target; therefore, the
freedom must be added.
SIMULATION STUDIES
j =1
n
Step 2: If ys = yd and us = ud , then the algorithm ends,
otherwise go to step 3.
Step 3: Set the priority of constraints adjustment and
target coordination according to the desire of
the operator. Solve goal-programming problem (14), get the solution.
Step 4: Judge the positive and negative error variables
from (14); if they are satisfying, then the algorithm ends. Otherwise, give the positive and
negative error variables that are receivable, then
go to step 3.
i = 1, 2, · · · , n (15)
j = 1, 2, · · · , m
(16)
= ymax
i = 1, 2, · · · , m (17)
= ymin
i = 1, 2, · · · , m (18)
ys = Gu s
(19)
umin ≤ us ≤ umax
(20)
d +, d − ≥ 0
(21)
Heavy oil fractionators shown in Fig. 1 can be regarded
as the benchmark processes for the Shell standard
control problem.
The three inputs of the process represent the product
draw rate from the top of the column (u1 ), the product
draw rate from the side of the column (u2 ), and the
reflux heat duty from the bottom of the column (u3 ).
The three outputs of the process include the draw
composition (y1 ) from the top of the column, the draw
composition (y2 ) from the side of the column, and the
reflux temperature at the bottom of the column (y3 ).
The two disturbances are the reflux heat duty for the
intermediate section of the column (d1 ) and the reflux
heat duty for the top of the column (d2 ).
where ω with super- and subscript denote the weighting
coefficients and can distinguish the relative level of
importance of each constraint or goal in each priority.
Using the simplex method in goal programming to solve
this goal-programming problem, we can get the steadystate target.
In fact, the soft constraint adjustment and target
coordination algorithm based on goal programming
consists of the following steps:
Step 1: Solve optimization problem (7); if Wmin = 0,
then CMMO is feasible, go to step 2; otherwise,
go to step 3.
 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 1.
problem.
Heavy oil fractionator-Shell column control
Asia-Pac. J. Chem. Eng. 2008; 3: 648–655
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
STEADY-STATE TARGET USING GOAL PROGRAMMING
Prett and Morari presented the following model[22] :


4.05e−27s 1.77e−28s 5.88e−27s
1
60s + 1
50s + 1 
 50s +
−18s

5.72e−14s 6.90e−15s 
G(s) =  5.39e

60s + 1
40s + 1 
 50s + 1
4.38e−20s 4.42e−22s
7.20
33s + 1
44s + 1
19s + 1


1.20e−27s 1.44e−27s
40s + 1 
 45s + 1
−15s

1.83e−15s 
Gd (s) =  1.52e

 25s + 1
20s + 1 
1.26
1.14
27s + 1
32s + 1
steady-state equation, and the optimization problem is
feasible. We employ MPC steady-state target calculation
and dynamic control. Figure 2 shows the corresponding
result.
Another example gives the control targets as y1 =
y2 = 0.3 and y3 = −0.3, and an output target changes
with no disturbance. We can see that the feasible
region is away from the desired working point. In order
to get a satisfactory result, it is necessary to adjust
constraints and coordinate targets. According to the
technical character of oil fractionators, assume that the
priority order of the control targets is
The inputs and outputs are constrained between −0.5
and 0.5. The inputs velocity constraints are 0.20. Both
models are sampled with a period of 4 min. Its steadystate equation is
4.05 1.77 5.88
1.20 1.44
ys = 5.39 5.72 6.90 us + 1.52 1.83 d
4.38 4.42 7.20
1.14 1.26
1)|y1 − 0.3| ≤ 0
In fact, the heavy oil fractionators are a subsystem of
the complex chemical process; thus, the desired working point comes from the local optimizer. Usually, the
global optimizer determines optimal steady-state settings for each unit in the plant. These may be sent to
local optimizers, which run more frequently than the
global optimizer. The local optimizer computes an optimal economic steady state and passes this information
to the MPC algorithm for implementation.[4]
An example of output target change with disturbance
d = [0.6 0.5]T is given below. The control objective is
y1 = y2 = 0.3 and y3 = −0.3, which satisfies the system
2)|y2 − 0.3| ≤ 0
3)|y3 + 0.3| ≤ 0
Suppose the combinational priority order of control
target and constraint is
1)|y1 − 0.3| ≤ 0
2)|y2 − 0.3| ≤ 0
3)y1
4)y2
5)y3
6)|y3 + 0.3| ≤ 0
then the mathematical description of this goalprogramming problem is given below
min P1 (d1− + d1+ ) + P2 (d2− + d2+ ) + P3 (d3+ + d4− )
(a)
(b)
Figure 2. (a and b) Model predictive control result at steady state.
 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2008; 3: 648–655
DOI: 10.1002/apj
653
654
S. LI, Y. ZHENG AND B. WANG
Asia-Pacific Journal of Chemical Engineering
+ P4 (d5+ + d6− ) + P5 (d7+ + d8− ) + P6 (d9− + d9+ )
s.t.
y1 + d1− − d1+ = 0.3
y2 + d2− − d2+ = 0.3
y1 + d3− − d3+ = 0.5
y1 + d4− − d4+ = −0.5
y2 + d5− − d5+ = 0.5
y2 + d6− − d6+ = −0.5
y3 + d7− − d7+ = 0.5
y3 + d8− − d8+ = −0.5
y3 + d9− − d9+ = −0.3
From this result, manipulating variables u to satisfy
the hard constraints of the system, both y1 and y2
reach their desired value, and y3 is close to its desired
value that can satisfy the requests of the operators. The
corresponding simulation result is shown in Fig. 3, and
the time taken by steady-state calculation is 0.109 s
(CPU: 1.8 GHz, Memory: 512 Mb, simulation tool:
Matlab).
When the result is compared with that shown in
Fig. 4, which uses the method that only constraints
softening of outputs are adopted in steady-state target
calculation, it is obvious that the result of steady-state
target calculation obtained by the method proposed in
this article is more satisfactory than that without target
coordination.
y1 = 4.05u1 + 1.77u2 + 5.88u3 + 1.20d1 + 1.44d2
y2 = 5.39u1 + 5.72u2 + 6.90u3 + 1.52d1 + 1.83d2
y3 = 4.38u1 + 4.42u2 + 7.20u3 + 1.14d1 + 1.26d2
−0.5 ≤ u1 ≤ 0.5
−0.5 ≤ u2 ≤ 0.5
−0.5 ≤ u3 ≤ 0.5
di+ , di− ≥ 0, i = 1, 2, · · · , 9
Then, using the simplex method in goal programming
to solve this optimization problem, we get a satisfying
result
us = [ 0.5
−0.1018
ys = [ 0.3
0.3
−0.2627 ]T
−0.15151 ]T
CONCLUSIONS
It can be concluded that soft constraints adjustment and
target coordination were investigated according to the
characteristics of constrained multiobjective and multifreedom in steady-state target calculation. When system
optimization is infeasible or the feasible region is away
from the desired working point, the soft constraints
adjustment and objectives coordination in MPC steadystate target calculation are solved along with a little calculation by transforming steady-state target calculation
into a goal-programming problem. Through constraints
adjustment and objectives coordination, the system optimization becomes feasible or the feasible region comes
near the desired working point. This method has strong
practical significance.
(a)
(b)
Figure 3. (a and b) Model predictive control result at steady state based
on local optimization.
 2008 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2008; 3: 648–655
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
STEADY-STATE TARGET USING GOAL PROGRAMMING
(a)
(b)
Figure 4. (a and b) Model predictive control result at steady state without
target coordination.
Acknowledgements
This work was supported by the National Nature Science Foundation of P. R. China under Grant 60774015
& 60534020, the High Technology Research and Development Program of China (Grant: 2006AA04Z173), the
Specialized Research Fund for the Doctoral Program of
Higher Education of China (Grant: 20060248001).
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