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# Time Optimal Control of a Binary Distillation Column.

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Dev Chem. Eng. Mineral Process, lO(l/2), p p 19-31,2002
Time Optimal Control of a Binary
Distillation Column
Rein Luus*
Department of Chemical Engineering, University of Toronto,
lime optimal control problems involving high dimensional systems are generally very
dificult to solve. Instead of using suboptimal control based on reduced system, iterative
dynamic programming is used directly to establish the time optimal control policy of a
binary distillation column described by 11 ordinary differential equations. The resulting optimal control policy that is bang-bang in nature was obtained very rapidly. &
minimum time of 2236 seconds obtained here is substantially better than 4120 seconds
obtained by a suboptimal control policy. Simulations show that the procedure may be
used for on-line optimal control of the distillation column, since the optimal control
policy can be established very fast on a personal computer.
Introduction
Time optimal control problems, where the desired state is to be reached in minimum
time, are important in the operation of engineering systems For example, in the startup
of a chemical reactor we may wish to reach the desired state of operation as fast as possible, or in the case of some upset in the operation we may wish to recover in minimum
time In the operation of a distillation column, some upset in the system may occur, and
we may want to recover the desired state of operation as fast as possible. Therefore time
optimal control is important in industry and has been studied by numerous researchers.
For linear systems one approach that has been used successfully is to convert the
continuous system to a system of difference equations by integrating the system over
a sampling time, and then to find the minimum number of sampling periods necessary
to reach the desired state. Such an approach with linear programming was used for
linear systems by Lapidus and Luus [l], Bashein [2], and by Rosen et a1 [3]. For
nonlinear systems a similar approach with sequential quadratic programming (SQP)
was used by Rosen and Luus [3, 41. Other approaches include the use of dynamic
programming and penalty functions [6] and the use of a suboptimal approach [7] For
time optimal control of high-dimensional systems the use of model reduction was used
by Wong and Luus [8, 91. More recently, Bojkov and Luus [lo, 111 used iterative
dynamic programming to solve time optimal control problems. By using a quadratic
penalty function with shifting terms, Luus [ 12, 131 showed that the time optimal control
problem can be solved successfully for several nonlinear systems that had been difficult
to solve by other methods
~~
*Authorfor correspondence (email luus@ecf utoronto cn)
19
The goal here is to test the viability of iterative dynamic programming for the solution of the time optimal control problem involved in transferring the operation of a
binary distillation column to its desired state of operation. The distillation column chosen is the one modelled by Davison [14] and used for the study of interaction of control
systems by Davison [ 151, and by Davison and Man [161. This model was used for model
reduction by Bosley and Lees [ 171, and for time suboptimal control based on reduced
models by Wong and Luus [8, 9). The model consists of 8 plates, plus a condenser
and a reboiler The control is achieved by changing the heat input to the reboiler and
the heat withdrawn from the condenser. There are therefore two control variables and
10 concentration variables, plus the pressure in the column, giving a total of 11 state
variables. The detailed development of the equations for the distillation column with
pressure variation is given in the appendix of the paper [ 141.
Time Optimal Control Problem
The model of the binary distillation column with pressure variation, as modelled by
Davison [14], is given by the differential equations:
3
= -1.4 x 10-22~+ 4.3 x 10-3z2
dt
h 2
dt
= 9.5 x 1 0 - 3 ~ 1- 1.38 x 10-222
3.0 x 1 0 - ~ 2~
3.0~x
+ 4.6 X 1 0 - 3 ~ 3+ 3.0 x ~ O - ~
U ~
dx3
= 9.5 X 10-3x2 - 1.41 x 10-223 + 6.3 x 1 0 - 3 ~ 4-
dt
dt
&6
dt
5.0 x 1 0 - ~ 2~
5.0~x
+ 5.0 x 10-5u2
8.0 x 1 0 - ~ 2-~5.0
~ x 10-6u1
+ 5.0 x ~
o - ~ u ~
- 3.12 x 10-2z5 + 1.5 x 10-2z48.0 x 1 0 - 4 ~ 1 1- 5.0 x ~ O - ~+U5.0~ x ~ O - ~ U
= 9.5 x 10-3z4
- 3.52 x 1 0 - 2 ~ 6+ 2.2 x 10-2~3
8.0 x 10-4~11
- 5.0 x 10-%~+ 5.0 x 10-5u2
= 2.02 X 10-225
dx7
- = 2.02 x 10-2x6- 4.22 x 10-2x7+ 2.8 x 1 0 - 2 ~ 3dt
8.0 x 1 0 - ~ 2-~5.0
~ x 10-%~+ 5.0 x
20
~
Time Optimal Control of a Binary Distillation Column
dZ8
-
= 2.02 x 10-2x7 - 4.82 x 1 0 - 2 ~ s 3.7 x 10-2x3 -
-
= 2.02 x 10-2z8
dt
dX9
dt
+
6.0 x 10-~211- 2.0 x 10% ~ + 5.0 x 1 0 % ~
- 5.72 x 1 0 - 2 ~ 9+ 4.2 x
3.0 x 1 0 - 4 ~ 1 1 4.0 x
+
~ O - ~ 4.0
U ~x
10-2z3
10%~
- - - 2.02 x ~ o - -~4.83
&lo
z ~x ~ o - ~ 2.0
x x~ ~
dt
kll
-
dt
(9)
+ 2.0 x 10- %~ (10)
+
+
= 2.55 x 1 0 - 2 ~ 1 2.55 x 10-2~10- 1.85 x 10-2~11
4.6 x 10-4u1
+ 4.6 x ~ o - ~ u ~ .
(1 1)
By using vector-matrix notation, these 11 equations can be written in a more compact form as:
dx
- = AX + Bu.
(12)
dt
The initial state is:
~ ( 0 ) = [-0.0836
-0.8901
-0.2730
-0.6102
- 0.6135 - 0.3105
-0.9104
0.1298
-0.7639
- 1.0
- 0.2946IT.
(13)
-
The state variables zi, i = 1, ..,10 represent normalized concentration deviations
from the desired states of the distillate, the 8 plates, and the bottoms, respectively The
state variable 211 is the normalized deviation of the pressure in the column from the
pressure at the desired state of operation. Normalization is carried out so that the desired
state is the origin The control variables 261 and zb2 are the normalized change of heat
input to the reboiler and the normalized heat output from the condenser, respectively.
These control variables are bounded by:
-1
5 uj 5 1,
j
= 1,2
(14)
and are normalized such that they are zero at the desired state of operation.
The time optimal control problem is to find the control policy u(t) in the time interval 0 5 t < t f that transfers the system from the given initial state to the origin in
minimum time. The performance index to be minimized is therefore chosen as:
I =tf.
(15)
Numerically, it is required that in a minimum value of the final time t f ,we satisfy the
constraint:
[Zi(tf)l5 €i,
i = 1,2,* .
*,
11,
(16)
21
where Q is some tolerance, such as the measurement error. For time suboptimal studies
for each state variable.
of this system, Wong and Luus [8] chose Q to be
Direct Approach to Time Optimal Control
To solve the optimal control problem, according to the suggestion of Luus [12, 131, we
choose the augmented performance index to be minimized as:
n
J = tf
+ B C(Zi(tf)
- siy,
i=1
and update the shifting terms si after every pass of the IDP optimization procedure
according to:
sf"
= si9
- zi(tf)i
(18)
where the superscript 9 is used to indicate the pass number, and z i ( t f )is the value of the
state variable at the end of the pass The product 28si provides sensitivity of the change
of the performance index with respect to the equality constraint. Unless sensitivity
information is available, the initial choice for si is taken to be zero. The effectiveness
of this type of penalty function to handle an equality constraint in the optimal control
of a fed-batch reactor was shown by Luus and Hennessy [18].
To measure the closeness of the final states to the origin, we use the norm of the
sum of absolute deviations, namely:
i=
1
In order to solve the time optimal control problem, we transform the continuous
control policy into a piecewise constant control problem by dividing the time interval
[0,t f ]into P stages, each of variable length, such that:
v(k) = t& - t k - 1 ,
k = 1,2,...,P.
(20)
Computationally. we impose the condition:
v(k) > 0,
k = 1,2, * . * , P ,
(21)
so that we do not deal with negative stage lengths.
Let us now introduce a normalized time variable T,by defining d~ in the time interval t k - 1
t < t k through the relationship:
<
dt = v(k)Pd.r
(22)
so that:
t&- t k - 1 = V ( k ) P ( T k - 71f-1).
Therefore:
22
(23)
Time Optimal Control of a Binary Distillation Column
and in the normalized time, the stages are of equal length and the final time is T = 1,so
that iterative dynamic programming (IDP) can be used easily. The only change is that
the differential equation now becomes:
dx
= v ( k ) P ( A x+ Bu)
dr
in the kth time interval.
Algorithm for IDP
We consider the stage length at each stage as an additional control variable and augment
the control vector to a (3 x 1)vector, but for clarity of presentation we keep it separate
We use KDP in a multi-pass fashion, where after a pass the region is restored to a fraction
1 of its value at the beginning of the previous pass. The algorithm can be presented in
eight steps as follows
1. Choose the number of time stages P, the number of grid points N, the number of
allowable values for control (including stage lengths) R at each grid point, the region
contraction factor 7,the region restoration factor 77, initial values for the control and the
stage lengths, the initial region sizes, the number of iterations to be used in every pass,
and the number of passes.
2. By choosing N values for control and the stage length, evenly distributed inside
the specified regions, integrate Eq. (25) from 7 = 0 to T = 1to generate N trajectories.
The N values for x at the beginning of each time stage constitute the N grid points at
each stage
3 Starting at stage P,corresponding to the normalized time 7 = (P - l)/P,for
each grid point generate R sets of values for control and stage length:
+ D\$(P)
(26)
v ( P ) = v'j(P) + w w j ( P )
(27)
u(P) = u'j(P)
where D is a 2 x 2 diagonal matrix with different random numbers between -1 and 1
along the diagonal and w is another random number between -1 and 1; u'j(P) is the
best value for control, and v*j(P) is the best value for the stage length, both obtained
for that particular grid point in the previous iteration; w j is the stage length at iteration
j . Integrate Eq (25) from (P - 1 ) / P to T = 1 once with each of the R allowable
values for control and stage length to yield x ( t f ) so that the performance index can be
evaluated Compare the R values of the augmented performance index and choose the
control and stage length which give the minimum value. The corresponding control and
stage length are stored for use in step 4.
4. Step back to stage P - 1, corresponding to time r = (P - 2 ) / P . For each grid
point generate R allowable sets of control and stage lengths. Integrate Eq. (25) from
(P- 2 ) / Pto (P- 1 ) / Ponce with each of the R sets. To continue integration, choose
the control and the stage length from step 3 that corresponds to the grid point that is
closest to the x at r = (P l ) / P .Now compare the R values of the augmented performance index and store the control policy and the stage length that yield the minimum
value.
-
23
R Luus
5. Step back to stage P - 2, and continue the procedure in the previous step. Continue until stage 1 corresponding to T = 0 with the given initial state as the grid point
is reached. Make the comparison of the R values of the performance index to give the
best control and the stage length for this stage We now have the best control policy and
the stage length for each stage in the sense of minimizing the augmented performance
index from the allowable choices.
6. In preparation for the next iteration, reduce the size of the allowable regions:
.i+'(k)=+(k),
Wj+'(k)
k=1,2,-**,P
= p J j ( k ) , k = 1,2,**.,P
(28)
(29)
where 7 is the region reduction factor and j is the iteration index. Use the best control
policy and the stage lengths from step 5 as the midpoint for the next iteration.
7. Increment the iteration index j by 1 and go to step 2 to generate another set of
grid points. Continue for the specified number of iterations.
8 Increment the pass number index by 1, set the iteration index j to 1 and go to step
2. Continue for the specified number of passes, and examine the results.
Computational Results
All computations were done in double precision. For integration of the differential
equations we used the subroutine DVERK of Hull et a1 1191 with a local error tolerance
of
to give reliable results. The listing of this subroutine is given in [13]. The
problem was solved in FORTRAN on a PentiumIIY600 personal computer.
To solve this time optimal control problem by IDP, we chose the region contraction
factor used after every iteration 7 = 0.70, region restoration factor 7) = 0.98, a single
grid point (N = l), R = 10 randomly chosen points, P = 10 stages and allowed 200
passes, each consisting of 20 iterations. The initial stage length for each stage was taken
to be 250 s, and an initial control policy of u1 = 0, u2 = 0 was taken for each stage.
The initial region sizes for control and stage length were taken to be 1.0.
The first run was performed with 0 = 2 x 10'. After 200 passes requiring 348 s of
computation time on a PentiumIIU600we obtained a final time of t f = 2305 with the
sum of absolute deviations S = 0.00396. All the state variables were within
of
the origin. In the resulting control policy, the first 9 stages were at u1 = -1, u2 = 1,
for a time of 2061.4 s. The last stage had u1 = 0.17391,~2= -0.20644 and a length
of 243 5 s
For the next run, the penalty function factor was increased to B = 2 x lo6. At
the end of 200 passes, S was decreased to 0.00036, but the final time was increased to
t f = 2410.8s. The first 8 stages had the control policy ul = -1,212 = 1. The gth stage
of length 241.02 had u1 = 0.33925,u2 = 1, and the last stage of length 246.91 had the
policy u1 = 0.02375,~2= -0.04342.
The third run with 0 = 2 x lo4 did not yield a final state that was within the required
tolerance. The convergence profiles for these three values of the penalty function factor
are shown in Figure 1. It is seen that B = 2 x lo5 is a good choice for the penalty
function factor.
24
Time Optimal Control of a Binary Distillation Coliunn
1E4
i
,
50
PASS
100
150
200
NUMBER
Figure 1. Convergence prof2es for different values
of the penalqfunctionfactor 8.
To refine the results, we now took as the starting condition the first stage of length
2000 s with '111 = -1,912 = 1, and the remaining 9 stages of length 33.3 s. The initial
control policy of u = 0 was chosen for these stages, and we used the penalty function
factor 8 = 2 x lo5. The run of 200 passes, each consisting of 20 iterations, took a
computation time of 135 s, yielding t i = 2236.4 and S = 0.001124 The final state
was x ( t / ) = [0.00030 0.00006 0.00008 -0.00009 -0.00021 0.00017 0.00005
-0.00011 0.00003 0.00002 O.OOOOO]T.
The resulting optimal control policy is given in Table 1. It can be seen that the
time optimal control policy is really a four-stagebang-bang policy, where the first stage
is of length 2016.98, the second stage is of length 174.20, the third stage is of length
12 41 and the fourth stage is of length 32.84 The state trajectories corresponding to
this control policy are given in Figures 2 4 With the exception of zll that exhibits
oscillatory behavior, the states approach the origin in a very smooth manner.
We have brought the system closer to the origin than had been required. Therefore
upon examination of the output, it was observed that already at the end of the 3gfhpass,
all the state variables are within loe3 of the origin. At the end of pass 39 requiring 22.7
s of computation time, the final time t i = 2167.4, S = 0.004485, and the final state is
x ( t / ) = [0.00016 -0.00031 -0.00070 -0.00094 -0.00043 0.00048 0.00072
0.00040 0.00020 0.00005 O.OOO1l]T. Here the control policy, as shown in Table
2, is not bang-bang.
It is interesting to note that the t / obtained here is almost one-half of the t f = 4120
25
Table 1. lime optimal control policy for the binary distillation
column giving t f = 2236.4 s with S = 0 001124.
Stage number
u1
1
-1.ooooo
2
3
4
5
6
7
8
9
10
-1
.ooooo
- 1.00000
1.00000
1.ooooo
1.ooooo
1.00000
1.ooooo
1.ooooo
-1 .ooooo
Stage length
2000.00
1.00000
10.43
6.55
1.00000
1.00000
34.85
35.59
1.ooooo
38.18
1.00000
37.31
1.00000
28.27
1.00000
12.41
-1.ooooo
-1.ooooo
32.84
112
1.ooooo
Table 2. Control policyfor the binary distillation
column giving tf = 2167.4 s with S = 0.004485.
Stage number
1
2
3
4
5
6
7
8
9
10
211
-1.00000
-1.00000
0.98650
1.ooooo
1.00000
1.00000
1.ooooo
1.ooooo
-1.00000
-1.ooooo
Stage length
2000 00
1.ooooo
23.66
1.00000
28.04
1.ooooo
24 40
1.00000
22.34
1.00000
24.56
1.ooooo
2 1.98
1.ooooo
24.06
-0.24856
27.72
-1.ooooo
31.19
u2
1.00000
obtained for this system by Wong and Luus [9] by the use of time suboptimal control
obtained from a reduced model.
On-line Time Optimal Control
For on-line time optimal control we are faced with the problem of how to control the
system while computations are being carried out. Since a good approximation to the
optimal control policy can be obtained by using a small number of passes and a small
number of random points, we can specify control action for the first time stage and then
use IDP to find the best control for the other stages. The first stage is taken as short
as possible, but it must still be long enough to enable time optimal calculations to be
carried out.
For this system we chose a total of 11 stages. We let the first stage be 60 s long and
kept u = 0 throughout the run. Thus, no optimization was carried out on this stage. For
the other 10 stages we chose each of them initially of length 240 s and the initial control
u = 0. Since we wanted the calculations to be executed in less than one minute, we
26
Time Optimal Control of a Binary Distillation Column
Figure 2. State trajectoriesfor X I ,22,x3. and x4.
-1 2
0
500
lcm
1500 2ooo
n m
1
2500
Figure 3. State trajectoriesfor 25,x6, xl,and xg.
27
R Luus
Figure 4. State trajectories for xg, 210, and 211.
chose R = 5, and allowed only 40 passes with the region restoration factor 7 = 0.95.
The other parameters in IDP were the same as before. The computation time for this
run was 52 s.
After 40 passes, we obtained the control policy in Table 3 with t f = 2273.4, and final state x(t,) = [-0.00009 -0.00030 -0.00071 -0.00086 -0.00025 0.00015
0.00047 0.00051 0.00033 0.00016 0.00008IT. Therefore, we should switch to
u1 = -1,112 = 1 after 60 s. We could continue with the control policy in Table 3
and bring the system to the vicinity of the origin (such that all variables are within
in magnitude) in a total time of 2273.4 s. However, we can improve the result
by carrying out another optimization run where we have for the initial control policy
u1 = -1,212 = 1 and a new initial state obtained after the system has been running
for 60 s This new initial state is x(0) = [-0.08192 -0.25390 -0.57503 -0.86554
-0.72484 -0.95424 -0.84941 -0.58713 -0.30062 -0.12744 -0.90243jT.
We therefore carried out the run with 11 stages, keeping the first stage control policy
at 211 = -1,212 = 1, and choosing its initial length to be 1741.66. The remaining 10
stages were chosen of initial length 45.8 and initial control policy u = 0. For this run
we chose R = 10, 7) = 0.98, and allowed 200 passes
After 200 passes, requiring a computation time of 180 s, we obtained the control
policy in Table 4, giving tf = 2182.5 and S = 0.001091. The final state is x ( t f ) =
(0.00030 0.00008 0.00010 -0.00008 -0.00024 0.00013 0.00008 -0.00006
0.00000 0.00001 O.OOOO1]T, so all the states are within 0.0003 of the origin. The
time required to take the system from the original initial state to the origin is therefore
60 + 2182.5 = 2242.5 s. It is interesting to note that this value of final time is only
28
Time Optimal Control of a Binary Distillation Column
6 s more than the final time obtained by the optimal control policy in Table 1 . This
simulation run supports the idea of using optimal control for batch reactors as outlined
by Luus and Okongwu [20].
Table 3. Control policy for the binary distillation column
where the control policy u = 0 is imposed f o r thejrst stage.
Stage number
1
2
3
4
5
6
7
8
9
10
11
u1
0.00000
-1.00000
-1.ooooo
-1.ooooo
-1 .ooooo
-1.ooooo
-1.ooooo
-1 .ooooo
-1.ooooo
-0.84869
0.04203
u2
0.00000
1.00000
1.ooooo
1.00000
1.00000
1.ooooo
1.00000
1.00000
1.ooooo
1.ooooo
-0.03460
Stage length
60 00
211 02
210 42
216 88
216 20
217.61
222 59
224 01
222 93
232.72
238.02
Table 4. On-line rime optimal control policy for the binary distillation
column a f e r the control policy for thejrst stage is obtained in 60 s
Stage number
1
2
3
4
5
6
7
8
9
10
11
211
212
-1.ooooo
-1.ooooo
1.ooooo
1.ooooo
-1.00000
1.00000
1.00000
1.ooooo
1.00000
1.00000
1.00000
1.ooooo
-1.00000
-1.00000
-1.00000
-1.ooooo
1 .ooooo
1.ooooo
1.00000
-0.60848
1.00000
-1.00000
Stage length
1742.00
34.34
37.20
40.97
33.99
31.51
36 16
52.64
59 57
68.91
45.21
Conclusions
The time optimal control of a binary distillation column was easily established by iterative dynamic programming Since the computation time is small, such optimization
can be done on-line and the method should find industrial application.
The minimum time to reach the desired state is substantially less than has been
reported in the literature. The state trajectories show that the origin is approached very
smoothly in spite of the bang-bang control, so there is no concem about the stability of
the system once the desired state has been reached.
29
R. Luus
Acknowledgement
Financial support from the Natural Sciences and Engineering Research Council of
Nomenclature
A
B
D
I
J
L
n
N
P
9
r
R
si
S
t
t/
%
U
V
20
xi
X
State coefficient matrix (11 x 11)
Control coefficient matrix (11 x 2)
Diagonal matrix of random numbers between -1 and 1
Performance index
Augmented performance index
Length of a time stage
Number of state variables
Number of grid points
Number of time stages
Pass number
Region vector over which allowable values for control are chosen
Number of randomly chosen values for control
Shifting term corresponding to constraint i
Sum of absolute values of deviations
Time
Final time of operation
j t h element of control vector u
Control vector (2 x 1)
Variable stage length
Region over which allowable stage lengths are chosen
ith state variable
State vector (n x 1)
Subscripts
f
Final time
i
Index
j
Index
Index pertaining to time stage
k
Superscripts
Best value obtained from previous iteration
j
Iteration step
4
Passnumber
Transpose
Greek letters
Region contraction factor by which the region is reduced after every iteration
7
e
Tolerance
71
Region restoration factor
0
Penalty function factor
7
Normalized time variable
Random number between -1 and 1
w
30
Time Optimal Control of a Binary Distillation Colwnn
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20 Luus,R and Okongwu, 0 N 1999 Towards practical optimal conml of batch reactors Chem Eng
J .75, 1-9
Received: 4 December 2000; Accepted afier revision: 20 April 2001.
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