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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5395_Библиотеки_им_академика_М_И_Перельмана
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306 A. Nikolakopoulou et al.
https://t.me/medicina_free
Fig. 5 A series of optimizations for the atropine synthesis startup for various values of T , nt= 1,
and for piecewise constant parametrizations indicate that the knee of pareto optimal solutions for
the competing objectives of total productivity and total waste was obtained for a weighting factor
of about w
= 100 for all T
a
Table 1 Dynamic
optimization constraints
Optimization variable Minimum value Maximum value
ui[mL/min], i = 1 −4 0 0.1
u5[mL/min] 0.1 1
u6[mL/min] 0 0.2
T1, T2[K] 300 400
tp/nt[min] 20 500
flowrates u1−u6, reactor temperatures T1, T2, and the length of each of the ntstages
t
− tp,p= 0,...,nt− 1. The input constraints are shown in Table 1.
p+1
After discretizing the optimization variables, (19) is an NLP that can be solved
iteratively. In every iteration, the system is simulated by solving an initial value
problem for a total time of T = t
− u6, T1, and T2are fixed to their last values and an additional 200 min of
u
1
− t0+ 200 min. After time t
n
t
, the inputs
n
t
simulation ensures that the states and outputs have reached steady state, that is, that
the process has transitioned from startup to its steady operation. The integral part of
the objective function g is integrated alongside the DAE system and h is calculated
at the end of each integration.
The nonlinear DAE system that simulates the upstream atropine synthesis in
each iteration has 9114 differential states and 29 algebraic states, resulting in
a very computationally intensive NLP unsuitable for real-time applications. The
computational time of the simulated operation of the plant from startup to steady
state in MATLAB is on the order of 10 s, when simulated on a laptop with Intel
i7-5600U CPU at 2.60 GHz and 16.0 GB RAM. A DO is a nonconvex problem in
general, hence converging to different local minima depending on the initial guesses
for the optimal solution vector. To address this, the optimization was initialized
with 100 random guesses, which were solved in parallel for a total time of 1–2
Fast Model Predictive Control of Modular Systems for Continuous... 307
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days in a system with two Intel Xeon Gold 6152 CPUs at 2.10 GHz and 262.0 GB
RAM. The DOs were carried out in MATLAB using an adaptive mesh method.
Reducing the computational time by using coarser spatial discretizations for the
tubular reactors resulted in suboptimal startup. To control dynamical operations in
real time, the DO problem can be solved offline and then the optimal trajectory
can be provided as a setpoint to a fast linear MPC algorithm implementable in real
time, thus suppressing the effects of disturbances and model uncertainties during
the operation. Implementation of this approach on the atropine synthesis case study
is presented in Sect. 3.2.
3 Control and Optimization Case Study Results
This section presents and discusses simulation results for the plant-wide control and
dynamic optimization of the upstream atropine synthesis.
3.1 Quadratic Dynamic Matrix Control Simulation Study
For the atropine synthesis control case study, QDMC was implemented and Wu,
p, and c were tuned while simultaneously screening various linear models. Design
parameters were selected to produce the most satisfactory closed-loop performance
across all tests, which resulted in a prediction horizon of 300 min and a control
horizon of 30 min. The prediction horizon was about 6–10 times the residence time
of the nominal operation and 1.2 times the maximum residence time of the system
(i.e., when all the flowrates are at their minimum possible value). This approach
was followed to approximate an infinite-horizon control. The control horizon was
chosen to be 1/10 of the prediction horizon. The weight matrix was W
for all MVs when controlling the production rate, and W
u
MVs when controlling the yield. The manipulated variables were constrained based
on the pump specifications and the scale of the plant. The rate of change of
the manipulated variables was set to 10% of their steady-state values to avoid
aggressive control moves that might wear off the pumps or result in instabilities. The
control simulation and the quadratic optimization were implemented in MATLAB.
For a closed-loop plant operation of 8 h, the computational time of a closed-loop
simulation was in the order of 6 min in a laptop with i7-5600U CPU at 2.60 GHz
and 16.0 GB RAM. Sections “Production Rate Control”, “Atropine Yield Control”
and “Multivariable Control” show that this highly nonlinear dynamical system was
successfully controlled using QDMC equipped with the selected step responses
discussed in Section “Linear Input-Output Model Construction”.
= diag{100}
u
= diag{50} for all
308 A. Nikolakopoulou et al.
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Linear Input-Output Model Construction
To control the upstream atropine synthesis plant with QDMC, multiple linear
models were screened in closed-loop studies and the different linear IO model construction approaches discussed in Section “Linear Input-Output Model Construction
Methodology” were tested. Three model construction approaches used in singlevariable closed-loop studies are presented below. The corresponding step responses
are shown in Figs. 3 and 4.
• Model 1 (M1) refers to the high steady-state gain model for the MVs. For the
production rate as the CV, step responses that exhibit gain sign changes are
excluded for these MVs. The resulting model denoted as M1-P is [−80%, 0,
−80%, 0, −80%, −80%] where the first element of the vector refers to u
the second element of the vector refers to u
, etc. and −80% refers to the step
2
response corresponding to a step change in the MV of −80% from its nominal
value. For yield as the only CV, the high-gain model denoted as M1-Y is [−20%,
−20%, −80%, −40%, −80%, −80%].
• Model 2 (M2) refers to the high steady-state gain/fast-dynamics model with the
contribution of MVs resulting in gain sign change eliminated. When the gain
magnitude varies a lot, such as u
in Fig. 3, the low-gain model is selected. Then
3
the models M2-P and M2-Y are [−40%, 0, +80%, 0, −80%, −80%] and [+80%,
−20%, 0, −40%, −80%, −80%] for the production rate and yield, respectively.
The models that can be chosen are not unique, because there is a tradeoff between
the fast dynamics and the high steady-state gain. The decision to eliminate the
contributions of u
to the yield output is made due to big variations in the gain
3
magnitude for that input.
• Model 3 (M3) refers to a similar model as M2 but does not eliminate the step
responses for any MVs. Instead, linear models with a low steady-state gain are
used, in cases where gain sign change occurs. These models are selected because
of their fast dynamics and their ability to describe the behavior of the plant for
most cases. Additionally, lower gains will maintain a not-too-sluggish control
action. Hence the models [−40%, +80%, +80%, +80%, −80%, −80%] and
[+80%, −20%, +80%, −40%, −80%, −80%] denoted as M3-P and M3-Y are
selected for the production rate and yield, respectively.
,
1
For multivariable control, the linear model interactions should be taken into
account when designing the QDMC model. For the atropine synthesis case study,
results are presented for
• Model 1 (M1-M) with [−80%, 0, −80%, 0, −80%, −80%] for the production
rate and [+80%, −80%, −80%, +80%, −80%, −80%] for the yield.
• Model 2 (M2-M) with [−40%, 0, +80%, 0, −80%, −80%] and [+80%,
+80%, +80%, +80%, −80%, −80%] for the production rate and yield as CVs,
respectively, and
• Model 3 (M3-M) [−40%, +80%, +80%, +80%, −80%, −80%] and [+80%,
+80%, +80%, +80%, −
80%, −80%] for the production rate and yield.
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The stepresponses for theyield in multivariable control were obtained by linearizing
the plant around the same operating conditions as those for the atropine production
rate in Fig. 3. These operating conditions are not the same as for single-variable
control of yield seen in Fig. 4. The resulting step responses for yield demonstrated
mild directional nonlinearities, and time delays, but no steady-state sign changes,
similarly to Fig. 4.
In the rest of this chapter, the same model notation is used to denote the QDMC
designed with the corresponding linear model, e.g., M1-P refers to QDMC with
M1-P as its predictive linear model for production rate as the CV, M1-Y refers to
QDMC with the predictive linear model M1-Y for yield as the CV, etc.
Production Rate Control
The closed-loop responses for a setpoint tracking scenario for the production rate
of atropine with up to +40% and −20% deviations from the nominal value, are
shown in Fig. 6a. The QDMC with the predictive models M2-P and M3-P resulted in
good closed-loop performance, reaching nearly zero steady-state error within 250–
300 min after the setpoint change. The MPC provided very slow setpoint tracking
for model M1-P, which is expected since using a high-gain linear model results in a
very slow response when the plant is operating in a lower gain region.
The closed-loop responses were also tested for several disturbance scenarios.
Some characteristic responses for temperature disturbances are shown in Fig. 6b.
The temperatures of the reactors 1 and 2 (Fig. 2) are held constant by keeping
a jacket at a constant temperature using resistances. However, a disturbance in
that constant temperature will need to be rejected by manipulating the flowrates,
since the physical system design does not allow for temperature manipulation on
a short time scale. Small step disturbances ±1
temperatures T
and T2. The controller successfully rejects each of the disturbances
1
◦
C and ±5◦C were applied to the
after about 250 min when the linear models M2-P and M3-P are used, while M1-P
is very slow at rejecting the disturbances.
Step disturbances in the MVs were also examined. The disturbance rejection in
one of the inputs, namely u
, for a constant disturbance equal to +40% and −40%
1
of the nominal value is shown in Fig. 6c. The results were similar for all other MV
disturbances; M2-P and M3-P were observed to perform better than M1-P under
the same circumstances. Finally, we examined scenarios where simultaneous MV
disturbances occur. Two of the scenarios are shown in Fig. 6d, e. The time scale of
the disturbance rejection as well as the model performance is similar to that of all
the previously discussed cases.
Overall, the linear models M2-P and M3-P resulted in controllers that give
significantly better closed-loop performance than M1-P. The controller using M1P as a predictive model has slower, more sluggish dynamics across all tests, as
expected. No significant difference in the closed-loop performance between models
M2-P and M3-P was observed. M1-P differs from M2-P with respect to the step
responses associated with u
and u2, with higher steady-state gain step responses
1
310 A. Nikolakopoulou et al.
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Fig. 6 Plant-wide control of atropine production rate based on QDMC equipped with three
different linear models. The disturbances are steps introduced at the 2 min mark, starting from
a steady-state operation. (a) Setpoint tracking of production rate. (b) Closed-loop response for a
disturbance in thesecond reactor temperature T
a disturbance in the input u
to simultaneous disturbances in the inputs u
respectively. (e) Closed-loop response to simultaneous disturbances in the inputs u
−40% of the nominal value for both inputs
of +1◦Cand−5◦C. (c) Closed-loop response for
,for+40% and −40% of the nominal value. (d) Closed-loop response
1
2
and u2,for−40% and +40% of the nominal value
1
and u4,for
3
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having been selected for M1-P. M2-P and M3-P differ with respect to the step
responses for u
and u4, with no step responses being used for the former model and
2
low-gain responses being selected forthe latter model for bothMVs. Eliminating the
step responses does not seem to have a large effect on the closed-loop performance
if the linear model selection corresponding to the remaining MVs is done carefully.
Additionally,M3-P performs satisfactorily even without eliminating these two MVs.
When opting to not exclude models with gain sign change, care must be taken to
select models with the appropriate gain. Constructing a model with high steadystate gain that also exhibits gain sign change behavior will likely result in very poor
closed-loop performance.
Atropine Yield Control
On-demand manufacturing can require operation in different conditions and/or
control of different CVs depending on the utilization of the plant. To this end, this
section evaluates the closed-loop performance of QDMC in an additional operating
region of the plant while controlling a different CV, yield.
The closed-loop responses are shown in Fig. 7a for setpoint tracking of the yield
of atropine of up to +15% and −35% deviation from the nominal value. All linear
controllers provide good setpoint tracking control due to the similarities between the
constructed linear models and the significantly reduced nonlinear effects observed
for yield as a CV compared to the production rate (Fig. 4).
Step disturbances in the reactor temperatures T
and T2by ±1 and ±5◦Cwere
1
considered. Some characteristic closed-loop responses are shown in Fig. 7b. M2-Y
and M3-Y provide a slightly slower closed-loop response in this case, by rejecting
the disturbance at approximately 150min and 200 min, respectively, whereas M1-Y
rejects the disturbance at about 100 min.
Figure 7c is a representative closed-loop response for a disturbance in an MV,
for +40% and −40% on the nominal value of u
. Similarly to the response for a
4
temperature disturbance, M1-Y provides a faster closed-loop response compared to
M2-Y and M3-Y. Scenarios of simultaneous MV disturbances were also examined,
with two of the scenarios shown in Fig. 7d, e. For a simultaneous disturbance in
u
and u3shown in Fig. 7d, M3-Y provides the fastest closed-loop response, being
1
able to reject the disturbance at about 100 min while M1-Y and M2-Y reject the
disturbance at about 150 min. For the simultaneous disturbance in u
and u4shown
3
in Fig. 7e, M3-Y provides again the fastest closed-loop response, being able to
reject the disturbance at about 200 min while M2-Y rejects the disturbance at about
250 min. M1-Y is not able to reject the disturbance within 350 min.
M1-Y and M2-Y differ with respect to MVs u
state gain model is used in M1-Y while u
is eliminated in M2-Y since it exhibits
3
and u3.Foru1a higher steady-
1
a very large magnitude difference between the various gains. M2-Y and M3-Y
differ only with respect to the model for u
, with M3-Y using a model with a
3
low gain. All of these models demonstrate good closed-loop performance for most
cases. M1-Y is faster at rejecting disturbances in temperature and single MVs. M3-
312 A. Nikolakopoulou et al.
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Fig. 7 Plant-wide control of atropine yield based on QDMC equipped with three different linear
models. The disturbances are steps introduced at the 2 min mark, starting from a steady-state
operation. (a) Setpoint tracking of yield. (b) Closed-loop response for a disturbance in the second
reactor temperature T
input u
,for+40% and −40% of the nominal value. (d) Closed-loop response to simultaneous
4
of +1◦Cand−5◦C. (c) Closed-loop response for a disturbance in the
2
disturbances in the inputs u
loop response to simultaneous disturbances in the inputs u
nominal value, respectively
and u3,for−40% of the nominal value for both inputs. (e) Closed-
1
and u4,for−40% and +5% of the
3
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Fig. 8 Plant-wide multivariable control of (a) atropine production rate and (b) yield based
on QDMC equipped with three different linear models. The simultaneous step disturbances in
the MVs u
introduced at the 2 min mark for a plant starting from steady-state operation
,for+40%, −40%, +40%, and −40% of the nominal value, respectively, are
1–u4
Y has the slowest response by a small margin for setpoint tracking and rejection
of disturbances in temperature and single MVs, but offers the fastest closed-loop
performance when there are multiple disturbances in the MVs. Overall, M2-Y has
the most consistent closed-loop performance.
Multivariable Control
An advantage of advanced control methodologies such as QDMC is their ability
to handle multivariable control. The production rate and yield of atropine were
simultaneously controlled in some extreme operating scenarios to further test
the linear model performance. The four MVs affecting the CVs the most were
simultaneously disturbed by 5 and 40% in increasing and decreasing directions. One
of these cases is shown in Fig.8. Simultaneous disturbance rejection in both CVs is
achieved by the controllers using models M2-M and M3-M at about 250 min.
M2-M and M3-M differ only with respect to the production rate model.
Eliminating the contributions of the MVs with step responses demonstrating a gain
sign change (u
and u4) for the production rate in M2-M seems to result in faster
2
closed-loop response. M1-M and M2-M have different models for both CVs. M1-M
uses the high-gain model for production rate while M2-M adopts lower gain step
responses for inputs u
and u3. Regarding the yield predictive model, M1-M adopts
1
a higher gain step response model compared to M2-M. Overall, M2-M outperforms
the other controllers. M1-M provides a much slower disturbance rejection having
a more sluggish, conservative control, compared to M2-M as expected, which is
in agreement with the observations from the previously examined cases that the
proposed models M2 (M2-P, M2-Y) and M3 (M3-P, M3-Y) provide an overall better
control of the plant compared to M1 (M1-P, M1-Y).
314 A. Nikolakopoulou et al.
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3.2 Control of Startup
Offline computation of the NLP formulated in Sect. 2.4 resulted in optimal trajec-
tories that were then used as a setpoint for QDMC to control the plant from the
point of no operation where the system is filled with the organic solvent to steady
state. The input-output formulation of QDMC was exploited to enable online control
implementation, since the associated QP is not a function of the state dimension.
Controlling startup which is a highly nonlinear region of operation with LMPC
can be challenging. Therefore, it is important to select a linear model that results in
good closed-loopperformance when appliedto a nonlinearsystem. The linearmodel
construction was informed by the insights gained from the analysis presented in
Sects. 2.3 and 3.1. The process was linearizedaround the final steady-state operating
conditions. QDMC was implemented for a prediction horizon of 300 min, a control
horizon of 30 min, and a weighting factor of W
Dynamic Optimization Results
Dynamic optimization results using piecewise constant and continuous piecewise
affine input vector parametrizations are denoted as pwc and pwa, respectively,
followed by the number of stages n
n
= 1 and nt= 2. The resulting solutions were very similar, with the optimization
t
pwc
returning a slightly better value for the objective function as seen in Fig. 9b.
nt=2
(Figs. 9 and 10). Results are presented for
t
The optimal values for the inputsseen in Fig.10 vary the most between formulations
for u
and u4.Bothu1and u3operate at their maximum value most of the time. The
3
buffer solution and organic solvent flowrates u
value of 0, which results in minimal waste. This result is an artifact of the model,
which does not capture the phenomena related to varying pH in the LLS unit in great
detail due to limited experimental data. Interestingly, the optimal temperature T
significantly lower for pwc
than for the other parameterizations, indicating that
nt=1
the objective function is not very sensitive to T
= diag{300}.
u
and u6are realized in their trivial
5
.
1
1
is
Quadratic Dynamic Matrix Control of Startup
Three MPC strategies for startup control were compared for each of the DO results
(Fig. 11). The first MPC strategy (labeled as “QDMC” in the plots) used the
dynamical trajectory obtained from the DO as a setpoint. Additionally, an approach
denoted as “QDMC+O” was investigated to examine if there is any significant
benefit from implementing the DO optimal inputs during the time period that there
is no product exiting from the system and hence no feedback to the MPC (this time
is from t = 0 to the residence time of the overall system). The third strategy labeled
as “QDMC SS” in the plots, was to use a steady-state setpoint to MPC (obtained as
the value of the DO solution at steady-state).
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Fig. 9 Dynamic optimization results for pwc
nt=1
,pwa
production rate during startup. (b) Objective function value
0.1
0.05
0
0 100 200
0.1
0.05
0
0 100 200
400
350
300
0 100 200
0.1
0.05
0
0 100 200
0.8
0.6
0.4
0.2
0 100 200
400
350
300
, and pwc
nt=1
0 100 200
nt=2
0.1
0.05
0
0 100 200
0.2
0.1
0
0 100 200
[18]. (a) Optimal
Fig. 10 Optimal inputs for the startup of upstream atropine synthesis for pwc
pwc
nt=2
[18]
nt=1
,pwa
nt=1
,and
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