integration of design and control: a robust control approach by nongluk chawankul peter l. douglas...
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Integration of Design and Control: A Robust Control Approach
by
Nongluk ChawankulPeter L. Douglas
Hector M. Budman
University of WaterlooWaterloo, Ontario, Canada
Process design
Background
Control performance depends on the controller and the design of the process.
Traditional design procedure: Step 1: Process design (sizing + nominal operating conditions) Step 2: Control design
Idea of integrating design and control:
Process control+ = Integrated approach
Background
Traditional design and control design
Integrated design and control design
Step 1: Process design
Step 2: Control design
Only one step design
Cost = capital cost(x) + operating cost(x) + cost related to closed loop system(x,y)
where x is design variable y is control tuning parameter
Objective Function (Cost)
Two steps design
Process constraints Equality constraints, h(x) = 0 Inequality constraints, g(x,y) 0
Min Cost(x,y) x,y
s.t. h(x) = 0g(x,y) 0
Cost = Capital cost(x) + operating cost(x)
where x is design variable
Min Cost(x) x
s.t. h(x) = 0g(x) 0
Optimum design
Optimum design
Design controller
Closed loop system
• Nonlinear Dynamic Model (difficult optimization problem)
• Variability cost not into cost function: Multi-objective optimization
• Decentralized Control : PI /PID
• Linear Nominal Model + Model Uncertainty (Simple optimization problem)
• Variability cost into cost function : One objective function
• extended to Centralized Control : MPC
Previous studies Our study
Integrated Design and Control Design
Case study
• Case study II: SISO MPC• Case study III: MIMO MPC
Feed
RR
Ethane Propane Isobutane N-Butane N-PentaneN-Hexane
A
XD*+
-
Q
Feed
RR
Ethane Propane Isobutane N-Butane N-PentaneN-Hexane
A1
MPCXD*
+-
Q
IMC or MPC
SISO system MIMO system
A2
-+
XB*
XD
XB
XD
XB
MIMO case study:
RadFrac model in ASPEN PLUS was used.
Different column designs, 19 – 59 stages were studied.
Product specifications Mole fraction of propane in distillate product = 0.783 Mole fraction of isobutane in bottom product = 0.1
Design variables are functions of nominal RR at specific product compositions.
Case Study III: MIMO MPC
U is a vector of design variables. C is a vector of control variables. Lm is a set of uncertainty.
Optimization
Minimize Cost(U,C) = CC(U) + OC(U) + max VC(U,C) U,C Lm
Such that h(U) = 0 (equality constraints)
g(U,C) 0 (inequality constraints)
Objective Function
Objective Function
• Capital Cost, CC
– Cost of sizing, e.g. number of stages N and column diameter D
– Capital cost for distillation column from Luyben and Floudas, 1994 ($/day)
• Operating Cost, OC
– Operating cost from Luyben and Floudas, 1994 ($/day)
)5.17.0(245))76.06(486324615(3.12 22 DNDNDCC
UCOPRRHDOC tax )(
where tax = tax factorHD = reboiler duty (GJ/hr)OP = operating period (hrs)UC = Utility cost ($/GJ)
Capital Cost (CC) and Operating Cost (OC)
Feed
RR
Ethane Propane Isobutane N-Butane N-PentaneN-Hexane
A1
MPCXD*
+-
Q
A2
-+
XB*
V1
V2
t
t
t
t
t
- Variability cost, VC = inventory cost
- sinusoid disturbance induces process variability
- consider holding tank to attenuate the product variation
Variability Cost, VC
Variability Cost (VC)
)()1()1()()()( ,
11, zzzzz MPCmnomnpMPCmnomnn
Y WKSMKSMIWGY
Assume, W is sinusoidal disturbance with specific d. (alternatively, superposition of sinusoids)
WGY )( jYTjez With phase lag
Re
Imtan 1
Consider worst case variability :
)2,1(maxty variabilimaximumm
YY
max,mm
Calculation of Variability Cost (VC) - 1
Related to maximum VC
Objective Function (-cont-)
1
1
sC
C
in
out
Apply Laplace transform
din
out
Q
VC
C
,
1
12
The product volume in the holding tank
inC
VQ in Q out
Cin Cout
dout
in Q
C
CV
Calculation of Variability Cost (VC) - 2
VC1 = W1P1V1(A/P,i,N) VC2 = W2P2V2(A/P,i,N)
VC = VC1+ VC2
Equality Constraints
Process Models: ASPEN PLUS simulations at specific product compositions
Equality Constraints: Process models -1
3800
3900
4000
4100
4200
4300
4400
4500
4600
1.6 1.8 2 2.2 2.4 2.6 2.8
RR
Boi
lup
rate
,Q (
lbm
ol/h
r)
ASPEN simulation
equation (6.1)
0
10
20
30
40
50
60
70
1.6 1.8 2 2.2 2.4 2.6 2.8
RR
Num
ber
of s
tage
s, N ASPEN simulation
equation (6.2)
3.35
3.4
3.45
3.5
3.55
3.6
3.65
1.6 1.8 2 2.2 2.4 2.6 2.8
RR
Col
umn
Dia
met
er, D
(m
)
ASPEN simulation
equation (6.3)
Q(RR) N(RR)
30
31
32
33
34
35
36
37
1.6 1.8 2 2.2 2.4 2.6 2.8
RR
Hea
t d
uty
in
reb
oil
er,
HD
(G
J/h
r)
ASPEN Simulation
equation (6.4)
HD(RR) D(RR)
Process Models: ASPEN PLUS simulations
960
965
970
975
980
985
990
1.6 1.8 2 2.2 2.4 2.6 2.8
RR
Dis
tilat
e ra
te (
lbm
ol/h
r)
ASPEN simulation (tol = 0.001)
equation (6.5)
ASPEN simulation (tol = 0.00001)
2235
2240
2245
2250
2255
2260
2265
1.6 1.8 2 2.2 2.4 2.6 2.8
RR
Bot
tom
rat
e (l
bmol
/hr)
ASPEN simulation (tol = 0.001)
equation (6.6)
ASPEN simulation (tol = 0.00001)
DF(RR)
BF(RR)
Equality Constraints: Process models - 2
Process Models: Input/Output Model for 22 system
First Order Model
ip
t
ipi eKy ,)(
, 1
yi
t
S1
S2
S3
Sn
1%
35%
y
0
y1
y2
time
-35%
+35%
+1%
-1%
2%1
2
%35
1
,
RRy
RRy
K RRp
2
%1
2
%40
1
,
Qy
Qy
K Qp
Process gains
In a similar fashion, time constants and dead time
p(RR) and p(Q)
(RR)
and
Kp1(RR) for paring xD-RR
Kp2(RR) for paring xB-RR
Kp3(Q) for paring xD-Q
Kp4(Q) for paring xB-Q
Equality Constraints: Process models - 3
0.12
0.14
0.16
0.18
0.2
0.22
1.6 2.1 2.6 3.1
RR
Proc
ess
gain
of
x D-R
R, K
p1
ASPEN PLUS simulation
equation (6.10)
0.027
0.028
0.029
0.03
0.031
0.032
0.033
1.6 2.1 2.6 3.1
RR
Proc
ess
gain
of
x B-R
R, K
p2
ASPEN PLUS simulation
equation (6.11)
0.45
0.5
0.55
0.6
0.65
0.7
1.9 2.1 2.3 2.5
Q' (lbmole/hr)
Proc
ess
gain
of
x D-Q
', K
p3
ASPEN PLUS simulation
equation (6.12)
0.11
0.12
0.13
0.14
0.15
0.16
1.9 2.1 2.3 2.5
Q' (lbmole/hr)
Pro
cess
gai
n o
f x B
-Q',
Kp
4
ASPEN PLUS simulation
equation (6.13)
Process gains for 2 2 system
Equality Constraints: Process models - 4
0
5
10
15
20
25
1.5 2 2.5 3 3.5
RR
Proc
ess
time
cons
tant
, p
1
(min
)
ASpen simulation
equation (6.16)
0
5
10
15
20
25
1.2 1.3 1.4 1.5 1.6 1.7
Q' (lbmole/hr)
Proc
ess
time
cons
tant
, p
2
(min
)
ASPEN simulation
equation (6.17)
4
4.5
5
5.5
6
6.5
1.6 1.8 2 2.2 2.4 2.6 2.8 3
RR
Dea
d t
ime,
(m
in)
ASPEN simulation
equation (6.18)
Process time constants: p(RR) and p(Q)
Process dead time: (RR)
Equality Constraints: Process models - 5
max,m
Model uncertainty
lowernuppernnn
m SorSSS
SS,,
nomn,
nomn,max, ,max
Time
y
Sn,upper
Sn,lower
Sn,nom
)(
)(
)(
)(
4max,
3max,
2max,
1max,
Q
Q
RR
RR
m
m
m
m
xD-RR
xB-RR
xD-Q
xB-Q
max,m
Equality Constraints: Process models - 6
00.0005
0.0010.0015
0.0020.0025
0.0030.0035
0.004
1.6 2.1 2.6 3.1
RR
Unc
erta
inty
(x
D-R
R)
ASPEN PLUS simulation
equation (6.22)0
0.001
0.002
0.003
0.004
0.005
1.6 2.1 2.6 3.1
RR
Unc
erta
inty
(x
B-R
R)
ASPEN PLUS simulation
equation (6.23)
0.11
0.16
0.21
0.26
0.31
0.36
1.9 2.1 2.3 2.5
Q' (lbmole/hr)
Unc
erta
inty
(x
D-Q
') ASPEN PLUS simulation
equation (6.24)
0.025
0.03
0.035
0.04
0.045
0.05
1.9 2 2.1 2.2 2.3 2.4 2.5
Q' (lbmole/hr)
Unc
erta
inty
(x
B-Q
')
ASPEN PLUS simulation
equation (6.25)
Model Uncertainty for 22 system
Equality Constraints: Process models - 7
Inequality Constraints
1. Manipulated variable constraint
Inequality Constraints- 1
εKu ).(MPC
)()()( zz MPCU WKGu
is a tuning parameter. Large less aggressive control
WezGRRp
TjUj
1
1max )(
maxmax uuu nom
WezGQp
TjUj
1
2max )(
Two manipulated variables Calculate RR and Q
and
2. Robust stability constraint (Zanovello and Budman, 1999)
Li
Mp
Kmpc T1T2 H N1
W1 W2
N2
Z-1I
+ + ++
++
+
-+N1
-
M
(k+1/k)u(k)
U(k)
U(k-1)
Z(k) w(k)
H H
Block diagram of the MPC and the connection matrix M
Z-1I
U(k) U(k+1)
M
w z
1))(( jM
Inequality Constraints- 2
Two different approaches
Integrated Method Traditional Method
0)(
0)(uch that
min
RRg
RRhs
OCCCRR
1))((..
min,
Mts
Robust Performance (Morari, 1989)
max
such that
UΔUU nom
1)( M
VCOCCCmLQRR
maxmin,,
Where U is manipulated variables
maxUΔUU nom
Results
Results - 1
Results from Integrated design and control design approach
w1 w2 RR* * 11 N D (m)
1 1 1.913 0.2350 3.65 26 3.392
5 1 1.911 0.2341 3.63 26 3.392
10 1 1.908 0.2338 3.62 27 3.391
15 1 1.753 0.2331 3.03 38 3.370
20 1 1.753 0.2332 3.03 38 3.370
1 5 1.912 0.1886 3.64 26 3.392
1 10 1.909 0.1848 3.62 26 3.391
1 15 1.906 0.1836 3.61 27 3.391
1 20 1.904 0.1830 3.60 27 3.390
w1 or w2 increases;
-RR* decreases smaller dead time
11 decreases interaction decreases as RR decreases
* decreases RS constraint is easy to satisfy as 11 decreases
Compare Results from Traditional and Integrated design and control design approaches.
Results - 2
600
700
800
900
1000
1100
1200
1300
0 5 10 15 20 25
w 1
Tot
al c
ost (
$/da
y)
Traditional approach
Integrated approach
600
800
1000
1200
1400
1600
1800
0 5 10 15 20 25
w 2
Tot
al c
ost (
$/da
y)
Traditional approach
Integrated approach
Results - 3
RR max RR* * N D (m) CC ($/day) OC ($/day) VC ($/day) TC ($/day)
2.633 1.913 0.2350 26 3.392 195.98 586.21 23.34 805.53
2 1.913 0.2349 26 3.392 195.98 586.21 23.34 805.53
1.9 1.854 0.2024 38 3.370 259.76 570.48 72.62 902.86
1.8 1.761 0.2021 39 3.370 262.60 570.20 93.04 925.84
Effect of RRmax on Total Cost (TC)
Conclusions
1- For the case ≠ 0, using the integrated method, the optimization tends to select smaller RR values which correspond to smaller dead time and smaller interaction.
2- The optimal design obtained using the integrated method resulted in a lower total cost as compared to the traditional method.
3- Limit on manipulated variable affects the closed loop performance and leads to more cost.
ProcessMPC
W (Sinusoid unmeasured disturbance)
yr=0-
+++u
)1()1()( kkk nn uSYMY
)/1()( kkk MPC εKu )1/()1( kkk MPCεKu
)/1()()1()/1( kkkkkk p WYMRε )1/()1()1/( kkkkk p WYMε
)()(11 zzz MPCnpMPCnn WKSMKSMIY
Substitute (k), u(k-1) into the first equation and apply z-transform
Calculation of Variability Cost (VC) -1
Process variability
Results - 1
Results from Integrated design and control design approach for = 0
w1 w2 RR* * 11 N D (m)
1 1 1.921 0.2503 3.68 26 3.393
5 1 1.955 0.2509 3.83 26 3.399
10 1 1.980 0.2518 3.95 25 3.404
15 1 2.012 0.2521 4.11 25 3.410
20 1 2.103 0.2527 4.63 24 3.431
1 5 1.960 0.2511 3.86 26 3.400
1 10 1.992 0.2516 4.01 25 3.406
1 15 2.102 0.2522 4.63 24 3.431
1 20 2.234 0.2540 5.57 23 3.465
w1 or w2 increases;
-RR* increases uncertainty decreases as RR increases
11 increases interaction increases as RR increases
* increases RS constraint is more difficult to satisfy as 11 increases
0
5
10
1.6 1.8 2 2.2 2.4 2.6 2.8
RR
Results - 4
Compare savings when = 0 and 0
05
1015202530
0 5 10 15 20 25
w 1
Sav
ings
(%
)
Series1
Series2 0 = 0
0
20
40
60
80
100
0 5 10 15 20 25
w 2
Savin
gs (
%)
Series1
Series2
= 0 0