Integration of Design and Control : Robust approach using MPC and PI controllers. N. Chawankul, H. M. Budman and P. L. Douglas Department of Chemical Engineering University of Waterloo. Outline. Introduction Objectives Methodology Case study Results Conclusions. Introduction.
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N. Chawankul, H. M. Budman and P. L. Douglas
Department of Chemical Engineering
University of Waterloo
Outline
Introduction
Objectives
Methodology
Case study
Results
Conclusions
Integrated Approach
1 Design+Control
min (capital +operating
+variability costs)
st
stability
actuator constraints
Traditional Approach
1 Design:
min (capital + operating costs)
2 Control for designed plant
stability
actuator constraints
performance specs:
Small Overshoot
Short Settling Time
Large closed loop bandwidth
disturbance
output
For a disturbance d (green):
What is the cost due to offspec product (blue)?
Nonlinear model: stability (Lyapunovdifficult)
variability (numerically difficult).
“Robust” linear model:
nonlinear model= family of linear models
family of linear models= nominal model
+model uncertainty (error)
Introduction
Previous Approaches
Our approach
• Linear Dynamic Nominal Model +
Model Uncertainty
(Simple optimization problem)
• Variability cost into cost function :
One objective function
• Centralized Control : MPC
• Nonlinear Dynamic Model
(difficult optimization problem)
• Variability cost not into cost function:
Multiobjective optimization
• Decentralized Control : PI /PID
 Objective Function
 Constraints
d
Simplified MPC block diagram
W
Sd
r
u
y
+
+
MPC
Process
+
+
+

u(k)
past
future
target
y(k)
y(k+1/k)
k
k1
k+1
k+n
k+3
k+2
Step Response Model, Sn
y
u
Sn
S6
S5
S4
1
S3
S2
S1
t
0
t
u
Nominal step response model, Sn,nom
1
t
0
1
Uncertainty,
Upper bound
Nominal step response
Actual Sn
Lower bound
t
SnSn,nom
W
(Sinusoid unmeasured disturbance)
r=0
y
u
+
MPC
Process
+
+

Substitute (k), u(k1) into the first equation and apply Ztransform
disturbance
output
Assume, W is sinusoidal disturbance with specific d. (alternatively, superposition of sinusoids)
Amplitude of output,y
With phase lag
Amplitude of disturbance,W
Consider worst case variability :
MinimizeCost(u,c) = Capital Cost + Operating Cost + Variability Cost
u,c
Such thath(u,c) = 0(equality constraints)
g(u,c) 0(inequality constraints)
u is a vector of design variables.
c is a vector of control variables.
Inequality Constraints 1
Consider the MPC controller gain, KMPC:
Manipulated variable constraint
where
is a manipulated variable weight
The infinity norm of
A is the amplitude of the disturbance.
Inequality Constraints 2
Block diagram of the MPC and the interconnection M
M
Z(k)
w(k)
U(k)
W1
W2
Z1I
(k+1/k)
u(k)
U(k1)
+
T2
N1
+
Li
Kmpc
T1
+
+
H
+

2. Robust stability constraint (Zanovello and Budman, 1999)
H
N1
+
+
Mp
H

+
N2
Z1I
U(k+1)
U(k)
M
z
w
Case study Distillation ColumnPreliminary study: SISO system
Depropanizer column from Lee, 1994
adjust reflux ratioto control the mole fraction of propane in distillate
RR
A
+
Feed

MPC
= 0.783 (propane)
XD*
Ethane
Propane
Isobutane
NButane
NPentane
NHexane
Q
(Equality Constraints)
Process Model
The mathematic expressions of the process variables (N, D, Q) as functions of RR
RadFrac model in ASPEN PLUS
different column design, 19 – 59 stages
design variables (number of stages and column diameter) are functions of nominal RR
Number of stages VS. RR
(Equality Constraints)
y
Sn
S3
S2
S1
t
Dynamic simulation using ASPEN DYNAMICS
Step change on RR by 10 % (19 59 stages)
Upper bound
Nominal step response
Lower bound
63.2 %
Objective Function
Cost = CC(u) + OC(u) + VC(u,c)
Annualized capital cost, CC, (Luyben and Floudas, 1994) ($/day)
Operating cost, OC ($/day)
where Q = reboiler duty (GJ/hr)
OP = operating period (hrs)
UC = Utility cost ($/GJ)
Variability Cost (Inventory cost)  1
V1
Variability cost, VC ($/day)
 assume sinusoid unmeasured disturbance, W
disturbance induces process variability
consider a holding tank to attenuate the product variation
calculate the volume of the holding tank
 calculate the loss due to the product held in the tank
V2
where P = product price, N = payoff period (10 years),
i = interest rate (10%) and V = volume of the holding tank
Variability Cost (Inventory cost)  2
The worst case variability:
A simple
mass balance
Feed disturbance
Distillation
Column
Cin
Cout
Holding
V
F out
F in
The required volume of the holding tank
spec
Two different approaches
Traditional Method
Robust Performance
(Morari, 1989)
W
RR
D (m)
N
Capital cost
($/day)
Operating cost
($/day)
Variability cost
($/day)
Total cost ($/day)
1
2.4
0.1849
5.8
31
551
654
68
1273
3
2.8
0.1705
6.23
28
530
703
168
1401
5
3.0
0.1653
6.5
27
529
726
257
1512
7
3.4
0.1357
6.9
25
538
774
284
1596
Results1
Results from Integrated Method: W is a product price multiplier.
W
Total Cost of integrated method ($/day)
Total Cost of traditional method ($/day)
Saving ($/day)
% Saving
1
1273
1297
24
1.8
3
1401
1481
80
5.4
5
1512
1665
153
9.2
7
1596
1849
253
13.7
Results2
Comparison using Traditional and Integrated methods
(Morari and Zafiriou, 1989) is used.
Gd(s)
d
+
r
y
+
+
C(s)
Gp(s)

+
y’

Gp(s)
F(s)
IMCbased PID parameters for
Results
Comparison using both methods
Conclusions
Ongoing work: Formulate the MIMO problem with MPC
Acknowledgement
Funding was provided by The Natural Sciences and Engineering Research Council (NSERC)