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
OutlineIntroduction
Objectives
Methodology
Case study
Results
Conclusions
Introduction
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
Integrated Approach
1- Design+Control
min (capital +operating +variability costs)
st stability actuator constraints
Variability Cost
• Variability Cost = Cost of Imperfect Control
For a disturbance d (green):
What is the cost due to off-spec product (blue)?
disturbance
output
Robust Control Approach
• To test stability and calculate variability cost we need a model.
Nonlinear model: stability (Lyapunov-difficult) variability (numerically- difficult).
“Robust” linear model: nonlinear model= family of linear models
family of linear models= nominal model +model uncertainty (error)
• Nonlinear Dynamic Model (difficult optimization problem)
• Variability cost not into cost function: Multi-objective optimization
• Decentralized Control : PI /PID
• Linear Dynamic Nominal Model + Model Uncertainty (Simple optimization problem)
• Variability cost into cost function : One objective function
• Centralized Control : MPC
Previous Approaches Our approach
Introduction
Objectives of the current work
Variability using MPC based on a nominal model and model error.
Cost of variability in one objective function together with the design cost.
Model uncertainty (as a function of design variables) into the objective function.
The robust stability criteria as a process constraint.
Compare the traditional method to integrated method.
Preliminary study on SISO system (distillation column) with MPC.
Methodology
• Model Predictive Control (MPC)
• Nominal Step Response and Uncertainty
• Process Variability • Optimization
- Objective Function- Constraints
MPC Controller
Sd
ProcessMPC
d
W
yr +-
++ ++
u(k)
k k+1 k+nk-1
k+2 k+3
targetpast future
y(k)y(k+1/k)
Simplified MPC block diagram
u
Nominal Step Response and Uncertainty
,.....1,1,1u ,...,,,0 321 SSSy
t
y
1
tS1
S2
S3
S4
S5
S6 Sn
Step Response Model, Sn
Nominal step response model, Sn,nom
2
,,,
lowernuppernnomn
SSS
Uncertainty, m
nomn
nomnnm S
SS
,
,
Actual Sn
mnomnn SS 1,
Upper bound
Nominal step response
Lower bound
tSn-Sn,nom
u
t
u1
-1
n
n
S
SSS
S3
2
1
uy
0
0
Process variability-1y = f(W)
ProcessMPC
W (Sinusoid unmeasured disturbance)
yr=0-
+++u
)1()1()( kuSkYMkY nn
)/1()( kkKku MPC )1/()1( kkKku MPC
)/1()()1()/1( kkWkYMkRkk p )1/()1()1/( kkWkYMkk p
)()(11 ZWKSZMKSMIZY MPCnpMPCnn
MPCnpMPCnn KSZMKSMIZWZY 11
)()(
Substitute (k), u(k-1) into the first equation and apply Z-transform
Process variability-2
MPCmnomnpMPCmnomnn KSZMKSMIZWZY
ZG )1()1()()(
)( ,11
,
Assume, W is sinusoidal disturbance with specific d. (alternatively, superposition of sinusoids)
ImRe)(ˆ
jjGAAAR TjeZ With phase lag
ReImtan 1
ImRe,
,1,
1,2
,12,11,1
jG
GG
GGGG
AR ij
pnpnn
p
pGGGA ,12,11,1 ...1 B output, theof Bound
Consider worst case variability :
1092
83
74
65
56
47
38
29
1
1092
83
74
65
56
47
38
29
10
1)*(1
bZbZbZbZbZbZbZbZcbZbaZaZaZaZaZaZaZaZaZaa
Gmm
mij
Bm
maxty variabilimaximum
disturbance
output
Amplitude of disturbance,W
Amplitude of output,y
max,mm
u is a vector of design variables. c is a vector of control variables.
MinimizeCost(u,c) = Capital Cost + Operating Cost + Variability Cost u,c
Such that h(u,c) = 0 (equality constraints)g(u,c) 0 (inequality constraints)
Optimization
Constraints
h(u,c) = 0 (equality constraints)- steady state empirical correlations
g(u,c) 0 (inequality constraints)- manipulated variable constraint- robust stability
1. Manipulated variable constraint
Inequality Constraints- 1
TT
pT
pTT
pMPC SSSK1
I.
ppMPCnpMPCnnnnpMPC IKSZMKSMIMK
ZWZu 11
)()(
Consider the MPC controller gain, KMPC:
where is a manipulated variable weight
TeWeu
AZWZu
Au j
j
,)()(
max)()(
max],0[The infinity norm of
)()(
ZWZu
maxuuunom
A is the amplitude of the disturbance.
Inequality Constraints- 2
2. Robust stability constraint (Zanovello and Budman, 1999)
Li
Mp
Kmpc T1 T2 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 interconnection M-
Z-1IU(k) U(k+1)
M
w z
1))(( jM
Case study- Distillation Column Preliminary study: SISO system
Feed = 0.783 (propane)
RR
Depropanizer column from Lee, 1994
adjust reflux ratio to control the mole fraction of propane in distillate
Ethane Propane Isobutane N-Butane N-PentaneN-Hexane
A
MPC XD*
+-
Q
Process Model
RadFrac model in ASPEN PLUS different column design, 19 – 59 stages design variables (number of stages and column diameter) are
functions of nominal RR
010203040506070
1 2 3 4 5 6 7
RR
Num
ber o
f sta
ges,
N
Number of stages VS. RR
432 ))(ln(006.1
))(ln(934.3
))(ln(554.0
)ln(23.247303.6
RRRRRRRRN
7523.22365.1 RRD
649.196777.6 RRQ
The mathematic expressions of the process variables (N, D, Q) as functions of RR
(Equality Constraints)
Input/Output Model
First Order Model
00.020.040.060.080.1
0.120.140.16
0 0.2 0.4 0.6 0.8t
Nominal step response
Upper bound
Lower bound
3452.0)log(0963.0 RRK 432 ))(ln(363.0
))(ln(747.4
))(ln(41.21
ln892.41322.24
RRRRRRRRp
Step change on RR by 10 % (19- 59 stages)
p
t
eKy
1
5432max, 0469.07913.09532.45541.141858.200143.11 RRRRRRRRRRm
y
t
S1
S2
S3
Sn
Dynamic simulation using ASPEN DYNAMICS
63.2 %
RRy
K
y
max,m
p
(Equality Constraints)
Cost = CC(u) + OC(u) + VC(u,c)
Annualized capital cost, CC, (Luyben and Floudas, 1994) ($/day)
Operating cost, OC ($/day)
)5.17.0(245))76.06(486324615(3.12 22 DNDNDCC
UCOPRRQOC )(
where Q = reboiler duty (GJ/hr)OP = operating period (hrs)UC = Utility cost ($/GJ)
Objective Function
Variability cost, VC ($/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
Variability Cost (Inventory cost) - 1
),,/(),( NiPAVPcuVC
where P = product price, N = payoff period (10 years), i = interest rate (10%) and V = volume of the holding tank
V1
V2
dout
in FCCV
1
2
The required volume of the holding tank
inC
F in F out
Cin Cout
The worst case variability:
BCm
in max
Distillation Column
Feed disturbance
A simple mass balance
Holding V
Variability Cost (Inventory cost) - 2
spec
Two different approaches
Integrated Method Traditional Method
maxmin..
min
RRRRRRts
OCCCRR
1))((..
min,
Mts
Robust Performance (Morari, 1989)
max
..
uuu
ts
nom
1)( M
VCOCCCmRR
maxmin,
Results-1
Results from Integrated Method: W is a product price multiplier.
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
Results-2
Comparison using Traditional and Integrated methods
W Total Cost of integrated method ($/day)
Total Cost of traditional method ($/day)
Saving ($/day) % Saving
1 1273 1297 24 1.8
3 1401 1481 80 5.4
5 1512 1665 153 9.2
7 1596 1849 253 13.7
IMC Control
1)(
seK
sGp
sp
p
1111)(
sssksc
FIDc
Internal Model Control, IMC
C(s) Gp(s)
F(s)
d++ +
+--
r
y’
y
IMC-based PID parameters for
(Morari and Zafiriou, 1989) is used.
Gd(s)
Gp(s)
Results
Comparison using both methods
P Integrated Method Traditional Method Saving ($/day)
% Saving
RR c (sec) Total cost ($/day)
RR c (sec) Total cost ($/day)
1 1.5711 166.44 2624.2 1.5711 214.832 2666.67 42.45 1.59
3 1.3481 136.702 3970.6 1.5711 214.832 4592.41 621.81 13.53
5 1.3358 121.163 4259.7 1.5711 214.832 6518.15 2258.45 34.65
6 1.3314 115.616 4367.4 1.5711 214.832 7481.02 3113.62 41.62
Conclusions
• single objective function
• linear dynamic model + model uncertainty
• MPC variability cost is explicitly incorporated in the objective function
• integrated approach results in lower costs - savings can be significant; >13% for high value products
On-going work: Formulate the MIMO problem with MPC
Acknowledgement
Funding was provided by The Natural Sciences and Engineering Research Council (NSERC)