REAL TIME OPTIMIZATION:A Parametric Programming Approach
Vivek Dua
YYou ou Only Solve OOnly Solve OncenceYYou ou Only Solve OOnly Solve Oncence
Parametric Programming
Given:a performance criterion to minimize/maximizea vector of constraintsa vector of parameters
Obtain: the performance criterion and the optimization
variables as a function of the parameters the regions in the space of parameters where
these functions remain valid
Parametric Optimization (POP)
Obtain optimal solution as a Obtain optimal solution as a function of parametersfunction of parameters
Obtain optimal solution as a Obtain optimal solution as a function of parametersfunction of parameters
s
0),( s.t.
),(min)(
θ
x
θxg
θxfz
n
x
)(x
)(x
Critical Region
An Example – Linear ModelAn Example – Linear Model
Crude Oil # 1
Crude Oil # 2
REFINERY
GasolineKeroseneFuel OilResidual
Objective: Maximize ProfitParameters:Gasoline Prod. Expansion (GPE)Kerosene Prod. Expansion (KPE)
Solve optimization problems at many points?Solve optimization problems at many points?Solve optimization problems at many points?Solve optimization problems at many points?
24,000 bbl/day 2,000 bbl/day 6,000 bbl/day
Current Max. Prod.
GPE
KPE
(Edgar and Himmelblau, 1989)(Edgar and Himmelblau, 1989)
Parametric Solution
Only 2 optimization problems solved!Only 2 optimization problems solved!Only 2 optimization problems solved!Only 2 optimization problems solved!
Profit = 4.66 GPE + 87.5 KPE + 286759
Crude#1 = 1.72 GPE – 7.59 KPE + 26207
Crude#2 = -0.86 GPE + 13.79 KPE + 6897if
-0.14 GPE + 4.21 KPE < 896.550 < GPE < 60000 < KPE (REGION #1)
Profit = 7.53 GPE + 30541
Crude#1 = 1.48 GPE + 24590
Crude#2 = -0.41GPE + 9836if
-0.14 GPE + 4.21 KPE > 896.550 < GPE < 6000KPE < 500 (REGION #2)
GPE
KPE
GPE
KPE
Region #2
Region #1
Real Time Optimization
OPTIMIZER
SYSTEM
System StateControl Actions
Model Predictive Control (MPC)
past future
target
outputmanipulated
variable
k k+1 k+pPrediction Horizon
Model Predictive Control
c
c
N
k
N
kNukuku
Nkukuu
Nkxxkxx
kukxfkx
kRukukQxkxy u
,...,1,0,)(
,...,1,0,)1(
))(),(()1(s.t.
)]()'([)]()('[min
maxmin
maxmin
0 0)(),...,(
Solve an optimization problem at each time interval k
Model Predictive ControlModel Predictive Control
min A quadratic and convex function of discretized state and control
variables
s.t. 1. Constraints linear in discretized state and control variables
2. Lower and upper bounds on state and control variables
Solve a QP at each time intervalSolve a QP at each time intervalSolve a QP at each time intervalSolve a QP at each time interval
Parametric Programming Approach
State variables Parameters
Control variables Optimization
variables
MPC Parametric Optimization problem
Control variables = F(State variables)
Multi-parametric Quadratic Programs
m
2
1
s.t.
min)(
n
TT
x
x
FbAx
Qxxxcz offunction linear are and x
Theorem 1:
Theorem 2:
quadratic and
convex ,continuous is )(z
icesector/matrconstant v ;matrix constant definite positive
smultiplier Lagrange ; parameters ; variablescontinuous
b,c,A,FQ
x
Critical Region (CR)
CR: the region where a solution remains optimal Feasibility Condition:
Optimality Condition:
CR: A polyhedron Obtain:
FbAx )(
0)( 1
2
CR
321rest CRCRCR CRCR
1CR
3CR
2CR
Real Time Optimization
POP
PARAMETRIC PROFILE
SYSTEM
System StateControl Actions
Function Evaluation!Function Evaluation!Function Evaluation!Function Evaluation!
OPTIMIZER
SYSTEMS
SYSTEMSYSTEM STATESTATE
CONTROLCONTROL ACTIONSACTIONS
Example
1,0,22
4142.10
0064.0
0609.0
9909.01722.0
0861.07326.0s.t.
][ min)(
1
1
02
1222
1
, 1
ku
xy
uxx
RuuQxxPxxxJ
kt
tt
ttt
ktTktkt
Tkt
kt
Tt
uut
tt
Explicit Solution
..
0267.00353.0
6341.2
0922.01259.01215.01044.06452.44155.3
if 2
2222
8291.65379.18291.65379.18883.69220.58883.69220.5
if 8883.69220.5
x
xx
u
1
2,4
1x
2x
Explicit Solution
6423.20357.0
3577.1
6953.44159.61220.00275.0
6953.44159.6 if 6423.0 6953.44159.6
0267.00353.0
6341.2
0922.01259.01215.01044.06452.44155.3
if 2
6423.20357.0
3577.1
6953.44159.61220.00275.06953.44159.6
if 6423.0 6953.44159.6
0524.00519.0 0924.00679.0
0922.01259.0 if 2
0519.00524.0 0922.01259.0
0924.00679.0 if 2
0267.00353.0
6341.2
0922.01259.01215.01044.06452.44155.3
if 2
2222
8291.65379.18291.65379.18883.69220.58883.69220.5
if 8883.69220.5
xx
x
xx
x
x
x
xx
u
1
2,4
3
5
6
7,8
9
Parametric Programming ApproachModel Predictive Control
Real Time Optimization Problem
Off-line Parametric Optimization ProblemMeasurements as Parameters
Control Variables as Optimization variables
Obtain Explicit Control Law(a) Explicit functions of measurements
(b) Critical Regions where these functions are valid
State-of-the-art Performanceon a simple computational hardware
1x
2x
Blood Glucose Control
IPXPdt
dX
VtUIIndt
dI
tDGGXGPdt
dG
b
b
32
1
1
)()(
)()(
Plasma Insulin I(t)
Plasma Glucose G(t)
Effective Insulin X(t)
Tissue
Liver
Exogenous Insulin U(t)
Clearance
Exercise, Meals D(t)
State variables: G(t), I(t), X(t) Control variable: U(t) Parameters: Pi, n
(Bergman et al., 1981)
ParametricGlucose Control
(Off-line)
MechanicalPump
PatientGlucoseSensor
Reference
Meals, Exercise
Insulin
an in-vivo glucose sensor a parametric ‘look-up function’ to manipulate the
insulin delivery rate given a sensor measurement a mechanical pump
Parametric Control of Blood Glucose
Control of AnesthesiaRESPIRATORY SYSTEM
1
5
4
2
1. Lungs and Heart
2. Vessel rich organs (e.g. liver)
3. Muscles
4. Others
5. Fat
DP, SNP Injection
Isoflurane uptake
Pharmacodynamic aspect
Pharmacokinetic aspect
3
Surgery under Anesthesia
0
10
20
30
40
50
60
70
80
90
100
0 100 200 300 400 500 600 700 800 900 1000
time (min)
MA
P (
mm
Hg
)
0
20
40
60
80
100
120
BIS
MAP BIS
0.6% Isoflurane
0.3 g/kg/min SNP 4.5 g/kg/min DP
20 mmHg MAP drop
DP stopped
Isoflurane,
SNP stopped
Control of Pilot Plant Reactor
Cooling Water
Product
Reactor&
Cooling Jacket
FeedTr
CaController
Ff
Tj
Control of Catalytic Converter
Clean Exhaust Gas Control the amount of Oxygen stored on the Catalyst to an
Optimal amount Use Converter Model as an inferential sensor Ensuring Minimum Energy Consumption and Maximum
Emissions Reduction
(Balenovic and Backx, 2001)
CATALYTIC CONVERTER
MODEL
CAR ENGINE
CATALYTIC
CONVERTERFuel
AirExhaust Gas Clean Gas
Parametric Control of Catalytic Converter
OC: (Fractional) Oxygen Coverage
EMF: Exhaust Mass Flowrate (kg/hr)
AFR: (Normalised) Air to Fuel Ratio OC
EM
F
-115.58 OC – EMF <= -84.77
-60.88 OC + EMF <= 33.67
70.90 OC + EMF <= 85.10
186.70 OC – EMF <= 44.10
AFR = -0.68 OC - 0.0059 EMF + 0.60
EMF
OC
AF
R
Concluding Remarks Real Time Optimization
Solve optimization problem at regular time intervals
Parametric Programming ApproachObtain optimal solution as a set of functions of
state variablesOptimality and satisfaction of constraints are
guaranteedFunction Evaluations!
PAROS plc: www.parostech.com
References Dua, P., Doyle III, F.J., Pistikopoulos, E.N. (2006) Model based blood glucose
control for type 1 diabetes via parametric programming, accepted for publication in IEEE Transactions on Biomedical Engineering.
Dua, P., Dua, V., Pistikopoulos, E.N. (2005) Model based drug delivery for anesthesia, Proceedings of the 16th IFAC World Congress, Prague, 2005.
Sakizlis, V., Kakalis, N.M.P., Dua, V., Perkins, J.D., Pistikopoulos, E.N. (2004) Design of robust model-based controllers via parametric programming, Automatica, 40, 189-201.
Dua, V., Bozinis, N. A., Pistikopoulos, E.N. (2002) A multiparametric programming approach for mixed-integer and quadratic process engineering problems, Computers & Chemical Engineering, 26, 715-733.
Pistikopoulos, E.N., Dua, V., Bozinis, N. A., Bemporad, A., Morari, M. (2002) On-line optimization via off-line parametric optimization tools, Computers & Chemical Engineering, 26, 175-185.
Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N. (2002) The explicit linear quadratic regulator for constrained systems, Automatica, 38, 3-20.