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.