Profit Based Control System Designmypages.iit.edu/~chmielewski/presentations/seminar/UIUC...Profit...

Profit Based Control System Design: A Globally Optimal Approach

Donald J. Chmielewski

Department of Chemical & Biological Engineering

Illinois Institute of Technology

*

BOP with

more profit

BOP with

less profit

OSSOP

EDOR’s due to

different controller

tunings

*

*

x

u

)(

)(

2

1

undesiredCB

desiredBA

k

k

iascapiscap

Rscap

Escap

iarm

Rarm

Larm

DC-DC

Converter

iabatibat

Rbat

Ebat

DC-DC

Converter

iafcifc

EfcDC-DC

Converter

Fuel

Cell

Power Bus

warm

Earm

kfc kbat kscap

SEPARATR

RISER

REGEN-TR

REG-CATY

FLUE-GAS

PRODUCTS

AIR

STEAM

STRP-STM

FEED-OIL

Department of Chemical and Biological Engineering


Current Research Topics

• Fuel Cells – Modeling, Design, and Control

• PEMFC, SOFC and On-board Fuel Processors (ATR)






• Stationary Power Plants – Modeling and Control

• Coal Fired Boilers with Oxygen Enrichment, IGCC






• Stationary Power Plants – Modeling and Control

• Coal Fired Boilers with Oxygen Enrichment, IGCC

• Control Theory

• Profit Based Controller Design

• Sensor and Actuator Selection



Outline

• Motivating Example

• Controller Tuning

• Economic Based Tuning

• Robust Formulation



Motivating Example (Non-isothermal Reactor)

F

F

CA, T

AA

Apin

AAAinA

CTkr

rCHVTTFdt

dTV

VrCCFdt

dCV

)(

)/()(

)(

Increase F Increased production rate



Motivating Example (Non-isothermal Reactor)

F

F

CA, T

AA

Apin

AAAinA

CTkr

rCHVTTFdt

dTV

VrCCFdt

dCV

)(

)/()(

)(

Increase F Increased production rate

Decrease F Increase T Increase reaction rate

Increase production



Motivating Example (Limited Operating Region)

Process Limitations:

(max))( TtT

(max))( FtF

- Catalyst protection or

onset of side reactions

- Pump limit or limit on

downstream unit




)()( )( spsp

c FTTKF

Process Limitations:

(max))( TtT

Possible Controller:

(max))( FtF

- Catalyst protection or

onset of side reactions

- Pump limit or limit on

downstream unit



Motivating Example (Performance in Time Series)

time

T(t)

F(t)

F(sp)

T(sp)

time

F(max)

T(max)



Motivating Example (Performance in Phase Plane)

)(tF

)(tT

*



Motivating Example (Elliptical Operating Region)

)(tF

)(tT

*




)()( )( spsp

c FTTKF

Controller:

),( )()( wTfF spsp

Steady-State Relation:



Motivating Example (Elliptical Operating Region)

)(tF

)(tT

*



Motivating Example (Steady-State Operating Line)

)(tF

)(tT

*



Motivating Example (Optimal Operating Point)

)(tF

)(tT

*

Decrease F

Increase T

Increase

conversion

Increase

production



Motivating Example (Optimal Operating Point: Another Possibility)

)(tF

)(tT

* Increase F

Increased

production rate



Motivating Example (Optimal Operating Point: Another Possibility)

)(tF

)(tT

*

Increase F

Increased

production rate



Motivating Example

(Suggests Different Controller Tuning)

)(tF

)(tT

*



Motivating Example (Less Aggressive Tuning)

time

F(t)

F(sp)

T(sp)

time

F(max)

T(max)

T(t)



Motivating Example (Need for Automated Tuning)

)(tF

)(tT

*

*



Motivating Example (Need for Automated Tuning)

)(tF

)(tT

*

* *

* *



Outline







Covariance Analysis (Open-Loop Case)

Process Model:

Steady State Covariance:

T

xz

T

w

T

xx

DD

GGAA

0

Plant

)()(

)()()(

tDxtz

twGtxAtx

)(tw )(tz

Gaussian white noise with covariance )(tww



Expected Dynamic Operating Region (EDOR)

EDOR

defined by:

*

1z

2z

2

22

2

21

2

12

2

11

z

11

22



Flexibility in EDOR Definition

a = 1 constraint

observance ~84% of time

a = 2 constraint


a = 3 constraint

observance ~99.5% of

time

*

1z

2z

11a

22a



Closed-Loop Covariance Analysis (Full State Information Case)

T

www

T

uxxuxz DDLDDLDD )()(

)()( tLxtu

0)()( T

w

T

xx GGBLABLA

wDuDxDz

wGBuAxx

wux

Process Model:

Controller:

Steady-State Covariance:

Plant )(tw

)(tx

L )(tu

)(tz



1z

2z

Closed-Loop EDOR

EDOR’s from different

controllers

* xLu 1

xLu 2



Constrained Closed-Loop EDOR

*

1z

2z

Constraints

) ( izi za



Does there exist L such that:

00100 i

columnthi

Constrained Controller Existence

T

www

T


0)()( T

w

T

xx GGBLABLA

zii

T

izii niz 1/22

a




00100 i

columnthi


T

www

T


0)()( T

w

T

xx GGBLABLA

zi

T

izii niz 12



If and only if there exist X>0 and Y such that:

00100 i

Constrained Controller Existence (Convex Condition)

0)(

)(

XYDXD

YDXDDDT

i

T

ux

uxi

T

i

T

wwwii

0)()( T

w

T GGBYAXBYAX

zii niz 12

1 YXLAnd controller is constructed as: Lxu



Implementation of (Pseudo-) Constrained Controller

*

1z

2z

EDOR based on

controller: Lxu

Constraints ) ( izi za



ziii

wux

TTT

ux

niztzz

wDuDxDtz

GwBuAxxts

dtRuuMxuQxx

1)(

)(

..

2min0

,

Model Predictive Control



zi

wux

TTT

ux

nitz

wDuDxDtz

GwBuAxxts

dtRuuMxuQxx

1)(

)(

..

2min0

,

MPC LQG



..

2min0

,

GwBuAxxts

dtRuuMxuQxx TTT

ux

LQG

Given A, B, Q, R and M, then the unconstrained controller is

01 TT MPBRMPBQPAPA

TMPBRL 1



Does there exist Q, R and M such that:

Constrained LQG Controller Existence

T

www

T


0)()( T

w

T

xx GGBLABLA

zi

T

izii niz 12

01 TT MPBRMPBQPAPA

TMPBRL 1



Constrained LQR Controller Existence (Convex Condition)

???





T

www

T


0)()( T

w

T

xx GGBLABLA

zi

T

izii niz 12



such that:

Constrained Minimum Variance (CMV) Controller

T

www

T


0)()( T

w

T

xx GGBLABLA

zi

T

izii niz 12

i

iiL

dix

min

,,0



Theorem 1 (Chmielewski & Manthanwar, 2004):

Controller Equivalence

The controller generated by CMV

is coincident with

the controller generated by some

LQG problem.



such that:

Pareto Frontier Interpretation of Minimum Variance Control

T

www

T

uxxuxz

DD

LDDLDD

)()(

0

)()(

T

w

T

xx

GG

BLABLA

z

T

izii ni 1

i

iiL

dix

min

,,0

Achievable

Performance

Levels

Unachievable

2

1

Pareto frontier



zi

T

izii niz 12

such that:

Pareto Frontier Interpretation of CMV Control

T

www

T

uxxuxz

DD

LDDLDD

)()(

0

)()(

T

w

T

xx

GG

BLABLA

2

1 1<z12

2<z22

i

iiL

dix

min

,,0



Example 1: Surge Tanks

V1 q2 V2 q1

q0

FT

FC

V1

q1(sp)

FT

FC

q2(sp)

0

1

11

01

00

00GBA

State Variables: x = [V1 V2]T Manipulated Variables: u = [q1 q2]

T

Disturbance w = [q0] Performance Output: z = [V1 V2 q1 q2]T

0

0

0

0

10

01

00

00

00

00

10

01

wux DDD




Objective Weights: d1 = d2 = d3 = d4 = 1 Performance Output: z = [V1 V2 q1 q2]T

Constraint: 4 < 2.252

-5 0 5

-150

-100

-50

0

50

100

150

Level in

Tan

k 1

(m

3)

Flow out of in Tank 1 (m3/min)

-2 -1 0 1 2

-150

-100

-50

0

50

100

150

Flow out of Tank 2 (m3/min)

Level in

Tan

k 2

(m

3)





Constraints: 4 < 2.252 and 1.52

-5 0 5

-150

-100

-50

0

50

100

150

Level in

Tan

k 1

(m

3)


-2 -1 0 1 2

-150

-100

-50

0

50

100

150


Level in

Tan

k 2

(m

3)





Constraints: 4 < 2.252, 1.52 and 0.752

-5 0 5

-150

-100

-50

0

50

100

150

Level in

Tan

k 1

(m

3)


-2 -1 0 1 2

-150

-100

-50

0

50

100

150


Level in

Tan

k 2

(m

3)




Constraints: 4 < 2.252

041.0039.0

687.0726.0L


014.0013.0

700.0712.0L


003.0003.0

701.0734.0L



ziii

wux

TTT

ux

niztzz

wDuDxDtz

GwBuAxxts

dtRuuMxuQxx

1)(

)(

..

2min0

,

Model Predictive Control



Inverse Optimality

01 TT MPBRMPBQPAPA

TMPBRL 1

Theorem 2 (Chmielewski & Manthanwar, 2004):

If there exists P > 0 and R > 0 such that

0

RPBRL

PBRLPAPARLLTT

TTT

PAPARLLQ TT )( PBRLM T

0

RM

MQT

and P and L satisfy

Then and are such that





003.0003.0

701.0734.0L

160160

160161Q

4448.81

8.81299R

8.588.20

8.602.20M

Thus, weights are available for MPC implementation.



GwBuAxxts

dtRuuMxuQxx TTT

ux

..

2min0

,

CMV Control

zi

T

izii niz 12

such that:

T

www

T

uxxuxz

DD

LDDLDD

)()(

0

)()(

T

w

T

xx

GG

BLABLA

i

iiL

dix

min

,,0

Achievable

Performance

Levels

Unachievable

2

1

Pareto frontier



Example 2: Inventory Control

V1

Starts

Inventory Demand

Delivery delay

IC

Inventory

Set-point



Example 2: Inventory Control

V1

Starts

Inventory Demand

Delivery delay

IC

Inventory

Set-point

Specific scenario:

Delivery Delay = 5 days

Demand Variance = 102



Inventory Control (Pareto Frontier)

2 4 6 8 10 1224

25

26

27

28

29

30

31

32

33

Std

. D

ev

. In

ve

nto

ry

Std. Dev. Starts

A

B

C

-10 -5 0 5 10-40

-30

-20

-10

0

10

20

30

40

Inv

en

tory

Starts

C

B

A

Point A (the point of minimum inventory variance)

is the basis of classic “safety stock” analysis



Multi-Echelon Inventory Control

(from Seferlis & Giannelos, 2004)



2-Echelon Inventory Control (Decentralized Control)

V1

Starts

Inventory

1 Demand IC

Inventory

Set-point

1

V1

Starts

IC

Inventory

Set-point

2

Inventory

2

Delivery delay = 3 Delivery delay = 5



2-Echelon Inventory Control (Centralized Control)

V1

Starts

Inventory

1 Demand

Delivery delay = 3

Inventory Set-

point 1

V1

Starts Delivery delay = 5

IC Inventory Set-

point 2

Inventory

2



2-Echelon Inventory Control (Pareto Frontier)

0 5 10 15 20 25 3024

25

26

27

28

29

30

31

Std

. D

ev

. In

ve

nto

ry T

an

k 2

Std. Dev. Inventory Tank 1

Decentralized



2-Echelon Inventory Control (EDORs)

-10 -5 0 5 10-30

-20

-10

0

10

20

30

Inv

en

tory

Starts

Tank 2

Decentralized

-15 -10 -5 0 5 10 15-20

-15

-10

-5

0

5

10

15

20

Inv

en

tory

Starts

Tank 1

Decentralized




Small “safety stock” increase at I2, leads to large “safety stock” decrease at I1

0 5 10 15 20 25 3024

25

26

27

28

29

30

31

Std

. D

ev

. In

ve

nto

ry T

an

k 2


Decentralized

Centralized A





-15 -10 -5 0 5 10 15-20

-15

-10

-5

0

5

10

15

20

Inv

en

tory

Starts

Tank 1

Centralized A

Decentralized

-10 -5 0 5 10-30

-20

-10

0

10

20

30

Inv

en

tory

Starts

Tank 2

Decentralized

Centralized A





0 5 10 15 20 25 3024

25

26

27

28

29

30

31

Std

. D

ev

. In

ve

nto

ry T

an

k 2


Decentralized

Centralized A

Centralized B





but starts variance size trend is unexpected

-15 -10 -5 0 5 10 15-20

-15

-10

-5

0

5

10

15

20

Inv

en

tory

Starts

Tank 1

Centralized A

Decentralized

Centralized B

-10 -5 0 5 10-30

-20

-10

0

10

20

30

Inv

en

tory

Starts

Tank 2

Decentralized

Centralized A

Centralized B



Example 3: Hybrid Fuel Cell Vehicle

FC Kfc

iafcifc

Vfc

iab

ib

Vb

Rb

Eb

ia

Ra

La

VaKb



Classic Control

FC Kfc

iafcifc

Vfc

iab

ib

Vb

Rb

Eb

ia

Ra

La

VaKb

Vehicle

kbatPmot

+-

kfc

x

Pfc

+-

x

Pbat

Pmot(sp)

Pbat(sp)

VfcFUEL CELL

VOLTAGE

CONTROLLER

Vfc(sp)

PI

PI



Separation of Time-Scales

FC Kfc

iafcifc

Vfc

iab

ib

Vb

Rb

Eb

ia

Ra

La

VaKb

5 10 15 20 25 30 35

-200

0

200

400

600

800

1000

Power Profles [W]

time, sec

Pload

(sp)

Battery

Fuel Cell

Armature

Vehicle

kbatPmot

+-

kfc

x

Pfc

+-

x

Pbat

Pmot(sp)

Pbat(sp)

VfcFUEL CELL

VOLTAGE

CONTROLLER

Vfc(sp)

PI

PI



Dynamics of PEMFC

Cooling

Air In

Jacket

Exhaust

Anode

In

(H2, H2O)

Ecell

H2

Cathode

In

(air)

Cathode

Exhaust

O2

H2O

N2

Solid Material Current Collector

Insulator

H2O

Anode

Exhaust

Polymer Membrane

Catalyst LayersGas Diffusion

Layers (GDLs)



Hybrid Fuel Cell Vehicle (Double Storage Configuration)

iascapiscap

Rscap

Escap

iarm

Rarm

Larm

DC-DC

Converter

iabatibat

Rbat

Ebat

DC-DC

Converter

iafcifc

EfcDC-DC

Converter

Fuel

Cell

Power Bus

warm

Earm

kfc kbat kscap



Supervisory Control

Vehicle

Power

System

+ -Pscap

(sp) Pscap

kscap

Supervisory

Controller

Pmotor

+ -Pbat

(sp) Pbat

kbat

+ -Pfc

(sp) Pfc

kfc

PI

PI

PI



Hybrid Fuel Cell Vehicle (Supervisory Model)

maxminmax

maxminmax

maxminmax

0

0

0

scapscapscapscapscapscapscap

batbatbatbatbatbatbat

fcfcfcfcfcfcfc

PPPEEPE

PPPEEPE

PPPPPPP

scapbatfcmot PPPP

fc

ratefcfc

sc

ratescsc

bat

ratebatbat CPPCEPCEP maxmaxmaxmaxmaxmax



Hybrid Fuel Cell Vehicle (Supervisory Model)

maxminmax

maxminmax

maxminmax

0

0

0

scapscapscapscapscapscapscap

batbatbatbatbatbatbat

fcfcfcfcfcfcfc

PPPEEPE

PPPEEPE

PPPPPPP

scapbatfcmot PPPP

fc

ratefcfc

sc

ratescsc

bat

ratebatbat CPPCEPCEP maxmaxmaxmaxmaxmax

Colored noise disturbance

(modeled from drive cycle data)



Hybrid Fuel Cell Vehicle (Drive Cycle Simulation 1)

0 200 400 600 800 1000-20

-15

-10

-5

0

5

10

15

20

25Power Out/ Motor In

Pow

er

(kW

)

time (sec)



Hybrid Fuel Cell Vehicle (EDOR Case 1)

-30 -20 -10 0 10 20 30-1.5

-1

-0.5

0

0.5

1

1.5x 10

4

Fu

el

Ce

ll P

ow

er

(W)

Fuel Cell Power Rate of Change (W/s) -3000 -2000 -1000 0 1000 2000 3000

-1

-0.5

0

0.5

1

x 107

En

erg

y i

n B

att

ery

(J

)

Power from Battery (W)

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000-5000

0

5000

En

erg

y i

n S

up

er

Ca

p (

J)

Power from Super Cap (W)



Hybrid Fuel Cell Vehicle (EDOR Case 2)

-30 -20 -10 0 10 20 30-1.5

-1

-0.5

0

0.5

1

1.5x 10

4

Fu

el

Ce

ll P

ow

er

(W)

Fuel Cell Power Rate of Change (W/s) -3000 -2000 -1000 0 1000 2000 3000

-1

-0.5

0

0.5

1

x 107

En

erg

y i

n B

att

ery

(J

)

Power from Battery (W)

-8 -6 -4 -2 0 2 4 6 8

x 104

-1

-0.5

0

0.5

1

x 105

En

erg

y i

n S

up

er

Ca

p (

J)

Power from Super Cap (W)



Hybrid Fuel Cell Vehicle (Drive Cycle Simulation 2)

0 200 400 600 800 1000-20

-15

-10

-5

0

5

10

15

20

25Power Out/ Motor In

Pow

er

(kW

)

time (sec)



Outline







Motivating Example

)(tF

)(tT

*

*

* *



EDOR

Constrained Operating Region

MV’s

CV’s

*

Constraints

Steady-State Operating Point



EDOR

Real-Time Optimization

*

*

Optimal

Steady-State

Operating Point

(OSSOP)

MV’s

CV’s Constraints

Steady-State Operating Point




),,(),,( pmshqpmsfs

Original Nonlinear Process Model:

(s,m,p,q) ~(state, mv, dist, performance) ~ (x,u,w,z)




),,(),,( pmshqpmsfs

Original Nonlinear Process Model:

maxmin

,,

),,(),,(0

s.t. )(min

iii

qms

qqqpmshqpmsf

qg

Real-Time Optimization (minimize profit loss):

(s,m,p,q) ~(state, mv, dist, performance) ~ (x,u,w,z)

RTO solution denoted as (sossop,mossop,possop,qossop)



Backed-off Operating Point (BOP)

EDOR

*

* *

Backed-off

Operating Point

(BOP)

MV’s

CV’s

Optimal

Steady-State

Operating Point

(OSSOP)



Steady-State BOP Selection (Bahri, Bandoni & Romagnoli, 1996)

0

),,(

),,(0s.t.

}{ maxmax s.t. )(mini],[,, maxmin

ii

iipppqms

qq

pmshq

pmsf

qqqg

Solve the following Semi-infinite Programming Problem

Extensions:

- Dynamic version in Bahri, et al, (1995)

- Linearized version in Contreras-Dordelly & Marlin (2000)



Linearized Perspective

maxmin

),,(

),,(

)(

iii

bop

qqq

pmshq

pmsfs

qg

Nonlinear Linear wrt OSSOP Linear wrt BOP

Deviation Variables w.r.t. OSSOP:

s’ = sbop – sossop m’ = mbop - mossop

p’ = pbop - possop q’ = qbop - qossop

maxmin '''

''''

''''

')(

iii

wux

q

ossop

qqq

pDmDsDq

GpBmAss

qgqg

Deviation Variables w.r.t. BOP:

x = s– sbop u = m - mbop

w = p - pbop z = q - qbop

maxmin

iii

wux

zzz

wDuDxDz

GwBuAxx



Assume controller L is given and calculate i :

T

www

T


0)()( T

w

T

xx GGBLABLA

z

T

izii ni 1

Stochastic BOP Selection (Loeblein & Perkins, 1999)



min2/1max2/1

maxmin

',','

''''

')''('

''0 s.t. 'min

iiiiii

iiiuxii

qqms

qqqq

qqqmDsDq

BmAsqg

Assume controller L is given and calculate i :

T

www

T


0)()( T

w

T

xx GGBLABLA

z

T

izii ni 1

Solve the following Linear Program:

Stochastic BOP Selection (Loeblein & Perkins, 1999)



Stochastic BOP Selection (EDOR within Constraint Set)

iiiiii qqqq '''' max2/1min2/1




iiiiii qqqq '''' maxmin



q'1 max

EDOR BOP

q'1 min

q'2 min q'2 max

q'2 - q'2min q'2

max - q'2


iiiiii qqqq '''' maxmin



Fixed Controller BOP Selection

Controller is fixed EDORs have fixed

sizes and shapes

Loeblein and Perkins (1999):

OSSOP

*

* *

x

EDOR

u



Peng et al. (2005):

Variable Controller EDORs have variable

sizes and shapes

*

* *

OSSOP EDOR

u

x

Variable Controller BOP Selection


Illinois Institute of Technology Peng et al. (2005)

*

BOP with

more profit

BOP with

less profit

OSSOP

EDOR’s due to

different controller

tunings

*

Profit Control (Simultaneous BOP and Controller Selection)




min2/1max2/1

maxmin

,,,',','

''''

')''('

''0 s.t. 'min

iiiiii

iiiuxii

q

Lqms

qqqq

qqqmDsDq

BmAsqg

zxi

T

www

T


0)()( T

w

T

xx GGBLABLA

z

T

izii ni 1

Peng et al. (2005)



min2/1max2/1

maxmin

,,',','

''''

')''('

''0 s.t. 'min

iiiiii

iiiuxii

q

YXqms

qqqq

qqqmDsDq

BmAsqg

i

0)(

)(

XYDXD

YDXDDDT

i

T

ux

uxi

T

i

T

wwwii

0)()( T

w

T GGBLAXBYAX


Peng et al. (2005)



min2/1max2/1

maxmin

,,',','

''''

')''('

''0 s.t. 'min

iiiiii

iiiuxii

q

YXqms

qqqq

qqqmDsDq

BmAsqg

i

0)(

)(

XYDXD

YDXDDDT

i

T

ux

uxi

T

i

T

wwwii

0)()( T

w

T GGBLAXBYAX

Peng et al. (2005)

Computational Aspects of Profit Control



Reverse-Convex Constraints

-2 -1.5 -1 -0.5 00

0.2

0.4

0.6

0.8

1

zss,i

i

(zss,i

+dmin,i

)2 (z

ss,i+d

max,i )

2

Feasible Region

2

1

max

11 )''( qq 2min

111 )''( qq

1'q

1



Global Solution

-2 -1.5 -1 -0.5 00

0.2

0.4

0.6

0.8

1

zss,i

i

Region 1

Region 2

Region 3

Region 4

Region 5

Based on Branch and Bound algorithm

1'q

1



Combinatorial Growth of Branch and Bound

-2 -1.5 -1 -0.5 00

0.2

0.4

0.6

0.8

1

zss,i

i

Region 1

Region 2

Region 3

Region 4

Region 5

1'q-2 -1.5 -1 -0.5 0

0

0.2

0.4

0.6

0.8

1

zss,i

i

Region 1

Region 2

Region 3

Region 4

Region 5

1'q



-2 -1.5 -1 -0.5 00

0.2

0.4

0.6

0.8

1

zss,i

i

Region 1

Region 2

Region 3

Region 4

Region 5

1'q-2 -1.5 -1 -0.5 0

0

0.2

0.4

0.6

0.8

1

zss,i

i

Region 1

Region 2

Region 3

Region 4

Region 5

1'q

-2 -1.5 -1 -0.5 00

0.2

0.4

0.6

0.8

1

zss,i

i

Region 1

Region 2

Region 3

Region 4

Region 5

1'q-2 -1.5 -1 -0.5 0

0

0.2

0.4

0.6

0.8

1

zss,i

i

Region 1

Region 2

Region 3

Region 4

Region 5

1'q

-2 -1.5 -1 -0.5 00

0.2

0.4

0.6

0.8

1

zss,i

i

Region 1

Region 2

Region 3

Region 4

Region 5

1'q

Combinatorial Growth of Branch and Bound



Heuristic Scheme

-2 -1.5 -1 -0.5 00

0.2

0.4

0.6

0.8

1

zss,i

i

(zss,i

+dmin,i

)2 (z

ss,i+d

max,i )

2

Feasible Region

1'q



Combinatorial Growth??

-2 -1.5 -1 -0.5 00

0.2

0.4

0.6

0.8

1

zss,i

i

(zss,i

+dmin,i

)2 (z

ss,i+d

max,i )

2

Feasible Region

1'q-2 -1.5 -1 -0.5 0

0

0.2

0.4

0.6

0.8

1

zss,i

i

(zss,i

+dmin,i

)2 (z

ss,i+d

max,i )

2

Feasible Region

2'q

-2 -1.5 -1 -0.5 00

0.2

0.4

0.6

0.8

1

zss,i

i

(zss,i

+dmin,i

)2 (z

ss,i+d

max,i )

2

Feasible Region

3'q -2 -1.5 -1 -0.5 00

0.2

0.4

0.6

0.8

1

zss,i

i

(zss,i

+dmin,i

)2 (z

ss,i+d

max,i )

2

Feasible Region

4'q



Mass-Spring-Damper Example

Mass

wf

rmax

r

rmin

160and11 fr

System Model:

wfv

r

v

r

1

0

1

0

32

10

force edisturbanc theis

and (MV) forceinput theis

velocity, theis position, mass theis

w

f

vr

System Constraints:

where




Mass

wf

rmax

r

rmin

160and11 fr

System Model:

wfv

r

v

r

1

0

1

0

32

10




w

f

vr

System Constraints:

where

Mas

s P

osi

tio

n

Upper Bound on Position

time

} EDOR


*

OSSOP




Mass

wf

rmax

r

rmin

160and11 fr

System Model:

wfv

r

v

r

1

0

1

0

32

10




w

f

vr

System Constraints:

where

Mas

s P

osi

tio

n


time

} EDOR


*

OSSOP

Mas

s P

osi

tio

n


time

}

EDOR


*

OSSOP



* BOP

Mass-Spring-Damper Example (Phase Plane)

r

f

Steady-State Line

*

OSSOP

Constraints



Mass-Spring-Damper Example (Phase Plane Solution)

4 6 8 10 12 14 16-0.5

0

0.5

1

Mass

Po

siti

on

(m

)

Input Force (N)

FSI Case

PSI Case

OSSOP

FSI Case:

Full State Information:

Controller is

PSI Case:

Partial State Information:

One Velocity Sensor.

Controller is

)()( tLxtu

)(ˆ)( txLtu



Mass-Spring-Damper Example (Impact of Constraints )

6 8 10 12 14 16 18 20

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Mass

Posi

tion (m

)

Input Force (N)

Case A

Case B

Case C



Discrete-time Simulation (Scatter Plot)

0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)

Input Force (N)



ziii

wux

TTT

ux

niztzz

wDuDxDtz

GwBuAxxts

dtRuuMxuQxx

1)(

)(

..

2min

maxmin

0,

MPC and the EDOR

0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)

Input Force (N)



0

1)(

)(

..

2min

maxmin

0,

i

ziiiii

wux

TTTT

ux

s

nisztzsz

wDuDxDtz

GwBuAxxts

ssdtRuuMxuQxx

Soft Constraints



MPC with Soft Constraints

0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)

Input Force(N)0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)

Input Force(N)

gm = 107 gf = 103

gm = 103 gf = 103



Flexibility in EDOR Definition

a = 1 constraint


a = 2 constraint


a = 3 constraint

observance ~99.5% of

time

*

1z

2z

11a

22a



Impact of EDOR Definition

0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)

Input Force (N)0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Input Force (N)

Mass P

osit

ion

(m

)

a = 1 a = 2



MPC with Soft Constraints EDOR = 2 std dev’s

0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)

Input Force(N)

gm = 107 gf = 103

gm = 103 gf = 103



Impact of EDOR Definition (Reduced Sensitivity to Soft Weights)

0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)

Input Force(N)

0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mass P

osit

ion

(m

)

Input Force (N)

0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Input Force (N)

Mass P

osit

ion

(m

)

gm = 107 gf = 103

gm = 103 gf = 103



Fluidized Catalytic Cracker (FCC) Example

Regenerator and Separator (dynamic):

Riser (pseudo steady state):

(adapted from Loeblein & Perkins, 1999)



FCC Example

Process Constraints:

Profit Function:

Fgs Fgl and Fugo are product flows

(gasoline, light gas and unconverted oil).

(adapted from Loeblein & Perkins, 1999)



Fixed LQG Controller

wDuDxDzGwBuAxxts

dtDzz

wux

T

ux

..

min0

,

),,( MRQRicL

DDDMDDDRDDDQ

ID

u

T

xu

T

ux

T

x



785 790 795 800 805 810 815

990

992

994

996

998

1000

Cyclo

ne T

em

pera

ture

(K

)

Separator Temperature (K)

0.0125 0.013 0.0135 0.014 0.0145 0.015

990

992

994

996

998

1000

Reg

en

era

tor

Tem

p (

K)

Coke Fraction in Separator

Fixed Controller

5 5.5 6 6.5 7 7.5 8 8.5

x 10-3

280

300

320

340

360

380

400

Cata

lyst

Flo

w (

kg

/s)

Fraction of Coke in Regenerator0 5 10 15 20

x 10-4

25

26

27

28

29

30

31

32

Inle

t A

ir (

kg

/s)

Oxygen Mass Fraction

Fixed Controller FCC (Loeblein & Perkins, 1999)

(adapted)



0 5 10 15 20

x 10-4

25

26

27

28

29

30

31

32

Inle

t A

ir (

kg

/s)

Oxygen Mass Fraction5 5.5 6 6.5 7 7.5 8 8.5

x 10-3

280

300

320

340

360

380

400

Cata

lyst

Flo

w (

kg

/s)

Fraction of Coke in Regenerator

0.0125 0.013 0.0135 0.014 0.0145 0.015

990

992

994

996

998

1000

Reg

en

era

tor

Tem

p (

K)

Coke Fraction in Separator

Fixed Controller Free Controller

785 790 795 800 805 810 815

990

992

994

996

998

1000

Cyclo

ne T

em

pera

ture

(K

)

Separator Temperature (K)

Free Controller FCC (Profit Control)



FCC Profit

Gross Profit Diff from OSSOP

($/day) ($/day) OSSOP $36,905 - Fixed Control $35,768 - $1,137



FCC Profit

Gross Profit Diff from OSSOP

($/day) ($/day) OSSOP $36,905 - Fixed Control $35,768 - $1,137 Profit Control $36,160 - $745



FCC Profit

Gross Profit Diff from OSSOP ($/day) ($/day) OSSOP $36,905 - EDOR = 1 std. dev. Fixed Control $35,768 - $1,137 Profit Control $36,160 - $745 EDOR = 2 std. dev. Fixed Control $34,631 - $2,274 Profit Control $35,416 - $1,489



CSTR’s in Series

CB

BA

k

k

2

1

(adapted from de Hennin & Perkins, 1991)



CSTR’s in Series


211, ccMF QQQQ

Manipulated Variables:

Disturbances:

inA

in

C

Tw

,

2

2

10

03w



CSTR’s in Series


KT 3501 KT 3502

smQQ MF /8.0 3

1

KTKT outcoutc 300330 ,2,1

smQsmQ MF /05.0/05.0 33

1

21

1

21

01.0

1.01.0

)(10

qq

QQ

CQQ

MF

BMF

aj

incja

aj

incja

UQ

TTQUq

UQ

TTQUq

2,

,222,

2

1,

,111,

12

)(2

2

)(2

Some Process Constraints:

Profit Function:



CSTR’s in Series

300 305 310 315 320 325 330294

295

296

297

298

299

300

301

Tem

pera

ture

fro

m J

acket

2 (

K)

Temperature from Jacket 1 (K)

344 345 346 347 348 349 350

325

330

335

340

345

350

Rea

cto

r 2

Tem

per

atu

re (

K)

Reactor 1 Temperature (K)

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

Feed

Flo

w (

m3/s

)

Makeup Flow (m3/s)

0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

Jacket

Flo

w 1

(m

3/s

)

Jacket Flow 2 (m3/s)



CSTR’s in Series


Disturbances:

a

inA

in

U

C

T

w ,

2

2

2

021.000

010

003

w



CSTR’s in Series

300 305 310 315 320 325 330294

295

296

297

298

299

300

301

Tem

pera

ture

fro

m J

acket

2 (

K)


No HEX Fault

HEX Fault

344 345 346 347 348 349 350

325

330

335

340

345

350

Rea

cto

r 2

Tem

per

atu

re (

K)


0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

Feed

Flo

w (

m3/s

)

Makeup Flow (m3/s)

0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

Jacket

Flo

w 1

(m

3/s

)




CSTR’s in Series

Disturbances:

a

inA

in

U

C

T

w ,

2

2

2

021.000

010

003

w Assume Ua is not white noise, but highly correlated colored noise

(slowly varying).



CSTR’s in Series

300 305 310 315 320 325 330294

295

296

297

298

299

300

301

Tem

pera

ture

fro

m J

acket

2 (

K)


No HEX Fault

HEX Fault White Noise

HEX Fault Highly Correlated

Noise

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

Feed

Flo

w (

m3/s

)

Makeup Flow (m3/s)

0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

Jacket

Flo

w 1

(m

3/s

)


344 345 346 347 348 349 350

325

330

335

340

345

350

Rea

cto

r 2

Tem

per

atu

re (

K)




CSTR Profit

Gross Profit Diff from OSSOP ($/day) ($/day) OSSOP $2,486 - Ua = 0 $2,342 - $144



CSTR Profit

Gross Profit Diff from OSSOP ($/day) ($/day) OSSOP $2,486 - Ua = 0 $2,342 - $144 Ua - white noise $2,176 - $310



CSTR Profit

Gross Profit Diff from OSSOP ($/day) ($/day) OSSOP $2,486 - Ua = 0 $2,342 - $144 Ua - white noise $2,176 - $310 Ua – slow varying $2,115 - $371



Outline







Uncertainty Characterization

Plant in the polytopic set:

W=co{Aj, Bj, Gj, Dxj, Duj j = 0 … NW}



Uncertainty and Stability

Plant in the polytopic set:

W=co{Aj, Bj, Gj, Dxj, Duj j = 0 … NW}

Sufficient condition (quadratic stability):

$ P > 0 s.t. AjP + PAjT < 0 j = 0…NW



2min2max

maxmin

,,',','

)''()''(

')''('

''0 s.t. 'min

iiiiii

iiiuxii

q

YXqms

qqqq

qqqmDsDq

BmAsqg

i

0)(

)(

XYDXD

YDXDT

i

T

ux

uxii

0)()( T

w

T GGBLAXBYAX




W

Njniqqqq

niqqqmDsDq

mBsAqg

ziiijiiij

ziiiuxii

q

YXqms

ij

...1...1)''()''(

...1')''('

''0 s.t. 'min

2min2max

maxmin

00

00

,,',','

W

Nj

ni

XYDXD

YDXDz

T

i

T

ujxj

ujxjiij

...1

...10

)(

)(

W

Nj

G

GYBXAYBXA

w

T

j

j

T

jjjj...10

)()(1

Profit Control with Robust Performance Conditions



Profit Control with Robust Performance Conditions

Robust BOP

OSSOP

EDOR’s

due to

uncertain

plant

model *



Inventory Control with Yield Uncertainty

V1

Starts

Inventory Demand

Delivery delay

IC

Inventory

Set-point Specific scenario:

Delivery Delay = 5 days

Demand Variance = 102

Yield Rate [0.8, 1.0]

Lost Orders



0.8 0.85 0.9 0.95 124.4

24.5

24.6

24.7

24.8

24.9

25

25.1

25.2

25.3

Yield

Std

. D

ev

. In

ve

nto

ry

Yield = 1.0

Yield = 0.8

Robust

Inventory Control with Yield Uncertainty



2-Echelon Inventory Control (Delivery Delay Uncertainty)

V1

Starts

Inventory

1 Demand

Delivery delay = 3

Inventory Set-

point 1

V1

Starts Delivery delay [4, 5]

IC Inventory Set-

point 2

Inventory

2



Inventory Control with Delivery Delay Uncertainty

0 0.2 0.4 0.6 0.8 125

26

27

28

29

30

Alpha

Std

. D

ev

. In

ve

nto

ry T

an

k 2

Robust

Alpha = 0

Alpha = 1



0 0.2 0.4 0.6 0.8 125

26

27

28

29

30

Alpha

Std

. D

ev

. In

ve

nto

ry T

an

k 2

Robust

Alpha = 0

Alpha = 1

0 5 10 15 20 25 3024

25

26

27

28

29

30

31

Std

. D

ev

. In

ve

nto

ry T

an

k 2


Centralized A

Robust

Alpha = 1

Alpha = 0

Inventory Control with Delivery Delay Uncertainty



2min2max

maxmin

,,,',','

)''()''(

')''('

''0 s.t. 'min

iiiiii

iiiuxii

q

YXqms

qqqq

qqqmDsDq

BmAsqg

i

a

0)(

)(

XYDXD

YDXDT

i

T

ux

uxii

Profit Control (Peak-to-Peak Formulation)

0)()(

IG

GXBYAXBYAXT

T

a

a



Conclusions

• Relationship between control system performance

and plant profit quantified.



Conclusions



• Enables profit guided control system design.



Conclusions




• Globally optimal search algorithm as well as

heuristic scheme proposed.



Conclusions






• Applicable to a broad set of applications from

a variety of disciplines.



Conclusions






• Applicable to a broad set of applications from

a variety of disciplines.

• Extendable it robust and peak-to-peak framework,

but conservatism a concern.



Acknowledgements

• Students and Collaborators:

Amit Manthanwar Jui-Kun Peng

Syed K. Ahmed Benjamin P. Omell

Wai-Kit Ong (UG)

Prof. Professor Durango-Cohen (IIT Business School)

Dr. Aymeric Rousseau (ANL)

• Funding:

Argonne National Laboratory

Illinois Clean Coal Institute

Undergraduate Research Center, IIT

Graduate and Armour Colleges, IIT

Chemical & Biological Engineering Department, IIT

Profit Based Control System Designmypages.iit.edu/~chmielewski/presentations/seminar/UIUC...Profit...

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