Optimizing Process Economic Performance Using Model Predictive …jbr · · 2012-03-09Optimizing...

Optimizing Process Economic Performance Using ModelPredictive Control

James B. Rawlings and Rishi Amrit

Department of Chemical and Biological EngineeringUniversity of Wisconsin–Madison

International Workshop on Assessment and Future Directions ofNMPC

Pavia, ItalySeptember 5, 2008

Rawlings and Amrit (UW) Economic MPC NMPC 2008 1 / 49

Outline

1 Historical overview and the current status

2 Tutorial example: unreachable setpoint MPC

3 Optimizing economic performance — linear dynamics

4 Turnpike theorems and nonlinear dynamics

5 Conclusions


The current hierarchical structure of plant operations

Validation

Planning and Scheduling

Reconciliation

Model UpdateOptimizationSteady State

Plant

Controller

Two layer structure

Steady-state layerI RTO optimizes steady state

modelI Optimal setpoints passed to

dynamic layer

Dynamic layerI Controller tracks the setpointsI Linear MPC


The current hierarchical structure of plant operations

Validation


Reconciliation


Plant

Controller

Two layer structure

Steady-state layerI RTO optimizes steady state

modelI Optimal setpoints passed to

dynamic layer

Dynamic layerI Controller tracks the setpointsI Linear MPC


The big(ger) picture — What is the goal?

The goal of optimal process operations is to maximize profit.— Helbig, Abel, and Marquardt (1998) . . . (−10 years)

Thus with more powerful capabilities, the determination ofsteady-state setpoints may simply become an unnecessaryintermediate calculation. Instead nonlinear, dynamic referencemodels could be used directly to optimize a profit objective.— Biegler and Rawlings (1991) . . . (−20 years)

In attempting to synthesize a feedback optimizing controlstructure, our main objective is to translate the economicobjective into process control objectives.— Morari, Arkun, and Stephanopoulos (1980) . . . (−30 years)



The goal of optimal process operations is to maximize profit.

— Helbig, Abel, and Marquardt (1998) . . . (−10 years)





The goal of optimal process operations is to maximize profit.— Helbig, Abel, and Marquardt (1998)

. . . (−10 years)











Thus with more powerful capabilities, the determination ofsteady-state setpoints may simply become an unnecessaryintermediate calculation.

Instead nonlinear, dynamic referencemodels could be used directly to optimize a profit objective.— Biegler and Rawlings (1991) . . . (−20 years)





Thus with more powerful capabilities, the determination ofsteady-state setpoints may simply become an unnecessaryintermediate calculation. Instead nonlinear, dynamic referencemodels could be used directly to optimize a profit objective.

— Biegler and Rawlings (1991) . . . (−20 years)





Thus with more powerful capabilities, the determination ofsteady-state setpoints may simply become an unnecessaryintermediate calculation. Instead nonlinear, dynamic referencemodels could be used directly to optimize a profit objective.— Biegler and Rawlings (1991)

. . . (−20 years)











In attempting to synthesize a feedback optimizing controlstructure, our main objective is to translate the economicobjective into process control objectives.

— Morari, Arkun, and Stephanopoulos (1980) . . . (−30 years)





In attempting to synthesize a feedback optimizing controlstructure, our main objective is to translate the economicobjective into process control objectives.— Morari, Arkun, and Stephanopoulos (1980)

. . . (−30 years)







Other approaches and relevant research

Self-optimizing controlSkogestad (2000); Aske et al. (2008)

Extremum seeking controlKrstic and Wang (2000); Guay and Zhang (2003); Guay et al. (2003);DeHaan and Guay (2004)

Reviews and industrial case studiesEngell (2007); Zavala and Biegler (2008); Backx et al. (2000), Zaninet al. (2002); Rotava and Zanin (2005)


Setpoints and unreachable setpoints

Consider the steady state of a dynamic model with state x , controlledinput u, and disturbance w

x(k + 1) = Ax(k) + Bu(k) + Bdw(k)

xs = (I − A)−1B︸︷︷︸G

us + (I − A)−1Bdws︸︷︷︸ds

xs = Gus + ds




x(k + 1) = Ax(k) + Bu(k) + Bdw(k)

xs = (I − A)−1B︸︷︷︸G


xs = Gus + ds




x(k + 1) = Ax(k) + Bu(k) + Bdw(k)

xs = (I − A)−1B︸︷︷︸G


xs = Gus + ds


Steady states — unconstrained system

xs

ds2 = 0

ds1 = 1

xs = Gus + ds

ds3 = −1Gxsp

us2 us3us1

us

For an unconstrained system with G 6= 0, any setpoint xsp with anydisturbance ds has a corresponding us .


Constraints and unreachable setpoints

xs

ds2 = 0

ds1 = 1

xs = Gus + ds

ds3 = −1Gxsp

us1usus2

us3

0 1

0 ≤ us ≤ 1

For a constrained system, the setpoint xsp may be unreachable for a givendisturbance ds . MPC is method of choice for this situation.


Constraints and unreachable setpoints

xs

xsp

us

0

0 ≤ ds ≤ G

0 ≤ us ≤ 1

xs = Gus + ds

ds ≤ 0

1

ds ≥ G

As the estimated disturbance changes with time, the setpoint may changebetween reachable and unreachable.


Steady-state problem definition

Stage cost:

L(x , u) = |x − xsp|2Q + |u − usp|2R Q,R > 0

Optimization: minu

L(x , u)

subject to: x = Ax + Bu u ∈ USolution: (x∗, u∗)



Stage cost:


Optimization: minu

L(x , u)

subject to: x = Ax + Bu u ∈ U

Solution: (x∗, u∗)



Stage cost:


Optimization: minu

L(x , u)

subject to: x = Ax + Bu u ∈ USolution: (x∗, u∗)


MPC problem definition — dynamic case

Cost function: V =N−1∑j=0

L(x(j), u(j))

Optimization: minu

V (u, x(0))

subject to:x+ = Ax + Bu u = {u(0), u(1), . . . u(N − 1)} u ∈ UStage cost:

targ–MPC: L(x , u) = |x − x∗|2Q + |u − u∗|2R –or–

sp–MPC: L(x , u) = |x − xsp|2Q + |u − usp|2R

Control law: u0(x) = u0(0, x)




L(x(j), u(j))

Optimization: minu

V (u, x(0))

subject to:x+ = Ax + Bu u = {u(0), u(1), . . . u(N − 1)} u ∈ U

Stage cost:







L(x(j), u(j))

Optimization: minu

V (u, x(0))








L(x(j), u(j))

Optimization: minu

V (u, x(0))






What closed-loop behavior is desirable? Fast tracking

xsp

x∗x

k

x(0)

x(0)Q � R (fast tracking)


What closed-loop behavior is desirable? Slow tracking

xsp

x∗x

k

x(0)Q � R (slow tracking)

x(0)


What closed-loop behavior is desirable? Asymmetrictracking

xsp

x∗x

k

x(0)Q � R (fast tracking)

x(0)


Why analysis? Unexpected closed-loop behavior

A finite horizon objective function may not give a stable controller!

How is this possible?

x1

x2

0

k

x1

x2

0

k

k + 1

x1

x2

0

k

k + 1

k + 2

x1

x2

0

k

k + 1

k + 2

closed-loop trajectory



A finite horizon objective function may not give a stable controller!How is this possible?

x1

x2

0

k

x1

x2

0

k

k + 1

x1

x2

0

k

k + 1

k + 2

x1

x2

0

k

k + 1

k + 2





x1

x2

0

k

x1

x2

0

k

k + 1

x1

x2

0

k

k + 1

k + 2

x1

x2

0

k

k + 1

k + 2





x1

x2

0

k

x1

x2

0

k

k + 1

x1

x2

0

k

k + 1

k + 2

x1

x2

0

k

k + 1

k + 2





x1

x2

0

k

x1

x2

0

k

k + 1

x1

x2

0

k

k + 1

k + 2

x1

x2

0

k

k + 1

k + 2





x1

x2

0

k

x1

x2

0

k

k + 1

x1

x2

0

k

k + 1

k + 2

x1

x2

0

k

k + 1

k + 2



Terminal constraint solution

Adding a terminal constraint ensures stability

May cause infeasibilityOpen-loop predictions not equal to closed-loop behavior

0

k

Φk

x1

x2

0

k

Φk

x1

x2

Vk+1 ≤ Vk − L(xk , uk)

k + 1

0

k

Φk

x1

x2


k + 1

k + 2

Vk+2 ≤ Vk+1 − L(xk+1, uk+1)



Adding a terminal constraint ensures stabilityMay cause infeasibility

Open-loop predictions not equal to closed-loop behavior

0

k

Φk

x1

x2

0

k

Φk

x1

x2


k + 1

0

k

Φk

x1

x2


k + 1

k + 2

Vk+2 ≤ Vk+1 − L(xk+1, uk+1)



Adding a terminal constraint ensures stabilityMay cause infeasibilityOpen-loop predictions not equal to closed-loop behavior

0

k

Φk

x1

x2

0

k

Φk

x1

x2


k + 1

0

k

Φk

x1

x2


k + 1

k + 2

Vk+2 ≤ Vk+1 − L(xk+1, uk+1)




0

k

Φk

x1

x2

0

k

Φk

x1

x2


k + 1

0

k

Φk

x1

x2


k + 1

k + 2

Vk+2 ≤ Vk+1 − L(xk+1, uk+1)




0

k

Φk

x1

x2

0

k

Φk

x1

x2


k + 1

0

k

Φk

x1

x2


k + 1

k + 2

Vk+2 ≤ Vk+1 − L(xk+1, uk+1)


Infinite horizon solution

The infinite horizon ensures stability

Open-loop predictions equal to closed-loop behavior

May be difficult to implement

x1

x2

0

Φk

k

x1

x2

0

Φk

k

k + 1

Vk+1 = Vk − L(xk , uk)

x1

x2

0

Φk

k

k + 1


k + 2

Vk+2 = Vk+1 − L(xk+1, uk+1)






x1

x2

0

Φk

k

x1

x2

0

Φk

k

k + 1


x1

x2

0

Φk

k

k + 1


k + 2

Vk+2 = Vk+1 − L(xk+1, uk+1)






x1

x2

0

Φk

k

x1

x2

0

Φk

k

k + 1


x1

x2

0

Φk

k

k + 1


k + 2

Vk+2 = Vk+1 − L(xk+1, uk+1)






x1

x2

0

Φk

k

x1

x2

0

Φk

k

k + 1


x1

x2

0

Φk

k

k + 1


k + 2

Vk+2 = Vk+1 − L(xk+1, uk+1)






x1

x2

0

Φk

k

x1

x2

0

Φk

k

k + 1


x1

x2

0

Φk

k

k + 1


k + 2

Vk+2 = Vk+1 − L(xk+1, uk+1)


Unreachable case — challenges for analyzing closed-loopbehavior

Sequence of optimal costs is not monotone decreasing

Infinite horizon cost is unbounded for all input sequences

Optimal cost is not a Lyapunov function for the closed-loop system

Standard nominal MPC stability arguments do not apply

Simulations indicate the closed loop is stable

How can we be sure?








How can we be sure?








How can we be sure?








How can we be sure?








How can we be sure?








How can we be sure?


Unreachable case — theoretical result

Theorem (Asymptotic Stability of Terminal Constraint MPC)

The optimal steady state is the asymptotically stable solution of theclosed-loop system under terminal constraint MPC. Its region of attractionis the steerable set.

(Rawlings, Bonne, Jørgensen, Venkat, and Jørgensen, 2008)


Example 1. Single input–single output system

G (s) =−0.2623

60s2 + 59.2s + 1

Sample time T = 10 sec

Input constraint, −1 ≤ u ≤ 1

Setpoint ysp = 0.25

Qy = 10,R = 0, S = 1,Q = C ′Qy C + 0.01I2

Horizon length N = 80

Periodic state disturbance dx = [17.1 1.77]′ which is estimated fromthe measurements


Disturbance estimation

As the estimated disturbance changes with time, the setpoint changesbetween reachable and unreachable.

xs

xsp

us

0

0 ≤ ds ≤ G

ds ≤ 0

1

ds ≥ G

0 ≤ us ≤ 1

xsp

k0

d(k)

x∗(k)


Disturbance estimation

As the estimated disturbance changes with time, the setpoint changesbetween reachable and unreachable.

xs

xsp

us

0

0 ≤ ds ≤ G

ds ≤ 0

1

ds ≥ G

0 ≤ us ≤ 1

xsp

k0

d(k)

x∗(k)


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 50 100 150 200 250 300 350 400Time (sec)

y

setpointtarget (y ∗)

y(sp-MPC)y(targ-MPC)

-1

-0.5

0

0.5

1

0 50 100 150 200 250 300 350 400Time (sec)

u

target (u∗)u(sp-MPC)

u(targ-MPC)


Summary of Example 1

Performance targ-MPC sp-MPC ∆(index)%Measure

Vu 0.016 2.2× 10−6 99.98Vy 3.65 1.71 53V 3.67 1.71 54


Example 2. Two input–two output system with noise

G (s) =

[1.5

(s+2)(s+1)0.75

(s+5)(s+2)

0.5(s+0.5)(s+1)

2(s+2)(s+3)

]

Sample time T = 0.25 sec

Input constraints −0.5 ≤ u1, u2 ≤ 0.5

Setpoint ysp = [0.337 0.34]′

Measurement and state noise


0.29

0.295

0.3

0.305

0.31

0.315

0.32

0.325

0.33

0.335

0.34

0 5 10 15 20 25Time (sec)

y1

setpointy1(sp-MPC)

y1(targ-MPC)

0.31

0.315

0.32

0.325

0.33

0.335

0.34

0.345

0 5 10 15 20 25Time (sec)

y2

setpointy2(sp-MPC)

y2(targ-MPC)


0.4

0.5

0 5 10 15 20 25

0.4

0.5

Time (sec)

u1

u1

u1(targ-MPC)u1(sp-MPC)

-0.48

-0.46

-0.44

0 5 10 15 20 25

-0.48

-0.46

-0.44

Time (sec)

u2

u2

u2(targ-MPC)u2(sp-MPC)


-0.1

0

0.1

0.2

0.3

0 5 10 15 20 25Time (sec)

y ∗1

d1

setpoint

target (y ∗1 )

d1

0

0.1

0.2

0.3

0 5 10 15 20 25Time (sec)

y ∗2

d2

setpoint

target (y ∗2 )

d2


Summary of Example 2

Performance targ-MPC sp-MPC ∆(index)%Measure (×10−3) (×10−3)

Vu 1.32 1.24 6.1Vy 4.4 0.48 89V 5.72 1.72 70


Optimizing economics: Current industrial practice

Validation


Reconciliation


Plant

Controller

Two layer structure

Drawbacks

I Inconsistent modelsI Re-identify linear model as

setpoint changesI Time scale separation may not

holdI Economics unavailable in

dynamic layer


Optimizing economics: Current industrial practice

Validation


Reconciliation


Plant

Controller

Two layer structure

DrawbacksI Inconsistent modelsI Re-identify linear model as

setpoint changesI Time scale separation may not

holdI Economics unavailable in

dynamic layer


Motivating the idea

-4 -2 0 2 4-4

-20

24

Profit

Input (u)

State (x)

Profit

-4 -2 0 2 4-4

-20

24

Profit

Input (u)

State (x)

Profit


Motivating the idea

-4 -2 0 2 4-4

-20

24

Profit

Input (u)

State (x)

Profit

-4 -2 0 2 4-4

-20

24

Profit

Input (u)

State (x)

Profit


Economics controller

Stage cost:eco–MPC: L(x , u) = any strictly convex function


L(x(j), u(j))

Optimization: minu

V (u, x(0))

subject to:x+ = Ax + Bu u = {u(0), u(1), . . . u(N − 1)} u ∈ UControl law: u0(x) = u0(0, x)

Asymptotic stability: (x(k), u(k)) −→ (xe , ue), the optimal economicsteady state for the chosen L(x , u).Requires terminal constraint, terminal controller, or infinite horizon.





L(x(j), u(j))

Optimization: minu

V (u, x(0))







L(x(j), u(j))

Optimization: minu

V (u, x(0))

subject to:x+ = Ax + Bu u = {u(0), u(1), . . . u(N − 1)} u ∈ U







L(x(j), u(j))

Optimization: minu

V (u, x(0))







L(x(j), u(j))

Optimization: minu

V (u, x(0))


Asymptotic stability: (x(k), u(k)) −→ (xe , ue), the optimal economicsteady state for the chosen L(x , u).

Requires terminal constraint, terminal controller, or infinite horizon.





L(x(j), u(j))

Optimization: minu

V (u, x(0))




Example

xk+1 =

[0.857 0.884−0.0147 −0.0151

]xk +

[8.565

0.88418

]uk

Input constraint: −1 ≤ u ≤ 1

Leco = α′x + β′u

α =[−3 −2

]′β = −2

Ltarg = |x − x∗|2Q + |u − u∗|2RQ = 2I2 R = 2

x∗ =[60 0

]′u∗ = 1


targ-MPCtarg-MPC60 65 70 75 80 85

x1

-2

0

2

4

6

8

10

x 2

targ-MPC eco-MPCtarg-MPC eco-MPC60 65 70 75 80 85

x1

-2

0

2

4

6

8

10

x 2


targ-MPCtarg-MPC60 65 70 75 80 85

x1

-2

0

2

4

6

8

10

x 2

targ-MPC eco-MPCtarg-MPC eco-MPC60 65 70 75 80 85

x1

-2

0

2

4

6

8

10

x 2


55

60

65

70

75

80

0 2 4 6 8 10 12 14

Sta

te

targ-MPC

-2

0

2

4

6

8

10

0 2 4 6 8 10 12 14

Sta

te

targ-MPC

-1

-0.5

0

0.5

1

0 2 4 6 8 10 12 14

Inpu

t

Time

targ-MPC


55

60

65

70

75

80

85

90

0 2 4 6 8 10 12 14

Sta

te

targ-MPCeco-MPC

-2

0

2

4

6

8

10

0 2 4 6 8 10 12 14

Sta

te

targ-MPCeco-MPC

-1

-0.5

0

0.5

1

0 2 4 6 8 10 12 14

Inpu

t

Time

targ-MPCeco-MPC


Introduction to Turnpike Theorems

It is exactly like a turnpike paralleled by a network of minor roads.

There is a fastest route between any two points; and if the originand destination are close together and far from the turnpike, thebest route may not touch the turnpike.

But if the origin and destination are far enough apart, it willalways pay to get on the turnpike and cover distance at the bestrate of travel, even if this means adding a little mileage at eitherend.

—Dorfman, Samuelson, and Solow (1958, p.331)




















Creating a turnpike example

Standard linear quadratic problem

x+ = Ax + Bu

L(x , u) = |Cx − ysp|2Q + |u − usp|2R Q > 0,R > 0

Choose an inconsistent setpoint

A = 1/2 B = 1/4 C = 1 Q = 1 R = 1

ys = Gus G = 1/2

usp = 0 ysp = 2


Creating a turnpike example

Standard linear quadratic problem

x+ = Ax + Bu

L(x , u) = |Cx − ysp|2Q + |u − usp|2R Q > 0,R > 0

Choose an inconsistent setpoint

A = 1/2 B = 1/4 C = 1 Q = 1 R = 1

ys = Gus G = 1/2

usp = 0 ysp = 2


Inconsistent Setpoint and Optimal steady state

u

(u∗, x∗)

(usp, xsp)

G

xOptimal steady state

usp = 0 xsp = 2

u∗ = 0.8 x∗ = 0.4


Optimal control problem

Cost function and dynamic model

V (x ,u) =N−1∑i=0

L(xi , ui ) s.t. x+ = Ax + Bu, x(0) = x

Optimal state and input trajectories

minu

V (x ,u) u0(x), x0(x)


Optimal control problem

Cost function and dynamic model

V (x ,u) =N−1∑i=0

L(xi , ui ) s.t. x+ = Ax + Bu, x(0) = x

Optimal state and input trajectories

minu

V (x ,u) u0(x), x0(x)


Optimal trajectory: xsp = 2, usp = 0

-1

-0.5

0

0.5

1

0 1 2 3 4

xN = 5

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4

t

ux0 = 1

x0 = −1



-1

-0.5

0

0.5

1

0 5 10 15 20 25 30

x N = 30

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

t

u x0 = 1

x0 = −1



-1

-0.5

0

0.5

1

0 10 20 30 40 50 60 70 80 90 100

x N = 100

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90 100

t

u x0 = 1

x0 = −1


Nonlinear Dynamics: Chemical Reaction

Chemical Reaction

A −→ B r = kcnA

k is the rate constant and n is the reaction order

Model

dcA

dt=

1

τ(cAf − cA)− kcn

A

dx

dt=

1

τ(u − x)− kxn τ = 10, k = 1.2, n = 2

x = cA reactor A concentrationu = cAf feed A concentration


Nonlinear Dynamics: Chemical Reaction

Chemical Reaction

A −→ B r = kcnA

k is the rate constant and n is the reaction order

Model

dcA

dt=

1

τ(cAf − cA)− kcn

A

dx

dt=

1

τ(u − x)− kxn τ = 10, k = 1.2, n = 2

x = cA reactor A concentrationu = cAf feed A concentration


Maximizing production rate in a CSTR

Input constraints

0 ≤ u(t) ≤ 31

T

∫ T

0u(t)dt = 1

T is the time interval considered

Maximize the average production rate

V (x(0), u(t)) = − 1

T

∫ T

0kxn(t)dt

The optimal control problem

minu(t)

V (x(0, u(t)) subject to model and constraints


Maximizing production rate in a CSTR

Input constraints

0 ≤ u(t) ≤ 31

T

∫ T

0u(t)dt = 1

T is the time interval considered

Maximize the average production rate

V (x(0), u(t)) = − 1

T

∫ T

0kxn(t)dt

The optimal control problem

minu(t)

V (x(0, u(t)) subject to model and constraints


Optimal Control

Optimal u and x

0

1

2

3

4

0 20 40 60 80 100

0

0.1

0.2

0.3

0.4

0.5

t

u x

Production rate, RB = kc2A

0

0.05

0.1

0.15

0.2

0.25

0.3

0 20 40 60 80 100t

c2A

cAs

〈c2A〉

c2As


Optimal Control

Optimal u and x

0

1

2

3

4

0 20 40 60 80 100

0

0.1

0.2

0.3

0.4

0.5

t

u x

Production rate, RB = kc2A

0

0.05

0.1

0.15

0.2

0.25

0.3

0 20 40 60 80 100t

c2A

cAs

〈c2A〉

c2As


The Big Picture: Convexity and Nonlinearity

Mean state and control

x =1

N

N∑i=1

xi u =1

N

N∑i=1

ui

Carlson et al. (1991, pp.51–52)




x =1

N

N∑i=1

xi u =1

N

N∑i=1

ui

Convexity of L and nonlinearity of f

L (x , u) ≤ 1

N

N∑i=1

L(xi , ui ) Jensen’s inequality (1906)

x+ = Ax + Bu linear dynamics





x =1

N

N∑i=1

xi u =1

N

N∑i=1

ui


L (x , u) ≤ 1

N

N∑i=1

L(xi , ui ) − γ (|xi − x , ui − u|)︸︷︷︸measure of convexity

x+ = f (x , u)︸︷︷︸measure of nonlinearity





x =1

N

N∑i=1

xi u =1

N

N∑i=1

ui


L (x , u) ≤ 1

N

N∑i=1

L(xi , ui ) − γ (|xi − x , ui − u|)︸︷︷︸measure of convexity

x+ = f (x , u)︸︷︷︸measure of nonlinearity



Conclusions

Overall goal: develop alternatives to the current two-layer approachto optimizing process economics

Require practical solutions with sound supporting theory

Nonlinear MPC provides one approach

Opportunities and Challenges . . .


Conclusions






Conclusions






Conclusions






Conclusions

OpportunitiesI Performance advantage

I Consistent model for optimizing performanceI Consistent statement of process objectivesI Optimization software well developedI Linear dynamics and convex objectives well supportedI Leverage from industrial implementation of linear MPCI This is a big opportunity!


Conclusions

OpportunitiesI Performance advantageI Consistent model for optimizing performance

I Consistent statement of process objectivesI Optimization software well developedI Linear dynamics and convex objectives well supportedI Leverage from industrial implementation of linear MPCI This is a big opportunity!


Conclusions

OpportunitiesI Performance advantageI Consistent model for optimizing performanceI Consistent statement of process objectives

I Optimization software well developedI Linear dynamics and convex objectives well supportedI Leverage from industrial implementation of linear MPCI This is a big opportunity!


Conclusions

OpportunitiesI Performance advantageI Consistent model for optimizing performanceI Consistent statement of process objectivesI Optimization software well developed

I Linear dynamics and convex objectives well supportedI Leverage from industrial implementation of linear MPCI This is a big opportunity!


Conclusions

OpportunitiesI Performance advantageI Consistent model for optimizing performanceI Consistent statement of process objectivesI Optimization software well developedI Linear dynamics and convex objectives well supported

I Leverage from industrial implementation of linear MPCI This is a big opportunity!


Conclusions

OpportunitiesI Performance advantageI Consistent model for optimizing performanceI Consistent statement of process objectivesI Optimization software well developedI Linear dynamics and convex objectives well supportedI Leverage from industrial implementation of linear MPC

I This is a big opportunity!


Conclusions

OpportunitiesI Performance advantageI Consistent model for optimizing performanceI Consistent statement of process objectivesI Optimization software well developedI Linear dynamics and convex objectives well supportedI Leverage from industrial implementation of linear MPCI This is a big opportunity!


Conclusions

ChallengesI Understanding the interplay between nonlinearity of model and

convexity of objective

I Managing the complexity of an optimal economic solutionI Developing theory, algorithms, tuning procedures, and realistic case

studies that support industrial implementationI This is a big challenge!


Conclusions


convexity of objectiveI Managing the complexity of an optimal economic solution

I Developing theory, algorithms, tuning procedures, and realistic casestudies that support industrial implementation

I This is a big challenge!


Conclusions


convexity of objectiveI Managing the complexity of an optimal economic solutionI Developing theory, algorithms, tuning procedures, and realistic case

studies that support industrial implementation

I This is a big challenge!


Conclusions


convexity of objectiveI Managing the complexity of an optimal economic solutionI Developing theory, algorithms, tuning procedures, and realistic case

studies that support industrial implementationI This is a big challenge!


Further Reading I

E. M. B. Aske, S. Strand, and S. Skogestad. Coordinator MPC for maximizing plantthroughput. Comput. Chem. Eng., 32:195–204, 2008.

T. Backx, O. Bosgra, and W. Marquardt. Integration of model predictive control andoptimization of processes. In Advanced Control of Chemical Processes, June 2000.

L. T. Biegler and J. B. Rawlings. Optimization approaches to nonlinear model predictivecontrol. In Y. Arkun and W. H. Ray, editors, Chemical Process Control–CPCIV,pages 543–571. CACHE, 1991.

D. A. Carlson, A. B. Haurie, and A. Leizarowitz. Infinite Horizon Optimal Control.Springer Verlag, second edition, 1991.

D. DeHaan and M. Guay. Extremum seeking control of nonlinear systems withparametric uncertainties and state constraints. In Proceedings of the 2004 AmericanControl Conference, pages 596–601, July 2004.

R. Dorfman, P. Samuelson, and R. Solow. Linear Programming and Economic Analysis.McGraw-Hill, New York, 1958.

S. Engell. Feedback control for optimal process operation. J. Proc. Cont., 17:203–219,2007.


Further Reading II

M. Guay and T. Zhang. Adaptive extremum seeking control of nonlinear dynamicsystems with parametric uncertainty. Automatica, 39:1283–1293, 2003.

M. Guay, D. Dochain, and M. Perrier. Adaptive extremum seeking control ofnonisothermal continuous stirred tank reactors with temperature constraints. InProceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii,December 2003.

A. Helbig, O. Abel, and W. Marquardt. Structural concepts for optimization basedcontrol of transient processes. In International Symposium on Nonlinear ModelPredictive Control, Ascona, Switzerland, 1998.

J. L. W. V. Jensen. Sur les fonctions convexes et les inegalites entre les valeursmoyennes. Acta Math., 30:175–193, 1906.

M. Krstic and H.-H. Wang. Stability of extremum seeking feedback for general nonlineardynamic systems. Automatica, 36:595–601, 2000.

M. Morari, Y. Arkun, and G. Stephanopoulos. Studies in the synthesis of controlstructures for chemical processes. Part I: Formulation of the problem. processdecomposition and the classification of the control tasks. Analysis of the optimizingcontrol structures. AIChE J., 26(2):220–232, 1980.


Further Reading III

J. B. Rawlings, D. Bonne, J. B. Jørgensen, A. N. Venkat, and S. B. Jørgensen.Unreachable setpoints in model predictive control. Accepted for publication in IEEETAC, April 2008.

O. Rotava and A. Zanin. Multivariable control and real-time optimization — anindustrial practical view. Hydrocarbon Processing, pages 61–71, June 2005.

S. Skogestad. Plantwide control: the search for the self-optimizing control structure. J.Proc. Cont., 10:487–507, 2000.

A. C. Zanin, M. Tvrzska de Gouvea, and D. Odloak. Integrating real-time optimizationinto the model predictive controller of the FCC system. Control Eng. Practice, 10:819–831, 2002.

V. M. Zavala and L. T. Biegler. The advanced step nmpc controller: optimality, stabilityand robustness. To appear in Automatica, 2008.


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