Economic optimization in Model Predictive Control

Economic optimization in Model Predictive Control

Rishi Amrit

Department of Chemical and Biological EngineeringUniversity of Wisconsin-Madison

29th February, 2008

Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 1 / 37

Outline

1 Incentives for process control

2 Preliminaries

3 Motivating the idea

4 Current work

5 Future work

6 Conclusions


Incentives for process control


Production specifications

Operational constraints / Environmental regulations

Safety

Economics






Safety

Economics






Safety

Economics






Safety

Economics


Incentives for process control Economic incentive

Economic Incentive

Production of a plant depends heavily on plant’s limitations andoperating constraints

Operating conditions keep changing plant production

Under all variations and restrictions, plant must do the best it can:Process optimization



Global production maximum

��

��

��

��

$ Loss

Process parameter Time

Profit ($)

Profit ($)$ Loss

��

��

��

��

$ Loss


Profit ($)

Profit ($)$ Loss

Higher profit expected whenband of variation is reduced

Allows operation at/near theoptimum for more time

Smoother operation =⇒Higher profit



Global production maximum

��

��

��

��

$ Loss


Profit ($)

Profit ($)$ Loss

��

��

��

��

$ Loss


Profit ($)

Profit ($)$ Loss

Higher profit expected whenband of variation is reduced





Maximum production at bound

��

��

��

��

Profit ($)$ Loss

Profit ($)

$ Loss


Higher profit when band ofvariation is reduced





$$ Savings !

Better ControlPoor Control

Profit

Higher fluctuations: Poor disturbance rejection

Forces the mean operating state to be away from optimum to meetthe constraints

Solution: Reduce fluctuations and go nearer to the optimal


Preliminaries Process model

Process model

Processu x

y

Process model governing process dynamics

dx

dt= f (x(t), u(t))

y(t) = g(x(t))

Steady state:

f (xs , us) = 0

ys = g(xs)



Process model

Processu x

y


dx

dt= f (x(t), u(t))

y(t) = g(x(t))

Steady state:

f (xs , us) = 0

ys = g(xs)



Process model

Processu x

y


dx

dt= f (x(t), u(t))

y(t) = g(x(t))

Steady state:

f (xs , us) = 0

ys = g(xs)



Objective translation

Economic objectives are translated into process control objectives

Notion of setpoints / targets

Economic profit functionΦ(x ,u)

Economic optimumu

x


Economic optimumu

xConstraints


Economic optimumu

xConstraints

Target

Steady state curvef (xs ,us) = 0


Economic optimumu

xConstraints

Target








Economic optimumu

x


Economic optimumu

xConstraints


Economic optimumu

xConstraints

Target



Economic optimumu

xConstraints

Target








Economic optimumu

x


Economic optimumu

xConstraints


Economic optimumu

xConstraints

Target



Economic optimumu

xConstraints

Target








Economic optimumu

x


Economic optimumu

xConstraints


Economic optimumu

xConstraints

Target



Economic optimumu

xConstraints

Target



Preliminaries Model Predictive Control

Model Predictive Control

Output target

Measured Input

Measured Output

Control/Prediction Horizon

k

Output target

Measured Input

Measured Output


Optimized Future Input Trajectory

REGULATIONk k + 1

uk

Output target

Measured Input

Measured Output



REGULATION

Predicted Output

k k + 1

uk

uk+1




Output target

Measured Input

Measured Output


k

Output target

Measured Input

Measured Output



REGULATIONk k + 1

uk

Output target

Measured Input

Measured Output



REGULATION

Predicted Output

k k + 1

uk

uk+1




Output target

Measured Input

Measured Output


k

Output target

Measured Input

Measured Output



REGULATIONk k + 1

uk

Output target

Measured Input

Measured Output



REGULATION

Predicted Output

k k + 1

uk

uk+1


Preliminaries MPC Optimization problem

Problem definition

Get to the steady economic optimum (target): Minimize the distancefrom the target (stage cost)

L(x,u) = (x − xt)′Q(x − xt) + (u − ut)′R(u − ut)

Minimize the stage cost summed over a chosen control horizon(number of moves into the future: N)

minu

N−1∑i=0

L(x,u)

subject to the process model

xk+1 = Axk + Buk


Preliminaries MPC Optimization problem

Problem definition

Get to the steady economic optimum (target): Minimize the distancefrom the target (stage cost)

L(x,u) = (x − xt)′Q(x − xt) + (u − ut)′R(u − ut)

Minimize the stage cost summed over a chosen control horizon(number of moves into the future: N)

minu

N−1∑i=0

L(x,u)

subject to the process model

xk+1 = Axk + Buk


Preliminaries Implementation strategies

Current practice: RTO

Validation

Plant

Controllers

Planning and Scheduling

Reconciliation

Model UpdateOptimizationSteady State

Real time optimization

Two layer structure used toaddress economically optimalsolutionRTO generated setpointspassed to lower level controllerControllers try to “track” thetargets provided to it

Drawbacks

Lower sampling rateAdaptation of operatingconditions is slowConsequence: Loss ineconomics




Validation

Plant

Controllers


Reconciliation




Drawbacks





Validation

Plant

Controllers


Reconciliation




Drawbacks



Motivating the idea

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

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

��

��

Profit ($)

$ Profit

Maximum profit

Global economic optimum not being a steady state introduces highpotential areas of transient operation

Translation of economic objective to control objective loses theinformation about maximum profit possible


Motivating the idea

Motivating the idea

What is not the primary objective of feedback control

Tracking setpoints or targetsTracking dynamic setpoint changes

Setpoints/Targets: Translating economic objectives to process controlobjectives

Process Economics

Steady state economics

Process Economics

Loss of economic information dueto two layer approach

Control objective:

L(x,u) = (x− xt)′Q(x− xt)

+(u− ut)′R(u− ut)


Motivating the idea

Motivating the idea




Process Economics


Process Economics


Control objective:




Motivating the idea

Motivating the idea




Process Economics


Process Economics


Control objective:




Motivating the idea

Motivating the idea




Process Economics


Process Economics


Control objective:




Motivating the idea

Motivating the idea




Process Economics


Process Economics


Control objective:

L(x,u) = −P(x,u)


Motivating the idea

Make money or chase target ?

Due to disturbances and constraints, the economic optimum is not asteady state in general

System stabilizes at the steady target estimated from the steady stateoptimization

During system transients, system may or may not pass through theeconomic optimum


Motivating the idea

The contest

The closer the system gets to the economic optimum, the moreprofitable it is

Who gets closest to the global economic optimum ?

Tracking controllers: Rush to the target (away from non steadyeconomic optimum)Tracking speed chosen through penalties, but still the objective remainsto drive away from non steady economic optimum !

Economics optimizing controller: Expected to get closer to theoptimum with eventual setting at the steady target


Motivating the idea

The contest



Tracking controllers: Rush to the target (away from non steadyeconomic optimum)

Tracking speed chosen through penalties, but still the objective remainsto drive away from non steady economic optimum !



Motivating the idea

The contest






Motivating the idea

The contest






Current work Quadratic economics

A motivating formulation

A

A, B

V

F , CAf

A→ B

Consider a CSTR

VdCA

dt= F (CAf − CA)− kCAV

VdCB

dt= F (CBf − FCB) + kCAV

States: CA,CB Input: F

The simplest form of profit:

P = αAF (CA − CAf ) + αBF (CB)

=[CA CB

] [αA

αB

]′F− αACAf F

αA: Cost of A αB : Cost of BState-Input Cross term !


Current work Quadratic economics

A motivating formulation

A

A, B

V

F , CAf

A→ B

Consider a CSTR

VdCA

dt= F (CAf − CA)− kCAV

VdCB

dt= F (CBf − FCB) + kCAV

States: CA,CB Input: F

The simplest form of profit:

P = αAF (CA − CAf ) + αBF (CB)

=[CA CB

] [αA

αB

]′F− αACAf F

αA: Cost of A αB : Cost of BState-Input Cross term !


Current work SISO Example

Example: Single input single output

Consider a linear system

xk+1 = 0.3xk + uk

Profit function: −3x2k − 5u2

k−2xkuk + 98xk + 80uk

Objective: Maximize Profit !

Scheme one:

Evaluate the best economic target at every sample time (RTO)Controller tracks the target given to it



Example: Single input single output

Consider a linear system

xk+1 = 0.3xk + uk




Scheme one:

Evaluate the best economic target at every sample time (RTO)Controller tracks the target given to it



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-4 -2 0 2 4 6 8 10

Sta

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Input

Cost contours

Steady state line

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-4 -2 0 2 4 6 8 10

Sta

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Input

Cost contours

Tracking contours

Steady state line

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-4 -2 0 2 4 6 8 10

Sta

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Input

Cost contours

Tracking contours

Steady state linetarg-MPC



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Input

Cost contours

Steady state line

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-4 -2 0 2 4 6 8 10

Sta

te

Input

Cost contours

Tracking contours

Steady state line

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-4 -2 0 2 4 6 8 10

Sta

te

Input

Cost contours

Tracking contours




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-4 -2 0 2 4 6 8 10

Sta

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Input

Cost contours

Steady state line

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-4 -2 0 2 4 6 8 10

Sta

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Input

Cost contours

Tracking contours

Steady state line

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-4 -2 0 2 4 6 8 10

Sta

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Input

Cost contours

Tracking contours







Scheme two:

Controller minimizes the negative of profit



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-4 -2 0 2 4 6 8 10

Sta

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Input

Cost contours

Tracking contours


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-4 -2 0 2 4 6 8 10

Sta

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Input

Cost contours

Tracking contours

Steady state linetarg-MPCeco-MPC

Performance Measures

targ-MPC eco-MPC ∆(index)%Lossa $642.6 $588.2 8.5

aReference: Maximum profit = 0



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-4 -2 0 2 4 6 8 10

Sta

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Input

Cost contours

Tracking contours


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-4 -2 0 2 4 6 8 10

Sta

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Input

Cost contours

Tracking contours






Current work Effect of disturbance

-30

-25

-20

-15

-10

-5

0

16 18 20 22 24 26 28 30

Pro

fit

Input

p = 5

p = 0

p = −5

-30

-25

-20

-15

-10

-5

0

16 18 20 22 24 26 28 30

Pro

fit

Input

p = 5

p = 0

p = −5

Disturbance model:xk+1 = Axk + Buk + Bdpk

Disturbance shifts the steadystate cost curve

The steady state target changes

System transients from previoustarget to the new target



-30

-25

-20

-15

-10

-5

0

16 18 20 22 24 26 28 30

Pro

fit

Input

p = 5

p = 0

p = −5

-30

-25

-20

-15

-10

-5

0

16 18 20 22 24 26 28 30

Pro

fit

Input

p = 5

p = 0

p = −5

Disturbance model:xk+1 = Axk + Buk + Bdpk

Disturbance shifts the steadystate cost curve

The steady state target changes

System transients from previoustarget to the new target



-4

-2

0

2

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Sta

te

Input

p = 0






-4

-2

0

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Sta

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Input

p = 5

p = 0






-4

-2

0

2

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Sta

te

Input

p = 5

p = 0

targ-MPC






-4

-2

0

2

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6

8

14 16 18 20 22 24 26 28

Sta

te

Input

p = 5

p = 0

targ-MPCeco-MPC






Random disturbance corrupts state evolution

All states assumed measured

-15

-10

-5

0

5

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0 5 10 15 20 25 30 35 40 45 50

Sta

te

Time

targ-MPCeco-MPC

eco-opt

-10

-5

0

5

10

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0 5 10 15 20 25 30 35 40 45 50

Inpu

t

Time

targ-MPCeco-MPC



Random disturbance corrupts state evolution


-15

-10

-5

0

5

10

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0 5 10 15 20 25 30 35 40 45 50

Sta

te

Time

targ-MPCeco-MPC

eco-opt

-10

-5

0

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Inpu

t

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targ-MPCeco-MPC






Current work Linear economics

Maximum throughput

Consider a typical profit function for the plant:

(−L) =∑

j

pPjPj −

∑i

pFiFi −

∑k

pQkQk

Pj : Product flows Fi : Feed flows Qk : Utility duties

Assume all feed flows set in proportion to throughput (F ), constantefficiency in the units and constant intensive variables

Fi = kF ,iF Pj = kP,jF Qk = kQ,kF

(−L) =

∑j

pPj kP,j −∑

i

pFi kF ,i −∑

k

pQkkQ,k

F = pF

p: operational profit per unit feed F processed



Maximum throughput


(−L) =∑

j

pPjPj −

∑i

pFiFi −

∑k

pQkQk




(−L) =

∑j

pPj kP,j −∑

i

pFi kF ,i −∑

k

pQkkQ,k

F = pF




Maximum throughput


(−L) =∑

j

pPjPj −

∑i

pFiFi −

∑k

pQkQk




(−L) =

∑j

pPj kP,j −∑

i

pFi kF ,i −∑

k

pQkkQ,k

F = pF




Economic optimum ⇐⇒ Maximizing throughput

Linear economics: Unconstrained problem unbounded

Constrained problem: Optimal solution lies on the process bounds

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Sta

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Input

Linear economic contours

Steady state line

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0 2 4 6 8 10

Sta

te

Input


Tracking contours

Steady state line

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0 2 4 6 8 10

Sta

te

Input


Tracking contours







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0 2 4 6 8 10

Sta

te

Input


Steady state line

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0 2 4 6 8 10

Sta

te

Input


Tracking contours

Steady state line

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0 2 4 6 8 10

Sta

te

Input


Tracking contours







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0 2 4 6 8 10

Sta

te

Input


Steady state line

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0 2 4 6 8 10

Sta

te

Input


Tracking contours

Steady state line

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0 2 4 6 8 10

Sta

te

Input


Tracking contours



Current work Example: Transient to steady state

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

Ltrack = ‖x − x∗‖2Q + ‖u − u∗‖2

R

Q = 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

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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



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targ-MPC

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0

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0 5 10 15 20 25 30 35 40 45 50

Sta

te

targ-MPC

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0.5

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0 5 10 15 20 25 30 35 40 45 50

Inpu

t

Time

targ-MPC



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0 5 10 15 20 25 30 35 40 45 50

Sta

te

targ-MPCeco-MPC

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Sta

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targ-MPCeco-MPC

-1

-0.5

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Inpu

t

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targ-MPCeco-MPC



Example: Effect of disturbance

Random disturbance affecting the state evolution


System started at the steady optimum with zero disturbance



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Sta

te

targ-MPC

-3-2.5

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Sta

te

targ-MPC

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-0.5

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Inpu

t

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targ-MPC



46485052545658606264

0 10 20 30 40 50 60 70 80 90 100

Sta

te

targ-MPCeco-MPC

-3-2.5

-2-1.5

-1-0.5

00.5

11.5

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Sta

te

targ-MPCeco-MPC

-1

-0.5

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0.5

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0 10 20 30 40 50 60 70 80 90 100

Inpu

t

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targ-MPCeco-MPC


Future work Theoretical Issues

Future work

Investigate economic models

Presented idea banks on a good economic measureTranslation of objectives needs deep investigationNeed to define a good representative of the process economics

Establish asymptotic stability and convergence properties for broaderclass of cost functions

Steady state cost maybe nonzero =⇒ Infinite horizon cost isunboundedCosts corresponding to the optimal input sequence may not bemonotonically decreasing


Future work Theoretical Issues

Future work

Investigate economic models

Presented idea banks on a good economic measureTranslation of objectives needs deep investigationNeed to define a good representative of the process economics

Establish asymptotic stability and convergence properties for broaderclass of cost functions

Steady state cost maybe nonzero =⇒ Infinite horizon cost isunboundedCosts corresponding to the optimal input sequence may not bemonotonically decreasing


Future work Software development

Update the software tools to handle the new class of problems

Efficient software tools critical to the evaluation of the new class ofproblemsThe existing tools handle quadratic objective functionsEconomics may not be quadratic and hence the tools have to becapable of handling more general cost functions


Future work

Set up the problem for a realistic scenario and test using industrialdata

Simulations, like the ones shown, just predict the possible advantagesof the new schemeThe idea must be tested for a physical system with well definedeconomics

Collaborate for the distributed version

Distributed control schemes allow more robust and flexible controlThe new scheme can be implemented in distributed scenario


Conclusions

Conclusions

Profit depends heavily on steady state economic optimization layer

A separate layer causes a loss in economic performance duringtransient

Opportunity to rethink distribution of functionality between layers

Merging the economics with the controller objective reduces the lossof economic information

Economic optimizing control expected to capture the potentialprofitable areas of operation


Conclusions

Conclusions







Conclusions

Conclusions







Conclusions

Conclusions







Conclusions

Conclusions







Documents

Economic optimization in Model Predictive Control