Post on 19-Jan-2022
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 2 / 37
Incentives for process control
Incentives for process control
Production specifications
Operational constraints / Environmental regulations
Safety
Economics
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 3 / 37
Incentives for process control
Incentives for process control
Production specifications
Operational constraints / Environmental regulations
Safety
Economics
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 3 / 37
Incentives for process control
Incentives for process control
Production specifications
Operational constraints / Environmental regulations
Safety
Economics
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 3 / 37
Incentives for process control
Incentives for process control
Production specifications
Operational constraints / Environmental regulations
Safety
Economics
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 3 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 4 / 37
Incentives for process control Economic incentive
Global production maximum
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$ Loss
Process parameter Time
Profit ($)
Profit ($)$ Loss
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$ Loss
Process parameter Time
Profit ($)
Profit ($)$ Loss
Higher profit expected whenband of variation is reduced
Allows operation at/near theoptimum for more time
Smoother operation =⇒Higher profit
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 5 / 37
Incentives for process control Economic incentive
Global production maximum
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$ Loss
Process parameter Time
Profit ($)
Profit ($)$ Loss
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$ Loss
Process parameter Time
Profit ($)
Profit ($)$ Loss
Higher profit expected whenband of variation is reduced
Allows operation at/near theoptimum for more time
Smoother operation =⇒Higher profit
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 5 / 37
Incentives for process control Economic incentive
Maximum production at bound
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Profit ($)$ Loss
Profit ($)
$ Loss
Process parameter Time
Higher profit when band ofvariation is reduced
Allows operation at/near theoptimum for more time
Smoother operation =⇒Higher profit
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 6 / 37
Incentives for process control Economic incentive
$$ 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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 7 / 37
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)
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 8 / 37
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)
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 8 / 37
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)
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 8 / 37
Preliminaries Process model
Objective translation
Economic objectives are translated into process control objectives
Notion of setpoints / targets
Economic profit functionΦ(x ,u)
Economic optimumu
x
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Target
Steady state curvef (xs ,us) = 0
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Target
Steady state curvef (xs ,us) = 0
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 9 / 37
Preliminaries Process model
Objective translation
Economic objectives are translated into process control objectives
Notion of setpoints / targets
Economic profit functionΦ(x ,u)
Economic optimumu
x
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Target
Steady state curvef (xs ,us) = 0
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Target
Steady state curvef (xs ,us) = 0
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 9 / 37
Preliminaries Process model
Objective translation
Economic objectives are translated into process control objectives
Notion of setpoints / targets
Economic profit functionΦ(x ,u)
Economic optimumu
x
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Target
Steady state curvef (xs ,us) = 0
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Target
Steady state curvef (xs ,us) = 0
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 9 / 37
Preliminaries Process model
Objective translation
Economic objectives are translated into process control objectives
Notion of setpoints / targets
Economic profit functionΦ(x ,u)
Economic optimumu
x
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Target
Steady state curvef (xs ,us) = 0
Economic profit functionΦ(x ,u)
Economic optimumu
xConstraints
Target
Steady state curvef (xs ,us) = 0
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 9 / 37
Preliminaries Model Predictive Control
Model Predictive Control
Output target
Measured Input
Measured Output
Control/Prediction Horizon
k
Output target
Measured Input
Measured Output
Control/Prediction Horizon
Optimized Future Input Trajectory
REGULATIONk k + 1
uk
Output target
Measured Input
Measured Output
Control/Prediction Horizon
Optimized Future Input Trajectory
REGULATION
Predicted Output
k k + 1
uk
uk+1
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 10 / 37
Preliminaries Model Predictive Control
Model Predictive Control
Output target
Measured Input
Measured Output
Control/Prediction Horizon
k
Output target
Measured Input
Measured Output
Control/Prediction Horizon
Optimized Future Input Trajectory
REGULATIONk k + 1
uk
Output target
Measured Input
Measured Output
Control/Prediction Horizon
Optimized Future Input Trajectory
REGULATION
Predicted Output
k k + 1
uk
uk+1
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 10 / 37
Preliminaries Model Predictive Control
Model Predictive Control
Output target
Measured Input
Measured Output
Control/Prediction Horizon
k
Output target
Measured Input
Measured Output
Control/Prediction Horizon
Optimized Future Input Trajectory
REGULATIONk k + 1
uk
Output target
Measured Input
Measured Output
Control/Prediction Horizon
Optimized Future Input Trajectory
REGULATION
Predicted Output
k k + 1
uk
uk+1
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 10 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 11 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 11 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 12 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 12 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 12 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 13 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 13 / 37
Motivating the idea
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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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 14 / 37
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)
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 15 / 37
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)
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 15 / 37
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)
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 15 / 37
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)
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 15 / 37
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) = −P(x,u)
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 15 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 16 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 17 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 17 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 17 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 17 / 37
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 !
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 18 / 37
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 !
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 18 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 19 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 19 / 37
Current work SISO Example
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
Input
Cost contours
Steady state line
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
Input
Cost contours
Tracking contours
Steady state line
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
Input
Cost contours
Tracking contours
Steady state linetarg-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 20 / 37
Current work SISO Example
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
Input
Cost contours
Steady state line
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
Input
Cost contours
Tracking contours
Steady state line
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
Input
Cost contours
Tracking contours
Steady state linetarg-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 20 / 37
Current work SISO Example
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
Input
Cost contours
Steady state line
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
Input
Cost contours
Tracking contours
Steady state line
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
Input
Cost contours
Tracking contours
Steady state linetarg-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 20 / 37
Current work SISO Example
Profit function: −3x2k − 5u2
k−2xkuk + 98xk + 80uk
Objective: Maximize Profit !
Scheme two:
Controller minimizes the negative of profit
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 21 / 37
Current work SISO Example
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
Input
Cost contours
Tracking contours
Steady state linetarg-MPC
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 22 / 37
Current work SISO Example
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
Input
Cost contours
Tracking contours
Steady state linetarg-MPC
4
6
8
10
12
14
16
18
20
22
-4 -2 0 2 4 6 8 10
Sta
te
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 22 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 23 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 23 / 37
Current work Effect of disturbance
-4
-2
0
2
4
6
8
16 18 20 22 24 26 28
Sta
te
Input
p = 0
Performance Measures
targ-MPC eco-MPC ∆(index)%Lossa $102.59 $48.722 52.5
aReference: Maximum profit = 0
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 24 / 37
Current work Effect of disturbance
-4
-2
0
2
4
6
8
14 16 18 20 22 24 26 28
Sta
te
Input
p = 5
p = 0
Performance Measures
targ-MPC eco-MPC ∆(index)%Lossa $102.59 $48.722 52.5
aReference: Maximum profit = 0
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 24 / 37
Current work Effect of disturbance
-4
-2
0
2
4
6
8
14 16 18 20 22 24 26 28
Sta
te
Input
p = 5
p = 0
targ-MPC
Performance Measures
targ-MPC eco-MPC ∆(index)%Lossa $102.59 $48.722 52.5
aReference: Maximum profit = 0
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 24 / 37
Current work Effect of disturbance
-4
-2
0
2
4
6
8
14 16 18 20 22 24 26 28
Sta
te
Input
p = 5
p = 0
targ-MPCeco-MPC
Performance Measures
targ-MPC eco-MPC ∆(index)%Lossa $102.59 $48.722 52.5
aReference: Maximum profit = 0
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 24 / 37
Current work Effect of disturbance
Random disturbance corrupts state evolution
All states assumed measured
-15
-10
-5
0
5
10
15
0 5 10 15 20 25 30 35 40 45 50
Sta
te
Time
targ-MPCeco-MPC
eco-opt
-10
-5
0
5
10
15
20
0 5 10 15 20 25 30 35 40 45 50
Inpu
t
Time
targ-MPCeco-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 25 / 37
Current work Effect of disturbance
Random disturbance corrupts state evolution
All states assumed measured
-15
-10
-5
0
5
10
15
0 5 10 15 20 25 30 35 40 45 50
Sta
te
Time
targ-MPCeco-MPC
eco-opt
-10
-5
0
5
10
15
20
0 5 10 15 20 25 30 35 40 45 50
Inpu
t
Time
targ-MPCeco-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 25 / 37
Current work Effect of disturbance
targ-MPC eco-MPC ∆(index)%Lossa $2537.6 $968.5 61.8
aReference: Maximum profit = 0
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 26 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 27 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 27 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 27 / 37
Current work Linear economics
Economic optimum ⇐⇒ Maximizing throughput
Linear economics: Unconstrained problem unbounded
Constrained problem: Optimal solution lies on the process bounds
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10
Sta
te
Input
Linear economic contours
Steady state line
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10
Sta
te
Input
Linear economic contours
Tracking contours
Steady state line
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10
Sta
te
Input
Linear economic contours
Tracking contours
Steady state linetarg-MPCeco-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 28 / 37
Current work Linear economics
Economic optimum ⇐⇒ Maximizing throughput
Linear economics: Unconstrained problem unbounded
Constrained problem: Optimal solution lies on the process bounds
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10
Sta
te
Input
Linear economic contours
Steady state line
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10
Sta
te
Input
Linear economic contours
Tracking contours
Steady state line
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10
Sta
te
Input
Linear economic contours
Tracking contours
Steady state linetarg-MPCeco-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 28 / 37
Current work Linear economics
Economic optimum ⇐⇒ Maximizing throughput
Linear economics: Unconstrained problem unbounded
Constrained problem: Optimal solution lies on the process bounds
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10
Sta
te
Input
Linear economic contours
Steady state line
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10
Sta
te
Input
Linear economic contours
Tracking contours
Steady state line
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10
Sta
te
Input
Linear economic contours
Tracking contours
Steady state linetarg-MPCeco-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 28 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 29 / 37
Current work Example: Transient to steady state
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 30 / 37
Current work Example: Transient to steady state
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 30 / 37
Current work Example: Transient to steady state
55
60
65
70
75
80
0 5 10 15 20 25 30 35 40 45 50
Sta
te
targ-MPC
-2
0
2
4
6
8
10
0 5 10 15 20 25 30 35 40 45 50
Sta
te
targ-MPC
-1
-0.5
0
0.5
1
0 5 10 15 20 25 30 35 40 45 50
Inpu
t
Time
targ-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 31 / 37
Current work Example: Transient to steady state
55
60
65
70
75
80
85
90
0 5 10 15 20 25 30 35 40 45 50
Sta
te
targ-MPCeco-MPC
-2
0
2
4
6
8
10
0 5 10 15 20 25 30 35 40 45 50
Sta
te
targ-MPCeco-MPC
-1
-0.5
0
0.5
1
0 5 10 15 20 25 30 35 40 45 50
Inpu
t
Time
targ-MPCeco-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 31 / 37
Current work Example: Transient to steady state
Example: Effect of disturbance
Random disturbance affecting the state evolution
All states assumed measured
System started at the steady optimum with zero disturbance
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 32 / 37
Current work Effect of disturbance
46
48
50
52
54
56
58
60
62
0 10 20 30 40 50 60 70 80 90 100
Sta
te
targ-MPC
-3-2.5
-2-1.5
-1-0.5
00.5
11.5
0 10 20 30 40 50 60 70 80 90 100
Sta
te
targ-MPC
-1
-0.5
0
0.5
1
0 10 20 30 40 50 60 70 80 90 100
Inpu
t
Time
targ-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 33 / 37
Current work Effect of disturbance
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
0 10 20 30 40 50 60 70 80 90 100
Sta
te
targ-MPCeco-MPC
-1
-0.5
0
0.5
1
0 10 20 30 40 50 60 70 80 90 100
Inpu
t
Time
targ-MPCeco-MPC
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 33 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 34 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 34 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 35 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 36 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 37 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 37 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 37 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 37 / 37
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
Rishi Amrit (UW-Madison) Economic Optimization in MPC 29th February, 2008 37 / 37