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Fundamentals of Economic Model Predictive Control
James B. Rawlings, David Angeli and Cuyler N. Bates
Dept. of Chemical and Biological Engineering,Univ. of Wisconsin-Madison, WI, USA
Dept. of Electrical and Electronic Engineering,Imperial College London, UK
CDC MeetingMaui, HI
December 10-14, 2012
Rawlings/Angeli/Bates Economic MPC 1 / 94
Outline
1 Introduction to MPC and economics
2 Stability of standard (tracking) MPC
3 Unreachable setpoints and turnpikes
4 Economic MPC
5 Dissipativity
6 Average constraints
7 Periodic terminal constraint
8 Conclusions and open research issues
Rawlings/Angeli/Bates Economic MPC 2 / 94
Optimizing economics: current industrial practice
Validation
Planning and Scheduling
Reconciliation
Model UpdateOptimizationSteady State
Plant
Controller
1 Two layer structureI Steady-state layer
F RTO optimizes steadystate model
F Optimal setpoints passedto dynamic layer
I Dynamic layerF Controller tracks the
setpointsF Linear MPC
Rawlings/Angeli/Bates Economic MPC 3 / 94
Optimizing economics: current industrial practice
Validation
Planning and Scheduling
Reconciliation
Model UpdateOptimizationSteady State
Plant
Controller
1 Two layer structureI Steady-state layer
F RTO optimizes steadystate model
F Optimal setpoints passedto dynamic layer
I Dynamic layerF Controller tracks the
setpointsF Linear MPC
Rawlings/Angeli/Bates Economic MPC 3 / 94
Optimizing economics: current industrial practice
Validation
Planning and Scheduling
Reconciliation
Model UpdateOptimizationSteady State
Plant
Controller
1 Two layer structure2 Drawbacks
I Inconsistent modelsI Re-identify linear model as
setpoint changesI Time scale separation may not
holdI Economics unavailable in
dynamic layer
Rawlings/Angeli/Bates Economic MPC 4 / 94
Optimizing economics: current industrial practice
Validation
Planning and Scheduling
Reconciliation
Model UpdateOptimizationSteady State
Plant
Controller
1 Two layer structure2 Drawbacks
I Inconsistent modelsI Re-identify linear model as
setpoint changesI Time scale separation may not
holdI Economics unavailable in
dynamic layer
Rawlings/Angeli/Bates Economic MPC 4 / 94
Steady-state optimization problem definition
Stage cost: `(x , u)
Optimization:
(xs , us) = arg minx ,u
`(x , u)
subject to: x = f (x , u), (x , u) ∈ Z
Rawlings/Angeli/Bates Economic MPC 5 / 94
Tracking MPC
← Past
Inputs
Setpoint
Future →
u
k = 0
yOutputs
One step of a closed-loop MPC trajectory
Rawlings/Angeli/Bates Economic MPC 6 / 94
Tracking MPC problem definition
Stage cost:
`t(x , s) = |x(k)− xs |2Q + |u(k)− us |2R + |u(k)− u(k − 1)|2S
Optimization:
minu
VN(x ,u) =N−1∑k=0
`t(x(k), u(k))
subject to
x+ = f (x , u)(x(k), u(k)) ∈ Z k ∈ I0:N−1x(N) = xs x(0) = x
Control law: u = κN(x)
Admissible set: XN
Rawlings/Angeli/Bates Economic MPC 7 / 94
Tracking MPC problem definition
Stage cost:
`t(x , s) = |x(k)− xs |2Q + |u(k)− us |2R + |u(k)− u(k − 1)|2SOptimization:
minu
VN(x ,u) =N−1∑k=0
`t(x(k), u(k))
subject to
x+ = f (x , u)(x(k), u(k)) ∈ Z k ∈ I0:N−1x(N) = xs x(0) = x
Control law: u = κN(x)
Admissible set: XN
Rawlings/Angeli/Bates Economic MPC 7 / 94
Why worry about stability for tracking MPC? Unexpectedclosed-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
Rawlings/Angeli/Bates Economic MPC 8 / 94
Why worry about stability for tracking MPC? Unexpectedclosed-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
Rawlings/Angeli/Bates Economic MPC 8 / 94
Why worry about stability for tracking MPC? Unexpectedclosed-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
Rawlings/Angeli/Bates Economic MPC 8 / 94
Why worry about stability for tracking MPC? Unexpectedclosed-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
Rawlings/Angeli/Bates Economic MPC 8 / 94
Why worry about stability for tracking MPC? Unexpectedclosed-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
Rawlings/Angeli/Bates Economic MPC 8 / 94
Why worry about stability for tracking MPC? Unexpectedclosed-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
Rawlings/Angeli/Bates Economic MPC 8 / 94
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
Vk+1 = Vk − L(xk , uk)
k + 2
Vk+2 = Vk+1 − L(xk+1, uk+1)
Rawlings/Angeli/Bates Economic MPC 9 / 94
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
Vk+1 = Vk − L(xk , uk)
k + 2
Vk+2 = Vk+1 − L(xk+1, uk+1)
Rawlings/Angeli/Bates Economic MPC 9 / 94
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
Vk+1 = Vk − L(xk , uk)
k + 2
Vk+2 = Vk+1 − L(xk+1, uk+1)
Rawlings/Angeli/Bates Economic MPC 9 / 94
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
Vk+1 = Vk − L(xk , uk)
k + 2
Vk+2 = Vk+1 − L(xk+1, uk+1)
Rawlings/Angeli/Bates Economic MPC 9 / 94
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
Vk+1 = Vk − L(xk , uk)
k + 2
Vk+2 = Vk+1 − L(xk+1, uk+1)
Rawlings/Angeli/Bates Economic MPC 9 / 94
Terminal constraint solution
Adding a terminal constraint ensures stability
May cause infeasibility
Open-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
Vk+1 ≤ Vk − L(xk , uk)
k + 1
k + 2
Vk+2 ≤ Vk+1 − L(xk+1, uk+1)
Rawlings/Angeli/Bates Economic MPC 10 / 94
Terminal constraint solution
Adding a terminal constraint ensures stability
May cause infeasibility
Open-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
Vk+1 ≤ Vk − L(xk , uk)
k + 1
k + 2
Vk+2 ≤ Vk+1 − L(xk+1, uk+1)
Rawlings/Angeli/Bates Economic MPC 10 / 94
Terminal constraint solution
Adding a terminal constraint ensures stability
May cause infeasibility
Open-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
Vk+1 ≤ Vk − L(xk , uk)
k + 1
k + 2
Vk+2 ≤ Vk+1 − L(xk+1, uk+1)
Rawlings/Angeli/Bates Economic MPC 10 / 94
Terminal constraint solution
Adding a terminal constraint ensures stability
May cause infeasibility
Open-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
Vk+1 ≤ Vk − L(xk , uk)
k + 1
k + 2
Vk+2 ≤ Vk+1 − L(xk+1, uk+1)
Rawlings/Angeli/Bates Economic MPC 10 / 94
Terminal constraint solution
Adding a terminal constraint ensures stability
May cause infeasibility
Open-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
Vk+1 ≤ Vk − L(xk , uk)
k + 1
k + 2
Vk+2 ≤ Vk+1 − L(xk+1, uk+1)
Rawlings/Angeli/Bates Economic MPC 10 / 94
Closed-loop stability of tracking MPC
Assumption: Model, cost and admissible set
1 The model f (·) and stage cost `(·) are continuous. The admissible setXN contains xs in its interior.
2 There exists a set Xf containing xs in its interior and K∞-functionγ(·) such that V 0
N(x) ≤ γ(|x − xs |) for x ∈ Xf .
Theorem: Stability of tracking MPC with terminal constraint
The steady-state target (xs , us) is an asymptotically stable equilibriumpoint of the closed-loop system
x+ = f (x , κN(x))
with region of attraction XN .
Rawlings/Angeli/Bates Economic MPC 11 / 94
Closed-loop stability of tracking MPC
Assumption: Model, cost and admissible set
1 The model f (·) and stage cost `(·) are continuous. The admissible setXN contains xs in its interior.
2 There exists a set Xf containing xs in its interior and K∞-functionγ(·) such that V 0
N(x) ≤ γ(|x − xs |) for x ∈ Xf .
Theorem: Stability of tracking MPC with terminal constraint
The steady-state target (xs , us) is an asymptotically stable equilibriumpoint of the closed-loop system
x+ = f (x , κN(x))
with region of attraction XN .
Rawlings/Angeli/Bates Economic MPC 11 / 94
Setpoints and unreachable setpoints
Consider the steady state of a linear dynamic model with state x ,controlled input 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
Rawlings/Angeli/Bates Economic MPC 12 / 94
Setpoints and unreachable setpoints
Consider the steady state of a linear dynamic model with state x ,controlled input 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
Rawlings/Angeli/Bates Economic MPC 12 / 94
Setpoints and unreachable setpoints
Consider the steady state of a linear dynamic model with state x ,controlled input 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
Rawlings/Angeli/Bates Economic MPC 12 / 94
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 .
Rawlings/Angeli/Bates Economic MPC 13 / 94
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.
Rawlings/Angeli/Bates Economic MPC 14 / 94
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.
Rawlings/Angeli/Bates Economic MPC 15 / 94
What closed-loop behavior is desirable? Fast tracking
xsp
x∗x
k
x(0)
x(0)Q � R (fast tracking)
Rawlings/Angeli/Bates Economic MPC 16 / 94
What closed-loop behavior is desirable? Slow tracking
xsp
x∗x
k
x(0)Q � R (slow tracking)
x(0)
Rawlings/Angeli/Bates Economic MPC 17 / 94
What closed-loop behavior is desirable? Asymmetrictracking
xsp
x∗x
k
x(0)Q � R (fast tracking)
x(0)
Rawlings/Angeli/Bates Economic MPC 18 / 94
Creating a turnpike example
Standard linear quadratic problem
x+ = Ax + Bu
`(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
Rawlings/Angeli/Bates Economic MPC 19 / 94
Creating a turnpike example
Standard linear quadratic problem
x+ = Ax + Bu
`(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
Rawlings/Angeli/Bates Economic MPC 19 / 94
Inconsistent setpoint and optimal steady state
u
(us , xs)
(usp, xsp)
G
xOptimal steady state
usp = 0 xsp = 2
us = 0.8 xs = 0.4
Rawlings/Angeli/Bates Economic MPC 20 / 94
Optimal control problem
Cost function and dynamic model
VN(x ,u) =N−1∑k=0
`(x(k), u(k)) s.t. x+ = Ax + Bu, x(0) = x
Optimal state and input trajectories
minu
V (x ,u) u0(x), x0(x)
Rawlings/Angeli/Bates Economic MPC 21 / 94
Optimal control problem
Cost function and dynamic model
VN(x ,u) =N−1∑k=0
`(x(k), u(k)) s.t. x+ = Ax + Bu, x(0) = x
Optimal state and input trajectories
minu
V (x ,u) u0(x), x0(x)
Rawlings/Angeli/Bates Economic MPC 21 / 94
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
Rawlings/Angeli/Bates Economic MPC 22 / 94
Optimal trajectory: xsp = 2, usp = 0
-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
Rawlings/Angeli/Bates Economic MPC 22 / 94
Optimal trajectory: xsp = 2, usp = 0
-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
Rawlings/Angeli/Bates Economic MPC 22 / 94
Introduction to Turnpike Literature
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)
Rawlings/Angeli/Bates Economic MPC 23 / 94
Introduction to Turnpike Literature
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)
Rawlings/Angeli/Bates Economic MPC 23 / 94
Introduction to Turnpike Literature
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)
Rawlings/Angeli/Bates Economic MPC 23 / 94
Introduction to Turnpike Literature
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)
Rawlings/Angeli/Bates Economic MPC 23 / 94
Unreachable case—challenges for analyzing closed-loopbehavior
Consider the MPC controller with the stage cost
`(x , s) = |x(k)− xsp|2Q + |u(k)− usp|2R + |u(k)− u(k − 1)|2S
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?
Rawlings/Angeli/Bates Economic MPC 24 / 94
Unreachable case—challenges for analyzing closed-loopbehavior
Consider the MPC controller with the stage cost
`(x , s) = |x(k)− xsp|2Q + |u(k)− usp|2R + |u(k)− u(k − 1)|2S
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?
Rawlings/Angeli/Bates Economic MPC 24 / 94
Unreachable case—challenges for analyzing closed-loopbehavior
Consider the MPC controller with the stage cost
`(x , s) = |x(k)− xsp|2Q + |u(k)− usp|2R + |u(k)− u(k − 1)|2S
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?
Rawlings/Angeli/Bates Economic MPC 24 / 94
Unreachable case—challenges for analyzing closed-loopbehavior
Consider the MPC controller with the stage cost
`(x , s) = |x(k)− xsp|2Q + |u(k)− usp|2R + |u(k)− u(k − 1)|2S
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?
Rawlings/Angeli/Bates Economic MPC 24 / 94
Unreachable case—challenges for analyzing closed-loopbehavior
Consider the MPC controller with the stage cost
`(x , s) = |x(k)− xsp|2Q + |u(k)− usp|2R + |u(k)− u(k − 1)|2S
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?
Rawlings/Angeli/Bates Economic MPC 24 / 94
Unreachable case—challenges for analyzing closed-loopbehavior
Consider the MPC controller with the stage cost
`(x , s) = |x(k)− xsp|2Q + |u(k)− usp|2R + |u(k)− u(k − 1)|2S
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?
Rawlings/Angeli/Bates Economic MPC 24 / 94
Unreachable case—stability result (linear model)
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 feasible set.
(Rawlings, Bonne, Jørgensen, Venkat, and Jørgensen, 2008)
Rawlings/Angeli/Bates Economic MPC 25 / 94
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 = 1,R = 0,S = 10−3
Horizon length N = 80
Periodic disturbance d = 2 with Gd = G and exact measurement
Rawlings/Angeli/Bates Economic MPC 26 / 94
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)
xs(k)
Rawlings/Angeli/Bates Economic MPC 27 / 94
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)
xs(k)
Rawlings/Angeli/Bates Economic MPC 27 / 94
-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 (ys)
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 (us)u(sp-MPC)
u(targ-MPC)
Rawlings/Angeli/Bates Economic MPC 28 / 94
Summary of Example 1
Performance targ-MPC sp-MPC ∆(index)%Measure (×10−3) (×10−3)
Vu 1.7× 10−2 2.2× 10−6 99.99Vy 6.98 3.27 53V 7.00 3.27 53
Vu =1
T
T−1∑0
|u(k)− usp|2R + |u(k)− u(k − 1)|2S
Vy =1
T
T−1∑0
|y(k)− ysp|Q2y
V = Vu + Vy
Rawlings/Angeli/Bates Economic MPC 29 / 94
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]′
Qy = 5I ,R = I , S = I
Horizon length N = 80
Periodic disturbance d = ±[0.03 − 0.03]′ with Gd = G andmeasurement and state noise
Rawlings/Angeli/Bates Economic MPC 30 / 94
0.33
0.331
0.332
0.333
0.334
0.335
0.336
0.337
0.338
0 5 10 15 20 25Time (sec)
y1
setpointy1(sp-MPC)
y1(targ-MPC)
0.335
0.336
0.337
0.338
0.339
0.34
0.341
0.342
0 5 10 15 20 25Time (sec)
y2
setpointy2(sp-MPC)
y2(targ-MPC)
Rawlings/Angeli/Bates Economic MPC 31 / 94
0.45
0.5
0 5 10 15 20 25
0.45
0.5
Time (sec)
u1
u1
u1(targ-MPC)u1(sp-MPC)
-0.48
-0.465
-0.45
0 5 10 15 20 25
-0.48
-0.465
-0.45
Time (sec)
u2
u2
u2(targ-MPC)u2(sp-MPC)
Rawlings/Angeli/Bates Economic MPC 32 / 94
-0.1
0
0.1
0.2
0.3
0 5 10 15 20 25Time (sec)
y1s
d1
setpoint
target (y1s)
d1
0
0.1
0.2
0.3
0 5 10 15 20 25Time (sec)
y2s
d2
setpoint
target (y2s)
d2
Rawlings/Angeli/Bates Economic MPC 33 / 94
Summary of Example 2
Performance targ-MPC sp-MPC ∆(index)%Measure (×10−4) (×10−4)
Vu 3.32 2.10 37Vy 1.63 0.04 98V 4.95 2.14 57
Rawlings/Angeli/Bates Economic MPC 34 / 94
Economic MPC: 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
Rawlings/Angeli/Bates Economic MPC 35 / 94
Economic MPC: 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
Rawlings/Angeli/Bates Economic MPC 35 / 94
Economic MPC definition (with terminal constraint)
Economic stage cost: `(x , u)
Optimization:
minu
VN,e(x ,u) =N−1∑k=0
`(x(k), u(k))
subject to
x+ = f (x , u) x(0) = x(x(k), u(k)) ∈ Z k ∈ [0 : N − 1]x(N) = xs
(1)
Control law: u = κN,e(x)
Admissible set: XN,e
Rawlings/Angeli/Bates Economic MPC 36 / 94
Economic MPC definition (with terminal constraint)
Economic stage cost: `(x , u)
Optimization:
minu
VN,e(x ,u) =N−1∑k=0
`(x(k), u(k))
subject to
x+ = f (x , u) x(0) = x(x(k), u(k)) ∈ Z k ∈ [0 : N − 1]x(N) = xs
(1)
Control law: u = κN,e(x)
Admissible set: XN,e
Rawlings/Angeli/Bates Economic MPC 36 / 94
Example
x+ = Ax + Bu
A =
[0.857 0.884−0.0147 −0.0151
]B =
[8.565
0.88418
]Input constraint: −1 ≤ u ≤ 1
`(x , u) = α′x + β′u
α =[−3 −2
]′β = −2
`t(x , u) = |x − xs |2Q + |u − us |2RQ = 2I2 R = 2
xs =[60 0
]′us = 1
Rawlings/Angeli/Bates Economic MPC 37 / 94
targ-MPCtarg-MPC60 65 70 75 80 85
x1
-2
0
2
4
6
8
10
x2
targ-MPC eco-MPCtarg-MPC eco-MPC60 65 70 75 80 85
x1
-2
0
2
4
6
8
10
x2
Rawlings/Angeli/Bates Economic MPC 38 / 94
targ-MPCtarg-MPC60 65 70 75 80 85
x1
-2
0
2
4
6
8
10
x2
targ-MPC eco-MPCtarg-MPC eco-MPC60 65 70 75 80 85
x1
-2
0
2
4
6
8
10
x2
Rawlings/Angeli/Bates Economic MPC 38 / 94
55
60
65
70
75
80
0 2 4 6 8 10 12 14
Sta
te1
targ-MPC
-2
0
2
4
6
8
10
0 2 4 6 8 10 12 14
Sta
te2
targ-MPC
-1
-0.5
0
0.5
1
0 2 4 6 8 10 12 14
Inpu
t
Time
targ-MPC
Rawlings/Angeli/Bates Economic MPC 39 / 94
55
60
65
70
75
80
85
90
0 2 4 6 8 10 12 14
Sta
te1
targ-MPCeco-MPC
-2
0
2
4
6
8
10
0 2 4 6 8 10 12 14
Sta
te2
targ-MPCeco-MPC
-1
-0.5
0
0.5
1
0 2 4 6 8 10 12 14
Inpu
t
Time
targ-MPCeco-MPC
Rawlings/Angeli/Bates Economic MPC 39 / 94
Closed-loop performance measures
Profitability:I Average asymptotic cost relative to steady state
limT→∞
1
T
T∑k=0
`(x(k), u(k))− `(xs , us)
I Net cost relative to steady state
∞∑k=0
`(x(k), u(k))− `(xs , us)
Stability:I Asymptotic convergence to optimal steady state
limk→∞
(x(k), u(k)) = (xs , us)
Rawlings/Angeli/Bates Economic MPC 40 / 94
How can profitability and stability be opposing goals?
We consider a nonlinear constant volume isothermal CSTRI State: CA
I Input: CAf
The following reactions take place:
A→ B r = kc2A
Economic stage cost:`(x , u) = −CB
Input constraints over horizon of N:
0 ≤ u(k) ≤ 31
N
N∑k=0
u(k) = 1
Rawlings/Angeli/Bates Economic MPC 41 / 94
Optimal control solution
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
u x
t
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
Rawlings/Angeli/Bates Economic MPC 42 / 94
Optimal control solution
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
u x
t
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
Rawlings/Angeli/Bates Economic MPC 42 / 94
Average economic performance of EMPC
When considering economic optimization as the objective of control,average economic performance is the more natural performancemeasure.
In tracking MPC, average closed-loop economic performance isguaranteed indirectly, via stability
limk→∞
(x(k), u(k)) = (xs , us)⇒ limT→∞
1
T
T∑k=0
`(x(k), u(k)) = `(xs , us)
Several methods are available for stabilizing tracking MPC includingthe addition of a terminal equality constraint, x(N) = xs , and aterminal penalty
Rawlings/Angeli/Bates Economic MPC 43 / 94
What does a terminal constraint accomplish for EMPC?
Theorem: Average economic performance of EMPC
The average asymptotic cost of the closed-loop system
x+ = f (x , κN,e(x))
satisfies
lim supT→+∞
∑Tk=0 `(x(k), u(k))
T + 1≤ `(xs , us)
Rawlings/Angeli/Bates Economic MPC 44 / 94
What this means . . .
The nominal average asymptotic cost of closed-loop EMPC is notworse than the best steady state
What this theorem does not say:I EMPC is stable under these conditionsI A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state?
∞∑k=0
`(x(k), u(k))− `(xs , us)
I Bounded above because V 0N(x) is bounded on XN .
I Not bounded below because the controller can outperform the beststeady state on average
I No proven relationship between closed-loop cost for tracking MPC vs.EMPC from a given initial state
Rawlings/Angeli/Bates Economic MPC 45 / 94
What this means . . .
The nominal average asymptotic cost of closed-loop EMPC is notworse than the best steady state
What this theorem does not say:I EMPC is stable under these conditions
I A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state?
∞∑k=0
`(x(k), u(k))− `(xs , us)
I Bounded above because V 0N(x) is bounded on XN .
I Not bounded below because the controller can outperform the beststeady state on average
I No proven relationship between closed-loop cost for tracking MPC vs.EMPC from a given initial state
Rawlings/Angeli/Bates Economic MPC 45 / 94
What this means . . .
The nominal average asymptotic cost of closed-loop EMPC is notworse than the best steady state
What this theorem does not say:I EMPC is stable under these conditionsI A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state?
∞∑k=0
`(x(k), u(k))− `(xs , us)
I Bounded above because V 0N(x) is bounded on XN .
I Not bounded below because the controller can outperform the beststeady state on average
I No proven relationship between closed-loop cost for tracking MPC vs.EMPC from a given initial state
Rawlings/Angeli/Bates Economic MPC 45 / 94
What this means . . .
The nominal average asymptotic cost of closed-loop EMPC is notworse than the best steady state
What this theorem does not say:I EMPC is stable under these conditionsI A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state?
∞∑k=0
`(x(k), u(k))− `(xs , us)
I Bounded above because V 0N(x) is bounded on XN .
I Not bounded below because the controller can outperform the beststeady state on average
I No proven relationship between closed-loop cost for tracking MPC vs.EMPC from a given initial state
Rawlings/Angeli/Bates Economic MPC 45 / 94
What this means . . .
The nominal average asymptotic cost of closed-loop EMPC is notworse than the best steady state
What this theorem does not say:I EMPC is stable under these conditionsI A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state?
∞∑k=0
`(x(k), u(k))− `(xs , us)
I Bounded above because V 0N(x) is bounded on XN .
I Not bounded below because the controller can outperform the beststeady state on average
I No proven relationship between closed-loop cost for tracking MPC vs.EMPC from a given initial state
Rawlings/Angeli/Bates Economic MPC 45 / 94
What this means . . .
The nominal average asymptotic cost of closed-loop EMPC is notworse than the best steady state
What this theorem does not say:I EMPC is stable under these conditionsI A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state?
∞∑k=0
`(x(k), u(k))− `(xs , us)
I Bounded above because V 0N(x) is bounded on XN .
I Not bounded below because the controller can outperform the beststeady state on average
I No proven relationship between closed-loop cost for tracking MPC vs.EMPC from a given initial state
Rawlings/Angeli/Bates Economic MPC 45 / 94
What this means . . .
The nominal average asymptotic cost of closed-loop EMPC is notworse than the best steady state
What this theorem does not say:I EMPC is stable under these conditionsI A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state?
∞∑k=0
`(x(k), u(k))− `(xs , us)
I Bounded above because V 0N(x) is bounded on XN .
I Not bounded below because the controller can outperform the beststeady state on average
I No proven relationship between closed-loop cost for tracking MPC vs.EMPC from a given initial state
Rawlings/Angeli/Bates Economic MPC 45 / 94
Industrial simulation example
FeedF1, X1, T1
F5
Condensate
L2
Separator
Evaporator
Condensate
F3
F4, T3
T201
Condenser
waterCooling
T200
F200
P100
LC
LT
ProductX2, T2
F2
T100F100
Steam
Evaporator system (Newell and Lee, 1989, Ch. 2)
Rawlings/Angeli/Bates Economic MPC 46 / 94
Evaporator system
Measurements: product composition X2 and operating pressure P2
Inputs: steam pressure P100, cooling water flow rate F200
The economic stage cost is the operating cost for electricity, steamand cooling water (Wang and Cameron, 1994; Govatsmark andSkogestad, 2001).
J = 1.009(F2 + F3) + 600F100 + 0.6F200
We consider the process subject to disturbances in feed flow rate F1,Feed composition C1, Circulating flow rate F3, feed temperature T1
and cooling water inlet temperature T200
We consider both tracking MPC and EMPC with a terminal stateconstraint
Rawlings/Angeli/Bates Economic MPC 47 / 94
Evaporator system
Measurements: product composition X2 and operating pressure P2
Inputs: steam pressure P100, cooling water flow rate F200
The economic stage cost is the operating cost for electricity, steamand cooling water (Wang and Cameron, 1994; Govatsmark andSkogestad, 2001).
J = 1.009(F2 + F3) + 600F100 + 0.6F200
We consider the process subject to disturbances in feed flow rate F1,Feed composition C1, Circulating flow rate F3, feed temperature T1
and cooling water inlet temperature T200
We consider both tracking MPC and EMPC with a terminal stateconstraint
Rawlings/Angeli/Bates Economic MPC 47 / 94
Evaporator system
Measurements: product composition X2 and operating pressure P2
Inputs: steam pressure P100, cooling water flow rate F200
The economic stage cost is the operating cost for electricity, steamand cooling water (Wang and Cameron, 1994; Govatsmark andSkogestad, 2001).
J = 1.009(F2 + F3) + 600F100 + 0.6F200
We consider the process subject to disturbances in feed flow rate F1,Feed composition C1, Circulating flow rate F3, feed temperature T1
and cooling water inlet temperature T200
We consider both tracking MPC and EMPC with a terminal stateconstraint
Rawlings/Angeli/Bates Economic MPC 47 / 94
-9-8-7-6-5-4-3-2-10
0 20 40 60 80 100
Dis
turb
ance
P2
(kP
a)
time (min)
2000
4000
6000
8000
10000
12000
14000
0 20 40 60 80 100
Cos
t($
/h)
time (min)
100
150
200
250
300
350
400
0 20 40 60 80 100
Inpu
tP
100
(kP
a)
time (min)
100
120
140
160
180
200
220
0 20 40 60 80 100
Inpu
tF
200
(kg/
min
)
time (min)
20
30
40
50
60
70
0 20 40 60 80 100
Out
putX
2(%
)
time (min)
30
35
40
45
50
0 20 40 60 80 100
Out
putP
2(k
Pa)
time (min)
eco-MPCtrack-MPC
Rawlings/Angeli/Bates Economic MPC 48 / 94
101520253035404550
0 10 20 30 40 50 60 70 80 90
Dis
turb
ance
time (min)
F1T1
T1004000
6000
8000
10000
12000
0 10 20 30 40 50 60 70 80 90
Cos
t($
/h)
time (min)
100
150
200
250
300
350
400
0 10 20 30 40 50 60 70 80 90
Inpu
tP
100
(kP
a)
time (min)
200
240
280
320
360
400
0 10 20 30 40 50 60 70 80 90
Inpu
tF
200
(kg/
min
)
time (min)
20
24
28
32
36
40
0 10 20 30 40 50 60 70 80 90
Out
putX
2(%
)
time (min)
50
52
54
56
0 10 20 30 40 50 60 70 80 90
Out
putP
2(k
Pa)
time (min)
eco-MPCtrack-MPC
Rawlings/Angeli/Bates Economic MPC 49 / 94
Evaporator system closed-loop economics
Performance comparison under economic and tracking MPC
Avg. operating cost ×10−6 ($/ hr)Disturbance eco-MPC track-MPC ∆ (%)
Measured 5.89 5.97 2.2Unmeasured 5.80 6.15 6.2
Rawlings/Angeli/Bates Economic MPC 50 / 94
Asymptotic stability and EMPC
Steady operation often desired by practitionersI Equipment not designed for strongly unsteady operationI Operator acceptance issue for unsteady operation
Stability analysisI Check that stability is consistent with the process model and control
objectivesI Or modify the control objectives (stage cost) given the process model
Rawlings/Angeli/Bates Economic MPC 51 / 94
Asymptotic stability and EMPC
Steady operation often desired by practitionersI Equipment not designed for strongly unsteady operationI Operator acceptance issue for unsteady operation
Stability analysisI Check that stability is consistent with the process model and control
objectivesI Or modify the control objectives (stage cost) given the process model
Rawlings/Angeli/Bates Economic MPC 51 / 94
Stabilizing assumption for EMPC
Storage: λ(x)
System: x+ = f (x , u)
Supply rate: s(x , u) Dissipation
Assumption: Dissipativity
The system x+ = f (x , u) is dissipative with respect to the supply rates : Z→ R if there exists a function λ : X→ R such that:
λ(f (x , u))− λ(x) ≤ s(x , u)
for all (x , u) ∈ Z. If ρ : X→ R≥0 positive definite exists such that:
λ(f (x , u))− λ(x) ≤ −ρ(x) + s(x , u)
then the system is said to be strictly dissipative.
Rawlings/Angeli/Bates Economic MPC 52 / 94
Optimality of steady-state operation
Optimal operation at steady-state
The system x+ = f (x , u) is optimally operated at steady-state if for alltrajectories (x,u), such that (x(k), u(k)) ∈ Z for all k ∈ N it holds that
lim infT→+∞
∑T−1k=0 `(x(k), u(k))
T≥ `(xs , us)
Suboptimal operation off steady-state
A system optimally operated at steady-state is suboptimally operated offsteady-state if for all trajectories (x,u), such that (x(k), u(k)) ∈ Z it holdsthat either
lim infT→+∞
∑T−1k=0 `(x(k), u(k))
T> `(xs , us)
orlim infk→+∞
|x(k)− xs | = 0.
Rawlings/Angeli/Bates Economic MPC 53 / 94
Optimality of steady-state operation
Optimal operation at steady-state
The system x+ = f (x , u) is optimally operated at steady-state if for alltrajectories (x,u), such that (x(k), u(k)) ∈ Z for all k ∈ N it holds that
lim infT→+∞
∑T−1k=0 `(x(k), u(k))
T≥ `(xs , us)
Suboptimal operation off steady-state
A system optimally operated at steady-state is suboptimally operated offsteady-state if for all trajectories (x,u), such that (x(k), u(k)) ∈ Z it holdsthat either
lim infT→+∞
∑T−1k=0 `(x(k), u(k))
T> `(xs , us)
orlim infk→+∞
|x(k)− xs | = 0.
Rawlings/Angeli/Bates Economic MPC 53 / 94
Dissipativity and Optimality
Dissipativity is closely related to optimal operation at steady-state
1 Dissipativity ⇒ Optimal operation at steady-state
2 Strict Dissipativity ⇒ Sub-optimal operation off steady-state
3 Gap between Dissipativity and Optimal steady state operation:∀ x , all trajectories (x,u) such that (x(k), u(k)) ∈ Z and x(0) = xsatisfy
infT≥1
T−1∑k=0
[`(x(k), u(k))− `(xs , us)
]> −∞
investigated by Mueller et al. (2013)
Rawlings/Angeli/Bates Economic MPC 54 / 94
Dissipativity and Optimality
Dissipativity is closely related to optimal operation at steady-state
1 Dissipativity ⇒ Optimal operation at steady-state
2 Strict Dissipativity ⇒ Sub-optimal operation off steady-state
3 Gap between Dissipativity and Optimal steady state operation:∀ x , all trajectories (x,u) such that (x(k), u(k)) ∈ Z and x(0) = xsatisfy
infT≥1
T−1∑k=0
[`(x(k), u(k))− `(xs , us)
]> −∞
investigated by Mueller et al. (2013)
Rawlings/Angeli/Bates Economic MPC 54 / 94
Dissipativity: an example
Consider the following system
x+ = αx + (1− α)u, α ∈ [0, 1)
With the following nonconvex stage cost
`(x , u) = (x + u/3)(2u − x) + (x − u)4
This system and stage cost are dissipative for α ∈ [12 , 1], but are notstrongly dual1 for any α. What does this tell us about the behavior ofEMPC?
1Strong duality requires constant λ such that λ′f (x , u) − λ′x ≤ `(x , u).Rawlings/Angeli/Bates Economic MPC 55 / 94
Dissipativity: an example
Consider the following system
x+ = αx + (1− α)u, α ∈ [0, 1)
With the following nonconvex stage cost
`(x , u) = (x + u/3)(2u − x) + (x − u)4
This system and stage cost are dissipative for α ∈ [12 , 1], but are notstrongly dual1 for any α. What does this tell us about the behavior ofEMPC?
1Strong duality requires constant λ such that λ′f (x , u) − λ′x ≤ `(x , u).Rawlings/Angeli/Bates Economic MPC 55 / 94
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 2 4 6 8 10 12 14 16 18
Sta
te
Time
α = 0.2 α = 0.4 α = 0.6
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14 16 18
Inpu
t
Time
α = 0.2 α = 0.4 α = 0.6
Rawlings/Angeli/Bates Economic MPC 56 / 94
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0 2 4 6 8 10 12 14 16 18
Sta
geC
ost
Time
α = .2α = .4α = .6
EMPC controller with N = 10, terminal equality constraint
Closed-loop EMPC is stable for α ≥ 1/2
Closed-loop EMPC is unstable for α < 1/2 and outperforms the beststeady state on average
Dissipativity is a tighter condition for EMPC stability than strongduality
Rawlings/Angeli/Bates Economic MPC 57 / 94
x
u
u = −3x
u = x/2
α
(a)
(b)
Qualitative picture of L0: global minima (a),(b); α-dependent period-2 solution(black dots). Adapted from Angeli et al. (2012).
Rawlings/Angeli/Bates Economic MPC 58 / 94
Stability theorem for EMPC
Theorem: Stability of EMPC
If the systemx+ = f (x , κN,e(x))
is strictly dissipative with respect to the supply rate
s(x , u) = `(x , u)− `(xs , us)
then xs is an asymptotically stable equilibrium point of the closed-loopsystem with region of attraction XN,e .
Nominal average asymptotic performance not worse than steadyoperation is always implied by stability
Rawlings/Angeli/Bates Economic MPC 59 / 94
Sketch of proof
Lyapunov-based proof, with rotated stage cost:
L(x , u) := `(x , u)− `(xs , us) + λ(x)− λ(f (x , u))
Notice that
N−1∑k=0
L(x(k), u(k))
=N−1∑k=0
`(x(k), u(k)) + λ(x(k))− λ(x(k + 1))− `(xs , us)
= λ(x(0))− λ(x(N))− N`(xs , us) +N−1∑k=0
`(x(k), u(k))
Optimal control is unaffected by cost rotation.
Rawlings/Angeli/Bates Economic MPC 60 / 94
Sketch of proof
Lyapunov-based proof, with rotated stage cost:
L(x , u) := `(x , u)− `(xs , us) + λ(x)− λ(f (x , u))
Notice that
N−1∑k=0
L(x(k), u(k))
=N−1∑k=0
`(x(k), u(k)) + λ(x(k))− λ(x(k + 1))− `(xs , us)
= λ(x(0))− λ(x(N))− N`(xs , us) +N−1∑k=0
`(x(k), u(k))
Optimal control is unaffected by cost rotation.
Rawlings/Angeli/Bates Economic MPC 60 / 94
Sketch of proof (continued)
Under dissipativity assumption rotated cost fulfills standard MPCconditions (positive semi-definite):
L(x , u) ≥ 0
In addition, under strict dissipativity, the rotated cost is positivedefinite:
L(x , u) ≥ ρ(x)
Cost-to-go can be used as a candidate Lyapunov function
V (x) := minu
N−1∑k=0
L(x(k), u(k))
subject to initial, terminal and dynamic constraints.
Rawlings/Angeli/Bates Economic MPC 61 / 94
Sketch of proof (continued)
Under dissipativity assumption rotated cost fulfills standard MPCconditions (positive semi-definite):
L(x , u) ≥ 0
In addition, under strict dissipativity, the rotated cost is positivedefinite:
L(x , u) ≥ ρ(x)
Cost-to-go can be used as a candidate Lyapunov function
V (x) := minu
N−1∑k=0
L(x(k), u(k))
subject to initial, terminal and dynamic constraints.
Rawlings/Angeli/Bates Economic MPC 61 / 94
Sketch of proof (continued)
Recursive feasibility:Feasibility of:
x = (x(0), x(1), . . . , x(N − 1)) u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of:
x = (x(1), x(2), . . . , x(N − 1), xs) u = (u(1), . . . , u(N − 1), us).
Along closed-loop solutions:
V (x+) ≤ V (x)− ρ(x).
Hence, ρ(x(k))→ 0 as k → +∞ and convergence to equilibriumfollows by compactness of Z.
Rawlings/Angeli/Bates Economic MPC 62 / 94
Sketch of proof (continued)
Recursive feasibility:Feasibility of:
x = (x(0), x(1), . . . , x(N − 1)) u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of:
x = (x(1), x(2), . . . , x(N − 1), xs) u = (u(1), . . . , u(N − 1), us).
Along closed-loop solutions:
V (x+) ≤ V (x)− ρ(x).
Hence, ρ(x(k))→ 0 as k → +∞ and convergence to equilibriumfollows by compactness of Z.
Rawlings/Angeli/Bates Economic MPC 62 / 94
Asymptotic averages
As convergence to equilibrium is not always enforced, asymptoticaverages of output variables need not equal their limit (which in factmay fail to exist)
Possibility of asymptotic average constraints different from pointwisein time constraints (pointwise in time constraints are typically morestringent)
Definition of asymptotic average:
Av [v ] = {w : ∃ {Tn}+∞n=1 : Tn → +∞ as n→ +∞
and limn→+∞
∑Tn−1k=0 v(k)
Tn= w }
Asymptotic average need not be a singleton:
0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, . . .
Notice that {1/3, 1/2} ⊂ Av[v ] ⊂ [1/3, 1/2].
Rawlings/Angeli/Bates Economic MPC 63 / 94
Asymptotic averages
As convergence to equilibrium is not always enforced, asymptoticaverages of output variables need not equal their limit (which in factmay fail to exist)
Possibility of asymptotic average constraints different from pointwisein time constraints (pointwise in time constraints are typically morestringent)
Definition of asymptotic average:
Av [v ] = {w : ∃ {Tn}+∞n=1 : Tn → +∞ as n→ +∞
and limn→+∞
∑Tn−1k=0 v(k)
Tn= w }
Asymptotic average need not be a singleton:
0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, . . .
Notice that {1/3, 1/2} ⊂ Av[v ] ⊂ [1/3, 1/2].
Rawlings/Angeli/Bates Economic MPC 63 / 94
Average constraints
Goals:
Modify economic MPC algorithm to guarantee:
(x(k), u(k)) ∈ Z for all k ∈ N;
Av[h(x , u)] ⊂ Y 3 h(xs , us);
Recursive feasibility
Remarks:
1 For technical reasons Y is assumed to be convex.
2 Average constraint does not imply limits on averages computed onfinite time windows.
For the following:
At each time t let variables z(k) and v(k) denote virtual (predicted)variables, and x(k), u(k) (k ≤ t) actual variables
Rawlings/Angeli/Bates Economic MPC 64 / 94
Average constraints
Goals:
Modify economic MPC algorithm to guarantee:
(x(k), u(k)) ∈ Z for all k ∈ N;
Av[h(x , u)] ⊂ Y 3 h(xs , us);
Recursive feasibility
Remarks:
1 For technical reasons Y is assumed to be convex.
2 Average constraint does not imply limits on averages computed onfinite time windows.
For the following:
At each time t let variables z(k) and v(k) denote virtual (predicted)variables, and x(k), u(k) (k ≤ t) actual variables
Rawlings/Angeli/Bates Economic MPC 64 / 94
Economic MPC with average constraints
A modified receding-horizon strategy:
minv,z
N−1∑k=0
`(z(k), v(k))
subject to:(z(k), v(k)) ∈ Z ∀ k ∈ {0, 1, . . . ,N − 1}
z(k + 1) = f (z(k), v(k)) ∀ k ∈ {0, 1, . . . ,N − 1}z(0) = x(t) z(N) = xs
N−1∑k=0
h(z(k), v(k)) ∈ Yt
where:
Yt+1 = Yt ⊕ Y {h(x(t), u(t))} Y0 = NY⊕ Y00
with ⊕, denoting set addition and subtraction, and Y00 is an arbitraryconvex compact set.
Rawlings/Angeli/Bates Economic MPC 65 / 94
Properties of the algorithm
Time-varying state-feedback
Recursive feasibility follows by standard argument: Feasibility of:
x = (x(0), x(1), . . . , x(N − 1)) u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of:
x = (x(1), x(2), . . . , x(N − 1), xs) u = (u(1), . . . , u(N − 1), us).
Additional constraint does not limit feasibility region (Y0 is arbitrarilylarge)
Asymptotic average constraints are guaranteed
Average performance not worse than best steady-state
Possibility of replacing terminal constraint with terminal penaltyfunction
Rawlings/Angeli/Bates Economic MPC 66 / 94
Properties of the algorithm
Time-varying state-feedback
Recursive feasibility follows by standard argument: Feasibility of:
x = (x(0), x(1), . . . , x(N − 1)) u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of:
x = (x(1), x(2), . . . , x(N − 1), xs) u = (u(1), . . . , u(N − 1), us).
Additional constraint does not limit feasibility region (Y0 is arbitrarilylarge)
Asymptotic average constraints are guaranteed
Average performance not worse than best steady-state
Possibility of replacing terminal constraint with terminal penaltyfunction
Rawlings/Angeli/Bates Economic MPC 66 / 94
Working principle
Show by induction:
Yt = tY + Y0 {t−1∑k=0
h(x(k), u(k))}
Hence:
N−1∑k=0
h(z(k), v(k)) +t−1∑k=0
h(x(k), u(k)) ∈ tY⊕ Y0
Taking tn → +∞ so that limit exists:
limn→+∞
∑tn−1k=0 h(x(k), u(k))
tn=
= limn→+∞
∑N−1k=0 h(z(k), v(k)) +
∑tn−1k=0 h(x(k), u(k))
tn∈ Y
Rawlings/Angeli/Bates Economic MPC 67 / 94
Working principle
Show by induction:
Yt = tY + Y0 {t−1∑k=0
h(x(k), u(k))}
Hence:
N−1∑k=0
h(z(k), v(k)) +t−1∑k=0
h(x(k), u(k)) ∈ tY⊕ Y0
Taking tn → +∞ so that limit exists:
limn→+∞
∑tn−1k=0 h(x(k), u(k))
tn=
= limn→+∞
∑N−1k=0 h(z(k), v(k)) +
∑tn−1k=0 h(x(k), u(k))
tn∈ Y
Rawlings/Angeli/Bates Economic MPC 67 / 94
Dissipativity for averagely constrained systems
Problem:
Asymptotic convergence of averagely constrained MPCSufficient conditions, possibility of relaxing dissipativity?
Dissipativity taking into account average constraints
Consider the modified supply function:
sa(x , u) = `(x , u)− `(xs , us) + λT [Ah(x , u)− b]
provided:Y = {y : Ay ≤ b}
Rawlings/Angeli/Bates Economic MPC 68 / 94
Dissipativity and Convergence for Averagely ConstrainedEMPC
Optimal steady-state operation
If a system is dissipative with respect to the supply rate sa(x , u) for somenon-negative λ then every feasible solution fulfills:
lim infT→+∞
T−1∑k=0
`(x(k), u(k))
T≥ `(xs , us)
Convergence of Averagely Constrained EMPC
If a system is strictly dissipative with respect to the supply rate sa(x , u) forsome non-negative λ then closed-loop solutions of Averagely ConstrainedEMPC asymptotically approach the best steady-state.
Rawlings/Angeli/Bates Economic MPC 69 / 94
Dissipativity and Convergence for Averagely ConstrainedEMPC
Optimal steady-state operation
If a system is dissipative with respect to the supply rate sa(x , u) for somenon-negative λ then every feasible solution fulfills:
lim infT→+∞
T−1∑k=0
`(x(k), u(k))
T≥ `(xs , us)
Convergence of Averagely Constrained EMPC
If a system is strictly dissipative with respect to the supply rate sa(x , u) forsome non-negative λ then closed-loop solutions of Averagely ConstrainedEMPC asymptotically approach the best steady-state.
Rawlings/Angeli/Bates Economic MPC 69 / 94
Enforcing convergence in Economic MPC
By modifying the economic stage cost `(x , u) as:
˜(x , u) = `(x , u) + γ(|x − xs |+ |u − us |)
in order to recover dissipativity or strong duality
By adding an auxiliary average constraint, such as:
Av[|x − xs |2] ⊂ {0}
or:Av[|xi − xsi |2] ⊂ {0} i = 1 . . . n;
Rawlings/Angeli/Bates Economic MPC 70 / 94
Enforcing convergence in Economic MPC
By modifying the economic stage cost `(x , u) as:
˜(x , u) = `(x , u) + γ(|x − xs |+ |u − us |)
in order to recover dissipativity or strong duality
By adding an auxiliary average constraint, such as:
Av[|x − xs |2] ⊂ {0}
or:Av[|xi − xsi |2] ⊂ {0} i = 1 . . . n;
Rawlings/Angeli/Bates Economic MPC 70 / 94
Case study: a reactor with parallel reactions
The reactions:R → P1 R → P2
The state-space model:
x1 = 1− 104x21 e−1/x3 − 400x1e−0.55/x3 − x1
x2 = 104x21 e−1/x3 − x2
x3 = u − x3
x1 is concentration of R, x2 is concentration of P1 (the desiredproduct), x3 is the temperature and u is the heat flux. P2 is the wasteproduct (not modeled).
Goal: maximize x2, viz.:
`(x , u) = −x2;
Known that best performance is not at steady state
Rawlings/Angeli/Bates Economic MPC 71 / 94
Case study: a reactor with parallel reactions
The reactions:R → P1 R → P2
The state-space model:
x1 = 1− 104x21 e−1/x3 − 400x1e−0.55/x3 − x1
x2 = 104x21 e−1/x3 − x2
x3 = u − x3
x1 is concentration of R, x2 is concentration of P1 (the desiredproduct), x3 is the temperature and u is the heat flux. P2 is the wasteproduct (not modeled).
Goal: maximize x2, viz.:
`(x , u) = −x2;
Known that best performance is not at steady state
Rawlings/Angeli/Bates Economic MPC 71 / 94
Closed-loop simulations
0.1
0.2
0.3
0.4
0 2 4 6 8 10
u
00.10.20.30.40.50.60.7
0 2 4 6 8 10
x1
IC 1IC 2IC 3
0
0.2
0 2 4 6 8 10
x2
0.05
0.1
0.15
0.2
0 2 4 6 8 10
x3
Closed-loop input and state profiles for economic MPC with a convex term anddifferent initial states
Rawlings/Angeli/Bates Economic MPC 72 / 94
Closed-loop simulations (2)
0.1
0.2
0.3
0.4
0 2 4 6 8 10
u
00.10.20.30.40.50.60.7
0 2 4 6 8 10
x1
IC 1IC 2IC 3
0
0.2
0 2 4 6 8 10
x2
0.05
0.1
0.15
0.2
0 2 4 6 8 10
x3
Closed-loop input and state profiles for economic MPC with a convergenceconstraint and different initial states
Rawlings/Angeli/Bates Economic MPC 73 / 94
Outperforming best steady-state
Terminal constraint instrumental in:1 guaranteeing recursive feasibility2 providing bound to asymptotic performance
Any feasible trajectory may be used as a terminal constraint
Idea:
Replace best equilibrium by best feasible periodic solution of givenperiod
Rawlings/Angeli/Bates Economic MPC 74 / 94
Best feasible periodic solution
Let xs ,us be:xs = [xs(0), xs(1), . . . , xs(Q − 1)]
us = [us(0), us(1), . . . , us(Q − 1)],
and assume that they belong to:
arg minx,u
Q−1∑k=0
`(x(k), u(k))
subject to:
x(k + 1) = f (x(k), u(k)) k ∈ {0, 1, . . . ,Q − 2}
x(0) = f (x(Q − 1), u(Q − 1))
(x(k), u(k)) ∈ Z k ∈ {0, 1, . . . ,Q − 1}We call xs ,us an optimal Q-periodic solution.
Rawlings/Angeli/Bates Economic MPC 75 / 94
EMPC with periodic terminal constraint
Solve at each time t the following optimization problem:
minv,z
N−1∑k=0
`(z(k), v(k))
subject to:(z(k), v(k)) ∈ Z ∀ k ∈ {0, 1, . . . ,N − 1}
z(k + 1) = f (z(k), v(k)) ∀ k ∈ {0, 1, . . . ,N − 1}z(0) = x(t) z(N) = xs(t mod Q)
Rawlings/Angeli/Bates Economic MPC 76 / 94
Features of the algorithm
Q periodic state feedback
Recursive feasibility: feasibility at time t of:
x = (x(0), x(1), . . . , x(N − 1)) u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of:
(x(1), . . . , x(N−1), xs(t mod Q)) (u(1), . . . , u(N−1), us(t mod Q))
at time t + 1.
Possibility of incorporating average constraints
Rawlings/Angeli/Bates Economic MPC 77 / 94
Average performance with periodic end constraint
Let V be the cost to go:
V (t, x) = minv,z
N−1∑k=0
`(z(k), v(k))
subject to:
(z(k), v(k)) ∈ Z ∀ k ∈ {0, 1, . . . ,N − 1}
z(k + 1) = f (z(k), v(k)) ∀ k ∈ {0, 1, . . . ,N − 1}z(0) = x z(N) = xs(t mod Q)
Along closed-loop solution:
V (t + 1, x(t + 1)) ≤ V (t, x(t))
− `(x(t), u(t)) + `(xs(t mod Q), us(t mod Q))
Rawlings/Angeli/Bates Economic MPC 78 / 94
Average performance with periodic end constraint
Let V be the cost to go:
V (t, x) = minv,z
N−1∑k=0
`(z(k), v(k))
subject to:
(z(k), v(k)) ∈ Z ∀ k ∈ {0, 1, . . . ,N − 1}
z(k + 1) = f (z(k), v(k)) ∀ k ∈ {0, 1, . . . ,N − 1}z(0) = x z(N) = xs(t mod Q)
Along closed-loop solution:
V (t + 1, x(t + 1)) ≤ V (t, x(t))
− `(x(t), u(t)) + `(xs(t mod Q), us(t mod Q))
Rawlings/Angeli/Bates Economic MPC 78 / 94
Average performance with periodic end constraint
Taking sums between 0 and T − 1 and dividing by T yields:
lim supT→+∞
∑T−1k=0 `(x(k), u(k))
T≤∑Q−1
k=0 `(xs(k), us(k))
Q
Average performance at least as good as optimal Q-periodic solution
Q and N may be different from each other and unrelated
The closed-loop system need not be asymptotically stable to theoptimal Q periodic solution
The optimal Q periodic solution need not be an equilibrium of theclosed-loop system
Rawlings/Angeli/Bates Economic MPC 79 / 94
Average performance with periodic end constraint
Taking sums between 0 and T − 1 and dividing by T yields:
lim supT→+∞
∑T−1k=0 `(x(k), u(k))
T≤∑Q−1
k=0 `(xs(k), us(k))
Q
Average performance at least as good as optimal Q-periodic solution
Q and N may be different from each other and unrelated
The closed-loop system need not be asymptotically stable to theoptimal Q periodic solution
The optimal Q periodic solution need not be an equilibrium of theclosed-loop system
Rawlings/Angeli/Bates Economic MPC 79 / 94
Ammonia synthesis (Jain et al., 1985)
Open loop stable equilibria
Open-loop periodic forcing improves average production rate1 Improvement of 24% by using sinusoidal inputs u(t) = c + A sin(ωt)2 Improvement of 25% by using square-waves of different amplitudes,
average and duty cycle
Rawlings/Angeli/Bates Economic MPC 80 / 94
Simulation of EMPC with terminal steady-state constraint
Improvement of 30 % inAmmonia production rate
Rawlings/Angeli/Bates Economic MPC 81 / 94
Best Q-periodic solutions
Discretization Ts = 0.1
Best Q-periodic solution for Q = 16
Average production rate: 50 % better than best square wave
Rawlings/Angeli/Bates Economic MPC 82 / 94
Closed-loop with Q periodic terminal constraint
Chaotic regime: 25 % better than periodic solutionOverall ≈ 140 % improvement over best steady state
Rawlings/Angeli/Bates Economic MPC 83 / 94
Terminal penalty formulations
Just as with tracking MPC, we can expand the feasible set XN byreplacing the terminal equality constraint with a terminal setconstraint and a terminal penalty
Admissible set:
XN,p ={x ∈ X | ∃u such that the trajectory (x(k), u(k)) satisfies
(x(k), u(k)) ∈ Z k ∈ I0:N−1, x(N) ∈ Xf }
Objective function:
VN,p(x ,u) =N−1∑k=0
`(x(k), u(k)) + Vf (x(N))
Rawlings/Angeli/Bates Economic MPC 84 / 94
Terminal penalty in EMPC
Assumption: Terminal penalty
There exists a terminal region control law κf : Xf → U such that
Vf (f (x , κf (x))) ≤ Vf (x)− `(x , κf (x)) + `(xs , us)
(x , κf (x)) ∈ ZN,p ∀x ∈ Xf
This assumption is identical in form to the tracking case, but `(x , u)has changed
Vf (x) need not be positive definite with respect to xs
Rawlings/Angeli/Bates Economic MPC 85 / 94
Terminal penalty in EMPC
Theorem: EMPC stability with terminal penalty
If the systemx+ = f (x , κN,p(x))
is strictly dissipative with respect to the supply rate:
s(x , u) = `(x , u)− `(xs , us)
then xs is an asymptotically stable equilibrium point of the closed-loopsystem with region of attraction XN .
Rawlings/Angeli/Bates Economic MPC 86 / 94
Case study: nonlinear CSTR
We consider a nonlinear constant volume isothermal CSTRI States: P0,B,P1,P2
I Inputs: inflow concentrations of P0,B
The following reactions take place:
P0 + B −→ P1
P1 + B −→ P2
Economic stage cost:`(x , u) = −CP1
The controller stage cost is modified according to:
˜(x , u) = `(x , u) + |x − xs |2Q + |u − us |2R
Rawlings/Angeli/Bates Economic MPC 87 / 94
Case study: nonlinear CSTR
We consider a nonlinear constant volume isothermal CSTRI States: P0,B,P1,P2
I Inputs: inflow concentrations of P0,B
The following reactions take place:
P0 + B −→ P1
P1 + B −→ P2
Economic stage cost:`(x , u) = −CP1
The controller stage cost is modified according to:
˜(x , u) = `(x , u) + |x − xs |2Q + |u − us |2R
Rawlings/Angeli/Bates Economic MPC 87 / 94
Representative open-loop trajectories
0
0.5
1
1.5
2
0 5 10 15
x1
0
2
4
6
0 5 10 15
x2
0
0.5
1
1.5
0 5 10 15
x3
Time (t)
0
0.4
0.8
0 5 10 15
x4
Time (t)
EcoR = 0
Str. dualtrack
Open-loop state profiles with different cost functions and arbitrary initial state.
Rawlings/Angeli/Bates Economic MPC 88 / 94
Average and net profits
Case avg profit net profit
Unstable Economic 0.46 ∞
StableDissipative 0.38 2.6Strongly dual 0.38 1.6Tracking 0.38 1.0
Average profit and net profit for open-loop system with different stage costs.
All stable schemes have about the same average profit (large N)
Net profit decreases with increase in tracking term weight
Rawlings/Angeli/Bates Economic MPC 89 / 94
Conclusions
The economic objective function of EMPC causes novel behaviorI EMPC may be unstable where MPC is stable
I Using a terminal penalty or terminal equality constraint guaranteesasymptotic average profit not worse than best steady state
I Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPCI Terminal penalty formulationI Average constraintsI Periodic terminal constraints
Rawlings/Angeli/Bates Economic MPC 90 / 94
Conclusions
The economic objective function of EMPC causes novel behaviorI EMPC may be unstable where MPC is stableI Using a terminal penalty or terminal equality constraint guarantees
asymptotic average profit not worse than best steady state
I Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPCI Terminal penalty formulationI Average constraintsI Periodic terminal constraints
Rawlings/Angeli/Bates Economic MPC 90 / 94
Conclusions
The economic objective function of EMPC causes novel behaviorI EMPC may be unstable where MPC is stableI Using a terminal penalty or terminal equality constraint guarantees
asymptotic average profit not worse than best steady stateI Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPCI Terminal penalty formulationI Average constraintsI Periodic terminal constraints
Rawlings/Angeli/Bates Economic MPC 90 / 94
Conclusions
The economic objective function of EMPC causes novel behaviorI EMPC may be unstable where MPC is stableI Using a terminal penalty or terminal equality constraint guarantees
asymptotic average profit not worse than best steady stateI Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPC
I Terminal penalty formulationI Average constraintsI Periodic terminal constraints
Rawlings/Angeli/Bates Economic MPC 90 / 94
Conclusions
The economic objective function of EMPC causes novel behaviorI EMPC may be unstable where MPC is stableI Using a terminal penalty or terminal equality constraint guarantees
asymptotic average profit not worse than best steady stateI Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPCI Terminal penalty formulation
I Average constraintsI Periodic terminal constraints
Rawlings/Angeli/Bates Economic MPC 90 / 94
Conclusions
The economic objective function of EMPC causes novel behaviorI EMPC may be unstable where MPC is stableI Using a terminal penalty or terminal equality constraint guarantees
asymptotic average profit not worse than best steady stateI Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPCI Terminal penalty formulationI Average constraints
I Periodic terminal constraints
Rawlings/Angeli/Bates Economic MPC 90 / 94
Conclusions
The economic objective function of EMPC causes novel behaviorI EMPC may be unstable where MPC is stableI Using a terminal penalty or terminal equality constraint guarantees
asymptotic average profit not worse than best steady stateI Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPCI Terminal penalty formulationI Average constraintsI Periodic terminal constraints
Rawlings/Angeli/Bates Economic MPC 90 / 94
Open research issues
Remove terminal constraints and costs by choosing sufficient N. SeeGrune (2012a,b) for recent results in this direction.
Generalized terminal state constraint. Terminate on the steady-statemanifold and move the end location dynamically to the best steadystate (Fagiano and Teel, 2012; Ferramosca et al., 2009)
Analyzing closed-loop performanceI What can be proven about net closed-loop performance of tracking
MPC relative to EMPC?I What is observed about differences in net closed-loop performance in
simulations?I What model, stage cost and disturbance characteristics cause large
performance differences between MPC and EMPC?
Rawlings/Angeli/Bates Economic MPC 91 / 94
Open research issues
Remove terminal constraints and costs by choosing sufficient N. SeeGrune (2012a,b) for recent results in this direction.
Generalized terminal state constraint. Terminate on the steady-statemanifold and move the end location dynamically to the best steadystate (Fagiano and Teel, 2012; Ferramosca et al., 2009)
Analyzing closed-loop performanceI What can be proven about net closed-loop performance of tracking
MPC relative to EMPC?I What is observed about differences in net closed-loop performance in
simulations?I What model, stage cost and disturbance characteristics cause large
performance differences between MPC and EMPC?
Rawlings/Angeli/Bates Economic MPC 91 / 94
Open research issues
Remove terminal constraints and costs by choosing sufficient N. SeeGrune (2012a,b) for recent results in this direction.
Generalized terminal state constraint. Terminate on the steady-statemanifold and move the end location dynamically to the best steadystate (Fagiano and Teel, 2012; Ferramosca et al., 2009)
Analyzing closed-loop performance
I What can be proven about net closed-loop performance of trackingMPC relative to EMPC?
I What is observed about differences in net closed-loop performance insimulations?
I What model, stage cost and disturbance characteristics cause largeperformance differences between MPC and EMPC?
Rawlings/Angeli/Bates Economic MPC 91 / 94
Open research issues
Remove terminal constraints and costs by choosing sufficient N. SeeGrune (2012a,b) for recent results in this direction.
Generalized terminal state constraint. Terminate on the steady-statemanifold and move the end location dynamically to the best steadystate (Fagiano and Teel, 2012; Ferramosca et al., 2009)
Analyzing closed-loop performanceI What can be proven about net closed-loop performance of tracking
MPC relative to EMPC?
I What is observed about differences in net closed-loop performance insimulations?
I What model, stage cost and disturbance characteristics cause largeperformance differences between MPC and EMPC?
Rawlings/Angeli/Bates Economic MPC 91 / 94
Open research issues
Remove terminal constraints and costs by choosing sufficient N. SeeGrune (2012a,b) for recent results in this direction.
Generalized terminal state constraint. Terminate on the steady-statemanifold and move the end location dynamically to the best steadystate (Fagiano and Teel, 2012; Ferramosca et al., 2009)
Analyzing closed-loop performanceI What can be proven about net closed-loop performance of tracking
MPC relative to EMPC?I What is observed about differences in net closed-loop performance in
simulations?
I What model, stage cost and disturbance characteristics cause largeperformance differences between MPC and EMPC?
Rawlings/Angeli/Bates Economic MPC 91 / 94
Open research issues
Remove terminal constraints and costs by choosing sufficient N. SeeGrune (2012a,b) for recent results in this direction.
Generalized terminal state constraint. Terminate on the steady-statemanifold and move the end location dynamically to the best steadystate (Fagiano and Teel, 2012; Ferramosca et al., 2009)
Analyzing closed-loop performanceI What can be proven about net closed-loop performance of tracking
MPC relative to EMPC?I What is observed about differences in net closed-loop performance in
simulations?I What model, stage cost and disturbance characteristics cause large
performance differences between MPC and EMPC?
Rawlings/Angeli/Bates Economic MPC 91 / 94
Open research issues
Tuning EMPC and robustnessI For nondissipative process/stage costs, how should the stage cost be
modified?
I How robust is EMPC to model errors and disturbances?I How can economic risk be incorporated into the controller?
Computational methods for implementing EMPC; strategies foradapting existing control hierarchies
Rawlings/Angeli/Bates Economic MPC 92 / 94
Open research issues
Tuning EMPC and robustnessI For nondissipative process/stage costs, how should the stage cost be
modified?I How robust is EMPC to model errors and disturbances?
I How can economic risk be incorporated into the controller?
Computational methods for implementing EMPC; strategies foradapting existing control hierarchies
Rawlings/Angeli/Bates Economic MPC 92 / 94
Open research issues
Tuning EMPC and robustnessI For nondissipative process/stage costs, how should the stage cost be
modified?I How robust is EMPC to model errors and disturbances?I How can economic risk be incorporated into the controller?
Computational methods for implementing EMPC; strategies foradapting existing control hierarchies
Rawlings/Angeli/Bates Economic MPC 92 / 94
Open research issues
Tuning EMPC and robustnessI For nondissipative process/stage costs, how should the stage cost be
modified?I How robust is EMPC to model errors and disturbances?I How can economic risk be incorporated into the controller?
Computational methods for implementing EMPC; strategies foradapting existing control hierarchies
Rawlings/Angeli/Bates Economic MPC 92 / 94
Further reading I
D. Angeli, R. Amrit, and J. B. Rawlings. On average performance and stability ofeconomic model predictive control. IEEE Trans. Auto. Cont., 57(7):1615–1626, 2012.
R. Dorfman, P. Samuelson, and R. Solow. Linear Programming and Economic Analysis.McGraw-Hill, New York, 1958.
L. Fagiano and A. R. Teel. Model predictive control with generalized terminal stateconstraint. In IFAC Conference on Nonlinear Model Predictive Control 2012, pages299–304, Noordwijkerhout, the Netherlands, August 2012.
A. Ferramosca, D. Limon, I. Alvarado, T. Alamo, and E. Camacho. MPC for tracking ofconstrained nonlinear systems. In IEEE Conference on Decision and Control (CDC),pages 7978–7983, Shanghai, China, 2009.
M. S. Govatsmark and S. Skogestad. Control structure selection for an evaporationprocess. Comput. Chem. Eng., 9:657–662, 2001.
L. Grune. NMPC without terminal constraints. In IFAC Conference on Nonlinear ModelPredictive Control 2012, pages 1–13, Noordwijkerhout, the Netherlands, August2012a.
L. Grune. Economic receding horizon control without terminal constraints. Automatica,2012b. Accepted for publication.
Rawlings/Angeli/Bates Economic MPC 93 / 94
Further reading II
A. K. Jain, R. R. Hudgins, and P. L. Silveston. Effectiveness factor under cyclicoperation of a reactor. Can. J. Chem., 63:166–169, February 1985.
M. Mueller, D. Angeli, and F. Allgower. On convergence of averagely constrainedeconomic MPC and necessity of dissipativity for optimal steady-state operation. InAmerican Control Conference, 2013.
R. B. Newell and P. L. Lee. Applied Process Control – A Case Study. Prentice Hall,Sydney, 1989.
J. B. Rawlings, D. Bonne, J. B. Jørgensen, A. N. Venkat, and S. B. Jørgensen.Unreachable setpoints in model predictive control. IEEE Trans. Auto. Cont., 53(9):2209–2215, October 2008.
F. Wang and I. Cameron. Control studies on a model evaporation process–constrainedstate driving with conventional and higher relative degree systems. J. Proc. Cont., 4(2):59–75, 1994.
Rawlings/Angeli/Bates Economic MPC 94 / 94
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