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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


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


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


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


Tracking MPC

← Past

Inputs

Setpoint

Future →

u

k = 0

yOutputs

One step of a closed-loop MPC trajectory


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


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)


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)


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 .


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


Steady states—unconstrained system

xs

ds2 = 0

ds1 = 1

xs = Gus + ds

ds3 = −1Gxsp

us2 us3us1

us

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


Constraints and unreachable setpoints

xs

ds2 = 0

ds1 = 1

xs = Gus + ds

ds3 = −1Gxsp

us1usus2

us3

0 1

0 ≤ us ≤ 1

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


Constraints and unreachable setpoints

xs

xsp

us

0

0 ≤ ds ≤ G

0 ≤ us ≤ 1

xs = Gus + ds

ds ≤ 0

1

ds ≥ G

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


What closed-loop behavior is desirable? Fast tracking

xsp

x∗x

k

x(0)

x(0)Q � R (fast tracking)


What closed-loop behavior is desirable? Slow tracking

xsp

x∗x

k

x(0)Q � R (slow tracking)

x(0)


What closed-loop behavior is desirable? Asymmetrictracking

xsp

x∗x

k

x(0)Q � R (fast tracking)

x(0)


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


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


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)


Optimal trajectory: xsp = 2, usp = 0

-1

-0.5

0

0.5

1

0 1 2 3 4

xN = 5

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4

t

ux0 = 1

x0 = −1



-1

-0.5

0

0.5

1

0 5 10 15 20 25 30

x N = 30

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

t

u x0 = 1

x0 = −1



-1

-0.5

0

0.5

1

0 10 20 30 40 50 60 70 80 90 100

x N = 100

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90 100

t

u x0 = 1

x0 = −1


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)


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?


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)


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


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)


-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)


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


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)


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)


-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


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


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


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


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


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


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


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


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)


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


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


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


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)


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




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


∞∑k=0

`(x(k), u(k))− `(xs , us)







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


∞∑k=0

`(x(k), u(k))− `(xs , us)





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)


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


-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


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


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


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


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.


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.


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)


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


-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


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).


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


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.


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.


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.


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].


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


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.


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))


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


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


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}


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.


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;


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


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


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


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


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.


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)


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))


(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


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))


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


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


Simulation of EMPC with terminal steady-state constraint

Improvement of 30 % inAmmonia production rate


Best Q-periodic solutions

Discretization Ts = 0.1

Best Q-periodic solution for Q = 16

Average production rate: 50 % better than best square wave


Closed-loop with Q periodic terminal constraint

Chaotic regime: 25 % better than periodic solutionOverall ≈ 140 % improvement over best steady state


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))


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


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 .


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


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.


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


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


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



Conclusions


asymptotic average profit not worse than best steady stateI Stability of EMPC requires dissipativity of process/stage cost



Conclusions



Many techniques from MPC can be applied to EMPC

I Terminal penalty formulationI Average constraintsI Periodic terminal constraints


Conclusions



Many techniques from MPC can be applied to EMPCI Terminal penalty formulation

I Average constraintsI Periodic terminal constraints


Conclusions



Many techniques from MPC can be applied to EMPCI Terminal penalty formulationI Average constraints

I Periodic terminal constraints


Conclusions





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?





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?






MPC relative to EMPC?

I What is observed about differences in net closed-loop performance insimulations?








simulations?








simulations?I What model, stage cost and disturbance characteristics cause large

performance differences between MPC and EMPC?



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




modified?I How robust is EMPC to model errors and disturbances?

I How can economic risk be incorporated into the controller?





modified?I How robust is EMPC to model errors and disturbances?I How can economic risk be incorporated into the controller?



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.


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.


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