Model Predictive Control: Basic Concepts - Penn Engineering

Model
Predictive
Control

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l:
Basic
Concepts

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2008/09 1 /94

Model
Predictive
Control

model‐based

optimizer

reference in

r(t) u

measu

A
model
of
the
process
is
used
to
of
the
process
to
optimize
the
co

©
2009
by
A.
Bemporad Controllo
di
Processo
e
dei
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l
(MPC)

process

nput output

u(t) y(t)

urements

o
predict
the
future
evolution

ontrol
signal


emi
di
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‐
A.a.
2008/09 2 /94

Receding
horizon
philoso

•
At
time t:
solve
an
optimal
control





problem
over
a
finite
future
horizon





of N steps:

•
Only
apply
the
first
optimal
move

•
At
time
t+1:
Get
new
measurements,
rep





 
Advantage
of
repeated
on

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Bemporad Controllo
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ophy

r(t) yt+k

Predicted
outputs

Manipulated ut+k
Inputs
t+N
t t+1

t+1 t+2 t+N+1

peat
the
optimization.
And
so
on
…


n‐line
optimization:
FEEDBACK!

emi
di
Produzione
‐
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2008/09 3 /94

Receding
Horizon
‐
Exam

•
MPC
is
like
playing
chess
!

•
“Rolling
horizon”
policies
are



also
used
frequently
in
finance


©
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A.
Bemporad Controllo
di
Processo
e
dei
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mples

emi
di
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‐
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2008/09 4 /94

Receding
Horizon
‐
Exam

‐
prediction
model how
vehicle
moves

‐
constraints drive
on
roads,
resp

‐
disturbances mainly
driver’s
inat

‐
set
point desired
location

‐
cost
function minimum
time,

minimum
distance,

‐
receding
horizon
mechanism

event‐based

(optimal
route
re‐planned
when
path


©
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Bemporad Controllo
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Processo
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mples

s
on
the
map

pect
one‐way
roads,
etc.

ttention
!

x
=
GPS
position
u
=
navigation
commands

,
etc. Fastest route

is
lost) Shortest route
Avoid motorways
Walking route
Bicycle route
Limited speed

emi
di
Produzione
‐
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2008/09 5 /94

Good
Models
for
(MPC)
C

Note:
computational
complexity
an
stability)
depend
on
chosen
model/

Good
models
for
MPC:
•
Descriptive
enough
to
capture
th



dynamics
of
the
system

TRAD

•
Simple
enough
for
solving
the
op

“Make
everything
as
simple
as
possi


—
Albert
Einstein

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Bemporad Controllo
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Control

nd
theoretical
properties
(e.g.

/objective/constraints

he
most
significant
 6 /94

DE
OFF

ptimization
problem

ible,
but
not
simpler.”

emi
di
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2008/09

MPC
in
Industry

•
History:
Computer
control
(“Man

Fluid
catalytic
cracking

(courtesy
of
Shell
/
M.
Morari)

©
2009
by
A.
Bemporad Controllo
di
Processo
e
dei
Siste

nual”
MPC)

Fluid
catalytic
cracking
(FCC)
is

the
most
important
conversion

process
used
in
petroleum

refineries.
It
is
widely
used
to

convert
the
high‐boiling

hydrocarbon
fractions
of

petroleum
crude
oils
to
more

valuable
gasoline,
olefinic
gases

and
other
products

(http://en.wikipedia.org/wiki/Catalytic_cracking)

emi
di
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2008/09 7 /94

MPC
in
Industry

•
History:
1979
Dynamic
Matrix
Co




(Motivation:
multivariable,
const

•
Present
Industrial
Practice
•
linear
impulse/step
response
models

•
sum
of
squared
errors





objective
function

•
executed
in
supervisory
mode

•
Particularly
suited
for
problems
w

•
many
inputs
and
outputs

•
constraints
on
inputs,
outputs,
states

•
varying
objectives
and
limits
on
variab




(e.g.
because
of
faults)

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ontrol
(DMC)
by
Shell
trained)

with 8 /94

bles


emi
di
Produzione
‐
A.a.
2008/09

MPC
in
Industry

Hierarchy
of
control
system
functio

Conventional

©
2009
by
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Bemporad Controllo
di
Processo
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dei
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ons:


MPC

emi
di
Produzione
‐
A.a.
2008/09 (Qin,
Badgewell,
1997)
9 /94

MPC
in
Industry

(snapshot
survey
conducted
in
mid‐1999)

“For
us
multivariable
con

Tariq
Samad,
Honeywell
(IEEE
Control
Syste

©
2009
by
A.
Bemporad Controllo
di
Processo
e
dei
Siste

(Qin,
Badgewell,
2003)

ntrol
is
predictive
control
” 10 /94

em
Society,
President)
(1997)

emi
di
Produzione
‐
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2008/09

MPC
in
Industry

Results
from
a
recent
survey
(November
7,
2005)
real‐time
optimization
in
a
set
of
US
industries:

Industrial
area
of
respondents
to
the
survey:

Pharmaceuticals 0.7
%
Machinery 0.7
%
0.7
%
Food
&
Beverage 0.7
%
Aerospace
Automotive 1.5
%
Mining
1.5
%
Cement
&
Glass 2.2
%
Metals 2.2
%

Plastics
&
Rubber 2.9
%
Electronics 5.1
%
Power 5.1
%

Pulp
&
Paper 5.1
%
Other
11.0
%
Oil
&
Gas
Chemicals 5 10 15 20 15.4
% 20.6
%
25 24.3

Refining
30 35
0

courtesy: Controllo
di
Processo
e
dei
Siste

©
2009
by
A.
Bemporad

)
about
the
use
of
MPC
techniques
/


Do
you
see
your
use
of
MPC
accelerating,

staying
constant,
or
declining?

Decreasing

Constant 28.8
%

Increasing 69.8
%
0 20 40 60 80 100

Does
your
company
use
MPC
?

No 19.0
%


%

Routinely 35.2
%

Sometimes 45.8
%

0 10 20 30 40 50 60 70 11/94

emi
di
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‐
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2008/09

MMoodedl
ePrl
ePdircetidvei
cCtoinvtero
lC
Tooonlbtorxol

•
MPC
Toolbox
3.0
(Bemporad,
Ricker,
Mo

–
Object‐oriented
implementation
(MPC

–
MPC
Simulink
Library

–
MPC
Graphical
User
Interface

–
RTW
extension
(code
generation)



[xPC
Target,
dSpace,
etc.]

–
Linked
to
OPC
Toolbox
v2.0.1

Only
linear
models
are
handled

http://www.mathworks.com/products/

©
2009
by
A.
Bemporad Controllo
di
Processo
e
dei
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l
Toolbox

orari,
1998‐today):

C
object)

/mpc/ 12 /94

emi
di
Produzione
‐
A.a.
2008/09

MPC
Simulink
Library

•
Single
MPC
and
multiple
swicthed
MPC
b
•
Reference/disturbance
preview
and
time

©
2009
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Bemporad Controllo
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blocks
supported 13 /94
e‐varying
limits
supported

emi
di
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2008/09

MPC
Graphical
User
Inter

©
2009
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Bemporad Controllo
di
Processo
e
dei
Siste

rface

emi
di
Produzione
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2008/09 14 /94

MPC
Tuning
Advisor

©
2009
by
A.
Bemporad Controllo
di
Processo
e
dei
Siste

emi
di
Produzione
‐
A.a.
2008/09 15 /94

Hybrid
Toolbox
for
MATL

Features:

•
Hybrid
models:
design,
simulation,
verifi

•
Control
design
for
linear
systems
w/
con




and
hybrid
systems
(on‐line
optimization

•
Explicit
MPC
control
(via
multi‐parametr

•
C‐code
generation

•
Simulink
library

http://www.dii.unisi

©
2009
by
A.
Bemporad Controllo
di
Processo
e
dei
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LAB (Bemporad,
2003‐2009)

fication
nstraints

n
via
QP/MILP/MIQP)
ric
programming)

2450+
download
requests
since
October
2004

i.it/hybrid/toolbox 16 /94

emi
di
Produzione
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2008/09

Basics
of
Constrained
Opt

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Processo
e
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timization

emi
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Produzione
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2008/09 17/94

Mathematical
Programm

In
general,
problem
is
difficult
t
use
software
tools

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Processo
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ming

to
solve 18 /94

emi
di
Produzione
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2008/09

Optimization
Software

• Taxonomy
of
most
known
solver
optimization
problems:

http://www-fp.mcs.anl.gov/o

• Network
Enabled
Optimization
S
solving
optimization
problems:

http://neos.mcs.anl.gov/neo

• Comparison
on
benchmark
prob

http://plato.la.asu.edu/ben

• Good
open‐source
optimization


http://www.coin-or.org/

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Bemporad Controllo
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rs,
for
different
classes
of


otc/Guide/SoftwareGuide/

Server
(NEOS)
for
remotely


os/solvers/

blems:

nch.html


software

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di
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2008/09 19 /94

Convex
sets

Convex
set

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Nonconvex
set

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2008/09 20 /94

Convex
function

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Bemporad Controllo
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Processo
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dei
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emi
di
Produzione
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2008/09 21 /94

Convex
Optimization
Pro

• Very
efficient
numerical
algorithm
• Global
solution
attained
• Extensive
useful
theory
• Often
occurring
in
engineering
pr
• Tractable
in
theory
and
practice

Excellent
reference
textbook:
“Convex
O
and
L.
Vandenberghe http://www.s

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2009
by
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Bemporad Controllo
di
Processo
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oblem

f
and
C
convex
ms
exist

roblems

Optimization”
by
S.
Boyd
 22 /94

stanford.edu/~boyd/cvxbook/

emi
di
Produzione
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2008/09

Polyhedra

• A
convex
polyhedron
is
the
inter




set
of
halfspaces
of
Rd
• A
convex
polytope
is
a
bounded

A2

A3x=b3


•
Hyperplane
representation:

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rsection
of
a
finite
d
convex
polyhedron

A 2x=b 2

A

1

A x=b1 


1

A3

emi
di
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2008/09 23 /94

Linear
Program

‐f

f’x

Slack
variables

©
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George
Dantzig

(1914
‐
2005)

Standard
form:

emi
di
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‐
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2008/09 24 /94

Linear
Program

trasformation
from
max
to
min:

Change
inequality
direction:

It
is
always
possible
to
using
“min”
and

©
2009
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o
formulate
LP
problems 25 /94
d
“·”
inequalities

emi
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2008/09