Model Predictive Control: Basic Concepts - Penn Engineering

MPC
vs.
Conventional
Con


Single
input/single
output
control
l

 





equivalent
performance
can
be
ob



simpler
control
techniques
(e.g.:
P

HOW


MPC
allows
(in
principle)


UNIFO

(i.e.
same
technique
for
wide
ra

–
reduce
training
–
reduce
cost
–
easier
design
maintenance


Satisfying
control
specs
and
walking
o

both
are
not
difficult
if
frozen
!

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ntrol

loop
w/
constraints:

btained
with
other

PID
+
anti‐windup)
WEVER
ORMITY
ange
of
problems)

on
water
is
similar
…
 /94

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MPC
Features

•
Multivariable
constrained
“non‐square”



(i.e.
#inputs
and
#outputs
are
different)
•
Delay
compensation

•
Anticipative
action
for
future
reference

•
“Integral
action”,
i.e.
no
offset
for
step‐

Price
to
pay:
•
Substantial
on‐line
computation
•
For
simple
small/fast
systems
other
tech




(e.g.
PID
+
anti‐windup)

•
New
possibilities
for
MPC:
explicit
piece

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”
systems
 52 /94


changes
‐like
inputs

hniques
dominate

ewise
linear
forms

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MPC
Theory

•
Historical
Goal:
Explain
the
succe
•
Present
Goal:
Improve,
simplify,
a
•
Areas:

•
Linear
MPC:




linear
model
•
Nonlinear
MPC:
nonlinear
mod
•
Robust
MPC:



uncertain
(linea
•
Hybrid
MPC:




model
integrati























and
constraints
•
Issues:
–
Feasibility
–
Stability
(Convergence)
–
Computations

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ess
of
DMC
and
extend
industrial
algorithms

del
ar)
model
ing
logic,
dynamics,


(Mayne,
Rawlings,
Rao,
Scokaert,
2000) 53 /94
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Feasibility

•
Feasibility:
Guarantee
that
the
QP
proble
•
Input
constraints
only:
no
feasibility
issue

•
Hard
output
constraints:

•
When
N<1
there
is
no
guarantee
th



will
remain
feasible
at
all
future
time
s

•
N=1


















infinite
number
of
con

•
Maximum
output
admissible
set
theo
(Gilbert,
Tan,
IEEE
TAC,1991),
(Kerrigan,
Maciejowski,CDC,2
(Chmielewski,
Manousiouthakis,
Sys.
Cont.
Letters,
1996)

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QUADRATIC

PROGRAM
(QP)

em
is
feasible
at
all
sampling
times
t
es
!

hat
the
QP
problem
 54 /94
steps
t
nstraints
!

ory:
N<1
is
enough

2000),


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Stability

•
Stability
is
a
complex
function
of
the
MPC



 N,
Q,
R,
P,
umin,
umax,
ymin,
ymax

•
Stability
constraints
and
weights
on
the
t




the
prediction
horizon
to
ensure
stability


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C
parameters

terminal
state
can
be
imposed
over


of
MPC

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Convergence
Result

(Keerthi


Proof:
Use
value
function
as
Lyapun

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and
Gilbert,
1988)(Bemporad
et
al.,
1994) 56 /94

nov
function

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Convergence
Proof

Global
optimum
is
not
needed
to

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Lyapunov
function

o
prove
convergence
! 57 /94

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Convergence
Result
(a
little

(Keerthi


Proof:
Use
value
function
as
Lyapun

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e
more
general)


and
Gilbert,
1988)(Bemporad
et
al.,
1994) 58 /94

nov
function

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Convergence
Proof

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optimum
is
no

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Lyapunov
function

ot
needed
to
prove
convergence
! 59 /94

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Convergence
Proof

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Convergence
Proof

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Stability
Constraints

1.
No
constraint,
infinite
output
horizon:


(Keerthi
and
Gilbert,
1988)
(Rawlings
and
Muske,
1993)

2.
End‐point
constraint:

(Kwon
and
Pearson,
1977)
(Keerthi
and
Gilbert,
1988)

3.
Relaxed
terminal
constraint:

(Scokaert
and
Rawlings,
1996)

4.
Contraction
constraint:

(Polak
and
Yang,
1993)
(Bemporad,
1998)

All
the
proofs
in
(1,2,3)
use
the
value
fun
as
a
Lyapunov
function

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nction V (t) = min J(U, t) 62 /94

U

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Predicted
and
Actual
Traj

•
Even
assuming
perfect
model
&
no
distu

prx(t)
x*(t+2|t+1) =x(t+2)

x*(t+2|t)

x*(t+1|t) =x(t+1) t+N

0t

•
Special
case:
for
infinite
horizon,
open





trajectories
coincide.
This
follows
by
B

optimal
state
x*(t)

x*(t)

0τ

optimal
input
u*(t)

0 τ
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jectories

urbances:

redicted
open‐loop
trajectories
may
be

different
from
actual
closed‐loop

trajectories

n‐loop
trajectories
and
losed‐loop

Bellman’s
principle
of
optimality.

emi
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Bellman
(1920
‐

1984)

63 /94

Input
and
Output
Horizon

• Input
horizon
Nu
can
be
shorter
than
ou

• Nu<N
=
less
degrees
of
freedom,
and
h

– Loss
of
performance
– Decreased
computation
time
(QP
is
smaller
– Feasibility
still
maintained
(constraints
are
s

typically
N

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ns

utput
horizon
N 64 /94
hence:


r)
still
checked
up
to
N)

Nu=1÷10

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MPC
and
LQR

•
Consider
the
MPC
control
law:


(Unconstraine

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Jacopo
Francesco

Riccati
(1676
‐
1754)

ed)
MPC
=
LQR 65 /94

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MPC
and
LQR

•
Consider
the
MPC
control
law:

•
In
a
polyhedral
region
around
the
origin
t




the
constrained
LQR
controller
with
weig

MPC
=
const

•
The
larger
the
horizon,
the
larger
the
reg

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Jacopo
Francesco

Riccati
(1676
‐
1754)

the
MPC
control
law
is
equivalent
to

ghts
Q,R

trained
LQR (Chmielewski,
Manousiouthakis,
1996)
(Scokaert
and
Rawlings,
1998)

gion
where
MPC=LQR

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Double
Integrator
Examp

•
System: sampling
+
Ts=1
s
•
Constraints:
•
Control
objective:
min

•
Optimization
problem

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ple

+

ZOH

s

LQ
gain

solution
of
algebraic
Riccati
equation

(cost
function
was
normalized
by

max
svd(H))

emi
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Example:
AFTI‐16

•
Linearized
model:

go
to
demo

/demos/linea
afti16.m

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ar/afti16.m (Hyb‐Tbx)
m (MPC‐Tbx)

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Example:
AFTI‐16

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Example:
AFTI‐16

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Example:
AFTI‐16


Unconstrained
MPC 
(=linear
contro


+
actuator
saturation
±25o


Saturation
needs
to
be
considered
i

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oller,
¼

LQR)


UNSTABLE
!!!

in
the
control
design
! 71/94

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Saturation

•
Saturation
needs
to
be
considered
in
the

linear v=Kx
controller

sa

•
MPC
takes
it
into
account
automatically
(

MPC v=f(x)=u
controller

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e
control
design plant x

u
at

(and
optimally)

u plant x
sat
72 /94
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Tuning
Guidelines

• Weights:
the
larger
the
ratio
Wy/WΔu
the
more
a
• Input
horizon:
the
larger
Nu,
the
more
“optimal”
• Output
horizon:
the
smaller
N,
the
more
aggres
• Limits:
controller
less
aggressive
if
¢umin,
¢umax

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try
to
set
Nu


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aggressive
the
controller
”
but
the
more
complex
the
controller

ssive
the
controller
x
are
small


as
small
as
possible
! 73 /94

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Scaling

• Humans
think
infinite
precision
...
• Computers
do
not
!
• Numerical
difficulties
may
arise
if
variab

Example:



y1
2 [‐1e‐4,1e‐4]
(V)






















y2
2 [‐1e4,1e4]

(Pa)

use
instead:
y1
2 [‐0.1,0.1]
(mV)
























y2
2 [‐10,10]

(kPa)













•
Ideally
all
variables
should
range
in
[‐1,1
with
y/ymax

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bles
assume
very
small
or
very
large
values

1].
For
example,
one
can
replace
y

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Observer
Design
in
MPC

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