Model Predictive Control: Basic Concepts - Penn Engineering

Observer
Design
for
M

r(t) Optimizer u(t)

Reference

x(t)

state
estimate

•
Full
state
x(t)
may
not
be
availa
•
Even
if
available,
noise
should
b
•
State
of
prediction
model
may
b




(e.g.:
model
reduction,
identifica

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2009
by
A.
Bemporad we
need
to
use


Controllo
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MPC y(t) measured

Plant outputs

Observer

able
be
filtered
out
be
different
from
plant
model
x(t)

ation)


a
state
observer
 76 /94

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2008/09

Model
for
Observer
De

u(k) MVs
(Manipulated
Variables)

v(k) MDs
(Measured
Disturbances) Plan
mod
nd(k) Unmeasured
 d(k)
disturbance

white
noise
 UMDs

innovations model (Unmeasured
Disturbances)

nm(k

white
nois
innovation

unmeasured
disturbance
model

•
Note:
meas.
noise
model
not
needed
for

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Bemporad Controllo
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esign

nt
 CVs
(Controlled
Variables) yu(k)
del (Unmeasured

Outputs)

k) m(k)Measurement zm(k)

++
MOs

y (k)noise

se
 model m
ns
(Measured
Outputs)

measurement
noise
model

r
optimization
! 77/94

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Observer
Design

•
Measurement
update

•
Time
update

•NOTE:
separation
principle
holds
!
(und


(Muske,
Meadows,
Rawlings,
ACC94)

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der
certain
assumptions)
 78 /94

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Kalman
Filter
Design

•
Full
model
for
designing
observer
gain
M

•
nd(k):
represents
source
for
modeling
e
•
nm(k):
represents
source
for
measurem
•
nu(k):
white
noise
on
all
inputs
u
added

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2009
by
A.
Bemporad Controllo
di
Processo
e
dei
Siste

M

Rudolf
Emil

Kalman
(1930
‐
)

errors
ment
noise
d
for
solvability
of
the
Riccati
equation

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2008/09 79 /94

Integral
Action
in
MPC

(and
not
only
in
MPC)

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Output
Integrators

u(k)

v(k) Pla
mo
nd(k) Unmeasured
 d(k)
disturbance

white
noise

innovations model

ni(k) O
Inte
white
noise

innovations n

•
Introduce
output
integrators
as
additio

•
Under
certain
conditions,
observer
+
co




steady‐state

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Bemporad Controllo
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ant
 + yu(k)
odel + zm(k)

Output
egrators

nm(k) Measurement ++ ym(k)

white
noise
 noise
 m(k)
innovations model

onal
disturbance
models
ontroller
provide
zero
offset
in


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Integrators
and
Steady‐St

•
More
generally,
add
integrators
on
states

n

iwnnh

•
Use
the
above
model
+
meas.noise
mode


(e.g.
Kalman
filter)

•
Main
idea:
observer
makes
MPC
makes

)



the
combination
makes

•
Explanation:














compensates
model

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tate
Offsets

s
+
outputs: Plant
 yu(k)
model
u(k) + ym(k)
v(k)
+

ni(k) Output d(k) Ddm
nhoitvea
ntoioisnes
 Integrators
Ddmd(k)


el
to
design
an
observer

(estimation
error)
(predicted
tracking
error)
(actual
tracking
error)

l
mismatch
in
steady‐state 82 /94

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Error
feedback

•
Idea:
add
integrals
of
measured
outputs
a




state‐feedback
case)

•
Extended
prediction
model:

•
Implementation:

•
Explanation:
if
closed‐loop
asymptotically




and
hence
y(k)
!
r(k)

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Bemporad Controllo
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as
additional
states
(similar
to
linear


(Kwakernaak,
1972)

y
stable
)
q(k)
!
cost. 83 /94

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Reference
Governor

•
Idea:
Apply
MPC
as
the
set‐point
gener



(e.g.:
PID)


actual
reference

r(k) w(k) Local
Feedba
desired Reference
reference Governor

estimated
plant
state
+
controller
s

•
Separation
of
problems:

‐
Local
feedback
designed
for
stability,
dist


without
taking
care
of
constraints

‐
Actual
reference
w(t)
generated
on‐line
b





Objective:


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Bemporad Controllo
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rator
to
linear
feedback
loops

(Bemporad,
1997)

control constrained
input outputs

u(k) c(k)
ack Plant
y(k)
¼
r(k)

measured
outputs

Primal
System

x(k)

state

turbance
attenuation,
good
tracking,
by
an
MPC
algorithm
to
take
care
of
constraints

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2008/09 84 /94

MPC
frequency
analysis


• Unconstrained
MPC
gain
+
linear
o
(=
2
d.o.f.
dynamic
controller)

• Closed‐loop
MPC
analysis
can
be
p
frequency‐domain
tools
(e.g.
Bod

u(t) Plant
x(t)

MPC
^xe(t)

In
MPC
Tbx:
ss(mpc)
or
tf(mpc)
return
the
LT
(=no
constraints)
MPC
object

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Bemporad Controllo
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Processo
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(inactive
constraints)

observer
=
linear
dynamical
system


performed
using
standard

de
plots
for
sensitivity
analysis)

t ym(t)

C
r(t)

TI
discrete‐time
form
of
the
linearized
 85 /94

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2008/09

Controller
matching
pro

•
Given
the
controller
u=Kx,
find
we




such
that

− I 0 ... 0





that
is,
the
MPC
controller
coincid




are
not
active

•
QP
matrices:

©
2009
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Bemporad Controllo
di
Processo
e
dei
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oblem

(Di
Cairano,
Bemporad,


eights
Q,R,P

foErC
Ct’h0e9)
MPC
problem




H−1F = Kfv

des
with
Kfv

when
the
constraints


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2008/09 86 /94

Controller
matching
pro

•
Open‐loop
process:


•
Constraints:
•
Desired
controller
(PID):


•
State‐space
form:

©
2009
by
A.
Bemporad controller
matching
based
on
inverse
LQR

Controllo
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Processo
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oblem
‐
Example

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Controller
matching
pro

%&'(&')*+,-

$!

#!

!

!#!

!$! " #! #"
!

01(&')&+,-

/!

$!

!

!$!

!/!

!.! " #! #"
!

what PID would
matched MPC

Note:
it’s
not
trivially
a
saturation
o
sat(PID)
leads
to
instability

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Bemporad Controllo
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oblem
‐
Example

%&'(&')*+,-

$! " #! #"
#!
01(&')&+,-
!
!#! " #! #"
!$!
add
output
constraint
y(k)≥-5
!

/!
$!

!
!$!
!/!
!.!

!

d apply

of
PID
controller.
In
this
case

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2008/09 88 /94

Example:
MPC
of
a
DC
Se

R

Ve µM Tr µs
JM

bM r

go
to
demo

/demos/linea
mpcmotor

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Bemporad Controllo
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Processo
e
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ervomotor

sT µL

JL

bL

ar/dcmotor.m (Hyb‐Tbx)
r.m (MPC‐Tbx)

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2008/09 89 /94

DC
Servomotor
‐
Model

R

Ve µM Tr µs
JM

bM r

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di
Processo
e
dei
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sT µL

JL

bL

Step
Response

0

0 5 Time
(sec)10 90 /9415

emi
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2008/09

DC
Servomotor:
Specs

R

Ve µM Tr µs
JM

bM r

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Bemporad Controllo
di
Processo
e
dei
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sT µL

JL

bL

emi
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2008/09 91 /94

DC
Servomotor:
Exercise

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DC
Servomotor:
MPC

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2008/09 93 /94

DC
Servomotor:
MPC

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2009
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Processo
e
dei
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emi
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2008/09 94 /94