8th IFAC Symposium on Mechatronic Systems
8th IFAC Symposium on Mechatronic Systems
Vienna,Sept. on
4-6,Mechatronic
2019
8th
IFACAustria,
Symposium
Systems
8th
IFAC
Symposium
Systems
Vienna,Austria,
Sept. on
4-6,Mechatronic
2019
Available online at www.sciencedirect.com
Vienna,
Sept.
4-6,
2019
Vienna,Austria,
Sept. on
4-6,Mechatronic
2019
8th
IFACAustria,
Symposium
Systems
Vienna, Austria, Sept. 4-6, 2019

ScienceDirect
IFAC PapersOnLine 52-15 (2019) 157–162

Data-based
feedback controller
tuning utilizing
Data-based
Data-based feedback
feedback controller
controller tuning
tuning utilizing
utilizing
collocatedandnon-collocatedsensors
collocatedandnon-collocatedsensors
Data-based
feedback
controller
tuningutilizing
collocated and non-collocated sensors
∗collocated and non-collocatedsensors

WataruWataru Ohnishi
Ohnishi ∗∗∗∗
Wataru Ohnishi
Ohnishi
Wataru∗WataruOhnishi ∗Hongo,
Tokyo, 113-8656, Japan,∗∗ The University of Tokyo, 7-3-1
The University Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Bunkyo-ku,
Tokyo,
113-8656, Japan,∗∗ The University ofofTokyo,
7-3-1
Hongo,
Bunkyo-ku,
Tokyo,
ohnishi@koseki.t.u-tokyo.ac.jp
The University of Tokyo,
7-3-1
Hongo,
Bunkyo-ku,
Tokyo, 113-8656,
113-8656, Japan,
Japan,
ohnishi@koseki.t.u-tokyo.ac.jp∗
ohnishi@koseki.t.u-tokyo.ac.jp
The University of Tokyo,
7-3-1
Hongo,
Bunkyo-ku,
Tokyo,
113-8656, Japan,
ohnishi@koseki.t.u-tokyo.ac.jp
ohnishi@koseki.t.u-tokyo.ac.jp
Abstract: In
In motion
motion control
control applications,applications, such
such asas machinemachine tools
tools andand high-precisionhigh-precision positioning
positioning stages
stagesAbstract:
Abstract:In
motioncontrolapplications,suchasmachinetools
andhigh-precision
positioning
stagesforthesemiconductor
or
flat
panel
industries,
itisbecoming
commontoplace
sensors
on
not
only
the
Abstract:
In motion control
suchitasand high-precision
positioning
stagesforthe semiconductor
or flat applications,panel industries,
is machine
becomingtools
commonto place sensors
on not only
thefor
the side
semiconductor
orside)flat applications,panel
industries,
itas
is machine
becoming
commonto placeplace
sensors
on not
not
only
the
motor
(collocated
but
also
the
load
side
(non-collocatedside).
The
load
side
information
Abstract:In
motioncontrol
such
tools
and
high-precision
positioning
stagesforthesemiconductor
or
flat
panel
industries,
itisbecoming
commontosensorson
only
the
motor side (collocated side) but also the load side (non-collocated side). The load side information
motor(collocated
side)
but
the
load
(non-collocated
The
load
side
information
enablescompensationoftransmission
dynamics,
such
as friction
frictionorside).
backlash.
However,dueonly
to theforthe side
semiconductor
flat
panel
industries,
is becoming
commontoplace
on not
the
motor
side
(collocatedorside)
but also
also
theloadit side
side
(non-collocatedside).
Thesensors
load
side
information
enablescompensationoftransmission
dynamics,
such
as
or
backlash.
However,due
toenablescompensationoftransmission
dynamics,
suchasfrictionorbacklash.
However,
due
to
the
phase
lag
of
the
load
side
frequency
characteristics,
itisdifficulttoincreasethefeedback
bandwidth.
motor
side
(collocated
side)
but
also
the
load
side
(non-collocatedside).
The
load
side
information
enablescompensationoftransmission
dynamics,
suchasfrictionorbacklash.
However,duetothephase lag of the load side frequency characteristics, it is difficult to increase the feedback bandwidth.
phase
lagcompensation
ofthistheproblem,
load side
side
frequency
characteristics,
it is
isas
difficulttodesign
increase
the However,feedback
bandwidth.
To
address
this
paper
proposesafeedback
controller
methodbasedonfrequency
enablesoftransmission
dynamics,
suchfrictionorbacklash.
due
to
the
phase
lag
of
the
load
frequency
characteristics,
itdifficulttoincreasethefeedback
bandwidth.
To address this problem, this paper proposes a feedback controller design method based on frequency
Toaddress
thisproblem,
thispaper
proposesanon-collocatedfeedback
controller
designmethodbased ononbandwidth.
frequency
responsedatathatutilizesbothcollocatedandsensorinformation.
Experimental
results
phase
lag
of
the
load
side
frequency
characteristics,
itisdifficulttoincreasethefeedbackTo
address
thisproblem,
thispaper
proposesafeedback
controller
designmethodbasedfrequency
response data that utilizes both collocated and non-collocated sensor information. Experimental results
responsethatutilizesbothcollocatedandinformation.
Experimental
results
show
thatdata
theproposedmethodeffectively
suppresses
the controller
inputsensor
disturbance.
Furthermore,
thefrequency
proposedTo
address
thisproposed
problem,
thisproposesanon-collocatedfeedback
designmethodbased onresponsedatathatutilizes
bothpaper
collocatedandnon-collocatedsensorinformation.
Experimental
results
show
that
the
methodeffectively
suppresses
the
input
disturbance.
Furthermore,theproposed
show
that
the
proposedmethodeffectively
suppresses
the
input
disturbance.
Furthermore,theproposed
optimization
requireslesscalculationtime
comparedwithnonlinear
optimization
withrandominitialresponsedatathatutilizesbothcollocatedandnon-collocatedsensorinformation.
Experimental
results
show that the requires
proposedlessmethodeffectively
disturbance.
Furthermore,
the proposed
optimization
calculation
timesuppresses
compared thewithinput
nonlinear
optimization
with random
initialoptimization
requireslesscalculation
timesuppresses
compared thewithinput
nonlinear
optimization
with random
randominitialvaluesandexhibits
betterperformance.
show
that
the
proposedmethodeffectively
disturbance.
Furthermore,theproposed
optimization
requireslesscalculationtime
comparedwithnonlinear
optimization
withinitialvalues and exhibits better performance.
values andand exhibits
exhibits
betterperformance.
optimization
requireslessperformance.
calculation time compared with nonlinear optimization with random initialvaluesbetter© 2019,and
IFAC(International
Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
valuesexhibits
betterperformance.
Keywords:
Additional
sensor,Concave-convex procedure,
procedure, Data
Data based
based tuning,tuning, Frequency
Frequency responses
responses
Keywords: Additional sensor, Concave-convex
Keywords: Additional
Additional sensor,
sensor, Concave-convex
Concave-convex procedure,
procedure, Data
Data based
based tuning,tuning, Frequency
Frequency responses
responses
Keywords:
Keywords:
Additional sensor, Concave-convex procedure,
based
Frequency
1. INTRODUCTIONINTRODUCTION
et al.,Data
2014)
andtuning,
SRC-P-PI
controlresponses(Hasegawa et al., 2017).
1.
et
al.,
2014)
and
SRC-P-PI
control(Hasegawaet al.,
2017).
1.
INTRODUCTION
et
al.,
2014)
and
SRC-P-PI
control(Hasegawa
The
SRC-P-PI
control
has
an
inner
proportional-integral(PI)
1. INTRODUCTION
et
al.,
2014)
and
SRC-P-PI
control(Hasegawa etet al.,
al., 2017).
2017).
The SRC-P-PI control has an inner proportional-integral(PI)
The
SRC-P-PI
control
has
an
inner
proportional-integral(PI)
1.
INTRODUCTION
velocity
feedback
with
SRC
and
an
outer
proportional(P)
posiet
al.,
2014)
and
SRC-P-PI
control(Hasegawaetal.,2017).
The
SRC-P-PI
control
has
an
inner
proportional-integral(PI)In high-precision motion control systems, such as machine velocity feedback with SRC and an outer proportional (P) posiIn
high-precision
motioncontrolsystems,suchasmachinevelocity
feedback
with
SRC
and
an
outer
proportional(P)
position
feedback.
However,theabovemethodsneedaparametric
The
SRC-P-PI
control
has
an
inner
proportional-integral(PI)In
high-precision
motioncontrolsystems,suchasmachinevelocity
feedback
with
SRC
and
an
outer
proportional(P)
positools
andpositioning
stagesforthesemiconductor
or
flat
panel
In high-precision
motionsuch oras flatmachinefeedback. However, the above methods need a parametrictools
and positioning
stagescontrol
for the systems,
semiconductor
panel tion
feedback.
However,theabovemethodsneed
aa parametricmodel
includingthefirstresonance
and
they
do
not
explicitly
velocity
feedback
with
SRC
and
an
outer
proportional(P)
positools
and positioning
positioning
stagesfor thethe systems,
semiconductoror
flat
panel tion
tion
feedback.
However,theabovemethodsneedparametricindustries,
it is becoming
increasingly
commonto place
sensorsInhigh-precision
motioncontrolsuch
as flatmachinetools
andstagesforsemiconductor
or
panel
model including the first resonance and they do not explicitly
industries,
itisincreasingly
commontosensorsmodel
includingtheresonance
and
they
do
explicitly
discussdesignmultipleresonances.
feedback.
However,theabovemethodsa parametricindustries,
it theis becoming
becoming
increasingly
commonto placeplace
sensorsmodel
includingthe first
firstwithresonance
andtheyneed
do notnot
explicitlyonnotandonly
drivingside(collocatedside) butalso
theload tion
tools
positioning
stagesforthe semiconductor
or flatpanel
industries,
itisbecoming
increasingly
commontoplace
sensors
discussdesign methods
methods
withmultipleresonances.
on
not
only
the
drivingside(collocated
side)
but
also
the
load
discussdesignmethods
withmultipleresonances.
model
includingthefirstresonance
and
they
do
not
explicitly
on
not
only
the
drivingside(collocated
side)
but
also
the
load
discussdesignmethods
withmultipleresonances.
sidenot(non-collocatedside)(Sakata
et al.,
2014;but
Oomen,
2018).
industries,
it theis becoming
increasingly
commontoplace
sensorson
only
drivingside(collocated
side)
also
the
load
Feedback
controller
tuningmethods
basedonfrequency
reside(non-collocatedside)(Sakata
et
al.,
2014;
Oomen,
2018).
Feedback
controller
tuningmethods
basedonfrequency
rediscussdesignmethods
withmultipleresonances.
side(non-collocatedside)(Sakata
et
al.,
2014;
Oomen,
2018).
There
are
two
common
ways
to
close
the
feedback
(FB)
loop,
on
not
only
the
drivingside(collocated
side)
but
also
the
load
side (non-collocatedside)(Sakata
et al.,
Oomen,
Feedback
controller
tuningmethods
basedonfrequency
reThereare two common
ways
to closethe2014;feedback(FB)2018).
loop, sponse
data,
such
as
the
Ziegler–Nichols
method,have
been
exFeedback
controller
tuningmethods
basedonfrequency
response
data,
such
as
the
Ziegler–Nichols
method,have
been
exThere(non-collocatedaretwo common
commonways
to closeclosethefeedback(FB)
loop, sponse data, such as the Ziegler–Nichols method, have been exnamely
semi-closed
loop
controlandfull-closed
loop
controlside
side)(Sakata
et al.,
2014;
Oomen,
2018).
There
are
two
ways
to
the
feedback(FB)
loop,
Feedback
controller
tuningmethods
basedonfrequency
response
data,
such
as
the
Ziegler–Nichols
method,have
beenextensively
studied(ZieglerandNichols,
1942;Åströmand
Murnamely
semi-closed
loop
controlandfull-closed
loop
controltensively
studied(ZieglerandNichols,
1942;Åströmand
Murnamely
semi-closed
loop
controlandfull-closed
loop
control(Yamaguchial., 2011).
Theto semi-closed
loop loop
controlsys- sponse
There
are
twoetcommon
close
feedback(FB)
loop,
namelysemi-closed
loopways
controland thefull-closed
controldata,such (Ziegler
as the Ziegler–Nichols
method,haveandbeen
extensively
studied(ZieglerandNichols, 1942;
1942;Åströmand
Murray,2008).
Furthermore,
optimization-basedtuningmethods,(Yamaguchietal.,2011).
The
semi-closed
loop
controlsystensively
studiedand
Nichols,Åström
Mur(Yamaguchi
et al.,al.,2011).
The
semi-closed
loop
controlsys- ray, 2008). Furthermore, optimization-based tuning methods,
tem
feeds
back
the2011).
sensorinformation
onlyloopfromthecontrol
motor
namelysemi-closed
loop control
and full-closed
loop(Yamaguchi
etThe
semi-closed
controlsysray,2008).
Furthermore,
optimization-basedtuningmethods,
suchas thegenetic
algorithm(Tang
et al.,
2001),
Nelder–Mead
tem
back
the
sensorinformation
only
from
the
motor
tensively
studied(ZieglerandNichols,
1942;Åströmand
Mur2008).
Furthermore,
optimization-basedtuningmethods,
tem feeds
feeds
back
the2011).
sensorloop
information
onlyloop
from
the
motor
side,
while
the
full-closed
controlsystemfeeds
back
the ray,(Yamaguchi
et
al.,
The
semi-closed
controlsyssuchasthegenetic
algorithm
(Tang
et
al.,
2001),
Nelder–Mead
tem
feeds
back
the
sensorinformation
only
from
the
motor
side, while the full-closed loop control system feeds back the
such
as
the
genetic
algorithm
(Tang
et
al.,
2001),
Nelder–Meadmethod(Leeetal.,1985),
andparticleswarm
optimizationray,2008).
Furthermore,
optimization-basedtuningmethods,
such
as
the
genetic
algorithm
(Tang
et
al.,
2001),
Nelder–Meadside, feeds
while
the full-closed
full-closed
loop
controlsystemfeeds
backthe method (Lee et al., 1985), and particle swarm optimization
sensorinformation
the
motor
sidesystem
as well
asthe
from
tem
back
the from
sensorinformation
only
from
motor
side,
while
the
loop
control
feeds
back
the
method(Leeetal.,1985),
andparticleswarm
optimization(Gaing,2004),
been
proposed.
Convexoptimization-based
sensorinformation
from
the
motor
side
as
well
as
from
the
as thegenetic(Tang
et al., 2001),
Nelder–Meadmethod(Lee
ethaveal.,algorithm1985),
andparticleswarm
optimization(Gaing,2004),
havebeen
proposed.
Convexoptimization-based
sensorinformation
from
the
motor
side
as close
well
as
from
the such
load
side
(Hu
al., from2014),thewhichis sidelocatedto from
the
point
side,
while
theetfull-closed
loop
controlsystemfeeds
back
sensorinformation
motor
as
well
as
the
(Gaing, 2004),
2004),
havebeen
proposed.
Convexoptimization-based
methods
forafixed
structurecontroller
have
been
proposed,
load
side
(Hu
et
al.,
2014),
whichislocated
close
to
the
point
method(Leeetal.,1985),
andparticleswarm
optimization(Gaing,have
been
proposed.
Convexoptimization-based
methods
forafixed
structurecontroller
have
been
proposed,
load
side
(Hu
et
al.,
2014),
whichislocated
close
to
the
point
of
interest(Butler,2011).
The
full-closed
loop
controlsystemsensorinformation
from
the
motor
side
as
well
as
from
the
load
side (Huet al.,2011).
2014),The
which
is locatedclose
to thesystem
point (Gaing,
methods2004),
for
aopen
fixedloop
structurecontroller
have
been
proposed,
includinganshaping
methodusing
adesiredopenofinterest(Butler,full-closed
loop
controlhavebeen
proposed.
Convexoptimization-based
methods
forafixed
structurecontroller
have
been
proposed,
including an open loop shaping method using a desired
open
of interest
interest(Butler,2011).
The
full-closed
loop
controlsystemgenerallyshows
betterperformance,
especiallyinthesteadyload
side
(Hu
et
al.,
2014),
whichislocated
close
to
the
point
of
(Butler,2011).
The
full-closed
loop
controlsystemgenerally shows better performance, especially in the steady- including
an
open
loop
shaping
methodusing
aa anddesiredopen
looptransferfunction(KarimiandGaldos,2010)asequenmethods
for
a
fixed
structurecontroller
have
been
proposed,
including
an
open
loop
shaping
methodusing
desiredopen
transfer function (Karimi and Galdos, 2010) and a sequengenerallyshows
betterperformance,
especiallyin thethetransmissteadystatebecauseit feeds
backinformation
thatloop
includesofinterest(Butler,2011).
The
full-closed
controlsystem loop
generallyshows
betterperformance,
especiallyinsteadylooptransferfunction(KarimiandGaldos,2010)
andasequential
linearization
methodusing
theconcave-convex
procedure
state
because
it
feeds
back
information
that
includestransmisincluding
an
open
loop
shaping
methodusing
adesiredopen
looptransferfunction(KarimiandGaldos,2010)
andasequential
linearization
methodusing
theconcave-convex
procedure
state
because
it
feeds
back
information
that
includestransmission part
dynamics,
suchasinformation
friction orespeciallybacklash.
However,
it is tial linearization method using the concave-convex procedure
generallyshows
betterperformance,inthetransmissteadystatebecauseit feeds
backthat includes(CCCP)
(Hast
et
al.,
2013;
Nakamuraetal.,2016).
For
sion
part
dynamics,
suchasfrictionorbacklash.
However,itisloop
transferfunction(KarimiandGaldos,2010)
andasequential
linearization
methodusing
theconcave-convex
proceduresion part
part
dynamics,
suchasinformation
frictionor backlash.
backlash.
However,it isis (CCCP) (Hast et al., 2013; Nakamura et al., 2016). For thedifficultfor
theit full
full
closed-loop
control
to
have
high
feedbackstatebecause
feeds
back
that
includestransmission
dynamics,
suchasfrictionorHowever,itdifficultfortheclosed-loop
controltohaveaa high
feedback(CCCP)(Hast etetmethodal., 2013;
2013;
Nakamura
et limitedal., 2016).
2016).
For thetheabove
mentioned
methods,
there
only
discussions
tial
linearization
using
theareprocedure(CCCP)(Hastal.,
Nakamura
et
al.,
For
the
above
mentioned
methods,
there
areconcave-convex
only
limiteddiscussionsdifficultforthe fullfull closed-loop
closed-loop
controlto have
have
aload
high
feedback
bandwidth
becausetheinformation
from
the
side
(nonsionpart
dynamics,
suchasfrictionorbacklash.
However,itisdifficultforthecontrol
toahigh
feedback
bandwidth because the information from the load side (non- above
mentioned
methods,
there
are
only
limiteddiscussions
on
how
to
utilizebothcollocatedandnon-collocatedsensor(CCCP)
(Hast
et
al.,
2013;
Nakamuraetal.,2016).
For
the
above
mentioned
methods,
there
are
only
limiteddiscussions
bandwidth
becausethe information
information
from
the
load
side
(non- on how to utilize both collocated and non-collocated sensor
collocatedcontainsa phasecontrol
lag (Haraet al.,
2008).
difficultforside)the fullclosed-loop
to haveaload
highfeedback
bandwidth
because
the
from
the
side
(nononhowtoutilizebothcollocatedandnon-collocatedsensorinformation
todesignfixed
structurecontrollers.
collocatedside)
contains
aa phase
lag
(Hara
et
al.,
2008).
above
mentioned
methods,
there areonly
limited discussions
on
how
to
utilizebothcollocatedandnon-collocatedsensorinformation
todesignfixed
structurecontrollers.
collocatedside)
contains
phase
lag
(Hara
et
al.,
2008).
bandwidth side)
becausethe informationfrom theloadside (non- information to design fixed structure controllers.
collocatedcontains
phase lag (Haraet al.,
2008).
how to utilize
bothcollocatedandnon-collocated sensor
Herein, aa method
methodcalleda self-resonanceself-resonancecancellation(SRC), on
information
to design
structurecontrollers.
Herein,calledcancellation(SRC),
collocatedside)
contains
a
phase
lag
(Hara
et
al.,
2008).
aafixedfeedback
controller
tuningThis paper
paper proposes
proposes
feedback
controller
tuning method
method with
with
Herein,aims
a method
methodcalledself-resonancecancellation(SRC), This
information
todesignfixed
structurecontrollers.
which
to achieve
high-feedback
bandwidthby utilizing
Herein,acalledself-resonancecancellation(SRC),
This
paper
proposesaa feedback
controller
tuninghighdisturbance
suppression
performanceforaa method
system
which
aims
to
achievehigh-feedback
bandwidthbyutilizing
This
paper
proposes
feedback
controller
tuninghighdisturbance
suppression
performanceforsystem with
with
which
aims
to
achievehigh-feedback
bandwidthbyutilizingboth collocatedandcalled
non-collocatedsensorinformation,is pro- high disturbance suppression performance for a method
Herein,a method
self-resonancecancellation(SRC),
which
aims
to
achievehigh-feedback
bandwidthbyutilizing
systemwithcollocatedandnon-collocatedsensors.
Thetuningproposedmethodboth
andnon-collocatedsensorinformation,is
paper
proposesa feedbackperformance
controllerhighdisturbance
suppression
fora methodsystemwithboth collocatedcollocatedandnon-collocatedsensorinformation,is propro- This
posed
(Sakataetal.,non-collocated2014).
In short,
short,
this
methodby1)
1)utilizing
modelswhichaims to etachievehigh-feedback
bandwidth
collocatedandnon-collocatedsensors.
The
proposedmethodboth
collocatedandsensorinformation,is
proposed(Sakataal.,2014).
In
this
methodmodelscollocatedandnon-collocatedsensors.
The
proposedmethodrequiresthefollowing
information:1)
singleinput
doubleouthighdisturbance
suppression
performanceforasystemcollocatedandnon-collocatedsensors.
The
proposedmethodthe following information: 1) single input double with
outposed(Sakataetandal.,non-collocated2014).
In short,
short,
this
methoda rigid1) modelsmodelsthesystemas aetfourthorder systemthatthis
contains
both
collocatedsensorinformation,
isbodypro- requires
posed(Sakataal.,2014).
In
method1)
requiresthefollowing
information:1)
singleinput
doubleoutputfrequency
response
data
and
2)
rigid
body
model
(nominal
thesystemasafourthorder
systemthat
contains
a
rigid
body
collocatedandnon-collocatedsensors.
The
proposedmethodrequiresthefollowing
information:1)
singleinput
doubleoutfrequency response data and 2) rigid body model (nominal
the systemsystemas aaetfourth
fourthordermode,systemthat
contains
a rigid
rigid
body
mode
and
first
resonance2)thatcalculates
theof putposed(Sakataal., 2014).
In short,
thismethod1)centermodelstheasorder
systemcontainsabody
put frequency
frequency
response
data and
andThe2) rigidrigid
bodyinput
model(nominalmass
anddampingcoefficient).
proposedcontroller
has
mode
and
first
resonance
mode,
2)
calculates
thecenterofrequiresthefollowing
information:1)
singledoubleout-aaput
response
data
2)
body
model
(nominalmass
anddampingcoefficient).
The
proposedcontroller
has
mode
and
first
resonance
mode,
2)
calculates
thecenterofgravitypositionby interior
division
ofcalculates
thetwo sensors(“resthesystemas a fourthordermode,system2)that
contains
a rigid
body
mode
and
first
resonancethecenterof putmassfrequency
and structure
dampingcoefficient).
The
proposedcontroller
has aacascadedwithacollocatedsidePI
velocity
controller
gravitypositionbyinterior
divisionofthetwo
sensors(“resresponse
data
and
2)
rigid
body
model
(nominalmass
anddampingcoefficient).
The
proposedcontroller
hasstructure with a collocated side PI velocity controller
gravityand
positionby interior
interior
division
ofcalculates
thetwo sensors
sensors(“resonancecancellation”),
and mode,3)
designs
high-bandwidth
feedmode
first resonance2) of
the centerof cascaded
gravitypositionbydivision
two(“resonancecancellation”),
and3)
designs
aa thehigh-bandwidth
feedcascadedwithaa collocatedPI
controller
with SRC
and a non-collocatedsideside
P position
controller.
mass
and structure
dampingcoefficient).
The
proposedcontroller
has Itastructurewithcollocatedside
PI velocity
velocity
controller
onancecancellation”),
and 3)
3)
designs
high-bandwidth
feedback
controller
imaginarysystem
without
the first(“resres- cascaded
gravityposition forby aninterior
division
ofaa thetwo sensors
with
SRC
and
aa non-collocatedsideP
positioncontroller.
Itonancecancellation”),
anddesigns
high-bandwidth
feedwithSRC
and
non-collocatedsideP
positioncontroller.
It
has
four
non-affine
tuningparametersoptimized
throughbackcontrollerforanimaginarysystem
withoutthefirstrescascaded
structurewithacollocatedsidePI
velocity
controller
with
SRC
and
a
non-collocatedsideP
positioncontroller.
Ittheback controller
controller
for anan imaginary
imaginarysystem
without
the first
firstres- has four non-affine tuning parameters optimized through themode.
However,this3)method
does
not ensure
position
onancecancellation”),
anddesignsa high-bandwidth
feedbackforsystem
withoutthereshaspole-placement
four
non-affine
tuning 2)
parametersoptimized
throughthe1)
method,sequential
linearization(CCCP)
onancemode.
However,
thismethoddoes
notensureposition
with
SRC
and
a
non-collocatedsideP
positioncontroller.
It
has
four
non-affine
tuningparametersoptimized
throughthe1)pole-placement
method,2)
sequential
linearization(CCCP)
onancemode.
However,
thismethoddoes
notensurepositionaccuracyof theHowever,
load
side
becauseit doesnotdirectlyback
controller
for animaginarysystem
without
the feedback
firstres- 1) pole-placement method, 2) sequential linearization (CCCP)
onancemode.
thismethoddoesnot
notdirectly
ensureposition
method,and3)
Nelder–Meadmethod.
accuracy
of
the
load
side
becauseit
does
feedbackhas
four
non-affine
tuningparametersoptimized
throughthe1)pole-placement
method,2)
sequential
linearization(CCCP)
and 3) Nelder–Mead method.
accuracyof thetheload side
side thisbecauseit doesdoes
notdirectly
feedback
theloadmode.
side
information.
To address
address
thisproblem,thefeedback
method method,onanceHowever,
methoddoesnot
notdirectly
ensurepositionaccuracyofload
because
it
the
load
side
information.
To
thisproblem,themethodmethod,and3)1) pole-placement
method, 2)method.
sequential linearization (CCCP)
andnonlinear
3) Nelder–Mead
Nelder–Meadmethod.
theextended
load side
side
information.
To address
address
this notproblem,thefeedback
methodistoaloadcomplementary
method(Sakata method,
accuracyof theside because
itfilter-based
does
directlyIn
general,optimization
problemstheloadinformation.
To
thisproblem,themethodIn general,
nonlinear
optimization
problems have
have strong
strong iniiniis
extendedtoacomplementary
filter-based
method(Sakatamethod,and3)
Nelder–Meadmethod.
isextendedtoacomplementary
filter-based
method(Sakata
In general,
general,nonlinear optimization
optimization
problemsthat
havestronginitheload side to
information.
To address
this problem,the method
tial
value
dependence.
We
also
demonstratetheproposedis extended
a complementary
filter-based
method(Sakata In
nonlinear
problems
havestronginitialvaluedependence.
We
also
demonstratethattheproposed
would like totothankProf. Fujimoto for his
help with themethod
experimental
setup.
tial
value
dependence.
We
also
demonstratethattheproposedis IIextended
a complementary
filter-based(Sakata
methodachieves
higherperformancewithshorter
calculationIn
general,nonlinear
optimization
problems
havestronginitialvaluedependence.
We
also
demonstratethattheproposed
would
like
to
thank
Prof.
Fujimoto
for
his
help
with
the
experimental
setup.
method achieves higher performance with shorter calculationThis
work like
was tosupported
by JSPSKAKENHI
Grant
Number
18H05902.setup.
would
like
thank Prof.
Prof.
Fujimoto
for hishis help
help
with
the experimental
experimental
setup.
methodhigherperformancewithII would
thankFujimoto
for
with
the
tial
valueachieves
dependence.
also demonstratethat thecalculation
proposedmethodachieves
higherWe
performancewith shorter
shorter
calculationThis
work was tosupported
by JSPSKAKENHI
Grant
Number
18H05902.

This
work
was
supportedby
KAKENHI
Grant
Number
18H05902.
This
work like
was tosupported
by JSPSJSPS
KAKENHI
Grant
Number
18H05902.setup.
I would
thank Prof.
Fujimoto
for his help
withthe experimental
method achieves higher performance with shorter calculationThis
work wassupportedby (International
JSPS KAKENHIFederation
Grant Number
18H05902.Control)
2405-8963© 20192019,
IFAC
of Automatic
Copyright©
IFAC
560 Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC
560
Peer
review
under
responsibilityofInternational
Federationof Automatic
Copyright
©
2019
IFAC
560
Copyright © 2019 IFAC
560 Control.
10.1016/j.ifacol.2019.11.667
Copyright © 2019 IFAC
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2019 IFAC MECHATRONICS
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1snon-collocatedLinear encoder(non-collocated)

Pv2

Table
r

Cp

+ −

Leaf spring

+ −

Cpi

d
u+
+

v2

α

collocated

Pv1

v1

p2

1−α +
p1
1

+ vsrcs

Linear motors

Carriage(a) P-PI control with SRC.

non-collocatedAir guide

Pv2
d
r
u+
Cpid
+ −
+

Linear encoder(collocated)

Fig. 1. High-precision positioning stage (Hara et al., 2008).
Magnitude [dB]collocated

Pv1

v1

1
s

p2
p1

Fig. 3. Block diagrams of the compared controller configurations.

-20
-40

A first-order nominal model Pn is
1
Pn (s) =Mn s + Bn
Mn = 0.412, Bn = 0.866.

-60

100

101

102

103

90

Phase [deg]1s(b) PID control.

0

-80
10-1

v2

(1)
(2)

0
-90

3. SRC-P-PI CONTROLLER DESIGN

-180
-270
10-1

100

101

102

In this section, the proposed SRC-P-PI controller design using
frequency response data is described. The block diagram representation is shown in Fig. 3(a).

103Frequency [Hz]Fig. 2. Frequency response of the plant (input current to velocities). The frequency response data and first order nominalmodel Pn are used for controller design.

The transfer function of the plant with SRC istime than the Nelder–Mead method with 1000 pattern randomly
selected initial values. This paper also compares the proposed
method with proportional-integral-derivative (PID) control forthe non-collocated position to show the effectiveness of the
SRC-P-PI control structure. The PID gains and the time constant of pseudo difference are optimized similar to the proposed
procedure: 1) pole-placement method, 2) sequential linearization method, and 3) Nelder–Mead method. Note that the PID
controller has four tuning parameters, which is the same number of tuning parameters as that for SRC-P-PI control.

where ωk and k denote the measured frequency point andthe data number, respectively. Hence, the open loop transfer
function is represented by 1 Pv1 ( jωk )

Lvsrc ( jωk , ρ, α) = [(1 − α) α]  [kvp kvi ]  1  , (4)
Pv2 ( jωk )jωk

2. EXPERIMENTAL SETUP
A one-axis high-precision positioning stage, as shown in Fig.
1, is used as an experimental setup. The input is a pair of linear
motors at the carriage. The output is two encoders located at
the carriage and table. There is a spring between the carriage
and table. Therefore, the encoder at the carriage is a collocatedside sensor, and the encoder at the table is a non-collocated side
sensor. Here, the point of interest is assumed as the table side
position.
Frequency responses from the input current to both the sides’
velocities are shown in Fig. 2. This is obtained through frequency domain identification (Pintelon and Schoukens, 2012).
The collocated side response Pv1 has an anti-resonance first anda resonance later. In contrast, the non-collocated side response
Pv2 has a resonance first followed by an anti-resonance.
561

Pvsrc ( jωk , α) = (1 − α)Pv1 ( jωk ) + αPv2 ( jωk ),(3)

where ρ = [kvp , kvi ] denotes the PI gain. It is not affine forρ and α; therefore, the linear optimization is not applicable.
Hence, the pole-placement and sequential linearization are applied with a fixed α first, and then, the Nelder–Mead method(Nelder and Mead, 1965; Lagarias et al., 1998), which is oneof the nonlinear optimization methods, is applied to optimize ρand α at the same time.
Step 1. Initial guess of α for the first resonance cancellation

The original idea of SRC (Sakata et al., 2014) “cancels” the firstresonance mode. The initial guess of α is obtained through an
optimization problem to make the gain flat between ωkmin andωkmax , where ωkmin and ωkmax are around the first resonance mode
frequency.
minimizeαsubject to jωk (1 − α)Pv1 ( jωk ) + jωk αPv2 ( jωk )∞k ∈ [kmin , kmax ]
0≤α≤1

(5)
(6)
(7)

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Wataru Ohnishi / IFAC PapersOnLine 52-15 (2019) 157–162

Step 2. Initial C pi gain calculation by the pole-placement

The open loop frequency response is L( jωk , ωv ) =P( jωk )C( jωk , ωv ). ωv is maximized by the dichotomy method
as follows:
maximize ωv(10)subject to rm − |L( jωk , ωv ) + σ| ≤ 0, ∀ k,(11)
where α = α src . rm and σ denote the circle condition (Maeda
and Iwasaki, 2014) for gain and phase margins. The prohibitedcircle in the Nyquist plot with a center at (−σ, j0) and radius rm
is given by
g2m − 1



σ
=



2gm (gm cos Φm − 1)
,
(12)



(gm − 1)2 + 2gm (1 − cos Φm )



r
=
m

2gm (gm cos Φm − 1)
where Φm and gm denote the phase and gain margins, respectively.
A designed case referred to as SRC-P-PI (Case 1) skips Steps 3
and 4 and directly applies Step 5.
Step 3. C pi optimization through sequential linearization
Step 3 optimizes C pi through sequential linearization. The gains
obtained in Step 2 are used as the initial gains. To linearize theproblem, α is fixed as α src .
The controller and open loop frequency responses are given as
C pi ( jωk , ρ) = ρ φ( jωk )(13)
L( jωk , ρ) = P( jωk )C pi ( jωk , ρ),(14)
1 
jωk ] denotes the basis function of the

The optimization problem is formulated as
maximize Ωgc
ρm Ωgc subject to − |1 + L( jωk , ρ)| ≤ 0, ∀ k jωk 

(15)
(16)

rm − |L( jωk , ρ) + σ| ≤ 0, ∀ k
(17)
0≤ρ
(18)
Ωgc in (15) is a constraint for the sensitivity function in (16).
Ωgc denotes a 0 dB cross-over frequency of the sensitivity function. By maximizing Ωgc , the gain of the sensitivity functionin the low-frequency range is minimized. Equation (17) is thecircle condition of the gain and phase margins. Equation (18) is
a constraint to obtain a stable controller.
Equations (16) and (17) are the differences in convex functions
and they are not convex. However, by applying a linearization
around the current solution point, the new feasible set becomes
562

Magnitude [dB]

40

20

0

-20
10-1

100

101

102

103

101

102

103

180

Phase [deg]This section calculates the PI gain by pole-placement for the
first order model formulated in (1). A transfer function of thevelocity PI controller iskvi(8)C pi (s) = kvp + .
sAccording to (1) and (8), the controller that places the velocity
closed-loop poles of the nominal model (1) at (s + ωv )2 is(Goodwin et al., 2000)
kvp = 2ωv Mn − Bn , kvi = ω2v Mn .
(9)where φ( jωk ) = [1,controller.

159

90
0
-90
-180
10-1

100

Frequency [Hz]Fig. 4. Initial value of α for SRC determined by the optimization (5)–(7).
a convex subset of the original feasible set (Yuille and Rangarajan, 2003; Hast et al., 2013; Nakamura et al., 2016). Equations(16) and (17) are relaxed as
m



(L j ( jωk , ρ j ) + 1)∗ Ωgc (L( jωk , ρ) + 1) ≤ 0, (19)
 −R

jωk 
|L j ( jωk , ρ j ) + 1|
(L j ( jωk , ρ j ) + σ)∗
rm − R(L( jωk , ρ) + σ) ≤ 0(20)|L j ( jωk , ρ j ) + σ|
where R and ∗ denote the real part and complex conjugate
respectively. L j ( jωk , ρ j ) denotes the open loop frequency response of the previous iteration. Therefore, by iterating theprocedure, the solution will converge to a saddle point or a local
minimum. This procedure is referred to as the concave–convex
procedure (CCCP) (Yuille and Rangarajan, 2003).
Ωgc itself is maximized by a dichotomy method. The feasibility
problem is solved by YALMIP (Löfberg, 2004) and MOSEK(MOSEK ApS, 2017).
A designed case referred to as SRC-P-PI (Case 2) skips Step 4
and directly applies Step 5.
Step 4. Nonlinear optimization through the Nelder–MeadmethodStep 4 applies the Nelder–Mead method to optimize C pi and α
at the same time by the following optimization problem:(21)
maximize Ωgc
ρ,α

m
 Ωgc subject to − |1 + L( jωk , ρ, α)| ≤ 0, ∀ k(22)
 jωk rm − |L( jωk , ρ, α) + σ| ≤ 0, ∀ k
(23)
0≤ρ
(24)
0≤α≤1
(25)
The C pi gains obtained in Step 3 and α src obtained in Step 1 areused as the initial values for the Nelder–Mead method.

The optimized α by the Nelder–Mead method is denoted as
αopt . A designed case referred to as SRC-P-PI (Case 3) applies
Step 5 after Step 4.
Step 5. C p maximization by dichotomy method
This step optimizes the outer loop position controller C p = k pp .
The open loop frequency response of the position controller2019 IFAC MECHATRONICS
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1
Magnitude [dB]

20

0
-0.5

-60
100

101

102

103

101

102

103

180

Case 1Case 2
Case 3

-1
-1.5
-1.5

-40

-80
10-1

Phase [deg]Imaginary axis

0.5

0
-20

-1

-0.5

0

0.5

90
0
-90
-180
10-1

1

Real axis

100

Frequency [Hz](a) Nyquist plot.

(b) Sensitivity function.

Fig. 5. Design results of the velocity loop (SRC-P-PI control).
L p ( jω, k pp ) is formulated as
L p ( jωk , k pp ) = k pp

( jωk )
C pi ( jωk ) Pv2jω
k

1 + Lvsrc ( jωk , ρ, α)(26)

k pp is maximized by the dichotomy method with the following
optimization:(27)
maximize k pp
subject to rm − |L p ( jωk , k pp ) + σ| ≤ 0, ∀ k(28)

4. EXPERIMENTAL VALIDATION
Experiments are performed using the high-precision positioning stage shown in Fig. 1. The gain and phase margins are set
as 6 dB and 30◦ , respectively. m in (16) is set as 2. The sampling
period is set as 400 µs, and the 1 A input disturbance is applied
at t = 0.0 s.
4.1 Design results of the proposed method
In Step 1, the optimal α, which minimizes the gain of the firstresonance mode, is obtained as α src = 0.484. According to Fig.
2, ωkmin = 22.4 × 2π [rad/s] and ωkmax = 32.6 × 2π [rad/s]. A
frequency response of the plant with SRC is shown in Fig. 4;
this shows that the gain and phase characteristics are almost flat
between ωkmin and ωkmax .
The design results of SRC-P-PI control are shown in Tab. 1
and Fig. 5. The frequency response from the control input tothe velocity is shown in Fig. 6. From Fig. 6, α = αopt in
SRC-P-PI (Case 3) results in incomplete cancellation of thefirst resonance. This is because, according to Fig. 2, the gainof the secondary resonance decreases when α approaches 1. In
addition, Fig. 5(a) shows that the circle of the first and second
resonances of SRC-P-PI (Case 3) have the same radius. As aresult, the bandwidth is increased, as shown in Fig. 5(b).
Experimental results of the step disturbance suppression are
shown in Fig. 7. According to Fig. 7(a) and Tab. 2, SRC-PPI (Case 3) shows the best tracking performance in terms ofinfinity and 2-norm tracking error.
4.2 Initial value dependency of Nelder–Mead method

placement method and sequential linearization. This section
compares the results of the Nelder–Mead method with random
initial values to demonstrate the effectiveness of the proposed
procedure.
1000 pattern randomly selected values in the range of 0 ≤α ≤ 1, 0 ≤ kvp ≤ 1, 0 ≤ kvi ≤ 10 are used as the initialvalues. The optimization results of Ωgc are shown in Fig. 8. The
maximum Ωgc case is referred to as “SRC-P-PI (NM1000)”.
Tab. 1 shows that SRC-P-PI (NM1000) converges a smaller Ωgc
when compared to the results of SRC-P-PI (Case 3).
The calculation time of both methods is shown in Tab. 3.
NM1000 (Parallel) is calculated using MATLAB’s parfor
to utilize multiple CPU cores. This shows that the proposed
method achieves better performance with a shorter calculation
time. Moreover, even when using the laptop computer, theproposed method completed the calculation in 26 s, and it isthus also practical in terms of calculation time. Fig. 9 and Tab. 2
show the experimental results of the input disturbance rejection.
Tracking performance of SRC-P-PI (Case 3) outperforms thatof SRC-P-PI (NM1000).
4.3 Comparison with PID control
This section shows a comparison with PID control shownin Fig. 3(b) to demonstrate the superiority of the controller
structure of the SRC-P-PI control. Both design methods use
frequency response data and low-order nominal models. The
gain margin and phase margin are set as the same as in the
SRC-P-PI control 6 dB and 30◦ , respectively. The weighting
m for sensitivity function is set as 3, considering the integratorsof the plant and controller. The parameter of the PID controller
is optimized through the pole placement method, the sequential
linearization method, and the Nelder–Mead method.
Fig. 10 and Tab. 2 show a comparison of the two methods. This
indicates that sufficient disturbance suppression performancecannot be obtained only with the non-collocated sensor. The
above result shows the superiority of the controller structure of
SRC-P-PI control and the effectiveness of the proposed tuning
method.
5. CONCLUSION

In the proposed method, the initial value of the Nelder–Meadmethod is given by the stepwise optimization with the pole
563

This paper proposed a tuning method for P-PI control utilizing
multiple sensors. An advantage of the proposed method is that2019 IFAC MECHATRONICS
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161

0.2

1.5

0

1

-0.2

0.5

-0.4

0

-0.6

-0.5

Magnitude [dB]

0
-20
-40

Disturbance
SRC-P-PI (Case1)
SRC-P-PI (Case2)
SRC-P-PI (Case3)

-60
-80
10-1

100

101

102

103

Phase [deg]

90

-1

-0.8

0
-90

SRC-P-PI (Case1)
SRC-P-PI (Case2)
SRC-P-PI (Case3)

-1

-180
-270
10-1

100

101

102

-1.2
-0.2

103

0

0.2

Frequency [Hz]

0.4

0.6

-1.5
0.8

-2
-0.2

0

0.2

Fig. 6. Frequency response of the plantwith SRC.

0.4

0.6

0.8

Time [s]Time [s](a) Tracking error.

(b) Disturbance and feedback current.

Fig. 7. Experimental results for step disturbance rejection (SRC-P-PI control).

1.5

0.5
40

1

35

0

25

Disturbance
SRC-P-PI (Case3)
SRC-P-PI (NM1000)

0.5

SRC-P-PI (Case 3)
SRC-P-PI (NM1000)
SRC-P-PI (NM1000, other trials)

30

20

-0.5

0

-1

-0.5
-1

15

-1.5

10
5

SRC-P-PI (Case3)
SRC-P-PI (NM1000)

-2
-0.2

0
0

200

400

600

800

0

0.2

0.4

0.6

-1.5
0.8

-2
-0.2

0

0.2

Time [s]1000

Trial number

0.4

0.6

0.8

Time [s](a) Tracking error.

(b) Disturbance and feedback current.

Fig. 8. Optimization results by Nelder– Fig. 9. Experimental results for step disturbance rejection (SRC-P-PI (Case 3,
Mead method.
NM1000)).

0.5

1.5

0

1

-0.5

0.5

-1

0

-1.5

-0.5

-2

-1

SRC-P-PI (Case 3)
PID

-2.5
-3
-0.2

Disturbance
SRC-P-PI (Case 3)
PID

0

0.2

0.4

0.6

-1.5
0.8

Time [s]

-2
-0.2

0

0.2

0.4

0.6

Time [s](a) Tracking error.

(b) Disturbance and feedback current.

Fig. 10. Experimental results for step disturbance rejection (SRC-P-PI and PID control).
Table 1. Design results (SRC-P-PI control).
α
kvp
kvi
ωv , Ωgc [rad/s]
k pp
Note
SRC-P-PI (Case 1)
0.484 17.3
194
ωv = 21.3 50.4
PP
SRC-P-PI (Case 2)
0.484 15.1
1710
Ωgc = 35.2 16.6
PP→CCCP
PP→CCCP→NM
SRC-P-PI (Case 3)
0.571 18.82160
Ωgc = 39.5 20.3
SRC-P-PI (NM1000)
0.01779.03
524Ωgc = 16.9 9.27
See sec. 4.2
PP: Pole Placement, CCCP: Concave–Convex Procedure, NM: Nelder–Mead method

564

0.8

2019 IFAC MECHATRONICS
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Wataru Ohnishi / IFAC PapersOnLine 52-15 (2019) 157–162

Table 2. Tracking error comparison of the experimental results in mm. Here, () denotes the relativevalue compared with SRC-P-PI (Case 3).
Methods
SRC-P-PI (Case 1)
SRC-P-PI (Case 2)
SRC-P-PI (Case 3)
SRC-P-PI (NM1000)
PID||e||212.7 (2.75)
6.73 (1.46)
4.61 (1.00)
26.5 (5.74)
37.9 (8.24)||e||∞
1.07 (2.05)
0.656 (1.26)
0.522 (1.00)
1.98 (3.78)
2.63 (5.03)

Table 3. Calculation time comparison.
CPU (# of cores)
MemoryNM1000NM1000 (Parallel)
SRC-P-PI (Case 3)

Laptop computer
Core i7 8550U (4 core)
12GB
13279 [s] (221 [min])4764 [s] (79 [min])26 [s]

Desktop computer
Core i9 9900K (8 core)
48GB
7753 [s] (129 [min])1305 [s] (22 [min])19 [s]

it does not require the high-order parametric model and only
utilizes the frequency response data and rigid body nominalmodel. Hence, the proposed method has the potential to address
the plant perturbations and unstructured dynamics by dealing
with multiple frequency response data measured under different
conditions.
The proposed method employs stepwise optimization with thepole-placement using a nominal model, sequential linearization
method, and the Nelder–Mead method. The bandwidth is improved by each step. Further, the proposed procedure obtains
high performance with less calculation time compared to the
Nelder–Mead method with 1000 pattern random initial values.
The effectiveness of the controller structure was demonstrated
through comparison with the PID controller optimized by thesame procedure.
The proposed method is practical because it requires only frequency response data and a low-order nominal model, and it isapplicable with multiple sensors. Considering that systems withmultiple encoders are becoming popular, this paper has proposed an effective way to utilize collocated and non-collocatedsensors.
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