Proceedings of the 12th IFAC Workshop on Time Delay Systems
Proceedings
of
12th
IFAC
Workshop
on
Proceedings
of the
the
12th
IFACMI,
Workshop
on Time
Time Delay
Delay Systems
Systems
June
28-30, 2015.
Ann
Arbor,
USA
Proceedings
of the
12th
IFAC Workshop
on Time Delay Systems
June
June 28-30,
28-30, 2015.
2015. Ann
Ann Arbor,
Arbor, MI,
MI, USA
USAAvailable online at www.sciencedirect.com
June 28-30, 2015. Ann Arbor, MI, USA
ScienceDirect
IFAC-PapersOnLine 48-12 (2015) 129–134
Performance
Comparison
of
Input-Shaped
Model
Performance
Comparison
of
Input-Shaped
Model
Performance
Comparison
of
Input-Shaped
Model
Performance
Comparison
of Input-Shaped
Model
Reference
Control
on
an
Uncertain
Flexible
System
Reference
Control
on
an
Uncertain
Flexible
System
Reference
Control
on
an
Uncertain
Flexible
System
Reference Control on an Uncertain Flexible System
∗
∗∗
Daichi
Fujioka
∗ William Singhose ∗∗
∗ William Singhose ∗∗
Daichi
Fujioka
Daichi
Fujioka
William
Singhose
Daichi Fujioka ∗ William Singhose ∗∗
∗
∗ The George W. Woodruff School of Mechanical Engineering
∗ The George W. Woodruff School of Mechanical Engineering
George
W.
School
Mechanical
Engineering
∗ The
Georgia
Institute
of
Technology,
GA
30332
USA
The
George
W. Woodruff
Woodruff
School of
ofAtlanta,
Mechanical
Engineering
Georgia
Institute
of
Technology,
Atlanta,
GA
30332
USA
Georgia
Institute
of
Technology,
Atlanta,
GA
30332
USA
(gtg369q@mail.gatech.edu)
Georgia
Institute
of
Technology,
Atlanta,
GA
30332
USA
(gtg369q@mail.gatech.edu)
∗∗
(gtg369q@mail.gatech.edu)
Woodruff
School
of
Mechanical
Engineering
∗∗ The George W.(gtg369q@mail.gatech.edu)
∗∗ The George W. Woodruff School of Mechanical Engineering
George
W.
School
Mechanical
Engineering
∗∗ The
Georgia
Institute
of
Technology,
Atlanta,
GA
30332
USA
The
George
W. Woodruff
Woodruff
School of
of
Mechanical
Engineering
Georgia
Institute
of
Technology,
Atlanta,
GA
30332
USA
Georgia
Institute
of
Technology,
Atlanta,
GA
30332
USA
Georgia Institute(Singhose@gatech.edu)
of
Technology,
Atlanta,
GA
30332
USA
(Singhose@gatech.edu)
(Singhose@gatech.edu)
(Singhose@gatech.edu)
Abstract: This
This paper
paper investigates
investigates input-shaped
input-shaped model
model reference
reference control
control (IS-MRC)
(IS-MRC) applied
applied to
to an
an
Abstract:
Abstract:
This
paper
investigates
input-shaped
model
reference
control
(IS-MRC)
applied
to
an
uncertain
nonlinear
double-pendulum
crane.
In
order
to
investigate
practical
implementation
issues,
Abstract:
This paper
investigates input-shaped
model
reference practical
control (IS-MRC)
applied
to anaa
uncertain
nonlinear
double-pendulum
crane.
In
order
to
investigate
implementation
issues,
uncertain
nonlinear
double-pendulum
crane.
In
order
to
investigate
practical
implementation
issues,
aa
simple
single-pendulum
is
used
as
the
reference
model.
The
proposed
controller
enhances
the
robustness
uncertain
nonlinear
double-pendulum
crane.
In
order
to
investigate
practical
implementation
issues,
simple
single-pendulum
is
used
as
the
reference
model.
The
proposed
controller
enhances
the
robustness
simple
single-pendulum
is
used
as
the
reference
model.
The
proposed
controller
enhances
the
robustness
to the
the uncertain
uncertain
difference
between
the
reference
model
and
the plant.
plant.
Single-enhances
and double-pendulum
double-pendulum
simple
single-pendulum
is used
as thethe
reference
model.
Theand
proposed
controller
the robustness
to
difference
between
reference
model
the
Singleand
to
the
uncertain
difference
between
the
reference
model
and
the
plant.
Singleand
double-pendulum
crane
dynamics
are
presented.
The
natural
frequencies
of
the
double-pendulum
crane
are
calculated
to
the
uncertain
difference
between
the
reference
model
and
the
plant.
Singleand
double-pendulum
crane
dynamics
are
presented.
The
natural
frequencies
of
the
double-pendulum
crane
are
calculated
crane
dynamics
are
presented.
The
natural
frequencies
of
the
double-pendulum
crane
are
calculated
and
utilized
to
design
input
shapers.
A
Lyapunov
control
law
with
guaranteed
asymptotic
stability
is
crane
dynamics
are presented.
The natural
frequencies
of law
the with
double-pendulum
crane are stability
calculated
and
utilized
to
design
input
shapers.
A
Lyapunov
control
guaranteed
asymptotic
is
and
utilized
to
design
input
shapers.
A
Lyapunov
control
law
with
guaranteed
asymptotic
stability
is
derived
using
only
the
first
mode
states
of
the
plant.
The
state
tracking,
control
effort
reduction,
and
and
utilized
to
design
input
shapers.
A
Lyapunov
control
law
with
guaranteed
asymptotic
stability
is
derived
using
only
the first
mode
states of
the plant.
The
state
tracking, control
effort
reduction, and
derived
using
only
mode
of
The
state
control
effort
and
oscillation
suppression
performances
of various
various
IS-MRC
designs
are tested
tested
via numerical
numerical
simulations
derived
using
only the
the first
first
mode states
states
of the
the plant.
plant.
Thedesigns
state tracking,
tracking,
control
effort reduction,
reduction,
and
oscillation
suppression
performances
of
IS-MRC
are
via
simulations
oscillation
suppression
performances
of
various
IS-MRC
designs
are
tested
via
numerical
simulations
and
experiments.
By
analyzing
the
results,
the
IS-MRC
design
that
has
the
largest
robustness
to
the
plant
oscillation
suppression
performances
of various
IS-MRC
designs
are the
tested
via numerical
simulations
and
experiments.
By
analyzing
the
results,
the
IS-MRC
design
that
has
largest
robustness
to
the
plant
and
experiments.
By
results,
IS-MRC
that
uncertainties
is demonstrated
demonstrated
tothe
provide
thethe
best
overalldesign
performance.
and
experiments.
By analyzing
analyzingto
the
results,
the
IS-MRC
design
that has
has the
the largest
largest robustness
robustness to
to the
the plant
plant
uncertainties
is
provide
the
best
overall
performance.
uncertainties
is
demonstrated
to
provide
the
best
overall
performance.
uncertainties
demonstratedFederation
to provideofthe
best overall
performance.
© 2015, IFACis(International
Automatic
Control)
Hosting by Elsevier Ltd. All rights reserved.
Keywords: Flexible
Flexible Manufacturing
Manufacturing Systems,
Systems, Lyapunov
Lyapunov Stability,
Stability, Model
Model Reference
Reference Control,
Control, Shaping
Shaping
Keywords:
Keywords:
Flexible
Manufacturing
Systems,
Lyapunov
Stability,
Model
Reference
Control,
Shaping
Filters,
Uncertain
Dynamic
Systems
Keywords:
FlexibleDynamic
Manufacturing
Systems, Lyapunov Stability, Model Reference Control, Shaping
Filters,
Uncertain
Systems
Filters,
Uncertain
Dynamic
Systems
Filters, Uncertain Dynamic Systems
1. INTRODUCTION
INTRODUCTION
1.
1.
INTRODUCTION
1. INTRODUCTION
Model reference
reference control
control
(MRC) is
is an
an adaptive
adaptive control
control strategy
strategy
Model
(MRC)
Model
reference
control
(MRC) is
isapplications,
an adaptive
adaptive control
control
strategy
that
is
widely
used
in
engineering
such
as
control
Model
reference
control
(MRC)
an
strategy
that
is
widely
used
in
engineering
applications,
such
as
control
that
is
widely
used
in
engineering
applications,
such
as
control
of
mechanical
oscillators
[Hovakimyan
et
al.,
1999]
and
robots
that
is
widely
used
in
engineering
applications,
such
as
control
of
mechanical
oscillators
[Hovakimyan
et
al.,
1999]
and
robots
of
mechanical
oscillators
[Hovakimyan
et
al.,
1999]
and
robots
[Ekrekli
and
Brookfield,
1997].
MRC
is
very
useful
when
deof
mechanical
oscillators
[Hovakimyan
et
al.,
1999]
and
robots
[Ekrekli
and
Brookfield,
1997].
MRC
is
very
useful
when
de[Ekrekli
and
Brookfield,
1997].
MRC
is
very
useful
when
designing
a
controller
for
systems
containing
time-varying
param[Ekrekli
Brookfield,
1997].containing
MRC is very
useful when
designing
aaand
controller
for
systems
time-varying
paramsigning
controller
for
systems
containing
time-varying
parameters [Chien
[Chien
and Fu,
Fu,
1992;
Huang
and Chen,
Chen,
1993; Abdullah
Abdullah
signing
a controller
for1992;
systems
containing
time-varying
parameters
and
Huang
1993;
etersZribi,
[Chien
and Fu,
Fu,
1992;
Huang and
and
Chen, 1979;
1993; Nijmeijer
Abdullah
and
2009]
and
nonlinearities
[Landau,
eters
[Chien
and
1992;
Huang
and
Chen,
1993;
Abdullah
and
2009]
and
nonlinearities
[Landau,
1979;
Nijmeijer
and Zribi,
Zribi,
2009]
andKim
nonlinearities
[Landau,
1979;
Nijmeijer
and
Savaresi,
1998;
et al.,
al., 2010].
2010].
MRC 1979;
has also
also
shown
and
Zribi,
2009]
and
nonlinearities
[Landau,
Nijmeijer
and
Savaresi,
1998;
Kim
et
MRC
has
shown
and
Savaresi,
1998;
Kim
et
al.,
2010].
MRC
has
also
shown
to
be
effective
in
controlling
time-delay
systems
[Basher
and
and
Savaresi,
1998;
Kim et al.,
2010]. MRC
has [Basher
also shown
to
be
effective
in
controlling
time-delay
systems
and
to
be
effective
in
controlling
time-delay
systems
[Basher
and
Mukundan,
1987;
Basher,
2010;
Santosh
and
Chidambaram,
to
be
effective
in
controlling
time-delay
systems
[Basher
and
Mukundan,
1987;
Basher,
2010;
Santosh
and
Chidambaram,
Mukundan,
1987;
Basher,
2010;
Santosh
and
Chidambaram,
2013].
Mukundan,
1987; Basher, 2010; Santosh and Chidambaram,
2013].
2013].
2013].
The performance
performance of
of MRC
MRC depends
depends on
on the
the formulation
formulation of
of aa
The
The performance
performance
of MRC
MRC
depends
on the
the
formulation
of aa
reference
model
that
properly
represents
the
controlled
plant
The
of
depends
on
formulation
of
reference
model
that
properly
represents
the
controlled
plant
reference
model
that
properly
represents
the
controlled
plant
[Sastry
and
Isidori,
1989;
Chen
and
Liu,
1994;
Kim
et
al.,
reference
model
that1989;
properly
represents
the
controlled
plant
[Sastry
and
Isidori,
Chen
and
Liu,
1994;
Kim
et
al.,
[Sastry
and
Isidori,
1989;
Chen
and
Liu,
1994;
Kim
et
al.,
2010].
This
is
not
always
possible
because
actual
systems
can
[Sastry
and isIsidori,
1989;possible
Chen and
Liu, actual
1994; systems
Kim et can
al.,
2010].
This
not
always
because
2010].
This
is
not
always
possible
because
actual
systems
can
be difficult
difficult
to
givenpossible
nonlinearities
and
uncertainties.
In
2010].
This to
is model
not always
becauseand
actual
systems can
be
given
nonlinearities
uncertainties.
In
be difficult
difficult
toofmodel
model
given
nonlinearities
and
uncertainties.
In
the
absence
accurate
reference
models,
MRC
has
degraded
be
to
model
given
nonlinearities
and
uncertainties.
In
the
absence
of
accurate
reference
models,
MRC
has
degraded
the
absence
of
accurate
reference
models,
MRC
has
degraded
state
tracking
performance.
Furthermore,
the
issue
of
exceeding
the absence
of accurate reference
models,
MRC has
degraded
state
tracking
Furthermore,
the
of
statemaximum
tracking performance.
performance.
Furthermore,
the issue
issue
of exceeding
exceeding
the
control effort
effortFurthermore,
must be
be considered
considered
because
signifstate
tracking
performance.
the
issue
of
exceeding
the
maximum
control
must
because
signifthe maximum
maximum
control
effort
must
be considered
considered
because
significant
mismatch
between
the
reference
model
and
actual
system
the
control
effort
must
be
because
significant
mismatch
between
the
reference
model
and
actual
system
icant
mismatch
between
the
reference
model
and
actual
system
can
saturate
the
actuators
and
also
degrade
the
controller’s
icant
mismatch
between the and
reference degrade
model and actual
system
can
the
can saturate
saturate
the actuators
actuators and
and also
also degrade
degrade the
the controller’s
controller’s
tracking
ability.
can
saturate
the
actuators
also
the
controller’s
tracking
ability.
tracking ability.
ability.
tracking
Previous
research approached
approached this
this challenge
challenge by
by enhancing
enhancing the
the
Previous
research
Previous
research
approached
this
challenge
by
enhancing
the
robustness
of
MRC
[Duan
et
al.,
2001].
Sun
et
al.
enhanced
Previous
research
approached
this
challenge
byetenhancing
the
robustness
of
MRC
[Duan
et
al.,
2001].
Sun
al.
enhanced
robustness
of
MRC
[Duan
et
al.,
2001].
Sun
et
al.
enhanced
the system
system stability
stability
robustness
via aa MRC
MRC
controller
comrobustness
of MRCand
[Duan
et al., 2001].
Sun et
al. enhanced
the
via
controller
comthe system
system
stability and
and robustness
robustness
via aa MRC
MRC
controller
composed
of
a
conventional
model
matching
feedback
and
a
linear
the
stability
and
robustness
via
controller
composed
of
aa conventional
model
matching
feedback
and
aa linear
posed
of
conventional
model
matching
feedback
and
linear
model
error
compensator
[Sun
et
al.,
1994].
Patino
and
Liu
posed
of
a conventional
model
matching
feedback
andand
a linear
model
error
compensator
[Sun
et
al.,
1994].
Patino
Liu
model
error
compensator
[Sun
et
al.,
1994].
Patino
and
Liu
proposed
a
controller
based
on
neural
networks
and
analyzed
model
error
compensator
[Sun
et al., networks
1994]. Patino
and Liu
proposed
aa controller
based
on
neural
and
analyzed
proposed
controller
based
on
neural
networks
and
analyzed
it for
for aa class
class
of first-order
first-order
continuous-time
nonlinear
dynamical
proposed
a controller
based
on neural networks
anddynamical
analyzed
it
of
continuous-time
nonlinear
it for
for aa class
class
of first-order
first-order
continuous-time
nonlinear
dynamical
systems
[Patino
and
Liu,
2000].
Pedret
et
al.
developed
a robust
it
of
continuous-time
nonlinear
dynamical
systems
systems [Patino
[Patino and
and Liu,
Liu, 2000].
2000]. Pedret
Pedret et
et al.
al. developed
developed aaa robust
robust
systems
[Patino
and
Liu,
2000].
Pedret
et
al.
developed
robust
MRC structure
structure based
based on
on aa right
right coprime
coprime factorization
factorization of
of the
the
MRC
MRC
structure
based
on
a
right
coprime
factorization
of
the
plant
along
with
an
observer-based
feedback
control
scheme
MRC
structure
based
on a right coprime
factorization
of the
plant
along
with
an
observer-based
feedback
control
scheme
plant
along
with
an
observer-based
feedback
control
scheme
combined
a prefilter
controller [Pedret
[Pedret
et al.,
2005].scheme
plant
alongwith
with
an observer-based
feedback
control
combined
combined with
with aaa prefilter
prefilter controller
controller [Pedret
[Pedret et
et al.,
al., 2005].
2005].
combined
with
prefilter
controller
et
al.,
2005].
While the
the robustness
robustness of
of MRC
MRC for
for the
the state
state tracking
tracking perforperforWhile
While the
the
robustness
of extensively,
MRC for
for the
the
stateless
tracking
performance
has
been
studied
much
emphasis
has
While
robustness
of
MRC
state
tracking
performance
has
been
studied
extensively,
much
less
emphasis
has
mance
has
been
studied
extensively,
much
less
emphasis
has
been
placed
on
the
issue
of
the
control
effort.
To
realize
both
mance
has
been
studied
extensively,
much
less
emphasis
has
been
placed
on
the
issue
of
the
control
effort.
To
realize
both
been
placed
on
the
issue
of
the
control
effort.
To
realize
both
excellent
tracking
performance
and
limited
control
effort,
we
been
placed
on theperformance
issue of the and
control
effort.
To realize
both
excellent
tracking
limited
control
effort,
we
excellent
tracking
performance
and
limited
control
effort,
we
investigated
a
control
method
called
input-shaped
model
refexcellent tracking
performance
and limited
controlmodel
effort, refwe
investigated
aa control
method
called
input-shaped
investigated
control
method
called
input-shaped
model
reference
control
(IS-MRC).
The called
controller
combines model
MRC with
with
investigated
a control
method
input-shaped
reference
control
(IS-MRC).
The
controller
combines
MRC
erence
control (IS-MRC).
(IS-MRC).
The controller
controller
combines
MRC
with
command-shaping
technique
called input
input
shaping MRC
[Yuanwith
and
erence
control
The
combines
aaa command-shaping
technique
called
shaping
[Yuan
and
command-shaping
technique
called
input
shaping
[Yuan
and
Chang,
2006,
2008;
Yu
and
Chang,
2010].
Input
shaping
has
a command-shaping
technique
called2010].
input shaping
[Yuan and
Chang,
2006,
2008;
Yu
and
Chang,
Input
shaping
has
Chang,
2006,
2008;
Yu
and
Chang,
2010].
Input
shaping
has
been
shown
to
reduce
system
vibration
and
improve
the
perChang,
2006,to2008;
Yusystem
and Chang,
2010].
Input
shaping
has
been
shown
reduce
vibration
and
improve
the
perbeen
shown
to
reduce
system
vibration
and
improve
the
performance
of totime-delay
time-delay
systems
with flexibility
flexibility
[Potter
and
been
shown
reduce system
vibration
and improve
the performance
systems
with
[Potter
and
formance of
of
time-delay
systems controllers
with flexibility
flexibility
[Potter
and
Singhose,
2012,
2014].
IS-MRC
had
been
develformance
of
time-delay
systems
with
[Potter
and
Singhose,
2012,
2014].
IS-MRC
controllers
had
been
develSinghose,
2012,
2014].
IS-MRC
controllers
had
been
developed
and
shown
to
be
effective
for
reducing
the
control
effort
Singhose,
2012, to
2014].effective
IS-MRCforcontrollers
hadcontrol
been developed
shown
reducing
the
effort
oped and
andand
shown
to be
be effective
effective
formaking
reducing
the
control
effort
[Fujioka
Singhose,
2014], and
and
the the
overall
controller
oped
and
shown
to
be
for
reducing
control
effort
[Fujioka
and
Singhose,
2014],
making
the
overall
controller
[Fujioka
and
Singhose,
2014],
and
making
the
overall
controller
insensitive
toSinghose,
the plant
plant 2014],
parameter
variations
and
system
order
[Fujioka
andto
and making
theand
overall
controller
insensitive
parameter
variations
system
order
insensitive [Fujioka
to the
the plant
plant
parameter
variations
and
system
order
difference
et
al.,
2015].
IS-MRC
can
be
designed
for
insensitive
to
the
parameter
variations
and
system
order
difference
[Fujioka
et
al.,
2015].
IS-MRC
can
be
designed
for
difference
[Fujioka
et
al.,
2015].
IS-MRC
can
be
designed
for
robustness
to
the
time-varying
parameters
and
nonlinear
dydifference [Fujioka
et al., 2015].parameters
IS-MRC can
benonlinear
designed dyfor
robustness
to
the
time-varying
and
robustness
to
the
time-varying
parameters
and
nonlinear
dynamics of
of the
the
[Fujioka and
andparameters
Singhose, 2015].
2015].
robustness
to plant
the time-varying
and nonlinear dynamics
namics of
of the
the plant
plant [Fujioka
[Fujioka and
and Singhose,
Singhose, 2015].
2015].
namics
plant
[Fujioka
Singhose,
In this
this paper,
paper, we
we extend
extend the
the past
past works
works by
by considering
considering an
an ununIn
In this
this paper,
paper,
we investigating
extend the
the past
past
works
by designs
considering
an ununcertain
plant
and
the
IS-MRC
that
reduce
In
we
extend
works
by
considering
an
certain
plant
and
investigating
the
IS-MRC
designs
that
reduce
certain
plant
and
investigating
the
IS-MRC
designs
that
reduce
the sensitivity
sensitivity
to investigating
parameter uncertainties
uncertainties
and
nonlinearities
in
certain
plant and
the IS-MRCand
designs
that reduce
the
to
parameter
in
the sensitivity
sensitivity
to the
parameter
uncertainties
and nonlinearities
nonlinearities
in
the
plant.
Then,
performance
of
the
proposed
controllers
the
to
parameter
uncertainties
and
nonlinearities
in
the
plant.
Then,
the
performance
of
the
proposed
controllers
the
plant.
Then,
the
performance
of
the
proposed
controllers
are
compared
and
analyzed.
In
Section
2,
the
IS-MRC
dethe plant.
Then,and
theanalyzed.
performance
of the proposed
controllers
are
In
Section
the
deare compared
compared
and
analyzed.
Inexemplary
Section 2,
2,singlethe IS-MRC
IS-MRC
design
scheme is
is and
explained
usingIn
and doubledoubleare
compared
analyzed.
Section
2,
the
IS-MRC
design
scheme
explained
using
exemplary
singleand
sign
scheme
is
explained
using
exemplary
singleand
doublependulum
crane
systems as
as
the reference
reference
model
and
plant.
sign
scheme
is explained
using
exemplarymodel
single-and
andthe
doublependulum
crane
systems
plant.
pendulum
crane
systems
as the
the
reference
modelcontrol
and the
thelaw
plant.
Various
input
shaper
designs
andreference
the Lyapunov
Lyapunov
are
pendulum
crane
systems
as
the
model
and
the
plant.
Various
input
shaper
designs
and
the
control
law
are
Various
input
shaper
designs
and
the
Lyapunov
control
law
are
also
presented.
In
Section
3,
the
performance
of
the
various
ISVarious
input shaper
designs
andperformance
the Lyapunov
control
law ISare
also
presented.
In
Section
3,
the
of
the
various
also
presented.
In
Section
3,
the
performance
of
the
various
ISMRCpresented.
designs are
are
measured
of the
the state
state
tracking,
also
In Section
3, in
theterms
performance
of the
variousconISMRC
MRC designs
designs are
are measured
measured in
in terms
terms of
of the
the state
state tracking,
tracking, conconMRC
designs
measured
in
terms
of
tracking,
con-
Copyright
IFAC 2015
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2405-8963 © 2015,
IFAC (International Federation of Automatic Control)
Copyright
© IFAC
2015
129
Copyright
IFAC responsibility
2015
129Control.
Peer review©
of International Federation of Automatic
Copyright
©under
IFAC 2015
129
10.1016/j.ifacol.2015.09.365
IFAC TDS 2015
130
June 28-30, 2015. Ann Arbor, MI, USA
Input
Shaper
Control
Law
Shaped
Command, vs
x
Trolley
Desired
States, xd
Model
Command
Signal, v
Daichi Fujioka et al. / IFAC-PapersOnLine 48-12 (2015) 129–134
Control
Signal, u
Plant
at
Plant
States, x
Datum
L1
Fig. 1. Model reference control block diagram
x
Trolley
θ1
vt , at
L2
Datum
θ2
L
θ
Hook, m1
Payload, m2
Fig. 3. Model of a double-pendulum crane
the cable length by ωm = g/L, where g is the acceleration
due to gravity.
Mass
Fig. 2. Model of a single-pendulum crane
trol effort reduction, and oscillation suppression via numerical
simulations and experimental data.
2. INPUT-SHAPED MODEL REFERENCE CONTROL
Figure 1 shows the block diagram of IS-MRC. The system
consists of four blocks; a reference model, an uncertain plant,
an input shaper, and a control law. A command signal v is
first sent through the input shaper to obtain a shaped signal vs .
The vs signal is then sent to the reference model to calculate
the desired states xd . The control law takes in vs , xd , and the
plant states x to formulate a control signal u that controls the
uncertain plant.
2.1 Reference Model and Uncertain Plant
In this work, the uncertain plant used in the MRC is a nonlinear
double-pendulum crane; however, the reference model is a
linearized single-pendulum crane. We utilize this one-mode
model because it is easier to implement and accurate real-time
measurement of a real crane payload is extremely difficult.
Furthermore, a single-pendulum is a very good representation
of a crane when it does not carry a payload.
Figure 2 shows a single-pendulum crane with a point-mass
payload. The trolley’s horizontal position is indicated by x, and
is moved with velocity vt and acceleration at . The point-mass
m is suspended via a massless cable of length L. The swing
angle of the payload measured with respect to the vertical axis
is represented by θ.
A state space representation of the model using a velocity input
is obtained by defining the second state x2 as the horizontal
swing distance of the mass −θL, and the first state x1 as the
integral of x2 . Assuming small swing angles, the state space
representation of the system and the associated desired states
xd becomes:
x˙d = ASP xd + BSP vt
(1)
0
1
x˙
xd,1
0
= d,1 =
+
v
2
1 t
xd,2
˙
−2ζm ωm xd,2
−ωm
where, ωm and ζm are the natural frequency and the damping
ratio of the reference model. The swing frequency is related to
130
A double-pendulum crane is shown in Figure 3. The trolley
moves with an acceleration input at , and its horizontal position
is indicated by x. The hook m1 is a point-mass suspended from
the trolley via a massless hoist cable of length L1 . The payload
m2 is a point-mass attached to the hook via a massless cable
of a fixed length L2 . The swing angle of the hook is measured
with respect to the position of the trolley, and is represented by
θ1 . The swing angle of the payload θ2 is measured with respect
to the swing of the hook. Damping of the swing is neglected.
The nonlinear equations of motion of the double-pendulum
crane in terms of θ1 and θ2 are:
(m1 + m2 ) L1 θ¨1 + gsin (θ1 ) + 2L˙1 θ˙1 +
2
m2 L2 θ¨1 + θ¨2 cos (−θ2 ) + θ˙1 + θ˙2 sin (−θ2 )
= at (m1 + m2 ) cos (θ1 )
(2)
L2 θ¨1 + θ¨2 + L¨1 sin (−θ2 ) + gsin (θ1 + θ2 ) −
L1 θ˙12 sin (−θ2 ) − θ¨1 cos (−θ2 ) + 2L˙1 θ˙1 cos (−θ2 ) (3)
= at cos (θ1 + θ2 )
2.2 Frequencies of the Double-Pendulum Crane
Input shapers are designed by specifying the vibration frequencies to suppress. Hence, the hook and the payload oscillation
modes are analyzed so that appropriate frequency ranges can be
determined. Assuming small swing angles and a constant hoist
cable length (L˙1 = L¨1 = 0), the linearized natural frequencies
of the double-pendulum crane are [Blevins, 1979]:
1
g
1
∓β
(1 + RM )
+
ω1,2 =
2
L1
L2
2
(4)
1
1
1
+
R
M
2
β = (1 + RM )
+
−4
L1
L2
L1 L2
where, RM = m2 /m1 is the mass ratio of the payload to the
hook.
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131
Table 1. Small-scale bridge crane parameters
vmax
0.20 m/sec
m1
0.69 kg
a
1.00 m/sec2
(m1 + m2 )max
5.00 kg
0.16 m/sec
(L1 + L2 )max
1.50 m
L˙1
Fig. 5. ω2 as a function of mass ratio and length ratio
2.3 Input Shaping
Fig. 4. ω1 as a function of mass ratio and length ratio
The crane parameters listed in Table 1 were utilized. They
correspond to our experimental bridge crane. Figure 4 shows
the first mode frequency ω1 as a function of RM and the cable
length ratio RL = L2 /L1 . The value of ω1 ranges between
2.61 rad/sec and 4.32 rad/sec (this mode roughly corresponds
to the single-pendulum case). This gives an overall variation of
±24.7% about the median value of 3.47 rad/sec.
The oscillation amplitude of the second mode compared to the
first mode can be very small, or even negligible, at certain
parameter settings. To obtain the practical range of ω2 that
needs to be suppressed, the relative swing contribution of
the two modes is examined by breaking the overall dynamic
response into ω1 and ω2 components. The linearized horizontal
swing of the payload to an impulse can be approximated as
[Singhose et al., 2008]:
x(t) = C1 sin(ω1 t + ψ1 ) + C2 sin(ω2 t + ψ2 )
(5)
where,
ω1 L1 1 + ω22 α(L1 + L2 )
C1 =
k
−ω2 L1 1 + ω12 α(L1 + L2 )
(6)
C2 =
k
−g (1 + RM )
,
k = βL1 g
α=
ω12 ω22 L1 L2
Here, the coefficients C1 and C2 represent the contributions of
each mode to the overall payload oscillation.
The ranges of RM and RL when the second mode oscillation
C2 becomes significant compared to the first mode C1 are
found by calculating the ratio C2 /C1 . The parameter ranges of
0 RM 0.69 and 0.23 RL 1.84 were found to give
C2 /C1 10%.
Figure 5 shows the second mode frequency ω2 as a function of
RM and RL utilizing the same parameter settings as in Figure
4. The solid portion of the graph indicates the region where
the second mode oscillation is significant compared to the
first mode oscillation. From the effective RM and RL ranges,
the frequency range that produces problematic second-mode
oscillation is found to be 4.32 rad/sec ω2 8.98 rad/sec.
This is a ±35.0% variation about the median value of 6.65
rad/sec.
131
Input shaping suppresses the command-induced oscillation by
convolving the command signal with a sequence of impulses
at the appropriate amplitudes and timings so that the resultant
oscillation from all the impulses sums to zero, or a small
value, [Smith, 1958; Singer and Seering, 1990]. Input shaping
makes MRC robust to plant uncertainty by decreasing the
complex behavior in the states. In this work, a zero-vibration
(ZV) shaper, a two-mode zero-vibration (ZV2M) shaper, a
three-mode zero-vibration (ZV3M) shaper, and a two-mode
specified insensitivity (SI2M) shaper are applied to suppress
the oscillations of the double-pendulum crane. The ZV shaper
is non-robust but the simplest type of the input shaper. The
ZV2M and ZV3M shapers are obtained by combining two or
three different ZV shapers into one shaper via convolution. The
SI2M shaper is the most insensitive to the parameter uncertainty
and variation, and is obtained through optimization. The details
on design constraints and derivation process can be found for
ZV [Singer and Seering, 1990], ZV2M [Singhose et al., 1997],
ZV3M [Hyde and Seering, 1991], and SI2M shaper [Singhose
et al., 2008].
Figure 6 shows the sensitivity curves of the input shapers. The
curves illustrate the shapers’ robustness to the natural frequency
of the system, [Singer and Seering, 1990]. The horizontal axis
shows the natural frequency in rad/sec. The vertical axis shows
the vibration percentage, which is defined as the vibration
caused by the input-shaper divided by the vibration from a unit
impulse.
The ZV shaper only limits the oscillation predicted from the
single-pendulum reference model mode by ωm and ζm . Thus,
as the actual frequencies of the plant deviate from ωm the
system vibration increases rapidly, [Smith, 1957].
Because the actual plant parameters remain unknown during
the controller design process, the value of ωm is selected based
on the possible ω1 range found in Section 2.2. To reduce the
control effort resulting from a modeling error, the damped
natural frequency ωd of the model was set
to the ω1 range
2 .
median. Then, ωm was found by ωd = ωm 1 − ζm
The ζm value was determined
from thedesired percent over
2 . Limited oscillation
shoot, MP = exp −ζm π/ 1 − ζm
in the reference response is necessary in order to reduce the
complexity of the state dynamics, which also helps reduce the
MRC control effort. However, choosing a large ζm would make
the reference model significantly different from the plant and
leads to a control effort increase. Considering the trade-off,
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50
40
Vibration [%]
Trolley
(Y-) Axis
ZV
ZV2M
ZV3M
SI2M
30
Hoist
20
Hook
10
V
tol
0
2
4
ωm
ω1,median
6
8
10
Payload
ωn [rad/sec]
ω2,median
Fig. 6. Sensitivity curves of the input shapers
Table 2. Design parameters of the input shapers
Shaper
ZV
ZV2M
ZV3M
SI2M
ωm
ωm
ωm
–
Target ω
–
ω2,median
ω2,median
ω2,range
–
–
ω1,median
ω1,range
ζm
ζm
ζm
–
Target ζ
–
0
0
0
–
–
0
0
Fig. 7. Bridge crane experimental setup
MP = 30% (ζm = 0.36) was selected to avoid actuator
saturation in the experimental setup.
The ZV2M shaper handles both the reference model mode plus
the second mode oscillation in the uncertain double-pendulum
plant. The median value of the possible ω2 range is used as
the representative value for the second mode. As shown in
Figure 6, the shaper does not suppress the system vibration
effectively as the natural frequency deviates significantly from
the designed frequency values. The ZV3M shaper reinforces
the ZV2M shaper by including the representative value for the
possible ω1 range. The shaper demonstrates larger robustness
to natural frequency changes. The SI2M shaper is designed to
suppress each frequency range to under a tolerable vibration
level of 5%. The zero-vibration constraint is relaxed to achieve
a higher robustness to the uncertainty, [Singhose et al., 1996]. In
all shaper designs, the damping ratios in the double-pendulum
crane are assumed zero. Table 2 summarizes the types of design
parameters utilized in each input shaper.
2.4 Lyapunov Control Law
The second method of Lyapunov is applied to formulate the
control law. Given the system order difference between the
single-pendulum reference model and the double-pendulum
plant, only the first mode states of the plant, x1 and x2 , are
utilized. The details of the derivation process can be found in
[Fujioka et al., 2015].
Defining a Lyapunov function of the error as V (e) = eT P e,
where e = xd − x, and making the derivative of the Lyapunov
function V̇ (e) always negative to ensure the asymptotical stability of the controller, the control law expression u becomes:
2
2
u = (−ωm
+ ω1,median
)x1 + (−2ζm ωm )x2 + v + Λ (7)
where,
132
Λ=













0.05vmax
if e1 P1,2 + e2 P2,2 > 0.05vmax
e1 P1,2 + e2 P2,2 if e1 P1,2 + e2 P2,2 0.05vmax
e1 P1,2 + e2 P2,2 −0.05vmax
if e1 P1,2 + e2 P2,2 < −0.05vmax
(8)
P1,2 and P2,2 are the entries in the P matrix that is derived
from the expression ATSP P + P ASP = −Q, where Q is a real,
positive-definite, symmetric matrix. The saturation condition is
enforced on the scalar constant Λ to minimize the chattering
in the signal. The cutoff was set to ±5% of the maximum
magnitude of the input velocity vmax . ω1,median is used as the
representation of the first mode in the double-pendulum crane
because the actual plant parameters and dynamics are uncertain.
−0.05vmax
3. SIMULATION AND EXPERIMENT
Figure 7 shows the bridge crane used to perform experiments.
The crane can move the overhead trolley and hoist the suspension cable, which carries a hook and a payload connected via
a rigging cable. A downward-pointing camera attached on the
trolley records the horizontal position of the hook. The position
of the payload is measured via a camcoder from side.
Numerical simulations and experimental data are used to analyze the proposed IS-MRC controllers on state tracking, control
effort reduction, and oscillation suppression performance. The
case studies were tested for motion of the trolley with a constant
L1 = 0.53 m. The parameters from the bridge crane in Table 1
and two settings of the rigging cable and payload, [L2 = 0.41
m; m2 = 0.23 kg] and [L2 = 0.64 m; m2 = 0.89 kg], were
utilized for the double-pendulum plant. The ZV-MRC, ZV2MMRC, ZV3M-MRC, and SI2M-MRC derived in Section 2 are
compared for their effectiveness.
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Daichi Fujioka et al. / IFAC-PapersOnLine 48-12 (2015) 129–134
133
Table 3. Performance comparison of IS-MRC
3.1 Performance Indices
Performance indices were defined to quantify and compare the
various IS-MRC designs. The state tracking index is:
2
(x2 − xd,2 ) dt
(9)
= 2
(x2 − xd,2 ) dtM RC,sim
The integral squared error measure between the desired and
actual hook swing is normalized via the measure from the
simulated MRC case. The performance of the MRC without
input shaping is treated as the datum to compare different ISMRC controllers. The index measures how closely the plant
tracks the states defined by the reference model. Lower values
indicate better tracking performance.
Monitoring the control effort for energy efficiency and saturation is an important consideration. The control effort index µ
that describes the energy usage is:
2
u dt
µ= 2
(10)
vref dt
The index is normalized via the energy of the raw reference
command vref , or v in (7).
The maximum value of the control signal Umax is also of
the interest, and is normalized via the maximum value of the
reference command vmax :
M ax( u(t) )
Umax =
(11)
vmax
In both control effort indices, lower values are favorable because they indicate less energy consumption for the same task.
The hook and payload oscillation reduction is also important
factors for consideration. The maximum swing amplitudes of
the hook ∆1,max and payload Θ2,max are:
M ax ( |L1 θ1 (t)| )
(12)
∆1,max =
M ax ( |L1 θ1 (t)| )M RC,sim
M ax ( |θ2 (t)| )
(13)
M ax ( |θ2 (t)| )M RC,sim
The measures are normalized by the values obtained from the
MRC without input shaping simulaion case. In both indices,
smaller values indicate better oscillation reduction.
Θ2,max =
3.2 Performance Comparison Analysis
Table 3 summarizes the simulation and experimental results for
each IS-MRC controller. In both test cases, the ZV-MRC resulted in the worst performance as measured by all indices. This
is because the ZV-MRC only eliminates the single-pendulum
mode in the reference model defined by ωm and ζm (and some
portion of ω1 because their values are close), and ignores the
second-mode oscillation in the double-pendulum plant. The
large payload swing in the plant induces unwanted oscillation
in the hook, which complicates the state tracking and consequently increases the energy usage by the ZV-MRC. Furthermore, the Umax value being larger than 1.00 indicates that
actuator saturation could occur while using the ZV shaper.
The ZV2M-MRC provided much improved performance over
the ZV-MRC, especially in , ∆1,max , and Θ2,max . The improvement occurs because the ZV2M shaper suppressed both
ωm and ω2,median oscillations and reduced the complex behavior of the states. This facilitated the state tracking of the hook
and, as a result, decreased the required control effort. The Umax
was found to be less than 1.00, meaning that saturation was
133
L2 = 0.41 m (RL = 0.78)
m2 = 0.23 kg (RM = 0.33)
ZV-MRC
(Simulation)
(Experiment)
ZV2M-MRC (Simulation)
(Experiment)
ZV3M-MRC (Simulation)
(Experiment)
SI2M-MRC (Simulation)
(Experiment)
L2 = 0.64 m (RL = 1.22)
m2 = 0.89 kg (RM = 1.28)
ZV-MRC
(Simulation)
(Experiment)
ZV2M-MRC (Simulation)
(Experiment)
ZV3M-MRC (Simulation)
(Experiment)
SI2M-MRC (Simulation)
(Experiment)
0.72
2.31
0.25
1.46
0.14
1.28
0.09
1.30
µ
0.82
0.93
0.80
0.90
0.76
0.87
0.73
0.85
Umax
1.01
1.18
0.94
1.15
0.93
1.09
0.92
1.12
∆1,max
0.75
1.97
0.48
1.16
0.31
0.79
0.27
0.70
Θ2,max
0.83
1.63
0.19
0.38
0.11
0.24
0.09
0.18
0.62
2.99
0.19
1.74
0.09
0.84
0.06
0.79
µ
0.82
0.85
0.80
0.81
0.76
0.78
0.73
0.75
Umax
1.03
1.08
0.98
0.99
0.95
0.98
0.93
0.96
∆1,max
0.75
1.71
0.67
1.29
0.38
0.85
0.37
0.89
Θ2,max
0.75
1.51
0.31
0.62
0.14
0.29
0.16
0.32
prevented. The ZV2M-MRC has a potential for improvement
because the representative ω2,median value used to design the
ZV2M shaper is not exact. In each case study, the ω2 values
in the uncertain double-pendulum plant were 6.53 rad/sec and
8.25 rad/sec respectively. Therefore, further reduction in the
hook and payload swing is possible.
The ZV3M-MRC and SI2M-MRC exhibited excellent performance in all indices. This is because both shapers consider
all oscillation modes in the control design; ωm of the singlependulum reference model, and ω1 and ω2 of the doublependulum plant. In general, the SI2M-MRC outperformed the
ZV3M-MRC; the indices of SI2M-MRC were lower than those
of ZV3M-MRC, except Θ2,max in the [L2 = 0.64 m; m2 =
0.89 kg] case. This is primary due to the difference in the shaper
design parameters. While the ZV3M reduced the oscillations of
the uncertain double-pendulum plant using the representative
values ω1,median and ω2,median , the SI2M shaper effectively
suppressed both modes of the oscillation under 5%. As a result,
the SI2M-MRC produced results that are more robust to the
plant uncertainties than the ZV3M-MRC.
The experimental results verified the simulation data because
they showed similar characteristics. The ZV3M-MRC and
SI2M-MRC produced the lowest performance indices, followed
by the ZV2M-MRC and then the ZV-MRC which produced the
worst values. The experimental index values were found to be
larger than the simulations, especially in the state tracking. One
of the sources for this deviation originated from the unmodeled
dynamics, rail friction, and uncertainty/error in the doublependulum model. On real systems, a more aggressive design of
MRC by using a larger ζm value may be desirable to improve
the performance. The largest source of the error, however, is
likely the sensor noise in the data which hindered the effectiveness of MRC. The data also contained offsets and jumps
that needed to be processed during the analysis. In addition, the
swing angle of the payload was analyzed via image processing,
making it difficult to achieve high accuracy.
4. CONCLUSIONS
This paper compared the performances of various input-shaped
model reference control (IS-MRC) designs applied to an uncertain double-pendulum crane. Various input shapers were designed using the practical ranges of the double-pendulum crane
natural frequencies and the single-pendulum crane reference
model parameters based on the best representation of the uncer-
IFAC TDS 2015
134
June 28-30, 2015. Ann Arbor, MI, USA
Daichi Fujioka et al. / IFAC-PapersOnLine 48-12 (2015) 129–134
tain plant. The Lyapunov control law using only the states associated with the first mode was designed for asymptotic stability.
Performance indices for the state tracking, energy usage, maximum control effort, and maximum hook and payload swings
were defined and measured for different IS-MRC designs using
both numerical simulations and experiments. The SI2M-MRC
achieved results that were more robust to the plant uncertainties
than the other types of IS-MRC designs because it suppressed
the hook and payload oscillations most effectively.
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