IFAC-PapersOnLine Volume 50 issue 1 2017 [doi 10.1016 j.ifacol.2017.08.127] Novella-Rodríguez, David F.; del Muro-Cuéllar, Basilio; Hernan -- Delayed Model Approximation and Control Design for Under

Proceedings of the 20th World Congress
Proceedings
of
20th
The
International
Federation
of Congress
Automatic Control
Proceedings
of the
the
20th World
World
Congress
Proceedings
of the 20th World
Congress Control
The
of
Toulouse,
France,Federation
July 9-14, 2017
The International
International
Federation
of Automatic
Automatic
Control
Available
online at www.sciencedirect.com
The
International Federation
of Automatic Control
Toulouse,
Toulouse, France,
France, July
July 9-14,
9-14, 2017
2017
Toulouse, France, July 9-14, 2017
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 1316–1321
Delayed Model Approximation and Control Design
Delayed Model Approximation and Control Design
Delayed Model
Approximation Systems
and Control Design
for Under-Damped
for Under-Damped Systems
for Under-Damped
Systems
∗
∗∗
David F. Novella-Rodrı́guez ∗ Basilio del Muro-Cuéllar ∗∗
∗ Basilio
∗∗ ∗∗
∗∗
David
F.
Novella-Rodrı́guez
David
Novella-Rodrı́guez
del
Muro-Cuéllar
German
Hernández-Hernández
Juandel
F. Muro-Cuéllar
Márquez-Rubio
∗∗
David F.
F.
Novella-Rodrı́guez ∗ Basilio
Basilio
Muro-Cuéllar
∗∗
∗∗
∗∗ Juandel
∗∗
German
Hernández-Hernández
F.
Márquez-Rubio
German
Hernández-Hernández
∗∗ Juan F. Márquez-Rubio ∗∗
German
Hernández-Hernández
Juan
F.
Márquez-Rubio
∗
Instituto Tecnológico de Estudios Superiores de Monterrey, ITESM Campus
∗
∗ Instituto Tecnológico de Estudios Superiores de Monterrey, ITESM Campus
Estudios
Superiores
de
Ciudad dede
México.
(email:
dnovellar@itesm.mx)
∗ Instituto Tecnológico
Instituto Tecnológico
Estudios
Superiores
de Monterrey,
Monterrey, ITESM
ITESM Campus
Campus
∗∗
Ciudad de
dede
México.
(email:
dnovellar@itesm.mx)
Ciudad
México.
(email:
dnovellar@itesm.mx)
Instituto
Politécnico
Nacional,
IPN,
ESIME
Culhuacan,
Santa
Ana 1000,
Ciudad
de
México.
(email:
dnovellar@itesm.mx)
∗∗
∗∗ Instituto Politécnico Nacional, IPN, ESIME Culhuacan, Santa Ana 1000,
Nacional,
IPN,
Santa
C.P.
44300, Politécnico
México, (email:
bdelmuro@ipn.mx,
german.h.team@gmail.com,
∗∗ Instituto
Instituto
Politécnico
Nacional,
IPN, ESIME
ESIME Culhuacan,
Culhuacan,
Santa Ana
Ana 1000,
1000,
C.P.
44300,
México,
(email:
bdelmuro@ipn.mx,
german.h.team@gmail.com,
C.P.
44300,
México,
(email:
bdelmuro@ipn.mx,
german.h.team@gmail.com,
jfcomr23@yahoo.com.mx
C.P. 44300, México, (email:
bdelmuro@ipn.mx,
german.h.team@gmail.com,
jfcomr23@yahoo.com.mx
jfcomr23@yahoo.com.mx
jfcomr23@yahoo.com.mx
Abstract: The presented paper introduces a simple methodology to obtain a reduced-order approximaAbstract:
The
presented
simple
methodology
to
obtain
aa reduced-order
approximaAbstract:
The
paper
introduces
simple
to
approximation
to high-order
systemspaper
with introduces
oscillatingaaadynamics.
The reduction
is based
on the frequency
domain
Abstract:
The presented
presented
paper
introduces
simple methodology
methodology
to obtain
obtain
a reduced-order
reduced-order
approximation
to
high-order
systems
with
oscillating
dynamics.
The
reduction
is
based
on
the
frequency
domain
tion
to
high-order
systems
with
oscillating
dynamics.
The
reduction
is
based
on
the
frequency
domain
characteristics
of
the
under-damped
high-order
process.
Classic
control
theory
tools
are
used
in
order
tion
to
high-order
systems
with
oscillating
dynamics.
The
reduction
is
based
on
the
frequency
domain
characteristics
of
the
under-damped
high-order
process.
Classic
control
theory
tools
are
used
in
order
characteristics
of
the
under-damped
high-order
process.
Classic
control
theory
tools
are
used
in
order
to
obtain
an
accurate
model
reduction.
Time-delay
term
plays
an
important
role
at
the
low-order
model
characteristics
of the model
under-damped
high-order
process.
Classic
control
theory
tools
are
used inmodel
order
to
obtain
an
accurate
reduction.
Time-delay
term
plays
an
important
role
at
the
low-order
to
obtain
an
accurate
model
reduction.
Time-delay
term
plays
an
important
role
at
the
low-order
model
since
it
allows
to
hold
the
phase
characteristics
of
the
original
system.
Finally,
a
simple
PID
tuning
to
obtain
an
accurate
model
reduction.
Time-delay
term
plays
an
important
role
at
the
low-order
model
since
it
allowshas
to
hold
the
phaseincharacteristics
the
original
system.
Finally,
aa simple
PID
tuning
since
it
to
hold
the
of
the
original
Finally,
PID
methodology
order to obtainof
gains from
the reduced
Some
since
it allows
allowshas
to been
hold proposed
the phase
phaseincharacteristics
characteristics
ofthe
thecontroller
original system.
system.
Finally,
a simple
simplemodel.
PID tuning
tuning
methodology
been
proposed
order
to
obtain
the
controller
gains
from
the
reduced
model.
Some
methodology
has
been
proposed
in
order
to
obtain
the
controller
gains
from
the
reduced
model.
Some
numerical
simulations
are
presented
obtaining
a
similar
performance
when
the
PID
controller
applied
to
methodology
has beenare
proposed
in order
to obtain
the controller
gains
from
the
reduced
model.
Some
numerical
simulations
presented
obtaining
a
similar
performance
when
the
PID
controller
applied
to
numerical
simulations
are
presented
obtaining
a
similar
performance
when
the
PID
controller
applied
the
high-order
plant.
numerical
simulations
are presented obtaining a similar performance when the PID controller applied to
to
the
high-order
plant.
the
high-order
plant.
the
high-order
plant.
© 2017,
IFAC (International
Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Keywords: Time-invariant systems, Linear systems, Systems with time-delays, Process control,
Keywords:
Time-invariant
Keywords:
Time-invariant
systems, Linear
Linear systems,
systems, Systems
Systems with
with time-delays,
time-delays, Process
Process control,
control,
N-dimensional
systems systems,
Keywords:
Time-invariant
N-dimensional
systems systems, Linear systems, Systems with time-delays, Process control,
N-dimensional
systems
N-dimensional systems
1. INTRODUCTION
contrary of the half-rule presented in Skogestad (2001, 2003),
1. INTRODUCTION
INTRODUCTION
contrary
of the
the half-rule
half-rule presented
presented
in Skogestad
Skogestad
(2001,
2003),
1.
contrary
of
in
2003),
the
proposed
given here
is based on(2001,
a frequency
1. INTRODUCTION
contrary
of themethodology
half-rule presented
in Skogestad
(2001,
2003),
the
proposed
methodology
given
here
is
based
on
a
frequency
the
proposed
methodology
given
here
is
based
on
a
frequency
analysis,
taking intogiven
account
the
characteristics
above
the
proposed
methodology
here
is
based
on
a
frequency
Systems with under-damped characteristics appear in such domain
domain analysis,
analysis,
taking
into account
account
thecoefficients
characteristics
above
domain
taking
into
the
characteristics
above
mentioned;
natural
frequency,
damping
and
resoSystems
with
under-damped
characteristics
appear
in
such
Systems
with
under-damped
characteristics
appear
in
such
domain
analysis,
taking
into
account
the
characteristics
above
different
areas.
For instance mechanical
systems,
Gawronski
mentioned;
natural
frequency,
damping
coefficients
and
resoSystems
with
under-damped
characteristics
appear
in
such
mentioned;
natural
frequency,
damping
coefficients
and
resonance
frequency.
A
time-delay
term
is
added
in
order
to
condifferentnoise
areas.
For
instancevibration,
mechanical
systems,
Gawronski
different
areas.
instance
mechanical
Gawronski
naturalA frequency,
damping
coefficients
and
reso(2010),
andFor
structural
Maosystems,
and Pietrzko
(2013) mentioned;
nance
frequency.
time-delay
term
is
added
in
order
to
condifferent
areas.
For
instance
mechanical
systems,
Gawronski
nance
frequency.
A
time-delay
term
is
added
in
order
to
consider
the
phase
lag
due
to
the
high
order
dynamics.
The
main
(2010),
noise
and
structural
vibration,
Mao
and
Pietrzko
(2013)
(2010),
structural
Mao
and
Pietrzko
(2013)
frequency.
A due
time-delay
termorder
is added
in order
to main
conand
An noise
et al.and
(2013),
largevibration,
civil structures,
Spencer-Jr.
et al. nance
sider
the
phase
lag
to
the
high
dynamics.
The
(2010),
noise
and
structural
vibration,
Mao
and
Pietrzko
(2013)
sider
the
phase
lag
due
to
the
high
order
dynamics.
The
main
of the lag
approximation
is to order
designdynamics.
suitable controllers
and An
An Quinonero
et al.
al. (2013),
(2013),
large
civilThenozhi
structures,
Spencer-Jr.
et al.
al. objective
and
et
large
civil
structures,
Spencer-Jr.
et
sider
the
phase
due
to
the
high
The
main
(1998);
et
al.
(2012);
and
Yu
(2013).
Flexobjective
ofaccount
the approximation
approximation
is to
to design
suitable
controllers
and
An Quinonero
et al. (2013),
civilThenozhi
structures,
et al. objective
the
is
controllers
intoof
the reduced order
model suitable
in order to
simplify
(1998);
et al.
al.large
(2012);
andSpencer-Jr.
Yuthat
(2013).
Flex(1998);
Quinonero
et
Thenozhi
and
(2013).
objective
the approximation
is to design
design
suitable
controllers
ible
structures
in motion
have specific
features
are Flexnot a taking
taking
intoofaccount
account
thePID
reduced
order
model
inconsidered
order to
to
simplify
(1998);
Quinonero
et al. (2012);
(2012);
Thenozhi
and Yu
Yuthat
(2013).
Flextaking
into
the
reduced
order
model
in
order
the
controller
tuning.
controllers
will
be
to give
ible
structures
in
motion
have
specific
features
are
not
a
ible
structures
in
motion
specific
that
are
not
into account
thePID
reduced
order model
inconsidered
order to simplify
simplify
secret
to an engineer.
Onehave
of them
is thefeatures
resonance-strong
am-aa taking
the
controller
tuning.
controllers
will
be
to give
give
ible
structures
in
motion
have
specific
features
that
are
not
the
controller
tuning.
PID
controllers
will
be
considered
a
perspective
of
the
approximation
method
effectiveness.
secret
to
an
engineer.
One
of
them
is
the
resonance-strong
amsecret
to
an
One
is
the
resonance-strong
amthe
controller of
tuning.
PID controllers
will beeffectiveness.
considered to
to give
plification
ofengineer.
the motion
atof
a them
specific
frequency,
called natural
a
perspective
the
approximation
method
secret
to
an
engineer.
One
of
them
is
the
resonance-strong
ama
perspective
of
the
approximation
method
effectiveness.
plification of
of
the motion
motion
at aaofspecific
specific
frequency, called
called
natural
plification
the
at
frequency,
natural
frequency.
Another
feature
large mechanical
structures
is a perspective of the approximation method effectiveness.
plification
of
the motion
at aofspecific
frequency, called
natural
frequency.
Another
feature
large
mechanical
structures
is
2. BACKGROUND
frequency.
Another
feature
of
large
mechanical
structures
is
that
they
can
be
mathematically
represented
as
systems
with
frequency.
Another
feature
of
large
mechanical
structures
is
2. BACKGROUND
BACKGROUND
2.
that they
they conjugated
can be
be mathematically
mathematically
represented
as systems
systemsmodal
with
that
can
represented
as
with
complex
poles.
Their
real
parts
(representing
2. BACKGROUND
that
they conjugated
can be mathematically
represented
as systemsmodal
with
complex
poles.
Their
real
parts
(representing
complex
conjugated
poles.
Their
real
parts
(representing
modal
damping)
are typically
small,
and
their
distance
from the
ori- In the literature, there are different works dealing with the probcomplex
conjugated
poles.
Their
real
parts
(representing
modal
In the
theofliterature,
literature,
thereaare
are
different
works
dealing
withathe
the
probdamping)
are typically
typically
small,
and
their
distance
from the
the
ori- lem
In
there
different
with
probcharacterize
large
order works
systemdealing
by means
simpler
damping)
are
small,
and
their
distance
from
origin
is
the
natural
frequency
of
a
structure,
Gawronski
(2010).
In
theofliterature,
thereaare
different
works
dealing
withathe
probdamping)
are
typically
small,
and
their
distance
from
the
orilem
characterize
large
order
system
by
means
simpler
gin is
is the
the
natural
frequency
of
structure,
Gawronski
(2010).
lem
of
characterize
aa large
order
system
by
means
simpler
preserving
the important
features
of theaa original
gin
natural
frequency
aaa structure,
Then,
a large
mechanical
or of
civil
structure Gawronski
consists on (2010).
several description,
lem
of
characterize
large
order
system
by
means
simpler
gin
is
the
natural
frequency
of
structure,
Gawronski
(2010).
description,
preserving
themargins.
important
features
of the
the original
Then, aa large
large mechanical
or civil
civil structure
structure
consists
ondifferent
several systems,
description,
the
important
features
of
suchpreserving
as stability
There
exist methods
based
Then,
or
consists
several
under-damped
systems connected
them
or in aon
description,
preserving
themargins.
important
features
of the original
original
Then,
a large mechanical
mechanical
or civil among
structure
consists
ondifferent
several systems,
systems,
such
as
stability
There
exist
methods
based
under-damped
systems
connected
among
them
or
in
a
such
as
stability
margins.
There
exist
methods
on
truncated
balanced
realizations
Moore
(1981),
Singularbased
Valunder-damped
systems
connected
among
them
or
in
a
different
perspective,
a
high-order
system
with
n
pairs
of
complex
consystems,
such
as
stability
margins.
There
exist
methods
based
under-damped
systems connected
among
themoforcomplex
in a different
on
truncated
balanced
realizations
Moore
(1981),
Singular
Valperspective,
a
high-order
system
with
n
pairs
conon
truncated
balanced
realizations
Moore
(1981),
Singular
Values
decomposition,
Krylov
Antoulas
and
Sorensen
(2001),
freperspective,
a
high-order
system
with
n
pairs
of
complex
conjugated
and
stable
poles,
representing
a
challenge
for
the
design
on
truncated
balanced
realizations
Moore
(1981),
Singular
Valperspective,
a
high-order
system
with
n
pairs
of
complex
conues
decomposition,
Krylov
Antoulas
and
Sorensen
(2001),
frejugated
andcontrol
stable poles,
poles,
representing aa challenge
challenge for
for the
the design
design quency
ues
decomposition,
Krylov
Antoulas
and
Sorensen
(2001),
fredomain
methods
Sootla
(2013),
etc.
All
the
mentioned
jugated
and
stable
representing
of
suitable
laws.
ues
decomposition,
Krylov
Antoulas
andetc.
Sorensen
(2001),
frejugated
andcontrol
stable poles,
representing a challenge for the design quency
quency
domain
methods
Sootla
(2013),
All
the
mentioned
of
suitable
laws.
domain
methods
Sootla
(2013),
etc.
All
the
mentioned
methods
obtain amethods
finite dimensional
approximation.
A
different
of
suitable
control
laws.
quency
domain
Sootla
(2013),
etc.
All
the
mentioned
of
suitable
control
laws.
In Skogestad (2001, 2003), first and second-order delayed ap- approach
methods obtain
obtain
finite dimensional
dimensional
approximation.
A different
different
methods
finite
approximation.
A
is the aaamethod
developed by
Skogestad (2003,
2001),
methods
obtain
finite dimensional
approximation.
A different
In Skogestad
Skogestad (2001,
(2001,
2003),
first
and second-order
second-order
delayed
apIn
and
proximations
of high2003),
order first
systems
are proposed delayed
in orderapto which
approach
is the
the method
method
developed
by
Skogestad
(2003,
2001),
approach
is
developed
by
Skogestad
(2003,
2001),
approximate
a
high
order
system
by
a
first
or
second
In
Skogestad
(2001,
2003),
first
and
second-order
delayed
apis the method
developed
by Skogestad
(2003,
2001),
proximations
of high
high order
order
systems
are
proposed
in order
order
to approach
proximations
of
systems
are
proposed
in
to
derive
PID
controllers
by
means
of
a
simple
analytically
prowhich
approximate
a
high
order
system
by
a
first
or
second
which
approximate
aa high
order
system
aa first
or
second
system
plus time-delay,
in order
to by
simplify
the
design
proximations
of high order
systems
are
proposed
in order
to order
which
approximate
high
order
system
by
first
or
second
derive
PID
controllers
by
means
of
a
simple
analytically
proderive
PID
controllers
by
of
analytically
cedure,
yielding
to acceptable
results
in the closed
loop proper- of
order
system
plus time-delay,
time-delay,
in methodology
order to
to simplify
the design
design
order
plus
in
order
the
PIDsystem
controllers,
the empirical
of Skogestad
is
derive
controllers
by means
means
of aa simple
simple
analytically
order
system
plus time-delay,
in methodology
order to simplify
simplify
the design
cedure,PID
yielding
to
acceptable
results
insystem.
the closed
closed
loop propercedure,
yielding
to
acceptable
results
in
the
loop
performance
for
the
high-order
closed
loop
However,
this
of
PID
controllers,
the
empirical
of
Skogestad
is
of
PID
controllers,
the
empirical
methodology
of
Skogestad
is
based
on
the
time
response
of
the
open-loop
high
order
system
cedure,
yielding
to
acceptable
results
in
the
closed
loop
perof
PID
controllers,
the
empirical
methodology
of
Skogestad
is
formance for
for
thebeen
high-order
closedforloop
loop
system. However,
However,
this based on the time response of the open-loop high order system
formance
the
high-order
closed
system.
this
procedure
have
developed
high-order
systems conbased
on
the
time
response
of
the
open-loop
high
order
system
and
is
briefly
presented
below.
formance
for
the
high-order
closed
loop
system.
However,
this
based
on
the
time
response
of
the
open-loop
high
order
system
procedure
have
been
developed
for
high-order
systems
conprocedure
have
been
developed
for
systems
sidering
only
simple
poles. Therefore,
the aforementioned
and is
is briefly presented
presented below.
procedure
have
beenreal
developed
for high-order
high-order
systems concon- and
and is briefly
briefly presented below.
below.
sidering
only
simple
real
poles.
Therefore,
the aforementioned
aforementioned
sidering
only
simple
real
poles.
Therefore,
the
work
does
not
consider
the
case
of
oscillating
processes,
i.e.
sidering
only
simple
real the
poles.
Therefore,
the aforementioned
2.1 Half Rule Method
work
does
not
consider
case
of
oscillating
processes,
i.e.
work
does
not
consider
the
case
of
oscillating
processes,
i.e.
large
mechanical
systemsthe
andcase
civilof
structures
hasprocesses,
not been con2.1 Half
Half Rule Method
Method
work
does not consider
oscillating
i.e. 2.1
large mechanical
mechanical
systems and
and civil
civil structures
structures
has not
not been
been concon2.1 Half Rule
Rule Method
large
systems
has
sidered.
large
mechanical
systems
and
civil
structures
has
not
been
conThe half rule method approximates a high order system to
sidered.
sidered.
The
half rule
rule method
methodplus
approximates
highmethod
order system
system
to
sidered.
half
approximates
high
order
to
The
aim of this work is to present a second order approxima- aThe
first-order
time delay. aaaThe
states the
half rulesystem
methodplus
approximates
highmethod
order system
to
The with
aim of
of
this work
work term
is to
to for
present
a second
second
order approximaapproximaaThe
first-order
system
time delay.
delay.
The
states the
the
The
aim
this
is
present
a
order
a
first-order
system
plus
time
The
method
states
tion
a
time-delay
high-order
oscillating
structures
following:
the
largest
neglected
time
constant
is
distributed
to
The
aim of
this work term
is to for
present
a second
order approximaafollowing:
first-orderthesystem
plus
time delay.
The method
states the
tion
with
a
time-delay
high-order
oscillating
structures
largest
neglected
time
constant
is
distributed
to
tion
with
aa time-delay
term
oscillating
following:
the
largest
time
constant
is
to
(typical
on mechanical
and civil engineering).
On the the
effective
andneglected
the smallest
constant
retained by
tion
withmodels
time-delay
term for
for high-order
high-order
oscillating structures
structures
following:
thedelay
largest
timetime
constant
is distributed
distributed
to
(typical
models
on mechanical
mechanical
and civil
civil engineering).
engineering).
On the
the the
the effective
effective
delay
andneglected
the smallest
smallest
time
constant
retained by
by
(typical
models
on
and
On
delay
and
the
time
constant
retained
(typical models on mechanical and civil engineering). On the the effective delay and the smallest time constant retained by
Copyright
1348Hosting by Elsevier Ltd. All rights reserved.
2405-8963 ©
© 2017
2017, IFAC
IFAC (International Federation of Automatic Control)
Copyright
©
2017
1348
Copyright
© under
2017 IFAC
IFAC
1348Control.
Peer
review
responsibility
of
International
Federation
of
Automatic
Copyright © 2017 IFAC
1348
10.1016/j.ifacol.2017.08.127
Proceedings of the 20th IFAC World Congress
Toulouse, France, July 9-14, 2017 David F. Novella-Rodríguez et al. / IFAC PapersOnLine 50-1 (2017) 1316–1321
the approximated time constant, Skogestad (2003, 2001). Let
the original system be:
G(s) =
j
−Tjinv + 1
i (τi0 s
+ 1)
e θ0 s ,
where the lags τi0 are ordered according to their magnitude, and
inv
Tj0
> 0 denote the inverse response time constants, (negative
numerator of the original system). Then, according to the half
rule method a first-order model can be obtained as:
1317
term. At this frequency, the resonance peak takes place, namely
the maximum magnitude of the frequency domain analysis. The
resonance peak is directly associated to the damping coefficient
of an under-damped system. Finally, the approximation of a
high order system by a reduced order systems have a significant difference in the phase characteristics. Every couple of
complex conjugated poles adds a lag of 180◦ on the high order
frequencies region. In order to compensate the mentioned phase
difference, a time-delay is introduced to the approximation.
Then, for the high-order system given by (1), it is proposed the
second order process with time delay given by:
−θs
ˆ = e
,
G(s)
τ1 s + 1
GLO (s) =
where:
(s2
K̄e−τ s
,
+ 2ζωn s + ωn2 )
(2)
Bode Diagram
50
i≥3
−50
−100
j
Phase (deg)
−180
θ
r
−360
−540
−2
10
−1
0
10
10
Frequency (rad/s)
1
10
2
10
Fig. 1. Oscillating system frequency analysis
3. APPROXIMATION FOR OSCILLATING SYSTEMS
The aim of this work is to compute a descriptive approximation
for the high order system:
m
K j=1 (s2 + 2ζj ωnj s + ωn2 j )
GHO (s) = n
,
2
2
i=1 (s + 2ζi ωni s + ωni )
ωr
−150
0
Is it worth to stress that the half rule approximation is based
on an empirical method. Moreover, the approach has been
developed for typical process control applications. Unstable
processes have not been considered, with the exception of integrating processes. Oscillating processes (with complex poles
or zeros) have also not been considered.
Based on these three terms it is possible to find the parameters
of the system approximation (2), namely the damping coefficient ζ, the natural frequency ωn and the time delay τ . Below,
a procedure to compute the reduced model is provided.
4. REDUCTION PROCEDURE
(1)
where m < n, ζi is the damping coefficient and ωni is the natural frequency of the corresponding under damped subsystem.
For this purpose, it is possible to consider the frequency response of the high order system. The chosen tool to perform
the reduction model is the Bode plot of the high-order system. Taking into account that the analyzed systems have oscillating modes, there are three important characteristics of the
frequency domain analysis to point out:
• Resonance frequency ωr ,
• Resonance peak Mr ,
• Phase margin θr .
Mr
0
Magnitude (dB)
and
τ1 = τ10 + τ20
inv
θ = θ0 +
τi0 +
Tj0
.
The movement of mechanical or civil structure depends on
several factors like the amplitude and other features such as
external perturbation motion, the dynamic properties of the
structure, the characteristics of the materials of the structure and
its foundation (basestructure interaction). A mechanical and/or
civil structure will have multiple natural frequencies, which are
equal to its number of Degree of Freedom (DOF), Thenozhi and
Yu (2013). If the frequency of the ground motion is close to the
natural frequency of the building, the resonance occurs. As a
result, the floors may move rigorously in different directions
causing inter-story drift, the relative translational displacement
between two consecutive floors. For this reason, the reduced
model should consider the resonance frequency as an important
Taking into account the parameters of the high-order system,
the procedure to compute the reduction parameters will be
presented in this section.
4.1 Damping coefficient ζ
From the Bode diagram of the high-order open-loop system it is
possible to compute the resonance peak, namely, the maximum
value of the magnitude plot in [dB]. Since the purpose of the
paper is to deal with oscillatory systems, i.e. systems with
complex conjugated poles, the magnitude Bode diagram always
presents a resonance peak for some frequency ωr .
Considering the frequency domain characteristics of an oscillatory system, it is possible to found the relation between the
resonance peak of the high order system and the damping coefficient of a second order process with the following equation
(or expressed graphically in Fig. 2):
Mr =
4.2 Natural frequency ωn
2ζ
1
1 − ζ2
(3)
As a second step it is possible to relate the resonant frequency of
the high order system with the natural frequency of the reducedorder system by means of the following equation:
1349
Proceedings of the 20th IFAC World Congress
1318
Toulouse,
France, July 9-14, 2017 David F. Novella-Rodríguez et al. / IFAC PapersOnLine 50-1 (2017) 1316–1321
4.5 Example
30
Let us consider a high order system with oscillating modes
given by the following transfer function:
Resonance peak Mr
25
20
GHO (s) =
15
(7)
This system can be separate as four under-damped subsystems
with the following parameters:
10
•
•
•
•
5
0
0.1
0.2
0.3
0.4
0.5
Damping coefficient ζ
0.6
0.7
0.8
Fig. 2. Relation: Damping coefficient and resonance peak.
ωr
,
ωn = 1 − 2ζ 2
(4)
using the damping coefficient computed before for the second
order model. As the damping ratio ζ approaches to zero, the
natural frequency of the reduction approaches to the resonant
frequency of the original system.
4.3 Proportional gain K̄
Once the parameters of the reduced model have been calculated, the value corresponding to the proportional gain should
be determined taking into account the DC gains of the high
order system and the reduced model, as follows:
K̄ = K
limω→0 GHO (jω)
.
limω→0 GLO (jω)
(5)
4.4 Time delay τ
Finally, a time delay is introduced to the reduced model in
order to compensate the phase difference due to the model
reduction. As a parameter of compensation, the gain margin
of the high order system is considered, then, the crossover
frequency ωgm is taking into account for this purpose (see Fig.
3 ), the difference between the phase of the reduced model
φLO (ωgm ) and the phase original model φHO(ωgm ) at this
frequency is related with the delay as follows:
φHO − φLO
τ=
57.3ωgm
= 0.5
= 0.6
= 0.65
= 0.4
ζ1
ζ2
ζ3
ζ4
ωn1 = 1.1180,
ωn2 = 2.2361,
ωn3 = 3.1623,
ωn4 = 1.4142,
Then, we proceed to compute the model reduction, namely a
second order under-damped system plus time delay. First, from
the frequency domain of the high-order system we can obtain
the resonance peak Mr = 3.66dB, which is equivalent to have
a damping coefficient ζ = 0.3513 and this peak occurs at the
resonance frequency ωr = 1.37rad/s. Taking into account the
previous values, it is possible to obtain the natural frequency
of the reduced model by means the equation (4), then, ωn =
1.0946. Finally a proportional constant of the system is given
by K = 1.198. Then we have a delay free reduced model given
by:
G(s) =
1.198
.
s2 + 0.7691s + 1.198
(8)
Comparing the phase plot of the Bode diagram of the original
system and the reduced model, the phase margin of the original
systems takes place at a frequency ω = 1.38rad/s. At this
frequency, the high order model has a phase of −284.5, and
for the reduced model the phase at this frequency is −123.62.
To compensate this pase difference it is necessary to introduce
a time delay, using the equation (6) we compute a delay τ =
2.0428.
In order to compare the frequency and time characteristics of
the original high-order systems and the model reduction, Fig. 4
shows the Bode diagram of both systems and Fig. 5 shows the
time response to a step input and a sinusoidal input respectively.
Bode Diagram
Gm = -3.64 dB (at 0.953 rad/s) , Pm = -97.7 deg (at 1.34 rad/s)
10
0
Magnitude (dB)
0
125
s 8 + a 7 s 7 + a 6 s 6 + a 5 s 5 + a 4 s4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0
(6)
-10
-20
-30
High-Order System
Second order plus time-delay system
-40
0
-90
Phase (deg)
whit the phase given in degrees.
-180
-270
-360
10 -1
High-order system
Reduced model
25
20
Magnitude (dB)
10 0
Frequency (rad/s)
30
Fig. 4. Frequency response: delayed system
15
10
5
0
-5
5. PID CONTROLLER TUNING
-10
0
Phase (deg)
-90
φ LO
-180
Phase to be
compensated
-270
φ HO
-360
3
4
Fig. 3. Phase compensation
5
Frequency
(rad/s)
6
7
8
9
10
11
12
13
Once a model reduction is obtained, a procedure to obtain
the parameters of a PID controller is proposed. Following
the methodology proposed by Skogestad Skogestad (2003);
Skogestad and Grimholt (2012), an Internal Model Control
approach is used to derive the controller gains based on the
reduced model. Let us consider the closed loop system given
in Fig. 6.
1350
Proceedings of the 20th IFAC World Congress
Toulouse, France, July 9-14, 2017 David F. Novella-Rodríguez et al. / IFAC PapersOnLine 50-1 (2017) 1316–1321
2
ωnd
,
K̄
ω2
KP = nd 2ζωn ,
K̄
2
ωnd
KI =
ω2 ,
K̄ n
KD =
Step Response
Amplitude
1.5
1
0.5
High-Order System
Second Order System plus Time-Delay
0
0
2
4
6
1319
8
10
12
14
16
Time (seconds)
Linear Simulation Results
2
(15)
Amplitude
1
and the first-order filter has a pole located at s = −(2ζd ωnd +
τ ).
0
-1
-2
0
2
4
6
8
10
12
14
16
18
20
Time (seconds)
5.1 Example
Fig. 5. Time response
The PID controller is given for the following equation:
CP ID (s) = KP +
KI
+ KD s.
s
(9)
The closed loop transfer function is:
Y (s)
GLO (s)C(s)
=
.
R(s)
1 + GLO (s)C(s)
(10)
Let us consider the low-order retarded approximation computed
in Section 4.5, given by the following transfer function:
1.198
e−2.0428s ,
G(s) = 2
s + 0.7691s + 1.198
the controller parameters are computed following the procedure
given in the previous section. An important issue on this work,
is to compare the response of the closed loop approximated
reduced system and the original high order system, both controlled by means of the PID tuning methodology presented previously. Fig. 7 provides such comparison. The continuous line
shows the closed loop performance of the PID controller used
for the original system. The dashed line indicates the closed
loop response of the low order approximation.
2.5
Original system
Reduced system
2
Fig. 6. Closed Loop System
In order to obtain the parameters of the controller, it follows
to specify a desired closed loop response, for this purpose, we
consider a second order desired response:
Y (s)
R(s)
d
1.5
1
2
ωnd
−τ s
= 2
,
2 e
s + 2ζd ωnd s + ωnd
(11)
0.5
note that the delay term is considered in the desired closed loop
response due to it is not a negligible factor. Then, considering
the closed loop system (10), and solving for the controller C(s)
we have:
C(s) =
1
GLO (s)
R(s)
Y (s)
1
(12)
,
d
−1
0
0
10
20
30
40
50
Fig. 7. Controller performance comparison
5.2 Application Example
Substituting (2) and (11) into (12), we obtain the following
infinite dimensional expression:
2
Let consider the mechanical systems shown in Fig. 8, with a
transfer function relating the input force F (s) and the position
X2 (s) corresponding to the mass m2 given by:
To obtain the controller parameters it is possible to consider
a first-order Taylor series approximation of the delay term in
equation (13), giving the following expression:
b1 s + k 1
X2 (s)
=
,
F (s)
m1 m2 s4 + αs3 + βs2 + (b1 k1 + b2 k1 )s + k1 k2
(16)
C(s) =
2
s2 + 2ζωn + ωn
K̄
C(s) =
ωnd
2 − ω 2 e−τ s
s2 + 2ζd ωnd s + ωnd
nd
2
s2 + 2ζωn + ωn
sK̄
2
ωnd
s + 2ζd ωnd + τ
,
.
(13)
(14)
it can be noted that C(s) consists in a PID in series with a firstorder filter, the PID controller gains are computed as follows:
with α = m1 (b1 + b2 ) + m2 b1 and β = m1 k1 + m2 (k1 +
k2 ) + b2 b1 . The system parameters are: m1 = m2 = 1kg,
k1 = k2 = 1N/m, b1 = 0.65N s/m and b2 = 1N s/m, the
example is analyzed in Nechak et al. (2015) with a different
reduction method. Applying the method described previously,
1351
Proceedings of the 20th IFAC World Congress
1320
Toulouse, France, July 9-14, 2017 David F. Novella-Rodríguez et al. / IFAC PapersOnLine 50-1 (2017) 1316–1321
a second order approximation is computed, obtaining the following result:
5.3 Disturbance Rejection
Disturbance rejection is one of the main issues of concern in
mechanical and civil engineering. As it can be noted in Fig.
7, the proposed PID control structure is not able to deal with
this problem. In order to obtain a satisfactory performance with
respect to the disturbance rejection, the feed-forward control
structure, shown in Fig. 11, could be used, Visioli (2006).
0.3853
GLO (s) = 2
e−0.319s
s + 0.3129s + 0.3853
Fig. 8. Mass spring damper fourth order system
Fig. 11. Feedforward action for load disturbance rejection task.
Step Response
Amplitude
1.5
Taking into account the model reduction the proposed feedforward controller is:
1
0.5
0
High-Order System
Approximation
0
5
10
15
20
25
30
35
Linear Simulation Results
and
Amplitude
4
T (s) =
2
0
-2
0
5
10
15
20
25
2
ωnh
−τ s
,
2 e
+ 2ζh ωnh s + ωnh
(17)
2
s2 + 2ζωn s + ωn2
K¯h ωnh
2 ,
2
K̄ωn2 s + 2ζh ωnh s + ωnh
(18)
H(s) = K¯h
40
Time (seconds)
30
Time (seconds)
Fig. 9. Response of the open-loop system to a step and a
sinusoidal input
The comparison of the open-loop response of the original
system and the reduced model are presented in Fig. 9. As it
can be seen the response for a step response (Fig. 9 up) is
similar for both systems. For a sinusoidal input oscillating to
the resonance frequency sin(ωr t), (Fig. 9, down). The PID
controller is designed in order to regulate the output X2 (s),
considering the tuning methods given in Section 5. The tuning
parameters are ωnd = 4ωn , with ωn being the reduction natural
frequency, and ζd = 1.2. Under these conditions, the closed
loop performance shown in the Fig. 10.
s2
The feedforward controller gains are: ζh = 2ζ, ωnh = 2ωn
and K¯h = 0.1 The obtained response are shown in Fig.
12. Dashed line indicates the closed-loop performance for the
control system shown in Fig. (6), it can be seen a large overshot
due to an unitary input disturbance introduced in the instant
t = 50s. On the other hand, the response of the system with
the feed-forward controller shown in Fig. 11 is denoted by the
solid line, where the effect of the disturbance is attenuated. Both
simulations were performed considering the high order system
(16).
2
1.8
1.6
1.4
1.2
1
1.2
0.8
0.6
1
0.4
0.8
PID controller
Feedforward PID controller
0.2
0
0.6
0
10
20
30
40
50
60
70
80
90
100
0.4
Fig. 12. Closed-loop response: Feedforward effect on the disturbance rejection task.
0.2
0
−0.2
Original system
Reduced system
0
20
40
Fig. 10. Closed loop Performance
60
80
100
5.4 Evaluation of the reduction model with different tuning
control methods
In order to validate the reduced order model, it is evaluated
using different tuning methods for PID controllers found in
1352
Proceedings of the 20th IFAC World Congress
Toulouse, France, July 9-14, 2017 David F. Novella-Rodríguez et al. / IFAC PapersOnLine 50-1 (2017) 1316–1321
the available literature O’Dwyer (2009). The PID controllers
were computed taking into account the second order reduced
model plus time delay obtained in the example given in Section
5.2 (except in the case of the Ziegler-Nichols method), the
tuning rules considered have been taken from different authors,
Schaedel (1997); Rivera and Jun (2000); Huang et al. (2005);
Jahanmiri and Fallahi (1997). The computed controller is applied to the original high order system.
In order to numerically evaluate the output control performance, it is possible to compute the Integral absolute error
(IAE) of the error signal, e(t) = r(t) − y(t):
IAE =
0
∞
|e(t)|dt,
Fig. 13 shows the obtained results in terms of the performance
index (IAE) using different tuning methods available in the
literature for second order systems, and applying the obtainedd
controller to the original high order system.
10
1. Proposed Method
2. Shaedel
3. Rivera and Jun
4. Ziegler−Nichols
5. Huang et al
6. Jahanmiri and Fallali
9
Integral Absolute Error IAE
8
7
6
5
4
3
2
1
0
1
2
3
4
Tuning Method
5
6
Fig. 13. Performance Index for different Tuning Methods
6. CONCLUSIONS
The main contribution of this paper is to propose a reduced
order model for high order systems with oscillating dynamics.
This class of systems appears commonly in the modeling of
large mechanical process or civil structures. The reduction
methodology is simple but it takes into account frequency
domain characteristics of the high order original system and
uses them to obtain an accurate model reduction. In order to
match the phase characteristics of the high order systems and
the low order simplification a time delay is introduced to the
model reduction.
The reduction of the model parameters allows us to simplify the
tuning of PID controllers for the system. Following the IMC
tuning rules it is possible to synthesize PID controller parameters in a simple and efficient way and use this controllers in the
original high order process. Numerical simulations show that
the controllers computed for the model reductions work with
an acceptable performance for the high order plant. Moreover,
different tuning strategies found in the current literature for
second order systems plus time-delay are tested numerically
on the high order system, obtaining satisfactory closed loop
performance.
1321
REFERENCES
An, F., Chen, W.D., and Shao, M.Q. (2013). Study on discrete
acceleration feedback control with time delay. Journal of
Vibration and Control, Online version, 1–19.
Antoulas, A.C. and Sorensen, D.C. (2001). Approximation of
large-scale dynamical systems: An overview. Int. J. Appl.
Math. Comput. Sci., 11(5), 1093–1121.
Gawronski, W.K. (2010). Advanced Structural Dynamics and
Active Control of Structures. Mechanical Engineering Series.
Springer.
Huang, H.P., Jeng, J.C., and Luo, K.Y. (2005). Auto-tune
system using single-run relay feedback test and model-based
controller design. Journal of Process Control, 15(6), 713 –
727.
Jahanmiri, A. and Fallahi, H. (1997). New methods for process
identification and design of feedback controller. Chemical
Engineering Research and Design, 75(5), 519 – 522.
Mao, Q. and Pietrzko, S. (2013). Control of Noise and Structural Vibration: A MATLAB-Based Approach. Springer.
Moore, B. (1981). Principal component analysis in linear
systems: Controllability, observability, and model reduction.
Automatic Control, IEEE Transactions on, 26(1), 17–32. doi:
10.1109/TAC.1981.1102568.
Nechak, L., Raynaud, H.F., and Kulcsár, C. (2015). Model
order reduction of random parameter-dependent linear systems. Automatica, 55, 95 – 107.
O’Dwyer, A. (2009). Handbook of PI and PID Controller
Tuning Rules. Imperial College Press.
Quinonero, F.P., Rubió-Massegú, J., Rossell, J., and Karimi,
H. (2012). Semiactive-passive structural vibration control
strategy for adjacent structures under seismic excitation.
Journal of the Franklin Institute, (349).
Rivera, D.E. and Jun, K.S. (2000). An integrated identification
and control design methodology for multivariable process
system applications. IEEE Control Systems, 20(3), 25–37.
Schaedel, H.M. (1997). A new method of direct PID controller
design based on the principle of cascaded damping ratios. In
1997 European Control Conference (ECC), 1265–1271.
Skogestad, S. (2003). Simple analytic rules for model reduction
and PID controller tunning. Journal of Process Control, 13,
291–309.
Skogestad, S. (2001). Probably the best simple PID tuning rules
in the world. In AIChE Annual Meeting. Reno, Nevada.
Skogestad, S. and Grimholt, C. (2012). PID Control in the
Third Millennium: Lessons Learned and New Approaches,
chapter The SIMC Method for Smooth PID Controller Tuning. Advances in Industrial Control. Springer.
Sootla, A. (2013). Semidefinite hankel-type model reduction
based on frequency response matching. Automatic Control,
IEEE Transactions on, 58(4), 1057–1062.
Spencer-Jr., B.F., Dyke, S.J., and Deoskar, H.S. (1998). Benchmark problems in structural control: Part i active mass driver
system. Earthquake Engineering and Structural Dynamics,
27, 1127–1139.
Thenozhi, S. and Yu, W. (2013). Advances in modeling and
vibration control of building structures. Annual Reviews in
Control, 37, 346–364.
Visioli, A. (2006). PID Control in the Third Millennium.
Lessons Learned and New Approaches. Advances in Industrial Control. Springer.
1353