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IFAC PapersOnLine 53-1 (2020) 195–200
New
New
New
New
tuning
strategy
for
series
cascade
tuning
strategy
for
series
cascade
tuning
strategy
for
series
cascade
control
structure
tuning
strategy
for series cascade
control
structure
control structure
control
structure ∗∗
Md. Imran Kalim ∗∗ Ahmad Ali ∗∗
Md. Imran Kalim ∗∗ Ahmad Ali ∗∗
Md. Imran Kalim Ahmad Ali ∗∗
Md.
Imran
Kalim ∗ Ahmad
Ali ∗∗ of Technology
∗
of
Electrical
Engineering,
Indian
Institute
∗ Department
Department
of
Electrical
Engineering,
Indian
Institute
of Technology
∗
∗
Department
of Electrical
Engineering,
Indian Institute of Technology
Patna,
Bihta-801106
India
(e-mail:
imran2025kalim@rediffmail.com).
Patna,
Bihta-801106
India
(e-mail:
imran2025kalim@rediffmail.com).
∗
∗∗
Department
of
Electrical
Indian
Institute of
Technology
Department
of
Engineering,
Indian
Patna,
Bihta-801106
IndiaEngineering,
(e-mail: imran2025kalim@rediffmail.com).
∗∗
Department
of Electrical
Electrical
Indian Institute
Institute of
of Technology
Technology
∗∗
∗∗
Patna,
Bihta-801106
India Engineering,
(e-mail:
imran2025kalim@rediffmail.com).
Patna,
Bihta-801106
India
(e-mail:
ali@iitp.ac.in).
Department
of
Electrical
Engineering,
Indian
Institute
of
Technology
Patna,
Bihta-801106
India
(e-mail:
ali@iitp.ac.in).
∗∗
Department
of Bihta-801106
Electrical Engineering,
Indian
Institute of Technology
Patna,
India (e-mail:
ali@iitp.ac.in).
Patna, Bihta-801106 India (e-mail: ali@iitp.ac.in).
Abstract:
Abstract: In
In the
the presence
presence of
of disturbances
disturbances associated
associated with
with the
the manipulated
manipulated variable,
variable, perforperforAbstract:
In
the presence
of disturbances
associated
with with
the manipulated
variable,
performance
of
a
single
loop
control
structure
can
be
improved
the
help
of
a
series
cascade
mance
of
a
single
loop
control
structure
can
be
improved
with
the
help
of
a
series
cascade
Abstract:
In
the approach
presence
of disturbances
associated
with
theismanipulated
variable,
perforcontrol.
A
design
for
a
series
cascade
control
system
proposed
in
this
paper.
The
mance
of
a
single
loop
control
structure
can
be
improved
with
the
help
of
a
series
cascade
control.
design
for astructure
series cascade
system
is the
proposed
in athis
paper.
The
mance
ofA
acontroller
singleapproach
loop
control
cancentroid
becontrol
improved
with
help of
series
cascade
secondary
is
tuned
by
finding
the
of
the
convex
stability
region
to
quickly
control.
A
design
approach
for
a
series
cascade
control
system
is
proposed
in
this
paper.
The
secondary
controller
is
tuned
by
finding
the
centroid
of
the
convex
stability
region
to
quickly
control.
Athe
design
approach
forbya finding
series
cascade
control
is proposed
inregion
this paper.
The
secondary
controller
is tuned
the centroid
of system
the
convex
stability
tovariable,
quickly
attenuate
disturbances
arising
in
the
inner
loop
before
they
influence
the
controlled
attenuate
the
disturbances
arising
in the inner
loop before
theyconvex
influence
the controlled
variable,
secondary
controller
is
tuned
by
finding
the
centroid
of
the
stability
region
to
quickly
attenuate
the
disturbances
arising
in
the
inner
loop
before
they
influence
the
controlled
variable,
whereas
the
primary controller
is
tuned
by
optimum
criterion
to achieve
whereas
the
controller
is in
tuned
by magnitude
magnitude
optimum
criterion
achieve improved
improved
attenuate
theprimary
disturbances
the inner
loop
before
they influence
theto
variable,
servo
response
in
the
outerarising
loop.
Servo
andmagnitude
regulatory
responses
for the
the
nominal
model,
as
whereas
the
primary
controller
is tuned
by
optimum
criterion
tocontrolled
achieve model,
improved
servo
response
in
the
outer
loop.
Servo
and
regulatory
responses
for
nominal
as
whereas
the
primary
controller
is
tuned
by
magnitude
optimum
criterion
to
achieve
improved
well
as
for
the
perturbed
model
of
the
plant
obtained
by
the
proposed
method
are
compared
servo
response
in
the
outer
loop.
Servo
and
regulatory
responses
for
the
nominal
model,
as
well
asresponse
for the perturbed
model
the plant
the proposed
method
are model,
compared
servo
in the
outer
loop.of
and obtained
regulatoryby
responses
forthat
the
nominal
as
well as
for recently
the perturbed
model
ofServo
the plant
obtained
byresults
the proposed
method
are compared
with
some
reported
tuning
strategies.
Simulation
show
the
proposed
tuning
with
some
recently
reported
tuning
strategies.
Simulation
results
show
that
the
proposed
tuning
well
as
for
the
perturbed
model
of
the
plant
obtained
by
the
proposed
method
are
compared
with
some
recently
reported
tuning
strategies.
Simulation
results
show
that
the
proposed
tuning
strategy
yields
improved
closed-loop
performance.
strategy
yields
improved
closed-loop
performance.
with
some
recently
reported
tuning strategies.
Simulation results show that the proposed tuning
strategy
yields
improved
closed-loop
performance.
© 2020, IFAC
(International
Federation ofperformance.
Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
strategy
yields
improved closed-loop
Keywords:
Keywords: Cascade
Cascade control,
control, PID
PID controller,
controller, All
All stabilizing
stabilizing controller,
controller, Magnitude
Magnitude optimum
optimum
Keywords: Cascade control, PID controller, All stabilizing controller, Magnitude optimum
criterion.
criterion.
Keywords: Cascade control, PID controller, All stabilizing controller, Magnitude optimum
criterion.
criterion.
1. INTRODUCTION
totuning
totuning has
has been
been applied
applied to
to the
the two
two loops
loops sequentially.
sequentially.
1.
1. INTRODUCTION
INTRODUCTION
Another
way
of
design
is
simultaneous
tuning
approach
totuning
has
been
applied
to
the
two
loops
sequentially.
Another
way
of
design
is
simultaneous
tuning
approach
1. INTRODUCTION
totuning
has
been
applied
to are
the tuned
two loops
sequentially.
Another
way
of
design
is simultaneous
tuning
approach
in
which
the
two
controllers
concurrently.
In
in
which
the
two
controllers
are
tuned
concurrently.
In
In
conventional
single
loop
control
structure,
disturbance
Another
way
of
design
is
simultaneous
tuning
approach
in
which
the
two
controllers
are
tuned
concurrently.
In
Jeng
(2014),
the
controllers
are
tuned
using
step
response
In
conventional
single
loop
control
structure,
disturbance
Jeng
(2014),
the
controllers
are
tuned
using
step
response
In
conventional
single
loop
control
structure,
disturbance
rejection
does
not
take
place
until
the
system’s
output
dein
which
thethe
twocontrollers
controllers
are
tuned
concurrently.
In
data
without
considering
the
process
model.
The
aim
rejection
does
not
take
place
until
the
system’s
output
deJeng
(2014),
are
tuned
using
step
response
In conventional
single
control
structure,
withoutthe
considering
the
model.step
The
aim is
is
rejection
does
take loop
place
until
the
system’s
output
de- data
viates
from
thenot
set-point.
That
is why
why
Franks disturbance
and
Worley
Jeng
(2014),
controllers
areprocess
tuned using
response
to
obtain
the
controller
parameters
such
that
the
two
viates
from
the
set-point.
That
is
Franks
and
Worley
data
without
considering
the
process
model.
The
aim
is
rejection
does
not
take
place
until
the
system’s
output
deto
obtain
the
controller
parameters
such
that
the
two
viates
from
the
set-point.
That
is
why
Franks
and
Worley
proposed aa cascade
cascade control
control scheme
scheme (Franks
(Franks and
and Worley to
data
without
the process
model.
Thethe
aim
is
obtain
the considering
controller
parameters
such
that
two
loops
behave
as
close
as
possible
to
the
assumed
referproposed
viates from
set-point.
That
is why(Franks
Franks
and cascade
behave
ascontroller
close as parameters
possible to such
the assumed
referproposed
a the
cascade
control
scheme
and
Worley loops
(1956)).
Fig.
1
shows
the
block
diagram
of
a
series
to
obtain
the
that
the
two
loops
behave
as
close
as
possible
to
the
assumed
reference
models
for
the
two
loops.
In
Veronesi
and
Visioli
(1956)).
Fig.
1
the
block
diagram
of
a
proposed
a cascade
control
scheme
(Franks
and cascade
Worley
ence models
for
the two
loops. IntoVeronesi
and Visioli
(1956)).
Fig.
1 shows
shows
theand
block
diagram
of
a series
series
cascade
control
scheme.
Gp1 (s)
G
(s)
the
funcbehave
as close
asprimary
possible
the assumed
refer(2011),
parameters
of
and
secondary
processes
and
Gp2
(s) are
are of
thea transfer
transfer
func- loops
control
scheme.
G
ence
models
for
the
two
loops. In
and
Visioli
p1 (s)
p2
(1956)).
Fig.
1
shows
the
block
diagram
series
cascade
(2011),
parameters
of the
the
primary
andVeronesi
secondary
processes
control
scheme.
G
p1 (s)
p2processes
tions
of
primary
and
secondary
whereas
primary
and
G
(s)
are
the
transfer
funcence
models
for
the
two
loops.
In
Veronesi
and
Visioli
p1
p2
were
first
estimated
by
evaluating
a
step
response
of
tions
of
primary
and
secondary
processes
whereas
primary
(2011),
parameters
of
the
primary
and
secondary
processes
control
G
andare
Gp2
(s) are the
transfer
funcwere
first
estimatedofby
evaluating
a step
response
of the
the
tions
of scheme.
primarycontrollers
and
secondary
processes
whereas
p1 (s)
and
secondary
represented
by
G
(s)
and
(2011),
parameters
the
primary
and
secondary
processes
c1primary
were
first
estimated
by
evaluating
a step
response
ofusing
the
system
with
roughly
tuned
PID
controllers.
Then
and
secondary
controllers
are
represented
by
G
(s)
and
c1
tions
of
primary
and
secondary
processes
whereas
primary
system
with
roughly
tuned
PID
controllers.
Then
using
and
secondary
controllers
are
represented
by
G
(s)
and
c1
G
(s)
respectively.
y
(s)
denotes
the
output
of
the
outer
were
first
estimated
by
evaluating
a
step
response
of
the
c2 (s) respectively. y1 (s) denotes the output of c1
system
with
roughly
tuned
PID
controllers.
Then
using
the
estimated
model,
the
two
controllers
have
been
tuned
G
the
outer
c2 secondary controllers
1
and
are isrepresented
by
Gyc1
(s)outer
and the
estimated
model, the
twoPID
controllers
have Then
been tuned
c2
loop
and
output
of
inner
represented
by
SetG
respectively.
y11 (s)loop
denotes
the output
of
the
system
with
roughly
tuned
controllers.
using
2 (s).
c2 (s)
by
using
internal
model
control
strategy.
Simultaneous
loop
and
output
of
inner
loop
is
represented
by
y
(s).
Setthe
estimated
model,
the
two
controllers
have
been
tuned
2
G
respectively.
y1 (s)
denotes
the output
outer
using
internal
model
control
strategy.
Simultaneous
c2 (s)for
2 (s).
point
outer
is
denoted
by
output
loop
output
ofloop
inner
is represented
byofythe
Set- by
the
estimated
model,
the
two
controllers
have
been
tuned
2
tuning
based
on
user
defined
settling
time
and
maximum
pointand
for the
the
outer
loop
isloop
denoted
by rr11 (s)
(s) whereas
whereas
output
by
using
internal
model
control
strategy.
Simultaneous
loop
and
output
of
inner
loop
is
represented
by
y
(s).
Settuning
based
on
user
defined
settling
time
and
maximum
point
for
the
outer
loop
is
denoted
by
r
(s)
whereas
output
2
1r2 (s)) for the inner
of
G
(s)
serves
as
the
set-point
(i.e.
by
using
internal
model
control
strategy.
Simultaneous
c1
1
tuning
based
on
user
defined
settling
time
and
maximum
using
genetic
algorithm
is
reported
in
of
Gc1for
(s)the
serves
asloop
the isset-point
(i.e.
for theoutput
inner overshoot
2 (s))
point
outer
denoted
by r1rr(s)
whereas
overshoot
using
genetic
algorithm
istime
reported
in Kaya
Kaya
2
loop.
is
manipulated
variable.
d
d
of
Gc1
(s) serves
the set-point
(i.e.
for and
the inner
based
on user
defined
settling
and maximum
c1u(s)
2 (s))
and
Nalbantoğlu
(2016).
Internal
model
control
approach
loop.
u(s)
is the
theas
manipulated
variable.
d11 (s)
(s)
and
d22 (s)
(s) tuning
overshoot
using
genetic
algorithm
is reported
in Kaya
of
G
(s)
serves
as
the
set-point
(i.e.
r
(s))
for
the
inner
and
Nalbantoğlu
(2016).
Internal
model
control
approach
c1u(s)
2outputs
1
2
are
the
load
disturbances
entering
the
of
G
loop.
is
the
manipulated
variable.
d
(s)
and
d
(s)
overshoot
using
genetic
algorithm
is
reported
in
Kaya
p1
1
was
used
tune
primary
and
are
loadis disturbances
entering
the outputs
of Gdp12 (s)
and
Nalbantoğlu
(2016). Internal
control
approach
loop.the
u(s)
thedisturbance
manipulated
variable.
d1 (s) Gand
(s)
was
used to
to simultaneously
simultaneously
tune model
primary
and secondary
secondary
are
the
loadwith
disturbances
entering
the
outputs
of
Gp1
2and
p1
and
G
(s)
transfer
functions
(s)
(s)
and
Nalbantoğlu
(2016).
Internal
model
control
approach
p2
d1
was
used
to
simultaneously
tune
primary
and
secondary
in Jeng
Two
of
freedom
and
Gp2 (s)
disturbance
transferthe
functions
Gof
(s)
and
are
the
loadwith
disturbances
entering
outputs
Gp1
(s) controllers
controllers
Jeng and
and Lee
Lee (2012).
(2012).
Two degree
degree
of
freedom
and
Gp2
(s)
with
disturbance
transfer
functions
Gd1
and
p2respectively.
d1 (s)
G
Series
cascade
control
scheme
is
used
was
usedcontrol
toin
tune primary
and
secondary
d2 (s)
d1
controllers
insimultaneously
Jengstructures
and Lee (2012).
Twobeen
degree
of
freedom
cascade
have
also
proposed
in
G
(s)
respectively.
Series
cascade
control
scheme
is
used
d2
and
G
(s)
with
disturbance
transfer
functions
G
(s)
and
cascade
control
structures
have
also
been
proposed
in
p2
d1
d2
when
the
dynamics
of
the
inner
loop
is
fast
so
that
there
G
(s)
respectively.
Series
cascade
control
scheme
is
used
controllers
in (2008),Alfaro
Jengstructures
and Lee (2012).
Twobeen
degree
of freedom
d2 the dynamics of the inner loop is fast so that there
Alfaro
et
al.
et
al.
(2009).
If
there
exists
when
cascade
control
have
also
proposed
ina
G
(s)
respectively.
Series
cascade
control
scheme
is
used
Alfaro
et
al.
(2008),Alfaro
et
al.
(2009).
If
there
exists
d2 bethe
will
rapid
attenuation
of
disturbances
in
the
inner
loop.
when
dynamics
of
the
inner
loop
is
fast
so
that
there
cascade
control
structures
have
also
been
proposed
ina
large
time
delay
in
the
outer
loop,
a
series
cascade
control
will
be
rapid
attenuation
of
disturbances
in
the
inner
loop.
Alfaro
et
al.
(2008),Alfaro
et
al.
(2009).
If
there
exists
a
when
the
dynamics
of
the
inner
loop
is
fast
so
that
there
large
time
delay
in
the
outer
loop,
a
series
cascade
control
will
be
rapid
attenuation
of
disturbances
in
the
inner
loop.
Tuning
methods
reported
for
single
loop
control
system,
Alfaro
et fails
al.
(2008),Alfaro
al.servo
(2009).
Ifcascade
there exists
a
large
time
delay
in
theacceptable
outeretloop,
a series
control
structure
to
give
and
regulatory
perTuning
methods
reported
for
single
loop
control
system,
will
be
rapid
attenuation
offor
disturbances
the inner
loop.
structure
fails
to in
give
acceptable
servo
and cascade
regulatory
perTuning
methods
reported
single(Krishnaswamy
loopincontrol
system,
e.g.
optimal
integral
error
method
et
al.
large
time
delay
the
outer
loop,
a
series
control
structure
fails
to
give
acceptable
servo
and
regulatory
perAn
cascade
control
structure which
e.g.
optimal
integral
error for
method
et al. formances.
Tuning
methods
reported
single(Krishnaswamy
loop
control
system,
Antoimproved
improved
cascadeservo
control
which
(1990)),
IMC
based
method
(Lee
e.g.
optimal
error method
et al. formances.
structure
fails
give acceptable
andstructure
regulatory
peruses
an
internal
model
control
for
the
inner
loop
a
(1990)),
IMCintegral
based tuning
tuning
method(Krishnaswamy
(Lee et
et al.
al. (2002)),
(2002)),
formances.
An improved
cascade
control
structure
e.g.
optimal
integral
error
method
(Krishnaswamy
et
al.
uses
an
internal
model
control
for
the
inner
loop and
andwhich
a PIPIdirect
synthesis
approach
(Raja
and
Ali
(2015))
etc.
have
(1990)),
IMC
based
tuning
method
(Lee
et
al.
(2002)),
formances.
An
improved
cascade
control
structure
which
PD
Smith
predictor
in
the
outer
loop
has
been
proposed
direct
synthesis
approach
(Raja
and
Ali
(2015))
etc.
have
uses
an
internal
model
control
for
the
inner
loop
and
a
PI(1990)),
IMC
based
tuning
method
(Lee
et
al.
(2002)),
PD
Smith
predictor
in
the
outer
loop
has
been
proposed
direct
synthesis
approach
(Raja
and
Ali
(2015))
etc.
have
been
extended
to
cascade
control
scheme.
There
exists
two
uses
an internal
model
control
for loop
the
inner
loop and
a PIPD
Smith
predictor
in
the
outer
has been
proposed
in
Kaya
et
al.
(2007)
for
such
class
of
processes.
been
extended
to
cascade
control
scheme.
There
exists
two
direct
synthesis
(Raja
and
Ali (2015))
etc. have
Kaya
et predictor
al. (2007) in
forthe
such
classloop
of processes.
been
extended
toapproach
cascade
control
scheme.
There
exists
two in
different
ways
design
cascade
control
structure.
The
PD
Smith
outer
has
been controller
proposed
in
Kaya
et al. (2007)
for such
class
of processes.
Determination
of
all
stabilizing
PI,
PD
and
different
ways to
to cascade
design a
acontrol
cascade
controlThere
structure.
been extended
scheme.
exists The
two
Determination
of
all
stabilizing
PI,
PD
and PID
PID controller
usual
approach
is
to
first
tune
the
secondary
controller
by
different
ways
to
design
a
cascade
control
structure.
The
in
Kaya et al.
(2007)
for
such
class
of processes.
parameters
is
reported
in
Ho
et
al.
(1997);
Tan
et
al.
usual
approach
is
to
first
tune
the
secondary
controller
by
Determination
of
all
stabilizing
PI,
PD
and
PID
controller
different
ways
tois design
atune
cascade
control
structure.
The
is reported
in Ho et al.
Tan
et controller
al. (2003,
(2003,
assuming
the
primary
controller
open
state.
The
next
usual
approach
to first
thein
secondary
controller
by parameters
Determination
of all(2013,
stabilizing
PI,(1997);
PDusing
and
PID
2005a,
2006);
Onat
2019).
By
the
above
reassuming
the
primary
controller
in
open
state.
The
next
parameters
is
reported
in
Ho
et
al.
(1997);
Tan
et
al. (2003,
usual
approach
is
to
first
tune
the
secondary
controller
by
2005a,
2006);
Onat
(2013,
2019).
By
using
the
above
reassuming
the
primary
controller
in
open
state.
The
next
step
is
to
tune
the
primary
controller
with
the
application
parameters
is reported
in Ho2019).
et al. (1997);
Tan
et above
al. (2003,
2005a,
2006);
Onata (2013,
Byregion
using
the
reported
techniques,
convex
stability
is
obtained
instep
is
to
tune
the
primary
controller
with
the
application
assuming
the primary
controller
incontroller
open
The
next ported
techniques,
a
convex
stability
region
is
obtained
instep
is previously
to tune
thetuned
primary
controller
withstate.
thetoapplication
of
the
secondary
the
inner
2005a,
2006);
Onat
(2013,
2019).
By
using
the
above
reported
techniques,
a
convex
stability
region
is
obtained
inside
the
stability
boundary
locus
in
the
plane
of
controller
of
the
previously
tuned
secondary
controller
to
the
inner
step
isThis
to tune
thetuned
primary
controller
with thetoapplication
side
the
stability
boundary
locus
in
the
plane
of
controller
loop.
approach
is
known
as
sequential
approach.
of
the
previously
secondary
controller
the
inner
ported
techniques,
a convex
stability
region
is of
obtained
inparameters.
Weighted
geometrical
center
(WGC)
method
loop.
This
approach
is
known
as
sequential
approach.
side
the
stability
boundary
locus
in
the
plane
controller
of
the This
previously
tunedis secondary
todesign
the inner
parameters.
Weighted
geometrical
center
(WGC)
method
Direct
synthesis
approach
has
been
used
to
the
loop.
approach
known
ascontroller
sequential
approach.
side
the
stability
boundary
locus
inthe
theconvex
plane
of
controller
and
calculation
of
centroid
from
stability
reDirect
synthesis
approach
has
been
used
to
design
the
parameters.
Weighted
geometrical
center
(WGC)
method
loop.
This
approach
is known
as (2015).
sequential
approach.
calculation
of centroid
from the
convex
stability
reDirect
synthesis
approach
has Ali
been
used to
design
the and
controller
settings
in
In
Vivek
parameters.
Weighted
geometrical
center
(WGC)
method
and
calculation
of centroid
from (2013)
the
convex
stability
rehas
been
reported
in
Onat
and
Onat
(2019)
controller
settingsapproach
in Raja
Raja and
and
(2015).
In design
Vivek and
and
Direct
synthesis
has Ali
been
used
to
the gion
gion
has
been
reported
in
Onat
(2013)
and
Onat
(2019)
controller
settings
in Raja
and
Ali
(2015).
In Vivek
and
Chidambaram
(2013),
a
typical
relay
feedback
based
auand
of centroid
from (2013)
the convex
stability
region calculation
has been reported
in Onat
and Onat
(2019)
Chidambaram
(2013),
a
typical
relay
feedback
based
aucontroller
settings
in Raja
and Ali
(2015).
In Vivek
Chidambaram
(2013),
a typical
relay
feedback
based and
au- gion has been reported in Onat (2013) and Onat (2019)
Chidambaram
a typical relay
feedback
basedControl)
au- Hosting by Elsevier Ltd. All rights reserved.
2405-8963 © 2020,(2013),
IFAC (International
Federation
of Automatic
Peer review under responsibility of International Federation of Automatic Control.
10.1016/j.ifacol.2020.06.033
Md. Imran Kalim et al. / IFAC PapersOnLine 53-1 (2020) 195–200
196
Fig. 1. Series cascade control scheme
respectively. Calculation of WGC is very tedious as it uses
all frequency points forming the stability boundary locus,
whereas calculation of centroid requires only few frequency
dependent points, namely cusp and corner points. It is also
reported in Onat (2019) that the controller parameters
corresponding to the centroid yields satisfactory closedloop performance. Furthermore, the centroid is located at a
safe distance from all the points on the stability boundary
locus. Hence, it will give improved set-point tracking and
yields faster disturbance rejection. Also, it will maintain
excellent robustness in case of process parameter variations. Hence, the controller settings for the inner loop is
obtained by finding the centroid of the convex stability
region.
For the output to exactly follow the set point, the closed
loop response should have infinite bandwidth and zero
phase shift which is practically not possible (Vrancic et al.
(2001)). The magnitude optimum criterion tries to maintain the closed loop magnitude frequency response curve
flat and close to unity over the widest possible frequency
range for any given plant and controller structure (Umland
and Safiuddin (1988)). This method is called as modulus
optimum (Åström and Hägglund (1995)), Betragsoptimum
(Kessler (1955)), magnitude optimum (Umland and Safiuddin (1988)). Actually, this criterion is based on optimizing the closed-loop amplitude response. Furthermore, it is
reported in the literature that the magnitude optimum
criterion results in superior tracking performance for a
large number of process models (Kessler (1955); Vrancic
et al. (2001); Vrančić et al. (1999)). Hence, in this proposed
work, the primary controller is tuned with the help of magnitude optimum criterion to achieve improved set-point
tracking performance and faster disturbance rejection in
the outer loop.
The paper is organized as follows: Tuning rules of primary
and secondary controllers are derived in Section 2. Simulation examples, results and comparisons are given in Section
3 while Section 4 provides the concluding remarks about
the proposed work.
2. TUNING OF GC1 AND GC2
2.1 All stabilizing PI controller for Gc2
In the proposed work, Gc2 is considered as a PI controller
whose transfer function is given below,
Gc2 (s) =
Nc2 (s)
Ki2
= Kp2 +
Dc2 (s)
s
(1)
where Kp2 and Ki2 are the proportional and integral gains
of the secondary controller respectively. Gc2 (s) is tuned by
finding the centroid of the convex stability region obtained
by plotting the stability boundary locus in Kp2 and Ki2
plane (Tan et al. (2003), Onat (2019)). Let, the transfer
function of the secondary process be,
Gp2 (s) =
Np2 (s) −τ2 s
e
Dp2 (s)
(2)
The characteristic polynomial of the inner loop is given by,
∆P (s) = 1 + Gc2 (s)Gp2 (s)
= Dp2 (s)Dc2 (s) + Np2 (s)Nc2 (s)e−τ2 s
Substituting s = jw and e
jsin(τ2 w)) in (3), we get
−τ2 s
= e
−jτ2 w
(3)
= (cos(τ2 w) −
∆P (jw) = Dp2 (jw)Dc2 (jw)
+ Np2 (jw)Nc2 (jw)(cos(τ2 w) − jsin(τ2 w)) (4)
Let R∆P and I∆P represent the real and imaginary parts
of ∆P (jw). Following are the three different ways in which
a root of any stable polynomial can cross the imaginary
axis (Tan (2009)).
(i) Real Root Boundary (RRB): A real root can cross the
imaginary axis at s = 0. RRB is obtained by substituting
s = 0 in (3) which gives Ki2 = 0 as the RRB line.
(ii) Infinite Root Boundary (IRB): A real root can cross
the imaginary axis at s = ∞. IRB is obtained by substituting s = ∞ in (3). There will not be any IRB line for
proper transfer function.
(iii) Complex Root Boundary (CRB): Roots on the imaginary axis can become unstable which implies that both
the real and imaginary parts of (4) will become zero
simultaneously.
The procedure to obtain complex root boundary is given
in Tan (2009) and is discussed below in brief.
Kp2 and Ki2 are obtained as functions of w by solving R∆P
= 0 and I∆P = 0. The complex root boundary in Kp2 and
Ki2 plane is obtained by varying w from 0 to wc , where wc
is the lowest value of frequency other than w = 0 at which
Ki2 = 0. Convex stability region in Kp2 and Ki2 plane is
the area bounded by CRB and RRB which contains all the
stabilizing values of Kp2 and Ki2 for the considered plant.
The cusp and corner points of the convex stability region
are identified as (Kp1
,Ki1
),(Kp2
,Ki2
).......(Kpm
,Kim
) and
(Kp1 ,Ki1 ),(Kp2 ,Ki2 )......(Kpn , Kin ), where m and n are
the number of cusp and corner points respectively. The
parameters of inner controller are obtained by finding
the centroid of the convex stability region as given below
(Onat (2019)),
n
m
) + ( l=1 (Kpl
)
( l=1 Kpl
(5)
Kp2 =
m
+
n
n
m
( l=1 Kil ) + ( l=1 (Kil )
Ki2 =
(6)
m+n
Md. Imran Kalim et al. / IFAC PapersOnLine 53-1 (2020) 195–200
2.2 Magnitude optimum criterion for Gc1
197
120
Cusp point of the convex stability region
100
G1 (s) =
Gc2 (s)Gp2 (s)
Gp1 (s)
1 + Gc2 (s)Gp2 (s)
(8)
.Let Gcl (s) be the closed loop transfer function from
reference to output. To achieve the magnitude optimum
criterion, the controller should satisfy the following two
conditions for as many q as possible (Vrancic et al. (2001)):
Gcl (0) = 1
dq |Gcl (jw)| = 0, q = 1, 2, ..., qmax
dwq
w=0
(9)
(10)
First equation can be satisfied by using a controller structure having an integral term. qmax is taken as 2 for a PI
controller and 3 for a PID controller. Assuming that the
plant model for the primary controller is given as follows:
1 + a1 s + a2 s2 + .... + am sm −τ1 s
e
(11)
1 + b1 s + b2 s2 + .... + bn sn
where Kr is the process steady state gain, τ1 is the process
time delay. (a1 ,..., am ) and (b1 ,..., bm ) are the process
parameters. Time delay term is replaced by its Taylor’s
series approximation. Multiplying (7) and (11), the open
loop transfer function becomes,
G1 (s) = Kr
c0 + c1 s + c2 s2 + .......
(12)
G1 (s)Gc1 (s) =
d0 s + d1 s2 + d2 s3 + .......
where, the coefficients ci and di (i = 0, 1, 2, 3...) are
functions of the plant and controller parameters which can
be obtained using the expressions given below (Vrancic
et al. (2001)):
di = bi
ci = Kr [Ki1 φi + Kp1 φi−1 + Kd1 φi−2 ]
φi =
i
(−1)i−l al
l=0
τ1i−l
(i − l)!
a0 = b0 = 1
ai = bi = 0, i < 0.
(13)
Necessary and sufficient condition for stability reported in
Hanus (1975) is as given below:
c0
>0
(14)
d0
The PID controller parameters can be obtained by solving
the following equations for n = 0 − 2 (Hanus (1975)).
2n+1
i=0
(−1)i ci d2n+1−i = 0.5
2n
(−1)i di d2n−i
(15)
i=0
By substituting (13) into (15), following expressions are
obtained for Kp1 , Ki1 and Kd1 (Vrancic et al. (2001)).
Stability
boundary
locus
i2
80
K
In the proposed work, Gc1 (s) is assumed as a PID controller with following transfer function:
Ki1
Gc1 (s) = Kp1 +
+ Kd1 s
(7)
s
where Kp1 , Ki1 and Kd1 are the proportional, integral
and derivative gains of the primary controller respectively.
From Fig. 1, G1 (s) can be calculated as,
CONVEX
STABILITY
REGION
60
(15.89,38.03)
Centroid of the convex
stability region
40
20
Corner points of the convex stability region
0
−20
−5
0
5
10
15
Kp2
20
25
30
35
Fig. 2. Stability boundary locus of inner loop of example
1
A23 − A1 A5
(16)
2(A1 A2 A3 + A0 A1 A5 − A21 A4 − A0 A23 )
A2 A3 − A1 A4
Ki1 =
(17)
2(A1 A2 A3 + A0 A1 A5 − A21 A4 − A0 A23 )
A3 A4 − A2 A5
Kd1 =
(18)
2(A1 A2 A3 + A0 A1 A5 − A21 A4 − A0 A23 )
where symbols A0 , A1 , A2 , A3 , A4 and A5 are known as
characteristic areas of the process. These areas can be
calculated using the following expressions.
A0 = Kr
j
l
τ
b
j−l
)
Aj = Kr (−1)j+1 (bj − aj ) + ( (−1)j+l 1
l!
Kp1 =
l=1
+
j−1
(−1)j+l−1 Al aj−l
(19)
l=1
3. SIMULATION EXAMPLES
In this section, three simulation examples are provided
to demonstrate the effectiveness of the proposed method.
Following commonly used parallel form of PID controller
is used in simulation.
Kd1 s
Ki1
+
(20)
Gc1 (s) = Kp1 +
s
tf 1 s + 1
where, tf 1 = 0.1Td1 , and Td1 = Kd1 /Kp1 .
Example 1.
In Jeng (2014), following higher order process has been
studied.
1
Gp1 (s) = Gd1 (s) =
(10s + 1)(4s + 1)(s + 1)2
1
e−0.1s
Gp2 (s) = Gd2 (s) =
(21)
1.9s + 1
For tuning of secondary controller, convex stability region
of the inner loop is obtained in Kp2 and Ki2 plane using
the procedure given in section 2.1, and is shown in Fig. 2.
It is having one cusp point and two corner points as shown.
Centroid of the convex stability region is calculated using
(5) and (6). By applying (16) to (18), controller settings
for outer loop are obtained. Table 1 shows the controller
tuning parameters for the method of Jeng (2014) and the
proposed technique. Closed loop response to unit step setpoint change at t = 0s, step input disturbances d2 = 30
at t = 80s and d1 = 1 at t = 160s is shown in Fig. 3 while
control signal is provided in Fig. 4. Performance measures
are given in Table 2. To show the robustness to parameter
Md. Imran Kalim et al. / IFAC PapersOnLine 53-1 (2020) 195–200
198
Table 1. Controller tuning parameters for various examples.
Example 1
Example 2
Example 3
Tuning
method
J.C.Jeng
Proposed
J.C.Jeng
Proposed
Kaya and Nalbantoglu
Proposed
Inner
Kp2
3.633
15.89
0.617
1.218
1.2214
0.79
loop
Ki2
3.043
38.03
0.0595
0.128
1.2214
5.73
Outer
Kp1
1.974
3.8113
0.088
0.1016
1
1.0548
loop
Ki1
0.1376
0.2690
0.0010136
0.0012
0.32106
0.4897
Kd1
6.025
11.219
2.27304
2.2703
0.6090
0.5899
Table 2. Performance measures for various examples.
Example 1
Example 2
Example 3
Tr (sec.)
Ts (sec.)
%Mp
IAE
ISE
ITAE
Tr (sec.)
Ts (sec.)
%Mp
IAE
ISE
ITAE
Tr (sec.)
Ts (sec.)
%Mp
IAE
ISE
ITAE
Servo
J.C.Jeng
10.5
21.6
0
7.268
4.807
38.46
390
201
0.12
98.95
75.4
6042
Kaya and Nalbantoglu
4.64
5.587
1.006
3.1298
2.485
5.687
Performance
Proposed
7.7
13.7
6
4.272
2.993
13.01
146.3
231
5.84
91.98
69.39
5364
Proposed
2.94
7.1
8.1
2.3
1.82
3.318
Regulatory(d2)
J.C.Jeng
51
38.3
6.064
1.164
121.3
425
53.1
80.82
24.98
14990
Kaya and Nalbantoglu
12.37
16.2
0.7874
0.06688
5.054
Performance
Proposed
7.8
2.7
0.3149
0.00485
4.827
334
24.5
36.01
5.478
6160
Proposed
6.72
16.3
0.2092
0.00817
0.8688
Regulatory (d1)
J.C.Jeng
50
31
7.257
1.517
153.8
408
60.7
98.77
41.67
17810
Kaya and Nalbantoglu
14.27
63.5
3.115
1.288
16.85
Performance
Proposed
38
20
3.714
0.4766
66.12
317.5
57.6
83.36
34.51
13790
Proposed
8.6
60.2
2.047
0.8686
8.273
1.5
1.5
y
1
1
0
0
y1
1
0.5
J.C.Jeng
Proposed
50
100
150
Time (seconds)
0.5
0
0
Control Signal Magnitude
Fig. 3. Closed-loop response of example 1
4
2
0
0
50
100
150
200
50
100
150
Time(seconds)
200
Fig. 5. Closed-loop response for the perturbed plant of
example 1
J.C.Jeng
Proposed
6
J.C.Jeng
Proposed
200
250
Time (seconds)
Fig. 4. Control signal of example 1
variations of the plant, the controller is applied to following
perturbed model of the plant.
1.2
Gp1 (s) =
(8s + 1)(3.2s + 1)(0.8s + 1)2
1.2
e−0.12s
Gp2 (s) =
(22)
1.65s + 1
Closed loop response for the perturbed plant is shown
in Fig. 5. So, the proposed method provides excellent
robustness with improved closed loop response. It can be
concluded from Fig. 3, Fig. 4, Fig. 5, and Table 2 that
the proposed method performs better than the strategy
reported in Jeng (2014).
Example 2.
Let us consider the following higher order process which
exhibits nonminimum-phase dynamics (Jeng (2014)).
10(−5s + 1)
Gp1 (s) = Gd1 (s) =
e−5s
(30s + 1)3 (10s + 1)2
3
e−3s
(23)
Gp2 (s) = Gd2 (s) =
13.3s + 1
The computed convex stability region is shown in Fig. 6.
Table 1 summarizes the controller settings for the inner
and outer loops for the design method of Jeng (2014)
and the proposed technique. Fig. 7 shows the closed loop
response to unit step set point change at t = 0s, step
input disturbances d2 = 0.4 at t = 700s and d1 = 0.1 at
t = 1400s. Control signal is shown in Fig. 8. For robustness
consideration, following perturbed model of the plant has
been considered.
12(−6s + 1)
Gp1 (s) =
e−6s
(24s + 1)3 (8s + 1)2
3.6
e−3.6s
Gp2 (s) =
(24)
10.64s + 1
It is observed from Fig. 7, Fig. 8, Fig. 9, and Table 2
that the proposed method yields improved closed loop
Md. Imran Kalim et al. / IFAC PapersOnLine 53-1 (2020) 195–200
0.4
Cusp point of the convex stability region
Stability
boundary
CONVEX
0.3
locus
STABILITY
0.25
REGION
Cusp point of the convex stability region
16
0.2
(1.218,0.128)
Centroid of the convex
stability region
0.1
CONVEX
STABILITY
REGION
8
(0.79,5.73)
Centroid of the convex
stability region
4
Corner points of the convex stability region
(Corner points of the convex stability region)
0.05
0
−0.05
−0.5
Stability
boundary
locus
12
Ki2
Ki2
0.35
0.15
199
0
0
0.5
1
K
1.5
2
2.5
−2
3
−1
−0.5
0
p2
0.5
K
1
1.5
2
2.5
p2
Fig. 6. Stability boundary locus of inner loop of example
2
Fig. 10. Stability boundary locus of inner loop of example
3
1.5
J.C.Jeng
Proposed
0.5
0
0
500
1000
1500
Time (seconds)
y1
y
1
1.5
1
0.5
0
0
20
40
60
80
Time(seconds)
100
120
Fig. 11. Closed-loop response of example 3
0.6
J.C.Jeng
Proposed
0.4
0.2
500
1000
1500
2000
Time(seconds)
Control Signal Magnitude
Control Signal Magnitude
Kaya and Nalbantoglu
Proposed
2000
Fig. 7. Closed-loop response of example 2
0
0
1
Fig. 8. Control signal of example 2
1.5
1
0.5
0
0
Kaya and Nalbantoglu
Proposed
20
40
60
80
100
120
Time (seconds)
2
Fig. 12. Control signal of example 3
1
0
0
1.5
J.C.Jeng
Proposed
0.5
500
1000
1500
Time(seconds)
2000
1
y1
y1
1.5
0.5
Fig. 9. Closed-loop response for the perturbed plant of
example 2
performance for nominal as well as perturbed model of
the plant.
Example 3.
As a third example, Following self regulating process has
been considered (Kaya and Nalbantoğlu (2016)).
1
Gp1 (s) = Gd1 (s) =
e−s
(s + 1)2
1
e−0.1s
Gp2 (s) = Gd2 (s) =
(25)
0.1s + 1
The convex stability region is shown in Fig. 10. Obtained
values of the controller parameters for the two loops are
given in Table 1 for the proposed method and the design
approach of Kaya and Nalbantoğlu (2016) . Fig. 11 shows
the closed loop response to unit step set-point change at
t = 0s, step input disturbances d2 = 1 at t = 40s and
0
0
Kaya and Nalbantoglu
Proposed
20
40
60
80
Time (seconds)
100
120
Fig. 13. Closed-loop response under noise for example 3
d1 = 1 at t = 80s. Control signal is provided in Fig. 12.
As can be seen from the graph that the proposed method
yields improved servo as well as regulatory responses as
compared to the method of Kaya and Nalbantoğlu (2016).
To demonstrate the effects of the noise constraint, the
system is subjected to white noise of standard deviation
σn = 0.005, band-limited by the Nyquist frequency corresponding to the controller sampling period h = 0.01.
Closed-loop response and corresponding control signal under noise are shown in Fig. 13 and Fig. 14 respectively. As
shown in the figures, the response by the proposed method
shows still better performance than that by the method of
Kaya and Nalbantoğlu (2016).
Md. Imran Kalim et al. / IFAC PapersOnLine 53-1 (2020) 195–200
200
Control Signal Magnitude
3
2
1
0
−1
−2
0
Kaya and Nalbantoglu
Proposed
20
40
60
80
100
120
Time (seconds)
Fig. 14. Control signal under noise for example 3
4. CONCLUSIONS
A tuning strategy for a series cascade control structure
has been presented in this paper. Simulation examples
have been provided to show the benefits of the proposed
technique. Closed-loop responses illustrate that the proposed tuning method yields better servo and regulatory
performances as compared to the recently reported tuning
strategies for the nominal as well as the perturbed model
of the plant. The improved results of the procedure are
demonstrated with respect to measurement noise. An extension of the present work can include the simultaneous
tuning of series cascade control system and also the incorporation of performance-robustness trade-off into the
parameter selection criterion.
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