IAES
Inter
national
J
our
nal
of
Robotics
and
A
utomation
(IJRA)
V
ol.
15,
No.
3,
September
2026,
pp.
690
∼
697
ISSN:
2722-2586,
DOI:
10.11591/ijra.v15i3.pp690-697
❒
690
Zer
oing
neur
odynamics
pr
oportional-integral-deri
v
ati
v
e
contr
oller
f
or
order
-2
system:
design,
analysis,
and
v
erication
Y
unong
Zhang,
J
unyan
Liu,
Zhonghua
Li
School
of
Intelligent
Systems
Engineering,
Shenzhen
Campus
of
Sun
Y
at-sen
Uni
v
ersity
,
Shenzhen,
China
Article
Inf
o
Article
history:
Recei
v
ed
Apr
6,
2026
Re
vised
Jul
24,
2026
Accepted
Aug
6,
2026
K
eyw
ords:
Nonlinear
control
Order
-2
system
Pole
placement
Proportional-inte
gral-deri
v
ati
v
e
Zeroing
neurodynamics
ABSTRA
CT
This
paper
presents
a
zeroing
neurodynamics
proportional-inte
gral-deri
v
ati
v
e
(PID)
controller
for
an
order
-2
single-input
single-output
(SISO)
af
ne-in-
control
nonlinear
system.
The
controller
is
deri
v
ed
from
zeroing
neurodynam-
ics
(ZN)
and
reformulates
PID
g
ains
through
ZN
parameters
,
establishing
an
e
xplicit
correspondence
to
the
closed-loop
poles
of
the
error
dynamics.
This
enables
syst
ematic
pole
placement
for
stability
and
performance
tuning,
elimi-
nating
empirical
trial-and-error
.
Theoret
ical
analysis
pro
v
es
e
xponential
con
v
er
-
gence
of
the
tracking
error
.
The
in
v
erted
pendulum,
a
benchmark
system
with
strong
geometric
nonlinearity
,
is
used
to
v
erify
the
controller
.
Simulations
with
lar
ge
initial
of
fsets
and
constant
references
sho
w
that
the
proposed
controller
maintains
stable
tracking
in
operating
conditions
where
the
con
v
entional
PID
controller
e
xhibits
substantial
performance
de
gradation.
The
resulting
frame-
w
ork
pro
vides
a
model-based
and
theoretically
grounded
alternati
v
e
for
nonlin-
ear
control.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Zhonghua
Li
School
of
Intelligent
Systems
Engineering,
Shenzhen
Campus
of
Sun
Y
at-sen
Uni
v
ersity
Shenzhen,
518107,
Guangdong,
China
Email:
lizhongh@mail.sysu.edu.cn
1.
INTR
ODUCTION
Order
-2
nonlinear
systems
constitute
a
fundamental
and
ubiquitous
class
of
systems
in
control
en-
gineering,
which
are
used
to
model
di
v
erse
ph
ysical
phenomena
ranging
from
electromechanical
de
vices
to
v
ehicle
dynamics
[1].
Precise
trajectory
tracking
control
for
such
systems
remains
challenging
because
of
their
inherent
nonlinearities.
While
adv
anced
control
techniques
such
as
optimal
control
[2],
rob
ust
control
[3],
and
sliding
mode
control
[4]
ha
v
e
emer
ged,
proportional-inte
gral-deri
v
ati
v
e
(PID)
control
remains
widely
used
in
industry
o
wing
to
i
ts
structural
simplicity
and
intuiti
v
e
tuning
[5]–[7].
Ho
we
v
er
,
its
performance
can
de
grade
signicantly
for
systems
with
strong
nonlinearities
or
comple
x
reference
signals,
primarily
because
it
does
not
e
xplicitly
use
model
kno
wledge
and
its
parameters
are
usually
selected
empirically
[8],
[9].
T
o
adapt
PID
controllers
for
nonlinear
systems,
v
arious
tuning
and
stability
frame
w
orks
ha
v
e
been
de-
v
eloped.
F
or
instance,
internal-model-control
tuning
methods
often
simplify
comple
x
nonlinear
dynamics
into
lo
w-order
linear
systems
with
time
delays,
potentially
de
grading
transient
performance
[10].
T
o
further
guar
-
antee
stability
without
o
v
er
-simplication,
parameter
-manifold
approaches
were
proposed
to
achie
v
e
global
or
semi-global
stabilization
for
order
-2
and
n
-dimensional
nonlinear
systems
[11]–[12].
Although
these
w
orks
J
ournal
homepage:
http://ijr
a.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
IAES
Int
J
Rob
&
Autom
ISSN:
2722-2586
❒
691
established
a
solid
theoretical
foundation
for
PID-based
nonlinear
control,
their
control
la
ws
remain
tied
to
classical
PID
structures
and
empirical
g
ain
selection.
This
paper
addresses
the
control
of
an
order
-2
SISO
af
ne-in-control
nonlinear
system
by
proposing
and
analyzing
a
zeroing
neurodynamics
PID
(ZN-PID)
controller
deri
v
ed
from
the
zeroing
neurodynamics
for
-
mula
[13]–[17].
The
ef
fecti
v
eness
of
the
ZN
formula
has
been
v
alidated
in
numerous
applications
[18]–[21].
The
core
contrib
ution
is
the
inte
gration
of
e
xact
model-based
linearization
[22]–[24]
with
a
reformulated
PID
structure.
The
con
v
entional
PID
g
ains
are
redened
through
ZN
parameters
and
directly
related
to
the
closed-
loop
pole
locations.
The
benchmark
in
v
erted
pendulum,
which
e
xhibits
strong
geometric
nonlinearity
,
is
se-
lected
as
the
representati
v
e
system.
The
zer
oing
neurodynamics
PID
controller
design
methodology
le
v
erages
recursi
v
ely
dened
zeroing
functions
to
construct
a
higher
-order
error
dynamic.
By
strate
gically
setting
a
specic
zeroing
function
to
zero
and
incorporating
the
e
xact
kno
wledge
of
the
system
dynamics,
we
deri
v
e
the
general
control
la
w
.
This
la
w
naturally
decomposes
into
a
model
compensation
term
responsible
for
linearizing
the
plant
dynamics
and
a
PID
control
term
ensuring
asymptotic
error
con
v
er
gence.
Crucially
,
the
parameters
of
the
zeroing
neurodynamics
PID
controller
(i.e.,
λ
1
,
λ
2
,
and
λ
3
)
are
transformed
into
equi
v
alent
PID
g
ains
(i.e.,
K
p
,
K
d
,
and
K
i
),
and
a
direct
correspondence
between
these
parameters
and
the
closed-loop
poles
of
the
resulting
order
-3
error
dy-
namics
is
established.
This
intrinsic
link
pro
vides
a
theoretically
grounded
alternati
v
e
to
the
often
cumbersome
trial-and-error
tuning
associated
with
con
v
entional
PID
controllers.
Simulation
studies
compare
the
proposed
controller
with
a
con
v
entionally
tuned
PID
controller
[25]
on
the
in
v
erted
pendulum.
The
results
sho
w
superior
tracking
by
ZN-PID
for
lar
ge
initial
of
fsets
and
constant
references.
In
particular
,
the
ZN-PID
controller
maintains
the
pendulum
equilibrium
under
high-amplitude
references,
whereas
the
con
v
entional
PID
controller
loses
balance
because
of
the
s
e
v
ere
geometric
nonlinearity
.
The
remainder
of
this
paper
is
structured
as
follo
ws.
Section
2
introduces
the
general
model
of
order
-2
systems
and
the
in
v
erted
pendulum
model.
Section
3
presents
the
ZN-PID
deri
v
ation
and
its
in
v
erted-pendulum
formulation.
Section
4
pro
v
es
the
e
xponential
stability
of
the
controller
.
Section
5
reports
comparati
v
e
simula-
tion
results.
Section
6
concludes
the
paper
.
Finally
,
the
main
contrib
utions
of
the
paper
are
listed
as
follo
ws.
−
Controller
deri
v
ation:
A
ZN
design
formula
is
used
to
deri
v
e
a
ZN-PID
controller
for
order
-2
systems.
−
PID
g
ains
r
eformulation:
The
analysis
establi
shes
a
correspondence
between
PID
g
ains
and
the
closed-loop
poles
of
the
order
-3
error
dynamics,
pro
viding
a
basis
for
pole-placement
tuning.
−
Stability
proof:
A
rigorous
proof
v
eries
the
e
xponential
stability
of
the
ZN-PID
controller
.
−
Comparati
v
e
simulations:
Comparati
v
e
simulation
e
xperiments
on
an
in
v
erted
pendulum
v
alidate
the
ef
fec-
ti
v
eness
of
the
proposed
controller
.
2.
ORDER-2
SYSTEM
AND
INVER
TED
PENDULUM
In
this
section,
the
general
form
of
an
order
-2
system
and
the
in
v
erted
pendulum
model
are
gi
v
en.
First,
the
general
form
of
a
SISO
af
ne-in-control
order
-2
system
is
described
as,
˙
x
1
(
t
)
=
x
2
(
t
)
,
˙
x
2
(
t
)
=
f
(
x
1
(
t
)
,
x
2
(
t
))
+
g
(
x
1
(
t
)
,
x
2
(
t
))
u
(
t
)
,
y
(
t
)
=
x
1
(
t
)
,
(1)
where
x
1
(
t
)
∈
R
,
x
2
(
t
)
∈
R
,
and
x
(
t
)
=
[
x
1
(
t
)
,
x
2
(
t
)]
T
are
state
v
ariables
of
the
system;
u
(
t
)
∈
R
denotes
the
control
input
of
the
system;
y
(
t
)
∈
R
denotes
the
output
of
the
syste
m;
f
:
R
×
R
→
R
and
g
:
R
×
R
→
R
\{
0
}
are
transformation
functions
of
state
v
ariables,
and
g
(
x
1
(
t
)
,
x
2
(
t
))
is
a
positi
v
e
or
ne
g
ati
v
e
coef
cient
term.
Gi
v
en
an
order
-2
system
described
by
(1)
and
a
desired
output
(reference
signal)
y
d
(
t
)
∈
R
,
our
goal
is
to
nd
a
u
(
t
)
as
input,
which
mak
es
the
output
tracking
error
e
(
t
)
=
y
d
(
t
)
−
y
(
t
)
approach
zero.
F
or
simplicity
,
in
the
remainder
of
this
section,
x
1
(
t
)
and
x
2
(
t
)
are
abbre
viated
as
x
1
and
x
2
,
respecti
v
ely
.
All
system
parameters
are
real
numbers.
The
in
v
erted
pendulum
system
[26]
is
formulated
as
an
order
-2
nonlinear
system
within
the
st
ructure
of
(1).
The
state
v
ariable
x
1
is
the
angular
displacement
of
the
pendulum
from
the
v
ertical
equilibrium,
and
x
2
is
the
angular
v
elocity
.
The
force
applied
to
the
cart
is
the
control
input
u
(
t
)
.
Then,
the
functions
f
(
x
1
,
x
2
)
and
Zer
oing
neur
odynamics
pr
oportional-inte
gr
al-derivative
contr
oller
for
or
der
-2
system
...
(Y
unong
Zhang)
Evaluation Warning : The document was created with Spire.PDF for Python.
692
❒
ISSN:
2722-2586
g
(
x
1
,
x
2
)
in
(1)
are
identied
as
f
(
x
1
,
x
2
)
=
(
M
+
m
)
g
a
sin
x
1
−
ml
sin
x
1
cos
x
1
x
2
2
l
M
+
m
sin
2
x
1
,
g
(
x
1
,
x
2
)
=
−
cos
x
1
l
M
+
m
sin
2
x
1
,
(2)
where
M
>
0
and
m
>
0
are
respecti
v
ely
the
mass
of
the
cart
and
the
pendulum;
l
>
0
is
the
pendulum
length;
g
a
is
the
gra
vitational
acceleration.
T
o
ensure
g
(
x
1
,
x
2
)
̸
=
0
,
the
angular
displacement
x
1
is
constrained
to
x
1
∈
(
−
π
/
2
,
π
/
2)
,
which
guarantees
cos
x
1
>
0
,
ensuring
the
non-singularity
of
g
(
x
1
,
x
2
)
.
3.
ZER
OING
NEUR
OD
YN
AMICS
PID
CONTR
OLLER
In
this
section,
the
general
ZN-PID
design
procedure
and
its
in
v
erted-pendulum
formulation
are
gi
v
en.
3.1.
General
f
orm
First,
according
to
[27],
we
dene
the
follo
wing
zeroing
functions
recursi
v
ely:
z
0
(
t
)
=
Z
t
0
e
(
τ
)d
τ
,
z
1
(
t
)
=
˙
z
0
(
t
)
+
λ
1
z
0
(
t
)
=
e
(
t
)
+
λ
1
Z
t
0
e
(
τ
)d
τ
,
z
2
(
t
)
=
˙
z
1
(
t
)
+
λ
2
z
1
(
t
)
=
˙
e
(
t
)
+
(
λ
1
+
λ
2
)
e
(
t
)
+
λ
1
λ
2
Z
t
0
e
(
τ
)d
τ
,
z
3
(
t
)
=
˙
z
2
(
t
)
+
λ
3
z
2
(
t
)
=
¨
e
(
t
)
+
(
λ
1
+
λ
2
+
λ
3
)
˙
e
(
t
)
+
(
λ
1
λ
2
+
λ
1
λ
3
+
λ
2
λ
3
)
e
(
t
)
+
λ
1
λ
2
λ
3
Z
t
0
e
(
τ
)d
τ
,
(3)
where
λ
1
,
λ
2
,
and
λ
3
are
the
controller
parameters
that
satisfy
λ
i
∈
C
and
Re(
λ
i
)
>
0
(
i
∈
{
1
,
2
,
3
}
);
z
0
(
t
)
,
z
1
(
t
)
,
z
2
(
t
)
,
and
z
3
(
t
)
are
order
-0
through
order
-3
zeroing
functions.
Besides,
e
(
t
)
,
˙
e
(
t
)
,
and
¨
e
(
t
)
represent
the
output
tracking
error
and
its
order
-1
and
order
-2
time
deri
v
ati
v
es,
respecti
v
ely
.
Based
on
the
state
equat
ions
of
the
general
form
of
the
order
-2
system
in
(1),
their
specic
e
xpressions
are
deri
v
ed
as
follo
ws:
e
(
t
)
=
y
d
(
t
)
−
y
(
t
)
=
y
d
(
t
)
−
x
1
(
t
)
,
˙
e
(
t
)
=
˙
y
d
(
t
)
−
˙
y
(
t
)
=
˙
y
d
(
t
)
−
x
2
(
t
)
,
¨
e
(
t
)
=
¨
y
d
(
t
)
−
¨
y
(
t
)
=
¨
y
d
(
t
)
−
˙
x
2
(
t
)
=
¨
y
d
(
t
)
−
f
(
x
1
,
x
2
)
−
g
(
x
1
,
x
2
)
u
(
t
)
.
(4)
Then,
to
simplify
the
form
of
the
deri
v
ed
control
la
w
and
to
f
acilitate
the
comparison
between
the
zeroing
neurodynamics
PID
controller
and
the
con
v
entional
PID
controller
,
we
dene
the
follo
wing
coef
cients:
K
d
=
λ
1
+
λ
2
+
λ
3
,
K
p
=
λ
1
λ
2
+
λ
1
λ
3
+
λ
2
λ
3
,
K
i
=
λ
1
λ
2
λ
3
.
(5)
If
the
parameters
λ
i
are
arbitrary
comple
x
numbers,
the
corresponding
coef
cients
K
d
,
K
p
,
K
i
might
also
become
comple
x,
rendering
the
controller
ph
ysic
ally
unrealizable.
T
o
address
this,
we
impose
an
additional
requirement:
if
an
y
λ
i
is
comple
x,
it
must
appear
in
a
comple
x-conjug
ate
pair
.
From
(5),
it
is
straightforw
ard
to
v
erify
that
this
ensures
all
coef
cients
are
real-v
alued,
thus
guaranteeing
the
ph
ysical
realizability
of
the
controller
.
In
addition,
to
a
v
oid
the
e
xplicit
inte
gral
item
in
the
control
input,
a
ne
w
state
v
ariable
x
3
(
t
)
which
satises
˙
x
3
(
t
)
=
e
(
t
)
and
x
3
(0)
=
0
is
introduced
into
the
original
order
-2
system
(1),
and
x
e
is
de
ned
as
[
x
1
,
x
2
,
x
3
]
T
.
Then
the
e
xtended
order
-2
system
is
de
v
eloped
as
belo
w:
˙
x
1
(
t
)
=
x
2
(
t
)
,
˙
x
2
(
t
)
=
f
(
x
1
,
x
2
)
+
g
(
x
1
,
x
2
)
u
(
t
)
,
˙
x
3
(
t
)
=
y
d
(
t
)
−
x
1
(
t
)
,
y
(
t
)
=
x
1
(
t
)
.
(6)
IAES
Int
J
Rob
&
Autom,
V
ol.
15,
No.
3,
September
2026:
690–697
Evaluation Warning : The document was created with Spire.PDF for Python.
IAES
Int
J
Rob
&
Autom
ISSN:
2722-2586
❒
693
Finally
,
by
setting
z
3
(
t
)
=
0
in
(3)
and
combining
it
with
(4)
and
(5),
we
obtain
the
general
form
of
the
zeroing
neurodynamics
PID
controller
as
follo
ws:
u
(
t
)
=
1
g
(
x
1
,
x
2
)
(
¨
y
d
(
t
)
−
f
(
x
1
,
x
2
)
+
K
d
(
˙
y
d
(
t
)
−
x
2
(
t
))
+
K
p
(
y
d
(
t
)
−
x
1
(
t
))
+
K
i
x
3
(
t
))
=
1
g
(
x
1
,
x
2
)
¨
y
d
(
t
)
−
f
(
x
1
,
x
2
)
+
K
d
˙
e
(
t
)
+
K
p
e
(
t
)
+
K
i
Z
t
0
e
(
τ
)d
τ
=
1
g
(
x
1
,
x
2
)
(
u
mp
(
t
)
+
u
pid
(
t
))
,
(7)
where
u
mp
(
t
)
=
¨
y
d
(
t
)
−
f
(
x
1
,
x
2
)
is
the
model
compensation
term;
u
pid
(
t
)
=
K
d
(
˙
y
d
(
t
)
−
x
2
(
t
))
+
K
p
(
y
d
(
t
)
−
x
1
(
t
))
+
K
i
x
3
(
t
)
is
the
PID
control
term.
The
follo
wing
is
a
detailed
discussion
for
components
in
the
proposed
controller
.
First,
by
le
v
erag-
ing
e
xact
kno
wledge
of
the
system
dynamics
f
(
x
1
,
x
2
)
,
the
model
compensation
term
u
mp
(
t
)
linearizes
the
nonlinear
order
-2
system
and
simplies
the
tracking
problem.
The
inclusion
of
¨
y
d
(
t
)
directly
embeds
the
de-
sired
trajectory’
s
acceleration
prole
into
the
control
input.
Then,
the
PID
control
term
u
pid
(
t
)
ensures
the
asymptotic
con
v
er
gence
of
the
tracking
error
.
Besides,
the
con
v
entional
PID
g
ains
are
ref
actored
into
the
tun-
ing
parameters
λ
i
(
i
∈
{
1
,
2
,
3
}
),
establishing
an
e
xplicit
correspondence
between
these
parameters
and
the
closed-loop
poles
of
an
order
-3
error
system.
This
point
is
to
be
discussed
in
detail
in
Section
4.
It
is
w
orth
noting
that,
unlik
e
optimization-based
control
strate
gies
or
online
learning
algorithms
t
hat
require
iterati
v
e
computation
at
e
v
ery
sampling
time,
the
ZN-PID
controller
only
in
v
olv
es
basic
arithmetic
op-
erations.
Its
computational
comple
xity
is
O
(1)
per
time
step
and
it
can
be
implemented
on
standard
embedded
microcontrollers.
3.2.
Zer
oing
neur
odynamics
PID
contr
oller
f
or
the
in
v
erted
pendulum
Combining
(2)
and
(7),
the
ZN-PID
controller
for
the
in
v
erted
pendulum
is
obtained
as
u
(
t
)
=
−
l
M
+
m
sin
2
x
1
cos
x
1
m
sin
x
1
cos
x
1
x
2
2
M
+
m
sin
2
x
1
+
¨
y
d
(
t
)
−
(
M
+
m
)
g
a
sin
x
1
l
M
+
m
sin
2
x
1
+
K
d
(
˙
y
d
(
t
)
−
x
2
(
t
))
+
K
p
(
y
d
(
t
)
−
x
1
(
t
))
+
K
i
x
3
(
t
)
!
.
(8)
4.
ST
ABILITY
AN
AL
YSIS
In
this
section,
we
pro
v
e
the
stability
of
the
ZN-PID
controller
.
Specically
,
t
he
st
ability
of
the
ZN-
PID
controller
is
described
by
the
follo
wing
theorem.
Theor
em
1.
When
the
states
of
system
(6)
and
parameters
in
functions
f
(
x
1
,
x
2
)
and
g
(
x
1
,
x
2
)
are
accurately
measured,
the
output
tracking
error
e
(
t
)
of
the
zeroing
neurodynamics
PID
controller
(7)
e
xponent
ially
con-
v
er
ges
to
zero.
Pr
oof
.
By
substituting
(7)
for
the
u
(
t
)
in
(6),
we
get
¨
e
(
t
)
+
K
d
˙
e
(
t
)
+
K
p
e
(
t
)
+
K
i
Z
t
0
e
(
τ
)d
τ
=
0
,
(9)
which
is
equi
v
alent
to
...
e
(
t
)
+
K
d
¨
e
(
t
)
+
K
p
˙
e
(
t
)
+
K
i
e
(
t
)
=
0
.
(10)
Then,
by
performing
Laplace
transform
[28]
on
(10),
we
obtain
(
s
3
+
K
d
s
2
+
K
p
s
+
K
i
)
E
(
s
)
=
(
s
2
+
K
d
s
+
K
p
)
e
(0)
+
(
s
+
K
d
)
˙
e
(0)
+
¨
e
(0)
.
(11)
Therefore,
the
e
xpression
of
E
(
s
)
is
E
(
s
)
=
(
s
2
+
K
d
s
+
K
p
)
e
(0)
+
(
s
+
K
d
)
˙
e
(0)
+
¨
e
(0)
s
3
+
K
d
s
2
+
K
p
s
+
K
i
.
(12)
Zer
oing
neur
odynamics
pr
oportional-inte
gr
al-derivative
contr
oller
for
or
der
-2
system
...
(Y
unong
Zhang)
Evaluation Warning : The document was created with Spire.PDF for Python.
694
❒
ISSN:
2722-2586
From
Routh-Hurwitz
stability
criterion
[29],
the
necess
ary
and
suf
cient
conditions
for
all
roots
of
the
char
-
acteristic
equation
Φ(
s
)
=
s
3
+
K
d
s
2
+
K
p
s
+
K
i
=
0
to
ha
v
e
ne
g
ati
v
e
real
parts
are:
K
d
>
0
,
K
p
>
0
,
K
i
>
0
,
and
K
d
K
p
−
K
i
>
0
.
Since
Re(
λ
i
)
>
0
(
i
∈
{
1
,
2
,
3
}
),
from
(5),
we
kno
w
that
all
coef
cients
K
d
,
K
p
,
and
K
i
are
strictly
positi
v
e,
and
K
d
K
p
−
K
i
=
(
λ
1
+
λ
2
+
λ
3
)(
λ
1
λ
2
+
λ
1
λ
3
)
+
λ
2
λ
3
(
λ
2
+
λ
3
)
>
0
.
Therefore,
all
roots
of
Φ(
s
)
=
0
ha
v
e
ne
g
ati
v
e
real
parts,
which
means
the
solution
e
(
t
)
is
a
linear
combination
of
e
xponentially
decaying
terms.
Thus,
we
ha
v
e
lim
t
→∞
e
(
t
)
=
0
,
pro
ving
the
e
xponential
stability
of
the
zeroing
neurodynamics
PID
controller
at
the
origin
of
the
augmented
error
dynamics.
□
In
f
act,
from
(12)
and
the
denitions
in
(5),
the
characteristic
polynomial
can
be
f
actorized
as
(
s
+
λ
1
)(
s
+
λ
2
)(
s
+
λ
3
)
=
0
,
which
means
that
the
ne
g
ati
v
es
of
the
ZN
parameters
λ
i
are
directly
the
poles
of
the
dynamics
equation
of
the
error
.
Consequently
,
unlik
e
the
classical
Zie
gler–Nichols
PID
tuning
method,
which
typically
determines
controller
g
ains
by
dri
ving
the
system
to
a
mar
ginally
stable
oscillatory
state
and
empirically
estimating
the
ultimate
g
ain
and
oscillation
period,
the
proposed
ZN-PID
frame
w
ork
establishes
an
e
xplicit
analytical
correspondence
between
controller
parameters
and
closed-loop
pole
locations.
Therefore,
the
tuning
process
is
transformed
from
heuristic
oscillation-based
g
ain
adjustment
i
nto
a
systematic
pole-placement
procedure
with
clear
interpretations
of
con
v
er
gence
rate
and
transient-response
beha
vior
.
In
this
sense,
the
proposed
method
a
v
oids
the
trial-and-error
nature
and
near
-instability
operation
commonly
associated
with
classical
Zie
gler–Nic
ho
l
s
tuning,
while
pro
viding
a
theoretically
grounded
and
more
predictable
PID
tuning
methodology
for
nonlinear
order
-2
systems.
5.
SIMULA
TIVE
EXPERIMENTS
AND
VERIFICA
TIONS
In
this
section,
we
use
MA
TLAB
programs
[30]
to
e
x
ecute
rele
v
ant
simulation
e
xperiments
to
v
erify
the
ef
fecti
v
eness
of
proposed
zeroing
neurodynamics
PID
controllers
compared
with
con
v
entional
PID
con-
trollers.
All
simulations
are
performed
using
MA
TLAB
R2022a
and
on
a
W
indo
ws
10
personal
computer
equipped
with
an
Intel
Core
i5-10300H
(2.50
GHz)
CPU
and
16
GB
RAM,
and
the
type
and
maximum
step
size
of
the
solv
er
are
ode15s
and
10
−
4
s,
respecti
v
el
y
.
The
general
form
of
control
input
of
con
v
entional
PID
controllers
is
described
as
u
PID
(
t
)
=
σ
g
[
e
K
d
(
˙
y
d
(
t
)
−
x
2
(
t
))
+
e
K
p
(
y
d
(
t
)
−
x
1
(
t
))
+
e
K
i
x
3
(
t
)]
,
where
σ
g
=
sgn(
g
(
x
1
,
x
2
))
∈
{−
1
,
+1
}
,
and
the
Zie
gler
-Nichols
method
or
other
e
xperience-based
parameter
tun-
ing
methods
is
used
to
obtain
the
PID
parameters.
Then,
we
compare
the
proposed
ZN-PID
controller
with
a
con
v
entional
PID
controller
on
the
in
v
erted
pendulum.
The
in
v
erted-pendulum
parameters
are
selected
as
M
=
1
,
m
=
0
.
1
,
g
a
=
9
.
8
,
and
l
=
0
.
25
.
F
or
the
sak
e
of
simplicity
,
the
units
of
these
parameters
are
omitted.
F
or
all
cases
belo
w
,
the
ZN-PID
parameters
are
λ
1
=
9
+
3i
,
λ
2
=
9
−
3i
,
and
λ
3
=
0
.
5
,
while
the
con
v
entional
PID
g
ains
are
e
K
p
=
100
,
e
K
d
=
10
,
and
e
K
i
=
50
.
The
tracking
ef
fects
of
tw
o
controllers
are
sho
wn
as
Figure
1.
First,
the
control
performance
in
dif
ferent
initial
states
is
sho
wn
as
Figure
1(a)
and
(b).
It
is
seen
that,
when
the
initial
condition
x
1
(0)
is
f
ar
from
π
/
2
,
the
nonlinearity
of
the
system
is
subtle,
thus
bo
t
h
controllers
achie
v
e
good
performance.
Ho
we
v
er
,
when
x
1
(0)
is
near
π
/
2
,
the
nonlinearity
of
the
system
is
ob
vious,
resulting
in
a
decrease
in
control
performance
of
the
con
v
entional
PID
controller
.
In
contrast,
the
proposed
zeroing
neurodynamics
PID
controller
maintains
stable
tracking,
highlighting
its
superiority
in
the
control
of
highly
nonlinear
systems.
Besides,
Figure
1(c)
and
(d)
sho
w
the
response
for
constant
signals
with
dif
ferent
amplitudes.
It
is
seen
that
as
the
amplitude
of
the
constant
reference
signal
increases,
the
performance
of
the
con
v
entional
PID
controller
deteriorates.
When
the
amplitude
e
xceeds
a
certain
v
alue,
the
in
v
erted
pendulum
system
under
con
v
entional
PID
control
e
xhibits
a
phenomenon
where
the
swing
angle
sharply
increases
and
then
sharply
decreases,
often
causing
the
pendulum
to
lose
balance
and
topple
in
actual
en
vironment.
In
contrast,
the
zeroing
neurodynamics
PID
controller
proposed
in
this
paper
illustrates
its
superior
performance
and
superiority
under
the
same
conditions.
Ev
en
with
strong
nonlinearity
brought
by
lar
ge
constant
reference
amplitudes,
the
zeroing
neurodynamics
PID
controller
maintains
stability
,
ef
fecti
v
ely
suppresses
osci
llations,
achie
v
es
rapid
and
smooth
con
v
er
gence
to
the
desired
angular
setpoint
with
minimal
o
v
ershoot.
T
able
1
summarizes
the
simulation
results.
The
ZN-PID
controller
achie
v
es
lo
wer
root
mean
squared
error
(RMSE)
for
the
la
r
ger
constant
references
and
the
lar
ger
initial
of
fset,
while
both
controllers
reach
zero
absolute
steady-state
error
(ASSE)
in
all
four
cases.
IAES
Int
J
Rob
&
Autom,
V
ol.
15,
No.
3,
September
2026:
690–697
Evaluation Warning : The document was created with Spire.PDF for Python.
IAES
Int
J
Rob
&
Autom
ISSN:
2722-2586
❒
695
0
1
2
3
4
5
Time
(s)
-0.2
0
0.2
0.4
0.6
0.8
System Output (rad)
ZN-PID
PID
Desired
(a)
0
1
2
3
4
5
6
Time
(s)
0
0.5
1
1.5
System Output (rad)
ZN-PID
PID
Desired
(b)
1
2
3
4
5
Time
(s)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
System Output (rad)
ZN-PID
PID
Desired
(c)
0
1
2
3
4
5
Time
(s)
-1
0
1
2
3
4
5
6
7
System Output (rad)
ZN-PID
PID
Desired
(d)
Figure
1.
Control
ef
fects
of
PID
and
ZN-PID
controllers
on
the
in
v
erted
pendulum
(a)
Zero
reference
with
x
e
(0)
=
[0
.
8
,
0
,
0]
T
,
(b)
Zero
reference
with
x
e
(0)
=
[1
.
5
,
0
,
0]
T
,
(c)
constant
reference
with
amplitude
0.92,
and
(d)
constant
reference
with
amplitude
1
T
able
1.
Quantitati
v
e
Comparisons
between
ZN-PID
Controller
and
Con
v
entional
PID
Controller
Reference
signal
Initial
state
RMSE
ASSE
PID
ZN-PID
PID
ZN-PID
y
d
(
t
)
=
0
.
92
x
e
(0)
=
0
0.156
0.145
0
0
y
d
(
t
)
=
1
x
e
(0)
=
0
0.843
0.161
0
0
y
d
(
t
)
=
0
x
e
(0)
=
[0
.
8
,
0
,
0]
T
0.104
0.129
0
0
y
d
(
t
)
=
0
x
e
(0)
=
[1
.
5
,
0
,
0]
T
0.373
0.221
0
0
F
or
simplicity
,
units
of
references,
states,
and
errors
are
omitted
in
the
abo
v
e
table.
6.
CONCLUSION
In
this
paper
,
a
zeroing
neurodynamics
PID
controller
for
an
order
-2
af
ne-in-control
nonlinear
sys-
tem
has
been
presented
and
applied
to
an
in
v
erted
pendulum.
The
ZN
construction
reformulates
the
PID
g
ains
through
parameters
that
correspond
directly
to
the
poles
of
the
order
-3
error
dynamics,
turning
controller
tuning
into
a
systematic
pole-placement
task.
Theoretical
analysis
pro
v
es
the
e
xponential
s
tability
of
the
zeroing
neu-
rodynamics
PID
controller
.
Comparati
v
e
simulations
with
a
con
v
entional
PID
controller
sho
w
that
the
proposed
controller
maintains
stable
tracking
for
lar
ge
initial
of
fsets
and
high-amplitude
constant
references,
where
the
con
v
entional
controller
e
xhibits
pronounced
de
gradation.
Future
w
ork
will
consider
hardw
are
v
alidations
and
e
xtensions
to
systems
with
incomplete
model
information.
FUNDING
INFORMA
TION
This
w
ork
is
supported
by
the
National
Natural
Science
F
oundation
of
China
with
number
62376290,
the
Natural
Science
F
oundation
of
Guangdong
Pro
vince
with
number
2024A1515011016,
and
the
Shenzhen
Science
and
T
echnology
Program
with
number
JCYJ20250604175409013.
A
UTHOR
CONTRIB
UTIONS
ST
A
TEMENT
This
journal
uses
the
Contrib
utor
Roles
T
axonomy
(CRediT)
to
recognize
indi
vidual
author
contrib
u-
tions,
reduce
authorship
disputes,
and
f
acilitate
collaboration.
Name
of
A
uthor
C
M
So
V
a
F
o
I
R
D
O
E
V
i
Su
P
Fu
Y
unong
Zhang
✓
✓
✓
✓
Jun
yan
Liu
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
Zhonghua
Li
✓
✓
C
:
C
onceptualization
I
:
I
n
v
estig
ation
V
i
:
V
i
sualization
M
:
M
ethodology
R
:
R
esources
Su
:
Su
pervision
So
:
So
ftw
are
D
:
D
ata
Curation
P
:
P
roject
Administration
V
a
:
V
a
lidation
O
:
Writing
-
O
riginal
Draft
Fu
:
Fu
nding
Acquisition
F
o
:
F
o
rmal
Analysis
E
:
Writing
-
Re
vie
w
&
E
diting
Zer
oing
neur
odynamics
pr
oportional-inte
gr
al-derivative
contr
oller
for
or
der
-2
system
...
(Y
unong
Zhang)
Evaluation Warning : The document was created with Spire.PDF for Python.
696
❒
ISSN:
2722-2586
CONFLICT
OF
INTEREST
ST
A
TEMENT
Authors
state
no
conict
of
interest.
D
A
T
A
A
V
AILABILITY
The
MA
TLAB
source
codes
of
simulati
v
e
e
xperiments
and
the
data
that
support
the
ndings
of
this
study
are
a
v
ailable
on
request
from
the
corresponding
author
,
Zhonghua
Li.
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BIOGRAPHIES
OF
A
UTHORS
Y
unong
Zhang
recei
v
ed
the
B.S.
de
gree
in
industrial
electrical
automa
tion
from
the
Huazhong
Uni
v
ersity
of
Science
and
T
echnology
,
W
uhan,
China,
in
1996,
the
M.S.
de
gree
in
control
theory
and
control
engineering
from
the
South
China
Uni
v
ersity
of
T
echnology
,
Guangzhou,
China,
in
1999,
and
the
Ph.D.
de
gree
i
n
mechanical
and
automation
engineering
from
the
Chinese
Uni
v
ersity
of
Hong
K
ong,
Hong
K
ong,
China,
in
2003.
He
is
currently
a
Professor
with
the
School
of
Intelli-
gent
Systems
Engineering,
Sun
Y
at-sen
Uni
v
ersity
,
Shenzhen,
China.
Before
joining
the
Sun
Y
at-sen
Uni
v
ersity
in
2006,
he
had
been
with
the
National
Uni
v
ersity
of
Sing
apore,
Sing
apore
(Research
Fel-
lo
w);
the
Uni
v
ersity
of
Strathclyde,
Glasgo
w
,
U.K.
(Research
Fello
w);
and
the
National
Uni
v
ersity
of
Ireland,
Maynooth,
Ireland
(Research
Fello
w
/
Research
Scientist)
since
2003.
His
main
research
interests
include
automation,
robotics,
neurodynamics,
computation,
and
optimization.
He
can
be
contacted
at
zh
ynong@mail.sysu.edu.cn.
J
unyan
Liu
recei
v
ed
the
B.S.
de
gree
in
intelligence
science
and
technology
from
Sun
Y
at-sen
Uni
v
ersity
,
Shenzhen,
China,
in
2024.
He
is
curre
ntly
w
orking
to
w
ard
the
M.S.
de
gree
in
Electronic
Information
with
the
School
of
Intelligent
Systems
Engineering,
Sun
Y
at-sen
Uni
v
ersity
,
Shenzhen,
China.
His
research
interests
include
control
theory
and
robotics.
He
can
be
contacted
at
liujy297@mail2.sysu.edu.cn.
Zhonghua
Li
recei
v
ed
the
Ph.D.
de
gree
in
control
science
and
engineering
from
the
South
China
Uni
v
ersity
of
T
echnology
,
in
2005.
He
is
currently
a
F
aculty
Member
with
the
School
of
Intelligent
Systems
Engineering,
Sun
Y
at-sen
Uni
v
ersity
,
Shenzhen,
China.
His
research
interests
include
the
Internet
of
Things,
edge
computing,
articial
intelligence,
and
big
data
applicati
ons.
He
can
be
contacted
at
lizhongh@mail.sysu.edu.cn.
Zer
oing
neur
odynamics
pr
oportional-inte
gr
al-derivative
contr
oller
for
or
der
-2
system
...
(Y
unong
Zhang)
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