非线性离散时间网络控制系统的T-S模糊模型事件触发控制器设计
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"这篇论文探讨了离散时间非线性网络控制系统的事件触发模糊控制器设计，特别是考虑了具有时变通信延迟的情况。文章提出了一种更一般的混合事件触发策略，并基于T-S模糊模型方法和时延系统理论，将原始的非线性NCS重构成一个新的事件触发型网络T-S模糊系统。"
在《应用软计算》期刊的一篇文章中，作者Songlin Hua等人深入研究了在网络控制系统的背景下，如何设计非线性离散时间系统的事件触发控制器。网络控制系统(NCSs)由于其节省资源和提高效率的特性，在现代工业自动化中得到了广泛应用。然而，它们也面临着诸如通信延迟和网络资源限制等挑战。
文章首先介绍了事件触发通信方案，这是一种有效的减缓网络负担的策略。与传统的周期性采样不同，事件触发机制仅在系统状态达到某些预设条件时才进行数据传输，从而减少了不必要的通信开销。作者提出了一种更通用的混合事件触发策略，这可能包括状态变化、误差阈值或其他性能指标触发的数据传输。
接着，利用Takagi-Sugeno (T-S) 模糊模型，文章将非线性系统转化为一系列线性子系统，简化了控制设计的复杂性。T-S模糊模型是一种强大的工具，它能够以结构化的方式近似非线性动态系统，通过一组线性规则来表示整个系统的行为。
考虑到通信延迟的影响，作者采用了时延系统的方法对NCSs进行分析。时变通信延迟是网络控制系统中的一个重要问题，因为它可能导致稳定性丧失和性能下降。通过将这些因素纳入考虑，研究人员能够为事件触发的网络控制设计提供稳定性保证。
文章进一步开发了一种协同设计算法，旨在同时优化控制策略和事件触发机制，以确保系统的性能和稳定性，同时减少网络负载。这种联合优化方法对于实现高效且可靠的网络控制至关重要。
关键词包括：网络控制系统、T-S模糊模型、事件触发通信方案，这表明该研究集中在解决网络控制环境中的关键问题，即如何通过事件触发机制有效地管理和控制通信，以及如何在存在延迟的情况下保持系统性能。
这项工作为非线性离散时间网络控制系统的控制设计提供了一个创新的解决方案，通过引入T-S模糊模型和事件触发通信，提高了系统的效率和鲁棒性，同时解决了通信延迟带来的挑战。这对于提升现代网络控制系统的性能和可靠性具有重要的理论与实践价值。
402
S.
Hu
et
al.
/
Applied
Soft
Computing
30
(2015)
400–411
is
denoted
by
B
x
.
min
(
A
)
and
max
(
A
)
are
used
to
denote
the
minimal
and
maximal
eigenvalue
of
matrix
A,
respectively.
2.
Problem
formulation
2.1.
Physical
plant
Consider
the
following
nonlinear
discrete-time
system
x(k
+
1)
=
f
(x(k))
+
g(x(k))u(k),
(1)
where
x(k)
∈
R
n
,
u(k)
∈
R
m
are
the
system
state
vector,
control
input,
respectively.
x(0)
x
0
is
the
initial
state.
We
assume
that
f(x(k))
and
g(x(k))
are
well
modeled,
which
implies
that
if
the
initial
condition
is
given,
the
precise
prediction
of
the
state
based
on
Eq.
(1)
can
be
derived.
In
this
article,
a
framework
of
the
considered
networked
control
system
(NCS)
consisting
of
a
discrete-time
non-
linear
system
(1),
a
smart
sensor,
an
event
generator,
a
controller,
and
an
actuator
is
schematically
depicted
in
Fig.
1.
This
architecture
represents
a
system
where
actuation
is
inexpensive,
but
transmis-
sion
of
sensor
measurements
to
the
controller
is
expensive.
This
might
be
the
case
if
measurements
are
transmitted
to
the
controller
by
sensors
in
a
sensor
network,
and
each
sensor
has
a
limited
power
budget.
The
type
of
NCS
with
ETS
depicted
in
Fig.
1
is
studied
hereinafter,
and
for
the
convenience
of
narration,
the
following
assumptions
fairly
standard
in
the
NCS
framework,
are
also
made
in
this
work:
Assumption
1.
The
state
variables
of
the
plant
are
available
for
feedback
controller;
Assumption
2.
The
sensor
is
clock-driven,
controller
and
actuator
(consists
of
a
zero-order
holder)
are
event-driven;
Assumption
3.
The
communication
channel
is
implemented
only
in
the
sensor-controller
side
while
the
controller
is
connected
directly
to
the
plant,
and
thus
only
the
sensor-to-controller
delay
k
is
considered,
moreover,
the
delay
k
is
bounded,
i.e.,
k
∈
[
m
,
M
]
,
where
m
and
M
(
0
≤
m
<
M
)
are
positive
integers.
Assumption
4.
The
data
packet
is
transmitted
in
single
packet
without
any
loss,
and
each
data
packet
is
time
stamped.
2.2.
Discrete-time
event
generator
The
function
of
the
event
generator
is
to
check
the
event-
triggering
condition
to
determine
whether
an
“event”
is
generated
or
not.
Once
an
event
is
triggered,
it
implies
that
the
genera-
tor
will
transmit
the
current
sampled
states
to
the
controller.
In
order
to
quantitatively
describe
such
scheme
mathematically,
let
us
denote
the
time
instant
when
an
event
happens
by
k
s
∞
s=0
with
k
s
<
k
s+1
,
k
s
∈
Z
+
.
Without
loss
of
generality,
it
is
assumed
that
the
first
event
is
generated
at
time
instant
k
0
0.
With
the
last
transmitted
state
x
(
k
s
)
at
the
triggering
instant
k
s
and
the
current
sensor
measurement
x
(
k
)
at
the
current
sampling
instant
k,
the
next
triggering
instant
k
s+1
can
be
determined
by
k
s+1
=
inf
k
k
>
k
s
|
[
x
(
k
)
−
x(k
s
)
]
T
W
[
x
(
k
)
−x(k
s
)
]
>
x
T
(
k
)
Wx
(
k
)
+
ε
,
(2)
where
∈
[
0,
1
)
and
ε
≥
0,
k
∈
Z
+
,
W
>
0
is
a
weighting
matrix
to
be
designed
later.
Remark
1.
According
to
the
ETS
in
(2),
define
two
sets
as
1
k
1
,
k
2
,
·
·
·
and
1,
2,
·
·
·
.
Then,
it
can
be
seen
that
1
⊆
.
Moreover,
for
given
parameters
∈
[
0,
1
)
,
W
>
0
and
ε
≥
0,
there
is
no
need
to
transmit
the
current
sensor
measurement
x
(
k
)
at
the
sampling
instant
k
except
those
instants
when
the
event-triggering
condition
(2)
is
satisfied.
In
other
words,
from
(2),
∀k
∈
[
k
s
,
k
s+1
−
1
]
,
it
yields
[
x
(
k
)
−
x(k
s
)
]
T
W
[
x
(
k
)
−
x(k
s
)
]
≤
x
T
(
k
)
Wx
(
k
)
+
ε.
(3)
which
implies
in
the
sensor
node,
only
part
of
the
sensor
mea-
surement
will
be
transmitted
to
the
controller
for
computation.
Compared
with
the
existing
time-triggered
(or
periodic)
transmis-
sion
schemes
used
in
[18,54,55],
the
communication
bandwidth
in
the
network
is
saved
and
the
computation
burden
of
the
controller
is
also
reduced.
Particularly,
in
wireless
sensor
networks,
this
data
transmission
strategy
may
save
the
transmission
energy,
and
thus
increase
the
lifespan
of
the
battery
of
the
sensor
nodes.
Remark
2.
It
is
worth
noting
that
the
considered
description
of
ETS
(2)
comprises
several
classes
of
ETSs
investigated
in
the
literature.
For
example,
if
=
0
and
ε
=
0,
the
ETS
(2)
is
reduced
to
the
classic
periodic
sampling
scheme,
that
is,
k
s+1
=
k
s
+
1,
or
equvialently
k
s+1
=
inf
k
k
|
k
>
k
s
+
1
;
if
=
0,
W
=
I,
and
ε
>
0,
the
ETS
(2)
is
reduced
to
the
so-called
absolute
ETS,
that
is,
k
s+1
=
inf
k
k
>
k
s
|
x
(
k
)
−
x(k
s
)
>
√
ε
,
which
is
simi-
lar
to
that
in
[22,25,27,28];
if
∈
(
0,
1
)
and
ε
=
0,
the
ETS
(2)
can
be
simplified
as
the
so-called
relative
ETS,
that
is,
k
s+1
=
inf
k
k
>
k
s
|
[
x
(
k
)
−
x(k
s
)
]
T
W
[
x
(
k
)
−
x(k
s
)
]
>
x
T
(
k
)
Wx
(
k
)
,
which
is
proposed
in
[6,14,32,38,53].
Furthermore,
notice
that
the
state
x
(
k
)
in
(2)
denotes
the
current
sensor
measurement,
if
we
use
the
last
transmitted
state
x
(
k
s
)
instead
of
the
state
x
(
k
)
in
(2),
then
the
ETS
(2)
can
be
formulated
as
k
s+1
=
inf
k
k
>
k
s
|
[
x
(
k
)
−
x(k
s
)
]
T
W
[
x
(
k
)
−
x(k
s
)
]
>
x
T
(
k
s
)
Wx
(
k
s
)
+
ε
,
(4)
where
the
parameters
,
ε
and
W
are
defined
as
in
(2).
Compared
with
(2),
under
the
ETS
(4),
the
event
generator
only
needs
to
com-
pute
the
threshold
x
T
(
k
)
Wx
(
k
)
at
the
triggering
instant
k
s
not
at
every
sampling
instant
k.
Therefore,
the
ETS
(4)
may
not
only
save
network
bandwidth
resources
than
that
(2),
but
also
reduce
the
computation
burdon
of
processor
in
NCSs
to
some
extent.
On
the
other
hand,
it
can
be
seen
that
when
∈
(
0,
1
)
and
ε
=
0,
the
ETS
(4)
is
reduced
to
the
so-called
PETS
below
k
s+1
=
inf
k
k
>
k
s
|
[
x
(
k
)
−
x(k
s
)
]
T
W
[
x
(
k
)
−
x(k
s
)
]
>
x
T
(
k
s
)
Wx
(
k
s
)
,
(5)
which
is
discussed
in
[17,31,33].
Based
on
the
above
discussions,
it
can
be
seen
that
the
proposed
ETS
(2)
is
more
general
than
some
existing
ETSs.
2.3.
Networked
T-S
fuzzy
control
with
discrete-time
event
generator
In
this
subsection,
a
networked
T-S
fuzzy-mode-based
con-
troller
is
designed
via
parallel
distributed
compensation
(PDC)
technique
[39]
to
stabilize
the
nonlinear
system
(1)
with
ETS
(2).
For
the
analysis
of
convenient
later,
we
first
use
T-S
fuzzy
model
to
approximate
the
nonlinear
system
(1).
Based
on
[36],
the
nonlinear
system
(1)
can
be
represented
by
some
local
linear
dynamic
systems
with
their
linguistic
description.
The
ith
rule
of
the
system
is
represented
by
a
T-S
fuzzy
model
Plant
rule
i
:
IF
1
(k)
is
F
i
1
and,
·
·
·,
and
g
(k)
is
F
i
g
,
THEN
x(k
+
1)
=
A
i
x(k)
+
B
i
u(k),
(6)
where
i
∈
S
=
1,
2,
·
·
·,
r
,
r
is
the
number
of
IF-THEN
rules;
j
(k),
j
∈
H
=
1,
2,
·
·
·,
g
represent
premise
variables;
F
i
j
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