BULLETIN OF THE POLISH ACADEMY OF SCIENCES
TECHNICAL SCIENCES, Vol. 65, No. 1, 2017
DOI: 10.1515/bpasts-2017-0005
Observer-based leader-following formation control
of uncertain multiple agents by integral sliding mode
D.W. QIAN1*, S.W. TONG 2, and C.D. LI 3
1
School of Control and Computer Engineering, North China Electric Power University, Beijing, 102206 China
2
College of Automation, Beijing Union University, Beijing, 100101 China
3
School of Information and Electrical Engineering, Shandong Jianzhu University, Jinan, 250101 China
Abstract. This paper investigates the formation control problem of multiple agents. The formation control is founded on leader-following approaches. The method of integral sliding mode control is adopted to achieve formation maneuvers of the agents based on the concept of graph
theory. Since the agents are subject to uncertainties, the uncertainties also challenge the formation-control design. Under a mild assumption that
the uncertainties have an unknown bound, the technique of nonlinear disturbance observer is utilized to tackle the issue. According to a given
communication topology, formation stability conditions are investigated by the observer-based integral sliding mode formation control. From
the perspective of Lyapunov, not only is the formation stability guaranteed, but the desired formation of the agents is also realized. Finally, some
simulation results are presented to show the feasibility and validity of the proposed control scheme through a multi-agent platform.
Key words: multi-agent system, formation control, integral sliding mode control, uncertainty, nonlinear disturbance observer, stability.
1. Introduction
Recently, multi-agent systems have been paid considerable attention [1–3]. The technology of multi-agent systems has been
praised as a novel paradigm for designing and implementing
many real systems, such as distributed sensor networks [4], robots [5–8], satellite flying [9] and aerial vehicles [10]. In reality,
a multi-agent system has some advantages over a single agent,
including but not limited to efficiency, reliability, extensibility,
robustness and flexibility [11, 12]. Increasing applications involve multiple agents that have to work together, which implies
the requirement of multiple agents coordination.
Among a variety of cooperative tasks, the consensus
problem becomes attractive because it integrates both graph
theory and control theory. The consensus problem contains
several typical control problems, i.e., cooperative control, formation control and flocking. See [13] for a complete review of
recent approaches to this area. As a branch, formation control
works on controlling agents in a multi-agent system to form
up and move in specified geometrical shapes [14]. The background of formation control originates from the real world. For
instance, the agents need to maintain some formations when
they move at disaster sites, warehouses and hazardous areas.
One of coordinated control schemes for multi-agent formations is defined as leader-following; one agent is designed
as the leader and other agents are followers. The sole leader is
self-commanded and moves along a predefined trajectory. The
followers track the leader. A cooperative task can be achieved
by the interactions between these followers and the leader. The
scheme in [4, 6, 9, 10] has been successfully applied to the anal*e-mail: [email protected]
ysis and design of multiple agent systems. However, the classic
leader-following scheme is centralized, meaning that the control
systems in [4, 6, 9, 10] heavily depend on the leader and suffer
from the “single point of failure’’ problem. With the development
of communication technology, it is desired to consider the communication topology in the scheme because the adaptability and
practicability of multiple agents can be strengthened [15, 16].
The leader-following scheme gains popularity in multi-agent
formations because its dynamics are experimentally modelled,
but the internal formation stability can be theoretically guaranteed. Adopting such a scheme, various control methods have
been developed for multi-agent formations, that is, robust control [17], dynamic output feedback method [18], adaptive fuzzy
approach [19], multi-step predictive mechanism [20], iterative
learning technique [21], and neural network-based adaptive design [22], to name but a few. A systematic review on this topic
is presented by Oh, Park and Ahn [23].
As a nonlinear feedback design tool, the sliding mode control (SMC) technique is popular because of its invariance [24],
meaning that matched uncertainties in an SMC system can
be suppressed by the invariance. Some SMC-based methods
have been addressed to solve the formation-control problem of
multi-agent systems, that is, first-order SMC [25, 26], terminal
SMC [27], backstepping SMC [28], etc. Previous contributions
[15, 25–28] have verified the feasibility of the SMC methodology for multi-agent formations.
Compared to other SMC methods, the integral SMC design
can guarantee an integral SMC system against matched uncertainties from the initial time, indicating that an entire response
of the system is of invariance. Some successful applications of
the integral SMC method have been reported in industries, for
example, power systems [29], hypersonic vehicles [30] as well
as multi-agent systems [15, 31, 32].
35
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D.W. Qian, S.W. Tong, and C.D. Li
Uncertainties exist everywhere. Each individual agent may is round with a radius of r. Use index i to refer to the robot in
suffer from uncertainties, i.e., external disturbances, unmodelled Fig. 1. The robot’s movement is based on two separately driven
dynamics and parameter perturbations. Originated from the un- wheels placed on either side of the robot’s body, where (xci, yci)
certainties of individual agents, formation dynamics of multi- are the center of this robot, (xLi, yLi) are the center of the left
agent systems become uncertain. In previous works on the SMC- wheel, (xRi, yRi) are the center of the right wheel and are the
based multi-agent formations [15, 25–28, 31, 32], uncertainties center of the robot’s body. Then, a vector qi = [xhi yhi θi]T can
are considered because they adversely affect the formation sta- be defined to describe the robot’s posture, where (xhi, yhi) are
bility. Two solutions can be summarized from those works. One the robot’s head in the inertial frame and θi denotes the oriensolution is to discuss the formation stability by means of graph tation angle of a mobile frame [15]. Under the assumption of
theory [15, 32]. The other is to analyze the formation stability pure rolling and non-slipping motion, the robot is subjected to
in light of Lyapunov’s theorem [25–28, 31]. To guarantee the [¡sinθi cosθi 0] ¢ [xci yci θi] = 0.
formation stability, both solutions have to assume that the forThe Lagrangian equation of the agent can be obtained as
mation uncertainties are bounded. Unfortunately, the boundary Li = Ki ¡ Pi, where Ki denotes the kinetic energy of the agent
of uncertainties is rather hard to exactly measure or know in and Pi means the potential energy. Concerning the agent, it can
advance. The lack of such a boundary may result in severe prob- only move in the horizon so that its potential energy remains
lems, i.e., decrease of the formation robustness, deterioration of unchanged, that is, Pi ´ 0. Then, Li can be written by
the formation performance as far as deficiency of the formation
stability. To obtain the important information, adaptive approx- Li = Kbi + Kli + Kri(1)
imation of the formation uncertainties is desired.
1 2
1
2
2
2
1 ¡
1 = ¡
2 ) +
̇ ci2 ) + technique
of nonlinearitdisturbance
(NDO) has In (1),InK(1),
m (x ̇ 2 + y
, liKli=
= 1¡12mmww((x
Ib2θ̇Ii2bθ, ̇ iK
ẋ ̇ 2lili + y
+ ẏ ̇ li2li) +
)+
To obtain theThe
important
information,
is desired observer
to adaptively
bi =Kbi
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1
1
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22
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̇
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1
been
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and
im+ ¡
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θ
and
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= ¡
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(x
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) + ¡
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θ
,
where
m
and
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2
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w
ri
w
w
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b
i
ri
ri
i
approximate
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uncertainties.
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) +2 I2bIθ̇wi2θ̇,i K, liwhere
and
Iẏbliare
)are
+
To obtain the important information, it is desired to adaptively 2 IwInθ̇i(1),
= 2 mm
w b(ẋand
b (wẋ(
bi ==2 m
2m
ci ri ẏciof
li +
proving
robustness
[33].
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applications
of
NDO
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the
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The technique
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and
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robot’s
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To obtain
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= wheel,
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To
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ainties.
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This
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Based
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Thisremains
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where
τ
=
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]
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T
i
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paperthe
touches
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where τiT=model
[τLi τRiof
] the
is the
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developed
toThis
achieve
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Based
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[15]
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] is M(q
the torque
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the
i =dynamic
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istouches
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of agents.
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By
the
Lagrangian
i )q̈matrix.
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i , q̇
i )q̇
i =the
i )τmethod,
i
observer
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)
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Lagrangian
method,
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formation
stability
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By
a
observer
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of
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developed
to
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and
B(q
)
is
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time-varying
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method,
i
Based on
the observer,
an integral
sliding sliding
mode controller
is theisdynamic
model M(q
of
the)agent
[15]
can
Based
on numerical
the observer,
an integral
mode
controller
the
model
of
agent
can
be formulated
by order
(3)
dynamic
τiTby
+theC(q
q̇[15]
)be
q̇iformulated
= B(q
(3)
i q̈
i[15]
i ,be
iformulated
i )B
comparison,
some
results
illustrate
that
the presentC(q
(qi ) in
where
the matrices
M(q
Based
on the
observer,
an
integral
sliding
modeofcontroller
isThe
thea dynamic
model
of the agent
i ),can
i , q̇i ) andby
formation
stability
is
presented
in
the
sense
Lyapunov.
By
developed
to
achieve
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maneuvers
of
agents.
2.
Modelling


developed to achieve the formation maneuvers of agents. The
M(q
τ
)
q̈
+
C(q
,
q̇
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q̇
=
B(q
)
(3)
i M(q
i
i C(q
i i , q̇ )q̇ i =i B(q )τ T
developed
todesign
achieve
the
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maneuvers
of Lyapunov.
agents.
The
ed
control
isnumerical
preferable.
(3)
m
mh
sin
θ
i )q̈i +i ),
i ii, q̇
i i )0 andi B
i (q
comparison,
some
results
illustrate
that
the
presentformation
stability
is
presented
in
the
sense
of
By
a
C(q
)
in
order
where
the
matrices
M(q
i
i
M(qi )q̈i + C(q
(3)
formation stability is presented in the sense of Lyapunov. By a
i , q̇i )q̇i = B(qi )τi

T (q


ormation
isissingle
presented
inresults
the
sense
of Lyapunov.
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a
̇ i) and
where
the matrices
2.1.
One
agent.
A multi-agent
system
contains
N iden),
BTi(q
)Tin
order
are
comparison,
some
numerical
illustrate
that thethat
presentC(qim
, C(q
q̇C(q
and
B)m
)B
order
where
matrices
M(q
ed
controlstability
design
preferable.
iin
i),
i )i, q
are the
determined
by i),M(q
,
0 and
mh
sin
θθi i de0
−mh
cos
comparison,
some
numerical
results illustrate
the present
,
q̇
(q
)
in
order
where
the
matrices
M(q


i
i
i
i
T

omparison,
some
numerical
results
the presentB (qi )mh
in sin
order
i ) and
tical
agents.
Displayed
in illustrate
Fig. 1, thethat
single
agent underwhere
consid-the matrices M(qi ), C(q
ed control
design
is
preferable.
m i , q̇m
0
θ

i
ed
control
design
is preferable.
2.d control
Modelling
0
mh
sin
θ
are
determined
by
0 0θi −mhmh
mcos
sin
design
 m  mh
Icosi θi ,
sinθθi i −mh
erationisispreferable.
a differential wheeled robot in the horizon. The robot

are determined
by
,, �
termined
by
θ
0
m
−mh
cos

i
 θ̇ cosby
 cos
θ ,
0 θi � −mhmcos θi −mh
 sin
2. Modelling
0are0determined
are determined
bymh
0 θi mh
m
−mh cos θi , I i 
 i mh
2.1.2.One
single agent A multi-agent system contains N idenModelling


sin
−mh
cos
I
θmh
θ
θ
θ
sin
−1
cos
i sin
i
2.
Modelling

i
i
I �.
1θi� −mh cos θi
θθθi ii −mh
2.
mhθ̇mh
icalModelling
agents.
Displayed
inmulti-agent
Fig. 1, the single
under
con-   0 0 −mh
θ̇i cos
sin
I
θi
and
i sin
2.1.
One single
agent A
systemagent
contains
N iden� cos
� −1
r �
θ
0 0 0mh0θ̇i cosmh

�
iθ̇ cos θ 
cos
θ
θ
sin
1
2.1.
One
single
agent
A
multi-agent
system
contains
N
ideni
i
i
i
2.1.
One
singlein
agent
A
multi-agent
contains
N iden�
� θ −1
sideration
is aDisplayed
differential
wheeled
robot
in system
the
horizon.
The
sin θθi sin
−1
cos1 θi cos
.
ical
agents.
Fig.
1,
the
single
agent
under
conθ̇i cos
θi θ̇0i sin
0 0 0
mh0
θand
−mh
i 1and
2.1.
One
single
agent
A multi-agent
system
contains
N
iden-coni
i .
r
1
tical
agents.
Displayed
in
Fig.
1,
the
single
agent
under
θ̇
θ
0
0
−mh
sin


i
i
θi 2rθsincos
θsin
−1
cosand
θi 1 ..
 θ̇i sin1 θi r
 con-  0 0 −mh
i θ
tical
agents.
Displayed
in
1,index
the
single
agent
under
i sin
robot
is round
with a in
radius
ofthe
r. Fig.
Use
i tohorizon.
refer
to
the
2
cos
θ
1
sideration
isDisplayed
a
differential
wheeled
robot
in
the
The
i
i
.
and
icalsideration
agents.
Fig.
1,
single
agent
under
conθ̇
θ
0
0
−mh
sin


=θm
00 (3),
0 i , I =i I0b + 2Irw + 2mw l + m
coshθ , msin
1 2mw , h is
is a differential
wheeledwheeled
robot inrobot
the horizon.
The The0 In
i b+
sideration
is robot’s
differential
thetwo
horizon.
cos θi sin θib i 1
0 00
0
robot
in
Fig.
1.with
The
movement
is
sepaideration
isround
a differential
wheeled
robot
inindex
thebased
horizon.
The
robot
is is
round
aaaradius
of
r.
Use
index
itoin
toon
refer
to
the0 0theInlength
2 2+ m and
2, m
0
between
the
agent’s
center
the
head
robot
with
radius
of
r.
Use
i
refer
to
the
2
+
2I
+
2m
l
h
=
m
+
, hthe
is
(3),
,
I
=
I
w wl + m
w b h2 , m b=22mb + 2m
robot
is round
with on
a radius
ofside
r. Use
index
i to refer
to the
,+hand
is2ml,wis
In (3), ,InI (3),
= Ib,+I 2I
2mwbb
rately
driven
wheels
placed
either
robot’s
body,
=bwthe
Ibb+
+2m
2I
+ 2m
l + and
mwblh + m
, m b=
2m
h isw ,
obot
isin
round
with
a radius
of movement
r.
Use index
iof
to the
refer
totwo
the
w + 2m
w + 2m
w
2w
2w
robot
Fig.
1.
The
robot’s
is
based
on
sepaIn
(3),
I = I
+ 2I
h
,
m = m
+ 2m
robot
in
Fig.
1.
The
robot’s
movement
is
based
on
two
sepab
distance
between
agent’s
center
one
wheel.
+
2I
+
2m
l
+
m
h
,
m
=
m
+
2m
,
h
is
In
(3),
,
I
=
I
wthe agent’s
w agent’s
w head
length
the
center
and lisisthethe
b between
b
bheadthe
robot in Fig. 1. The robot’s movement is based on two sepathe the
length
between
center
and
theand
and
l isand
the
the
between
the agent’s
center
and the
head
obot
indriven
movement
is side
based
on
two
where
(xFig.
yciwheels
)The
arerobot’s
the
center
of
this
robot,
(x
,robot’s
ysepaare
the
ci , 1.
Li
Li ) body,
h islength
the
length
between
theand
agent’s
center
headl and
l is
rately
on
either
the
body,
rately
driven
wheels
placed
on
either
ofofside
the
robot’s
thebody,
length
between
the
center
the
head
andand
lwheel.
isthe
theunexplained,
From
Fig.
1,agent’s
two
symbols
of
the
agent
are
rately
drivenplaced
wheels
placed
onside
either
of
the
robot’s
distance
between
the
agent’s
center
and
one
wheel.
distance
between
the
agent’s
center
and
one
distance
between
the agent’s
centercenter
and one
wheel.
ately
driven
wheels
placed
on
either
side
of
thecenter
robot’s
body,
center
of
the
left
wheel,
(x
,
y
)
are
the
of
the
right
the
distance
between
the
agent’s
and
one
wheel.
where
,
y
)
are
the
center
this
robot,
(x
,
y
)
are
the
Ri
Ri
where
(xci(x
,
y
)
are
the
center
of
this
robot,
(x
,
y
)
are
the
ci
ci
Li
Li
distance
between
the
agent’s
center
and
one
wheel.
ci (xci , yci ) are the center of this robot,
Li (xLiLi , yLi ) are the
thatFrom
is,
the
velocity
and
the
rotation
angular
velociwhere
From
Fig.
1, linear
two
symbols
ofvithe
agent
unexplained,
Fig.
1, 1,
two
symbols
of
the
agent
are
unexplained,
From
Fig.
two
symbols
ofof
the are
agent
where
(xand
yare
)left
are
the
center
of,, this
robot,
(xcenter
, yLiThen,
) are
ci
Licenter
From
two
symbols
agentare
areunexplained,
unexplained,
wheel
the
center
of
robot’s
body.
a right
vector
center
the
left
wheel,
(xRi
the
T the
Rithe
Fig.
1,Define
two Fig. 1,
symbols
of
the
agent
are
unexplained,
center
ofci ,of
the
yyRi
are
the
ofthethe
the
right
Ri
center
of wheel,
the
left (x
wheel,
(x))Riare
, yRi
) are
the of
center
of
the From
right
that
is,
the
linear
velocity
v
and
the
rotation
angular
velocity
ω
ξ
ω
ξi ex.
a
vector
=
[v
]
.
Then,
q̇
=
T
(q
)velocii
i
i
i
i
i
i
thatthat
is, the
linear
velocity
vivviand
rotation
angularvelociis,
the
linear
velocity
and the
the
rotation
angular
T wheel,
�
enter
the
)to
are
the center
the right
Ri ,ofyRi
that
is,velocity
the
linear
velocity
the
rotation
angular
velocity
wheel
are
thecenter
center
the
robot’s
body.
Then,
aThen,
vector
T i. and
q
θleft
can
be(x
defined
describe
theof
robot’s
posture,
[xofhi
yand
that
is,ωthe
linear
vξii �and
the
rotation
angular
velocii ] the
i=
hiare
wheel
and
of
the
robot’s
body.
Then,
a
vector
wheel
and
are
the
center
of
the
robot’s
body.
a
vector
T
ty
ω
ξ
.
Define
a
vector
=
[v
]
Then,
q̇
=
T
(q
)
exT
i
i
i
i
i
i
ty ωtyω
ξcos
a�vector
.. θThen,
exa vector
Then,
==
(q(q
0q̇q̇i�i exθ[v
� i = T(q
wheel
and
the
center
ofrobot’s
the robot’s
body.
Then,
a vector
i . Define
i =
i]]sin
i )iξ)iξexi . iω
iξT=
ihere
i[vii]iTω
.aDefine
Define
ω
.ωiThen,
)ξTi Texists,
qi [x
θ)Ti ]Tare
be
defined
to
describe
the
robot’s
posture,
=
[xyhiqare
T
hi
ty ωi . ists,
Define
vector
=
[v�i �θξωi = [v
q̇�i 0= iTq ̇ (q
where
(x
,iy=
yThi
head
intothe
inertial
and
i ] . iThen,
i )ξ�
i .i Substituting
θbe
can be defined
describe
the frame
robot’s
posture,
ycan
=cos
here
TT�(qaξi )ivector
θ
to
describe
the
robot’s
posture,
=
i ] defined
hican
hi the
i ][x
hiy hi
hiθ
sin
θ
i
i
qqii=
]
can
be
defined
to
describe
the
robot’s
posture,
[x
T
sin
0
θ
θ
cos
hi hi
cos
where
(x
) are
robot’s
head
inhead
the inertial
frame
and
sinθθiθi ii 0hsin
cos θii .i Substituting
10 . Substituting
=T cos) θ=i −h
ists,
here
T (qi )T T
hii , yorientation
θwhere
the
angle
of
a mobile
[15].
Under
where
ythe
are
the robot’s
in
the inertial
frame
andists,
sin
i denotes
ists,
hi , the
hi ) robot’s
)hi(xare
head
in inertial
the frame
inertial
frame
and
T here
T
. Substituting
) i=
here
T (qi(q
−h sin θ−h
h cos
1.. Substituting
θi h cos
where
(x(x
, ,yhiythe
)hiare
the robot’s
head
ina mobile
the
frame
and
i sin
Substituting
(q
)
=
T
(q
) = q ̇ i = T(qi)ξi
ists,
here
T
hihi
θ
denotes
orientation
angle
of
frame
[15].
Under
i
i
1
θ
θ
i
i
i
(3)
yields
−h
sinθθii 1h cos θi 1
θthe
orientation
angle
of a mobile
frame
[15].
Underq̇i = T (qi )ξi into
assumption
thatthe
pure
rolling
and
non-slipping
motion,
the
i denotes
−h
sin
h
cos
θ
denotes
orientation
angle
of
a
mobile
frame
[15].
Under
i
i
θθhe
denotes
the
orientation
angle
of
a
mobile
frame
[15].
Under
q̇
=
T
(q
)
ξ
into
(3)
yields
i the assumption that pure rolling and non-slipping motion, the
i
i iT (q )ξ into (3) yields
the assumption
that
and
i=
i into
i
robot
is subjected
to
[−rolling
sin
θipure
θnon-slipping
θi ] =motion,
cosrolling
· [x
0.themotion,
q̇ii )=ξq̇into
(q(3)
ξyields
(3))ξ̇yields
i 0]non-slipping
ci ynon-slipping
ci motion,
q̇i =the
T (q
(3)
herobot
assumption
that
pure
rolling
and
the
i)
i yields
he
assumption
that
pure
and
i Tinto
(4)
M̃(q
Fig.
1.
Schematic
diagram
of
a
single
agent
isrobot
subjected
to
[−
sin
θ
θ
θ
cos
0]
·
[x
y
]
=
0.
i + C̃(qi , q̇i )ξi = B̃(qi )τi
i
i
ci
ci
i
is subjected to [− sin θi cos θi 0] · [xci yci θi ] = 0.
ξ̇
)
+i C̃(q
(4)
M̃(q
i
i
i , q̇i )ξi = B̃(qi )τi
(4)
M̃(q
robot
subjected
θi cos
obot isissubjected
to to
[−[−
sinsin
θi cos
θi 0]θ·i [x0]ci·y[x
=θ0.i ] = 0.
i )ξ̇i + C̃(qi , q̇i )ξi = B̃(qi )τi
ciciθiy] ci
ξ̇
ξ
τ
)
+
C̃(q
,
q̇
)
=
B̃(q
(4)
M̃(q
ξ̇
ξ
τ
)
+
C̃(q
,
q̇
)
=
B̃(q
)
M̃(q
i B̃(q
i i
i i i iand
i
ii ii order are
i i determined (4)
by
where
M̃(q
i ), iC̃(q
i , q̇i )B̃(q
i ) inare
C̃(q
,),q̇C̃(q
order
determined
by
where
M̃(q
i ), M̃(q
i ) and
i ) inB̃(q
,
q̇
)
and
)
in
order
are
determined
by
where
T (q )M(q )T(q
i T
iT i
i
T (q
T
),
T
(q
)[M(q
)
Ṫ(q
)
+
C(q
,
q̇
)T(q
)]
and
),
C̃(q
,
q̇
)
and
B̃(q
)
in
order
are
determined
by
where
M̃(q
i
i
i
i
i
i
i
i
i
i
i
i
i
),
C̃(q
,
q̇
)
and
B̃(q
)
in
order
are
determined
by
where
M̃(q
T
)M(q
)T(q
),
T
(q
)[M(q
)
Ṫ(q
)
+
C(q
,
q̇
)T(q
)]
and
36
Bull.)i +
Pol.
Tech.
65(1) 2017
i i� i )T(q
i�T (qii )[M(q
i TT (q
i )M(q
i T
i Ac.:, iq̇
�i i ),
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C(q
i
i
i
i )T(qi )] and
T (q )[M(q
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�
T
T
TT (qi )M(q
)T(q
),
T
)
Ṫ(q
)
+
C(q
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q̇
)T(q
)]
and
i
i
i
i
i
i
i
i
T (qi )M(q
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T
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T (qi )B(q
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.
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TT (q
i )B(qi )−1
In (4),
C̃(q
,
q̇
)
=
0
where
0where
2 ×is2 azero
−1
1
iC̃(q
i i , q̇,i )2×2
2×2 is00a2×2
In (4),
=
0
where
2 ×matrix.
2 zero matrix.
2×2
In (4),
C̃(q
q̇
)
=
0
i
i
2×2
2×2 is a 2 × 2 zero matrix.
w
w
w
b symbols
bagent are bunexplained,
ht
yi
From
Fig.
1, two
of the
the and
ht
distance
between
the
agent’s
center
oneare
wheel.
the
that
is,From
the
linear
velocity
and
rotation
angular
velocioaresepathat
is,
the
linear
vvii and
angular
Fig.
1,velocity
twothesymbols
ofthe
therotation
agent
unexplained,
the
length
between
agent’s
center
and
the head
and lvelociis the Assumption 2: Both δxi and δyi are slowly time-varying, inht
the
Fu
or
T
that
is,
the
linear
velocity
v
and
the
rotation
angular
velociright
or
1,vector
two
agent
are
se body,
ty ω
ωFrom
ωof
ξii exDefine
vector
=i [v
]T the
Then,
q̇�iiangular
=T
Tunexplained,
(qivelociexi))ξ
that
is, Fig.
the
velocity
viiicenter
and
rotation
ty
ξξsymbols
ω
.. Define
aalinear
..and
Then,
q̇
=
(q
between
agent’s
one wheel.
�the
iidistance
ii = [v
ii]Tthe
or
ght
lower
�
�
δ̇
δ̇
≃
0
and
≃
0.
dicating
e,
xi
yi
vector
tythat
ωty
Define
a vector
[v ωω
q̇�i =
(qi))ξξvelociT. . Then,
e,
are
the
i . is,
i ξ= =
i ] sin
i extheFig.
linear
velocity
the
rotation
angular
� �ξsymbols
ω
a vector
]the
TT(q
θviiii[vand
θThen,
cos
From
two
agent
i . Define
i θ
i of isin
i i exe,
�i .=unexplained,
00 q̇are
cos
T (q ) 1,
ctor
ii
Tθ
Assumption
1
indicates
the uncertainties are bounded by an is a s
osture,
T
Substituting
=
ists,
here
T
dde right
ty
ω
ξ
ω
ξ
.
Define
a
vector
=
[v
]
.
Then,
q̇
=
T
(q
)
exi
sin
00 angular
θviθi iand
cosicos
.�i Substituting
ists, that
here
i is,Tthe
i leader-following
ithe
i i
ii
i ) = Observer-based
linear
velocity
rotation
velocisinθθ
�
T (q
formation
control
of
uncertain
multiple
agents
by
integralFrom
slidingAssumption
mode
d
−h
sin
h
cos
1
θ
θ
T
ure,
i
i
.
Substituting
(q
)
=
ists,
here
T
me
and ists, here T i (qi ) = −h sin θi h cos
unknown constant, which is mild.
2, δxi and R (N−
θi 1 q̇ .=Substituting
ervector
er
ty ωi . Define
ξcos
a vector
[v
]Tcos
. θThen,
sin
0�i T (qi )ξi exθ
θ
i =
ii ω
ih
−h
sin
h
cos
1
θ
i
i
−h
sin
1
θ
θ
i
i
�
T
i
er
and
q̇iiists,
=T
T (q
(q
)ξ into
(3)
yields
δyi seem almost constant. Considering the technical contents,
eUnderq̇
. Substituting
(qi(3)
) =yields
here
=
eosture,
ii )ξiiTinto
θi θi 0 1
−hcos
sinθiθi hsin
cos
q̇i =q̇ists,
Ti (q
)(q
ξi iinto
yields
=T
)T
ξiT into
(3)
yields
e and
on,
the
ihere
me
der
Assumption 2 hints the designed observer could evaluate or
.
Substituting
(q(3)
)
=
i
q̇iii ))ξξhii cos
+ C̃(q
= B̃(q
(4) δyi seem almost constant. Considering the technical contents,
C̃(q
B̃(q
M̃(q
i(3)
i ,, θ
−h sin
θi ii1))ττii (4)
+
q̇
=
(4)
))ξ̇ξ̇ii yields
i
i
into
Under q̇i = T (qi )ξi M̃(q
the
calculate
δxi and2 δindicates
δxi
faster
than the observer
change rates
ξ̇i C̃(q
+ C̃(q
, iq̇)iξ)iξi==B̃(q
B̃(qii))ττii
(4)
M̃(q
yi munch
ξ̇ii)+
(4)
M̃(q
i )(3)
i , iq̇
Assumption
that
the designed
could of
evalA
q̇i M̃(q
= T (q),i )ξC̃(q
yields
on, the
i into
,
q̇
)
and
B̃(q
)
in
order
are
determined
by
where
and
.
In
this
sense,
and
could
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δ
δ
δ
i
i
i
i
yi
xi
yi
),
C̃(q
,
q̇
)
and
B̃(q
)
in
order
are
determined
by
where
M̃(q
q̇
̇ ii)i +
where M̃(q
M(q
C̃(q
and
B̃(q
)i )i in
order
are
determined
by(4) uate or calculate δxi and δyi much faster than the change rates
i
i iii, q
i ξ̇
i iq̇
ξ
τ
C̃(q
,
)
=
B̃(q
)
)
M̃(q
ii),
i
i
i
i
),
C̃(q
,
)
and
B̃(q
in
order
are
determined
by
where
Ti )(q
C̃(q
B̃(qi )Ṫ(q
in
are
determined
by
where
M̃(qii),
i , Tq̇T
TTT (q
(qTiiTT)M(q
)M(q
)T(q
),
)[M(q
++
C(q
q̇ )T(q
)T(qi)]
)] and
and
ξ̇TTand
C̃(q
)ξiorder
B̃(q
(4)
)(q
M̃(q
constants
by the
observer.
i(q
i )τiiii,,,, q
T
T
)[M(q
Ṫ(q
))) + C(q
+
C(q
q̇
)T(q
and
(q
T
of δxi and
δyi. In
this sense, δxi and δyi could be treated as almost
�),
� � i ,ii ))iq̇i)Ṫ(q
i )T(q
ii�
ii+
iii=
ii )]and
i)T(q
i),
i)[M(q
(qi)M(q
),TiT
(q
)iṪ(q
C(q
q̇ ̇ iiii)T(q
i )M(q
i )T(q
i�
i )[M(q
i )+
i, q̇
i )]
�
TTwhere
(qTi )M(q
)T(q
),
T
(q
)[M(q
)
Ṫ(q
)
C(q
)T(q
)]
and
i i ), C̃(q
i� i ,1 q̇i )i1and
i
i
i
i
i
B̃(q
)
in
order
are
determined
by
M̃(q
i
�
constant
by
the
observer.
1, q̇T i )1and
M̃(q
1 . B̃(qi ) in order are determined by
i ),
Twhere
11 C̃(q
1 ),1iT1
T)B(q
(q
)=
TTTT(q
Ṫ(qi ) + C(qi , q̇i )T(qi )] 2.2.
and Communication topologies Re-consider the multi-agent
1 i )[M(q
.�.. i))Ṫ(q
T
i )M(q
i )T(q
T(q
T (q
T(q
ii )B(q
ii ) =
i ))T(q
1rr= ri�
Th
(qii)B(q
)M(q
),
T
(q
−1
1
i
i
i
i ) + C(qi , q̇i )T(qi )] and
.� i
)
=
TT (qTi )B(q
−1
1)[M(q
−1
1
�
i
r
2.2.
Communication
topologies.
Re-consider
the
multi-agent
1
1
system
with
N
agents.
The
communication
topologies
of
such
−1
1
diag{
10 where
1 where
1=
InTT(q
(4),
C̃(q
)B(q
)ii,,C̃(q
=q̇ii,i1ri)), q
TIn
=
02×2
isis aa 222×
× 22 zero
zero matrix.
matrix. system with N agents. The communication topologies of such
TIn
(4),
q̇
02×2
i(q
2×2
2×2
(4),
00=2×2
where
zero
matrix.
i̇ )) = 0
In
(4),iC̃(q
where
02×2
matrix.
.. 0
)B(q
2£2
2£2isisa a 2£2
iC̃(q
i )q̇=
a system can be modelled via the theory of algebraic graph [15,
r=i −1
1
In
(4),
C̃(q
,
q̇
)
0
where
0
is
a
2
×
2
zero
matrix.
−1
1, 2,
i
i that
2×2
2×2
The
indicates
M̃(q
=B̃(q
B̃(q
)ττiiiican
can be
deduced
The fact
fact fact
indicates
M̃(q
)1ξ̇)ξ̇=
=
B̃(q
))τ
can
be
deduced
from a system can be modelled using the theory of algebraic graph
i ̇i = B̃(q
The
fact
indicates
that
M̃(q
can
be deduced
deduced from
from
The
indicates
thatthat
M̃(q
from
ii)ξ̇i ii)ξ
ii )iiτ
32]. Define a directed graph G = (V , E ) composed of a vertex
−1
−1
The
fact
indicates
that
M̃(q
ξ̇
τ
)
=
B̃(q
)
can
be
deduced
from
¡1
i
i
i
i
can b
In
(4),
C̃(q
,
q̇
)
=
0
where
0
is
a
2
×
2
zero
matrix.
(4).
Provided
det[M(q
)]
=
�
0,
the
assumption
means
M̃
(q
)
−1
In
(4),
C̃(q
,
q̇
)
=
0
where
0
is
a
2
×
2
zero
matrix.
(4).
Provided
det[M(q
i
i
)]
=
�
0,
the
assumption
means
M̃
(q
)
i
i
2×2
2×2
(4). Provided
det[M(q
=0, 0,the
theassumption
assumption
means
(q(q
a directed
i
i ii )]i)] 6
2×2
2×2
i) is
(4). Provided
det[M(q
�=
meansM̃M̃−1
ii ) V[15, 32].
set
and an Define
edge set
E , wheregraph
V =G = (V, E)
{v1 , v2 , · ·composed
· , vN }, E of
⊆
(4).
Provided
det[M(q
)]
=
�
0,
the
assumption
means
M̃
(q
)
is
invertible
such
that
the
the
equations
of
motion
describing
The
fact
indicates
that
M̃(q
ξ̇
τ
)
=
B̃(q
)
can
be
deduced
from
i
i
The
fact
indicates
M̃(q
τ
=
B̃(q
)
can
be
deduced
from
invertiblesuch
such that
that the
the
describing
the a vertex set V and an edge set E, where V = fv1, v2, ¢¢¢, vNg,
i of
i motion
is invertible
invertible
such
that
the the
theiiequations
equations
of
motion
describing
i
iof
imotion
is
the
equations
describing
V × V , the node vi denotes the ith agent and i = 1, 2, · · · , N.
−1 (q
the
behaviour
of
the
agent
be
formulated
by
is
invertible
such
that
the
the
equations
ofby
motion
describing
(4).
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det[M(q
)]
�=can
0,
the
assumption
means
M̃−1
(q
(4).
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det[M(q
=
�
the
assumption
means
M̃
behaviour
of
the
agent
can
be
formulated
i)]
i ) i ) E µ V£V, the node vi denotes the ith agent and i = 1, 2, ¢¢¢, N.
the
behaviour
of
the
agent
can
be
formulated
by
i
the behaviour of the agent can be formulated by
This
paper
investigates
thethe
directed
is invertible
such
that
the
the
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ofofmotion
theisbehaviour
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formulated
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This
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(5)
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i
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system.
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ith agent
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L
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B̃(q
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behaviourofofthe
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i ))B̃(q
system.
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ith agent
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(5)
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ibe
ii )τii (5)
thethe
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formulated
−1
ξ̇i T=(qM̃
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(qi )B̃(q
(5)
is defined
as
N
=
{v
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V
:
(v
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v
)
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}.
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ordered
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Reconsider q̇i =
i
j
i
j
i )ξi −1
bors is defined as Ni = fvj 2 V : (vi, vj) 2 Eg. The ordered pair
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(5)
ξ=ii M̃
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iequations of
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ithat
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Reconsider
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means that
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i
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such
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Reconsider
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ithagent,
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ned as of the agent at its head
hi
graph
cos[θẋihi[ẋẏ−h
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(6)
TθiH[v


� H[v
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i ωi ] T(6)
hi ] sin
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as
e agent where H�
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11
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i i
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.
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= coscos
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N£N
N×N
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R
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A
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−h
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θ
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b̄i =
h
cos
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θ
.
.
i
i
..
hrespect
cos θ to time t yields
sin
i θwith
 .
i
Differentiating
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t,nent
it can where
..
.. 
pt
Hẍ=
Differentiating
(6)
respect
to. time t yields ρ1i
.
v̇with


.
pt
xi
hi
−1
rank(
(6)θ(6)
respect
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yields
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� Differentiating
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h=respect
cos
θ to
H
M̃
(qtotime
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pt
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�
�
�
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(6)�with respect ito time
ti yields
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� ÿ�hi �� v̇v̇�xi v̇�
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aN1 aN2 · · · aNN
ρ
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hi
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Differentiating
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with
respect
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time
t
yields
−1
xi v̇ = HM̃−1 (q )B̃(q )τ +
1i
hi ẍ =
ept
(7)
i
i
i
xi
1i
hi
ρ
v̇
ẍ
(7)
=
=
H
M̃
τ
(q
)
B̃(q
)
+
−1
xi
i )B̃(q
i i + �ρ1i
−1 (q
� (7)
�
�
�ÿÿhi
= M̃
HM̃
1)
iB̃(qi )i )ττi i+
yi
(7) where
= =�v̇v̇yi
=H
(qi )Bull.
2i
hi
1)
ρρ2i
ai j indicates
thethe
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thethe
pair
v jj);
); 8 (v
∀ (vi, v
v j ) ∈ 3. F
aij indicates
weight
pair(v(v
i ,i, v
i , j) 2 E,
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v̇
Pol. Ac.: Tech.
XX(Y)
2016where
yi
2i
ẍ
ρ
v̇
1) (1)
ρ
ÿhi
v̇
xi
1i
hi
yi
2i
hi
−1
(7)
aii = 0.
=
= HM̃ (qi )B̃(qi )τi +
E(7)
, ∃9 a
ai ijj = 1;
= 1; 8 (v
∀ (vi, v
v j/ E,
) ∈
/9 a
E ,ij = 0
∃ ai jand
= 0;
and aii = 0.
Assum
i , j) 2
(1)
ÿhi
v̇yi
ρXX(Y)
2i
Bull.
Pol.
Ac.:
Tech.
2016
The
degree
matrix
of
G
is
a diagonal
matrix,
determined
by
The
degree
matrix
of
G
is
a
diagonal
matrix,
determined
the m
Bull.
Pol.
Ac.:
Tech.
XX(Y)
2016
Bull. Pol. Ac.: Tech. XX(Y)Example
2016
of article
Bull. Pol. Ac.: Tech. XX(Y) 2016 D = diagfd1, d2, ¢¢¢, dNg 2 RN£N. In the diagonal
N×N
matrix,
d
is
the
i
by
D
=
diag{d
trolle
,
d
,
·
·
·
,
d
}
∈
R
.
In
the
diagonal
N
2
Example
ofin-degree
article of 1v , formulated
N
2
N Acwhere
ρ
= ¡v
ω
sin
θ
¡ hω
cosθ
and
ρ
= ¡v
ω
cos
θ
¡
by
d
= ∑
a
(i = 1, 2, ¢¢¢, N).
2
Bull.
Pol.
Ac.:
Tech.
XX(Y)
2016
1i
i
i
i
i
2i
i
i
i
i
i
ij
i
j=1
diAssumption
is the in-degree
vi , multi-agent
formulated by
di = ∑
ai j leader
4: Inofthe
system,
thej=1sole
where ρ21i = −vi ωi sin θi − hωi cos θi and ρ2i = vi ωi cos θmatrix,
as we
i−
cordingly, the Laplacian matrix of G can be defined
by
2 sin iθsinθ
i.
2 cos θ and ρ = v ω cos θ −
Th
(i
=
1,
2,
·
·
·
,
N).
Accordingly,
the
Laplacian
matrix
of
G
can
cannot
receive
any
information
from
the
followers.
hω¡ hω
.
Assumption
4:
In
the
multi-agent
system,
the
sole
leader
where
sin
−
h
=
−v
ρ
ω
θ
ω
i
i control
i
i actions
i ]T = HM̃
i i
i
i Define
1i
2i ¡1(q
i [uxi u
the
L = D ¡ A 2 RN£N. Proven in [13],
L has at least one zero eiyiT
i)τi +
N×N
−1 (qi)B̃(q
2
be
defined
by
L
=
D
−
A
∈
R
.
Proven
in
[13],
L
has
Consider
Assumption
3.
For
the
zero
eigenvalue,
an
eigena
pre
Define
the
control
actions
[u
u
]
=
H
M̃
)
B̃(q
)
+
τ
cannot receive
information
from the
followers.
hωi + [ρ
sin θi . ρ ]T. (7) can have the formxi ofyi
i
i i
genvalue;
all otherany
eigenvalues
are located
at the
open right-half
1i 2i
at least
one
zero
eigenvalue
as
well
as
all
other
eigenvalues
areeigenvector
of
L
is
1
[Define
. (7)
can have
a form
ρ1i ρ2i ]Tthe
,
that
is,
L
1
=
0
holds
true, where
1Nas=th
Consider
Assumption
3.
For
the
zero
eigenvalue,
an
control
actions
[uxiofuyi ]T = HM̃−1 (qi )B̃(qi )τi +
N
N
N
plane if G is connected.
N×1
�
ing. A
located
at the
right-half
plane
[1,
1,ofopen
·L
· · , is1]
R N×1
[0,
0, true,
· · · , where
0]T ∈ 1R
vector
1TN∈
[ρ1i ρ2i ]T . (7) can �have a�form� of � �
, that
is,
Land
1ifN 0G=Nis0=Nconnected.
holds
N =
ẍ
v̇
u
T of the
N×1
T
N×1
xi�
xi�
hi�
�
�
�
Further,
rank(L
)
=
N
−
1
for
the
simple
zero
eigenvalue
[13].
Assumption
3:
G
multi-agent
system
has
a
spanning
[1, 1, · · · , 3:1]G of
∈ the
Rmulti-agent
and 0Nsystem
= [0,has
0, a spanning
· · · , 0] ∈
Assumption
tree,R in. the
=
=
v̇xiv̇yi
uxiuyi
ẍhiÿhi
tree,
indicating
that
the
zero
eigenvalue
of
L
is
simple.
other
Without
losszero
of generality,
theL Nth
agent in the multi-agent
indicating
that the
eigenvalue of
is simple.
Further,
=
=
Example
Example
Example
Example
Example
Example
Example
ofofof
article
of
article
ofarticle
of
article
of
article
article
article rank(L ) = N − 1 for the simple zero eigenvalue [13].
system
named
leader andthe
other
− 1 agents
followers.
ÿhi
v̇yi
uyi
Withoutisloss
of generality,
NthNagent
in the are
multi-agent
Consider the agent’s
uncertainties,
Finally,
the agent can be
Consider the agent’s
uncertainties,
Finally, the agent can
Assumption
4:
In
the
multi-agent
system,
the
sole
leader
cannot
Consider
Assumption
4,
that
is,
a
=
0
(i
=
1,
2,
·
·
· , N) and
2
2
2
2
2
2
2
system
is
named
leader
and
other
N
−
1
agents
are
followers.
Ni
Bull.
Pol.
Ac.: Tech.
XX(Y)
2016
Assumption
Assumption
Assumption
Assumption
Assumption
Assumption
4:4:4:4:
In
4:
In
4:
In
the
4:
In
the
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the
the
In
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the
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the
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the
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the
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the
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sole
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sole
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sole
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where
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where
where
where
where
where
sin
hθih−
−
hi−
hω
−
cos
cos
cos
−cos
−
−v
−v
=
=
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vρ2i=
vi=
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(N−1)1
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where
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that
is,
aa
= 0(i = 1, 2, ¢¢¢, N)
and
the
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and
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the
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constant
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yi
vxixivvxivkδ
vxiyivk ∙ δ
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(10)
Assumption
xixi
xi
yi, where δxi > 0 and
hihi
hi xi xi vand
yi


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graph
G
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Apparently,
δyi > 0δ̇are
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and
dicating
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Both
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and
are
Further,
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dd11dd1d1dare
−a
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xi 
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v̇xixiv̇
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+
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1(N−1)
1(N−1)
1(N−1)
1(N−1)
1(N−1)
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≃
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xi
xi
xiδ
xi
xi
xi
==
(8)
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
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 are bounded
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−a
−a



A G ∈
a−a
subgraph
of
G
.
The
weighted
adjacency
matrix
Assumption
the
by
are
described
by
a
directed
graph
G
.
Apparently,
δ̇xi ≃
0
0.
dicating
−a
−a
−a
−a
·−a
···−a
·2(n−1)
−a
−a
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−a
−a
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−a
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−a
and
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21
2121
212121 dd22dd
2d2d
2d
2 2· · ···· · ·−a
2N
2N
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2N2N
2N
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2(n−1)
2(n−1)
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2(n−1)
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ẏhihiẏBoth
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yiδyivv
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hi
Assumption
2:
δ
xi
yi
R
of
G
is
defined
by

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






A ∈
unknown
constant,
which
is
mild.
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2,
δ
and
a
subgraph
Assumption 1 indicates the uncertainties are bounded byxian is
.. .. .. .. .. .. .. of.. .. G.. ..... .. The
.. . .. .. ..weighted
.. ..
.. .. .. .. ..matrix
.. 
..
.. .. .. . .. .. .. .. .. adjacency
=



̇ xi ' 0v̇v̇yi
̇ yiyiv̇ ' 0.
L
L
L
=
L
=
L
L
=
=
L
=
=
v̇
v̇
v̇
v̇
δ
δ
δ
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δ
δ
u
u
u
+
u
+
u
+
u
+
u
+
+
+
L
cating
δ
and
δ
(10)
(N−1)×(N−1)
yi
yi
yi
yi
yi
yi
yi
yi
yi
yi
yi
yi
yi
yi
yi
yi
yi
yi
.
.
.
.
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δyi seemconstant,
almost constant.
Considering
the technical
of G is defined by
unknown
which is mild.
From Assumption
2, contents,
δxi and R



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a
a
·
·
·
a
11
12
1(N−1)
 −a
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(N−1)1

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2yiδare
hints
the
designed
observer
could evaluate
−a
−a
−a
−a
−a
−a
· · ···· · · ···d·d·N−1
ddN−1
dN−1
dN−1
dN−1
−a
−a
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Assum
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3.1.3.1.
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3.1.
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Consider
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NDO-based
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Consider
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(15)
[33,
34].
system
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[33,
[33,
34].
34].
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[33, 34].
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3.1.
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Consider
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The degree matrix –of G is determined
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T
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xtrolled
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v̇xixixixixi
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BNth
) ) (15)
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· · · , N − 1). Accordingly,
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xi
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ximatrix
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xi +
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T
T
T
T
(15)
(15)
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designed
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Define
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designed
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designed
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Define
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tiate
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designed
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Define
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and
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e
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Dea
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xi −
into
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2 into
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account.
account.
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Wecan
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obtain
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dN−1
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tiate
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tosubstitute
time
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the
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e
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e
tiate the
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e
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T
T
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Design
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e
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Similarly, assuming that the subgraph G is itself ainto
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We
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Subs
We
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xixi
xiof
xi−
=
−
=
−
−
uuuxixixi=
can
obtain
(16).
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ddd
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T
system
(14)
and
design
its
NDO-based
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(15)
[33,
34].
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T
ρ
ρ
We
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graph,Similarly,
L 1N−1assuming
= 0N−1 exists
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that thewhere
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G=
is itself
can
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≃
−
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˙T−
xid =
xiλ
uxi =
T xixixi
graph,
L 1–N−1
+
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xxxixixi+
+
λλ−
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exists where 1TN−1
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+
LLL
(B
(B
LL
LTxiṗTxiTxixL
+BBBxixixiuuuxixixi)))
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xi
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ė=
=
≃xixiṗxi−
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xiδ̇d xi=
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=
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2 ėxi ˙=
Rgraph,
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. More−
≃
−
−
L
ẋL
−L
B
p
−
L
(B
ṗ
=
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(N−1)×1
xi xi T˙TT xi
xiT xi xiT xiT xixi
xixi xxi + Axi xxi + Bxi uxi )
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xi
c
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d
xi 1
xi
R2 R(N¡1)£1and
0
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L
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uxiu) )
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ṗxixi
−
L
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−
LL
(A
x−
+
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L
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(A
x+
xxixixiLṗ+
B
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+
BB
Bxi+
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xi
xi
xi
xi
xiB
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xid =ė
xi
xi
xi
xi
xi
xixiT+
xixi
xixi
=
−
u
xi
xi
xi
xi
xi
xi
and
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2 R
.
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u
xi
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−
−
xi
xi
p
(B
x
+
=
over, define
a
matrix
B
=
diag{
b̄
,
b̄
,
·
·
·
,
b̄
},
xi xi TT xi xiT xi xi
xi xi
xi xi
– a matrix B = diag{b̄ 1 , b̄ 2 ,–· · · , b̄ N−1
– },
– where
(16)
(16)
(16) ρxi (d¯i + db̄d¯
=
+TTL
LTxiTxiT(B
xxi xiTLxiTxTTxi + ATTxi
δ̂xiλxi
over,
define
=
1
2 B = diagfb
N−1, b
TTppxixi+
xi + Bxi uxi )
Tx
a matrix
= diagfb
, b
, ¢¢¢, b
g,
where
, ¢¢¢,
1
2
N¡1
1
2
p
+
L
(B
L
x
+
A
x
+
u
)
=
λ
T
−
L
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x
+
B
u
+
B
δ
)
there existsxi xi λ
b̄ = aiN (i = 1, 2, · · · , N − 1). Apparently,
=
xi
xiL
xixixi
−
))+
+
(B
L
+
Ax
+
Bu
=
=
LLL
xxixxxi
+
L
(B
(BA+
L
LxiB
xδxi+
+
+B
Ax
Axuxi+
+
Buxixixi)))
λλλxi
δ̂xixixixi−
pxixixi((δ̂(δ̂−
+
L
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xLB
+
xxixixxixxi
)Bu
xi−xi
xi
xi
xi
xi
–
–
xi
xi
xi
xi
xi
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xi
xi
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+
xi
xi xi
xi uxixixixi
xixixi
xi ) xixixixi
aiN Apparently,
(i = 1, 2, there
· · · , exists
N − 1).rank(L
Apparently,
there exists =
b̄ii b–=
T Lxixi(A
+ B ) = rank(L) =
N¡1g.
TIn (15),
b̄i
−
L
(A
x
+
B
u
+
B
δ
)
T
Txi
Txi
is
the
internal
state
variable
of
the
observer,(16)
δ̂(16)
p
xi
xi
xi
xi
xi
xi
rank(L
+ B) = rank(L
) = N − 1.
xi
T
T
T
T
−
L
(A
x
+
B
u
+
B
δ
)
TB
xi
xi
xi
xi
−
Lxi
(A
x−
+
+
B
)))xiT x+ Ax
L
(A
(A
xxixi
+
B
uu)uxi
+
BB
)x
δδδδik
xi λ
δ̂−xi
−
xi
xixi
xi
xi
xi
xi
xi+
xi
ikxi
ik
−xixi−
L
(A
xxixixixxi
uL
+
−
=
(−=
−
L
xxixi+
)TxiBB
+
(B
L
+
BuBu
δ̂L
rank(L
+ B) = rank(L ) = N − 1.
xixi
−
xi
(16)
(
L
x
+
L
(B
L
+
Ax
+
)
λ
δ̂
xi
xi
xi
xi
xi
xixi
xi (16)
xi )
= N ¡ 1.
(16)
2×1
xi
xi
xi
xi
xi
xi
xi
xi
¯
xi
xi
is the approximation
of δTxi and
vector Lxi ∈ R (16)
is di + b̄i ρρ
T the gain
T
T(δ̂xi − LT
T xxi
=
)
+
L
(B
L
x
+
Ax
+
Bu
)
λ
xi
xi
xi
xi
xi
=λxidesigned
(δ̂xi=
−
+
L
(B
L
xeexixixi
Ax
TδδTδ
xi
Te
T=
=
−
)xithe
=
−
xi
xixi+
xi )is positive.
=λλL
−
−
)xi
)=
=xi−
−
λ(xixiδ̂xiT((such
δ̂(xδ̂−
δ̂xixixi−
λλ
λxiB
xi
xi
xi)L
xixixi
that
constant
LikTxiBu
+
uxidd+
+
B
)BAx
x(A
uL
dB
=λ−
+
(B
L
+
xiδ
xi xi + Buxi )
xi
xiL
xi
xixi
xi
xi
xiL
xix)B
xiλ
xixi(A
xixix+
xixi+
xiikxix)δxi
3. Formation
Formation Design
Design
T
cxi
T
3.
)
−
L
(A
x
+
B
u
+
B
δ
−The
LThe
(Asolution
x
+
B
u
+
B
)
δ
xi
xi
xi
xi
xi
ik
Define
an
estimation-error
variable
e
δxixixixid−
δ̂xi , differen=
xi
xi
xi
xi
xi
ik
xi
xi
T
xi
The
solution
of
(16)
is
e
λ
=
exp(−
t)e
(0),
where
of−
(16)
(16)
eeexi+
=
exp(−
exp(−λ
t)e
t)e
(0),
where
where−
−
−
dλ
xi
xixixi
)xiis
=
e=
δxiB
λ
dd(0),
ddxi
dB
xi
xi
=λ−xisolution
(=
−(xiδ̂δof
=
−
δ̂xixiλ(A
λis
L
xxixi
uxixi−
¯i −
xixi
xi) +
xidδik )
3. Formation
design the formation-control problem of tiate
ρ
(
d
+ρ
b̄ρ
Assumption
5: Considering
d
xi
the
error
variable
withat
respect
to
time to
t,
substitute
(15)
λ
(0)
isis
the
initial
condition
at
t
=
0.
Owing
to
>
0,
this
eλeexixixixidd(d(0)
λ
λ
is
the
initial
initial
condition
condition
at
t
t
=
=
0.
0.
Owing
Owing
to
>
>
0,
0,
this
this
Assumption 5: Considering the formation-control problem=of
xi
xi
xi
−
)
=
e
δ̂(0)
δthe
λ
(
−
)
=
−
e
λ
δ̂
δ
λ
xi=
xi
xi
xi
xi
xi
xi
xi
xi
d
Desi
d
The
solution
(16)
e=
=variable
exp(−
(0),2where
where
the multi-agent system, the leader is assumed to be well conxid exp(−
into
derivative
of
eis
and
Assumptions
1xi(0),
and
into
=
−
) of
=
−
eis
λthe
δthe
dexponenxiλd xie
fact
indicates
that
the
estimation-error
variable
eexixixixidt)e
exponenThe
solution
of
(16)
λxieλt)e
fact
fact
indicates
indicates
that
the
estimation-error
estimation-error
variable
isisis
xi (δ̂
xithat
xi
xixidd take
b̄i c−
theAssumption
multi-agent5:system,
the leader
is assumed to be well conddxi
xi−
d exponen−
Considering
the formation-control
e
solution
of
(16)
is
e
λ
λ
(0)
is
the
initial
condition
=
exp(−
at
t
t)e
=
0.
(0),
Owing
where
to
>
0,
this
− followe
trolled by a developed
technology
and the controllerproblem
is The
known
xi
xi
xi
xi
xi
account.
We
can
obtain
(16).
The
solution
of
(16)
is
e
=
exp(−
λ
t)e
(0),
where
d
d
d
tially
convergent
to
0
as
t
→
∞.
tially
tially
convergent
convergent
to
to
0
0
as
as
t
t
→
→
∞.
∞.
xi
xi
xi
e
λ
(0)
is
the
initial
condition
at
t
=
0.
Owing
to
>
0,
this
¯
d
d
xi
xi
trolled
by
a
developed
technology
and
the
controller
is
known
(
d
+
b̄ρ
ρ
of the multi-agent system, the leader is assumed to be
The
solution
of
(16)
= exp(¡λ
(0),
where
exi (0) xi i ρ
xixit)e
solution
ofthat
(16)
is
eexixiOwing
λ>xixit)e
= exp(−
(0), where
fact
indicates
the
eto
exiwell
λvariable
(0) eisdThe
the
initial
condition
at t estimation-error
=is 0.
0,
this
xi
xi
dat t =to
d d λis exponenas well as the leader’s its dynamic [17].
d
(0)
is
the
initial
condition
0.
Owing
>
0,
this
˙
xi
xi
fact
indicates
that
the
estimation-error
variable
e
is
exponenT
d
xi
ascontrolled
well as the
its dynamic
[17].
byleader’s
a developed
technology;
the controller’s and the
is tially
the
initial
t = 0.
to
λ xi > 0,
this
fact
in- u κxi= −
ėthat
=
δ̇initial
δ̂ xi condition
−condition
≃to−0ṗat
Latxi∞.
ẋt xiOwing
xidthe
xiestimation-error
xi −
exifact
λcoordinate
(0)
is
= 0.
Owing
toedcoordinate
0, this
convergent
as
tcontroller
→
the
variable
exidesign
is exponenxiis>exponenxi
ddesign
d3.2.
The Nth agent has
been named leader, its duty isfact
to indicates
track
3.2.
Integral
SMC-based
design
To
coordinate
3.2.
Integral
Integral
SMC-based
SMC-based
controller
controller
To
To
−
indicates
that
the
estimation-error
variable
xi
tially
convergent
to
0
as
t
→
∞.
leader’s
is known
dicates
that
the
estimation-error
variable
e
is
exponentially
d
The Nthdynamic
agent has
been [17].
named leader, its duty istially
to track
¯i +κκκxib̄x
where
where
where
xi
T
T
convergent
to
0
as
t
→
∞.
ρ
(
d
fact
indicates
that
the
estimation-error
variable
e
is
exponenxi
xi
p
+
L
(B
L
x
+
A
x
+
B
u
)
=
λ
the
leader-following-based
multi-agent
system,
a
trackingthe
the
leader-following-based
leader-following-based
multi-agent
system,
trackingxi xi to 0
xit →
xi
xi xi system,
xi xi ada trackinga predefined
trajectory
and its control
problem
can
bea pretreated
xi
ximulti-agent
tially
convergent
as
∞.
The Nth trajectory
agent identified
the leader
has to can
trackbe
convergent
to
0
as
t ! �
.
a predefined
and itsascontrol
problem
treated
3.2.
Integral
SMC-based
controller
design
The
eq
Theeq
e
tially
convergent
0ith
tfollower
→ ∞.
Tto
error
variable
of
the
ith
agent
defined
by
error
error
variable
variable
of
of
the
the
ith
follower
follower
agent
agent
is)defined
defined
by
byTo coordinate The
as the tracking-control
problem
ofproblem
a single
robot.
Concernwher−
−SMC-based
L
xas
uxidesign
+
Bxi δisis
xi + Bxi
xiTo
trajectory and its
controlof
canrobot.
be treated
xi (Axi
3.2. Integral
controller
design
To coordinate
asdefined
the tracking-control
problem
a single
ConcerntheSMC-based
leader-following-based
multi-agent
system,
a tracking3.2. as
Integral
controller
coordinate
(16)
Replace
δ
Replace
where κxiReplace
>wher
0 is
N−1
ing Assumption
5, the leader can
be treated
as a nominal
one
N−1
N−1
T
T
tracking-control
of a single
robot.
3.2.
Integral
SMC-based
controller
design.
ToTo
coordinate
3.2.leader-following-based
Integral
design
coordinate
Thp
the
multi-agent
system,
aby
trackingerror
variable
ofL
the
follower
defined
the leader-following-based
multi-agent
system,
=
(δ̂SMC-based
xith
)+
Lcontroller
(B
LTxi xdesign
+trackingAxxi To
+
Bu
)
λ=
ingtheAssumption
5, theproblem
leader can
be treated
as aConcerning
nominal
one
xxxixagent
xiSMC-based
xi −
xi −
xiais
xi
xi
xi
(22).
The
whe
(22).
(22).
The
Th
˙
3.2.
Integral
controller
coordinate
e
(v
−
v
)]
a
[(x
x
−
d
)
+
ρ
e
e
(v
(v
−
−
v
v
)]
)]
=
=
a
a
[(x
[(x
−
−
x
x
−
−
d
d
)
)
+
+
ρ
ρ
xi
xi
x
j
xi
i
j
hi
h
j
xi
xi
xi
xi
x
x
j
j
xi
xi
i
j
i
j
hi
hi
h
h
j
j
∑
∑
∑
iijij j
in the formation-control
that is, δxN = δyN =
0. variable
The
5, the leaderproblem,
can be treated
one
leader-following-based
multi-agent
system,
a tracking-error
Th
δ̂
The equation
thethe
leader-following-based
multi-agent
system,
wher
error
of the−ith
follower
agent
is
defined
variable
ofLj=1
the
ith
agent
isby
by a trackingRepla
T
(17)
in Assumption
the formation-control
problem,
that is,asδa nominal
0. inerror
The
(17)
(17)
N−1
xN = δyN =
j=1
j=1
(A
xxifollower
+
Bximulti-agent
uagent
) defined
the
leader-following-based
system,
abytrackingxifollower
xi + Bis
xi δdefined
−
xi
˙
T
the formation-control
that is,
δxNthey
= δyNare
= 0.
The other
variable
ofethe
ith
by
other
N − 1 agents actproblem,
as followers
and
equipped
with
xik is defined
error
variable
of
the
ith
follower
agent
(22).
in
(21
−)vx j )]Replace δ̂xi Repla
a−
−dxxxhx j)−
db̄i jρ) +(vρxi (v
xi
N−1
i j [(x
xi+=
other N − 1 agents act as followers and they are equipped with
hi−
∑
Th
N−1
b̄
(x
x
−
+
−
v
+
+
b̄
b̄
(x
(x
−
−
x
x
−
d
d
)
)
+
+
b̄
b̄
(v
(v
−
−
v
v
)
)
ρ
ρ
error
variable
of
the
ith
follower
agent
is
defined
by
ii δ
xN
hi
hN
i−
ii i xixixi xixixi
xN
xN
hi
hi
hN
hN
N ¡ 1 agents act as followers and they are equipped with the e = =aλxi[(x
iN
iN
iN
x
(δ̂xi−
)
=
−
e
λ
(17)
xi
xi
xi
x
can
xh j −
d ) + ρ d(v − v j )](vxi − vx j )] (22). Then, ṡ(22).
xiRep
xi
hi aij=1
∑
N−1
exi =i jx ∑
j [(x
hi −i jxh j −xidi jxi) + ρxxxi
Repla
designed formation
controllers
to achieve
formation
maneuvers
(17)
x x dddxx position
erThe
and
d
is
the
pre-defined
relative
between
the
ith
the designed
formation
controllers
to achieve
formation
ma- where
N−1
j=1
>
0
a
pre-defined
constant,
is
the
pre-defined
>
>
0
0
is
is
a
a
pre-defined
pre-defined
constant,
constant,
is
is
the
the
pre-defined
pre-defined
ρρρxixiexixi
where
3where
(17)
(22
iN
+
b̄
(x
−
x
−
d
)
+
b̄
(v
v
)
ρ
i
j
i
j
i
j
(v
−
v
)]
=
a
[(x
−
x
−
d
)
+
ρ
j=1
solution
ofi ji (16)
λxit)e
i xi
xi
xi xid (0),
xN
−
hihi ishN
xi
x j where
iNi jexp(−
hejxid x=
∑between
3relative
of the
system.
exi
(vxithe
−tovjth
=hithe
xhb̄ith
−atfollower
d(v
)−
+0.vρxNOwing
between
the
follower
and
the
jth
followfollower
and
the
neuvers
ofmulti-agent
the multi-agent
system.
relative
relative
position
position
between
the
ith
follower
and
the
jth
followxi
xλ
j )]followi j [(x
hix −
jith
(17) (22).
∑
i xi
j=
ṡxi =
+
b̄position
(x
−j=1
xahN
−leader.
dcondition
)the
+
)and
e
(0)
initial
t
>
0,
this
iis
i ρxxi
x
iN
xi
xi
d
xhN(17)
− dwith
+respect
b̄iρxi (vdxito
>b̄i0(xishi a−pre-defined
is vthe
pre-defined
where ρxi+
(17) the
Concerning
the follower
ith follower
agent
(17)
xN )t,
j=1
iN )constant,
ij−
Differentiate
exi in
Concerning
the ith
agent
(i =(i = 1, 2, ¢¢¢, N ¡ 1),
1, 2, · · · , N − 1), fact
time
substitute
indicates
estimation-error
variable
exithe
is exponenxx is
d vxNjth
+that
b̄i (xthe
−
xhN the
−xddiith
)follower
+
) followρxxi (vand
position
between
0 is
a pre-defined
theb̄
where
its equations of motion in (8) are decoupled in the x-axis
and ρxi4 > relative
ipre-defined
xi −
−
hiconstant,
iN
j
44 tially
its equations of motion in (8) are decoupled in the x-axis and
x-axis
subsystem
into
of
e)xi and consider
−
xfollower
−
d∞.
) +derivative
b̄iρxidjth
−
vxNpre-defined
>
0+
isb̄ai (x
pre-defined
constant,
is
the
ρxi convergent
where
toith
0(13)
as
t→
hi
hN
iNthe
i(v
j xi
relative
between
the
and the
followy-axis. Consequently, its formation design can be divided
into position
x
pre-defined
d j isthe
thejth
pre-defined
ρxi > 0 is
where
y-axis.
Consequently, its formation design can be divided into
Assumption
5ayields
relative
position
between
the ith constant,
follower and
followtwo parts, that is, design for the x-axis and design for the y-axis.
4ρxi > 0 is a pre-defined constant, dixj iis
thethepre-defined
where
relative
position
between
the
ith
follower
and
jth
follow3.2.
Integral
SMC-based
controller
design
To
coordinate
two parts,
that
is,
design
for
the
x-axis
and
design
for
the
y-axis.
x
Here the formation design for the x-axis is taken into account
where
ρxiN−1
> 0between
is a pre-defined
constant,
dand
the pre-defined
4
where
ij is the
relative
position
the
ith
follower
jth
followtheė leader-following-based
trackingHerefirst.
the From
formation
design
the x-axis
is uncertainties
taken into account
(uxi − usystem,
(aδxi
− δx j )]
ai j [(v
) +multi-agent
ρxifollower
ρxijth
4 relative
(8), the
x-axisfor
subsystem
with
can be
between
ith
follower
xand
j ) +the
xi = position
xi − vx jthe
∑
errord xvariable
of the ith follower
is defined
by the ith fol- (18)The
at first.
From
written
as (8), the x-axis subsystem with uncertainties can4 and
the pre-defined
relativeagent
position
between
iN isj=1
4
Replac
N−1
be written by
lower and
leader.
+the
b̄i [(v
vxN ) + ρxi (uxix− uxN ) + ρxi δxi ]
xi −
e
(v
−
v
)]
=
a
[(x
−
x
−
d
)
+
ρ
xi
xi
x
j
i
j
xi
hi
h
j
∑
ij
Differentiate exi in (17) with respect
to time t, substitute the (22). T
ẋhi
vxi
(17)
j=1the x-axis subsystem of the ith agent,
(13) x-axis
=
(13)
Concerning
an intesubsystem (13) into the derivative of exi and consider
x
uxi + δxi
v̇xi
+
b̄
(x
−
x
−
d
)
+
b̄
(v
−
v
)
ρ
i surface
i xi xi
xN observer output
hi
hN with the NDO-based
gral sliding-mode
 −a
 −a
21

.. 21

L=
..
L =


 −a(N−1)1
 −a(N−1)1
0
0
d
d22
..
..
···
···
.
. .. .
.
···
···
0
0
d
d
(9)
(9)
) ∈
) ∈
ned
ned
onal
onal
1 ai j
1 ai j
can
can
has
has
are
are
ning
ning
d
d
d
d
Assumption 5 to obtain
Further, (13) can be re-written by the following state-space
representation.
38
ẋxi = Axi xxi + Bxi uxi + Bxi δxi
(14)
0 1
, Bxi = [0 1]T and uxi is
where xxi = [xhi vxi ]T , Axi =
iN
[11] isρxidefined
> 0 is by
a pre-defined constant, dixj is the pre-defined
where
relative position between the ith follower
and the jth follow-
4
+ δ̂xiTech. 65(1) 2017
sxi = exi + cxiBull.ePol.
xi dt Ac.:
Brought to you by | Gdansk University of Technology
Authenticated
cxi > 0 is constant.
Download Date | 3/2/17 11:22 AM
(19)
here
The derivative of the sliding-mode surface variable with re-
N−1
its decoupled
equations of
in (8)
are decoupled
in the x-axis
and the x-axis
subsystem
(13)consider
into the derivative ofxi exi and consider
ny-axis.
(8) are
in motion
theformation
x-axis
and
x-axis
(13)
into
derivative
of exi and
Consequently,
its
design
cansubsystem
betaken
divided
into
Assumption
5
yields
Here
the
formation
design
for
the
x-axis
is
into
account
ė
a
[(v
−
v
=
xi
x j ) + ρxi (uxi − ux j ) + ρxi (δxi − δx j )]
xi
∑ i 5j yields
y-axis.
Consequently,
its formation
design can 5beyields
divided into Assumption
formation
design
can(8),
be the
divided
into
Assumption
(18)
two
parts,
that
is, design
for
the
x-axis
and
design
for
the y-axis. can
at
first.
From
x-axis
subsystem
with
uncertainties
j=1
N−1
two parts,
is, design
for the x-axis and design for the y-axis.
or
the the
x-axis
and that
design
for the
y-axis.
N−1
N−1into account
Here
formation
design
for
the
x-axis
is
taken
be
written
by
−−vxvjxN
) ++
−−uxujxN
)+
ρxiρ(u
ρxiρ(xiδδxixi−
δx j )]
xi xi
xi xi
[(v
)j+
design
forthe x-axis is taken into account ėxi =ėxi∑=a+i jb̄[(v
Here
the formation
)+
− δv)xxby
+xiρ(u
ρxi (δ] xi −
δx j )]
n for
the
x-axis
is taken
into
account
xi −
jj ))]integral
xi (u
xi − uxmode
Observer-based
formation
sliding
ėxi =uncertainties
ai j [(v
vx jcontrol
) + ρxiof(uuncertain
ux j∑
)multiple
+iaρi xij [(v
(δagents
(18)
xi −can
xi −
xi
∑
at
first.
From
(8),
the x-axis
subsystemleader-following
with
j=1
ẋ
v
xi
(18)
hi
at first. From
(8), the x-axiscan
subsystem
with
uncertainties
can
j=1
(18)
xis
subsystem
with
uncertainties
j=1
(13)
=
Concerning
the
x-axis
subsystem
of
the
ith
agent,
an
intebe written by
+ b̄i [(vxi − vxN ) + ρxi (uxi − uxN ) + ρxi δxi ]
v̇xi be written by
− vxN ) + ρwith
− uNDO-based
]
uxi + δxi + b̄i [(vxi − vxN ) + ρxi (uxi gral
sliding-mode
output
xi (uxithe
xN ) + ρxi δxi observer
− uxN
)+
+b̄ρi [(v
ẋhi xi δxi
xi ] surface
vxi
ẋ
v
xi
hi
(13)
=
Concerning
the
x-axis
subsystem
of
the
ith
agent,
an
intevxi
[11]
is defined the
by x-axis subsystem of the ith agent, an inteFurther,
(13) v̇can be re-written
the following state-space
Concerning
u=xi + δby
(13)
=
the x-axis(13)
subsystem
of the ithsurface
agent,with
an intexi
gral sliding-mode
the NDO-based
observer output
δxi
uxixi +Concerning
v̇xi
representation.
uxi + δxi
gralNDO-based
sliding-mode
surfaceoutput
with the NDO-based observer output
gral sliding-mode surface with
the
observer
s
=
e
+
c
(19)
e
dt
+
δ̂
xi
xi
xi
xi
xi
[11]
is
defined
by
Further, (13) can be re-written by the following state-space
Further,
(13) can
beAre-written
defined
bystate-space
ẋxi state-space
δxi
=
u[11]
+isBfollowing
(14) [11] is defined by
-written
by
the
following
xi xxi + Bxiby
xi the
xi
representation.
representation.
here
cxi > 0 issconstant.
(19)
exi dt + δ̂xi
xi = exi + cxi
s
=
e
+
c
exi dt + δ̂xi
(19)
0+ B1 δ
xithe sliding-mode
xi (19)xi
s(14)
exi dtderivative
+ δ̂xi
xi = exi + cxi The
T
T
ẋ
=
A
x
+
B
u
of
surface
variable
with
rexi
xi
xi
xi
xi
xi
xi
=
[x
v
]
,
A
=
,
B
=
[0
1]
and
u
is
where
x
xi
xi
xi
xi
xi
hi
ẋ
δ
=
A
x
+
B
u
+
B
(14)
xi
xi
xi xi
xxi + Bxi uxi + Bxi δxi
xi (14)
xi 0xi
0
here
c
>
0
is
constant.
spect
to
time
t
can
be
written
by
xi
here cxi > 0 is constant.
0 1 here cxi > 0Tis constant.
T
The derivative of the sliding-mode surface
with re0 ,B
1 xi = [0 1] andT uxi is
= [xhi vinput.
where0the
xxi 1control
xi ] , AxiTT=
˙ variable
Thesurface
derivative
of the
sliding-mode
surface
variable with(20)
reThe
theusliding-mode
variable
with
re[x vxi ] ,and
Axi u0=xi is0
, Bderivative
xi = [0 1] ofand
xi is
=
ė
+
c
e
+
δ̂
ṡ
= where x,xiB=
xi
xi
xi
xi
xi
xi =hi[0 1]
spect to time t can be written by
0 spect
0 to time t can be written
spect
to
time
t
can
be
written
by
0 0
by
the control
input.
3.1.
NDO-based
˙ gives
Substituting (18) and (19) into (20)
the control
input. observer design Consider the x-axis sub(20)
ṡxi = ėxi + cxi exi + δ̂xi ˙
˙
system (14) and design its NDO-based observer (15) [33,
34].
=
ė
+
c
e
+ δ̂xi
(20)
ṡ
xi
xi
xi
xi
=
ė
+
c
e
+
δ̂
(20)
ṡ
xi
xi
xi
xi
xi
N−1
3.1. NDO-based
observer
design
Consider
the x-axis subT
T
T
Substituting
(18)
and
(19)
into
(20)
gives
3.1. ṗNDO-based
design
Consider
the
B observer
pxi − Lxisub(B
+ A xxi +
Bxix-axis
u ) subvx j )(19)
+ ρxiinto
(uxi (20)
− ux gives
ṡSubstituting
er design
xi = −Lxithe
xi Lxi xxiSubstituting
j ) + ρxi (δxi − δx j )]
xi = gives
xi −and
∑ ai j [(v(18)
(18)xi34].
and (15)
(19) into (20)
system
(14)Consider
and designxiitsx-axis
NDO-based
observerxi (15)
[33,
j=1
system
(14)
and
design
its
NDO-based
observer
(15)
[33,
34].
s NDO-based
(15)
[33, 34].
N−1
T
=
δ̂xi observer
N−1
T pxi + Lxi xxi
−L
Bxi pTxi − LTxi (Bxi TLTxi xxi +T Axi xxi N−1
+ Bxi uxi )
ṗxi T=
ρxiρ(δδxixi−
δx j )]
−−uxujxN
)+
−−vxvjxN
) + ρxiρ(u
ṡxi = ∑ a+i jb̄[(v
xi xi
xi xi
[(v
)+
xi
T
pxiuxi−) Lxi (Bxi Lxi xṡxixi+=A∑
B
u
)
ρxi (δ] xi −
δx j )]
− δv)xx j+
) +xiρ(u
=
ṡ
xi xi
xi xxiai+
xi
xi
xi (u
xi − ux j ) + xi
xi
xi −
Lxi (Bxi Lxi xxiṗxi+=
Axi−L
xxixi+BB
ux j∑
) +iaρi xij [(v
(δxi
vx j ) + ρxi (uxi −j=1
(15)
j [(vxi −
j )]
(21)
Txi is the internal
(15)
state
variable
of
the
observer,
δ̂
p
N−1
j=1
xi
(15)
+
L
x
δ̂xi =Inp(15),
j=1
xi
xi T
xi +
x )+ρ δ ]
2×1
=
p
L
x
δ̂
xi
xi
xi
xi
+
b̄
ρ
[(v
−
v
)
+
(u
−
u
ai vj [(x)hixi+−ρxxih(u
is
is the approximation of δxi and the gain vector Lxi ∈ R
i + cxixi ∑xN
xi(vxixi − vx j )]
xi
j − dxN
i j uxN ) +
+ b̄ [(vδxi
ρxi δxi ]
−
xN
xi xi −
x is the pre-defined relative position
+ b̄observer,
+ ρith
i [(vbetween
xi − vxNδ̂) the
xi (uxi − uxN ) + ρi xij=1
xi ]
(21)
er and
mathe that
internal
state variable
ofTxi B
the
In (15),
pxi xisdsuch
designed
the
constant
λ
=
L
is
positive.
iN
N−1
xi
xi
xi
(21)
(21)
is
the
internal
state
variable
of
the
observer,
δ̂
In
(15),
p
er
and
d
is
the
pre-defined
relative
position
between
the
ith
aN−1
xi
xi
nal
state
variable
of
the
observer,
δ̂
x
iN
2×1
N−1
xi
follower
and
the
leader.
Fig.
2.
Structure
of
the
control
scheme
˙
x
+
c
ρ
a
[(x
−
x
−
d
)
+
(v
−
v
)]
is
the
approximation
of
and
the
gain
vector
L
∈
R
is
δ
x
x
xi
i
j
xi
xi
x
j
2×1
xi
xi
hi
h
j
∑
Define
an
estimation-error
variable
exidvector
−Lδ̂xixi∈, differen= δxibetween
eris and
diNand
is the
the leader.
position
ith
a- andfollower
xhN
−xdhiNj i−
)j +dib̄j )i c+
) + δ̂xi
cb̄xii cxixi∑
[(x
ρxixi(v
(vxixi −
−vvxN
the
gain
Rhithe
ish j − dixj ) + ρ+xij=1
the
approximation
of δ2×1
xi ρ
j)]
x j )]
xi and
hi −
cxi ∑
ai j [(x
− xthe
(v
−(xvhiaxij−
gain
vector
Lpre-defined
isrelative
TB
xi 1), thesuch
−
Differentiate
exixi ∈inR
(17)
to+
time
t, substitute
designed
that
the
constant
λxiwith
= Lrespect
positive.
xi Tis
x
tiate
the
error
variable
with
respect
to
time
t,
substitute
(15)
xi
j=1
follower
and
the
leader.
T
designed
such
the
constant
λrelative
er=and
diN
isthat
the
pre-defined
between the ith
),ma(17)
withtherespect
toBposition
time
t, substitute
j=1
xi = Lxi
xi is positive.
xi in
λDifferentiate
Lxisubsystem
B
positive.
xix-axis
xi is e
snstant
and
(13)
into
of
and
consider
Design the following xx-axis formation-control law
˙ for the ith
Define
an
estimation-error
variable
ederivative
δ̂xieet,,xidifferen−
= δtoxi time
into
the
derivative
of
e
and
take
1
and
2 into
xidAssumptions
Differentiate
e
),
in
(17)
with
respect
substitute
thex
xi
+ b̄i cxi (xhi − xhN − d˙ iN ) +
xi
d
x b̄i cxi ρxi (vxi − vxN ) + δ̂xi ˙
follower
and
the
leader.
d
x-axis
subsystem
(13)
into
the
derivative
of
and
consider
Define
an
estimation-error
variable
e
δ
δ̂
−
,
differen=
xi
xi
xi
xi
rror
variable
e
δ
δ̂
−
,
differen=
ρ
c
(x
−
x
−
d
)
+
b̄
c
(v
−
v
)
+
b̄
d
into
follower
agent
Assumption
5
yields
xi obtain
xi respect
xidvariable
i xi xi xi
xN + δ̂xi
δ̂xi iN
xconsider
b̄i cexi (xand
xN ) +hN
hi −(15)
hN − diN ) + b̄i cxi ρxi (vixixi− vhi
tiate
error
with
to
time
t,+substitute
account.
We5can
(16).
d 1), the
x-axis
subsystem
(13)
into
the
derivative
of
xi
Differentiate
e
−
in
(17)
with
respect
to
time
t,
substitute
the
o
Assumption
yields
tiate
the
error
variable
with
respect
to
time
t,
substitute
(15)
xi
ith
respect
to
time
t,
substitute
(15)
Design the following N−1
x-axis formation-control law for the ith
axis.the derivative
into
and
takeT Assumptions
1 of
and
intoconsider
N−1
Designcthe
following
Assumption
5of
yields
˙ exi1dofand
and
s.oount
x-axis
subsystem
(13)
the derivative
e1xi2and
and
Design the
following
formation-control
the ithformation-control
cxi ρxi + law
1 forx-axis
1 for the ith
into
the
derivative
eṗxid2−into
and
take
Assumptions
2 x-axis
intofollower
xi ρxi + law
N−1
and
take
Assumptions
into
agent
ė
δ̇
δ̂
=
−
≃
−
L
ẋ
xi
xi
xi
xi
xi
ė
a
[(v
−
v
)
+
ρ
b̄i (vxi − vxN )
a
(v
−
v
)
−
=
−
u
xixi (uxi − ux j ) + ρxi (δxi − δx j )]
d =
account.
We
can
obtain
(16).
i
j
xi
x
j
xi
i
j
xi
x
j
xi
∑
∑
follower
agent
s.
into ėxiAssumption
nt
yields
follower
We
can5xiobtain
ux j ) + ρagent
= N−1
ai j [(v
−T vx j )(16).
+Tρxi (uxi −
ρxi (d¯i + b̄i ) j=1
ρxi (d¯i + b̄i )
xi (δxi − δx j )] (18)
16).
(18)
∑
s can account.
j=1
N−1
+xiA−xi xuxix j+
uxi(δ)xi − δx j )] (18)
=˙λxiaipj xi[(v+xiL−xiv(B
nt
axis.
cxi ρxi + 1
cxi ρxi + 1
n ė =
ėxiδ̇=−j=1
)TL
+ẋxiρxxixi(u
) +Bxi
ρxi
xL
jxi
N−1
˙ ṗ−
δ̂+
−
cxi ρxi +b̄1i (vxi − vxN )
cxi
+∑
1 N−1
ρ
T(u − u ) + ρcxiδρxi] + 1 N−1 (18)
N−1
xi
xi ∑
xi b̄≃[(v
xi −
xi
xi
xi
ai j (v
vx j ) −
uxi = −
xi+
c
+
1
ρ
Td
xi −
v
)
ρ
xi
ė
δ̇
δ̂
=
−
≃
−
ṗ
−
L
ẋ
i
xN
xi
xi
xN
xi
xi
1
T
b̄
n
j=1
xi
xi
xi
xi
xi
i
¯
ai−
−
vx(jδ̂)ρ−
−
ount
−
Lxi ẋxi ėxid +=b̄−i∑
j (v
xi
(di + b̄¯i )δ̂xij=1
(dδ̂¯ix+
LTxixia(A
δxi−
∑
u−xρj )xi+
+xixixiρ+xi−B(uxiuxixN
δx ja)]i j (vxi −uxivx=
b̄
) xi−
=
(v
v
)
xi−
xi
[(v
vxxN
)+
+vBxρjxixi)u(u
))+
δxiρ¯]xi (δxi −∑
i−
j [(v
xixi
jρ
i
xi
xN
xi
a
−
)¯b̄i ) b̄i (vxi − vxN )
−
T−
i
j
xi
∑
(
d
+
b̄
)
ρ
ρ
xi+
xi (j di + b̄i )
Lxi (B
+T A
B)xi uxiρ)ρxiδ(d]i + b̄i ) j=1 (16)(18)
=λxi p+xi b̄+j=1
(db̄¯ii i+ b̄ii ) j=1
ρ
¯ixi
¯i + b̄i
T xi)x+
xivL
xi ρ
xi x−
xi +
d
d
s
can
T
[(v
−
(u
u
(13)
Concerning
the
x-axis
subsystem
of
the
ith
agent,
an
intep
+
L
(B
L
x
+
A
x
+
B
u
)
=
λ
j=1
xN
xi xi
xi
T xi xi
Txi
T xi
xi xi
xixN
L3)xi xxi + A
+ iB
uxithe
) L
xi
xi xxi=
Concerning
the
subsystem
ith
integral
−
L
xofxithe
+ xi
Ax
+ Buan
xi ) +
xi )an
Concerning
x-axis
subsystem
the
ithxiagent,
agent,
inte-N−1
1 N−1N−1
T λxi (xiδ̂xixi
b̄i
xi xx-axis
xi (Bxi Lxiof
−
L
(A
x
+
B
u
+
B
δ
)
gral
sliding-mode
surface
with
the
NDO-based
observer
output
1
T
b̄
xi
xi
xi
xi
xi
xi
+
b̄
[(v
−
v
)
+
(u
−
u
)
+
]
ρ
ρ
δ
N−1
i
xi
δ̂
a
−
−
i
xi
xN
xi
xi
xN
xi
xi
1
b̄
xicxi ¯
sliding-mode
the+
NDO-based
output
[11]
∑ ∑i j (aδ̂ixij (−δ̂xiδ̂−x jx)δ̂x j )
− Lxi
xxi +with
Bwith
BxiNDO-based
δxiof) −
3)xi uxi +
thesurface
x-axis
subsystem
the observer
ithi observer
agent,
anoutput
inteT (A
xisurface
xi u
xi
BConcerning
δxi ) −
¯xii−
(16)
gral
the
δ̂xidi−+ b̄¯i aj=1
xisliding-mode
+
b̄
d
δ̂
δ̂
δ̂
a
(
−
−
)
L
)
(A
x
+
B
u
+
B
δ
i
i
j
xi
x
j
xi
xi
xi
xi
xi
ik
∑
−
[11]
is
defined
by
xi
¯
(16)
T
T
T
hi − xh j − di j )
pace
∑
b̄¯i
di +
di + b̄i ji(xj=1
¯the
(16)
=λis
(defined
xxisurface
)+
(B
+T Axof
Bub̄observer
δ̂xi − Lby
+an
b̄i inted¯ioutput
gral
sliding-mode
with
T L
TLthe
xi
xisubsystem
xi d+
xi
i+
i )Bu
(13) [11]
the
x-axis
ith
agent,
xiby
xi
xi xxiNDO-based
isTConcerning
defined
ρ
j=1
xi (di + b̄i ) j=1
=
(
−
L
x
)
+
L
(B
L
x
+
Ax
+
)
λ
δ̂
(22)
xi
xi
xi
xi
xi
xi
xi
+e LTxi (B
L
x
+
Ax
+
Bu
)
xi
xi
xi
N−1
xi is
xi− δxi ) xi= −λxi exi
xi =
cxi
Txiλxi (δ̂xiby
N−1
sliding-mode
surface
NDO-based
observer
output −
ce [11]gral
x
N−1 (19)
− Ldefined
u=
B+xiwith
δcxiikd )the
e
e
dt
+
δ̂
c
T xi + Bsxixi
xi x
xi +
N−1
xi
xi
xi
xi
a
(x
−
x
−
d
)
xi (A
cxi
∑ i j ahii j (xhi h−j xxh j −i j dixj )1
−
Lxi (Axisby
xxi=+eB
ik )+
−
uxi + B[11]
+uxicexi+B=xi
exiδexp(−
dt
xi
xiδik )is
xixi
(ddix¯ij )+b̄b̄¯i ci )xi j=1 (x∑
defined
a(19)
−δ̂λxixi(19)
(14)
xi
i j (xhi − xh jρ−
pace
The
solution
ofxi (16)
is
t)e
where
−
xid
xid (0), ∑
hi − xhN − diN ) + ¯
∑ ai j ux j
¯
ρ
(
d
+
b̄
xi
i
=λxi (δ̂xi − δxi ) =sxi−=
λxieexixid+ cxi exi dt + δ̂ρxixi (di + b̄i ) j=1 (19)
4)
ρxi (d¯i + b̄ii )) j=1
di + b̄i j=1
here
c
>
0
is
constant.
=
(
−
)
=
−
e
λ
δ̂
δ
λ
xi
xi
xi
xi
xi
xi
e
λ
(0)
is
the
initial
condition
at
t
=
0.
Owing
to
>
0,
this
−
xi
d
d xid cxi > 0 is constant.
4)λxi exihere
1 N−1N−1
b̄i cxi N−1
x
=sliding-mode
exi=+exp(−
cxi eλvariable
(19) −
dt
+ δ̂b̄(0),
sestimation-error
xi
xisurface
xie
κ
1 ∑ ai j ux j
c
b̄
The
derivative
of
the
variable
with
resolution
of
(16)
is
e
t)e
where
xi
i
xi
(x
−
x
−
d
)
+
fact
indicates
that
the
is
exponenwhere
c
> 0
is
constant.
1
c
x
xi
xi
xi
u(14)
is here
xi
hi
hN
xi
i
xi
iN
xiThe
d
d
d
x − ¯
cexp(−
>
0λisxit)e
constant.
The
solution
(16)sliding-mode
is exid = exp(−
λxit)e
where
xiderivative
sgn(s
(x
+
The
of
the
surface
variable
with
rexid (0),(x
xix)hN − diN )d¯
hix −
is
e
=
(0),
where
∑ ai j ux j
(
d
+
b̄
)
ρ
+
b̄
−
−
x
−
d
)
+
a
u
is
xi
xi
xi
i
i
i
i
i
j
j
hi
hN
∑
iN
¯
¯
d
d
spect
to time
t can
beas
The
derivative
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surface
with
exid (0)tially
λ(variable
is
theconvergent
initial
condition
at→t sliding-mode
= by
0. Owing
to
>variable
0,i ) this
to the
0condition
tthe
∞.
ρ¯ixi+(db̄ii + b̄i )
b̄i j=1
di + j=1
xi¯i +
d
b̄
ρ
d
The
derivative
of
sliding-mode
surface
with
rexi
e
λ
(0)
is
the
initial
at
t
=
0.
Owing
to
>
0,
this
here
c
>
0
is
constant.
spect
to
time
t
can
be
written
by
j=1
xi
is at xit d=respect
xi to time
ition
0. that
Owing
to tλcan
0,written
this variable
xi >be
κxi
by
fact indicates
the
estimation-error
exid is exponen(22)
˙surface
κxi sgn(sxi )
spect
tovariable
time
t that
can
be
written
by
−
fact
indicates
the
estimation-error
variable
e
is
exponenκ
The
derivative
of
the
sliding-mode
variable
with
rexi
xi
e
is
exponen=
ė
+
c
e
+
δ̂
(20)
ṡ
d
umation-error
is
xi
xi
xi
xi
xi
xi
˙
¯
xi
sgn(s
−
)
d
xi
tially
convergent
to
0
as
t
→
∞.
ρ
(
d
+
b̄
)
sgn(s(20)
−
3.2. Integral
SMC-based
controller
To coordinate
xi i
i
xi )
ė→
exi + δ̂˙design
xi +
xi ρ (d¯ + b̄ )
convergent
as=twritten
∞.cxiby
spect
to time t to
can0ṡxibe
i)
is b̄predefined
and sgn(·) is the sign function.
where κxiρxi>(d0¯i +
→ ∞. tially
xi i a i trackingsub(22)
the
leader-following-based
multi-agent
system,
ṡ
=
ė
+
c
e
+
δ̂
(20)
(20)
Substituting
(18)
and
(19)
into
(20)
gives
xi
xi
xi
xi
xi
˙
b-34]. Substituting (18) and (19) into (20) gives˙
(22)
(22)
=
−
(
−
)
can
be
drawn
from
(16).
δ̂
λ
δ̂
δ
The
equation
xi
xi
xi
xi
3.2.
Integral
SMC-based
controller
design
To
coordinate
error
variable
of
the
ith
follower
agent
is
defined
by
=
ė
+
c
e
+
δ̂
(20)
ṡ
bxi
xi
xi
xi
xi
].
3.2.
Integral
SMC-based
controller
design
To
coordinate
N−1 (18)
where
κxi0
> 0
is predefined
and
sgn(¢)isisthe
thesign
sign function.
function.
is predefined
and
sgn(·)
κxi >
Substituting
(19) into (20)
gives a tracking- where
ed
controller
design
To and
coordinate
> in
0˙sign
is(21)
predefined
sgn(·)
the signufunction.
where
κisxiδ̂˙the
the
leader-following-based
multi-agent
system,
N−1
by the(δequation
andis
replace
(21) by
0δis
predefined
andReplace
sgn(·)
where
κ+
̂ ̇ function.
xi
xi in (16).
].
̂ xiand
xi ρ>
N−1
ρ
δ
a
[(v
−
v
)
+
(u
−
u
)
(
−
)]
=
subthe ṡleader-following-based
multi-agent
system,
a
trackingSubstituting
(18)
and
(19)
into
(20)
gives
The
equation
δ
= ¡λ
¡ δ
)
can
be
drawn
from
xi
xi
x
j
xi
xi
x
j
xi
i
j
xi
x
j
∑
xi
xi
xi
ed
multi-agent
system,
a
tracking(18)
and
(19)
into
(20)
gives
=˙ −
(δ̂xi − δxi ) by
can be drawn from
(16).
δ̂xixican
λxire-arranged
The
equation
x defined
N−1
ρ
ρ
δ
δ
(u
−
u
)
+
(
−
)]
a
[(v
−
v
)
+
ṡxi =Substituting
error
variable
of
the
ith
follower
agent
is
by
˙
(15)
i
j
xi
x
j
xi
xi
x
j
xi
xi
x
j
(22).
Then,
ṡ
̂
̇
be
∑
e
(v
−
v
)]
a
[(x
−
x
−
d
)
+
=
ρ
j=1
=
−
(
−
)
can
be
drawn
from
(16).
δ̂
λ
δ̂
δ
The
equation
xi
xi
xδ̂j xi = −λxi (δ̂xi −
xi of
ij
hi
h j agent
xi from
xi
xi
xi replace uxi in (21) with
Replace
δxi in
with
the equation
and
∑xithe
i j The
be(21)
drawn
(16).
δxi )˙ can
equation
variable
follower
is defined
by
ollower
is defined
by
5)34]. error
ai j [(vj=1
− vith
ṡagent
xi = j=1
x j ) + ρxi (uxi − ux j ) + ρxi (δxi − δx j )] (17)
∑
δ̂
in
(21)
by
the
equation
and
replace
u
in
(21)
by
Replace
˙
N−1
xi
xi
(22).
Then,
s ̇ i can
re-arranged
N−1
N−1
+ b̄i [(v
uxN ) + ρδ̂˙xixiδxiin] (21) by the equation
δ̂replace
(21)
by
the equation and
replace uxi in (21) by
Replace
5)
xi − vxN ) + ρxi (u
xi −Replace
xi xin
j=1
and
ube
xi in¯(21) byby
xρxi
ρ−
−
uρ
)xi+
ṡxi=
=b̄∑
aiN−1
−
vhxρ
) (u
+
x(u
(22).
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can
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+
δ
−
u
)
+
]
[(v
v
)
+
i−
j [(v
xi−
j−
xi
xρ
j (v
xiv(δxi) − δx j )]
∑
e
(v
−
v
)]
a
[(x
x
d
)
+
ρ
xi
i
xi
xN
xi
xi
xN
xi
x
(
d
+
b̄
)e
−
=
ρ
ρ
ṡ
(21)
xi
xi
x
j
xi
j
hi
j
+
b̄
(x
−
x
−
d
)
+
b̄
i
j
xi
i
i
xi
xi
xi
i Then,
xi
xN
∑ a i j ex j d
hi
hNxh j −
r,−
δ̂xxi − d x ) + ρ
iNd(22).
d by
exi
=N−1
[(x
(15)
be re-arranged
by Then, ṡxi can be re-arranged
xi (vxi
xi ṡ−
x j )](17)
ij j)]
xi[(v
(v
− viaxxN
xi vcan
i j) + ρ
(21) (22).
b̄j=1
) +hiρ−
hj
ij
xi (uxi − u
ij=1
xi ∑
(23)
j=1
x xN ) + ρxi δxi ]
N−1
×1
xi is
(17)
N−1
j=1ai j [(xhi −(17)
x xiis−the
xh j −constant,
vx j )]
xi0 ∑
(21)
xdi j ) + ρxid(v
N−1
>cN−1
isaxia[(x
pre-defined
pre-defined
where+ρxic+
x jρ
N−1
i
j
¯
ρ
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(u
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u
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+
]
+
b̄
[(v
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v
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+
xi
ρ
−
x
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d
)
+
(v
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v
)]
is
i
xN
xi
xi
xN
xi
xi
xi
i
j
xi
xi
x
j
hi
h
(
d
+
b̄
)e
−
a
e
ṡ
=
ρ
ρ
+ b̄∑
− xhN − diN ) + xb̄ixjiρxi (vxi − vxN )
i xid xi ) −xiλ∑
j xδjxi
xi
i (xj=1
hi b̄
)x j
x
xii −
+ b̄i)exid − ρxixi(δ̂∑
ai jde(23)
ṡ xi=ρ−i κ(dxi¯sgn(s
+
(x)hi−−xxhN
− ddiNfollower
) + b̄ ρxi (v
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and
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+ b̄(21)
ṡxxivjxN
xis
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) ++
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−
vji[(x
cxi (vj=1
aibetween
−−vthe
)]=jth
d
i )exid − ρxi ∑ ai j exix j d xi i
(23)
xi N−1
xN hi
xi xi
h j −ith
j=1
r,hNδ̂−
∑
i j ) + ρixi (v
xi iN
˙
(23)
x
x
j=1
(23)
erenj=1
xcixij ρis
>
0
is
a
pre-defined
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the
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ρ
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x
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d
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xi
x
i
xi
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xi
xi
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xi
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iN
x
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a
[(x
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x
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+
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is constant,
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d
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ixij − vxN ) + δ̂xi
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xi (vthe
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− κxi sgn(sxi ) − λBull.
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xi (δ̂Pol.
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+
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Design
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ith
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Bull. Pol. Ac.: Tech. XX(Y) 2016
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Tech. XX(Y)in2016
4
System
structure.
The2016
system Bull.
structure
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Pol. Ac.:
Tech. XX(Y)
N−1 x-axis formation-control law for the ith3.3.Bull.
follower
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xi −the
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xi − vx j ) − c
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xi
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uxi = − ρxi (d¯i + b̄i ) j=1
∑ ai j (vxiN−1
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) 1j=1N−1
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i
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+
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2017
39
∑
j=1
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i
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− d¯i + b̄i δ̂xi − d¯i + b̄i j=1
∑
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Brought
to
you
by
|
Gdansk
University
of
Technology
6)
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di + b̄b̄i ci xi di + b̄1i j=1
Authenticated
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−∑¯ai j (xhi∑
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− xahi jj (−δ̂xidx ix−
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Download Date | 3/2/17 11:22 AM
a
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T i,jT multi-agent
T u
T xic
i ·iN
the
sliding-mode
surface
The
deT u
T=
TTvu
T··,·uiN
i j∈dR
i jmined
iN[v]v
d variable
mploys
δ̂subsystem
generate
and
T
T
T
T
T
xi
the
augmented
vector
of
z
in
the
multi-agent
system
xi tointegral
xi , xxistruct
written
by
z
z
·
·
·
]
by
v
=
[v
·
·
·
v
]
u
=
[u
·
i
on
maneuvers
of
its
x-axis
subsystem.
mined
by
v
=
mined
by
v
=
[v
v
·
·
·
v
]
,
v
·
·
·
[u
v
]
uTN
T, ∆
T, e =
T on
TT. [e Here
1x-axis
2to[δgenerate
N−1
ˆ i = ˆ[δ̂swritten
T1 ezyi
TN−1
T ˆ,∈∆
2]T·v·2, ·, zu
1 ,T1N−1
2 2, d ·N−1
ˆR
1T
N−1
1, 2(N−1)×1
2i ,subsysagent
tocontrol
achieve
formation
maneuvers
of
its
subsystem.
signed
scheme
employs
s
δ̂
the
,
x
and
u
δ
δ̂
]
]
=
[u
u
]
,
∆
=
T
ocated
at
the
feedback
channel.
From
Fig.
2,
it
feeds
the
T
=
[s
s
]
,
∆
e
e
d
a
xi
xi
xi
by
z
=
[z
]
.
xi
yi
xi
yi
i
xi
i
xi
yi
i
Furthermore,
the
closed-loop
stability
of
the
x-axis
T
T
T
T
2(N−1)×1
i
xi
yi
i
i
i
i
i
i
j
iN
u
δ
δ
δ̂
δ̂
]
,
∆
=
[
,
e
=
[e
=
[u
u
]
,
∆
=
[
D.W.
S.W.
C.D.
Li
d] T
signedtocontrol
sxi ,[sxxixisand
to the
generate
the
sSimilarly,
.Tδ̂Qian,
viTong,
,following
uxi , and
∆
,
∆
,
e
,
e
,
d
,
d
and
s
1
2
N−1
inally, uxi is applied
the ith scheme
followeremploys
T
T
T
T
T
T
T
xi
yi
i
xi
yi
i
xiT
i
xi
yi
i
xiHere
i =
yi ]define
i
i
i
i
i
j
iN
i
written
by
z
=
[z
z
·
·
·
z
]
∈
R
.
T
T
T
T
T
T
T
T
T
ˆ
ˆ
ˆ
ˆ
y T ∆ vectors
d v[∆
=2d∆y[v
] ,] ,][∆
2and
N−1
=
[∆
·2·T]∆
·1T· ·∆v·N−1
=
·ˆ· ·∆are
∆
2×1
∆diN
=
∆2∆yaugmented
∆=
,their
·∆·ˆT·Concerning
∆N−1
= [∆] [1∆
,ˆvectors
·yN−
∆ˆ=
∆ˆ ·2x·∆
i x∆
xi
yi
1
2
final
control
input
u
.
Finally,
u
is
applied
to
the
ith
follower
e
e
]
,
d
=
[d
d
]
,
=
[d
=
[e
x
T
NDO-based
observer
to
estimate
the
approximation
δ̂
.
Fur1
2×1
∈
R
.
Correspondingly,
xi
xi
yi
i
j
i
xi
1
N−1
1
N
xi
tem
can
be
extended
to
the
multi-agent
system.
i their
j i j augmented
iN] are
d ∈uxi
d. Correspondingly,
the
v
=TdiN
[v
eid Similarly,
= T[eTvectors
[deTi j di j=
] T, vectors
uxi . Finally,
the ith follower
Ris applied
deterT eT
T Td
Tj =
T T[d
on maneuvers final
of itscontrol
x-axis input
subsystem.
xieTdiN
yidefine
iN
Tfollowing
T Te=
T d ∆to=
T ]xi
d[e
Similarly,
define
vectors
v·eii·iN
[v
·· ·xie·,[v
[ed]T1=
T,,]TT
=
[e
eTT2∈d e·R
e, =
, that
ui =
δ ,]T ∆
δ̂e=yi ][e
, T∆,ˆ i ∆ˆ=eT, [δ̂emined
eeTTyiN−1
]]viTthe
=its
[uxi
uyi]of
[δ the
Tv
exi=
[edefine
e=
,1following
[e
·Td·e=··(N−1)
e=
T] , coni
1,by
N−1
2TdeT2·ddu
agent
formation
x-axis
hermore, the tracking-error
variable
exi and
integral
2· ·=
N−1
d·]·T2×1
T1ji2, u
T
v
·
·
·
u
·
·
uxiTNd
(N−
d 1dd[u
agent,
z
=
[x
y
]
such
ˆ
smaneuvers
[s
,
,
d
d
and
s
T
i
hi
hi
1
2
N−1
1
2
ˆ
agent to
to achieve
achieve
formation
maneuvers
of. its
its
x-axis
subsystem.
i mined
xi syiby
iN−1
i
i
i
i
iN
i
v THere
=
[vT1vxiisubsystem.
v, T2uyi·iith
· ·xvfollower
]
,
u
=
[u
·
·
·
u
]
,
u
δ
δ
δ̂
δ̂
]
,
∆
=
[
]
,
e
=
[e
=
[u
u
]
,
=
[
y T s
y∆
T
T
T
d[s
x
T
=
s
]
.
Here
v
,
u
,
∆
,
∆
,
e
,
e
,
d
,
d
a
xi
yi
i
xi
yi
i
i
xi
yi
i
ˆ
1
2
N−1
i
xi
yi
i
i
i
i
i
i
i
j
iN
T
T
T
T
T
T
T
udvector
δ]xi[d,δ]TTyi,d] dTx,of
δ̂]dTyisub=Tx-axis
[δ̂[d
=d
[eˆxi
==[d
[uthe
=T=
[Tstability
d]ˆ TT,· ·ˆe·T
eidR
=
eThe
, di j the
= [d
diaugmented
, id
=
[d
]dthe
and
Tuclosed-loop
Td·,iN
T
i·=
xi can
i iN
xiof
yiTd]T
2×1[e
yiTd ] ded
·T· ∆
[d
struct the located
integralatsliding-mode
surface As
variable
s xi
·d∆·Nyythe
,dbe
iN
the feedback channel.
shown
ind. Fig. 2,
j] d
Nd
[d
·TTd
d
augmented
multi-agent
system
i·deterTit· feeds
T their
Ti,j Furthermore,
T
T=,i1 [di2
∆
=
[∆
∆in
·i,i(N−1)
· Ti(N−1)
∆
∆ˆ]dTT=
=d
[2N
∆
·=
·yyT(N−
∆]xiTTTN[
∆d·2xxiN·d·(N−1)
∈
vectors
are
2×1
i1
i2z
1NiN
i=
i·i2
ˆ∈
ˆ
ˆ
ˆ
i1
1N
2N
i(N−1)
e
e
]
d
,
=
[d
d
=
[e
1
2
N−1
1
x
∆ =. xiCorrespondingly,
[∆
∆
·
·
∆
]
∆
=
[
∆
·
·
·
∆
]
∆
yi
i
j
i
xi
R
.
Correspondingly,
their
augmented
vectors
are
i j di j ] , diN T = T[diN diN
dT =T [exi
2 T δT ̂ N−1
N−1
e[z
eto2d
] Tand
,multi-agent
=
ˆzcan
yiddthe
i j2(N−1)×1
T
Td
T T[dsystem.
T]
T∆by
on thescheme
NDO-based
observer
approximation
be
Concerning
jv,
i jT ∆,
iN
signed control
employs
sxi , xtoxiestimate
toxiby
generate
the
and
T=
si δ̂=xi the
[s
syi ]vTT .1=THere
vxiiwritten
,. ui ,system
, ueand
e=
,TT·T[e
d·[su·1iT1sTddjTzT, TN−1
ˆ]T[e
ˆe,
i , T∆
i=
iid,=
zsi[e
∈
R
.e· ·v,
e=
·T[v
·v,
mined
[v
[u
·=
TN−1
T,.i]]]TTHere
T
s1=extended
[s
···]u·es·Tand
∆,
e,deTe2,dddu
d,iN
an
d2=
s]vTs=
[s
sTiT2∆
· ∆ˆsu,
.e1iHere
u,
∆,
du,
i,,·d··d
s···2T2·.··iN
·N−1
seTN−1
.,·,u1·Here
∆,
∆,
e,
,·eNiN
du
id
NT
1· evT
2 · ·]·T v
N−1 ] ed, and
11by
22]eT
e
=
[e
e
·
·
,
e
·
(N−
mined
v
·
]
,
u
=
·
N−1
d[u
Tv
2£1
s
[s
s
Here
v
,
,
,
,
e
,
T
1
1
2
N−1
1
2
i
xi
yi
i
i
i
i
i
i
j
2×1
ˆ
1
2
N−1
1
2
(N−1)
d syi
d] . are
Furthermore,
the tracking-error
variable
and
its integral
the ith
follower
agent,
zdetersuch
dvhii y
si 2(N−1)×1
= 2(N−1)×1
[s
Here
, hiTTu]i , 2 R
∆i ,v ∆=
eviddthe
,]TTTd, i jT, diNT NTaa
i = [x
final control
input uxi . Finally,
uxi is applied
to=thee.xiith
∈R
Correspondingly,
their
augmented
vectors
Tfollower
T
T
TSimilarly,
T
Tdefine
T
T
xithe
i , e[vi ,that
T
T
T
T
ˆ
ˆ
ˆ
ˆ
2(N−1)×1
define
following
vectors
2×1
∆
[∆
∆
·
·
·
∆
]
,
∆
=
[
∆
·
·
·
∆
]
,
∆
T
T
T
∈
R
.
i
xi
yi
TT
TT,
T 1T
T2Td·i2
==
[d
,,their
= be
d
ˆdvectors
ˆ · · · d·(N−
R
. d ∆ˆcan
iR
∈
.∆· ·2· ·d∈
1T d2T ·s· ·T.dThe
N−1
.d[∆
Correspondingly,
∆vector
··N−1
· i(N−1)
∆]TN−1
=[dwritten
∆ˆ N
2×1
i1
1N[∆
TT·d
T , ]T]] system
[d
=
dx-axis
construct
the integral
sliding-mode
surface
variable
of
i =by
1 ·the
1T2N∆2 · · are
i in
TT Naugmented
R[d
their
augmented
vectors
are
i1
i2 [vxi1T vi(N−1)
1N
2N
agent to achieve
formation
maneuvers
of its
subsystem.
mined
vT =
· ]vaugmented
]udT N, ]=
=
·e]T·T,umulti-agent
(N−1)N
T
T [u
T. zuCorrespondingly,
T u∈
ˆ
2]uT·i,· =
N−1
1δ
2δ·ϒ
N−1
TT=
T
Tρ
Tρ
T, ρ
T
T2(N¡1)£1
T
T
T
T
e
=
[e
e
·
·
·
e
e
·
·
]
,
T ,mined
T∆[e
Te
TxiT
δ̂
δ̂
∆
[
]
,
e
=
[e
e
]
,
[u
=
[
T
T
T
d
̂
,
,
·
·
·
,
ρ
ρ
ρ
Define
=
diag
ˆ
yi
i
xi
yi
i
xi
yi
xi
yi
i
1
2
N−1
1
2
,
,
·
·
·
,
,
·
ρ
ρ
ρ
ρ
Define
ϒ
=
Define
diag
ϒ
=
diag
by
v
=
[v
v
·
·
·
v
]
,
u
=
[u
u
·
·
·
u
(N−1)
T x1
T u,
T
e2 ¢¢¢ z
=s∆,
[e
ev,
=u
[e
x1
designed control scheme employs sxi, and
x and δxi[s
z = [z
][s
T generate
Tthe T
T . asHere
dˆN¡1
dˆ 2 R
and
=Te,
s
·i ,ˆ· ·Td[vsyNeN−1
∆,
∆,1x(N−1),
e,ex(N−1),
e2ddx1
,·2T·d·iy(N−1
,·ey1·(N−
dy(N
d]y1
1 z
y1
NN
1T.dN−1
N−1
1T
Tto
TT ······∆sT
T2T,]] ·s.·, ·Here
11eTdev
2·d·
mined
by
=
v
v
,
=
[u
u
u
ˆ
ˆ
y
s
v,
u,
∆,
,
and
d
T
x
T
x
T
∆ xi= s =[∆
∆
]
,
∆
=
[
∆
·
∆
]
∆
1
2
N−1
1
2
N
1T 22
N−1
T
T
T
T
T
T
T
T and
1
N−1
1
2
N−1
e
=
[e
e
]
,
d
=
[d
d
]
,
d
=
[d
d
]
T
T
T
T
T
T
T
i
xi
yi
i
j
iN
ˆ
ˆ
ˆ
ˆ
2(N−1)×1
[dtoi1Tthe
di2T ith
· · ·follower
d T T]d , dSimilarly,
d2N
· ·T1d·T.following
dcx1
Tthe
T∆
final control input uxi. Finally, uxi isdapplied
define
vectors
= [v
=
c d∈
=
diag
,T ,x(N−1)
ΛT,∆ˆΛ
,]TN−1
·,· ··]]·,TTdiag
c,x(N−1)
i·jc
iN
cNx1
·, ·[[u·∆
i = 2(N−1)×1
Nd =
∆[dR
∆
∆ˆiN,,=
=
id
xicc
yi,] T
cd
diag
c=
cvx(N−1)
, v
c[d
y(N−1)
T=
T [∆
y1
=
·c,··x1·icd··,jy1
2(N−1)N
= e[d
[∆
∆eT(N−1)
] ·,,,
∆ˆid,T1TTc∆
∆cTNN
∆ˆ ·T22· ·····d··T(N−
1N
1N
Rits[ex-axis
e ∈=
·. · · ei(N−1)
] , =[ued u ]T=
]]y1
,eT =[e
i(N−1)
T ∆i [e
T
T
T ∆ =y(N−1)
2 · ̂ · δ
N−1
T̂dˆyi∆
·̂ i2
1T eT
2 subsystem.
N−1 T
1
2vyidi1
T·, 1·u
Ti =[δ
T yi 1]2N
agent to achieve formation maneuvers
of
,
∆
=[δ
δ
]
∆
,
e
]
,
e
=[e
e
,
d
T
s
=
[s
s
]
.
Here
,
,
∆
,
∆
,
e
,
e
,
d
,
d
and
s
xi
yi
i
xi
xi
i
xi
yi
i
xi
ˆ
e
=
[e
e
·
·
·
e
]
,
e
=
[e
e
·
·
·
e
i
xi
yi
i
i
i
i
i
i
i
j
iN
i
Tx(n−1)
diag
,y12T,...,
λsx1
λd,x1T,y1
λand
κTdy(N
λs ]x1
λλTy1
λ=x(n−1)
d, ρu,
dy(n−1)
and
sDefine
=2.control
[sT1 scheme.
sT2 scheme.
· ·ϒ· of
s Tthe
. diag
v, u,
∆,
∆,
eρ
Tλde
T,T.,λ
y(n−1)
,TdT·...,
λ,,x1
λN−1
κN
2Td ,· ·d· ,e
, · ·[e
·ˆ ,1T1, ̂ dand
ρ,x1e...,
ρλex(N−1),
Define
=
y(n−1)
Fig.
2. 2.
Structure
of
the
control
ediag
[e
esTN−1
THere
Fig.
scheme.
x, ρyand
[s
·Ndiag
sx(n−1)
v,
∆,
d vy1
Fig.
Structure
the
,,T,Here
·iN=
· = [d
·e,=
ρ· y(N−1)
= ]control
i ρ(N−
11· d
N−1
(N−
=
[dStructure
· dN−1
]R
, d2×1
d.ρNx1
d, T2N
·,s·iNyiT2their
d]ϒ
]T]diag
diof
d
]T[d
,,dT1N
and
. Here
u∆,
,deter∆de,
dd
·(N−1)N
ie2,ϒ,
x(N−1),
ij = [d
i = [s
xi s
yi] κ
i,}.
i, ∆Here
iT
i, eT
ij =
ijy1
iN
∈
Correspondingly,
augmented
vectors
are
i1 di2 · · T
T
T
i(N−1)
2(N−1)×1
diag{
·
·
,
κ
κ
κ
,
c,
Λ
diag{
,
,
·
·
·
,
κ
κ
κ
κ
,
}Λ
x1
y1
x(N−1)
y(N−1)
diag{
,
,
·
·
·
,
κ
κ
κ
κ
,
}.
Here
ϒ,
c,
x1
y1
x1
y1
=
[d
d
·· ··x1d
dy(N−1)
=, c[d
·· ·· d
x(N−1)
R
.T
2(N−1)×1
T. T·· c
T d
T , ··y(N−1)
2£1i1
cd
=
diag
Λ
,Λ
c
,=·]]·Ttheir
·,, , caugmented
2N
y1x(N−1)
i(N−1)
x(N−1)
y(N−1)
T v
TT
TT
TdN
T1N
∈
du,
=
[d
d·i2
d
[d
dT(N−
d
3.4. Stability analysis. In order to ∈
consider
the
formation
,x(N−1)
s∆,
vectors
T s
Ty1., Here
c s ==[s
diag
csTx1 ,sta,d
c]mined
·e· i·,,dby
cijv,
,ii R
cˆ[v
NuT=
iNvand
i 2 R
i1.,·Correspondingly,
i2,vN−1
1N d]T2N
y(N−1)
=
·
]
,
u
=
[u
·
·
·
u
,
i(N−1)
(N−
2(N−1)×2(N−1)
and
·
·
·
∆,
e,
e
d
and
s
2(N−1)×2(N−1)
i
N
2(N−1)×2(N−1)
d
1
2
1
2
N−1
1 2ϒ
R
T, T T
T .T
T
T
T
RN¡1
κ
∈
, ρTy1κ, ∈
· ,sRρ
Define
=N−1
diag are
diag
, TT1λTssy1TT2ϒ
,ρ1T··y(N−1)
...,
λx(N−1),
λˆ.x(n−1)
λu = [u
κN
bility of the x-axis subsystem,
the following
determined
by
v
u
¢¢¢ u
and
=
[s
··κ··2T. ¢¢¢ v
ssκ∈
]]diag
u,
∆,
∆,
e,
e, d ,, d
T=..] ,,Here
x1
y(n−1)
T· ··Define
T
T
Te
,ˆ Tρ1u,
·ˆˆ N¡1
, and
ρ
ρ]]x(N−1),
=TN−1
diag
λassumption
λx(n−1)
,ρλx1y(n−1)
and
Fig. 2. Structure
ofR
the
control
2(N−1)×1
ˆ Tx1v,
y12,∆,·ˆ·∆,
y1 , ...,about
and
sT ∆
=
[sv = [v
Here
v,
e,
dii ,, ρd
dy(N
x1 , .λscheme.
N
∆
=
[∆
·
·
,
∆
=
[
∆
,d augmen
∆
ructure of the
control
scheme.
∈
2
T1 ∆
T From
T 1 ̂ ] above
T ̂ TN−1
T
Tand
T · ·T· ∆N−1
T the
T
,the
2
N−1
1
2
̂
̂
the
vectors
matrices,
2(N−1)×1
analysis
Inconsidered.
order
to
consider
the
formation
staFrom
the
above
vectors
and
matr
the3.4.
estimation-error
variable
is
∆ = [∆
∆
¢¢¢ ∆
]
,
∆
= [∆
∆
¢¢¢ ∆
]
,
e = [e
e
¢¢¢ e
]
3.4. Stability
analysis
In
order
to
consider
the
formation
stadiag{
,
,
·
·
·
,
κ
κ
κ
κ
,
}.
Here
ϒ,
c,
From
the
above
vectors
and
matrices,
augm
c diag{
=to
diag
c
,
Λ
=
,
c
,
·
·
·
,
c
,
c
1
2
1
2
1
2
3.4.Stability
Stability
analysis
In
order
consider
the
formation
staN¡1
N¡1
N¡1
x1
y1
x1· · · y1
∈
R
.
x(N−1)
y(N−1)
T
T
T
T
T
T
T
T
2(N−1)×1
x(N−1)
y(N−1)
,
,
,
κ
κ
κ
κ
,
}.
Here
ϒ,
c,
Λ
and
c
=
diag
c
,
ΛΛ
,
c
,
·
·
·
,
c
,
c
x1
y1
x(N−1)
y(N−1)
y1
followingϒassumption
e about
=e ρ = [e
[e, 1ρTey1
·· ··R
·· ,e
]T, ,dvector
ex1dTe dcan
= ¢¢¢ d
evector
·by
· = [d
ee(N−1)
]T written
,
. = [d
x(N−1)
y(N−1)
TN−1
T
T1
T
, be[ebe
ρ2(N−1)×2(N−1)
ρvector
Define
=the following
diag
22T,∈
2Td, ·d
e
¢¢¢ e
]
]
d
tracking-error
written
x1
bility
of of
thethe
x-axis
subsystem,
the
d written
x(N−1),
y(N−1)
tracking-error
can
be
by
bility
of thediag
x-axis
subsystem,
assumption
about
1
d
i
N
(N¡1)
d
i1
i2
i(N¡1)
1N
2N
tracking-error
e
can
by
bility
x-axis
subsystem,
the
following
assumption
about
, λy1 ,..., λx(n−1)
λy(n−1)
κT =λT. =diag
and
,T λTT ϒ
κ∈
R
x1 2(N−1)×2(N−1)
ρ
ρ ,, T·· ·· ·· ,, and
ρT
ρ
Define
ture of theAssumption
control scheme.
TT
,...,
κ
κ > 0
∈λR
. ,d· ·,·=
Tλx1
T λu,
T∆,,, ∆
y(n−1)
ρdx1
ρy(N
2.variable
the
control
scheme.
6: jexi Fig.
j < e
where iscexi*of
is variable
constant
but
un],T diag
¢¢¢ s
] .x(n−1)
Here
ed, di,ρdx(N−1),
[d
dTi2
· y(N−1)
· ·Define
ds = [s
]2y1
,ϒ
dΛN=
=
[d,v,
·ρ ̂ ·,y1
·e,d(N−1)N
]
x1
y1
x(N−1),
y(N
=
diag
c· x1
,1 sϒ,
Λ
=diag
,, considered.
cκ
cx(N−1)
cand
thethe
estimation-error
considered.
xi ,Structure
N
N¡1
i ,¢¢¢ d
(N¡1)N
estimation-error
the
estimation-error
variable
is
considered.
is
y1
i1
1N
2N
i(N−1)
diag{
,
,
·
·
κ
κ
κ
,
}.
Here
c,
and
x1
y1
From
the
above
vectors
and
matrices,
the
augm
x(N−1)
y(N−1)
2(N¡1)£1
3.4.
Stability
analysis
In
order
to
consider
the
formation
stac
=
diag
c
,
Λ
,
c
,
·
·
·
,
c
,
c
diag{
,
,
·
·
·
,
κ
κ
κ
κ
,
}.
Here
ϒ,
c,
x1
y1
∗
∗
x1+
y1
From
above
vectors
and matrices,
the
=[(L
B)
⊗augmented
Ix12 ](z
−
d·i ·)dˆ+
ϒ[(L
B)
⊗d⊗
]v
x(N−1)
y(N−1)
s In orderknown.
to consider
the formation
∗ is, constant
∗but
s 2 R
.Tand
B)
⊗ Id+
−
+
∗, where
∗<
ceconstant
=
diag
cκ⊗
ΛΛ
cu,
,−
·+
, e,
cx(N−1)
cdy(N−1)
+
B)
](z
+
+
B)
I,2ϒ[(L
]v
i)
iI)2+
2,](z
Assumption
6: 6:|exi|eAssumption
| <|staexieddiag
y1
6:
|e
e0...,
where
>
x(N−1)
y(N−1)
,xidλed|the
,>
λ2(N−1)×2(N−1)
=can
Assumption
<
,the
where
0d λis
constant
but
and, eλsand
=
[sT10 sisT2 tracking-error
· ·e· s=[(L
]Tbut
. Here
v,eI, 2=[(L
∆,
∆,
eϒ[(L
x1exi
y1>
d xid
x(n−1)
y(n−1)
i ,by
N and s
xidd∈
xi
xiassumption
xiabout
d, cture of the control
scheme.
N−1
d
d
d
(
vector
e
be
written
κ
R
.
2(N−1)×2(N−1)
bility
of
the
x-axis
subsystem,
following
d
diag
λ
λy1
λ
κ
and
,λ
ϒ = diagfρ
, ρ,y1...,
, ¢¢¢, ρ
g, c = diagfc
tracking-error
e can2(N−1)×1
writtenκ
bsystem, the unknown.
following
assumption
about diag{
∈−
R
.x(n−1)
x1 ,, ϒ,
x(N¡1), ρ
x1,
Fig.
2.
Structure
of
the
control
diag
...,
λ−
λN−1
λy(N¡1)
κn
and
unknown.
, κy1scheme.
, ·vector
· · , κx(N−1)
κx1
,beκDefine
}.by
Here
Λ
and
(B
⊗λx1⊗
I2,c,
)(1
⊗−⊗
z(B
dy(n−1)
−) −
ϒ(B1
⊗d⊗
IN2 )v
x1(B
y1
unknown.
x(n−1)
y(n−1)
I
⊗
)(1
−
)(1
ϒ(B1
⊗
z
−
I
)
−
)v
Nz,−
NId2)N
n ϒ(
N−1
N−1
y(N−1)
Fig.
2.
Structure
of
the
control
scheme.
∈
R
.
N
N−1
N
2
N−1
2
the
estimation-error
variable
is
considered.
From
the
above
vectorscy1and
matrices,
the
augmented
Theorem
1:
Concerning
the staith follower
agent,
consider
its x-axis
, ¢¢¢, c
, cy(N¡1)
g,κy1
Λ = diagfλ
, λ y(n¡1)
In orderistoconsidered.
consider
the
formation
diag{
,the
,, ·· ·· ·· ,, vectors
κ
κ
κ
}.
Here
c,
x1 , λ y1,, ¢¢¢, λ
x(n¡1)
ariable
g ϒ,
above
y(N−1)
and
matrices,
3.4.
Stability
analysis
Inthe
order
to
consider
the
sta2(N−1)×2(N−1)
Theorem
1: 1:Concern
the
consider
itdiag{
κx1
κy1
κx(N−1)
κϒ[(L
}.
Here
ϒ,augm
c, Λ
Λ
Theorem
Concern
Theorem
1:∈
ithfollower
Concern
follower
the
agent,
ith.formation
follower
consider
agent,
it- x(N¡1)
consider
itx1 ,diag
∗ agent,
x(N−1)
y(N−1)
eFrom
=[(L
+
B)
⊗,I2ρ, κ
](z
−· ·d·g.i,),ρ
+
+
B)
⊗ Ithe
κ∗ith
R
2 ]v
,
,
ρ
ρ
Define
ϒ
=
∗
∗
Assumption
6:
|e
|
<
e
,
where
e
>
0
is
constant
but
e
=[(L
+
B)
⊗
I
](z
−
d
)
+
ϒ[(L
+
B)
⊗
I
]v
subsystem
dynamic
(13),
take
Assumptions
1–6
into
account,
and
κ = diagfκ
, κ
, ¢¢¢, κ
Here
ϒ,
c,
Λ
and
2(N−1)×2(N−1)
x1
y1
tracking-error
vector
e
can
be
written
by
xi
stem,
the
following
assumption
about
i
x(N−1),
y(N−1)
2
2
x1
y1
x(N¡1)
y(N¡1)
xi
xi
d
| < exid , where
e
>
0
is
constant
but
d
d
is
a
2
×
2
identity
matrix
and
⊗
means
the
Kronec
where
I
κ
∈
R
.
2(N−1)×2(N−1)
tracking-error
vector
e
can
be
written
by
is
a
2
×
2
identity
matrix
and
⊗
where
I
bility
of
the
x-axis
subsystem,
the
following
assumption
about
2
xi
s x-axis
subsystem
dynamic
(13),
take
Assumptions
1–6
inis
a
2
×
2
identity
matrix
and
⊗
means
the
Kron
where
I
2
2
sobserver
x-axis
subsystem
dynamic
(13),
takeκ 2 R
Assumptions
1–6thein-augmented
2(N¡1)£2(N¡1)
s x-axis
dynamic
Assumptions
inκ c∈ R
. (25)
From
the sliding-mode
above
vectors
and
matrices,
the dNDO-based
(15),(13),
definetake
In order
todesign
consider
thesubsystem
formation
staunknown.
c ⊗1–6
=
diag
, z) −
Λ
=the
,Both
· ·⊗
· ,above
,v⊗
cy(N−1)
⊗
)(1
zNare
−and
dconcerned
ϒ(B1
⊗
Iaugm
able
is considered.
x1 ,. c−
y1(B
N
na
N−1
2 )vag
x(N−1)
product.
Both
of
zc22Nproduct.
and
are
to
the
leader
−
(B
⊗ (15),
Ithe
)(1
zthe
−
dN(15),
)−
ϒ(B1
IIof
)v
product.
z
Both
of
and
and
vNto
are
the
concern
leader
the
estimation-error
variable
is
considered.
Nv
From
the
vectors
to
account,
design
the
NDO-based
observer
define
N stan N−1
2 (15),
N−1
N−1
3.4.
Stability
analysis
In
order
to
consider
formation
Nmatrices,
to
account,
design
the
NDO-based
observer
define
the
N
N theconcerned
to
account,
design
the
NDO-based
observer
define
the
From
the
above
vectors
and
matrices,
the
augm
∗
∗
surface
(19)
and
utilize
the
integral
SMC-based
control
law
(22).
From
the
above
vectors
and
matrices,
augmented
track3.4.
Stability
analysis
In
order
to
consider
the
formation
stae
=[(L
+
B)
⊗
I
](z
−
d
)
+
ϒ[(L
+
B)
⊗
I
]v
tracking-error
vector
e
can
be
written
by
ystem,
the
following
assumption
about
Theorem
1:
Concern
the
ith
follower
agent,
consider
iti
2
2
T
T
T
<
,
where
e
>
0
is
constant
but
∗
∗
e
=[(L
+
B)
⊗
I
](z
−
d
)
+
ϒ[(L
+
B)
⊗
I
]v
diag
,
,
...,
λ
λ
λ
λ
κ
=
,
and
T
T
ernexithe
ith
follower
agent,
consider
iti
2
2
x1
y1
determined
by
z
=
[x
y
]
and
v
=
[v
v
]
.
xi
x(n−1)
y(n−1)
by
zand
=
yhN
] vyN
and
|esubsystem,
| <and
,surface
where
exiintegral
>
0 strucis
constant
but vector
Nvector
Nv[x
xN
Fig.
2. Structure
of6:sliding-mode
the
control
scheme.
tracking-error
can
be
d
dAssumption
surface
(19)
utilize
the
SMC-based
hN
hN
bility
of
x-axis
the
following
assumption
about
determined
zNdetermined
=identity
[xeehN
yhN
] written
=means
[v
] Kron
.vN =
N and
xid(19)
(19)
and
utilize
the
integral
SMC-based
hN
(25)
Nby
xNyN
xi
sliding-mode
surface
utilize
the
SMC-based
Thesliding-mode
closed-loop
control
system
ofeand
the
x-axis
ing-error
e Ican
as
d (13),
d integral
isbeaby
2written
matrix
⊗
the
where
vector
can
be
written
by
of the
the
x-axis
subsystem,
the
following
assumption
about
able is considered.
sbility
x-axis
subsystem
dynamic
take
Assumptions
1–6
inaI2system
2)(1
×
identity
⊗
the
where
Isubsystem
–2
diag{
,Nx-)system
,tracking-error
·Differentiating
· ·of,means
κmatrix
κ−y1and
κ2(B
κ2in
,×Kronecker
}.
Here
ϒ,−
c,
Λtime
and
2 is
ynamic
(13),
take
Assumptions
1–6
in−
(B
⊗
⊗
z
−
d
ϒ(B1
⊗
I
)v
x1
x(N−1)
y(N−1)
N
n
N−1
N−1
2
unknown.
control
law
(22).
The
closed-loop
control
of
the
xDifferentiating
e
(25)
with
respect
to
time
t
gives
*
the
estimation-error
variable
is
considered.
control
law
(22).
control
The
closed-loop
law
(22).
The
control
closed-loop
system
of
control
the
the
xe
Differentiating
in
(25)
with
respect
e
in
(25)
to
with
t
respect
gives
t
−
⊗
I
)(1
⊗
z
−
d
)
ϒ(B1
⊗
I
∗ Fig. 2
turedto
asymptotically
stableiseifconsidered.
κxi > (λ
+ ρ⊗xi bI(15),
)e](z
N
=[(L
+xiB)
di ) +2(N−1)×2(N−1)
ϒ[(L +
B) ⊗ I2Both
]v of
2 zNN−1
2 )vn a
xi .−
i2
product.
and vNNare concerned
toN−1
the leader
the
estimation-error
variable
<
e∗xid ith
, where
ein
>
0 is
is(15),
constant
but
account,
design
the
NDO-based
observer
define
the
product.
Both
of
z
and
v
are
concerned
to
the
leader
agent,
follower
agent,
consider
itxiTheorem
∗
∗
N
N
e the
NDO-based
observer
define
the
e
=[(L
+
B)
⊗
I
](z
−
d
)
+
ϒ[(L
+
B)
⊗
I
]v
d
(25)
κ
∈
R
.
i
2
2
1:
Concern
the
ith
follower
agent,
consider
itaxis
subsystem
structured
by
Fig.
2
is
asymptotically
stable
Assumption
6:
|e
|| <
where
eexi
>
0 Fig.
is
but
axis
structured
2 is stable
asymptotically
stable
∗ ,, Fig.
=[(L
+by
B)
I2[x
](zhN−–ydhNi )]–T+and
ϒ[(L
⊗ vI2yN]v]T .
axis
subsystem
structured
by
– determined
xi
dsubsystem
T and
Tz.N⊗=
vN+=B)
[vxN
Assumption
6:1–6
|e∗xi
< eeand
where
>by
is=constant
constant
d determined
dby
sliding-mode
surface
(19)
utilize
the
integral
SMC-based
is∗is
a∗xi
×
⊗N–emeans
the
where
I22(B
xiin−
⊗2asymptotically
⊗hNzmatrix
−
ϒ(B1
⊗
z.0Nidentity
[x
yhN
]dbut
vwhere
= N−1
[v I
v=[(L
amic
(13),utilize
take
Assumptions
12 )(1
22
N
N ) − + B
ni) + ϒ[(L
N−1
d (13),
dI
xN
yN
(19) Stability
and
the
integral
SMC-based
e = [(L
) ](z ¡ d
+ B
) dthe
I
]v
+and
B)
⊗
I⊗
+−
ϒ[(L
+
B)
⊗2 ]v
I2+
]u
is
aI⊗2Kronecker
2])v
×
2+
identity
matrix
and
⊗⊗B)
means
Iėvectors
ė
B)
⊗
ė
I
=[(L
+
B)
I
]v
+
ϒ[(L
+
⊗
I the
ϒ[(L
]uI Kron
+
∗d.>. (λ xi+
2 ]v
22=[(L
2
sif
x-axis
subsystem
dynamic
take
Assumptions
1–6
inProof:
Select
a Lyapunov
candidate
function
V = ¡
s
Differif κ
λ
ρ
>
(
+
)e
b̄
From
the
above
matrices,
augmented
unknown.
3.4.
analysis
In
order
to
consider
the
formation
staif
κ
ρ
)e
.
b̄
2
xi
xi
xi
i
xi
−
(B
I
)(1
z
)
−
ϒ(B1
κ
λ
ρ
>
(
+
)e
b̄
xi
xi
xi
i
xi
xi
xi
i
xi
2 system of the xN
N ) − ϒ(B1
n
2 )(1
N−1
N−1t2⊗
2 )v
xid of
(25)
d xi
unknown.
the
ith follower
agent,
consider
it-dd xcontrol
law
(22).
The
closed-loop
control
Differentiating
e
in
(25)
with
respect
to
time
gives
product.
Both
zN eand
v
are
concerned
to
the
leader
agent,
−
(B
⊗
I
⊗
z
−
d
⊗
I
)v
–are
– product.
N
N
N
na
NDO-based
observer
(15),
define
the
2
N−1
N−1
2
e
closed-loop
control
system
of
the
Differentiating
in
(25)
with
respect
to
time
t
gives
Both
of
z
and
v
concerned
to
the
leader
1tracking-error
21 . 2Difentiate
V
with
respect
to Proof
time
t.: the
The
derivative
ofabout
Vcandidate
can
be
N+) ¡ ϒ(B
NI 1]∆ − (B1
1 2 e can
to
account,
design
the NDO-based
observer
(15),
define
the
vector
be
written
by
Theorem
1:
Concern
ith
follower
agent,
consider
itbility
of
the
x-axis
subsystem,
the
following
assumption
e ¡ (B
I
](1
z
¡ d
I
)v
(
+
ϒ[(L
B)
⊗
⊗
I
)v
Proof
:
Select
a
Lyapunov
candidate
function
V
=
s
T
T
+
ϒ[(L
+
B)
⊗
I
]∆
−
(B
Select
a
Lyapunov
function
V
=
s
.
Dif2
N¡1
N
N
N¡1
2
n
N
2
N−1
2
is
a
2
×
2
identity
matrix
and
⊗
means
the
Kronecker
where
I
+
ϒ[(L
+
B)
⊗
I
]∆
−
(B1
⊗
I
)v
N
Proof
:
Select
a
Lyapunov
candidate
function
V
=
s
.
Dif2
N−1
2
2
Theorem
1:
Concern
the
ith
follower
agent,
consider
itxi
2
amic
(13),
take
Assumptions
1–6
inaxis
subsystem
structured
by
Fig.
2
is
asymptotically
2 y2hNxi]stable
xi
determined
by
z
=
[x
and
v
=
[v
v
]
.
T
T
2
N
N
xN
yN
9)
andbyutilize
the
integral
SMC-based
hN
ured
Fig.
2
is
asymptotically
stable
determined
zN2=identity
[xhN yhN
] andand
vN ⊗
=means
[vxN vyN
] Kron
.
written
as V̇ = swith
Replace
s ̇ xi by
(23).
The
derivative
of SMC-based
V can
aaby22 ×
matrix
the
where
II2 ėis
sliding-mode
(19)
and
utilize
the
integral
xi s ̇ xi.surface
s
x-axis
subsystem
dynamic
(13),
take
Assumptions
1–6
inhe
estimation-error
variable
is
considered.
∗
ferentiate
V
respect
to
time
t.
The
derivative
of
V
be
is
×
2
identity
matrix
and
⊗
means
the
Kron
where
ferentiate
V
with
respect
to
time
t.
The
derivative
of
V
can
be
=[(L
+
B)
⊗
I
]v
+
ϒ[(L
+
B)
⊗
I
]u
product.
Both
of
z
and
v
are
concerned
to
the
leader
agent,
2
ferentiate
V
with
respect
to
time
t.
The
derivative
of
V
can
be
sif x-axis
dynamic
(13), takeė =[(L
Assumptions
1–6
in22 )u
NDO-based
(15),
the
κxi >system
λxi +∗define
ρ
(subsystem
.∗
+ B)
⊗
Ithe
+x-+
ϒ[(L
+
⊗−gives
I−2d]u
ϒ(B1
⊗
IB)
i )e
−
ϒ(B1
Iare
)u
I2the
)ut2 Ngives
closed-loop
control
ofxi b̄the
Differentiating
eN(15),
in
(25)
to⊗
time
N ϒ(B1
2 ]v respect
xixhasobserver
the
of
eNwith
=[(L
B)
IB)
+
ϒ[(L
+v⊗
⊗
Irespect
N2 ]v N−1
d NDO-based
N−1
2−
product.
zzeNN−1
concerned
to
leader
aa
control
(22).
closed-loop
control
system
of
Differentiating
inand
(25)
to ⊗
time
2 ](ztBoth
id .Assumption
Nwith
Tof
T i.)of
to
account,
design
the
observer
define
the
6:form
|exi
|law
where
eby
>
0ṡby
is
constant
but
written
byd by
V̇<V̇
=eSMC-based
sdxi,sṡwritten
. xiThe
Replace
ṡxi
(23).
The
derivative
of
Vand
product.
Both
of
and
v
are
concerned
to
the
leader
written
=
=
s
ṡ
.
Replace
by
ṡ
.
(23).
Replace
The
ṡ
derivative
by
(23).
The
V
derivative
of
V
xi
xid V̇
determined
by
z
=
[x
y
]
v
=
[v
v
]
xi
1
N
N
2
to
account,
design
the
NDO-based
observer
(15),
define
the
xi
xi
xi
N
N
xN
yN
9)
and
utilize
the
integral
hN
hN
xi
xi
(25)
T
T
1
ed
by
Fig.
2
is
asymptotically
stable
2
where
I
is
a 2£2
identity
matrix
and
©
means
the
Kronecker
+
ϒ[(L
+
B)
⊗
I
]∆
−
(B1
⊗
I
)v
Proof
:
Select
a
Lyapunov
candidate
function
V
=
s
.
DifN
2
2
N−1
2
zzNNd=
yyhN ]]T and
vN)v=
[v
vyN ]]T ..
xistable
(26)
+ ϒ[(L SMC-based
+2B)
⊗
I(B
−determined
⊗ Izby
axis
subsystem
byofFig.
2 is
asymptotically
punov candidate
function
= structured
sxithe
. (19)
Difsliding-mode
surface
and
utilize
the
integral
hN
unknown.
2 ]∆⊗
2N)v
the
form
ofVof
−ϒ[(L
I(B1
)(1
)[x
−by
ϒ(B1
⊗leader
determined
by
=
[x
[vT xN
2the
form
Nconcerned
2to
N−1
N−1
Nwhere
Nu=
xN
yN[uxN uy
has
the
form
sliding-mode
surface
(19)
utilize
thederivative
integral
SMC-based
hN
hN
formulated
[uand
uvI2yN
]Tnby
.]agent,
where
uN−1
ė =[(L
+e B)
⊗
]vcan
+
+
B)
⊗
INand
]u
closed-loop has
control
system
ofhas
Differentiating
in (25)
respect
time
t2⊗
gives
is
formulated
by
isu]v=
=
[u
.u⊗N vI=
where
u
N
xN
product.
Both
of
zuNis
vN−
are
to
the
NuIN
∗xferentiate
with
toand
time
t. The
of Iof
V2with
be
Nformulated
xN
yN
ė
=[(L
+
B)
⊗
+
ϒ[(L
+
B)
2
2 ]u
ifConcern
κxiderivative
λVxithe
ρ
> (law
+
)e
.
b̄respect
ectTheorem
to time t.1:The
of
V
can
be
control
(22).
The
closed-loop
control
system
the
x−
ϒ(B1
⊗
I
)u
Differentiating
e
in
(25)
with
respect
to
time
t
gives
ith
follower
agent,
consider
itxi
i
N−1
N
2
T e inN−1
T
xi
N−1
N−1
d
−
ϒ(B1
⊗
I
)u
control
law
(22).
The
closed-loop
control
system
of
the
xDifferentiating
(25)
with
respect
to
time
t
gives
N−1 determined
2 VN
ed
by
Fig.
2
is
asymptotically
by
z
= [x
y
]
and
v
= [v
v
]
.
1 s2xistable
The
augmented
integral
sliding-mode
surface
vector
is i
N
hN
hN
N
xN
yN
The
augmented
integral
The
augmented
sliding-mode
integral
surface
sliding-mode
vector
written
by
V̇
=
ṡ
.
Replace
ṡ
by
(23).
The
derivative
of
(26)
xi
xi
+
ϒ[(L
+
B)
⊗
I
]∆
−
(B1
⊗
I
)v
1
¯
nov
candidate
function
V
=
.
Dif2
¯
is
a
2
×
2
identity
matrix
and
⊗
means
the
Kronecker
where
I
N
2
N−1
2
Replace
ṡ
by
(23).
The
derivative
of
V
¯
=s
ρ
ρ
(
d
+
b̄
)e
−
a
e
[
subsystem
structured
by
Fig.
2
is
asymptotically
stable
2
s x-axis subsystem
dynamic
(13),
take
Assumptions
1–6
inxi
xi V̇ axis
V̇
=s
ρ
ρ
(
d
+
b̄
)e
−
a
e
V̇
=s
[
+
ϒ[(L
+
B)
⊗
I
]∆
−
(B1
⊗
I
)vN
2
Proof
:
Select
a
Lyapunov
candidate
function
V
=
s
.
Difxi
i
i
xi
xi
i
j
x
j
xi
∑
ρ
ρ
(
d
+
b̄
)e
−
a
e
[
xi
i
i
xi
xi
i
j
x
j
xi
ė
=[(L
+
B)
⊗
I
]v
+
ϒ[(L
+
B)
⊗
I
]u
2
N−1
2
∑
∑
d
d
xi
i
i
xi
xi
i
j
x
j
xi
d
dis asymptotically
2d xistable
2 (25) with
axis
subsystem
structured
by
Fig.
2
Differentiating
e
in
respect
to
time
t
gives
2
d
d
mulated
by
T
mulated
by
mulated
by
has
the
ofρV can
∗be. j=1
B)
II2u]v
ϒ[(L
+
time t. The
derivative
Tu.Nė
=
[uleader
]agent,
.⊗
product.
Both
ofuzyNN ]and
v=[(L
are +
concerned
to
j=1 of
N+
xN u
if
κ
> form
(λNDO-based
+
b̄respect
Nformulated
j=1
otoaccount,
design
the
(15),
the
ferentiate
Vxi of
with
to time
t. The
derivative
Vby
i )e∗xiobserver
ė is
=[(L
+
B) ⊗
⊗by⊗
+
ϒ[(L
+yNB)
B)
⊗ II22 ]u
]u
the
=
[uxNwhere
where
udefine
− formulated
ϒ(B1
⊗
Ican
)uNNbe
2 ]v
N is
N−1
2u
1 xi
iffunction
κxi
ρ
+
d.
−
ϒ(B1
I
)u
xi > (λVxi =
xi2b̄.i )e
N
N−1
(26)
xi
N−1
2
+
ϒ[(L
+
B)
⊗
I
]∆
−
(B1
⊗
I
)v
T
T
nov candidate
s
DifN
d
2
N−1
2
eplace
ṡ
by
(23).
The
derivative
of
V
xiṡ . Replace
N−1
1 2 of V by zNThe
determined
=
[x
y
]
and
v
=
[v
v
]
.
2
ˆ
xi
augmented
integral
sliding-mode
surface
N
xN
yN
sliding-mode
surface
(19)
and
utilize
the
integral
SMC-based
hN
hN
ˆ
s
=
e
+
c
edt
+
∆
written
by
V̇
=
s
ṡ
by
(23).
The
derivative
e +⊗
c I2vector
edtN + ∆(ˆi
ϒ[(Lis+
B)
⊗
−
)v
−−
(xiδ̂−xi
κxixi[κρsgn(s
λxixi
δcandidate
::xixiSelect
aaxi)i )e
Lyapunov
.sliding-mode
Difsgn(s
(δ̂xi − δxiV
)]=
λ function
¯i xi+V)xib̄−
sfor=
e+
+s ∆=N−1
surface +
vector
xiλ−
xiδ)]xixixi
sgn(s
−
(κδ̂xi
−
)]i)j e−xaugmented
xixi
V̇ Proof
=s
ρ−
(dof
aThe
⊗ cII22 ]∆
]∆edt
− (B1
(B1
Proof
Select
Lyapunov
candidate
function
Vintegral
= 122 ss2xi
Difj d xi
t−
to ρtime
t.aThe
can
∑
N−1 ⊗ I2 )vN
dbe
–]T . – by + ϒ[(L +–B) –
xi .)u
i (22).
j ehas
x jderivative
xi ∑
−
ϒ(B1
⊗
I
mulated
is
formulated
by
u
=
[u
u
where
u
T
d
control
law
The
closed-loop
control
system
of
the
xDifferentiating
e
in
(25)
with
respect
to
time
t
gives
2 Nbee ̇xN
the form
of
N
yN + B
ferentiate
V
to
time
t.
of
mulated
= [(L
+ B
by uIN) =
[u]u u ] .
where
u) j=1
N−1
N−1
N−1 N−1
N is Iformulated
2]v + ϒ[(L
ferentiate
V with
with respect
respect
time
t. NThe
Thebyderivative
derivative
of V
V can
can
be
I
−
ϒ(B1
eplace
ṡj=1
The derivative
of2Vis to
N 2 xN yN
N−1 ⊗
2 )u
xi by (23).
Differentiating
−
ϒ(B1
⊗
I
)u
N
¯
N−1
2
axis N−1
subsystem
structured
by
Fig.
asymptotically
stable
N−1
¯
–
–
–
s
in
(27)
with
respect
time
t yields
The
augmented
integral
sliding-mode
surface
vector
is
for¯
written
by
V̇
=
s
ṡ
.
Replace
ṡ
by
(23).
The
derivative
of
V
Differentiating
s
Differentiating
in
(27)
with
respect
s
into(27)
to
with
t respect
yields
to it
=s
[
ρ
ρ
κ
λ
(
d
+
b̄
)e
−
a
e
−
sgn(s
)
+
e
]
=s
=s
[(ρxiδ̂ρxixi
λxiaugmented
(−
d∑
+
− ρThe
sgn(s
+
exi) d ] I ]∆ ¡ (B
xi
xi xixi
xi
ixi
jb̄iixi
xiaxi
∑
[by
ρxixisgn(s
(iV̇d¯i=
+isxib̄xi
−
ed xixi(23).
eκof
]e ̇ + ϒ[(L
iδ
xisgn(s
i j)e+
xidxiλ−
xiddV
xi )The
∑derivative
integral
sliding-mode
surface
vector
d xi
d xi
(26)
xixi+
j)e
d
d −xiκxi
written
ṡ−
.λddxi
Replace
ṡxi
s(27)
=
eB)
+
∆ˆtime
−
dis
+ B
1+N¡1
c⊗ II22edt
)v
xi)i )e
xi
xia
ˆT B)
)]iby
−
∗[κ
xi−
xixi
2+ ϒ[(L
N+
ė
=[(L
+
⊗
I
]v
]u
formulated
by
u
=
[u
u
]
.
where
u
s
=
e
c
edt
+
∆
V̇
=s
ρ
ρ
(
d
+
b̄
)e
a
e
2
(xiδ̂xixi∑
λfxiρκ
N
N
xN
yN
>−(aδλixijxie)]x+jdρ
)e
.
b̄
j
x
j
xi
i
i
xi
xi
j=1
∑
T
j=1
j=1
xi
i
mulated
by
d
has
of
d
d
formulated
u
[uxN u
]T .
where
– uNby
mulated
j=1
has the
thexiform
form
of
N−1
e ̇ ¡ ϒ(B
˙ˆ )uˆ˙ is for- by
isvector
by
uNN+=
=
uyN
where
uNce is
j=1
N−1
yN ] .
1 2
TheVaugmented
integral
sliding-mode
surface
ˆ˙ I[u2 xN
1=ė
Iformulated
∆
N−1
ṡ
=ė
+
+
∆
N−1
N−1
N¡1
2
N
ṡ
=ė
+
ce
N−1
(26) vector i
+
ϒ[(L
+
B)
⊗
I
]∆
−
(B1
⊗
)v
Proof
:
Select
a
Lyapunov
candidate
function
=
s
.
Difṡ
+
ce
+
∆
N−1
N
2
N−1
−δ̂ρxi−∑
a)]i j ex jd =sxi [ρxi (d¯¯i + b̄i )exi − ρxi N−1
2 xiκ sgn(ss =
The
augmented
sliding-mode
surface
s inintegral
(27)
with
respect
to∆ˆtime
t yields
λ
aab̄[iλ)e
)e+
] +Differentiating
e)ewith
+
edt
∆ˆto
¯∑
je
xied −
xi
xi b̄
xice−
xirespect
¯∑
The
augmented
integral
sliding-mode
surface
vector
Differentiating
sbe
in
(27)
time
txiyields
xi ρxiδxi
d−
dρ
mulated
by
κ
λ
−
a j exiV̇
−
sgn(s
)
+
e
]
¯
=
|s
|
+
[
e
+
(
d
+
−
a
]s
λ
ρ
ρ
=s
ρ
ρ
(
d
+
b̄
)e
−
e
[
s
=
e
+
c
edt
+
=
|s
|
+
+
(
d
+
a
e
]s
κ
ρ
xi
xi
sgn(s
)
−
(
−
)]
λ
δ̂
δ
xi
xi
xi
xi
xi
i
xi
xi
i
j
xi
xi
i
j
x
j
xi
i
i
xi
xi
xi
¯
∑
=
−
|s
|
+
[
e
+
(
d
+
b̄
)e
−
a
e
]s
κ
λ
ρ
ρ
∑
xi
xi
xi
xi
xi
xi
xi
i
xi
i
xi
xi
xi
xi
i
i
i
j
xi
xi
xi
xi
i
j
xi
ferentiate
respect
to
time
t.
The
derivative
of
V
can
d
d
¯
¯
¯
¯
∑
xi
xi
xi
xi
d
d
d
d
j=1∑ V iwith
¯ ⊗i
¯
d L
V̇ =sxi [ρxi (di + b̄i )exid dd− ρxi j=1
ai j ex jd d dd
¯
¯
¯
¯
]v
+
ϒ[(
L
+
B)
⊗
I
]u
=[(
L
+
B)
⊗
I
ddd ]v
+
ϒ[(
L
+
B)
]v
⊗
+
I
ϒ[(
]u
L¯ + B)
=[(
+
B)
⊗
I
=[(
L
+
B)
⊗
I
∑
mulated
by
2
2
−
ϒ(B1
⊗
I
)u
2
N
N−1
2
2
2
j=1
j=1
j=1
mulated
by
j=1
N−1
˙ˆ ¯ = [u
j=1 derivative of ˙Vj=1
written
= sxi ṡxi . Replace ṡxi by (23). N−1
The
ˆformulated
e+
c edt
(27)
¯L
uN+
is∗∆
+
uyN
]T−
.L
ṡ =ė
+ϒ[(
ceby
+
¯(B1
¯⊗
¯ N−1 ⊗
)]i jV̇
δ̂ρxixi−∑δxiby
ˆ N−1
N B)
xN
¯u∆
¯⊗
¯edt
∗exi∗ ]
∗ ∗+
∗ s∗ =with
+
ϒ[(
L
+
B)
I
]∆
−
(
B1
Differentiating
s
in
(27)
respect
to
time
t
yields
∗where
ϒ[(
B)
I22)v
(B1
κ
λ
−(has
a
e
−
sgn(s
)
+
ṡ
=ė
ce
+
∆
N−1
Nt−N
2
¯
¯
ˆˆItime
∗
+
+
⊗
I
]∆
⊗
I]∆
¯
xi
xi
xi
xi
¯
2
N−1
2 )v
Differentiating
s
in
(27)
with
respect
to⊗∆
yields
¯
¯
d
d
≤
−
|s
|
+
[
e
+
(
d
+
b̄
)e
−
e
]s
κ
λ
ρ
ρ
d
ss =
ee ]+
+
T .c +N−1
ρxixixi|s
ρxixiρ
λxixid]se−
(dxii||+
b̄xi[[))λ
−((κ+
exiixixi)e
−
sgn(s
]isdaugmented
|s
[iλjb̄
edxi
(i dxii ed+xiadb̄)where
)e
≤xi
+κ−
ρxi(24)
sgn(s
−
−
)]
δ̂δ̂xi
the form
of =s
xi]s
xixi[κ
i )e
xixi
xi
xi
+
eedλ
((i|δδdd+
+
≤
−
ρ
¯xi
xi∑
xi
i+
i exid ]sxi integral
xi
xi
xi
xi
xi
ii a
xi
xixi
d−
dρxi
formulated
by
u
=
[u
u
u
d xi
(
=
+
c
edt
+
∆
The
sliding-mode
surface
vector
is
for|s
+
+
+
b̄
)e
−
e
=
−
κ
λ
ρ
ρ
d
d
d
N
N
xN
yN
sgn(s
−
−
)]
λ
¯
xi
xi
xi
xi
xi
i
xi
xi
i
j
xi
d
d
d
∑
xi
xi
xi
xi
j=1
¯
¯
¯
¯
d
d
ai j exid ]sdxi ∗ j=1
ρxi (di + b̄i )exid − ρxi ∑N−1
id +N−1
+
ϒ[(
L
+
I2⊗
Lϒ(
B)
⊗
II22]v
¯+
¯L
¯I⊗
¯N−1
¯c[(
¯⊗
¯)u
¯ ⊗ I=[(
¯ ∗⊗
−2 ]u
ϒ(
B1
⊗⊗
)u
+
c[(
L
+B)
B)
I]u
](z
−−
di¯)d+i )B¯
B1
+2 ](z
c[(
L
+
ϒ[(
L¯ +by
B)
=[(+Lρ¯+b̄sˆ˙B)
I2 ]v
N ϒ(
−
B1
I
)u
+
+
B)
I
Nsurface
N2forN−1
2−
N−1
2⊗
∗s+
j=1
N−1
mulated
=
−
κ
λ
ρ
|s
|
+
(
+
)e
b̄
=
−
=
−
κ
λ
ρ
κ
λ
|s
|
+
(
+
|s
)e
|
s
(
)e
s
b̄
The
augmented
integral
sliding-mode
vector
is
Differentiating
in
(27)
with
respect
to
time
t
yields
xi
xi
xi
xi
i
xi
j=1
˙ˆ ¯
−V̇ρxi
ai j exi¯ − κ)e
exied ] xixi ixixi
xi ce +
xi ∆i xid xi
N−1
=ė +
N−1
xi sgn(s
xixi
xi ρxixi¯) + λa
d xiṡd
d xi
ṡ =ė++ϒ[(
ceL
+
¯+¯∆
=s∑
−
N−1
i jxi
i xi
xi [ρxi (di d+ b̄=s
ss+
in
(27)
respect
to
time
tI−
yields
∑
∗dx j−
∗−
¯L
¯B)
κ
λ
((ddxia¯i|∗+
b̄
+
] ¯ Differentiating
Iwith
]∆]v
−cϒ(
(cϒ(
B1
⊗
)v
¯¯B1
¯
d∗ ρxi ∑¯ ai j exi+
¯sgn(s
¯∗xi ))mulated
xid[[κρ
xi|s
i )e
xiρ
xi e
xiby
N
2Icϒ[(
N−1
¯B)
¯⊗⊗⊗
¯N−1
+
cϒ[(
L
B)
I+
]v
⊗
I]v
d −
d(]B1
j=1
Differentiating
inB)
(27)
respect
toN−1
time
+
cϒ[(
+
L
+¯B1
B)
⊗
II22⊗
)v
∗=s
ϒ[(
L
+
⊗
I
]∆
−
⊗
I
)v
Ncϒ(
∗
+
[
e
(
d
+
b̄
)e
e
≤
−
+
−
]s
λ
ρ
d
2with
2t)v
ρ
ρ
κ
λ
+
b̄
)e
−
a
e
sgn(s
+
e
+ ρ+xiρ
(dxi¯i(+
b̄
)e
−
e
]s
ρ
N
2
N−1
2
¯
¯
∗
xi
xi
xi
xi
i
i
xi
i
xi
xi
xi
i
i
xi
xi
i
j
xi
xi
xi
xi
¯
N B1N
2−−
2yields
∑
i
xi
xi
i
j
xi
xi
xi
xi
∑
d
d
d
¯
¯
¯
¯
di + b̄i )e
]s
d+
(λixie(xi|xiλj=1
+
)e
]|s
|
≤[−
κ−xiκρ+
ρ
b̄
d xi
dxixi
diκ
d)e
d
+
+
]|s
|
≤[−
λ
ρ
b̄
=d −
+
[
e
+
(
d
+
b̄
)e
−
a
e
]s
λ
ρ
ρ
xi|s
i
xi
+
)e
]|s
|
≤[−
ρ
b̄
xi
xi
xi
xi
xi
xi
xi
i
xi
]v
+
ϒ[(
L
+
B)
⊗
I
]u
=[(
L
+
B)
⊗
I
xi
xi
xi
i
i
xi
xi
i
j
xi
xi
xi
id
j=1
∑
xi
xi
˙
¯
¯
¯
¯
2
2
d d d
d
d
d
(28)
d
d
ˆ
]v
+
ϒ[(
L
+
B)
⊗
I
]u
=[(
L
+
B)
⊗
I
¯
¯
¯
j=1
j=1
2⊗ I2 ](z − di )
ṡ =ė + ce + ∆
N−1
ϒ(
B1
⊗2 I2⊗
)uzN +
B)
¯ N−1
¯−⊗−
N−1
¯eB
j=1⊗ I2 )u(24)
¯ c[(
L¯ + B)
I=
−
¯+
| +δ(xiλ)]xi + ρxi b̄i )e∗xid sxi − ϒ(¯B1
B
⊗
Ic2∆
dNIL
+)++
∆˙ˆN−1
2 ](z
(24)
(24)
−
c(
⊗
Ii2))(1
⊗
z−
−
d2)N)(1
∆˙ˆ ⊗(27)
zN − dN ) + ∆
˙ˆ˙ˆd)(1
NB
N−1
ṡI =ė
=ė
+
ce
(27)
N +
sc(
+
edt
+−∆ˆc(
¯N−1
¯c[(N−1
)−−−
(|s
λxiκxixi
δ̂∗xixia−
++ρρxixib̄−
)e
sxii )e∗xixi =
¯κi ∗xi
N⊗
N−1
xidsgn(s
id
+
ϒ[(
L
+
B)
⊗
I
]∆
−
(
B1
⊗
)v
(
+
b̄
e
]s
ρ
ṡ
+
ce
+
∆
N−1
¯
¯
¯
N
2
2
¯
¯
i
j
xi
xi
∑
¯
¯
¯
¯
∗
∗
∗
d
+
ϒ[(
L
+
B)
⊗
I
]∆
−
(
B1
⊗
I
)v
¯
¯
¯
+ ρxi (di + b̄i )exidd ≤
−
e
]s
ρ
d
2
N−1
2
¯
¯
xi
i
xi
]v
+
ϒ[(
L
+
B)
⊗
I
]u
=[(
L
+
B)
⊗
I
+
cϒ[(
L
+
B)
⊗
I
]v
−
cϒ(
B1
⊗
IN2 )vN
xi
¯ d∑
¯ N−1
2 ⊗ I¯)v ¯
=
|s
−ρ
]s
κxij=1
b̄b̄i )e
]s
2
N−1
d | + [λxi exid + ∗ρxi (d¯
i⊗
je
i2eB)
xixi
+xi
cϒ[(
I2d]v
− cϒ(
xi¯a
¯¯ + B)
¯¯ ⊗
d −L
N
+xi
≤[−
λ+xi [+
=−
−κ
+ xiρdxi]|s
(dxiii |+
+−
ad)u
]sxi
λxiρexixib̄di )e
ρxi
+
ϒ[(
L
II2 ]u
=[(
L¯2−
+¯d
B)
⊗ (28)
I2 ]v
xi |s
xi(|N−1
i )e
xiB1
xi+
i j exi
xi
¯ +B1
¯ ⊗=[(
∑
¯
b̄i )e∗xi∗d ]|sxi | Bull. Pol.
d¯ N−1
d c[(
ϒ(
⊗
I
+
L
B)
I
](z
)
]v
+
ϒ[(
L
+
B)
⊗
]u
L
+
B)
⊗
I
¯
¯
N
i
2
2
j=1
2⊗ I ](z − di )
¯]j=1
¯ N−1 s⊗inI˙2(27)
Ac.:
Tech.
2016
−
B1
⊗2 Ito
+ tc[(
L + B)
∗
Bull.
Pol.
Ac.:xiTech.
XX(Y)
2016
Nz
N−1
2 )u
Bull.
Pol.
Ac.:
2016
with
time
yields
∗ b̄=
+ )ϒ[(
⊗ IDifferentiating
]∆ − (24)
(B1
xi b̄ρ
i )e(xi
xi (b̄d¯
¯⊗
=s
[ρsxi
ρ∗xixiTech.
λ¯xi¯+
−
−
sgn(s
+L
exiB)
xi¯
N ϒ(
¯i e|s
xidρ
i+
i )e
xidκd
−
λai j e+
|∑
+XX(Y)
(XX(Y)
c(
BL
I2B)
)(1
⊗
−
) +⊗
∆˙ˆ I )v2
dI )(1 2
xi
xi b̄κi )e
+
ˆ )v−
Ntime
¯¯respect
¯¯(28)
¯¯dtNN−1
N−1
xid sxi xi −∗c(B
xi ddi +
i )e
⊗
⊗
z
−
d
)
+
∆
xid − ρxi
xid ]sxi xi ∗(24)
∗⊗N−1
+
ϒ[(
+
⊗
I
]∆
−
(
B1
N
N
Differentiating
s
in
(27)
with
respect
to
yields
¯
¯
2
2
¯
¯
+
cϒ[(
L
+
B)
I
]v
−
cϒ(
B1
⊗
I
)v
≤
−
|sxi || j=1
+ [λ
e∗ d +
((dd¯i +
b̄
]s
κ
∗ ¯− ρxi d¯i exi
+
ϒ[(
− (B1N−1
⊗ I22⊗)vINN2 )vN
¯ ⊗⊗I2I]∆
¯ N−1
NL
2 cϒ[(
¯ N−1
¯ ⊗ I+
B)
d −N−1
≤
−κ
+ ∗ρ
b̄ii )e
)e
]s2xi
κxi
λxi
ρxi
ρxi d⊗i eI∗xi2dd)u
)e∗xid ]|s∗xi |
2 ]v − cϒ(B1
ϒ(xi
−¯dLi¯)++B)
xi |s
xi(λ+ [+
xiρexi
xi]|s
i |+ −
xi
N + c[(L + B)
2 ](z
xib̄di )e
xiB1
˙
+
≤[−
d
xi
xi
xi
xi
¯
¯
)e
s
b̄
ˆ
xi
xi i xid xi
d ∗ N−1
−
ϒ(
⊗ I2 )uN +
c[(L¯ + B)
I2 ](z − di )
ṡ =ė + ce + ∆˙ˆ
¯ N−1
¯˙ ⊗
¯ ⊗¯I2 )(1
−
ϒ(B1
B1
Bull.
Pol.
Ac.:
2016
̂̇ )v
∗xi sxi− c(B
=
−
κ
λ
ρ
|| +
((XX(Y)
+
N + c[(L + B)
N−1 ⊗ I2 )u
⊗
z
−
d
)
+
∆
xi |s
xiTech.
xi(24)
xi b̄
i )e
¯
¯
¯
ˆ ⊗ I2 ](z − di )
N
N
N−1
Y) 2016
5
̇
̇
s
= e
+ ce + ∆
d
¯
=
−
κ
λ
ρ
|s
+
+
)e
s
b̄
+
cϒ[(
L
+
B)
⊗
I
]v
−
cϒ(
B1
⊗
I
(24)
−
c(
B
⊗
I
)(1
⊗
z
−
d
)
+
∆
∗= − κxi |sxi | + [λxi exi xi+ xi
xib̄i )exixi −
i ρxixid ∑
xi ai j exi ]sxi
N
2
N−1
2
N
N
2
N−1
(
d
+
ρ
xi
i
¯¯¯++B)
¯B)
¯
¯⊗⊗
¯ ⊗
)exid ]|sxi |
d
d∗
d
L
−
B1
⊗ I2 )vN
]v cϒ[(
+ ϒ[(
=[(L¯+–B)
I2+
2 ]u
¯ N−1
–¯
LL
+ B)
⊗–II) 22I]v
]v
− cϒ(
cϒ(
B1
∗ ]|sxi |j=1
((λ
≤[−
κ
ρ
b̄i )e
N−1 ⊗ I2 )vN
xi +
xi +
xi b̄
˙ˆ –) + Icϒ[(
xi
̇
d ]|sxi | − c(B
¯
+ B
]v + ϒ[(L
+ B
I
]u
s
= [(L
+
+
)e
≤[−
κ
λ
ρ
xi
xi
xi
i
2
2
(24)
⊗
I
)(1
⊗
z
−
d
)
+
∆
xi
N + ϒ[(
N
2
N−1
¯ ⊗
¯ N−1
˙ˆ˙
016
5 ⊗Iz2 )v−
¯¯−
∗ d
(IB1
N dN ) + ∆
Bull.
2 ]∆
¯XX(Y) 2016
¯∗
(24) L¯–+ B)
B
⊗
– −I
– N−1⊗
N−
2 )(1
≤ − κxi |sxi | +
[λxiPol.
e∗xidAc.:
+ ρTech.
ˆ (28)
xi (di + b̄i )exid − ρxi di exid ]sxi
(24)
− c(
c(
B
⊗
I
)(1
⊗
z
d
)
+
∆
N
N
2
N−1
+ B
) I
]∆ ¡ (B
1
I
)v
s ̇ + ϒ[(L
N
¯ N−1 ⊗ I2 )u2N + c[(L¯N¡1
¯ ⊗2I2 ](z
(28)
−
ϒ(
B1
+
B)
−
d
)
∗
i
–
–5 –
016= − κxi |sxi | + (λxi + ρxi b̄i )exi sxi
̇ ¡ ϒ(B¯1N¡1¯ I2)uN + c[(L¯ + B) I2](z ¡ di)
s
Bull. Pol. Ac.: Tech.dXX(Y) 2016
Bull. Pol. Ac.:∗Tech. XX(Y)
2016
+ cϒ[(L–+ B)
⊗ I2 ]v − cϒ(B1
⊗ I2 )vN
–
–
– N−1
≤[−κxi + (Pick
λxi +upρxiκb̄xi > (λ
* such that V̇ < 0 exists. Coni )exid xi]|s
xi | xi bi)exi
+ ρ
I2)vN
s ̇ ¡ cϒ[(L + B) I2]v + cϒ(B1N¡1
˙
(24) sub− c(B¯–⊗ I2 )(1N−1 ⊗ zN − dN ) + ∆ˆ ̇
cerning V ¸ 0, the closed-loop control system of the x-axis
s ̇ ¡ c(B I )(1 z ¡ d ) + ∆ ̂
d
d
d
d
d
d
d
d
d
d
d
d
system is asymptotically stable in Lyapunov sense.
□
2
N¡1
N
N
5
Bull. Pol. Ac.: Tech. XX(Y) 2016
40
Bull. Pol. Ac.: Tech. 65(1) 2017
Brought to you by | Gdansk University of Technology
Authenticated
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Design
control
lawu uofofthe
themulti-agent
multi-agentsystem
systemasas
Design
thethe
control
law
∗
the formation
formation stability,
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shouldbe
be
the
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value of
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assigned.
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On this
seem no benefits earned from
−1
[(L+ +
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−d)
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SMCsliding
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such
an
scheme.
However, exixid d orig+2(N−1)
(I2(N−1)
B)
cϒ)[(L++
inatedfrom
from the
the scheme
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2 ]v
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as
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Nmulti-agent
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xishould
dd coningto
to Theorem
Theorem 1,
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ing
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could
conassigned.OnOn
On
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assigned.
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−
c(B
⊗
I–2 )(1
z−
−
d)N{c[(L
) –−ϒ(B1
ϒ(B1
⊗
)u
–B)
–
N−1
N−1
2−
c(B
I2[(L
)(1
⊗⊗
zI⊗
−
⊗
)u
[(L
+
B)
I−1
+) B)
B)
Isystem
−Nas
d)
uu = ϒ
=−
−−1
ϒ¡1
[(L
+
+
B)
]]d
{c[(L
+
+
B)
⊗
I⊗
−
d)
d)
u−
u
=
=
−
ϒ
ϒ⊗
tribute
to
the
decrease
chattering
phenomenon
as
asas origN
N
N
N−1
N−1
2I](z
2I]N
22¡1
2I](z
2I2](z
tribute
to
decrease
the
chattering
phenomenon
aswell
well
should
be
Design
the
control
law
of{c[(L
the
multi-agent
the
formation
stability,
aintegral
conservative
value
of However,
e∗xiHowever,
Take
Assumption
6of
into
account.
The
tracking-error
variable
[(L
+ B
) ⊗⊗
I
+ B
I⊗
such
an
NDO-based
integral
SMC
scheme.
However,
e
such
such
anthe
an
NDO-based
NDO-based
integral
SMC
SMC
scheme.
scheme.
e
e
origorig2] u fc[(L
2](z ¡ d) +
xi
xi
xi
d
dd dd
ˆ+
the
performance.
* improvement
+
I]∆
κ+
+
ϒ[(L
the
ofscheme
the
formation
performance.
On the
this
aspect,
there
seem
nothat
benefits
earned
from
–κ
–
eassigned.
is constant
but
unknown,
itconvergent
isconvergent
hard
to deter2ˆ]∆
+
+ ϒ[(L
+
⊗+
Icϒ)[(L
+
(I
]vB) ⊗ I ](z − d)
+⊗
+sgn(s)}
B)
I2]v ¡
+u+
(I
+
cϒ)[(L
+
B)
B)
⊗
⊗
I⊗
−1B)
−1
xiimprovement
inated
from
the
scheme
ismeaning
exponentially
convergent
as
proven,
inated
inated
from
from
the
scheme
is is
exponentially
exponentially
asas
proven,
proven,
2cϒ)[(L
2]v
2I]v
2(N−1)
2(N−1)
2(N−1)
[(L
+
B)
⊗
Isgn(s)}
{c[(L
+
=(I
−2(N¡1)
ϒB)
+ cϒ)[(L
+ B
I
u + (I
2 ] ) 2
2
such
an
NDO-based
integral
SMC
scheme.
e
orig∗ However,
∗
∗
mine
κ
in
Theorem
1
as
well
as
κ
in
Theorem
2.
To
guarantee
xi
ik
d
indicating
that
small
value
of
could
be
chosen.
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indicating
that
that
a small
aa small
value
value
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exieeddxixicould
could
bebe
chosen.
chosen.
AccordAccord– − 1)
isa+
a2(N
2(N
×
−
identitymatrix.
matrix.
(29),
I2(N−1)
(29) the
(29)
(29)
−I(I
(I
+
cϒ)(B1
I22)v
−−
(I
+
+
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⊗
I⊗
I2(N
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)v
d
dconvergent
is
−
1)
×
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−]v
In In
(29),
2(N−1)
N1)1)identity
N
N
N−1
N−1
N−1
2 I
* as
+
(I2(N−1)
cϒ)[(L
+⊗
⊗
I)v
2(N−1)
2(N−1)
inated
from thestability,
scheme
is
exponentially
proven,
formation
a conservative
value
of edesign
should
be con + cϒ)(B
1N¡1
B)
u ¡ (I
xidesign
2(N¡1)
2 2N ¡(29) 4.
4.
Simulation
Results
ing
to
Theorem
1,
such
a
kind
of
formation
design
could
coning
ing
to
to
Theorem
Theorem
1,
1,
such
such
a
kind
a
kind
of
of
formation
formation
could
could
conExampl
Simulation
∗
Replacing
the
control
lawu uinin(28)
(28)byby(29)
(29)gives
gives
Replacing
the
control
law
indicating In
that
a small
value
of seem
exid could
be chosen.
Accord–
assigned.
this
aspect,
there
no benefits
earned
from
−
c(B
I2)(1
)(1
−Nd
d)NNI−
−Nϒ(B1
ϒ(B1
I2)u
)uN +
−−
c(B
c(B
⊗
⊗
I⊗
⊗ ⊗
z⊗ z
−
d
)2))v
−
ϒ(B1
⊗
⊗
I⊗
−–
(I
+N−1
cϒ)(B1
⊗
N
N(29)
Nzz−
NN−1
N
2)(1
N−1
N−1
2)u
2I)(1
N−1
N−1
N−1
2I)u
Example
of
article
tribute
to
the
decrease
of
the
chattering
phenomenon
as
well
as
tribute
tribute
to
to
the
the
decrease
decrease
of
of
the
the
chattering
chattering
phenomenon
phenomenon
as
as
well
well
as
as
u ¡ c(B
2(N−1)
I
¡ d
) ¡ ϒ(B
1
I
2
N¡1
N
N
N¡1
Tosuch
demonstrate
performance
of
presented
ingdemonstrate
toanTheorem
1,the
such
a−v
kind
of formation
design
could
con2 cos
NDO-based
integral
SMC
scheme.
However,
exi*2i control
origperformance
ofhωthe
the
presented
control
˙ˆ 2 N(30) To
where
sin
−
and
cos
=
=
v
ρ
ω
θ
θ
ρ
ω
θ
˙
ˆ
i
i
i
i
i
i
i−
1i
i
–
–
ˆ
ˆ
ˆ
ˆ
ˆ
ṡ
=
ϒ[(L
+
B)
⊗
I
](∆
−
∆)
−
κ
sgn(s)
+
∆
the
improvement
of
the
formation
performance.
the
the
improvement
improvement
of
of
the
the
formation
formation
performance.
performance.
− c(B
⊗B)
I⊗
z−
d−
−sgn(s)
ϒ(B1N−1
2+
=ϒ[(L
ϒ[(L
+
B)
⊗
I]∆
](∆
∆)
+ ∆⊗ I2 )u(30)
∆⊗
+
sgn(s)}
+
ϒ[(L
+
B)
∆
κ
sgn(s)}
κκsgn(s)}
+ṡ+
ϒ[(L
+
+
B)
⊗
I⊗
I]I∆
N
N )κ
N
2 )(1
2N−1
222] ̂ ]+
tribute
to
the
decrease
of
the
chattering
phenomenon
as
well
as
2 cosfrom
2ρ
scheme,
this
section
some
simulations
on
a
multiinated
the
scheme
is
exponentially
convergent
proven,
+ B
) I
+ κsgn(s)g
u + ϒ[(L
scheme,
this
section
implements
some
simulations
on
a
multiAssumption
4:
In
the
multi-age
h
where
sin
−
h
and
.
cos
−
=
−v
=
v
ρ
ω
θ
ω
θ
ω
sin
θ
ω
θ
2
i i
i
i
i
1i
i 2i i i i
* performance.
thei improvement
ofdiscusses
the
formation
T = According
−1
2 sin θ suppose
indicating
thatDefine
a small
value
ofthe
eactions
could
chosen.
∆ˆ−+
κ1)
sgn(s)}
ϒ[(L
+isB)
⊗
Imulti-agent
robot
platform
and
The
platform
isis(q
comxiresults.
2 ]multi-agent
robot
platform
and
results.
The
comcannot
any
Theorem
Concern
the
system,
that
h
the
control
[ube
.
ureceive
HM̃information
ω
τi +
i
xi
yi ]platform
i )B̃(qi )from
Theorem
2:+I2:
Concern
the
system,
suppose
that
is
a
2(N
−
1)
×
2(N
−
1)
identity
matrix.
In
(29),
I
is
a
a
2(N
2(N
−
1)
×
×
2(N
2(N
−
−
1)
1)
identity
identity
matrix.
matrix.
InIn
(29),
(29),
I
i
2(N−1)
2(N−1)
2(N−1)
T .Results
Tfour
−1
In
(29),
I
is
a 2(N¡1) £ 2(N¡1)
identity
matrix.
Reto
Theorem
1,
this kind
of
formation
design
could
contribute
4.
Simulation
Results
4.
4.
Simulation
Simulation
Results
posed
of
mobile
wheeled
robots.
The
robots
are
iden2(N¡1)
Consider
Assumption
3.
For
the
[
]
(7)
can
have
a
form
of
ρ
ρ
Define
the
control
actions
[u
u
]
=
H
M̃
(q
)
B̃(q
)
+
τ
communication
graph
has
ain
spanning
tree.
Thematrix.
sta- posedxi ofyi four1imobile
2i
i
i i robots. The robots are idenits its
communication
graph
has
directed
spanning
tree.
The
staReplacing
the
control
law
udirected
in
(28)
by
(29)
gives
Replacing
Replacing
thethe
control
u
in
(28)
by
(29)
(29)
gives
islaw
aua2(N
−(28)
1)by
×
2(N
−gives
1)
identity
In (29),
Icontrol
2(N−1)law
T . (7) can have a form
placing the control
law u in (28) by (29)
gives
to
the
decrease
of
the
chattering
phenomenon,
as
well
as
the
�
�
�
�
�
�
4.
Simulation
Results
vector
of
L
is
1
tical
(See
Fig.
1),
where
three
follower
agents
in
order
are
[
]
of
ρ
ρ
,
that
is,
L
1
N =0
1i is
2igives
(See
Fig. 1), thethe
follower
agents
inN ordercontrol
are
bility
leader-following
formation
control
isguaranteed
guaranteedifif tical
Replacing
the control law
u in (28)control
by
(29)
bility
of of
thethe
leader-following
formation
To
demonstrate
theperformance
performance
ofthe
thepresented
control
ToTo
demonstrate
demonstrate
performance
of
of
the
presented
v̇
u
ẍ
Tpresented
N×1 control
˙
˙
˙
improvement
of
the
formation
performance.
xi
xi
hi
�
�
�
�
�
�
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
[1,
1,
·
·
·
,
1]
∈
R
and
0
numbered
by 1,
1, 2theandperformance
and the =of
sole
isis numN =
– ++
–
=
ϒ[(L
+of
B)
I22](∆
](∆
−
∆)
−
sgn(s)
+
∆ (30)
(30)
ṡ =
ṡṡparameters
=
ϒ[(L
ϒ[(L
B)
B)
⊗each
⊗
I⊗
−−
∆)
∆)
−−
κ
sgn(s)
κκsgn(s)
+designed
+
∆
(30)
= agent
numbered
by
3implements
sole
leader
agentcontrol
controller
parameters
of
each
follower
agent
are
designed
2I](∆
˙ˆ̂̇ ∆(30)
To
demonstrate
theleader
presented
thethe
controller
follower
agent
are
scheme,
this
section
implements
some
simulations
on
multischeme,
scheme,
this
this
section
section
implements
some
some
simulations
simulations
on
on
anummultia1a for
multi ̂
v̇
u
ẍ
s ̇ = ϒ[(L
+ B
) I
](∆ ¡ ∆
) ¡ κsgn(s) + ∆
xi
xi
hi
ˆ
ÿ
v̇
u
2
Further,
rank(L
)
=
N
−
the
si
yi
yi
hi
(30)=bered
bered
by=
4.
The
physical
parameters
of
these
agents
are
picked
by
4.
The
physical
parameters
of
theseThe
agents
picked
Theorem
1.ṡ = ϒ[(L + B) ⊗ I2 ](∆ − ∆) − κ sgn(s) + ∆
scheme,
this
section
implements
some
simulations
onplatform
aare
multibyby
Theorem
1.2:
robot
platform
and
discusses
the
results.
The
platform
is comcomrobot
robot
platform
platform
and
and
discusses
discusses
the
the
results.
results.
The
platform
is is
comTheorem
2:Concern
Concern
the
multi-agent
system,
suppose
that
Theorem
Theorem
2:
Concern
thethe
multi-agent
multi-agent
system,
system,
suppose
suppose
that
that
ÿ
v̇
u
Without
loss
of
generality,
the
N
yi
yi
hi
′ (t) = �s�
uprobot
fromplatform
[16], Consider
listed
bythe
l =agent’s
0.0265m,
hhThe
=
rr=
0.02m,
and
discusses
the results.
platform
is=the
comuncertainties,
Finally,
agent
can be
up
from
[16],
listed
0.0265m,
= 0.04m,
0.04m,
0.02m,
Proof
: Define
a Lyapunov
candidate
function
V′tree.
Theorem
Concern
the
system,
suppose
that
Proof
: Define
a 2:
Lyapunov
candidate
function
Vsuppose
(t)
=The
�s�
posed
of
four
mobile
wheeled
robots.
The
robots
are
idenposed
posed
ofof
four
four
mobile
mobile
wheeled
wheeled
robots.
robots.
The
The
robots
robots
are
idenidenTheorem
2:
Concern
thehas
multi-agent
system,
that
2its
Simulation
results
2staits
communication
graph
has
directed
spanning
tree.
The
sta-m4.
itsits
communication
communication
graph
graph
has
a directed
aamulti-agent
directed
spanning
spanning
tree.
The
stasystem
is
named
leader
and
other N
−4
22 are
−4
′
=
0.018kg,
I
=
1.44
×
10
kg·m
and
posed
of
four
mobile
wheeled
robots.
The
robots
are
idendescribed
by
Consider
the
agent’s
uncertainties,
Finally,
the
agent
can
be
′
w
b
b
m
=
0.018kg,
m
=
0.007kg,
=
1.44
×
10
kg·m
and
its
communication
graph
has
a
directed
spanning
tree.
The
stahere
�·�
means
2-norm.
Differentiate
V
(t)
with
respect
to
w
b
graph Differentiate
has a directed
spanning
tree.
The
stability
here
�·�communication
means
2-norm.
Vcontrol
(t)
with
respect
to if ifif btical
tical
(See
Fig.
1),
where
three
follower
agents
in
order
are
tical
(See
(See
Fig.
Fig.
1),1),
where
where
three
follower
follower
agents
agents
order
2of

three
 inin
Consider
Assumption
4,order
thatare
is,are
aNi
2 of
bility
the
leader-following
formation
control
is
guaranteed
bility
bility
of
thethe
leader-following
leader-following
formation
formation
control
is by
is
guaranteed
guaranteed
−6
−6
22. The
(See×
1),
where
three
follower
agents
in order
are
described
TsTṡ ṡ if IIwtical
=
1.44
×Fig.
10
communication
topology
ofof
this
vleader
ẋand
bility
of
the leader-following
control
is′ (t)
guaranteed
′ canformation
=
1.44
10
kg·m
communication
topology
this
xi
hithe
s
′
′
of
the
leader-following
formation
control
is
guaranteed
if
the
To
demonstrate
the
performance
of
presented
control
scheme,
w



time
t.
The
derivative
of
V
.
be
written
by
V̇
=
numbered
by
1,
2
and
3
the
sole
leader
agent
is
numnumbered
numbered
by
by
1,
1,
2
2
and
and
3
3
and
and
the
the
sole
sole
leader
agent
agent
is
is
numnumLaplacian
matrix
of
G
can
be
w
time
t.controller
The
derivative
of V ofcan
befollower
written
by
V̇ (t)
=designed
the
controller
parameters
of
each
follower
agent
are
designed
thethe
controller
parameters
parameters
of
each
each
follower
agent
agent
are
are
designed
�s�2ẋ.


is numnumbered
by
1, 2 is
and
3 and
the
sole
leader
agent
�s�
vxi system
the controller
parameters
of each
follower
agent
are
designed
multi-agent
system
illustrated
in
Fig.
3.
2 by
hi multi-agent
v̇
u
+
δ
Fig.
3.
controller
parameters
of each
follower
agent
are
designed
this
section
implements
some
simulations
on
a multi-robot
plat



xi
xi
xi
′

bered
by
4.
The
physical
parameters
of
these
agents
are
picked(8)
bered
by4.
by
4.The
4.
The
physical
physical
parameters
of
these
these
agents
agents
areare
picked
picked
′ by
Replacing
ṡ in
by(30)
(30)yields
yields
by
Theorem
1. 1.derivativeofofV V
byby
Theorem
Theorem
1.the
1.
=these
 of
parameters
bered
 v̇  form
of
 picked
 The
bered
by
physical
parameters
agents
are
Replacing
in
the
by ṡTheorem
dcomposed
−a
···
1 
12
uand
+ δdiscusses
Theorem
1. derivative
theby
results.
is=
of
up
 xi  up
 listed
xifrom
xi[16],
h =

platform
ẏThe
v=
′V
′′(t)
yi
hi
up
from
[16],
listed
by
l
=
0.0265m,
h
0.04m,
r
=
0.02m,
from
[16],
listed
by
l
=
l
=
0.0265m,
0.0265m,
h
0.04m,
0.04m,
r
=
r
=
0.02m,
0.02m,
′
=
(8)
Proof
:
Define
a
Lyapunov
candidate
function
V
(t)
=
�s�
Proof
Proof
:
Define
:
Define
a
Lyapunov
a
Lyapunov
candidate
candidate
function
function
V
(t)
=
=
�s�
�s�




up
from
[16],
listed
by
l
=
0.0265m,
h
=
0.04m,
r
=
0.02m,

0
T
Proof
:
Define
a
Lyapunov
candidate
function
V
(t)
=
�s�
2
2
2
T s Define a Lyapunov candidate function V (t) = ksk
Proof:
mobile
wheeled
robots. Thev̇ robots are
(Fig. 1),
d22 22 · · ·
 ẏhi22  four
 mm
21 −4

v0.018kg,
s�·�
yi0.018kg,
−4
2−4
˙ˆ respect
′V
′′(t)
δyi×
u×yiidentical
+−a
m==
mbbm
=
0.018kg,
=
0.007kg,
=
1.44
×
10−4
kg·m
and
=
=
=
0.007kg,
0.007kg,
Ib1.44
II=
1.44
1.44
×
1010
kg·m
kg·m
and
and
yi
˙ˆ∆}
ˆ −−κκV
′here
w
wmw
m
mw agents
=
0.007kg,
Ib =
10
kg·m
and
bb 0.018kg,
bb =

�·�
means
2-norm.
Differentiate
(t)
with
respect
to with
here
�·�
2-norm.
2-norm.
Differentiate
(t)
with
with
respect
to
ˆ ∆)
0′(t)
{ϒ[(L
=
+
B)
⊗
I](∆
](∆
−∆)
sgn(s)
+
V̇=′here
here
�·�
means
2-norm.
Differentiate
VV
(t)
with
respect
toto
2Differentiate
{ϒ[(L
+
B)
⊗
I
−
sgn(s)
+
∆}
V̇ here
.
..
2 means
2 k¢k
2means
2-norm.
Differentiate
V
with
respect
to
three
follower
in
order
marked
as
1,
2
and
3
and
.
2means
2
2

−6
2
−6
−6
2
2
.
−6
2
�s�
L
=
v̇
δ
u
+
T
T
T
T
T
yi
yi
yi
�s�
I
=
1.44
×
10
kg·m
.
The
communication
topology
of
this. .
I
I
=
=
1.44
1.44
×
×
10
10
kg·m
kg·m
.
The
.
The
communication
communication
topology
topology
of
of
this
this
. parameters
×
10 agent
kg·m
. δTheare
communication
topology
of this.
2
ss ṡss ṡṡ Iww =
′(t)
 terms.
′′ ′0can
w1.44
wsole
′ (t)
0′(t) = 2 t.
where
δ
and
the
uncertain
time
derivative
of
.
can
be
written
by
(t)
=
time
time
t.time
t.t.The
The
derivative
derivative
of
of
V
.
.
can
be
be
written
written
by
by
V̇
(t)
=
time
The
derivative
of′VVV
can
written
.
the
leader
numbered
as
4.
The
physical
t.The
The
derivative
written
by
V̇′V̇V̇
==
xi
yi

�
�
�s��s�
�s�
�s�
T
22 22 multi-agent
system
is illustrated
Fig.
∗3.Fig.
multi-agent
system
is1:
illustrated
in
Fig.
3.
multi-agent
multi-agent
system
system
is is
illustrated
illustrated
inδin
Fig.
3.
3.�
� ≤ δ ∗ ,−a
· · ·0
−a
sT s ṡ in
where
�from
δxi and δyi are the
δxiin
and
δyi(N−1)1
δxi∗ >
uncertain
terms.
� [16],
≤
(N−2)2
′V
′′0′ by
̇ ṡin
Replacing
sin
the
derivative
of
Vby
by
(30)
yields
of
these
agents
are
picked
up
by
l = 0.0265m,
xilisted
yi where
Replacing
in
the
derivative
of−
V
by
(30)
yields
� Assumption
�
Replacing
the
derivative
of
V
(30)
yields
Replacing
Replacing
ṡṡ in
the
the
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derivative
of
of
V
(30)
(30)
yields
yields
{ϒ[(L
+
B)
⊗
I
]e
κ
sgn(s)
+
Λe
}
2
d
d
{ϒ[(L
+
B)
⊗
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]e
−
κ
sgn(s)
+
Λe
}
≃≃
∗
∗
∗
2
d
d
�
�
δyi ,>where
0 are constant
Assumption 1: �δxi � ≤ δxih = 0.04m,
≤r = 0.02m,
and δyi and
δxim>
0 but unknown.
0
0
0
m w = 0.007kg,
b = 0.018kg,
�s��s�
2 T2TT sT
2
2
¡4
¡6 are slowly
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Assumption
2:
Both
and
δκyi∗sgn(s)
> 0 are
but unknown.
δ
δ
and
time-varying,
in–
–
ss TsT{ϒ[(L
I
= 1.44£10
kg
∙
m
and
I
= 1.44£10
kg
∙
m
.
The
comxi
yi
ˆ
T
b
w
˙
˙
˙
′ ′′ V̇s′ =
T
+
B)
⊗
I
](∆
−
∆)
−
+
∆}
Fig.3.
3. Communication
Communication topology
topology of
ˆ−−
ˆ ∆)
ˆ∆)
ˆ ∆}
ˆˆ
2](∆
sgn(s)
Fig.
of the
the multi-agent
multi-agentplatform.
platform.
(L+
+ B
)⊗
sI22s](∆
e−
s {ϒ[(L
sgn(s)
{ϒ[(L
+B)
B)
=�κ �{ϒ[(L
−
−
sgn(s)
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+
B)
⊗
I⊗2I](∆
==
∆)
κ
sgn(s)
κκsgn(s)
++
∆}
V̇ V̇V̇≤
ded−
Assumption
2:∆}
Both δxi and
δyi are slowly
time-varying,
incommunication topo
munication
topology
multi-agent
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is illustrated
+
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δ̇yi ≃ 0. Further,
≃ this
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dicating
2
κ�s�
�21�s�
≤−−
��s�
�s�
221T �s� + �Λ�1 1 �s�
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are described
by a directed
Assumption 1 indicates lowers
the uncertainties
are bounded
by an
s �s�2 2
(31)
–
–
sT≃
ssTT T T {ϒ[(L
(31)
+ B) ⊗ I2 ]ed − κ sgn(s)
+ Λed } 1 indicates
is
a
subgraph
of
G
.
The
weighte
As
shown
in
Fig. 3,
the
communication
graph
G
forms
From
Fig.
3,
the
communication
graph
G
forms
a
standard
unknown
constant,
which
is
mild.
From
Assumption
2,
Assumption
the
uncertainties
are
bounded
by
an
δ
and
s[(L
[(L
+
B)
⊗
I
]e
xi
From
Fig.
3,
the
communication
graph
G
forms
a
standard
(L
+ B
)
{ϒ[(L
+
B)
⊗
I
]e
−
κ
sgn(s)
+
Λe
}
≃ {ϒ[(L
{ϒ[(L
+
B)
B)
⊗
⊗
I
I
]e
]e
−
−
κ
sgn(s)
κ
sgn(s)
+
+
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Λe
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≃≃
2
�s�
d
s
+
B)
⊗
I
]e
d dd
22 22dd dd
2
(N−1)×(N−1) of G is defined by
�ϒ�
�s�
�s�
�s�
++
�ϒ�
R
a standard
spanning
tree,
where
the
adjacency
and
Laplacian
δ
seem
almost
constant.
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the
technical
contents,
unknown
constant,
which
is
mild.
From
Assumption
2,
δ
and
spanning
tree,
where
the
adjacency
and
Laplacian
matrices
are
21 221
yi
xi
T
T
spanning
tree,
where
adjacency
andmulti-agent
Laplacian
matrices are
�s�
Fig. 3.
Communication
topology of the
platform.
s �s�
sgn(s)
2 2 + �Λ�T sTT edδ seem almost constant. Considering

T1T sgn(s)
matrices
are
Assumption
hints
the designed
observer
could
evaluate or
the
technical2by
contents,
≤ − �sκT�sssgn(s)
yi
Fig.
3.formulated
Communication
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of
the
multi-agent
platform.
Fig.
Fig.
3.
3.
Communication
Communication
topology
topology
ofof
thethe
multi-agent
multi-agent
platform.
1se
formulated
by
s
e
sgn(s)
s
e
formulated
by
d
d
d
T
a11platform.
a12
···
T
�s�
s1�Λ�
12(N−1)
+
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�111 1 �s�2+s +
≤−
�Λ�
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1 11 ∗ ∗2Assumption 2 hints the designed 
calculate
observer
could
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or
δ
δ
and
munch
faster
than
the
change
rates
of δxi
2(N−1)
xi
yi



(31)
≤
−
�
κ
�
+
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e
�s�
�s�
�s�
�s�
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�s�


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1
1
≤ − �κ �1 �s�+s�Λ�
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22 1 + B)
astandard
···
 abe
From
Fig.
21 −1 aas
⊗ I22 ]ee2dd 2d2calculate δxi and (31)
In1 rates
this sense,
and22δGyi forms
could
δthe
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than
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−1
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00 and
113,change
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(31)
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2
A
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.
..
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From
Fig.
3,
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G
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From
From
Fig.
Fig.
3,
3,
the
the
communication
communication
graph
graph
G
G
forms
forms
a
standard
a
standard
spanning
tree,
where
the
adjacency
and
Laplacian
matrices
are
s
[(L
+
B)
⊗
I
]e
s
s
[(L
[(L
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B)
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⊗
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I
I
]e
]e
+ B�s�
) 2 2 22d dd and δyi . In this sense, δxi and δyicould
by 
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as almost
..
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.
11 00.. 00 
1 constants
0be treated
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
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1
0
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+�ϒ�
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2 ]1
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B)
⊗⊗
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11 ++
2(N−1)
∗
formulated
by
2(N−1)
spanning
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where
the
adjacency
and
Laplacian
matrices
are
spanning
tree,
where
where
the
the
adjacency
adjacency
and
and
Laplacian
Laplacian
matrices
matrices
Aspanning
and
LL =
A
= tree,
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by the observer. A
∗e
 areare

T
=
=
�s�
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e
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a
·
·
·
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0 1 
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11�s�
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11 2
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+
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ee∞-norm.
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
1 ∞means
11 d �∞
1�·�
1�·�
Notethat
that
(31),
e=
dd
Note
In In
(31),
e∗d−
−1
122 ¯−1
0−1
00 00
00topologies
011 0
000 be
d =
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d�
d−1
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I2means
−1
−1
2
−1
1
0
1
1
1
�s�
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�s�
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modelled
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[15,
system
of
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¯
¯
2 22[(L + B)∞�s�
2]1222(N−1)
∗ with N agents. The communication
diag{
d
,
d
,
·
·
·
,
},
whe
A =
 N−1
 and L =  – 1
2
ed [(L
1,·�ϒ�
(11)such
such
that
[(L
+B)
B)⊗⊗ via
–
–inin(11)



=






ii =
0 0(i (i
==
1,+
· ·T·,·T1T,N N
−−1)1)
that
+
aii a=
from
 GG

the
0
0
1
−1
0
0
0
1
Further,
communication
subgraph
can
be
derived
32].
Define
a
directed
graph
(V
,
E
)
composed
of
a
vertex
Further,
the
communication
subgraph
can
be
derived
from
a
system
can
be
modelled
the
theory
of
algebraic
graph
[15,
�s�
Further,
the
communication
subgraph
G
can
be
derived
from
−1
1
0
0
−1
−1
1
1
0
0
0
0
1
0
0
0
1
1
0
0
0
0
0
0
· · , N − 1). Accordingly,
  t

 
 
s [(L
[(L
+B)
B)
I22T]1
s s[(L
B)
I⊗22I]1
(L+
+ B
)⊗b̄⊗
1,
2, 
T2(N−1)
2(N−1)
∗
I22(N−1)
]]1
∈R
R2(N−1)
]12(N−1)
=
b̄+
and
L
and
and
L
==
A
A
A
==
E ⊆

·Vare



N−1b̄N−1
N−1
1 11b̄1· · · · ·b̄N−1
I2 ]1
] 2(N−1)
∈2(N−1)
bb∗ ==graph
=+
[b̄�ϒ�
set
set
ELmatrices
,matrices
where
=determined
{v
32].
aLet
directed
=whose
(V
,=
Eadjacency
) composed
ofedge
avertex
,whose
and
Laplacian
, v02 , · · by
· ,by
vN},
G,G,whose
adjacency
are
determined
e∗d∗d . . Let
+
�ϒ�
e∗deDefine
+
�ϒ�
1[b̄1b̄
0=
0are
001as
0
00V0 and
000 0and
0an
G
adjacency
Laplacian
matrices
determined
by
∗
can
be
defined
D
−
A
,
formulate

 








0
0
1
−1
0
0
0
1
1
−1
0
1
0
0
0
1
1
−1
In (31), ed = �ed �∞�s�
where
�·�
means
∞-norm.
Note
that
�s�
�s�
∞
22 can
V
the ith agent
i
= 1, 2, · · · , N.
set
V and an edge
, vV2 ,, ·the
· · ,node
v
⊆
max{
b̄N−1
}=
�B�
.2(31)
canbebe
re-arranged
by set E , where V = {v
1×

i denotes
N },vE
max{
b̄1 ,b̄1·,· ·· ,· ·b̄,N−1
}=
�B�
re-arranged
by


∞ .∞(31)
 and 
− 1) in (11)Vsuch
that
[(L
+v B)
⊗ the
aii = 0∗ (i∗∗= 1, · · · , N
0 12
0be
0 0d0¯01in0from
0 01,00investigates
0communication
000 0paper
01=
0 0·0· ·subgraph
00−1
00 0multi-agent
0−a
2
Further,
theThis
G
can
derived
the
directed
graph
the
×
V
,
the
node
denotes
ith
agent
and
i
2,
,
N.
2
−1
i
=
where
�·�
means
∞-norm.
Note
that
In
(31),
=
�e[�e
�d∞
��∞where
�·�
means
means
∞-norm.
Note
that
InIn
(31),
(31),
edee=
T�·�
T ∈ ∞-norm.
2(N−1) . Note
∗ that
∞·where
db̄
d�e
Tb̄s1
∞
∞
∞
d=
d�s�
I
·
·
]
R
Let
b
=
]1
b̄
b̄

1


–
–
�s�

2
1
1
N−1
N−1
*
2(N−1)



s

2(N−1)
G
,
whose
adjacency
and
Laplacian
matrices
are
determined
by
1 dk� where
system.
theL
ith=
agent
whose
paper
investigates
graph
multi-agent
k¢k� means
�
-norm.
Note thatthe directed
′ In (31), ed = ke
∗
2(N−1)
A=
=
′=
∗such
−a
d¯2
0Consider
0  and
0021 of neighbors
 11in the
 −1
11collection
A
=
 A
 L
−1
≤0(i
��κ=
�· 1·1,
eThis
V̇
– [(L
–+

=
0−
=
·�Λ�
, �Λ�
N−
−
1)
in
(11)
such
that
[(L
+
B)
⊗
=
0−
=
1,
·,1·b̄··+
·,··+
,N
N
−
1)
in∞in
(11)
that
that
[(L
+
B)
B)
⊗⊗
aV̇
1 1)
�κ(ib̄(i
esuch
dsuch
ii≤
iiaaii
)can
·1,
.(11)
(31)
bethat
re-arranged
by
Further,
the
communication
subgraph
Gcan
can
be
derived
from
Further,
the
communication
G
be
derived
from
Further,
the
communication
subgraph
G
be
derived
from
dcan

i = {vsubgraph


1 ,1 �s�
N−1 } =1 �B�
is
defined
as
N
∈
V
:
(v
,
v
∈
E
}.
The
ordered
system.
Consider
the
ith
agent
whose
collection
of
neighbors
L
=
amax{
in
(11)
[(L
+ B
) © I
]
�s�
�s�
j
i
j
ii = 0(i = 1, ¢¢¢, N ¡ 1)
2
.
.. pair
�s�
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px3
−0.4formation
−0.2
0.4
0.6
1 ith px1
00
00Follower
Follower 1
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Follower 3
Leader 4
x(m)
x is
−0.5
−0.5
follower and
the leader.
e multi-agent
the pre-defined
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ith
ers
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formation ma- er and diN
eeposition
e
e
e
e
ee 3
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e
e
0
0
1 epy1
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4 epy3
5
px1
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px1
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(a) 0.5 0.8px2
py1
py2
py3
1
Follower 2 −0.2
Leader
−0.4
0.2
0.4
0.6e in (17)
1 to time t, substitute the
Differentiate
gFollower
the ith
follower
agent (i =Follower
1,0.5follower
2,30· · · , Nand
−
1),
the4 leader.
m.
with
respect
Time(s)
xi
−0.5
x(m)−0.5
−0.5
−0.5
(c)
11
22 (d)
44
55
00subsystem
11
22 time
44
55 of e 00and consider
3the
(b) 33
of0.2motion
in0.4
(8)
in0.8
the
x-axis 1andexi in
Differentiate
agent
(i = 1,
2, ·are
· · , decoupled
N0.6− 1), (b)
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(13)
into3t,
derivative
(17)
with respect
to
substitute
the
(a)
xi 1
0.5
Time(s)
2
Time(s)
Time(s)
Time(s)
x(m) 0.5
0 e and consider
equently,
its formation
design
be divided
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e decoupled
in the x-axis
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Assumption
into the derivative
5 yields (b)
of
(c)
(c)
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at
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designcan
for the
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ion
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e5 yields e 11
e
Assumption
e
e22
e
px1
px2
py1
py2 0
py3
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0 N−1 px3
0
mation
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forfor
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is −0.5
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account
-axis and
design
thex-axis
y-axis.
−0.5
ė
(u
−
u
)
+
(
−
)]
a
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=
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ρ
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xi
x
j
xi
xi
x
j
xi
xi
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xi
e
e
e
e
e
e
e5
e 2
0 e
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4 (18) 5 vx1
vx2
vx3
0 px2 1
3 00
4
px1
px3 2
0 subsystem
m
the isx-axis
with
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he(8),
x-axis
taken
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j=1
ėxi =
aican
ρxi (uxi − upy1x j ) + ρxi (δpy2xi − δx j )] py3 Time(s)00−1
j [(vTime(s)
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x j) +
−2
−0.5
e
e
e
e
e
e
e
e
e
e
e
e
e
e (18)5
0
ysystem
can2
px3
vy1
1
2vy1
3 vy2
4 vy3
0 + b̄ [(v
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3 vx2
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vy2
vy3 5
0 1 epy1
3py2 j=14 epy3 5(b)
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with uncertainties
i
xi
xNvx1) + ρxi (uvx2
xi − uxN )vx3+ ρxi(c)
Time(s)
Time(s)
1
Time(s)
2
−0.5
−2
−2
−1
Time(s)
Time(s)
Time(s)
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hi
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2 (b) 3
00
1
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44
55
xN ) +
xi δxi3]3
4
5 11 uthe
22 ρ(c)
44 of the
55 ith agent,
(d) 33
(13)
=
Concerning
x-axis
subsystem
an1inte- 2Time(s)
Time(s)
Time(s)
vxi v̇xi 1
Time(s)
2
Time(s)
uxi + δ(13)
0.5
0.5
xi
(e)
(f)
(g)
gral sliding-mode
surface
with
the NDO-based
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subsystem of
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an inte(c)
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0ith
0 Concerning the xx-axis
(e)
(d)
uxicontrollers
+ δxi
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the
pre-defined
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ith
on
maer
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d
2to achieve formation
iN
[11]0.5
iswith
defined
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gral sliding-mode
the
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evy1
evy3
0.5
evx1 surface
evx2
e NDO-based
0.5 e0vy2
13) can be re-written
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0
00.5 leader.vx3
follower
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gent system. 0
[11]
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defined
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ux1
ux2
ux3
nn.by the following state-spacee −1 e
evy1 −2
evy2
evy3
e
Differentiate
follower agent
· , N0 − 1),
substitute
vx2 1
vx32
030 the 4 (19) 5
1 exi dtt,+
2δ̂xi
0 (i = 1, 2, · · vx1
3 00exi 4in (17)5with
sxi =respect
exi0 + cxito time
−0.5
−0.5
Time(s)
−2
uux1 derivative
uux2
uux3
Time(s)
−1
e
e
e
n in
are
decoupled
in
the
x-axis
and
x-axis
subsystem
(13)
into
the
of
e
and
consider
u2uy1
evx3
ẋ(8)
δ
=
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x
+
B
u
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(14)
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xi
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3 uuy2y2
4 uuy3y3 5
xi
xi xi0
xi 1xi
xi2 vy1
xi
(19)5 (e)
vy3 5 sxi = exi
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x3
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2 δ̂
3 x2
4
y1
3 vy2 4
−0.5
(d)here
−0.5
−0.5
Time(s)
−0.5
Time(s)
formation
design
can
be
divided
into
Assumption
5
yields
c
>
0
is
constant.
δ
Bits
u
+
B
(14)
−2
Time(s)
xi
xi xi
xi xi
00
11
22
44
55
11
2 0.5 3
4
5
0
11
22 0.5
33
44
55
(f)33
4
(e)
5T x-axis0 and0design
(d)here
Tcxiand
gn
the
y-axis.
> 0uxiis isconstant.
The 0derivative
of the
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variable with re- Time(s)
Time(s)
Time(s)
[xhifor
vxithe
] , Axi =
,for
BxiTime(s)
=
[0
1]
N−1
Time(s)
Time(s)
Time(s)
Time(s)
0esign
1 for the x-axis
0.5sliding-mode
0.5
(g)
0 u into
(g)
taken
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toxitime
be
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ėxi = ofspect
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ux jby
) + ρwith
−of
)] presented control
ai j [(v
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+ ρsurface
δx jthe
, Bxi = [0 1]0Tisand
0−variable
Fig.
Simulation
results
scheme. (a) Formation mane
xi4.
xi
xi (δxire0 The derivative
xi is
0nput.
0 subsystem
(18)
x-axis
with
uncertainties
can
0.5
spect
to
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t
can
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written
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j=1
u
u
u
uof
u(e)
uy3 of evyi , (f) Curves of uxi , (g) Curv
0
˙
Curves
of
e
,
(d)
Curves
e
,
Curves
0
x1
x2
x3
pyi
vxi
y1
y2
Fig. 4. Simulation
of the presented
control
(a) control
formation
theδ̂xiCartesian
coordinate
(b) curves
of epxi, (c) system,
Fig.
4.4. Simulation
results
of
presented
scheme.
Formation
maneuvers
in
the
coordinate
=maneuvers
ėxi + c(a)
ein
+
(20)
ṡxi−0.5
Fig.results
Simulation
results
ofthe
thescheme.
presented
control
scheme.
(a)
Formation
maneuvers
insystem,
theCartesian
Cartesian
coordinate
system,((
u−0.5
ux2 e , (d)
ux3curves
+ b̄iof
[(vexivxi−
vxNcurves
) + ρxiof(u˙exivyi
−
uxN
) u+y2 ρxixiofxiδuxixi]u,y3(g)
uy1
, (e)
,, (f)
curves
curves
of uyi(i = 1, 2, 3)
x1curves of
pyi
0
0
1
2
3
4
5
Curves
of
e
,
(d)
Curves
of
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,
(e)
Curves
of
e
(f)
Curves
of
u
,
(g)
Curves
of
u
(i
=
1,
2,
3).
=
ė
+
c
e
+
δ̂
(20)
ṡ
0
1
2
3
4
5
of the
epyi
, (d) Curves
, (e)
(f) Curves
of uxixi , (g) Curves of uyiyi (i = 1, 2, 3).
vyi
xi
xi Curves
xi xi of e
xi
−0.5
observer
Consider
sub- of evxi
pyix-axis
vxi
vyi ,(19)
−0.5
ẋased
vxidesign Curves
hi u
Substituting
(18) 2and
into (20) gives
Time(s)
uy3 Time(s) 0the x-axis
x3 =
1
5 an inte(13)
Concerning
subsystem3of the 4ith agent,
0 x-axis
1 sub2 uy1
3 uy2[33,
4 34].
5
and
design
its
NDO-based
observer
(15)
ign
Consider
the
(g)
Time(s)
Substituting (18)(f)and (19)
into
(20)
gives
v̇xi
u−0.5
xi + δxi
Time(s)
N−1
gral
sliding-mode
surface
with
the
NDO-based
observer
output
34].2 (f) 3
1
4
5
(g)
1.5
4-based
5observer
T
T 0 (15)T[33,
Lxi Bxi pxi − LxiConsider
(Bxi Lxi xxithe
+ith
AxiTime(s)
xxi + Bxi uagent
) [11]
ux j ) + ρthe
ai jcontrol
[(vxi − vx j ) + ρFig. 4
ṡxi =
xi
N−1
xi (uxi −
xi (δ
xi − δx j )] results of the presented confollower
(i = 1, 2, 3).
Take
its
displays
simulation
is
defined
by
e Tre-written
by theresults
following
state-space
g.
4.
Simulation
of
the
presented
control
scheme.
(a)
Formation
maneuvers
in
the
Cartesian
coordinate system, (b) Curves of e pxi , (c)
(15)
(g) ṡxi = into
Lxi xxiT+ Axidesign
x + Bof
ρxiparameters
(uxi − ux j ) +ρtrol
ai jaccount.
[(vxi − vxSome
+
xi uthe
xi ) x-axis subsystem
xi (δscheme
xi − δx j )]by the multi-agent system. In Fig. 4a, the agents
j )1.5
1.5j=1
+ Lxiofxof
esults
the,xipresented
(a)
Formation
maneuvers
in
the
Cartesian
coordinate
system,
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Curves of e pxi , (c)
irves
xie pyi
1
(d) Curvescontrol
of
evxischeme.
, (e) Curves
of
e
,
(f)
Curves
of
u
,
(g)
Curves
of
u
(i
=
1,
2,
3).
(15)
vyi be predefined,
xi+
yi + ρdiamond-shaped
j=1
+ sb̄xii [(v
ρe2,
(u
vxN
=xiue−
c=
+−
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of the
integral
SMC-based
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should
form
up
formation from string one
xi
xi δxi ] (19)
xithat
xi) +
xi dt
xiuxN )the
Curves
of
e
,
(e)
Curves
of
e
,
(f)
Curves
of
u
,
(g)
Curves
of
(i
1,
3).
vxi
vyi
xi
yi
)A
Formation
maneuvers
in
the
Cartesian
coordinate
system,
(b)
Curves
of
e
,
(c)
pxi
(21)
δvariable
xxi +internal
Bis,
+state
Bxiand
xi ucxixi = 5
xi κxi = 1.
+ b̄i [(v
δxi ] moving in straight lines; filled
(uxi N−1
− uxNits
) + ρxiwhile
− vxN ) + ρ11xiobserver,
of Concerning
the(14)
observer,
theδ̂xixiNDO-based
triangles denote the initial
xi isxi the
0.5
of uyi (i = 1, 2, 3). 2×1
es
uxi, of
(g)
T
here
0 is error
constant.
xi > and
+ cxisuch athat
ρ(21)
xh j − dixjof
) +the
ximation
and
theisgain
e of
variable
of
observer,
δCurves
δ̂vector
gain
vector
chosen
byisctrial
positions
agents
i j [(xhi −
xi (v
xi − vxand
j )] filled circles indicate the agents’
xithe
xi as L = [0 6]
xi ∈ R N−1
Follower 1
F
0 1 T
x sliding-mode surface variable with reT BT is positive.
Theaisiderivative
0.5
cxi(17)
[(xhiby−0.5
xh of
−the
dj=1
− vx j )] in the dynamic process. In order to demonstrate the
hAthat
constant
LThe
the
= thevector
=2×1
[0
and
uxi isρ+
λxi = L,xiBλ∈ = 6.
1.1.
jset
xi R
j The
xi gain
xi
xi in
i uncerj ) + ρxi (vxipositions
xi1]isxiconstant
1.5
0
T 0 is0positive.
Follower
Follower
Follower
33 0
Leader
44
spect
can be written by formation
j=1 to time
−0.4
−0.2 the
0.2
0.4
Follower11
Follower
Follower
Leader
˙ 22 bond
λestimation-error
x
tain term
of the x-axis
is designed
by δxit = 0.02£rand().
agents
together
at
1.5variable
xi = Lxi Bxi
exid =subsystem
δxi − δ̂xi , differen) + b̄imaneuver,
cxi ρxi (vxi −the
vxNdash
) + δ̂lines
+ b̄i cxi (xhi − xhN − diN
xi
x(m)
0
˙
xa uniformly
Here
rand()
is to
a MATLAB
command
that generates
the
same
At the 0.4
outset,
the 0.6
leader
agent
moves toward
oriable
variable
with
respect
time t, 1substitute
exid =
δxi −
δ̂xi , differen−0.4
00 δ̂
0.8
11
δ̂xi 0.2
d0 iN
)+
−
)˙xi+moment.
+ b̄i c(15)
(a) 0
0.5
−0.4
−0.2
0.2
0.4
0.6
0.8
xi (xhi − xhN −
i c−0.2
xi
xN
=
ėρxixi+(vcxixix-axis
e v+
(20)
ṡxib̄following
Design
the
formation-control
thethe
ith
x(m)
random
the closed
[¡1 1].
Without
thexi right-half
plane according
designed trajectory. The
1 take
x(m)law forto
vative
edistributed
pect tooftime
substitute
(15)number 1inand
Assumptions
2 into interval
xid t,and
erver
design
Consider
the x-axis
sub- SMC-based
0.5
follower
agent
Design
theSubstituting
following
x-axis
formation-control
for theagents
ith (a)
(a) 0.5
0.5
loss
integral
controller
and(19)
theintothree
follower
move
towards the left-half plane in order
(18)
and
(20) law
gives
0.5
canAssumptions
obtain
(16).of
ake
1generality,
and 2 intothe 0.5
0
n its NDO-based
observer
(15)
[33,
34].
follower
agent
0.5
NDO-based
observer
of
the
y-axis
subsystem
remain
the
same
to
form
up
the
desired
diamond.
N−1
e
e
e
N−1
˙
Follower 2
Leader 4
cxiFollower
cxi ρFollower
ρxi + 13 0
T
px1
px2
px3
xi + 1 1
−−δ̂L
− ṗLas
L+
00 Follower
xi T≃
xiT −
xi x + Bparameters
Follower
1u
2
Follower
3xi − vx j )Leader
4
0(vxi − vxN )
b̄
a
(v
−
=
−
xi ẋ
corresponding
of
the
x-axis
subsystem.
ConsidN−1
i
j
xi
i
(B
x
A
u
)
ρ
ρ
δ
δ
(u
−
u
)
+
(
−
)]
=
a
[(v
−
v
)
+
ṡ
−0
xi xi
xi xi 0
ij
xiρ (dx¯j + b̄ )xi cexixiρxi +x 1
je
xi xieρ (dx¯j + b̄−0.5
xi
xi xi xi xi
c
+
1
ρ
xi
xi
e
e
e
)
xi iv0.2) −
i
xi i 0.8i 0
epy1 2
epy2
epy3
e
eb̄ (v 0.6
ethe
py2
px1
px2
(15)
xi +
3
4py3
−0.2 agents,
0aboth
0the
=
u)xi
ering
motor
load
these
errors1 andpy1
the velocity
errors
of 5
px1 0.4In Fig. 4b-e,
px2
px3 ) position
j=1
i j (vxiu−
x j uui j=1
i xi − vpx3
xN
xi and
LTxi (B
LTxi x−0.4
+A
+
BxiFollower
uofxi−0.4
1 xi
30 −follower
xi Follower
xiFollower
xi2xxi−0.2
¯i +
¯
0.2b̄4 i )
0.4
0.6 ρx(m)
0.8
1
(
d
+
b̄
)
ρxi (dLeader
Time(s)
−0.5
xi i N−1
i
−0.5
−0.5
−0.5
and uyi ∙ 0.5. + b̄j=1
agent in the
x00and y11directions
x(m)
22 (b)are
33 illustrated.
44
55
) + B δto)uxi ∙ 0.5
1each
TAxi xxi + Bare
b̄i) +
xi uxiconfined
−
uxN0.53)3follower
+ ρxi δ4xi
00 vxN
11 ρxi (uxi
22(a)
i [(vxi −
4 ]− δ̂ 55)
0.5 0.8
u0.4
xi xxi + Bxi0.5
xi
xi xi0.6
1
xi (A0.2
δ̂
a
(
δ̂
−
−
N−1
i
j
xi
xi
x
j
(a)
1communicationTime(s)
Time(s)
Consider
the
following
formation
task.
The
leader
agent
According
to
the
designed
topology
in
Fig. 3,
Time(s)
0.5
(21)
Time(s)
¯
¯
(16)
1
b̄
i
nternal
variable
ofTthe observer, δ̂xi
di +)(b)
b̄i j=1
N−1 di + b̄i
B δ )Tstatex(m)
δ̂xi −agents
atracking
−xi )¯follower
(c)
i j (δ̂xi − δ̂x jxthe
(b) position errors of follower
)(a)
+ gain
LTxi0.5along
(Bvector
x(16)
+∈
AxRxiline.
+ Bu
δ̂ofxixiδ−xiLand
a straight
The
1 are defined(c)
by epx1 = [xh1 ¡
2×1
xi Lxi
xi
¯i + b̄1keep
xi xximoves
22
+
c
a
[(x
−
x
−
d
)
+ 0ρxi (vxix − vx j )]
the
L
is
d
+
b̄
d
xi
i
j
xi
xi
hi
h
j
1
i
i
i
i
j
0
j=1
T
N−1
x
y
0
0
0 TBuxi
BTxiconstant
L
x
+
Ax
+
)
c
the
leader
and
form
up
into
a diamond-shaped
formation.
The
¡ (x
¡
d
)] + [x
¡ (x
¡ d
)]
and
e
= [y
¡ (y
xi
xi
xi
h2
h1
h4
py1
h1
h2 ¡ d12)] +
xi
j=1
12
14
x
λB
=uxiL+
epy1
epy2
epy3
)
δikpositive.
e
e
e
xi xi
a
(x
−
x
−
d
)
xi (Axi xxi +
xi BBxixiis
i
j
px1 N−1 −px2
px3 e
y
x
e
e
e
hi
h
j
e
e
i
j
e
e
e
vx1
vx2
vx3 =
py3
cxi px3 as follows.
px1 leader px2
straight0trajectory of the
is presented
yh1py2 ¡ (yh4 ¡ d
epx2 = xh2 ¡ (xh1 ¡ d 21), epy2
d¯ixa Car+ b̄−ix)dpy1
14)].˙ Similarly,
x + [
00 ρxi (In
00
Bxi δik ) variable
j=1
differen−0.5
a
(x
−
)
−
xid = δxi − δ̂xi , −0.5
i
j
ρ
c
(x
−
x
−
d
)
+
b̄
c
(v
−
v
+
b̄
y) + δ̂xi
xee
hi
h
j
−1
i
j
e
ee ¡
−−0.5
δ̂n-error
λxieecoordinate
i
xi
i
xi
xi
xi
xN
−0.5
hi
hN
iN
−
xi − δe xi ) =tesian
xid
evy2
¯iepy3
epy1
epy2
0
2
5 vy1
system,
initial
= x3 ¡ (x4h21 ¡ d 32
),
ρ1xi (4dcoordinates
+ 2b̄i )5 j=1 3of the0 leader
0the
4
5evx1
h1 ¡ d
h3 ¡ (y
vy3 5
2vy1epy3 = y
3 vy2
4 h2vy3
e 2= yh2e ¡ (y
e3
e 21),5epx3
1 are
41evx3
vx2
N−1 h30
2
3
e withpx3respect to0 time t,1 substitute
(15)
vx1
vx3
y vx2
Time(s)
1
c
b̄
Time(s)
Time(s)
i
xi
x = vx1
Time(s)
located
[0 exp(−
m 0.6λxim].
Correspondingly,
in
the formation-control
¡ d
¡ v
, e = vy1 ¡ vy2 + vy1 ¡ vy4,
Time(s)
−1
Design5 its
thevelocities
following
x-axis
for
the ith
x2 + v
x1 ¡ v
32), evx1
ion
of (16)
is −0.5
exiat
=
(0), where
−10
xN−1
+ law
i j ux j x4 vy1
exi
2xiinto
−2
hN
(c) a−2
1 1 and
2t)e
d0
d 3
(d) 33
11 ¡1 (x
22hi1 −(c)
33 − diN4)
55
(b)¡1 −
4 d and 5take Assumptions
¯i e+
¯im∙s
00 , e11 = v22 ¡ v
4
5
b̄4i c0.2
0ρ0.1
4
(b)
xi m∙s
d
(
d
b̄
x+ b̄i ) , Time(s)
x-direction
and
y-direction
are
set
by
and
e
= v
¡ v
,
= v
¡ v
xi
i
2
follower
agent
vx2
x2
x1
vy2
y2
y1
vx3
x3
x2 and 4evy3 = 5
j=1
1
Time(s)
λxit)e1xidat(0),
=(16).
exp(−
condition
t = where
0. Owing to λxi−> 0, this (xhi − xhN
a
u
2 − diN ) +
Time(s)
ininitial
i
j
x
j
Time(s)
0.5
Time(s)
¯
¯
0.5
d
(
d
+
b̄
)
ρ
+
b̄
respectively.
order
into
diamond
= v
errors
can
ux2 converge
ux3
(c) isup
xi ithe desired
i
κxiin this i (d)
y3 ¡ vy2. From Fig. 4b-e, these definedux1
(d)i j=1
(e)
s tthat
theOwing
estimation-error
exponen= 0.
to 2λxi >In
0,variable
this to eform
xid
N−1
(e)
sgn(s
−
)
c
c
+
1
+
1
ρ
ρ
T coordinate
xi
xi
xi
xi
xi
0.5
system, the initial
coordinates
has been achieved. This indi¯+
0.50ρ1,(follower
ṗxi −toL
κxi −of follower
0
0u as theu desired
gent
0xi ẋasxi t →exi0∞.
-error
variable
is exponenb̄i (vxiformation
aib̄j (v
− vtox jzero
)−
−0.5
v0.5
i ) xiuux1
xN0 )
0
d
ux2x2
sgn(s
− uxi¯=
)b̄i )xi [0dei m ¯ei +xim],
¯i +ux3presented
evy2
evy3
x3b̄ei ) evy1
ex1
evy2
(
d
(
d
ρ
2 and follower 3 inevx1order are
at
1.1
0.8
cates
that
the
control
scheme
can realize the formaxi
xi
e [0 ρm vy1
vy3
T
j=1
e placed
e
(22)
vx1b̄i )
vx2
vx3
ρ
(
d
+
vx2
vx3
xi
i
Bxi Lxi xxi + Axi xxi +
B
u
)
xi xi
0
00
00
m] and
[0 m 0.3design
m],evy1
respectively.
Their
of the multi-agent
system in spite of uncertain−0.5
−1 To coordinate
e
evy3relative coordinations
N−1 tion−2maneuver (22)
−1
−0.5
l SMC-based
controller
e
b̄i 3 κ−2xi0 >
0 3 vy21 4
2 5 where
4 0 1is1predefined
5
uuy3 5
vx3u + B δ )
0
1
2
0
1
2
3
4 law
5uuy12each follower
and
sgn(·)
is
the
sign
function.
0
1
3uuy2y2
4 agent
2
3
4
5
+
B
xi
xi
xi
xi
in
order
are
assigned
as
[¡0.2
m 0.2
m],
[¡0.4
m 0
m]
and
ties.
Further,
the
formation-control
of
δ̂
a
(
δ̂
δ̂
−
−
)
−
y1
y3
i
j
xi
xi
x
j
Time(s)
ollowing-based
multi-agent
a tracking- d¯ + b̄−0.5
troller design To
coordinatesystem,
Time(s)
(16)
Time(s)
Time(s)
Time(s)
−0.5
−2
+ b̄δ̂˙i j=1
i−0.5
−0.5from (16).
where
predefined
andd¯isgn(·)
is the
sign
4
T m [¡0.2
m]Bu
with
regard
to>the
leader
shown
(d)
(3δ̂3xifunction.
− in
) can55be drawn
λ(e)
δ4xi4Fig. 4f-g.
The
equation
0 ¡0.2
2 (d)
3 κxi
40 is
5 i agent.
xi
00
11 xi =22−is
elti-agent
follower
is
defined
system,
tracking) +the
LTxi5ith
(B
Axxi1+
(e)
xiof
xi Lxi xxi a+agent
xi ) by
0
1
2
44
55
0
1
2 (f) 33
˙
Time(s)
˙xiN−1
Time(s)
0.5by
0.5The equation δ̂ xi Replace
=−
−
be
drawn
fromand
(16).
λ0.5
δ(21)
Time(s)
Time(s)
xi (δ̂
xi ) can
0.5
agent
is
defined
Time(s)
δ̂
in
by
the
equation
replace
u
in
(21)
by
c
xi
xi
xi
u
u
u
u
u
u
x
N−1
(e) x2 ˙
+ Bxi uxi + Bxi δik )
x1
x3 −
x1
x2
x3a (x
(g)
−Simulation
x(f)
(f)
i j hi
h j −udi j )inby
(g) control approach [16]. (
x
Fig.
5.replace
results
¯i +
Then,
ṡxi can
by
and
be
re-arranged
(21) byof the adaptive sliding mode
−0vx j )] δ̂xi in (21)ρ(22).
= u ai j [(xhi − x0.5
xi
b̄equation
h0j − di j ) + ρxi (vxi Replace
xi (dthe
i ) j=1
0
x3 λ e 42
Bull.
Pol.
Ac.:
Tech. 65(1) of
2017
0
x
=
−
(17)
xi
xi
Curves
of
e
,
(c)
Curves
of
e
,
(d)
Curves
of
e
,
evyi , (f) Cur
(22). Then, ṡxi can
be re-arranged
by
pxi
pyi (a) Formation maneuvers
vxi (e) Curves
− dj=1
5.5. Simulation
control
approach
[16].
in
N−1
i j ) + ρxid(vxi − vx j )] Fig.
umode
uy2
uy3 u
N−1
Fig.
Simulationresults
resultsof
ofthe
theb̄adaptive
adaptivesliding
slidingmode
control
approach
[16].
Formation
maneuvers
inthe
theCartesian
Cartesianco
c
u(a)
uy3
y1
y1
y2
1
c
(17)
x
Brought
to
you
by
|
Gdansk
University
of
Technology
i
xi
0
¯
x b̄i )exi − ρxi
−0.5
(
d
+
a
e
=
ρ
ṡ
+
b̄
(x
−
x
−
d
)
+
b̄
(v
−
v
)
ρ
−0.5
N−1
xi
i
i
j
x
j
xi
16) isi ehixid =hNexp(−
t)e
of
,
(c)
Curves
of
e
,
(d)
Curves
of
e
,
(e)
Curves
of
e
,
(f)
Curves
of
u
,
(g)
Curves
of
u
(i
=
1,
2,
3).
ixidxi(0),
xi ewhere
xN
−
(x
−
x
−
d
)
+
a
u
−0.5
iNλxiCurves
d
d
i
j
x
j
pxi
pyi
vxi
vyi
xi
yi
hi
hN
−0.5
Curves
of
e
,
(c)
Curves
of
e
,
(d)
Curves
of
e
,
(e)
Curves
of
e
,
(f)
Curves
of
u
,
(g)
Curves
of
u
(i
=
1,
2,
3).
iN
pxi3 uy2 4
pyi
vxi
vyi
xi
u
uy3
Authenticated yi
¯
x
(23)
+b̄b̄i0i)e
) 4xi −1ρxi
¯i¯3i+
j=1 5
1 ṡxi =
25ρρxixi((dd
5
2ai j ex j0d3di + b̄1i4 j=1
) + b̄iρat
) 1 to λxix 2y1> 0,0this
ondition
t xi=−
0.v 0Owing
xi (v
2
3
4
5
iN
d
Download
Date
| 3/2/17
11:22 AM
−0.5xN
0 is a pre-defined
constant,
di j Time(s)
is the pre-defined
Time(s)
Time(s)κ
(23)
Time(s)
j=1
4
5 x
xi
estimation-error
variable
e
is
exponen−
κ
sgn(s
)
−
λ
(
δ̂
−
δ
)
xi
0
1
2
3
4
5
(f)
xi
xi
xi
xi
xi
d
(g)
ion between
follower and the jth follow-− (f)
onstant,
d isthe
theith
pre-defined
sgn(s8 )
uy
ux
uuyy
uuxxevy(m/s)
evy(m/s)
vx
∑
evy(m/s)
(m/s)
eevy
vy(m/s)
e (m/s)
epy(m)
(m)
eepx
px(m)
(m/s)
e (m)
eevx
vx(m/s)
py
epx(m)
evx(m/s)
∑
u
uy u
x
y
uy
∑
ux
evx(m/s)
evy(m/s)
epy(m)
epx(m)
0.5
∑
uy
epy(m)
evy(m/s)
vy
evx(m/s)
(m/s)
eevy(m/s)
epy(m)
∑
∑
uuyy
u
vy
∑
ux
e (m/s)
∑
uuxx
evy(m/s)
evx(m/s)
e (m)
(m)
eepy
py(m)px
(m)
eepx
px(m)
∑
∑ ∑
(m/s)
eevxvx(m/s)
epy(m)
px
e (m)
∑
∑
∑
∑
∑
uy
uy
ux
ux
evx(m/s)
evy(m/s)
epx(m)
epy(m)
∑
∑
∑
y(m)
y(m)
∑
∑
y
y(m)
y(m)
y(m)
∑
∑∑
u scheme.
u
u
x2 u
x3 u
Fig. 4.x1 uSimulation
results
of the presented control
(a) Formation
maneuvers in the Cartesian coordinate system, (
uy
y1u
y2u
y3u
Time(s)
x1
x2
x3 Time(s)
y2
0
−0.5
Curves of
e pyi , (d)
Curves
of e−0.5
, (e) Curves of evyiy1, (f) Curves
of uy3xi , (g)
−0.5
−0.5
(g) Curves of uyi (i = 1, 2, 3).
of2 e2 vyi , (f)
(f)evxi
vxi0, 0(e) Curves
1 1
3 3Curves
4 4 of u5xi 5, (g) Curves of uyi (i = 1, 2, 3).
u,y2(d)4 Curves
uy3 5 of
0 0
1 1Curves
2 u2y1of e3pyi
3
4
5
Time(s)
−0.5
Time(s)
Time(s)
Time(s)
(g)(g)
0
1
4control5 scheme. (a) Formation
Simulation
results
of 2the(f)presented
maneuvers in the Cartesian
5
1.5 coordinate system, (b)
(f)3
ux3
Curves of e pxi , (c)
y(m)
y(m)
y(m)
g. 4.
Time(s)
leader-following
formation
uncertainofmultiple
agents
integral sliding mode
rves of e pyi , (d) Curves ofObserver-based
evxi
of evyi , (f) Curves
ofcontrol
uxi , (g)of Curves
uyi (i =
1, 2,by3).
(g), (e) Curves
1.5
ults
of of
thethe
presented
control
scheme.
Formation
ininthe
(b) Curves of e e pxi, (c)
esults
presented
control
scheme.(a)(a)
Formationmaneuvers
maneuvers
theCartesian
Cartesiancoordinate
coordinatesystem,
system,
, (c)
1.5
1 (b) Curves of pxi
rves
of
e
,
(e)
Curves
of
e
,
(f)
Curves
of
u
,
(g)
Curves
of
u
(i
=
1,
2,
3).
vxi
vyi
xi
yi
Curves of emaneuvers
of Cartesian
evyi , (f) Curves
of uxi ,system,
(g) Curves
of uyi (iof=e1,
2, 3).
vxi , (e) Curves
Formation
in the
coordinate
(b) Curves
pxi , (c)
1
1
(a)
0.5
of uxi , (g) Curves of uyi (i = 1, 2, 3).
1
−1 −10
0 0
vx3
−2
0
0.50.5
1
2 evy1
2
Time(s)
Time(s)
0.5
(d)(d)
2
3
uTime(s)
ux1 Time(s)
ux2 ux2
x1
(e) 0
−1
−10
20
5
e
e e
vx3 vx3
3 3 evy2 4 4
2
4
e
−0.5
−0.5
ux2 00
0.50.5
ux15
ux3 ux3
vx3
4
epy3
epy3
1
e3 3
evx2
4 4eevx3
vx2
Time(s)
Time(s)
Time(s)2
22 (c)(c)33
Time(s)
Time(s)
(d)
(f) 0
(d)
vx1
vx2
0 0
0.5
0
5
11
vx3
44
5
ux1
euvy1
e
x1
1 1
ux311
vy1
2 2
uux2
e e
3 3
Time(s)
Time(s)
0.5
22 (e)(e)
33
1
4 4
44
evy1
Time(s)
Time(s)
(f)
(f) 0
Fig. 5. Simulation
results of
px3
epy2
3
epy2
4eepy3
py3
Time(s)
e5
4
vx1
evy1
2evy1
1
11
evy3
5
4
4
Time(s)
Time(s)
(c)
(c)
e
vx3
e
3 evy2
vy2
4
Time(s)
Time(s)
22 (d) 33
Time(s)
Time(s)
ux1
ux2
(g)
(e)
(e)
−0.5
5
5
e
vx2
4
2
11
uuy12
y1
44
55
ux3
3 uuy2
4
33
44
Time(s)
22 (f)
y2
Time(s)
Time(s)
Time(s)
(g)
(g) control
mode
0
−2
evy3
evy3 5
5
1
0
py
e
px2
2 (d)
3
2 (b) 3
epy3
0.5
0.50
2
30
−0.5
0
5 5Time(s) 0
(e)
−0.5
−0.5
55
00
epy1
epy1 2
1
1
evy2
e
px1
1
−2
−20
0
0.5
55
e
1
02
3
0
Time(s)
−1 0
55
(c) 0
uux3
e e
x3 vy3
vy3
−2 x2vy2vy2
0
Leader 4
epy22
2
e (m)
e (m)
epy(m)
(m)px
epy
5
5
epy1
−0.5
epy2e
py2
0
epy1e
py1
1 1 eevx12 2
2
Time(s) −20−2
0 0
(d)
evy35 5
px3
4
4
4
(b)
(b)
e
e 0.5
3
3
3
Time(s)
Time(s)0
11
0.8 0
−0.5 0
0
−0.5
−0.50
0
1
1
1
vy
30
0−0.5
Time(s)−0.5
0 0
5 5(b)
4
1
11epx2
0 0
(c)
2
2
e
epx3
px3
0.8
0.8
0.4
x(m)
(a) 0.5
e (m/s)
2
4 4epy3
vx1
−1 evx2evx2
0
ux
5
Time(s)
Time(s)
1
Time(s)
2 (b)(b) 3
Time(s)
0
(e)
(c)
e e
vx1 vx1
1 1
1
3epy23
1
1
Leader
4 4
Leader
0.6
vy
1 1
epy12 2
e e
px3 px3
0.60.6
epy(m)
px
0
0 02
e
e e
px2 px2
ex(m)
e
x(m)
epx1 0.8
epx2
px1
(a)
0.8
0.5 px2
evx(m/s)
(m/s)
eevy(m/s)
−0.5
0
1 1
px1
Follower
3 3
Follower
0.4
0.6
0.6
Fo
0.5
uy
5
e (m/s)
vx
evx(m/s)
evy(m/s)
4
e
Follower 3 00.5
Leader
4
0.2
Leader 4
Follower 3
0.4
0.5
0.4
x(m)
x(m)
(a)
(a) 0.5
Follower 2
2 2 0.2
Follower
0 0Follower
0
0.40.4
−0.5
x(m)
x(m)
−0.50
(a)(a) 0.500.5
1
e e−0.5
px1 px1
Follower 1
0.2
0.2
Follower 1
−0.2 Follower 3 0
Follower 2
ux
px3
0.8
0
0
0
vy
py
0
−0.5
−0.5
0 0
0.6
0.20.2
Leader
4
(b)
e (m)
epx(m)
e (m)epx(m)
x(m)
(a) 0.5
0 0
e
Follower
1 1
Follower
−0.2
0 0
−0.4
Follower 2 −0.2
−0.4
−0.2 Follower
0.5 30 0
0.5
0.5
0.4
0.2
0
−0.4
evx(m/s)
Follower 1
0.5
0.5
epx(m)
(m)
epx
0.50.5
−0.2
−0.2
e (m/s)
0.5
0
Follower 2
−0.4
Follower 1
Follower 1
0
0
−0.4
−0.4
(m/s)
eevxvx(m/s)
epy(m)
epy
(m)
y(m)
y(m)
1 1
1
uuxxe (m/s)
vy
evy
(m/s)
y(m)
1.51.5
0.5
0.5
uuyy
1.5
uuy3
y3
5
0
−0.5
0
55
u
x
ux
uy
5
−0.5
0
Fig. 5
0 0
uy
uy
ux3
uy
the adaptive sliding
approach [16]. (a
uy1
uy2
uy3
Curves
of
e
,
(c)
Curves
of
e
,
(d)
Curves
of
e
,
(e)
Curves
of evyi , (f) Curv
Fig. 5. Simulation
results−0.5
of the adaptive
sliding
mode control
approach
[16]:
(a) formation
maneuvers
the Cartesian
coordinate system,
pxi
pyi (a) in
vxi
uy1 −0.5
Fig.
5.
results
adaptive
sliding
control
[16].
Formation
maneuvers
umode
uy2uy2 approach
uy3uy3
Fig.
5. Simulation
Simulation
resultsofof
ofe the
the
adaptive
sliding
mode
control
approach
[16].
(a)
Formation
maneuvers in
inthe
the Cartesian
Cartesianco
co
y1
(b)
curves
of
e
,
(c)
curves
,
(d)
curves
of
e
,
(e)
curves
of
e
,
(f)
curves
of
u
,
(g)
curves
of
u
(i = 1, 2, 3)
pxi
vxi 5
yi
0
1
2 pyi −0.5
3 −0.5 4
−0.5
−0.5
0 vyi
1of e ,2 (f) xiCurves
3
4u , (g)
5 Curves of u (i = 1, 2, 3).
0
Curves
of
eepxi
,
(c)
Curves
of
e
,
(d)
Curves
of
e
,
(e)
Curves
of
pyi
vxi
vyi
xi
yi
Curves
of
,
(c)
Curves
of
e
,
(d)
Curves
of
e
,
(e)
Curves
of
e
,
(f)
Curves
of
u
,
(g)
Curves
of
u
(i
=
1,
2,
3).
pxi
pyi
vxi
vyi Time(s)
xi
yi
5
0 0
1 1
2 2
3 3
4 4
5 Time(s)
0 0
0.5
uy1
Time(s)
Time(s)
(f)(f)
1
2the
illustrates
uy2
uy3
(f)
3
4 results5 of
simulation
0 0
1 1
the adaptive8fuzzy
2 2
3 3
Time(s)
Time(s)
(g)(g)
4 4
55
(g)
certainties when forming up the agents, a control scheme that
g. 5. Simulation results of theTime(s)
adaptive sliding mode control approach [16].
(a) Formation maneuvers in the Cartesian coordinate system, (b)
sliding-mode
control
approach
[16]
by
the
same
multi-agent
integratesin
the
technique
of NDO-based
observer
and
esults
of
the
adaptive
sliding
mode
control
approach
[16].
(a)
Formation
maneuvers
in
the
Cartesian
coordinate
system,
(b) the methodBull.
ults
of
the
adaptive
sliding
mode
control
approach
[16].
(a)
Formation
maneuvers
Cartesian
(b)
8
8 of e pyi ,(g)
Bull. Pol.
Pol. AA
rves of e pxi , (c) Curves
(d) Curves of evxi , (e) Curves of evyi , (f) Curves of
uthe
Curvescoordinate
of uyi (i =system,
1, 2, 3).
xi , (g)
system.
These
results
in
Fig. 5
are
adopted
for
performance
of
integral
SMC
is
addressed.
According
to
a given
communicaCurves
of
e
,
(d)
Curves
of
e
,
(e)
Curves
of
e
,
(f)
Curves
of
u
,
(g)
Curves
of
u
(i
=
1,
2,
3).
rves of e pyipyi
, (d) Curves of evxivxi
, (e) Curves of evyivyi
, (f) Curves of uxi ,xi(g) Curves of uyi yi(i = 1, 2, 3).
comparisons
and ourmaneuvers
motivationinisthe
to highlight
superiority
tion(b)
topology, the theoretical analysis proves that the formation
approach [16].
(a) Formation
Cartesianthe
coordinate
system,
the presented
control
scheme.
of evyi , (f)ofCurves
of uxi , (g)
Curves
of uyi As
(i =in1,Fig. 5a,
2, 3). the approach control of the multi-gent system in the presence of uncertainties
can also realize the same formation maneuver as the formation is of asymptotic stability. The control scheme has achieved the
Bull. Pol. Ac.: Tech. XX(Y) 2016
in Fig. 4a. However, our NDO-based integral sliding control formation maneuvers
a multi-robot
platform.
Bull.by
Pol.
Ac.:Tech.
Tech.XX(Y)
XX(Y)
2016The simulaBull.
Pol.
Ac.:
2016
method has better control performance in Fig. 4b-e via the com- tion results have demonstrated the effectiveness of the control
parisons in Fig. 5b-e.
Bull. Pol. Ac.: Tech. XX(Y)scheme.
2016
Concerning the approach in [16], a fuzzy inference system
This paper has presented some theoretical results, as well as
(FIS) is designed to resist the uncertainties such that the control some numerical results for the multi-robot platform. In order to
performance is subject to a number of fuzzy logic. The uncer- focus on the motivation of control design, some difficulties in
tainties in this paper are formulated by 0.02 £ rand(), compared application, such as communication delays and collisions bewith the expression of 0.005 £rand() in [16]. With the limited tween agents, are not considered during the control design. The
number of fuzzy rules, it is difficult for FIS to achieve better no-communication-delay and no-collision conditions are mild
performance against the variations of uncertainties.
enough for small-scale formations but they are rather idealized
for large-scale formations. In order to take the presented control
scheme into practical account, we continue to investigate this
area and further works are in progress.
5. Conclusions
This paper has investigated the formation-control problem of
multiple agents. The agents under consideration are wheeled
mobile robots. The formation mechanism is leader-following.
The uncertainties originated from each individual agent result
in the formation uncertainties of the multi-agent system. It is
mildly assumed that the formation uncertainties are bounded
by an unknown boundary. In order to resist the formation un-
Acknowledgements. This work was partially supported by the
National Natural Science Foundation of China Project under
grants 60904008 and 61473176, the Natural Science Foundation of Shandong Province for Outstanding Young Talents in
Provincial Universities (ZR2015JL021) and the International
Science and Technology Cooperation Project between China
and Denmark under grant No.2014DFG62530.
43
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D.W. Qian, S.W. Tong, and C.D. Li
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