are designed to solve the containment control problem
by using the algebra graph theory and the backstepping
design method.
(2) Compared with the traditional backstepping design with
tuning functions, the design procedure proposed in this
paper reduces the complexity of the virtual controls and
the distributed controllers. At each intermediate step,
the virtual update law is not introduced, and the actual
update law for the unknown parameter is obtained at the
nal step, which makes the controller design and per-
formance analysis of the closed-loop system much more
simple.
(3) Distributed adaptive controllers combined with σ -
modication are designed with the relative state infor-
mation, which guarantee that all the followers will expo-
nentially converge to the convex hull spanned by the
leaders’ outputs with adjustable tracking errors and all
the signals in the closed-loop system remain global uni-
formly ultimately bounded.
2. Notation
The following notation will be used throughout the paper. |·|
is the absolute value of a real number; · is the Euclidean
norm of a vector; λ
min
(·) is the minimum singular value of a
matrix. Let G = (V
f
, E, A)beaweighteddigraphoftheN fol-
lowers with a vertex set V
f
= {1, … , N},adirectededgeset
E࣪V
f
×V
f
, and a weighted adjacency matrix A = [a
ij
] ࢠ R
N × N
with a
ij
> 0if(j, i) ࢠ E,anda
ij
= 0otherwise.Notea
ii
=
0. The set of neighbours of node i is N
i
={j ∈ V
f
: ( j, i) ∈
E, i = j}.ThediagonalmatrixD = diag{d
1
, d
2
,…, d
N
} ࢠ
R
N × N
is degree matrix of G,inwhichd
i
=
j∈N
i
a
ij
for i =
1, … , N. The Laplacian of the weighted digraph G is dened as
L = D − A.
Consider the directed topology consisting of N followers and
M leaders, which is dened with directed graph
¯
G = (
¯
V ,
¯
E),
where
¯
V = V
f
V
l
, V
f
= {1, 2, … , N}, V
l
= {1, 2, … , M}, and
the edge set
¯
E ⊂ (V
f
×V
f
V
f
×V
l
). Similarly, we dene a
diagonal matrix B ∈ R
N×M
= diag(
M
j=1
b
1 j
,...,
M
j=1
b
Nj
) to
be the leader adjacency matrix and a weighted matrix B
h
=
[b
ij
] ࢠ R
N×M
associated with
¯
G,whereb
ij
> 0ifnodej in V
l
is a neighbour of node i in V
f
,andb
ij
= 0 otherwise. The set
V
l
is globally reachable in
¯
G if for every node i in V
f
,onend
anodej in V
l
,suchthatthereisapathin
¯
G from node j to
node i.
Denition 2.1 (Yoo, 2013): The set ࣮R
n
is said to be con-
vex if for any x
1
, x
2
ࢠ ,thepointλx
1
+(1− λ)x
2
is in
for any λ ࢠ [0, 1]. The convex hull Co (X ) for a set of points
X = {x
1
,…,x
n
} is the minimal convex set containing all points
in X andisdenedasCo(X ) ={
n
i=1
λ
i
x
i
|x
i
∈ X,λ
i
∈ R > 0,
n
i=1
λ
i
= 1}.
The following lemmas are crucial for the distributed con-
troller design.
Lemma 2.1 (Lin & Qian, 2002): For any real-valued continuous
function f(x, y),wherexࢠ R
m
,yࢠ R
n
,therearesmoothscalar
function a(x) ࣙ 0,b(y) ࣙ 0,c(x) ࣙ 1,d(y) ࣙ 1,suchthat
f (x, y)
≤ a(x) + b(y),
f (x, y)
≤ c(x)d(y).
Lemma 2.2 (Xie & Li, 2009): Let x
1
,x
2
,…,x
n
, p be positive real
numbers, then
(x
1
+ x
2
+···+x
n
)
p
≤ max{n
p−1
, 1}(x
p
1
+ x
p
2
+···+x
p
n
).
Lemma 2.3 (Li & Zhang, 2015): If Assumption 3.2 holds, then
λ
i
=
N
s=1
a
is
+
M
j=1
b
ij
> 0,i= 1,…,N.
Proof: By Assumption 3.2 and the denition of globally reach-
able, one can get the conclusion.
Lemma 2.4 (Li et al., 2016): Under Assumption 3.2,theithele-
ment of r
d
= (L+B)
−1
B
h
h(t) can be written as r
di
=
M
j=1
c
ij
h
j
with the nonnegative constant c
ij
satisfying
M
j=1
c
ij
= 1, i =
1, 2,...,N, where h(t) = (h
1
,…,h
M
)
T
.
3. Problem formulation
Consider the following high-order nonlinear multi-agent sys-
tems. The followers’ dynamics are described as follows:
˙
x
ij
= x
i, j+1
+ f
ij
(
¯
x
ij
,θ
i
),
˙
x
i,n
i
= u
i
+ f
i,n
i
(
¯
x
i,n
i
,θ
i
), (1)
y
i
= x
i1
, i = 1, 2,...,N,
where
¯
x
ij
= (x
i1
,...,x
ij
)
T
∈ R
j
, u
i
(t) ࢠ R, y
i
(t) ࢠ R are
the state vector, the controller, the output, respectively. θ
i
ࢠ
R
m
is an unknown constant vector. The nonlinearities f
ij
are
unknown and smooth with f
ij
(0, … , 0, θ
i
) = 0, i = 1, 2, … , N,
j = 1, 2, … , n
i
.
Remark 3.1: Considering that parameter uncertainties do exist
in practical systems due to changes in operating points, compo-
nent faults, plant deteriorations (Lin & Qian, 2002), the high-
order uncertain nonlinearly parameterised multi-agent systems
(1) considered n this paper is of practical importance.
The following mass–spring mechanical system gives a
practical example of such uncertain systems with nonlinear
parameterisation.
Example 3.1: By Newton’s law, the equation of motion for the
mass–springmechanicalsystemisgivenby
m
¨
y + F
sp
(y) = u, (2)
where m > 0 is the mass, without loss of generality, assume
that m = 1, F
sp
(y) is the restoring force of the spring, u is an
external force which serves as the controller. As discussed in
the Khalil (1992),therestoringforceofthespringcanbemod-
elled as F
sp
(y) = ky(
q
i=0
a
i
y
i
),wherek, a
i
and q are unknown
parameters. Let x
1
= y, x
2
=
˙
y, then (2) can be transformed into
the form:
˙
x
1
= x
2
,
˙
x
2
= u − F
sp
(x
1
). (3)
Clearly, the transformed system (3) with unknown parameters
is the case of dynamic system (1), and this example will be used
for simulation in Section 5.
INTERNATIONAL JOURNAL OF CONTROL 2301