t , for any t.
• We denote x 1( k) = O[ x 2( k)] , if there exist positive constants m 1 , m 2 and k 0 such that x 1( k) ≤
m 1 max k ≤ k x 2( k ) + m 2, ∀ k > k 0.
• We denote x 1( k) = o[ x 2( k)] , if there exists a discrete-time function α( k) satisfying lim k→∞ α( k) =
0 and a constant k 0 such that x 1( k) ≤ α( k) max k ≤ k x 2( k ) , ∀ k > k 0 .
• We denote x 1( k) ∼ x 2( k) if they satisfy x 1( k) = O[ x 2( k)] and x 2( k) = O[ x 1( k)] .
For convenience, in the followings we use O[1] and o[1] to denote bounded sequences and
sequences converging to zero, respectively. In addition, if sequence y( k) satisfies y( k) =
O[ x( k)] or y( k) = o[ x( k)], then we may directly use O[ x( k)] or o[ x( k)] to denote sequence y( k) for convenience.
According to Definition 5.2, we have the following lemma
Lemma 5.2. According to the definition on signal orders in Definition 5.2, we have following
properties:
(i)
O[ x 1( k + τ)] + O[ x 1( k)] ∼ O[ x 1( k + τ)] , ∀ τ ≥ 0 .
(ii)
x 1( k + τ) + o[ x 1( k)] ∼ x 1( k + τ) , ∀ τ ≥ 0 .
(iii)
o[ x 1( k + τ)] + o[ x 1( k)] ∼ o[ x 1( k + τ)] , ∀ τ ≥ 0 .
(iv) o[ x 1( k)] + o[ x 2( k)] ∼ o[| x 1( k)| + | x 2( k)|] .
(v) o[ O[ x 1( k)]] ∼ o[ x 1( k)] + O[1] .
(vi) If x 1( k) ∼ x 2( k) and lim k→∞ x 2( k) = 0 , then lim k→∞ x 1( k) = 0 .
(vii) If x 1( k) = o[ x 1( k)] + o[1] , then lim k→∞ x 1( k) = 0 .
(viii) Let x 2( k) = x 1( k) + o[ x 1( k)] . If x 2( k) = o[1] , then lim k→∞ x 1( k) = 0 .
The following lemma is a special case of Lemma 4.4 in Ma (2009).
Lemma 5.3. Consider the following iterative system
X( k + 1) = A( k) X( k) + W( k)
(5.20)
where W( k) = O[1] , and A( k) → A as k → ∞ . Assume that ρ( A) is the spectral radius of A, i.e.
ρ( A) = max{| λ( A)|} and ρ( A) < 1 , then we can obtain
X( k + 1) = O[1].
(5.21)
250
Discrete Time Systems
5.6 Proof of Theorem 5.1
In the following, the proof of mathematic rigor is presented in two steps. In the first step, we
prove that ˜ xj( k) → 0 for all j = 1, 2, . . . , N, which leads to x 1( k) − x∗( k) → 0 such that the hidden leader follows the reference trajectory. In the second step, we further prove that the
output of each agent can track the output of the hidden leader such that the control objective
is achieved.
Step 1: Denote ˜ θj( k) Δ
= ˆ θj( k) − θ j( k), especially ˜ bj 1( k) Δ= ˆ bj 1( k) − bj 1. For convenience, let
˜
Δ
b = ˆ b − b
denotes the original estimate of b
j 1
j 1
j 1, where ˆ
bj 1
j 1 without further modification.
From the definitions of ˆ b and ˆ b
( t), b
j 1
j 1, since ˆ
bj 1( t) = max(ˆ bj 1
j 1) and bj 1 ≥ bj 1, obviously we
have
˜ b 2 ( k) ≤ ˜ b 2( k).
j 1
j 1
(5.22)
Consider a Lyapunov candidate
Vj( k) = ˜ θj( k) 2
(5.23)
and we are to show that Vj( k) is non-increasing for each j = 1, 2, . . . , N, i.e. Vj( k) ≤ Vj( k − 1).
Noticing the fact given in (5.22), we can see that the minor modification given in (5.11) will not
increase the value of Vj( k) when ˆ b ( k) < b
j 1
j 1, therefore, in the sequel, we need only consider
the original estimates without modification. Noting that
ˆ θj( k) − ˆ θj( k − 1) = ˜ θj( k) − ˜ θj( k − 1)
(5.24)
the difference of Lyapunov function Vj( k) can be written as
Δ Vj( k) = Vj( k) − Vj( k − 1)
= ˜ θj( k) 2 − ˜ θj( k − 1) 2
= ˆ θj( k) − ˆ θj( k − 1) 2 + 2 ˜ θτ(
j k − 1)[ ˆ
θj( k) − ˆ θj( k − 1)].
(5.25)
Then, according to the update law (5.10), the error dynamics (5.8) and (5.9), we have
ˆ θj( k) − ˆ θj( k − 1) 2 + 2 ˜ θτ(
j k − 1)[ ˆ
θj( k) − ˆ θj( k − 1)]
μ 2 ˜ y 2( k)
2 μ
( k)
μ
( k)
≤
j j
−
j ˜
y 2 j
= − j(2 − μj) ˜ y 2 j
Dj( k − 1)
Dj( k − 1)
Dj( k − 1)
.
Noting 0 < μj < 2, we see that Δ Vj( k) is guaranteed to be non-positive such that the
boundedness of Vj( k) is obvious, and immediately the boundedness of ˆ θj( k) and ˆ bj 1( k) is
guaranteed. Taking summation on both sides of the above equation, we obtain
∞
˜ y 2( k)
∑ μ
j
j(2 − μj)
≤ V
D
j(0)
(5.26)
k=0
j( k − 1)
which implies
˜ y 2( k)
j
1
lim
= 0, or ˜ y
2
j( k) = αj( k) D ( k − 1)
(5.27)
k→∞ D
j
j( k − 1)
Decentralized Adaptive Control of Discrete-Time Multi-Agent Systems
251
with αj( k) ∈ L 2[0, ∞).
Define
¯
Yj( k) = [ xj( k), YT( k)] T
j
(5.28)
where Yj( k) is a vector holding states, at time k, of the j th agent’s neighbors. By (5.5) and (5.7),
we have
uj( k) = O[ ¯ Yj( k + 1)]
φj( k) = O[ ¯ Yj( k)]
(5.29)
then it is obvious that
1
D 2 ( k − 1) ≤ 1 + φ
j
j( k − 1) + | uj( k − 1)|
= 1 + O[ ¯ Yj( k)].
(5.30)
From (5.27) we obtain that
˜ yj( k) = o[1] + o[ ¯ Yj( k)], j = 1, 2, . . . , N
(5.31)
Using o[ X( k)] ∼ o[ x 1( k)] + o[ x 2( k)] + . . . + o[ xn( k)], we may rewrite the above equation as
˜
Y( k) ∼ diag( o[1], . . . , o[1])( GA + I) X( k)
+[ o[1], . . . , o[1]] T
(5.32)
where I is the n × n identity matrix. Substituting the above equation into equation (5.19), we
obtain
X( k + 1) = (Λ GA + diag( o[1], . . . , o[1])( GA + I)) X( k)
+[ x∗( k + 1) + o[1], o[1], . . . , o[1]] T.
Since
(Λ GA + diag( o[1], . . . , o[1])( GA + I)) Y( k) → Λ GA
(5.33)
as k → ∞, noting ρ(Λ GA) < 1, according to Lemma 5.1 and
[ x∗( k + 1) + o[1], o[1], . . . , o[1]] T = O[1]
(5.34)
from Lemma 5.3, we have
X( k + 1) = O[1].
(5.35)
Then, together with equation (5.32), we have ˜
Y( k) = [ o[1], . . . , o[1]] T, which implies
˜ yj( k) → 0 as k → ∞, j = 1, 2, . . . , N
(5.36)
which leads to x 1( k) − x∗( k) → 0.
Step 2: Next, we define a vector of the errors between each agent’s output and the hidden
leader’s output as follows
E( k) = X( k) − [1, 1, . . . , 1] Tx 1( k) = [ e 11( k), e 21( k), . . . , en 1( k)] T
(5.37)
252
Discrete Time Systems
where ej 1( k) satisfies
e 11( k + 1) = x 1( k + 1) − x 1( k + 1) = 0,
ej 1( k + 1) = xj( k + 1) − x 1( k + 1) = ¯ xj( k + 1) − x 1( k + 1) + ˜ xj( k + 1), j = 2, 3, . . . , N.
(5.38)
Noting that except the first row, the summations of the other rows in the sub-stochastic matrix
Λ GA are 1, we have
[0, 1, . . . , 1] T = Λ GA[0, 1, . . . , 1] T
such that equations in (5.38) can be written as
E( k + 1) = Λ GX( k) − Λ GA[0, 1, . . . , 1] Tx 1( k + 1)
+
(5.39)
diag(0, 1, . . . , 1) ˜
Y( k).
According to Assumption 5.1, we obtain
E( k + 1) = Λ GA( X( k) − [0, 1, . . . , 1] Tx 1( k))
+[0, 1, . . . , 1] T( x 1( k) − x 1( k + 1))
+[
(5.40)
o[1], . . . , o[1]] T
= Λ GE( k) + [ o[1], . . . , o[1]] T.
Assume that ρ is the spectral radius of Λ GA, then there exists a matrix norm, which is denoted
as · p, such that
E( k + 1) p ≤ ρ E( k) p + o[1]
(5.41)
where ρ < 1. Then, it is straightforward to show that
E( k + 1) p → 0
(5.42)
as k → ∞. This completes the proof.
6. Summary
The decentralized adaptive control problems have wide backgrounds and applications in
practice.
Such problems are very challenging because various uncertainties, including
coupling uncertainties, parametric plant uncertainties, nonparametric modeling errors,
random noise, communication limits, time delay, and so on, may exist in multi-agent systems.
Especially, the decentralized adaptive control problems for the discrete-time multi-agent
systems may involve more technical difficulties due to the nature of discrete-time systems
and lack of mathematical tools for analyzing stability of discrete-time nonlinear systems.
In this chapter, within a unified framework of multi-agent decentralized adaptive control,
for a typical general model with coupling uncertainties and other uncertainties, we
have investigated several decentralized adaptive control problems, designed efficient local
adaptive controllers according to local goals of agents, and mathematically established the
global properties (synchronization, stability and optimality) of the whole system, which in
turn reveal the fundamental relationship between local agents and the global system.
Decentralized Adaptive Control of Discrete-Time Multi-Agent Systems
253
7. Acknowledgments
This work is partially supported by National Nature Science Foundation (NSFC) under Grants
61004059, 61004139 and 60904086. And this work is also supported by Program for New
Century Excellent Talents in University, BIT Scientific Base Supporting Funding, and BIT Basic
Research Funding. We would like to thank Ms. Lihua Rong for her careful proofreading.
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