∑
=
j∈N ¯
g
γ
i
ij xj( t) (letting ¯
gij
i gij) as follows:
Ai( q−1) xi( t + 1) = Bi( q−1) ui( t) + wi( t + 1) + ∑ ¯ gijxj( t) (4.2)
j∈N i
with Ai( q−1) = 1 + ∑ ni a
b
j=1 ijq− j, Bi( q−1) = bi 1 + ∑ mi
j=2 ijq− j+1 and back shifter q−1.
4.2 Local controller design
For Agent i, we can rewrite its dynamic model as the following regression model
x
φ
i( t + 1) = θ T
i
i( t) + wi( t + 1)
(4.3)
where θ i holds all unknown parameters and φ i( t) is the corresponding regressor vector.
Then, by the following LS algorithm
ˆ
θ i( t + 1) = ˆθ i( t) + σi( t) Pi( t)φ i( t)[ xi( t + 1) − φ T( t) ˆθ
i
i( t)]
Pi( t + 1) = Pi( t) − σi( t) Pi( t)φ i( t)φ T( t) P
(4.4)
i
i( t)
σi( t)
= [1 + φ T( t) P
i
i( t)φ i( t)]−1
we can obtain the estimated values ˆ
θ i( t) of θ i at time t. For Agent i, to track a given local
re f
reference signal x
( t)
x∗( t), with the parameter estimate ˆ
θ
i
i
i( t) given by the above LS
algorithm, it can then design its adaptive control law ui( t) by the “certainty equivalence”
principle, that is to say, it can choose ui( t) such that
ˆ
θ T( t)φ
( t + 1)
(4.5)
i
i( t) = x∗
i
where x∗( t) is the bounded desired reference signal of Agent i, i.e. Agent i is to track the
i
deterministic given signal x∗( t).
i
Consequently we obtain
ui( t) =
1
{ (
ˆ
x∗ t + 1)
b
i
i 1 ( t)
+[ˆ ai 1( t) xi( t) + · · · + ˆ ai, p ( t) x
i
i( t − pi + 1)]
(4.6)
−[ˆ bi 2( t) ui( t − 1) + · · · + ˆ bi, q ( t) u
i
i( t − qi + 1)]
− ˆ¯g T( t) ¯
X
i
i( t)}
where ˆ¯g i( t) is a vector holding the estimates ˆ¯ gij( t) of gij ( j ∈ N i) and ¯
X i( t) is a vector holding
the states xij( t) ( j ∈ N i).
In particular, when the high-frequency gain bi 1 is known a priori, let ¯θ i denote the parameter
vector θ i without component bi 1, ¯
φ i( t) denote the regression vector φ i( t) without component
ui( t), and similarly we introduce notations ¯ ai( t), ¯ Pi( t) corresponding to ai( t) and Pi( t), respectively. Then, the estimate ¯
θ i( t) at time t of ¯θ i can be updated by the following
algorithm:
¯
θ i( t + 1) = ¯θ i( t) + ¯ σi( t) ¯ Pi( t) ¯φ i( t)
×[ xi( t + 1) − bi 1 ui( t) − ¯φ T( t) ¯θ
i
i( t)]
¯
(4.7)
Pi( t + 1) = ¯ Pi( t) − ¯ σi( t) ¯ Pi( t) ¯
φ i( t) ¯φ T( t) ¯ P
i
i( t)
¯ σi( t)
= [1 + ¯φ T( t) ¯ P
i
i( t) ¯
φ i( t)]−1
240
Discrete Time Systems
When the high-frequency gain bi 1 is unknown a priori, to avoid the so-called singularity
problem of ˆ bi 1( t) being or approaching zero, we need to use the following modified ˆ bi 1( t),
denoted by ˆ bi 1( t), instead of original ˆ bi 1( t):
⎧
⎨ ˆ bi 1( t)
if |ˆ bi 1( t)| ≥
1
√
ˆ
log r
b
i ( t)
i 1( t) = ⎩
(4.8)
ˆ bi 1( t) + sgn(ˆ bi 1( t))
√
if |ˆ b
log r
i 1( t)| <
1
√
i ( t)
log ri( t)
and consequently the local controller of Agent i is given by
ui( t) =
1
{ (
ˆ
x∗ t + 1)
ˆ b
i
i 1 ( t)
+[ˆ ai 1( t) xi( t) + · · · + ˆ ai, p ( t) x
i
i( t − pi + 1)]
(4.9)
−[ˆ bi 2( t) ui( t − 1) + · · · + ˆ bi, q ( t) u
i
i( t − qi + 1)]
− ˆ¯g T( t) ¯
X
i
i( t)}.
4.3 Assumptions
Assumption 4.1. (noise condition) { wi( t), F t} is a martingale difference sequence, with {F t} being a sequence of nondecreasing σ-algebras, such that
sup E[| wi( t + 1)| β|F t] < ∞, a. s.
t≥0
for some β > 2 and
t
lim 1 ∑ | wi( k)|2 = Ri > 0, a. s.
t→∞ t k=1
Assumption 4.2. (minimum phase condition) Bi( z) = 0, ∀ z ∈ C : | z| ≤ 1 .
Assumption 4.3. (reference signal) { x∗( t)} is a bounded deterministic signal.
i
4.4 Main result
Theorem 4.1. Suppose that Assumptions 4.1—4.3 hold for system (4.1). Then the closed-loop system
is stable and optimal, that is to say, for i = 1, 2, . . . , N, we have
t
lim sup 1
[| x
t ∑
i( k)|2 + | ui( k − 1)|2] < ∞,
a. s.
t→∞
k=1
and
t
lim 1 ∑ | xi( k) − x∗( k)|2 = Ri,
a. s.
t→∞ t
i
k=1
Although each agent only aims to track a local reference signal by local adaptive controller
based on recursive LS algorithm, the whole system achieves global stability. The optimality
can also be understood intuitively because in the presence of noise, even when all the
parameters are known, the limit of
t−1
J
1
i( t) Δ
= lim
∑ | xi( k + 1) − x∗( k + 1)|2
t→∞ t
i
k=0
cannot be smaller than Ri.
Decentralized Adaptive Control of Discrete-Time Multi-Agent Systems
241
4.5 Lemmas
Lemma 4.1. Under Assumption 4.1, we have | wi( t)| = O( di( t)), where { di( t)} is an increasing sequence and can be taken as tδ (δ can be any positive number).
Proof. In fact, by using Markov inequality, we obtain that
∞
∞
∑ P(| w
E[| wi( t+1)| β|F t]
i( t + 1)|2 ≥ t 2 δ|F t) ≤ ∑
tβδ
< ∞
t=1
t=1
holds almost surely. By applying the Borel-Cantelli-Levy lemma, immediately we have
| wi( t + 1)| = O( tδ), a. s.
Lemma 4.2. If ξ( t + 1) = B( z) u( t), ∀ t > 0, where polynomial (q ≥ 1 )
B( z) = b 1 + b 2 z + · · · + bqzq−1
satisfies
B( z) = 0, ∀ z : | z| ≤ 1,
(4.10)
then there exists a constant λ ∈ (0, 1) such that
t+1
| u( t)|2 = O( ∑ λt+1− k| ξ( k)|2).
(4.11)
k=0
Proof. See Ma et al. (2007b).
Lemma 4.3. Under Assumption 4.1, for i = 1, 2, . . . , N, the LS algorithm has the following properties
almost surely:
(a)
˜
θ T( t + 1) P−1( t + 1) ˜θ
i
i
i( t + 1) = O(log ri( t))
(b)
t
∑ αi( k) = O(log ri( t))
k=1
where
δi( t) Δ= tr( Pi( t) − Pi( t + 1))
σi( k) Δ= [1 + φ T( k) P
i
i( k)φ i ( k)]−1
α
(4.12)
i( k) Δ
= σi( k)| ˜θ T( k)φ
i
t( k)|2
t
r
φ T
i( t) Δ
= 1 + ∑
( k)φ
i
i( k)
k=1
Proof. This is a special case of (Guo, 1994, Lemma 2.5).
Lemma 4.4. Under Assumption 4.1, for i = 1, 2, . . . , N, we have
t
t
∑ | x
1
i( k)|2 → ∞, lim inf
|
t ∑
xi( k)|2 ≥ Ri > 0, a. s.
(4.13)
k=1
t→∞
k=1
t
Proof. This lemma can be obtained by estimating lower bound of ∑ [ xi( k + 1)]2 with the help
k=1
of Assumption 4.1 and the martingale estimation theorem. Similar proof can be found in Chen
& Guo (1991).
242
Discrete Time Systems
4.6 Proof of Theorem 4.1
To prove Theorem 4.1, we shall apply the main idea, utilized in Chen & Guo (1991) and Guo
(1993), to estimate the bounds of signals by analyzing some linear inequalities. However, there
are some difficulties in analyzing the closed-loop system of decentralized adaptive control
law. Noting that each agent only uses local estimate algorithm and control law, but the agents
are coupled, therefore for a fixed Agent i, we cannot estimate the bounds of state xi( t) and
control ui( t) without knowing the corresponding bounds for its neighborhood agents. This is
the main difficulty of this problem. To resolve this problem, we first analyze every agent, and
then consider their relationship globally, finally the estimation of state bounds for each agent
can be obtained through both the local and global analysis.
In the following analysis, δi( t), σi( k), αi( k) and ri( t) are defined as in Eq. (4.12).
Step 1:
In this step, we analyze dynamics of each agent. We consider Agent i for i =
1, 2, . . . , N. By putting the control law (4.9) into (4.3), noting that (4.5), we have
xi( t + 1) = θ Tφ
i
i( t) + wi( t + 1)
= x∗( t + 1) − ˆθ T( t)φ
φ
i
i
i( t) + θ T
i
i( t) + wi( t + 1)
= x∗( t + 1) + ˜θ T( t)φ
i
i
i( t) + wi( t + 1)
By Lemma 4.1, we have | wi( t)|2 = O( di( t)). Noticing also
| ˜θ