Dp( k)
:= α( k) → 0
(89)
Noting that lk− n+1 ≤ k − 2 n + 1 by (7) and considering Assumption 3.1, (5) in Lemma 3.3, we
see that Dp( k) in (27) satisfies
1
D 2 p ( k) = O[ y( k + 1)]
(90)
From (89) and (90), we have
1
1
a
2
2
p ( k)| ˜
y( k + 1| k)| = α 12 ( k) Dp ( k) = o[ Dp ( k)] = o[ O[ y( k + 1)]]
(91)
From the definition of dead zone in (28), when | ˜ y( k + 1| k)| > λ Δ Z( k − n + 1) , we have ap( k)| ˜ y( k + 1| k)| = | ˜ y( k + 1| k)| − λ Δ Z( k − n + 1) > 0
Discrete-Time Adaptive Predictive Control with Asymptotic Output Tracking
225
while when | ˜ y( k + 1| k)| ≤ λ ˆ cp( k − n + 2) Δ Z( k − n + 1) , we have ap( k)| ˜ y( k + 1| k)| = 0 ≥ | ˜ y( k + 1| k)| − λ Δ Z( k − n + 1) .
In summary, the definition of dead zone in (28) guarantees the following inequality
| ˜ y( k + 1| k)| ≤ ap( k)| ˜ y( k + 1| k)| + λˆ cp( k − n + 2) Δ Z( k − n + 1) (92)
which together with (91), boundedness of the parameter estimates, and the definition of
Δ s( k, m) in (31) yields
| ˜ y( k + 1| k)| ≤ o[ O[ y( k + 1)]] + λc 1Δ s( k, 1)
(93)
with c 1 = 1. Now, let us analyze the two-step prediction error:
˜ y( k + 2| k)= ˆ y( k + 2| k) − y( k + 2)
= ˜ y( k + 2| k + 1) + ˘ y( k + 2| k)
(94)
where
˜ y( k + 2| k + 1) = ˆ y( k + 2| k + 1) − y( k + 2)
˘ y( k + 2| k) = ˆ y( k + 2| k) − ˆ y( k + 2| k + 1)
(95)
From (93), it is easy to see that
| ˜ y( k + 2| k + 1)| ≤ o[ O[ y( k + 2)]] + λc 1Δ s( k, 2)
(96)
From (17), and (20), it is clear that ˘ y( k + 2| k) in (95) can be written as
˘ y( k + 2| k) = ˆ y( k + 2| k) − ˆ y( k + 2| k + 1)
= ˆ θT(
1 k − n + 3)[Δ ˆ
φ( k + 1| k) − Δ φ( k + 1)]
(97)
Using (93) and the Lipschitz condition of Δ φi(·) (or equivalently φi(·)) with Lipschitz
coefficient Li, we have
Δ ˆ φ( k + 1| k) − Δ φ( k + 1) ≤ L 1| ˜ y( k + 1| k)| ≤ o[ O[ y( k + 1)]] + λc 1 L 1Δ s( k, 1) (98)
which yields
| ˘ y( k + 2| k)| ≤ o[ O[ y( k + 1)]] + λL 1 bθ Δ
1
s( k, 1)
(99)
From (94), (96) and (99), it is clear that there exists a constant c 2 such that
| ˜ y( k + 2| k)| ≤ o[ O[ y( k + 2)]] + λc 2Δ s( k, 2)
(100)
Continuing the analysis above, for l-step estimate error ˜ y( k + l| k), we have
˜ y( k + l| k) = ˆ y( k + l| k) − y( k + l)
= ˘ y( k + l| k) + ˜ y( k + l| k + 1)
(101)
where
˜ y( k + l| k + 1)= ˆ y( k + l| k + 1) − y( k + l)
˘ y( k + l| k)= ˆ y( k + l| k) − ˆ y( k + l| k + 1)
(102)
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Discrete Time Systems
For ( l − 1)-step estimate error ˜ y( k + l − 1| k), it can be seen that there exist constants ˜ cl−1 and
˘ cl−1 such that
| ˜ y( k + l − 1| k)|≤ o[ O[ y( k + l − 1)]] + λ˜ cl−1Δ s( k, l − 1)
| ˘ y( k + l − 1| k)|≤ o[ O[ y( k + l − 2)]] + λ˘ cl−1Δ s( k, l − 2) (103)
From (25) and (102), it is clear that ˘ y( k + l| k) can be expressed as
l−1
˘ y( k + l| k) = ∑ ˆ θT(
i
k − n + l + 1)[Δ ˆ φ( k + l − i| k) − Δ ˆ φ( k + l − i| k + 1)]
(104)
i=1
From (102), we have
ˆ y( k + l − i| k) − ˆ y( k + l − i| k + 1) = ˘ y( k + l − i| k) (105)
According to the Lipschitz condition of φ(·) and (105), the following equality holds:
l−1
l−1
∑ Δ ˆ φ( k + l − i| k) − Δ φ( k + l − i) ≤ max{ Lj}1≤ j≤ l−1 ∑ | ˘ y( k + l − i| k)|
(106)
i=1
i=1
From (93),(101)-(106), it follows that there exist constants cl such that
| ˜ y( k + l| k)| ≤ o[ O[ y( k + l)]] + λclΔ s( k, l) which completes the proof.
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14
Decentralized Adaptive Control
of Discrete-Time Multi-Agent Systems
Hongbin Ma1, Chenguang Yang2 and Mengyin Fu3
1 Beijing Institute of Technology
2 University of Plymouth
3 Beijing Institute of Technology
1,3 China
2 United Kingdom
1. Introduction
In this chapter, we report some work on decentralized adaptive control of discrete-time
multi-agent systems. Multi-agent systems, one important class of models of the so-called
complex systems, have received great attention since 1980s in many areas such as physics,
biology, bionics, engineering, artificial intelligence, and so on. With the development of
technologies, more and more complex control systems demand new theories to deal with
challenging problems which do not exist in traditional single-plant control systems.
The new challenges may be classified but not necessarily restricted in the following aspects:
• The increasing number of connected plants (or subsystems) adds more complexity to the
control of whole system. Generally speaking, it is very difficult or even impossible to
control the whole system in the same way as controlling one single plant.
• The couplings between plants interfere the evolution of states and outputs of each plant.
That is to say, it is not possible to completely analyze each plant independently without
considering other related plants.
• The connected plants need to exchange information among one another, which may bring
extra communication constraints and costs. Generally speaking, the information exchange
only occurs among coupled plants, and each plant may only have local connections with
other plants.
• There may exist various uncertainties in the connected plants. The uncertainties may
include unknown parameters, unknown couplings, unmodeled dynamics, and so on.
To resolve the above issues, multi-agent system control has been investigated by many
researchers. Applications of multi-agent system control include scheduling of automated
highway systems, formation control of satellite clusters, and distributed optimization of
multiple mobile robotic systems, etc. Several examples can be found in Burns (2000); Swaroop
& Hedrick (1999).
Various control strategies developed for multi-agent systems can be roughly assorted into
two architectures: centralized and decentralized. In the decentralized control, local control
for each agent is designed only using locally available information so it requires less
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Discrete Time Systems
computational effort and is relatively more scalable with respect to the swarm size. In
recent years, especially since the so-called Vicsek model was reported in Vicsek et al. (1995),
decentralized control of multi-agent system has received much attention in the research
community (e.g. Jadbabaie et al. (2003a); Moreau (2005)). In the (discrete-time) Vicsek model,
there are n agents and all the agents move in the plane with the same speed but with different
headings, which are updated by averaging the heading angles of neighor agents. By exploring
matrix and graph properties, a theoretical explanation for the consensus behavior of the
Vicsek model has been provided in Jadbabaie et al. (2003a). In Tanner & Christodoulakis
(2005), a discrete-time multi-agent system model has been studied with fixed undirected
topology and all the agents are assumed to transmit their state information in turn. In
Xiao & Wang (2006), some sufficient conditions for the solvability of consensus problems
for discrete-time multi-agent systems with switching topology and time-varying delays have
been presented by using matrix theories. In Moreau (2005), a discrete-time network model
of agents interacting via time-dependent communication links has been investigated. The
result in Moreau (2005) has been extended to the case with time-varying delays by set-value
Lyapunov theory in Angeli & Bliman (2006). Despite the fact that many researchers have
focused on problems like consensus, synchronization, etc., we shall notice that the involved
underlying dynamics in most existing models are essentially evolving with time in an
invariant way determined by fixed parameters and system structure. This motivates us to
consider decentralized adaptive control problems which essentially involve distributed agents
with ability of adaptation and learning. Up to now, there are limited work on decentralized
adaptive control for discrete-time multi-agent systems.
The theoretical work in this chapter has the following motivations:
1. The research on the capability and limitation of the feedback mechanism (e.g. Ma (2008a;b);
Xie & Guo (2000)) in recent years focuses on investigating how to identify the maximum
capability of feedback mechanism in dealing with internal uncertainties of one single system.
2. The decades of studies on traditional adaptive control (e.g. Aström & Wittenmark (1989);
Chen & Guo (1991); Goodwin & Sin (1984); Ioannou & Sun (1996)) focus on investigating
how to identify the unknown parameters of a single plant, especially a linear system or
linear-in-parameter system.
3. The extensive studies on complex systems, especially the so-called complex adaptive systems
theory Holland (1996), mainly focus on agent-based modeling and simulations rat