Double-Integrator Leader-Follower
Networks: Sufficient Conditions for
Connectivity Maintenance
PHILIPP KÖHLER
Master’s Degree Project
Stockholm, Sweden August 2014
XR-EE-RT 2014:018
Double-Integrator Leader-Follower Networks:
Sufficient Conditions for Connectivity Maintenance
Philipp Köhler
Master Thesis
February 2014 — August 2014
Automatic Control Laboratory
School of Electrical Engineering, KTH Royal Institute of Technology, Sweden
and
Institute for Systems Theory and Automatic Control
Universität Stuttgart, Germany
Supervisor & Examiner
Dr. D. V. Dimarogonas
KTH Stockholm
Examiner
Prof. Dr.-Ing. F. Allgöwer
Universität Stuttgart
Abstract
In this thesis, a set of sufficient conditions that guarantee consensus towards a
pre-specified target state in double-integrator leader-follower networks are derived.
Since only the leader agents are aware of the global objective and proximity based
communication between all agents is considered, the follower agents must not lose
contact to the leaders.
In a first step, it is shown that such consensus seeking networks converge to the
leader induced target state, as long as the interconnection graph is connected. A
connectivity analysis framework is then established to make statements on the
interconnection of any two initially connected agents during evolution of the system.
This framework is subsequently used to state conditions which ensure preservation
of all inter-agent links – and thus keeping the graph connected. These sufficient
conditions put constraints on the magnitude of the goal attraction force experienced
by the leaders as well as on the ratio of leader and follower agents in the network.
Various different network topologies are examined, starting from an initially complete
graph structure and extending to incomplete graphs.
The theoretical results are illustrated by numerous computer simulations highlighting
the relevance and effectiveness of the presented conditions.
iii
Contents
List of symbols
vii
List of figures
x
1 Introduction
1.1 Multiagent systems . . . . .
1.2 Consensus seeking networks
1.3 Connectivity maintenance .
1.4 Leader-follower networks . .
1.5 Goal of this work . . . . . .
1.6 Outline . . . . . . . . . . . .
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2 Background
2.1 Graph theoretic preliminaries . . . . . . . . . . . . . . . . . . . . . .
2.2 Bounded confidence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Leader-follower consensus
3.1 System and consensus protocol .
3.2 Introduction of leader agents . . .
3.2.1 Target velocity consensus .
3.2.2 Target position consensus
3.2.3 Switching graph structure
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11
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4 Inherent connectivity preservation
23
4.1 Connectivity analysis framework . . . . . . . . . . . . . . . . . . . . . 24
4.1.1 Explicit calculations . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Leaderless complete graph . . . . . . . . . . . . . . . . . . . . . . . . 31
v
Contents
4.3
4.4
4.5
4.6
Complete graph with leaders . . . . . . .
Connector agents and incomplete graphs
Complete subgraphs with leaders (I) . .
Complete subgraphs with leaders (II) . .
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6 Conclusion and Outlook
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Relation to single-integrator networks . . . . . . . . . . . . . . . . . .
6.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
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73
75
A Extensions concerning higher dimensions
A.1 Kronecker product . . . . . . . . . . . . . .
A.2 Target velocity consensus . . . . . . . . . . .
A.3 Target position consensus . . . . . . . . . .
A.4 Connectivity analysis via Lyapunov equation
A.5 Critical region estimate . . . . . . . . . . . .
77
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79
80
5 Experimental results
5.1 Effect of parameters . . . . . . . . . . .
5.2 Simulations . . . . . . . . . . . . . . . .
5.2.1 Preliminaries . . . . . . . . . . .
5.2.2 Complete leaderless graph . . . .
5.2.3 Complete graph target velocity .
5.2.4 Complete graph target position .
5.2.5 Complete connector graph . . . .
5.2.6 Complete subgraphs with leaders
References
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84
List of symbols
|·|
Absolute value of a real number or cardinality of a set
k·k
Euclidean norm of a vector
n
Dimensionality of the regarded space Rn
In
Identity matrix of dimension n
1
Vector of all ones
N
Set of all agents
N
Number of agents
Ni
Neighbour set of agent i
Na
Arbitrary subset of agents a
Nf
Set of all follower agents
Nl
Set of all leader agents
∆
Sensing range of the agents
G
Interconnection graph
E
Edge set
L
Laplacian matrix of a graph
AG
Adjacency matrix of a graph
DG
Degree matrix of a graph
xi
Position of agent i
vi
Velocity of agent i
i
State vector [xi , vi ]T of agent i
x
Stack vector of positions of all agents
v
Stack vector of velocities of all agents
vii
List of symbols
xd
Target position
vd
Target velocity
xij
Inter-agent position difference between agent i and j
vij
Inter-agent velocity difference between agent i and j
ij
Inter-agent state difference between agent i and j
∗ij
Radius of maximum ball of inter-agent distances enclosed in Ωc∗
Vij
Inter-agent Lyapunov function
Ωc
Invariant set of (inter-)agent states
c
Value defining the boundary of Ωc
x̄
Average position of all agents
v̄
Average velocity of all agents
Q
Arbitrary positive definite matrix
P
Solution matrix to the Lyapunov equation
A
Linear share of inter-agent dynamics
ᾱij
General additional term of inter-agent dynamics
λmax (·)
Largest eigenvalue of a matrix
λmin (·)
Smallest eigenvalue of a matrix
f (sd )
Goal attraction force
F (sd )
Goal attraction potential
Uij
Artificial inter-agent potential
W
Collective Lyapunov function
η
Parameter to adapt the goal attraction function (fmax )
viii
List of Figures
1.1
Thematic classification . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1
Sample multiagent network
9
4.1
4.2
4.3
4.4
Sample of an ellipsoidal shaped contour line of Vij . . . . .
Network topology with connector agents . . . . . . . . . .
Network topology of complete subgraphs with leaders (I) .
Network topology of complete subgraphs with leaders (II) .
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28
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46
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
Differently shaped invariant sets Ωc∗ . . . . . . . . . . . . . . . . .
Bound on kαij k for absolute damping . . . . . . . . . . . . . . . . .
Bound on kαij k for relative damping . . . . . . . . . . . . . . . . .
Goal attraction potential F (ksd k) . . . . . . . . . . . . . . . . . . .
Initial inter-agent distances and invariant set Ωc∗ . . . . . . . . . .
Complete leaderless graph. Simulation results . . . . . . . . . . . .
Complete graph, target velocity. Initial inter-agent potentials . . . .
Complete graph, target velocity. Simulation results . . . . . . . . .
Complete graph, target position. Simulation results . . . . . . . . .
Complete graph, target position. Initial inter-agent potentials . . .
Connector agent topology, conditions satisfied. Initial inter-agent
potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connector agent topology, conditions satisfied. Simulation results .
Connector agent topology, conditions violated. Initial inter-agent
potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connector agent topology, conditions violated. Simulation results .
Complete subgraphs with leaders (II), conditions satisfied. Initial
inter-agent potentials . . . . . . . . . . . . . . . . . . . . . . . . . .
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51
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63
5.12
5.13
5.14
5.15
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. 64
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ix
List of Figures
5.16 Complete subgraphs with leaders (II), conditions satisfied. Simulation
results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.17 Complete subgraphs with leaders (II), fmax violates conditions. Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.18 Complete subgraphs with leaders (II), conditions satisfied. Simulation
results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.1
x
Three levels establishing the main goal . . . . . . . . . . . . . . . . . 72
1
Introduction
This thesis work is devoted to the analysis of leader-follower networks of consensus
seeking double-integrator agents. In particular, the problem of connectivity maintenance in such systems is addressed. This first chapter provides an introduction to
this field of research.
1.1
Multiagent systems
During the last decades, a growing interest of control theory community in multiagent
systems has emerged. A multiagent system literally consists of many distributed
dynamical (sub)systems that are called agents. Great progress in wireless communication technology and robotics leading to continually smaller, cheaper and more
powerful mobile robots explain the growing interest in such networked systems.
There are various real-world problems, such as exploration, surveillance and other
cooperative tasks, that can be tackled efficiently utilising a group of many simple
agents. Having access to numerous interacting agents opens up a wide range of new
opportunities. However, there arise also challenges that need to be met by adequate
system theoretic tools.
Centralised control of a system of agents contains several difficulties, e.g. growing
1
1 Introduction
complexity with the number of agents, susceptibility to errors and communication
links between all agents and the central unit. For this reason, distributed control
mechanisms are applied, where every agent acts according to local information
obtained from neighbouring agents. This rules out the drawbacks of centralised
control, but also makes it more difficult to ensure that the entire group achieves the
global objective.
The way in which all agents can access and exchange information can be described
by a communication graph in which agents are represented by vertices (or nodes)
and active communication links by connecting edges. To analyse the behaviour
of the entire system and to design control algorithms, some assumptions on the
characteristics of information exchange have to be made. A common and most easily
manageable model of communication is that of undirected (information over an edge
can be exchanged bidirectional), undelayed and undisturbed communication. The last
two points represent continuous communication or a sufficiently large communication
bandwidth and enables formulation of information updates in differential equations.
The achievement of many group objectives naturally relies on a consistent "opinion" of all agents. For this reason, consensus algorithms are an extensively studied
field in modern control theory; group objectives and applications that rely on consensus algorithms are e.g. sensor networks [1–3], attitude alignment [4, 5] and formation
control [6, 7], just to name a few. Of particular interest in this work is flocking [8–11]
and the rendezvous problem [12, 13].
1.2
Consensus seeking networks
The basic idea of a consensus algorithm is to make the states of all agents in a
network converge to the same value. All consensus seeking agents usually obey
similar dynamics.
For the simplest case of single-integrator agents (thus their state is updated according
to a first order differential equation), the consensus protocol presented by [14] achieves
convergence of the entire network to a common state. The applied consensus protocol
is a control law that only takes information from neighbouring agents into account.
There has been a vast amount of work devoted to this basic single-integrator consensus
protocol, see [15, 16] for surveys.
It is natural to extend this consensus protocol to double-integrator dynamics since
adequate modelling of many real world systems require double-integrator dynamics.
[8, 17, 18] propose a consensus protocol for double-integrators by including a velocity
2
1.3 Connectivity maintenance
alignment term into the control law. Networks of double-integrator agents can be
shown to achieve consensus motion and flocking behaviour under certain assumptions.
One crucial assumption to prove any form of consensus (both in the singleintegrator and in the double-integrator setup) is that the network’s interconnection
graph is connected at any time. It is plausible that this assumption has to be made,
since a disconnected network obviously is not likely to achieve consensus. However,
this assumption is extreme and has to be viewed critically, since the communication
graph in most real world systems is not static. This fact is based on technical
limitations of the communication and sensing hardware used. It is challenging to
make statements on network connectivity for a state dependent graph throughout
the systems evolution.
1.3
Connectivity maintenance
Immediately the interconnection graph between agents in a multiagent system is
not assumed to be static, but rather state dependent, the issue of connectivity
maintenance becomes essential to the correct functioning of algorithms relying on
connectivity. For this reason, the topic of connectivity maintenance of state dependent
interconnection graphs has gained much attention. One approach is to focus on
global connectivity measures such as the crucial eigenvalue of the graph Laplacian, as
for example in [19]. On the contrary, local connectivity measures rather correspond
to each individual inter-agent link. There are various studies regarding networks
of homogeneous agents that achieve connectivity preservation directly through the
control law. Some examples are [20–23]. Many authors apply infinite potentials
to the control that prevent existing edges from braking [24, 25]. On the contrary,
there is the work [13, 26] which establishes connectivity maintenance of a network
of single-integrator leader-follower agents obeying a standard consensus protocol.
The authors present metrics on parameters and initial conditions of the leaderfollower network, that inherently ensure connectivity preservation. This approach is
particularly appealing, since it does not involve any modification of the standard
consensus protocol and additionally even incorporates leader agents.
3
1 Introduction
Consensus seeking
control
Connectivity
maintenance
Leader-follower
interaction
Figure 1.1: Enclosing fields of research from multiagent control.
1.4
Leader-follower networks
The idea behind leader-follower networks is to select some agents to be leaders or
anchors. These agents should steer the entire group towards a common goal, which
only these agents are aware of. The benefit of having only some leader agents lies in
the fact that only those agents need to be equipped with specific sensors able to tell
e.g. global absolute positions. [8] introduces the concept of virtual leaders that are
statically connected to all other agents. [27] relaxes this assumption and shows that
not all followers have to stay in contact with the leader to reach the common goal.
Investigating the interaction of leaders and followers can generally be seen as a
controllability problem of multiagent networks [28–31].
A majority of the existing studies on leader-follower consensus rely on a virtual leader
or leader agents, that do not abide by the same consensus protocol as the follower
agents. More often the (virtual) leaders are assumed to be statically connected to
the follower agents. This however, as well as the fact that the leader agents do not
obey the same (consensus) control law as the followers, is quite unnatural. In [13], all
agents obey the same control law, with the leaders being equipped with an additional
goal attraction term.
1.5
Goal of this work
As mentioned in the introduction, this work analyses the interaction of leader and
follower double-integrator agents in a proximity based interconnection network. The
goal of this thesis is to find sufficient conditions on parameters and initial conditions
4
1.6 Outline
of the system, that inherently ensure preservation of the initial connectivity of the
state dependent interconnection graph. A standard consensus protocol [17] is applied
to both leader and follower agents and convergence towards the common goal induced
by the leader agents will be put forward.
The point of departure for this work is [13], which examines the same problem
for the case of single-integrator agents. However, the presence of double-integrator
dynamics demands a totally different approach. This work fuses the three mentioned
fields of multiagent system research in a consensus seeking, connectivity preserving
leader-follower network (Figure 1.1).
1.6
Outline
This thesis work is organized as follows: Initially, basic background information is
given in Chapter 2. Following that, the well-known consensus protocol for doubleintegrator agents is extended by leader agents and convergence under the assumption
of a connected graph is shown in Chapter 3. Chapter 4 establishes a connectivity
analysis framework for the given class of double-integrator leader-follower networks.
The framework is then applied to find sufficient conditions for connectivity preservation of several initial topologies of leader-follower networks. Theoretical results
are supported by numerous simulative experiments in Chapter 5. A set of sufficient
conditions are analysed numerically to give an idea of the effect of the crucial system parameters. Chapter 6 summarises the results, highlights the relation to the
single-integrator case and proposes further future work.
5
2
Background
2.1
Graph theoretic preliminaries
One fundamental element of multiagent systems is communication between agents.
Graphs can be used to model how agents can exchange information. In the following,
a few basic facts from algebraic graph theory are introduced, a thorough presentation
can be found in [32]. In the representation of the group topology as a graph, nodes
correspond to agents and edges to active communication links. Denote the graph by
G = (V, E), with V = (1, ..., N ) being the set of vertices (or nodes) and E the set of
edges connecting the vertices. If there is an edge between two vertices, they are called
adjacent or neighbours and the edge set is given by E = {(i, j) ∈ V ×V | i, j adjacent}.
For every node i, its neighbour set is defined as
Ni = {j ∈ V | (i, j) ∈ E},
thus the set containing all its adjacent nodes. A graph is called complete if any two
vertices are neighbours. A path from vertex i to vertex j is a sequence of distinct
neighbouring vertices starting from i and ending at j. The graph G is said to be
connected if there exists a path between any two vertices i, j ∈ V.
7
2 Background
In general edges are directed, thus information can only flow in one direction over
edge (i, j). However, in the work at hand only undirected graphs are considered on
which (i, j) ∈ E ⇔ (j, i) ∈ E holds. Any graph can be represented by the adjacency
matrix AG , in which the element aij = 1 if edge (i, j) ∈ E or else aij = 0. The degree
matrix DG is a diagonal matrix with diagonal elements dii equal to the number of
vertices that are connected to node i, i.e. the cardinality of its neighbour set |Ni |.
Many important properties of the graph structure are captured by the Laplacian
matrix L which is defined as L = DG − AG . For the case under consideration
of an undirected graph, the Laplacian is symmetric and positive semidefinite, i.e.
L = LT 0. If the graph G is connected, its Laplacian has a single zero eigenvalue
(with all other eigenvalues being positive) and the associated eigenvector is the vector
of ones 1. The Laplacian for the sample graph in Figure 2.1 is


2 −1 −1 0
0


−1 3
0 −1 −1


L=
1
0
0

−1 0


2 −1
 0 −1 0
0 −1 0 −1 2
Considering only a share of the graph G yields a subgraph G 0 which consists of a
subset of nodes V 0 ⊂ V and edges E 0 ⊂ E.
2.2
Bounded confidence
This section introduces a mechanism that is referred to as bounded confidence or
limited sensing range. The notation already illustrates its close connection to real
world phenomena. The term bounded confidence originates from analysis and
modelling of opinion dynamics [33]. In this context it represents the fact that
individuals adapt their opinion by communication with other individuals, but only if
both their opinions do not differ too much.
The term limited sensing range stems from the technical implementation of multiagent
systems, for example, as robot swarms. There it is technical limitations, such as
limited operating distance of (wireless) communication systems or sensors, that allow
two robots to communicate only if they are close enough.
Even though the origin of these two phenomena is quite diverse, they obviously have
the same effect in a graph theoretic description, namely a changing topology of the
8
2.3 Dimensionality
4
2
1
5
∆
3
Figure 2.1: Sample multiagent network with each agent’s sensing range and all
existing links sketched. Obviously the graph is connected. Agent 1 and links to its
neighbour set N1 = {2, 3} are coloured red. The orange links between agents 2, 4, 5
illustrate that the subgraph consisting of these nodes and edges is a complete graph.
communication graph depending on agent states. In particular, edges are dynamically
removed and created, since two vertices should only be connected if their positions
are within a certain distance ∆, i.e. E = {(i, j) ∈ V × V | kxi (t) − xj (t)k ≤ ∆}.
The value ∆ is called the sensing range. This also implies a different notion of the
neighbour set of each agent i ∈ V
Ni = {j ∈ V | kxi (t) − xj (t)k ≤ ∆}.
As mentioned before, this definition makes the communication graph dependent
on the states of the agents and therefore time varying, G = G(x(t)). Analysis
of multiagent systems with state dependent communication graphs is particularly
challenging, since many characteristics of the system behaviour rely heavily on the
graph structure. Obviously, for example, a set of agents whose communication graph
is not permanently connected are not likely to achieve any form of consensus.
Figure 2.1 shows a sample network of agents and visualises some of the notation.
2.3
Dimensionality
This work treats the evolution of a group of agents in 2D, thus on the plane. For this
reason, all state variables (such as position x or velocity v) are from R2 . However,
most results can easily be extended to higher dimensions. Actually, to keep notation
simple, all following calculations are even carried out for the one-dimensional case,
9
2 Background
i.e. n = 1. Adaptations for higher dimensions and according calculations are put
together in Appendix A. If any calculation requires special treatment or validation
for the higher dimensional case, there is a reference to the appropriate extension in
the appendix.
10
3
Leader-follower consensus in
double-integrator networks
In this chapter an analysis of a network consisting of double integrator agents is
shown. The goal of the group is to reach a form of consensus, a condition that has to
be defined more precisely for the case of double integrator dynamics. Additionally,
all agents should be guided to a predefined target state by introducing some leader
agents. Only these agents are aware of the common goal. Controls of all follower
agents are based solely on local information; information exchange between agents is
based on bounded confidence, i.e. regarding agents in the plane, one agent can only
assess state information from agents within a certain radius.
The following begins with existing consensus algorithms and extends them by the
introduction of leaders. Most existing results on consensus seeking double integrator
networks are only valid under the assumption that the communication graph is
continually connected. Dealing with agents of a limited sensing range, this is quite a
strong assumption. For this reason, this work aims to find conditions for the initial
configuration of the leader-follower network, which inherently ensure preservation
of connectivity while the system evolves. This guarantees consensus of the leaderfollower network without any assumptions on connectivity (see Chapter 4.1).
11
3 Leader-follower consensus
3.1
System and consensus protocol
Consider N agents evolving in Rn . Since double-integrator agents are regarded, the
dynamics of each agent is described by the following model:
ẋi = vi
i ∈ N = {1, ..., N }
(3.1)
v̇i = ui ,
All the agents act according to a consensus protocol introduced by [17] and [18],
which achieves common (flocking) motion of all agents (while avoiding collisions).
The control input for every agent i ∈ N is given as
!
X ∂Uij
+ γ(vi − vj ) ,
ui = −
∂x
i
j∈N
(3.2)
i
with Uij being an artificial potential function.
Definition 3.1. Let xij = xi − xj be the position difference between two agents. The
artificial potential Uij : R → R is defined as a non-negative, differentiable, radially
unbounded function of the distance kxij k between two agents i and j, such that Uij
attains its unique minimum when agents i and j are located at a desired distance
and it must hold either
• Uij (kxij k) → ∞ as kxij k → 0, or
∂U (kx k)
• ij∂xij ij kxij k=0 < ∞ (Uij differentiable at 0 w.r.t. applied norm).
This general notion of an artificial potential based on inter-agent positions enables
all pairs of agents to be kept at a desired distance. Thus the applied control should
achieve either position consensus (if Uij has its minimum at 0) or flocking with
collision avoidance (if Uij (kxij k) → ∞ for kxij k → 0). However, an important
special case of the potential is to chose Uij = 12 kxij k2 . With this choice of Uij ,
the above consensus protocol stated by [10] is equal to the one introduced by [17].
The definition of the artificial potential used by the first author achieves collision
avoidance between agents and keeps them at a desired distance, whereas the latter
potential leads to position consensus of all agents. Note that by this choice of Uij
the consensus protocol even becomes linear.
In the following, the focus is set on the special case Uij = 12 kxij k2 as the main interest
of this work is not flocking but rather consensus of the swarm (towards a common
target position or velocity). Along with this comes the fact that the consensus
12
3.2 Introduction of leader agents
protocol (3.2) used as a basis for all following adaptations becomes linear in the
agent’s states, i.e.
!
X
ui = −
(xi − xj ) + γ(vi − vj ) .
(3.3)
j∈Ni
The following convergence result of leader-follower networks in this chapter can easily
be generalized to an arbitrary potential function Uij as defined above. This is not
possible for the connectivity analysis as it is carried out in the subsequent chapters.
3.2
Introduction of leader agents
This section introduces a number of leaders to the network of agents. Since leader
and follower agents are distinguished, every agent belongs either to the subset of
leaders N l or to the subset of followers N f , with Nl = |N l | and Nf = |N f |. Due to
the fact that N l ∩ N f = ∅, it holds N = Nl + Nf .
All agents obey basically the same consensus protocol as before (3.2), but with the
leader agents being equipped with an extra term referred to as goal attraction force,
which should pull them towards the common target.
Dealing with double integrator agents in leader-follower networks requires a
precise definition of consensus.
Definition 3.2. A group of N double integrator agents is said to achieve position consensus if they all converge to the same position x̂ and remain there, i.e.
limt→∞ xi = x̂ and limt→∞ vi = 0 ∀i ∈ N .
If a target position xd is to be reached (e.g. induced by some leader agents), the
group is said to achieve target position consensus if x̂ = xd .
Definition 3.3. A group of N double integrator agents is said to achieve velocity
consensus if they all converge to the same velocity v̂, i.e. limt→∞ vi = v̂ ∀i ∈ N . All
inter-agent distances should additionally converge to a desired distance.
If a target velocity vd is to be reached (e.g. induced by some leader agents), the
group is said to achieve target velocity consensus if v̂ = vd .
Sometimes the term position consensus is casually used in the context of velocity
consensus if the desired inter-agent distance is 0. Velocity consensus is also called
flocking if all agents keep a desired inter-agent distance.
First, consider the case in which the common target is consensus motion at the
target velocity vd . The resulting control law for all agents i ∈ N is modified to
13
3 Leader-follower consensus
become
ui = −
X ∂Uij
j∈Ni
∂xi
+ γ(vi − vj ) + f¯i (vi − vd )
(3.4)
and is referred to as relative damping protocol. This common notion using the barnotation for leader and follower agents is in accordance to the following definitions
and was chosen to make formulation in the following steps more consistent and clear.
Clearly the term f¯i should not affect follower agents but should pull all leader agents
towards the common goal. Therefore a general notion of the goal attraction force is
defined in the following.
Definition 3.4. f¯ is the general notion of the goal attraction force that enables
formulation of leader and follower dynamics in a consistent way
f¯i (sd ) =

0
∀i ∈ N f
f (s ) ∀i ∈ N l
d
,
with f being a goal attraction force.
Definition 3.5. Let sd = s − d be the difference of the state variable s ∈ Rn to a
desired value d ∈ Rn towards which the goal attraction force should pull the leader
agents. Thus sd is the state s expressed in a reference frame centred at the target
state d.
Define the goal attraction force f : Rn → Rn such that

−∇ F (ks k) ks k > 0
sd
d
d
f (sd ) =
0
ksd k = 0
(3.5)
and denote with fmax the maximum absolute value it can take, i.e. kf (sd )k ≤ fmax .
The scalar valued function F : [0, ∞) → R called goal attraction potential is a
differentiable class K function (positive definite, strictly increasing, continuous) of
the distance ksd k.
dF
1
Continuity of f (sd ) is guaranteed by demanding limksd k→0 dks
< ∞ to ensure
d k ksd k
limksd k→0 ∇sd F (ksd k) = 0.
The goal attraction potential that is dependent on the distance to the target
state ksd k can be designed to suit the application. For this purpose, the following
fact might be helpful.
14
3.2 Introduction of leader agents
Proposition 3.1. For any goal attraction potential F (ksd k) it holds
dF .
k∇sd F (ksd k)k = dksd k Proof. This fact is proven by simply applying the chain rule for the calculation of
gradients, thus writing ∇F (k · k) as ∇(F ◦ k · k). This yields
∇sd F (ksd k) =
dF
sd
·
.
dksd k ksd k
Taking the norm concludes the proof.
Formulating a control protocol that guides all agents towards a common target
position xd requires several changes. Firstly a damping term −γvi is added in each
agents steering law. This helps to make all agents remain finally on the target
position. Secondly, the goal attraction function must be chosen to aim towards the
common target position and becomes f (xi − xd ). The resulting control law for all
agents following the above notation then becomes
ui = −γvi −
X ∂Uij
j∈Ni
∂xi
+ γ(vi − vj ) + f¯i (xi − xd ).
(3.6)
Thus it is the standard relative damping protocol extended by an absolute damping
term.
In some cases it can be useful to rely only on absolute velocity damping and drop the
velocity alignment term completely. This version of the consensus protocol commonly
known as absolute damping protocol will be used in some later analysis and can be
stated as
X ∂Uij + f¯i (xi − xd ).
(3.7)
ui = −γvi −
∂x
i
j∈N
i
The absolute damping protocol is not only useful for analysis, but has also practical
advantages. Since this control law relies only on the position of neighbouring agents,
it is much easier to implement in real world applications. Only a mechanism to
measure relative distances is needed, in contrast to also measuring relative velocities.
3.2.1
Target velocity consensus
It has been shown that a network of follower agents implementing the consensus
protocol (3.2) achieves asymptotic velocity consensus and dependent on the choice of
15
3 Leader-follower consensus
the artificial potential function Uij either stable flocking with collision avoidance [18] or
consensus motion [17]. In the following, these results are extended by the introduction
of the leader agents that should guide the group to move at a target velocity vd . The
proof of target velocity consensus becomes analogous to the leaderless consensus
proof in [10]. For this reason, only those parts of the proof involving the effect of
leader agents are explained in detail.
Consider a network of leader and follower agents obeying (3.4). First assume
that the communication graph G is connected and static at all times.
Reformulate agent dynamics by the change of coordinates wi = vi − vd , which
translates the velocity coordinate into an error coordinate with respect to the target
velocity vd induced by the leader agents. This yields the following system dynamics
for all agents i ∈ N
ẋi = wi + vd
ẇi = v̇i
=−
X ∂Uij
j∈Ni
∂xi
+ γ(wi − wj ) + f¯i (wi ).
(3.8)
For every agent i its total artificial potential Ui stemming from the neighbour related
artificial potentials can be calculated by summing up Uij for all its neighbours which
gives
X
Ui =
Uij (kxij k).
(3.9)
j∈Ni
Consider now the following positive definite function of pairwise inter-agent distances
xij = xi − xj and velocity errors wi
N
1X
(Ui + wiT wi ).
2 i=1
(3.10)
Λ = {(wi , xij ) | W ≤ c}
(3.11)
W =
The level sets of W
define compact sets in the space of agent velocity errors wi and pairwise agent
distances xij . This is because the set {xij , wi such that W ≤ c, for c > 0} is closed
by continuity. To see boundedness, note that from the connectivity assumption a
path connecting two connected nodes i and j has length of at most (N − 1). Thus
√
kxij k ≤ Uij−1 (c(N − 1)). Similarly wiT wi ≤ c yields kwi k ≤ c.
16
3.2 Introduction of leader agents
By symmetry of Ui w.r.t. xij and xij = −xji
∂Uij
∂Uij
∂Uij
=
=−
∂xij
∂xi
∂xj
(3.12)
the derivative of agent i’s potential Ui along the system trajectories of (3.8) is now
N
N
d X1
1XX T
Ui =
ẋ ∇x Uij
dt i=1 2
2 i=1 j∈N ij ij
i
=
1
2
N
X
X
(ẋi − x˙j )T ∇xij Uij
i=1 j∈Ni
N
1XX
=
ẇi T ∇xij Uij − ẇj T ∇xij Uij
2 i=1 j∈N
(3.13)
i
=
=
1
2
N
X
X
ẇi T ∇xi Uij + ẇj T ∇xj Uij
i=1 j∈Ni
N
X
X
T
ẇi ∇xi Uij =
N
X
wiT ∇xi Ui .
i=1
i=1 j∈Ni
Summation over all N agents and symmetry of Uij make the last step possible.
Theorem 3.2. Consider a system of leader and follower agents of dynamics (3.1),
each of them steered by the control law (3.4). Assuming a static connected communication graph, the depicted network of agents achieves target velocity consensus.
Proof. Begin by calculating the time derivative of W which gives
N
Ẇ =
=
N
X
1X
U̇i +
wiT ẇi
2 i=1
i=1
N
X
wiT ∇xi Ui
i=1
=−
N
X
+
N
X
!
wiT
−
i=1
X
i=1 j∈Ni
wiT γ(wi
− wj ) +
X
γ(wi − wj ) − ∇xi Ui + f¯i (wi )
j∈Ni
N
X
wiT f¯i (wi ).
i=1
All agent velocities and the goal attraction force are now written in stack vector form w = [w1 , ..., wN ]T and f¯ = [f¯i (w1 , d), ..., f¯N (wN , d)]T . Thereby the term
PN P
T
T
i=1
j∈Ni wi (wi −wj ) can be rewritten as w Lw in accordance with the properties
17
3 Leader-follower consensus
of the Laplacian matrix L. The above expression then becomes
Ẇ = −γwT Lw +
N
X
wiT f¯i (wi )
i=1
T
= −γw Lw −
X
wiT ∇wi F (kwi k)
i∈N l
= −γwT Lw −
X
i∈N l
= −γwT Lw −
X
i∈N l
wiT
wi
d
F (kwi k)
dkwi k
kwi k
kwi k
d
F (kwi k).
dkwi k
Both these terms are obviously non-positive. Since by assumption the graph G is
connected, the Laplacian L is negative semidefinite and has exactly one eigenvalue
equal to 0. The corresponding eigenvector is 1, so −γwT Lw is strictly negative unless
w belongs to span(1), i.e. all velocity error vectors wi are equal. The second term is
also negative unless wi = 0, ∀i ∈ N l , by definition of the goal attraction potential.
Thus Ẇ is negative semidefinite w.r.t. the state variables wi and xij , it holds
Ẇ ≤ 0
Ẇ = 0 ⇔ wi = 0, ∀i ∈ N .
It follows immediately that ẋij = 0, ∀(i, j) ∈ N × N . Application of LaSalle’s
invariance principle establishes convergence of all system trajectories to S = {w |
Ẇ = 0} and therefore vi → vd , ∀i ∈ N for t → ∞.
From this stand point, the proof can be continued exactly as in [10] to show that the
inter-agent distances in S are such that the potential Ui of each agent is minimised.
For the commonly examined quadratic inter-agent potential Uij = 12 (xi − xj )2 it is
clearly seen that the dynamics in S degenerate to
ẇ ≡ 0 = −Lx
and thus x belongs to span(1) (which implies xij = 0).
Slight extensions to the calculations for higher dimensions can be found in
appendix A.2.
18
3.2 Introduction of leader agents
3.2.2
Target position consensus
Now consider the case where the group of agents is to be steered to a common
position, thus having all agents remaining finally on the target position xd induced
by the leaders. The outline is exactly the same as in the previous section, but some
changes are necessary to achieve the specified task.
Set the neighbouring potential Uij to be 12 (xi − xj )2 to achieve position consensus
of the agents and use the steering law (3.6) to include an absolute damping term.
Again a change of coordinates is introduced describing the position of each agent as
the error to the target position si = xi − xd . The system becomes
s˙i = ẋi − x˙d = vi
X ∂Uij
v̇i = −γvi −
+ γ(vi − vj ) + f¯i (xi − xd )
∂s
i
j∈Ni
X
= −γvi −
((si − sj ) + γ(vi − vj )) + f¯i (si ).
(3.14)
j∈Ni
Theorem 3.3. Consider a system of leader and follower agents of dynamics (3.1),
each steered by (3.6). Assuming a static connected communication graph, the depicted
network of agents achieves asymptotic convergence to the common position target xd
induced by the leader agents.
Proof. We only sketch this proof here, since it follows strictly the one laid out in the
previous section. Choose as a Lyapunov function candidate
1
W =
2
1
=
2
N
X
!
Ui +
viT vi
+
i=1
N
X
i=1
XZ
i∈N l
si
−f (ξ)dξ
0
(3.15)
!
Ui + viT vi
+
X
F (ksi k)
i∈N l
which is positive definite w.r.t. all regarded inter-agent positions xij = sij , all
velocities and the leader position errors (according to the Definition 3.5 of F and
positive definiteness of the Euclidean norm). The level sets of W
Λ = {(vi , sij , si )|W ≤ c}
(3.16)
define compact sets in the space of agent velocities vi , pairwise agent distances sij
and agents target position errors si as before.
19
3 Leader-follower consensus
By taking the time derivative, the convergence result is straight-forward.
Ẇ = −
=−
!
N
X
X
i=1
j∈Ni
viT γ(vi − vj ) + γviT vi
+
X
i=1
j∈Ni
viT f¯i (si ) +
i=1
!
N
X
N
X
viT γ(vi − vj ) + γviT vi
+
N
X
X
s˙i T ∇si F (ksi k)
i∈N l
viT f¯i (si ) −
i=1
X
viT f (si )
i∈N l
= −γv T Lv − γv T v
(3.17)
which is negative semidefinite due to the properties of L. Ẇ = 0 implies v = 0, since
L is positive semidefinite. Regarding trajectories staying identically in S = {Ẇ = 0}
gives also v̇i ≡ 0 and ṡi ≡ 0. Using these properties yields
v̇i ≡ 0 ≡ −
X
(si − sk )
∀i ∈ N f
(sj − sk ) + f (sj )
∀j ∈ N l
k∈Ni
v̇j ≡ 0 ≡ −
X
k∈Nj
Now examine two cases: First assume that for all leader agents j ∈ N l it holds
f (sj ) = 0 which in turn implies sj ≡ 0 ∀j ∈ N l . This yields in stack vector form
v̇ ≡ 0 ≡ −Ls and thus si ≡ sj ≡ 0 ∀i, j ∈ N .
In contrast, assume now that f (sj ∗ ) 6= 0 for some leaders j ∗ ∈ N l . For these agents
j ∗ it can be written f (sj ∗ ) = − dksdFj∗ k ksj1∗ k sj ∗ . Let η̄ be a diagonal matrix with
dF
1
> 0 as diagonal elements η̄j ∗ j ∗ in lines according to leader agents j ∗ . This
dksj ∗ k ksj ∗ k
yields in stack vector form
v̇ ≡ 0 ≡ −Ls − η̄s
(3.18)
≡ −(L + η̄)s.
By the following lemma this can only be satisfied by s ≡ 0.
Both cases put together provides
v̇ ≡ 0 ⇔ s ≡ 0.
(3.19)
Thus si ≡ 0 ∀i ∈ N must hold for all trajectories staying identically in S which
by LaSalle’s invariance principle concludes the proof of convergence of all agents to
the common target position.
Slight extensions to the calculations for higher dimensions can be found in appendix A.3.
20
3.2 Introduction of leader agents
Remark. The theorem also holds under application of the absolute damping protocol
(3.7). The proof can be carried out strictly analogously.
Lemma 3.4. Let L be the Laplacian matrix of a connected graph and D a diagonal
matrix with diagonal elements dii ≥ 0 of which at least one is positive. Then the sum
L + D has only 0 as its Null Space.
Proof. Let x ∈ null(L + D), i.e. (L + D)x = 0. This implies
xT (L + D)x =
X
(xi − xj )2 + xT Dx = 0.
(i,j)∈E
Both terms are non-negative. Thus the entire expression vanishes only if xi =
xj ∀(i, j) ∈ E and all xi = 0 for which dii =
6 0. As G is assumed to be connected, it
holds xi = xj ∀i, j ∈ V and by this, xi = 0 ∀i ∈ V.
3.2.3
Switching graph structure
Both presented convergence results of consensus seeking leader-follower networks
were obtained under the assumption of a static connected graph. However, the core
problem to be treated in this work, namely connectivity maintenance, is based on
the limited sensing range of the agents. Therefore the interconnection topology is by
no means static but rather depends on the state of the system, G = G(x(t)).
Since all agents act on local information only (according to the current interconnection), the abrupt changes in the communication graph introduce discontinuities
in the applied control. The system dynamics become discontinuous and stability
should be investigated using differential inclusions [34] and stability theorems for
nonsmooth systems [35–37]. Convergence of the switched system can be shown using
the same arguments as in the static case, but correct mathematical formulation and
treatment of discontinuities is quite involved. The convergence proof for the system
at hand with consensus protocol (3.2) has been presented in a detailed fashion by [11]
under careful examination of the discontinuities introduced by the state-dependent
switching of the communication graph. The introduction of leader agents does
not introduce additional discontinuities, and is basically only an extension of the
convergence results for leaderless networks. For this reason, a similar proof to that
of Theorem VI.2 in [11] shows that previous Theorems 3.2 and 3.3 also hold under a
switching, but always connected, graph structure G(t).
21
4
Inherent connectivity preservation
In the previous chapter it has been shown that networks of leader and follower agents
asymptotically achieve consensus towards a common target in the velocity or position
domain induced by the leader agents. This becomes possible by a small adaptation
of the well-established consensus algorithm (3.2) that adds a goal attraction force
to the leader agents. The fundamental assumption made to guarantee consensus is
however that the interconnection graph G is connected at all times.
This is a strong assumption which can not be fulfilled easily regarding the switching
inter-agent connections arising from the agent’s limited sensing range.
In introducing Section 1.3, a literature review on the issue of connectivity preservation
in dynamic multiagent systems was given. Previous work on connectivity analysis of
the consensus algorithm under consideration uses the collective potential (kinetic
energy plus "tension" of inter-agent links) of the whole network as connectivity criterion. Connectivity preservation is achieved by applying potentials in the consensus
protocol that become infinite at the braking distance |xi − xj | = ∆ of an existing link,
as for example in [25]. In [38] the infinite potentials are replaced by bounded ones,
what in turn requires rather strong restrictions on initial conditions to guarantee
connectivity preservation.
The work at hand in contrast does not enforce connectivity preservation directly
23
4 Inherent connectivity preservation
through (an adapted) control law, but rather indirectly by sufficient conditions on
parameters and initial conditions of the network. These conditions ensure that the
initially connected graph remains connected at all times. Having found conditions
for connectivity preservation, we can refer to the results of the previous chapter that
claim convergence of connected graph networks.
Not being equipped with a special kind of inter-agent potentials, no significant
statements on connectivity can be made by looking at the collective potential of the
network. For this reason we establish a framework to investigate inter-agent connectivity in double-integrator networks on a more microscopic scale. This framework
will enable us to investigate connectivity of any two agents in the network during its
evolution.
The ideas laid out here create the basis for all subsequent investigations of different
graph topologies of leader-follower networks. Depending on the initial topology
setup, we get as a result sufficient conditions that ensure maintenance of connectivity. In combination with the previous convergence theorems, these conditions
state guaranteed target velocity/position consensus for several initial leader-follower
interconnection topologies.
4.1
Connectivity analysis framework for
double-integrator networks
As mentioned before, the work [13] addresses this problem for the case of singleintegrator consensus seeking leader-follower networks. Constraints on the magnitude
if the goal attraction force and the ratio of leaders and followers ensure connectivity
preservations. The authors propose a framework to analyse connectivity between
two arbitrary agents. To keep every initial link of interconnected pairs of agents up,
the derivative of their distance must be negative at the the braking distance ∆, i.e.
2 dδij
≤ 0, with δij = kxi − xj k and (i, j) ∈ E.
dt δij =∆
But facing double-integrator dynamics, this elegant and plausible approach is not
sufficient, since next to position, velocity also needs to be taken into consideration.
Thus it is most natural to examine the relative position xij = xi − xj as well as
the relative velocity vij = vi − vj of two connected agents (i, j) ∈ E. The according
24
4.1 Connectivity analysis framework
dynamical system of inter-agent dynamics becomes
ẋij = vij
v̇ij = ui − uj
(i, j) ∈ E.
(4.1)
For connectivity preservation we obviously need to ensure that kxij k ≤ ∆ holds at all
times. This can be guaranteed if we find a compact set Ω that is invariant under the
system dynamics (4.1). Additionally we need to ensure that the set is symmetrically
bounded by ∆ in xij -direction.
Proposition 4.1. Let Ω = {(xij , vij ) | kxij k ≤ ∆} be a compact set that is invariant
under dynamics (4.1). Then two interconnected agents stay connected at all times if
their initial conditions are chosen from that set.
A slightly more restrictive formulation can be stated with respect to the average
position and velocity of the network. In some cases this alternative formulation
facilitates the analysis, if the average values of the network are known. Therefore
consider the following dynamical system describing the relative error of each agent
P
(ei = xi − x̄ and pi = vi − v̄) w.r.t. to the average value of all agents (x̄ = N1 N
i=1 xi
P
N
and v̄ = N1 i=1 vi )
ėi = ẋi − x̄˙ = vi − v̄ = pi
ṗi = v̇i − v̄˙ = ui − v̄˙
i ∈ N = {1, ..., N }
(4.2)
Proposition 4.2. Let Ω = {(ei , pi ) | kei k ≤ ∆2 } be a compact set that is invariant
under dynamics (4.2). Then all agents whose initial conditions are chosen from that
set stay connected at all times.
Proof. This proposition can be verified easily, since the distance between two agents
each chosen from Ω is kxi − xj k = kei + x̄ − ej − x̄k ≤ kei k + kej k ≤ ∆.
In this context, the terms persistent (complete) connectivity and proper (complete)
connectivity are defined.
Definition 4.1. A group of agents N a that are initially (completely) connected
are said to be persistently (completely) connected if their interaction graph remains
(completely) connected under evolution of the system according to its respective
dynamics.
25
4 Inherent connectivity preservation
Definition 4.2. A group of agents N a that are initially (completely) connected
are said to be properly (completely) connected, if their interaction graph remains
(completely) connected under evolution of the isolated system according to its
dynamics (thus irrespective of any influence outside from N a , such as additional
agents or any goal attraction force).
Note that persistent (complete) connectivity implies proper (complete) connectivity.
To realise the importance of the definition of proper connectivity consider two connected follower agents. In contrast to the single integrator case, initial connectivity
of the two agents is not necessarily preserved, even though both agents are consensus
seeking. This is because the statement of initial connectivity targets only initial
positions of the agents. Proper connectivity on the contrary involves initial conditions
of position and velocity and thus can be seen as the double-integrator counterpart
of the statement in terms of Proposition 4.1. The notion of proper connectivity
will become particularly beneficial when formulating conditions for connectivity
maintenance of an incomplete graph.
It is stressed that the following framework is based on the special choice Uij =
1
(xi − xj )2 for the artificial potential in (3.2). The results in this chapter do not
2
extend easily to an arbitrary artificial potential Uij .
In the following, connectivity conditions for inter-agent dynamics of some special
structure are stated, which are helpful in investigating connectivity in leader-follower
networks.
Assume that the inter-agent dynamics (4.1) are of the following form:
˙ij =
ẋij
v̇ij
!
!
0
= Aij +
αij
| {z }
(4.3)
ᾱij
with A being a Hurwitz matrix and ᾱij a general additional term. To ensure
connectivity preservation it is necessary to find a set Ωc according to Proposition 4.1
plus conditions to render it invariant.
The subsequent procedure uses basic invariance arguments. It is noted however
that this procedure basically confirms and exploits input-to-state stability (ISS) [39,
Theorem 4.19] of the system of inter-agent dynamics with respect to the "input" ᾱij .
The basic assumption for the following is that the matrix A is Hurwitz. This
26
4.1 Connectivity analysis framework
allows to use the Lyapunov equation
AT P + P A = −Q
(4.4)
which has an unique positive definite solution P for every positive definite matrix
Q. This supplies us with a positive definite Lyapunov function for the given system,
namely
Vij (ij ) = Tij P ij 0.
(4.5)
Derivation along the system trajectory yields
V̇ij (ij ) = (Aij + ᾱij )T P ij + Tij P (Aij + ᾱij )
= T (AT P + P A) + 2Tij P ᾱij
= −Tij Qij + 2Tij P ᾱij .
Using the relation
λmin (S)xT x ≤ xT Sx ≤ λmax (S)xT x
for any positive definite matrix S = S T 0 gives
V̇ij ≤ −λmin (Q)kij k2 + 2λmax (P )kij kkᾱij k
λmax (P )
= −λmin (Q)kij k kij k − 2
kᾱij k
λmin (Q)
λmax (P )
kᾱij k.
≤ 0 for kij k ≥ 2
λmin (Q)
| {z }
(4.6)
β
Thus the set Ωc = {ij | Tij P ij ≤ c} is invariant under system dynamics (4.3) as long
as kᾱij k ≤ β1 kij k holds for at least ∀ij ∈ ∂Ωc . The higher-dimensional clarification
can be found in A.4.
In terms of input-to-state stability (ISS), we have Vij being a positive definite function
of the state ij . Its derivative along the system trajectories V̇ij (ij ) is rendered negative
definite for ∀kij k ≥ βkᾱij k = ρ(kᾱij k), with ρ obviously being a class K function.
Choosing c = c∗ such that
max kxij k ≤ ∆,
∗
T
ij P ij =c
(4.7)
the set Ωc∗ = {ij | Tij P ij ≤ c∗ } fulfils all the requirements needed to apply
27
4 Inherent connectivity preservation
vij
Tij P ij = c∗
∆
xij
k∗ij k
Figure 4.1: Ellipsoidal contour line of a typical inter-agent Lyapunov function Vij
calculated from the Lyapunov equation in the ij -plane. The critical value ∗ is
obtained from the maximum ball around the origin that is enclosed by the particular
contour line Vij = c∗ .
Proposition 4.1. Invariance for all ij inside Ωc∗ is achieved by calculating k∗ij k from
the maximum ball around the origin that is still enclosed by the ellipse Tij P ij = c∗
and consequently requiring
1
λmin (Q)
kᾱij k = kαij k ≤ k∗ij k =
β
2λmax (P )
s
c∗
.
λmax (P )
(4.8)
Estimations incorporating maximum or minimum eigenvalues of a matrix can
be very conservative, especially when their magnitude differs substantially. To
circumvent this issue, the special structure of ᾱij = (0, αij )T is exploited. Let in the
following
!
P1 P2
P =
(4.9)
P2 P3
be the symmetric positive definite solution of the Lyapunov equation (4.4). This
28
4.1 Connectivity analysis framework
yields the estimation
V̇ij = −Tij Qij + 2Tij P ᾱij
!
!
P P
0
1
2
≤ −λmin (Q)kij k2 + 2 xij vij
P2 P 3
αij
!
P
2
2
= −λmin (Q)kij k + 2αij xij vij
P3
!
p
P22 + P32
≤ −λmin (Q)kij k kij k − 2kαij k
λmin (Q)
p
P22 + P32
≤ 0 for kij k ≥ 2kαij k
λmin (Q)
(4.10)
and thus puts again a condition on kαij k to render Ωc∗ invariant
λmin (Q)
λmin (Q)
kαij k ≤ p 2
k∗ij k = p 2
2
2 P 2 + P3
2 P2 + P32
s
c∗
.
λmax (P )
(4.11)
Find some clarification of this relaxation for n > 1 in appendix A.5.
All these results are put together in the following theorem.
Theorem 4.3. Two interconnected agents whose dynamics can be written as (4.3)
stay interconnected at all times if their initial conditions are chosen from the set
Ωc∗ = {Tij P ij ≤ c∗ }, with P being the unique solution of the Lyapunov equation
(4.4) for any positive definite matrix Q, c∗ such that (4.7) holds and kαij k bounded
by (4.11).
Corollary 4.4. The condition (4.11) on kαij k in the foregoing theorem can be
replaced by condition (4.8).
Remark. The characteristic values used in the proof of this theorem are visualised in
Figure 4.1. Some of the estimations can be very restrictive and might render the
system unrealisable in practice.
Remark. Regarding (all) pairs of agents in a network, this theorem can be used to
establish persistent (complete) connectivity from (initial) proper (complete) connectivity.
Remark. If we are dealing with several (completely connected) groups of agents at
the same time, we add subindices to refer to the respective variables Aa , αaij , Pa , Qa ,
c∗a , ∗aij , Ωac∗ of agent group N a .
29
4 Inherent connectivity preservation
Which of the both conditions (4.8) or (4.11) is less restrictive depends on the
matrix P (and therefore indirectly on the matrices A and Q). Numerical examples in
Section 5.1 for the networks studied in the following show that the latter condition
is much easier to fulfil. Additionally, this explicit formulation gives more insight into
the actual system parameters that influence the inequality (by means of P2 and P3 ),
which are analytically calculated for certain special cases below.
4.1.1
Explicit calculations
A few of the values used in Theorem 4.3 such as c∗ and the matrix P obtained from
the Lyapunov equation do themselves not directly reflect parameters of the networked
system. This makes a comprehensive analysis of the system difficult. The following
explicit calculations are therefore given in order to provide a better understanding of
the relation of the characteristic parameters of the system.
It will later prove interesting to further investigate matrices A of the kind
A=
0
1
−N1 −γN2
!
(4.12)
which is Hurwitz for all N1 , γN2 > 0 with eigenvalues
λ1,2 =
−γN2 ±
p
γ 2 N22 − 4N1
2
√
which are both negative if γN2 > 2 N1 , or else respectively have negative real parts.
With the choice Q = I the symmetric, positive definite solution P of the Lyapunov
equation for this particular matrix A is
P =
P1 P2
P2 P3
!
=
γN2
2N1
+
N1 +1
2γN2
1
2N1
1
2N1
1
2γN2
+
1
2N1 γN2
!
.
(4.13)
The calculation of the value c∗ is not trivial. To obtain the value c∗ we can
either use a matrix algebraic fact from [39, Sample Lecture 13] or directly solve the
optimization problem (4.7) which is shown in the following.
Instead of requiring kxij k ≤ ∆, this condition can be squared to become xTij xij ≤ ∆2 .
Thus setting c∗ such that it holds
max xTij xij = ∆2 .
∗
T
ij P ij =c
30
(4.14)
4.2 Leaderless complete graph
Denoting the objective with g(ij ) and the equality constraint with h(ij ) = 0 gives
g(ij ) = xTij xij
h(ij ) = Tij P ij − c∗
(4.15)
= P1 xTij xij + P3 vijT vij + 2P2 vijT xij − c∗ = 0.
The extremal value of g(ij ) in dependency of c∗ is calculated next. Since at any
extremal point of the objective function the Karush-Kuhn-Tucker (KKT) conditions
[40] must hold, the following equations are valid at the extremal point ˆij = (x̂ij , v̂ij )T
2x̂ij
0
∇g(ˆij ) =
!
2P1 x̂ij + 2P2 v̂ij
=λ
2P3 v̂ij + 2P2 x̂ij
!
= λ∇h(ˆij )
(4.16)
h(ˆij ) = P1 x̂Tij x̂ij + P3 v̂ijT v̂ij + 2P2 v̂ijT x̂ij − c∗ = 0.
This gives v̂ij = − PP23 x̂ij and thus
x̂Tij x̂ij
P22
= c∗ .
P1 −
P3
(4.17)
Note that this expression contains the extremal value of the objective function that
should not exceed xTij xij ≤ ∆2 . To respect that condition under the given constraint
choose
P22
∗
c = P1 −
∆2 .
(4.18)
P3
The particular choice of A from above leads to
∗
c =
4.2
N1 + 1
γN2
+
2(N1 + 1)
2γN2
∆2 .
(4.19)
Maintaining a leaderless complete graph
A first step is intended to investigate the most basic network configuration to find
conditions for connectivity preservation: a complete graph without leader agents.
This section can be seen as an introductory example and therefore does not use the
entire framework as lied out in the section before. It rather uses a straightforward
Lyapunov approach, which gives a better understanding of the ideas used in order to
prove connectivity maintenance in complete graphs.
Under consideration of a complete graph structure, agent dynamics exhibit certain
31
4 Inherent connectivity preservation
special properties which can be used for the investigation. The state of the center
of the agents network obviously can be described by calculating all agents average
position and velocity, i.e.
x̄ =
N
1 X
xi
N i=1
N
1 X
v̄ =
vi .
N i=1
Assuming a complete graph structure (thus Ni = N ∀i ∈ N ), the system (3.1) under
consensus protocol (3.3) can be rewritten as
ẋi = vi
v̇i = −
N
X
i ∈ N = {1, ..., N }
(xi − xj ) + γ(vi − vj )
(4.20)
i=1
= −N (xi − x̄) − N γ(vi − v̄)
Hence the dynamics of every single agent does not directly depend on the other agent
states, but merely indirectly by referring to the center. This simplifies the analysis
of the system enormously, since the dynamics of the center can be expressed quite
simple as the following lemma states.
Lemma 4.5. For a (leaderless) network of agents whose connection graph is complete,
P
the average velocity of all agents v̄ = N
i=1 vi stays constant over time, the average
PN
agents position x̄ = i=1 xi grows linearly in time with (constant) growth rate v̄.
Remark. For graph structures that are not complete but connected at all times, these
statements only hold for large t (see e.g. [17]).
Proof. The derivative of the average velocity under the dynamics of the complete
graph network (4.20) vanishes:
N
N
1 X
1 X
v̄˙ =
v̇i =
−N (xi − x̄) − N γ(vi − v̄)
N i=1
N i=1
=−
N
X
i=1
=0
32
xi − γ
N
X
i=1
vi + N x̄ + N γv̄
4.2 Leaderless complete graph
For the average position we similarly get
x̄˙ =
N
N
1 X
1 X
ẋi =
vi = v̄.
N i=1
N i=1
The center of the network is therefore known to evolve according to the simple
dynamics
x̄˙ = v̄
(4.21)
v̄˙ = 0.
In the proceeding analysis, this fact is exploited by introducing a change of coordinates
and regarding each agents error with respect to the center of the network. Denote the
position error by ei = xi − x̄ and the velocity error by pi = vi − v̄. The transformed
system (4.20) describing the error dynamics of agent i is then given by
ėi = ẋi − x̄˙ = vi − v̄ = pi
ṗi = v̇i − v̄˙ = −N ei − N γpi .
i ∈ N = {1, ..., N }
(4.22)
Lemma 4.6. All agents i evolving according to dynamics (4.20) asymptotically
converge to the common center of the network (4.21), i.e. the position and velocity
error of each agent vanish.
Remark. The proof for this lemma is actually simple, since the system (4.22) is linear
and can easily shown to be exponentially stable. However, the proof is stated using
a Lyapunov function because one will be needed in the following to define closed
level-sets.
Proof. The claim is proven through stability analysis of the error system (4.22)
by using a standard Lyapunov approach with the candidate Lyapunov function
Vi (ei , pi ) = 12 (N kei k2 + kpi k2 ) 0. Derivation of Vi along the trajectories of the
system gives
V̇i (ei , pi ) = N eTi pi − N eTi pi − N γpTi pi = −N γpTi pi 0.
(4.23)
Thus V̇i (ei , pi ) is only negative semidefinite.
Asymptotic convergence of the system to the steady state 0 (and thus asymptotic
convergence of all agents towards the center of the system described by (4.21)) is
shown by LaSalle’s invariance principle. Therefore it has to be shown that the set
33
4 Inherent connectivity preservation
S = {(ei , pi ) | V̇i (ei , pi ) ≡ 0} contains no other than the trivial solution ei ≡ 0, pi ≡ 0.
V̇i (ei , pi ) ≡ 0 directly leads to pi ≡ 0, which again according to (4.22) requires ei to
be 0 at all times.
Of course this is not a new result, since convergence of the consensus protocol at
hand is a well known fact (even in non-complete but connected graph structures).
Regarding however the special case of a complete graph supplies us with an explicit
formulation of the network’s center at all times. This in turn allows us to formulate the
error of each agent to the final configuration. By doing so, we can show convergence
for every single agent to that common goal by an agent-wise Lyapunov approach.
The key feature of the agent-wise formulation is that we gain more insight to the
network system dynamics on a microscopic scale. Expanding on that, we can now
state requirements on the initial conditions of the network, under which the complete
graph stays connected at all times.
Theorem 4.7. N agents initially connected by a complete graph structure evolving under dynamics (4.20) stay completely connected at all times if for all initial
2
conditions ei0 = xi0 − x̄ and pi0 = vi0 − v̄ it holds Vi (ei0 , pi0 ) ≤ c∗ = N ∆8 .
Proof. By virtue of Lemma 4.6 we know that the set Ω∗c = {(ei , pi ) | Vi (ei , pi ) ≤ c∗ }
2
is invariant under system dynamics (4.22). It is compact and by choosing c∗ = N ∆8
also bounded by ∆2 in ei direction. Thus Proposition 4.2 can be applied for every
agent i ∈ N .
Remark. A relaxation of the condition stated by Theorem 4.7 requires that the
pairwise potential between all agents i and j does not exceed N ∆2 , i.e.
Vi (ei , pi ) + Vj (ej , pj ) ≤ N ∆2
∀i, j ∈ N .
The calculations with respect to the network center in this section have been
carried out to demonstrate its practicability with the complete graph at hand.
However, in the following we tend to focus on inter-agent dynamics instead of
regarding error dynamics with respect to the network center. For the case at hand,
a similar result can be obtained by examining the inter-agent dynamics (4.1) and
applying Proposition 4.1: Regarding the difference between two agents xij = xi − xj
and vij = vi − vj gives the following inter-agent dynamics for every two agents
i, j ∈ N , i 6= j
ẋij = vij
(4.24)
v̇ij = −N xij − N γvij .
34
4.3 Complete graph with leaders
Obviously, this corresponds to the error dynamics 4.22. Thus the same Lyapunov
equation as before can be used to carry out the same proof, but dependent on the
position and velocity difference of each agent pair, i.e. Vij (xij , vij ) = 12 (N x2ij + vij2 ).
Corollary 4.8. N agents initially connected by a complete graph structure evolving
under dynamics (4.20) stay completely connected for all times if for all initial inter2
agent distances xij0 , vij0 it holds Vij (xij0 , vij0 ) ≤ c∗ = N ∆2 .
Proof. This proof goes analogous to the one of Theorem 4.7 with using Proposition 4.1
to define the invariant set Ωc∗ .
As mentioned above, these results are quite narrow and not generic. However,
they give a good understanding of the ideas and approaches used throughout the
following connectivity analysis.
Essentially, the examined case (a complete graph without leaders) is included as a
special case in the following problem.
4.3
Maintaining a complete graph with leaders
In the previous chapter it has been shown that the leader-follower network in a
connected graph structure achieves asymptotic consensus. In order to find sufficient
conditions for connectivity, the complete graph is taken as starting point again, thus
extending the foregoing section by leader agents. However, in this section from the
beginning inter-agent dynamics for any two connected agents are considered and the
framework from Section 4.1 is used.
Given a complete graph, this still yields a quasi-autonomous description relying only
on the difference between the two particular regarded agents and not directly on the
whole network. For notational simplicity, investigations are first confined to the case
of leaders heading for a desired velocity vd , thus applying control law (3.4).
Following the notation from the previous sections, the dynamics become for all agents
i∈N
ẋi = vi
(4.25)
v̇i = −N (xi − x̄) − N γ(vi − v̄) + f¯i (vi − vd ).
Rewriting the system in terms of inter-agent dynamics yields for every two agents
i, j ∈ N , i 6= j
ẋij = vij
v̇ij = −N xij − N γvij + f¯i (vi − vd ) − f¯j (vj − vd ).
(4.26)
35
4 Inherent connectivity preservation
This system can be decomposed into a linear, autonomous share and the influence of
the goal attraction function, concentrated to the variable ᾱij :
˙ij =
ẋij
v̇ij
!
!
!
0
1
0
=
ij +
−N −N γ
f¯i (vi − vd ) − f¯i (vj − vd )
|
{z
}
{z
}
|
(4.27)
ᾱij
A
The aim is now to find a set Ωc and conditions on kᾱij k to render it invariant
according to Proposition 4.1.
This point marks the dead end of a standard Lyapunov approach as carried out
in the leaderless case before. The derivative of the Lyapunov function used in the
previous section Vij = 12 (N xij + vij ) is only negative-semidefinite and is independent
of the state variable xij
V̇ij = −N γvijT vij + vijT ᾱij
≤ −N γkvij k2 (1 − θ) − kvij k (N γkvij kθ − kᾱij k)
≤ −N γkvij k2 (1 − θ)
for
kvij k ≥
(4.28)
kᾱij k
N γθ
with 0 < θ < 1. This condition that should render V̇ij negative definite does not
depend on the norm of the entire state kij k. Thus it is not suitable to apply the
ISS and invariance arguments established in the basic framework in Section 4.1.
For this reason return to the Lyapunov equation, which supports a proper
Lyapunov function candidate, that has a negative definite derivative.
The linear share of the dynamics denoted by matrix A is asymptotically stable since
it is of type (4.12). Thus Theorem 4.3 can be applied to determine bounds on
kαij k = kf¯i (vi − vd ) − f¯i (vj − vd )k under which complete connectivity is preserved.
For this, three different cases must be examined, depending on which class the two
considered agents belong to. Ωc∗ , c∗ and k∗ij k are defined by Theorem 4.3.
Follower-follower In this case αij = 0 and thus requirement (4.11) is fulfilled
for any two follower agents. Ensuring proper complete connectivity of the
initial graph, thus the correct choice of initial conditions from Ωc∗ , suffices to
guarantee persistent connectivity of any two followers in the complete graph
by virtue of Theorem 4.3.
Leader-follower Regarding the interconnection of a leader and a follower agent
λmin (Q)
gives kαij k ≤ fmax and thus the additional condition fmax ≤ √
k∗ij k.
2
2
2
36
P2 +P3
4.4 Connector agents and incomplete graphs
Leader-leader Analogously, the interconnection of two leader agents results in a
stronger condition constraining the goal attraction function, since in this case
λmin (Q)
kαij k ≤ 2fmax leads to the condition fmax ≤ √
k∗ij k.
2
2
4
P2 +P3
These conditions retain all initial links up (and thus keep the network graph complete
at all times). From Theorem 3.2 it is known that velocity consensus for the applied
control is achieved as long as the network graph remains connected. Concluding
these findings results in the following theorem.
Theorem 4.9. Consider a group of leader and follower agents of dynamics (4.25).
The network is persistently completely connected, if the magnitude of the goal attracλmin (Q)
tion force is bounded by the upper bound fmax ≤ √
k∗ij k and the initial graph
2
2
4
P2 +P3
is properly completely connected.
Remark. Since the result relies solely on an upper bound of the attraction force fmax ,
the theorem applies to both target velocity as well as target position with a slight
change of matrix A, according to the absolute damping in the control law needed to
apply Theorem 3.3 for target position consensus.
Remark. For a network that has only one leader, consideration of leader-leader
connectivity can be let out entirely and the condition on fmax relaxes to fmax ≤
√ 12 2 k∗ij k.
2
P2 +P3
For a network without any leaders at all, the condition on fmax can be dropped
completely and it remains to ensure proper complete connectivity. This is an
alternative approach to Theorem 4.7 for a leaderless complete graph.
4.4
Connector agents and incomplete graphs
In this section, the concept of connector agents is introduced. Connector agents
act as chain link between different complete subgraphs and in so doing enable
conditions preserving connectivity of the overall network. This concept is later used
to investigate some special topologies that put the focus back on leader-follower
interaction.
Analysis of incomplete subgraph interconnection is drastically simplified if the
absolute damping protocol (3.7) is used. In addition, the following sections focus
mainly on consensus w.r.t. a target position, i.e. all agents should converge to
the common position target and finally rest there which can be accomplished using
37
4 Inherent connectivity preservation
N1
N ĉ
Nc
N3
N2
Figure 4.2: Topology of interconnections between the group of connectors N c and
K = 3 groups of agents N k . All agents within a set are properly completely connected.
All initial links should be maintained.
absolute damping. For simplicity, this introduction of connector agents studies
follower networks only.
Lemma 4.10. Consider two neighbouring agents i ∼ j and the set of agents N a .
Let Nka denote the subset of agents from N a that are neighbouring agent k. Then
the following estimation holds
X
X
a
(xi − xk ) −
(xj − xk )
≤ |N |∆.
k∈Nia
k∈Nja
Proof. To see this, define the set of common neighbours Nca = Nia ∩ Nja , thus all
those agents from N a which are simultaneously neighbouring agent i and j. The
given term can then be rewritten to
X
X
X
(x
−
x
)
(x
−
x
−
x
+
x
)
+
(x
−
x
)
−
i
j
k
k
i
k
j
k
k∈Nca
a
a
k∈Ni \Nca
k∈Nj \Nca
X
X
≤ |Nca |kxi − xj k +
kxi − xk k +
kxj − xk k
k∈Nia \Nca
k∈Nja \Nca
≤ |Nca |∆ + (|Nia | − |Nca |)∆ + (|Nja | − |Nca |)∆
≤ |N a |∆.
38
4.4 Connector agents and incomplete graphs
This calculation uses the fact that the distance between neighbouring agents is always
smaller than ∆ as well as the obvious fact that the number of individual neighbours
of agents i and j (|Nia | − |Nca | + |Nja | − |Nca |) plus the number of common neighbours
|Nca | can not exceed the total number of agents |N a | in the target set.
Theorem 4.11. Let the set of connectors N c and the sets N k , k = 1, ..., K be K + 1
non-overlapping sets containing all N agents of dynamics (3.1) with absolute damping
protocol (3.7). Let all agents be follower agents, thus f¯i = 0 ∀i ∈ N . If
1. ∀k = 1, ..., K, each union N c ∪ N k is properly completely connected and
2. N ≤
λmin (Qĉ )
q
2∆ Pĉ2 +Pĉ2
2
k∗ĉ k + |N ĉ |
3
with N ĉ = N c ∪ argminN k {|N k |}, then the overall graph remains connected at all
times.
Proof. The investigated topology is illustrated in Figure 4.2. All pairs of connector
agent group N c and any other group N k are singularly properly completely connected
by assumption. This assumption (namely ensuring complete connectivity preservation
of the separate complete graphs) can be satisfied following the previous Theorem 4.9.
Next, it must be shown that the presence of additional agents (not included in the
respective group pair) does not break this complete connectivity. This is due to the
assumption that proper connectivity of the regarded group refers to the isolated
group only. Indeed possible interconnections to all other agents have to be considered,
since as time passes the network contracts while forming consensus.
Examine the group pair N ĉ containing the smallest number of agents. This is
obviously the most easy group to pull apart. Formulate the inter-agent dynamics of
two agents in this group, i.e. i, j ∈ N ĉ , as
ẋij = vij
v̇ij = −γvij −
X
(xi − xk ) +
k∈Ni
X
(xj − xk )
k∈Nj
ĉ
= −|N |xij − γvij −
X
(xi − xk ) +
k∈Ni \N ĉ
|
X
(4.29)
(xj − xk ) .
k∈Nj \N ĉ
{z
αĉij
}
Note that this formulation is again of type (4.12) with the influence of the agents not
in the considered group (N \N ĉ ) put to the additional term αĉij . Thus Theorem 4.3
ensures connectivity preservation of the considered group in the presence of additional
39
4 Inherent connectivity preservation
agents as long as their influence (kαĉij k) remains below a certain bound (4.11) which
is
s
λmin (Qĉ )
c∗ĉ
kαĉij k ≤ q
.
(4.30)
2 Pĉ22 + Pĉ23 λmax (Pĉ )
Apply Lemma 4.10 to upper bound kαĉij k by
kαĉij k ≤ |N \N ĉ |∆ = (N − |N ĉ |)∆.
(4.31)
Thus, a limit on the total number of agents in the network
N≤
λmin (Qĉ )
q
k∗ĉ k + |N ĉ |
2∆ Pĉ22 + Pĉ23
(4.32)
ensures persistent complete connectivity of the smallest pair N ĉ of the connectoragent groups.
What remains to show is that this condition not only ensures connectivity preservation
of this specific group pair, but also of all possible other pairs. This can be realised
easily by including only |N ĉ | agents of any group pair N c ∪ N k into the linear share
of their inter-agent dynamics. Consequently, again N − |N ĉ | agents are treated as
"distracting" agents and their effect is put into the additional term αkij (even though
some of them belong to the complete graph of the considered group pair). This
can be done for all groups of agents, since it holds |N ĉ | ≤ |N c ∪ N k | ∀k = 1, ..., K
because of N ĉ = N c ∪ argminN k {|N k |}. This proves persistent complete connectivity
for all group pairs N c ∪ N k , k = 1, ..., K.
4.5
Maintaining complete subgraphs with leaders
(I)
In this section, the leader-follower interaction is studied again, but with the aim to
relax the requirement of complete graph connectivity to the case of only completely
connected subgraphs. Assume the group of leaders as well as the group of followers
to be each properly completely connected. These two proper complete subgraphs are
connected by several connector agents. Those in turn should form a proper complete
graph consisting of some of the leader and some of the follower agents. Note the
investigated topology in Figure 4.3. Again, the goal is to find conditions under which
40
4.5 Complete subgraphs with leaders (I)
Nf
Nc
Nl
Figure 4.3: Topology of the interconnection between leader, follower and connector
agents. All agents within a set are properly completely connected. All initial links
should be maintained.
the entire network remains connected – thus establishing persistent connectivity of
the three subgraphs under consideration of all agents present.
The focus is on consensus w.r.t. a target position, i.e. all agents should converge to
the common position target and finally rest there. To accomplish that, the absolute
damping protocol (3.7) is used in this section. Note that the way connector agents
are arranged in this setup differs to the leaderless setup described in the previous
section – here the connector group is not completely connected to the other groups.
The system is again formulated in terms of inter-agent distances xij = xi − xj and
vij = vi − vj which yields for any two interconnected agents i, j ∈ N , i 6= j
ẋij = vij
v̇ij = −γvij
−
X
(xi − xk ) + f¯i (xi − xd )
k∈Ni
+
X
(4.33)
(xj − xk ) − f¯j (xj − xd ).
k∈Nj
In the following, three kinds of agent interconnection are distinguished corresponding
to the three subgraphs.
41
4 Inherent connectivity preservation
Follower-follower
Consider the interconnection between two arbitrary follower agents i, j ∈ N f . Their
inter-agent dynamics are
ẋij = vij
v̇ij = −γvij −
X
(xi − xk ) +
k∈Ni
= −Nf xij − γvij −
X
(xj − xk )
(4.34)
k∈Nj
X
(xi − xk ) +
k∈Nil
X
(xj − xk ).
k∈Njl
The dynamics can again be separated into an autonomous, linear part and the
influence due to additional neighbouring leader agents. Note that it is indeed
necessary to regard possible connections to all other (leader) agents. This is because
the assumption of having only some agents as connector agents is only valid initially.
As time passes, the number of links between the leader and follower group should
increase, since consensus should finally be reached. Rewriting the system in its
decomposed form yields
!
!
0
0
1
P
P
ij +
.
˙ij =
− k∈N l (xi − xk ) + k∈N l (xj − xk )
−Nf −γ
j
i
{z
}
|
|
{z
}
Af
(4.35)
ᾱf
Connectivity preservation between follower agents is guaranteed if the requirements
of Theorem 4.3 are fulfilled. Thus it has to be shown that kᾱf k remains below a
certain bound.
This bound can be retrieved from Lemma 4.10 by investigating the worst case
scenario. It can easily be seen that kᾱf k is bounded by Nl ∆. Also intuitively this is
the worst case, depicting one of the two regarded followers being connected to all
leader agents, whereas the other one has no leader connection at all. This result is
valid throughout time, even when the two groups of agents approach each other and
more links between leaders and followers are formed (because common neighbours
relax the above estimation). Put together, condition (4.11) from Theorem 4.3 places
a limit on the number of leader agents
λmin (Qf )
q
Nl ≤
2∆ Pf22 + Pf23
42
s
c∗f
.
λmax (Pf )
(4.36)
4.5 Complete subgraphs with leaders (I)
Designing a network of agents, the parameters to adapt to ensure complete connectivity between follower agents and to receive a (valid) upper bound on the number
of leaders, are the damping constant γ and the number of followers Nf .
Leader-leader
Next we investigate what it takes to keep the leader subgraph persistently complete.
Having a majority of follower agents on the one hand, we have the goal attraction
function that acts on all leaders and thus somehow unifies them on the other hand.
Considering the inter-agent dynamics and reformulating the system as before yields
Al =
0
1
−Nl −γ
!
(4.37)
and
αlij = −
X
k∈Nif
(xi − xk ) +
X
(xj − xk ) + f¯i (xi − xd ) − f¯j (xj − xd )
(4.38)
k∈Njf
for all agents i, j ∈ N l . Now assume a quadratic goal attraction potential F (kxi −
xd k) = 12 η(kxi − xd k)2 . This allows to simplify the expression
f¯i (xi − xd ) − f¯j (xj − xd )
= − η(xi − xd ) + η(xj − xd )
(4.39)
= − ηxij .
This term can now be put to the linear share of the inter-agent dynamics. The
matrix Al then becomes
!
0
1
Al =
.
(4.40)
−(Nl + η) −γ
The remaining parts in αl can be bounded, as in the follower-follower case using
Lemma 4.10, by the maximum number of potential follower agent neighbours to
kαlij k ≤ Nf ∆.
On first sight this seems impossible to fulfil, since a condition that looks similar
to the one in the follower-follower case arises (which would thus not be compliable,
Nl < Nf < Nl ). However, here the additional term η resulting from the quadratic
goal attraction function appears in Al , which also affects Pl and renders the condition
feasible.
43
4 Inherent connectivity preservation
Hence the crucial parameter in this case is the strength of the goal attraction potential
η.
Leader-follower / connector-connector
To keep the overall graph connected, and to achieve consensus towards the common
goal induced by the leader agents, conditions to keep the group of leaders connected
to the group of followers are proposed. This is where the group of connector agents
becomes essential, since a complete interconnection structure between them is to be
preserved. Thus consider the inter-agent dynamics of a leader and a follower agent
from the group of connector agents N c which are assumed to be properly completely
connected. Their inter-agent dynamics can be stated as follows:
Ac =
0
1
−Nc −γ
!
(4.41)
and
αcij = −
X
k∈Ni
(xi − xk ) +
\N c
X
k∈Nj
(xj − xk ) + f¯i (xi − xd )
(4.42)
\N c
for agents from the group of connectors i ∈ N l ∩ N c , j ∈ N f ∩ N c .
Again we apply the same mechanism as before to find a bound on kαcij k which turns
out to be
kαcij k ≤ (Nf + Nl − 2Nc ) + fmax .
(4.43)
The value fmax sets the bound of the goal attraction force and calculates to
fmax
λmin (Qc )
p
≤
2∆ Pc22 + Pc23
s
c∗c
− (Nf + Nl − 2Nc ).
λmax (Pc )
(4.44)
With all parameters chosen according to the previous analysis, the only remaining
design parameter is the number of connector agents. Setting Nc finally returns a
maximum value for fmax .
Note that the assumption of a quadratic goal attraction function is crucial for this
procedure, thus fmax depends on the maximum distance any leader agent may ever
be from the target. Implementing this restriction is very conservative and hard to
verify in terms of initial conditions.
Concluding, we can state the following theorem to ensure persistent connectivity
of the entire graph of leader and follower agents from an initially incomplete graph.
44
4.6 Complete subgraphs with leaders (II)
Theorem 4.12. Consider a group of leader and follower agents such that the initial
interconnection graph among all leader respectively follower agents is proper complete.
Additionally, require the group of connector agents consisting of some leader and
follower agents also be properly completely connected.
Then the following conditions ensure connectivity preservation of the entire graph at
all times and, by Theorem 3.3, convergence to the common target position:
1. The goal attraction potential is of quadratic form F (kxi − xd k) = 12 η(kxi − xd k)2
2. A number of leader agents such that (4.36) holds
3. η such that kαl k ≤ Nf ∆ fulfilled
4. f̄i ≤ fmax (4.44) must hold for all times.
4.6
Maintaining complete subgraphs with leaders
(II)
Having faced some severe difficulties in the previous section, an alternative approach
that also utilises the idea of connector agents is proposed. Consequently, the
fundamental assumption is again a complete connector graph structure. Additionally,
require all leader agents to belong to that group. This assumption is more restrictive
than the one considered in the section before, but the resulting conditions result in
being much more powerful and flexible.
Investigate a network of leader and follower agents, in which the group of followers
makes up a proper complete graph. Some of the follower agents belong to the group of
connector agents. Denote this subset of follower agents as N fc ⊂ N c ⊆ N . All leaders
belong to the group of connector agents, thus N l ⊂ N c and the interconnection
graph of all connector agents is assumed to be proper complete. Figure 4.4 illustrates
this initial topology. Connectivity preservation of the whole graph is again achieved
by enforcing preservation of all initially existing links. In the following, the set N c
contains only those agents that initially belong to the group of connectors.
This is as an application of the general connector framework presented in Section 4.4,
extended by leader agents. A closer look reveals, how the groups have to be defined
to match the definitions of Theorem 4.11. However, due to the presence of leader
agents, the theorem can not be applied directly and calculations are redone with
focus on the two types of agents rather than focusing on the central connector group
as in the previous theorem.
45
4 Inherent connectivity preservation
Nc
Nf
Nl
Figure 4.4: Topology of the interconnection between leader, follower and connector
agents. All agents within a set are properly completely connected. All initial links
should be maintained.
Follower-follower
Analysing the interconnection between two arbitrary follower agents can be carried
out exactly the same as in the foregoing section. The only requirement to keep the
subgraph of followers complete at all times is a bound on the number of leader agents
(4.36).
Connector-connector
The second and only other type of agent interconnections to be regarded in this
topology is the interconnection of any two connector agents. The elements making
up their dynamics in decomposed form are
Ac =
0
1
−Nc −γ
!
(4.45)
and
αcij = −
X
k∈Ni \N c
(xi − xk ) +
X
(xj − xk ) + f¯i (xi − xd ) − f¯j (xj − xd ).
(4.46)
k∈Nj \N c
An upper bound for kαcij k is again obtained regarding the "worst case" as in
Lemma 4.10, which in this case is two connected leader agents (thus f¯i , f¯j 6= 0) with
one of them being connected to all follower agents N f . This scenario provides the
following bound
kαcij k ≤ (Nf − Nfc )∆ + 2fmax
(4.47)
46
4.6 Complete subgraphs with leaders (II)
which sets an upper bound on fmax
fmax
λmin (Qc )
≤ p 2
4 Pc2 + Pc23
s
∆(Nf − Nfc )
c∗c
−
.
λmax (Pc )
2
(4.48)
Note that this calculation does not require the goal attraction potential F to be
quadratic as before, which makes this approach much more practical to use, since F
can be designed to be bounded for all positions.
Theorem 4.13. Consider a group of leader and follower agents of initial interconnection topology such that the interconnection graph among the follower agents
is a proper complete graph. Let the group of all leader agents and their follower
neighbours also be properly completely connected.
Then the following conditions ensure connectivity preservation of the whole network
at all times and, by Theorem 3.3, convergence of all agents to the common target
position:
1. A maximum number of leader agents (4.36)
2. An adequate number of connector agents and bound on fmax such that (4.48)
is fulfilled.
47
5
Experimental results
This chapter visualises the presented theoretical results. First, several calculations
are made to point out the effect of the basic system parameters. Complex expressions
of those basic parameters in the presented conditions make their influence incomprehensible, why a (numerical) investigation is favourable. Following that, various
computer simulations of the investigated network topologies are shown.
5.1
Effect of parameters
This section is devoted to clarify the effect of the most basic system parameters
such as the damping factor γ or the size of the considered group of agents N on the
conditions proposed in the foregoing sections. Closer scrutiny is necessary since many
of the foregoing results and conditions are based on the solution to the Lyapunov
equation (4.4) and complex calculations involving it. Even though the solution P
can still be formulated explicitly in a compact form as in (4.13), more involved
calculations, relying for example on eigenvalues of this matrix, mask the influence of
the basic parameters on the outcome.
The key concept of the proposed connectivity analysis is to formulate inter-agent
dynamics and separate them into a linear share and an additive term (4.3). Connec-
49
5 Experimental results
tivity preservation is achieved by bounding this additional term αij by an adequate
bound, such that the set Ωc∗ remains invariant. Two ways of calculating such a bound
have been proposed, (4.8) and (4.11). The latter is usually much less restrictive,
which is why the estimate
λmin (Q)
kαij k ≤ p 2
2 P2 + P32
s
c∗
∗
(N, γ)
=: αij
λmax (P )
is used to bound kαij k throughout this work. As noted before, this bound on kαij k
implies directly the sufficient conditions found to ensure connectivity, such as the
maximum number of leader or follower agents and the bound on the magnitude of
the goal attraction force fmax . Thus, it would be appealing to get a feeling of the
effect of the basic system parameters on these conditions. However, kαij k is bounded
by a rather complex function of these parameters, which is therefore analysed in the
following.
Since all subsequent steps rely on the solution matrix P , first the choice of the
positive definite matrix Q used for solving the Lyapunov equation is discussed. In
order to get an insight into favourable characteristics of Q, numerical optimisation
of the matrix Q with respect to the resulting bound on kαij k have been carried out.
Results differ according to the considered consensus protocol which substantially
modifies the linear share of the dynamics included in the matrix A. Referring back
to the general notation of A in (4.12), the case N1 = N2 = N reflects the case of a
relative damping protocol, N2 = 1 refers to absolute damping. Those two cases are
the most relevant, since they are used for velocity and position consensus (in the
complete connector case) respectively.
For the absolute damping protocol, the optimal choice of the matrix Q is Q∗ = κI,
wherein κ is arbitrary. For simplicity choose κ = 1. This result is consistent regarding
variations of both parameters γ and N . Only if γ is chosen extraordinarily large
(γ > 10N ), the resulting bound on kαij k can be enhanced by adding off-diagonal
elements to Q.
A similar effect can be observed for the relative damping protocol: Roughly
speaking, for N > γ the optimal Q∗ is similar to the (arbitrarily scaled) identity
matrix. For an increasing damping γ however, the optimal matrix Q∗ degenerates to
a more complex form and the difference of the resulting bound on kαij k compared
to the one obtained from Q = I is significant.
Generally speaking, the damping value for the regarded multiagent system dynamics
50
5.1 Effect of parameters
8
P1
P2
P0
7
6
5
4
3
2
vij
1
0
−1
−2
−3
−4
−5
−6
−7
−8
−1
0
xij
1
(a) N = 3, γ = 1
−1
0
xij
1
(b) N = 5, γ = 1
−1
0
xij
1
(c) N = 3, γ = 5
Figure 5.1: Boundary Tij P ij = c∗ of the invariant set Ωc∗ for different matrices P .
P1 and P2 are obtained from the Lyapunov equation under the relative damping
control for Q1 = I and Q2 = [N 0; 0 γ] respectively. P0 = [N 0; 0 1] represents the
simple Lyapunov function from Section 4.2.
51
5 Experimental results
should be as low as possible to make the system exhibit an agile and dynamic
behaviour. Therefore it is reasonable to assume thatγ is in general of lower magnitude
than the number of agents N . Consequently, for most cases, the choice Q = I for
solving the Lyapunov equation provides us with a sound bound on kαij k. This
motivates the use of Q = I not only in terms of optimality but also simplifies the
analysis and qualifies the explicit calculations in Section 4.1.1 carried out for Q = I.
Note that this reasoning is strongly focused on a high bound on kαij k. For a particular
application however, different criteria might be decisive, such as the shape of the
invariant set Ωc∗ . To give an impression, the resulting shape of Ωc∗ is shown for
different choices of Q and different parameters in Figure 5.1.
Focusing on the value of the upper bound on kαij k, again the two consensus
protocols must be distinguished.
∗
First consider the absolute damping protocol. Experiments reveal that αij
(N, γ) ≤
∗
min{γ, N }. This conjecture is confirmed by some analytical investigations. αij
(N, γ)
calculates to
s
4N 3 γ 3
(2γ + 2N γ)
(5.1)
− 21
p
p
· (2N + N 2 − 2N + γ 2 + 1 N 2 + 2N + γ 2 + 1 + N 2 + γ 2 + 1)
which has the following limits
∗
lim αij
(N, γ) = γ,
r
(5.2)
N →∞
∗
lim αij
(N, γ) =
γ→∞
N3
< N.
N +1
(5.3)
∗
Additionally it can be shown numerically that αij
(N, γ) is monotonically increasing
∂
∂ ∗
∗
in both directions, N and γ, i.e. ∂N αij > 0 and ∂γ αij > 0. This approves validity of
∗
the conjecture. To get an impression of the actual shape of the function αij
(N, γ),
Figure 5.2 shows this function for a range of parameter values for absolute damping.
52
5.1 Effect of parameters
∗
(N, γ)
αij
20
10
0
30
20
γ
10
0 0
10
5
20
15
25
30
N
Figure 5.2: The bound on kαij k obtained from (4.11) for the absolute damping
∗
protocol as a function αij
of the basic parameters N and γ.
Results for the relative damping protocol can be obtained similarly for
calculated to
∗
αij
(N, γ)
s
4N 6 γ 3
2γN 2 + 2γN
− 12
p
p
· 2N + N 2 γ 2 + N 2 − 2N + 1 N 2 γ 2 + N 2 + 2N + 1 + N 2 + N 2 γ 2 + 1
(5.4)
∗
Experiments for this setup foretell that αij (N, γ) ≤ N . Using the analytical calculation, the limits can again easily be determined to
∗
lim αij
(N, γ) = ∞,
N →∞
r
∗
lim αij
(N, γ) =
γ→∞
N3
< N.
N +1
(5.5)
(5.6)
∂
∂ ∗
∗
αij
αij > 0, the validity of the conjecture can again
With monotonicity ∂N
> 0 and ∂γ
be approved. A visualisation of the analysed function for relative damping can be
seen in Figure 5.3.
∗
Summing up, the bound αij
(N, γ) grows proportionally with N . This result is
53
5 Experimental results
∗
αij
(N, γ)
30
20
10
0
30
20
γ
10
0 0
5
10
20
15
25
30
N
Figure 5.3: The bound on kαij k obtained from (4.11) for the relative damping
∗
protocol as a function αij
of the basic parameters N and γ.
intuitive: The greater the group of completely connected agents is, the greater the
(possibly) distracting additional influence αij can be without splitting the group.
Figure 5.3 shows that under the relative damping protocol a rather small damping
value γ suffices to earn a high bound on kαij k. The bound obtained for the absolute
damping protocol however is much more sensitive to the value of γ, which can clearly
be seen in Figure 5.2. Thus, using the absolute damping protocol requires a higher
value for γ to receive an adequate bound on the magnitude of αij .
5.2
Simulations
In this section, the validity of the theoretical results is shown in simulations. Although the conditions derived in Chapter 4 are mostly only sufficient conditions for
connectivity maintenance, we also show experimental set ups that demonstrate their
relevance by showing loss of connectivity by only slight "violation" of the sufficient
conditions.
First, a few preliminaries that are shared by all the simulations are stated, followed
by several simulations regarding each investigated scenario.
54
5.2 Simulations
3
µ=1
µ=5
µ = 10
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
ksd k
2
2.5
3
Figure 5.4: Goal attraction potential F (ksd k) (5.7) for different values of µ in solid
lines and norm of the according goal attraction force kf (sd )k in dashed lines.
5.2.1
Preliminaries
All agents are assumed to have a sensor range of ∆ = 1. Follower agents are
represented by dots whereas leader agents are represented by triangles. The position
of the agents and their current velocity vectors (scaled) are shown at various specific
timestamps. For some agents their sensing range is also plotted as a dashed circle to
illustrate existing agent interconnections. To follow the evolution of the agents, the
position trajectory of each agent is also included in the plots. A position target is
denoted by an asterisk, a velocity target is shown by a red velocity vector.
Since most above results require the goal attraction force to be bounded by some
value fmax , the following goal attraction potential is chosen
Z
2η
arctan(µksd k) dksd k
π
1
2η
2
2
=
ksd k arctan(µksd k) −
log(µ ksd k + 1) .
π
2µ
F (ksd k) =
(5.7)
55
5 Experimental results
Note that the gradient ∇sd F (ksd k) is not defined at ksd k = 0, but it holds
1
2ηµ
dF
=
< ∞,
ksd k→0 dksd k ksd k
π
lim
(5.8)
why F (ksd k) is a valid goal attraction potential according to Definition 3.5. In
addition it is easily adjustable. From Proposition 3.1 it is known that the norm
of ∇sd F (ksd k) for this goal attraction potential is bounded by the same value as
2η
arctan(ksd k) which is smaller than η for all ksd k > 0. This makes it easy to limit
π
the maximum goal attraction force kf (sd )k ≤ fmax = η by setting this parameter.
The parameter µ determines how fast the goal attraction force grows in close proximity
to the target and is set to µ = 10. Note the goal attraction potential F (ksd k) and
norm of the according goal attraction force kf (sd )k for different values of µ in
Figure 5.4.
Unless otherwise stated, the damping factor γ is set to γ = 1.
The choice Q = I is made for solving the Lyapunov equation throughout the
simulations.
5.2.2
Complete leaderless graph
Let us first consider the case of a complete graph without any leaders. For this
topology, the only condition stated by corollary 4.8 to keep the graph complete is
the correct choice of initial conditions. Remember that all initial conditions have to
be chosen from the invariant set Ωc∗ . In this presented scenario, a network of two
agents is investigated with identical initial conditions for xy and vy , why it satisfies
to consider inter-agent distances in the (xx , vx )-plane. Figure 5.5 shows this plane
with initial inter-agent distances and boundary of two invariant sets Ωc∗ . These
sets are obtained from two different Lyapunov functions. For the leaderless case,
the simple Lyapunov function proposed by Corollary 4.8 Vij (ij ) = 12 (N x2ij + vij2 ) is
sufficient to determine an invariant set Ωc∗ . Note that this simple Lyapunov function
is independent of the damping value γ. For comparison, the level set of the general
Lyapunov function Vij (ij ) = Tij P ij obtained as a special case of Theorem 4.9 is
shown.
It is obvious that the initial conditions are not chosen in accordance with the
corollary and the theorem for γ = 0.1. Loss of connectivity can therefore be observed
in Figure 5.6a. However, Figure 5.6b shows that connectivity maintenance and
convergence of the system can still be achieved with a sufficiently high value for the
56
5.2 Simulations
2
2
V1
V2
ij0
1.5
1.5
0.5
0.5
vij
1
vij
1
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−1
−0.5
0
xij
0.5
V1
V2
ij0
1
−2
−1
(a) γ = 0.1
−0.5
0
xij
0.5
1
(b) γ = 1.0
Figure 5.5: Shape of the set Ωc∗ in the (xx , vx )-plane for the standard Lyapunov
function V1 = 12 (N x2ij + vij2 ) and V2 = Tij P ij and initial condition ij0 = [xij0 , vij0 ].
relative velocity damping γ. Figure 5.5b shows that the invariant set provided by
Theorem 4.9 already foretells that outcome, since ij0 ∈ Ωc∗ = {ij | Tij P ij ≤ c∗ }.
5.2.3
Complete graph target velocity
Introducing leader agents and a goal requires compliance of additional conditions
apart from choosing initial conditions correctly. Theorem 4.9 sets an upper bound
on the maximum goal attraction force fmax to ensure consensus towards a target
state while keeping the complete network connectivity.
A total number of four agents is studied in this example, of which one is a leader
agent that should guide the group towards the target velocity vd = [0, 2]T . All agents
obey the relative damping protocol. Theorem 4.9 is used to calculate the bound
on fmax to 2.368. Initial conditions are chosen from the invariant set Ωc∗ , i.e. all
inter-agent potentials fulfil the condition Tij P ij ≤ c∗ .
Exemplary, at this point numerical results from explicit calculations are presented.
57
5 Experimental results
2
1.5
x2
1
0.5
0
−0.5
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
0
0.5
1
x1
(a) γ = 0.1
2
1.5
x2
1
0.5
0
−0.5
−2.5
−2
−1.5
−1
−0.5
x1
(b) γ = 1.0
Figure 5.6: Evolution of N = 2 agents with same initial conditions according to
Figure 5.5 with different values for the relative damping γ.
58
5.2 Simulations
The linear share of the inter-agent dynamics is due to the relative damping protocol
0
1
−N −γN
A=
!
=
!
0
1
−4 −4
with the according matrix P obtained from solving the Lyapunov equation for Q = I
γ
2
P =
N +1
2γN
1
2N
+
1
2N
N +1
2γN 2
!
=
9
8
1
8
1
8
5
32
!
(see (4.13)). The value c∗ calculates to
P22
41
=
c = ∆ P1 −
P3
40
∗
2
which gives a value for k∗ij k
k∗ij k
r
=
c∗
≈ 0.9479.
λmax (P )
Thus, we can finally calculate the bound on fmax for the case of a single leader to
1
k∗ij k ≈ 2.3685.
fmax ≤ p 2
2 P2 + P32
Initial conditions in stacked vector form x = [x1x , x1y , ..., xNx , xNy ]T , v = [v1x , v1y , ..., vNx , vNy ]T
are
x0 = [−0.55, 0, −0.05, .30, .42, 0, 0.20, −0.20]T
v0 = [0, 0, −1.20, 1.00, −0.90, 0, 0.10, 0]T .
To verify that they all lie within the invariant set Ωc∗ , we need to check that all
normalized initial inter-agent potentials Vij (ij0 ) = c1∗ Tij0 P ij0 are below 1. The
numerical values are presented in matrix form, where the element (i, j) depicts the
inter-agent potential between agent i and j.


0
0.6125 0.9415 0.6978


0.6125

0
0.5963
0.8941
1 T


=
P
ij
0
ij
0.9415 0.5963
0
c∗
ij
0
0.1695


0.6978 0.8941 0.1695
0
59
5 Experimental results
This matrix is of particular interest, since the planar trajectory plots can only tell,
which agents are initially connected (at t = 0). However, no statements about proper
connectivity can be made. In the following, this matrix of normalised inter-agent
potentials is presented as colour coded block image (as in Figure 5.7) to give a quick
impression of which agents are properly connected.
1
0.000
0.612
0.941
0.697
2
0.612
0.000
0.596
0.894
3
0.941
0.596
0.000
0.169
4
0.697
0.894
0.169
0.000
1
2
3
4
Figure 5.7: Normalised inter-action potentials at the initial state t = 0. For any
considered group of agents, all links with an initial potential smaller than 1 are
proper links. Thus the initial graph is proper complete.
The resulting trajectories are shown in Figure 5.8. For fmax = 2.3 no link breaks
occur and the complete graph structure is preserved. Velocity consensus is finally
achieved. Setting fmax = 4.5 however leads to loss of complete graph interconnection
and finally even detaches the leaders from the followers completely.
5.2.4
Complete graph target position
Leading the group of agents to a target position, while keeping complete connectivity
among the agents, works basically same as the previous target velocity case. The
only difference is that a damping term is added to the relative damping protocol
to guarantee convergence to the common target position (see Theorem 3.3). With
respect to this inter-agent dynamics, the bound on fmax calculates in the same way
as before to fmax ≤ 2.649. Again only one leader agent is considered which relaxes
the condition on fmax . All initial conditions are chosen in accordance with the
invariant set Ωc∗ (see Figure 5.10). The target position to be reached is xd = [1.5, 1]T .
The simulations shown in Figure 5.9 again demonstrate that the graph structure is
preserved if fmax is chosen within the given bound. Otherwise the graph structure
(and even connectivity) is lost. Of course this happens not in general, since all
conditions provided in the work at hand are sufficient conditions.
60
5.2 Simulations
1
x2
0.5
0
−0.5
−1 −0.8−0.6−0.4−0.2 0
0.2 0.4 0.6 0.8
x1
1
1.2 1.4 1.6 1.8
2
1
1.2 1.4 1.6 1.8
2
(a) fmax = 2.3
1
x2
0.5
0
−0.5
−1 −0.8−0.6−0.4−0.2 0
0.2 0.4 0.6 0.8
x1
(b) fmax = 4.5
Figure 5.8: Evolution of N = 4 agents with suitable initial conditions in a proper
complete graph structure. Consensus towards a target velocity vd = [0, 2]T under
preservation of the complete graph structure is achieved for a properly bounded goal
attraction force fmax .
61
5 Experimental results
1.5
x2
1
0.5
0
−1 −0.8−0.6−0.4−0.2 0
0.2 0.4 0.6 0.8
x1
1
1.2 1.4 1.6 1.8
2
1
1.2 1.4 1.6 1.8
2
(a) fmax = 2.6
1.5
x2
1
0.5
0
−1 −0.8−0.6−0.4−0.2 0
0.2 0.4 0.6 0.8
x1
(b) fmax = 4.5
Figure 5.9: Evolution of N = 4 agents with suitable initial conditions in a proper
complete graph structure. Consensus towards the target position xd = [1.5, 1]T under
preservation of the complete graph structure is achieved for a properly bounded goal
attraction force fmax .
62
5.2 Simulations
1
0.000
0.048
0.870
0.983
2
0.048
0.000
0.612
0.995
3
0.870
0.612
0.000
0.959
4
0.983
0.995
0.959
0.000
1
2
3
4
Figure 5.10: Normalised inter-action potentials at the initial state t = 0 in the target
position consensus scenario.
i
group
x i0
vi0
1
2
3
4
N1
[0, 0]T
[a1 , −0.02]T
[a1 , 0.02]T
[a2 , 0]T
[0, c]T
[b1 , c]T
[b1 , c]T
[b2 , c]T
Nc
N2
Table 5.1: Initial conditions and definition of agent groups with constants a1 ≈ 0.74,
a2 ≈ 1.74, b1 ≈ 0.97 and b2 ≈ 0.63. c = 0.75 for visualisation purposes.
5.2.5
Complete connector graph
The results shown so far were based on an initial complete graph structure. This
requirement is now removed and the effectiveness of connector agents to maintain
connectivity among incomplete subgraphs is demonstrated. The general approach
for this laid out in Section 4.4 treats the case of a network consisting of follower
agents only.
For the present simulations, a damping value of γ = 1.457 is chosen. Table 5.1 presents
initial conditions of the agents and definition of the different groups. To guarantee
persistent connectivity of the whole graph, the conditions of Theorem 4.11 have to
be fulfilled. Firstly, each union of agent group N k with the group of connectors N c
has to be properly completely connected. This can easily be verified by calculating
the normalised initial inter-agent potentials of these subgroups as in Figure 5.11.
Secondly, setting N ĉ = N c ∪ N 1 gives an upper bound on the total number of agents
N ≤ 4. Thus all sufficient conditions for connectivity preservation are met in this
scenario. Successful position consensus under connectivity preservation can be seen
in Figure 5.12.
Consider in contrast the following scenario with agent i = 4 duplicated (and
63
5 Experimental results
1
0.000
0.999
0.999
2
0.000
0.001
0.998
2
0.999
0.000
0.001
3
0.001
0.000
0.998
3
0.999
0.001
0.000
4
0.998
0.998
0.000
1
2
3
2
3
4
(a) N c ∪ N 1 = N ĉ
(b) N c ∪ N 2
Figure 5.11: Initial inter-agent potentials of all possible group pairs. Every union of
connector agents plus another group is properly completely connected.
0.6
x2
0.4
0.2
0
−0.2
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x1
Figure 5.12: Successful convergence of N = 4 agents in the connector setup. Agents
denoted by a diamond are connector agents.
slightly moved in y-direction for visualisation purposes) as presented in Table 5.2.
Everything else remains unchanged, fulfilling again all requirements (see Figure 5.13).
However, the total number of agents is now N = 5 > 4 and therefore condition 2
of Theorem 4.11 is violated. Loss of graph connectivity is apparent in Figure 5.14,
the properly completely connected graph N ĉ = N c ∪ N 1 is broken by the strong
influence of the agent group N 2 .
5.2.6
Complete subgraphs with leaders
Let us now look at the case of a non-complete graph of leader and follower agents.
The basic topology investigated in this subsection consists of a complete follower
subgraph, a complete leader subgraph and the connector subgraph, that is as well
complete and contains some follower and all leader agents.
Sufficient conditions to achieve convergence of all agents towards a common target
64
5.2 Simulations
i
group
xi 0
vi0
1
2
3
4
5
N1
[0, 0]T
[a1 , −0.02]T
[a1 , 0.02]T
[a2 , 0.02]T
[a2 , −0.02]T
[0, c]T
[b1 , c]T
[b1 , c]T
[b2 , c]T
[b2 , c]T
Nc
N2
Table 5.2: Initial conditions and definition of agent groups with constants a1 ≈ 0.74,
a2 ≈ 1.74, b1 ≈ 0.97 and b2 ≈ 0.63. c = 0.75 for visualisation purposes.
2
0.000
0.001
0.999
0.998
1
0.000
0.999
0.999
3
0.001
0.000
0.998
0.999
2
0.999
0.000
0.001
4
0.999
0.998
0.000
0.001
3
0.999
0.001
0.000
5
0.998
0.999
0.001
0.000
2
3
1
2
c
3
1
(a) N ∪ N = N
ĉ
4
c
(b) N ∪ N
5
2
Figure 5.13: Initial inter-agent potentials of all possible group pairs. Every union of
connector agents plus another group is properly completely connected.
0.6
x2
0.4
0.2
0
−0.2
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x1
Figure 5.14: Evolution of N = 5 agents in the connector setup. Agents denoted by a
diamond are connector agents. Connectivity of the graph is lost since the sufficient
conditions are not met.
65
5 Experimental results
position for this setup are stated by Theorem 4.13. Compared to the afore mentioned
cases preserving a complete graph structure, the conditions for this setup are more
complex. Some careful design procedure is required to determine parameters that
describe a particular scenario in which all requirements of the theorem are met.
Since the requirements rely on parameters that in turn depend on each other, the
following design sequence is proposed to choose them in a practical and straight
forward way. Remember that all connector-based approaches do not use velocity
alignment between agents, but instead have an absolute damping term.
1. Choose the number of follower agents N f and a value for the absolute damping
γ. By (4.36) this puts an upper bound on the number of leaders in the network.
Adapt both mentioned parameters to receive a desired bound on the number
of leaders and set N l in accordance.
2. Having fixed the number of leaders and followers, it then is natural to form
the group of connector agents N c . Since all leader agents belong to the group
of connectors by assumption, it simply remains to select some followers N fc
that should belong to that group as well. The cardinality of that group Nfc
is the last remaining Figure that is needed to determine the bound on fmax .
Adapt Nfc such that this bound calculated by (4.48) is positive.
3. The key to guaranteed connectivity preservation is again to keep initially complete subgraphs complete, i.e. all follower as well as all connector agents should
initially be interconnected by a proper complete graph. Finally ensure that
initial conditions of all agents (respectively their initial inter-agent distances)
are from the according invariant sets. Hence respect the sets Ωc∗f and Ωc∗c for
follower’s respectively connector’s initial conditions.
In practice it turns out that the number of connector agents Nc = Nl + Nfc must
always be similar to the number of followers to render the bound on fmax positive.
Figure 5.16 shows a simulation of such a scenario of N = 10 agents. A total
number of Nf = 7 follower agents and a damping factor of γ = 9 permits Nl = 3
leader agents and make 4 of the follower agents connector agents. This leads to a
bound on the maximum goal attraction force of fmax ≤ 1.012. For the simulation
fmax = 1 is chosen and the target position is set to xd = [0.4, 0.4]T .
The sensing ranges of all leaders at their initial positions are shown to highlight that
the initial interconnection graph is not complete. It is easy to spot the group of
66
5.2 Simulations
1
0.000
0.249
0.095
0.710
0.821
0.945
0.997
2
0.249
0.000
0.515
0.233
0.899
0.495
0.911
3
0.095
0.515
0.000
0.878
0.544
0.973
0.787
4
0.710
0.233
0.878
0.000
0.773
0.083
0.703
5
0.821
0.899
0.544
0.773
0.000
0.528
0.067
6
0.945
0.495
0.973
0.083
0.528
0.000
0.426
7
0.997
0.911
0.787
0.703
0.067
0.426
0.000
1
2
3
4
5
6
7
(a) N f
4
0.000
0.773
0.083
0.703
0.966
0.706
0.971
5
0.773
0.000
0.528
0.067
0.803
0.968
0.752
6
0.083
0.528
0.000
0.426
0.525
0.375
0.562
7
0.703
0.067
0.426
0.000
0.548
0.634
0.423
8
0.966
0.803
0.525
0.548
0.000
0.085
0.082
9
0.706
0.968
0.375
0.634
0.085
0.000
0.094
10
0.971
0.752
0.562
0.423
0.082
0.094
0.000
4
5
6
7
8
9
10
(b) N c = N l ∪ N fc
Figure 5.15: Initial inter-agent potentials of followers and connectors (consisting of all
leaders plus some followers). Both these groups are properly completely connected.
67
5 Experimental results
0.8
0.6
0.4
x2
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−0.8 −0.6 −0.4 −0.2
0
x1
0.2
0.4
0.6
0.8
1
Figure 5.16: Evolution of N = 10 agents in an initially connected but not complete
graph structure. Sufficient conditions for complete subgraphs with leaders are
fulfilled, consensus towards the target position xd = [0.4, 0.4]T under preservation of
the connector graph structure is achieved.
68
5.2 Simulations
connector agents; namely the leader agents plus the four followers in their marked
sensing ranges. Links exist also between all the follower agents. Additionally, all
inter-agent distances are well chosen according to the requirements, which is shown
in Figure 5.15. The evolution of the trajectories shows characteristic behaviour: All
agents first gather in some common center region and then slowly progress towards
the common goal. This characteristic is typical for the approach of keeping complete
subgraphs together with the help of connectors. It is based on rather restrictive
conditions to be met and can be observed similarly for different parameter values
and initial conditions.
In particular, the absolute damping protocol is responsible for this behaviour. Lacking
a velocity alignment term makes position alignment the dominant effect. The rather
high value for the absolute damping γ reinforces this effect since all non-positionconsensus-seeking velocity components are immediately damped to zero. Furthermore,
the goal attraction force is required to remain quite low. Because of that, a significant
effect of the goal attraction is not observed before the effort towards position consensus
has sufficiently ebbed away.
This discussion and a glimpse on the simulation indicate the conservativeness
of the sufficient conditions that are posed for the connector agent case. The slow
convergence towards the target position is one special drawback of these restrictions.
Figure 5.17 for example shows a simulation of the same scenario with the target
position and the maximum allowed goal attraction force changed to xd = [0.9, 0.7]T
and fmax = 5. The entire group of agents still achieves convergence towards the goal
and even the follower and connector subgraphs remain complete at all times, i.e. all
objectives are met just as before.
Even though the sufficient conditions are rather conservative, they are a powerful
tool to guarantee connectivity preservation of an initially non-complete graph and
convergence of the whole group towards an arbitrary position goal. Observe for
example the simulation in Figure 5.18, where only a minority of the follower agents
is linked initially to the leader agents. The achievement of the objective is still
guaranteed.
Removal of the velocity alignment term in the consensus protocol (thus in the
control law of each agent) can also be seen as an advantage. In the (technical)
realisation of the system, each agent needs simply to measure the distance to its
neighbouring agents and no longer their velocity. When it comes to choosing sensor
hardware, this can be a major advantage.
69
5 Experimental results
0.8
0.6
0.4
x2
0.2
0
−0.2
−0.4
−0.6
−0.8
−1 −0.8 −0.6 −0.4 −0.2
0
x1
0.2
0.4
0.6
0.8
1
Figure 5.17: Evolution of N = 10 agents in an initially connected but not complete
graph structure. Conditions of the complete connector case are fulfilled, apart from
the bound on fmax . Nevertheless position consensus towards xd = [0.9, 0.7]T is
achieved.
1.2
1
x2
0.8
0.6
0.4
0.2
0
−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2
x1
0
0.2
0.4
0.6
0.8
Figure 5.18: Evolution of N = 13 agents in an initially connected but not complete
graph structure. Conditions of the complete connector case are fulfilled. Position
consensus towards xd = [0, 0.9]T is achieved with a only minority of follower agents
connected to the group of leaders.
70
6
Conclusion and Outlook
This chapter summarises the main results of the work, draws a comparison of the
presented approach to the case of single-integrator dynamics and finally proposes
some directions for further research.
6.1
Conclusions
The main goal of this work was to elaborate guaranteed consensus of multiagent
networks to a pre-specified target induced by several leader agents. Inspired by [13],
sufficient conditions on parameters and initial conditions of the leader-follower
network were found to achieve the goal. The main effect of these conditions is
to preserve connectivity of the interconnection graph of the network – the basic
assumption of all consensus proofs. In contrast to previous work, maintenance of
the connectivity is not achieved directly through a special control law, but rather
indirectly by restrictions on the initial configuration of the network. The underlying
connectivity analysis framework is therefore particularly interesting for analysis
purposes of existing multi-agent systems.
The main contributions of this work can be seen as three levels building upon
each other to achieve the overall goal: convergence of the leader-follower network, a
71
6 Conclusion and Outlook
sufficient conditions
for connectivity maintenance
connectivity analysis framework
leader-follower consensus towards target
Figure 6.1: The main goal of this work is established in three levels.
connectivity analysis framework and through this, conditions for a variety of network
configurations that establish the goal. These main contributions and according
theorems are recapitulated in the following.
The base is laid by extending the well known consensus protocol for doubleintegrator agents to leader-follower networks. Two cases are worthwhile to be
examined in the context of double-integrator dynamics, namely consensus towards
a target velocity and a target position. Theorem 3.2 respectively Theorem 3.3
confirm convergence of the entire group of agents towards the leader-induced target.
For this however, it is still assumed that the communication graph is connected at
all times. As mentioned above, this is the core issue tackled by this work.
Being equipped with a control protocol shown to achieve target velocity/position
consensus, the matter of guaranteed connectivity is approached on the second level.
Chapter 4 elaborates a framework analysing connectivity in the concerned networks.
This framework is the main contribution of the study at hand and is used later to
synthesise sufficient conditions for a variety of initial network configurations.
The core concept of the framework is to formulate inter-agent dynamics and to regard
complete subgraphs of the network. Connectivity preservation is guaranteed for all
inter-agent distances from the set Ωc∗ – hence Ωc∗ is both invariant under the system
dynamics of the entire network and bounded in such a way that all agents with initial
inter-agent distances from that set remain connected. Theorem 4.3 defines the set
Ωc∗ and provides conditions to render it invariant for a very general formulation of
the problem in question.
The topmost level finally applies the proposed connectivity analysis framework to
synthesise conditions that ensure connectivity maintenance for various initial network
configurations. Together with the consensus results from the first level, the overall
objective of this work is achieved for several network configurations.
The configuration assumed in Theorem 4.9 is an initially properly completely
72
6.2 Relation to single-integrator networks
connected network of agents, it is therefore a complete graph of leaders and followers
with all inter-agent distances chosen from the set Ωc∗ . Naturally, the goal attraction
force applied on the leaders has to be bounded to ensure preservation of (complete)
connectivity. This is the only additional requirement stated by this theorem.
The assumption of an initial complete graph structure is relaxed by introduction
of connector agents in Theorem 4.11. This theorem studies a follower network
consisting of several complete subgraphs. These complete subgraphs are each analysed individually resulting in conditions, that ensure connectivity preservation and
convergence of the entire initially incomplete graph.
Initially stated in a general and leaderless fashion, the idea of connector agents is
refined and applied to a leader-follower network that is not completely connected.
Theorem 4.13 hence formulates conditions for connectivity maintenance for the
least restrictive initial configuration in terms of connectivity (namely completeness
only of the subgraphs of leaders and followers).
This lengthy list of theorems demonstrates the success of the work at hand in
fulfilling the specified goal for a variety of network configurations. Following the
sufficient conditions on initial conditions, the ratio of leaders and followers and the
magnitude of the goal attraction force guarantees convergence of all agents to the
common target. Restricting these values is quite intuitive and has its counterpart in
the single-integrator case, which is discussed below.
The effectiveness and validity of the theoretical results is shown in various simulations
for agents evolving in the plane.
6.2
Relation to single-integrator networks
The idea of sufficient conditions for connectivity maintenance and consensus in leaderfollower networks was first presented by [26] and [13]. These works on leader-follower
networks of single-integrator agents served as a starting point for the work at hand.
Although the approach and the connectivity analysis frameworks differ substantially
in the case of single- respectively double-integrator dynamics, the resulting sufficient
conditions exhibit surprising analogies. This section is devoted to highlight parallels
of the two problems.
Extension of the applied standard consensus protocol to leader-follower interaction
is straightforward in both cases. Negative (semi-) definiteness of the derivative of the
networks collective potential is guaranteed by the properties of the goal attraction
function. In both cases, a generic goal attraction function with similar properties is
73
6 Conclusion and Outlook
presumed.
In [13], two network configurations are investigated: The complete graph case and
the incomplete graph case (consisting of complete leader and follower subgraphs).
Their basic idea for the proposed connectivity analysis framework is to inspect the
dδ 2
derivative of the quadratic distance δij = |xi − xj | of two connected agents, i.e. dtij .
Connectivity preservation of existing links is achieved by requiring this derivative to
be smaller than or equal to zero at the braking distance δij = ∆. Such an approach
is not suitable for double-integrator dynamics. This is because the braking condition
can no longer be expressed position dependent only, but must take velocity of the
agents into account as well. On these grounds, the invariant set Ωc∗ of inter-agent
distances (of position and velocity) is introduced, to limit "inter-agent energy" of
connected agents that does not allow those links to brake.
Let us first compare the case of an initial complete interconnection graph structure
(see [13, Theorem 3] v. Theorem 4.9). In the single-integrator case, it is only the
conditions on the goal attraction function, that ensure connectivity maintenance.
These are an upper bound as well as increasing monotony. The upper bound is, as
in the double-integrator case, dependent on the number of agents in the network.
Monotony is used to show that the goal attraction force always brings two leaders
closer to another – a fact that does not hold for double-integrator agents. In addition
to that, in the case of double-integrators, initial complete connectivity does not
suffice to keep the network together. As pointed out above, the initial conditions
of the agents (incorporating position and velocity) have to be chosen appropriately.
This fact is captured by the notion of proper (complete) connectivity.
Analysis of complete subgraphs naturally becomes more involved in both cases
(see [13, Theorem 4] v. Theorem 4.12, Theorem 4.13). Analogous to the previous
discussion, in the double-integrator case the complete subgraphs have initially to be
properly completely connected. Naturally, a bound on fmax must be applied in both
cases.
Leader-leader connectivity in the single-integrator case can be established by a
specific shape of the goal attraction force (namely being a convex function). "A lower
bound on the number of links connecting the leader and follower subgraphs" [13]
has also to be complied. The correspondent requirement in the double-integrator
case is the number of connector agents. The concept of connector agents (which
form a complete graph) is necessary to apply the connectivity analysis framework to
leader-follower links; thus guaranteeing preservation of initial leader-follower links by
keeping their inter-agent distances inside an appropriate invariant set Ωc∗ .
74
6.3 Future work
6.3
Future work
The results of this thesis demonstrate the practicability of sufficient conditions for
inherent connectivity preservation. Although this work delivers sufficient conditions
for a variety of initial network configurations, there is room for further investigation.
A major improvement would be to adapt the presented connectivity analysis
framework in such a way that it is capable of analysing connectivity among agents
which are not arranged in a complete (sub-)graph. This would allow study of
a much broader class of initial network configurations. However, this requires a
completely different approach, since incomplete (sub-)graphs do not exhibit the
practical properties exploited in the present approach.
Assuming there was such an analysis framework not relying on the linear inter-agent
dynamics, one could also go ahead and again apply an arbitrary inter-agent potential
Uij . This would widen the approach to capture flocking motion of agents.
In cooperation goes study of the incomplete graph case and velocity consensus. In
the work at hand, the absolute damping control has been applied to achieve position
consensus from an initially incomplete graph structure. Extension of these results
to the case of a target velocity is not trivial, since the influence of additional agent
links can then not be bounded (or only very conservatively).
The connectivity analysis framework at hand should generalise quite straight
forward to capture any form of additional disturbances such as network communication delays, sensor measurement errors or quantisation effects. This is, because
the ISS approach applied to establish invariance of the set Ωc∗ naturally guarantees
invariance for bounded inputs influencing inter-agent dynamics. Artefacts of the
mentioned kind can be included to the respective term αij . Generalisation of the
underlying consensus proofs however has to be regarded separately.
Another direction of future research is the application of the proposed scheme to
different control protocols and control objectives.
75
A
Extensions concerning higher dimensions
In this appendix various extension that are necessary to extend the results of this
work to higher dimensions are put together. Let as before xi = [xi1 , ..., xin ]T ∈ Rn
and vi = [vi1 , ..., vin ]T ∈ Rn be the position and velocity of agent i. The stack vector
form is used to put all agent’s positions and velocities in two vectors x and v, with
x = [x11 , ..., x1n , ..., xN1 , ...xNn ]T ∈ RN n and v = [v11 , ..., v1n , ..., vN1 , ...vNn ]T ∈ RN n .
This form of stacking is also used for any other variable shared by all agents.
Use in the following the notation vx , vy , ... to refer to all components of a stacked
vector v in (x, y, ...) direction, with (x, y, ...) each denoting one of the n directions of
the regarded space Rn .
A.1
Kronecker product
Many of the following extensions are based on the Kronecker product of two matrices
A = aij and B = bij denoted by A ⊗ B. Let A and B be of size n × m and
p × q, then their Kronecker product is the np × mq matrix with the block structure:
A ⊗ B = (aij B). Refer to [41] for a precise definition and properties.
Notice for example that the regarded linear consensus protocol (3.3) for double-
77
A Extensions concerning higher dimensions
integrator agents (3.1)
ẋi = vi
!
v̇i = −
X
(xi − xj ) + γ(vi − vj )
,
∀i ∈ N
(A.1)
j∈Ni
can compactly be rewritten for the whole network using stack vector form and the
Laplacian L to
! "
!
#
!
ẋ
0
I
x
=
⊗ In
.
(A.2)
v̇
−L −γL
v
This Kronecker multiplication with the identity matrix decomposes the system into
its linear, decoupled subsystems in n directions.
A.2
Target velocity consensus
To apply LaSalle’s invariance principle, notice that any sums of the following form
can be rewritten
!
N
X
X
viT γ(vi − vj ) = −γv T (L ⊗ In )v.
(A.3)
i=1
j∈Ni
Demanding this term to be zero yields
0 = −γv T (L ⊗ In )v = −γvxT Lvx − γvyT Lvy − ...
(A.4)
Exploit the properties of L as in the one-dimensional case to get vi1 = vj1 , ..., vin =
vjn ∀i, j ∈ N and thus equality of all vectors vi .
A.3
Target position consensus
First note that the goal attraction force f is a so called central force [42], since
according to Definition 3.5 it can be written as
f (sd ) = −
78
dF sd
dksd k ksd k
(A.5)
A.4 Connectivity analysis via Lyapunov equation
and hence the goal potential F (ksd k) is indeed a potential for the conservative field
f (sd ). The integral
Z
sd
−f (ξ)dξ
(A.6)
0
is therefore path independent and evaluates to F (ksd k), which is according to the
definition of F and the positive definiteness of the Euclidean norm again positive
definite w.r.t. sd .
For applying the LaSalle argument later in the proof, analogously to the proof of
target velocity consensus rewrite
−
!
N
X
X
i=1
j∈Ni
viT γ(vi − vj ) + γviT vi
= −γv T (L ⊗ In )v − γv T v
(A.7)
and solve
0 = −γv T (L ⊗ In )v − γv T v
= −γvxT Lvx − γvyT Lvy − ... − γvxT vx − γvyT vy − ...
(A.8)
with similar arguments by vi1 = 0, ..., vin = 0 ∀i ∈ N and thus v = 0 ∀i ∈ N .
Following the procedure of the original proof, let η̄j ∗ j ∗ = dksdFj∗ k ksj1∗ k and thus
∇sj∗ F (ksj ∗ k) = η̄j ∗ j ∗ sj ∗ ∀j ∗ ∈ N l for which f (sj ∗ ) > 0. This yields
0 = −[(L + η̄) ⊗ In ]s


(L + η̄)sx



= − (L + η̄)sy 

..
.
(A.9)
which as in the one-dimensional equivalent results in sx = sy = ... = 0 and therefore
implies s = 0.
A.4
Connectivity analysis via Lyapunov equation
Start from the specific inter-agent dynamics of the form (4.3) and formulate the
higher dimensional equivalent as
˙ij =
ẋij
v̇ij
!
!
0
= [A ⊗ In ]ij +
.
αij
| {z }
(A.10)
ᾱij
79
A Extensions concerning higher dimensions
The matrix [A ⊗ In ] remains Hurwitz, since the eigenvalues of a Kronecker multiplication are all combinatorial products of the eigenvalues of both matrices alone.
Thus in the case at hand, only the multiplicity of the two negative eigenvalues of A
is multiplied by n. Use the Lyapunov equation
[A ⊗ In ]T P̃ + P̃ [A ⊗ In ] = −[Q ⊗ In ]
(A.11)
which still has a unique positive definite solution P̃ for every positive definite matrix
Q. A closer look reveals that P̃ = [P ⊗ In ] and thus all further results hold as in
the one-dimensional case (e.g. same eigenvalues of the incorporated matrices). This
is again due to the decomposition property of a Kronecker multiplication with the
identity matrix.
A.5
Critical region estimate
It needs to be clarified how the structure of the additional term ᾱij can be exploited
to estimate the maximum value of its norm in higher dimensions. In this case,
estimation (4.10) needs a more careful examination due to αij ∈ Rn with n > 1.
V̇ij = −Tij [Q ⊗ In ]ij + 2Tij [P ⊗ In ]ᾱij
"
!
#
!
P
P
0
1
2
= −λmin (Q)kij k2 + 2 xTij vijT
⊗ In
P2 P3
αij
!
Pα
2 ij
= −λmin (Q)kij k2 + 2 xTij vijT
P3 αij
"
!#
kP α k 2
2 ij
≤ −λmin (Q)kij k kij k −
λmin (Q) kP3 αij k p
P22 + P32
≤ 0 for kij k ≤ 2kαij k
λmin (Q)
(A.12)
The crucial step in order to perform this estimation is to take the norm of sub-elements
of the vector
! !
P α kP α k 2 ij 2 ij
(A.13)
=
.
P3 αij kP3 αij k 80
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