Multi-agent Consensus using Generalized Cyclic Pursuit Strategies A Thesis Submitted For the Degree of Doctor of Philosophy in the Faculty of Engineering by Arpita Sinha Department of Aerospace Engineering Indian Institute of Science BANGALORE – 560 012 July 2007
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Multi-agent Consensus using Generalized CyclicPursuit Strategies
2.9 Trajectories of a team of 5 agents with all positive gains (Case I) . . . . . 36
2.10 Trajectories of a team of 5 agents with one gain zero and all other gainspositive such that Theorem 2.2 is satisfied (Case II) . . . . . . . . . . . . 37
2.11 Trajectories of a team of 5 agents with the gain of first agent is negativeand other gains positive such that the gains satisfy Theorem 2.2 (Case III) 37
2.12 Trajectories of a team of 5 agents with the gain of first agent is negativeand other gains positive, such that Theorem 2.2 is not satisfied and theagents do not converge to a point (Case IV). . . . . . . . . . . . . . . . . 38
2.13 Trajectories of a team of 5 agents with two negative gains. Theorem 2.2is not satisfied and the agents do not converge to a point (Case V). . . . 39
3.2 Trajectories of a swarm of 12 agents when all gains positive (Case I) . . . 64
3.3 Trajectories of a swarm of 12 agents when the gain of one of the agent isnegative while the others are positive (Case II) . . . . . . . . . . . . . . . 64
3.4 Invariance property of the reachable point, Zf = (−16.47, 2.52) for a givenbasic pursuit sequence and different weights (Case III) . . . . . . . . . . 65
3.5 Invariance property of the reachable point, Zf = (−16.47, 2.52) for a givenweight and different basic pursuit sequences (Case IV) . . . . . . . . . . 66
4.1 Trajectories of a swarm of 12 agents when the gains of all the agents arepositive (Case I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Trajectories of a swarm of 12 agents when the gain of one of the agent isnegative and the other gains are positive such that Theorem 4.2 is satisfied(Case II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3 Trajectories of the agents converging to Zf = (60, 60, 60) (Case III) . . . 87
4.4 Trajectories of the agents converging to Zf /∈ Co(Z0) (Case IV) . . . . . 88
LIST OF FIGURES xvii
4.5 Trajectories of the agents under centroidal cyclic pursuit (CCP) and gen-eralized centroidal cyclic (GCCP) (satisfying some properties) demon-strating the pursuit sequence invariance of the rendezvous point Zf =(64.3, 41.3, 58.7) (Case V) . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6 Trajectories of the agents under CCP and GCCP demonstrating that therendezvous point is not pursuit sequence invariance (Case VI) . . . . . . 90
5.1 The trajectories of 5 agents when the gains of the agents satisfies Condition(i) of Theorem 5.2 (Case I) . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2 The trajectories of 5 agents when the gains of the agents satisfies Condi-tion(ii) of Theorem 5.2 (Case II) . . . . . . . . . . . . . . . . . . . . . . . 109
5.3 The trajectories of 5 agents when the gains of the agents satisfies Condi-tion(iii) of Theorem 5.2 (Case III) . . . . . . . . . . . . . . . . . . . . . . 109
5.4 The trajectories of 5 agents when the gains of the agents satisfies none ofthe conditions of Theorem 5.2 (Case IV) . . . . . . . . . . . . . . . . . . 110
5.7 Trajectories of a swarm of agents when one gain is negative and Theorem5.5 is satisfied (Case VII) . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.8 Trajectories of a swarm of agents when three gain are negative and The-orem 5.5 is not satisfied (Case IX) . . . . . . . . . . . . . . . . . . . . . . 114
5.9 Directed motion with combination of stable and unstable gains . . . . . . 115
6.4 Angle calculation for a general polygon of n sides . . . . . . . . . . . . . 122
6.5 Representation of the range of φ in polar coordinate . . . . . . . . . . . . 124
6.6 A representation of the ranges of φ and ρ for different agents . . . . . . . 124
6.7 Range of ρi for Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.8 Trajectories of n = 5 agents for Case I ( • - initial position, N - finalposition of the UAVs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.9 Range of ρi for Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
LIST OF FIGURES xviii
6.10 The roots of (6.19) for Cases II, III, and IV . . . . . . . . . . . . . . . . 131
6.11 Trajectories of n = 5 agents for Case II (• - initial position, N - finalposition of the UAVs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.12 Range of ρi for Case III . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.13 Trajectories of n = 5 agents for Case III (• - initial position, N - finalposition of the UAVs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.14 Range of ρi for Case IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.15 Trajectories of n = 5 agents for Case IV (stable equilibrium) (• - initialposition, N - final position of the UAVs) . . . . . . . . . . . . . . . . . . 134
6.17 Roots of (6.19) for Case V . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.18 Trajectories of n = 5 agents for case V for different initial conditions (• -initial position, N - final position of the UAVs) . . . . . . . . . . . . . . . 135
7.1 Trajectories of the agents with speed saturation . . . . . . . . . . . . . . 141
7.2 Trajectories of the agents with appropriate selection of gains such thatthe speed do not saturate and rendezvous occur at Zf = (0, 0) . . . . . . 143
7.3 Trajectories of the agents with fixed turn rate . . . . . . . . . . . . . . . 144
7.4 Rendezvous of the agents with fixed turn rate and speed saturation . . . 145
7.5 Directed motion of the agents with fixed turn rate and speed saturation . 146
7.6 Circular motion of the agents with latax saturation . . . . . . . . . . . . 147
A − Matrix representing the equation of motion of the agents.
ai − Lateral acceleration of the non-holonomic agent i.
Co(Z0) − Convex hull of a Z0.
Cp − Finitely generated cone
d − Dimension of the space.
ki − Controller gain of agent i.
ki − Lower bound on ki.
K − Gain matrix.
n − Number of agents.
P(Z0) − Reachable set.
Rn − n-dimensional space.
Rp − pth eigenvalue of A.
ui − Control of the holonomic agent i.
vi − Unit velocity vector of the holonomic agent i.
vmi − Maximum speed of holonomic agent i.
Vi − Constant speed of non-holonomic agent i.
w − Set of weights for computing the centroid in CCP and GCCP.
xi(t) − Position of the agent i at time t along a given axis.
Zi(t) − Position of the agent i at time t in d-space.
Z0 − Set of initial positions of the agents.
Zf − Rendezvous/reachable point.
xix
Notation and Abbreviations xx
Zic − The weighted centroid that agent i follows.
Zf (Z0) − Reachable set.
Γ − Set of all possible weights.
Γ − Subset of Γ.
ξij − Elements of Adj(χ)
χ − Pursuit sequence matrix.
ηi − Weight for calculating the centroid in CCP and GCCP.
φ − Deviation between the LOS and the non-holonomic agent orienntation.
ρ − Radius of the circle traversed by non-holonomic agent i (Chaper 6).
ρ(s) − Characteristic equation of A
θγδ − Direction of motion of the holonomic agent i in (γ, δ)−plane.
Adj(A) − Adjoint of A.
Rank(A) − Rank of A.
trace(A) − Trace of A.
BCP − Basic cyclic pursuit.
BPS − Basic pursuit sequence.
BPS − Set of all possible basic pursuit sequences.
CCP − Centroidal cyclic pursuit.
GCCP − Generalized centroidal cyclic pursuit.
PS − Pursuit sequence.
LOS − Line of sight.
Chapter 1
Introduction
Multi-agent systems are groups of intelligent agents, interacting with each other and
the environment, to accomplish certain tasks that are difficult to achieve by a single
agent. Multi-agent systems are characterized by limited information gathering and pro-
cessing capability of each agent, decentralized control, and asynchronous computation.
Research on multi-agent systems started in the late 1980s as a subfield of Distributed
Artificial Intelligence. Presently, multi-agent system theory is applied widely from au-
tonomous vehicles to e-commerce. The advantages of multi-agent systems are well-
established in the literature. Firstly, multi-agent systems are robust. Since each agent
is autonomous, multi-agent systems degrade gracefully with agent failure. They are
also scalable. Another advantage, mainly from the robotics research point of view, is the
“performance/cost ratio”. A single robot is much costlier than many simple robots. This
is also true in the case of sensor networks. However, multi-agent systems also introduce
many new challenges like the coordination and communication that should exist between
the agents to perform a task. The issues of information exchange and design of control
strategies for coordination of agents are current topics of research interest.
1
Chapter 1. Introduction 2
1.1 Multi-agent consensus
A major part of the researches on multi-agent systems, with reference to autonomous
agents, are related to the consensus problem. Consensus, as defined in [1], implies
reaching an agreement regarding a certain quantity of interest that depends on the
states of all the agents. The quantity of interest can be position, direction of motion, the
relative distance between the agents, or some other functions of the states. Consensus of
multi-agent systems can be found in nature, e.g., flock of flying birds, schools of fishes or
herd of land animals. Reynolds [2] and Viscek et al. [3] are pioneers in modeling these
types of behaviours, which were later used in achieving consensus of groups of agents
like robots, UAVs, or satellites.
To reach a consensus, each agent in a multi-agent system should have information
about certain state(s) of all or some of the agents. The flow of information defines
the interaction topology in multi-agent systems, which can be represented by a directed
graph G = (V,E) where V = {1, 2, · · · , n} represents the agents and E ⊆ V × V defines
the connection between the agents. Thus, the neighbors of agent i are Ni = {j ∈ V :
(i, j) ∈ E}. In continuous time consensus protocol, if xi(t) denotes the information state
of the ith agent at time t, then the dynamics of agent i is given as
xi(t) =∑j∈Ni
aij(t){xj(t)− xi(t)} (1.1)
where, aij(t) are the time varying weighting factors. For the group of n agents, the
dynamics can be written as x = −Lx where L = [lij], the graph Laplacian, is given as
lij =
−1, j ∈ Ni, i 6= j;
|Ni|, i = j.(1.2)
where, |Ni| is the cardinality of Ni. Consensus is achieved when xi = xf , ∀i, or, in other
words, ||xi − xj|| → 0,∀j 6= i as t →∞.
Chapter 1. Introduction 3
The consensus problems are solved using concepts from algebraic graph theory [4]
and matrix theory [5]. Convergence results are obtained from spectral analysis of the
graph Laplacian. It is shown in [6], [7] that for a time invariant information exchange
topology, the consensus is reached if, and only if, the topology has a spanning tree. The
Fiedler eigenvalue [8] of G gives a measure of the rate of convergence of the consensus
protocols [9].
The consensus protocol becomes more practical and challenging under dynamic in-
formation exchange topology. Jadbabaie et al. [10] show that consensus can be reached
in a switching network if the union of the information exchange graphs is connected most
of the time. This result is further extended in [6], [11], [12]. The other aspects studied in
this problem are reaching consensus under communication delays [9] and under relative
information uncertainties [13]. A comprehensive study of the consensus and cooperative
control of multi-agent system can be found in [1], [14].
The consensus algorithms are used in several applications as described below:
Flocking : Flocking of a group of mobile agents are obtained by aligning the velocities of
all the agents while maintaining a certain distance between them and avoiding collision
with each other and with obstacles. This implies a consensus in the velocity of all the
agents. Jadbabaie et al. [10] proved the convergence results for a group of agents using
distributed control laws, where the information topology changes with time. In [15],
different flocking algorithms that are scalable and has obstacle avoidance capabilities are
proposed. Other flocking algorithms and their stability analysis are addressed in [16],
[17], [18], [19], [20], [21].
Formation control : In formation control problems, the relative positions of the agents are
maintained. It is shown in [1] that the distributed formation control can be considered
as a consensus problem. The different approaches to formation control can be broadly
classified as leader-follower, behaviour-based, and rigid body type formations. These
strategies have been reviewed in [22]. The stability of the formation for leader-follower
and virtual leader strategy has been studied in [23] and [24], respectively. In [25], the
Chapter 1. Introduction 4
effect of communication topology on the stability of agent formation is studied using
the Nyquist criterion. Tabuada et al. [26] obtained the feasibility of motion of a rigid
formation given the kinematics and inter-agent constraints. A lower dimensional control
system is also obtained for a formation to move on a given feasible trajectory.
Rendezvous : Rendezvous of a group of mobile agents implies reaching a consensus in
position of all the agents. Lin et al. [27], [28] considered the agents to move towards
rendezvous through a “stop-and-go” strategy, which can be synchronous or asynchronous.
In [29], [30], [31], the concept of rendezvous of agents include the notion that all the agents
should reach the rendezvous point at the same time. A robust algorithm for rendezvous
of a group of agents under switching communication topology and communication failure
is studied in [32].
Distributed sensor fusion: In a sensor network, the measurements taken by each node
are often corrupted with noise. In [33], [34], distributed Kalman filters are designed that
allow each node of the sensor network to track the average of all sensor measurements.
This is called the consensus filter. The stability properties of this filter is studied in
[34]. In [35], data fusion in the presence of package loss in the communication channel
is discussed.
Coupled oscillators : Synchronization of the frequency of coupled nonlinear oscillators
can be considered as a nonlinear extension of a consensus problem and is analyzed by
linearizing about the equilibrium point. The classic Kuramoto model of coupled non-
linear oscillators assumes identical oscillators and all-to-all connection. In [36], [37], the
stability of the coupled oscillator is studied when the natural frequency of the oscillators
are different and the interconnection between the agents are not all-to-all. The sufficient
conditions for synchronization and desynchronization of the nonlinear coupled oscillator,
in terms of the eigenvalues of the graph Laplacian, is obtained in [38]. Papachristodoulou
et al. [39] studied the synchronization problem under variable time delays.
Chapter 1. Introduction 5
1.2 Cyclic pursuit as consensus protocol
Cyclic pursuit of a group of agents implies that there exists a predefined cyclic connection
between the agents and each agent follows its predecessor, called its leader. The problem
of cyclic pursuit originated from a mathematical study of the path traveled by 3 dogs,
placed at the three vertices of an equilateral triangle, chasing one another in a cyclic order
along the instantaneous line of sight, with constant speed. Edouard Lucas posed this
problem in 1877 [Nouvelles Correspondance Mathematique 3 (1877)] and, in 1880, Henri
Brocard showed that the dogs follow a logarithmic spiral [Nouvelles Correspondance
Mathematique 6 (1880)]. Several researchers [40], [41], [42] generalized this problem
where n bugs are considered and studied the conditions for mutual capture of the bugs.
Mutual capture implies that the bugs reached a consensus in position. Other problems
looked at in the pursuit literature includes the study of evolution of the path traveled by
a trail of ants from one point to another [43], stability of the regular geometries of cyclic
pursuit [44], forward-time (when an agent moves towards its leader) and reverse-time
(when an agent moves away from its leader) cyclic pursuit [45], cyclic pursuit games [46]
where the evader and pursuer moves on a cyclic graph.
Bruckstein et al. [47] studied the evolution of the cyclic pursuits for ants, crickets and
frogs. Ants represent the continuous time cyclic pursuit with varying speeds, while crick-
ets and frogs represent discrete time cyclic pursuit with constant speeds. The possible
outcomes of these pursuits − collision, limit cycle, equilibrium states, periodic motion −are studied. In another paper, Bruckstein et al. [48] studied the linear and cyclic pursuit
on grids, where the ants are allowed to move from one grid point to another.
The application of cyclic pursuit to multi-agent systems was demonstrated by Mar-
shall et al. [49] where two types of agents are considered − holonomic agents that do
not have any motion constraints and nonholonomic agents, like the wheeled robots, that
have turn rate constraints. Cyclic pursuit for holonomic agents give rise to linear cyclic
pursuit and for non-holonomic agents, to nonlinear cyclic pursuit.
Chapter 1. Introduction 6
For linear cyclic pursuit, it is assumed that each agent i knows the position xi+1(t)
of its leader and the pursuit law is given as
xi(t) = κ(xi+1(t)− xi(t)) (1.3)
where, κ is the gain of all the agents. Consensus is reached when all the agents converge
to a point. Marshall et al. [49], [50] proved that for every initial condition, the centroid
of the agents remains constant and the agents exponentially converge to the centroid.
Nonlinear cyclic pursuit is also studied in [49], where each agent is homogenous,
that is, the agents have same speed and gain, and each agent knows the position and
orientation of its leader. Equilibrium is reached when the agents form a stable polygon
in space. This can be considered as a formation control problem. The stability of
the formation is obtained by linearizing the system about the equilibrium point and
evaluating the eigenvalues of the linearized system. In [50], the speeds of the agents
are assumed to be proportional to the distance between an agent and its leader and the
limits on the constant of proportionality (or the gains) are found for stable formation.
These results are experimentally verified in [51].
Lin et al. [52] used linear cyclic pursuit laws to obtain different formations of the
agents, like line formation or triangle formation and derived the conditions for collision
avoidance during rendezvous. They also studied the rendezvous of the agents under
limited field of view. In [7], the feasibility of obtaining different formations of the agents
having motion constraints are discussed. Smith et al. [53] used hierarchical cyclic pursuit
and compared the rate of convergence with the traditional cyclic pursuit scheme. Linear
cyclic pursuit concept was also applied to Euclidean curve shortening [54].
Chapter 1. Introduction 7
1.3 Generalization of cyclic pursuit
Certain generalizations of the basic cyclic pursuit, as described in the literature surveyed
in Section 1.2, is of interest and forms the subject matter of this thesis. In cyclic pursuit, a
group of n agents, ordered from 1 to n, are considered. A cyclic connection exists between
the agents with each agent following its predecessor. The sequence in which each agent
pursues another is called the Pursuit Sequence (PS) of the agents. The basic pursuit
sequence is BPS= {1, 2, . . . , n} which implies that the agents are following each other in
the sequence 1 → 2 → · · · → n → 1. Assume a pursuit sequence BPS={p1, p2, . . . , pn}where, pi ∈ {1, 2, . . . , n}, ∀i and pi 6= pj,∀i, j. This is a generalization in terms of
the pursuit sequence. An agent pi, instead of following the agent pi+1, can follow a
point which is the weighted centroid of the remaining n− 1 agents. Let, the weights be
w = [η1, η2, . . . , ηn−1] where an agent pi associates the weight ηj with the agent pi+j (mod
n) while calculating the centroid. The weight w can be same or different for different
agents. Then, the pursuit sequence of a group of agents are given by (BCP, {wi}ni=1).
With this, the following definitions follow:
Definition 1.1 (Basic cyclic pursuit) If a group of agents follow one another in a
cyclic order, they execute a basic cyclic pursuit (BCP).
For basic cyclic pursuit, the elements of w or the weights ηi, are only 0 and 1, such that
the cyclic structure is preserved. Note that any arbitrary distribution of 0 or 1 may not
preserve the cyclic structure.
Definition 1.2 (Centroidal cyclic pursuit) In a group of agents, if each agent fol-
lows a point that is the weighted centroid of the other agents and the weights used by each
of the agents are the same, then the agents are said to execute centroidal cyclic pursuit
(CCP).
Thus, the weight w is same for all the agents and the pursuit sequence is PS = (BPS, w).
Chapter 1. Introduction 8
Z2
Z1
Z4Z3
(a) Basic cyclic pursuit
x
xx
xZ2
Z1
Z4Z3
(b) Generalized centroidal cyclic pursuit
Figure 1.1: Generalization of cyclic pursuit
Definition 1.3 (Generalized centroidal cyclic pursuit) In a group of agents, if each
agent follows a point that is the weighted centroid of the other agents and the weights
used by different agents are different, then the agents execute generalized centroidal cyclic
pursuit (GCCP).
Here, we have a set of n weights that each of the agents follow and hence the pursuit
sequence is PS = (BPS, {w1, . . . , wn})
The basic cyclic pursuit and generalized centroidal cyclic pursuit is illustrated in
Figure 1.1. These generalized cyclic pursuit laws are studied in this thesis.
Another generalization that we consider in this thesis is the concept of using hetero-
geneous agents. A heterogeneous group of agents will have different speeds and controller
gains. In reality, a group of agents cannot be identical in all respects. Thus, it is log-
ical to study heterogeneous systems. Moreover, heterogeneity gives more flexibility in
controlling the behaviour of the agents.
1.4 Contributions and organization of the thesis
(i) Generalization of the concept of cyclic pursuit − BCP, CCP and GCCP
(ii) Analysis of cyclic pursuit laws for heterogenous agents.
Chapter 1. Introduction 9
(iii) Conditions for stability of different generalized linear cyclic pursuit laws.
(iv) Characterizing stable behaviour under generalized linear cyclic pursuit laws.
(v) Characterizing unstable behaviour under generalized linear cyclic pursuit laws.
(vi) Invariance properties of generalized linear cyclic pursuit laws.
(vii) Equilibrium formation of heterogenous agents under nonlinear basic cyclic pursuit.
(viii) Behaviour of cyclic pursuit laws under realistic constraints.
This thesis is organized according to the sequence of generalization of the cyclic
pursuit strategies. The analysis is carried out for heterogeneous agents, that is, agents
with different gains and speeds. The heterogeneity of the agents are utilized to obtain
different behaviours of the agents. Initially, the holonomic agents are studied, followed
by the study of non-holonomic agents.
Holonomic agents under cyclic pursuit strategies give rise to a linear system of state
equations. In Chapter 2, linear basic cyclic pursuit is analyzed for a group of heteroge-
neous agents. The agents converge to a point when the linear system is stable. The point
of convergence, called the reachable point or the rendezvous point, can be controlled by
the controller gains of the agents. Thus, we show that, with heterogeneous agents, the
rendezvous point can occur at any desired point. The stability and the rendezvous point
also exhibit some invariance properties with respect to the pursuit sequence of the agents,
which allow changing the connection between the agents while executing the same goal.
In Chapter 3, we formulate and analyze linear centroidal cyclic pursuit. The agents
under this strategy follow a group of other agents instead of only one of them. The
behaviour of the agents are similar to basic cyclic pursuit. A stable system results in
rendezvous of the agents, where the rendezvous points are functions of the controller
gains. The invariance of stability and rendezvous point for centroidal cyclic pursuit are
addressed and compared with basic cyclic pursuit.
Chapter 1. Introduction 10
The behaviour of a stable linear system under generalized centroid cyclic pursuit is
studied in Chapter 4. Generalized cyclic pursuit gives the flexibility that each agent can
select independently the group of agents it will follow. In this case, the stability and
rendezvous point depends on the pursuit sequence of the agents. Thus, the invariance
properties of the system do not hold in general except under certain conditions.
In Chapter 5, we shift our attention to the analysis of unstable linear system under
different cyclic pursuit strategies. The instability of the system is utilized to obtain
directed motion of the agents. The direction of motion changes with the pursuit sequence,
but there exists a point, called the asymptote point that remains invariant to pursuit
sequences. All the asymptotes of the directed motion passes through this point. An
alternate approach to obtain directed motion is also proposed.
Chapter 6 focuses on the non-holonomic agents that gives rise to nonlinear cyclic
pursuit. At equilibrium, the agents under basic cyclic pursuit exhibit circular motion
about a point. The radius of the circles are different for heterogeneous agents. The
equilibrium formation and the necessary conditions for equilibrium are studied.
In Chapter 7, cyclic pursuit strategies are applied to coordinate a group of au-
tonomous vehicles and to model the behaviour of the biological organisms like the fish
schools. These applications require imposition of realistic constraints to the basic cyclic
pursuit strategies. The behaviour of the autonomous vehicles like the robots and UAVs
and the schools of fishes under realistic cyclic pursuit are observed through simulation.
Chapter 8 concludes the thesis with a summary of the work done and some discussions
on the future promising directions of research.
Chapter 2
Rendezvous using linear Basic
Cyclic Pursuit
In this chapter, the behavior of a swarm of heterogenous agents in linear cyclic pursuit
is analyzed. The agents follow basic cyclic pursuit (BCP) laws as discussed in Chapter
1. The trajectories of the agents are studied as a function of the controller gains of
the agents. The conditions for rendezvous, under which the agents converge to a point,
called the reachable point, are obtained. The complete set of reachable points, called the
reachable set, is characterized. The possible points at which convergence or rendezvous
can occur are also obtained. Some interesting properties of the reachable point are
discussed.
2.1 Problem formulation
Linear cyclic pursuit between n agents indexed from 1 to n, in a d dimensional space, is
formulated as follows: The position of the agent i at any time t ≥ 0 is given by
Zi(t) = [z1i (t) z2
i (t) . . . zdi (t)]
T ∈ Rd, i = 1, 2, . . . , n. (2.1)
11
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 12
The equation of motion of agent i is
Zi = ui (2.2)
where ui is the control of agent i.
For basic cyclic pursuit, we assume the pursuit sequence (discussed in Section 1.3)
to be BPS=(1, 2, . . . , n). Then, ui is given as
ui = ki
[Zi+1(t)− Zi(t)
](2.3)
where, ki is the gain of the agent i. Let
k = {ki}ni=1 (2.4)
define the set of gains of all the agents. Thus, the equation of motion of the agent i is
given by
Zi(t) = ki
[Zi+1(t)− Zi(t)
](2.5)
From (2.5), it can be seen that, for every agent i, each coordinate zδi , δ = 1, · · · , d,
of Zi, evolves independently in time. Hence, these equations can be decoupled into d
identical linear system of equations and can be represented as
X = AX (2.6)
where
A =
−k1 k1 0 · · · 0
0 −k2 k2 · · · 0...
kn 0 0 · · · −kn
(2.7)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 13
The characteristic polynomial of A is
ρ(s) =n∏
i=1
(s + ki)−n∏
i=1
ki (2.8)
We can expand (2.8) as
ρ(s) = sn + Bn−1sn−1 + Bn−2s
n−2 . . . + B2s2 + B1s + B0 (2.9)
where the coefficients B0 and B1, which we will need for our analysis later, can be
obtained directly from (2.8), as
B0 = 0 (2.10)
B1 =n∑
i=1
n∏
j=1,j 6=i
kj (2.11)
This shows that there is exactly one eigenvalue of A at the origin, provided not more
than one gain is zero. The stability of the linear system, given in (2.6), is analyzed in
the next section.
2.2 Stability analysis
We prove stability using the Gershgorin’s disc theorem [5] which is stated below:
Theorem 2.1 (Gershgorin’s Theorem) Let A = [aij] ∈ Mn, and let
Ri(A) ≡n∑
j=1,j 6=i
|aij|, 1 ≤ i ≤ n
denote the deleted absolute row sums of A. Then, all the eigenvalues of A are located
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 14
in the union of n discs
n⋃i=1
{z ∈ C : |z − aii| ≤ Ri(A)
}≡ G(A)
Therefore, for a n × n square matrix A, we can draw n circles with centers at the
diagonal elements of A, i.e., aii, i = 1, 2, . . . , n, and with radius equal to the sum of the
absolute values of the other elements in the same row, that is,∑
j 6=i |aij|. Such circles
are called Gershgorin’s discs. All the eigenvalues of A lie in the region formed by the
union of all the n discs.
Theorem 2.2 The linear system, given by (2.6), is stable if and only if the following
conditions hold
(a) At most one ki is negative or zero, that is, at most for one i, ki ≤ 0 and kj > 0,
∀j, j 6= i.
(b)∑n
i=1
(∏nj=1,j 6=i kj
)> 0
Proof. From (2.11), it can be seen that Condition (b) implies B1 > 0. First, we prove
the “if” part of the theorem, that is, if both the Conditions (a) and (b) hold, then the
system is stable. Consider the following cases:
Case 1: All the gains are positive.
Condition (b) is satisfied. For ki > 0,∀i, the Gershgorin’s discs of A are shown in Figure
2.1. It can be seen that A does not have any eigenvalue on the right-hand side of the
s-plane and on the imaginary axis except at the origin. At the origin, there is only
one eigenvalue. Hence, the system is stable (in the sense that the output will remain
bounded).
Case 2: One gain is zero and other gains are positive.
Condition (b) is satisfied. Let, ki = 0 and kj > 0,∀j, j 6= i. Then the Gershgorin’s discs
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 15
-k2-kn
-k1
Complex Plane
jω
σ
Figure 2.1: Gershgorin Discs of A when all the gains are positive
for the A matrix will be similar to Case I (Figure 2.1) and all the roots of A will lie either
on the left hand side of the s-plane or at the origin. From (2.11), B1 6= 0, therefore,
there is only one root of A at the origin. Hence, the system is stable.
Case 3: One gain is negative and other gains are positive.
Let ki < 0 and kj > 0,∀j, j 6= i. Then, for Condition (b) to be satisfied, ki > ki where
ki = −∏n
j=1,j 6=i kj∑nl=1,l 6=i
∏nj=1,j 6=i,l kj
(2.12)
It is to be shown that, given the gains kj > 0,∀j, j 6= i, if ki > ki, then the system is
stable. We prove this by contradiction. Using (2.11), we can rewrite B1, as a function
of ki, as
B1 = ki
(n∑
l=1,l 6=i
n∏
j=1,j 6=i,l
kj
)+
n∏
j=1,j 6=i
kj (2.13)
Let us plot B1 as a function of ki (Figure 2.2). The system should be stable in [ki,∞).
Let β ≤ α ≤ 0 and the system be unstable in [β, α] ⊆ [ki,∞) which implies that there
are some roots of A on the right-hand side of the s-plane. The Gershgorin’s discs of A is
shown in Figure 2.3. Since the root locus is continuous and the roots of A should always
remain within the Gershgorin’s disc, at ki = α, at least two roots of A should be at the
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 16
B1
kiαβ
(0,0)k_
i
Figure 2.2: Condition (b) of Theorem 2.2 as a function of the gain ki
-k2-kn -k1
Complex Plane
jω
σ-ki
Figure 2.3: Gershgorin Discs of A when only one gains is negative
origin (since, one root of A is always at the origin). This requires B1 = 0 in (2.9). But,
from (2.11), B1 6= 0 for ki = α > ki. This leads to a contradiction and hence the system
is stable.
The “only if” part is proved by contradiction. Assume the system is stable but any
one or both the conditions do not hold. We consider the following cases separately.
Case 1: Two or more gains are zero.
When two or more gains are zero and others are either positive or negative, B1 = 0 in
(2.9), which implies that more than one root is at the origin, and hence the system is
unstable.
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 17
-kn -k1
Complex Plane
jω
σ-kj
α
-ki
Figure 2.4: Gershgorin Discs of A when α changes from 0 to 1
Case 2: Two or more gains are negative.
Assume only two gains are negative. More than two negative gains can be proved simi-
larly. Let, ki < 0, kj < 0 and kl ≥ 0, ∀l, l 6= i, j. Consider a matrix
A =
−k1 αk1 0 · · · 0
0 −k2 αk2 · · · 0...
αkn 0 0 · · · −kn
(2.14)
where, α ∈ [0, 1]. The characteristic polynomial of A is
ρ(s) =n∏
i=1
(s + ki)− αn
n∏i=1
ki (2.15)
When α = 0, the eigenvalues of A are −ki,∀i. The corresponding Gershgorin’s discs are
points (circles of zero radius) at (−ki, 0), i = 1, 2, · · · , n. When α = 1, A = A, and the
characteristic polynomial is ρ(s). The Gershgorin’s discs, as α varies from 0 to 1, are
shown in Figure 2.4.
Now, as α goes from 0 to 1, by continuity of the root locus, the root locus starting
from (−ki, 0) and (−kj, 0) remain within the discs centered at (−ki, 0) and (−kj, 0) with
radius α|ki| and α|kj|. At α = 1, these roots are either still on the right hand side or
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 18
are both at the origin or one at the origin and the other at the right hand side of the s
plane. All these yield an unstable system.
Case 3 :∑n
i=1
(∏nj=1,j 6=i kj
)≤ 0
This case implies B1 ≤ 0. If B1 = 0, then two roots are at the origin and if B1 < 0,
then one root is at the origin and at least one root on the right hand side of the s plane.
Hence, the system cannot be stable. ¤
Therefore, when the system is stable, there is one and only one eigenvalue of A at
the origin. The solution of (2.6), in the frequency domain, is
X(s) = (sI − A)−1X(t0) (2.16)
Expanding the ith component of X(s)
xi(s) =1
ρ(s)
n∑q=1
biq(s)xq(t0) , i = 1, . . . , n (2.17)
where, ρ(s) is the characteristic polynomial of A and biq(s),∀i, are functions of k and can
be expressed as
biq(s) =
∏nl=1,l 6=q (s + kl), q = i;
∏q−il=1 kl
∏nl=q−i+1,l 6=q (s + kl), q > i;
∏n−q+i−1l=1,i6=q kl
∏nl=n−q+i (s + kl), q < i.
(2.18)
Let the non-zero eigenvalues of A be Rp = (σp + jωp), p = 1, · · · , n − 1, where n is the
number of distinct eigenvalues of A, with the pth eigenvalue having algebraic multiplicity
of np. Let Sr = {Rp|ωp = 0} be the set of real eigenvalues and Si = {Rp|ωp 6= 0} be the
set of complex conjugate eigenvalues. Then, taking inverse Laplace transform of (2.17),
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 19
we get
xi(t) = xf +∑p∈Sr
{n∑
q=1
(np∑
r=1
aipqrt
r−1
)xq(t0)
}eσpt
+∑p∈Si
[n∑
q=1
{np∑
r=1
aipqrt
r−1 cos(ωpt) + ai∗pqrt
r−1 sin(ωpt)
}xq(t0)
]eσpt (2.19)
where, for the pth eigenvalue, Rp
aipqr =
1
r!
dr
dsr
[{s− (σp + jωp)
}np biq(s)
ρ(s)
] ∣∣∣∣s=Rp
(2.20)
and ai∗pqr is the complex conjugate of ai
pqr, and when Rp = 0
xf =
∑nq=1(1/kq)xq∑n
q=1(1/kq)(2.21)
When the system is stable, i.e., σp < 0,∀p, as t → ∞, xi(t) = xf ,∀i. This implies that
all the the agents will converge to the point xf . In the next section, the rendezvous of
the agents is analyzed.
2.3 Rendezvous and Reachable point
For a stable system, the agents will converge to a point. We analyze the point of con-
vergence in the following theorem.
Theorem 2.3 (Reachable Point) If a system of n-agents, with equation of motion
given in (2.5), have their initial positions at Z0 = {Zi(t0)}ni=1, and gains k that satisfies
Theorem 2.2, then they converge to a point Zf given by,
Zf =n∑
i=1
{(1/ki∑n
j=1 1/kj
)Zi(t0)
}=
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(2.22)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 20
where Zf is called a reachable point or the rendezvous point of this system of n agents.
Proof. Summing (2.5) for all n, in the sense of mod n, we get
n∑i=1
Zi(t)
ki
=n∑
i=1
(Zi+1(t)− Zi(t)) = 0 (2.23)
⇒n∑
i=1
Zi(t)
ki
= constant (2.24)
for all time, t. Then, considering the initial position Zi(t0) and final position Zi(tf ) of
the agent i, we can write
n∑i=1
Zi(t0)
ki
=n∑
i=1
Zi(tf )
ki
(2.25)
When the system is stable, all the agents converge to a point, that is, Zi(tf ) = Zf , ∀i.Thus
n∑i=1
Zi(t0)
ki
=n∑
i=1
Zf
ki
= Zf
n∑i=1
1
ki
(2.26)
⇒ Zf =
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(2.27)
from which we get (2.22) or the rendezvous point. ¤
We can compare (2.19) with (2.22) and observe that xf is same as one of the coordi-
nates of Zf .
Now, let us denote
Zf (Z0, k) =
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(2.28)
as the reachable point obtained from the initial positions Z0, and gains k that satisfy
Theorem 2.2. Then, the set of reachable points (called the reachable set), at which
rendezvous can occurs, starting from the initial point Z0, be denoted as Zf (Z0) and
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 21
defined as,
Zf (Z0) =
{Zf (Z
0, k)∣∣∣ ∀k satisfying Theorem 2.2
}(2.29)
Thus for every Z ∈ Zf (Z0), there exists a k satisfying Theorem 2.2, such that Zf (Z
0, k) =
Z. Hence, the point of convergence of the n agents, given their initial positions, can be
controlled by a judicious selection of the gains k.
Next, the region in Rd, where a rendezvous of n agents is possible, is obtained. Let
Co(Z0) be the convex hull of Z0. A finitely generated cone [55] can be defined as,
Definition 2.1 (Finitely generate cone) A cone C is finitely generated by vectors a1, . . . ,
am, if C consists of all the vectors of the form
x = λ1a1 + λ2a2 + . . . + λmam (2.30)
with λ1 = 1, λi ≥ 0 for i = 2, . . . , m. This cone C has a vertex at a1.
With this, we define a cone Cp as follows:
Definition 2.2 A cone Cp is finitely generated by the vectors [Zp(t0) − Zi(t0)], i =
1, · · · , p− 1, p + 1, · · · , n if Cp consist of all vector of the form
Z = Zp(t0) +n∑
i=1,i6=p
λi(Zp(t0)− Zi(t0)) (2.31)
where λi ≥ 0, i = 1, . . . , p− 1, p + 1, . . . , n. Cp has a vertex at Zp(t0).
Theorem 2.4 Consider a system of n agents, with equation of motion given in (2.5)
and initial positions at Z0. A point Z is reachable if and only if,
Z ∈ Co(Z0)⋃ { n⋃
p=1
Cp
}= P(Z0) (say) (2.32)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 22
that is, Zf (Z0) = P(Z0).
Proof. First, we show that Zf (Z0) ⊆ P(Z0). Let Z ∈ Zf (Z
0). Then, by definition of
Zf (Z0), there exists a k, satisfying Theorem 2.2, such that
Z =
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(2.33)
holds. We will show that Z ∈ P(Z0). Consider the following cases.
Case I : Let ki > 0, ∀i. Then, (2.33) can be written as
Z =n∑
i=1
(1/ki)∑nj=1(1/kj)
Zi(t0) (2.34)
Thus, Z is a convex combination of Zi(t0), i = 1, ..., n. Hence, Z ∈ Co(Z0) and so
Z ∈ P(Z0).
Case II : Let one of the gains kp < 0 and the remaining gains ki > 0, ∀i, i 6= p. Then,
from (2.34),
Z =n∑
i=1,i6=p
(1/ki)∑nj=1(1/kj)
Zi(t0) +(1/kp)∑nj=1(1/kj)
Zp(t0) (2.35)
⇒ Z
n∑i=1
1
ki
=n∑
i=1,i6=p
1
ki
Zi(t0) +1
kp
Zp(t0) (2.36)
⇒ Z
n∑i=1
1
ki
− Zp(t0)n∑
i=1
1
ki
=n∑
i=1,i 6=p
1
ki
Zi(t0) +1
kp
Zp(t0)− Zp(t0)n∑
i=1
1
ki
(2.37)
⇒{
Z − Zp(t0)} n∑
i=1
1
ki
=n∑
i=1,i 6=p
1
ki
{Zi(t0)− Zp(t0)
}(2.38)
Since only kp < 0, and ki > 0,∀i, i 6= p, we have
n∏i=1
ki < 0 (2.39)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 23
From Condition (b) of Theorem 2.2, for stable systems
n∑i=1
( n∏
j=1,j 6=i
kj
)> 0 (2.40)
Dividing the above equation by 2.39, we get
n∑i=1
1
ki
< 0 (2.41)
Let
n∑i=1
1
ki
= −1
c(2.42)
where, c > 0. Then, from (2.38),
−1
c
{Z − Zp(t0)
}=
n∑
i=1,i6=p
1
ki
{Zi(t0)− Zp(t0)
}(2.43)
⇒ Z − Zp(t0) =n∑
i=1,i 6=p
− c
ki
{Zi(t0)− Zp(t0)
}(2.44)
⇒ Z = Zp(t0) +n∑
i=1,i6=p
c
ki
{Zp(t0)− Zi(t0)
}(2.45)
Then, from (2.31), Z ∈ Cp and so Z ∈ P(Z0).
Case III : Let one of the gains kp = 0 and the remaining gains ki > 0,∀i, i 6= p. Then,
(2.33) can be written as,
Z =
∑ni=1
(∏nj=1,j 6=i kj
)Zi(t0)∑n
i=1
∏nj=1,j 6=i kj
(2.46)
Putting kp = 0 in the above equation,
Z =
∏nj=1,j 6=p kjZp(t0)∏n
j=1,j 6=p kj
= Zp(t0) (2.47)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 24
Thus, Z ∈ Co(Z0) and so Z ∈ P(Z0).
Therefore, from Case I-III above, if Z ∈ Zf (Z0), then Z ∈ P(Z0). Therefore,
Zf (Z0) ⊆ P(Z0).
Now, to show that P(Z0) ⊆ Zf (Z0) it has to be shown that for any point Z ∈ P(Z0),
there exists k such that (2.33) holds.
We denote int{P(Z0)} as the interior of the set P(Z0) and define it as: α ∈int{P(Z0)} if there exists an ε > 0 such that, for all β satisfying de(α, β) < ε, β ∈ P(Z0),
where de(α, β) is the Euclidean distance between α and β. Then, boundary of P(Z0),
denoted by ∂{P(Z0)}, is defined as: α ∈ ∂{P(Z0)} if α does not belong to int{P(Z0)}.
Thus, P(Z0) can be partitioned as
P(Z0) = P1(Z0) ∪ P2(Z
0) ∪ P3(Z0) (2.48)
where,
P1(Z0) = int{P(Z0)} (2.49)
P2(Z0) =
{Zi(t0)
∣∣∣Zi(t0) ∈ ∂{P(Z0)}}
(2.50)
P3(Z0) = ∂{P(Z0)} \ P2(Z
0) (2.51)
Then, P1(Z0) is the interior of P(Z0), P2(Z
0) is the set of vertices of the convex set
P(Z0), and P3(Z0) is the boundary of P(Z0) without the vertices. We will consider
these sets separately.
Case I : Z ∈ P1(Z0). We have the following cases:
Case Ia: Let Z ∈ int{Co(Z0)}. Then, there exists αi, i = 1, . . . , n,∑n
i=1 αi = 1 with αi >
0,∀i such that
n∑i=1
αiZi(t0) = Z (2.52)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 25
Let
ki =c
αi
, i = 1, 2, . . . , n (2.53)
where, c > 0 is any positive constant. Thus, ki > 0,∀i, and
n∑i=1
1
ki
=1
c(2.54)
Replacing αi by c/ki in (2.52),
Z =n∑
i=1
(c
ki
)Zi(t0) =
n∑i=1
(1/ki
1/c
)Zi(t0) =
n∑i=1
{1/ki∑n
j=1 1/kj
}Zi(t0) (2.55)
The above equation is the same as (2.33) and all the gains satisfy Theorem 2.2. Therefore,
Z ∈ Zf (Z0)
Case Ib: Let Z ∈ int{Cp} for some p. Then, there exist βi > 0, i = 1, 2, . . . , n, such that
Then, I0 = [xmin, xmax] is the closed interval that contains the initial positions of all the
agents. It will be shown that xi(t) ∈ I0,∀i, ∀t. Consider the case when xp(t) ∈ ∂{I0},that is, at the boundary of I0, for some p and some t. The gain kp can be positive, zero
or negative. We consider these cases separately.
Case I : xp(t) ∈ ∂{I0} and kp ≥ 0
From (2.6), xp(t) = kp(xp+1 − xp). Assume xp(t) = xmin. Then, xp+1 ≥ xp, and so,
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 31
xp(t) ≥ 0. Therefore xp(t) ∈ I0. Similarly, when xp(t) = xmax, xp+1 ≤ xp, and thus
xp(t) ≥ 0. Therefore xp(t) ∈ I0
Case II : xp(t) ∈ ∂{I0} and kp < 0
Here, we prove here that, if kp < 0, then xp /∈ ∂{I0}. From (2.21), we can write
xf
n∑i=1
1
ki
=n∑
i=1,i6=p
1
ki
xi(t) +1
kp
xp(t) (2.68)
⇒ xp(t) =
(∑ni=1 1/ki
1/kp
)xf +
n∑
i=1,i6=p
(− 1/ki
1/kp
)xi (2.69)
Let cp =(Pn
i=1 1/ki
1/kp
)and ci =
(− 1/ki
1/kp
). Then,
xp(t) = cpxf +n∑
i=1,i6=p
cixi (2.70)
Since∑n
i=1 1/ki < 0 (from (2.41)), it can be seen that ci > 0,∀i and also∑n
i=1 ci = 1.
Hence, xp(t) is a convex combination of the xf and xi,∀i, i 6= p. Therefore, xp(t) cannot
lie on the boundary of I0, that is, xp(t) /∈ ∂{I0} if kp < 0.
Therefore, xi(t) ∈ I0,∀i, ∀t.
Now, if the pursuit sequence switches, the characteristic polynomial of A, given by
(2.8), does not change. Hence, the eigenvalues of A remain the same. In (2.19), Rp will
be same but aipqr will change if the pursuit sequence changes. Let, for pursuit sequence
Consider the pursuit sequence as BPS1 = (1, 2, 3, 4, 5).
Case I : We show that when the gains of all the agents are positive, they converge
within Co(Z0). The gains of the agents are shown in Table 2.2. For this set of gains,∑n
i=1
∏j=1,j 6=i kj = 16704 > 0 and therefore Theorem 2.2 is satisfied and the system
is stable. From Theorems 2.3 and 2.4, the agents will converge at Zf given by (2.22).
Satisfying the initial conditions and gains, we get Zf = (4.55, 1.6) ∈ Co(Z0). This is
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 36
−10 −5 0 5 10 15−10
−5
0
5
10
15
1
3
5
2
4Co(Z0)
Figure 2.9: Trajectories of a team of 5 agents with all positive gains (Case I)
verified in the trajectories of the agents shown in Figure 2.9.
Case II : When one gain is zero and others are all positive, we show that the agents
converge to a point. Let, the gains of the agents be as shown in the Table 2.2 Here,∑n
i=1
∏j=1,j 6=i kj = 5760 > 0 and thus the system is stable. The rendezvous point can
be calculated from (2.22) and it is equal to Zf = (10,−1). The trajectories are shown
in Figure 2.10 and we can verify the rendezvous point.
Case III : When the gain of only one agent is negative, but the system satisfies
Theorem 2.2, the agents converge outside the convex hull Co(Z0). The gains selected
in this case is shown in Table 2.2. These gains satisfy the conditions in Theorem 2.2,
since∑n
i=1
∏j=1,j 6=i kj = 3024 > 0. Thus, the system is stable. Also from (2.12),
k1 = −2.86 < k1 . From Theorems 2.3 and 2.4, the rendezvous occurs at Zf given
by (2.22). For the initial conditions and the gains, we get Zf = (17.5,−4.6) ∈ C1.
The simulation result is shown in Figure 2.11. We observe that the rendezvous point
Zf /∈ Co(Z0) and the figure illustrates that Zf ∈ C1.
Case IV : We consider the gain of one of the agent to be negative and the others
to be positive such that the system is not stable according to Theorem 2.2. Then, the
agents should not converge to a point. The gains of the agents are shown in Table 2.2.
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 37
−10 −5 0 5 10 15−10
−5
0
5
10
15
2
5
1
3
4Co(Z0)
Figure 2.10: Trajectories of a team of 5 agents with one gain zero and all other gainspositive such that Theorem 2.2 is satisfied (Case II)
−10 −5 0 5 10 15 20−10
−5
0
5
10
15
3
4
1
5
2
C1
Figure 2.11: Trajectories of a team of 5 agents with the gain of first agent is negativeand other gains positive such that the gains satisfy Theorem 2.2 (Case III)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 38
−20 0 20 40 60−30
−10
10
20
21
5
3
4
Figure 2.12: Trajectories of a team of 5 agents with the gain of first agent is negativeand other gains positive, such that Theorem 2.2 is not satisfied and the agents do notconverge to a point (Case IV).
Theorem 2.2 is not satisfied in this case, since∑n
i=1
∏j=1,j 6=i kj = −2448 < 0. Also, from
(2.12), k1 = −2.86 > k1. The trajectories of the agents are shown in Figure 2.12 and we
observe that rendezvous does not occur.
Case V : Consider that the gains of more than one agent is negative. Then, the
system will be unstable according to Theorem 2.2. The gains of the agents are shown
in Table 2.2. These gains violate Condition (a) of Theorem 2.2. Figure 2.13 shows the
trajectories of the agents and we observe that the agents do not converge to a point.
2.5.2 Computation of controller gains for a rendezvous point
Pursuit sequence BPS1 = (1, 2, 3, 4, 5) is considered for these cases.
Case VI : Let the desired rendezvous point be Zf = (0, 0). For the given the ini-
tial positions (in Table 2.2), Zf ∈ int{Co(Z0)}. Therefore, the gains of the agents
can be calculated using (2.52) and (2.53). One of the set of αis that satisfy (2.52) is
[0.07, 0.08, 0.095, 0.39, 0.37] . Assuming c = 1, the gains of the agents are given in
Table 2.2. The trajectories of the agents are shown in Figure 2.14(a) and we observe
that the agents converge to the point Zf = (0, 0). If we assumed c = 2, the gains of the
agents will be doubled, but the trajectories remain the same as shown in Figure 2.14(b)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 39
−40 −20 0 20 40 60−40
−20
0
20
40
12
5
4
3
Figure 2.13: Trajectories of a team of 5 agents with two negative gains. Theorem 2.2 isnot satisfied and the agents do not converge to a point (Case V).
−10 0 10 15−10
0
10
15
3
2
1
4
5
(a) c = 1−10 0 10 15
−10
0
10
15
3
2
1
4
5
(b) c = 2
Figure 2.14: Trajectories of the agents converging to a desired points Zf = (0, 0) ∈Co(Z0) (Case VI)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 40
−10 0 10 20 25−10
0
10
15
1
2
4
5
3
Figure 2.15: Trajectories of the agents converging to desired points Zf = (20,−5) /∈Co(Z0) (Case VII)
Case VII : Let us consider the desired rendezvous point to be Zf = (20,−5). For
the given initial position of the agents, Zf /∈ Co(Z0). Since Z2(t0) ∈ int{Co(Z0)}, all
points in R2 are reachable. Note that Zf ∈ C1 also. So we can select a negative gain for
the first agent. One of the sets of βis that satisfy (2.56) is [−2.16, 0.6, 0.03, 0.43, 0.1].
Using (2.59), the gains of the agents are calculated with c = 1 and the values are given in
Table 2.2. Figure 2.15 shows that the trajectories of the agents converging at the desired
point Zf = (20,−5).
2.5.3 Pursuit sequence invariance properties
To demonstrate the invariance properties, we consider two different pursuit sequences
BPS1 = (1, 2, 3, 4, 5)
BPS2 = (1, 3, 5, 2, 4)
Case VIII : We demonstrate the invariance of stability and rendezvous point with
respect to the pursuit sequence when all the gains are positive. Consider the gains same
as in Case I. Figure 2.16 show the trajectories of the agents for the two different pursuit
sequences BPS1 and BPS2. We observe, from Theorem 2.2, that the system is stable for
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 41
−10 0 10 15−10
0
10
15
2
1
5
4
3
(a) Pursuit sequence BPS2
−10 −5 0 5 10 15−10
−5
0
5
10
15
1
3
5
2
4
(b) Pursuit sequence BPS1
Figure 2.16: Trajectories of the agents, with all positive gains, for different pursuitsequences (Case VIII)
both the pursuit sequences and the rendezvous point is the same and is given by (2.22).
However, the trajectories of the agents are different.
Case IX : Here, we demonstrate the invariance property when one gain is negative
and all the others are positive. We select the same gains as in Case III. The system
is stable and has the same rendezvous point for both the pursuit sequences, BPS1 and
BPS2 as shown in Figure 2.17. The figure also shows that the trajectories of the agents
are different for the two pursuit sequences.
2.5.4 Finite and infinite switching of pursuit sequences
We consider the following pursuit sequences
BPS1 = (1, 2, 3, 4, 5)
BPS2 = (1, 3, 5, 2, 4)
BPS3 = (1, 4, 2, 5, 3).
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 42
−10 −5 0 5 10 15 20−10
−5
0
5
10
15
2
1
4
5
3
(a) Pursuit sequence BPS2
−10 −5 0 5 10 15 20−10
−5
0
5
10
15
3
4
1
5
2
(b) Pursuit sequence BPS1
Figure 2.17: Trajectories of the agents, with the gain of first agent negative, for differentpursuit sequences (Case IX)
Case X : Invariance properties with finite switching is demonstrated when the gains of
all the agents are positive. We consider the same gains as in Case I. At t = 0, the pursuit
sequence is BPS1. At t = 0.05, it switches to BPS2 and at t = 0.15, it switches to BPS3.
The trajectories of the agents are shown in Figure 2.18(a). Comparing with Figure
2.18(b), we observe that the rendezvous point is the same and hence the rendezvous
point is invariant under finite switching.
Case XI : We demonstrate the invariance properties with respect to the finite switch-
ing when the gain of one of the agent is negative and the other gains are positive.
Consider the same gains as in Case III. The pursuit sequence switches from BPS1 to
BPS2 to BPS3 at t = 0.05 and t = 0.15, respectively. The simulation is shown in Figure
2.19 and it can seen that the agents converge to the same point without switching (Figure
2.19(b)) and with finite switching (Figure 2.19(a)).
Case XII : We demonstrate the invariance properties with respect to infinite switching
of the pursuit sequence. Infinite switching is simulated by switching between the three
pursuit sequences BPS1, BPS2 and BPS3 repeatedly with the time between consecutive
switching ∆t = 0.02. The controller gains are the same as in Case I and the trajectories
are shown in Figure 2.20. The figure verifies the stability and rendezvous point invariance
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 43
−10 −5 0 5 10 15−10
−5
0
5
10
15
4
2
5
3
1
(a) Finite switching−10 −5 0 5 10 15
−10
−5
0
5
10
15
1
3
5
2
4
(b) No switching
Figure 2.18: Trajectories of the agents, with all positive gains, when the pursuit sequenceswitches as BPS1 → BPS2 → BPS3 (Case X)
−10 −5 0 5 10 15 20−10
−5
0
5
10
15
2
1
5
3
4
(a) Finite switching−10 −5 0 5 10 15 20
−10
−5
0
5
10
15
3
4
1
5
2
(b) No switching
Figure 2.19: Trajectories of the agents, with the gain of the first agent negative, whenthe pursuit sequence switches as BPS1 → BPS2 → BPS3 (Case XI)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 44
−10 −5 0 5 10 15−10
−5
0
5
10
15
1
2
4
5
3
(a) Infinite switching−10 −5 0 5 10 15
−10
−5
0
5
10
15
1
3
5
2
4
(b) No switching
Figure 2.20: Trajectories of the agents, with all positive gains, when the pursuit sequenceswitches infinitely at regular time intervals as BPS1 → BPS2 → BPS3 → BPS1 → BPS2
→ . . . (Case XII)
with respect to infinite switching.
2.6 Conclusions
In this chapter, the stable behaviour of a group of heterogenous agents under linear
cyclic pursuit is studied. The conditions for stability are obtained. The stable group of
agents converge to a point, called the reachable or rendezvous point, which is obtained
as a function of the gains and the initial positions of the agents. The reachable set
is determined for a given initial position of the agents. It is found that the stability,
reachable points and the reachable sets are not affected by the basic pursuit sequence
that the agents follow. These invariance properties are also proved to be valid for finite
and infinite switching of the basic pursuit sequences.
The analysis done in this chapter assumes that the agents follow a basic cyclic pursuit,
where one agent follows another in a cyclic manner. This can be generalized further
where an agent can follow a point in the convex combination of the other agents. This is
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 45
called a centroidal cyclic pursuit. In the next chapter, the behaviour of the agents under
centroidal cyclic pursuit is studied.
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 46
Chapter 3
Rendezvous using linear Centroidal
Cyclic Pursuit
In basic cyclic pursuit, addressed in Chapter 2, an agent pursues another agent according
to the basic pursuit sequence of the agents. In this chapter, we consider a generalized
pursuit strategy where an agent follows a point which is a convex combination of the
positions of the other agents. This strategy is called centroidal cyclic pursuit. Similar
to the basic cyclic pursuit, the stability of the centroidal cyclic pursuit, and the be-
haviour of the stable system of agents, are analyzed and several invariance properties are
demonstrated.
3.1 Problem formulation
Consider a group of n agents, ordered from 1 to n, in a d dimensional space, as in Chapter
2. The position of the agent i at any time t ≥ 0 is Zi(t) = [z1i (t) z2
i (t) . . . zdi (t)]
T ∈ Rd.
For centroidal cyclic pursuit, the agent i follows a point, Zic, which is the weighted cen-
troid of the other agents’ position (Figure 3.1). Let the weights be w = (η1, η2, . . . , ηn−1)
with∑n
i=1 ηi = 1 and ηi ≥ 0,∀i. Then, assuming a basic pursuit sequence BPS=(1, 2, . . . ,
47
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 48
x
x
x x
x
xx
Z3
Z1
Z2Z1
Z2Z4
Z3
Z3cZ3c
Z2c Z4c
Z1c
Z2cZ1c
Figure 3.1: Centroidal cyclic pursuit
n), agent i will follow agent i + j (mod n) with weights ηj. For a general basic pursuit
sequence BPS= (p1, p2, . . . , pn), where pi ∈ {1, 2, . . . , n}, ∀i, and pi 6= pj, agent pi will
follow agent pi+j with weights ηj.
Thus, the equation of motion of agent i can be written as
Zi(t) = ui(t) = ki[Zic(t)− Zi(t)] (3.1)
where, assuming BPS=(1, 2, . . . , n), Zic is given by
Then, Γ defines the set of all possible weights that can be used with each basic pursuit
sequence to obtain the centroidal cyclic pursuit. Now, if ηi = 1 for some i, then either
it will correspond to one of the basic pursuit sequences, or it will destroy the cyclic
structure. For example, when n > 2 and n is even, η2 = 1 will give rise to two distinct
cycles instead of one. Hence, in this chapter, the weights are assumed to belong to the
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 49
set
Γ =
{w = (η1, η2, . . . , ηn−1)
∣∣∣n∑
i=1
ηi = 1, 0 < ηi < 1
}(3.4)
Note that, Γ ⊆ Γ. Here, we assume that ηi 6= 0,∀i. The cases when one of the ηi = 1
and the cyclic structure is maintained, is already discussed in Chapter 2.
Now, (3.1) can be decoupled into d identical linear system of equations, as in Section
(2.6), and we can write
X = AX (3.5)
where
A =
−k1 η1k1 η2k1 · · · ηn−1k1
ηn−1k2 −k2 η1k2 · · · ηn−2k2
...
η1kn η2kn η3kn · · · −kn
(3.6)
The above expression can be written as A = Kχ, where
K =
k1 0 0 · · · 0
0 k2 0 · · · 0...
0 0 0 · · · kn
(3.7)
where K is the gain matrix and
χ =
−1 η1 η2 · · · ηn−1
ηn−1 −1 η1 · · · ηn−2
...
η1 η2 η3 · · · −1
(3.8)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 50
is the weight matrix of the centroidal cyclic pursuit, called the pursuit sequence matrix.
The characteristic polynomial of A can be written as
ρ(s) = sn + Bn−1sn−1 + Bn−2s
n−2 . . . + B2s2 + B1s + B0 (3.9)
This is similar to (2.9) in Chapter 2, but the values of the coefficients are different.
Below, we evaluate B0 and B1 which are required for later analysis.
Since the columns of A are linearly dependent, A is singular and B0 = 0. Further,
Rank(A) = Rank(χ) as Rank(K) = n. Consider a matrix χii formed by removing the
ith row and ith column of χ. The Gershgorin’s discs of χii will have center at (−1, 0)
and radius less than one, since ηi > 0,∀i. Thus, χii does not have any eigenvalue at the
origin and so is of full rank. Therefore, Rank(A) = Rank(χ) = n− 1 and B1 6= 0.
The expression for B1 can be obtained using the L’Hospital rule as
B1 = lims→0
det(sI − A)
s= lim
s→0
[ d
ds
{det(sI − A)
}](3.10)
The derivative of the determinant of any n × n matrix P (y) with respect to y, is
ddy{det(P )} and is the sum of the n determinants obtained by replacing in all possi-
ble ways the elements of one row (column) of P by their derivatives with respect to y
[56]. If mij represents the minor of the (i, j) − th element of A, then from (3.10), as
s → 0
B1 = (−1)n−1
n∑i=1
mii
= (−1)n−1trace(Adj(A))
= (−1)n−1trace {Adj(χ) Adj(K)} (3.11)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 51
Let the adjoint of χ be denoted by,
Adj(χ) =
ξ11 ξ12 ξ13 · · · ξ1n
ξ21 ξ22 ξ23 · · · ξ1n
...
ξn1 ξn2 ξn3 · · · ξnn
(3.12)
Theorem 3.1 The adjoint of the pursuit sequence matrix χ for centroidal cyclic pursuit
is given by
Adj(χ) = (−1)n+1|ξ|1n×n (3.13)
for some ξ, that is in (3.12), ξij = ξ, with the sign of ξ determined by n.
Proof. It is to be shown here that all the elements of Adj(χ) are identical, that is,
ξij = ξ, ∀i, j.
First, we show that ξil = ξjl,∀i, j, and for a given l, where ξil is the cofactor of the
(l, i)-th element of χ, that is, ξil = (−1)i+l|χli|. Here, χij is the matrix obtained by
removing the ith row and jth column of χ.
Consider two matrices, χli and χl(i+1), i = 1, . . . , n − 1. For both of these matrices,
only the ith column is different and the other columns are the same. We can perform
an elementary column transformation on χli, such that the ith column is replaced by the
sum of all the columns of χli (including the ith column). Let the new matrix be χli. Note
that χli will have the same determinant as χli. Now, since the sum of the elements of a
row of χ is zero, the ith column of χli will be equal to the ith column of χl(i+1) with a
negative sign. Then, χli has the same elements as χl(i+1) except that the sign of all the
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 52
elements of the ith column of these two matrices are opposite. Thus,
|χli| = |χli| = −|χl(i+1)| (3.14)
⇒ (−1)l+i|χli| = (−1)l+i+1|χl(i+1)| (3.15)
⇒ ξil = ξ(i+1)l (3.16)
Since the above equation is true for all i, ξil = ξjl,∀i, j and for a given l.
Since the rows of χ also sum to zero, it can be similarly shown that ξli = ξlj, ∀i, j,for a given l. Hence, ξij = ξ, ∀i, j.
Now, the determinant of a matrix is equal to the product of its eigenvalues, so ξii
is the product of the eigenvalues of χii. From the Gershgorin’s disc theorem, all the
eigenvalues of χii have negative real parts. Therefore, if n is odd, ξii > 0 and if n is even,
ξii < 0.1 Since ξ = ξii, ξ > 0, if n is odd and ξ < 0, if n is even. Thus, each element of
Adj(χ) in (3.12) can be written as (−1)n+1|ξ| and hence we get (3.13). ¤
Below we give an example to illustrate Theorem 3.1.
Illustrative example: Consider a matrix
χ =
−1 η1 η2 η3
η3 −1 η1 η2
η2 η3 −1 η1
η1 η2 η3 −1
(3.17)
Then,
χ11 =
−1 η1 η2
η3 −1 η1
η2 η3 −1
, χ12 =
η3 η1 η2
η2 −1 η1
η1 η3 −1
(3.18)
1This is true in general because, even if the eigenvalues are imaginary, the imaginary roots come incomplex conjugate pairs and will contribute a positive value.
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 53
Here, the second and third columns of χ11 and χ12 are the same. Consider the matrix
χ11 = χ11e(I) =
−1 η1 η2
η3 −1 η1
η2 η3 −1
1 0 0
1 1 0
1 0 1
=
−η3 η1 η2
−η2 −1 η1
−η1 η3 −1
(3.19)
since∑3
i=1 ηi = 1. As |e(I)| = 1, |χ11| = |χ11| = −|χ12|. This shows that ξ11 = ξ21.
Then, from (3.11), we can write
B1 = (−1)n−1
n∑i=1
{(−1)n+1|ξ|
n∏
j=1,j 6=i
kj
}
= |ξ|n∑
i=1
{n∏
j=1,j 6=i
kj
}(3.20)
Using the above, the stability of the system is analyzed in the next section.
3.2 Stability analysis
We state and prove a theorem identical to Theorem 2.2 in Chapter 2
Theorem 3.2 The linear system, given by (3.5), is stable if and only if the following
conditions hold
(a) At most one ki is negative or zero, that is, at most for one i, ki ≤ 0 and kj > 0,
∀j, j 6= i.
(b)∑n
i=1
( ∏nj=1,j 6=i kj
)> 0
Proof. From (3.20), Condition (b) implies B1 > 0 as |ξ| > 0. Since ηi > 0, ∀i and∑n
i=1 ηi = 1, the Gershgorin’s discs of A (for centroidal cyclic pursuit, CCP given in
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 54
(3.6)), are the same as that of A (for basic cyclic pursuit, BCP given in (2.7)). Thus,
this proof follows in the similar lines as Theorem 2.2.
First, we assume that Conditions (a) and (b) hold. We have to show that the system
is stable. Consider the following cases:
Case 1: All the gains are positive.
When all the gains are positive, Condition (b) is satisfied and all the Gershgorin’s discs
lie on the left hand side of the s plane. Therefore, none of the eigenvalues of A can have
positive real parts and, since B1 6= 0, there is only one eigenvalue of A at the origin.
Hence, the system is stable.
Case 2: One gain is zero and other gains are positive.
In this case, Condition (b) is satisfied and the Gershgorin’s disc remains the same as in
Case 1. Also, B1 6= 0 and thus, the system is stable.
Case 3: One gain is negative and other gains are positive.
For Condition (b) to be satisfied, we can find the lower bound on the gain, ki, given by
ki = −∏n
j=1,j 6=i kj∑nl=1,l 6=i
∏nj=1,j 6=i,l kj
(3.21)
Note that, this equation is the same as (2.12) in Chapter 2. It has been proved in
Theorem 2.2 that, given the gains kj > 0,∀j, j 6= i, if ki > ki, then the system is stable.
We omit the details of the proof here.
To prove the “only if” part, assume that the system is stable but any one or both
the conditions do not hold. Consider the different possible cases:
Case 1: More than one gain is zero.
From (3.20), B1 = 0 and therefore ρ(s), given in (3.9), will have more than one root at
the origin. Hence the system is unstable.
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 55
Case 2: More than one gain is negative
Consider a matrix, similar to (2.14)
A =
−k1 αη1k1 αη2k1 · · · αη(n−1)k1
αη(n−1)k2 −k2 αη1k2 · · · αη(n−2)k2
...
αη1kn αη2kn αη3kn · · · −kn
(3.22)
The Gershgorin’s discs of A when α = 0 are points at (−ki, 0),∀i. When α = 1, A = A.
From the continuity of the root locus and the necessity of the roots to be within the
Gershgorin’s disc, it can be argued, similar to Theorem 2.2, that as α goes from 0 to 1,
more than one root will always remain on the right hand side of the s plane. Therefore,
at α = 1, there will be more than one roots on the right hand side or at the origin.
Hence, the system is unstable.
Case 3 :∑n
i=1
( ∏nj=1,j 6=j kj
)≤ 0.
This case implies B1 ≤ 0 and hence, the system can not be stable ¤
Thus, the conditions for stability of centroidal cyclic pursuit (CCP) are same as that
for basic cyclic pursuit (BCP). When the system is stable, there is one and only one
eigenvalue of A at the origin. As in (2.19), the solution of (3.5) can be written as
xi(t) = xif +
∑p∈Sr
{ n∑q=1
( np∑r=1
aipqrt
r−1)xq(t0)
}eσpt
+∑p∈Si
{ n∑q=1
( np∑r=1
aipqrt
r−1 cos(ωpt) + ai∗pqrt
r−1 sin(ωpt))xq(t0)
}eσpt (3.23)
where, xif corresponds to the zero eigenvalue, and ai
pqr and ai∗pqr are functions of χ and
K, and are complex conjugates. This equation is the same as (2.19), except that the
values of the coefficients aiprq and ai∗
prq are different as they now depend on the weights
w. When the system is stable, i.e., σp < 0,∀p, as t →∞, xi(t) = xif ,∀i.
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 56
The eigenvector corresponding to the zero eigenvalues is v = 11×n. Since the Rank(A)
= n − 1, the dimension of the null space N (A) = 1 and it is spanned by v. Therefore,
at equilibrium, X = 0, which implies AX = 0 and the nontrivial solution of this is
X = cv = c11×n for some c. Hence, at equilibrium x1 = x2 = . . . = xn, or in other word,
the agents will converge to a point. Therefore, xif = xf ,∀i. In the next section, the point
of rendezvous is analyzed.
3.3 Rendezvous and Reachable point
When the system is stable, the agents converge to a point at equilibrium. We state and
prove the rendezvous point theorem that is identical to Theorem 2.3.
Theorem 3.3 (Reachable Point) If a system of n-agents, with equation of motion
given in (3.1), have their initial positions at Z0 = {Zi(t0)}ni=1 and gains matrix K, that
satisfies Theorem 3.2, then they converge to a point Zf given by,
Zf =n∑
i=1
{( 1/ki∑nj=1 1/kj
)Zi(t0)
}=
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(3.24)
where Zf is called a reachable point or the rendezvous point of this system of n agents.
Proof. This theorem is identical to Theorem 2.3 and the proof follows in the same line.
Summing (3.1) for all n, in the sense of mod n, we get
n∑i=1
Zi(t)
ki
=n∑
i=1
(Zic(t)− Zi(t))
=n∑
i=1
[n−1∑j=1
ηjZi+j(t)− Zi(t)
](3.25)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 57
Changing the order of the summations and assuming ηn = −1,
n∑i=1
Zi(t)
ki
=n∑
j=1
[n∑
i=1
ηn−j+iZj(t)
](3.26)
=n∑
j=1
[Zj(t)
n∑i=1
ηn−j+i
](3.27)
Since∑n
i=1 ηn−j+i = 0,
n∑i=1
Zi(t)
ki
= 0 (3.28)
⇒n∑
i=1
Zi(t)
ki
= constant (3.29)
Then, considering the initial position Zi(t0) and final position Zi(tf ) of the agent i,
n∑i=1
Zi(t0)
ki
=n∑
i=1
Zi(tf )
ki
(3.30)
When the system is stable, all the agents converge to a point, Zi(tf ) = Zf , ∀i. Thus,
n∑i=1
Zi(t0)
ki
=n∑
i=1
Zf
ki
= Zf
n∑i=1
1
ki
(3.31)
⇒ Zf =
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(3.32)
Hence, we get (3.24) ¤
From (3.23), the final value of xif = xf ,∀i. Therefore, using (3.24), we can write
xf =
∑nq=1(1/kq)xq(t0)∑n
q=1(1/kq)(3.33)
Comparing (3.24) with (2.22), it can be seen that, given the initial positions of
the agents and the gains, the reachable point is the same for both basic cyclic pursuit
(BCP) and centroidal cyclic pursuit (CCP). The weights w ∈ Γ do not play any role in
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 58
determining the rendezvous point. Thus, the reachable set remains the same for BCP
and CCP. To make the agents converge to a desired point within the reachable set, the
gains can be selected as discussed in Theorem 2.4. Any weight w ∈ Γ can be chosen.
The difference in the two strategies, BCP and CCP, are reflected in the trajectory of
the agents, which changes with the weight w. This leads automatically to the invariance
properties of the reachable point and will be discussed in the next section.
3.4 Invariance properties
Similar to Section 2.4, we study the invariance properties of the system (3.1) with respect
to the pursuit sequence of the agents. For centroidal cyclic pursuit, the pursuit sequence
can change in two ways − (i) by changing the weights while keeping the basic pursuit
sequence same or (ii) by changing the basic pursuit sequence while keeping the weights
same. We show that the stability and rendezvous point do not change with
(a) Different sets of weights for a given basic pursuit sequence.
(b) Different basic pursuit sequence for a given set of weights.
(c) Different sets of weights and basic pursuit sequence.
Theorem 3.4 The stability of the linear centroidal cyclic pursuit is pursuit sequence
invariant.
Proof. It can be seen from Theorem 3.2 that the stability of the system depends only
on the gains K. Thus, given a set of stable gains, the system is stable for any basic
pursuit sequence and any weights. Therefore, under Conditions (a), (b) and (c) stated
above, the stability of the system remains unchanged and hence, it is pursuit sequence
invariant. ¤
Theorem 3.5 The reachable point and thus the reachable set of a linear centroidal cyclic
pursuit is pursuit sequence invariant.
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 59
Proof. Consider (3.24), which gives the expression for the reachable or the rendezvous
point. This equation depends only on the initial positions of the agents and their gains
and is independent of the basic pursuit sequence and the weights of the agents. Hence,
the reachable point is pursuit sequence invariant. Similarly, the reachable set is also
pursuit sequence invariant. ¤
Now, let us consider switching of pursuit sequence. Again, we can have three different
types of pursuit sequence switching
a) Switching of the weights while the same basic pursuit sequence is followed.
b) Switching of the basic pursuit sequence keeping the weights same.
c) Switching both the weights and the basic pursuit sequence.
The definition of finite and infinite switching of pursuit sequence is the same as given in
Section 2.4.
Theorem 3.6 (Stability with finite switching) The stability of the linear centroidal
cyclic pursuit is invariant under finite switching of pursuit sequences.
Proof. It is proved in Theorem 3.4 that if a system of n agents is stable for a given basic
pursuit sequence and weights, then it is stable for all the basic pursuit sequences and
weights. Therefore, if the pursuit sequence switches, it implies that the switch occurs
between two stable systems. Since the number of switches are finite, the system, after
the last switch, is stable and hence the stability is invariant under finite switching of
pursuit sequences. ¤
Theorem 3.7 (Reachability with finite switching) The reachable point of linear cen-
troidal cyclic pursuit is invariant under finite switching of pursuit sequences.
Proof. Let the switching of pursuit sequences occur at t1, · · · , tm, m < ∞ such that
0 < t1 < · · · < tm < ∞. During tj ≤ t < tj+1, let the pursuit sequence be (BPSj, wj).
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 60
The switching invariance property is proved by showing that the reachable point, given
in (3.24), remains the same after a switch of either the basic cyclic pursuit or the weights
or both. At t = tj, the positions of the agents are Zi(tj),∀i. If there are no further
switchings, then the reachable point, from (3.24), is
Zf =
∑ni=1 Zi(tj)/ki∑n
i=1 1/ki
(3.34)
Let, at tj+1, the weight wj switch to wj+1 while BPSj = BPSj+1. The position of agent
i at tj+1 is Zi(tj+1). For t ≥ tj+1, the pursuit sequence is (BPSj+1, wj+1). If there are
no more switching of connections, let the reachable point be Z ′f . Now, from (3.24) and
(3.29), we have
Z ′f
n∑i=1
1
ki
=n∑
i=1
Zi(tj+1)
ki
=n∑
i=1
Zi(tj)
ki
= Zf
n∑i=1
1
ki
(3.35)
This shows that Z ′f = Zf . It can be similarly shown that if at tj+1, BPSj switches to
BPSj+1 while wj = wj+1, Zf will remain as the rendezvous point. This is also true if
both basic pursuit sequence and weights changes. Hence, when there is one switching
of connection, the rendezvous point does not change. This can be extended to a finite
number of switchings to show that the reachable point remains unchanged after the final
switch tm. ¤
Theorem 3.8 (Stability with infinite switching) The stability of the linear centroidal
cyclic pursuit is invariant under infinite switching of pursuit sequences.
Proof. The stability is proved similar to Theorem 2.9. We analyze the system along
one direction, since from Section 3.1, we have seen that the system can be decoupled
and analyzed separately along each direction. Consider (3.23). We define
xmax = max{
xf , x1(t0), x2(t0), . . . , xn(t0)}
(3.36)
xmin = min{
xf , x1(t0), x2(t0), . . . , xn(t0)}
(3.37)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 61
which are the same as in (2.66) and (2.67). We prove that xi(t) ∈ I0 = [xmin, xmax],∀i, ∀t.Now, as long as xi(t) ∈ int{I0}, that is, interior of I0, xi(t) ∈ I0. We consider the case
when xi(t) ∈ ∂{I0}, that is, on the boundary of I0, for some i and some t. Consider the
following cases:
Case I : xp(t) ∈ ∂{I0} and kp ≥ 0.
Assume xp(t) = xmin. From (3.5), xp(t) = kp(xpc−xp). Since, xpc is the weighted centroid
of the remaining n−1 agents, xpc ≥ xp, and thus xp(t) ≥ 0. Hence, xp(t) ∈ I0. Similarly,
when, xp(t) = xmax, xpc ≤ xp, and thus xp(t) ≤ 0. Hence, again xp(t) ∈ I0.
Case II : xp(t) ∈ ∂{I0} and kp < 0.
It has been shown in the proof of Theorem 2.9 that, when kp < 0, xp can be expressed
as a convex combinations of xf and xi,∀i, i 6= p. This is true even for CCP. Hence,
xp /∈ ∂{I0}. Therefore, xi(t) ∈ I0,∀i, ∀t.
Now, if either the basic pursuit sequence or the weights change, the characteristics
equation will be different and so will be the eigenvalues of A. But, from the continuity of
the root locus, given ε > 0, ∃ δ > 0 such that if 0 < ε ≤ ηi ≤ 1− ε < 1,∀i, then the real
part of all the eigenvalues σp < −δ,∀p, except for the one at the origin. Let, for basic
pursuit sequence BPSj and weights wl, the coefficients aipqr be represented as a
Case III : We demonstrate the stability and rendezvous point invariance with respect
to the weights used by the agents for a given basic pursuit sequence. We consider pursuit
sequence BPS1 and two different sets of the weights w1 and w2. The gains are same as
in Case II. The agents converge to the point Zf = (−16.47, 2.52) as seen in Figure 3.4
Case IV : The invariance properties for a given weight but different pursuit sequences
are demonstrated. Consider the basic pursuit sequences BPS1 and BPS2 and weight
w1. Assuming the gains are same as in Case II, the agents converge to the point Zf =
(−16.47, 2.52) for both the pursuit sequence as seen in Figure 3.5.
Case V : We demonstrate that the stability and the rendezvous point is invariant
under finite switching of the pursuit sequences and weights. The gains are taken to be
the same as in Case II. At t = 0, the pursuit sequence is (BPS1, w1). The pursuit sequence
switches to (BPS1, w2) at t = 0.05, and to (BPS2, w1) at t = 0.2. The trajectories are
shown in Figure 3.6(a). We observe that the reachable point Zf = (−16.47, 2.52) remains
the same as in the case of no switching (Figure 3.6(b)).
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 66
−20 −10 0 10 20−15
−10
0
10
15
(a) Pursuit sequence (BPS1, w1)−20 −10 0 10 20
−15
−10
0
10
15
(b) Pursuit sequence (BPS2, w1)
Figure 3.5: Invariance property of the reachable point, Zf = (−16.47, 2.52) for a givenweight and different basic pursuit sequences (Case IV)
−20 −10 0 10 20−15
−10
0
10
15
(a) Finite switching−20 −10 0 10 20
−15
−10
0
10
15
(b) No switching
Figure 3.6: Invariance property of the reachable point, Zf = (−16.47, 2.52) with switch-ing of pursuit sequence from (BPS1, w1) → (BPS1, w2) → (BPS2, w1) (Case V)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 67
−20 −10 0 10 20−15
−10
0
10
15
(a) Infinite switching−20 −10 0 10 20
−15
−10
0
10
15
(b) No switching
Figure 3.7: Invariance property of the reachable point, Zf = (−16.47, 2.52) with infiniteswitching from (BPS1, w1) → (BPS1, w2) → (BPS2, w1) → (BPS1, w1) → (BPS1, w2)→ . . . (Case VI)
Case VI : We demonstrate that the stability and the rendezvous point is invariant
under infinite switching of the pursuit sequence. The gains of the agents are same as in
Case II. The pursuit sequences are switched from (BPS1, w1) → (BPS1, w2) → (BPS2,
w1) → (BPS1, w1) → (BPS1, w2) → . . . and the time between each switch ∆t = 0.02.
The trajectories are shown in Figure 3.7 and we observe the invariance property.
3.5.3 Computation of controller gains for a rendezvous point
Here, we demonstrate that to make the agents converge to a desired point, we can select
the gains as illustrated in Theorem 2.4.
Case VII : In this case, we consider 5 agents with initial positions and gains as in
Case VI of Section 2.5. We want the agents to converge at Zf = (0, 0). Assume the
pursuit sequence as BPS=(1, 2, 3, 4, 5) and weights as w = (0.5, 0.5, 0, 0). Considering
the same gains as in Case VI of Section 2.5, the trajectories are shown in Figure 3.8.
We observe the trajectories are different but the rendezvous occurs at Zf = (0, 0). This
shows that the computations give in Chapter 2 are sufficient to determine the required
controller gain for rendezvous at the specified point.
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 68
−10 0 10 15−10
0
10
15
2
1
3
5
4
Figure 3.8: Trajectories of 5 agents using centroidal cyclic pursuit (CCP) converging atZf = (0, 0) (Case VII)
3.6 Conclusions
In this chapter, the stability, rendezvous and invariance properties of the centroidal
cyclic pursuit (CCP) are studied. The stability, reachable point and reachable sets are
the same for both the basic and centroidal cyclic pursuit case. We observe the invariance
properties of stability and rendezvous point with respect to pursuit sequence. In the
next chapter, we generalize centroidal cyclic pursuit by relaxing the requirement that
the weights w used by each agent is the same and consider different weights.
Chapter 4
Rendezvous using linear Generalized
Centroidal Cyclic Pursuit
This chapter generalizes the cyclic pursuit strategies discussed in the previous two chap-
ters. Here, the agents follow a centroidal cyclic pursuit but the weights used by each
agent to compute the centroid are different. The stability, reachable/rendezvous point,
reachable set and the invariance properties are studied under this generalized centroidal
cyclic pursuit (GCCP) strategy.
4.1 Problem formulation
A group of n agents are considered in a d dimensional space as in Chapters 2 and 3.
They are ordered from 1 to n. At any time t, the agents are at positions Zi(t) =
[z1i (t) z2
i (t) . . . zdi (t)]
T ∈ Rd. Each agent i follows a point Zic which is the weighted
centroid of the position of the remaining n − 1 agents, as discussed in Chapter 3. The
agent i uses a weight wi ∈ Γ where Γ is the set defined in (3.4) and is given by
Γ =
{w = (η1, η2, . . . , ηn−1)
∣∣∣n∑
i=1
ηi = 1, 0 < ηi < 1
}(4.1)
69
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit70
Assuming the basic pursuit sequence BPS=(1, 2, . . . , n), the state equation of the agent
i can be written as
Zi = ui = ki(Zic − Zi) (4.2)
where
Zic = ηi1Zi+1 + . . . + ηi
n−iZn + ηin−i+1Z1 + . . . + ηi
n−1Zi+1 =n−1∑j=1
ηijZi−j (4.3)
Here, the summation is mod n and the weights wi = (ηi1, η
i2, . . . , η
in−1) ∈ Γ. Thus, the
set of weights wi that the agent i uses may be different from the set of weights wj used
by agent j. This is a generalization of the concept of centroidal cyclic pursuit and hence
is called generalized centroidal cyclic pursuit (GCCP).
Now, for each agent i, (4.2) can be decoupled along each coordinate and as in (2.6)
and (3.5), we will have, for all the agents, d identical linear systems of equations, given
by
X = AX (4.4)
where,
A =
−k1 η11k1 η1
2k1 · · · η1n−1k1
η2n−1k2 −k2 η2
1k2 · · · η2n−2k2
...
ηn1 kn ηn
2 kn ηn3 kn · · · −kn
(4.5)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit71
Equation (4.4) can be analyzed similar to (3.5). We can write A = Kχ where
K =
k1 0 0 · · · 0
0 k2 0 · · · 0...
0 0 0 · · · kn
(4.6)
is the gain matrix and
χ =
−1 η11 η1
2 · · · η1n−1
η2n−1 −1 η2
1 · · · η2n−2
...
ηn1 ηn
2 ηn3 · · · −1
(4.7)
is the weight matrix or the pursuit sequence matrix of genralized centroidal cyclic pursuit
(GCCP).
We can write the characteristic polynomial of A as
ρ(s) = sn + Bn−1sn−1 + Bn−2s
n−2 . . . + B2s2 + B1s + B0 (4.8)
This has the same form as (2.9) and (3.9). Similar to Section 3.1, we will evaluate B0
and B1 for further analysis.
The sum of the elements of the rows of χ is zero. Therefore, using Gershgorin’s disc
theorem, we have Rank(χ) = n − 1, as shown in Section 3.1. Thus, Rank(A) = n − 1
and B0 = 0, B1 6= 0. The expression for B1 can be obtained using the L’Hospital rule
B1 = lims→0
det(sI − A)
s= lim
s→0
[ d
ds
{det(sI − A)
}](4.9)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit72
Let, mij represents the minor of the (i, j)-th element of A, then as s → 0
B1 = (−1)n−1
n∑i=1
mii = (−1)n−1trace(Adj(A))
= (−1)n−1trace{
Adj(χ) Adj(K)}
(4.10)
Let, the adjoint of χ be
Adj(χ) =
ξ11 ξ12 ξ13 · · · ξ1n
ξ21 ξ22 ξ23 · · · ξ1n
...
ξn1 ξn2 ξn3 · · · ξnn
(4.11)
Theorem 4.1 The adjoint of the pursuit sequence matrix χ, for generalized centroidal
cyclic pursuit, is given by
Adj(χ) = (−1)n+11n×1
[|ξ1| |ξ2| . . . |ξn|
](4.12)
for some ξj that is in (3.12), ξij = ξj,∀i, j, with the sign of each ξj determined by n.
Proof. Since the sum of the columns of χ = 0, as shown in Section 3.1, the elements of
a column of Adj(χ) are identical, that is, ξij = ξj,∀i, j. However, since the sum of the
elements of a column of χ need not necessary be zero, the elements of a row of Adj(χ)
are not necessarily identical.
Again, from Gershgorin’s Disc Theorem, it can be proved as in Section 3.1 that, if n
is odd, ξi > 0 and if n is even, ξi < 0. Thus, ξi,∀i have the same sign and each row of
Adj(χ) can be written as (−1)n+11n×1
[|ξ1| |ξ2| . . . |ξn|
]. Hence, we get (4.12). ¤
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit73
Then, from (4.10), we can write
B1 = (−1)n+1
n∑i=1
{(−1)n+1|ξi|
n∏
j=1,j 6=i
kj
}
=n∑
i=1
|ξi|{ n∏
j=1,j 6=i
kj
}(4.13)
B1 has similar form as in (3.20). Now, since χ is singular,
Adj(χ).χ = 0 (4.14)
Using (4.4) and (4.14), let us evaluate
(−1)n+1[|ξ1| |ξ2| . . . |ξn|
]K−1X
= (−1)n+1[|ξ1| |ξ2| . . . |ξn|
]K−1(Kχ)X
= (−1)n+1[|ξ1| |ξ2| . . . |ξn|
]χ X
= 01×nX = 0 (4.15)
Simplifying (4.15),
n∑i=1
|ξi|xi
ki
= 0 (4.16)
⇒n∑
i=1
|ξi|xi
ki
= constant (4.17)
In the next section, the conditions for stability of generalized centroidal cyclic pursuit
(GCCP) is analyzed.
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit74
4.2 Stability analysis
The stability of the system, under generalized centroidal cyclic pursuit, is analyzed sim-
ilar to the analysis done in Sections 2.2 and 3.2.
Theorem 4.2 The linear system, given by (4.4), is stable if and only if the following
conditions hold
(a) At most one ki is negative or zero, that is, at most for one i, ki ≤ 0 and kj > 0,
∀j, j 6= i.
(b)∑n
i=1 |ξi|( ∏n
j=1,j 6=i kj
)> 0
Proof. This proof is similar to Theorem 2.2 and 3.2. From (4.13), Condition (b) implies
B1 > 0. The Gershgorin’s discs of A are similar to that in Theorem 3.2, since the sum
of the elements of the rows of χ are zero for both the strategies.
First, we prove the ’if’ part, that is, if Conditions (a) and (b) are satisfied, then the
system is stable. Three different cases are considered.
Case 1: All the gains are positive.
When ki > 0,∀i, Condition (b) is automatically satisfied. The Gershgorin’s discs of A
are centered at (−ki, 0) with radius −ki, i = 1, . . . , n. Therefore, all the eigenvalues of
A has negative real part except only one at the origin, since B0 = 0 and B1 6= 0. Hence
the system is stable.
Case 2: One gain is zero and other gains are positive.
This case is similar to Case 1. All the Gershgorin’s discs lie on left half of the s plane,
and B0 = 0 and B1 6= 0. Hence, the system is stable.
Case 3: One gain is negative and other gains are positive.
Let ki < 0 and kj > 0, ∀j, j 6= i. Then, for Condition (b) to be satisfied, ki > ki where
ki = − |ξi|∏n
j=1,j 6=i kj∑nl=1,l 6=i |ξl|
∏nj=1,j 6=i,l kj
(4.18)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit75
This equation is similar to (2.12) and (3.21). It can be shown that, given the gains
kj > 0,∀j, j 6= i, if ki > ki, then the system is stable as discussed in the proof of
Theorem 2.2. We omit the details of the proof here.
The “only if” part is proved by contradiction. Assume the system is stable but any
one or both the conditions do not hold. We consider the following cases separately.
Case 1: More than one gain is zero
When two or more gains are zero, B1 = 0, which implies more than one root at the
origin, and hence the system is unstable.
Case 1: More than one gain is negative.
Consider a matrix
A =
−k1 αη11k1 αη1
2k1 · · · αη1(n−1)k1
αη2(n−1)k2 −k2 αη2
1k2 · · · αη2(n−2)k2
...
αηn1 kn αηn
2 kn αηn3 kn · · · −kn
(4.19)
where α varies from 0 to 1. This matrix is similar to (3.22). At α = 0, A has more than
one eigenvalues on the right hand side. From the continuity of the root locus and the
Gershgorin’s disc theorem, as α varies from 0 to 1, there will be more than one root of
A on the right hand side or the origin of the s plane, and hence the system is not stable.
This proof is illustrated in Theorem 2.2.
Case 3 :∑n
i=1 |ξi|( ∏n
j=1,j 6=j kj
)≤ 0
This case implies B1 ≤ 0 and hence, the system is unstable. ¤
Therefore, as in basic cyclic pursuit (BCP) and centroidal cyclic pursuit (CCP), when
the system under generalized centroidal cyclic pursuit (GCCP) is stable, there is one and
only one eigenvalue of A at the origin and the others are all on the left hand side of the
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit76
s plane. Thus, we can write the solution of (4.4), similar to (2.19) and (3.23), as
xi(t) = xif +
∑p∈Sr
{ n∑q=1
( np∑r=1
aipqrt
r−1)xq(0)
}eσpt
+∑p∈Si
{ n∑q=1
( np∑r=1
aipqrt
r−1 cos(ωpt) + ai∗pqrt
r−1 sin(ωpt))xq(0)
}eσpt (4.20)
where again xif corresponds to the zero eigenvalue and ai
pqr and ai∗pqr are complex conjugate
and are functions of χ and K. When the system is stable, i.e., σp < 0,∀p, as t → ∞,
xi(t) = xif ,∀i. Now, the eigenvector corresponding to the zero eigenvalue is v = 11×n
and it spans the null space of A, as Nullity(A) = 1. Thus, at equilibrium, the solution
of X = AX = 0 is X = c11×n for some c. Therefore, all the agents will converge to a
point and xif = xf , ∀i. In the next section, the point of rendezvous is analyzed.
4.3 Rendezvous and Reachable point
We analyze the rendezvous of a stable system similar to Sections 2.3 and 3.3
Theorem 4.3 (Reachable Point) If a system of n-agents, with equations of motion
given in (4.2), have their initial positions at Z0 = {Zi(t0)}ni=1, gain matrix K and pursuit
sequence matrix χ, that satisfies Theorem 4.2, then they converge to a point Zf given by,
Zf =n∑
i=1
{( |ξi|/ki∑nj=1 |ξj|/kj
)Zi(t0)
}=
∑ni=1(|ξi|/ki)Zi(t0)∑n
i=1 |ξi|/ki
(4.21)
where Zf is called a reachable point or the rendezvous point of this system of n agents.
Proof. Equation (4.17) holds for all the directions. Thus, in general, we can write, ∀t
n∑i=1
|ξi|ki
Zi(t) = constant (4.22)
⇒n∑
i=1
|ξi|ki
Zi(t0) =n∑
i=1
|ξi|ki
Zi(tf ) (4.23)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit77
where, Zi(t0) and Zi(tf ) are the initial and final position of the ith agent, respectively.
When the system is stable, all the agents converge to a point, and Zi(tf ) = Zf , ∀i. Thus,
n∑i=1
|ξi|ki
Zi(t0) =n∑
i=1
|ξi|ki
Zf = Zf
n∑i=1
|ξi|ki
(4.24)
⇒ Zf =
∑ni=1(|ξi|/ki)Zi(t0)∑n
i=1 |ξi|/ki
(4.25)
Hence, we get (4.21). ¤
The reachable point Zf , given in (4.21), is different from (2.22) and (3.24). Here,
the gains ki are multiplied by a factor 1/|ξi|. However, since 1/|ξi| > 0, the reachable
set remains the same as for basic cyclic pursuit and centroidal cyclic pursuit. This is
discussed in the next theorem.
Given the initial positions of the agents Z0 and the weight matrix χ, let us define the
reachable set as
Zf (Z0) =
{Zf (Z
0, k, χ)∣∣∣ ∀k satisfying Theorem 4.2
}(4.26)
Here, Co(Z0) and Cp has the same definition as in Section 2.3.
Theorem 4.4 Consider a system of n agents with equation of motion given in (4.2) and
initial positions at Z0. A point Z is reachable if and only if,
Z ∈ Co(Z0)⋃ { n⋃
p=1
Cp
}= P(Z0) (4.27)
that is, Zf (Z0) = P(Z0).
Proof. The proof follows similar to Theorem 2.4. First, we show that Zf (Z0) ⊆ P(Z0).
Let Z ∈ Zf (Z0). Then, by definition of Zf (Z
0), there exists a gain matrix K, satisfying
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit78
Theorem 4.2, such that,
Z =
∑ni=1(|ξi|/ki)Zi(t0)∑n
i=1 |ξi|/ki
(4.28)
holds. We will show that Z ∈ P(Z0). Consider the following cases.
Case I : Let ki > 0, ∀i. Then, (4.28) can be written as
Z =n∑
i=1
(|ξi|/ki)∑nj=1(|ξj|/kj)
Zi(t0) (4.29)
Thus, Z is a convex combination of Zi(t0), i = 1, ..., n. Hence, Z ∈ Co(Z0) and so
Z ∈ P(Z0).
Case II : Let one of the gains kp < 0 and the remaining ki > 0, ∀i, i 6= p. Then, from
(4.29),
Z =n∑
i=1,i6=p
(|ξi|/ki)∑nj=1(|ξj|/kj)
Zi(t0) +(|ξp|/kp)∑nj=1(|ξj|/kj)
Zp(t0) (4.30)
⇒ Z
n∑i=1
|ξi|ki
=∑
i=1,i 6=p
|ξi|ki
Zi(t0) +|ξp|kp
Zp(t0) (4.31)
⇒ Z
n∑i=1
|ξi|ki
− Zp(t0)n∑
i=1
|ξi|ki
=∑
i=1,i6=p
|ξi|ki
Zi(t0) +|ξp|kp
Zp(t0)− Zp(t0)n∑
i=1
|ξi|ki
(4.32)
⇒{
Z − Zp(t0)} n∑
i=1
|ξi|ki
=n∑
i=1,i6=p
|ξi|ki
{Zi(t0)− Zp(t0)
}(4.33)
Since kp < 0 and ki > 0, ∀i, i 6= p, we have
n∏i=1
ki < 0 (4.34)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit79
From Condition (b) of Theorem 4.2, a stable system will have
n∑i=1
|ξi|( n∏
j=1,j 6=i
kj
)> 0 (4.35)
Dividing the above equation by∏n
i=1 ki < 0, we get
n∑i=1
|ξi|ki
< 0 (4.36)
Let
n∑i=1
|ξi|ki
= −1
c(4.37)
where c > 0. Then, from (4.33)
−1
c
{Z − Zp(t0)
}=
n∑
i=1,i 6=p
|ξi|ki
(Zi(t0)− Zp(t0)) (4.38)
⇒ Z − Zp(t0) =n∑
i=1,i6=p
−c|ξi|ki
(Zi(t0)− Zp(t0)) (4.39)
⇒ Z = Zp(t0) +n∑
i=1,i 6=p
c|ξi|ki
(Zp(t0)− Zi(t0)) (4.40)
Then, from (2.31), Z ∈ Cp and so Z ∈ P(Z0).
Case III : Let one of the gains kp = 0 and the remaining ki > 0, ∀i, i 6= p. We can write
(4.28) as
Z =
∑ni=1 |ξi|(
∏nj=1,j 6=i kj)Zi(t0)∑n
i=1 |ξi|(∏n
j=1,j 6=i kj)(4.41)
Putting kp = 0 in the above equation
Z =|ξp|(
∏nj=1,j 6=p kj)Zp(t0)
|ξp|(∏n
j=1,j 6=p kj)= Zp(t0) (4.42)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit80
Thus, Z ∈ Co(Z0) and so Z ∈ P(Z0).
Therefore, from Cases I−III, if Z ∈ Zf (Z0), then Z ∈ P(Z0) or Zf (Z
0) ⊆ P(Z0).
Now, to prove P(Z0) ⊆ Zf (Z0), we will show that for any point Z ∈ P(Z0), there
exists K such that (4.28) holds. We can partition P(Z0), similar to (2.48), as
P(Z0) = P1(Z0) ∪ P2(Z
0) ∪ P3(Z0) (4.43)
where,
P1(Z0) = int{P(Z0)} (4.44)
P2(Z0) =
{Zi(t0)
∣∣∣Zi(t0) ∈ ∂{P(Z0)}}
(4.45)
P3(Z0) = ∂{P(Z0)} \ P2(Z
0) (4.46)
We will consider these sets separately.
Case I : Z ∈ P1(Z0). We have the following cases:
Case Ia: Let Z ∈ int{Co(Z0)}. Then, there exists αi, i = 1, . . . , n,∑n
i=1 αi = 1 with αi >
0,∀i such that
n∑i=1
αiZi(t0) = Z (4.47)
Let
ki =c|ξi|αi
, i = 1, 2, . . . , n (4.48)
where, c > 0 is any positive constant. Thus, ki > 0, ∀i, and
n∑i=1
|ξi|ki
=1
c(4.49)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit81
Replacing αi by (c|ξi|)/ki in (4.47),
Z =n∑
i=1
(c|ξi|ki
)Zi(t0) (4.50)
=n∑
i=1
( |ξi|/ki
1/c
)Zi(t0) (4.51)
=n∑
i=1
{|ξi|/ki∑n
j=1 |ξj|/kj
}Zi(t0) (4.52)
The above equation is the same as (4.28) and all the gains satisfy Theorem 4.2. Therefore,
Z ∈ Zf (Z0)
Case Ib: Let Z ∈ int{Cp} for some p. Then, there exist βi > 0, i = 1, 2, . . . , n, such that
Case I : We demonstrate that when the gains of all the agents are positive, rendezvous will
occur within Co(Z0). Consider the gains given in Table 4.1. Based on χ given in (4.64),
we compute the adjoint matrix given in (4.65). Thus, we get∑n
i=1 |ξi|( ∏n
j=1,j 6=i kj
)=
1.17×108 > 0. Therefore, Theorem 4.2 is satisfied. From Theorem 4.3 and 4.4, the agents
converge to the point Zf given in (4.21). For the given initial conditions, pursuit sequence
matrix and gains, we compute Zf = (60.12, 12.47, 58.40) ∈ Co(Z0). The trajectories of
the agents are shown in Figure 4.1, which verifies the rendezvous point.
Case II : Here, we consider one gain to be negative and all others gains to be positive
such that Theorem 4.2 is satisfied. We show that the agents converge to a point. The
gains of the agents are given in Table 4.1, where the agent 11 has a negative gain. For
this set of gains,∑n
i=1 |ξi|( ∏n
j=1,j 6=i kj
)= 1.47× 107 > 0 and hence the system is stable
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit86
0
50
100
0
50
1000
50
100
Figure 4.1: Trajectories of a swarm of 12 agents when the gains of all the agents arepositive (Case I)
0
50
100
−50
0
50
1000
50
100
Figure 4.2: Trajectories of a swarm of 12 agents when the gain of one of the agent isnegative and the other gains are positive such that Theorem 4.2 is satisfied (Case II)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit87
0
50
100
0
50
1000
50
100
Figure 4.3: Trajectories of the agents converging to Zf = (60, 60, 60) (Case III)
as the gains satisfy Theorem 4.2. The lower bound k11 = −0.43 < k11. From Theorems
4.3 and 4.4, Zf = (89.64,−34.05, 90.09) /∈ Co(Z0). The simulation, given in Figure 4.2,
confirms the reachable point.
4.5.2 Computation of controller gains for a rendezvous point
Case III : Let the desired rendezvous point be Zf = (60, 60, 60) ∈ Co(Z0). To satisfy
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit89
0
50
100
0
50
1000
50
100
(a) CCP with pursuit sequence (BPS, w)
0
50
100
0
50
1000
50
100
(b) GCCP with pursuit sequence (BPS, χ)
Figure 4.5: Trajectories of the agents under centroidal cyclic pursuit (CCP) and gener-alized centroidal cyclic (GCCP) (satisfying some properties) demonstrating the pursuitsequence invariance of the rendezvous point Zf = (64.3, 41.3, 58.7) (Case V)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit90
0
50
100
0
50
1000
50
100
(a) CCP Zf = (64.3, 41.3, 58.7)
0
50
100
0
50
1000
50
100
(b) GCCP Zf = (60.12, 12.47, 58.40)
Figure 4.6: Trajectories of the agents under CCP and GCCP demonstrating that therendezvous point is not pursuit sequence invariance (Case VI)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit91