Multi-agent formation control for target tracking and ...1091904/...A friendly thank goes to...

INOM EXAMENSARBETE ELECTRICAL ENGINEERING,AVANCERAD NIVÅ, 30 HP

, STOCKHOLM SVERIGE 2017

Multi-agent formation control for target tracking and circumnavigation missions

ANTONIO BOCCIA

KTHSCHOOL OF ELECTRICAL ENGINEERING

Multi-agent formation control for target tracking andcircumnavigation missions

ANTONIO BOCCIA

M.Sc. ThesisStockholm, Sweden 2017

TRITA-EE 2017:030ISSN 1653-5146ISRN KTH/xxx/xx--yy/nn--SEISBN x-xxxx-xxx-x

KTH School of Electrical EngineeringSE-100 44 Stockholm

SWEDEN

Akademisk avhandling som med tillstand av Kungl Tekniska hogskolan framlagges till of-fentlig granskning for avlaggande av examensarbete i reglerteknik i .

© Antonio Boccia, February 24th, 2017

Tryck:

Abstract

In this thesis, we study a problem of target tracking and circumnavigation with anetwork of autonomous agents. We propose a distributed algorithm to estimate theposition of the target and to drive the agents to rotate around the target while forminga regular polygon and keeping a desired distance from it. We formally show that thealgorithm attains exponential convergence of the agents to the desired polygon if thetarget is stationary, and bounded convergence if the target is moving with boundedspeed. Numerical simulations in Matlab-Simulink and ROS corroborate the theoreticalresults and demonstrate the resilience of the network to addition and removal of agents.

Sammanfattning

I denna avhandling studeras ett problem av malfoljning och omsegling med ennatverk av sjalvstyrande agenter. Vi foreslar en distribuerad algoritm for att upp-skatta positionen av malet och for att driva agenterna att rotera runt malet, medan debildar en regelbunden polygon och haller ett onskat avstand fran malet. Vi formelltvisar att algoritmen uppnar exponentiell konvergens av agenterna till den onskadepolygonen om malet ar stillastaende, och avgransad konvergens om malet ror sig medavgransad hastighet. Numeriska simuleringar i Matlab/Simulink och ROS bekraftarden teoretiska resultat och demonstrerar natverkets spanstighet till addition och avfly-ttning av agenter.

Contents

Contents v

Acknowledgments vii

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Statement of contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Technical preliminaries 52.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Elements of graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Agreement in directed networks . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Elements of non linear control theory . . . . . . . . . . . . . . . . . . . . . . 62.5 Properties of matrices defined on unit-norm vectors . . . . . . . . . . . . . . 82.6 Persistency of excitation condition . . . . . . . . . . . . . . . . . . . . . . . 92.7 Singular perturbations on the infinite interval . . . . . . . . . . . . . . . . . 10

3 Control algorithm 123.1 Dynamics of the network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Collective bearing-only circumnavigation . . . . . . . . . . . . . . . . . . . . 133.3 Estimate algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Circumnavigation of a stationary target with known position 174.1 Problem statement and main result . . . . . . . . . . . . . . . . . . . . . . . 174.2 Proof of the convergence of the distance errors . . . . . . . . . . . . . . . . 184.3 Geometric properties of the network . . . . . . . . . . . . . . . . . . . . . . 204.4 Proof of the convergence of the counterclockwise angles for a network of two

agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5 Proof of the convergence of the counterclockwise angles for a network of

N ≥ 2 agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.6 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.7 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Circumnavigation of a stationary target with estimated position 335.1 Problem statement and main result . . . . . . . . . . . . . . . . . . . . . . . 33

v

Contents

5.2 Proof of the convergence of the estimate errors . . . . . . . . . . . . . . . . 355.3 Proof of the convergence of the distance errors . . . . . . . . . . . . . . . . 405.4 Proof of the convergence of the counterclockwise angles . . . . . . . . . . . 415.5 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.6 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Circumnavigation of a mobile target with estimated position 496.1 Problem statement and main results . . . . . . . . . . . . . . . . . . . . . . 496.2 Proof of the bounded convergence of the estimate errors . . . . . . . . . . . 516.3 Proof of the bounded convergence of the distance errors . . . . . . . . . . . 536.4 Proof of the bounded convergence of the counterclockwise angles errors . . 546.5 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.6 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 Circumnavigation in switching networks 627.1 Switching control algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.2 Implementation in ROS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

8 Conclusions 758.1 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.2 Future developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Bibliography 76

vi

Acknowledgments

I thank my advisor Mario di Bernardo, for having introduced me to the world of nonlinearcontrol and for having given me the opportunity to join the interesting Erasmus experience. Iam grateful to my examiner Karl Henrik Johansson for having welcomed me at KTH and forhaving provided me interesting advices for my thesis work. I thank my supervisor AntonioAdaldo, for his huge availability and for the attention and professionality in revisioning mywork.

I am grateful to my marvellous parents, Giuseppe and Maria Carmela, to have alwayssupported me and for the exemplary teachings in every aspect of my life: this thesis isdedicated to you.

A special thank goes to my soul sister Simona, for having proved me how can someonebe near even if she is thousands kilometers away: I am glad to have finally met you.

I thank my nearest neighbors Salvatore and Francesco, for the funny moments spentin the last twenty years, and I am grateful to the loyal and always present friend of a lifeAndrea.

I thank my friends and colleagues Marco and Vincenzo F., for the perfect synchronizationin finishing our academic path together, as we started together six years ago.

I am grateful to my friends and colleagues that are living an academic experience abroadAgostino, Vincenzo S., Dario, Carmela and Alberto, for the personal growth derived fromthe exchange of views with them.

A friendly thank goes to Massimiliano, Matteo, Umar, Paul and Pedro for having sharedgood moments in Stockholm in the last six months.

Last, but not least, I am grateful to Armando, for his perfect timing.

vii

Chapter 1

Introduction

In this thesis, we present a multi-agent control for a network of agents in order to performa circumnavigation task around a mobile target.

In the most general scenario, every agent in the network needs an estimator to localizethe target, and a controller to approach it and to start circling around it; the multi-agentbehavior achieved by the agents is the equidistance of them on a desired circle centred atthe target position, thus the formation of a regular polygon.

In this work, we propose an estimate algorithm and a control algorithm to drive theagents of a directed network to achieve the collective behavior described above. The behav-ior of the closed-loop system is analyzed in three different scenarios. In the first scenario, theposition of a stationary target is known to the network; this scenario allows us to separatethe study of the formation on the circle from the study of the exponential convergence of theestimate. In the second scenario, an estimator is introduced to localize a stationary target;the results achieved in the previous scenario on the convergence and the formation on thedesired circle are extended. In the third scenario, the mobility of the target is added, andboth analytical proofs and simulations in Matlab-Simulink show that the circumnavigationtask and the polygonal formation are reached only through a bounded convergence of thesystem variables.

In the last part of this work, we present a modified version of the distributed controlalgorithm, that is adopted in order to simulate switching networks, where the addition andthe removal of agents are considered.

1.1 Motivation

The problem of target tracking and circumnavigation finds applications in numerous fields.A first motivation for driving a network of agents to circle around a mobile target is given bysurveillance missions, where the agents (e.g., a team of UAVs, like the drone in Figure 1.1)are controlled to circumscribe an object (e.g., a building or an infrastructure) that they haveto monitor. An interesting application is the escorting and patrolling mission by a networkof unicycles, or more generally by a group of autonomous vehicles. A new challengingapplication is the tracking of an underwater target by a network of AUVs (see Figure 1.2),where the vehicles are employed to study, and possibly reproduce, the behavior of small seaanimals.

1

1.2. Literature review

Figure 1.1: Unmanned aerial vehicle.

Figure 1.2: Autonomous underwater vehicle.

1.2 Literature review

The design of a control algorithm that drives an agent to approach a target and to follow acircle trajectory around it has been studied in [1–5]. The solutions proposed for the singleagent [5], have been extended to multi-agent systems, where a great attention has beengiven to the formation of a regular polygon inscribed in a desired circle, centered at thetarget position [6–9]. This type of formation is optimal to solve triangulation problemsand it is a good solution to control agents that cannot stop moving, such as UAVs. Forexample, an application of the circumnavigation to escorting and patrolling missions isanalyzed in [10], assuming that a unicycle vehicle can measure both the bearing anglesformed with its neighbor and the distance from a target object.

In a large number of applications, the position of the target is unknown to the agents,so that a localization procedure is required to achieve the tracking. In [11] a peer-to-peercollaborative localization is studied for a network of sensors, while in [12] a stereo-vision-type estimation is realized by the leading agent, sending its visual measurements of thetarget to its followers.

The problem of using the information obtained from an identification process of unknowncharacteristics of a system to update a control algorithm is known in the literature as the

2

1.3. Statement of contribution

dual problem [13,14]. In [15] and [5], the circumnavigation of a target with unknown positionis formally modeled as a dual problem. In particular, in [15] the dual problem is solvedusing distance measurements, while in [5] it is solved using bearing measurements. In [16]the robot and landmark localization problem is solved using an association of data frombearing-only measurements.

1.3 Statement of contribution

Similarly to the solution adopted in [5] and [9], we propose a distributed control algorithmbased on bearing measurements, and an estimator to localize a mobile target. Specifically, inthis thesis we propose a different control strategy where every agent has a tangential motiondepending on its estimated distance from the target. Moreover, in our algorithm every agentupdates its control signal on the basis of information it received by other agents within adefined communication radius, irrespectively of the distance from the target. Adoptingthis control strategy leads to some major improvements. Firstly, the angular velocity ofthe agents about the target does not grow unbounded when the desired distance from thetarget is small. Secondly, we formally prove the exponential convergence of the agents to aregular polygon.

The control algorithm is simulated in ROS [17], where each simulated agent is im-plemented as a separate ROS node. The simulations also demonstrate the resiliency ofthe algorithm to addition and removal of some agents, showing that the agents rearrangethemselves to form a different polygon when one agent enters or leaves the network. Eachagent starts in a monitoring position, and when it receives the first bearing measurement ifeffectively enters the network and approaches the desired circle.

1.4 Thesis outline

The rest of the thesis is organized as follows.

Chapter 2: Technical preliminariesIn Chapter 2, we introduce some technical definitions and results that are used in this thesis.

Chapter 3: Control algorithmIn Chapter 3, we present the estimator and the distributed control algorithm adopted inthis thesis work, giving the objectives of the designed network control.

Chapter 4: Circumnavigation of a stationary target with knownpositionIn Chapter 4, we analyze the simplest scenario for the circumnavigation, showing the asymp-totic convergence of the network on the desired circle and the formation through simulationsin Matlab-Simulink.

3

1.4. Thesis outline

Chapter 5: Circumnavigation of a stationary target with estimatedpositionIn Chapter 5, we introduce the estimate of the target position in the analysis of the algorithmfor the circumnavigation, extending the results achieved for the formation in the previouschapter.

Chapter 6: Circumnavigation of a mobile target with estimated positionIn Chapter 6, we analyze the scenario of a network tracking a target moving with a slowdrift, showing the bounded convergence of the system variables.

Chapter 7: Circumnavigation in switching networksIn Chapter 7, we introduce a modified version of the control algorithm and simulationsperformed in ROS, related to the scenario of switching networks.

Chapters 8: Conclusions and future developmentsIn Chapter 8, we present a summary of the results, and discuss directions for future research.

4

Chapter 2

Technical preliminaries

The aim of this chapter is to provide the general notional and the technical concepts fromthe areas of algebraic graph theory and non linear control theory that are used to derivethe main results presented in this thesis.

2.1 Notation

The set of the positive integers is denoted N, while N0 = N ∪ 0.The metric spaces of real numbers are denoted Rn, where n is the dimension of the

space.For n ∈ N, the vector made up of n unitary elements is denoted 1n, the vector made up

of n null elements is denoted 0n, and the n-by-n identity matrix is denoted In.The set of the symmetric matrices in Rn×n is denoted Sn, the set of the positive semidef-

inite matrices in Rn×n is denoted Sn≥0, and the set of positive definite matrices in Rn×n isdenoted Sn>0.

The set of the unit-norm vectos in R2 is denoted S1.The operator [·]i denotes the i-th element of a vector.The operator [·]ij denotes the element at the i-th row and at the j-th column of a matrix.The operator ‖·‖ denotes the Euclidean norm of a vector and the corresponding induced

norm of a matrix.The operators ∧ and ∨ denote the logical and and the logical or respectively.

2.2 Elements of graph theory

In this section, we review the main concepts of graph theory that are used to model multi-agent systems; the mentioned results can all be found in [18].

The multiagent system analyzed in this thesis can be described by a directed and un-weigthed graph, representing the topology of the interconnections among the agents in thenetwork.

Definition 2.2.1 (Digraph). An unweighted directed graph or digraph is a tuple G =(V, E), where V = 1, · · · , n with n ∈ N, E = e1, · · · , em ⊆ V ×V. The elements ofthe set V are called the vertices of the graph, while the elements ek = (j, i) ∈ E, withj, i ∈ V, i 6= j and k ≤ m, are called the edges of the graph.

When an edge ek exists between the vertices i and j we call them adjacent, and the edgeis said to be incident with i and j.

5

2.3. Agreement in directed networks

A directed path of length l in the graph G is a sequence of distinct vertices v1, · · · , vlsuch that vk and vk+1 are adjacent for k = 1, · · · , l − 1.

Definition 2.2.2 (Strongly connected and weakly connected diagraph). A diagraph isstrongly connected if, between every pair of vertices, there is a directed path.

We can draw a digraph with circles representing the vertices and arrows connecting thevertices, representing the edges: for the edge (j, i) ∈ E , j is said to be the tail (the vertexfrom the which the arrow starts), while i is said to be the head (the vertex where the arrowsarrives).

The Laplacian matrix of a digraph is defined as

L = ∆−A,

where ∆ is the degree matrix and the A is the adiacency matrix. The adiacency matrix isdefined as

[A]ij =

1 if (j, i) ∈ E ,0 otherwise,

while the degree matrix is defined as

[∆]ii = d(i)in ∀i ∈ V,

where d(i)in is the in degree of the vertex i, defined as the number of incoming edges.

2.3 Agreement in directed networks

In this section we consider the problem of the agreement or consensus for directed networks,that are used in Chapter 4.

Consider a consensus protocol over a directed network of N agents (i.e., a network thatcan be represented with a digraph); denoting the r-dimensional state vector of every agentas xi(t) ∈ Rr, and collecting everyone of this vectors in the state vector of the networkx(t) ∈ RrN , the overall system can be described by

x = −Lx(t).

We mention the following theorem, providing the conditions for the convergence of a directednetwork to the average consensus (i.e., the common value reached by the agreement protocolthat is the average value of the initial nodes):

limt→∞

x(t) = 1N

1Nr1TNrx0.

Theorem 2.3.1 (Theorem 3.17 in [18]). The agreement protocol over a diagraph reachesthe average consensus for every initial condition if and only if the diagraph is strongly-connected.

2.4 Elements of non linear control theory

Definition and conditions for the existence of limit cyclesIn this section, we give the definition of limit cycle and we present a theorem that providesa sufficient condition to prove the existence of limit cycles; we present this concept in orderto prove the exponential convergence of the agents to a desired circle in Chapter 3 .

6

2.4. Elements of non linear control theory

In order to define a limit cycle, we need to give the definition of ω-limit set and α-limitset.

Definition 2.4.1 (ω-limit set). Let φ(t;x0) be the trajectory of a n-dimensional non linearsystem rooted in x0 ∈ Rn ; a point ω(x0) ∈ Rn is said to be an ω-limit point of the trajectoryof the system φ(t;x0) if there exists a sequence of time instants t0, t1, . . . , tk such that

limk→∞

φ(tk;x0) = ω(x0).

We define the ω-limit set for the trajectory φ(t;x0) as the set of all the ω-limit points

Ωφ(x0) = ω(x0) ∈ Rn.

Definition 2.4.2 (α-limit set). Let φ(t;x0) the trajectory of a n-dimensional non linearsystem rooted in x0 ∈ Rn ; a point α(x0) ∈ Rn is said to be an α-limit point of the trajectoryof the system φ(t;x0) if there exists a sequence of time instants t0, t1, . . . , tk such that

limk→∞

φ(−tk;x0) = α(x0).

We define the α-limit set for the trajectory φ(t;x0) as the set of all the α-limit points

Aφ(x0) = α(x0) ∈ Rn.

Definition 2.4.3 (Periodic orbit, Definition in [19]). Consider the autonomous system

x = f(x(t)), x ∈ Rn, n ≥ 2.

A non-constant solution to this system, x(t), is periodic if there exists a T ≥ 0 such that

x(t+ T ) = x(t), ∀t.

The image of the periodicity interval [0, T ] under x in the state space Rn is called the periodicorbit or cycle.

Definition 2.4.4 (Limit cycle, Definition in [19]). A limit cycle is defined as a periodicorbit Γ for which there exists at least a point x? /∈ Γ such that

Γ = Ωφ(x?) ∨ Γ = Aφ(x?).

In non linear control theory it is important to estabilish whether a system exhibits alimit cycle or not; in [20] are presented methods to assest the existence of limit cycles,among the which the following theorem provides a sufficient condition for the existence.

Theorem 2.4.1 (Poincare-Bendixon Theorem, Theorem 7.3 in [20]). Suppose that R isa compact and invariant subset of the plane and consider the planar non linear systemx = f(x), where f is a continuosly differentiable vector field on an opet set containing R.

If R doesn’t contain any equilibria of the planar system, then R contains a periodic orbit.

Lyapunov stability theoryIn this section, we recall two theorems to study the stability of a dynamic system with theLyapunov theory.

7

2.5. Properties of matrices defined on unit-norm vectors

Theorem 2.4.2 (Lyapunov Theorem for the global asymptotical stability, Theorem 4.2in [21]). Consider an autonomous twodimensional system x = f(x) and let x = 0 be anequilibrium point for this system. Let V : R → R2 be a continuosly differentiable functionsuch that

V (0) = 0, V (x) > 0, ∀x 6= 0,‖x(t)‖ → ∞ =⇒ V (x)→∞,V (x) < 0, ∀x 6= 0.

Then x = 0 is globally asymptotically stable.

Definition 2.4.5 (Positively invariant set, in [21]). Consider the twodimensional dynamicsystem x = f(x) and let φ(t0;x0) be a trajectory of the system rooted at x0 at time t0: wedefine a set M ⊂ R2 positively invariant with respect to the system if

φ(t0;x0) ∈M =⇒ φ(t;x0) ∈M ∀t ≥ t0

Theorem 2.4.3 (La Salle’s theorem, Theorem 4.4 in [21]). Consider the twodimensionaldynamic system x = f(x) and let Ω ⊂ D ⊂ R2 be a positively invariant set for the system.Let V : D → R a continuosly differentiable function such that V (x) ≤ 0 in Ω. Let E be theset of points in Ω where V (x) = 0. Let M the largest invariant set in E; then every solutionstarting in Ω approaches M as t→∞.

2.5 Properties of matrices defined on unit-norm vectors

Both the control algorithm and the estimate presented in the next chapter are based on theupdating of unit-norm vectors.

Let ϕ(t) ∈ S1; defining ϑ(t) as the counterclockwise angle between the vector and thex-axis of a reference frame, ϕ(t) can be represented as

ϕ(t) =[cosϑ(t)sinϑ(t)

].

In this section, some relevant properties of a matrix ϕ(t)ϕT (t) are presented.

Lemma 2.5.1. Let ϕ(t) ∈ S1. The following properties hold:

• ϕ(t)ϕT (t) ∈ S2≥0

• The eigenvalues of ϕ(t)ϕT (t) are 0 and 1

• Defining ϕ(t) ∈ S1 as the unit-norm vector obtaining rotating ϕ(t) clockwise, thefollowing equality holds

I2 − ϕ(t)ϕT (t) = ϕ(t)ϕT (t). (2.1)

Proof. From the definition of the vector ϕ(t) we have

ϕ(t)ϕT (t) =[

cos2 ϑ(t) sinϑ(t) cosϑ(t)sinϑ(t) cosϑ(t) sin2 ϑ(t)

], (2.2)

hence the matrix is symmetric. Furthermore, we have that for every vector z ∈ R2 thequadratic form with the considered matrix as hull is positive semidefinite

zT (ϕ(t)ϕT (t))z = (z1 sinϑ(t) + z2 cosϑ(t))2 ≥ 0 ∀z ∈ R2,

8

2.6. Persistency of excitation condition

hence the matrix is also positive semidefinite.Since the the characteristic polynomial of the matrix is

λ(λ− 1),

the eigenvalues of the matrix are 0 and 1.Because of (2.2), we have that

I2 − ϕ(t)ϕT (t) =[

1− cos2 ϑ(t) − sinϑ(t) cosϑ(t)− sinϑ(t) cosϑ(t) 1− sin2 ϑ(t)

]=[

sin2 ϑ(t) − sinϑ(t) cosϑ(t)− sinϑ(t) cosϑ(t) cos2 ϑ(t)

].

Since the unit-norm vector ϕ(t) ∈ S1 is represented as

ϕ(t) =[

sinϑ(t)− cosϑ(t)

],

we have that

I2 − ϕ(t)ϕT (t) =[sinϑ(t) − cosϑ(t)

] [ sinϑ(t)− cosϑ(t)

]= ϕ(t)ϕT (t),

therefore the equality (2.1) holds.

2.6 Persistency of excitation condition

In this section, we recall the definition of the persistency of excitation condition from [22].

Definition 2.6.1 (Persistency of excitation condition). A vector ϕ : R≥0 → R2 is persis-tently exciting (p.e.) if there exist ε1, ε2, T > 0 such that

ε1I2 ≤∫ t0+T

t0

ϕ(t)ϕT (t)dt ≤ ε2I2, (2.3)

for all t0 ≥ 0. Condition (2.3) is called persistency of excitation condition.

Condition (2.3) requires that the vector ϕ rotates sufficiently in the plane that theintegral of the semipositive definite matrix ϕϕT is uniformly definite positive over anyinterval of some positive length T .

From [22] we have that the p.e. condition has another interpretation, and it can beexpressed in an equivalent scalar form:

ε1 ≤∫ t0+T

t0

(UT ϕ(t))2dt ≤ ε2, (2.4)

for all t0 ≥ 0 and for every constant unit-norm vector U ∈ R2: (2.4) appears as a conditionon the energy of ϕ in all the directions.

Given the previous definition, we recall the following convergence theorem:

Theorem 2.6.1 (P.E. and Exponential Stability, Theorem 2.5.1 in [22]). Let ϕ : R≥0 → R2

be piecewise continuous; if ϕ is P.E., then the system

x(t) = −kϕ(t)ϕT (t)x(t)

is globally exponentially stable.

9

2.7. Singular perturbations on the infinite interval

2.7 Singular perturbations on the infinite interval

In this section we recall the theorem from [23] that is used in Chapter 5.Consider the following perturbed system

x = f(t, x, y, ε), x(t0) = x0,

εy = g(t, x, y, ε), y(t0) = y0,(2.5)

where x(t) ∈ Rk, y(t) ∈ Rj , f = (f1 · · · , fk), g = (g1, · · · , gj) and ε is a small positiveparameter.

To analize the behavior of (2.5) for ε → 0+ and for t0 ≤ t < ∞, we introduce thefollowing systems.

The degenerate system (2.6) is obtained considering the perturbed system at ε = 0.x = f(t, x, y, 0), x(t0) = x0,

0 = g(t, x, y, 0), y(t0) = y0.(2.6)

In this section, we introduce two parameters γ and λ that have respectively the same roleof parameters α and β in [23]. The following system is written by first making a stretchingtransformation of the independent variable s = t−γ

ε , and then setting ε to 0.dxds = 0,dyds = g(γ, x, y, 0).

(2.7)

Since the only solution of dxds = 0 is x = λ and it is constant, we can rewrite the system (2.7)in the following form:

dy

ds= g(γ, λ, y, 0), y(0) = y0, (2.8)

where γ and λ are treated as parameters; the system (2.8) is called the boundary-layersystem.

Let I = [0,∞), SR = (x, y) ∈ Rk+j : ‖x‖+ ‖y‖ ≤ R, and let SR|x and SR|y representthe restrictions of SR to Rk and Rj .

Assume that f and g satisfy the following conditions.

I. The system (2.6) has x = 0, y = 0 as a solution for all t0 ≤ t < ∞. Therefore thesystem (2.6) can be written in a more convenient form as

x = f(t, x, 0j , 0), x(t0) = x0. (2.9)

II. f, g, fx, fy, gx, gy, gt are continuosly differentiable in I×SR×[0, ε0], where [0, ε0] is a setof values for ε. Here fx denotes the matrix with components ∂fi

∂xl, with i, l = 1, · · · , k,

and similarly for gy,gx,fy.

III. It holds thatg(t, x, 0j , 0) = 0 ∀(t, x) ∈ I × SR|x.

IV. The function f is continuous at y = 0, ε = 0, uniformly in (t, x) ∈ I × SR|x, andf(t, x, 0j , 0) and fx(t, x, 0j , 0) are bounded on I × SR|x.

V. The function g is continuous at ε = 0, uniformly in (t, x, y) ∈ I × SR, and g(t, x, y, 0)and its derivatives with respect to t and the components of x and y are bounded onI × SR.

10

2.7. Singular perturbations on the infinite interval

VI. We denote K the class of all continuous, strictly increasing, real-valued functionsd(r), r ≥ 0, with d(0) = 0, and S the class of all continuous, stricly decreasing,nonnegative, real-valued functions σ(s), 0 ≤ s <∞, for which σ(s)→ 0 as s→∞.The zero solution of (2.9) is uniform-asymptotically stable. That is, if φ(t, t0, x0) isthe solution of (2.9), there exist d ∈ K and σ ∈ S such that

‖φ(t, t0, x0)‖ ≤ d(‖x0‖)σ(t− t0) for all ‖x0‖ ≤ R and 0 ≤ t0 ≤ t <∞.

VII. The zero solution of (2.8) is uniform-asymptotically stable, uniformly in the parame-ters (γ, λ) ∈ I ×SR|x. That is, if y = ψ(s, y0, γ, λ) is the solution of (2.8), there existl ∈ K and ρ ∈ S such that

‖ψ(t, y0, γ, λ)‖ ≤ l(‖y0‖)ρ(s),

for all ‖y0‖ ≤ R, 0 ≤ s <∞ and (γ, λ) ∈ I × SR|x.

Theorem 2.7.1 (Theorem in [23]). Let conditions above be satisfied; then, for sufficientlysmall ‖x0‖ + ‖y0‖ and ε, the solution of (2.5) exists and converges to the solution of (2.6)as ε→ 0+ uniformly on any closed subset of t0 < t <∞.

11

Chapter 3

Control algorithm

In this chapter, we present the distributed control algorithm used in this work to control anetwork of agents, in order to drive it to a desired configuration.

In Section 3.1, we present some relevant definitions for the network we analyze in thisthesis; in Section 3.2 and in Section 3.3 we introduce the control algorithm and the estimatealgorithm adopted in this work, for the most general scenario of mobile target with estimatedposition.

3.1 Dynamics of the network

We consider a directed network of N agents indexed as 1, · · · , N , with agent i at positionyi(t) ∈ R2 and a target at position x(t) ∈ R2; the network is represented by a digraphG = (V, E) and the agents are modelled as simple integrators

yi(t) = ui(t) i ∈ V, (3.1)

where ui(t) is the control action applied to the agent i.Below we recall some definitions from [9].

Definition 3.1.1 (Counterclockwise angle). The counterclockwise angle βij(t) at time t isthe angle subtended at x(t) by yi(t) and yj(t) (see Figure 3.1).

Definition 3.1.2 (Counterclockwise neighbor). We define the agent j the counterclockwiseneighbor of the agent i if βij(t) is the smallest counterclockwise angle among all βik(t) fork ∈ V \i.

From the previous definition it is convenient to define the counterclockwise neighborhoodfunction νi(t) ∈ 1, ..., N \ i returning the label of the agent that at time t is thecounterclockwise neighbor of agent i, subtended at x(t).

It is important to notice that the definition of the neighborhood function allows theintroduction and the removal of an agent in the network; this scenario is analyzed in Chap-ter 7. For this purpose, consider that at time t0 νi(t0) = j.

If at time t1 the agent k is added to the network, and it is such that βik(t) < βij(t) thenνi(t1) = k; on the other hand, if at time t2 there is a fault in the network and the agents jand k are removed, it holds that νi(t2) = r where r is the agent forming the third smallestcounterclockwise angle with i in the original network.

12

3.2. Collective bearing-only circumnavigation

Figure 3.1: Counterclockwise angles of the network.

Since the network considered in this thesis can be dynamic, and the counterclockwiseneighbor is time-variant, it is convenient to denote the counterclockwise angle between agenti and its neighbor as βi(t) = βiνi(t); from the previous notation, we have that

βi(t) = βij(t) if j = νi(t).

3.2 Collective bearing-only circumnavigation

Definition 3.2.1 (Bearing vector). The bearing vector for the agent i is defined as the unit-norm vector ϕi(t) ∈ S1 in the direction from the agent to the target; defining the distancefrom the agent and the target as Di(t) = ‖x(t)− yi(t)‖, we have

ϕi(t) = x(t)− yi(t)Di(t)

. (3.2)

In order to introduce the chosen control law, we consider the unit vector ϕi(t) ∈ S1,obtained by a π/2 clockwise rotation of ϕi(t), like shown in Figure 3.2. The control objective

13

3.2. Collective bearing-only circumnavigation

Figure 3.2: Bearing vectors for the single agent: ϕic is the vector ϕi(t).

is formally written as

limt→∞

Di(t) = D∗, (3.3)

limt→∞

βi(t) = 2πN, (3.4)

for all i ∈ 1, · · · , N. In order to achieve (3.3) and (3.4), we propose a distributed controlalgorithm, based on measurements of the bearing vectors and on the estimate of the distancebetween the agent and the target.

For the control law, we set:

ui(t) = kd(Di(t)−D∗)ϕi(t) + kϕDi(t)(α+ βi(t))ϕi(t) (3.5)

where Di(t) = ‖xi(t)− yi(t)‖ is the estimated distance between the agent i and the target,and xi(t) represents the estimate of the target position computed by the agent i at time t;kd and kϕ are positive gains and D∗ is the desired distance from the target.

Note that the control signal ui(t) is made up of two contributions: a radial termkd(Di(t) − D∗)ϕi(t) drives the agent towards the desired circle, and a tangential termkϕDi(t)(α+ βi(t))ϕi(t) makes the agent circumnavigate the target while attaining the de-sired formation with the other agents. Differently from [5, 9], we let the tangential termdepend on the estimated distance from the target, Di(t), in order to avoid high angularvelocities when the desired distance from the target is small. Another important propertyof control law (3.5) is that ui(t) is always nonzero. In fact, since ϕi(t) and ϕi(t) are or-thogonal, and since α + βi(t) > 0, we have that ui(t) = 0 would require Di(t) − D∗ = 0

14

3.3. Estimate algorithm

Figure 3.3: Geometric illustration of the estimator.

and Di(t) = 0, which is not possible since D∗ > 0. This property also implies that theclosed-loop system has no equilibria.

3.3 Estimate algorithm

The chosen algorithm is based on an estimate of the target position, according to thefollowing dynamics

˙xi(t) = −ke(I2 − ϕi(t)ϕTi (t))(xi(t)− yi(t)), i ∈ V (3.6)

where I2−ϕi(t)ϕTi (t) is a projection matrix onto a plane perpendicular to the vector ϕi(t).The goal of the estimator is to make the estimation error

xi(t) = xi(t)− x(t)

converge to the null vector, so that the agent can localize the target. As shown in Figure 3.3,xi(t) moves in a direction orthogonal to the bearing vector; in this way the term xi(t)−yi(t)is rotated so that its direction converges to the direction of ϕi(t), with a rate dependent onthe estimation gain ke.

Since, from Lemma (2.1),

I2 − ϕi(t)ϕTi (t) = ϕi(t)ϕTi (t), ∀i ∈ V

the dynamics of the estimator can be rewritten in terms of the estimation error in thefollowing way

˙xi(t) = −keϕi(t)ϕTi (t)xi(t)− x(t), ∀i ∈ V

15

3.4. Summary

3.4 Summary

In this section, we have described the network of agents we will consider in this work,highlighting remarkable characteristics.

Subsequently, we have introduced the distributed control algorithm and the estimate al-gorithm we will analyze in this thesis; furthermore, we have both given a detailed motivationof the choise of the controller and a geometrical interpretation of the estimator.

In Chapter 7, we propose a modified version of (3.5), in order to achieve the controlobjectives in dynamic networks.

16

Chapter 4

Circumnavigation of a stationarytarget with known position

In this chapter, we assume that every agent in the network knows the position of a stationarytarget; for this purpose there is no need of a localization algorithm.

Since in Chapter 5 we are extending the results achieved in this chapter, this scenario isuseful because it allows to separate the study of the convergence and the formation on thedesired circle from the analysis of the estimate convergence. In Section 4.1, we introducethe control algorithm adopted for this case and we give the main result for this scenario; inSections 4.2, 4.3, 4.5, we give proofs that are used to prove the main result in Section 4.6; inthe end, in Section 4.7, we show the results of simulations performed in Matlab-Simulink.

4.1 Problem statement and main result

In this scenario, every agent of the network has the following dynamics:

yi(t) = kd(Di(t)−D∗)ϕi(t) + kϕDi(t)(α+ βi(t))ϕi(t), (4.1)

where kd, kϕ, α are positive constant.For every agent we define the error on the distance between itself and the target ∆i(t) =

Di(t)−D∗ and the error on the counterclockwise angle βi(t) = βi(t)− 2πN .

The circle of desired radius D∗ centered at the stationary target position x is denotedC(x,D∗).

The main result of this chapter is formalized in the following theorem.

Theorem 4.1.1. Consider a network of N autonomous agents under control law (4.1);then the agents converge to the desired circle C(x,D∗) while forming a regular polygon; i.e.,they achieve the control objective (3.3) and (3.4).

The proof of Theorem 4.1.1 derives from the following sections.In Section 4.2, we prove that the circle C(x,D∗) is an attractive limit cycle for the

trajectories of the network; subsequently in Section 4.3, we present remarkable geometricrelations for the network that are used to prove the asymptotic convergence of the counter-clockwise angles in Section 4.5. Finally in Section 4.6, we use the results obtained in theprevious sections to formalize the proof of Theorem 4.1.1.

17

4.2. Proof of the convergence of the distance errors

4.2 Proof of the convergence of the distance errors

It is convenient to rewrite the system (4.1) using as polar coordinates Di(t) and ϕi(t).

Lemma 4.2.1. Consider a network of N agents under control law (4.1): the closed-loopsystem can be written using as polar coordinates Di(t) and ϕi(t) as follows:

Di(t) = −kd(Di(t)−D∗),ϕi(t) = −kϕ(α+ βi(t))ϕi(t).

(4.2)

Proof. Since Di(t) = ‖x − yi(t)‖ and the target is stationary, differentiating Di(t) w.r.t.time we have that

Di(t) = 1Di(t)

(x− yi(t))T (−yi(t)).

Expressing yi(t) as (4.1) and remembering that x − yi(t) = Di(t)ϕi(t), we can rewrite theprevious equation as

Di(t) = ϕTi (t)[−kd(Di(t)−D∗)ϕi(t)− kϕDi(t)(α+ βi(t))ϕi(t)].

Exploiting the orthogonality between ϕi(t) and ϕi(t), we obtain the following expressionfor the dynamics of Di(t):

Di(t) = −kd(Di(t)−D∗).Differentiating (3.2) w.r.t. time, we write the equation for the dynamics of the bearingvector as

ϕi(t) = d

dt

(x− yi(t)Di(t)

)= − yi(t)

Di(t)− (x− yi(t))

D2i (t)

[1

Di(t)(x− yi(t))T (−yi(t))

].

Since x− yi(t) = Di(t)ϕi(t), we have

ϕi(t) = − yi(t)Di(t)

− ϕi(t)Di(t)

(−ϕTi (t)yi(t)).

Expressing yi(t) like in (4.1) and exploiting the orthogonality between ϕi(t) and ϕi(t), theprevious equation is rewritten as

ϕi(t) = −kϕ(α+ βi(t))ϕi(t).

In order to apply the control law (4.1), we need that the bearing vector ϕi(t) is welldefined for any t ≥ t0, or, in other words, that Di(t) > 0 for all t ≥ t0. Therefore, we willbegin by showing that Di(t) > 0 is always guaranteed.

Lemma 4.2.2. Consider a network of N agents under control law (4.1): it holds that

Di(t) > 0 ∀t ≥ t0, ∀i ∈ V .

Proof. From Lemma 4.2.1 we have that the dynamics of the distance is

Di(t) = −kdDi(t) + kdD∗.

Since the solution of the previous equation is

Di(t) = Di(t0)e−kd(t−t0) +D∗(1− e−kd(t−t0)).

and D∗ is a positive constant, Di(t) is always positive, for all the agents in the network.

18


Figure 4.1: Trapping region in the plane for the single agent.

Note that the tangential velocity kϕDi(t)(α + βi(t)) is always not null, because bothα and the counterclockwise angle βi(t) are positive and for Lemma 4.2.2 Di(t) is alwayspositive; furthermore ϕi(t) ∈ S1, hence it is never null. Therefore system (4.2) has noequilibria and consequently the agent approaches to the target and keeps on rotating aroundit, without stopping on the circle.

Since the dynamics of the distances Di(t) in (4.2) are decoupled, it is possible to analyzethem independently; in this way we prove the existence of an attractive limit cycle for asingle agent so that the proof could be automatically extended to the network.

In order to prove the existence of a limit cycle for the single agent we use Theorem 2.4.1:it is sufficient to find an invariant region in R2, that works like a trapping region for thetrajectories of the single agent in the network.

The idea is to find the maximum radius Di,max ∈ R and the minimum radius Di,min ∈R defining a region in the plane (see Figure 4.1), containing no equilibria, so that thetwodimensional vector field of the system (4.1) points inside this region:

Di,min ∈ R : Di(t) > 0,Di,max ∈ R : Di(t) < 0.

Using the equations of the closed-loop system (4.2), we have thatDi(t) < D∗ =⇒ Di(t) > 0,Di(t) > D∗ =⇒ Di(t) < 0.

Thus the closed orbit C(x,D∗) is an invariant set for the trajectories of the agent i, andsince there are no equilibria of (4.2) in R2, it is an attractive limit cycle for the single agent.

Now consider the vector

D(t) =[D1(t) . . . DN (t)

]T ∈ RN .

Since every element of the vector converge to the same closed orbit with radius D∗ andcentered at the target position, it holds that

limt→∞

D(t) = D∗1N ,

19

4.3. Geometric properties of the network

where the equality is element-wise.Consequently, the invariant region C(x,D∗) is an attractive limit cycle for the closed-

loop system (4.1) and, because of Definition 2.4.4, the network converges asymptotically tothis circle.

Furthermore, from the definition of ∆i(t) in Section 4.1, we write the dynamics of thedistance error

∆i(t) = −kd∆i(t), (4.3)from which it can be noticed that the convergence is also exponential, with a rate of con-vergence depending on the approach term gain.

Hence the agents of the network exponentially converge to the desired circle C(x,D∗) ast→∞.

4.3 Geometric properties of the network

In this section, we present remarkable relations between angles defined in the network, thatare used to rewrite the dynamics of the counterclockwise angles as a consensus equation.Recalling that βi(t) = βi,νi(t)(t) whenever νi(t) is defined, let us consider the generic coun-terclockwise angle βij(t) from ϕi(t) to ϕj(t). First we define ϑij(t) as the counterclockwiseangle between the vectors ϕi(t) and ϕj(t), for i, j ∈ V with j = νi(t), defined according tothe following scalar product:

ϕTi (t)ϕj(t) = cosϑij(t). (4.4)Similarly, the counterclockwise angle βij(t) is defined according to the following scalarproduct:

ϕTi (t)ϕj(t) = cosβij(t). (4.5)

Lemma 4.3.1. Consider the network described by (3.1); then

cosϑij(t) = − cosϑji(t), ∀i ∈ V, j = νi(t), ∀t ≥ t0. (4.6)

Proof. In order to prove (4.6) we analyze all the four possible scenarios depending on thevalue of the angle βij(t).

If 0 ≤ βij(t) ≤ π2 , then, from Figure 4.2 we have that

(2π − ϑij(t)) + βij(t) = π2 ,

(2π − ϑji(t)) = π2 + βij(t),

hence,(2π − ϑij(t)) = π − (2π − ϑji(t)).

If π2 ≤ βij(t) ≤ π, then, from Figure 4.3,

ϑij(t) = βij(t)− π2 ,

ϑji(t) = 2π − π2 − βij , (t)

hence,ϑij(t) = π − ϑji(t).

If π ≤ βij(t) ≤ 3π2 , then, from Figure 4.4,

ϑij(t) = βij(t)− π2 ,

ϑji(t) = 2π − π2 − βij(t),

20


Figure 4.2: Relation between ϑij(t) and ϑji(t): 0 ≤ βij(t) ≤ π2 .

Figure 4.3: Relation between ϑij(t) and ϑji(t): π2 ≤ βij(t) ≤ π.

21


Figure 4.4: Relation between ϑij(t) and ϑji(t): π ≤ βij(t) ≤ 3π2 .

hence,ϑij(t) = π − ϑji(t).

If 3π2 ≤ βij(t) ≤ 2π, then, from Figure 4.5,

(2π − ϑij(t)) = (2π − βij(t)) + π2 ,

(2π − ϑji(t)) + (2π − βij(t)) = π2 ,

hence,(2π − ϑij(t)) = π − (2π − ϑji(t)).

From the previous analysis, it holds that

cosϑij(t) = − cosϑji(t), ∀i ∈ V, j = νi(t), ∀t ≥ t0,

and because of (4.4) we have

ϕTi (t)ϕj(t) = −ϕTj (t)ϕi(t), ∀i ∈ V, j = νi(t), ∀t ≥ t0.

Lemma 4.3.2. Consider the network described by (3.1); then

cosϑij(t) = sin βij(t), ∀i ∈ V, j = νi(t), ∀t ≥ t0. (4.7)

22


Figure 4.5: Relation between ϑij(t) and ϑji(t): 3π2 ≤ βij(t) ≤ 2π.

Proof. Similarly to what done in Lemma 4.3.1, in order to prove (4.6) we analyze all thefour possible scenarios depending on the value of the angle βij(t).

If 0 ≤ βij(t) ≤ π2 , then, from Figure 4.6,

βij(t) = π − π

2 − (2π − ϑij(t)),

hence,βij(t) = ϑij(t)−

3π2 .

If π2 ≤ βij(t) ≤ π, then, from Figure 4.7,

βij(t) = 2π − (π − ϑij(t))−π

2 ,

hence,βij(t) = ϑij(t) + π

2 .

If π ≤ βij(t) ≤ 3π2 , then, from Figure 4.8,

βij(t) = 2π − π

2 − (π − ϑij(t)),


2 .

If 3π2 ≤ βij(t) ≤ 2π , then, from Figure 4.9,

βij(t) = π + π

2 + [π − (2π − ϑij(t))],

23


Figure 4.6: Relation between ϑij(t) and βij(t): 0 ≤ βij(t) ≤ π2 .

Figure 4.7: Relation between ϑij(t) and βij(t): π2 ≤ βij(t) ≤ π.

24


Figure 4.8: Relation between ϑij(t) and βij(t): π ≤ βij(t) ≤ 3π2 .

Figure 4.9: Relation between ϑij(t) and βij(t): 3π2 ≤ βij(t) ≤ 2π.

25

4.4. Proof of the convergence of the counterclockwise angles for a network of two agents


2 .

In all the scenarios it holds that

sin βij(t) = cosϑij(t), ∀i ∈ V, j = νi(t), ∀t ≥ t0.

4.4 Proof of the convergence of the counterclockwise angles for anetwork of two agents

In this section, we use the Lyapunov stability theory to prove the convergence of the errorβij(t) to 0 for a network of N = 2 agents; starting from the results in Lemma 4.3.1 andLemma 4.3.2, we analize the monotony of the derivative of a chosen Lyapunov function.

The goal is to prove that under control law (4.1), the counterclockwise angles βij(t)converge asymptotically to π; the idea is to introduce a vector Φ(t) defined as

Φ(t) =2∑i=1

ϕi(t).

If we manage to prove that limt→∞ Φ(t) = 0, this means that the bearing vectors of thetwo agents are opposite and so that β12(t) = β21(t).

Lemma 4.4.1. Consider a direct network of 2 agents described by (3.1) under control law(4.1): defining the vector Φ(t) as

Φ(t) =2∑i=1

ϕi(t) = ϕ1(t) + ϕ2(t),

it holds that it asymptotically converges to 0 as t→∞.

Proof. The following Lyapunov function is chosen

V (Φ(t)) = 12‖Φ(t)‖2 ≥ 0. (4.8)

In order to study the monotony of the Lyapunov function, we compute its derivative

V (Φ(t)) = ΦT (t)Φ(t) = (ϕT1 (t) + ϕT2 (t))(ϕ1(t) + ϕ2(t)).

Exploiting the orthogonality of the vectors ϕi(t) and ϕi(t) and using (4.2), the derivativeof V (Φ(t)) can be written as

V (Φ(t)) = −kϕ(α+ β21(t))ϕT1 (t)ϕ2(t)− kϕ(α+ β12(t))ϕT2 (t)ϕ1(t).

Because of (4.4) and (4.6), the previous function is

V (Φ(t)) = kϕ(β12(t)− β21(t)) cosϑ12(t).

For Theorem 2.4.2, a sufficient condition for the asymptotical convergence of βij(t) to π isthe following

kϕ(β12(t)− β21(t)) cosϑ12(t) < 0, (4.9)

26

4.5. Proof of the convergence of the counterclockwise angles for a network of N ≥ 2 agents

which is equal to require that the terms β12(t)− β21(t) and cosϑ12(t) have opposite signs,since kϕ is positive.

Using (4.7) and writing β21(t) as 2π − β12(t), the condition is rewritten as

(β12(t)− π) sin β12(t) < 0.

Since (β12(t)− π) > 0⇒ sin β12(t) < 0,(β12(t)− π) < 0⇒ sin β12(t) > 0,

condition (4.9) is verified for all β12(t) 6= 0, π. Therefore

V (Φ(t)) < 0⇐⇒ βij(t) 6= 0, π.

Consequently, for a network of two agents, once they get on the circle, they are arrangedequidistant; since the chosen Lyapunov function (4.8) is radially unbounded, the convergenceof the counterclockwise angles to π is globally asymptotical.

4.5 Proof of the convergence of the counterclockwise angles for anetwork of N ≥ 2 agents

We want to prove that under control law (4.1), the counterclockwise angles βi(t) convergeto 2π

N , i.e. the agents form a regular polygon on the desired circle.In this case, it is not convenient to use a Lyapunov function like (4.8), because for a

network of N > 2 agents it is not straightforward to verifiy a condition like (4.9).In order to prove the convergence of the counterclockwise angles in this scenario, we

look at their dynamics.

Lemma 4.5.1. Consider a strongly-connected network of N ≥ 2 agents under controllaw (4.1); the dynamics of the error on the counterclockwise angle βi(t) is the following.

˙βi(t) = −kϕ(βi(t)− βνi(t)(t)),

for every agent i ∈ V.

Proof. Without loss of generality, we can look at the dynamics of the counterclockwise angleβij(t) between the agent i and its counterclockwise neighbor at time t, j; we have that

βi(t) = βij(t) if j = νi(t),βj(t) = βjk(t) if k = νj(t).

From (4.5) we haveβij(t) = − 1√1−(ϕT

i(t)ϕj(t))2

[ϕTi (t)ϕj(t) + ϕTi (t)ϕj(t)], if 0 ≤ βij(t) ≤ π,

βij(t) = 1√1−(ϕT

i(t)ϕj(t))2

[ϕTi (t)ϕj(t) + ϕTi (t)ϕj(t)], if π < βij(t) ≤ 2π.(4.10)

It is worth noticing that, because of (4.5), the square root in the denominator in (4.10) isequal to sin βij(t) if 0 ≤ βij(t) ≤ π, while it is equal to − sin βij(t) if π < βij(t) ≤ 2π,therefore the evolution of the counterclockwise angle can be written as

βij(t) = − 1sin βij(t)

[−kϕ(α+ βij(t))ϕTi (t)ϕj(t)− kϕ(α+ βjk(t))ϕTj (t)ϕi(t)],

27

4.5. Proof of the convergence of the counterclockwise angles for a network of N ≥ 2 agents

with i ∈ V, j = νi(t), k = νj(t).Using Lemma (4.6) and Lemma (4.7) we have

βij(t) = −kϕ(βij(t)− βjk(t)), ∀i ∈ V, j = νi(t), k = νj(t), ∀t ≥ t0. (4.11)

Since βij(t) = βij(t) − 2πN the dynamics of the error on the counterclockwise angle is the

following:

˙βij(t) = −kϕ(βij(t)− βjk(t)), ∀i ∈ V, j = νi(t), k = νj(t), ∀t ≥ t0.

We can write equivalently the previous equation like follows:

˙βi(t) = −kϕ(βi(t)− βνi(t)(t)).

From the equation above, applied for all the agents of the network, the closed-loop systemcan be written as

˙β(t) = −kϕ L(t)β(t), (4.12)where

β(t) =[β1(t) . . . βN (t)

],

and [L(t)]ij = 1 if j = i,

[L(t)]ij = −1 if j = νi,

[L(t)]ij = 0 otherwise.

Note that L(t) is the Laplacian matrix of the digraph associated to a time-varying graphG = (V, E(t)), with V = 1, . . . , N and E(t) = (νi(t), i) : i ∈ V. This type of graph isknown in the literature as a cycle digraph, and it is a connected graph [18].

First we are going to complete the proof in the case that νi(t) is constant in [T,∞) forall the agents; then, we are going to show that the proof is easily extended to the case thatsome agents change their counterclockwise neighbor.

If νi is constant for all i ∈ 1, . . . , N, then L(t) is constant; since the network of agentsdescribed by (4.10) is directed and strongly-connected, from Theorem 2.3.1 it reaches theaverage consensus, therefore

limt→∞

β(t) = 0N .

Now consider the case that some agent i changes its counterclockwise neighbor νi(t)at some time τ ≥ T (cfr. Figure 4.10). Without loss of generality, let j = νi(τ−) =limt→τ− νi(t) and k = νi(τ+) = limt→τ+ νi(t).

Since we consider the case of static network, this change cannot be caused by someagent entering or exiting V, but must be due βik(t) becoming as small as βij(t) for t = τ ,which also implies that k = νj(τ−), j = νk(τ+) and νk(τ−) = νj(τ+). Therefore, L(τ+)is obtained by L(τ−) by simply permuting the jth row (resp., column) with the kth row(resp., column).

Moreover, since βij(τ) = βik(τ), βi(t) is continuous at t = τ . Conversely, βj(t) andβk(t) switch their values at τ , in fact: limt→τ− βj(t) = limt→τ− βjk(t) = 0, limt→τ+ βj(t) =limt→τ− βk(t), and limt→τ+ βk(t) = limt→τ+ βkj(t) = 0. Therefore, the vector β(τ+) isobtained by the vector β(τ−) by simply permuting the jth element with the kth element.Hence, the dynamics of β(t) are not affected (up to a permutation of two indexes) if someagents change their counterclockwise neighbor, and we can conclude that, also in this case,βi(t)→ 2π/N , hence βi(t)→ 0, for all i ∈ V.

28

4.6. Proof of the main result

x

yi

yj

yk

x

yi

yj

yk

Figure 4.10: Example of an agent i changing its counterclockwise neighbor. Left: t = τ−

and νi(τ−) = j. Right: t = τ+ and νi(τ+) = k.

i 1 2 3 4 5yi1(t0) 1 0 5 6 4yi2(t0) 3 0 1 3 5

Table 4.1: Initial positions of the agents in the simulation of Section 4.7.

4.6 Proof of the main result

Using the results achieved in Section (4.2), we have that all the agents of the networkconverge to the desire circle C(x,D∗).

Using Lemma 4.5.1 the closed-loop system can be rewritten as (4.12); since the consid-ered network is directed and strongly-connected, from Theorem 2.3.1 all the errors on thecounterclockwise angles asymptotically converge to 0.

4.7 Numerical simulations

To illustrate the effectivness of the control algorithm proposed in this chapter, we apply itto a simulated network of N = 5 agents modelled according to (4.1).

The simulation is set on the time interval [0, 30]. The initial positions of the agents inthe plane are indicated in Table 4.1, where yi1 and yi2 are the coordinates of the agent i inthe plane.The target position is [1, 1]T . The gain kd and kϕ in (4.1) are set to 1, while theangular velocity term α is set to 2; the desired radius D∗ of the circle is 1.

Figure 4.11 shows the evolution of the trajectories of the network under control algo-rithm (4.1), converging to the circle C(x,D∗): after reaching the circle, the agents keep onrotating on it at equisitant positions, as testified by the steady-state value of the angles βiin Figure 4.13.

4.8 Summary

In this chapter, we have proposed a distributed control algorithm to drive a network ofagents to converge on a desired circle and to form a regular polygon during the rotation.

After proving that the bearing vectors are always well-defined, it has been proved thatthe desired circle is an attractive limit cycle for the trajectories of the system.

29

4.8. Summary

yi1

-2 -1 0 1 2 3 4 5 6

yi2

-1

0

1

2

3

4

5

Figure 4.11: Trajectories of the agents under control law (4.1). The black circles representthe positions of the agents at the end of the simulation, while the blue squares representthe positions of the agents at the beginning of the simulation; the red circle represents thestationary position of the target.

Subsequently, using geometric considerations and results from the graph theory and thenetwork control, it has been shown that the dynamics of the angles βi(t) can be written asa consensus equation, therefore the agents achieve the average consensus on the circle.

In the end, simulations performed in Matlab-Simulink have testified the results achieved.

30

4.8. Summary

t

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

∆1(t)

∆2(t)

∆3(t)

∆4(t)

∆5(t)

Figure 4.12: Distance errors of the agents under control law (4.1).

31

4.8. Summary

t

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

1.5

β1(t)

β2(t)

β3(t)

β4(t)

β5(t)

Figure 4.13: Errors on the counterclockwise angles under control law (4.1).

32

Chapter 5

Circumnavigation of a stationarytarget with estimated position

In this chapter, we analyze the behavior of a network of N agents tracking a stationarytarget, whose position is localized through an estimator.

The core of the analysis is the convergence of the estimate of the target position; startingfrom this result, we will extend the results about the convergence and the formation on thedesired circle achieved in Chapter 4.

In Section 5.1, we modify the distributed control algorithm proposed in Chapter 4 on thebasis of the estimator introduced in Chapter 3, in order to solve a dual problem; furthermorewe give the main result of this chapter.

In Sections 5.3 and 5.4 we provide results used to prove the main theorem in Section 5.5;in the end, in Section 5.6 we show the results of simulations performed in Matlab-Simulink.

5.1 Problem statement and main result

In this scenario, the target position is unknown to the agents, which use the followingestimate algorithm to localize the target.

˙xi(t) = −ke(I2 − ϕi(t)ϕTi (t))(xi(t)− yi(t)),xi(t0) = xi0.

(5.1)

The geometric interpretation of the previous updating law for the estimate is given inChapter 2.

Therefore algorithm (4.1) is modified according to (5.1) as follows:yi(t) = kd(Di(t)−D∗)ϕi(t) + kϕDi(t)(α+ βi(t))ϕi(t),Di(t) = ‖xi(t)− yi(t)‖,

(5.2)

where kd, kϕ, α are positive constants.The errors on the distance and on the counterclockwise angle are defined as in the

previous chapter, but in this scenario we need to define also the error on the estimate

xi(t) = xi(t)− x,

and the error between the actual and the estimated distance

Di(t) = Di(t)− Di(t).

33

5.1. Problem statement and main result

In order to prove the convergence of the estimate we need to write the equation ruling thedynamics of the estimate error.

Lemma 5.1.1. Consider the estimate law (5.1) for a single agent of the network; thedynamics of the estimate error is written as

˙xi(t) = −keϕi(t)ϕTi (t)xi(t). (5.3)

Proof. From (2.1) we can write (5.1) as follows:

˙xi(t) = −keϕi(t)ϕTi (t)[(xi(t)− x) + (x− yi(t))].

Since x − yi(t) = Di(t)ϕi(t), exploiting the orthogonality between ϕi(t) and ϕi(t) we canrewrite the previous equation in the following way.

˙xi(t) = −keϕi(t)ϕTi (t)(xi(t)− x).

Hence, because of the stationarity of the target, we obtain that the error dynamics is givenby (5.3).

Similarly to what done in the previous chapter, we write the equations ruling the dy-namics of the network using as polar coordinates Di(t) and ϕi(t).

Lemma 5.1.2. Consider the directed network under (5.2); the closed-loop system is writtenin polar coordinates Di(t), ϕi(t) as

Di(t) = −kdDi(t) + kdD∗,

ϕi(t) = −kϕ Di(t)Di(t) (α+ βi(t))ϕi(t).(5.4)

Therefore, since ∆i(t) = Di(t)−D∗, we write the dynamics of the distance error as

∆i(t) = −kd∆i(t) + kdDi(t), (5.5)

amd the dynamics of the error on the counterclockwise angle as

˙βi(t) = −kϕDi(t)Di(t)

(α+ βi(t)) + kϕDνi(t)(t)Dνi(t)(t)

(α+ βνi(t)(t)). (5.6)

Proof. Similarly to the proof of Lemma (4.2.1), we can write the dynamics of the distanceDi(t) as

Di(t) = ϕTi (t)[−kd(Di(t)−D∗)ϕi(t)− kϕDi(t)(α+ βi(t))ϕi(t)].

Exploiting the orthogonality between the vectors ϕi(t) and ϕi(t), we can rewrite the previousdynamics as

Di(t) = −kd(Di(t)−D∗).

Since Di(t) = Di(t) − Di(t), we can express the dynamics of the distance from the targetas in (5.5).

We write the equation for the dynamics of the bearing vector as

ϕi(t) = d

dt

(x− yi(t)Di(t)

)= − yi(t)

Di(t)− (x− yi(t))

D2i (t)

[1

Di(t)(x− yi(t))T (−yi(t))

].

34

5.2. Proof of the convergence of the estimate errors

Since x − yi(t) = Di(t)ϕi(t), exploiting the orthogonality between ϕi(t) and ϕi(t), theprevious equation is rewritten as

ϕi(t) = −kϕDi(t)Di(t)

(α+ βi(t))ϕi(t).

Similarly to Lemma 4.5.1, without any loss of generality, we can write the dyamics of βij(t)as

βij(t) = − kϕsin βij(t)

[Di(t)Di(t)

(α+ βij(t))ϕTi (t)ϕj(t) + Dj(t)Dj(t)

(α+ βjk(t))ϕTj (t)ϕi(t)],

where βi(t) = βij(t) if j = νi(t),βj(t) = βjk(t) if k = νj(t).

From Lemma 4.3.1 and Lemma 4.3.2, we rewrite the dynamics of βij(t) = βij(t) − 2πN as

in (5.6).

In order to guarantee that the distances of the agents from the target are never null, wegive the following assumption:

Assumption 5.1.1. For every agent of the network it holds that

Di(t0) > 0 and ‖xi(t0)‖ < D∗

The main result of this chapter is formalized by the following theorem.

Theorem 5.1.1. Consider a network of N autonomous agents under estimate law (3.6)and control law (5.2). If Assumption 5.1.1 holds, the agents converge to the desired cir-cle C(x,D∗) while forming a regular polygon; i.e., they achieve the control objective (3.3)and (3.4).

Theorem (5.1.1) states that, introducing an estimate algorithm to localize the target, weobtain the same results for the convergence of both the distances and the angles achievedfor the previuos scenario.

The proof of Theorem 5.1.1 derives from the following sections.In Section 5.2, we prove the exponential convergence of the error xi(t) for every agent

of the network.Subsequently, in Section 5.3 and in Section 5.4, we use Theorem 2.7.1 to show the multi-

time scale behavior of the system, in order to prove the convergence of the agents on thecircle C(x,D∗) and similarly the consensus of the angles βi(t).

Finally in Section 4.6, we use the results obtained in the previous sections to formalizethe proof of Theorem 5.1.1.

5.2 Proof of the convergence of the estimate errors

In this section an analysis of convergence for the estimate of the position of the target isgiven: first we use the Lyapunov theory to prove that the estimate error xi(t) convergesasymptotically to 0, for all the agents of the network, and subsequently we show that theconvergence is also exponential using Theorem 2.6.1.

35


Asymptotic convergence of the estimate errorThe goal is to prove the convergence of the estimate of the target’s position computed bythe agent i to the real position; in order to prove that, we consider a Lyapunov function ofthe estimate error for the single agents.

Lemma 5.2.1. Consider a network of N agents adopting estimate algorithm (5.1): theestimate error norm ‖xi(t)‖ asymptotically converges to 0 as t→∞.

Proof. We consider the following Lyapunov function, defined on the estimate error.

V (xi(t)) = 12‖xi(t)‖

2 ≥ 0.

Its derivative with respect to the time is

V (xi(t)) = xTi (t) ˙xi(t). (5.7)

We can rewrite (5.7) using (5.3),

V (xi(t)) = −ke‖xTi (t)ϕi(t)‖2 ≤ 0.

Since the derivative of the Lyapunov function is negative semi-definite, Theorem 2.4.3 isused to prove asymptotic stability: system (5.3) asymptotically converges to the equilibriumxi(t) = 0 if xi = 0, Di ∈ R, ϕi ∈ S1 is the biggest invariant set in E for the system

˙xi(t) = −keϕi(t)ϕTi (t)xi(t),Di(t) = −kdDi(t) + kdD

∗ + kdDi(t),ϕi(t) = −kϕ Di(t)Di(t) (α+ βi(t))ϕi(t),

whereE = (xi, Di, ϕi) ∈ R2 × R× S1 : xTi (t)ϕi(t) = 0

is the set containing all the points (xi, Di, ϕi) ∈ R2 × R × S1 where the derivative of theLyapunov function vanishes.

From Definiton 2.4.5 and from the definition of E, a subset A ⊆ E is invariant if thefollowing necessary and sufficient condition is satisfied

d

dt(xTi ϕi) = 0 ∀(xi, Di, ϕi) ∈ A ∀t ≥ t0. (5.8)

Condition (5.8) means that the system trajectories starting in A at t0 will remain in thesame set for all the future time, and it is written as

d

dt(xTi (t)ϕi(t)) = ˙xTi (t)ϕi(t) + xTi (t) ˙ϕi(t) = 0 ∀(xi, Di, ϕi) ∈ A ∀t ≥ t0.

From the dynamics of ϕi(t) in (5.4), we write the dynamics of ϕi(t) as

˙ϕi(t) = kϕDi(t)Di(t)

(α+ βi(t))ϕi(t). (5.9)

Using (5.9), we rewrite condition (5.8) as

−keϕTi (t)xi(t) + kϕDi(t)Di(t)

(α+ βi(t))xTi (t)ϕi(t) = 0 ∀(xi, Di, ϕi) ∈ A ∀t ≥ t0.

36


Since A belongs to E, its elements satisfy the condition xTi (t)ϕi(t) = 0, hence condi-tion (5.8) is written as

Di(t)Di(t)

xTi (t)ϕi(t) = 0 ∀t ≥ t0.

From the previous equation, we have that a state of the system (xi, Di, ϕi) ∈ R2 × R× S1

belongs to the set A if

xTi (t)ϕi(t) = 0 ∧ [xTi (t)ϕi(t) = 0 ∨ Di(t) = 0] ∀t ≥ t0.

If there exists t∗ ≥ t0 such that Di(t∗) = 0, the estimate xi(t∗) coincides with the positionof the agent yi(t∗): from(5.2), we have that the agent moves away from the target, withouthaving a circumnavigation motion; furthermore, from (3.6) it holds that there ˙xi(t∗) = 0,since xi(t∗)− yi(t∗) = 0.

If Di(t∗) = 0, then using (3.6) and (5.2) we have ˙xi(t∗) − yi(t∗) = −D∗ϕi(t∗), whichimplies that there exists a t ∈ [t∗,∞) such that Di(t) = ‖xi(t) − yi(t)‖ > 0. Hence, sincecondition Di(t) = 0 is not satisfied ∀t ≥ t0, the states of the system where this condition isverified don’t belong to A.

Therefore, a necessary condition for the invariance of A is

xTi (t)ϕi(t) = 0 ∧ xTi (t)ϕi(t) = 0 ∀(xi, Di, ϕi) ∈ R2 × R× S1 ∀t ≥ t0,

but since ϕi(t) and ϕi(t) are orthogonal, the only state satisfying this condition is xi =0, Di ∈ R, ϕi ∈ S1 .

In fact, from (5.3) we have

xi(t0) = 0⇐⇒ xi(t) = 0 ∀t ≥ t0.

Since xi(t) = 0, Di ∈ R, ϕi ∈ S1 is the only invariant set in E, the system (5.3)is asymptotically stable and the estimate error converges to 0 asymptotically for all theagents of the network, thus

limt→∞

xi(t) = 0 ∀i ∈ V .

Exponential convergence of the estimate errorSimilarly as what achieved in [5] for a single agent, we prove that the convergence of theestimate error is also exponential for all the agents of the network.

Lemma 5.2.2. Consider a network of N agents adopting estimate algorithm (5.1): theestimate error norm ‖xi(t)‖ exponentially converges to 0 as t→∞.

Proof. In order to prove exponential stabilty for system (5.3), according to Theorem 2.6.1 weneed to prove that the vector ϕi(t) is persistently exciting, i.e. that there exist ε1, ε2, T > 0such that

ε1 ≤∫ t0+T

t0

(UT ϕi(t))2dt ≤ ε2, (5.10)

for all t0 ≥ 0 and for every constant unit-norm vector U ∈ S1.Let ϑUϕi(t) the clockwise angle between an arbitrary constant unit-norm vector U ∈ S1

and ϕi(t); the persistency of excitation condition can be written as

ε1 ≤∫ t0+T

t0

(cosϑUϕi(t))2dt ≤ ε2,

37


Figure 5.1: Relations between vectors U , ϕi(t) and ϕi(t).

Since cos2 ϑUϕi(t) ≤ 1 we have that∫ t0+T

t0

cos2 ϑUϕi(t)dt ≤ T.

So an upper bound ε2 satisfying condition (5.10) exists and it is equal to T ; in order toprove that a lower bound ε1 for the integral of the condition exists we use a reduction adabsurdo.

The vector ϕi(t) is not P.E. if for all T there exist t0 and a unit norm vector U ∈ S1

such that ∫ t0+T

t0

(UT ϕi(t))2 = 0

since the integral is always upper-bounded by T .Suppose by contradiction that

∫ t0+Tt0

(UT ϕi(t))2dt = 0 for some t0 ≥ 0 and some U ∈ S1.Then, we have UT ϕi(t) = 0 for all t ∈ [t0, t0 + T ], or equivalently |UTϕi(t)| = 1 for allt ∈ [t0, t0 + T ], which by continuity implies ϕi(t) = ϕi(t0) for all t ∈ [t0, t0 + T ].

From the dynamics of ϕi(t) in (5.4), we notice that a necessary condition for ϕi(t) tobe constant in [t0, t0 + T ] is that Di(t) = 0 in the same interval.

However, if Di(t0) = 0, xi(t0) = yi(t0); then, using (3.6) and (5.2) we have ˙xi(t0) −yi(t0) = −D∗ϕi(t0), which implies that there exists a t ∈ [t0, t0 + T ] such that Di(t) =‖xi(t) − yi(t)‖ > 0. Hence, we must conclude that there exists ε1 > 0 such that ε1 ≤∫ t0+Tt0

(UT ϕi(t))2dt, and therefore, ϕi(t) is p.e. The previous proof is valid for each agentindependently; therefore, we can state that ϕi(t) is p.e. for every agent. Hence, by Theo-rem 2.6.1, xi(t) exponentially converges to zero, for all i ∈ V.

Starting from Lemma (5.2.2), we give a proof of the exponential convergence of |Di(t)|.

38


Figure 5.2: Geometric relation between Di(t), Di(t) and xi(t).

Lemma 5.2.3. Consider a network of N agents under estimate law (3.6) and controllaw (5.2); the error norm |Di(t)| exponentially converges to 0 as t→∞, for every i ∈ V.

Proof. Observe that, by the triangular inequality (see Figure 5.2), we have

‖x− yi(t)‖ ≤ ‖yi(t)− xi(t)‖+ ‖xi(t)− x‖,

which since Di(t) = ‖x− yi(t)‖, Di(t) = ‖xi(t)− yi(t)‖, and Di(t) = Di(t)− Di(t), can berewritten as

Di(t) ≤ ‖xi(t)‖. (5.11)

Since for Lemma (5.2.2) ‖xi(t)‖ exponentially converges to 0 ∀i ∈ V, it holds that |Di(t)|exponentially converges to 0 ∀i ∈ V.

Lemma 5.2.4. Consider a network of N agents under estimate law (3.6) and controllaw (5.2); then, under Assumption 5.1.1, the distance from the target is always positive forevery agent.

Proof. For simplicity we set t0 = 0: integrating the dynamics of ∆i(t) in (5.5), and recallingthat ∆i(0) = Di(0)−D∗, we can write

Di(t) = D∗ + (Di(0)−D∗)e−kdt +∫ t

0e−kd(t−τ)kd(Di(τ))dτ. (5.12)

Using (5.11), and observing from (5.3) that the norm of the estimate error is non-increasing,we have

Di(t) ≤ ‖xi(0)‖ (5.13)

for all t ≥ 0. Using (5.13) in (5.12), and computing the integral explicitly, we have

Di(t) ≥ D∗ + ∆i(0)e−kdt − ‖xi(0)‖(1− e−kdt). (5.14)

39


Adding and subtracting D∗e−kdt from the right-hand side of (5.14), we have

Di(t) ≥ Di(0)e−kdt + (D∗ − ‖xi(0)‖)(1− e−kdt). (5.15)

Finally, using (5.15) under Assumption 5.1.1, we have

Di(t) > 0 ∀t ≥ 0 ∀i ∈ V .

5.3 Proof of the convergence of the distance errors

In this section we prove the convergence of the agents on the circle C(x,D∗), starting fromthe results achieved about the exponential convergence of the estimate error. The dynamicsof ∆i(t) in the scenario with the adoption of the estimate algorithm,

∆i(t) = −kd∆i(t) + kdDi(t),

differs from the dynamics (4.3) in the scenario of known target for an additive term Di(t),depending on the estimate; from Lemma 5.2.3, this term represents a vanishing perturba-tion.

In the following Lemma, we use Theorem 2.7.1 to show that the solution of (5.5) con-verges to the solution of (4.3), as the perturbation Di(t) vanishes.

Lemma 5.3.1. Consider the system∆i(t) = −kd(∆i(t)− Di(t)),˙xi(t) = −keϕi(t)ϕTi (t)xi(t),

(5.16)

that can be seen as a system of the type∆i(t) = f(t,∆i(t), xi(t), |Di(t0)|),|Di(t0)| ˙xi(t) = g(t,∆i(t), xi(t), |Di(t0)|),

where the error norm at the initial time |Di(t0)| is treated as a positive parameter; it holdsthat the solution of the system (5.16) converges to the solution of the degenerate system

∆i(t) = f(t,∆i(t), xi(t), 0) = −kd∆i(t) (5.17)

as |Di(t0)| → 0 and for t0 ≤ t <∞.

Proof. Let I = [0,∞); in order to give a proof of Lemma (5.3.1), we have to check if theconditions of Theorem (2.7.1) are satisfied.

1. The system (5.17) has ∆i(t) = 0 as a solution, for all t0 ≤ t <∞.

2. f(t,∆i(t), xi(t), |Di(t0)|) and its derivatives with respect to the components of ∆i(t)and xi(t), and g(t,∆i(t), xi(t), |Di(t0)|) and its derivatives with respect to t and thecomponents of ∆i(t) and xi(t) are continuous.

3. The function g(t,∆i(t), xi(t), |Di(t0)|) satisfies

g(t,∆i(t), 02, 0) = 0, ∀(t,∆i(t)) ∈ I × R.

40

5.4. Proof of the convergence of the counterclockwise angles

4. f(t,∆i(t), 0, 0) is continuous uniformly in (t,∆i(t)) ∈ I × R and bounded on I × R,since its solution is exponentially stable; its derivative with respect to ∆i(t) is boundedbecause it is constant:

df(t,∆i(t), 0, 0)d∆i(t)

= −kd.

5. Since g(t,∆i(t), xi(t), 0) = 0, its derivatives with respect to any variable are bounded.

6. Since the system (5.17) is exponentially-stable, its solution is such that

|∆i(t)| ≤ c|∆i(t0)|e−kd(t−t0).

where c > 0. Therefore there exist a strictly increasing, real-valued and continuosfunction d(‖∆i(t0)‖), such that d(0) = 0, and a strictly decreasing, real-valued, non-negative and continuous function σ(t− t0) such that

|∆i(t)| ≤ d(|∆i(t0)|)σ(t− t0),

for all t0 ≤ t < ∞ and |∆i(t0)|: hence the zero-solution of system (5.17) is uniform-exponentially stable.

7. Consider the boundary layer system:

dxi(s)ds

= g(γ, λ, xi, 0) = 0, (5.18)

where s = t−γ|Di(t0)| is a stretching transformation of the time and ∆i(s) = λ is the

constant solution of d∆i(s)ds = 0. The solution of

dxi(s)ds

= −ke|Di(t0)|ϕi(s)ϕTi (s)xi(s)

is‖xi(s)‖ ≤ k‖xi(0)‖e−res,

where k > 0 and re is the rate of convergence. Therefore there exist a strictly increas-ing, real-valued and continuos function l(‖xi(0)‖), such that l(0) = 0, and a strictlydecreasing, real-valued, nonnegative and continuous function ρ(s) such that

‖xi(s)‖ ≤ l(‖xi(0))‖)ρ(s),

for all 0 ≤ s < ∞ and ‖xi(0)‖, and for all(γ, λ) ∈ R× R. Hence the system (5.18) isexponentially-stable uniformly in the parameters (γ, λ) ∈ I × R.

5.4 Proof of the convergence of the counterclockwise angles

In this section we prove the consensus achieved from the agents on the angles βi(t), startingfrom the results achieved about the exponential convergence of the distance in the previoussection: we want to show that βi(t) → 0 as ∆i(t) → 0, since Lemma 5.3.1 states thatthe convergence of the agents on the circle implies the convergence of the estimate, too.From (5.6), the dynamics of the vector β(t) is written as follows:

˙β(t) = −kϕ L β(t) + δβ(t), (5.19)

41


where

[δβ(t)]i = kϕ

[−α

(Di(t)Di(t)

−Dνi(t)(t)Dνi(t)(t)

)+(βi(t)

Di(t)Di(t)

− βνi(t)(t)Dνi(t)(t)Dνi(t)(t)

)].

The dynamics (5.19) differs from the dynamics (4.5) in the scenario of known target for anadditive term δβ(t), depending on the estimate and on the error ∆i(t).

Since from Lemma 5.2.3 limt→∞ Di(t) = 0 ∀i ∈ V, from Lemma 5.3.1 limt→∞Di(t) =D∗ ∀i ∈ V, it holds that δβ(t) is a vanishing perturbation.

In the following Lemma, we use Theorem 2.7.1 to show that the solution of (5.19)converges to the solution of (4.5), as the perturbation δβ(t) vanishes.

Notice that the convergence of the agents on the circle C(x,D∗) implies the vanishingof δβ(t), from what achieved in the previous section.

Lemma 5.4.1. Consider the system ˙β(t) = −kϕ L β(t) + δβ(t),∆(t) = −kd∆(t).

(5.20)

The system (5.20) can be seen as a system of the type ˙β(t) = f(t, β(t),∆(t), ‖∆(t0)‖),‖∆(t0)‖∆(t) = g(t, β(t),∆(t), ‖∆(t0)‖),

where ∆(t) =

∆1(t)· · ·

∆n(t)

and ‖∆(t0)‖ is a positive parameter; it holds that the solution of the

system (5.20) converges to the solution of the degenerate system

˙β(t) = f(t, β(t),∆(t), 0) = −kϕ L β(t) (5.21)

as ‖∆(t0)‖ → 0 and for t0 ≤ t ≤ ∞, where L is the Laplacian matrix of the network.

Proof. Let I = [0,∞); we have to verify the conditions of Theorem 2.7.1.

1. The system (5.21) has β = 0 as solution for t0 ≤ t <∞.

2. The functions f(t, β(t),∆(t), ‖∆(t0)‖) and g(t, β(t),∆(t), ‖∆(t0)‖), and their deriva-tives with respect to t and to the components of β(t) and ∆(t) are continuous.

3. The function g(t, β(t),∆(t), ‖∆(t0)‖) satisfies

g(t, β(t), 0N , 0) = 0, ∀(t, β) ∈ I × RN .

4. The function f(t, β(t), 0N , 0) is continuous uniformly in (t, β) ∈ I ×RN and boundedon I × RN , because the system

˙β(t) = f(t, β(t), 0N , 0) = −kϕ L β(t) (5.22)

is exponentially-stable, hence its solution is bounded.

42


Let fi be the i-th component of f ; we denote fβ(t, β(t), 0N , 0) the matrix with com-ponents ∂fi(t,β(t),0N ,0)

∂βjk, such that

∂fi(t, β(t), 0N , 0)∂βjk

=

kϕ if j = νi(t),−kϕ if i = j,0 otherwise.

Since the components of fβ(t, β(t), 0N , 0) are bounded, the matrix is bounded.

5. The function g(t, β(t),∆(t), 0) is continuous uniformly in (t, β,∆) ∈ I × RN ; since

g(t, β(t),∆(t), 0) = 0N ∀t ≥ t0,

it is bounded with all its derivatives with respect to any variable.

6. Since the system (5.22) is exponentially-stable, its solution is such that

‖β(t)‖ ≤ b‖β(t0)‖e−rβ(t−t0),

where b > 0 and rβ is the rate of convergence of the system.Therefore there exist a strictly increasing, real-valued and continuos function d(‖β(t0)‖),such that d(0) = 0, and a strictly decreasing, real-valued, nonnegative and continuousfunction σ(t− t0) such that

‖β(t)‖ ≤ d(‖β(t0)‖)σ(t− t0),

for all t0 ≤ t < ∞ and ‖β(t0)‖, i.e. the zero solution of (5.22) is uniformly-exponentially stable.

7. Making the stretching transformation of t, s = t−γ‖∆(t0)‖ , and setting ‖∆(t0)‖ to 0, we

obtain the boundary-layer system

d∆(s)ds

= g(γ, λ,∆, 0) = −kd‖∆(t0)‖∆(s), ∆(0) = ∆0, (5.23)

where β = λ is the only solution of

dβ(s)ds

= 0.

Since system (5.23) is exponentially-stable, its solution is such that

‖∆(s)‖ ≤ δ‖∆(0)‖e−r∆s,

where δ > 0 and r∆ is the rate of convergence of the system.Therefore there exists a strictly increasing, real-valued and continuos function l(‖∆(0)‖),such that l(0) = 0, and a strictly decreasing, real-valued, nonnegative and continuousfunction ρ(s) such that

‖∆(s)‖ ≤ l(‖∆(0)‖)ρ(s),for all 0 ≤ s <∞ and ‖∆(0)‖, i.e. the zero solution of (5.23) is exponentially-stableuniformly in the parameters (γ, λ) ∈ R× RN .

43


i 1 2 3yi1(t0) 1 0 5yi2(t0) 3 0 1


i 1 2 3xi1(t0) 1.5 2.5 2xi2(t0) 0.5 1.5 1.8

Table 5.2: Initial estimates in the simulation of Section 5.6.


Used the results achieved in Section 5.2 we have that the estimate error norm exponentiallyconverges to 0 for all the agents of the network; from Lemma 5.3.1 it holds that the dynamicsof the error distances in the scenario of estimated target converge to the dynamics of theerror distances in the scenario of known target. Therefore, using the results in Section 4.2,it holds that every agent converges to the desidered circle C(x,D∗), that is an attractivelimit cycle also for system (5.4).

Starting from the convergence on the limit cycle, using Lemma 5.4.1, we have that theagents reach the consensus, forming a regular polygon inscribed in C(x,D∗).


In order to illustrate the effectivness of the control algorithm proposed in this chapter, weapply it to a simulated network of N = 3 agents modelled according to (5.2).

The simulation is set on the time interval [0, 30]. The initial positions of the agents inthe plane are indicated in Table 5.1, where yi1 and yi2 are the coordinates of the agent iin the plane; the initial estimate of the target position, made by every agent, are indicatedin Table 5.2, where xi1(t) and xi2(t) are the components of the estimate made by agent i .The target position is [1, 1]T .

The gain kd and kϕ in (5.2) are set to 1 the angular velocity term α is set to 2 while theestimate gain ke is set to 7; the desired radius D∗ of the circle is 1.

Figures 5.3 and 5.5 illustrate the trajectories of the agents in the plane and the con-vergence on C(x, 1); Figure 5.6 shows the exponential convergence of the angles βi(t) to2πN = 2π

3 , and Figure 5.4 shows the exponential convergence of the estimate error to 0.

5.7 Summary

In this chapter, we have proposed a distributed control algorithm to drive a network ofagents to localize a stationary target and to achieve the circumnavigation task.

Firstly, the convergence of the estimate error has been proved, starting from the persis-tence of excitation condition; subsequently, the results about the circumnavigation achievedin the previous chapter have been extended to this scenario, showing the multi-time scalebehavior of the multi-agent system. In the end, simulations performed in Matlab-Simulinkhave testified the results achieved.

44

5.7. Summary

yi1

-1 0 1 2 3 4 5

yi2

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 5.3: Trajectories of the agents under control law (5.2). The black circles representthe positions of the agents at the end of the simulation, while the blue squares representthe positions of the agents at the beginning of the simulation; the red circle represents thestationary position of the target.

45

5.7. Summary

t

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

‖x1(t)‖

‖x2(t)‖

‖x3(t)‖

Figure 5.4: Estimate errors of the agents under control law (5.2).

46

5.7. Summary

t

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

∆1(t)

∆2(t)

∆3(t)

Figure 5.5: Distance errors of the agents under control law (5.2).

47

5.7. Summary

t

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

1.5

β1(t)

β2(t)

β3(t)

Figure 5.6: Errors on the counterclockwise angles of the agents under control law (5.2).

48

Chapter 6

Circumnavigation of a mobile targetwith estimated position

In this chapter, we analyze the behavior of a network of agents tracking a mobile target;the estimate algorithm and the distributed control algorithm presented in Chapter 5 areadopted to solve respectively a localization problem, and a tracking and formation problem.

In Section 6.1, we introduce the problem statement and the main result of this chapter;subsequently, in Sections 6.2, 6.3 and 6.4 we prove that the system variables converge toballs centered at 0, with defined radii depending on the parameters of the control algorithm.In the end, in Section 6.6, we present the results of simulations performed in Matlab-Simulink.

6.1 Problem statement and main results

In this scenario, the target moves in the plane with a velocity whose norm is upper-boundedas

‖x(t)‖ ≤ εT ∀t ≥ t0, (6.1)and the agents are subjected to control law (5.2).

In order to guarantee that the agents are always able to circumnavigate the target, weshall need the following technical assumption that the desired circumnavigation distance islarge enough with respect to the unknown motion of the target.

Assumption 6.1.1. The desired distance from the target satisfies D∗ > εT /(kϕα).

The agents in the network use algorithm (3.6) to estimate the position of the mobiletarget; since the estimate error is defined as

xi(t) = xi(t)− x(t) ∀t ≥ t0,

we have that xi(t) has the dynamics described in the following Lemma.

Lemma 6.1.1. Consider estimate law (5.1) for a single agent of the network; in the scenarioof mobile target, the dynamics of xi(t) is written as

˙xi(t) = −keϕi(t)ϕTi (t)xi(t)− x(t). (6.2)

Proof. From Lemma 5.1.1 we can write (3.6) as follows:˙xi(t) = −keϕi(t)ϕTi (t)(xi(t)− x).

49

6.1. Problem statement and main results

Since the target is mobile with velocity x(t), from the definition of xi(t) we obtain that thedynamics of xi(t) is (6.2).

Similarly to what done in the previous chapters, we write the equations of the systemusing as polar coordinates Di(t) and ϕi(t).

Lemma 6.1.2. Consider the network under (5.2) in the scenario of mobile target; theclosed-loop system is written in polar coordinates as

Di(t) = −kd(Di(t)−D∗) + ϕTi (t)x(t),ϕi(t) = −kϕ Di(t)Di(t) (α+ βi(t))ϕi(t) + ϕi(t)ϕTi (t)x(t)

Di(t) .(6.3)

Therefore, since ∆i(t) = Di(t)−D∗, we write the dynamics of the distance error as

∆i(t) = −kd∆i(t) + kdDi(t) + ϕTi (t)x(t), (6.4)

and the dynamics of the error on the counterclockwise angle as

˙βi(t) =− kϕDi(t)Di(t)

(α+ βi(t)) + kϕDνi(t)(t)Dνi(t)(t)

(α+ βνi(t)(t))

+ xT (t)(ϕi(t)Di(t)

−ϕνi(t)(t)Dνi(t)(t)

).

(6.5)

Proof. Similarly to the proof of Lemma (4.2.1), we can write the dynamics of the distanceDi(t) as

Di(t) = d

dt(‖x(t)− yi(t)‖) = ϕTi (t)x(t)− ϕTi (t)yi(t).

Writing yi(t) as in (5.2), we have

Di(t) = ϕTi (t)x(t)− ϕTi (t)[kd(Di(t)−D∗)ϕi(t) + kϕDi(t)(α+ βi(t))ϕi(t)].

Exploiting the orthogonality between the vectors ϕi(t) and ϕi(t), we can rewrite the previousdynamics as

Di(t) = −kd(Di(t)−D∗) + ϕTi (t)x(t).

Since Di(t) = Di(t)− Di(t), we can express the dynamics of the distance error as in (6.4).We write the equation for the dynamics of the bearing vector as

ϕi(t) = d

dt

(x(t)− yi(t)Di(t)

)= x(t)− yi(t)

Di(t)

− (x(t)− yi(t))D2i (t)

[1

Di(t)(x(t)− yi(t))T (x(t)− yi(t))

].

Since x(t) − yi(t) = Di(t)ϕi(t), exploiting the orthogonality between ϕi(t) and ϕi(t), theprevious equation is rewritten as

ϕi(t) = ϕi(t)ϕTi (t)x(t)Di(t)

− kϕDi(t)Di(t)

(α+ βi(t))ϕi(t).

50

6.2. Proof of the bounded convergence of the estimate errors

Similarly to Lemma 4.5.1, without any loss of generality, we can write the dyamics of βij(t)as

βij(t) =− kϕsin βij(t)

[Di(t)Di(t)

(α+ βi(t))ϕTi (t)ϕj(t) + Dj(t)Dj(t)

(α+ βjk(t))ϕTj (t)ϕi(t)]

+ x(t)[ϕi(t)Di(t)

− ϕj(t)Dj(t)

],

where βi(t) = βij(t) if j = νi(t),βj(t) = βjk(t) if k = νj(t).

From Lemma 4.3.1 and Lemma 4.3.2, we rewrite the dynamics of βij(t) = βij(t) − 2πN as

in (6.5).

In order to guarantee that the distances of the agents from the target are never null, wegive the following assumption:

Assumption 6.1.2. For every agent of the network it holds that

bx‖xi(t0)‖+ εTrx

+ εTkd

< D∗,

where bx > 0 and rx > 0.

The main result of this section is formalized as the following theorem.

Theorem 6.1.1. Consider a network of N autonomous agents under estimate law (3.6)and control law (5.2), with ‖x(t)‖ ≤ εT . Under Assumption 6.1.1 and Assumption 6.1.2,the agents converge to an annulus of radii D∗ − ε∆ and D∗ + ε∆, containing C(x(t), D∗)and they are in a formation such that, as t→∞, ‖β(t)‖ ≤ Uβ/kϕ, where kϕ is the controlgain for the tangential term in (5.2), and Uβ is given by (6.12).

In the next sections we give proofs of the bounded convergence of the system variables,that are used to prove Theorem 6.1.1 in Section 6.5.

6.2 Proof of the bounded convergence of the estimate errors

In order to prove the bounded convergence of the estimate error for every agent, we recallthe following proposition from [5]:

Proposition 6.2.1 (Proposition 1 in [5]). If the coefficient matrix A(t) is continuous forall t > 0 and constants r > 0, b > 0 exist such that for every solution of the homogeneousdifferential equation p(t) = A(t)p(t) one has

‖p(t)‖ ≤ b‖p(t0)‖e−r(t−t0), 0 ≤ t0 < t <∞,

then for each f(t) bounded and continuous on [0,∞), every solution of the nonhomogeneousequation

p(t) = A(t)p(t) + f(t), p(t0) = 0is also bounded for t ∈ [0,∞). In particular, if ‖f(t)‖ ≤ Kf < ∞, then the solution of theperturbed system satisfies

‖p(t)‖ ≤ b‖p(t0)‖e−r(t−t0) + Kf

r(1− e−r(t−t0)).

51

6.2. Proof of the bounded convergence of the estimate errors

Using Proposition 6.2.1, we give the following Lemma.

Lemma 6.2.1. Consider a network of N agents under (3.6) and (5.2), in the scenarioof mobile target; then the estimate error norm ‖xi(t)‖ converges exponentially to a ballcentered at zero, of radius εx = εT

rx, where rx is the rate of convergence of xi(t).

Proof. According to Proposition 6.2.1 the first step is to prove that the homogeneous formof (6.2), that is

˙xi(t) = −keϕi(t)ϕTi (t)xi(t), (6.6)

is exponentially-stable. Similarly to what done in Lemma 5.2.2, according to Theorem 2.6.1we need to prove that the vector ϕi(t) is persistently exciting, hence that there existε1, ε2, T > 0 such that

ε1 ≤∫ t0+T

t0

cos2 ϑUϕi(t)dt ≤ ε2,

for all t0 ≥ 0 and for every constant unit-norm vector U ∈ S1. Since cos2 ϑUϕi(t) ≤ 1 wehave that ∫ t0+T

t0

cos2 ϑUϕi(t)dt ≤ T ∀t ≥ t0.

So an upper bound ε2 satisfying the condition (5.10) exists and it is equal to T ; the nextstep is to prove that a lower bound ε1 for the integral in the condition exists.

For this purpose, like in Lemma 5.2.2 we use a reduction ad absurdum: the vector ϕi(t)is not P.E. if for all T > 0 there exist t0 and a constant unit-norm vector U ∈ S1 such that∫ t0+T

t0

(UT ϕi(t))2 = 0,

since the integral is always upper-bounded by T .Suppose by contradiction that

∫ t0+Tt0

(UT ϕi(t))2dt = 0 for some t0 ≥ 0 and some U ∈ S1.Then, we have UT ϕi(t) = 0 for all t ∈ [t0, t0 + T ], or equivalently |UTϕi(t)| = 1 for allt ∈ [t0, t0 + T ], which by continuity implies ϕi(t) = ϕi(t0) for all t ∈ [t0, t0 + T ].

From the dynamics of ϕi(t) in (6.3), we notice that ϕi(t) is constant in [t0, t0 + T ] ifand only if kϕ(α+βi(t))Di(t) = ϕTi (t0)x(t) in the same interval. However, if this conditionis satisfied, then, Di(t) ≤ εT /(kϕα). Consequenlty, using (3.6) and (5.2), we have that thecomponent of ˙xi(t)− yi(t) along ϕi(t0) is

( ˙xi(t)− yi(t))Tϕi(t0) = D∗ − ϕTi (t)x(t)kϕ(α+ βi(t))

≥ D∗ − εTkϕα

,

which is strictly positive under Assumption 6.1.1. Therefore, we can write

(xi(t)− yi(t))Tϕi(t0) ≥ (xi(t0)− yi(t0))Tϕi(t0) +(D∗ − εT

kϕα

)(t− t0)

which, by Assumption 6.1.1, implies that (xi(t) − yi(t))Tϕi(t0) (and consequenlty Di(t))eventually becomes larger than εT /(kϕα), thus contradicting the condition (α+βi(t))Di(t) =ϕi(t0)T x(t). Therefore, there exists a t ∈ [t0, t0 + T ] such that kϕ(α + βi(t))Di(t) 6=ϕTi (t)x(t). Hence, we must conclude that there exists ε1 > 0 such that ε1 ≤

∫ t0+Tt0

(UT ϕi(t))2dt,and therefore, ϕi(t) is p.e. The previous proof is valid for each agent independently; there-fore, we can state that ϕi(t) is p.e. for every agent. Since the homogeneous equation (6.6)

52

6.3. Proof of the bounded convergence of the distance errors

is exponentially stable and the target velocity is bounded from (6.1), from Proposition 6.2.1the following inequality holds:

‖xi(t)‖ ≤ bx‖xi(0)‖e−rxt + εTrx

(1− e−rxt)

for all t ≥ t0 and for all i ∈ V.Therefore

limt→∞

‖xi(t)‖ ≤εTrx, i ∈ V

hence the estimate error norm exponentially converges to a ball of radius εx = εTrx

.

Lemma 6.2.2. Under Assumption 6.1.2, under estimate law (3.6) and control law (5.2),the distance from the target Di(t) is always positive.

Proof. For simplicity we set t0 = 0. Integrating the dynamics of ∆i(t) in (6.3), we can write

Di(t) = D∗ + (Di(0)−D∗)e−kdt +∫ t

0e−kd(t−τ)[kdDi(τ) + xT (τ)ϕi(τ)]dτ.

Using (5.11) and (6.1), the previous equation is lower-bounded as follows.

Di(t) ≥ D∗ + (Di(0)−D∗)e−kdt −∫ t

0e−kd(t−τ)[kd‖xi(τ)‖]dτ − εT

kd(1− e−kdt).

Using Lemma 6.2.1 we have

Di(t) ≥ −bx‖xi(0)‖e−rxt − εTrx

(1− e−rxt) ≥ −bx‖xi(0)‖ − εTrx.

Therefore it holds that

Di(t) ≥ Di(0)e−kdt +(D∗ − εT

kd− εTrx− bx‖xi(0)‖

)(1− e−kdt).

Using Assumption 6.1.2 we have that

Di(t) > 0 ∀t ≥ 0 ∀i ∈ V

6.3 Proof of the bounded convergence of the distance errors

Starting from the bounded convergence of the estimate error, we give a proof of the boundedconvergence of the distance error for every agent of the network.

Lemma 6.3.1. Consider a network under (5.2) in the scenario of mobile target; then thedistance error converges exponentially to a ball of radius εT ( 1

kd+ 1

rx), where rx is the rate

of convergence of the estimate error.

Proof. For simplicity we set t0 = 0. Integrating the dynamics of ∆i(t) in (6.3), we can write

∆i(t) = ∆i(0)e−kdt +∫ t

0e−kd(t−τ)[kdDi(τ) + xT (τ)ϕi(τ)]dτ.

53

6.4. Proof of the bounded convergence of the counterclockwise angles errors

From (5.11) and noting that |xT (t)ϕi(t)| ≤ εT , the previous equation is upper-bounded asfollows.

|∆i(t)| ≤ |∆i(0)|e−kdt +∫ t

0e−kd(t−τ)[kd‖xi(τ)‖+ εT ]dτ. (6.7)

From Lemma 6.2.1, the estimate error norm can be bounded such that the distance errornorm is bounded as follows.

|∆i(t)| ≤ |∆i(0)|e−kdt +∫ t

0e−kd(t−τ)

[kdbx‖xi(0)‖e−rxτ + kd

εTrx

(1− e−rxτ ) + εT

]dτ.

Inequality (6.7) is rewritten as

|∆i(t)| ≤|∆i(0)|e−kdt +(−kdbx‖xi(0)‖

rx − kd+ kdεTrx(rx − kd)

)e−(rx+kd)t

−(εTrx

+ εTkd

)e−kdt + εT

rx+ εTkd.

Since the first three terms of the previous inequality go to 0 as t→∞, we have that

limt→∞

|∆i(t)| ≤ εT(

1kd

+ 1rx

)for every agent in the network.

6.4 Proof of the bounded convergence of the counterclockwiseangles errors

In this section, starting from the results achieved for the convergence of the estimate errorand the distance error, we give a proof of the bounded convergence of the counterclockwiseangle errors βi(t) for every agent of the network.

Lemma 6.4.1. Consider the directed network under (5.2) in the scenario of mobile target;then the norm ‖β(t)‖ converges exponentially to a ball of radius Uβ centered at 0, where Uβis given by (6.12).

Proof. For the agent i we denote

βi(t) = βiνi(t) = βiνi(t)(t)−2πN.

From (6.5), we write the dynamics of βi(t) as

βi(t) =− kϕ(βi(t)− βνi(t)(t))− kϕα(Di(t)Di(t)


)

+ kϕβi(t)(

1− Di(t)Di(t)

)− kϕβνi(t)(t)

(1−

Dνi(t)(t)Dνi(t)(t)

)



).

(6.8)

54

6.4. Proof of the bounded convergence of the counterclockwise angles errors

Since Di(t) = Di(t)− Di(t), we rewrite (6.8) as

βi(t) =− kϕ(βi(t)− βνi(t)(t)) + kϕα

(Di(t)Di(t)


)

+ kϕ

(βi(t)

Di(t)Di(t)


)



).

We can write the dynamics of the counterclockwise angles as

˙β(t) = −kϕ L β(t) + fα(t) + fβ(t) + fT (t), (6.9)

where L is the Laplacian matrix of the network, β(t) is the vector containing the errors onthe counterclockwise angles, and the last three terms of (6.9) are defined as follows

[fα(t)]i =kϕα(Di(t)Di(t)


),

[fβ(t)]i =kϕ

(βi(t)

Di(t)Di(t)


),

[fT (t)]i =xT (t)(ϕi(t)Di(t)


).

The homogeneous form of (6.9)˙β(t) = −kϕ L β(t)

is exponentially stable, for Theorem 2.3.1.According to Proposition 6.2.1, in order to prove bounded convergence of (6.9), we need

to prove the boundness of fα(t) + fβ(t) + fT (t).Since |Di(t)| ≤ ‖xi(t)‖, |βi(t)| < 2π and |xT (t)ϕi(t)| ≤ εT , we have the following time-

varying upper-bounds:

|[fα(t)]i| ≤kϕα(‖xi(t)‖Di(t)

+‖xνi(t)(t)‖Dνi(t)(t)

),

|[fβ(t)]i| ≤kϕ2π(‖xi(t)‖Di(t)

+‖xνi(t)(t)‖Dνi(t)(t)

),

|[fT (t)]i| ≤εT(

1Di(t)

+ 1Dνi(t)(t)

).

From Lemma 6.3.1, we have that, as t→∞,

D∗ − ε∆ ≤ Di(t) ≤ D∗ + ε∆, (6.10)

where ε∆ = εT

(1kd

+ 1rx

). From Lemma 6.2.1, we have that

limt→∞

‖xi(t)‖ ≤ εx, (6.11)

where εx = εTrx

.

55


i 1 2 3yi1(t0) 1 3.5 0yi2(t0) 3 1 -1


Using (6.10) and (6.11), we have that, as t→∞,

‖fα(t)‖ ≤2√Nkϕα

εx(D∗ − ε∆) ,

‖fβ(t)‖ ≤4π√Nkϕ

εx(D∗ − ε∆) ,

‖fT (t)‖ ≤2√N

εT(D∗ − ε∆) .

From Proposition 6.2.1, we have that there exists bβ > 0 and rβ = kϕ s.t.

‖β(t)‖ ≤ bβ‖β(t0)‖e−rβ(t−t0) + Uβrβ

(1− e−rβ(t−t0)) ∀t ≥ t0,

whereUβ =

√N

[(2kϕαεx + 4πkϕεx + 2εT ) 1

(D∗ − ε∆)

]. (6.12)

Therefore we have thatlimt→∞

‖β(t)‖ ≤ Uβrβ.


From Section 6.2 we have that, for every agent, the estimate error converges to a ballcentred at 0 of radius εT

rx; from Section 6.3 we have that, for every agent, the distance error

converges to a ball centred at 0 of radius εT(

1kd

+ 1rx

), while from Section 6.4 we have that,

for every agent, the error on the angle converges to a ball centred at 0 of radius Uβ .


In order to illustrate the effectivness of the control algorithm proposed in this chapter, weapply it to a simulated network of N = 3 agents modelled according to (5.2).

The simulation is set on the time interval [0, 50]. The initial positions of the agents inthe plane are indicated in Table 6.1, where yi1 and yi2 are the coordinates of the agent i inthe plane; the initial estimate of the target position, made by every agent, are indicated inTable 6.2, where xi1(t) and xi2(t) are the components of the estimate made by agent i .

The target moves according to the following lawx1(t) = 0.1 cos 0.05tx2(t) = 0.1 sin 0.05t

56

6.7. Summary

i 1 2 3xi1(t0) 2 0.5 2xi2(t0) 1.6 2.5 2.5

Table 6.2: Initial estimates in the simulation of Section 6.6.

yi1

-2 -1 0 1 2 3 4

yi2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Figure 6.1: Trajectories of the agents under control law (5.2) and with mobile target. Theblack circles represent the positions of the agents at the end of the simulation, while theblue squares represent the positions of the agents at the beginning of the simulation; thetrajectory of the target is represented in red.

The gain kd and kϕ in (5.2) are set to 1 the angular velocity term α is set to 1 while theestimate gain ke is set to 5; the desired radiusD∗ of the circle is 1; notice that, approximatingr with ke = 5, and recalling that, for the chosen trajectory of the target, εT = 0.07, D∗satisfies Assumption 6.1.2.

In Figures 6.2–6.5 we can clearly see the persistent oscillations caused by the target’sunknown motion.

6.7 Summary

In this chapter, we have proposed a distributed control algorithm to drive a network ofagents to localize a mobile target and to achieve the circumnavigation task.

First we have shown that, differently from the previous scenarios, the localization is not

57

6.7. Summary

t

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

‖x1(t)‖

‖x2(t)‖

‖x3(t)‖

Figure 6.2: Estimate errors of the agents under control law (5.2) and with mobile target.

achieved completely, since the convergence of the estimate error is only bounded. Further-more, we have proved that the agents converge to an annulus of the desired circle and thatthe difference between their angular distances on the circle is bounded. In the end, we haveshown the achieved results through simulations performed in Matlab-Simulink.

58

6.7. Summary

t

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

|∆1(t)|

|∆2(t)|

|∆3(t)|

Figure 6.3: Distance errors of the agents under control law (5.2) and with mobile target.

59

6.7. Summary

t

0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

2

β1(t)

β2(t)

β3(t)

Figure 6.4: Errors of the counterclockwise angles of the agents under control law (5.2) andwith mobile target.

60

6.7. Summary

t

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

‖β(t)‖

Figure 6.5: ‖β(t)‖ under the control law (5.2) and with mobile target.

61

Chapter 7

Circumnavigation in switchingnetworks

The decentralized control algorithm presented in the previous chapters requires that everyagent collects measurements of the vector ϕi(t) from its counterclockwise neighbor, in orderto update the control signal at every instant of time. The computation of the neighbor isdone on the basis of information received by all the other agents, therefore, actually thereis an all-to-all communication in the network.

In the applications in which the agents are controlled to perform a surveillance missionin a broad area, this solution might require the employment of high-performing sensors,capable of detecting the bearing signals sent by the rest of the network at great distances;furthermore, the control action could be not updated properly, due to communication prob-lems that could born.

In order to use the control algorithm proposed in this work without the need of expensiveand complex hardware on the vehicles, in this chapter we present a modified version of (5.2),driving the agents to move according to information coming from a subset of the otheragents; in particular, in order to update the control signal, every agent collects the bearingvector of the agents whose position belongs to a well-defined communication radius. Forthis purpose, we formalize the problem statement for this scenario in Section 7.1, whilein Section 7.2, we present the implementation in ROS of the algorithm presented in thischapter. Eventually, in Section 7.3, we show the results of simulations performed in ROS,for a dynamic network with addition and removal of agents.

7.1 Switching control algorithm

In this chapter, we consider a directed network, where the agents have a communicationradius ρ > 0, and can exchange the bearing measurements with other agents within thatradius. We need to give a different definition of counterclockwise neighbor and βi(t): forthis purpose, we denote as N i(t) the set of the agents laying within the communicationradius of agent i, namely N i(t) = j ∈ 1, . . . , N : ‖yj(t)− yi(t)‖ ≤ ρ. Let βij(t) ≥ 0 bethe positive counterclockwise angle from ϕi(t) to ϕj(t). The agent j ∈ N i(t) that attainsthe minimum βij(t) is called the counterclockwise neighbor of agent i at time t, and it isdenoted as νi(t). Moreover, we let

βi(t) = βiνi(t)(t) if N i(t) is nonempty,βi(t) = 0 if N i(t) is empty.

(7.1)

62

7.1. Switching control algorithm

Then, the control objective is formally written as

limt→∞

Di(t) = D∗, (7.2)

limt→∞

βi(t) = 2πN, (7.3)

for all i ∈ 1, . . . , N. In order to achieve (7.2) and (7.3), we adopt the estimator (3.6) andthe distributed control algorithm (5.2).

In order to guarantee that, whatever is the number N of agents in the network, once itgets on the desired circle C(x,D∗), every agent can receive information from at least oneneighbor, we give the following assumption.

Assumption 7.1.1. The communication radius of the agents satisfies ρ > 2D∗.

Now we can formalize the main result of this section, giving the following theorems thatextend the results achieved in Chapter 5 and Chapter 6.

Theorem 7.1.1. Consider a network of N autonomous agents tracking a stationary targetunder estimate law (3.6) and control law (5.2), where the counterclockwise angle is defined asin (7.1). If Assumption 5.1.1 and Assumption 7.1.1 hold, the agents converge to the desiredcircle C(x,D∗) while forming a regular polygon; i.e., they achieve the control objective (3.3)and (3.4). Moreover the control algorithm (5.2) is robust to the introduction and the removalof agents in the network.

Proof. The proofs of the exponential convergence of the estimate and of the distance er-ror are the same of Theorem 5.1.1, since every agent has an approaching motion towardsthe target that is independent from the other agents in the network. From the proof ofLemma 4.5.1, we have that the dynamics of β(t) are not affected (up to a permutation oftwo indexes in L) if some agents change their counterclockwise neighbor, due to the enteror the exit of agents in the communication radii, or in the whole network; from this consid-eration it holds that the agents in the network reach the average consensus on the anglesβi(t).

Theorem 7.1.2. Consider a network of N autonomous agents under estimate law (3.6)and control law (5.2), with ‖x(t)‖ ≤ εT ; the counterclockwise angle is defined as in (7.1).Under Assumption 6.1.1, Assumption 6.1.2 and Assumption 7.1.1 , the agents converge toan annulus of radii D∗−ε∆ and D∗+ε∆, containing C(x(t), D∗) and they are in a formationsuch that, as t→∞, ‖β(t)‖ ≤ Uβ/kϕ, where kϕ is the control gain for the tangential termin (5.2), and Uβ is given by (6.12). Moreover the control algorithm (5.2) is robust to theintroduction and the removal of agents in the network.

Proof. The proofs of the bounded convergence of the estimate and of the distance error arethe same of Theorem 6.1.1, since every agent has an approaching motion towards the targetthat is independent from the other agents in the network. From the proof of Theorem 7.1.1,it holds that in the scenario of stationary target the agents form a regular polygon on thecircle C(x,D∗); similarly to what done in the proof of Theorem 6.1.1, we can prove that theconsensus on the angles βi(t) is bounded.

63

7.2. Implementation in ROS

7.2 Implementation in ROS

In this section, we describe the implementation of the distributed control algorithm in ROS;we associate to every agent of the network four different ROS nodes written in Python, im-plementing respectively the control algorithm, the simulator of the agent motion, the sensorsimulator and the estimator. This nodes exchange information through the publication oftopics and the subscription to topics. The inter-agent communication is managed by acloud, implemented on a separate ROS node, that is useful also to manage the addition andthe removal of agents from the network.

In order to illustrate the communication system with the adoption of the remote cloud,we will describe the implementation of the ROS nodes Controller.py and Cloud.py,implementing respectively the control algorithm and the cloud.

ControllerWe start describing the ROS node Controller.py, implementing the control algorithmand publishing the message containing the control signal (thus, the velocity) ui(t); this nodesubscribes to the topics bearing measurement, estimate and position publishedby other ROS nodes. In order to update the control algorithm, in this node the angle βi(t)is computed at every iteration. The function in Listing 7.1 returns the counterclockwiseangle βij(t) between agent i and agent j, computed on the basis of the information on theposition that agent i receives by agent j; it is worth noticing that βij(t) can be equivalentlycomputed using the information on the bearing vector of the other agent.

1 de f Counte rc l ockwi s e ang l e ( p o s i t i o n , n e i g h b o r p o s i t i o n ) :y i=np . array ( [ p o s i t i o n [0]− t a r g e t p o s i t i o n [ 0 ] , p o s i t i o n [1]− t a r g e t p o s i t i o n[ 1 ] , 0 . 0 ] )

3 y j=np . array ( [ n e i g h b o r p o s i t i o n [0]− t a r g e t p o s i t i o n [ 0 ] , n e i g h b o r p o s i t i o n[1]− t a r g e t p o s i t i o n [ 1 ] , 0 . 0 ] )n i=np . l i n a l g . norm( y i )

5 n j=np . l i n a l g . norm( y j )sp=np . inne r ( y i , y j )

7 vp=np . c r o s s ( y i , y j )c o s b e t a=sp /( n i ∗ n j )

9 s i n b e t a=vp [ 2 ] / ( n i ∗ n j )beta=math . atan2 ( s i n b e t a , c o s b e t a )

11 i f beta <0:beta=beta+2∗math . p i

13 r e turn beta

Listing 7.1: Counterclockwise angle function in Controller.py.

The distributed control algorithm is updated as in Listing 7.2: in particular the functiondescribed in 7.1 is called to compute the angles βij(t) for all the agents j ∈ Ni(t) and thevelocity of agent i is published on the topic vel.

1 whi le not rp . i s shutdown ( ) :LOCK. a c q u i r e ( )

3 i f s t o p p u b l i s h :rp . s igna l shutdown ( ” agent planner removed” )

5 #Bearing vec to r in the t a n g e n t i a l d i r e c t i o nph i bar=np . array ( [ bearing measurement [1 ] ,− bearing measurement [ 0 ] ] )

7 agent beta =[ ]#Counterc lockwise ang le

9 f o r name in agent names :i f not a g e n t p o s i t i o n s [ name ] i s None and np . l i n a l g . norm( p o s i t i o n−

a g e n t p o s i t i o n s [ name ] )<=rho :

64


11 agent beta . append ( Counte rc l ockwi se ang l e ( p o s i t i o n , a g e n t p o s i t i o n s[ name ] ) )beta=0

13 i f l en ( agent beta ) >0:beta=min ( agent beta )

15 #Estimated d i s t a n c ee s t d i s t = np . l i n a l g . norm( est imate−p o s i t i o n )

17 #Control a lgor i thmv e l = K d∗ bearing measurement ∗( e s t d i s t−DESIRED DISTANCE)+K f i ∗ e s t d i s t ∗ph i bar ∗(ALPHA+beta )

19 #Ve loc i t y messagevel msg = gms . Vector ( x=v e l [ 0 ] , y=v e l [ 1 ] )

21 LOCK. r e l e a s e ( )# P ub l i ca t i o n o f the v e l o c i t y message

23 ve l pub . pub l i sh ( vel msg )RATE. s l e e p ( )

Listing 7.2: Control algorithm in Controller.py.

In order to manage the addition of agents in the network, the ROS Services AddAgentand AddMe are called: as shown in Listing 7.3, the service AddMe is called by agent i assoon as it enters the newtork, to notify its addition to the cloud.rp . w a i t f o r s e r v i c e ( ' /AddMe ' )

2 add me proxy=rp . Serv iceProxy ( ' /AddMe ' , dns . AddAgent )add me proxy . c a l l ( node name )

Listing 7.3: Call to the ROS Service AddMe in Controller.py.

Consequently the cloud calls the service AddAgent, whose handler function is shown inListing 7.4; for every agent j ∈ V \i, a subscriber to the position topic of the new agent iis created, and both the list agent names, containing the names of all the agents in thenetwork, and the list agent positions, containing the positions received by the otheragents, are updated (see Listing 7.4). Since the list agent positions is updated with theposition of the new agent in the newtork, the computation of the angle βi(t), and thereforethe updating of the control law, takes into account the presence of the new agent, as shownin Listing 7.2.

1 # S u b s c r i b e r s to the p o s i t i o n s o f the other agentsagent names =[ ]

3 a g e n t p o s i t i o n s = de f a g e n t c a l l b a c k (msg , name) :

5 g l o b a l a g e n t p o s i t i o n sLOCK. a c q u i r e ( )

7 a g e n t p o s i t i o n s [ name ] = [ msg . x , msg . y ]LOCK. r e l e a s e ( )

9 # Handler o f the s e r v i c e ”AddAgent”p o s i t i o n s u b s c r i b e r s =

11 de f add agent handler ( req ) :g l o b a l agent names

13 g l o b a l a g e n t p o s i t i o n sg l o b a l p o s i t i o n s u b s c r i b e r s

15 p o s i t i o n s u b s c r i b e r s [ req . name]= rp . Subsc r ibe r (name= ' / '+req . name+ ' / p o s i t i o n ' ,

17 d a t a c l a s s=gms . Point ,c a l l b a c k=a g e n t c a l l b a c k ,

19 c a l l b a c k a r g s=req . name ,q u e u e s i z e =1)

21 LOCK. a c q u i r e ( )agent names . append ( req . name)

23 a g e n t p o s i t i o n s [ req . name]=None

65


LOCK. r e l e a s e ( )25 r e turn dns . AddAgentResponse ( )

27 rp . S e r v i c e ( ' AddAgent ' , dns . AddAgent , add agent handler )

Listing 7.4: Handler of the ROS Service AddAgent in Controller.py.

In order to manage the removal of agents from the network, the ROS Services Re-moveAgent and RemoveController are called: consequently to the call of RemoveAgentby an external ROS node, the subscribers to the position of agent i are unregistered (seeListing 7.5), hence agent i is not considered anymore to update the control signals of theother agents in the network, as implemented in Listing 7.2.

1 de f remove agent handler ( req ) :g l o b a l p o s i t i o n s u b s c r i b e r s

3 g l o b a l agent namesg l o b a l a g e n t p o s i t i o n s

5 LOCK. a c q u i r e ( )p o s i t i o n s u b s c r i b e r s [ req . name ] . u n r e g i s t e r ( )

7 agent names . remove ( req . name)de l a g e n t p o s i t i o n s [ req . name ]

9 LOCK. r e l e a s e ( )re turn dns . RemoveAgentResponse ( )

11rp . S e r v i c e ( ' RemoveAgent ' , dns . RemoveAgent , remove agent handler )

Listing 7.5: Handler of the ROS Service RemoveAgent in Controller.py.

Furthermore, the cloud calls the service RemoveController, whose handler function isin Listing 7.6, setting the boolean stop publish to True and shutting the ROS nodeController.py down, as implemented in Listing 7.2.de f r e m o v e c o n t r o l l e r h a n d l e r ( req ) :

2 g l o b a l s t o p p u b l i s hLOCK. a c q u i r e ( )

4 s t o p p u b l i s h=TrueLOCK. r e l e a s e ( )

6 r e turn dns . RemoveAgentResponse ( )

8 rp . S e r v i c e ( ' RemoveControl ler ' , dns . RemoveAgent , remove p lanner handler )

Listing 7.6: Handler of the ROS Service RemoveController in Controller.py.

CloudIn this section, the ROS node Cloud.py is described. The remote cloud has a monitoringfunction for the network, updating the list of the names of all the agents in the network,agent names. When agent i enters the network, the function add me handler is calledand the cloud adds its name in the list agent names, creating proxies for the call ofthe services AddAgent and RemoveAgent associated to agent i. Furthermore, proxiesfor the call to the services RemoveController, RemoveSensor, RemoveVehicle andRemoveEstimate are created, in order to shut down the ROS nodes associated to agenti, in the case it is removed from the network.

Subsequently to the enter of agent i in the network, the cloud calls the service AddAgentfor all the agents in the network, notifying both the enter of agent i to the agents whosename is in agent names and the presence of the other agents to agent i.

66


Consequently to the call to the ROS service RemoveAgent by an external ROS node,the function remove handler is called and the name of the removed agent is deletedby the list agent names, and the services RemoveController, RemoveSensor, Re-moveVehicle and RemoveEstimate are called in order to shut down the ROS nodesassociated to the removed agent.agent names =[ ]

2 a d d a g e n t p r o x i e s =r emove agent prox i e s =

4 r e m o v e s e n s o r p r o x i e s =r em o v e e s t i m at e p ro x i e s =

6 r e m o v e v e h i c l e p r o x i e s =r e m o v e c o n t r o l l e r p r o x i e s =

8LOCK=thd . Lock ( )

10rp . w a i t f o r s e r v i c e ( ' AddAgentArtist ' )

12 p l o t t e r p r o x y=rp . Serv iceProxy ( ' AddAgentArtist ' , dns . AddAgent )rp . w a i t f o r s e r v i c e ( ' RemoveAgentArtist ' )

14 p lo t t e r proxy remove=rp . Serv iceProxy ( ' RemoveAgentArtist ' , dns . RemoveAgent )

16 #Handler o f the s e r v i c e ”AddMe”de f add me handler ( req ) :

18 LOCK. a c q u i r e ( )a d d a g e n t p r o x i e s [ req . name]= rp . Serv iceProxy ( req . name+ ' /AddAgent ' , dns .AddAgent )

20 r emove agent prox i e s [ req . name]= rp . Serv iceProxy ( req . name+ ' /RemoveAgent ' ,dns . RemoveAgent )r e m o v e s e n s o r p r o x i e s [ req . name]= rp . Serv iceProxy ( req . name+ ' /RemoveSensor ' ,dns . RemoveAgent )

22 r em o v e e s t i m at e p ro x i e s [ req . name]= rp . Serv iceProxy ( req . name+ ' /RemoveEstimate ' , dns . RemoveAgent )r e m o v e v e h i c l e p r o x i e s [ req . name]= rp . Serv iceProxy ( req . name+ ' /RemoveVehicle' , dns . RemoveAgent )

24 r e m o v e c o n t r o l l e r p r o x i e s [ req . name]= rp . Serv iceProxy ( req . name+ ' /RemoveControl ler ' , dns . RemoveAgent )f o r name in agent names :

26 a d d a g e n t p r o x i e s [ name ] . c a l l ( req . name)a d d a g e n t p r o x i e s [ req . name ] . c a l l (name)

28 agent names . append ( req . name)LOCK. r e l e a s e ( )

30 p l o t t e r p r o x y . c a l l ( req . name)re turn dns . AddAgentResponse ( )

32rp . S e r v i c e ( 'AddMe ' , dns . AddAgent , add me handler )

34# Handle o f the s e r v i c e ”Remove”

36 de f remove handler ( req ) :LOCK. a c q u i r e ( )

38 agent names . remove ( req . name)de l a d d a g e n t p r o x i e s [ req . name ]

40 f o r name in agent names :r emove agent prox i e s [ name ] . c a l l ( req . name)

42 p lo t t e r proxy remove . c a l l ( req . name)de l r emove agent prox i e s [ req . name ]

44 r e m o v e c o n t r o l l e r p r o x i e s [ req . name ] . c a l l ( req . name)r e m o v e s e n s o r p r o x i e s [ req . name ] . c a l l ( req . name)

46 r em o v e e s t i m at e p ro x i e s [ req . name ] . c a l l ( req . name)r e m o v e v e h i c l e p r o x i e s [ req . name ] . c a l l ( req . name)

48 de l r e m o v e c o n t r o l l e r p r o x i e s [ req . name ]de l r e m o v e s e n s o r p r o x i e s [ req . name ]

50 de l r e m o v e v e h i c l e p r o x i e s [ req . name ]

67

7.3. Numerical simulations

i 1 2 3 4 5yi1(t0) 2 3 4 5 6yi2(t0) 4 4 4 4 4

Table 7.1: Initial positions of the agents in the simulation with a stationary target.

de l r e m ov e e s t i m at e p ro x i e s [ req . name ]52 LOCK. r e l e a s e ( )

re turn dns . RemoveAgentResponse ( )54

rp . S e r v i c e ( 'Remove ' , dns . RemoveAgent , remove handler )56

rp . i n i t n o d e ( ' c loud ' )58 rp . sp in ( )

Listing 7.7: Cloud.py.


In this section we show the results of simulations of switching networks, perfomed in ROS:the first simulation is related to the scenario of a dynamic network of agents tracking astationary target, while the second simulation is related to the scenario of mobile target.

In both cases, the agents wait for the detection of ϕi(t) in monitoring positions, and theystart moving as soon as they receive the bearing measurements; the monitoring positionsbelong to a shore line.

In order to model the initial monitoring state of the network, we define for every agent anactivation time ai, representing the instant when agent i receives the bearing measurementsand consequently enters the network, and a removal time ri, representing the instant whenagent i is removed from the network, due to a fault.

Tracking a stationary targetFor the first simulation, we consider a network of N = 5 agents and the shore line ischosen as yi,2 = 4, where yi = [yi,1, yi,2]T ; in particular the initial positions of the agentsare indicated in Table 7.1. The reference distance is D∗ = 1 and the stationary target’sposition is [1, 1]T . The gains of the control law (5.2) are chosen as kd = 1, kϕ = 0.3, α = 1,while the estimate gain is chosen as ke = 0.7. The activation times are chosen as a1 = 2,a2 = 8, a3 = 20, a4 = 35, a5 = 38, while the removal time for the agent 2 is r2 = 55.

Figure 7.1 illustrates the trajectories of the agents in the plane, Figure 7.3 shows theconvergence of the angles βi(t), and Figure 7.2 shows the convergence of the estimate error.From Figure 7.3 we can see how the angles βi(t) converge to 2π/N each time a new agent isadded or removed. The spikes in this figure are due to the agents entering or leaving eachothers’ communication radii, as well as to them entering or leaving the network.

Tracking a mobile targetIn the second simulation, we consider a dynamic network of N = 4 agents, tracking atarget with a slow motion, whose kinematics is x(t) = 0.5[cos 0.05t, sin 0.05t]T , with initialposition [0, 0]T . The reference distance is D∗ = 1; notice that, approximating rx with

68

7.3. Numerical simulations

yi1(t)0 1 2 3 4 5 6

yi2(t)

0

1

2

3

41

2

3

4

5

Figure 7.1: Trajectories of the networked agents in the simulation with a stationary target.The black circles represent the position of agents 1, 3, 4, and 5 at the end of the simulation(agent 2 is removed at t = 55); the red circle represents the stationary target.

i 1 2 3 4yi1(t0) 2 3 4 5yi2(t0) 4 4 4 4

Table 7.2: Initial positions of the agents in the simulation with a moving target.

ke = 0.7, and recalling that, for the chosen trajectory of the target, εT = 0.07, D∗ satisfiesAssumption 6.1.2. The agents wait for the detection of ϕi(t) in monitoring positions,belonging to the the line yi2 = 4; in particular the initial positions of the agents areindicated in Table 7.2. The gains of the control law are chosen as kd = 2, kϕ = 0.2, α = 1,while the estimate gain is chosen as ke = 0.7. The activation times are a1 = 2, a2 = 5,a3 = 20, a4 = 40, while the removal time for the agent 2 is r2 = 55.

The results of this simulation are shown in Figures 7.4–7.6. In Figures 7.5–7.6 we canclearly see the persistent oscillations caused by the target’s unknown motion.

69

7.4. Summary

t

0 10 20 30 40 50 60 70

‖xi(t)‖

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1

2

3

4

5

Figure 7.2: Estimate errors of the networked agents in the simulation with a stationarytarget.

7.4 Summary

In this chapter, a modified version of the controller presented previously is introduced; theproposed distributed control algorithm is based on the information that the agent receivesfrom the neighbor located within its communication radius. The implementation in ROSof the control algorithm is described, dwelling on the inter-agent communication using aremote cloud. In the end, the results of simulations performed in ROS are presented.

70

7.4. Summary

t

0 10 20 30 40 50 60 70

βi(t)

0

1

2

3

4

5

6

1

2

3

4

5

Figure 7.3: Angles βi(t) of the networked agents in the simulation with a stationary target.

71

7.4. Summary

yi1(t)-3 -2 -1 0 1 2 3 4 5 6

yi2(t)

-3

-2

-1

0

1

2

3

4

1

2

3

4

x(t)

Figure 7.4: Trajectories of the networked agents in the simulation with a moving target.The black circles represent the position of agents 1, 3, 4 at the end of the simulation (agent2 is removed at t = 55), while the red circle represents the position of the target at the endof the simulation.

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7.4. Summary

t

0 10 20 30 40 50 60 70 80

βi(t)

0

1

2

3

4

5

6

1

2

3

4

Figure 7.5: Angles βi(t) of the networked agents in the simulation with a moving target.

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7.4. Summary

t

0 10 20 30 40 50 60 70 80

‖xi(t)‖

0

0.5

1

1.5

2

2.5

3

1

2

3

4

Figure 7.6: Estimate errors of the networked agents in the simulation with a moving target.

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Chapter 8

Conclusions

8.1 Summary of the results

We introduced a decentralized control algorithm and an estimate algorithm for a directednetwork, in order to achieve the circumnavigation task around a target.

We started from the analysis of the network behavior in the scenario of stationary target,with a position known to all the agents; considering the system written in polar coordinates,we proved the existence of an attractive limit cycle, corresponding to the desired circlecentered at the target position. Using geometric properties of the considered network, wewrote a consensus equation involving the counterclockwise angles, proving the equidistanceof the agents on the circle.

Furthermore, we introduced an estimator for every agent, in order to drive the systemto localize the target in the scenario where its position is unknown to the network; a proofof the exponential convergence of the estimate has been given and, starting from it, theexponential convergence of the agents and their equidistance on the desired circle havebeen proved.

Finally we proved that, in the scenario of mobile target, the convergence of the estimateand of the agents on the circle is not asymptotic but only bounded.

Subsequently, we showed the results achieved in the scenario of dynamic networks, withthe adding and the removal of agents: simulations in ROS have been presented and thecode written has been described.

8.2 Future developments

A challenging extension of the proposed algorithm, that has been carrying on, provides anintermittent and event-triggered inter-agent communication, that is possibly based on theexchange of data on a shared repository hosted on a cloud.

We proposed an estimator adotped by the single agent, on the basis of its estimate ofthe target position; a different solution is the exploitation of the multi-agent communicationto improve the localization of the target, in order to reuse the information on the estimatecomputed by the other agents.

An interesting development is the design of a thridimensional circumnavigation algo-rithm, on the basis of the planar control law described in this work, in order to improve theperformance and adopt it better to the applications involving the UAVs.

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