Licentiate Thesis in Electrical Engineering

**Cooperative Control of **

**Leader-follower Multi-agent Systems ** **under Transient Constraints**

**FEI CHEN**

**Stockholm, Sweden 2020**

**kth royal institute **
**of technology**

**Cooperative Control of **

**Leader-follower Multi-agent Systems ** **under Transient Constraints**

**FEI CHEN**

Licentiate Thesis in Electrical Engineering KTH Royal Institute of Technology Stockholm, Sweden 2020

Academic Dissertation which, with due permission of the KTH Royal Institute of Technology, is submitted for public defence for the Degree of Licentiate of Engineering on Friday the 23rd of October 2020, at 10:00 a.m. in Harry Nyquist, Malvinas väg 10, Stockholm.

© Fei Chen

ISBN: 978-91-7873-659-1 TRITA-EECS-AVL-2020:52

Printed by: Universitetsservice US-AB, Sweden 2020

**Abstract**

Signiﬁcant research has been devoted to the problem of distributed consensus or formation control of multi-agent systems in the last decades. These distributed control strategies are designed for all agents and sometimes it may be redundant and costly since the desired tasks may be fulﬁlled by steering part of the agents through the appropriately designed local control strategy while the other agents can just follow some standard distributed control protocol. Therefore, the leader-follower framework is considered in this thesis, which is meant in the sense that a group of agents with external inputs are selected as leaders in order to drive the group of followers in a way that the entire system can achieve consensus or target formation within certain transient bounds. The followers are only guided through their dynamic couplings with the steered leaders and without any additional control eﬀort.

The ﬁrst part of the thesis deals with consensus or formation control for leader- follower multi-agent systems in a distributed manner using a prescribed performance strategy. Both the ﬁrst and second-order cases are treated. Under the assumption of tree graphs, a distributed control law is proposed for the ﬁrst-order case when the decay rate of the performance functions is within a suﬃcient bound. Then, two classes of tree graphs that can have additional followers are investigated. For the second-order case, we propose a distributed control law based on a backstepping approach for the group of leaders to steer the entire system achieving the target formation within the prescribed performance bounds. In the second part, we further discuss the results for general graphs with cycles, which are extended based on the previous results of tree graphs. The extension of general graphs with cycles has more practical applications and oﬀers a complete theory for undirected graphs.

In the last part of the thesis, we derive necessary and suﬃcient conditions for the leader-follower graph topology in order to achieve the desired formation while satisfying the prescribed performance transient bounds. The results developed in this thesis are further veriﬁed by several simulation examples.

**Sammanfattning**

Under de senaste ˚artionden har en avsev¨ard m¨angd forskning har till¨agnats

˚at distribuerad reglering f¨or konsensus eller formationspositionering av ﬂer-agent system. Dessa distribuerade regleringsstrategier ber¨aknas f¨or alla agenter, vilket ibland kan b˚ade vara ¨overﬂ¨odigt och kostsamt eftersom en uppgift kan klaras av genom att endast styra en del av agenterna genom en l¨ampligt designad lokal regleringsstrategi medan resten av agenterna forts¨atter f¨olja deras ursprungliga styrningsprotokoll. I denna avhandling anv¨ands d¨arf¨or ett ledare-f¨oljare ramverk i den bem¨arkelsen att agenter som f˚ar insignaler fr˚an externa k¨allor v¨aljs som ledare f¨or att styra f¨oljarna till ett tillst˚and d¨ar konsensus eller formationspositionering uppn˚as f¨or hela systemet, inom vissa kortvariga gr¨anser. F¨oljarna p˚averkas endast genom deras dynamiska sammankoppling med ledarna, utan ytterligare styrning.

Avhandlingens f¨orsta del behandlar distribuerad konsensus- eller formationsre- glering av ﬂer-agents ledare-f¨oljare system genom en f¨oreskriven prestanda. B˚ade f¨orsta och andra ordningens fall behandlas. En distribuerad styrningslag f¨oresl˚as, under antagandet att n¨atverket ¨ar ett tr¨ad, f¨or f¨orsta ordningens fall, d¨ar prestanda- funktionen avtar tillr¨ackligt fort. D¨arefter unders¨oks tv˚a klasser av tr¨adn¨atverk som kan ha ytterligare f¨oljare. F¨or andra ordningens fall f¨oresl˚ar vi en distribuerad styrningslag, som baseras p˚a Backstepping-metoden, f¨or en grupp av ledare som uppn˚ar formationsm˚alet med f¨oreskriven prestanda. I den andra delen behandlar vi generella grafer med cykler, d¨ar det utvidgade resultatet baseras p˚a resultatet f¨or tr¨adn¨atverk. Generella grafer med cykler har mer praktiska till¨ampningar och erbjuder en mer fullst¨anding teori f¨or oriktade grafer. I avhandlingens sista del h¨arleder vi n¨odv¨andiga och tillr¨ackliga villkor f¨or ledare-f¨oljare-topologin f¨or att uppn˚a ett ¨onskat formationsm˚al samtidigt som en f¨oreskriven prestanda med ko- rtvariga gr¨anser uppfylls. Dessutom bekr¨aftas resultaten som utvecklas i denna avhandling genom ett ﬂertal simuleringsexempel.

**Acknowledgements**

This thesis could not have been accomplished without the support and help from many people I have met during the past few years. First and foremost, I would like to express my sincere appreciation to my supervisor Prof. Dimos Dimarogonas, for the opportunity to work on such an interesting research topic and to work in such a wonderful department. I truly appreciate his constant support, encouragement, and insightful advice. His earnest attitude towards work also inﬂuences me a lot. I would also like to thank my co-supervisor, Prof. Jana Tumova, for the guidance and interesting discussions.

I would also express my appreciation to all my colleagues at the Division of Decision and Control Systems for creating such a friendly environment, and for their continuous support and encouragement. Especially, I would like to thank my oﬃce mates Joana, Mina, Hanxiao, Rodrigo, and Yuchao for creating the coolest oﬃce.

I would like to thank Pian, Yulong, Xiao, Wei, Adrian, Lars, Peter, Chris, Pedro, Pouria, Maria, Soﬁe, Andrea, Dimitris, Alex, Rodrigo, and Yuchao, for interesting discussions and feedback about my research. The help of Rijad for translating the abstract into Swedish and Joana, Pushpak, Yu, and Rodrigo, for proof-reading this thesis has been highly appreciated. Many thanks also to all the professors and administrators of the department, for creating such a stimulating working environment.

I am also grateful to my former advisors Prof. Soﬁe Haesaert, Prof. Alessandro Abate, and Prof. Siep Weiland, for guiding me to the world of formal methods. They made me more determined in my research interest and what I want to do. I owe a special thanks to Prof. Karl Henrik Johansson for being the advance reviewer and Prof. Stephan Trenn for being the opponent.

I would also like to thank all my friends in Netherlands and China, for the time we spent together.

Finally, I want to thank all the support and love I have received from my family. I am deeply indebted to my life partner Zhe for her love, encouragement and patience during these years of physical distance; I am also grateful to our parents for their unconditional love and support.

*Fei Chen*
September, 2020.

### Contents

**Abstract** **iii**

**Sammanfattning** **v**

**Acknowledgements** **vii**

**1** **Introduction** **1**

1.1 Motivation . . . . 1

1.2 Literature Overview . . . . 3

1.3 Thesis Outline and Contributions . . . . 5

**2** **Preliminaries** **9**
2.1 Graph Theory . . . . 9

2.2 Leader-follower Multi-agent Systems . . . . 9

2.3 Prescribed Performance Control . . . . 12

**3** **Control of First-order Leader-follower Multi-agent Systems**
**under Prescribed Performance Guarantees** **15**
3.1 Introduction . . . . 15

3.2 System Description . . . . 16

3.3 Problem Statement . . . . 17

3.4 Control Strategy . . . . 17

3.5 Simulations . . . . 25

3.6 Conclusions . . . . 31

**4** **Second-order Formation Control for Leader-follower Multi-**
**agent Systems with Prescribed Performance** **33**
4.1 System Description . . . . 33

4.2 Problem Statement . . . . 34

4.3 Control Strategy . . . . 35

4.4 Simulations . . . . 41

4.5 Conclusions . . . . 42 ix

x Contents

**5** **Leader-follower Multi-agent Systems with Cycles** **45**

5.1 Introduction . . . . 45

5.2 Problem Statement . . . . 45

5.3 Control Strategy . . . . 46

5.4 Simulations . . . . 50

5.5 Conclusions . . . . 53

**6** **Necessary and Suﬃcient Conditions for Leader-follower For-**
**mation Control with Prescribed Performance** **55**
6.1 Introduction . . . . 55

6.2 Problem Statement . . . . 56

6.3 Necessary and Suﬃcient Conditions on Graph Topology . . . . . 56

6.4 Examples . . . . 63

6.5 Conclusions . . . . 65

**7** **Summary and Future Research** **67**
7.1 Summary . . . . 67

7.2 Future Work . . . . 68

**Bibliography** **71**

### Chapter 1

### Introduction

**1.1** **Motivation**

A multi-agent system is a system comprising of a number of interacting intelligent agents, which has been used to study and to simulate complex systems in diﬀerent domains due to its wide applications in robotics, cooperative control [1], formation control [2], social networks [3], network security [4], and transportation systems [5], etc. As a key topic in multi-agent systems, consensus has been well studied in the last few decades. Consensus means to reach an agreement regarding a certain quantity of interest that depends on the state of all agents and a consensus protocol is an interaction rule that speciﬁes the information exchange between an agent and all of its neighbors on the network [6]. Formation control [2] is another topic that has been widely investigated in the ﬁeld of multi-agent systems, which is characterised as achieving or maintaining desired geometrical patterns via the cooperation of multiple agents.

Besides the classical consensus or formation control purpose for multi-agent systems, the mobile robot team may also need to fulﬁll some additional transient behavior due to some performance requirements or the constraints in the space, e.g., collision avoidance in the work space or collision avoidance between agents. Some recent research further considers the transient constraints while the multi-agent systems are steered by the distributed control laws to the target average consensus [7]

or platoon formation [8]. However, these distributed control laws are designed for all agents and sometimes it may be redundant and costly since the target consensus or formation together with the transient behavior may be fulﬁlled by steering part of the agents through the suitably designed local control strategy while the other agents just follow some standard distributed control protocol. Therefore, the leader-follower framework is considered in this thesis, which is meant in the sense that a group of agents with external inputs are selected as leaders in order to drive the group of followers in a way that the entire system can achieve consensus or target formation within certain transient bounds. The followers are only guided through their dynamic couplings with the steered leaders and without any additional control eﬀort.

1

2 Introduction

Motivated by the above discussion, the goal of the thesis is to propose a systematic distributed control strategy for the leaders such that the controlled leader-follower multi-agent system can achieve consensus or formation target while satisfying all the predeﬁned transient performance bounds. The role of the leaders is to guide the followers and maintain the whole team within the transient performance bounds besides the consensus or formation purpose using the designed distributed control strategy. Besides applying the classical results of consensus control or formation control for leaderless multi-agent systems, we will also use the results of transient approaches in the research of leaderless multi-agent systems, and investigate how these results can ﬁt in the leader-follower framework. These transient approaches include Prescribed Performance Control (PPC) [9] and Control Barrier Functions (CBF) [10]. Both of PPC and CBF have been studied recently for leaderless multi- agent systems, but have not been investigated in the leader-follower setup. We will focus on PPC in this thesis, which is proposed in order to prescribe the evolution of system output or the tracking error within some predeﬁned region.

In this thesis, we consider the classical consensus or formation control in a leader- follower framework using transient approaches. More speciﬁcally, we aim to design the leaders only such that the entire system achieves consensus or target formation within certain performance bounds with the designed prescribed performance control strategy. We apply a PPC law only to the leaders while the followers will just follow the leaders by obeying some standard agreement protocol without any further control and knowledge of the prescribed team bounds. Compared with existing work of PPC for multi-agent systems [7], we apply a PPC law only to the leaders while most of the related work applies PPC to all the agents to achieve consensus or target formation. Unlike other leader-follower approaches using PPC [11], in which the multi-agent system only has one leader and the leader is treated as a reference for the followers, we focus on a more general framework in the sense that we can have more than one leader and the leaders are designed in order to steer the entire system achieving consensus or target formation within the prescribed performance bounds.

The diﬃculties of this general leader-follower framework are due to the combination of uncertain topologies, leader amount and leader positions. In addition, the leader can only communicate with its neighbouring agents.

In practical applications (e.g., cooperative control of mobile robots), a group of agents may have advanced actuation or computation capabilities compared with other agents. How to intelligently utilize their advanced capabilities to lower the cost or avoid redundant eﬀort has become signiﬁcantly more crucial. Therefore, how to design the distributed control strategy for the leader-follower multi-agent systems attracts great practical interest. It is also very challenging to apply the existing transient approaches to the leader-follower setup and to analyse the leader-follower graph topology.

1.2. Literature Overview 3

**1.2** **Literature Overview**

This section provides an overview of the literature related to the topics covered in this thesis. Recent research that has been done in the leader-follower framework can be divided into two parts. The ﬁrst part deals with the controllability of leader- follower multi-agent systems while the second part targets leader selection problems.

Besides, we also review the literature regarding the transient approach used in this thesis, i.e., Prescribed Performance Control (PPC). Finally, some related work on the leader-follower graph topology is discussed. In the following, the literature review will be reﬁned in the beginning of each corresponding section when needed.

We ﬁrst brieﬂy summarise some classical results of consensus and formation control for leaderless multi-agent systems, which will be the basis to further discuss the consensus or formation control of leader-follower multi-agent systems using transient approaches. The study of consensus is a key topic in multi-agent systems due to its wide applications in robotics and cooperative control [1]. Consensus is achieved when a group of agents converge to a common value. The ﬁrst-order consensus protocol was introduced in [12], while second-order consensus was investigated in [13].

A survey on consensus problems was given in [6]. Formation control [2] of multi-agent systems has attracted great interest due to its wide applications in coordination of multiple robots. A formation is characterised as achieving or maintaining desired geometrical patterns via cooperation of multiple agents. In this thesis, we focus on relative position-based formation control methods, which were summarised in [14], where both the ﬁrst and second-order relative position-based formation protocol are discussed. These are extended from the ﬁrst-order [12] and second-order consensus protocol [13], respectively.

*Controllability of leader-follower multi-agent systems was ﬁrst investigated in [15]*

by deriving conditions on the network topology, which ensures that the network can be controlled by a particular member which acts as a leader. In [16, 17], the authors identify necessary conditions for the controllability of the corresponding leader- follower networks using equitable partitions of graphs. Controllability conditions for leader-follower multi-agent systems with double integrator dynamics and their connection with graph topology properties are addressed in [18]. [19] investigates the controllability problem of multi-agent systems deﬁned by undirected signed graphs through almost equitable partition and provides a necessary condition for the controllability of the network. In [20], a suﬃcient controllability condition together with optimal control techniques for driving the leader-follower multi-agent systems between speciﬁc positions are developed. The work in [21–23] mainly focuses on investigating lower or upper bounds for controllable subspaces of leader-follower networks, which is quite useful in leader selection for achieving the controllability of any given network. That is, what is the smallest set of leaders that should be manipulated such that the overall system is controllable. Another interesting work [24] regarding leader selection with controllability guarantees ﬁrst shows that the dimension of the controllable subspace is invariant under the addition or removal of edges between leaders, then a controllable structure can be obtained via connecting

4 Introduction

the leaders of disjoint sub-graphs that are all controllable under their corresponding leaders.

*Leader selection aims to optimally choose a set of leaders that guarantees the*
controllability of the leader-follower multi-agent systems or maximizes a system
performance metric. This performance metric includes energy-related metric, robust-
ness metric, and transient performance metric, etc. There exists lots of literature
working on the controllability of leader-follower multi-agent systems, which has been
summarised in the last section, for example [17, 19, 22]. Although some of these
work does not consider leader selection problems, they derive some conditions on
the graph topology of the leader-follower multi-agent system, which will impose
implicitly some criteria on how to choose the leaders among the agents to make
some conditions hold, e.g., some matrix properties. Some other work involves the
problem of how to choose the leaders among the agents such that the leader-follower
multi-agent system can achieve the optimal performance in terms of some predeﬁned
performance metrics. For instance, in [25] several standard measures of controllabil-
ity, which are deﬁned based on the control Gramian, are utilized as the performance
metric. The leader selection problem is ﬁrst explicitly tackled in [26]. Here, the
authors consider the setting where leaders remain an identical, constant value and
*the followers are subject to stochastic disturbance, then k leaders are selected to*
minimize the steady-state variance of the derivation. Some eﬃcient algorithms are
*proposed in [27] to select k leaders with respect to performance metrics including*
coherence and convergence rate in the speciﬁc path graphs and ring graphs. In [28],
the authors investigate the problem of optimally assigning a predetermined number
of leaders to guarantee an optimal performance that is measured in terms of the*H*_{2}
norm of the network. The leader selection for tracking and the task of driving the
centroid of the agents to a given reference point is considered in [29].

*Prescribed performance control (PPC) was originally proposed in [9], with the*
aim to prescribe the evolution of system output or the tracking error within some
predeﬁned region. For example, an agreement protocol that can additionally achieve
prescribed performance for a combined error of positions and velocities is designed
in [7] for multi-agent systems with double integrator dynamics, while PPC for
multi-agent average consensus with single integrator dynamics is presented in [30].

In [31], the authors consider the formation control problem for nonlinear multi- agent systems with prescribed performance guarantees and connectivity constraints.

In [32], the authors use PPC for large vehicle platoons formation while avoiding connectivity breaks and collisions with neighboring vehicles. [33] employs PPC on a class of pure feedback systems. Applying PPC under temporal logic speciﬁcations is presented in [34, 35]. The authors investigate PPC for nonlinear systems subject to a subset of signal temporal logic speciﬁcations in [34], while a distributed cooperative manipulation problem satisfying a given metric interval temporal logic (MITL) speciﬁcation under PPC framework is studied in [35]. Funnel control, which uses a similar idea as PPC was ﬁrst introduced in [36] for reference tracking. In [37], the authors utilize funnel control for uncertain nonlinear systems that have arbitrary strict relative degree and input-to-state stable internal dynamics.

1.3. Thesis Outline and Contributions 5

There is some related research work on deriving topological conditions or number of leaders to ensure network connectivity for leader-follower multi-agent systems.

For example, in [38], the authors investigate the leader-to-follower ratio needed to maintain connectivity in leader-follower multi-agent networks with proximity based communication topology in one dimensional case, while an extension to two dimensional is presented in [39]. In [40], the authors propose a novel approach to consensus and connectivity maintenance. This work provides indirect metrics on parameters and initial conditions for a single-integrator leader-follower proximity- based network, which inherently ensures connectivity maintenance and consensus of leader-follower multi-agent systems. The extension to double-integrator case is discussed in [41] accordingly.

**1.3** **Thesis Outline and Contributions**

In this Section, we provide the outline of the thesis and indicate the contributions of each chapter.

**Chapter 2**

In Chapter 2, we introduce notation and preliminaries that are used throughout this
thesis. These preliminaries include graph theory, consensus and formation control of
leader-follower multi-agent systems and Prescribed Performance Control (PPC). The
thesis is then divided into four main parts in Chapters 3*− 6, which are elaborated*
in details next.

**Chapter 3**

In Chapter 3, we address the problem of distributed control for leader-follower multi- agent systems with ﬁrst-order dynamics under prescribed performance guarantees.

A group of agents with external inputs are selected as leaders in order to drive the group of followers in a way that the entire system can achieve consensus or target formation within certain prescribed performance transient bounds. Under the assumption of tree graphs, a distributed control law is proposed when the decay rate of the performance functions is within a suﬃcient bound. Then, two classes of tree graphs that can have additional followers are investigated, that is the speciﬁc chain and star graphs. This part is based on the following contributions:

• Fei Chen and Dimos V. Dimarogonas, “Consensus Control for Leader-follower Multi-agent Systems under Prescribed Performance Guarantees”, The 58th IEEE Conference on Decision and Control (CDC), Nice, France, 2019.

• Fei Chen and Dimos V. Dimarogonas, “Leader-follower Formation Control with Prescribed Performance Guarantees”, IEEE Transactions on Control of Network Systems, 2020. (Accepted)

6 Introduction

**Chapter 4**

In Chapter 4, we discuss the problem of distributed control for second-order leader- follower multi-agent systems with prescribed performance guarantees. Under the assumption of tree graphs, we propose a distributed control law based on a back- stepping approach for the group of leaders to steer the entire system to a target formation within certain prescribed performance transient bounds for the whole team. In particular, under the leader-follower framework, PPC is used in order to achieve the target formation along with the prescribed performance guarantees. This part is based on the following contributions:

• Fei Chen and Dimos V. Dimarogonas, “Second Order Consensus for Leader- follower Multi-agent Systems with Prescribed Performance”, 8th IFAC Work- shop on Estimation and Control of Networked Systems (NECSYS), Chicago, USA, 2019.

• Fei Chen and Dimos V. Dimarogonas, “Leader-follower Formation Control with Prescribed Performance Guarantees”, IEEE Transactions on Control of Network Systems, 2020. (Accepted)

**Chapter 5**

We have investigated consensus or formation control for leader-follower multi-agent systems with transient constraints under the assumption of tree graphs in Chapter 3 and Chapter 4. In this chapter, we further discuss general graphs with cycles and propose a distributed control law for general leader-follower graphs with cycles when the decay rate of the performance functions is within a suﬃcient bound and the convergence beneﬁts of the cycles are discussed. The extension of general graphs with cycles has more practical applications compared with our previous results for tree graphs and oﬀers a complete theory for undirected graphs. The challenge of general graphs with cycles is that we cannot use the positive deﬁniteness of the edge Laplacian for convergence analysis directly, thus we need to partition the graph into a spanning tree and the remaining edges that complete the cycles. This part is based on the following contributions:

• Fei Chen and Dimos V. Dimarogonas, “Further Results on Leader-follower Multi-agent Formation Control with Prescribed Performance Guarantees”, 59th IEEE Conference on Decision and Control (CDC), Jeju Island, Republic of Korea, 2020.

**Chapter 6**

In Chapter 6, we address the problem of deriving necessary and suﬃcient conditions for leader-follower multi-agent systems in order to achieve the desired relative position-based formation while satisfying prescribed performance guarantees. In this chapter, general graphs with cycles are considered and we derive the necessary and

1.3. Thesis Outline and Contributions 7

suﬃcient conditions on the leader-follower graph topology under which the target formation together with the the prescribed performance guarantees can be fulﬁlled.

This part is based on the following contributions:

• Fei Chen and Dimos V. Dimarogonas, “Further Results on Leader-follower Multi-agent Formation Control with Prescribed Performance Guarantees”, 59th IEEE Conference on Decision and Control (CDC), Jeju Island, Republic of Korea, 2020.

• Fei Chen and Dimos V. Dimarogonas, “Necessary and Suﬃcient Conditions for Leader-follower Formation Control with Prescribed Performance”, 2020.

(In preparation)
**Chapter 7**

In Chapter 7, we conclude the thesis and discuss the directions for future research.

### Chapter 2

### Preliminaries

In this chapter, the notation that will be used hereafter as well as the necessary background, are provided.

**2.1** **Graph Theory**

An undirected graph [42]*G = (V, E) comprises of the vertices set V = {1, 2, . . . , n}*

and the edges set *E = {(i, j) ∈ V × V | j ∈ N**i**} indexed by e*1*, e*_{2}*, . . . , e**m*. Here,
*m =**|E| is the number of edges and N**i* denotes the agents in the neighbourhood of
*agent i that can communicate with i.*

*The adjacency matrix* *A of G is the n × n symmetric matrix whose elements a**ij*

*are given by a**ij* *= 1, if (i, j)**∈ E, and a**ij**= 0, otherwise. The degree of vertex i is*
*deﬁned as d**i*=

*j**∈N**i**a**ij**. Then the degree matrix is Δ = diag(d*_{1}*, d*_{2}*, . . . , d**n*). The
*graph Laplacian of**G is L = Δ − A.*

*A path is a sequence of edges connecting two distinct vertices. A graph is*
*connected if there exists a path between any pair of vertices.*

By assigning an orientation to each edge of*G we can deﬁne the incidence matrix*
*D = D(G) = [d**ij*] *∈ R*^{n}^{×m}*. The rows of D are indexed by the vertices and the*
*columns are indexed by the edges with d**ij**= 1 if the vertex i is the head of the edge*
*(i, j), d**ij* =*−1 if the vertex i is the tail of the edge (i, j) and d**ij* = 0 otherwise. Based
on the incidence matrix, the graph Laplacian of*G can be described as L = DD** ^{T}*. In

*addition, L*

*e*

*= D*

^{T}*D is the so called edge Laplacian [42] and c*

*ij*denotes the elemnts

*of L*

*e*.

**2.2** **Leader-follower Multi-agent Systems**

In this thesis, we consider a multi-agent system with vertices *V = {1, 2, . . . , n}.*

*Without loss of generality, we suppose that the ﬁrst n**f* agents are selected as
*followers while the last n**l*agents are selected as leaders with respective vertices set
*V**F* =*{1, . . . , n**f**}, V**L*=*{n**f**+ 1, . . . , n**f**+ n**l**} and n = n**f**+ n**l*.

9

10 Preliminaries

*Let p**i**, v**i**∈ R be the respective position and velocity of agent i, where we only*
consider the one dimensional case, without loss of generality. Speciﬁcally, the results
can be extended to higher dimensions with appropriate use of the Kronecker product.

**2.2.1** **Relative Position-based Formation**

We aim to design a control strategy for the leader-follower multi-agent system such that it can achieve the following target relative position-based formation

*F := {p | p**i**− p**j**= p*^{des}_{ij}*, (i, j)**∈ E},* (2.1)
*where p = [p*_{1}*, . . . , p**n*]^{T}*and p*^{des}_{ij}*:= p*^{des}_{i}*− p*^{des}*j* *, (i, j)* *∈ E is the desired relative*
*position between agent i and agent j, which is constant and denoted as the diﬀerence*
*between the absolute desired spositions p*^{des}_{i}*, p*^{des}_{j}*∈ R. Here, p*^{des}*ij* is only needed to
*be known and p*^{des}_{i}*, p*^{des}* _{j}* are deﬁned with respect to an arbitrary reference frame
and do not need to be known.

*In the ﬁrst-order case, the state evolution of each follower i**∈ V**F* is governed by
the ﬁrst-order formation protocol:

˙

*p**i*=*−*

*j**∈N**i*

*(p**i**− p**j**− p*^{des}*ij* *).* (2.2)

*The state evolution of each leader i**∈ V**L* is governed by the ﬁrst-order formation
*protocol with an external input u**i* *∈ R:*

*p*˙*i*=*−*

*j**∈N**i*

*(p**i**− p**j**− p*^{des}*ij* *) + u**i**.* (2.3)

*In the second-order case, the state evolution of each follower i**∈ V**F* is governed
by the second-order formation protocol:

˙
*p**i**= v**i*

*˙v**i*=*−*

*j**∈N**i*

*(p**i**− p**j**− p*^{des}*ij* *) + (v**i**− v**j*)

*.* (2.4)

*The state evolution of leader i* *∈ V**L* is governed by the second-order formation
*protocol with an external input u**i* *∈ R:*

˙
*p**i**= v**i*

*˙v**i*=*−*

*j**∈N**i*

*(p**i**− p**j**− p*^{des}*ij* *) + (v**i**− v**j*)

*+ u**i**.* (2.5)

*Let us denote p = [p*_{1}*, . . . , p**n*]^{T}*, v = [v*_{1}*, . . . , v**n*]^{T}*, p*^{des}*= [p*^{des}_{1} *, . . . , p*^{des}* _{n}* ]

^{T}*∈ R*

*as the respective stack vector of absolute positions, velocities and target posi- tions and*

^{n}*u = [u*

*n*

*+1*

_{f}*, . . . , u*

*n*

_{f}*+n*

*l*]

^{T}*∈ R*

^{n}*is the control input vector including*

^{l}2.2. Leader-follower Multi-agent Systems 11

the external inputs of leader agents in (2.3), (2.5). Denote ¯*p = [¯**p*_{1}*, . . . , ¯**p**m*]* ^{T}*,

¯

*v = [¯**v*_{1}*, . . . , ¯**v**m*]^{T}*, ¯**p** ^{des}* = [¯

*p*

^{des}_{1}

*, . . . , ¯*

*p*

^{des}*]*

_{m}

^{T}*∈ R*

*as the respective stack vector of relative positions, relative velocities and target relative positions between the pair*

^{m}*of communication agents for the edge (i, j) = k*

*∈ E, where ¯p*

*k*

*p*

*ij*

*= p*

*i*

*− p*

*j*

*, ¯*

*v*

*k*

*v*

*ij*

*= v*

*i*

*− v*

*j*

*, ¯*

*p*

^{des}

_{k}*p*

^{des}*ij*

*= p*

^{des}

_{i}*− p*

^{des}*j*

*, k = 1, 2, . . . , m. It can be then veriﬁed that*

*Lp = D ¯*

*p and ¯*

*p = D*

^{T}*p. In addition, if ¯*

*p = 0, we have that Lp = 0. Similarly, it*

*holds that Lv = D¯*

*v, ¯*

*v = D*

^{T}*v and Lp*

^{des}*= D ¯*

*p*

*, ¯*

^{des}*p*

^{des}*= D*

^{T}*p*

*.*

^{des}By stacking (2.2), (2.3), the dynamics of the ﬁrst-order leader-follower multi- agent system is rewritten as:

Σ_{1}: ˙*p =**−L(p − p*^{des}*) + Bu.* (2.6)
Similarly, stacking (2.4) and (2.5), the dynamics of the second-order leader-follower
multi-agent system is rewritten as:

Σ_{2}:

*p*˙

*˙v*

=

0*n* *I**n*

*−L −L*

*p**− p*^{des}*v*

+

0*n**×n**l*

*B*

*u,* (2.7)

*where L is the graph Laplacian and B =*

_{0}

*nf ×nl*

*I*_{nl}

*.*

*In the sequel, we denote x = p**− p*^{des}*= [x*_{1}*, . . . , x**n*]* ^{T}* as the shifted absolute

*position vector with respect to p*

*. Accordingly, ¯*

^{des}*x = ¯*

*p*

*− ¯p*

*= [¯*

^{des}*x*

_{1}

*, . . . , ¯*

*x*

*m*]

*is denoted as the shifted relative position vector with respect to ¯*

^{T}*p*

*.*

^{des}**2.2.2** **Consensus**

In this thesis, consensus can be regarded as a special case of the relative position-
based formation*F as in (2.1) by setting p*^{des}*ij* *= 0,**∀(i, j) ∈ E, that is, consensus is*
reached when all the leader and follower agents converge to a common value. More
*speciﬁcally, in the ﬁrst-order case, the state evolution of each follower i* *∈ V**F* is
governed by the ﬁrst-order consensus protocol:

˙

*p**i*=*−*

*j**∈N**i*

*(p**i**− p**j**).* (2.8)

*The state evolution of each leader i**∈ V**L* is governed by the ﬁrst-order consensus
*protocol with an external input u**i**∈ R:*

˙

*p**i*=*−*

*j**∈N**i*

*(p**i**− p**j**) + u**i**.* (2.9)

*In the second-order case, the state evolution of each follower i**∈ V**F* is governed
by the second-order consensus protocol:

˙
*p**i**= v**i*

*˙v**i*=*−*

*j**∈N**i*

*((p**i**− p**j**) + (v**i**− v**j**)) .* (2.10)

12 Preliminaries

*The state evolution of leader i* *∈ V**L* is governed by the second-order consensus
*protocol with an external input u**i* *∈ R:*

˙
*p**i**= v**i*

*˙v**i*=*−*

*j**∈N**i*

*((p**i**− p**j**) + (v**i**− v**j**)) + u**i**.* (2.11)

Similarly, by stacking (2.8), (2.9), the dynamics of the ﬁrst-order leader-follower multi-agent system is rewritten as:

Σ_{1}: ˙*p =**−Lp + Bu.* (2.12)

By stacking (2.10) and (2.11), the dynamics of the second-order leader-follower multi-agent system is rewritten as:

Σ_{2}:

˙
*p*

*˙v*

=

0*n* *I**n*

*−L −L*

*p*
*v*

+

0*n**×n**l*

*B*

*u,* (2.13)

*where L is the graph Laplacian and B =*

_{0}

*nf ×nl*

*I*_{nl}

*.*

In the sequel, we will mainly focus on relative position-based formation control for both ﬁrst and second-order leader-follower multi-agent systems and treat consensus as a special case when needed. The results proposed later can be applied for consensus control directly.

**2.3** **Prescribed Performance Control**

The aim of PPC is to prescribe the evolution of the relative position ¯*p**i**(t) within*
some predeﬁned region described as

¯

*p*^{des}_{i}*− ρ*_{¯x}_{i}*(t) < ¯**p**i**(t) < ¯**p*^{des}_{i}*+ ρ*_{¯x}_{i}*(t),* (2.14)
or equivalently, to prescribe the evolution of the shifted relative position ¯*x**i**(t) within*

*− ρ**¯x**i**(t) < ¯**x**i**(t) < ρ*_{¯x}_{i}*(t).* (2.15)
(2.14) and (2.15) are equivalent since ¯*x = ¯**p**− ¯p** ^{des}* (while in component format

*also). Here ρ*

_{¯x}

_{i}*(t) :*R

_{+}

*→ R*

_{+}

*\ {0}, i = 1, 2, . . . , m are positive, smooth and strictly*decreasing performance functions that introduce the predeﬁned bounds for the shifted relative positions. One example choice is

*ρ*_{¯x}_{i}*(t) = (ρ*_{¯x}_{i0}*− ρ*_{¯x}_{i∞}*)e*^{−l}^{xi}^{¯} ^{t}*+ ρ*_{¯x}_{i∞}*.* (2.16)
*with ρ*_{¯x}_{i0}*, ρ*_{¯x}_{i∞}*and l*_{¯x}_{i}*positive parameters and ρ*_{¯x}* _{i∞}* = lim

*t*

*→∞*

*ρ*

_{¯x}

_{i}*(t) represents the*maximum allowable tracking error at steady state.

2.3. Prescribed Performance Control 13

Normalizing ¯*x**i**(t) with respect to the performance function ρ*_{¯x}_{i}*(t), we deﬁne the*
modulated error as ˆ*x*¯*i**(t) and the corresponding prescribed performance region**D*_{¯x}* _{i}*
as:

ˆ¯

*x**i**(t) =* *x*¯*i**(t)*

*ρ*_{¯x}_{i}*(t)**,* (2.17)

*D*_{¯x}_{i}* {ˆ¯x**i*: ˆ*x*¯*i**∈ (−1, 1)}.* (2.18)
*Then the modulated error is transformed through a transformed function T*_{¯x}* _{i}* that

*deﬁnes the smooth and strictly increasing mapping T*

_{¯x}*:*

_{i}*D*

_{¯x}

_{i}*→ R, T*

_{¯x}*(0) = 0. One example choice is*

_{i}*T*_{¯x}* _{i}*(ˆ

*x*¯

*i*) = ln

1 + ˆ*x*¯*i*

1*− ˆ¯x**i*

*.* (2.19)

The transformed error is then deﬁned as

*ε*_{¯x}* _{i}*(ˆ

*x*¯

*i*

*) = T*

_{¯x}*(ˆ*

_{i}*x*¯

*i*) (2.20)

*It can be veriﬁed that if the transformed error ε*

_{¯x}*(ˆ*

_{i}*x*¯

*i*) is bounded, then the modulated error ˆ

*x*¯

*i*is constrained within the region (2.18). This also implies the error ¯

*x*

*i*evolves within the predeﬁned performance bounds (2.15). Diﬀerentiating (2.20) with respect to time, we derive

*˙ε*_{¯x}* _{i}*(ˆ

*x*¯

*i*) =

*J*

*T*

_{xi}_{¯}(ˆ

*x*¯

*i*

*, t)[ ˙¯*

*x*

*i*

*+ α*

_{¯x}

_{i}*(t)¯*

*x*

*i*] (2.21) where

*J**T*_{xi}_{¯} (ˆ*x*¯*i**, t)* *∂T*_{¯x}* _{i}*(ˆ

*x*¯

*i*)

*∂ ˆ**x*¯*i*

1

*ρ*_{¯x}_{i}*(t)* *> 0* (2.22)
*α*_{¯x}_{i}*(t)** −**ρ*˙_{¯x}_{i}*(t)*

*ρ*_{¯x}_{i}*(t)* *> 0* (2.23)

*are the normalized Jacobian of the transformed function T*_{¯x}* _{i}* and the normalized
derivative of the performance function, respectively.

### Chapter 3

### Control of First-order Leader-follower Multi-agent Systems under Prescribed Performance Guarantees

In this chapter, we address the distributed formation control problem of ﬁrst-order leader-follower multi-agent systems under prescribed performance guarantees. Under the assumption of tree graphs, a distributed control law is proposed when the decay rate of the performance functions is within a suﬃcient bound. Then, two classes of tree graphs that can have additional followers are investigated.

**3.1** **Introduction**

We consider consensus or formation control for ﬁrst-order leader-follower multi- agent systems in this chapter and consensus will be treated as a special case of the relative position-based formation control. We are interested in how to design control strategies for the leaders such that the leader-follower multi-agent system achieves a relative position-based formation within certain performance bounds. Compared with the existing work of PPC for multi-agent systems, e.g., [7], we apply a PPC law only to the leaders while most of the related work applies PPC to all the agents to achieve here tasks such as consensus or formation. The beneﬁt is to lower the cost of the control eﬀort since the followers will follow the leaders by obeying ﬁrst-order formation protocols without any additional control and knowledge of the prescribed team bounds. Unlike other approaches for leader-follower multi-agent systems using PPC [11], in which the multi-agent system only has one leader and the leader is treated as a reference for the followers, we focus on a more general framework in the sense that we can have more than one leader and the leaders are designed in order to steer the entire system achieving the target formation within the prescribed performance bounds. The diﬃculties in this work are due to the combination of uncertain topologies, leader amount, and leader positions. In addition, the leader can only communicate with its neighbouring agents. The contributions of this chapter

15