Consensus on the Special Orthogonal Group: Theory and Applications to Formation Control

Transcription

1 Dissertation for Doctor of Philosophy Consensus on the Special Orthogonal Group: Theory and Applications to Formation Control Byung-Hun Lee School of Mechatronics Gwangju Institute of Science and Technology 2017

2 박사학위논문 특수직교군에서의컨센서스 : 이론과편대제어응용 이병훈 기전공학부 광주과학기술원 2017

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5 To my lovely wife

6 PHD/ME Byung-Hun Lee. Consensus on the Special Orthogonal Group: Theory and Applications to Formation Control. School of Mechatronics p. Advisor: Prof. Hyo-Sung Ahn. Abstract This dissertation studies specific issues related to the synchronization of a set of agents evolving on nonlinear manifold, more particularly S 1 and SO(3). The dissertation describes convergence analysis of the existing models and then addresses novel control algorithms. The objective of the dissertation can be summarized by the following question: Given N agents evolving on the non-convexity configuration space, how can we design individual agent s control law to get synchronization behavior with relative quantities of interest? Technical applications of the above question include orientation alignment problem in formation or flocking system and the attitude control of rigid bodies. The dissertation addresses the difficulty of global synchronization in existing control algorithm and suggestion of novel control strategy for global synchronization. The main contributions of this dissertation are as follows: An extensive study of synchronization on the circle is addressed; (a) highlighting difficulties encountered for synchronization and (b) proposing novel control strategy to overcome these difficulties. With virtual agents evolving on convexity configuration space, actual agents achieve the consensus on the non-convexity configuration space. Application of the proposed consensus algorithm to the formation control or network localization problem allows global convergence of the system to the desired configurations. The dissertation addresses a distributed control law for autonomous attitude synchronization based on the relative attitude. i

7 The author mainly focuses on the control strategy for synchronization behavior on the nonconvexity configuration space. We make virtual agents(called auxiliary variable in the dissertation) evolving in Euclidean space. Then, the motion of virtual agent is transformed into the motion of agent on the non-convexity configuration space. It implies that the actual agent follows the motion of virtual agent. Consequently, the control objective is achieved by designing control input of virtual agent. This strategy is advantageous to analyze the convergence property in vector space, as well as avoid the difficulty of stability analysis in nonlinear manifold. While the dissertation shows that this control strategy is applied to particular manifolds (i.e. S 1 and SO(3)), one may consider the extension of the result to various manifolds of Lie groups. c 2017 Byung-Hun Lee ALL RIGHTS RESERVED ii

8 PHD/ME 이병훈. 특수직교군에서의컨센서스 : 이론과편대제어응용. 기전공학부 p. 지도교수 : 안효성. 국문요약 이논문은 S 1 혹은 SO(3) 같은비선형메니폴드 (nonlinear manifold) 에서움직이는다중개체 (multi-agents) 들의동기화 (synchronization) 와관련한문제를다루고있다. 논문은기존에존재하는동기화모델의시스템해석을기반으로새로운제어알고리즘의설계를서술한다. 논문에서다루고자하는주제를요약하면아래질문으로정리할수있다. 비컨백시티짜임새공간 (non-convexity configuration space) 에서정의한 N개의개체를생각해보자. 각개체가상대적인편차 (relative quantity) 정보를이용해서동기화를이루도록하는제어입력을설계할수있을까? 이런문제를적용할수있는기술적인분야는편대제어나군집비행에서각개체의방위각을정렬하는문제나강체 (rigid body) 들의자세제어 (attitude control) 문제가있다. 논문은먼저기존에존재하는제어알고리즘이가지는동기화의한계를설명하고, 전역적인영역에서동기화가가능한새로운제어법칙을제시한다. 논문의주요결과는다음과같다. 원 (circle) 상에서동기화문제에대한깊은연구 : (a) 원상에서동기화문제를해석할때발생하는어려움을설명하고, (b) 이런어려움을극복하는새로운제어전략을제시함. 컨백시티짜임새공간 (convexity configuration space) 에서정의한가상개체를이용하여, 비컨백시티짜임새공간에서개체들의동기화를이룸. 제안한컨센서스알고리즘을편대제어나네트워크상의위치추정문제에적용하여, 전체시스템의안정성을전역적으로증명함. 상대적인자세편차 (relative attitude) 정보를이용하여자세동기화 (attitude synchronization) 를이루는분산적인제어법칙을제시함논문전체를관통하는주요결과는비켄벡시티짜임새공간에서의동기화를목적으로제어전략을설계하는것이다. 이제어전략의간단한설명은다음과같다. 먼저유클 iii

9 리디안공간 (Euclidean space) 에서가상개체를만들고, 가상개체의움직임을비컨백시티짜임새공간상에움직이는실제개체의움직임으로변환해준다. 이방법은실제개체가가상개체의움직임을따라가도록만들어준다. 그러므로가상개체의움직임을잘설계한다면제어목적을완수할수있다. 이런전략의이점은비선형메니폴드에서안정성을해석할때발생하는어려움을피할수있을뿐만아니라, 시스템의수렴성을백터공간에서해석하는것이가능하다는점이다. 이논문에서는이러한제어전략을특정메니폴드 ( 즉, S 1 와 SO(3)) 에만적용하였지만, 만약다른메니폴드에도적용이가능하다면논문의결과를확장할수있을것이다. c 2017 이병훈 ALL RIGHTS RESERVED iv

10 Contents Abstract (English) Abstract (Korean) List of Contents List of Figures i iii v viii 1 Introduction Introduction and Literature Review Synchronization of a group of agents Literature reviews on Synchronization problem Literature reviews on orientation alignment in formation control problem Summary and Outline Mathematical Background Notations Fundamentals of graph theory Laplacian matrix and Consensus Special Orthogonal Group Consensus Protocol using Relative Quantities Consensus on vector space Orientation alignment problem Relative quantity on the circle Discontinuities on dynamics Asymptotic pattern under orientation alignment law Analysis of convergence property Calculation of asymptotic pattern and simulation results Concluding remark Further analysis on the orientation alignment problem v

11 3.4.1 Manifold characterization An example of three agents Analysis on potential function Concluding remark Modified coupling function for global synchronization Problem Statement Characteristic of the critical point Potential shaping Numerical Results Concluding remark Global Synchronization on the Circle Introduction Consensus in S Problem Statement Estimation of positions on the circle Stability analysis Concluding remark Consensus protocol using auxiliary variables Estimation method using auxiliary variables Feedback controller from the estimated state Simulation result Concluding remark Orientation Estimation in SO(d) Introduction Novel Method for Orientation Estimation Orientation estimation in SO(d) Analysis of the Stability Non-existence of the solution Numerical example Formation control and Network localization Problem statement vi

12 5.3.2 Formation Control in 3-D space Network Localization in 3-D Space Simulation for formation control problem Simulation for network localization problem Concluding remark Consensus on SO(3) Introduction Preliminaries The property of SO(3) Obtaining the relative position on SO(3) Consensus on SO(3) Stability Analysis Simulation results Concluding remark Conclusion Summary and Contribution Future work Appendix A Nonlinear Control System 130 A.1 Comparison Principle A.2 Solution of Bernoulli Equation Bibliography 132 Acknowledgements 140 vii

13 List of Figures 1.1 Synchronization on the circle Illustration of attitude synchronization problem Balanced state of three agents: when agents are distributed over whole circle, attractive forces for each agent are canceled An overview of relations between the main chapters. Chapter 3 and Chapter 4 study the synchronization behavior in S 1, while Chapter 5 and Chapter 6 study the synchronization behavior in SO(3) Simple directed graph (left) and its incidence matrix (right) The motion of agents under the consensus algorithm(i.e (2.1) and (3.2)) is illustrated. Agents move towards neighbors or the interior of their convex hull. Consequently, convex hull of vertices progressively shrinks until reaching a consensus Illustration of arc of length(γ) and geodesic distance(θ N1 ) Interaction topology for obtaining relative orientation angle : k and j sense relative angle δ kj and δ jk, respectively. then, j transmits the information of δ jk to k to calculate the relative quantity θ jk Mechanical analog of the orientation alignment system with directed sensing graph(left) and discontinuities on dynamics(right)(modified from F. Dörfler et. al. [7]) An example of directed graph which is connected viii

14 3.6 Asymptotic patterns on unit circle for the first case(left) and second case(right) Simulation result for the Case 1) (left) and Case 2) (right) Undirected graph of a cyclic structure (left) and a tree structure (right) in the case of three agents An example of a bijective mapping between an open subset of R 2 and S An example of the splay state in the case of 6 agents Illustration of an off-diagonal entry of the Hessian matrix(solid line) and a modified Hessian matrix (dashed line) respectively A topology for interactions of 12 agents Comparison of the dynamics in the case of that initial states are arbitrary determined in the neighborhood of splay state Convergence rates for the modified function with various values of η are illustrated. A dashed line indicates the convergence rate of classical system Block diagram for the synchronization method based on the local information θ jk. The position of kth agent and the estimated position on the circle are denoted by θ k and ˆθ k respectively Expression of θ k in complex plane Undesired equilibrium set(:= null(h ) ) of (4.5) and its orthogonal space in C N The exponential map of θ S 1 in complex plane(left) and its one-to-one mapping in two dimensional vector space(right) ix

15 4.5 Figure illustrates the tangent mapping from M to S 1. Smooth function ϕ is defined by ϕ(x) = x, where x M Projection of vector field g(x) onto the unit circle is shown. Tangent mapping dϕ x (g(x)) is defined in the algebraic expression dϕ x (g(x)) = g(x) T r/ x. The term R π 2 denotes a rotation matrix of the angle π Sensing topology of ten agents on unit circle for simulation Synchronization of ten agents on the unit circle. Note that θ k = θ k ± 2π S Induced norm of R jk I 2 for all (j, k) E. R jk SO(2) identifies the relative quantity of the state θ jk S Interaction graph of 9 agents for simulation Induced norm of B i Ci T C for all i V The local coordinate frame of i-th agent is rotated from the global reference frame ( g Σ) by C i. In the figure, {e i,1, e i,2, e i,3 } is the set of orthonormal bases of i-th local frame i Σ Interaction graph of six agents Difference of the measured displacement and the desired displacement of neighboring agents. The matrix C denotes the common rotation matrix Simulation for formation control via orientation estimation in 3 dimensional space Actual formation of multi-agents and sensing topology Error of the estimation : difference between displacement of estimated positions and measured displacement. Note that C is a common rotation matrix. 105 x

16 5.9 Simulation for the network localization via orientation estimation Block diagram of the control strategy for consensus on SO(3). The rotation matrix C i SO(3) denotes the attitude of i-th agent Vision camera model(left) and block diagram of pose synchronization with the visual motion observer for ith agent(right). Feature points of j-th agent with respect to the i are denoted by p ijk, k {1, 2, 3} Sensing topology of five agents A figure illustrates the behavior of axis-angle representation of C i, denoted by ωi c. All trajectories of ωi c converge to a common value which identifies a common rotation C Induced norm of C i C for all i V xi

17 Chapter 1 Introduction In the present chapter, an overview of the dissertation is described without mathematical details. The first section introduces the consensus issue in cooperative control problem and literature reviews. Main problems considered in the dissertation are introduced by examples. The second section describes the outline of this dissertation. The main idea of each chapter is also summarized in the second section. 1.1 Introduction and Literature Review A cooperative control for a network of systems interacting via local coupling has attracted researcher s interest during the last decade. The consensus algorithm is one of the most actively studied topics in the networked multi-agent systems. In the consensus problem, the distributed strategies achieve the convergence of states to a common value Synchronization of a group of agents Consider N agents which must agree on a relative quantity of interest. When agents move towards each other, they may reach the same value as time goes to infinity. Ex.A. Synchronization in the vector space : Imagine N particles in d-dimensional vector space which move toward each other. The quantity of interest is a distance between two particles. The distance corresponds to the attractive force between neighboring 1

18 particles. The attractive force draws particles closer together and particles reach the same position as time goes to infinity(see Fig.3.1). Ex.B. Synchronization on the circle : Imagine that multi-vehicles are supposed to reach a common decision on a direction of motion in the plane. In this example, the quantity of interest could be the relative orientation of neighbor s head direction. Then, each vehicle turns its direction to neighbors to reach a common direction; one vehicle may turn clockwise while another vehicle may turn counter-clockwise. The orientation of vehicle s head direction is considered as the position on the circle as shown in Fig.1.1. Figure 1.1. Synchronization on the circle. Ex.C. Attitude synchronization : Consider a set of spacecrafts evolving in 3-dimensional space. Each spacecraft controls its own attitude to agree on the neighbor s attitude. The quantity of interest is the relative information of neighboring attitude. It is illustrated in Fig.1.2. The space on which the variables evolve is called the configuration space. The examples illustrate the importance of the configuration space. For synchronization in vector space, the 2

19 Figure 1.2. Illustration of attitude synchronization problem. particle always moves toward the interior of convex hull of all particles(see Fig.3.1). Then, particles asymptotically reach a common position. For synchronization on the circle, the situation may be similar when agents are close together. However, when agents on the circle are distributed over the whole circle, the convergence of agents is not guaranteed. According to the initial patterns of agents or sensing topology, agents are balanced 1 on the circle or agents move constantly in one direction. For instance, three agents which draw each other through the attractive force are balanced on the circle when agents are distributed over the whole circle. This is shown in Fig Unlike the point on vector space, it is impossible to distinguish a highest and lowest value on the circle. Let us consider the motion of agent on the circle. The agent which moves constantly in one direction can eventually meet the point where the agent passed. It causes a difficulty of analysis of dynamic behavior on the circle. This feature is also shown in SO(3). Consequently, the analysis of synchronization behavior is not as clear on the circle or SO(3) as on the vector space. 1 All agents lay on the same distribution 3

20 Figure 1.3. Balanced state of three agents: when agents are distributed over whole circle, attractive forces for each agent are canceled Literature reviews on Synchronization problem In this dissertation, a consensus algorithm on the circle is considered. Technical application of this problems is the synchronization of coupled oscillators. Synchronization problem in the network of coupled agents have attracted researcher s interest during the last decades. It is a popular topic in various scientific discipline such as engineering, physics, chemistry and biology. Scientists are interested in the collective behavior of insects, birds or cells. Synchronization in such systems is associated with the alignment of rotational directions [10, 55]. Engineers consider the network of coupled oscillators. Technical applications of the coupled model for synchronization include planar vehicle coordination, electric application in the power system and clock synchronization in the decentralized computing system [4, 50, 51, 56]. While the global convergence property of the classical consensus algorithm is analyzed in vector space, the analysis of it on the circle is quit intractable and at least very dependent on the communication graph [49]. A typical method for modeling of the coupled oscillators is Kuramoto model [17]. Kuramoto model provides the continuous- 4

21 time evolution of phase variable in the network of coupled oscillators. Various systems based on the Kuramoto model are described in previous literatures. For the collective behavior of multi-agent, Sepulchre et al. [50] proposed a Lyapunov-based stabilizing feedback control law under the assumption of all-to-all communication among agents. In [51], enhanced control law based on the method in [50] is proposed so that the collective behavior of multi-agent achieves stabilization in the case of time-varying and directed communication topology. Synchronization phenomena in the network of coupled oscillators is described in [4]. The paper provides a condition that predicts synchronization as a function of the parameters and the topology of the underlying network. There were several studies for global convergence analysis on the coupled oscillators. Under the assumption of that graph is fixed and undirected, [44] proposed the sinusoidal coupling model which is derived from the gradient of the modified potential function. The modified potential function is designed such that the only stable equilibrium corresponds to synchronization. The graph information including the number of nodes is required to construct this potential. In the result of [46], Gossip algorithm is proposed for the global synchronization on the circle. At each time instant, a given agent selects randomly one (or none) of its neighbors. The update is then taken as if this neighbor was the one, disregarding the information from others. The expected synchronization in Gossip algorithm can be shown to be independent of the initial condition but it is not easy to characterize which graph property favors a faster convergence. In [47], the novel dynamics for the synchronization is proposed in the networks of coupled oscillators. In the paper, auxiliary variables which are communicated by agents are introduced for recovering the synchronization properties of 5

22 vector spaces for almost all initial conditions. The proposed dynamic consensus algorithm exploits the embedding of the algorithm in a linear space. Under the assumption of that the position of agents on the circle is known, the algorithm computes an update in the embedding space and projects the result onto the manifold [47] Literature reviews on orientation alignment in formation control problem Formation control of mobile autonomous agents is one of the most actively studied topics in distributed multi-agent systems. Depending on the sensing capability and controlled variables, various problems for the formation control have been studied in the literature [36]. In consensus-based control law, known as the displacement-based approaches, agents measure the relative positions of their neighbors with respect to a common reference frame. Then, agents control the relative position for the desired formation [5, 24, 34, 40]. Fundamental requirement for the displacement-based approach is that all agents share a common sense of orientation. For the unknown common reference frame, decentralized estimation of a common reference frame was studied in 2-dimensional space by using the gossip algorithm in [8]. In [13], passivity-based attitude synchronization using relative orientation in SE(3) is investigated without sharing the common reference frame among agents. Sun and Anderson [52] combined the ideas of displacement-based approach and distancebased approach; a part of agents is supposed to control relative position vectors associated with desired direction, while others are tasked to control the distance of inter-agent for the desired formation shape. The objective of this approach is to control the formation to be both desired direction and desired shape. Zelazo et al. [58] proposed the decentralized estimation 6

23 of position in the absence of common reference frame. Oh and Ahn [34, 35] proposed a consensus protocol for local reference frame to be aligned with each other. They found the displacement-style formation control strategies based on orientation alignment, under the distance-based setup (i.e., local reference frames are not aligned with each other). In this approach, agents are allowed to align the orientation of their local coordinate systems by exchanging their relative bearing measurements. By controlling orientations to be aligned, the proposed formation control law allows agent to control relative positions simultaneously. Similar strategy using the consensus protocol is found in [28]. In the paper, control objective is to reach a desired formation with specified relative positions and orientations in the absences of a global reference frame. These strategies ensure asymptotic convergence of the agents to the desired formation under the assumption that interaction graph is uniformly connected and initial orientations belong to an open interval [0, π) [28, 34, 35]. In Chapter 5, we propose a novel formation control strategy via the global orientation estimation. The orientation of each agent is estimated based on the relative angle measurement and auxiliary variables assigned in the complex plane. Unlike the result of [28, 34, 35], in this new approach, consensus property, which is based on the Laplacian matrix with complex adjacency matrix, is applied to allow the global convergence of the auxiliary variables to the desired points. In this way, we show that the global convergence of formation to desired one for the multi-agent system having misaligned frames could be achieved. Consequently, the result remedies the weak point of previous works where the range of initial orientation is constrained to [0, π) [28, 34, 35]. Details of the work are described in Chapter 5 7

24 1.2 Summary and Outline The dissertation consists of seven chapters, including the current chapter, the next chapter on mathematical background, followed by four main chapters showing main technical results, and the last chapter providing conclusions. The following is a brief outline of the thesis structure as well as the content in each chapter. 1. Chapter 1 presents a general introduction to the research background, literature reviews in the research field and research problems to be discussed in this dissertation. 2. Chapter 2 provides some theoretical preliminaries on graph theory, consensus property of Laplacian matrix and special orthogonal group that will be frequently applied in the discussions of technical results in later chapters. 3. Chapter 3 focuses on the analysis of synchronization behavior. Fundamental distinction between circle and vector space is represented to analyze the convergence property on the circle. Under the consensus protocol based on the relative quantities, asymptotic pattern of agent on the circle is characterized. Global convergence property is also analyzed in the perspective of gradient decent algorithm. In the last section of Chapter 3, modified coupling function based on the gradient decent algorithm is proposed to achieve global synchronization on the circle. This chapter is partially adapted from [23]. 4. Chapter 4 aims to establish the control strategy to achieve global synchronization on the circle. The main result of this chapter includes dynamic controller using auxiliary variables for global convergence property. Contribution of this work is global syn- 8

25 chronization behavior based on local information. This chapter is adapted from [21] and [19]. 5. In Chapter 5, estimation of unknown parameters in SO(d) is discussed. Main result of this chapter is the extension of the proposed method in Chapter 4 to d-dimensional space. The estimation method for the element of SO(d) is also combined with the formation control and network localization problem. In both formation and localization problems, the weak point of existing algorithm in the case of that global reference frame is not shared with each other is remedied; convergence property of the system is globally analyzed. The related issue is described in subsection This chapter is adapted from [20] and [22]. 6. Chapter 6 provides the control strategy for the attitude synchronization based on the estimation method which is addressed in Chapter 5. Based on the control strategy which is proposed in Chapter 4, attitude control using relative information is discussed. Global synchronization property is also analyzed in this chapter. 7. Chapter 7 presents a brief summary of main results and contributions of this dissertation, and also addresses some possible directions for future research. The results of both Chapter 3 and Chapter 4 are associated with synchronization on the circle while results of both Chapter 5 and Chapter 6 are associated with attitude synchronization. An overview of relations between main chapters is shown in Fig

26 Figure 1.4. An overview of relations between the main chapters. Chapter 3 and Chapter 4 study the synchronization behavior in S 1, while Chapter 5 and Chapter 6 study the synchronization behavior in SO(3). 10

27 Chapter 2 Mathematical Background 2.1 Notations In this dissertation, we use the following notation. Let S 1 denote the unit circle. The product of d circles is denoted by T d = S 1 S 1, called d-torus. Given N column vectors x 1, x 2... x N R d, x denotes the stacking of the vectors, i.e. x = (x 1, x 2,..., x N ). The matrix I d denotes the d-dimensional identity matrix. Given two matrices A and B, A B denotes the Kronecker product of the matrices. The entry of j-th row and i-th column in the matrix A is denoted by [A] jk. When E x denotes a set, cardinality of E x is denoted by E x. We have i = 1 usually. 2.2 Fundamentals of graph theory In the multi-agent systems, a network of interconnected agents is represented by a graph. Definition 2.1 A directed graph G(V, E, A)(short G) is composed of a finite set V of vertices, a set E V V of edges which are ordered pairs of vertices (j, k) with j, k V, and a set A = [a ij ], i, j V of non-negative weighted values on edges. Definition 2.2 An undirected graph G(V, E, A) is a digraph in which (k, j) E whenever (j, k) E for all j, k V. 11

28 In the case of that all weighted values on edge are equivalent to one, we omit the notation A for the graph (i.e. use G(V, E) instead of G(V, E, A)). Consider a directed graph with m edges and n vertices, denoted by G(V, E). The neighbor set N k of node k is defined as N k := {j V : (j, k) E}. The matrix relating the nodes to the edges is called the incidence matrix I R n m whose entries are defined as [I(G)] jk = +1 if j is the terminal node of edge e k 1 if j is the initial node of edge e k 0 otherwise where e k denotes the k-th edge. An example of the incidence matrix is represented in Fig The incidence matrix can represent the difference between adjacent nodes: Figure 2.1. Simple directed graph (left) and its incidence matrix (right) z k = x j x i where x i R is the value of i V, and x j R is the value of adjacent node j N i. A vector form of z k using incidence matrix can be written as follows: z = I(G) T x From the definition of the incidence matrix, the null space of the transpose of I(G) contains span{1}, where 1 is the vector with all entries equal to one with appropriate dimensions. 12

29 The rank of the incidence matrix of connected graph is n 1 [57]. The reduced incidence matrix, denoted by Ī, of a connected graph G is obtained by deleting the n-th row from the matrix I. The following results are described in [29]. Lemma 2.1 Let G be a graph with n vertices. Then G is connected if and only if the rank of its incidence matrix I is n 1. Proof: (necessary) Since G is connected, the sum of any r rows of I must contain at least one non-zero entry if r < n. This implies no r rows are linearly dependent if r < n. The result is now rank(i) = n 1. (sufficient) If rank(i) = n 1, there are no r < n rows which add up to the all-zero row. This implies that there are no r vertices which are not connected to the other n r vertices. Hence G is connected. Lemma 2.2 If B is any non-singular square submatrix of I then the determinant of B is ±1. Proof: A matrix A is said to be totally unimodular if the determinant of every square submatrix of A has value 1, 0 or 1. The sufficient condition for the totally unimodular matrix is described in [11]. By using the result in [11], we know that the incidence matrix I is the totally unimodular matrix. Therefore, the determinant of non-singular submatrix of I is 1 or 1. From the above lemmas, we notice that there exists a nonsingular submatrix of I whose determinant is ±1. Now, we derive the number of spanning subtrees of G from these subma- 13

30 trices. Lemma 2.3 If B is a submatrix of order n 1 of I, then B is a non-singular matrix if and only if the edges corresponding to the columns of B determine a spanning subtree of G Proof: If Ḡ denotes the spanning subgraph of G whose n 1 edges correspond to the columns of B and the incidence matrix of Ḡ is denoted by Ī, then B is derived from Ī and the reduced incidence matrix of Ḡ. Hence, from the Lemma 2.1, following three statements are equivalent : B is non-singular rank(b) = n 1 Ḡ is connected. Moreover, Ḡ is a tree since rank(b) = n 1 and Ḡ is connected. A German physicist, Kirchhoff, found the number of spanning trees in a graph by using the reduced incidence matrix which is known as matrix-tree theorem. Theorem 2.1 (Matrix-Tree Theorem) If Ī is a reduced incidence matrix of the graph G, then the number of spanning trees of G equals the determinant of (Ī ĪT ). Proof: Before giving the proof, the Cauchy-Binet formula states that if R and Q are matrices of size p by q where p q, then det(rq T ) = det(r(s)) det(q(s) T ) where R(s) and Q(s) is all p p submatrices derived from columns of R and Q corresponding to the index set s respectively. The set s is defined as the subset of {1,..., q} and its cardinality is p. [12]. By using this formula, we find that det(ī ĪT ) = det(b) det(b T ) = {det(b)} 2 = 1 14

31 where B is any (n 1) (n 1) submatrices of Ī. The required result now follows from Lemma Laplacian matrix and Consensus The element of A is a non-negative value, denoted by a kj. The value of a kj is positive if (j, k) E; otherwise it is zero. For the graph G(V, E, A), the Laplacian matrix L R V V is defined as [L] kj = k N i a kj a kj, k = j, k j Consider N agents whose interaction graph is G. Let x k R be a state of k-th agent and x R N denote a stacked vector, defined by x := (x 1,..., x k ). Dynamic model of interconnected agents whose interaction graph is G is represented as follows: ẋ = Lx (2.1) For each agent, the dynamic model satisfies ẋ k (t) = a kj (x j (t) x k (t)) (2.2) j N k The following theorem is derived from the result in [41]. Theorem 2.2 The equilibrium set E N := {[x 1, x N ] T R N : x i = x j i, j = 1,... N} of the system (2.1) is globally exponentially stable if G has a spanning tree. Further, x(t) exponentially converges to a finite point in E N. A matrix is called reducible matrix if it can be placed into block upper-triangular form by simultaneous row and column permutations. A Laplacian matrix derived from the undirected 15

32 connected graph G is not reducible matrix which is called irreducible matrix. Consider a spectrum of the Laplacian matrix L. The following inequality is derived from Ger sgorin disc of L. N λ [L] ii [L] ij (2.3) j=1,j i Since L is a diagonally dominant matrix, the eigenvalue is a nonnegative value. The following theorem is given in [12]. Theorem 2.3 Let A be an irreducible matrix. A boundary point λ of Ger sgorin region of A can be an eigenvalue of A only if every Ger sgorin circle passes through λ. Since Laplacian matrix L has zero eigenvalue which is a boundary point of Ger sgorin region of L, Ger sgorin circle of L passes through zero. Let us consider any (N 1) (N 1) submatrices of L denoted by L. L is an irreducible matrix, and there is at least a Ger sgorin circle of L such that it does not pass through zero. According to the Theorem 2.3, we can see that L does not have zero eigenvalue. The following subsection provides definitions for consensus on Lie groups and properties of Lie group actions. Lie groups on manifolds typically arise in situations involving some kind of symmetry. 2.4 Special Orthogonal Group The special orthogonal group SO(d) is the group of rotations in R d. Equivalently, it can be viewed as the set of positively oriented orthonormal bases of R d. The orientation of a rigid body in R d is represented as the element of SO(d). The SO(d) is a compact and connected 16

33 Lie group. The rotation matrix Q SO(d) is characterized by a real d d orthogonal matrix Q with determinant equal to +1. Then, SO(d) is further characterized as follows [42] : Group multiplication g 1 g 2 G corresponds to Q 1 Q 2 SO(d) for any Q 1, Q 2 SO(d), inverse g 1 G corresponds to Q 1 = Q T SO(d) and identity g e corresponds to I d. The tangent space to SO(d) at the identity I d, i.e. its Lie algebra, is the space of skew-symmetric d d matrices so(d), which has dimension d(d 1)/2: for d d skew-symmetric matrices, only d(d 1)/2 entries are independent. Lie algebra of SO(3) has the following form, 0 ω(3) ω(2) ω(3) 0 ω(1) ω(2) ω(1) 0 so(3) [ ] [ ] ω(1) ω(2) ω(3) R 3 (2.4) where ω R 3 can be considered as the vector field of rotation group. The equation (2.4) illustrates the relation of bijective map between 3-dimensional vector and so(3). The exponential map of ξ so(3) is an entry of SO(3): Q = e ξ SO(3). If ξ(t) so(3) is a function of time, the derivative of Q(t) is written as follows : where d ξ(t) so(3) is the angular velocity of Q(t). dt d dt Q(t) = Q(t) d ξ(t), (2.5) dt Lie algebra obeys Lie bracket which is defined by a non-associative multiplication : i.e. [ξ 1, ξ 2 ] = ξ 1 ξ 2 ξ 2 ξ 1 for any ξ 1, ξ 2 so(3). 17

34 Chapter 3 Consensus Protocol using Relative Quantities 3.1 Consensus on vector space A classical linear consensus algorithm describes the behavior of N agents locally exchanging information about their state x k R d, k V. The dynamics of x k is as follows: ẋ k (t) = a jk (t)(x j (t) x k (t)) (3.1) j N k where a jk (t) is the weight on edge. Let x be a stacked vector defined as x = (x 1,..., x N ) R dn. The vector form of (3.1) is written as ẋ = (L(t) I d )x (3.2) Theorem 3.1 [30] Suppose that G(t) = (V, E(t), A(t)) is uniformly connected and L(t) is bounded piecewise continuous in time. Then, the equilibrium set E x = {x R dn : x j = x k, k, j V} of (3.2) is uniformly exponentially stable. Further, the state x(t) converges to a finite point E x. This result is the extension of the result in Theorem 2.2. Consensus algorithm in vector space has been studied in previous literatures including [14, 30, 31, 37, 38]. The proof of Theorem 3.1 essentially relies on the convexity of the control law. The position of each agent k for 18

35 t > τ always lies in the convex hull of the x j, j V. The permanent contraction of this convex hull allows to conclude that the agents end up at a consensus value shown in Fig.3.1. Figure 3.1. The motion of agents under the consensus algorithm(i.e (2.1) and (3.2)) is illustrated. Agents move towards neighbors or the interior of their convex hull. Consequently, convex hull of vertices progressively shrinks until reaching a consensus. 3.2 Orientation alignment problem Since the position on the circle is considered as phase of oscillators, a consensus problem on the circle usually identifies the synchronization problem of coupled oscillators. Typical method for modeling the coupled oscillators is provided by Kuramoto [17] as follows : θ k = ω k + K N N sin(θ j θ k ) (3.3) j=1 The coupled oscillator dynamics (3.3)(called Kuramoto model) supposes a complete interaction graph and uniform weights a kj = K/N. In an ingenious analysis, Kuramoto showed that synchronization occurs in the model (3.3) if the coupling gain K exceeds a certain threshold K critical function of the distribution of the natural frequencies ω i [17]. Kuramoto s 19

36 original work initiated a broad stream of research. In particular, oscillator networks relevant to the control system include coordination of multi agents and electric power network with synchronous generators. For those oscillator networks, performance analysis of frequency or phase synchronization has been studied in many literatures(coordination of multi agents : [48, 50, 51], electric power network : [4, 6, 53]). In the flocking or formation control of multi-agent systems, orientation alignment problem analyzes networks of identical oscillator as the extension of consensus protocol (2.2) and (3.1) [32, 33, 43, 45]. Oh and Ahn [32, 33] provides the orientation alignment law based on the consensus protocol using geodesic distance in S 1. The geodesic distance between two positions θ k and θ j S 1 is the minimum of the counter-clockwise and the clockwise lengths connecting θ k and θ j (see Fig.3.2). The orientation alignment law is as follows : θ k (t) = a kj (t)θ jk (t), k V (3.4) j N k where a kj (t) > 0. Let arc be a connected subset of S 1. An arc of length containing all agents {θ 1,..., θ N } is denoted by γ as shown in Fig.3.2. If a value of γ is less than π, the consensus protocol (3.4) is represented as follows : θ k (t) = a kj (t)(θ j (t) θ k (t)). (3.5) j N k It is obviously shown that (3.5) has the identical form of (3.1). Due to the contraction property of convex hull, θ k converges to a common value if the interaction topology is uniformly connected and a kj (t) is bounded and piecewise continuous in time [33]. Lemma 3.1 There exists a constant θ S 1 such that θ k exponentially converges to θ under the alignment law (3.4) if γ is less than π, interaction graph G is uniformly connected 20

37 Figure 3.2. Illustration of arc of length(γ) and geodesic distance(θ N1 ). and a kj (t) is bounded and piecewise continuous in time Relative quantity on the circle Consider N agents whose interaction graph G is connected. Orientation of the k-th agent with respect to the global coordinate system( g Σ) is denoted by θ k S 1. Orientation alignment law is written in (3.4). In practical perspective, it is hard to measure the relative quantity θ jk in (3.4) in the case that the GPS-like sensors are not available. For this reason, we assume that the agents can obtain such relative-angles by exchanging their local information with each other. As shown in Fig.3.3, the agent k measures a relative angle δ kj (t) for all j N i. The neighboring agents j also measure the relative-angle δ jk (t) and transmit the data to agent k. Using measured data δ jk (t) and δ kj (t), the i-th agent calculates relative orientation angles of its neighbors as follows : θ jk (t) = PV(δ kj (t) δ jk (t)) (3.6) where the function PV : R S 1 is defined as PV(δ kj (t) δ jk (t)) [(δ kj (t) δ jk (t) + π)mod 2π] π. In this manner, sensing and communication have different directions. Sens- 21

38 ing topology has undirected graph while communication link is corresponding to the directed graph. Figure 3.3. Interaction topology for obtaining relative orientation angle : k and j sense relative angle δ kj and δ jk, respectively. then, j transmits the information of δ jk to k to calculate the relative quantity θ jk Discontinuities on dynamics A mechanical analog of the consensus protocol (3.4) for the fixed and directed sensing topology is represented as a spring network system shown in Fig This spring network is assumed to consist of a group of particles moving on unit circle with no colliding. Some springs are on the wall which is fixed. An angle of each particle is represented as θ i R. Inertial and damping coefficients of particles are M k > 0 and D k > 0 respectively. Stiffness of each spring is k m > 0. We suppose that torques acting on particles are k m θ ji = k m PV(θ j θ k ). For a specific spring network illustrated in Fig. 3.4, the dynamics of mechanical analog 22

39 15 10 θ 1 θ Discontinuity t Figure 3.4. Mechanical analog of the orientation alignment system with directed sensing graph(left) and discontinuities on dynamics(right)(modified from F. Dörfler et. al. [7]) is as follows: M i θk + D i θk = j N i k m θ jk, i {0, 1, 2}, m {1, 2, 3} where θ 0 is an angle of the end point of spring which is connected on the wall. The limit of small mass M k and uniformly high viscous damping D = D k results in M k /D 0, and coupled strengths are a m = k m /D. Let us suppose that min(k 1, k 2 ) k 3 and arc length γ of all agents is larger than π. Then discontinuities on dynamics occur as shown in Fig.3.4, since the strength between 1-th and 2-th particles is weaker than the strengths between other pair of particles. θ 1 = a 1 θ 01 + a 3 θ 21 θ 2 = a 2 θ 02 + a 3 θ 12 The values of θ 0k and θ jk are always opposite in a sign under the case of γ > π (i.e. θ 0k θ jk < 0). Since min( a 1 θ 01, a 2 θ 02 ) > a 3 θ 21 = a 3 θ 12, a geodesic distance of θ 1 and θ 2 increases until all angles are under the condition of γ π. Once the pattern of angle values is changed 23

40 to the case of γ π, a sign of values of θ 21 and θ 12 is changed in the opposite way. The mechanical analog in Fig.3.4 illustrates the basic principle for phenomena of discontinuities in the orientation alignment system (3.4) with fixed and directed interaction graph. Particles on a unit circle move in opposite directions when the attractive force of two particles is weaker than the repulsive force of them. If particle s direction does not change until a shortest length between two particles becomes π, discontinuities on the dynamics of them occur. Same phenomena may occur in the orientation system (3.4). 3.3 Asymptotic pattern under orientation alignment law In this section, consider the orientation alignment law with uniformly weighted and fixed interaction graph G(V, E) as follows : θ k = θ jk, k V = {1,... N} (3.7) j N k where θ jk S 1 is defined in (3.6) Analysis of convergence property The dynamics (3.7) is piecewise continuous dynamics due to the discontinuous points in function PV( ). The dynamics of Θ = (θ 1,... θ N ) is written as Θ(t) = Q Θ(t) (3.8) where Θ(t) is a stacked vector form of every relative quantities θ jk (j, k) E. Matrix Q R N E is derived from the interaction topology. In the given directed graph, each 24

41 column and row of the matrix Q is corresponding to the edge and vertex respectively. 1, if e j is an outgoing edge from vertex k [Q] ij = 0, otherwise Let the frequency Θ be denoted by w. By assuming Θ(t) is continuous with respect to the time, time derivative of w is written as : ẇ(t) = Q Θ(t) = QI T w(t) (3.9) where I is a incidence matrix of the given interaction graph. From the definition of PV( ), Θ is represented as Θ(t) = PV(I T Θ(t)) (3.10) In the assumption that p(t) is continuous for some time interval, ṗ(t) is equivalent to E T w. Lemma 3.2 Assume that p(t) is continuous for t > T, where T > 0 is a finite value. For the directed graph G = (V, E), the frequency variable w is synchronized asymptotically in the dynamics (3.9). w w as t where w = [w, w..., w ] T. Proof: The Laplacian matrix in directed graph is defined as L = D A where D is a outdegree matrix and A is adjacency matrix. For the definition of matrix Q and I, the following relationship is true : D = QQ T, A = Q(I T + Q T ) 25

42 L = D A = QI T. Therefore, the dynamic system (3.9) is rewritten as: ẇ(t) = Lw(t), t > T (3.11) Since matrix L has zero row sum and all eigenvalues are positive real values except for a zero eigenvalue, the dynamics (3.9) satisfies the consensus property. Even though the general solution of w is hard to find, there exist boundaries for the norm of w. The Euclidean norm of Θ(t) follows the inequality : Θ(t) 2 = M(η)I T e iθ 2 η max I T e iθ 2 where M(η) = diag(η 1,...η G ) and η i is a factor to convert the norm of e iθ j e iθ k to a geodesic distance of θ j θ k. By the definition of η i, a maximum value of η i is as follows: η i = θ jk (2 sin (θ jk /2)) η max = π 2, θ jk π, (j, k) E where i {1, 2,..., G }. Using the above inequalities, the algebraic maximum values of the Euclidean norm of w is as follows: w 2 2 = Q Θ 2 2 = Θ T Q T Q Θ σ(q T Q) Θ 2 2 σ(q T Q)[η 2 max(e iθ ) II T (e iθ )] η 2 max σ(q T Q) σ(ii T ) e iθ 2 2 = η 2 max σ(q T Q) σ(ii T )N where σ( ) is a maximum singular value of the matrix and N is the number of vertices. By using the definition of I, the matrix L I = II T is a Laplacian matrix of undirected sensing topology. From the above inequality, we know that maximum value of w 2 depends on the 26

43 sensing topology. Consequently, due to the fact that w 2 = w N, a synchronized value satisfies the following inequality. w η max σ(qt Q) σ(l I ) (3.12) where η max is equivalent to π/2. There is another boundary of the norm of w which is subject to the number of the outdegree of vertices. Since the magnitude of PV( ) is bounded( PV( ) < π), each frequency w i (t) in (3.7) is bounded as follows: w i (t) = j N i θ ji (t) N i π where N i is the cardinality of the set of adjacent vertices. From the Lemma 3.2, all frequencies make a consensus as time goes to infinity. Thus, the value of consensus is bounded below by a minimum value of the boundaries for frequencies. By using the definition of the outdegree matrix D, the minimum eigenvalue is equivalent to the minimum cardinal number of the set N i. Therefore, the synchronized value w is bounded as follows: w πλ min (D) (3.13) where λ min is the minimum eigenvalue of the matrix. From the above inequalities (3.12) and (3.13), a necessary condition for the boundary of synchronized values is described as ( w min πλ min (D), π ) σ(qt Q) σ(l I ). 2 There is a sufficient condition for convergence of Θ to the equilibrium. In the connected graph G, we suppose that the root of a spanning tree is denoted by θ 1. 27

44 Corollary 3.1 For the directed graph G = (V, E), the time derivative of θ k, k V converges asymptotically to zero by assuming that θ jk is continuous with respect to the time, if the graph is connected and a convergence protocol is designed as follows: θ 1 (t) = 0 θ k (t) = j N k θ jk (t), k V \ 1 (3.14) Proof: From the equation (3.9) and (3.10), the double integral model of θ i is represented as Θ = L Θ, where L is the Laplacian matrix, and a dynamics satisfies the consensus property. Zero value of θ 1 results in that Θ goes to zero as time goes to infinity. An equilibrium set of the dynamics (3.8) is denoted by S = {θ k S 1, k V : j N i θ jk = 0}. The S is a union of the desired and undesired equilibrium set denoted by D = {θ k S 1, k V : θ jk = 0, j, kv} and U = {θ k S 1, k V : θ jk 0, j, kv} respectively. The undesired equilibrium set is determined by the graph. Under the proposed protocol (3.14), any initial points reach the equilibrium set S asymptotically since the set of discontinuities is the set of measure zero and the system is stable. From equation (3.8) and (3.10), the dynamics is represented as follows : Θ(t ) = QI T Θ(t ) = LΘ(t ) = 0 Consider the steady state solution Θ. By mapping a value of angle θ j to complex numbers(θ j e iθ j ), LΘ(t ) is represented as follows : e ilθ(t ) = 0 = e ic 28

45 where c R V is defined as c = 2π[α 1 α 2...α V ] T and α k Z could be any integer values. If Θ(t ) has unique solution which is [θ 1 θ 1...θ 1 ] T, undesired equilibrium set does not exist in the given sensing topology. Steady state solution has the following relationship: LΘ(t ) = c, (3.15) where Θ(t ) = (θ 1, θ 2..., θ N ) is in S. Under the dynamics (3.14), Laplacian matrix L in (3.15) is written as : L = l 21. L r (3.16) l n1 where L r R ( G 1) ( G 1) is a reduced matrix. Consider eigenvalues of matrix L r. The characteristic equation of L is as follows: det(l λi L ) = λ det(l r λi Lr ) = 0 where I L R G G and I Lr R ( G 1) ( G 1) are identity matrices. From the above equation, it is obviously shown that the matrix L r has the same eigenvalues as the matrix L except for a zero eigenvalue. Since the Laplacian matrix of connected graph has positive eigenvalues except for one zero eigenvalue, the reduced matrix L r is a non-singular matrix. By rearranging linear equations, (3.15) can be written in the reduce form as follows L r Θ(t ) = c (3.17) where Θ = (θ 2,..., θ N ) and c = (c 2,... c N ) R N 1. Consequently, in the assumption that p(t) has no discontinuities, Θ(t ) can be calculated if sensing topology and c are known. 29

46 3.3.2 Calculation of asymptotic pattern and simulation results Figure 3.5. An example of directed graph which is connected. In this section, we calculate the asymptotic pattern of agents on the circle under the dynamics (3.14). Numerical simulation is also achieved to verify the estimation of asymptotic pattern. Interaction topology is shown in Fig By using the incidence matrix and the initial value of each node, an equilibrium point is calculated. The incidence matrix of the graph shown in Fig.3.5 is as follows: I = The row and column indices of the incidence matrix are denoted respectively by the indices 30

47 of vertices and edges. The matrix Q derived from the protocol (3.14) is written as Q = The Laplacian matrix is calculated by using matrix multiplication of Q with E T. L = QI T = For the verification of various asymptotic patterns, two cases with respect to the initial values are considered. Case 1) Θ(t 0 ) = [ ] T Case 2) Θ(t 0 ) = [ ] T The values of Θ R V at an initial time t 0 are in [0, 2π). From equation (3.16), the reduced matrix L r is easily derived. The term c in (3.15) is calculated by using initial phase variables Θ(t 0 ). For the Case 1), entries of c are equivalent to zero. The solution for (3.17) is 31

48 Θ(t ) = [ ] T. For the Case 2), entries of c are calculated in similar way. c = [ 2π π 2π 2π ] T By rearranging linear equation (3.15), equation (3.17) is written as: θ 2 (t ) θ 3 (t ) = θ 4 (t ) θ 5 (t ) In the sequel, computational result is as follows: [ Θ(t ) = ] T We know that phase variables θ k go to undesired equilibrium set as time goes to infinity. Note that and are same phase since a value of the radian has the duplicated values in the range of 2π. Asymptotic patterns in terms of e iθ for each case are shown in Fig Simulations have been conducted to verify computational results for two cases. For the first case, a left figure in Fig.3.7 shows that θ k, k V converges to θ 1 = asymptotically. A right figure in Fig represents that Θ(t) converges to the undesired equilibrium point which is exactly same with the computational result for the second case Concluding remark In this section, the property of orientation alignment law (3.7) is studied. From the analysis of the dynamics, θ k, k V asymptotically achieves a consensus under the alignment 32

49 Figure 3.6. Asymptotic patterns on unit circle for the first case(left) and second case(right). Figure 3.7. Simulation result for the Case 1) (left) and Case 2) (right). 33

50 law (3.7). The consensus value for θ k is bounded by a value depending on the interaction graph and number of agents. If a state of root node is static, derivative of θ k converges zero and asymptotic pattern of θ k can be estimated from the interaction topology and initial position of θ k. Under the orientation alignment law (3.7) or (3.4), the asymptotic pattern of θ k is heavily dependent on the interaction graph and initial value of states. 3.4 Further analysis on the orientation alignment problem In this section, we make a further analysis on the consensus protocol (3.4) using the geodesic distance. While the consensus protocol (3.5) is a linear system under the condition of γ < π, in general, the dynamics of (3.4) is a discontinuous system since the value of θ jk is discontinuous in the field of real numbers(i.e. lim θjk π θ jk lim θjk π + θ jk). It makes an analysis of the dynamics of (3.4) complicated. Our objective is to determine the existence of multiple equilibria on the dynamics. If we consider the consensus protocol (3.4) as a gradient descent algorithm, a potential function of the form V = j N k f(θ j, θ k ) can be used, where typically f is zero when θ k = θ j and is otherwise positive. By searching multiple stationary points in V, we aim to show that anti-consensus is not a consequence of the particular algorithm but an automatic consequence of using a gradient descent law. Further, we find that the number of stationary points in V is associated with the number of spanning trees in the graph. 34

51 3.4.1 Manifold characterization It seems obvious that an orientation alignment problem of N agents should be regarded as lying in S N. While it is true, this observation is unhelpful for analyzing the number of critical points. In this N-dimensional space, no critical point of the potential function for designing a gradient descent law (3.4) is an isolated critical point, but there is a continuum of critical points. To facilitate an analysis of critical points of potential function, a dimension of the space is regarded as N 1 in the problem of N agents. We illustrate a reason for this approach. Let G(V, E) be an undirected, connected, and uniformly weighted graph. An edge denotes (j, k) E, and a potential function V : S N R is as follows: V (θ 1,..., θ N ) = 1 θjk 2 (3.18) 2 (j,k) E Let θjk be a value for the minimum value of V and C = {θ jk R, (j, k) E : θ V = 0, H(V ) > 0} be a set of geodesic distances for the local minima of V. Let θ V and H(V ) denote the gradient and the Hessian of V respectively. If there are θk and θ j such that the geodesic distance of these angles is θjk, the geodesic distance between θ k + φ and θ j + φ is also θjk. Thus, values of (θ k, θ j ) pairs for θji are the continuum of points. Suppose that a value of θ 1 is invariant(θ 1 (t) = c). Then, a potential function V : S (N 1) R with the identical graph topology is as follows: V (θ 2,..., θ n ) = 1 2 k,j 1,(k,j) E θ 2 jk + (l,1) E θ 2 l1 (3.19) If a set of geodesic distances for the local minima of V is defined as C = {θ jk R, (j, k) E : θ V = 0, H(V ) > 0}, it is clear that C = C. Since the θ 1 is a constant, the value of θ l 35

52 for θl1 is an isolated value. If the value of θ l is an isolated value, the values of neighbors of l-th node have also isolated values for C. Since the graph is connected, there is no continuum of angles for local minima of V. For this reason, searching the critical points of V is simpler than searching them of V. This approach can be generalized in the formation problem. The following result is derived from Kendall s main result quoted from [2]. Theorem 3.1 The space of the agreement problem of N agents in the unit circle can be represented as S (N 1). Proof: Regard the position of individual agent in S 1 as being given by complex numbers z k = e iθ k. Then the formation described by (z1, z 2,..., z N ) is the same as the formation with coordinates e iφ (z 1, z 2,..., z N ) by rotational invariance, where φ S 1. Therefore, identification process factors out an angle parameter. In the following subsection, we analyze a potential function of the form (3.19) for finding the critical points An example of three agents We consider a case of three agents for the alignment problem. The sensing topology of three agents is categorized by a tree and a cyclic structure as shown in Fig With slight abuse of notation, let θ j θ i denote the geodesic distance θ ji. Suppose that θ 1 = 0, and the potential function V c : S 2 R for the cyclic sensing topology is represented as V c (θ 2, θ 3 ) = 1 2 ( θ θ 2 θ θ 3 2 ) (3.20) 36

53 Figure 3.8. Undirected graph of a cyclic structure (left) and a tree structure (right) in the case of three agents. While the term θ i θ j 2 is non-smooth when θ j θ i = ±π, we can imagine that an infinitesimal perturbation of V c at critical points cannot change the eigenvalue of the Hessian of V c. Thus, we can search for the existence of critical points by using the gradient and the Hessian matrix. The gradient and the Hessian matrix of V c in the smooth manifold are as follows: θ V c = 2θ 2 θ 3 θ 2 + 2θ 3, H(V c) = where H(V c ) is positive definite. Thus, what we need to do is to find all points such that θ V c = 0. The gradient of V c can be written in terms of the Hessian of V c as follows: θ V c = H(V c ) θ 2 θ 3 = θ 2 θ 3, = 0 In the field of real numbers, there is a unique solution for this equation. However, the angle values in real numbers have different values for same angles such as θ = θ + 2π. For this property, we represent the angle value as a multi-valued function : θ k (w k ) := w k + 2n k π, where w k ( π, π) is a principle value and n k Z could be any integer value. By substituting this multi-valued function for the angle values, the equation is rewritten as 37

54 follows: where α k 2 1 w 2 + 2πn w 3 + 2πn w 2 w 3 = 0 = 2π α 2 α 3 Z is obtained by the linear combination of n k, k = {2, 3}. Possible solutions can be determined by the values of α k. Under the constraint w k ( π, π), possible combinations of (α 2, α 3 ) are as follows: (0, 0), ( 1, 1) and (1, 1). For each combination, solutions of (w 2, w 3 ) for θ V c = 0 are as follows : 0, 0 2π 3 2π 3, 2π 3 2π 3 Thus, we notice that the gradient descent law of the function V c cannot achieve a desired minimum in the global sense due to the local minima. We now consider the tree structure of three-agent case which is shown in Fig A potential function V t : S 2 R for the tree structure with θ 1 = 0 is as follows: V t (θ 2, θ 3 ) = 1 2 ( θ θ 3 2 ) In a similar manner, the existence of critical points of V t is determined by using the gradient and the Hessian matrix. θ V t = θ 2 θ 3, H(V t) = where H(V t ) is positive definite in the smooth manifold. By substituting angles with multivalued functions defined as θ k (w k ) := w k + 2πn k, the equation θ V t = 0 is represented as

55 follows : w 2 w 3 = 2π 1 0 w 2 + 2πn w 3 + 2πn 3 = 0 α 2 α 3 where α k Z is obtained by the linear combinations of n k, k = {2, 3}. Under the constraints of w k ( π, π), solution of (w 2, w 3 ) is (0, 0). It indicates that the origin of domain is a unique minimum of potential function V t Analysis on potential function While the potential function V : S (N 1) R is not smooth in all domains, our objective is to search for critical points in the smooth domain. From the definition of V in (3.19), the gradient of V can be represented as a linear map of angle variables: θ V = H(V )Θ, (3.21) where Θ = [θ 2,..., θ N ] T. Since θ k can be considered as a multi-valued function, we substitute θ k with w k + 2πn k, n k Z, k {2,..., N} where w k ( π, π). The equation (3.21) is rewritten as follows: θ V = H(V )(w + 2πn) (3.22) where w = [w 2 w N ] T and n = [k 2 k N ] T. Since the value of θ V is defined on S (n 1), each entry of θ V should be represented as the multi-valued function in the field of real numbers : [ θ V ] k = φ k + 2π n k, n k Z, i {2,... n}, where φ k ( π, π). 39

56 Therefore, the equation θ V = 0 is rewritten as θ V = 2π k, where n = [ n k n N ] T. By using this term and (3.22), the gradient of V is represented as θ V = 0 = 2π n = H(V )(w + 2πn) Let I and Ī denote the incidence matrix of the graph G(V, E) and its reduced incidence matrix which is obtained by deleting the 1-th row, respectively. Then, H(V ) is equivalent to Ī ĪT. Since the rank of Ī is N 1, H(V ) Z (N 1) (N 1) is a nonsingular matrix and positive definite. Possible solutions for w are calculated as w = 2πH(V ) 1 ( n H(V )n) = 2πH(V ) 1 ñ. (3.23) ( π < w i < π, i = {2,..., N}) Since all entries of H(V ) are integer values, all entries of H(V )n are also integer values. Thus, ñ Z (N 1) could be any integer values. Possible solutions are determined from possible combinations of ñ such that solutions are in a set ( π, π). We note that H(V ) is an invariant and positive definite matrix. Thus, the existence of solution w is enough to give a sufficient condition for the local minimum of V. We consider the number of critical points in the perspective of the graph G(V, E) from the following result. Theorem 3.2 Suppose that the graph G(V, E) is connected and a corresponding potential function V : S n R is defined as (3.18). The number of continuums of minima in V is one if the graph G(V, E) is a tree. Proof: We consider the potential function V : S (N 1) R of the form (3.19) for the problem of N-agent by using Theorem We aim to find the number of solutions of 40

57 (3.23). Since ñ Z (N 1) is an integer vector, a sufficient condition for a unique solution of w is that all entries of H(V ) 1 in (3.23) are integer values. The Hessian matrix H(V ) is equivalent to the multiplication of the reduced incidence matrix: H(V ) = Ī ĪT. From the Theorem.2.1, the graph is a tree structure if and only if the determinant of Ī ĪT is 1. This implies all entries of an inverse matrix of H(V ) are integer values. Therefore, the number of possible solutions w for (3.23) is one (i.e. w = [0 0] T S (n 1) ). It completes the proof. If the graph is a tree, there is an open subset of R (N 1) which has a bijective relationship with the smooth V. Let V x : R (N 1) R denote a function with the quadratic form. Suppose that V x is designed as the form of (3.19) with x 1 = 0. V x(x 2,..., x N ) ( i,j 1,(i,j) E (x i x j ) 2 + ) (j,1) E (x j x 1 ) 2 = 1 2 Let L denote the open subset of R (N 1), and L is defined as L = {x k, x j R, (k, j) E, k, j 1 : x k x j < π}. If we consider the domain of V x as L, there is a bijective function between L and S (N 1). For example, the bijective mapping between R 2 and S 2 is shown in Fig The Morse theory is a tool for examining multiple stationary points on any smooth manifold. The key result of the Morse theory is a collection of equality/inequality relations between the number of minima, saddle points and maxima of a smooth function f. We aim to verify the critical points of the smooth V by using the key result of Morse theory. However, there are nonsmooth points θ k θ j = ±π in V. Since the nonsmoothness rules out the use of Morse theory, we consider V x defined on R (N 1) instead of V. Suppose that the manifold 41

58 Figure 3.9. An example of a bijective mapping between an open subset of R 2 and S 2. is R N, the collection of relations between the number of critical points is as follows [1] : m 0 1 m 1 m 0 1 m 2 m 1 + m 0 1. N k=0 ( 1)k m k = 1 where m k is the number of critical points and the index k indicates the number of negative eigenvalues of the Hessian. m 0, m N and m k, k {1,... N 1} are the number of minima, maxima and saddle points respectively. Since the Hessian of V x is a positive definite and the gradient is zero at the origin, the number of critical points satisfies m 0 = 1, m j = 0, j {1,..., N}. We now consider a cyclic structure. Theorem 3.3 Suppose that the graph G(V, E) is connected and a corresponding potential function V : S N R is defined as (3.18). There are multiple continuums of minima, if the graph G(V, E) includes a cyclic structure. Proof: Let Ī denote the reduced incidence matrix of the graph G(V, E). By Theorem. 2.1, the number of spanning trees is equivalent to the determinant of Ī ĪT. Thus, if the 42

59 graph G(V, E) includes a cyclic structure, det ( Ī ) ĪT > 1. Due to H(V ) = Ī ĪT, there exists at least one entry of H(V ) 1 which is not an integer 1. This implies that there exist possible combinations of ñ for (3.23) except for ñ = 0. It completes the proof. From the result, we conclude that the existence of multiple equilibria is the intrinsic consequence of the sensing topology Concluding remark In this section, the orientation alignment law (3.4) is analyzed in the perspective of gradient decent algorithm. By searching multiple stationary points of the potential function, we show that the anti-consensus is not a consequence of the particular algorithm but automatic consequence of using a gradient descent law. The number of stationary points is dependent on the interaction graph. For the undirected graph with a unique spanning tree, the stationary point corresponding to the local minimum of potential function is unique. It indicates that the orientation alignment law (3.4) achieves a global synchronization if the graph has a spanning tree. When the interaction graph has cyclic structure, there are multiple local minima of a potential function. The number of critical points always satisfies the equality/inequality relations which are originated from the Morse theory. 3.5 Modified coupling function for global synchronization Due to the non-convexity of configuration space, the synchronization property of the orientation alignment law (3.4) is dependent on the graph or initial values of states. In a control 1 Suppose that all entries in H(V ) 1 are integer values. Then, the determinant of H(V ) 1 must be an integer value. It contracts to the fact that det ( H(V ) 1) 1 = det(h(v )) is not an integer. 43

60 framework, a natural question is then whether the consensus protocol can be modified to enforce better synchronization properties. Several design methods for global phase synchronization are described in [49]. First method, proposed in [46], is to add randomness in the link selection of the consensus algorithm. The idea follows Gossip algorithm. The method ensure asymptotic convergence while the convergence rate may be slow. Second method is to increase the amount of exchanged information among agents by using auxiliary variable [47]. This work assumes that the orientation of local frame is known for each agent. For third method, Sarlette [44] proposed a shaped potential such that its gradient system achieves the phase synchronization in almost global region. In this section, we studied the potential shaping which removes local minimum on the manifold. While a Hessian of the shaped potential, proposed in [44], is not defined on some points, we propose a well-defined Hessian so that we can analyze the characteristic of every critical points. By using the Hessian, the modified potential function is constructed on the smooth manifold. We prove that the proposed potential function has a unique minimum value which corresponds to the phase synchronization. While there exist many critical points of the proposed potential function, its gradient system converges to a synchronized point in almost entire region Problem Statement The interaction among N agents is modeled by the undirected graph G(V, E) with vertices V = {1,..., N} and undirected edges (k, j) E. Let a phase of each agent be denoted by θ k. The phase of each agent is controlled to achieve the synchronization by minimizing 44

61 an potential function(e : T N R) : E(Θ) = 1 2 n (1 cos(θ k θ j )), (3.24) k=1 (k,j) E where E(Θ) is a nonnegative value and smooth. Since a minimum value of the potential function is zero with synchronization θ 1 = = θ N, it is natural that a gradient descent law is used to achieve the synchronization of θ k as follows: Θ = grade. The dynamics of each agent is as follows : θ k = sin(θ j θ k ). (3.25) j N k This nonlinear consensus system is identical to the classic Kuramoto oscillator in the undirected graph with uniform weights. If the length of the shortest arc is considered as a Lyapunov function, an analysis of the dynamics relies upon the contraction property. Following result states that the convex hull of all states is decreasing [7, 25]. Theorem 3.4 [7](Contraction in Open Semicircle) Consider the nonlinear consensus system (3.25) with a connected graph G(V, E). Each trajectory originated in Arc N (γ) for γ [0, π[ achieves exponential phase synchronization, that is, Θ(t) θ avg 1 N 2 Θ(0) θ avg 1 N 2 e λpst where λ ps = λ 2 (L)sinc(γ) and θ avg = N i=1 θ k(0)/n. In the case of that a minimum length of the arc containing all agent is larger than π, the analysis of trajectories for the system (3.25) is complicated due to the non-convexity property. In 45

62 Figure An example of the splay state in the case of 6 agents. addition, there are many undesired equilibriums. Assume that a pair of agents has a length of π while others have length of zero. Such phases are undesired equilibrium points. Other sets of undesired equilibrium points include a splay state: all agents are evenly distributed on the unit circle(see Fig. 3.10). We consider trajectories around equilibrium points corresponding to the splay state. A Jacobian matrix of the dynamics (3.25) has following entries : m N k cos(θ m θ k ), k = j [J(Θ)] kj = cos(θ j θ k ), (k, j) E, (3.26) 0, (k, j) / E where J(Θ) is a symmetric matrix and its row sum is zero. Let equilibrium points at splay state be denoted by Θ b. In the case of N 4, J(Θ b ) becomes a nonnegative matrix due to the positiveness of diagonal entries ( cos(β) < 0, β ( π, π]). Therefore, trajectories 2 in small neighborhood of Θ b are unstable when N 4. Let us consider a case of N > 4. Suppose that the geodesic distance between k and j for any (k, j) E is less than π. Then, 2 off-diagonal elements are positive values except for zero. This implies the Jacobian matrix is negative semidefinite. From Theorem 2.2, we have a following result: Lemma 3.1 Suppose that a splay state corresponds to equilibriums (Θ b ) on the dynamics 46

63 (3.25) and the geodesic distance between k and j for any (k, j) E is less than π 2. Then, there exist a small neighborhood of Θ b such that Θ(t) exponentially converges to Θ b + Θ. Proof: Let Θ := Θ Θ b. Consider a Jacobian matrix of nonlinear system for Θ. A linearized system at the origin of Θ is written as : Θ = J(Θ b ) Θ According to the definition of Jacobian matrix (3.26), off-diagonal entries have nonnegative values because the geodesic distance between k and j for any (k, j) E is less than π. It 2 implies that J(Θ b ) can be considered as the Laplacian matrix. According to the Theorem 2.2, trajectories around the origin of Θ are exponentially stable. In the following section, we modify the control law to transform the undesired stable equilibrium point of a splay state into the unstable equilibrium. Then, every trajectory achieves synchronized state except for the trajectories originated from stable manifold of the undesired equilibrium points Characteristic of the critical point Consider critical points of smooth function (3.24) defined on a n-dimensional space. Critical points of smooth function (3.24) are identical to the equilibrium points of dynamics (3.25). To identify the characteristic of critical point, we can consider a Hessian matrix of (3.24) at the critical points. The entries of Hessian matrix for smooth function (3.24) are 47

64 defined as follows: [H] kj = 2 E(Θ) θ i θ j = j N k cos(θ m θ k ) cos(θ j θ k ), k = j, (k, j) E 0, (k, j) / E (3.27) We can see that the Hessian matrix has always zero eigenvalue since its row sum is zero. When the Hessian is a singular matrix at a critical point, the critical point is not isolated but there is a continuum of critical points. We call it a degenerate critical point. To facilitate an analysis for the critical point, we need to factor out the degeneracy of critical points, and find out the characteristic of critical points on n 1 dimensional space. The approach is derived from the following result. Lemma 3.2 The space of the synchronization for n agents on the unit circle can be represented as T (N 1). Proof: Regard the positions of individual agent in S 1 as being given by complex numbers z k = e iθ k. Then the formation described by (z1, z 2,..., z N ) is the same as the formation with coordinates e iφ (z 1, z 2,..., z N ) by rotational invariance, where φ S 1. Therefore, identification process factors out an angle parameter. Now, we can consider a smooth function Ē : TN 1 R to facilitate an analysis for the critical point of the smooth function (3.24). A Hessian matrix of Ē is a (N 1) (N 1) submatrix of the Hessian matrix of E. Let a value of θ 1 be fixed at zero. Then, the Hessian of Ē is simply calculated by reducing both first row and first column vectors of H. By using the Hessian of Ē, we can identify the characteristic of critical points corresponding to the splay states. From Theorem 2.2, we can derive the following result. 48

65 Theorem 3.5 Suppose that there are critical points of Ē such that positions of agents on the unit circle are corresponding to the splay state and a geodesic distance of any pairs (i, j) E is less than π. Then, such critical points are corresponding to local minimum 2 values of Ē. Proof: A sufficient condition for minimum values of any smooth function is the positiveness of Hessian matrix. To identify the spectrum of Hessian matrix, we consider the result of Theorem We know that Hessian matrix H of E is a positive semidefinite. A zero eigenvalue of H is the boundary point of Ger sgorin region of H. Then, every Ger sgorin cirlce of H passes through zero(theorem. 2.2). Consider any (N 1) (N 1) submatrix of H. There is at least one Ger sgorin cirlce of the submatrix which doesn t pass through zero point. This implies the submatrix is nonsingular. This completes the proof. The existence of a local minimum value on the smooth manifold implies that a gradient descent method as control law does not achieve a synchronization globally. We desire to remove local minimum values of Ē by modifying Hessian matrix. According to the definition of Hessian matrix of Ē, Hessian matrix is a negative definite when off-diagonal entries at critical points have only positive values. Variance of an off-diagonal entry with respect to the geodesic of (i,j) is represented in Fig Let θ (see Fig. 3.11) be the longest length among geodesic distances of pairs in the splay state. Since cos(θ ) is a negative value, all off-diagonal entries have negative values. Now, let us consider f( ) (see Fig.3.11) which is a modified sinusoidal function with f(±η) = 0. Suppose that the term of cos( ) in (3.27) is replaced with f( ). If the value of η is set as smaller than the shortest length of two agents in the splay state, all off-diagonal entries have positive values. This implies critical points 49

66 corresponding to the splay state are maximum values of smooth function defined on the n 1 dimensional space. In the following subsection, a modified potential function for the synchronization is con cos(θ j θ i ) f(θ j θ i ) η η θ* (θ j θ i ) Figure Illustration of an off-diagonal entry of the Hessian matrix(solid line) and a modified Hessian matrix (dashed line) respectively.. structed so that none of critical points is corresponding to the local minimum of the smooth function except for the synchronization Potential shaping We propose a modified potential function such that an off-diagonal entry of Hessian matrix satisfies the characteristic of f( ) illustrated in Fig The modified potential function defined on T n R is as follows: E M = 1 2 N z(θ jk, η) (3.28) k=1 (k,j) E 50

67 z(θ jk, η) = 2(π η) π ( ) π [1 cos (θ 2(π η) jk + (π 2η) ( ) π (θ 2(π η) jk (π 2η) ( ) 2(π η) 2η cos π θ π π 2η jk 2(π η) [1 cos π, θ jk (η, π], θ jk [ π, η), θ jk [ η, η] (3.29) where θ jk θ j θ k, and η is a graph-dependent parameter constrained by (0, π ]. To remain N smoothness and positiveness of E M, the term z(θ jk, η) is designed as (3.29). Now, let us consider the characteristic of critical points of the smooth function (3.28). It is obvious that a synchronized point(i.e. θ 1 = θ 2 = = θ N ) is a minimum value of E M We show that there is none of minimum for E M except for the synchronized point according to the following theorem. Theorem 3.6 Consider a smooth function E M : T N R R + defined in (3.28). Suppose that η (0, π ]. Then, there is a unique minimum value of smooth function (3.28). Moreover, N the minimum corresponds to the synchronized point as θ 1 = = θ N. Proof: Let us consider the critical point on the open semicircle. From the derivation (3.32), the gradient of E M has non-zero value on the open semicircle except for the synchronized point. Now, we have to find the characteristic of critical points which are not on the semicircle. If agents are not on the semicircle, there are always m-agents which are far more than η from N m agents in the sense of geodesic distance. Hessian matrix of E M can be written by using a proper permutation of row and column as follows: H = H a H T c H c H b 51

68 where H a R m m, H b R (n m) (n m) and H c R m (n m). H c is defined as follows : [H c ] kj = ( ) π 2(π η) cos π 2(π η) θ jk + (π 2η) ( ) π 2(π η) cos π 2(π η) θ jk (π 2η), θ jk (η, π], θ jk [ π, η) 0, (k, j) / E According to the definition of H c, each element of H c is 0 or a positive value. Due to the connectivity of graph G, there is at least one positive entry in H c. Now, let us consider eigenvalues of H. Except for zero eigenvalue, other eigenvalues can be denoted by λ 2 λ 3 λ N. A corresponding eigenvector for zero eigenvalue is 1 N which denote a vector with ones as its element. The eigenvalue λ 2 can be calculated by Rayleigh quotient as follows : (Theorem in [12]) Let v R N be [ 1 m 1 m 1 (N m) x T Hx min x 1 N x T x = λ 2 1 (N m) ]T which is perpendicular to 1 N. Rayleigh quotient using v is written as v T Hv v T v = 1 v 2 vt H a H T c H c H b v = ( 1 1 v 2 m 2 1T mh a 1 m + 1 (N m) 2 1 (N m)h b 1 (N m) ) 2 m(n m) 1T mh c 1 (N m). (3.30) Since H has zero row sum, a value of 1 T mh a 1 m and 1 (N m) H b 1 (N m) is equivalent to the value of 1 T mh c 1 (N m). Let 1 T mh c 1 (N m) be denoted by α. A value of α is always positive value according to the definition of H c. Then, (3.30) is rewritten as v T Hv v T v = 1 ( ) 1 v 2 m (N m) + 2 α < 0 2 m(n m) 52

69 θ k z(θ jk, η) = ( ) π sin (θ 2(π η) jk + (π 2η)) ( ) π sin (θ 2(π η) jk (π 2η)) ( ) sin π 2η θ jk, θ jk (η, π], θ jk [ π, η), θ jk [ η, η] (3.32) This implies λ 2 is a negative value. This completes the proof. Now, a gradient descent law is proposed to minimize a value of smooth function (3.28) with η (0, π ]. Dynamics for each agent is designed as follows: N θ k = grade M = j N k θ k z(θ jk, η) (3.31) θ i z(θ ji, η) is defined in (3.32). Following lemma states that the gradient descent law achieves the convergence to critical points of smooth function (3.28). Lemma 3.3 Let a real function be V (x) defined on N-dimensional smooth manifold. A gradient descent law achieves asymptotically stability for the critical points of V (x). Proof: Time derivatives of V (x) is as follows: d dx V (x) = dt dt V (x) = ( V (x))2 0 A value of V (x) decreases as time goes to infinity until reaching equilibriums. It completes the proof. According to the Theorem. 3.6, we show that there is a unique stable equilibrium point on the dynamics (3.31). It does not imply that the gradient descent law achieves the global stability since there are still many unstable equilibrium points. In other word, if Jacobian 53

70 matrices of unstable equilibrium points have at least a negative eigenvalue, there exist trajectories in some subspaces of T N such that trajectories converge to the undesired equilibriums. However, from Lemma 3.3, we may infer that every trajectory outside the stable manifold of undesired equilibriums converges to a synchronization asymptotically since every undesired equilibrium point is unstable(see Theorem 3.6) Numerical Results Figure A topology for interactions of 12 agents We compare the performance of a classical coupling function (3.25) and the modified one (3.31). Let us consider 12 agents which are interconnected by the undirected and connected graph G as shown in Fig Initial states of Θ are randomly assigned. Simulation results are shown in Fig Fig represents a radian value of each agent with respect to the time. Trajectories of the classic coupling model converge to the undesired equilibrium while the modified coupling model achieves the phase synchronization. We compared the convergence rate for the modified function according to the different values of γ. It is illustrated in Fig A dashed line in Fig indicates the convergence rate of classical system. A value of λ ps in the figure is depending on the arc length containing 54

71 θ(t) [rad] θ(t) [rad] t(tec) t(tec) (a) Classic coupling function (b) Modified coupling function with η = π 12 Figure Comparison of the dynamics in the case of that initial states are arbitrary determined in the neighborhood of splay state. all agents. In this simulation, λ ps is due to λ 2 (L) = and γ = As shown in Fig. 3.14, the convergence speed is dependent on the value of η. As η goes to zero, a rapid change of control signal occurs around synchronized points This rapid change of control signal gives almost finite time convergence around synchronized point(see the case of η = in Fig. 3.14) Concluding remark We study undesired stable equilibriums in the network of coupled agents on the unit circle. The equilibriums on the dynamics using gradient descent method correspond to the critical points of smooth function. The Hessian matrix of smooth function represents characteristic of critical points. We propose a modified smooth function by modifying the Hessian matrix of original smooth function; the modified function has a unique minimum value corre- 55

72 3.5 3 Θ θ avg 1 n Θ(0) θ avg 1 n 2 e λ ps t η=2.5 η=1.5 η=1 η= t(sec) Figure Convergence rates for the modified function with various values of η are illustrated. A dashed line indicates the convergence rate of classical system. sponding to the synchronization. Because there still exist many critical points in the modified smooth function, the gradient descent law for the phase synchronization does not guarantee global convergence. However, if initial states are not on a stable manifold of the undesired equilibrium point, the dynamics achieves synchronization asymptotically. We also numerically verified the convergence of the proposed method and compared it with the classic system. 56

73 Chapter 4 Global Synchronization on the Circle 4.1 Introduction In this chapter, a novel consensus protocol on the circle is proposed. While there are some solutions to recover almost global convergence on the unit circle, the existence of generalized dynamics is sill questioned in the case of that the information via local coupling is only available as the measurements. This chapter may answer the question. The position of agent on the unit circle can be considered as the position vector in the vector space. We consider the dynamics of the position of agents in vector space instead of the unit circle. Under the assumption of that the information can be exchanged by agents, we add the auxiliary variables defined in the vector space. The objective of the auxiliary variables is to design a feedback controller from the estimated value. From the convexity of vector space, global convergence property of the auxiliary variable can be analyzed under the proposed consensus dynamics. The use of the auxiliary variable is initially motivated by the result of [47]. The assumption of the global information as the measurement in [47] is relaxed to an assumption of local information obtained from the interaction with neighbors. We show that auxiliary variables estimate the position of agents on the circle under the proposed estimation law. Then, we desire to design the consensus protocol based on the estimated value. By embedding the proposed consensus protocol to the unit circle, we project the position of agents in vector 57

74 Figure 4.1. Block diagram for the synchronization method based on the local information θ jk. The position of kth agent and the estimated position on the circle are denoted by θ k and ˆθ k respectively. space onto the circle. It implies that the dynamics of the agents on the circle can be obtained by designing the dynamics of auxiliary variables Consequently, we do not have to analyze the dynamics of the agent directly on the circle. Instead, the global convergence property of the agents is analyzed in the vector space. The proposed control strategy is illustrated by the block diagram shown in Fig Consensus in S 1 The consensus protocol (2.2) is interpreted as the distributed method based on the relative quantities of states i.e(x j x k ), j N k. Suppose that there are N agents on the unit circle whose interaction graph is given by G = (V, E, A). The state of each agent is denoted by θ k S 1. Let us denote the geodesic distance 1 between θ j and θ i as θ ji which is defined by 1 Geodesic distance refers to the shortest path of two points in curved space. While the straight line between two points is unique and identical to the shortest path in vector space, the straight line between two points in 58

75 θ ji := PV(θ j θ i ), where PV(θ j θ i ) = [(θ j θ i +π)mod 2π] π. Note that θ jk θ j θ k in general. The consensus protocol based on the relative quantities in S 1 is as follows [28, 34]: θ k = a kj θ jk, i V (4.1) j N k In Euclidean space, the convex hull of a set of points in n-dimensions is invariant under the consensus algorithm of (2.2). The permanent contraction of this convex hull allows to conclude that the agents end up at a consensus value [14,49]. In S 1, convergence property of the consensus algorithm based on the relative quantities of states is analyzed in similar way, while the convergence of states to the consensus value is not possible for all initial values of states. In other words, there is a subset such that convex hull of the set of points is invariant. Then, a sufficient condition for the permanent contraction of a convex hull is stated in the following theorem. Theorem 4.1 ( [7]) Consider the coupled oscillator model (4.1) with a connected graph G(V, E, A). Suppose that a convex hull of all initial values is within an open semi circle. Then, this convex hull is positively invariant, and each trajectory originating in the convex hull achieves an exponential synchronization. The fundamental difference between a consensus protocol (2.2) in vector space and another protocol (4.1) in S 1 is the non-convex 2 nature of configuration spaces like the circle S 1 is not unique. In other words, there are always two paths from one point to another point on the unit circle. Therefore, the difference of two points does not guarantee the shortest path among them. For instance, suppose that θ j = 2 3 π and θ k = 2 3 π. Then, the geodesic distance of two points is θ jk = 2 3π, although the difference of two points is equivalent to θ j θ k = 4 3 π. 2 Non-convexity is the opposite of convexity. Convexity property implies that the future, updated value of any agent in the network is a convex combination of its current value as well as the current values of its neighbors. 59

76 or sphere. For this reason, a global convergence analysis of the consensus algorithm in the space is quite intractable and at least very dependent on the communication graph [49]. In this chapter, we embed θ k S 1 in vector space. Then, we estimate the embedding of θ k by using auxiliary variables defined on the Euclidean space. The estimated value is used to design a feedback controller for θ k. Although consensus algorithm plays still important role in the proposed method, unlike the consensus protocol (4.1), the proposed method achieves global convergence of the estimated variables. This is described in the next section Problem Statement Consider N agents on the unit circle which are modeled by θ k = ω k, k V. (4.2) We assume that geodesic distance of neighbors denoted by θ jk, j N k is measured by k. The objective of the control is then stated as follows: Problem 4.1 Suppose that N agents whose interaction graph G = (V, E, A) has a rootedout branch are modeled by (4.2). Design a control law ω k such that θ k θ as t for θ S 1 and k V based on the measurements θ jk, j N k. A SO(2) is diffeomorphic with the circle group. Using the rotation angle θ as a parameter, rotation matrix denoted by R can be parametrized as follows: R = cos(θ) sin(θ) sin(θ) cos(θ) It is thus clear that max{x 1, x 2,..., x N } is a non-increasing function of time( [31]). (4.3) 60

77 Figure 4.2. Expression of θ k in complex plane. Note that R k SO(2) corresponds to θ k S 1. Then, the geodesic distance θ jk S 1 corresponds to R jk = R j R T k. 4.2 Estimation of positions on the circle We consider the estimation method for θ k in this section. The orientation is assumed to be static(i.e. θk = 0). Let me consider θ k as an angle of complex number which can be expressed in the polar coordinate. The complex number with the unit radius is represented as z k := e iθ k for all k V which is illustrated in Fig Let z be a stacked column vector defined as z = (z 1, z 2,..., z N ). If we can estimate z k instead of θ k by using local interaction based on the measurement θ jk, θ k is naturally determined by z k = θ k. Let ẑ k C, k V be an auxiliary variable defined on complex plane. Estimation problem can be stated as follows: Problem 4.2 Suppose G = (V, E, A) is the interaction graph of the agents. For a common 61

78 complex value α C, design an estimation law such that ẑ k z k + α as t for all k V based on the measurements θ jk, j N k. Under the assumptions of Problem 4.2, we propose the following estimation law: ẑ k (t) = (e iθjkẑ j (t) ẑ k (t)), k V. (4.4) j N k Here, (4.4) can be written in the vector form as follows: ẑ(t) = Hẑ(t) (4.5) where ẑ = (ẑ 1,..., ẑ N ) and H is defined as N k, [H] kj = e iθ jk, k = j j Nk (4.6) 0, j / N k where N k denotes a cardinality of the set Stability analysis The property of eigenvalues of H can now be analyzed in the following result. Proposition 4.1 Zero is a simple eigenvalue of H with a corresponding eigenvector z if and only if the associated digraph has a rooted-out branch. Moreover, every eigenvalue except for zero eigenvalue has strictly negative real part. Proof: Let us define a N N diagonal matrix as D z = diag(z 1, z 2... z N ). Since D z is a nonsingular matrix, the similarity transformation is achieved as follows : H := D 1 z HD z. 62

79 Each entry of H is represented such as: [ H] kj = [D 1 z HD z ] kj = [Dz 1 ] k [HD z ] j = [Dz 1 ] kk [H] k [D z ] j = [Dz 1 ] kk [H] kj [D z ] jj (4.7) where [H] k and [H] k are k-th row vector and k-th column vector of H respectively. Then, diagonal entries of H are invariant with respect to the one of H while nonzero off-diagonal entry of H is written as [ H] kj = e iθ k e iθ jke iθ j = 1. This implies H can be considered as Laplacian matrix with an uniformly weighted graph. According to the result from Theorem 2.2, all eigenvalues of H have strictly negative real part except for a simple zero eigenvalue with the corresponding eigenvector ξ = [1, 1,..., 1] T. The linear transformation of ξ shows that z is an eigenvector with respect to the zero eigenvalue of H as follows: Hξ = D 1 z HD z ξ = D 1 Hz = 0. This completes the proof. z The result implies that equilibrium set of the dynamic (4.5) is written as E := {ˆq C N : ˆq = span{z}}. Since the origin is not represented by the angle in polar coordinate, the convergence of z to the origin is not desirable. Therefore, the desired equilibrium set is defined as S := E \ {0}. For a square complex matrix A, define the column space of matrix A as C(A). Then, we have the relationship C(A) = null(a ), where null(a ) denotes the orthogonal space of null space of A. Now, we consider the convergence of the dynamics (4.5). Based on the consensus property mentioned in the above, the following theorem provides conditions for the convergence of ẑ(t). Theorem 4.2 For the dynamics (4.5), there exists a finite point z E such that ẑ(t) globally exponentially converges if and only if G has a rooted-out branch. Further, z is in S if 63

80 and only if an initial value ẑ(t 0 ) is not in C(H). Proof: Let us consider a coordinate transformation as ẑ(t) = D z v(t), where D z = diag(z 1, z 2..., z N ). Then, (4.5) is written as v(t) = D 1 z HD zˆq(t) := H ˆq(t). As mentioned in the Proposition 4.1, H has zero row sum with zero eigenvalue corresponding to the right eigenvector ξ = [1,, 1] T. From the Theorem 2.2, there exists an equilibrium set E q := {ξ k C, k V : ξ 1 = = ξ N } such that ˆq(t) globally exponentially converges to E q. Then, there exist a finite point ˆq E q and constant k q, λ q > 0 such that ˆq(t) ˆq k q e λq(t t 0) ˆq(t 0 ) ˆq. (4.8) By using the coordinate transformation, (4.8) is written as D 1 z (ẑ(t) z ) k q e λq(t t0) D 1 (ẑ(t 0 ) z ), z where z := D zˆq. Based on the definition of D z, D z = Dz 1. Then, we have D 1 z (ẑ(t) z ) 2 = ( Dz 1 (ẑ(t) z ) ) ( D 1 z (ẑ(t) z ) ) = (ẑ(t) z ) (ẑ(t) z ) = ẑ(t) z 2 This implies that Dz 1 (ẑ(t) z ) = (ẑ(t) z ). Consequently, (4.8) is represented as follows: ẑ(t) z k v e λv(t t 0) ẑ(t 0 ) z. (4.9) Since D z is a linear map between E and E v, z is in E. Now, we consider a solution of ẑ(t). The solution of ẑ(t) is as follows : ẑ(t) = e H(t t 0)ẑ(t 0 ). The Jordan form is obtained 64

81 by the similarity transformation as follows: P 1 HP = J. Let P and P 1 be represented as P = [p 1 p N ] and P 1 = [w 1 w N ] respectively. We assume that p N is the right eigenvector of the zero eigenvalue. Then, w N is the left eigenvector of the zero eigenvalue. Since every nonzero eigenvalue of H has negative real part, the state transition matrix e J(t t 0) has the following form as t. e J(t t0) (4.10) From (4.10), a steady state solution of ẑ(t) is as follows: lim ẑ(t) = lim P t t ej(t t 0) P 1 ẑ(t 0 ) = p N w N ẑ(t 0 ) This implies that ẑ(t) converges to the origin if and only if ẑ(t 0 ) is perpendicular to the w N which is the left eigenvector of the zero eigenvalue. Then, it follows that w N ẑ(t 0 ) = 0, if and only if ẑ(t 0 ) is in null(h ). Since null(h ) = C(H), it completes the proof. Remark 4.1 Let us consider the probability that the initial value of auxiliary variable ẑ(t 0 ) is in null(h ). We assume that initial value ẑ(t 0 ) is the random variable which has probability density function(pdf) of uniform distribution. Therefore, the volume of target space with respect to the total space is associated with the probability. The target space is null(h ). The basis of arbitrary coordinate system in total space C N is denoted by (b 1, b 2, b N ) as illustrated in Fig Let b N be a basis for null(h ) and other basis be orthogonal to b N. The volume of the target space is calculated as follows : V H = 1db 1 1db 2 1db N, T 1 T 2 T N 65

82 Figure 4.3. Undesired equilibrium set(:= null(h ) ) of (4.5) and its orthogonal space in C N. where T i is the interval. Since b N is orthogonal to null(h ), T N approaches to zero. It implies that V H approaches to zero regardless of the value of other terms. Consequently, the probability of ẑ(t 0 ) = c null(h ) is equivalent to the zero. From this algebraic analysis, it may be said that the proposed approach guarantees almost global convergence Concluding remark In this section, a novel estimation method for unknown orientation angles based on the only relative angle measurements is proposed. The proposed method ensures the global convergence, when the interaction graph has a rooted-out branch. We add the auxiliary variables to estimate the orientation in vector space instead of the circle. Then, the use of auxiliary variables allows to estimate the positions on the circle globally. In Chapter 5, We 66