Decentralized Control of Multi-agent Systems: Theory and Applications

Transcription

1 Dissertation for Doctor of Philosophy Decentralized Control of Multi-agent Systems: Theory and Applications Kwang-Kyo Oh School of Information and Mechatronics Gwangju Institute of Science and Technology 2013

2 박사학위논문 다중에이전트시스템의분산제어 : 이론및응용 오광교 정보기전공학부 광주과학기술원 2013

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5 Dedicated to my family.

6 Ph.D/SM Kwang-Kyo Oh. Decentralized Control of Multi-agent Systems: Theory and Applications. School of Information and Mechatronics p. Advisor: Prof. Hyo-Sung Ahn. Abstract Multi-agent systems have recently attracted a significant amount of research interest due to their advantages such as flexibility, scalability, robustness, and cost-effectiveness. Advances in control, computation and communication provide the enabling technology for the construction of such systems. In this dissertation, we focus on formation control and consensus in multi-agent systems. We study various formation control problems based on local and partial measurements. First, we propose a decentralized formation control law based on inter-agent distances for singleintegrator modeled agents in the plane and show a rigid formation is locally asymptotically stable under the proposed law. We also show that a general n-dimensional rigid formation of double-integrator modeled agents is locally asymptotically stable under a gradient-like control law. Second, we propose a formation control strategy based on orientation estimation for single-integrator modeled agents in the plane under the assumption that the agents maintain their own local reference frames, whose orientations are not aligned with each other. The proposed strategy is applicable to network localization. Third, we propose a formation control strategy based on position estimation for single- and double-integrator modeled agents in n-dimensional space. Under the proposed strategy, the position is estimated up to translation in general. The proposed control strategy is successfully applied to formation control of unicycles. Finally, we propose a formation control strategy based on orientation and position estimation for single-integrator modeled agents in the plane. The proposed strategy allows the agents to overcome the lack of information by sharing their local information. We also study consensus of a class of heterogeneous linear agents and disturbance attenuation in a consensus network of identical linear agents. First, we derive a sufficient condition for consensus of linear agents under the assumption that the dynamics of each agent contains single-integrator and the rest part of the dynamics is positive real. Based on the sufficient i

7 condition, stability of the load frequency control network of a power grid can be analyzed. Second, we study disturbance attenuation problems in a consensus network of identical linear agents. Under the assumption that the agents are under exogenous disturbances, we take the H norm of the transfer function from the disturbances to the disagreement vector of the network as a metric for the disturbance attenuation performance. We show that the disturbance attenuation performance is enhanced by increasing the algebraic connectivity when the agents satisfy a certain property. Procedures to design the interconnection topology and the decentralized controller for the agents are provided. c 2013 Kwang-Kyo Oh ii

8 Ph.D/SM 오광교. 다중에이전트시스템의분산제어 : 이론및응용. 정보기전공학부 p. 지도교수 : 안효성. 국문요약 다중에이전트시스템은유연성, 확장성, 강인성, 비용의효율성등의장점으로인하여최근많은연구가이루어지고있다. 또한, 제어, 연산및통신기술의진보로인해서다중에이전트 (multi-agent) 시스템의구현이점차현실화되고있다. 이러한배경에근거하여, 본학위논문은다중에이전트시스템에있어서의전형적인제어문제라할수있는포메이션 (formation) 제어와컨센서스 (consensus) 에초점을둔다. 먼저, 포메이션제어와관련해서는, 여러가지상이한가정하에서분산포메이션제어법칙을제안하고제안된제어법칙하에서의원하는포메이션에대한안정성을분석한다. 첫째, 거리기반포메이션제어문제를해결하기위한새로운제어법칙을제안하고원하는포메이션이제안된법칙하에서점근적으로 (asyptotically) 안정하다는것을보인다. 또한, 거리기반포메이션제어문제에서주로이용되어온그래디언트기반제어법칙 (gradient-based control law) 하에서강성인 (rigid) 포메이션이국부적으로점근적으로안정함을보인다. 기존의연구와달리, 본논문에서는일반적인 n 차원공간에서이중적분기로표현되는시스템의포메이션또한안정함을보인다. 둘째, 각에이전트가자신의국부좌표계의방향을추정하도록하여모든에이전트들이공통의방향감각을공유하도록함으로써거리기반포메이션제어문제를변위기반문제로다루는기법을제안한다. 제안된제어기법은거리와방향센서를가지는센서네트워크의측위 (localization) 기법에응용될수있다. 셋째, 각에이전트가이웃하는에이전트들의상대적인위치를측정하는경우에있어서, 위치를추정하는기법을제안하고그기법을기반으로한포메이션제어법칙을제안한다. 제안된기법은유니사이클 (unicycle) 로모델링되는실제적인모바일에이전트에적용이가능하다. 넷째, 국부좌표계의방향과위치를동시에추정하는기법을제안하고그기법에기반한포메이션제어법칙을제안한다. 제안된기법은, 거리기반포메이션제어문제에서가정되는매우제한된정보만이측정가능한상황에서에이전트들이정보를서로공유함으로써문제를해결해나가는것이라할수있다. iii

9 한편, 컨센서스와관련해서는특정한구조를가지는상이한 (heterogeneous) 선형에이전트들이컨센서스에도달하기위한조건을제시하고또한동일한 (identical) 선형에이전트로구성된시스템에서외란의영향을줄이기위한설계기법을제안한다. 먼저, 각에이전트들의동적방정식이주파수영역에서단일적분기와실수부분이양의값을가지는 (positive real) 전달함수로표현되는경우에, 그에이전트들이컨센서스에도달한다는것을보인다. 그동안의컨센서스와관련된연구결과들이대부분동일한선형에이전트모델을가정하였다는점에서이결과는의미가있다. 또한, 주어진조건에기반하여전력시스템의주파수제어시스템의안정성을분석할수있다. 다음으로, 동일한선형에이전트들이외란의영향하에있다는가정하에서, 외란의영향을줄이기위해서에이전트들의상호연결 (interconnection) 과관련한그래프를설계하고또한분산제어법칙을설계하는기법을제안한다. 특히, 각에이전트가특정한특성을가지는경우에, 에이전트들의연결성 (connectivity) 가커질수록외란의영향이줄어들수있음을보인다. c 2013 오광교 iv

10 Contents Abstract (English) i Abstract (Korean) iii List of Tables ix List of Figures xi 1 Introduction Background Literature review Formation control Consensus Contributions Outline Preliminaries Algebraic graph theory Graph rigidity Distance-based formation control considering inter-agent distance dynamics Introduction Formation control considering inter-agent distance dynamics Problem statement v

11 3.2.2 Three-agent case N-agent case Simulation results Conclusion Distance-based formation under the gradient control law Introduction Undirected formations of single-integrators Problem statement Gradient control law Stability analysis Undirected formations of double-integrators Problem statement Gradient-like law and stability analysis Simulation results Conclusion Formation control based on orientation estimation Introduction Preliminaries Formation control based on orientation estimation Problem statement Control strategy and stability analysis: static graph case vi

12 5.3.3 Control strategy and stability analysis: switching graph case Application to network localization Problem statement Network localization based on orientation estimation Simulation results Conclusion Formation control based on position estimation Introduction Formation control based on position estimation: single-integrator case Problem statement Control strategy and stability analysis Application to unicycle-like mobile robots Formation control based on position estimation: double-integrator case Problem statement Control strategy and stability analysis Reduced-order position estimation Simulation results Conclusion Formation control based on orientation and position estimation Introduction Formation control based on orientation and position estimation vii

13 7.2.1 Problem statement Control strategy and stability analysis Simulation results Conclusion Consensus of networks of a class of linear agents Introduction Preliminaries Main result Examples Illustrating examples Load frequency control network of synchronous generators Conclusion Disturbance attenuation in consensus networks Introduction Preliminaries Problem statement Graph design problem Controller design problem Graph design for disturbance attenuation Decomposition of the consensus network Graph design viii

14 9.5 Controller design for disturbance attenuation Design of decentralized control networks Design of distributed control networks Examples Graph design example Controller design example Conclusion Conclusion Summary of results Future works Bibliography 187 ix

15 List of Tables 9.1 Result for the graph design example Result for the decentralized controller design example x

16 List of Figures 1.1 Typical self-organized collective behavior found in biological systems Projection of k d b i /4 onto the column space of A i Simulation result of the three-agents under (3.19) The interaction graph and the formation trajectory of the ten-agents under (3.19) Sensing graph for five-agents Simulation result for five single-integrators Simulation result for five double-integrators Measurement of relative orientation angle The interaction graph for the six single-integrators Simulation result of the six single-integrators under (5.34) and (5.38): static interaction graph case Simulation result of the six single-integrators under (5.34) and (5.38): switching interaction graph case Block diagram for formation control based on a position estimator Unicycle-like mobile robot The interaction graph for the six-agents Simulation result of the six single-integrators under (6.5) and (6.7) Simulation result of the six single-integrators under the existing displacementbased formation control law xi

17 6.6 Simulation result of the six double-integrators under (6.20) and (6.22) Simulation result of the six double-integrators under (6.24) and (6.26) Simulation result of the six-unicycles under (6.5) and (6.12) Measurement of relative orientation angle The interaction graph for the simulation Simulation result of six single-integrator modeled agents having the static interaction graph Interconnection of two first-order and two second-order systems (Systems 1 and 2 are first-order and systems 3 and 4 are second-order) Consensus of two first- and two second-order systems Node model for LFC network of synchronous generators Nyquist plots for G i (s) of synchronous generators with typical parameters Four types of graphs xii

18 Chapter 1 Introduction 1.1 Background For decades, scientists have revealed that collective behavior discovered in various fields are based on relatively simple mechanisms. Those collective behavior is considered selforganized in the sense that it appears from interactions among neighboring individuals rather than an intervention of a central coordinator or external command, and thus, it is achieved in decentralized and parallel ways. Such self-organized collective behavior is particularly ubiquitous in biological systems (Strogatz, 2003). One typical example is the flocking behavior of birds (Figure 1.1a), fish (Figure 1.1b), penguins, ants, and bees. It is remarkable that such visually complex flocking behavior arises from simple rules of individuals in flocks. For instance, fireflies scattering over an extensive area are flashing in unison by the stimulation via the sight of flashing of neighbors (Buck, 1935). Winfree (1967) showed that phase-dependent mutual influences among the insects gave rise to such striking synchronization. Based on time-series analysis of frames of videotaped samples, Partridge (1981) revealed that the behavior of a school of saithe, which is a kind of fish, was governed by several simple rules. According to Partridge (1981), the individuals of the fish school tend to match both swimming direction and speed of their neighbors, while they are not greatly affected by average velocity of the school. 1

19 (a) A flock of birds. (b) A school of fish. Figure 1.1: Typical self-organized collective behavior found in biological systems. Those scientific results indicate that local interaction, i.e., interaction among neighboring individuals, rather than all-to-all or global interaction, underlies self-organized collective behavior. This idea has also been verified by simulation. Reynolds (1987) has proposed a simple model having a set of basic rules, i.e., separation, alignment, and cohesion, for the simulation of self-organized flocking behavior. With the three simple rules, the flock moves in an extremely realistic way, creating complex motion and interaction that would be extremely hard to create otherwise. Another model is found in Vicsek et al. (1995). Vicsek et al. (1995) have proposed a discrete-time model of autonomous dynamical systems, i.e., point masses, which are all moving in the plane with same speed but with different heading angles. The heading angle of each system is updated by means of a local rule based on the average of its own heading angle plus the angles of its neighbors. The neighbors of each system are those systems that are in the pre-specified circle centered at the system. Though each system does not know the average heading angle of the collection of all the systems, it has been revealed that the heading angles of the systems aligned eventually. 2

20 Obviously, the results in Reynolds (1987); Vicsek et al. (1995) seems to indicate possibility of implementing artificial systems achieving collective behaviors. While various issues have yet to be resolved, we may expect the following benefits from such artificial systems: Scalability: Such a system is scalable because each individual system has its own local rule based on interaction with neighboring systems. Due to the absence of a centralized coordinator, the number of individual systems is not restricted. Cost effectiveness: Centralized coordinator would require huge computation capability and high communication capacity while interaction among neighboring individual systems could be implemented cost effectively. Robustness: Even when some individual systems malfunction, the overall system might work since individual systems behave based on interactions among neighboring individual systems. Inspired by self-organized collective behavior discovered in natural systems, a significant amount of research efforts have recently been focused on the implementation of engineering systems that are capable of achieving global tasks based on local control laws, opening a new research field, i.e., multi-agent systems. Advances in control, computation and communication make it possible to construct the multi-agent systems. Multi-agent systems are studied in various disciplines such as computer science and social science, but there is no universally accepted definition for multi-agent systems (Wooldridge, 2002). In this dissertation, from a control theoretic view point, a multi-agent system is understood as a collection of dynamical systems interacting with each other. Accordingly, an 3

21 agent is understood as a dynamical system. There is a variety of research topics on multi-agent system from a control theoretic point of view. The research topics in the literature can be categorized as follows (Mesbahi & Egerstedt, 2010; Ren & Cao, 2011): Consensus/synchronization: Consensus/synchronization means to stabilize a certain quantity of interest that depends on the state of all agents to a common value. The emphasis is on the interaction topology of a multi-agent system in consensus while the emphasis is on the dynamics of individual agents in synchronization rather than the interaction topology of the multi-agent system. Formation control: Formation control refers to the stabilization of the state of a multiagent system to form a certain geometrical configuration through local interaction among agents. In a general problem formulation, formation control covers the consensus/synchronization. For instance, rendezvous, which is a special case of formation control, can be regarded as consensus/synchronization. The two terms are regarded as the same in this dissertation. Distributed estimation: Global information is often crucial for the control of multiagent systems. Distributed estimation refers to estimating the global information in cooperative manners. Distributed task assignment: For a sensor/robotic network, local tasks should be allocated to each individual agent in a distributed fashion. Distributed task assignment includes task/resource allocation, coverage control, and scheduling. 4

22 Etc.: There is a variety of other research topics on multi-agent systems. Analysis and prediction of social networks and epidemics are typical examples. In this dissertation, we are mainly focused on formation control and consensus of multiagent systems because formation control and consensus problems cover general control problems in multi-agent systems. 1.2 Literature review Formation control Consider the following N-agents: ẋ i = f(x, u i ), i = 1,..., N, (1.1) where x i R n, x = [x T 1 x T N ]T, u i R r, and f i : R nn R r R n. For the multi-agent system, the global task is defined by the following M-constraints: g i (x) = g i (x ), i = 1,..., M, (1.2) where g i : R nn R, for some x R nn. Then the formation control problem is generally stated as follows: Problem (A general formation control problem) For the multi-agent system (1.1), design a control law u = [u T 1 u T N ]T such that the set E x = {x R nn : g i (x) = g i (x ), i = 1,..., M} (1.3) becomes asymptotically stable under the control law. 5

23 As already mentioned, consensus can be regarded as a special sort of formation control. For instance, Problem is reduced into a consensus problem, when the formation constraints (1.2.1) are given by x 1 = x 2 = = x N. (1.4) Problem is also called a rendezvous problem when the formation constraints are given by (1.4). The assumption that there is no centralized controller for the multi-agent system (1.1) raises primarily two issues on Problem The first issue is associated with available information for agents, which can be summarized by the following two questions (Summers et al., 2011): What variables are measured? What variables are controlled? Depending on controlled variables, a variety of formation control problems can be formulated. Formation control problems in the literature can be mainly categorized into two classes of approaches: Position-based approaches: In this approach, the position of each agent is controlled based on sensed positions and relative displacements among agents. Displacement-based approaches: Relative displacements among agents are controlled based on sensed relative displacements with respect to the global reference frame in these approaches. 6

24 Distance-based approaches: Relative distances among agents are controlled based on sensed relative displacements with respect to local reference frames of agents, which are not necessarily aligned, in these approaches. Etc.: angle-based approaches and pure distance-based approaches. Beside the two classes of approaches, one may formulate formation control problems based on appropriate assumptions on sensed variables and controlled variables. The second issue is interaction topology among agents. That is, under the assumption that sensed variables and controlled variables are determined, which agents senses and control the variables? By interaction topology, local tasks are assigned to the agents. Appropriately designed interaction topology allows consistency between the global task of a multi-agent system and the local tasks of the agents, i.e., the global task is achieved by the completion of all the local tasks. In the following, we review existing results on displacement- and distance-based approaches in the literature. Displacement-based approaches We consider the multi-agent system described by (1.1). Under displacement-based formation control problem setup, each agent senses and controls relative-displacements of its neighboring agents with respect to the global reference frame. That is, the sensed variables for agent i is given as x ji = x j x i (1.5) 7

25 for all its neighboring agents j. The global task for the multi-agent system is to stabilize x to satisfy the formation constraints given by desired relative-displacements, x j x i = x j x i (1.6) for some x and for all i, j = 1,..., N. The local task of agent i is automatically defined from the global task, i.e., stabilization of x i to satisfy x j x i = x j x i for all its neighboring agents. Graphs have turned out quite useful to modeling multi-agent systems. The multi-agent system (1.1) is modeled as a graph on N-vertices. Each edge in the graph is understood as the existence of interaction between two agents corresponding to the vertices incident to the edge. That is, at least one of the two agents senses and controls the relative-displacement between them. Since this graph describes the interaction topology of the multi-agent system, we refer to the graph as the interaction graph of the multi-agent system. Appropriate interaction topology for the multi-agent system is conveniently characterized by properties of the graph. It is known that the existence of a spanning tree, which is formally defined in Chapter 2, guarantees the consistency between the local tasks of the agents and the global task. Formation control problems are reduced into consensus problems if the dynamics of the agents are linear. Thus the results on consensus are directly applied to formation control problems. Olfati-Saber & Murray (2004) have provide a necessary and sufficient condition for interaction topology of single-integrator modeled agents for average consensus, i.e. x i (t) N k=1 x k(0) as t for all i = 1,..., N. Similar problems have been also addressed in Jadbabaie et al. (2003); Lin et al. (2004). Ren et al. (2004) has shown 8

26 that, for single-integrator modeled agents, the existence of a spanning tree in the interaction graph is a necessary and sufficient condition for consensus, i.e. x i (t) x j (t) 0 as t for all i, j = 1,..., N. Moreau (2004) has shown that a condition for the interaction graph, called uniform connectivity, is sufficient for consensus of single-integrator modeled agents. Consensus of double-integrator modeled agents has been studied in Ren & Atkins (2007). According to Ren & Atkins (2007), the existence of a spanning tree in the interaction graph of the double-integrators is a necessary but not sufficient for consensus. Consensus of identical linear time-invariant systems has been addressed in Scardovi & Sepulchre (2009). Displacement-based formation control problems for nonlinear agents have also been studied in Dimarogonas & Kyriakopoulos (2008); Dong & Farrell (2008); Lin et al. (2005); Ren & Atkins (2007). Distance-based approaches For the multi-agent system (1.1), under distance-based problem setup, each agent senses relative-displacements of its neighboring agents with respect to its local reference frame. The local reference frames of the agents are not aligned, thus each agent maintains its own local reference frames. That is, the sensed variables of agent i are given as x i ji = x i j x i i, (1.7) where superscript i denotes the the variables are with respect to the local reference frame of agent i, for all neighboring agents j of agent i. The global task for the multi-agent system is to stabilize x to satisfy the formation con- 9

27 straints given by desired relative-distances, x j x i = x j x i, (1.8) where denotes the Euclidean norm, for some x and for all i, j = 1,..., N. The local task of agent i is then defined from the global task, i.e., stabilization of x i to satisfy x i j x i i = x j x i = x j x i for all its neighboring agents. It has been known that when the interaction graph of agents is undirected, rigidity (Asimow & Roth, 1979; Laman, 1970) of the interaction graph ensures consistency between the local tasks and the global task. That is, if the interaction graph is rigid, the formation of the agents is uniquely determined up to congruence at least locally by the stabilization of each relative-distance to its desired value. When the interaction graph is directed, the notion of graph rigidity is generalized to the notion of persistence (Hendrickx et al., 2007). Persistence includes rigidity while requiring an additional condition called constraint consistency. A noticeable early work on distance-based formation control problems is found in Baillieul & Suri (2003). Baillieul & Suri (2003) have proposed a gradient-based control law, which has been adopted in most of other works, and analyzed stability of acyclic minimally persistent formations. Later on, the gradient-based control law has been adopted in most of existing results on distance-based formation control. Krick et al. (2009) have proved local asymptotic stability of undirected rigid formations and acyclic persistent formations under the gradient-based control law, based on the center manifold theory. Dimarogonas & Johansson (2010) have shown that undirected/directed tree formations are globally asymptotically stable whereas such formations are not uniquely determined up to congruence by the completion of the local tasks. Local asymptotic stability of cyclic and acyclic minimally 10

28 persistent formations has been addressed in Summers et al. (2011); Yu et al. (2009), respectively. Global asymptotic stability of triangular formations has been studied in Cao et al. (2007, 2011, 2008); Dörfler & Francis (2010) Consensus For the past decade, consensus of multiples of systems have attracted a significant amount of research interest due to its broad applications in various areas (see Olfati-Saber et al. (2007); Ren et al. (2007) and the references therein). Theoretical study on consensus has been particularly focused on linear system networks. Olfati-saber and Murray have provided a necessary and sufficient condition for the underlying graph topology of single-integrator modeled agents to achieve average consensus (Olfati-Saber & Murray, 2004). Ren and Atkins has shown that, for single-integrator modeled agents, the existence of a spanning tree in the underlying graph is a necessary and sufficient condition for consensus (Ren et al., 2004). Moreau has shown that the uniform connectivity of the underlying graph is sufficient to achieve consensus for single-integrator modeled agents (Moreau, 2004). Consensus of double-integrator modeled agents has been studied in Ren & Atkins (2007). According to Ren & Atkins (2007), the existence of a spanning tree in the interaction graph of the doubleintegrators is a necessary but not sufficient for consensus. Consensus of identical linear time-invariant systems has been addressed in Fax & Murray (2004); Scardovi & Sepulchre (2009); Tuna (2009). Recently, Wang & Elia (2010) have studied a single-integrator network interconnected by dynamic edges and provided a sufficient condition for consensus based on the diagonal 11

29 dominance of the complex interconnection matrix. Since the edges of the network have dynamics, one may call such networks as dynamic consensus networks. Another work on dynamic consensus networks is found in Moore et al. (2011). Motivated by thermal process in a building, Moore et al. (2011) have proposed general single-integrator dynamic consensus network model and provided a sufficient condition for consensus also based on the diagonal dominance condition. A load frequency control (LFC) network of synchronous generators can be described as a dynamic consensus network of single-integrators. In an LFC network, each generator control system can be represented as a node whose variable is the phase variation of its voltage, and the power exchange among the generators can be represented as the interconnection of the network. Though generator control systems are nonidentical high-order systems and the interconnection is diffusive (Tuna, 2009) in that power exchange between two nodes is proportional to the phase differences of the voltages of the nodes (Kundur et al., 1994), we may address the LFC network as a dynamic consensus network of single-integrators by transforming the network dynamics appropriately. While the great majority of the existing results is based on the ideal assumption that there exist no external disturbances to individual systems, physical systems are usually affected by some disturbances in reality. Liu et al. have studied design of a dynamic output feedback controller for a undirected network of identical linear-time invariant systems under exogenous disturbances (Liu & Jia, 2010; Liu et al., 2009). Considering the transfer function matrix from the disturbance vector to the disagreement vector of the network, they have formulated a H suboptimal problem. Based on the symmetry of the Laplacian of the undirected graph, they have decomposed the overall equation for the network into indepen- 12

30 dent systems with the same system order as that of individual systems, and then provided an LMI condition for the decomposed systems to solve the H suboptimal problem. Li et al. have considered a undirected network of identical linear-time invariant systems under disturbances assuming that some individual systems can measure their own states, and then provided LMI conditions to find state feedback controllers solving the H 2 and H suboptimal problems (Li et al., 2011). Meanwhile, most of the existing results have been mainly focused on decentralized feedback control gain matrix design to ensure the disturbance attenuation performance in a consensus network. Though the disturbance attenuation property is dependent not only on the feedback controller of the network but also on the graph associated with the network, less attention has been paid to the graph design. 1.3 Contributions The contributions of this dissertation are summarized as follows. First, we propose a new decentralized formation control law under distance-based setup. While most of existing control laws are gradient-based laws, which are focused on the dynamics of agents, the proposed law is focused on edge dynamics. Accordingly, the proposed law, which is designed considering inter-agent distance dynamics, shows performance comparable to existing control laws. Particularly, the proposed law shows a good property when applied to three-agent formations. Second, we present local asymptotic stability analysis results of undirected and directed formations of single-integrator modeled agents, and show that undirected formations of double-integrator modeled agents are also locally asymptotically stable based on the topolog- 13

31 ical equivalence of a Hamiltonian system and a gradient-like system, which is an extension of the existing results. While lots of systems including vehicles can be modeled as singleintegrators, vehicles are inherently double-integrator systems, which verifies the importance of the proposed stability analysis. Third, we propose a displacement-based formation control law via orientation alignment under distance-based setup. Though agents maintain their own local reference frame under distance-based setup and thus they exploit some directional information under distancebased setup, the directional information of each agent cannot be used to achieving global tasks, which causes the lack of information, because the agents have nonidentical orientation angles. Such lack of information links to the difficulties of formation control under distance-based setup. To overcome the obstacle, we propose an orientation alignment law, which allows the agents to align their local reference frames and utilize their directional information to stabilizing their states. That is, the lack of information is overcome by means of cooperation among the agents. Under the proposed control law, conditions of interaction topology for achieving desired formations is relaxed and the region of attraction is clearly provided. Forth, we propose a position-based formation control law via distributed observer under displacement-based setup. Obviously, desired formations would be far more readily achievable if each agent knows its own state. Though it is not possible for the agents to estimate their exact states, they can estimate their states up to translation based on the cooperation among the agents. Then the translated states of the agents can be utilized for achieving the desired formation. It is shown that the performance of formation control can be en- 14

32 hanced under the proposed control law especially when the distributed observer reaches to the steady-state. Fifth, by combining the proposed orientation alignment law and distributed observer, we proposed a position-based control law under distance-based control law. That is, by means of cooperation among agents, the agents obtain state information as well as directional information under distance-based setup, where relative-distances and only locally meaningful directional information are available. Sixth, we investigate conditions for the consensus of nonidentical linear agents interconnected by diffusive coupling. We present sufficient conditions for the consensus of a network of positive real (PR) agents multiplied by a single-integrator. Individual agents of the network may have nonidentical dynamics and different system orders. It is shown that a connected network of weakly strictly positive real (WSPR) agents multiplied by a singleintegrator reaches consensus. Further, a condition for the consensus of a connected network of PR agents multiplied by a single-integrator is also provided. Based on the conditions, we can check consensusability of an output diffusively coupled linear agent network and an LFC network of synchronous generators. Since many load frequency controllers have been designed without consideration of the stability of the overall network in the literature, the sufficient conditions presented in this paper might be useful for the design of load frequency controllers. Finally, we study two problems related with the disturbance attenuation of undirected consensus networks of identical linear agents that are under exogenous disturbances. We address the H suboptimal problem for a given identical linear agent network to ensure the 15

33 disturbance attenuation performance under the assumption that the topology of the network is given but the edge weights are variables belonging to a convex set. We show that the H suboptimal problem, which is the design of the edge weights of the graph, is solved by maximizing of the second eigenvalue of the graph Laplacian under some condition, which can be readily checked by solving an LMI feasibility problem. Since the disturbance attenuation performance is a highly nonlinear function of the edge weights of the graph, it is generally intractable to solve the H suboptimal problem. In this regard, the proposed approach might be useful in practice. We also consider an identical linear agent network with existing interconnection, which might be regarded as the physical interconnection. For the consensus network, we formulate two H suboptimal problems based on decentralized and distributed controllers, respectively, and provide algorithms for the design of decentralized and distributed controllers to ensure the given disturbance attenuation performance. When the network has certain properties, the decentralized controller is readily designed by solving an LMI feasibility problem and maximizing the second eigenvalue of the graph Laplacian. The distributed controller is also designed by solving an LMI feasibility problem. 1.4 Outline The outline of the remainder of this dissertation is as follows. In Chapter 2, we review some mathematical backgrounds used throughout the dissertation. Basic notions on graphs, algebraic graph theory, and graph rigidity are summarized. From Chapter 3 to Chapter 7, we study decentralized formation control problems under various assumptions on available information for agents. In Chapter 3, a new decentralized 16

34 formation control law is proposed under distance-based setup, and stability of undirected formations is analyzed under the proposed control law. In Chapter 4, we analyze stability of formations under existing gradient-based control laws. First, we present another proof for local asymptotic stability of undirected formations of single-integrator modeled agents under the existing laws, based on the Lyapunov direct method. Second, we provide a proof for local asymptotic stability of undirected formations of double-integrator modeled agents under the existing laws, based on the topological equivalence of a Hamiltonian system and a gradient-like system. Third, we present a new proof for local asymptotic stability of cycle-free persistent formations of single-integrator modeled agents under the existing laws. In Chapter 5, we propose a displacement-based formation control law via orientation alignment under distance-based setup. We propose an orientation alignment law based on cooperation among the agents, and then show that a displacement-based formation control law is effectively applied to stabilizing the formation of the agents. In Chapter 6, we propose a position-based control law via distributed observer under displacement-based setup. Then the formation of the agents is stabilized based on the estimated states. In Chapter 7, we, combining the results in Chapter 6 and Chapter 7, propose a positionbased control law via distributed observer and orientation alignment under distance-based setup. Under the proposed control law, agents align their orientations while estimating their positions, based on cooperation. The agents eventually obtain translated and rotated version of their states, and the formation of the agents is stabilized based on the estimated states. 17

35 Chapter 8 and 9 are mainly focused on consensus problems. In Chapter 8, we investigate conditions for consensus of a class of hetetogeneous linear agents. We assume that the dynamics of each agent contains a single-integrator and the rest of the dynamics is positive real. In Chapter 9, we study disturbance attenuation in a consensus network of identical linear agents. We show that the disturbance attenuation performance is enhanced by increasing the algebraic connectivity of the network when the agents satisfy a certain condition. In Chapter 10, we conclude this dissertation discussing future works. 18

36 Chapter 2 Preliminaries In this chapter, we provide mathematical notions used throughout this dissertation. First, basic graph notions and algebraic graph theory are summarized. Graphs turned out to be useful for the description of multi-agent systems. Details of algebraic graph theory are found in Godsil & Royle (2001); Merris (1994). Then, we provide notions of graph rigidity, which lies in the foundation of distance-based control. Details of graph rigidity are found in Asimow & Roth (1979); Laman (1970). 2.1 Algebraic graph theory Suppose that a group of agents interact with each other by means of sensing and/or communication. It has been known that the interaction among the agents is naturally modeled by a graph in such a case. Each vertex of the graph represents each agent of the group, each edge between two-vertices represents the interaction strength between two-agents corresponding to the vertices. An weighted directed graph G is defined as three-tuple (V, E, W), where V denotes the set of vertices, E V V denotes the set of edges, and W : E R + denotes the mapping assigning positive real numbers to the edges. Self-edges, i.e., (i, i) for some i V are not allowed in this dissertation since graphs are used to modeling the interaction among agents. When every weight of a graph G = (V, E, W) is identical, we often denote the graph by G = 19

37 (V, E). An weighted undirected graph is understood as a special type of weighted directed graphs. An weighted directed graph is called undirected if, for all (i, j) E, (j, i) E and w ij = w ji, where w ij and w ji denote the weights assigned to the edges (i, j) and (j, i), respectively. When all the weights of an directed weighted graph G = (V, E, W) are one, we often denote the graph by G = (V, E). We next summarize basic notions on graphs in the followings. Though such notions are defined for an weighted directed graph, they can be directly defined for an weighted undirected graph analogously. Definition (Basic notions on graphs I) Consider an weighted directed graph G = (V, E, W). Neighbor: Vertex j is a neighbor of vertex i if (i, j) E. Neighborhood: The neighborhood N i of vertex i is the set of its all neighbors. Parent: Vertex j is a parent of vertex i if (i, j) E. Child: Vertex i is a child of vertex j if (i, j) E. Definition (Basic notions on graphs II) Directed path: A directed path is a sequence of edges of the form (i 1, i 2 ), (i 2, i 3 ),.... Directed cycle: A directed cycle is a directed path that starts and ends at the same vertex. Connectedness: A directed graph is connected if there is a directed path between any pair of vertices. 20

38 Strong connectedness: A directed graph is strongly connected if there is a directed path from every vertex to every other vertex. Completeness: A directed graph is complete if there is an edge from every vertex to every other vertex. Tree: A directed tree is a directed graph in which every vertex has exactly one parent except for one vertex called the root of the graph. Subgraph: A graph G = (V, E, W ) is a subgraph of G = (V, E, W) if V V, E E, and W W. Directed spanning tree: A directed graph G = (V, E, W ) is a directed spanning tree of a directed graph G = (V, E, W) if G is a subgraph of G, G is a directed tree, and V = V. Suppose that a directed graph G = (V, E, W) has N-vertices and M-edges. The adjacency matrix W = [w ij ] R N N of the graph G is defined as w ij, (i, j) E, w ij := 0, (i, j) E, where w ij is the weight assigned to edge (i, j). Note that a ii = 0 for all i V since self-edges are not allowed. The in-degree d in i and out-degree d out i d in i := d out i := N w ij, j=1 N w ji. j=1 of a vertex i are defined as 21

39 The directed graph G = (V, E, W) is balanced if d in i = d out i for all i V. The out-degree matrix D = [d ij ] R N N of G is defined as N j=1 d ij := w ji, i = j, 0, i j. Then the graph Laplacian matrix L = [l ij ] of G is defined as L := D W. The elements of the graph Laplacian matrix L are defined as k N i w ik, i = j, l ij = w ij, i j. Graph Laplacian matrices have the following properties: Theorem (Properties of graph Laplacian matrices I) Consider a graph Laplacian matrix L R N N. Then the followings are true. The row sums of L are all zero. There is a zero eigenvalue of L with the associated eigenvector 1 N. The matrix L is diagonally dominant, i.e., l ii N j=1,j i l ij for all i = 1,..., N. All nonzero eigenvalues of L are in the open left-half complex plane. The matrix exponential e Lt has row sum equal to one for all t, i.e., e Lt is a stochastic matrix. A graph Laplacian matrix of a connected graph has the following properties: 22

40 Theorem (Properties of graph Laplacian matrices II) Suppose that L R N N is the graph Laplacian matrix of a connected graph. Then the followings are true. The zero eigenvalue of L is algebraically simple. There is a zero eigenvalue of L with the associated eigenvector 1 N. For a vector x = [x 1 x N ] T R N, the solution of ẋ = Lx satisfies x i x for all i = 1,..., N for some constant x. Consider a directed graph G = (V, E), where V = N and E = M. The incidence matrix H = [h ij ] R M N of G is defined as h ij := 1, if j is the sink vertex of i, 1, if j is the source vertex of i, 0, otherwise. The outgoing edge matrix O = [o ij ] R N M of G is defined as 1, if vertex i has outgoing edge j, o ij := 0, otherwise. The edges of a directed graph G = (V, E) can be partitioned such that E = E d E + E, where E d, E + and E are disjoint and (i, j) E + implies (j, i) E. We define Ē as Ē = E d E +. The incidence matrix of the graph is then obtained as H = [H T d, HT +, H T +] T and the outgoing edge matrix as O = [O d, O +, O + H T +], where H + and O + are the incident and outgoing edge matrix corresponding to E +. We defined H and Ō as H := [H T d, H T +] T, Ō := [O d, O + ]. 23

41 2.2 Graph rigidity Given an undirected graph G = (V, E), where V = {1,..., N} and E = {1,..., 2M}, let p i R n be assigned to i V. Then p = (p 1,..., p N ) R nn is called a realization of G in n-dimensional space. Further, the pair (G, p) is called a f framework of G in n-dimensional space. Two frameworks (G, p) and (G, q) in n-dimensional space are equivalent if p i p j = q i q j for all (i, j) E; they are congruent if p i p j = q i q j for all i, j V. The edges of the graph G can be partitioned such that E = E + E, where E + and E are disjoint and (i, j) E + implies (j, i) E. The incidence matrix of the graph is then obtained as H = [H+, T H+] T T and the outgoing edge matrix as O = [O +, O + H+], T where H + and O + are the incident and outgoing edge matrix corresponding to E +. By ordering all edges in E + in some way, an edge function g G : R nn R M associated with (G, p) is defined as g G (p) := 1 2 (..., p i p j 2,...), (i, j) E +, (2.1) where M is the cardinality of E +. The rigidity of frameworks is then defined as follows (Asimow & Roth, 1979): Definition A framework (G, p) is rigid if there exists a neighborhood U p of p R nn such that g 1 G (g G(p)) U p = g 1 K (g K(p)) U p, where K is the complete graph on N-vertices. In other words, the framework (G, p) is rigid if there exists a neighborhood U p of p R nn such that, for any ξ U p, if g G (ξ) = g G (p), then (G, ξ) is congruent to (G, p). Further, if the framework (G, p) is rigid in R nn, then it is globally rigid. 24

42 As mentioned in Dörfler & Francis (2009), the link e = (e 1,..., e M ) R nm of a framework (G, p) is obtained as e := (H + I n )p. (2.2) Note that e 1,..., e M are not independent but are located in the column space Im(H + I n ). The space Im(H + I n ) is referred to as the link space associated with the framework (G, p). We define a function v G : Im(H + I n ) R M/2 as v G (e) := 1 2 ( e 1 2,..., e M 2 ), which corresponds to the edge function g G parameterized in the link space. That is, g G (p) = v G ((H + I n )p). Defining D as D(e) := diag(e 1,..., e M ), we obtain g G (p) p = v G(e) e e p = [D(e)]T (H + I n ). Let G = (V, E) be a directed graph. Suppose that for a framework (G, p), desired squared distances d ij for all (i, j) E are given. An edge (i, j) E is active if p i p j 2 = d ij. A position p i of a vertex i V is fitting for the distances if there is no p i R 2 such that {(i, j) E : p i p j 2 = d ij} {(i, j) E : p i p j 2 = d ij}. (2.3) This means that p i is one of the best positions to satisfy the constraints for edge lengths. Then the framework (G, p) is fitting for the desired distances if all the vertices of E are at fitting positions for the desired distances. Then the persistence of frameworks is defined as follows (Hendrickx et al., 2007): 25

43 Definition Let G = (V, E) be a directed graph. A framework (G, p) is persistent if there exists ɛ > 0 such that every realization p fitting for the distance set induced by p and satisfying d(p, p ) < ɛ, where d(p, p ) = max i V p i p i, is congruent to p. That is, if (G, p) is persistence, then there exists a neighborhood of p such that every realization p fitting to p is congruent to p in the neighborhood. As a generalization of rigidity, which is defined for undirected graphs, to directed graphs, persistence formalizes not only rigidity but also constraint consistency. Intuitively, constraint consistency means that if each vertex satisfies its out-going edge length constraints, then the congruency of the framework is achieved. If G contains no cycles and (G, p) is persistent, then the framework is cycle-free persistent. Cycle-free persistent frameworks are also referred to as leader/follower structured rigid frameworks in the literature (Cao et al., 2011). For simplicity, we assume that vertices of a cycle-free framework are ordered as follows: j N k j < k. It has been known that a cycle-free persistent framework can be constructed by a Henneberg sequence (Hendrickx et al., 2007). 26