Coupled Dynamic Systems: From Structure Towards Stability And Stabilizability. Zhiyun Lin

Transcription

1 Coupled Dynamic Systems: From Structure Towards Stability And Stabilizability by Zhiyun Lin A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto Copyright c 2006 by Zhiyun Lin

2 Abstract Coupled Dynamic Systems: From Structure Towards Stability And Stabilizability Zhiyun Lin Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2006 In this thesis, we study stability and stabilizability problems in the framework of coupled dynamic systems. Particular attention is given to the class of coupled dynamic systems whose equilibrium set is described by all states having identical state components. Central to the stability and stabilizability issues of such systems is the graph describing the interaction structure that is, who is coupled to whom. A central question is, what properties of the interaction graphs lead to stability and stabilizability? The thesis initiates a systematic inquiry into this question and provides rigorous justifications. Firstly, coupled linear systems and coupled nonlinear systems are investigated. Necessary and sufficient conditions in terms of the connectivity of the interaction directed graphs are derived to ensure that the equilibrium subspace is (globally uniformly) attractive for systems with both fixed and dynamic interaction structures. We apply the results to several analysis and control synthesis problems including problems in synchronization of coupled Kuramoto oscillators, biochemical reaction network, and synthesis of rendezvous controllers for multi-agent systems. Secondly, the stabilizability problem of coupled kinematic unicycles is investigated when only local information is available. Necessary and sufficient graphical conditions are obtained to determine the feasibility of certain formations (point formations and line formations). Furthermore, we show that under certain graphical condition, stabilization of the vehicles to any geometric formation is also feasible provided the vehicles have a common sense of direction. ii

3 Acknowledgements During the last four years as a PhD student in the Systems Control Group, University of Toronto, I have had the chance to meet and interact with many intelligent and inspiring people. This journey would have been much harder and a lot less fun without them. For this, I must sincerely thank all the wonderful people I have met. First of all, my deepest gratitude must go to my PhD advisors, Professor Bruce Francis and Professor Manfredi Maggiore, who are, by all means, devoted teachers, caring mentors, and role models to me. Their knowledge, insight, vision, inspiration, and encouragement have guided me through the wonder land of science and have taken me to the frontier of scientific inquiry. Their pleasant personality and enthusiasm for the research have certainly made such a journey extremely enjoyable. I will always be indebted to them for everything they have taught and given to me. Secondly, it is with great pleasure that I acknowledge Professor Mireille E. Broucke and express my gratitude for her help and critical comments on my thesis. I would like to thank Professor Edward J. Davison and Professor Gabriele M. T. D Eleuterio for serving on my committee. Also, I would like to thank Professor P. S. Krishnaprasad of the Department of Electrical and Computer Engineering at the University of Maryland, for serving as an external appraiser. In addition, thanks are also due to my friends in and around the Systems Control Group for lots of good conversation and for helping to provide an exciting research environment. It is always a blessing to be surrounded by a big group of intelligent and pleasant people for there is never lack of excellent research partners and wonderful friends. Finally, I am forever indebted to my loving wife, Diyang, and our lovely little girl, Ivy. It is them that make doing all things worthwhile. iii

4 Contents 1 Introduction Literature Review Thesis Outline Thesis Contributions Digraphs and Matrices Introductory Graph Theory Digraphs, Neighbors, Degrees Walks, Paths, Cycles Connectedness Operations on Digraphs Dynamic Digraphs Undirected Graphs A Result on Digraphs Nonnegative Matrices and Graph Theory Nonnegative Matrices, Adjacency Matrices, and Digraphs Irreducible Matrices Primitive Matrices Stochastic Matrices SIA Matrices iv

5 2.2.6 Wolfowitz Theorem Generator Matrices and Graph Theory Metzler Matrices and M-Matrices Generator Matrices, Graph Laplacians, and Digraphs The Transition Matrix of a Generator Matrix The Zero Eigenvalue of Generator Matrix H(α, m) Stability Coupled Linear Systems Introduction Problem Formulation Coupled Linear Systems with Fixed Topology Cyclic Coupling Structure Generalization Coupled Linear Systems with Dynamic Topology Symmetric Coupling Structure Generalization Examples and Discussion Time Dependent Switching Time and State Dependent Switching State Dependent Switching Switched Positive Systems Coupled Nonlinear Systems Introduction Problem Formulation Mathematical Preliminaries Convex Set and Tangent Cone v

6 4.3.2 The Dini Derivative The Invariance Principle Coupled Nonlinear Systems: Fixed Topology Coupled Nonlinear Systems: Dynamic Topology Set Invariance and Uniform Stability Uniform Attractivity Examples and Further Remarks Applications Synchronization of Coupled Oscillators Biochemical Reaction Network Water Tank Network Synthesis of Rendezvous Controllers Coupled Kinematic Unicycles Introduction Problem Formulation Kinematic Models Information Flow Digraph Stabilizability of Vehicle Formations Stabilization of Point Formations Main Result Simulation Example Stabilization of Line Formations Main Result Simulation Example Stabilization of Any Geometric Formations Main Result Simulation Example vi

7 6 Conclusions and Future Work Thesis Summary Future Work Appendix 181 A Supplementary Material 181 A.1 Set Stability and Attractivity A.2 Averaging Theory Bibliography 185 vii

8 List of Figures 2.1 Digraphs Walk, semiwalk, path, and cycle Aperiodic and d-periodic digraphs Digraphs with different connectivity Digraph and its opposite digraph Digraphs and their union Induced subdigraph, strong component, and closed strong component A switching signal σ(t) Digraphs G 1 and G Undirected graph and bidirectional digraph Associated digraph and adjacency matrix Gershgorin discs Associated digraphs (renumbering the nodes) Associated digraphs of (irreducible and reducible matrices) A walk from v i to v i is either a cycle or is generated by a number of cycles Associated digraphs of (non-primitive and primitive matrices) An associated digraph having two closed strong components Associated digraphs of SIA matrices Associated digraphs of a generator matrix A digraph viii

9 2.21 Gershgorin discs of a generator matrix Shifting s units Interaction digraphs A switching signal σ(t) Interaction digraphs representing two communication links Cone-like fields of view The initial condition of three robots Trajectories of five agents and the interaction digraphs The disk-like field of view Initial locations Range is Range is Range is The interaction digraphs G 1 and G Time responses and switching signal Asymptotically stable trajectory A switching signal The interaction digraphs The set S, lin(s), ri(s), and rb(s) Tangent cones T (x 1, S) and T (x 2, S) are obtained by translation of T (x 1, S) and T (x 2, S) to the origin Properties of tangent cones to convex sets Two examples of vector fields f i satisfying assumption A Illustrations for B a (x) and f i (x) A point q in M but not in Ω Some examples of vector fields fp i satisfying assumption A ix

10 4.10 Illustration for the equaled point Illustration for notations Illustration for Lemma Illustration for Lemma A possible element in O ς (i, p, δ) The time interval [t, t + T ] A distribution of agents at time t The interaction digraphs G p, p = 1, 2, Time evolution of three coordinates not tending to a common value A smooth function g(y) Time evolution of two coordinates not tending to a common value Three interaction digraphs G p, p = 1, 2, Synchronization of three oscillators with a dynamic interaction structure A tank of water Two identical coupled tanks A network of water tanks The smallest enclosing circle Wheeled vehicle Frenet-Serret frame Local information The information flow digraph Trajectories of ten unicycles in the plane The information flow digraph A uniform distribution on a line Vehicles in formation A common sense of direction A circle formation x

11 A.1 Stability, attractivity, and global attractivity with respect to Ω xi

12 Chapter 1 Introduction Suppose that a large group of soldiers are scattered in a foggy battlefield, where visibility is limited to only, say 20 meters. For instance, a soldier may (faintly) see three other soldiers, but he might lose sight of them if he moves even slightly. Under such a circumstance, is it possible for the soldiers to gather, silently, at a single location? [5] This thesis answers this and related questions using a formal model of coupled dynamic systems. A coupled dynamical system is one composed of subsystems (or agents) with coupling, that is, the states of certain agents affect the time-evolution of others. In this thesis, special attention is given to the class of coupled dynamic systems for which the equilibrium set is described by all states having identical state components. Stability and stabilizability problems of these systems with respect to the equilibrium set are the main issues. Central to these problems is the interaction structure among them that is, who is coupled to whom. The main goal of the thesis is to determine how the interaction structure affects stability and stabilizability of the systems in a continuous time setup. For instance, the problem of gathering a swarm of robots in a small region or a point on the plane is within this framework involving the structure of the couplings among them. Suppose ideally that each robot has infinite range visibility and each one takes an 1

13 Chapter 1. Introduction 2 action to move towards certain neighbors all the time. Thus, it gives rise to a fixed (static) interaction topology. However, the more interesting and realistic situation is when each robot has a limited field of view. Consequently, it brings a dynamic interaction topology if robots may come into and go out of view of each other. Our primary interest is in the connectivity properties of interaction structures that permit the solvability of the gathering problem. This problem also arises in the more general notion of consensus or agreement: A group of autonomous and distributed automata should come to agree on a piece of information. In addition, it is linked to the formation problem, i.e., the problem of arranging multiple robots in a certain spatial configuration. In a different context, a system of coupled oscillators is another example in this framework where the coupling structure plays a major role in synchronization. 1.1 Literature Review The model we study encompasses, or is closely related to, several models reported in the literature. A prominent and well-studied example concerns synchronization, a phenomenon everywhere in nature and finding several applications in physics and engineering. Synchronous motion was probably first reported by Huygens ( [46], 1673). The subject of synchronization has received huge attention in recent decades. For example, arrays of chaotic systems are studied in [14, 15, 85, 121, 122]. For coupled nonlinear oscillators, a seminal study to understand synchronization was done by Kuramoto in [59]; the work is reviewed in [104, 105]; more recently, the problem has been reinvestigated from the viewpoint of system control in [49, 101]. In addition, synchronization of mechanical systems is dealt with in [79]. Another interesting example concerns swarming, an emergent collective behavior observed for a variety of organisms in nature such as ants, fishes, birds, and bacteria.

14 Chapter 1. Introduction 3 Through simple local agent interactions, desired cooperative behaviors emerge. Biologists have been working on understanding and modelling of group behavior for a long time. See for example [21], [83], and references therein (some of which date back to the 1920 s). The work by Breder [21] is one of the early efforts to develop mathematical models of the schooling behavior in fish. He suggests a simple model composed of attraction and repulsion components. Recently, in [113], Vicsek et al. proposed a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles that they usually call self-driven or self-propelled particles), and study the collective behavior due to their interaction, where they assume that particles are moving with constant absolute velocity and at each time step each one travels in the average direction of motion of the particles in its neighborhood with some random perturbation. In their paper, Vicsek et al. provide a variety of interesting simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent s set of nearest neighbors changes with time as the system evolves. An earlier model was introduced by Reynolds [88] in Reynolds wrote a program called boids [87] that simulates the motion of a flock of birds; they fly as a flock, with a common average heading, and they avoid colliding with each other. Each bird has a local control strategy there s no leader broadcasting instructions yet a desirable overall group behavior is achieved. The local strategy of each bird has three components: 1) separation, steer to avoid crowding local flock mates; 2) alignment, steer towards the average heading of local flock mates; 3) cohesion, steer to move toward the average position of local flock mates. Recently, Jadbabaie et al. [48] studied the second of these strategies and proved mathematically, under some conditions, that all agents will eventually move in a common direction. Thus, a local strategy produces a global objective. Besides, in [38,39,68], stability of synchronous and asynchronous swarms with a fixed communication topology is studied, where stability is used to characterize the cohesiveness of a swarm.

15 Chapter 1. Introduction 4 More recently, coordination and control of multi-agent/multi-vehicle systems in the framework of coupled dynamic systems have attracted increasing interest in the field of system control and robotics. Several researchers began investigating distributed algorithms for multi-agent systems in the early 1990s. In [106] a group of simulated robots forms approximations to circles and simple polygons using the scenario that each robot orients itself to the furthest or nearest robot. In [4, 5, 80, 108], some distributed algorithms are proposed with the objective of getting the robots to congregate at a common location (rendezvous). These algorithms were extended to various synchronous and asynchronous stop-and-go strategies in [24, 64, 65]. In addition to these modelling and simulation studies, research papers focusing on the detailed mathematical analysis of coupled dynamic systems began to appear. Theoretical development of information consensus and agreement among a network of agents is made in discrete time [48,75], in continuous time [12,43,66,76,82,86,95], and in a quantized data communication setup [51,100]. Stabilization of vehicle formations with linear dynamics is studied in [33, 34, 62, 92 94] using potential functions. For vehicles with nonholonomic constraints, achievable equilibrium formations are explored in [53, 67, 70 72, 109, 110, 123, 124]. Other relevant references on formation control are [27 31, 35, 74, 81, 116, 117]. More detailed discussion of these references is postponed to appropriate chapters. 1.2 Thesis Outline The main body of the thesis consists of four chapters. Digraphs and matrices. (Chapter 2) Coupled linear systems. (Chapter 3) Coupled nonlinear systems. (Chapter 4) Coupled kinematic unicycles. (Chapter 5)

16 Chapter 1. Introduction 5 Chapter 2 essential covers the necessary preliminary work on graph theory and matrix theory. It is broken into three main sections. Section 2.1 (especially subsections 2.1.3, 2.1.4, 2.1.5, and 2.1.7) is necessary and important for the whole thesis. Section 2.2 and section 2.3 (especially subsections 2.2.5, 2.2.6, 2.3.3, 2.3.4) are very important in the presentation of Chapter 3. Section 2.3 (especially subsection 2.3.5) is useful in Chapter 5. Since all the materials in Chapter 2 are related very tightly and can be treated as an independent subject from the remaining chapters, which contribute to the research areas of graph theory and nonnegative matrix theory, we keep them in one chapter. Chapter 3 formulates the problem for coupled linear systems and studies their stability and attractivity properties in terms of interaction structures. Chapter 4 generalizes the problem of Chapter 3 to coupled nonlinear systems and studies their stability and attractivity properties in terms of interaction structures. Chapter 5 formulates the stabilizability problem for coupled kinematic unicycles and studies its necessary and/or sufficient conditions for stabilization of vehicle formations in terms of information flow structures. 1.3 Thesis Contributions Coupled dynamic systems pose many interesting research problems that have remained open. This thesis concentrates on a particular class of continuous-time coupled dynamic systems. The objective is to ensure the asymptotic coincidence of all states of the subsystems. These systems are often encountered in physics, biology, and engineering applications such as synchronization, swarming, and multi-vehicle cooperation. By formulating in a mathematical way, we explore two issues from the structure point of view: the stability and stabilizability problems of coupled dynamic systems with respect to an invariant set. Rather than focusing on a single application area, we consider a general formalism for such problems and perform a systematic inquiry into it. We not

17 Chapter 1. Introduction 6 only investigate coupled dynamic systems with static (time-invariant) coupling structure, but also study coupled dynamic systems with dynamic (time-varying) coupling structure. The reason for that is a large number of real systems from different fields give rise to a dynamic interaction structure. The contribution of each chapter can be summarized as follows. Chapter 2: In this chapter we collect relevant common notions in graph theory and define several new terms in order to better suit our development. Concentrating on the deeper connections between nonnegative matrices and directed graphs, and between generator matrices and directed graphs, we develop and present several new results for digraphs, nonnegative matrices, and especially generator matrices, which not only are important in the presentation of the remaining chapters, but also may be of independent interest in the areas of graph theory and nonnegative matrix theory. Chapter 3: In this chapter we study coupled linear systems with static and dynamic interaction structure. Precursors for this chapter are [48, 75, 88, 113]. In [113], Vicsek et al. propose a simple but compelling discrete-time model of a group of autonomous agents and provide a variety of interesting simulation results showing that if each agent updates its heading based on its neighbors headings, then all agents will eventually move in the same direction. The Vicsek model turns out to be a special version of an earlier model introduced by Reynolds in [88]. Later, Jadbabaie et al. study this model by assuming that the interaction structure can be modelled as an undirected graph and provide a sufficient graphical condition to explain the observed behaviors. With different motivations, a few investigations [12,82,86,95] have been working on the consensus seeking problem in a continuous time setup. Indeed, they are related in some way, but there is a major difference. That is, the discretization of a continuous-time system at the switching times gives

18 Chapter 1. Introduction 7 rise to infinitely many transition matrices governing the trajectory evolution, while in a discrete time setup, it is assumed that the system matrix switches among a finite family of matrices. In this chapter, we investigate coupled linear systems in continuous time with different coupling structures. This chapter presents a most general result in continuous time so far, namely, a necessary and sufficient condition for the coupled linear system to be globally attractive with respect to an equilibrium subspace, where the interaction structure is time-varying and directed. The proof technique is borrowed from [48] and redeveloped in the thesis in order to better suit our application (dealing with directed graphs and infinite many different matrices). This generalization is one of the important insights contributed by the thesis. Chapter 4: In this chapter we study coupled nonlinear systems which are the generalization of the linear systems studied in Chapter 3. A key previous relevant paper is [77], where a nonlinear discrete-time interconnected system is studied. A central assumption in that paper is that each subsystem s updated state at next step is constrained to be a strict convex combination of the current states of itself and its neighbors. Necessary and sufficient conditions on the interconnection topology guaranteeing convergence of the individual agents states to a common value is thereby obtained. We develop in this chapter a continuous-time counterpart of the results in [77] on discrete-time coupled systems. However, continuous time is more challenging than discrete time; for example, existence and/or uniqueness of solutions may fail, Lyapunov-like functions may not be differentiable, and switching may cause Zeno phenomena. These and related technical difficulties make lots of questions remain unknown in coupled nonlinear continuous-time systems. Adapting the assumption in [77] for discrete-time systems, we impose some reasonable assumptions on the vector fields of the individual systems, which are satisfied in a variety of interesting real systems. Under these assumptions, we show that the coupled nonlinear continuous-time system with dynamic coupling structure is uni-

19 Chapter 1. Introduction 8 formly attractive with respect to an equilibrium subspace if and only if certain graphical condition holds, which is the same as the one in the discrete-time counterpart. For the coupled system with fixed coupling structure, under some less conservative assumptions, we also obtain a necessary and sufficient condition guaranteeing its attractivity. Our proof is using entirely different tools from [77]. It is recognized that nonsmooth analysis serves as a main technique. Our main results are applied to the analysis of synchronization of coupled Kuramoto oscillators with time-varying coupling, to the analysis of a biochemical reaction network and a water tank network, and to the synthesis of rendezvous controllers for multi-agent systems which provides the first solution for solving the rendezvous problem in continuous time. Chapter 5: In this chapter we investigate the stabilizability problem of group vehicle formations. Unlike the coupled dynamic systems studied in Chapter 3 and 4, the subsystems (vehicles) are indeed dynamically decoupled. However, they are coupled through information flow and output feedback controllers in order to achieve certain desired objectives. This naturally gives rise to the question: Under what conditions does there exist a controller based on local available information to solve certain problems such as stabilizing a formation? This chapter is the first work solving this problem and establishes necessary and sufficient conditions for stabilization of point formations and line formations. In addition, under the assumption that the group of vehicles have a common sense of direction, stabilization of vehicles to any geometric formation is also feasible provided the condition for a point formation holds. Chapter 6: In this chapter we review the results presented in this thesis and discuss future research.

20 Chapter 2 Digraphs and Matrices The main purpose of this chapter is to provide a mathematical foundation, based on the theory of graphs and nonnegative matrices. We shall strive for rigor in presentation and shall not discuss the applicability of the concepts in the real world. This is postponed for the later chapters where we apply the results developed here to various aspects of system analysis. In this chapter we begin by surveying some basic notions from graph theory [10, 36] and develop a very important result about connectivity of digraphs. Next, we explore the theory of nonnegative matrices with emphasis on the deeper connections between nonnegative matrices and directed graphs. An excellent reference on nonnegative matrices is [17]. We develop several new results on this topic, which will become useful in the proof of a key result in Chapter 3. These results can be treated as a complement to the results in [17] for independent interest. Finally, we study in detail the exponential, zero eigenvalues, and stability issues of generator matrices. Several new results are derived that will be important in the presentation of Chapter 3 and 5. 9

21 Chapter 2. Digraphs and Matrices Introductory Graph Theory Digraphs, Neighbors, Degrees A directed graph (or just digraph) G consists of a non-empty finite set V of elements called PSfrag replacements nodes and a finite set E of ordered pairs of nodes called arcs (see Fig. 2.1). We call V the node set and E the arc set of G. We will often write G = (V, E), which means that V and E are the node set and arc set of G, respectively. v 1 v 2 v 1 v 1 v 3 v 3 v 4 (a) (b) (c) v 2 v 3 v 2 Figure 2.1: Digraphs. For an arc (u, v) the first node u is its tail and the second node v is its head. We also say that the arc (u, v) leaves u and enters v. The head and tail of an arc are its end-nodes. A loop is an arc whose end-nodes are the same node. An arc is multiple if there is another arc with the same end-nodes. A digraph is simple if it has no multiple arcs or loops. For example, consider the digraphs represented in Fig Here, digraph (a) is simple; digraph (b) has multiple arcs, namely, (v 3, v 1 ); and digraph (c) has a loop, namely, (v 2, v 2 ). In what follows, unless otherwise specified, a digraph G = (V, E) is always assumed to be simple. The local structure of a digraph is described by the neighborhoods and the degrees of its nodes. For a digraph G = (V, E) and a node v in V, we use the following notation: N + v = {u V {v} : (v, u) E}, N v = {u V {v} : (u, v) E}.

22 Chapter 2. Digraphs and Matrices 11 The sets N v + and Nv are called the out-neighborhood and in-neighborhood of v, respectively. We call the nodes in N v + and Nv the out-neighbors and in-neighbors of v. The out-degree, d + v, of a node v is the cardinality of N v +. Correspondingly, the in-degree, d v, of a node v is the cardinality of Nv. In symbols, d + v = N v + and d v = Nv. As an illustration, consider digraph (a) in Fig. 2.1, in which we have, for the node v 1, N + v 1 = {v 2, v 3 }, N v 1 = {v 4 }, and d + v 1 = 2, d v 1 = Walks, Paths, Cycles A walk in a digraph G is an alternating sequence W : v 1 e 1 v 2 e 2 e k 1 v k of nodes v i and arcs e i such that e i = (v i, v i+1 ) for every i = 1, 2,..., k 1. We say that W is a walk from v 1 to v k. The length of a walk is the number of its arcs. Hence the walk W above has length k 1. A semiwalk in a digraph G is an alternating sequence v 1 e 1 v 2 e 2 e k 1 v k of nodes and arcs such that e i = (v i, v i+1 ) or e i = (v i+1, v i ) for every i = 1, 2,..., k 1. If the nodes of a walk W are distinct, W is a path. If the nodes v 1,..., v k 1 are distinct and v 1 = v k, W is a cycle. Since paths and cycles are special cases of walks, the length of a path and a cycle is already defined. Cycles of length 1 are loops. A digraph without cycles is said to be acyclic. These concepts are now illustrated. For the digraph in Fig. 2.2, v 1 (v 1, v 3 )v 3 (v 3, v 4 )v 4 (v 4, v 5 )v 5 is not only a walk but also a path from v 1 to v 5, and v 1 (v 1, v 3 )v 3 (v 3, v 2 )v 2 (v 2, v 1 )v 1

23 PSfrag replacements Chapter 2. Digraphs and Matrices 12 v 1 v 5 v 3 v 4 v 2 v 6 Figure 2.2: Walk, semiwalk, path, and cycle. is not only a walk from v 1 to v 1 but also a cycle. However, v 1 (v 1, v 3 )v 3 (v 3, v 2 )v 2 (v 2, v 1 )v 1 (v 1, v 3 )v 3 is just a walk from v 1 to v 3 which is neither a path nor a cycle. In addition, v 1 (v 1, v 3 )v 3 (v 3, v 4 )v 4 (v 6, v 4 )v 6 is not a walk but a semiwalk from v 1 to v 6. Nevertheless, all the walks are semiwalks. When a digraph G is not acyclic, the period d of G is defined as the greatest common divisor of all the lengths of cycles in G. We call the digraph d-periodic if d > 1 and aperiodic if d = 1. For each node v i in G, let S i be the set of all the lengths, m k i, of walks from v i to v i and define d i = g.c.d. {m k i }, the greatest common divisor of all the lengths, m k i S i the period of the node v i. We call the node v i d i -periodic if d i > 1 and aperiodic if d i = 1. When we say that a node v i is d i -periodic or aperiodic, there has to be a cycle through v i. As an example, three digraphs are given in Fig Clearly, the digraph (a) is aperiodic since it has a loop. For the digraph (b), there are two cycles: one of them is of length 3 and the other is of length 4, so it is also aperiodic. However, for the digraph (c), there are still two cycles but the lengths are 3 and 6, respectively, so it is 3-periodic. Now we look at the node v 1 in the digraph (a). The lengths of walks from v 1 to v 1 are 2, 4, 6,..., meaning that the walk starts at v 1 and ends at v 1 but it could repeatedly traverse v 2 and v 1 for any positive integer times, so the node v 1 has a period 2, which is different

24 Chapter 2. Digraphs and Matrices 13 v 1 v 4 v 5 v 5 v 6 v 3 v 2 v 2 v 4 v 7 v 2 v 1 v 3 v 1 v 3 (a) (b) (c) Figure 2.3: Aperiodic and d-periodic digraphs. from the period of the digraph. However, if the digraph is strongly connected, then the period of every node is the same as the period of the digraph, which we will show later on. For example, in the digraph (c), we choose any node, say v 4. The lengths of walks from v 4 to v 4 are 6, 9, 12, 15,..., which are actually nonnegative combinations of the lengths of two cycles, 3 and 6. Hence, the node v 4 is of period 3, equaling the period of the digraph Connectedness One of the most important graph theoretic concepts is that of connectedness. We now introduce some of the ideas concerned with this aspect of digraph structure. For a digraph G, if there is a walk from one node u to another node v, then v is said to be reachable from u, written u v. If not, then v is said to be not reachable from u, written u v. In particular, a node v is reachable from itself by recalling that the sequence v is a trivial walk of length 0. A node v which is reachable from every node of the digraph G is called a globally reachable node of the digraph. A node v from which every node of the digraph G is reachable is called a centre node of the digraph. A digraph G is fully connected if for every two nodes u and v there are an arc from u to v and an arc from v to u; G is strongly connected if every two nodes u and v are

25 Chapter 2. Digraphs and Matrices 14 mutually reachable; G is unilaterally connected if for every two nodes u and v at least one is reachable from the other; G is quasi strongly connected (QSC) if for every two nodes u and v there is a node w from which u and v are reachable; G is weakly connected if every two nodes u and v are joined by a semiwalk (disregarding the orientation of each PSfrag replacements arc). A digraph G is disconnected if it is not even weakly connected. It is easy to see that G is strongly connected if and only if every node of G is a globally reachable node, or equivalently every node of G is a centre node. Clearly, a digraph consisting of only one node is always strongly connected since the node is reachable from itself. v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 (a) (b) (c) v 2 v 1 v 2 v 3 v 1 v 3 v 1 v 2 v 3 (d) (e) (f) Figure 2.4: Digraphs with different connectivity. Fig. 2.4 shows: (a) a fully connected digraph, (b) a strongly connected digraph, (c) a unilaterally connected digraph, (d) a quasi strongly connected digraph, (e) a weakly connected digraph, (f) a disconnected digraph. Clearly, every fully connected digraph is strongly connected, every strongly connected digraph is unilaterally connected, every unilaterally connected digraph is QSC, and every QSC digraph is weakly connected, but the converses of these statements are not true in general. Hence these kinds of connectedness for digraphs are overlapping Operations on Digraphs We first study the operation of taking the converse of any digraph. We shall see that this operation, which involves reversing the direction of every arc of a given digraph, sets

26 Chapter 2. Digraphs and Matrices 15 the stage for a powerful principle called directional duality. This principle will enable us to establish certain theorems without effort once we have proved other corresponding PSfrag replacements theorems. Note that the digraphs of Fig. 2.5 (a) and (b) are related to each other in a particular way: either one can be obtained from the other simply by reversing the directions of all arcs. Given a digraph G, its opposite digraph G is the digraph with the same node set formed by exchanging the orientations of all arcs in G. v 2 v 4 v 1 v 3 v 1 v 3 v 2 v 4 (a) (b) Figure 2.5: Digraph and its opposite digraph. Next we study the operation of taking the union of any two or more digraphs which PSfrag replacements have the same node set. If G = (V, E) and G = (V, E ) are digraphs with the same node set V, then their union G G is the digraph with arc set E E. That is, G G = (V, E E ). Fig. 2.6 provides an example of union operation for two digraphs, G and G, with the same node set {v 1, v 2, v 3 }. v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 = G G G G Figure 2.6: Digraphs and their union. Finally, it is sometimes appropriate to examine just part of a digraph. This can be done in the following way. For a digraph G = (V, E), if U is a nonempty subset of V, then the digraph (U, E (U U)) is termed the induced subdigraph by U. A strong component of a digraph G = (V, E) is a maximal induced subdigraph of G which is strongly connected

27 Chapter 2. Digraphs and Matrices 16 (maximal induced subdigraph is not unique in general). If G 1 = (V 1, E 1 ),..., G k = (V k, E k ) are the strong components of G = (V, E), then clearly V 1 V k = V (recall that a digraph with only one node is strongly connected). Moreover, we must have V i V j = φ for every i j as otherwise all the nodes in V i V j are reachable from each other, implying that the nodes of V i V j belong to the same strong component of G. In other words, every node belongs to exactly one strong component of G. On the other hand, for a digraph G = (V, E), a nonempty node set U V is closed if the node v is not reachable from u for all u U and v V U. In other words, there is no arc leaving from the node set U. In particular, U = V is closed. A strong component G 1 = (V 1, E 1 ) of a digraph G is closed if V 1 is closed in G. In fact, an induced subdigraph having a minimal closed node subset in G is a strong component of G. PSfrag replacements Fig. 2.7 provide examples of induced subdigraphs, G 1, G 2, and G 3, of the first digraph G, where G 1 is not a strong component, G 2 is a strong component but it is not closed, and G 3 is a closed strong component. v 1 v 2 v 3 v 4 G v 2 v 4 v 1 v 2 v 4 v 3 v 3 G 1 G 2 G 3 Figure 2.7: Induced subdigraph, strong component, and closed strong component Dynamic Digraphs We introduce the notion of a dynamic digraph, which is a digraph whose connectivity changes over time. Consider a set of n nodes, V, and a set of all possible arc sets with n

28 Chapter 2. Digraphs and Matrices 17 nodes, {E p, p Q}, where Q is the set of indices. A dynamic digraph G σ(t) = ( ) V, E σ(t) is a digraph together with a piecewise constant function σ : R Q, which is called a switching signal. Given a dynamic digraph G σ(t) = ( ) V, E σ(t), we denote by G ([t1, t 2 ]) the union digraph whose arcs are obtained from the union of the arcs in G σ(t) over the time interval [t 1, t 2 ], that is, G ([t 1, t 2 ]) = V, t [t 1,t 2 ] Also, it is convenient to introduce a very important concept here, which is a property of connectedness for dynamic digraphs. A dynamic digraph G σ(t) is uniformly quasi strongly connected (UQSC) (uniformly strongly connected) if there exists T > 0 such that for all t, the union digraph G([t, t + T ]) is quasi strongly connected (strongly connected). Example 2.1 Consider a dynamic digraph G σ(t), where the switching signal σ(t) is a periodic piecewise constant signal σ : R P = {1, 2} depicted in Fig. 2.8, and G 1, G 2 are shown in Fig It can be easily verified that there is a T = 3 such that for all t, the union digraph G([t, t + T ]) = G 1 G 2 is QSC. Hence the dynamic digraph G σ(t) is UQSC. E σ(t) Undirected Graphs For completeness, we review some concepts for undirected graphs. Undirected graphs form in a sense a special class of directed graphs (symmetric digraphs) and hence problems that can be formulated for both directed and undirected graphs are often easier for the latter. An undirected graph G = (V, E) consists of a non-empty finite set V of elements called nodes and a finite set E of unordered pairs of nodes called edges. We can simply treat an undirected graph G as a bidirectional digraph by replacing each edge (u, v) of G with the pair of arcs (u, v) and (v, u). Thus, they are completely

29 Chapter 2. Digraphs and Matrices 18 PSfrag replacements 2 σ(t) 1 PSfrag replacements t Figure 2.8: A switching signal σ(t). v 1 v 2 v 2 v 3 v 3 v 1 G 1 G 2 Figure 2.9: Digraphs G 1 and G 2. PSfrag replacements the same in the sense of connectedness. An example is given in Fig Throughout the thesis, we will use bidirectional digraphs instead of undirected graphs when such a structural description is necessary. v 1 v 2 v 3 v 1 v 2 v 3 Undirected graph Bidirectional digraph Figure 2.10: Undirected graph and bidirectional digraph. Furthermore, it is worth pointing out that for bidirectional digraphs, the four kinds of connectedness we introduced in the previous section, namely, strongly connected, unilaterally connected, quasi strongly connected, and weakly connected, are equivalent,

30 Chapter 2. Digraphs and Matrices 19 and they are all referred to as connected in the context of undirected graphs A Result on Digraphs In this subsection we present and prove a fundamental result on connectedness of digraphs, which will become extremely important in the necessity proofs of the main results presented in Chapter 3, 4, and 5. Theorem 2.1 For a digraph G = (V, E), the following statements are equivalent: (a) The digraph G is QSC; (b) The digraph G has a centre node; (c) The opposite digraph G has a globally reachable node; (d) The opposite digraph G has only one closed strong component. In the above theorem, the conditions (a), (b), and (c) are equivalent because the same property is stated using different terminologies. The condition (d) provides a new useful characterization of this property. Due to lack of an appropriate term in graph theory describing this property, we first introduced the notion of a globally reachable node and then proved the equivalence of the conditions (c) and (d), which is also proved independently in [77] with logically contrapositive form. Later on, we found the notion of quasi strongly connected in [16] and became aware of the equivalent conditions (a) and (b) presented in [16] are just directional dual properties of (c) and (d). The proof of the equivalence of (a) and (b) can be found in [16], page 133, and so is omitted. In order to prove the remaining, the next preliminary result is needed, which shows the existence of a closed strong component for any digraph. The proof of the lemma also provides an algorithm to find a closed strong component.

31 Chapter 2. Digraphs and Matrices 20 Lemma 2.1 A digraph G = (V, E) has at least one closed strong component. Furthermore, if a nonempty set U V is closed in G, then G has a closed strong component G c = (V c, E c ) satisfying V c U. Proof: We prove the first assertion by means of a constructive algorithm. Select any node, say v 1, in V. Let V 1 be the set of nodes from which v 1 is reachable and let V 1 be the set of nodes which are reachable from v 1. Recall that every node is reachable from itself. So both V 1 and V 1 contain element v 1. Check whether V 1 V 1. If so, then the induced subdigraph G 1 by V 1 is a closed strong component of G. To see this, firstly, notice that every two nodes u, v V 1 V 1 are mutually reachable since u v 1 v and v v 1 u. So the induced subdigraph G 1 by V 1 is strongly connected. On the other hand, for all v V 1 and u V V 1, v u since otherwise v 1 v u and u V 1. Hence, V 1 is closed, and the induced subdigraph by V 1 + {u} is not strongly connected since u is not reachable from any other node in V 1. Therefore, G 1 is a maximal induced subdigraph which is strongly connected. In conclusion, G 1 is a closed strong component. If instead the condition above is false, select any node, say v 2, in V V 1. Let V 2 be the set of nodes from which v 2 is reachable and let V 2 be the set of nodes which are reachable from v 2. Thus V 2 must be a subset of V V 1 since otherwise v 1 is reachable from v 2. Check whether V 2 V 2. If so, the induced subdigraph by V 2 is a closed strong component of G by the same argument as above. If it is not, repeat this procedure again until this condition holds. The digraph G has a finite number of nodes and V k is getting smaller each step by noting that V k V V 1 V k 1. So eventually the condition must hold. Indeed, V m has only one element v m at some step m if the condition is not satisfied before step m. Thus V m = {v m } V m by recalling

32 Chapter 2. Digraphs and Matrices 21 that v m is also an element of V m. Therefore the closed strong component of G will have been constructed. If a nonempty node set U V is closed in G, we let the induced subdigraph by U be G u = (U, E u ). By the first assertion, we know that the digraph G u has at least one closed strong component, say G c = (V c, E c ). Obviously V c U. It remains to show that G c is also a closed strong component of G. Clearly, G c is also an induced subdigraph of G. Moreover, V c is closed in G. Therefore, G c is the maximal induced subdigraph which is strongly connected. Combining with the fact that V c is closed in G, it follows that G c is a closed strong component of G Proof of Theorem 2.1: (b) (c) By the definition of opposite digraph, immediately we know that a centre node of the digraph G is a globally reachable node of the opposite digraph G. (c) = (d) If the opposite digraph G has a globally reachable node, let V 1 be the subset of V consisting all the globally reachable nodes and let G 1 be the induced subdigraph by V 1, then we claim that G 1 is the only closed strong component in G. The set V 1 may equal V or be a strict subset of V. In the first case, V 1 = V, clearly G 1 = G is strongly connected and so it is the unique closed strong component of G. In the second case, V 1 V, we have that v is not reachable from u for all u V 1 and v V V 1, implying V 1 is closed. (To see this point, suppose by contradiction that there are u V 1 and v V V 1 such that v is reachable from u. Notice that u is a globally reachable node. So v is also globally reachable, which contradicts that v / V 1 is not a globally reachable node.) For any two distinct nodes u, v V 1, there is a walk from u to v in G since both nodes are globally reachable. Furthermore, since V 1 is closed, this walk cannot go through any node not in V 1 and must be in the induced subdigraph G 1. That means G 1 is strongly connected.

33 Chapter 2. Digraphs and Matrices 22 Moreover, since V 1 is closed and no node in V V 1 is reachable from any node in V 1, no node can be added to make the induced subdigraph strongly connected. This implies G 1 is the maximal induced subdigraph which is strongly connected. Hence it is a closed strong component of G. Finally we show it is the unique one in G. Suppose by contradiction that there is another closed strong component in G, say G 2 = (V 2, E 2 ). Recall that V 1 V 2 = φ. Since V 2 is closed by assumption, for any node v V 1 and any node u V 2, v is not reachable from u, which contradicts the fact that v is a globally reachable node. (c) = (d) If the opposite digraph G has only one closed strong component, say G 1 = (V 1, E 1 ), we claim that every node in V 1 is globally reachable. Suppose by contradiction that there is a node v V 1 which is not globally reachable. Let V 2 be the set of nodes from which v is reachable and let V 3 the set of nodes from which v is not reachable. Then for any node u V 2 and any node w V 3, w u since otherwise w u v. Notice that V 2 V 3 = V. So it follows that V 3 is closed. Let G 3 = (V 3, E 3 ) be the induced subdigraph by V 3. Then G 3 has a closed strong component by Lemma 2.1, which is also a closed strong component of G since V 3 is closed. Furthermore, this closed strong component is not the same as G 1 since it does not have the node v while G 1 has the node v. Therefore, G has two closed strong components, a contradiction. 2.2 Nonnegative Matrices and Graph Theory We shall deal in this section with square nonnegative matrices E = (e ij ), i, j = 1,..., n; i.e., e ij 0 for all i, j, in which case we write E 0. If, in fact, e ij > 0 for all i, j, we shall put E 0 and call it positive. This definition and notation apply to row vectors x T and column vectors x. We shall use the notation E 1 E 2 to mean E 1 E 2 0, where E 1, E 2, and 0 are square nonnegative matrices of compatible dimensions.