ARTICLE IN PRESS. Dynamic graphs. Electrical Engineering Department, Santa Clara University, Santa Clara, CA 95053, USA

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1 Nonlinear Analysis: Hybrid Systems ( ) Dynamic graphs D.D. Šiljak Electrical Engineering Department, Santa Clara University, Santa Clara, CA 95053, USA Received 26 July 2006; accepted 4 August 2006 Abstract Dynamic graphs are defined in a linear space as a one-parameter group of transformations of the graph space into itself. Stability of equilibrium graphs is formulated in the sense of Lyapunov to study motions of positive graphs in the nonnegative orthant of the graph space. Relying on the isomorphism of graphs and adjacency matrices, a new concept of dynamic connective stability of complex systems is introduced. A dynamic interaction coordinator is added to complex interconnected system to ensure that the desired level of interconnections between subsystems is preserved as a connectively stable equilibrium of the overall system despite uncertain structural perturbations. It is shown how the coordinator can be designed to adaptively adjust the interconnection levels in order to assign a prescribed state of the complex multi-agent system as a stable equilibrium point. The equilibrium assignment is achieved by the action of the coordinator which solves an optimization problem involving a two-time-scale system; the coordinator action is slow compared to the fast dynamics of the agents. Polytopic connective stability of the multi-agent systems with a coordinator is established by the concept of vector Lyapunov functions and the theory of M-matrices. c 2007 Elsevier Ltd. All rights reserved. Keywords: Graphs; Dynamic graphs and adjacency matrices; Positive dynamic graphs; Complex interconnected systems; Interconnection coordinator; Multi-agent systems; Equilibrium assignment problem; Two-time-scale optimization problem; Adaptive coordination 1. Introduction Graph theory provides a mathematical model for studying interconnections among elements in natural and manmade systems [1]. Initially, interactions were limited to binary relations among the individual elements represented by vertices of the graph. Subsequently, functions were associated with graphs that assign a real number to each edge of a graph in order to quantify the relation between any pair of elements in a given system [2]. While graphs have been traditionally studied as static objects, in applications of graph theory to social systems, it has long been recognized that graphs should change in time to better reflect the reality of social interactions in groups and organizations. In 1940, Radcliffe-Brown [3] wrote:... a consideration of the continuity of social structure through time, a continuity which is not static like that of a building, but a dynamic continuity (my italics), like that of organic structure of a living body. A natural way to introduce changes in graphs is to weight vertices and/or edges of a graph by their existence probability [4]. A generic problem in the context of probabilistic graphs has been to find the probability that a graph remains connected despite random drop out of vertices and edges. Tel.: address: DSiljak@scu.edu X/$ - see front matter c 2007 Elsevier Ltd. All rights reserved. doi: /j.nahs

2 2 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) Dynamic graphs have been introduced in stability studies of complex dynamic systems to study the effect of uncertain interconnections between the subsystems on the stability of the overall system [5,6]. The notion of connective stability was introduced within the context of Lyapunov stability to determine conditions under which a complex system remains stable despite structural perturbations whereby subsystems are disconnected and reconnected during the evolution of the systems in the state space. Dynamic changes of graphs have been subsequently described by integer-valued functions of integral arguments with operations defined by means of integer-valued recurrence relations [7]. Stochastic models of dynamic graphs have been introduced in the stability analysis of multi-controller schemes for reliability enhancement, where graphs representing failure and maintenance of controllers have appeared as states of a Markov process [8]. In control design, when input and output vertices are added to the system graph, the minimal requirement is that graph be input-reachable, that is, we want that each state variable of the system be reached by a path from at least one input. Either by accident or design, lines of communication and control may be disconnected, which raises the question of vulnerability [6]: Does a removal of an edge or vertex from the system graph destroy input-reachability? Graph-theoretic procedures have been developed [9] to identify the minimal number of edges which are essential for preserving input-reachability and structural controllability. Numerous methods and techniques have been developed to study static properties of complex control systems that go beyond reachability, including structural controllability, structurally fixed modes, and almost invariant subspaces, as well as structural decompositions of hierarchical and weak coupling variety [10]. Recently, there has been a growing interest in using system graphs in active roles to manipulate interconnections in a network of agents to achieve desired properties and performance of the overall multi-agent system. A dynamic adjacency matrix was added to a multi-agent system in [11] to set the interconnection levels among the agents that result in a preassigned equilibrium. Time- and state-dependent changes of adjacency matrices were used by Feddema et al. [12] to determine if and how communication failures affect input-reachability and stability of multiple cooperative robotic vehicles. Formation control graphs have been introduced by Tanner et al. [13] to model the control-related interconnections among the agents and achieve input-to-state stability in a string of leader-following agents. Fax and Murray [14] made the use of formation graphs to obtain information exchange strategies which are robust to changes in the communication structure. Assuming that state agreement among agents is reached if states of the agents converge to a common location in the state space, Lin et al. [15,16] have shown that such a state can be achieved by a switching control if the interconnection graph is sufficiently connected over time. The main objective of this paper is to initiate a study of graphs in a linear vector space, and describe a dynamic graph as a one-parameter group of transformations of the graph space into itself. By introducing a metric into the graph space, we define stability of equilibrium graphs in the sense of Lyapunov. Since numerous applications of graphs to natural and man-made systems require graphs to have positive weights, we define positive dynamic graphs that live in the nonnegative orthant of the graph space. In particular, we select the standard Lotka Volterra model to illustrate stability properties of positive graphs whose edges act as interacting species in a population model. We also show how a preassigned positive equilibrium can be inserted into a given model and then stabilized by feedback. When a nonlinear complex system is given as an interconnection of a number of subsystems (e.g., agents) having local inputs, we introduce the interconnection coordinator which is an add-on device that modifies the interconnections to maintain prespecified levels of coupling among the subsystems despite uncertain perturbations in the interconnection structure. The coordinator is modeled as a dynamic graph having the equilibrium that corresponds to the desired coupling strengths. By placing a region of attraction of the equilibrium graph inside the connective stability region, the proper functioning of the coordinator is ensured by the connective stability of the overall complex (multi-agent) system. Finally, by minimizing a suitably chosen functional in the graph space, we derive an adaptive coordination law to solve the equilibrium assignment problem in complex (multi-agent) systems. The coordinator applies the law to drive the coupling strengths among the subsystems (agents) until a prescribed constant state vector becomes the desired stable equilibrium of the overall complex system. In order to prevent interference of the coordinator dynamics with the dynamics of the subsystems (agents) during the on-line adaptation process, the complex system is set as a twotime-scale system; the coordinator, which contains a small parameter, moves by an order slower than the individual agents. Polytopic connective stability of the two-time-scale system is established using the concept of vector Lyapunov functions and the theory of M-matrices.

3 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) 3 2. Dynamic graphs We consider a weighted directed graph (weighted digraph, or simply, graph) D = (V, E) which is an ordered pair, where V is a nonempty and finite set of N vertices (points) and E is a set of edges (lines). The vertices {v 1, v 2,..., v N } are connected by edges (v j, v i ), each edge being oriented from v j to v i, i, j N = {1, 2,..., N}. To each edge (v j, v i ) we assign a weight e i j if edge (v j, v i ) D, while e i j = 0 if (v j, v i ) D. With digraph D we associate the isomorphic concept of N N adjacency (interconnection) matrix E = (e i j ). In the following, we will take advantage of this isomorphism and use D and E interchangeably to fit the context. Our first objective is to define a space D of graphs with a fixed number N of vertices, as a linear space over the field F of real numbers. For any D 1, D 2 D, there is a unique graph D 1 + D 2 D called the sum of D 1 and D 2, and for any D D and any α F there is a unique graph αd D which is the multiplication of the graph D by a scalar α. When we choose in (2) α = 0, then αd = 0 is the zero graph, D = 0 D, which consists of N disconnected vertices V, that is, its set of edges is empty (E = ). The two operations that define D as a linear space, can be interpreted in the context of the linear space E of adjacency matrices. For any two N N matrices E 1 = (ei 1 j ), E 2 = (ei 2 j ) the sum is (e 1 i j ) + (e2 i j ) = (e1 i j + e2 i j ) E, and for any N N matrix E = (e i j ) E and a scalar α F, we recall the multiplication of the matrix E by a scalar α, α(e i j ) = (αe i j ) E. Finally, we note that the zero element of space E is the N N zero matrix E = 0 E. Since we are interested in motions of graphs and their stability, we need to add a norm to space D. We proceed formally and define a graph norm ν(d) by the following properties: (i) ν(d) > 0, D D (D 0), (ii) ν(αd) = α ν(d), D D, α F (iii) ν(d 1 + D 2 ) ν(d 1 ) + ν(d 2 ), D 1, D 2 D. Again, we can use the space of matrices E, which is isomorphic to the abstract space D, to interpret the properties of the norm ν(d). This is of interest in computations, because we can use the matrix norm ν : R N N R + in R N N defined as (i) ν(e) > 0, E R N N (E 0), (ii) ν(αe) = α ν(e), E R N N, α F (iii) ν(e 1 + E 2 ) ν(e 1 ) + ν(e 2 ), E 1, E 2 R N N. Once we choose a norm, we can define a metric in D, which provides a distance between any two graphs D 1 and D 2 in D, ρ(d 1, D 2 ) = ν(d 1 D 2 ), D 1, D 2 D. (7) For adjacency matrices we consider the matrix space E as R N N with the metric ρ(e 1, E 2 ) = ν(e 1 E 2 ), E 1, E 2 E. (8) We now use the recipe from [17] and propose an axiomatic definition of dynamic graphs as mappings of the abstract space D into itself. In this way, we provide a more general setting for analysis of dynamic graphs than it would be available by using differential equations as in [6,11]. (1) (2) (3) (4) (5) (6)

4 4 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) Let us now consider the space D and a family of mappings Φ(t, D), which to any graph D D and any parameter (time) t R assigns a graph Φ D. Definition 1. A dynamic graph D is a one-parameter mapping Φ : R D D of the space D into itself satisfying the following axioms: (i) Φ(t 0, D 0 ) = D 0, t 0 R, D 0 D (ii) Φ(t, D) is continuous t R, D D (iii) Φ(t 2, Φ(t 1, D)) = Φ(t 1 + t 2, D), t 1, t 2 R, D D. (9) The first axiom establishes D(t 0 ) = D 0 as the initial graph. The second axiom requires continuity of mapping Φ(t, D) with respect to all t and D, which includes continuity with respect to t 0 and D 0. Finally, the third axiom establishes dynamic graph D as a one-parameter group Φ(t, D) of transformations of the space D into itself. In applications of dynamic graphs, adjacency matrices play a prominent role, and we state: Definition 2. A dynamic adjacency matrix E is a one-parameter mapping Ψ : R R N N R N N of space R N N into itself satisfying the following axioms: (i) Ψ(t 0, E 0 ) = E 0, t 0 R, E 0 R N N (ii) Ψ(t, E) is continuous t R, E R N N (iii) Ψ(t 2, Ψ(t 1, E)) = Ψ(t 1 + t 2, E), t 1, t 2 R, E R N N. (10) In dynamic graph analysis, Φ(t, D) is called a motion of graph D, while Ψ(t, E) is a motion of the adjacency matrix E. Of particular interest will be the constant graph motions defined by Φ(t, D e ) = D e, t R. (11) A graph D e satisfying this condition is called the equilibrium graph. Similarly, an equilibrium adjacency matrix E e is defined by Ψ(t, E e ) = E e, t R. (12) In the following section, we consider stability of equilibrium graphs to determine if a given equilibrium graph is stable or unstable. An unstable equilibrium graph does not represent a viable steady-state structure that can emerge from graph motions in its neighborhood. 3. Stability of graphs The shape and character of motions around the equilibrium graphs and adjacency matrices are of interest primarily because they can tell us if a particular structure represented by a graph (or adjacency matrix) is going to persist in time. For this purpose, we introduce the notion of stability of dynamic graphs in the sense of Lyapunov. With a slight abuse of notation, we use D(t, D 0 ) instead of Φ(t, D 0 ) and state the following: Definition 3. An equilibrium graph D e D is: (i) stable if for every ε > 0 and t 0 R there is a δ > 0, such that implies ρ(d 0, D e ) < δ ρ(d(t, D 0 ), D e ) < ε, t t 0 ; (13) (14) (ii) asymptotically stable if it is stable and for every t 0 R there is an η > 0, such that ρ(d 0, D e ) < η (15)

5 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) 5 implies lim ρ(d(t, D 0), D e ) = 0; t (iii) globally asymptotically stable if the limit holds for arbitrarily large (fixed) η, that is, if it holds for all D 0 D. Since stability properties of dynamic graphs are conveniently studied using adjacency matrices, we need the following variant of Definition 3: Definition 4. An equilibrium adjacency matrix E e R N N is: (i) stable if for every ε > 0 and t 0 R there is a δ > 0, such that (16) implies ρ(e 0, E e ) < δ ρ(e(t, E 0 ), E e ) < ε, t t 0 ; (17) (18) (ii) asymptotically stable if it is stable and for every t 0 R there is an ζ > 0, such that implies ρ(e 0, E e ) < ζ lim ρ(e(t, E 0), E e ) = 0; t (iii) globally asymptotically stable if the limit holds for arbitrarily large (fixed) ζ, that is, if it holds for all E 0 R N N. Time evolutions of dynamic graphs can be defined (abstractly) by the equation (19) (20) D : D = G(t, D), (21) where D represents a tendency of graph D to change in time, while G(t, D) is the transition function which defines how D depends on graph D(t) at time t. In terms of the corresponding adjacency matrix E, Eq. (21) can be written as E : d E = F(t, E), dt where we choose d/dt for E, but other choices, such as, E(t + 1) E(t), / t, etc., are possible. We assume that the matrix function F : R R N N R N N is sufficiently smooth, so that equation describing matrix E has a unique solution Ψ(t, E 0 ) for all (t 0, E 0 ) R R N N. In order to convert a matrix differential equation describing E into the standard vector form, we vectorize the N N matrix E = (e i j ) by stacking up the rows of E to form a vector differential equation (22) E : ė = g(t, e). (23) To show how the vector e R M and, thus, function g : R R M R M, are obtained from the matrix E, we first convert E to a binary matrix Ē = (ē i j ) R N N by the rule ē i j = 1, e i j 0 = 0, e i j 0. (24) We recall that the matrix Ē is the fundamental interconnection (adjacency) matrix, which serves to describe the nominal structure of a given graph D; it acknowledges the fact that some nodes of the graph D stay disconnected (one way or both ways) from other nodes throughout the dynamic process (for details see Šiljak [6]). Now, we line up all N rows of Ē to get a binary vector ē R N 2 defined as ē = (ē 11, ē 12,..., ē 1N ; ē 21, ē 22,..., ē 2N ;... ; ē N1, ē N2,..., ē N N ). (25)

6 6 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) Fig. 1. Dynamic graph. Then, we delete all fixed zeros from ē preserving the order of the nonzero elements, re-enumerate the remaining nonzero elements in natural order to get a new shorter vector ē R M and, finally, remove the overbars to get the vector e = (e 1, e 2,..., e M ) T, where superscript T denotes transpose. Once the vector e is set, the function g(t, e) follows from the matrix function F(t, E). Example 1. To illustrate the concept of dynamic graphs at this early stage of development, let us consider a simple example of a dynamic graph D shown in Fig. 1 having the adjacency matrix [ ] 0 e12 (t) E(t) =. (27) e 21 (t) 0 To vectorize this matrix we write the fundamental adjacency matrix [ ] [ ] 0 ē Ē = = ē and get the vector ē = (0, ē 12, ē 21, 0) T = (0, 1, 1, 0) T. Now, we remove the zeros from the vector ē and overbars from its elements, re-enumerate the nonzero elements as e 12 (t) = e 1 (t), e 21 (t) = e 2 (t), and get the state vector e = (e 1, e 2 ) T R 2, so that N 2 = 4, but M = 2. We assume that the functional dependence between the lines of graph D, that is, elements e 12 and e 21 of the matrix E, are such that Eq. (23), which describes the evolution in time of the dynamic matrix E, is given by two scalar equations E : ė 1 = 3e 1 + 4e 2 1 e e 1e 2 ė 2 = 2.1e 2 + e 1 e 2 (29) which can serve as a model for the predator prey interaction: e 1 a prey and e 2 a predator [18]. Although the interactions among the edges of a graph can be chosen in arbitrary ways, in selecting the predator pray model we want to indicate the possibility that interactions among the edges of a graph may mimic the interactions among species in a population community. A portrait of the motions of E is given in Fig. 2. To compute the equilibriums of E, we solve the algebraic equations, 3e 1 + 4e 2 1 e e 1e 2 = 0 2.1e 2 + e 1 e 2 = 0 and get the equilibrium states [ ] [ ] [ ] [ ] e1 e 0 =, e e =, e e =, e e = 0 which are denoted by stars in Fig. 2. We observe that the first two equilibria are asymptotically stable, while the remaining two are not. These facts can be verified by linearization of E at the equilibrium states. The corresponding equilibrium adjacency matrices E e 1 = [ ], E e 2 = [ ] [ ] [ ], E3 e 0 0 =, E e = 3 0 (26) (28) (30) (31) (32)

7 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) 7 Fig. 2. State portrait. Fig. 3. Equilibrium graphs. have binary representations [ ] [ ] [ ] [ ] Ē1 e 0 0 =, Ē e =, Ē e =, Ē e =. (33) 1 0 Since the first two equilibria e1 e and ee 2 are asymptotically stable, the two equilibrium matrices Ee 1 and Ee 2 correspond to emerging interconnection structures from the dynamic process represented by adjacency matrices Ē1 e and Ē2 e or, equivalently, by the first and second graph, De 1 and De 2, in Fig. 3. Since connectedness is the fundamental feature of graphs [1], we observe that D e 1 is disconnected, while graph De 2 is strongly connected (i.e., strong). The third and forth graph, D e 3 and De 4, of Fig. 3, which cannot persist in a real world applications, are (strongly) unilateral. An interesting observation, which can be made in Fig. 2, is that dynamic

8 8 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) graph D can start as a strongly connected graph D0 1 at the initial condition e1 0 = (4, 3)T, but end as a disconnected equilibrium graph D e 1 at the origin ee 1 = 0. On the other hand, if graph D starts at D2 0 arbitrarily close to the equilibrium D e 4, which is an unilateral graph, it would end at equilibrium De 2 which is a strongly connected graph. Example 2. Let us now look at the Example 1 from a different point of view, which was proposed in [6]. We can use parametric differential equations S : ẋ 1 = 3x 1 + 4x 2 1 x e 21(t, x)x 2 ẋ 2 = 2.1x 2 + e 12 (t, x)x 1 (34) to define a dynamic adjacency matrix [ ] 0 e E(t, x) = 12 (t, x) e 21 (t, x) 0 where x R 2 is the parameter vector. If we choose e 12 (t, x) = x 1 and e 21 (t, x) = x 2, we would obtain the same diagram of Fig. 2 illustrating the motions of the graph D of Example 1. An interesting aspect of the form of S is that each variable e 12 (t, x) and e 21 (t, x) can be viewed as either input or output of S, as well as uncertain functions representing structural perturbations of S. 4. Positive graphs In a wide variety of applications, graphs are required to have nonnegative weights [1,6]. In the context of dynamic graphs this restriction means that if the graph trajectory starts with nonnegative weights it stays with nonnegative weights for all time. This further implies that the corresponding dynamic adjacency matrix E, as defined in Eq. (23), would be a positive dynamic system [19,20,6]. Recently, breaking with tradition, such systems were termed nonnegative systems [21,22]. We assume that function g(t, e) in (23) is continuous with respect to t and e, and is sufficiently smooth so that for each initial condition e 0 R M, there is a unique solution e(t; t 0, e 0 ) of (23) defined for all t t 0. We also recall the notation (35) R M + = {e RM : e i 0, i M} (36) of the positive (nonnegative) orthant in the state space R M of the dynamic adjacency matrix E, where M = {1, 2,..., M}. Then, Definition 5. Adjacency matrix E is a positive dynamic system if e 0 R + M implies e(t; t 0, e 0 ) R + M for all t t 0, that is, R + M is an invariant set of E. To characterize the invariance of R M + in terms of the function g(t, e), we state the following [19]: Theorem 1. If the function g(t, e) is nonnegative, that is, whenever g i (t, e 1,..., e i 1, 0, e i+1,..., e M ) 0, (t, e) R R M +, i M (37) e j 0, j = 1,..., i 1, i + 1,..., M, (38) then to every initial condition e 0 R M +, there corresponds a nonnegative solution e(t; t 0, e 0 ) R M +. Proof. Let us associate with dynamic matrix E the auxiliary system E ε : ė 1 = g 1 (t, e 1,..., e M ) + ε ė 2 = g 2 (t, e 1,..., e M ) + ε ė M = g M (t, e 1,..., e M ) + ε (39)

9 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) 9 where ε > 0. Set R + M is invariant with respect to the auxiliary system E ε since at every point of intersection of a solution of E ε with the boundary of R + M, the components of the solution (which are zero) are increasing. When we let ε 0, the solutions of the auxiliary system E ε become solutions of the original system E. Since R + M is a closed set, the theorem follows. The dynamic adjacency matrix of Example 1, which is described by Eq. (29), is a positive system, as can be verified by Theorem 1 and directly from the motions in Fig. 2. Eq. (29), which describe a predator prey interaction between a prey species, density e 1, and its predator, density e 2, belongs to the class of Lotka Volterra models in population biology [23]. In terms of the graph in Fig. 1, which is represented by Eq. (29), this means that the rate at which the weight (prey) e 12 = e 1 is decreasing is proportional to the product e 21 e 12 of weights e 21 and e 12 (densities of predator and prey). For details regarding the interpretation of the sign and magnitude of population interactions (see Logofet [24] and references therein). We view the lines of a graph as interacting species in a population community, and their weights as size of the individual species. This type of interaction between the weights of the lines of a graph arises in modeling of a large variety of natural and man-made systems [6]. Now, we want to generalize Example 1 for graphs with M lines (edges), and choose the model E : ė = R(Pe + Qv + r) where, as before, e R M is the state of the dynamic matrix E. The system matrices R = diag{e 1, e 2,..., e M }, P = ( p i j ), and Q = (qi j ) are constant and have appropriate dimensions. The presence of inputs v R L and r R L represents a departure from the standard Lotka Volterra model. Constant vector r R L will be used to set equilibrium matrix E e at a desired location in the positive orthant R M + = {e RM : e i > 0, i M}, while input v will be used to stabilize it. We start the stability analysis of dynamic matrix E : ė = R(Pe + r). without the input v. Equilibriums of E are determined by the equation R(Pe + r) = 0. Since we are interested in equilibria e e within R + M, we follow [25] and represent ee as [ e e e e ] = α, e e β where e e α Rk +, 0 k M, and ee β = 0. We note that if Eq. (41) has a positive solution ee, the solution is unique and given by e e = P 1 r. To consider all equilibria in R + M, we introduce the vector q = Pe e + r and impose the decomposition [ ] [ rα Pαα r =, P = r β P βα ] P αβ, P ββ q = Eq. (42) implies q α = 0 and we get q β = P βα e e α + r β resulting in system equation E : ė = R[P(e e e ) + r], which is written explicitly in terms of the equilibrium e e we are interested in. [ qα (40) (41) (42) (43) (44) q β ]. (45) Remark 1. Since E is a positive system and R + M is an invariant space, the global asymptotic stability condition (iii) in Definition 4 is restricted to R+ M. In other words, the equilibrium ee R+ M is required to be stable and attractive with respect to R+ M, so that e 0 R+ M implies e(t; t 0, e 0 ) R+ M and e(t; t 0, e 0 ) e e as t. This is R+ M -stability [25]. (46) (47)

10 10 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) To establish R+ M -stability of equilibrium ee R+ M, we can use the Volterra-type Lyapunov function [26], V (e) = k δ i [e i ei e ei e ln(e i/ei e )] + i=1 M i=k+1 δ i e i, (48) where δ i s are all positive numbers. The function V : R+ M {ee } R + M is continuously differentiable and positive definite on R+ M, that is, V (e) > 0 for all e RM + {ee }, and V (e e ) = 0. Furthermore, to get R+ M -stability we note that V (e) + when, either e i 0 for some i = 1, 2,..., k, or e +. By computing the Lyapunov derivative V (e) E with respect to (47), we get V (e) E = k δ i (e i ei e )ė i/e i + i=1 M i=k+1 δ i ė i ( e e e) T Π ( e e e ) + e T β βe β (49) where e β = (e k+1, e k+2,..., e M ) T, β = diag{δ k+1, δ k+2,..., δ M }, Π = 1 2 (PT + P), (50) and = diag{δ 1, δ 2,..., δ k }. By applying the Lyapunov method, which requires that V (e e ), V (e) E > 0, we arrive at the following [25]: Theorem 2. The equilibrium e e of the dynamic adjacency matrix E is R+ M -stable if there exists a positive diagonal matrix, such that the matrix Π is positive definite, and q β is a nonpositive vector. Remark 2. We note that nonpositivity of vector q β in Theorem 2 takes place whenever the equilibrium e e is the solution of equation Pe = r, that is, e e = P 1 r. From Theorem 1 we conclude that if e e R + M, then all one needs for stability of the equilibrium e e is the existence of which makes Π positive definite. In order to test efficiently the stability condition of Theorem 2, we can use the theory of M-matrices (e.g., Kaszkurewicz and Bhaya [27]). We first recall the class of matrices Z M = {A R M M : a i j 0, i j; i, j M}. (52) The set M M of M-matrices is a subset of Z M, and we choose only a few characterizations of the set M M : Theorem 3. Let A Z M. Then, the following statements are equivalent: (i) A M M (ii) A is Hurwitz (iii) There exist numbers d i > 0, i M, such that d i a ii > (iv) All leading principal minors of A are positive (v) There exists a matrix = diag{δ 1, δ 2,..., δ M }, δ i > 0, i M, such that the matrix Π = A T + A is positive definite. M d j a i j, i M To establish a link between M-matrices of Theorem 3 and the stability Theorem 2, let us consider a case when the matrix P of the edge community of a given graph D is such that P Z M. This means that P has nonnegative off-diagonal elements and, therefore, is a Metzler matrix, which was introduced in modeling of the competitive equilibrium of multiple market systems (e.g., Arrow [28]). In terms of multiple market systems, we view the edges j i (51) (53)

11 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) 11 of graph D as competing commodities and their weights as prices. Since the off-diagonal elements of a Metzler matrix are nonnegative, we have a gross-substitute case, implying that a weight increase (decrease) on an edge of the corresponding graph D would cause a weight increase (decrease) on all existing edges of D (for details see Šiljak [6]). From Theorem 3 we have the following corollary to Theorem 2: Corollary 1. Let P be a Metzler matrix, that is P Z M. Then, there exists a positive diagonal matrix, such that the matrix Π is positive definite if and only if P M M. If, in addition to P M M, we have that q β is a nonpositive vector, then the equilibrium e e of the dynamic adjacency matrix E is R M + -stable. We note that the defining matrix set Z M requires off-diagonal elements p i j to be nonpositive, which is too restrictive in modeling of multispecies communities with predation (as in Example 1 above), where an increase in the predator population causes a decrease in the prey species [23], requiring negative off-diagonal elements in the community matrix P. Negative off-diagonal elements in the context of competitive equilibrium imply the existence of complementary goods and appearance of the Morishima matrix [6,29]. In dynamic models of graphs, this fact means that an increase of weight e j causes a decrease of weight e i if p i j < 0. To capture the possibility of negative elements appearing in arbitrary patterns within the matrix P, we introduce the comparison matrix ˆP = ( ˆp i j ) R M M as ˆp i j = p i j, i = j = p i j, i j (54) which is the negative of McKenzie s diagonal form [6]. It has been shown by McKenzie that if matrix ˆP is quasidominant diagonal, that is, satisfies (iv) of Theorem 3, then P is Hurwitz. The connection of McKenzie s form and the Lyapunov part (v) of Theorem 3 was established by Moylan and Hill [30]. We re-state the result to fit our context: Theorem 4. If ˆP M M, then there exists a positive diagonal matrix such that the matrix Π = (P T + P), (55) is positive definite. Now, we go back to the original model (40) of E to explore the role of the input v in the stabilization of a positive equilibrium e e located in R+ M. At the end of this section we will show how the constant bias vector r can be used to set a stable equilibrium at a desired location. We start with the auxiliary linear system L : ė = Pe + Qv, (56) and seek a state feedback control law v = K e (57) which makes the function V (e) = e T e (58) with a diagonal matrix of (51) a Lyapunov function for the closed-loop system L K : ė = (P + QK )e. (59) Sufficient conditions for this kind of stability are well known, and can be expressed as a pair of matrix inequalities, > 0 (P + QK ) T + (P + QK ) > 0. (60) With the change of variables L = K Y, where Y = 1, the solution for the gain matrix K can be formulated as a Linear Matrix Inequalities (LMI) problem [31],

12 12 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) Y > 0 P T Y + Y P + QL + L T Q T < 0 [ κl I L T ] [ ] Y I < 0; > 0. L I I κ Y I The gain matrix is computed as (61) K = LY 1, and its norm is implicitly constrained by the last two inequalities above, which imply that K κ L κ Y. This constraint is necessary in order to prevent unacceptably high gains that an unconstrained optimization may otherwise produce (see Šiljak and Stipanović [32]). We should also note that in this context we can replace state feedback by static output feedback [33]. If we want the stronger stability result of Remark 1 with Metzler matrix (P + BK ), then proving R M + -stabilizability of system L K is equivalent to feasibility of the following LMI problem [31]: Y > 0 P T Y + Y P + QL + L T Q T < 0 (PY + QL) i j 0, i j. If we wish to limit the size of gain matrix K, we can add to this set of inequalities the last two inequalities of (61). Once we solve either of the two LMI problems above for the gain matrix K, we obtain a quadratic form (58) as a Lyapunov function for the system L K. This further means that the Lyapunov matrix equation P K T + P K = 2Π for the closed-loop matrix P K = P + QK has the solution as a positive diagonal matrix. By Theorem 2, this fact implies that the equilibrium e e of the closedloop Lotka Volterra equation E K : ė = R(P K e + r) is globally asymptotically stable with respect to the positive orthant R+ M. Finally, we note that we can use vector r to set the (stable) equilibrium e e of E K anywhere within the invariant region R+ M. In terms of the dynamic graph D K, which corresponds to the adjacency matrix E K we conclude that we can establish any set of fixed weights for the free edges of the graph D K as a stable equilibrium. This possibility will play an interesting role in the study of complex dynamic system carried out in the next section. There is a wealth of information about the behavior of positive systems that can be used in the context of positive graphs. Besides the theory of competitive equilibrium in economics and population biology discussed above, positive systems have been used as models in areas as diverse as interactions in social groups [34], arms race [35 37], compartmental and chemical systems [38 40], impulsive and hybrid systems [21], and biological and physiological models [22]. For a recent comprehensive treatment of positive systems, see [27]. 5. Complex dynamic systems At the outset of modeling complex systems, which are composed of interconnected subsystems, directed graphs have been introduced [5,6] to define and interpret the interconnection structure underlying the dynamics of the interacting subsystems. Subsystems were associated with vertices while interconnections with edges of the graph. In order to allow for accidental and intentional changes in the interconnection structure, which are always present in real world applications, the graph was assumed to vary as a function of time and the state of the system. To capture the effect of changing structure on the stability of large complex systems, the concept of connective stability was introduced as Lyapunov s stability under structural perturbations [5,6]. A system is considered connectively stable (62) (63) (64) (65) (66)

13 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) 13 if it remains stable despite perturbations whereby subsystems are disconnected and reconnected in uncertain ways during its motion. Our objective in this section is to initiate a systematic study of dynamic interconnections within complex systems using the proposed definition of dynamic graphs, which is rooted in the concept of connective stability. This will considerably broaden the modeling of complex systems by giving the dynamic interconnections the same level of importance that the dynamic subsystems have in shaping the behavior of complex dynamic systems. A dynamic graph (interconnection matrix) of a complex system can now be used as a basis for coordination of interconnection strength among the individual subsystems in a smooth fashion until interconnection levels reach preassigned steadystate values. A model for a complex system is now represented as a pair of coupled systems S : ẋ = f (t, x, u, E) C : ė = g(t, e, v, x) where x R n is the state of the system S, u R m is its input, and E = (e i j ) is an N N interconnection matrix. The vectors e R M and v R L are the state and input of the coordinator C. We assume that the infinitesimal transition functions f (t, x, u, E) and g(t, e, v, x) are sufficiently smooth, so that solution (x, e)(t; t 0, x 0, e 0, u, v) of interconnected system S&C exists and is unique for all initial conditions (t 0, x 0, e 0 ) and fixed piecewise continuous inputs u(t) and v(t). At this stage of development, it is necessary to add more structure to the system S&C. We consider S as an interconnected system S : ẋ i = a i (t, x i, u i ) + b i (t, x, E), i N (68) composed of N subsystems (67) S i : ẋ i = a i (t, x i, u i ), i N (69) with states x i R n i, so that the state x = (x1 T, xt 2,..., xt N )T R n of the system S is a concatenation of (disjoint, not overlapping) subsystems states. Each subsystem S i has its own input u i R m i, as a part of the overall input u = (u T 1, ut 2,..., ut N )T of system S, and N = {1, 2,..., N}. A standard role of the input u i is to stabilize the corresponding subsystem S i using only the locally available state (or output). There are well-known methods to make the decentralized control strategy produce a connectively stable overall system S [10,32]. We now want to discuss the less understood coordination task and assume that each subsystem S i : ẋ i = a i (t, x i ), i N (70) has an equilibrium at the origin (x e i = 0) which is stable, or has been stabilized by decentralized feedback. To assign the proper role to the elements e i j of the adjacency matrix E appearing in the interconnections b i (t, x, E) of S, we recall [6] the formulation of interconnections, b i (t, x, e) = b i (t, e i1 x 1, e i2 x 2,..., e i N x N ), i N. (71) According to this form, the elements e i j (t, x) are functions e i j : R R n [0, ê i j ], which are continuous in both arguments and belong to intervals [0, ê i j ], where ê i j are positive numbers. The elements represent the strength of coupling between the individual subsystems, including disconnections. Example 2 provides an illustration of the proposed form of system S. We also assume that interconnection functions b i (t, x, E) are such that the overall system S has the equilibrium at the origin (x e = 0), that is, b i (t, 0, E) = 0 for all t R and all E such that 0 E Ê with element-by-element inequalities (or, equivalently, E [0, Ê]). The infinitesimal transition function g(t, e, v, x) of the coordinator C has no special form; its construction is guided only by our desire to change the structure of the complex system S in a suitable way. We want to maintain interactions between subsystems S i at preassigned levels that are defined by elements e e i j of the equilibrium matrix Ee. We also want the equilibrium E e to be reached from all initial matrices E 0 within a permissible region containing E e, without disturbing the stability of the overall system S. Recalling the original definition of connective stability [6], we observe that the coordinator C produces dynamic structural perturbations of the interconnected system S. Our task is to define

14 14 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) a suitable concept of connective stability of equilibrium (0, e e ) of S&C, which can guarantee that the perturbations do not destroy its stability. We first assume that the input v has been utilized to set the equilibrium e e within its region of attraction Ω C R M +, where the equilibrium vector e e corresponds to the equilibrium matrix E e. Then, the function g(t, e, v, x), takes the form g(t, e, x) with g(t, e e, 0) = 0 for all t R. Finally the complex system S&C becomes S : ẋ i = a i (t, x i ) + b i (t, e i1 x 1, e i2 x 2,..., e i N x N ), C : ė = g(t, e, x). and we state i N (72) Definition 6. Complex system S is dynamically connectively stable if the equilibrium x e = 0 is globally asymptotically stable for all e 0 Ω C. To provide an interesting interpretation of the new idea of connective stability advanced by Definition 6, let us consider the joint system S&C described as S : ẋ i = a i (t, x i ) + b i (t, e i1 x 1, e i2 x 2,..., e i N x N ), C : ė = R(Pe + r). where system S remains unchanged, while the coordinator is described by a Lotka Volterra model with a stable matrix P (or stabilized by feedback). We first note the absence of the state x in the description of coordinator C, which implies a hierarchical structure of the joint system S&C [10]. Therefore, we can establish stability of the equilibrium e e R+ M of C and have a Volterra-type Lyapunov function to prove it. Now, what is left to do is to show stability of the equilibrium x e = 0 of S, which can withstand the structural perturbation caused by motion of the coordinator C restricted to the region of attraction Ω C R+ M ; this is the dynamic connective stability of S formulated in Definition 6. To derive conditions for the new kind of connective stability, we start by showing ordinary connective stability of the interconnected system S following the standard recipe [10], which is based on the Matrosov Bellman concept of vector Lyapunov functions [41,42]. Let us associate with each subsystem S i a scalar Lyapunov function V i : R R n i R +, such that V i (t, x i ) is a continuous function in both arguments and satisfies a Lipschitz condition in x i, that is, for some κ i > 0, V i (t, x i ) V i(t, x i ) κ i x i x i, i N We further assume that the function V i (t, x i ) satisfies the inequalities (73) t R x i, x i R n i. (74) φ 1i ( x i ) V i (t, x i ) φ 2i ( x i ) D + V i (t, x i ) Si φ 3i ( x i ), (t, x i ) R R n i (75) where φ 1i, φ 2i K, φ 3i K are Hahn s comparison functions (e.g., Šiljak [6]), and D + V i (t, x i ) Si is the Dini derivative computed with respect to S i. As for the interconnections b i (t, x, e), we impose the following bounds: b i (t, e i1 x 1, e i2 x 2,..., e i N x N ) N ê i j φ 3 j ( x i ), i N (76) j=1 where we deviated from the standard recipe [10] and, instead of the binary N N fundamental interconnection matrix Ē = (ē i j ), we use a more flexible N N bounding matrix Ê = (ê i j ), where ê i j are positive real numbers (ê i j 0 if and only if the corresponding element ē i j = 1). In order to state the condition for connective stability of system S, we define the constant N N test matrix Ŵ = (ŵ) as ŵ i j = 1 ê ii κ i i = j = ê i j κ i i j. (77)

15 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) 15 Fig. 4. Stability regions. We note that matrix Ŵ has nonpositive off-diagonal elements, that is, Ŵ Z N and state without proof a straightforward modification of Theorem 2.5 of [10]: Theorem 5. The equilibrium x e = 0 of system S is connectively stable if matrix Ŵ is an M-matrix. To establish the dynamic connective stability of system S we first recall [10] that connective stability means that the equilibrium x e = 0 of S is (globally and asymptotically) stable for all E [0, Ê]. The next step is to convert this matrix statement to a statement involving state vector e of the coordinator C. By vectorizing the matrix Ê R N N, we obtain the vector ê R M. Then, connective stability implies that system S is stable for all e such that 0 e i ê i, i N. Now, Theorem 5 implies that the system S is connectively stable provided the uncertain vector e(t, x) is contained within the region (box), Ω S = {e R M : e i [0, ê i ], i N}. (78) The containment will obviously take place if a region of attraction Ω C of the equilibrium e e is itself contained within the connective stability region Ω S, and initial states e 0 are restricted to the region Ω C. Having the Volterra function V (e) available as a Lyapunov function for the coordinator C, we can compute a region of attraction as Ω C = {e R M + : V (e) < θ}, (79) where θ is a positive number. We finally arrive at the following: Theorem 6. The equilibrium x e = 0 of system S is dynamically connectively stable if Ω C Ω S. In Fig. 4, we illustrate the meaning of the inclusion Ω C Ω S, by borrowing the region of attraction Ω C of the equilibrium e e 2 = (2.10, 1.98)T from Example 1 (this region was computed by Parker and Chua in [18]). Now, if we establish the connective stability of an interconnected system S with respect to the rectangle Ω S specified by the fixed interconnection matrix Ê in Fig. 4, then the system is dynamically connectively stable with respect to region Ω C situated inside the rectangle Ω S. That is, e 0 Ω C implies e(t; t 0, e 0 ) Ω C Ω S for all t > t 0, and dynamic connective stability follows from the connective stability of system S. Remark 3. Theorem 6 poses an interesting numerical problem: How to maximize the sizes of regions Ω C and Ω S while retaining the inclusion condition Ω C Ω S. The larger the sizes of the two regions the larger the class of uncertain interconnections that can be neutralized by the coordinator. We should also note that over the years a large number of extensions and applications of the connective stability concept have been proposed. A wide variety of structural perturbations have been studied, starting as unknown but bounded functions of time and state [5,6]. Subsequently, modeling of uncertain interconnection matrices has (80)

16 16 D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) been broadened to include stochastic elements [8,43], expanding the size of complex systems [44,45], discontinuous interconnection matrices [46,47], impulsive and hybrid systems [48,21], singular perturbations [11], matrix Lyapunov functions [49,50], polytopic uncertainty sets [51], discrete systems [52], fuzzy sets [53], and hybrid systems [54, 55]. Of particular interest are the methods and techniques that have been invented for the construction of robust decentralized control laws which can produce connectively stable overall interconnected systems (see books by Šiljak [6,10], and Jamshidi [56]), as well as the recent papers [32,57]. 6. Multi-agent systems: The coordinator Let us now turn to the application of the concept of dynamic graphs to the control of multi-agent systems. One of the interesting distinctions of such systems within the general class of interconnected systems is the accessibility of interconnections to control action. We consider an interconnection of agents (scalar subsystems) which all have local control inputs and are nonlinearily connected to each other. The inputs are used to decentrally stabilize each individual agent. Then, a coordinator is attached to the system as an add-on control device to adaptively adjust the interconnection levels among the agents until a prescribed state of the multi-agent system becomes a stable equilibrium state. The principal difference between the coordinator role in this context and the role it played in the preceding section, is a departure from a passive action of maintaining a set of constant interaction levels among the subsystems to an active role of changing the interconnections among the agents until a desirable equilibrium is set for the overall multi-agent system Interconnected system Let us consider a multi-agent system n S : ẋ i = ax i + e i j h j (x j ) + u i, j=1 y i = h i (x i ) i N (81) which is composed of n agents (scalar subsystems) S i : ẋ i = a i x i + u i, i N y i = h i (x i ) (82) where x, u, y R n are the state, input, and output of S, a i is a positive number, e i j s are elements of the n n interconnection matrix E, and h i (x i ) is the sigmoid function h i : R ( 1, 1), which is a diffeomorphism having the following properties: (i) h i (0) = 0, 1 < h i ( ) < 1 (ii) h i ( ) is a monotonically increasing odd function (iii) h i ( ) and h 1 i ( ) are C 1 and dh 1 i (y i ) dy i ρ 0 and N = {1, 2,..., n}. We note that the subsystems are unstable and we use a decentralized state feedback u i = k i x i + r i (83) (84) to stabilize each subsystem independently. This is a popular approach in control of complex dynamic systems for at least two reasons. First, a decentralized feedback law uses the locally available state, which is suitable either for feasibility or economic reasons (for example, subsystems may be geographically far from each other). Second, the control strategy is shown to be robust with respect to uncertain interconnections among the subsystems [10]. We choose the feedback gain k i = 1 + a i and leave the reference input r i (bias) free but constant.

17 We consider the closed-loop system ARTICLE IN PRESS D.D. Šiljak / Nonlinear Analysis: Hybrid Systems ( ) 17 S : ẋ i = x i + y i = h i (x i ) n e i j h j (x j ) + r i, j=1 C : ė i j = µg i j (e, y, y e ), i, j N i N (85) where we added a coordinator C with a coordination law g i j : R n2 R n R n R. We shall use the coordinator to drive the interconnection vector e until a desired fixed output y e is achieved as an asymptotically stable equilibrium of S. To accomplish the equilibrium assignments objective, we should establish dynamic connective stability of the interconnected two-time-scale system S&C for bounded interconnection weights e i j by choosing a suitable coordination rate (small parameter) µ (0 < µ 1). We note that we shall use matrix E R n n and vector e R n2 interchangeably to fit the context; they are equivalent, with corresponding components related by the formula e i j = e j+(n i). The system S&C was used in [11] as a model for artificial neural networks with learning, which relied on the standard notion of connective stability involving a unit hypercube of uncertain matrices E = (e i j ). By replacing the hypercube by a convex polytope, we use the new concept of polytopic connective stability [51] which can significantly increase the flexibility of coordinator C in reshaping interconnections of the system S. We bear in mind that the use of the broader concept is made possible by the two-time-scale property of the system, which allows us to consider interconnection matrices within a polytope as constant ( frozen ) in time. We begin our analysis by representing the system S in the output space, S : ẏ = B(y)[ h(y) + Ey + r] C : ė = µg(e, y, y e ) (86) where [ dh 1 i (y i ) B(y) = diag{b 1, b 2,..., b n }, b i = dy i h(y) = [h 1 1 (y 1), h 1 2 (y 2),..., h 1 n (y n)] T ] 1 and r = (r 1, r 2,..., r n ) T. For a sufficiently small coordination rate µ, the interconnection matrix E is considered frozen, and each equilibrium ȳ = (ȳ 1, ȳ 2,..., ȳ n ) of S satisfies the equation h 1 i (ȳ i ) + n e i j ȳ j + r i = 0, i N. (88) j=1 We need the following assumptions: (A 1 ) Each quasi steady-state ȳ(e) is a function of the frozen matrix E living in a convex polytope (87) P = conv {E k }, which is a convex hull of fixed vertex matrices E k, k V = {1, 2,..., v}. (A 2 ) For each E P, there exist isolated equilibria ȳ(e) of the system S. (89) As pointed out in [11], using the coordinator C to set an equilibrium of system S at a desired location y e may cause mutual interference of coordinator and system dynamics causing instability of the interconnected system S&C. A satisfactory separation of the two dynamics is achieved by choosing a sufficiently small coordination rate µ that guaranties the existence of an integral manifold [58] containing the slow dynamics and the desired equilibria.