On system state equipartitioning and semistability in network dynamical systems with arbitrary time-delays

Systems & Control Letters 57 (2008) 670 679 wwwelseviercom/locate/sysconle On system state equipartitioning and semistability in network dynamical systems with arbitrary time-delays VijaySekhar Chellaboina a,1, Wassim M Haddad b,, Qing Hui b,2, Jayanthy Ramakrishnan a,3 a MABE Department, University of ennessee, Knoxville, N 37996-2210, United States b School of Aerospace Engineering, Georgia Institute of echnology, Atlanta, GA 30332-0150, United States Received 7 July 2006; received in revised form 21 January 2008; accepted 27 January 2008 Available online 20 March 2008 Abstract In many applications involving multiagent systems, groups of agents are required to agree on certain quantities of interest In particular, it is important to develop consensus protocols for networks of dynamic agents with directed information flow, switching network topologies, and possible system time-delays In this paper, we use compartmental dynamical system models to characterize dynamic algorithms for linear and nonlinear networks of dynamic agents in the presence of inter-agent communicatioelays that possess a continuum of semistable equilibria, that is, protocol algorithms that guarantee convergence to Lyapunov stable equilibria In addition, we show that the steady-state distribution of the dynamic network is uniform, leading to system state equipartitioning or consensus hese results extend the results in the literature on consensus protocols for linear balanced networks to linear and nonlinear unbalanced networks with time-delays c 2008 Elsevier BV All rights reserved Keywords: Nonnegative systems; Compartmental models; Network systems; Networks with time-delays; Semistability; State equipartition; Consensus protocols 1 Introduction Modern complex dynamical systems are highly interconnected and mutually interdependent, both physically and through a multitude of information and communication networks By properly formulating these systems in terms of subsystem interaction involving energy/mass transfer, the dynamical models of many of these systems can be derived from mass, energy, and information balance considerations that involve dynamic states whose values are nonnegative Hence, it follows from physical considerations that the state trajectory of such systems remains in the nonnegative orthant of the state his research was supported in part by the National Science Foundation under Grant ECS-0551947 and the Air Force Office of Scientific Research under Grant FA9550-06-1-0240 Corresponding author el: 1 404 894 1078; fax: 1 404 894 2760 E-mail addresses: chellaboina@utkedu (V Chellaboina), wmhaddad@aerospacegatechedu (WM Haddad), qing hui@aegatechedu (Q Hui), jayanthy@utkedu (J Ramakrishnan) 1 el: 1 865 974 5308; fax: 1 865 974 5274 2 el: 1 404 894 3474; fax: 1 404 894 3474 3 el: 1 865 974 6751; fax: 1 865 974 5274 space for nonnegative initial conditions Such systems are commonly referred to as nonnegative dynamical systems in the literature 1,2] A subclass of nonnegative dynamical systems are compartmental systems 2 5] Compartmental systems involve dynamical models that are characterized by conservation laws (eg, mass and energy) capturing the exchange of material between coupled macroscopic subsystems known as compartments Each compartment is assumed to be kinetically homogeneous, that is, any material entering the compartment is instantaneously mixed with the material of the compartment he range of applications of nonnegative systems and compartmental systems includes biological and physiological systems 5,6], chemical reaction systems 7,8], queuing systems 9], large-scale systems 10], stochastic systems (whose state variables represent probabilities) 9], ecological systems 11], economic systems 12], demographic systems 5], telecommunications systems 13], transportation systems, power systems, thermodynamic systems 14], and structural vibration systems, to cite but a few examples A key physical limitation of compartmental systems is that transfers between compartments are not instantaneous and realistic models for capturing the dynamics of such systems 0167-6911/$ - see front matter c 2008 Elsevier BV All rights reserved doi:101016/jsysconle200801008

V Chellaboina et al / Systems & Control Letters 57 (2008) 670 679 671 should account for material, energy, or information in transit between compartments Hence, to accurately describe the evolution of the aforementioned systems, it is necessary to include in any mathematical model of the system dynamics some information of the past system states In this case, the state of the system at a given time involves a piece of trajectories in the space of continuous functions defined on an interval in the nonnegative orthant of the state space his of course leads to (infinite-dimensional) delay dynamical systems 15,16] Nonnegative and compartmental models are also widespread in agreement problems iynamical networks with directed graphs and switching topologies 17,18] Specifically, distributed decision-making for coordination of networks of dynamic agents involving information flow can be naturally captured by compartmental models hese dynamical network systems cover a very broad spectrum of applications including cooperative control of unmanned air vehicles, distributed sensor networks, swarms of air and space vehicle formations 19, 20], and congestion control in communication networks 21] In many applications involving multiagent systems, groups of agents are required to agree on certain quantities of interest In particular, it is important to develop consensus protocols for networks of dynamic agents with directed information flow, switching network topologies, and possible system time-delays In this paper, we use compartmental dynamical system models to characterize dynamic algorithms for linear and nonlinear networks of dynamic agents in the presence of inter-agent communicatioelays that possess a continuum of semistable equilibria, that is, protocol algorithms that guarantee convergence to Lyapunov stable equilibria In addition, we show that the steady-state distribution of the dynamic network is uniform, leading to system state equipartitioning or consensus From a practical viewpoint, it is not sufficient to only guarantee that a network converges to a state of consensus since steady-state convergence is not sufficient to guarantee that small perturbations from the limiting state will lead to only small transient excursions from a state of consensus It is also necessary to guarantee that the equilibrium states representing consensus are Lyapunov stable, and consequently, semistable hese results extend the results in the literature on consensus protocols for linear balanced networks to linear and nonlinear unbalanced networks with time-delays 2 Mathematical preliminaries In this section we introduce notation, several definitions, and some key results concerning linear nonnegative dynamical systems with time-delay 22,23] that are necessary for developing some of the main results of this paper Specifically, for x R n we write x 0 (resp, x 0) to indicate that every component of x is nonnegative (resp, positive) In this case, we say that x is nonnegative or positive, respectively Likewise, A R n m is nonnegative 4 or positive if every entry 4 In this paper it is important to distinguish between a square nonnegative (resp, positive) matrix and a nonnegative-definite (resp, positive-definite) matrix of A is nonnegative or positive, respectively, which is written as A 0 or A 0, respectively Furthermore, for A R n n we write A 0 (resp, A > 0) to indicate that A is a nonnegativedefinite (resp, positive-definite) matrix In addition, rank( A) denotes the rank of a matrix A, spec(a) denotes the spectrum of A, ( ) denotes transpose, and ( ) D denotes the Drazin generalized inverse Recall that for a diagonal matrix A R n n the Drazin inverse A D R n n is given by A D (i,i) = 0 if A (i,i) = 0 and A D (i,i) = 1/A (i,i) if A (i,i) 0, i = 1,, n 24, p 227] Let R n and Rn denote the nonnegative and positive orthants of R n, that is, if x R n, then x R n and x Rn are equivalent, respectively, to x 0 and x 0 Finally, let e R enote the ones vector of order n, that is, e = 1,, 1], and let e i R enote the elementary vector of order n with 1 in the ith location and 0 s elsewhere he following definition introduces the notion of a nonnegative (resp, positive) function Definition 21 Let > 0 A real function x : 0, ] R n is a nonnegative (resp, positive) function if x(t) 0 (resp, x(t) 0) on the interval 0, ] he next definition introduces the notion of essentially nonnegative matrices and compartmental matrices Definition 22 (12]) Let A R n n A is essentially nonnegative if A (i, j) 0, i, j = 1,, n, i j A is compartmental if A is essentially nonnegative and A e 0 In the first part of this paper, we consider linear, time-delay dynamical systems G of the form ẋ(t) = Ax(t) A di x(t τ i ), x(θ) = η(θ), τ θ 0, t 0, (1) where x(t) R n, t 0, A R n n, A di R n n, τ i R, i = 1,,, τ = max i {1,,nd }τ i, η( ) C {ψ( ) C( τ, 0], R n ) : ψ(θ) 0, θ τ, 0]} is a continuous vector-valued function specifying the initial state of the system, and C( τ, 0], R n ) denotes a Banach space of continuous functions mapping the interval τ, 0] into R n with the topology of uniform convergence Note that the state of (1) at time t is the piece of trajectories x between t τ and t, or, equivalently, the element x t in the space of continuous functions defined on the interval τ, 0] and taking values in R n, that is, x t C( τ, 0], R n ), where x t (θ) x(t θ), θ τ, 0] Furthermore, since for a given time t the piece of the trajectories x t is defined on τ, 0], the uniform norm x t = sup θ τ,0] x(t θ), where denotes the Euclidean vector norm, is used for the definitions of Lyapunov and asymptotic stability of (1) For further details, see 15,16] In addition, note that since η( ) is continuous it follows from heorem 21 of 15, p 14] that there exists a unique solution x(η) defined on τ, ) that coincides with η on τ, 0] and satisfies (1) for all t 0 Finally, recall that if the positive orbit γ (η(θ)) of (1) is bounded, then γ (η(θ)) is precompact 25], that is, γ (η(θ))

672 V Chellaboina et al / Systems & Control Letters 57 (2008) 670 679 can be enclosed in the union of a finite number of ε-balls around elements of γ (η(θ)) he following theorem gives necessary and sufficient conditions for asymptotic stability of a linear time-delay nonnegative dynamical system G given by (1) For this result, the following definition and proposition are needed Definition 23 he linear time-delay dynamical system given by (1) is nonnegative if for every η( ) C, the solution x(t), t 0, to (1) is nonnegative Proposition 21 (22,23]) he linear time-delay dynamical system G given by (1) is nonnegative if and only if A R n n is essentially nonnegative and A di R n n, i = 1,,, is nonnegative heorem 21 (22,23]) Consider the linear time-delay dynamical system G given by (1) where A R n n is essentially nonnegative and A di R n n, i = 1,,, is nonnegative If there exist p, r R n such that p 0 and r 0 (resp, r 0) satisfying ( ) n d 0 = A A di p r, (2) then G is Lyapunov (resp, asymptotically) stable for all τ 0, ) Conversely, if G is asymptotically stable for all τ 0, ), then there exist p, r R n such that p 0 and r 0 satisfying (2) Next, we consider a subclass of nonnegative systems, namely, compartmental systems As noted in the Introduction, compartmental dynamical systems are of major importance in biological systems, physiological systems, chemical reaction systems, ecological systems, economic systems, power systems, telecommunications systems, and network systems Definition 24 (22,23]) he linear time-delay dynamical system (1) is called a compartmental dynamical system if A R n n is essentially nonnegative, A di R n n, i = 1,,, is nonnegative, and A A di is a compartmental matrix Note that the linear time-delay dynamical system (1) is compartmental if A and A d A di are given by a ki, i = j, A (i, j) = k=1 0, i j, (3) { 0, i = j, A d(i, j) = a i j, i j, where a ii 0, i {1,, n}, denotes the loss coefficients of the ith compartment and a i j 0, i j, i, j {1,, n}, denotes the transfer coefficients from the jth compartment to the ith compartment he following results are necessary for developing some of the main results of this paper Proposition 22 (2]) Let A R n n be essentially nonnegative and assume there exists p R n such that A p 0 hen A is semistable, that is, Re λ < 0, or λ = 0 and λ is semisimple, where λ spec(a) Corollary 21 Let A R n n be an essentially nonnegative matrix such that A = A If there exists p R n such that A p 0, then A 0 Proof he proof is a direct consequence of Proposition 22 by noting that if A is symmetric, then semistability implies that A 0 Lemma 21 Let X R n n and Z R m m be such that X = X and Z = Z, and let Y R n m be such that Y = Y Z D Z hen ] X Y M Y 0 (4) Z if and only if Z 0 and X Y Z D Y 0 Proof Define ] In Y Z D 0 I m and note that det 0 Now, noting that M 0 if and only if M 0, and M In Y Z = D ] ] ] X Y I n 0 0 I m Y Z Z D Y I m X Y Z = D Y ] 0 0 Z 0, the result follows immediately 3 Semistability and equipartition of linear compartmental systems with time-delay In this section, we present sufficient conditions for semistability and system state equipartition for linear compartmental dynamical systems with time-delay Note that for addressing the stability of the zero solution of a time-delay nonnegative system, the usual stability definitions given in 15] need to be slightly modified In particular, stability notions for nonnegative dynamical systems need to be defined with respect to relatively open subsets of R n containing the equilibrium solution x t 0 For a similar definition see 2] In this case, standard Lyapunov Krasovskii stability theorems for linear and nonlinear time-delay systems 15] can be used directly with the required sufficient conditions verified on R n he following lemma is needed for the main theorem of this section Lemma 31 Let A R n n and A di R n n, i = 1,,, be given by (3) Assume that (A A di)e = 0 hen there exist nonnegative-definite matrices Q i R n n, i = 1,,, such that A A (Q i A di QD i A di ) 0 (5) Proof For each i {1,, }, let Q i be the diagonal matrix defined by Q i (l,l) m=1,l m A di (l,m), (6)

V Chellaboina et al / Systems & Control Letters 57 (2008) 670 679 673 and note that it follows from (6) and the definition of the Drazin inverse that (A di Q i )e = 0 and Q i Qi D A di = A di, i = 1,, Since A and Q i, i = 1,,, are diagonal and (A A di)e = 0 it follows that A Q i = 0 Hence, Me = 0, where A A Q i A d1 A d2 A d M A d1 Q 1 0 0 A dnd 0 0 Q nd (7) Now, it follows from Corollary 21 that M 0 and since Q i Qi D A di = A di, i = 1,,, it follows from Lemma 21 that M 0 if and only if (5) holds For the next result, recall that the equilibrium solution x t x e to (1) is semistable if and only if x e is Lyapunov stable and lim t x(t) exists heorem 31 Consider the linear time-delay dynamical system given by (1) where A and A di, i = 1,,, are given by (3) Assume that (A A di) e = (A A di)e = 0 and rank(a A di) = n 1 hen for every α 0, αe is a semistable equilibrium point of (1) Furthermore, x(t) α e as t, where α = e η(0) e A di η(θ)dθ n τ i e A di e (8) Proof It follows from Lemma 31 that there exist nonnegative matrices Q i, i = 1,,, such that (5) holds Now, consider the Lyapunov Krasovskii functional V : C R given by V (ψ( )) = ψ (0)ψ(0) ψ (θ)a di QD i A di ψ(θ)dθ, and note that the directional derivative of V (x t ) along the trajectories of (1) is given by V (x t ) = 2x (t)ẋ(t) x (t)a di QD i A di x(t) x (t τ i )A di QD i A di x(t τ i ) = 2x (t)ax(t) 2x (t) A di x(t τ i ) x (t)a di QD i A di x(t) x (t τ i )A di QD i A di x(t τ i ) x (t)q i x(t) 2x (t)a di x(t τ i ) (9) x (t τ i )A di QD i A di x(t τ i )] = Q i x(t) A di x(t τ i )] Q D i Q i x(t) A di x(t τ i )] 0, t 0 (10) Next, let R {ψ( ) C : Q i ψ(0) A di ψ( ) = 0, i = 1,, } and note that since the positive orbit γ (η(θ)) of (1) is bounded, γ (η(θ)) belongs to a compact subset of C, and hence, it follows from heorem 32 of 15] that x t M, where M denotes the largest invariant set contained in R Now, since A Q i = 0, it follows that R ˆR {ψ( ) C : Aψ(0) A diψ( ) = 0} Hence, since rank(a A di) = n 1 and (A nd A di)e = 0, it follows that the largest invariant set ˆM contained in ˆR is given by ˆM = {ψ C : ψ(θ) = αe, θ τ, 0], α 0} Furthermore, since ˆM R ˆR, it follows that M = ˆM Next, define the functional E : C R by E(ψ( )) = e ψ(0) e A di ψ(θ)dθ, (11) and note that Ė(x t ) 0 along the trajectories of (1) hus, for all t 0, E(x t ) = E(η( )) = e η(0) e A di η(θ)dθ, (12) which implies that x t M E, where E {ψ( ) C : E(ψ( )) = E(η( ))} Hence, since M E = {α e}, it follows that x(t) α e, where α is given by (8) Finally, Lyapunov stability of αe, α 0, follows by considering the Lyapunov Krasovskii functional V (ψ( )) = (ψ(0) αe) (ψ(0) αe) (ψ(θ) αe) A di QD i A di (ψ(θ) αe)dθ and noting that V (ψ) ψ(0) αe 2 2 Note that if = n 2 n, A d = A d, and (A A d)e = 0, then (1) can be rewritten as ẋ i (t) = a i j x i (t) x j (t τ i j )], x(θ) = η(θ), τ θ 0, t 0, (13) where i = 1,, n, and τ i j 0, τ], i j, i, j = 1,, n, which implies that the rate of material transfer from the ith compartment to the jth compartment is proportional to the difference x j (t τ i j ) x i (t) Hence, the rate of material transfer is positive (resp, negative) if x j (t τ i j ) > x i (t) (resp, x j (t τ i j ) < x i (t)) Eq (13) is an information flow balance equation that governs the information exchange among coupled subsystems and is completely analogous to the equations of

674 V Chellaboina et al / Systems & Control Letters 57 (2008) 670 679 thermal transfer with subsystem information playing the role of temperatures Furthermore, note that since a i j 0, i j, i, j = 1,, n, information energy flows from more energetic (information rich) subsystems to less energetic (information poor) subsystems, which is consistent with the second law of thermodynamics requiring that heat (energy) must flow in the direction of lower temperatures 4 Semistability and equipartition of nonlinear compartmental systems with time-delay In this section, we extend the results of Section 3 to nonlinear compartmental systems with time-delay Specifically, we consider nonlinear time-delay dynamical systems G of the form ẋ(t) = f (x(t)) f d (x(t τ 1 ),, x(t τ nd )), x(θ) = η(θ), τ θ 0, t 0, (14) where x(t) R n, t 0, f : R n R n is locally Lipschitz continuous and f (0) = 0, f d : R n R n R n is locally Lipschitz continuous and f d (0,, 0) = 0, τ = max i {1,,nd } τ i, τ i 0, i = 1,,, and η( ) C = C( τ, 0], R n ) is a continuous vector-valued function specifying the initial state of the system Note that since η( ) is continuous it follows from heorem 23 of 15, p 44] that there exists a unique solution x(η) defined on τ, ) that coincides with η on τ, 0] and satisfies (14) for all t 0 In addition, recall that if the positive orbit γ (η(θ)) of (14) is bounded, then γ (η(θ)) is precompact 25] he following definitions generalize the notions of essential nonnegativity and nonnegativity to vector fields Definition 41 (2]) Let f = f 1,, f n ] : D R n, where D is an open subset of R n that contains R n hen f is essentially nonnegative if f i (x) 0 for all i = 1,, n and x R n such that x i = 0, where x i denotes the ith element of x f is compartmental if f is essentially nonnegative and e f (x) 0, x R n Definition 42 (26]) Let f = f 1,, f n ] : D R n, where D is an open subset of R n that contains R n hen f is nonnegative if f i (x) 0 for all i = 1,, n and x R n Note that if f (x) = Ax, where A R n n, then f ( ) is essentially nonnegative if and only if A is essentially nonnegative, and f ( ) is nonnegative if and only if A is nonnegative Definition 43 (23]) he nonlinear time-delay dynamical system G given by (14) is nonnegative if for every η( ) C, where C {ψ( ) C : ψ(θ) 0, θ τ, 0]}, the solution x(t), t 0, to (14) is nonnegative Proposition 41 (23]) Consider the nonlinear time-delay dynamical system G given by (14) If f ( ) is essentially nonnegative and f d ( ) is nonnegative, then G is nonnegative For the remainder of this paper, we assume that f ( ) is essentially nonnegative and f d ( ) is nonnegative so that for every η( ) C, the nonlinear time-delay dynamical system G given by (14) is nonnegative Next, we consider a subclass of nonlinear nonnegative systems, namely, nonlinear compartmental systems Definition 44 he nonlinear time-delay dynamical system (14) is called a compartmental dynamical system if F( ) is compartmental, where F(x) f (x) f d (x, x,, x) Note that the nonlinear time-delay dynamical system is compartmental if f ( ) and f d = f d1,, f dn ] are given by f i (x(t)) = a ji (x(t)), f di (x(t τ 1 ),, x(t τ nd )) = a i j (x(t τ i j )), (15) where a ii (x( )) 0, x( ) C, a ii (0) = 0, i {1,, n}, denotes the instantaneous rate of flow of material loss of the ith compartment, a i j (x( )) 0, x( ) C, i j, i, j {1,, n}, denotes the instantaneous rate of material flow from the jth compartment to the ith compartment, τ i j, i j, i, j {1,, n}, denotes the transfer time of material flow from the jth compartment to the ith compartment, and a ii ( ) and a i j ( ) are such that if x i = 0, then a ii (x) = 0 and a ji (x) = 0 for all i, j = 1,, n, and x R n Note that the above constraints imply that f ( ) is essentially nonnegative and f d ( ) is nonnegative he next result generalizes heorem 31 to nonlinear time-delay compartmental systems of the form ẋ(t) = f (x(t)) f di (x(t τ i )), x(θ) = η(θ), τ θ 0, t 0, (16) where f : R n R n is given by f (x) = f 1 (x 1 ),, f n (x n )], f (0) = 0, f di : R n Rn, i = 1,,, and f d (0) = 0 Furthermore, we assume that f i ( ), i = 1,, n, are strictly decreasing functions heorem 41 Consider the nonlinear time-delay dynamical system given by (16) where f i ( ), i = 1,, n, is strictly decreasing and f i (0) = 0 Assume that e f (x) nd f di(x)] = 0, x R n, and f (x) f di(x) = 0 if and only if x = αe for some α 0 Furthermore, assume there exist nonnegative diagonal matrices P i R n n, i = 1,,, such that P P i > 0, Pi D P i f di (x) = f di (x), x R n, i = 1,,, (17) f di (x)p i f di (x) f (x)p f (x), x R n (18) hen, for every α 0, αe is a semistable equilibrium point of (16) Furthermore, x(t) α e as t, where α satisfies nα τ i e f di (α e) = e η(0) e f di (η(θ))dθ (19)

V Chellaboina et al / Systems & Control Letters 57 (2008) 670 679 675 Proof Consider the Lyapunov Krasovskii functional V : C R given by ψi (0) V (ψ( )) = 2 P (i,i) f i (ζ )dζ 0 f d i (ψ(θ))p i f di (ψ(θ))dθ (20) Since, f i ( ), i = 1,, n, is a strictly decreasing function it follows that V (ψ) 2 n P (i,i) f i (δ i ψ i (0))]ψ i (0) > 0 for all ψ(0) 0, where 0 < δ i < 1, and hence, there exists a class K function α( ) such that V (ψ) α( ψ(0) ) Now, note that the directional derivative of V (x t ) along the trajectories of (16) is given by V (x t ) = 2 f (x(t))pẋ(t) f di (x(t))p i f di (x(t)) f di (x(t τ i ))P i f di (x(t τ i )) = 2 f (x(t))p f (x(t)) 2 f (x(t))p f di (x(t τ i )) f di (x(t))p i f di (x(t)) f di (x(t τ i ))P i f di (x(t τ i )) f (x(t))p f (x(t)) 2 f (x(t))p Pi D P i f di (x(t τ i )) f d i (x(t τ i ))P i P D i P i f di (x(t τ i )) = P f (x(t)) P i f di (x(t τ i ))] P D i P f (x(t)) P i f di (x(t τ i ))] 0, t 0, (21) where the first inequality in (21) follows from (17) and (18), and the last equality in (21) follows from the fact that f (x)p f (x) = f (x)p Pi D P f (x), x R n Next, let R {ψ( ) C : P f (ψ(0)) P i f di (ψ( )) = 0, i = 1,, } and note that since the positive orbit γ (η(θ)) of (16) is bounded, γ (η(θ)) belongs to a compact subset of C, and hence, it follows from heorem 32 of 15] that x t M, where M denotes the largest invariant set (with respect to (16)) contained in R Now, since e ( f (x) nd f di(x)) = 0, x R n, it follows that R ˆR {ψ( ) C : f (ψ(0)) f di (ψ( )) = 0} = {ψ( ) C : ψ(θ) = αe, θ τ, 0], α 0}, which implies that x t ˆR as t Next, define the functional E : C R by E(ψ( )) = e ψ(0) e f di (ψ(θ))dθ, (22) and note that Ė(x t ) 0 along the trajectories of (16) hus, for all t 0, E(x t ) = E(η( )) = e η(0) e f di (η(θ))dθ, (23) which implies that x t ˆR E, where E {ψ( ) C : E(ψ( )) = E(η( ))} Hence, ˆR E = {α e}, it follows that x(t) α e, where α satisfies (19) Finally, Lyapunov stability of αe, α 0, follows by considering the Lyapunov Krasovskii functional ψi (0) V (ψ( )) = 2 P (i,i) ( f i (ζ ) f i (α))dζ α f di (αe)]dθ, f di (ψ(θ)) f di (αe)] P i f di (ψ(θ)) and noting that V (ψ) 2 n P (i,i) f i (α) f i (αδ i (ψ i (0) α))](ψ i (0) α) > 0, for all ψ i (0) α, where 0 < δ i < 1 heorem 41 establishes semistability and state equipartition for the special case of nonlinear compartmental systems of the form (15) where f ( ) and f di ( ), i = 1,, n, satisfy (17) and (18) For general n-dimensional nonlinear compartmental systems with time-delay and vector fields given by (16) it is not possible to guarantee semistability and state equipartition However, semistability without state equipartition may be shown For example, consider the nonlinear time-delay compartmental dynamical system given by ẋ 1 (t) = a 21 (x 1 (t)) a 12 (x 2 (t τ 12 )), x 1 (θ) = η 1 (θ), τ θ 0, t 0, (24) ẋ 2 (t) = a 12 (x 2 (t)) a 21 (x 1 (t τ 21 )), x 2 (θ) = η 2 (θ), τ θ 0, t 0, (25) where x 1 (t), x 2 (t) R, t 0, a 12 : R R and a 21 : R R satisfy a 12 (0) = a 21 (0) = 0 and a 12 ( ) and a 21 ( ) are strictly increasing, τ 12, τ 21 > 0, τ = max{τ 12, τ 21 }, and η 1 ( ), η 2 ( ) C = C( τ, 0], R ) Note that (24) and (25) can have multiple equilibria with all the equilibria lying on the curve a 21 (u) = a 12 (v), u, v 0 It follows from the conditions on a 12 ( ) and a 21 ( ) that all system equilibria lie on the curve y = a 1 12 (a 21(x)) in the (x, y) plane, where a 1 12 ( ) denotes the inverse function of a 12 ( ) Consider the functional E : C C R given by E(ψ 1, ψ 2 ) = ψ 1 (0) ψ 2 (0) a 12 (ψ 2 (θ))dθ τ 12 a 21 (ψ 1 (θ))dθ τ 21 (26)

676 V Chellaboina et al / Systems & Control Letters 57 (2008) 670 679 Now, it can be easily shown that the directional derivative of E(ψ 1, ψ 2 ) along the trajectories of (24) and (25) is identically zero for all t 0, which implies that, for all t 0, E(x 1t, x 2t ) = E(η 1, η 2 ) = η 1 (0) η 2 (0) τ 12 a 12 (η 2 (θ))dθ τ 21 a 21 (η 1 (θ))dθ (27) Next, consider the functional V : C C R given by V (ψ 1, ψ 2 ) = 2 ψ1 (0) 0 a12 2 (ψ 2(θ))dθ τ 12 a 21 (θ)dθ 2 ψ2 (0) 0 a 12 (θ)dθ τ 21 a 2 21 (ψ 1(θ))dθ, (28) and note that the directional derivative of V (ψ 1, ψ 2 ) along the trajectories of (24) and (25) is given by V (x 1t, x 2t ) = a 21 (x 1 (t)) a 12 (x 2 (t τ 12 ))] 2 a 12 (x 2 (t)) a 21 (x 1 (t τ 21 ))] 2 (29) Now, using similar arguments as in the proof of heorem 41 it follows that (x 1 (t), x 2 (t)) (α, a 1 12 (a 21(α ))) as t, where α is the solution to the equation α a 1 12 (a 21(α )) (τ 12 τ 21 )a 21 (α ) = η 1 (0) η 2 (0) τ 12 a 12 (η 2 (θ))dθ τ 21 a 21 (η 1 (θ))dθ, (30) and (α, a 1 12 (a 21(α ))) is a Lyapunov stable equilibrium state he above analysis shows that all two-dimensional nonlinear compartmental dynamical systems of the form (24) and (25) are semistable with system states reaching equilibria lying on the curve y = a 1 12 (a 21(x)) in the (x, y) plane o demonstrate the utility of heorem 41 we consider a nonlinear two-compartment time-delay dynamical system given by ẋ 1 (t) = a i (x 1 (t)) a i (x 2 (t τ i ))], x 1 (θ) = η 1 (θ), τ θ 0, t 0, (31) ẋ 2 (t) = a i (x 1 (t τ i )) a i (x 2 (t))], x 2 (θ) = η 2 (θ), τ θ 0, (32) where a i : R R, i = 1,,, are such that for every i = 1,,, a i (x 1 ) a i (x 2 )](x 1 x 2 ) > 0, x 1 x 2, (33) and a i (0) = 0, i = 1,, If x 1 and x 2 represent system energies, then (31) and (32) capture energy flow balance between the two compartments, and (33) is consistent with the second law of thermodynamics; that is, energy flows from the more energetic compartment to the less energetic compartment 14] Furthermore, since a i (0) = 0, (33) implies that a i ( ), i = 1,,, is strictly increasing Now, note that (31) and (32) can be written in the form of (16) with a i (x 1 ) f (x) = n d, a i (x 2 ) ] ai (x f di (x) = 2 ), i = 1,, n a i (x 1 ) d, (34) which implies that f j (x j ), j = 1, 2, are strictly decreasing Next, with P i = I n, i = 1,,, (17) and (18) are trivially satisfied, and hence, it follows from heorem 41 that x 1 (t) x 2 (t) 0 as t Next, we consider nonlinear compartmental time-delay dynamical systems of the form ẋ i (t) = a ji (x i (t)) a i j (x j (t τ i )), x(θ) = η(θ), τ θ 0, t 0, (35) where i = 1,, n, a i j : R R, i j, i, j {1,, n}, are such that a i j (0) = 0 and a i j ( ), i j, i, j = 1,, n, is strictly increasing Note that since each transfer coefficient a i j ( ) is only a function of x j and not x, the nonlinear compartmental system (35) is a nonlinear donor controlled compartmental system 27] In this case, (35) can be written in the form given by (16) with = n, f i (x i ) = f di (x) = e i a ji (x i ), a i j (x j ), i = 1,, n (36) j=1 Next, with P i = e i ei, i = 1,, n, so that P = I n, it follows that (17) is trivially satisfied and (18) holds if and only if 2 ] 2 a i j (x j )] a ji (x i ), x R n (37) In the case where n = 2, (37) is trivially satisfied, and hence, it follows from heorem 41 that x 1 (t) x 2 (t) 0 as t In general, (37) does not hold for arbitrary strictly increasing functions a i j ( ) However, if a i j ( ) = σ ( ), i j, i, j = 1,, n, where σ : R R is such that σ (0) = 0 and strictly increasing, (37) holds if and only if σ (x j )] 2 ] 2 σ (x i ), x R n (38)

V Chellaboina et al / Systems & Control Letters 57 (2008) 670 679 677 In this case, since 0 (n 1) σ 2 (x i ) (n 2) (n 1) 2 = (n 2) n σ 2 (x i ) (σ (x i ) σ (x j )) 2, σ (x i )σ (x j ) (38) holds, and hence, it follows from heorem 41 that x i (t) x j (t) 0 as t, where i j, i, j = 1,, n Next, we specialize heorem 41 to nonlinear time-delay compartmental systems of the form ẋ(t) = A ˆσ (x(t)) A di ˆσ (x(t τ i )), x(θ) = η(θ), τ θ 0, t 0, (39) where ˆσ : R n R n is given by ˆσ (x) = σ (x 1 ), σ (x 2 ),, σ (x n )], where σ : R R is such that σ (u) = 0 if and only if u = 0, and A and A di, i = 1,,, are as given by (3) heorem 42 Consider the nonlinear time-delay system given by (39) where σ : R R is such that σ (0) = 0 and σ ( ) is strictly increasing Assume that (A A di) e = (A A di)e = 0 and rank(a A di) = n 1 hen for every α 0, αe is a semistable equilibrium point of (39) Furthermore, x(t) α e as t, where α satisfies nα σ (α ) τ i e A di e = e η(0) e A di ˆσ (η(θ))dθ (40) Proof It follows from Lemma 31 that there exists Q i, i = 1,,, such that (5) holds with Q i given by (6) Now, since A = Q i = PD i = P 1, where P = P i, it follows from (5) that, for all x R n, 0 2 ˆσ (x)a ˆσ (x) ˆσ (x) (Q i A di Qi D A di ) ˆσ (x) = f (x)p f (x) f di (x)p i f di (x), where f (x) = A ˆσ (x) and f di (x) = A di ˆσ (x), i = 1,,, x R n Furthermore, since PD i P i A di = A di, i = 1,,, it follows that Pi D P i f di (x) = f di (x), i = 1,,, x R n Now, the result is an immediate consequence of heorem 41 by noting that e f (x) f di(x)] = 0 and f (x) nd f di(x) = 0 if and only if x = αe for some α 0 5 he consensus problem iynamical networks In this section, we apply the results of Sections 4 and 5 to the consensus problem iynamical networks 17,18,21,28,29] he consensus problem appears frequently in coordination of multiagent systems and involves finding a dynamic algorithm that enables a group of agents in a network to agree upon certain quantities of interest with directed information flow subject to possible link failures and time-delays As in 17], we use directed graphs to represent a dynamical network and present solutions to the consensus problem for networks with balanced graph topologies (or information flow) 17] and unknown arbitrary time-delays Specifically, let G = (V, E, A) be a weighted directed graph (or digraph) denoting the dynamical network (or dynamic graph) with the set of nodes (or vertices) V = {1,, n} denoting the agents, the set of edges E V V denoting the direction of information flow, and a weighted adjacency matrix A R n n such that A (i, j) = a i j > 0, i, j = 1,, n, if ( j, i) E, and a i j = 0 otherwise he in-degree and out-degree of node i are, respectively, defined as deg in (i) n j=1 a ji and deg out (i) n j=1 a i j We say that the node i of a digraph G is balanced if and only if deg in (i) = deg out (i), and a graph G is called balanced if and only if all of its nodes are balanced, that is, n j=1 a i j = n j=1 a ji, i = 1,, n Furthermore, we denote the value of the node i {1,, n} at time t by x i (t) R he consensus problem involves the design of a dynamic algorithm that guarantees system state equipartition, that is, lim t x i (t) = α R for i = 1,, n he consensus problem is a dynamic graph involving the trajectories of the dynamical network characterized by the dynamical system ẋ(t) = u(t), x(0) = x 0, t 0, (41) where x(t) x 1 (t),, x n (t)] is the state of the network and u(t) u 1 (t),, u m (t)] is the input to the network with components u i (t) only depending on the states of the nodes i and its neighbors Specifically, the consensus problem deals with the design of an input u(t) such that x(t) converges to αe as t, where α R Due to the presence of directional constraints on information flow and system timedelays, u i (t) is constrained to the feedback form u i (t) = f i (x i (t), x j 1 (t τ i j 1 ),, x j mi (t τ i j mi )), where τ i j k > 0, j k N i { j {1,, n} : ( j, i) E}, are unknown constant time-delays between nodes i and j k For notational convenience we additionally define the parameters τ i j 0 if ( j, i) E As an example, consider the dynamical network given in Fig 51 where V = {1, 2, 3}, E = {(1, 2), (2, 3), (1, 3), (3, 1)}, with adjacency matrix A such that a 13, a 21, a 31, and a 32 > 0, and with the remaining elements being zeros In this case, the input to the network is given by u 1 (t) = f 1 (x 1 (t), x 3 (t τ 13 )), u 2 (t) = f 2 (x 2 (t), x 1 (t τ 21 )), u 3 (t) = f 3 (x 3 (t), x 2 (t τ 32 ), x 1 (t τ 31 )), so that for i = 1, 2, 3, ẋ i (t) is only dependent on the states (values) of the nodes that are accessible by node i and with τ i j denoting the communicatioelay from node j to node i Next, we apply heorems 31 and 42 to present linear and nonlinear solutions for the consensus problem Specifically,

678 V Chellaboina et al / Systems & Control Letters 57 (2008) 670 679 Fig 51 Dynamic network first we choose f i (x(t)) = a ji x i (t) so that the closed-loop system is given by ẋ i (t) = a i j x j (t τ i j ), i = 1,, n, (42) a ji x i (t) for all i = 1,, n, or, equivalently, ẋ(t) = Ax(t) A dl x(t τ l ), l=1 a i j x j (t τ i j ), x i (θ) = η i (θ), τ θ 0, t 0, (43) x(θ) = η(θ), τ θ 0, t 0, (44) where n 2, A R n n, and A dl R n n, l = 1,,, with ] n 1 A = diag a j1,, a jn, (45) j=2 j=1 A d((i 1)n j) = a i j e i e j, and τ ((i 1)n j) = τ i j, i, j = 1,, n Note that if ( j, i) E, then A d((i 1)n j) = 0, which implies that the algorithm is consistent with the directional constraints Furthermore, it can be easily shown that (A A d ) e = 0, where A d l=1 A dl, and rank(a A d ) = n 1 if and only if for every pair of nodes (i, j) V there exists a path from node i to node j 30] Here, we assume that the adjacency matrix A is chosen such that (A A d )e = 0 so that the linear time-delay closed-loop dynamical system (44) satisfies all the conditions of heorem 31 Hence, it follows from heorem 31 that the dynamical network given by (44) solves the consensus problem, that is, lim t x i (t) = lim t x j (t) = α, i, j = 1,, n, Fig 53 Linear consensus algorithm i j, where α is given by (8) Alternatively, it follows from heorem 42 that the nonlinear dynamical network given by ẋ(t) = A ˆσ (x(t)) A di ˆσ (x(t τ i )), x(θ) = η(θ), τ θ 0, t 0, (46) also solves the nonlinear consensus problem where σ ( ) and ˆσ ( ) satisfy the conditions in heorem 42 In this case, lim t x i (t) = lim t x j (t) = α, i, j = 1,, n, i j, where α is a solution to (40) Note that if σ (θ) = θ, (46) specializes to (44) Although both (44) and (46) solve the same network consensus problem, the nonlinear function σ ( ) within ˆσ ( ) may be used to enhance the performance of the dynamic algorithm or satisfy other constraints For example, choosing σ (θ) = tanh(θ) we can constrain bandwidth information from one agent to another o illustrate the two algorithms given by (44) and (46) consider the dynamical network given by the graph shown in Fig 52 17] where a i j and τ i j denote the weight and the time-delay for each edge shown Here, we choose a (i, j) = 1 if (i, j) E so that (A A d )e = 0 In addition, it can be easily shown that rank(a A d ) = n 1 = 9 With x 0 = 1 2 3 4 5 6 7 8 9 10], Figs 53 and 54 demonstrate the agreement between all nodes for the algorithms given by (44) and (46), respectively, with σ (θ) = tanh(θ) in (46) Finally, Figs 55 and 56 show the control input versus time for both linear and nonlinear consensus algorithms Note that the maximum amplitude of the linear consensus algorithm is about six times that of the nonlinear consensus algorithm and, as Fig 52 Balanced dynamic network

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