A Nonlinear Perron-Frobenius Approach for Stability and Consensus of Discrete-Time Multi-Agent Systems

A Nonlinear P erron-F rob enius Approac h for Stabilit y and Consensus of Discrete-Time Multi-Agen t Systems ? Diego Deplano a , Mauro F rancesc helli a , Alessandro Giua a a Dep artment of Ele ctric al and Ele ctr onic Engine ering, University of Cagliari, Italy Abstract In this paper w e prop ose a nov el metho d to establish stability and, in addition, conv ergence to a consensus state for a class of discrete-time Multi-Agen t System (MAS) ev olving according to nonlinear heterogeneous local interaction rules whic h is not based on Lyapuno v function arguments. In particular, w e fo cus on a class of discrete-time MASs whose global dynamics can b e represen ted b y sub-homogeneous and order-preserving nonlinear maps. This pap er directly generalizes results for sub-homogeneous and order-preserving linear maps which are shown to b e the counterpart to sto chastic matrices thanks to nonlinear Perron-F rob enius theory . W e provide sufficient conditions on the structure of lo cal interaction rules among agents to establish conv ergence to a fixed p oint and study the consensus problem in this generalized framework as a particular case. Examples to show the effectiveness of the metho d are provided to corrob orate the theoretical analysis. Key wor ds: Multi-agent systems; Consensus; Nonlinear Perron-F rob enius theory; Order-preserving maps; Stability analysis. ? This w ork was supp orted in part b y the Italian Ministry of Research and Education (MIUR) with the gran t “CoNet- DomeSys”, code RBSI14OF6H, under call SIR 2014 and b y Region Sardinia (RAS) with pro ject MOSIMA, RASSR05871, FSC 2014-2020, Annualita’ 2017, Area T ematica 3, Linea d’Azione 3.1. Email addr esses: diego.deplano@diee.unica.it (Diego Deplano), mauro.franceschelli@diee.unica.it (Mauro F ranceschelli), giua@unica.it (Alessandro Giua). Preprin t submitted to Automatica 25 July 2019 1 In tro duction The study of complex systems where lo cal interactions b etw een individuals give rise to a global collective b ehavior has aroused muc h interest in the con trol communit y . Such complex systems are often called Multi-Agent Systems (MAS), consisting of multiple interacting agents with m utual interactions among them. A topic that captured the atten tion of many researchers is the consensus problem [19], where the ob jective is to design local interaction rules among agents such that their state v ariables conv erge to the same v alue, the so called agreement or consensus state. A MAS can b e modeled as a dynamical system. In the discrete time linear case, classical P erron-F rob enius Theory is crucial in the conv ergence analysis. Indeed, in one of the most p opular works in this topic ([10]), the authors established criteria for con vergence to a consensus state for MAS whose global dynamics can be represen ted by linear time-v arying systems with non-negativ e ro w-sto c hastic state transition matrices, whic h are ob ject of study of the classical Perron-F robenius Theory . The notable asp ect of this work was to exploit suc h theory and graph theory instead of Lyapuno v theory , allowing to study systems for which finding a common Lyapuno v function to establish conv ergence is difficult or even imp ossible. Particularly , as it later b ecame clear by the work in [20], it allo w ed studying switched linear systems for whic h there do es not exist a common quadratic Ly apunov function. Along this line of though t, in this pap er w e aim to exploit nonlinear Perron-F rob enius theory [14], a generalization of non-negativ e matrix theory , to address nonlinear in teractions in MASs without Ly apunov based argumen ts. It follo ws that a MAS mo deled by a non-negativ e row-stochastic matrix is a particular case of the prop osed general theory . The literature on nonlinear consensus problems is v ast. It is mostly comp osed by particular nonlinear consensus proto cols which offer adv antages such as finite-time conv ergence [7,6], resilience to non-uniform time-delays [25] and man y more. These proto cols are usually pro ved to conv erge to the consensus state via ad hoc Ly apunov functions. Most of approaches whic h aim to establish conv ergence to consensus for some class of nonlinear MAS falls in the general conv exity theory of [17], i.e., each agent’s next state is strictly inside the conv ex h ull spanned by the state v alue of its neigh b ors. W e mention the w ork in [27], which is the con tinuous-time coun terpart to the result of Moreau in [17], where the authors iden tify a class of non-linear interactions denoted as sub-tangent and establish necessary and sufficient conditions on the netw ork top ology for conv ergence to consensus. Our approach sharply differs from the previous literature. W e identify a class of functions (whic h comprises also non-negativ e ro w-sto chastic matrices) whic h w e pro ve to ha v e a special con vergence properties in the positive orthan t R n ≥ 0 . In particular, we take inspiration from nonlinear P erron-F rob enius theory and considered order-preserving and sub-homogeneous nonlinear maps. F urthermore, the approach presented in this pap er differs significantly from the preliminary results presented in [5] in b oth the statement of the theorems, lemmas, and their pro of. The main contribution of this paper is threefold. First, w e pro vide sufficient conditions for stabilit y of a class of nonlinear discrete-time systems represented by positive, sub-homogeneous, type-K order preserving maps. Second, w e prop ose a sufficien t condition on the structure of heterogeneous lo cal interaction rules among agents whic h guaran tees that the global model of the MAS falls in to the considered class of nonlinear discrete-time systems. Third, w e prop ose a sufficient condition which links the topology of the netw ork and the structure of the local interaction rules to guarantee the ac hiev emen t of a consensus state, i.e., the netw ork state in which all state v ariables hav e the same v alue. Our results are a generalization to nonlinear discrete-time system of non-negative matrix theory applied to multi-agen t systems, in so doing our results do not exploit Lyapuno v function argumen ts. This pap er is organized as follows. In Section 2 we present our notation and background material on multi-agen t systems, order-preserving and sub-homogeno eus maps and recall the concept of p erio dic fixed-points. In Section 3 w e state our main results which consists in the statemen t of three main theorems. In Section 4 we discuss the pro of of our results, we first discuss and list the required tec hnical lemmas and then present the pro of of each theorem in separate subsections. In Section 5 w e present examples of application of our theoretical results. Finally , in Section 6 w e give our concluding remarks. 2 Bac kground In this work we prop ose no v el to ols to p erform stability analysis (consensus as a sp ecial case) of MASs whose state up date is represented b y p ositive , or der-pr eserving and sub-homo gene ous maps. In this section w e define a mo del of autonomous nonlinear MASs in discrete-time, its asso ciated graph and present the abov e mentioned properties whic h define the class of MAS under study . 2 2.1 Multi-agent systems W e consider a MAS comp osed by a set of agen ts V = { 1 , . . . , n } , which are mo deled as autonomous discrete-time dynamical systems with scalar state in R ≥ 0 = { x ∈ R : x ≥ 0 } . Agents are in terconnected and up date their state as follo ws x 1 ( k + 1) = f 1 ( x 1 ( k ) , . . . , x n ( k )) . . . x n ( k + 1) = f n ( x 1 ( k ) , . . . , x n ( k )) , (1) where k ∈ { 0 , 1 , 2 , . . . } is a discrete-time index. In tro ducing the aggregate state x = [ x 1 , . . . , x n ] T ∈ R n , system (1) can b e written as x ( k + 1) = f ( x ( k )) (2) where f : R n ≥ 0 → R n ≥ 0 is differentiable. Hence, in this w ork we consider p ositive systems [22]. Positivit y is a term with different meanings in different contexts, here by p ositive system w e denote a system (and the asso ciated map) with state that evolv es in R n ≥ 0 . Definition 1 (Positiv e systems and maps) System (2) is c al le d p ositive if f maps non-ne gative ve ctors into non-ne gative ve ctors, i.e., f : R n ≥ 0 → R n ≥ 0 . Corr esp ondingly, map f is said to b e p ositive.  W e now asso ciate to the map f a graph G ( f ) whic h captures the pattern of interactions among agen ts and denote it as infer enc e gr aph [16]. Let G = ( V , E ) b e a graph where V = { 1 , . . . , n } is the set of no des representing the agents and E ⊆ V × V is a set of directed edges. A directed edge ( i, j ) ∈ E exists if no de i sends information to no de j . T o eac h agen t i is asso ciated a set of no des called neighbors of agen t i defined as N i = { j ∈ V : ( j, i ) ∈ E } . A dir e cte d p ath b et w een t wo no des p and q in a graph is a finite sequence of m edges e k = ( i k , j k ) ∈ E that joins no de p to no de q , i.e., i 1 = p , j m = q and j k = i k +1 for k = 1 , . . . , m − 1. A no de j is said to b e r e achable from no de i if there exists a directed path from no de i to no de j . A no de is said to b e glob al ly r e achable if it is reachable from all no des i ∈ V . Definition 2 (Inference graph) Given a map f its infer enc e gr aph G ( f ) = ( V , E ) is define d by a set of no des V and a set of dir e cte d e dges E ⊆ {V × V } . A n e dge ( i, j ) ∈ E fr om no de i to no de j exists if ∂ f i ( x ) ∂ x j 6 = 0 x ∈ R n ≥ 0 \ S, wher e S is a set of me asur e zer o in R n .  2.2 Or der-pr eserving maps The set of R n ≥ 0 is a partially ordered set with resp ect to the natural order relation ≤ . F or u, v ∈ R n ≥ 0 , we can write the partial ordering relations as follows u ≤ v ⇔ u i ≤ v i ∀ i ∈ V , u  v ⇔ u ≤ v and u 6 = v , u < v ⇔ u i < v i ∀ i ∈ V . The partial ordering ≤ yields an equiv alence relation ∼ on R n ≥ 0 , i.e., x is equiv alent to y ( x ∼ y ) if there exist α, β ≥ 0 such that x ≤ αy and y ≤ β x . The equiv alence classes are called p arts of the cone of non-negative real v ectors and the set of all parts is denoted b y P . It can be shown (see [1]) that the cone R n ≥ 0 has exactly 2 n parts, whic h are given by P I = { x ∈ R n ≥ 0 | x i > 0 , ∀ i ∈ I and x i = 0 otherwise } , with I ⊆ { 1 , . . . , n } . W e define a partial ordering on the set of parts P given by P I 1  P I 2 if I 1 ⊆ I 2 . Maps which preserve such a vector order are said to b e or der-pr eserving . Next, we provide a formal definition of three kinds of order-preserving maps present in the current literature. 3 Definition 3 (Order-preserv ation) A p ositive map f is said to b e • Or der-pr eserving, if ∀ x, y ∈ R n ≥ 0 it holds x ≤ y ⇔ f ( x ) ≤ f ( y ) . • Strictly or der-pr eserving, if if ∀ x, y ∈ R n ≥ 0 it holds x  y ⇔ f ( x )  f ( y ) . • Str ongly or der-pr eserving, if ∀ x, y ∈ R n ≥ 0 it holds x  y ⇔ f ( x ) < f ( y ) .  The next remark is in order to clarify the context of the contribution of this pap er. Remark 4 F or line ar maps, or der-pr eservation and p ositivity ar e e quivalent pr op erties and c orr esp ond to non- ne gative matric es. Sinc e this is not the c ase for gener al nonline ar maps, in this work we c onsider p ositive nonline ar maps which ar e also or der-pr eserving.  No w, w e are ready to in tro duce the definition of typ e-K or der-pr eserving maps, sho wn next, whic h pla ys a piv otal role in the characterization of the class of nonlinear systems in whic h we are interested and which will b e discussed at length in the pro ofs of our results. Definition 5 (Type-K Order-preserv ation) A p ositive map f is said to b e typ e-K or der-pr eserving if for any x, y ∈ R n ≥ 0 and x  y it holds ( i ) x i = y i ⇒ f i ( x ) ≤ f i ( y ) , ( ii ) x i < y i ⇒ f i ( x ) < f i ( y ) , for al l i = 1 , . . . , n , wher e f i is the i -th c omp onent of f .  As it will be shown later, such a prop erty is sufficient but not necessary for classical order-preserv ation. Ho w ev er, since it is easily identifiable from the sign structure of the Jacobian matrix, it allows to easily establish order- preserv ation of a given function. F urthermore, it constrain ts the b ehavior of the system, prev en ting the system from ev olving with p erio dic tra jectories and thus helping in pro ving conv ergence to a steady state. 2.3 Sub-homo gene ous maps Order-preserving dynamical systems and nonlinear Perron-F robenius theory are closely related. In the theory of order-preserving dynamical systems, the emphasis is placed on strong order-preserv ation. F or discrete-time strongly order-preserving dynamical systems one has generic conv ergence to perio dic tra jectories under appropriate conditions [21]. An extensiv e o verview of these results w as given by Hirsc h and Smith [9]. On the other hand, in nonlinear P erron- F rob enius theory one usually considers discrete-time dynamical systems that need not b e strongly order-preserving, but satisfy an additional concav e assumption and obtain similar results regarding perio dic tra jectories [13]. The conca v e assumption of interest in this pap er is sub-homogeneity . Definition 6 (Sub-homogeneity) A p ositive map is said to b e sub-homo gene ous if αf ( x ) ≤ f ( α x ) for al l x ∈ R n ≥ 0 and α ∈ [0 , 1] .  Order-preserving and sub-homogeneous maps arise in a v ariety of applications, including optimal control and game theory [2], mathematical biology [23], analysis of discrete ev ent systems [8] and so on. 4 2.4 Perio dic p oints Concluding this section, we recall some basic concepts on p erio dic p oints which are instrumental to state our main results. Consider the state tra jectory of the system in eq. (2). A p oint x ∈ R n is called a p erio dic p oint of map f if there exist an integer p ≥ 1 suc h that f p ( x ) = x . The minimal such p ≥ 1 is called the p erio d of x under f . If f ( x ) = x , we call x a fixed point of f . A fixe d p oint is a p erio dic point with p erio d p = 1. Fixed points of a map are equilibrium p oints for a dynamical system. W e denote F f = { x ∈ X : f ( x ) = x } the set of all fixed p oints of map f . The tr aje ctory of the system in eq. (2) with initial state x is given by T ( x, f ) = { f k ( x ) : k ∈ Z } . If f is clear from the con text, w e simply write T ( x ) to denote its tra jectory , where x is the initial state. If x is a p erio dic point, we sa y that T ( x ) is a p erio dic tra jectory . W e denote the limit set of a p oint x of map f as ω ( x, f ) (or simply ω ( x ) if f is clear from the context), which is defined as ω ( x ) = \ k ≥ 0 cl ( { f m ( x ) : m ≥ k } ) , with cl ( · ) denoting the closure of a set, i.e., the set together with all of its limit p oints. If x is a fixed p oint it follows that the set ω ( x ) is a singleton, i.e., a set con taining a single p oint. 3 Main results In this section we state and clarify the main results of this pap er, while the following sections are dedicated to their pro of. F or p ositive maps whic h are also order-preserving and sub-homogeneous, existing results (see next section for insights) do not provide an y condition to ensure conv ergence to a fixed p oint, but only to p erio dic p oin ts [13] when the initial state is strictly p ositive, i.e., x ∈ R n + . F urthermore, to the b est of our knowledge, no result provides any information ab out tra jectories whose initial state lies in the b oundary of R n ≥ 0 . Our aim is thus to fill this v oid considering the previously defined class of order-preserving maps, called typ e-K or der-pr eserving , for which we pro ve conv ergence to a fixed p oint for any initial state x ∈ R n ≥ 0 (and not only for x ∈ R n + ). This result is given in next theorem. Theorem 7 (Conv ergence) L et a p ositive map f b e sub-homo gene ous and typ e-K or der-pr eserving. If f has at le ast one p ositive fixe d p oint in R n + then al l p erio dic p oints ar e fixe d p oints, i.e., the set ω ( x ) is a singleton and ∀ x ∈ R n ≥ 0 : lim k →∞ f k ( x ) = ¯ x wher e ¯ x is a fixe d p oint of f .  Using this technical result, another of our contributions is a sufficient condition on the heterogeneous lo cal in teraction rules under whic h a MAS is stable, i.e., its state con v erges to a fixed p oint. Fixed p oints are synonymous for equilibrium p oints, while in the literature the term fixed p oints is widely used in the con text of iterated maps, the term equilibrium p oint is usually preferred in the context of discrete-time dynamical systems. This result is given in Theorem 8, whose statement is shown next. Theorem 8 (Stability) Consider a MAS as in (2) with at le ast one p ositive e quilibrium p oint. If the set of differ entiable lo c al inter action rules f i , with i = 1 , . . . , n , satisfies the next c onditions: ( i ) f i ( x ) ∈ R ≥ 0 for al l x ∈ R n ≥ 0 ; ( ii ) ∂ f i /∂ x i > 0 and ∂ f i /∂ x j ≥ 0 for i 6 = j ; ( iii ) αf i ( x ) ≤ f i ( αx ) for al l α ∈ [0 , 1] and x ∈ K ; then the MAS c onver ges to one of its e quilibrium p oints for any p ositive initial state x (0) ∈ R n ≥ 0 .  5 As a sp ecial case, we also study the consensus problem for the considered class of MAS. W e prop ose a sufficient condition based on the result in Theorem 8 so that, for any initial state in R n + , the MAS asymptotically reaches the consensus state, i.e., all state v ariable conv erge to same v alue. The prop osed sufficient condition is graph theoretical and based on the inference graph G ( f ). The condition is satisfied if there exists a globally reac hable no de in graph G ( f ) and the consensus state is a fixed p oint for the considered MAS. This result is giv en in the next theorem. Theorem 9 (Consensus) Consider a MAS as in (2) . If the set of differ entiable lo c al inter action rules f i , with i = 1 , . . . , n , satisfies the next c onditions: ( i ) f i ( x ) ∈ R ≥ 0 for al l x ∈ R n ≥ 0 ; ( ii ) ∂ f i /∂ x i > 0 and ∂ f i /∂ x j ≥ 0 for i 6 = j ; ( iii ) αf i ( x ) ≤ f i ( αx ) for al l α ∈ [0 , 1] and x ∈ R n ≥ 0 ; ( iv ) f i ( x ) = x i if x i = x j for al l j ∈ N in i ; ( v ) Infer enc e gr aph G ( f ) has a glob al ly r e achable no de; then, the MAS c onver ges asymptotic al ly to a c onsensus state for any initial state x (0) ∈ R n ≥ 0 .  In the remainder of the pap er, we discuss the pro of of our main results in Theorem 7, 8 and 9. 4 Pro of of main results W e b egin by clarifying the relationships among the differen t kinds of order-preserv ation. Remark 10 Str ong or der-pr eservation ⇒ T yp e-K or der-pr eservation ⇒ strict or der-pr eservation ⇒ or der-pr eservation.  Ev ery conv erse relationship in Remark 10 do es not hold. Given x, y ∈ R , let f : R 2 → R 2 , we ha ve the following coun ter-examples: • f ( x, y ) = [1 , 1] T is order-preserving but not strictly; • f ( x, y ) = [ y , x ] T is strictly order-preserving but not type-K; • f ( x, y ) = [ √ x + y , y ] T is type-K order-preserving but not strongly . Usually , to verify order-preserv ation is not an easy task. F or differentiable con tin uous-time systems ˙ x = f ( x ) a sufficien t condition to ensure order-preserv ation is given by Kamk e [24,12]. The Kamke condition usually exploited in the analysis of contin uous time systems is shown next. Lemma 11 (Kamke Condition) [24,12] The map f of a c ontinuous-time system ˙ x = f ( x ) is or der-pr eserving if its Jac obian matrix is Metzler, i.e., ∂ f i /∂ x j ≥ 0 for i 6 = j .  As a counterpart to Lemma 11, for discrete-time systems we prop ose a sufficient condition to ensure type-K order- preserv ation of a map f , instrumen tal to the analysis of discrete-time systems, whic h w e denote Kamke-like condition. Prop osition 12 (Kamke-lik e condition) The map f of a discr ete-time system x ( k + 1) = f ( x ( k )) is typ e-K or der-pr eserving if its Jac obian matrix is Metzler with strictly p ositive diagonal elements, i.e., if ∂ f i /∂ x i > 0 and ∂ f i /∂ x j ≥ 0 for i 6 = j . (3) 6 Pro of. Let x ∈ R n and, without lack of generality , y = x + εe j where ε > 0 and e j denotes a canonical vector with all zero v alues but the j -th which is 1, thus x  y . If (3) holds, then (1) If i 6 = j then y i = x i + ε 0 = x i and ∂ f i ( x ) ∂ x j = lim ε → 0 f i ( x + εe j ) − f i ( x ) ε ≥ 0 , whic h implies that f i ( x ) ≤ f i ( x + εe j ) = f i ( y ), i.e., condition ( i ) of Definition 5. (2) If i = j then y i = x i + ε 1 > x i and ∂ f i ( x ) ∂ x i = lim ε → 0 f i ( x + εe i ) − f i ( x ) ε > 0 , whic h implies that f i ( x ) < f i ( x + εe i ) = f i ( y ), i.e., condition ( ii ) of Definition 5. Since 1) ⇒ (3) and 2) ⇒ (3), the pro of is complete.  Ha ving clarified how to verify the type-K order-preserving prop erty for a discrete-time system, w e mo ve on in the next subsection to discuss a significan t property of order-preserving maps whic h are also sub-homogeneous, i.e., non-expansivness with resp ect to the so-called Thompson’s metric. 4.1 Non-exp ansive maps Dynamical systems defined by order-preserving and sub-homogeneous maps are non-expansive under the Thompson’s metric. Here, we introduce the concepts of non-exp ansiveness and Thompson ’s metric and give a few useful lemmas. Definition 13 (Thompson’s metric [26]) F or x, y ∈ R n ≥ 0 define M ( x/y ) = inf { α ≥ 0 : y ≤ αx } , with M ( x/y ) = ∞ if the set is empty. By me an of function M ( y /x ) , Thompson ’s metric d T : R n × R n → [0 , ∞ ] is define d for al l ( x, y ) ∈ ( R n × R n ) \ (0 , 0) as fol lows d T ( x, y ) = log(max { M ( x/y ) , M ( y /x ) } ) with d T (0 , 0) = 0 .  Definition 14 (Non-expansiveness) A p ositive map f is c al le d non-exp ansive with r esp e ct to a metric d : R n × R n → R ≥ 0 , if d ( f ( x ) , f ( y )) ≤ d ( x, y ) for al l x, y ∈ R n ≥ 0 .  The next result taken from [1] but stated in reference to p ositive cones K . In this pap er we alw ays consider as a particular case the cone of non-negative vectors, i.e., K = R n ≥ 0 , which is a solid, closed and con v ex cone. Lemma 15 [1] L et K b e a solid close d c onvex c one 1 in R n . If f : K → K is an or der-pr eserving map, then it is sub-homo gene ous if and only if it is non-exp ansive with r esp e ct to Thompson ’s metric d T .  1 A set K ∈ R n is called a c onvex c one if αK ⊆ K for all α ≥ 0 and K ∩ ( − K ) = { 0 } . The conv ex cone K is close d if it is a closed set in R n and it is solid if it has a non-empty interior. 7 Suc h a prop erty allows one to prov e detailed results concerning the b ehavior of dynamical systems, see [1,15,18] and also Chapter 8 in [14] and reference therein. Thompson’s metric d T and the sup-norm k·k ∞ defined by k x k ∞ = max i k x i k , are closely related thanks to the following lemma. Lemma 16 [26] The c o or dinate-wise lo garithmic function L : R n + → R n is an isometry fr om ( R n + , d T ) to ( R n , k·k ∞ ) , with R + = { x ∈ R : x > 0 } .  By Lemma 16, if a positive map f can be restricted to R n + = int ( R n ≥ 0 ) and if it is order preserving and sub- homogenous, then g : R n → R n giv en by g = l og ◦ f ◦ exp , is a sup-norm non-expansive map that has the same dynamical prop erties as f . The dynamics of sup-norm non-expansive maps is widely known. In fact, there exists the follo wing result, which is the simplified v ersion of Theorem 4.2.1 in [14]. Lemma 17 [26] If f : R n → R n is a sup-norm non-exp ansive then only one of the next c ases c an o c cur: ( i ) ∀ x ∈ R n tr aje ctories T ( x ) ar e unb ounde d; ( ii ) ∀ x ∈ R n tr aje ctories T ( x ) ar e b ounde d.  4.2 Pr o of of The or em 7 As p ointed out in the previous section, for p ositive maps f which are also order-preserving and sub-homogeneous it is p ossible to establish the b oundedness of any tra jectory with initial states x (0) ∈ R n + and entirely enclosed in R n + . On the contrary , there are still no results for tra jectories with initial state x (0) in the b oundary of R n ≥ 0 . When f satisfies the additional prop erty of type-K order preserv ation, Theorem 7 giv es a sufficient condition for the b oundedness of any tra jectory with initial state x (0) ∈ R n ≥ 0 and conv ergence of such tra jectories to a fixed p oint of map f . The pro of of Theorem 7 requires the preliminary discussion of sev eral lemmas which are needed to pro ve that t yp e-K order-preserv ation as opp osed to simple order-preserv ation, together with other prop erties, is sufficient to extend result to tra jectories starting at any p oint in R n ≥ 0 and exclude the existence of p erio dic tra jectories. Lemma 18 L et a p ositive map f b e typ e-K or der-pr eserving. F or al l x ∈ R n ≥ 0 it holds that f k i ( x ) > 0 for al l i such that x i > 0 and k ≥ 1 . Pro of. F or any x ∈ R n ≥ 0 let I ( x ) ⊂ { 1 , . . . , n } b e suc h that x i > 0 for i ∈ I ( x ) and x i = 0 otherwise. Since 0 ≤ x , b y t yp e-K order-preserv ation of f follows f ( 0 ) ≤ f ( x ). More precisely it holds f i ( x ) > f i ( 0 ) ≥ 0 for i ∈ I ( x ) and f i ( x ) ≥ f i ( 0 ) ≥ 0 otherwise, implying I ( x ) ⊆ I ( f ( x )). By induction, I ( x ) ⊆ I ( f k ( x )), i.e., f k i ( x ) > 0 for all i ∈ I ( x ), completing the pro of.  Lemma 19 L et a p ositive map f b e sub-homo gene ous and typ e-K or der-pr eserving. F or al l x ∈ R n ≥ 0 ther e exists a p art P ∈ P ( R n ≥ 0 ) and an inte ger k 0 ∈ Z such that f k ( x ) ∈ P for al l k ≥ k 0 . Pro of. Since b y Lemma 15 f is non-expansiv e under the Thompson’s metric d T , then x ∼ y implies f ( x ) ∼ f ( y ). This can b e easily prov ed by noticing that d T ( f ( x ) , f ( y )) ≤ d T ( x, y ) < ∞ since x ∼ y . This means that f maps parts into parts, i.e., for all x ∈ R n ≥ 0 and x 0 ∈ [ x ] = P I 0 it holds f ( x 0 ) ∈ [ f ( x )] = P I 1 . By Lemma 18 it follows P I 0  P I 1 and therefore [ x ]  [ f ( x )]. Generalizing, w e sa y that f k ( x ) ∈ P I k with k ∈ Z and I k ⊆ I k +1 ⊆ { 1 , . . . , n } . There exists k 0 ∈ Z such that I k = I k 0 for all k > k 0 and thus P k = P k 0 . This completes the pro of.  8 Lemma 20 L et a p ositive map f b e sub-homo gene ous and typ e-K or der-pr eserving. If f has a p ositive fixe d p oint ¯ x ∈ R n + , then for al l x ∈ R n ≥ 0 the tr aje ctory T ( x ) is b ounde d. Pro of. By Lemma 16 it follows that g = log ◦ f ◦ exp is a sup-norm non-expansive map that has the sam e dynamical prop erties as f for all x ∈ R n + . By Lemma 17 we know that one of the tw o cases can o ccur: ( i ) all tra jectories T (log( x ) , g ) are unbounded; ( ii ) all tra jectories T (log( x ) , g ) are b ounded. Since f has a fixed p oint x f ∈ R n + , such that f ( x f ) = x f , then x g = log ( x f ) is a fixed p oint of g , i.e., g ( x g ) = x g . The tra jectory T (log ( x f ) , g ) is obviously b ounded and therefore case ( ii ) holds. By Lemma 19, w e can partition R n ≥ 0 in tw o disjoint sets S 1 , S 2 suc h that if for x there exists k 0 ∈ Z such that f k 0 ( x ) ∈ R n + , then x ∈ S 1 , otherwise x ∈ S 2 . W e analyze these tw o cases. 1) F or all x ∈ S 1 , by Lemma 19, it holds that f k ( x ) ∈ R n + for all k ≥ k 0 . Let x 0 = f k 0 ( x ). Since case ( ii ) holds T (log( x 0 ) , g ) is b ounded, b ecause of the isometry also T ( x 0 , f ) is b ounded, and therefore also T ( x, f ). W e conclude that for all x ∈ S 1 tra jectories T ( x, f ) are b ounded. 2) F or all x ∈ S 2 , by Lemma 19, there exists k 0 ∈ Z such that f k ( x ) ∈ P I with I ( x ) ⊂ N = { 1 , . . . , n } for all k ≥ k 0 . Without loss of generality , here we assume I = { 1 , . . . , m } , where m < n . Let x = [ z T 1 , z T 2 ] T with z 1 ∈ R m ≥ 0 and consider the following m -dimensional map f ∗ : R m + → R m + defined by f ∗ i ( z 1 ) = f i ( z 1 , z 2 ) , z 2 = 0 , with i ∈ I ( x ). It is not difficult to chec k that f ∗ is still sub-homogeneous and t yp e-K order preserving. Accordingly , g ∗ = log ◦ f ∗ ◦ exp is a sup-norm non-expansiv e map that has the same dynamical prop erties as f ∗ for all x ∈ R m + . The main p oint now is to prov e that if ( ii ) o ccurs then all tra jectories T (log ( z 1 ) , g ∗ ) are also b ounded. T o this aim, w e first need to show that for all i ∈ I ( x ) it holds g ∗ i ( z 1 ) ≤ g i ( z 1 , z 2 ) . (4) Since b oth the exp onential and the logarithmic functions are strictly increasing, (4) is equiv alent to f ∗ i ( z 1 ) ≤ f i ( z 1 , z 2 ) . (5) By definition, (5) holds if z 2 = 0. If z 2 6 = 0, for any x = [ z T 1 , z T 2 ] T consider ¯ x = [ z T 1 , ¯ z T 2 ] T suc h that ¯ z 2 = 0 . Since f is order-preserving, for all i ∈ I it holds that f i ( ¯ x ) ≤ f i ( x ), which is equiv alent to write f i ( z 1 , ¯ z 2 ) ≤ f i ( z 1 , z 2 ) . By definition, f ∗ i ( z 1 ) = f i ( z 1 , ¯ z 2 ). Therefore, f ∗ i ( z 1 ) ≤ f i ( z 1 , z 2 ) for all z 2 6 = 0, i.e., (5) and (4) hold. Supp ose that ( ii ) o ccurs and there exist ˆ z 1 ∈ R m + suc h that T (log( ˆ z 1 ) , g ∗ ) is un b ounded. By (4) it is clear that given ˆ x = [ ˆ z T 1 , z T 2 ] T the tra jectory T (log( ˆ x ) , g ) is also unbounded, contradicting ( ii ). Let x 0 = f k 0 ( x ). Since all tra jectories T (log ( x 0 ) , g ) are b ounded, b ecause of the isometry also T ( x 0 , f ) is b ounded, and therefore also T ( x, f ). W e conclude that for all x ∈ S 2 tra jectories T ( x, f ) are b ounded.  Lemma 21 [11] L et a p ositive map f b e sub-homo gene ous and typ e-K or der-pr eserving. If for al l x ∈ K the tr aje ctory T ( x ) is b ounde d, then for al l x ∈ R n ≥ 0 , ω ( x ) is a singleton.  Finally , after restating the Theorem 7 for conv enience of the reader, we presen t a compact pro of based on the results presen ted in this section and Lemma 3.1.3 in [14]. Theorem 7 (Conv ergence) L et a p ositive map f b e sub-homo gene ous and typ e-K or der-pr eserving. If f has at le ast one p ositive fixe d p oint in R n + then al l p erio dic p oints ar e fixe d p oints, i.e., the set ω ( x ) is a singleton and ∀ x ∈ R n ≥ 0 : lim k →∞ f k ( x ) = ¯ x wher e ¯ x is a fixe d p oint of f . Pro of of Theorem 7. By Lemma 20 it follows that all tra jectories T ( x ) are b ounded for all x ∈ R n + . By Lemma 21 it follows that all p erio dic p oints are fixed p oints, i.e., the set ω ( x ) is a singleton. By Lemma 3.1.3 in [14], since f 9 is contin uous all tra jectories are b ounded and all p erio dic points are fixed p oin ts it holds lim k →∞ f k ( x ) = ¯ x where ¯ x is a fixed p oint of f .  4.3 Pr o of of The or em 8 By means of the tec hnical result in Theorem 7 w e pro v e our second main result, a sufficien t condition on the structure of the heterogeneous lo cal interaction rules of the MAS under consideration so that the global map (possibly unkno wn due to an unkno wn net work top ology) is p ositiv e, t yp e-K order preserving and sub-homogeneous map, thus falling within the class of systems considered in Theorem 12. W e also restate the theorem for con v enience of the reader. Theorem 8 (Stabilit y) Consider a MAS as in (2) with at le ast one p ositive e quilibrium p oint. If the set of differ entiable lo c al inter action rules f i , with i = 1 , . . . , n , satisfies the next c onditions: ( i ) f i ( x ) ∈ R ≥ 0 for al l x ∈ R n ≥ 0 ; ( ii ) ∂ f i /∂ x i > 0 and ∂ f i /∂ x j ≥ 0 for i 6 = j ; ( iii ) αf i ( x ) ≤ f i ( αx ) for al l α ∈ [0 , 1] and x ∈ K ; then the MAS c onver ges to one of its e quilibrium p oints for any p ositive initial state x (0) ∈ R n ≥ 0 . Pro of of Theorem 8. W e start the pro of b y establishing equiv alence relationships betw een the prop erties ( i ) − ( iii ) of the lo cal interaction rules of the MAS listed in the statement of Theorem 8 and prop erties (a)-(c) shown next: ( a ) f is p ositive; ( b ) f is type-K order-preserving; ( c ) f is sub-homogeneous; W e now prov e all equiv alences one b y one. • [( i ) ⇔ ( a )] Condition ( i ) implies that f maps a p oint of R n ≥ 0 in to R n ≥ 0 and is, therefore, a p ositiv e map (see Definition 1). • [( ii ) ⇒ ( b )] due to Prop osition 12 (Kamke-lik e condition). • [( iii ) ⇔ ( c )] by Definition 6 of a sub-homogeneous map, sub-homogeneity can b e verified element-wise for ma p f , thus the equiv alence follows. Th us, if conditions ( i ) to ( iii ) hold true for all lo cal interaction rules f i with i = 1 , . . . , n , since b y assumption map f has at least one p ositive fixed p oint, w e can exploit the result in Theorem 7 to establish that for all p ositive initial conditions, the state tra jectories of the MAS conv erge one of its p ositive equilibrium p oints.  4.4 Pr o of of The or em 9 T o pro v e our third main result, namely Theorem 9, w e need to introduce tw o technical lemmas. The first lemma, sho wn next, states sufficient conditions under which the elements along the rows of the Jacobian matrix of a map f computed at a consensus p oint c 1 sum to one. Lemma 22 L et a map f b e p ositive and differ entiable. If the set of fixe d p oints F f of map f satisfies F f ⊇ { c 1 , c ∈ R ≥ 0 } , i.e., the set of fixe d p oints c ontains at le ast al l p ositive c onsensus states, then J f ( c 1 ) 1 = 1 ∀ c ∈ R + , wher e J f ( c 1 ) denotes the Jac obian of f evaluate d in c 1 . 10 Pro of. Since f is differentiable, w e can apply directly the definition of directional deriv ative in a p oin t x ∈ R n ≥ 0 along a vector v ∈ R n obtaining J f ( x ) v = lim h → 0 f ( x + hv ) − f ( x ) h . No w w e ev aluate this expression in a consensus p oint x = c 1 ∈ F f , and along the direction v = 1 which is an in v arian t direction of f . W e obtain J f ( c 1 ) 1 = lim h → 0 f ( c 1 + h 1 ) − f ( c 1 ) h , = lim h → 0   c 1 + h 1 −   c 1 h = 1 , th us proving the statement.  Next, we in tro duce a critical lemma needed to prov e our third main result. In particular, we show that if there exists a fixed p oint of map f differen t from a consensus p oint, then there exists a consensus p oin t such that the Jacobian of map f computed at that consensus p oint has a unitary eigenv alue with multiplicit y strictly greater than one. Lemma 23 L et f b e p ositive, sub-homo gene ous, typ e-K or der-pr eserving and have a set of fixe d p oints F f such that F f ⊇ { c 1 , c ∈ R ≥ 0 } . If ther e exists a fixe d p oint ¯ x ∈ R n ≥ 0 such that ¯ x 6 = c 1 , ∀ c ∈ R ≥ 0 then ther e exists ¯ c ( ¯ x ) > 0 such that the Jac obian matrix J f (¯ c ( ¯ x ) 1 ) of map f c ompute d at ¯ c ( ¯ x ) 1 has a unitary eigenvalue with multiplicity strictly gr e ater than one. Pro of. Let ¯ x = [ ¯ x 1 , . . . , ¯ x n ] T ∈ R n ≥ 0 b e a fixed p oint of map f and let c 1 , c 2 ∈ R ≥ 0 b e such that c 1 = min i =1 ,...,n ¯ x i , c 2 = max i =1 ,...,n ¯ x i . W e define three sets I min ( ¯ x ) = { i : ¯ x i = c 1 } , I max ( ¯ x ) = { i : ¯ x i = c 2 } , I ( ¯ x ) = { i : ¯ x i 6 = c 1 , c 2 } . Consider a p oint y such that the i -th comp onent is defined by y i =  c 1 if i ∈ I min ( ¯ x ) c 3 otherwise (6) and such that c 1 1  y  ¯ x  c 2 1 . (7) By (6) and (7) it follows that y ≤ c 3 1 . (8) 11 • Now, we prov e that f ( y ) ≤ y . (9) Since map f is type-K order preserving, from (7) it follows c 1 ≤ f i ( y ) ≤ ¯ x i and from (8) f i ( y ) ≤ c 3 for i = 1 , . . . , n . F or i ∈ I min ( ¯ x ), by definition ¯ x i = c 1 and th us f i ( y ) = c 1 , otherwise for i ∈ I ( ¯ x ) ∪ I max ( ¯ x ) by (7) ¯ x i ≥ y i = c 3 and it follows c 1 ≤ f i ( y ) ≤ c 3 . Thus, (9) holds. • Now, we prov e that c 3 c 2 ¯ x i ≤ f i ( y ) ≤ c 3 , i = 1 , . . . , n . (10) Since f is order-preserving and sub-homogeneous, then f is non-expansiv e under the Thompson’s metric (see Def- inition 13) b y Lemma 15. No w, b y exploiting the definition of non-expansive map, w e compute an upp er bound to d T ( ¯ x, f ( y )). It holds d T ( ¯ x, f ( y )) ≤ d T ( ¯ x, y ) = log (max { M ( ¯ x/y ) , M ( y / ¯ x ) } ) where M ( ¯ x/y ) = inf { α ≥ 0 : y ≤ α ¯ x } = max i y i x i = 1 , M ( y / ¯ x ) = inf { α ≥ 0 : ¯ x ≤ αy } = max i x i y i ≤ c 2 c 3 . Since c 2 ≥ c 3 , it holds d T ( ¯ x, f ( y )) ≤ c 2 c 3 . (11) No w, we compute a low er b ound to d T ( ¯ x, f ( y )), where d T ( ¯ x, f ( y )) = log (max { M ( ¯ x/f ( y )) , M ( f ( y ) / ¯ x ) } ) and M ( ¯ x/f ( y )) = inf { α ≥ 0 : f ( y ) ≤ α ¯ x } = max i f i ( y ) x i = 1 , M ( f ( y ) / ¯ x ) = inf { α ≥ 0 : ¯ x ≤ αf ( y ) } = max i x i f i ( y ) ≥ c 2 c 3 . Th us max { M ( ¯ x/f ( y )) , M ( f ( y ) / ¯ x ) } ≥ c 2 c 3 , therefore d T ( ¯ x, f ( y )) ≥ c 2 c 3 . (12) By inequalities (11) and (12) it follows that d T ( ¯ x, f ( y )) = max i x i f i ( y ) = c 2 c 3 , th us proving that the inequality in (10) holds true. • Due to Theorem 7 it holds lim k →∞ f k i ( y ) = ¯ y i . Now, we prov e that ¯ y i =              c 1 if i ∈ I min ( ¯ x ) , c 3 if i ∈ I max ( ¯ x ) , c 1 if i ∈ I ( ¯ x ) and ∃ k ∗ : f k ∗ i ( y ) < f k ∗ − 1 i ( y ) , c 3 otherwise . (13) 12 In (13) three cases may o ccur: (1) If i ∈ I min ( ¯ x ) then ¯ x i = c 1 and by (7) it follo ws f i ( y ) = c 1 . (2) If i ∈ I max ( ¯ x ) then ¯ x i = c 2 and by (10) it follo ws f i ( y ) = c 3 . (3) If i ∈ I ( ¯ x ), by (9) tw o cases may o ccur: (a) There exists k ∗ > 0 such that f k ∗ i ( y ) < y i . In this case, by type-K order-preserv ation it holds that f k i ( y ) < f k − 1 i ( y ) ∀ k ≥ k ∗ + 1 and therefore lim k →∞ f k i ( y ) = c 1 . (b) Otherwise f k i ( y ) = f k − 1 ( y ) ∀ k > 0 and therefore lim k →∞ f k i ( y ) = y i = c 3 . Th us, by (13) w e pro ved that for an y fixed p oint ¯ x different from a consensus p oint c 1 there exists a fixed p oin t ¯ y with elements corresp onding to either c 1 or c 3 and such that I ( ¯ y ) = ∅ . No w, consider a p oint z such that its i -th comp onent is defined as follows z i =  c 1 if i ∈ I min ( ¯ y ) c 4 if i ∈ I max ( ¯ y ) (14) with c 4 ∈ [ c 1 , c 3 ]. By (13) and (14), we can conclude that z is fixed p oint, i.e., f ( z ) = z , for all v alues of c 4 in the in terv al c 4 ∈ [ c 1 , c 3 ]. Now, let v ( ¯ x ) b e a vector such that v i ( ¯ x ) =    0 if i ∈ I min ( ¯ x ) 1 if i ∈ I max ( ¯ x ) 0 or 1 if i ∈ I ( ¯ x ) , (15) Th us, by (15) the p oint c 1 1 + hv ( ¯ x ) is a fixed p oint of map f for all h ∈ [0 , c 3 − c 1 ]. Thus, it follows that f ( c 1 1 + hv ( ¯ x )) = c 1 1 + hv , h ∈ [0 , c 3 − c 1 ] . Since v ( ¯ x ) 6 = 1 , it holds (b y reasoning along the lines of Lemma 22) that the Jacobian of map f computed at c 1 1 has a right eigenv ector equal to v ( ¯ x ), i.e., J f ( c 1 1 ) v ( ¯ x ) = v ( ¯ x ). By Lemma 22 it holds that the Jacobian of f satisfies J f ( c 1 ) 1 = 1 for all c > 0. Thus, if there exists ¯ x 6 = c 1 then there exists ¯ c ( ¯ x ) = min i =1 ,...,n ¯ x i = c 1 suc h that matrix J f (¯ c ( ¯ x ) 1 ) has a untira y eigen v alue with multiplicit y strictly greater than one, thus proving the statement of this lemma.  Finally , we recall for conv enience of the reader and detail a pro of of our third and last main result, i.e., Theorem 9, whic h provides sufficient conditions for asymptotic conv ergence to the consensus state. Theorem 9 (Consensus) Consider a MAS as in (2) . If the set of differ entiable lo c al inter action rules f i , with i = 1 , . . . , n , satisfies the next c onditions: ( i ) f i ( x ) ∈ R ≥ 0 for al l x ∈ R n ≥ 0 ; ( ii ) ∂ f i /∂ x i > 0 and ∂ f i /∂ x j ≥ 0 for i 6 = j ; ( iii ) αf i ( x ) ≤ f i ( αx ) for al l α ∈ [0 , 1] and x ∈ R n ≥ 0 ; ( iv ) f i ( x ) = x i if x i = x j for al l j ∈ N in i ; ( v ) Infer enc e gr aph G ( f ) has a glob al ly r e achable no de; then, the MAS c onver ges asymptotic al ly to a c onsensus state for any initial state x (0) ∈ R n ≥ 0 . Pro of of Theorem 9 W e start the proof b y establishing the relations betw een prop erties ( i ) − ( v ) and the following: 13 ( a ) f is p ositive; ( b ) f is type-K order-preserving; ( c ) f is sub-homogeneous; ( d ) F f = { c 1 : c ∈ R ≥ 0 } . W e go through all equiv alences one by one. (1) [( i ) ⇔ ( a )] See Pro of of Theorem 8. (2) [( ii ) ⇔ ( b )] See Prop osition 12. (3) [( iii ) ⇔ ( c )] See Pro of of Theorem 8. (4) [( i − v ) ⇒ ( d )] The pro of of this implication is given b elo w. Condition ( iv ) implies that the consensus space c 1 is a subset of the set of fixed p oints F f of map f , i.e., F f ⊇ { c 1 : c ∈ R ≥ 0 } . By Theorem 22 the Jacobian matrix J f ( c 1 ) ev aluated at a consensus p oint is row-stochastic, i.e., J f ( c 1 ) 1 = 1 . By the definition of inference graph (see Definition 2), it holds that G ( f ) = G ( J f ( c 1 )). Th us G ( J f ( c 1 )) has a globally reac hable no de by hypothesis and is ap erio dic b ecause condition ( ii ) ensures a self-lo op at each no de. No w, we are ready to prov e by contradiction that [( i − v ) ⇒ ( d )]. In particular, if there exists a fixed p oint ¯ x 6 = c 1 , then by Lemma 23 the Jacobian matrix J f ( c 1 ) has a unitray eigenv alue with multiplicit y strictly greater than one. On the other hand, by the widely kno wn Theorem 5.1 in [4], if G ( J f ( c 1 )) has a globally reachable no de and is ap erio dic then J f ( c 1 ) has a simple unitary eigenv alue with corresp onding eigenv ector equal to 1 , unique up to a scaling factor c . This is a contradiction, therefore it do es not exist a fixed point ¯ x suc h that ¯ x 6 = c 1 with c > 0. Th us, w e conclude that the set of fixed p oints of map f satisfies F f = { c 1 , c ∈ R ≥ 0 } . Finally , if conditions ( a ) to ( c ) are satisfied, then Theorem 8 the MAS conv erges to its set of fixed p oints F f . If ( d ) is satisfied, the F f con tains only consensus p oints and th us the MAS in (2) conv erges to a consensus state for all x ∈ R n ≥ 0 .  5 Examples In this section we pro vide examples to corrob orate our theoretical analysis of the conv ergences prop erties of discrete- time, nonlinear, p ositive, type-K order-preserving and sub-homogeneous MAS. Example 24 As a first example we consider a susceptible-infected-susceptible (SIS) epidemic model [3] describ ed b y the following x i ( k + 1) = f i ( x ( k )) (16) = x i + h   δ i (1 − x i ) − x i X j ∈N i β ij (1 − x j )   . Suc h mo del w as originally deriv ed to describ e the propagation of an infectious diseases ov er a group of individuals. Eac h group is sub divided according to susceptible and infectious. Individuals can b e cured and reinfected many times, there is not an immune group. Giv en n groups, let x i ( k ), y i ( k ) b e the p ortion of, resp ectively , susceptibles and infectious of group i at time k , it is clear that x i ( k ) , y i ( k ) ≥ 0 and x i ( k ) + y i ( k ) = 1 for any k . Thus, it is sufficien t to consider the dynamics of one of them to completely describ e the system. In mo del (16) v ariables ha v e the following meaning: • β ij ≥ 0 are the infectious rates; 14 • δ i ≥ 0 is the healing rate; • h ≥ 0 is the sampling rate. W e now ev aluate conditions ( i ) − ( iv ) of Theorem 8 to establish the conv ergence of the asso ciated MAS to a p os itiv e fixed p oint. Due to space limitations, w e omit all steps and give directly conditions under which the theorem holds. • First we notice that x i ( k ) b elongs to [0 , 1] for all k . It is guaranteed that for all x i ( k ) ∈ [0 , 1] also x i ( k + 1)+ ∈ [0 , 1] if and only if (17) and (18) hold, hδ i ≤ 1 , h X j 6 = i β ij ≤ 1 , (17) hβ ii ≤   s 1 − h X j 6 = i β ij + p hδ i   2 . (18) W e conclude that for any x ∈ [0 , 1] n ⊂ R n ≥ 0 then f i ( x ) ∈ [0 , 1] ⊂ R ≥ 0 , thus proving that condition ( i ) holds. • Condition ( ii ) holds if and only if the next inequality holds hβ ii < 1 − hδ i − h X j 6 = i β ij . (19) • Condition ( iii ) holds if and only if the next inequality holds δ i ≥ X j 6 = i β ij . (20) • Condition ( iv ) is satisfied since ¯ x = 1 ∈ R n ≥ 0 is a p ositive fixed p oint. One can prov e that (17), (18), (19), (20) are equiv alent to (21). Let β = P j 6 = i β ij , hδ i + hβ < 1 − hβ ii , hβ ≤ 0 . 5 . (21) If (21) holds, then conditions of Theorem 8 are satisfied, and we conclude that the MAS con v erges to a fixed p oint for all x ∈ [0 , 1] n . F or a MAS describ ed by graph G 1 in Figure 1a, a numerical simulation is giv en in Figure 2a. Example 25 Consider a MAS describ ed by graph G 2 in Figure 1b and nonlinear lo cal interaction rule x i ( k + 1) = f i ( x ( k )) (22) = x i ( k ) + ε i X j ∈N i atan( x j ( k ) − x i ( k )) . W e no w ev aluate conditions ( i ) − ( v ) of Theorem 9 to establish the conv ergence of the asso ciated MAS to consensus state. 1 2 3 4 5 6 7 8 (a) Graph G 1 . 1 2 3 4 5 6 (b) Graph G 2 . Fig. 1. Graphs of Examples 1 and 2. 15 • Condition ( i ) holds ∀ x ∈ R n ≥ 0 . • Condition ( ii ) holds if and only if ε i ∈  0 , |N i | − 1 i for i = 1 , . . . , n . • Condition ( iii ) holds ∀ x ∈ R n ≥ 0 . • Condition ( iv ) is satisfied since ¯ x = c 1 with c ∈ R ≥ 0 is a solution for x = f ( x ). • Condition ( v ) is satisfied since graph G has a globally reachable no de. Th us, the conditions of Theorem 9 are satisfied, and we conclude that the MAS in (22) con v e rges to a consensus state. A numerical simulation is given in Figure 2b. 6 Conclusions and future works In this pap er we presen ted three main results related to a class of nonlinear discrete-time m ulti-agen t systems represen ted by a state transition map which is p ositive, sub-homogeneous and type-K order preserving. The first result establishes that a general discrete-time dynamical system conv erges to one of its equilibrium p oints asymptotically if its corresp onding state transition map is p ositive, sub-homogeneous and type-K order preserving. The second result pro vides sufficient conditions for a set of nonlinear, discrete-time heterogeneous local in teraction rules which define the MAS to establish stability of the MAS, indep endently of its graph top ology (which is con- sidered unknown) by exploiting our first main result. Finally , the third result pro vides sufficient conditions for a set of nonlinear, discrete-time heterogeneous lo cal in teraction rules whic h define the MAS to establish asymptotic con v ergence to a consensus state if the inference graph of the MAS has a globally reachable no de. This pap er generalizes results for discrete-time linear MAS whose state transition matrix is sto c hastic to the nonlinear case thanks to nonlinear P erron-F rob enius theory . 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