Hindawi Publishing Corporation International Journal of Differential Equations Volume 2016, Article ID 9192127, 9 pages http://dx.doi.org/10.1155/2016/9192127

Research Article Static Consensus in Passifiable Linear Networks

Ibragim A. Junussov

Department of Control Engineering, Czech Technical University in Prague, Karlovo namesti 13, 121 35 Praha 2, Czech Republic

Correspondence should be addressed to Ibragim A. Junussov; [email protected]

Received 29 December 2015; Revised 21 March 2016; Accepted 27 March 2016

Academic Editor: Jinde Cao

Copyright © 2016 Ibragim A. Junussov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Sufficient conditions of consensus (synchronization) in networks described by digraphs and consisting of identical deterministic SIMO systems are derived. Identical and nonidentical control gains (positive arc weights) are considered. Connection between admissible digraphs and nonsmooth hypersurfaces (sufficient gain boundary) is established. Necessary and sufficient conditions for static consensus by output feedback in networks consisting of certain class of double integrators are rediscovered. Scalability for circle digraph in terms of gain magnitudes is studied. Examples and results of numerical simulations are presented.

1. Introduction available for measurement and when it is not. In the last case observers are constructed. Control of multiagent systems has attracted significant inter- Analysis of consensus with scalar coupling strengths [10, est in last decade since it has a great technical importance [1– 11] is fruitful in a sense that conditions on gains (which 5] and relates to biological systems [6]. depend on connection topology and single agent properties) Inconsensusproblemsagentscommunicateviadecen- givemoreinsighttoproblem.Alotofworksontopicconsider tralized controllers using relative measurements with a final dynamic couplings; however, for certain type of connections goal to achieve common behaviour (synchronization) which it might happen that tunable parameters will exceed upper can evolve in time. Many approaches have been developed for bound on possible control gains, that is, not meet physical different problem settings. limitations. So, necessary and sufficient conditions on con- Laplace , its spectrum, and eigenspace play crucial sensus achievement for different connection types in terms role in description and analysis of consensus problems. of coupling strengths are needed. Laplace matrix has broad applications, for example, [7]. Celebrated Kalman-Yakubovich-Popov Lemma (Positive Not all possible digraph topologies can provide consensus Real Lemma) establishes important connection between pas- over dynamical networks. Admissible digraph topologies and sivity (positive-realness) of transfer function 𝜒(𝑠) and matrix connection with algebraic properties of Laplace matrix have relations on its minimal state-space realization (𝐴, 𝐵, 𝐶);see been found in [8]. Tree structure analysis and properties of [12, 13]. Positive Real Lemma is a basis for Passification Laplace matrix spectrum of digraphs are also studied by these Method [14, 15] (“Feedback Kalman-Yakubovich Lemma”) authors. Work [9] contains examples of out-forests as well as which answers question when a linear system can be made useful graph theoretical concepts and can be recommended passive, that is, strictly positive real (SPR) by static output as an entry reading to the research of these authors on feedback. Powerful idea of rendering system into passive by algebraic digraph theory and consensus problems. feedback has been also studied for nonlinear systems, for Concept of synchronization region in complex plane for example, [13, 16, 17]. networks consisting of linear dynamical systems is intro- In consensus-type problems considering SPR agents with duced in [10]. In [11] this concept is used for analysis of stable (Hurwitz) matrix 𝐴 leads to a synchronous behaviour synchronization with leader. Problem is solved using Linear when all states go to zero. The latter is undesirable in essence, Quadratic Regulator approach in cases when full state is since such behaviour can be reached by local control without 2 International Journal of Differential Equations communication. So, instead of SPR systems it is possible 𝑁∈N. It is assumed hereafter that graphs do not have self- to consider passifiable systems, with an opportunity that loops, that is, for any vertex, 𝛼∈V arc(𝛼, 𝛼) ∉ E. a study is extendable to nonlinear systems. Also, Passifi- Digraphiscalleddirectedtreeifallitsverticesexcept cation Method allows avoiding constructing observers for one (called root) have exactly one parent. Let us agree that reaching full-state consensus by output feedback. Observers in any arc(𝛼, 𝛽) ∈ E vertex 𝛽 is parent or neighbour. implementation increases dimension of overall phase space Directed spanning tree of a digraph G is a directed tree and complexity of a dynamic network. However, finding a formed of all digraph G vertices and some of its arcs such that passification matrix (vector 𝑔)islongstandingopenproblem; there exists path from any vertex to the root vertex in this still set of passifiable systems is nonempty. tree. Existence of directed spanning tree has connection to InthispaperPassificationMethodisusedtosynthesizea principal achievement of synchronization in consensus-like decentralized control law and to derive sufficient conditions problems. of full-state synchronization by relative output feedbacks in A digraph is called weighted if to any pair of vertices networks described by digraphs with Linear Time Invariant 𝛼, 𝛽 ∈ E number 𝑤(𝛼, 𝛽) ≥0 is assigned such that dynamical nodes in continuous time. Assumptions made on networktopologyareminimal.Synchronousbehaviouris 𝑤(𝛼,𝛽)>0 if (𝛼, 𝛽) ∈ E, described, including case of nonidentical gains. It is deter- (1) mined that boundary of sufficient gain region geometrically 𝑤(𝛼,𝛽)=0 if (𝛼, 𝛽) ∉ E. is a hypersurface in corresponding gain space. For certain 1 three-node network this geometrical observation connects A digraph in which all nonzero weights are equal to will be algebraic properties of Laplace matrix with constructed referred to as unit weighted. hypersurface. Namely, Jordan block appears in a direction of An A(G) is 𝑁×𝑁matrix whose 𝑖th, 𝑗 𝑤(𝛼 ,𝛼 ), 𝑖,𝑗=1,...,𝑁 a cusp (nonsmooth) extremal point of the hypersurface. th entry is equal to 𝑖 𝑗 . Necessary and sufficient conditions for static consensus Laplace matrix of digraph G is defined as follows: by output feedback in networks consisting of certain class 𝐿 (G) = (A (G) ⋅ 1 ) − A (G) . of double integrators have been rediscovered. Conditions are diag 𝑁 (2) givenintermsofLaplacematrixspectrum. Matrix 𝐿(G) always has zero eigenvalue with corresponding Scalabilityinacircledigraphsintermsofgain(coupling right eigenvector 1𝑁 :𝐿(G)⋅1𝑁 =0⋅1𝑁. By construction strength)isstudied.Itisshownthatcommongaininlarge and Gershgorin Circle Theorem all nonzero eigenvalues of 𝑁 cycle digraphs consisting of double integrators should grow 𝐿 have positive real parts. Let us denote by V(𝐿) ∈ R left not slower than quadratically in number of agents. eigenvector of 𝐿 which is corresponding to zero eigenvalue T Results of numerical simulations in 3- and 20-node and scaled such that V(𝐿) ⋅ 1𝑁 =1.Itisknownthatvector double-integrator networks are presented. V(𝐿) describes synchronous behaviour if reached. Neighbouring papers, which influenced this work, are Suppose that a digraph has directed spanning tree. A set citedbeforeConclusions. of digraph vertices is called Leading Set (“basic bicomponent” intermsof[9])ifsubdigraphconstructedofthemisstrongly 2. Theoretical Study connected and no vertex in this set has neighbours in the remaining part of digraph. Nonzero components of V(𝐿) and 2.1. Preliminaries and Notations. Notations, some terms of only them correspond to vertices of Leading Set. Definition of , and Passification Lemma are listed in this basic bicomponent is wider and applicable for digraphs with section. no directed spanning trees. For illustration, by [21], there are 16 different types of 2.1.1. Notations. Notation ‖⋅‖2 stands for Euclidian norm. digraphs which can be constructed on 3 nodes. 12 of them For two symmetric matrices 𝑀1, 𝑀2 inequality 𝑀1 >𝑀2 have directed spanning tree; among these, 5 digraphs have means that matrix 𝑀1 −𝑀2 is positive definite. Notation Leading Set with 3 nodes, 2 digraphs have Leading Set with 2 T col(V1,...,V𝑑) stands for vector (V1,...,V𝑑) . nodes, and 5 digraphs have Leading Set with 1 node. of size 𝑑 is denoted by 𝐼𝑑. Vector 1𝑑 = (1,1,...,1) is vector of size 𝑑 and consisting of ones. Vector 0𝑑 is defined 2.1.3. Passification Lemma. Problem of linear system passifi- similarly. Matrix diag(V1,...,V𝑑) is whose 𝑖th cation is a problem of finding static linear output feedback element on main diagonal is V𝑖, 𝑖=1,...,𝑑; other entries are which is making initial system passive. It was solved in zeroes. Notation ⊗ standsforKroneckerproductofmatrices. [14, 15] for nonsquare SIMO and MIMO systems including Definition and properties of Kronecker product, including case of complex parameters. Brief outline of SIMO systems eigenvaluesproperty,canbefoundin[18,19].Directsumof passification is given below. matrices [20] is denoted by ⊕. Let 𝐴, 𝐵, 𝐶 be real matrices of sizes 𝑛×𝑛, 𝑛×1, 𝑛×𝑙 T −1 accordingly. Denote by 𝜒(𝑠) =𝐶 (𝑠𝐼 − 𝐴) 𝐵, 𝑠 ∈ C.Let 𝑙 T 2.1.2. Terms of Graph Theory. ApairG =(V, E),whereV is vector 𝑔∈R . If numerator of function 𝑔 𝜒(𝑠) is Hurwitz set of vertices and E ⊆ V × V issetofarcs(orderedpairs) with degree 𝑛−1and has positive coefficients then function T is called digraph (directed graph). Let V have 𝑁 elements, 𝑔 𝜒(𝑠) is called hyper-minimum-phase. International Journal of Differential Equations 3

Lemma 1 (Passification Lemma [14, 15]). The following state- 2.3. Static Identical Control. Denote ments are equivalent. 𝑟 (𝐿) = min Re 𝜆𝑖, 𝑙 T 𝜆𝑗=0̸ (8) (1) There exists vector 𝑔∈R such that function 𝑔 𝜒(𝑠) is

hyper-minimum-phase. where 𝜆𝑗 are eigenvalues of 𝐿. Under assumption (A2) zero T −1 𝐿 (2) Number 𝜘0 = sup 1 Re(𝑔 𝜒(i𝜔)) is positive 𝜘0 > eigenvalue is simple. By properties of other eigenvalues 𝜔∈R 𝑟(𝐿) 0 and for any 𝜘>𝜘0 there exists 𝑛×𝑛real matrix lieinopenrighthalfofcomplexplane,so is positive T 𝐻=𝐻 >0satisfying the following matrix relations: number. Suppose that assumption (A1) holds with known vector 𝑙 𝐻𝐴 +𝐴T 𝐻<0, 𝑔∈R . Consider the following static consensus controller ∗ ∗ 1 with gain 𝑘∈R , 𝑘>0which is the same for all agents: 𝐻𝐵=𝐶𝑔, (3) 𝑢 (𝑡) =−𝑘𝑔T𝑦 (𝑡) , 𝑖=1,...,𝑁, T T 𝑖 𝑖 (9) 𝐴∗ =𝐴−𝜘𝐵𝑔 𝐶 . where relative output 𝑦𝑖(𝑡) hasbeendefinedintheprevious 𝑥(𝑡) = (𝑥 (𝑡),...,𝑥 (𝑡)) 2.2. Problem Statement and Assumptions. Consider a net- section. Denote col 1 𝑁 . 𝑁 work consisting of agents modelled as linear dynamical Theorem 2. systems: Letassumptions(A1)and(A2)hold.Thenforall 𝑘 such that

𝑥̇𝑖 (𝑡) =𝐴𝑥𝑖 (𝑡) +𝐵𝑢𝑖 (𝑡) , 𝜘0 𝑘> (10) T (4) 𝑟 (𝐿) 𝑦𝑖 (𝑡) =𝐶 𝑥𝑖 (𝑡) , controller (9) ensures achievement of goal (6) in dynamical 𝑛 𝑙 where 𝑖 = 1,...,𝑁, 𝑥𝑖 ∈ R is state vector, 𝑦𝑖 ∈ R is output network (4); asymptotical behaviour in the case of goal achieve- 1 or measurements vector, 𝑢𝑖 ∈ R isinputorcontrol,and𝐴, 𝐵, mentisdescribedbythefollowingconsensusvector: 𝐶 are real matrices of appropriate size. By associating agents 𝑐 𝑡 = 𝐴𝑡 (V 𝐿 T ⊗𝐼)𝑥 0 . with 𝑁 vertices of unit weighted digraph G and introducing ( ) exp ( ) ( ) 𝑛 ( ) (11) set of arcs one can describe information flow in the network. For 𝑖=1,...,𝑁let us introduce notation for relative outputs Proof. Closed loop system (4) and (9) can be rewritten in the following form: 𝑦 (𝑡) = ∑ (𝑦 (𝑡) −𝑦 (𝑡)), 𝑖 𝑖 𝑗 (5) 𝑥̇ (𝑡) =(𝐼 ⊗𝐴−𝑘𝐿⊗𝐵𝑔T𝐶T)𝑥(𝑡) . 𝑗∈N𝑖 𝑁 (12) 𝑃 where N𝑖 is a set of 𝑖th agents neighbours. Consider nonsingular matrix (real or complex) such that Problem is to design controllers which use relative out- 0 0T puts and ensure achievement of the state synchronization 𝑁−1 −1 Λ=( )=𝑃 𝐿𝑃, (13) (consensus) of all agents: 0𝑁−1 Λ 𝑒

(𝑥 (𝑡) −𝑥 (𝑡))=0, 𝑖,𝑗=1,...,𝑁. (𝑁−1)×(𝑁−1) (𝑁−1)×(𝑁−1) 𝑡→∞lim 𝑖 𝑗 (6) where Λ 𝑒 ∈ R or Λ 𝑒 ∈ C .All eigenvalues of Λ 𝑒 have positive real parts. By considering T −1 T In the case of synchronization achievement asymptotical first (zero) columns of matrices 𝑃Λ = 𝐿𝑃 and (𝑃 ) Λ = T −1 T behaviour of all agents will be described by the same time- 𝐿 (𝑃 ) we can accept that first column of 𝑃 is 1𝑁 and first −1 T dependant consensus vector which is denoted hereafter by row of 𝑃 is V(𝐿) . −1 𝑐(𝑡): Let us apply coordinate transformation 𝑧(𝑡) = (𝑃 ⊗ 𝐼𝑛)𝑥(𝑡) and rewrite (12): (𝑥 (𝑡) −𝑐(𝑡))=0, 𝑖=1,...,𝑁. 𝑡→∞lim 𝑖 (7) 𝑧̇1 (𝑡) =𝐴𝑧1 (𝑡) , (14) Let us make the following assumption about dynamics of a ̇ T T single agent. 𝑧𝑒 (𝑡) =((𝐼𝑁−1 ⊗𝐴)−𝑘(Λ𝑒 ⊗𝐵𝑔 𝐶 )) 𝑧𝑒 (𝑡) , (15)

𝑙 𝑛 𝑛 (A1) There exists vector 𝑔∈R such that transfer function where 𝑧=col(𝑧1,𝑧𝑒), 𝑧1 ∈ R ,or𝑧1 ∈ C .Notethatzero T T T −1 𝑔 𝜒(𝑠) =𝑔 𝐶 (𝑠𝐼𝑛−𝐴) 𝐵 is hyper-minimum-phase. solution of (15) is globally asymptotically stable iff goal (6) is achieved. Now let us make an assumption on graph topology. For simplicity let 𝑃, Λ 𝑒,and𝑧(𝑡) be real till the end of 𝑘 0<𝜀 <1 G proof. For any fixed satisfying (10) there exists 𝑠 (A2) Digraph has at least one directed spanning tree. such that 𝐿 𝜘 Zero eigenvalue of Laplace matrix has unit multiplicity 𝜀 𝑘> 0 . iff this assumption holds [8]. 𝑠 𝑟 (𝐿) (16) 4 International Journal of Differential Equations

Eigenvalues of matrix (Λ 𝑒 −𝜀𝑠𝑟(𝐿)𝐼𝑁−1) have positive real the following consideration of synchronization gain region to parts. Therefore, according to [22], there exists (𝑁−1)×(𝑁−1) lower dimension 𝑁−1. T ̂ ̂󸀠 󸀠 󸀠 real matrix 𝑄=𝑄 >0such that the following Lyapunov Let us denote by 𝑘=(𝑘1,...,𝑘𝑁) and by 𝑘 =(𝑘1,...,𝑘𝑁) ̂ inequality holds: point which is projection of point 𝑘 on unit sphere S

T 󸀠 (Λ 𝑒 −𝜀𝑠𝑟 (𝐿) 𝐼𝑁−1) 𝑄+𝑄(Λ 𝑒 −𝜀𝑠𝑟 (𝐿) 𝐼𝑁−1) >0. (17) 𝑘=𝑘⋅̂ 𝑘̂ , 𝑘>0,

We can rewrite the last inequality: 𝑁 (24) 󸀠 2 ∑ (𝑘𝑖 ) =1, T 𝑖=1 Λ 𝑒𝑄+𝑄Λ𝑒 >2𝜀𝑠𝑟 (𝐿) 𝑄. (18)

T where scalar common gain 𝑘 is radius vector magnitude of By assumption (A1) there exists 𝐻=𝐻 >0such that (3) 󸀠 𝑘̂ 𝑘̂ 𝑘̂ O ={(𝑘,...,𝑘 )∈R𝑁 | is true with 𝜘=𝜀𝑠𝑘𝑟(𝐿),since𝜘>𝜘0. Considering following point .Points , lie in orthant 1 𝑁 𝑘𝑖 >0, 𝑖=1,...,𝑁}. Lyapunov function, 󸀠 󸀠 󸀠 󸀠 Denote 𝐾 = diag(𝑘1,𝑘2,...,𝑘𝑁).Laplacematrices𝐿 and T 𝐾󸀠𝐿 𝑉(𝑧𝑒 (𝑡))=𝑧𝑒 (𝑡)(𝑄⊗𝐻) 𝑧𝑒 (𝑡) , (19) correspondtothesamedigraphswhichdifferonlyinarc weights. Equation for closed loop system (4) and (23) can be and derivating it along the nonzero trajectories of (15), we rewritten as follows: obtain 󸀠 T T 𝑥̇ (𝑡) =(𝐼𝑁 ⊗𝐴−𝑘(𝐾𝐿⊗𝐵𝑔 𝐶 )) 𝑥 (𝑡) . (25) 𝑑 T 𝑉(𝑧𝑒 (𝑡))=𝑧𝑒 (𝑡) 𝑀𝑧𝑒 (𝑡) , (20) 𝑑𝑡 By repeating proof of Theorem 2 we can formulate the following result. where T T Theorem 4. Letassumptions(A1)and(A2)hold.Thenforall 𝑀=𝑄⊗(𝐴 𝐻+𝐻𝐴)−𝑘(Λ𝑒𝑄+𝑄Λ𝑒) 󸀠 𝑘𝑖 =𝑘⋅𝑘𝑖 such that T T ⊗(𝐶𝑔𝑔 𝐶 ) 𝑁 2 𝜘 ∑ (𝑘󸀠) =1, 𝑘> 0 T T T 𝑖 𝑟(𝐾󸀠𝐿) (26) ≤𝑄⊗(𝐴 𝐻+𝐻𝐴)−2𝑘𝜀𝑠𝑟 (𝐿) 𝑄⊗(𝐶𝑔𝑔 𝐶 ) (21) 𝑖=1

=𝑄⊗((𝐴T −𝜘𝐶𝑔𝐵T)𝐻+𝐻(𝐴−𝜘𝐵𝑔T𝐶T)) controller (23) ensures achievement of goal (6) in dynamical network (4); asymptotical behaviour in the case of goal achieve- T =𝑄⊗(𝐴∗𝐻+𝐻𝐴∗)<0. mentisdescribedbythefollowingconsensusvector:

󸀠 T Matrix relations (3) have been used here. Last inequality 𝑐 (𝑡) = exp (𝐴𝑡) (V (𝐾 𝐿) ⊗𝐼𝑛)𝑥(0) . (27) concludes the proof. Let us denote by K ⊂ O region in orthant such that for Assumptions of this theorem are relaxed in comparison any point (𝑘1,...,𝑘𝑁)∈K in this region control (23) ensures with Theorem 2 from [23]. Proof of Theorem 2 also provides achievement of the goal (6) in network (4), (23). Consider the the following auxiliary result. following region:

Lemma 3. Let assumption (A2) hold. Controller (9) ensures 󸀠 󸀠 𝜘 K ={𝑘∈̂ O | 𝑘=𝑘⋅̂ 𝑘̂ , 𝑘̂ ∈ S,𝑘> 0 } achievement of goal (6) in dynamical network (4) if and only if 𝑟 𝑟(𝐾󸀠𝐿) (28) all eigenvalues of matrix

which is subset of K : K𝑟 ⊂ K. Let us consider closed part 𝑅=(𝐼 ⊗𝐴)−𝑘(Λ ⊗𝐵𝑔T𝐶T) 𝑁−1 𝑒 (22) of unit sphere have negative real parts. In the case of goal achievement ̂󸀠 󸀠 S ={𝑘 ∈ S |𝑘 ≥𝜀, 𝑖=1,...,𝑁}, 𝜀>0. (29) asymptotical behaviour is described by (11). 𝜀 𝑖

Point on S𝜀 determines ray (half-line) in O with initial point 2.4. Nonidentical Control and Gain Region. Let the initial at the origin. According to Theorem 4, by moving along G 𝐿 digraph be unit weighted. Let us fix Laplace matrix and this ray from origin, that is, increasing 𝑘,wewillreachK𝑟. 𝑘 >0 consider static control with nonidentical gains 𝑖 : Consider map T 𝑢 (𝑡) =−𝑘𝑔 𝑦 (𝑡) , 𝑖=1,...,𝑁. 󸀠 𝜘0 󸀠 󸀠 𝑖 𝑖 𝑖 (23) ℎ:𝑘 󳨃󳨀→ ⋅𝑘,𝑘∈ S , 𝑟(𝐾󸀠𝐿) 𝜀 (30) Without loss of generality we can assume that network does not have a leader (formally: cardinality of Leading which is continuous as a composition of continuous maps Set is more than 1), since in leader case we can reduce ([20], continuous dependence of matrix eigenvalues on International Journal of Differential Equations 5

parameters). Image of this map is a subset of boundary digraph which is consisting of 𝑁 nodes 𝑆𝑗 with exactly 𝑁 𝜕K𝑟; therefore, by continuity of map ℎ,boundary𝜕K𝑟 is a arcs: 𝑁 hypersurface in R . Further, let us consider induced map (𝑆1,𝑆2)∪⋅⋅⋅∪(𝑆𝑗,𝑆𝑗+1)∪⋅⋅⋅∪(𝑆𝑁−1,𝑆𝑁) 󸀠 󵄩 󸀠 󵄩 󸀠 (34) ℎ :𝑘 󳨃󳨀→ 󵄩ℎ(𝑘 )󵄩 ,𝑘∈ S . 𝜌 󵄩 󵄩2 𝜀 (31) ∪(𝑆𝑁,𝑆1).

𝐶 Domain S𝜀 is compact, so we can apply Weierstrass Eigenvalues of 𝐿𝑁 are evenly located at circle in complex Extreme Value Theorem and arrive at the following lemma. plane [25]: 2𝜋 Lemma 5. Map ℎ:S𝜀 →ℎ(S𝜀)⊂𝜕K𝑟 is continuous. Map 𝜆 =1− (i ⋅𝑗⋅ ), ℎ : S → R1 𝑗 exp 𝑁 𝜌 𝜀 is continuous and has minimum and maximum. (35) 𝑗=0,...,𝑁−1, i2 =−1. Generally, hypersurface 𝜕K𝑟 is not smooth in all its points. Alternatively, part of a simplex of appropriate dimen- Theorem 6. sioncanbetakeninsteadofsphereparttoserveasthedomain Controller (33) ensures achievement of goal (6) in dynamical network consisting of 𝑁 double integrators (32) for maps ℎ and ℎ𝜌. Pairwise ratios of nonidentical gains and common gain connected in directed cycle if and only if define homogeneous coordinates in orthant. Common gain 𝑘 1 𝜋 𝑘> 2 . coefficient relates to reachability of consensus and to speed of 2cot 𝑁 (36) convergence but it does not influence consensus vector. Also, 𝐶 consensus vector can be changed only by gain ratios variation Proof. Let us diagonalize 𝐿𝑁.Matrix𝑅 from Lemma 3 in our within Leading Set of agents; see [24]. case is block diagonal: 𝑅=𝑅 ⊕𝑅 ⊕⋅⋅⋅⊕𝑅 , 3. Double-Integrator Networks 1 2 𝑁−1 (37) where 3.1. Agents Description. Suppose that each agent 𝑆𝑖 in a net- work is modelled as follows: −𝑘𝜆𝑗 −𝑘𝜆𝑗 𝑅𝑗 =( ), 𝑗=1,...,𝑁−1. (38) 10 𝑥̇𝑖 =𝐴𝑥𝑖 +𝐵𝑢𝑖,

T So, matrix 𝑅 is stable iff matrices 𝑅𝑗 are stable for all 𝑗= 𝑦𝑖 =𝐶 𝑥𝑖, 1,...,𝑁−1.Characteristicpolynomialof𝑅𝑗 is 𝑖=1,...,𝑁, 2 𝑓𝑗 (𝑧) =𝑧 +𝑘𝜆𝑗𝑧+𝑘𝜆𝑗, 𝑗=1,...,𝑁−1. (39) 00 𝐴=( ), 𝑘𝜆 =𝛼 + i𝛽 𝛼 ,𝛽 ∈ R 𝑗 = 1,...,𝑁−1 10 (32) Let 𝑗 𝑗 𝑗, 𝑗 𝑗 , .Takingin account (35) we can obtain 2 𝑗𝜋 𝐵=( ), 𝛼 =2𝑘 2 , 0 𝑗 sin 𝑁

0.5 𝑗𝜋 𝑗𝜋 (40) 𝐶=( ). 𝛽𝑗 =−2𝑘(sin )(cos ), 0.5 𝑁 𝑁 𝑗=1,...,𝑁−1. T T −1 For 𝑔=1transfer function 𝑔 𝜒(𝑠) =𝐶 (𝑠𝐼2 −𝐴) 𝐵 = (𝑠 + 2 Now let argument of 𝑓𝑗(𝑧) runonimaginaryaxisandlet 1)/𝑠 is hyper-minimum-phase. It can be shown that number us decompose this polynomial on real and imaginary parts: 𝜘0 =1. First and second components of 𝑥𝑖 can describe (or can be 1 𝑓𝑗 (i𝜔) =𝜑𝑗 (𝜔) + i𝜓𝑗 (𝜔) ,𝜔∈R , (41) interpreted as) velocity and position. Single system (32) can 1/𝑠2 be viewed as double integrator with transfer function and where 𝑗=1,...,𝑁−1and proportionally differential (PD) control applied to it. 𝑔=1 2 Since , static consensus controller (9) has the 𝜑𝑗 (𝜔) =−𝜔 −𝛽𝑗 ⋅𝜔+𝛼𝑗, following form: (42) 𝜓𝑗 (𝜔) =𝛼𝑗 ⋅𝜔+𝛽𝑗. 𝑢𝑖 (𝑡) =−𝑘𝑦𝑖 (𝑡) , 𝑖=1,...,𝑁. (33) According to Hermite-Biehler Theorem, polynomial 𝑓𝑗(𝑧) is stable iff both of the following conditions are satisfied: 3.2. Necessary and Sufficient Conditions on Consensus. Let 𝐶 us denote by 𝐿𝑁 Laplacematrixofunitweightedcycle (i) roots of 𝜑𝑗(𝜔) and 𝜓𝑗(𝜔) are interlacing; 6 International Journal of Differential Equations

(ii) is positive Theorem 9. Consider network 𝑆 consisting of 𝑁 agents (32). G 󸀠 󸀠 Let digraph , describing information flow, contain at least one 𝜑𝑗 (𝜔0)⋅𝜓𝑗 (𝜔0)−𝜑𝑗 (𝜔0)⋅𝜓𝑗 (𝜔0)>0 (43) directed spanning tree. Let all nonzero eigenvalues of Laplace matrix 𝐿(G) be denoted by 𝜆𝑗,1≤𝑗≤𝑁−1. Controller for at least one value of argument 𝜔0. (33) ensures achievement of goal (6) in dynamical network consisting of 𝑁 double integrators (32) if and only if Wronskianispositivefor𝜔0 = 0, 𝑗 = 1,...,𝑁−.Root 1 2 interlacing property is equivalently transformable to sin (arg (𝜆𝑗)) 𝑘> max . (48) 1≤𝑗≤𝑁−1 1 2 𝑗𝜋 Re 𝜆𝑗 𝑘> , 𝑗=1,...,𝑁−1. (44) 2cot 𝑁 𝑁−1 Proof. Using similar argumentation as in proofs of Theorems Right parts of these inequalities reach maximum when 8 and 6 we arrive at study of polynomial (39) stability with 𝑗=1(also when 𝑗=𝑁−1) and this concludes proof.

Therefore, for a large increasing number of agents 𝑁 gain 𝛼𝑗 =𝑘⋅Re (𝜆𝑗), 2 𝑘 should grow as 𝑁 : 𝛽𝑗 =𝑘⋅Im (𝜆𝑗), (49) 𝑁2 𝑘∼ ,𝑁󳨀→∞. (45) 2𝜋2 𝑗=1,...,𝑁−1.

It is possible to conclude that consensus in large cycle Wronskian property does hold for 𝜔0 =0.For𝑗=1,...,𝑁− digraphs is hard to achieve, at least for agents (32), since 1 root interlacing property is equivalent to trigonometric arbitrary high gains are not physically realizable. inequality On the other hand, it is worth noting that cycle digraph is the graph with minimal number of edges which is deliv- 𝛼 +𝛽 ( (𝜆 )) > 2 ( (𝜆 )) ering average consensus among all its nodes; it is strongly 𝑗 𝑗 tan arg 𝑗 tan arg 𝑗 (50) connected. or Remark 7 (see [24]). Minimality in edges number causes 2 2 simple relations on nonidentical gains and left eigenvector 𝑘 Re (𝜆𝑗)(1+tan (arg (𝜆𝑗))) > tan (arg (𝜆𝑗)) . (51) 𝐶 V(𝐾𝐿𝑁) components for agents in form (4):

𝑘1V1 =𝑘2V2 =⋅⋅⋅=𝑘𝑁V𝑁. (46) (𝑘 , V ) In other words, all pairs 𝑗 𝑗 lie on the same hyperbola. 4. Examples and Numerical The following result can be obtained by repeating proof Simulations Results of Theorem 6. 4.1. Three-Node Digraph and Gain Region. Consider digraph Theorem 8. 𝑆 𝑁 Consider network consisting of agents (32). shown in Figure 1 with dynamic nodes described in Sec- Let a digraph, describing information flow, contain directed tion 3.1. By Lemma 5 distance from origin to 𝜕K𝑟 reaches spanning tree. Let Laplace matrix of the digraph have real minimum. Let us draw 𝜕K𝑟.First,let𝜀, 𝛿 ∈ R.Let𝜀>0 󸀠 󸀠 spectrum. Controller (33) ensures achievement of goal (6) in be small, and 𝜀≤𝛿≤1−𝜀,𝑘2 =𝛿,𝑘3 =1−𝛿.Let𝐿 be dynamical network consisting of 𝑁 double integrators (32) if 󸀠 󸀠 unit weighted. Eigenvalues of matrix diag(0, 𝑘2,𝑘3)𝐿 are real: and only if {0,𝛿,2−2𝛿}. Using Theorem 8 we conclude that K ={𝑘2 > 0, 𝑘3 >0}.Any𝛿 ∈ [𝜀, 1 −𝜀] determines angle 𝑘>0. (47) 1−𝛿 𝑘󸀠 𝑘 Proof. For diagonalizable Laplace matrix with real spectrum 𝛾 𝛿 = = 3 = 3 , ( ) arctan arctan 󸀠 arctan (52) statement is following from the well-known fact that poly- 𝛿 𝑘2 𝑘2 nomial (39) with real coefficients is stable iff its coefficients are positive. For nondiagonalizable Laplace matrix 𝐿 let and radius vector 𝜌(𝛿) = 1/min𝛿∈[𝜀,1−𝜀]{𝛿, 2 − 2𝛿}. Note that us transform it to Jordan form. Expansion of matrix 𝑅− pair (𝜌(𝛿), 𝛾(𝛿)) is polar coordinates of boundary 𝜕K𝑟.We 𝑧𝐼 𝑅 −𝑧𝐼 determinant shows that only determinants 𝑗 2 can conclude that minimum on 𝜌(𝛿) is realized on a point across main diagonal are forming (factorizing) characteristic for which 𝛿=2/3and 𝑘2 :𝑘3 =2:1.Boundary𝜕K𝑟 is polynomial of 𝑅. 󸀠 presented in Figure 2. For all (𝑘2,𝑘3) = 𝑘 ⋅ (2, 1) matrix 𝐾 𝐿 Note that undirected graphs (i.e., digraphs with symmet- is similar to 𝐿 ric ) have real spectrum and some class of digraphs have real 2𝑘 1 spectrum too, for example, directed path graphs [26]. 0⊕( ). (53) We can formulate similar result for general digraphs. 02𝑘 International Journal of Differential Equations 7

S 25 S1 k2 2

20

15 (i) e k3 k3 10

5

0 S3 0 5 10 15 20 25 t Figure 1: Digraph of 3 nodes. (1) Figure 3: Performance with identical (𝑒 (𝑡); black line) and (2) nonidentical (𝑒 (𝑡); blue line) gains.

5 S16 S17

4.5

S2 4 S1 S3

3.5 S12 S10 S S 3 13 4

S11 3 2.5 k

S9 S14 S5 2 S S 15 20 S18 S 1.5 8 S6 S 1 7

0.5

0 S 0 1 2 3 45 19 k2 Figure 4: Digraph of 20 agents.

Figure 2: Gain region boundary 𝜕K𝑟. Let us choose the following initial conditions:

𝑥1 (0) = col (2, −2) , Let us consider two cases: 𝑥2 (0) = col (−7, 3) , (54) (1) (1) (1) 𝛿=1/2, identical gains 𝑘2 =𝑘3 = 0.527 ⋅ 𝑘; 𝑥3 (0) = col (1, −3) . (2) (2) 2 (2) 𝛿=2/3, nonidentical gains 𝑘2 = (2/3) ⋅ 𝑘,3 𝑘 = Denote by 𝑒(𝑡) = ∑𝑖=1 ‖𝑥𝑖(𝑡) −𝑖+1 𝑥 (𝑡)‖2 sum of error norms (1/3) ⋅ 𝑘. (1) or disagreement measure: 𝑒 (𝑡) error in the first case and 𝑒(2)(𝑡) 𝑘 𝑘=3/2= error in the second case. Results of 25 sec. simulations By Theorem 4 common gain is as follows: are shown in Figure 3. 𝜘 /𝑟(𝐾(2)𝐿) 𝐾(2) = (0, 𝑘(2),𝑘(2)) 0 , diag 2 3 . Identical gains are cho- Note that consensus vector (27) does not change for all (1) (1) (2) (2) sen such that ‖(𝑘2 ,𝑘3 )‖2 ≈ ‖(𝑘2 ,𝑘3 )‖2. (𝑘2,𝑘3)∈K since subsystem 𝑆1 is leader. 8 International Journal of Differential Equations

20 It is rediscovered that cycle digraph connection with 10 nonidentical actuation of nodes causes nonidentical impact on synchronous behaviour. Reachability of synchronization 0 corresponds to positive scalar—common gain. By construct- −10 ing boundary of sufficient gain region in 3-node digraph it is found that Jordan block of Laplace matrix (which affects −20 transient dynamics) appears in a direction of extremal point. −30 Comparison of dynamics is a subject of a future study. Geometrical interpretations which might be useful in theory −40 and applications were developed. −50

−60 Competing Interests

−70 The author declares that he has no competing interests. −20 −15 −10 −5 0 5 10 15 20 25 Figure 5: Trajectories of 20 agents on the same phase plane. Acknowledgments Trajectories of 𝑆1,...,𝑆10 colored blue, 𝑆11,...,𝑆15 colored red, and 𝑆16,...,𝑆20 colored green. The work was supported by the Ministry of Education, Youth and Sports of the Czech Republic within Project no. CZ.1.07/2.3.00/30.0034 (support for improving R&D teams and the development of intersectoral mobility at CTU in 4.2. Twenty-Node Digraph and Nonidentical Control. Let us Prague).TheauthorwouldliketothankA.L.Fradkovfor consider digraph shown in Figure 4 consisting of 20 agents problemstatementandZ.Hurak,A.Selivanov,andI.Herman 𝑆 ,...,𝑆 1 20 described in Section 3.1. This dodecahedron- for useful discussions. like digraph has Leading Set consisting of dynamic nodes 𝑆1,...,𝑆10 which are connected in directed circle. Let us choose ] =𝑘1 =𝑘2 =⋅⋅⋅=𝑘10 and 𝜇=𝑘11 = References 𝑘12 = ⋅⋅⋅ = 𝑘20. According to Theorem 6 gain ] should be ] > 0.5 ⋅ 2(𝜋/10) ≈ 4.74. [1] F. Bullo, J. Cortes,´ and S. Mart´ınez, Distributed Control of chosen cot For faster convergence Robotic Networks, Princeton University Press, 2009. ] let us choose ] = 5.5. Simulations show that 𝜇 can be ] 𝜇=1 [2]Y.Cao,W.Yu,W.Ren,andG.Chen,“Anoverviewofrecent chosen considerably less than .Letuschoose ,andlet progress in the study of distributed multi-agent coordination,” agents have different initial conditions. Results of numerical IEEE Transactions on Industrial Informatics,vol.9,no.1,pp. simulation show that such nonidentical gain choice provides 427–438, 2013. achievement of consensus. All trajectories of 20 agents on the [3]M.MesbahiandM.Egerstedt,Graph Theoretic Methods in same phase plane are shown in Figure 5. Multiagent Networks, Princeton University Press, 2010. 𝜇=] Numerical simulations also show that by choosing [4]R.Olfati-Saber,J.A.Fax,andR.M.Murray,“Consensusand and applying Theorem 9 for resulting Laplace matrix one can cooperation in networked multi-agent systems,” Proceedings of 𝜇=] > 4.74 obtain lower bound approximation . the IEEE, vol. 95, no. 1, pp. 215–233, 2007. [5]W.Ren,R.W.Beard,andE.M.Atkins,“Asurveyofconsensus 5. Reference Remarks problems in multi-agent coordination,” in Proceedings of the American Control Conference (ACC ’05),pp.1859–1864,Port- Assumptions of Theorem 2 are relaxed in comparison with land,Ore,USA,June2005. Theorem 2 from [23]. Proof of these results uses coordi- [6] C. W. Reynolds, “Flocks, herds, and schools: a distributed nate transformation as in [27]. Lemma 3 partially succeeds behavioral model,” Computer Graphics,vol.21,no.4,pp.25–34, Theorem 3 from [25]. Theorem 9 is a trigonometric form of 1987. Theorem 1 from [28] with different proof, which is potentially [7] I. Gutman and D. Vidovic, “The largest eigenvalues of adjacency extendable to higher orders of state space. and Laplacian matrices, and ionization potentials of alkanes,” Indian Journal of Chemistry—Section A,vol.41,no.5,pp.893– 896, 2002. 6. Conclusions [8]R.P.AgaevandP.Y.Chebotarev,“Thematrixofmaximal outgoing forests of a digraph and its applications,” Automation By means of Passification Method sufficient conditions of and Remote Control,vol.61,no.9,pp.15–43,2000. consensus with identical and nonidentical gains are derived. [9] P.Chebotarev and R. Agaev, “The forest consensus theorem and Synchronous behaviour (consensus vector) is described; it asymptotic properties of coordination protocols,”in Proceedings can be affected by nonidentical gains (nonidentity in actu- of the 4th IFAC Workshop on Distributed Estimation and ation) within Leading Set. Gain asymptote in growing cycle Control in Networked Systems, pp. 95–101, Koblenz, Germany, digraphs which have lowest communication cost for reaching September 2013. average consensus and consisting of double integrators is [10]Z.Li,Z.Duan,G.Chen,andL.Huang,“Consensusof studied. multiagent systems and synchronization of complex networks: International Journal of Differential Equations 9

a unified viewpoint,” IEEE Transactions on Circuits and Systems. I. Regular Papers,vol.57,no.1,pp.213–224,2010. [11] H. Zhang, F. L. Lewis, and A. Das, “Optimal design for synchro- nization of cooperative systems: state feedback, observer and output feedback,” IEEE Transactions on Automatic Control,vol. 56, no. 8, pp. 1948–1952, 2011. [12] B. Brogliato, B. Maschke, R. Lozano, and O. Egeland, Dissi- pative Systems Analysis and Control—Theory and Applications, Springer,NewYork,NY,USA,2007. [13] P. Kokotovic´ and M. Arcak, “Constructive nonlinear control: a historical perspective,” Automatica,vol.37,no.5,pp.637–662, 2001. [14] A. L. Fradkov, “Quadratic Lyapunov functions in the adaptive stability problem of a linear dynamic target,” Siberian Mathe- matical Journal,vol.17,no.2,pp.341–348,1976. [15] A. L. Fradkov, “Passification of non-square linear systems and feedback Yakubovich-Kalman-Popov Lemma,” European Journal of Control,vol.9,no.6,pp.577–586,2003. [16]C.I.Byrnes,A.Isidori,andJ.C.Willems,“Passivity,feedback equivalence, and the global stabilization of minimum phase nonlinear systems,” IEEE Transactions on Automatic Control, vol. 36, no. 11, pp. 1228–1240, 1991. [17] P. V. Kokotovic and H. J. Sussmann, “A positive real condition for global stabilization of nonlinear systems,” Systems & Control Letters, vol. 13, no. 2, pp. 125–133, 1989. [18] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, NY, USA, 196 0. [19] M. Marcus and H. Minc, A Survey Of Matrix Theory And Matrix Inequalities, Allyn and Bacon, Boston, Mass, USA, 1964. [20] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1990. [21] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, USA, 1969. [22] S.Boyd,L.Ghaoui,E.Feron,andV.Balakrishnan,Linear Matrix Inequalities in System and ,vol.15ofStudies in Applied ,SIAM,1994. [23] A. Fradkov and I. Junussov, “Synchronization of linear object networks by output feedback,” in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC ’11),pp.8188–8192,Orlando,Fla,USA, December 2011. [24] R. P. Agaev and P. Y. Chebotarev, “A cyclic representation of discrete coordination procedures,” Automation and Remote Control,vol.73,no.1,pp.161–166,2012. [25] J. A. Fax and R. M. Murray, “Information flow and cooperative control of vehicle formations,” IEEE Transactions on Automatic Control,vol.49,no.9,pp.1465–1476,2004. [26] I. Herman, D. Martinec, Z. Hurak, and M. Shebek, “Scaling in bidirectional platoons with dynamic controllers and propor- tional asymmetry,” IEEE Transactions on Automatic Control, http://arxiv.org/abs/1410.3943. [27] C. Yoshioka and T. Namerikawa, “Observer-based consensus control strategy for multi-agent system with communication time delay,” in Proceedings of the 17th IEEE International Conference on Control Applications (CCA ’08),pp.1037–1042, SanAntonio,Tex,USA,September2008. [28] W. Yu, G. Chen, and M. Cao, “Some necessary and sufficient conditions for second-order consensus in multi-agent dynami- cal systems,” Automatica,vol.46,no.6,pp.1089–1095,2010. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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