Static consensus in passifiable linear networks

Sufficient conditions of consensus (synchronization) in networks described by digraphs and consisting of identical determenistic 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.


Introduction
Control of multi-agent systems has attracted significant interest in last decade since it has a great technical importance [4,14,17] and relates to biological systems [18].
In consensus problems agents communicate via decentralized controllers using relative measurements with a final goal to achieve common behaviour (synchronization) which can evolve in time. Many approaches have been developed for a different problem settings.
In [22] performance in undirected graphs are studied. In [21] adaptive coupling strengths are considered. Communication between all agents except leader (if its presented) assumed to be bidirectional.
Conception of synchronization region in complex plane for a networks consisting of linear dynamical systems is introduced in [12]. In [23] this conception is used for analysis of synchronization with leader. Problem is solved using Linear Quadratic Regulator approach in cases when full state is available for measurement and when its not. In last case observers are constructed.
Laplace matrix and its spectrum plays crucial role in description and analysis of consensus problems. It has broad applications, e.g. [10]. Conditions which are causing unit multiplicity of digraph Laplace matrix zero eigenvalue have been found in [1]. Analysis of tree strucure and Laplace matrices spectrum of digraphs are also studied by these authors. Work [5] contains examples of out-forests as well as useful graph theoretical concepts and can be recommended as an entry reading to the research of these authors on algebraic digraph theory and consensus problems.
Scalability is one of the questions which should be considered. Control of platoons -one dimensional formations of automated vehicles, is a field related to consensus problems. It is known that in platoons string instability effect (amplified response of trailing vehicles to a leader disturbance while platoon length is growing) is taking place [19,20,16]. This effect can be avoided by adding certain level of centralization [16] or, possibly, by considering more sophisticated communication strategy and nonlinear control.
In this paper results of Passification Method [8,9] are used to synthesize a decentralized control law and to derive sufficient conditions of full state synchronization by relative outputs in a networks described by digraphs with dynamical nodes modelled as linear systems. Assumptions made on network topology are minimal, both leader and leaderless cases are treated.
Paper is organized as follows. In Section 2.1 some preliminaries are given. Problem statement and assumptions are given in Sect. 2 2 Theoretical study

Preliminaries and notations
In this section notations, some terms of graph theory and Passification Lemma are listed.

Notations
Notation . 2 stands for Euclidian norm. For two symmetric matrices M 1 , M 2 inequality M 1 > M 2 means that matrix M 1 − M 2 is positive definite. Notation col(v 1 , ..., v d ) stands for vector (v 1 , ..., v d ) T . Identity matrix of size d is denoted by I d . Vector 1 d = (1, 1, . . . , 1) is vector of size d and consisting of ones. Vector 0 d is defined similarly. Matrix diag(v 1 , . . . , v d ) is square matrix whose i-th element on main diagonal is v i , i = 1, . . . , d; other entries are zeroes. Notation ⊗ stands for Kronecker product of matrices. Definition and properties of Kronecker product, including eigenvalues property, can be found in [2,13].

Terms of graph theory
It is assumed hereafter that graphs does not have a self-loops, i.e. for any vertex α ∈ V arc (α, α) / ∈ E. Digraph is called directed tree if all it vertices except one (called root) have exactly one parent Let us agree that in any arc (α, β) ∈ E vertex β is parent or neighbour. Directed spanning tree of a digraph G is a directed tree formed of all digraph G vertices and some of its arcs such that there exists path from any vertex to the root vertex in this tree.
A digraph is called weighted if to any pair of vertices α, β ∈ E number w(α, β) ≥ 0 is assigned such that: A digraph in which all nonzero weights are equal to 1 will be referred as unit weighted.
An adjacency matrix A(G) is N ×N matrix whose i−th, j−th entry is equal to w(α i , α j ), i, j = 1, . . . , N.
Laplace matrix of digraph G is defined as follows: Matrix L(G) always has zero eigenvalue with corresponding right eigenvector 1 N : L(G) · 1 N = 0 · 1 N . By construction and Gershgorin Circle Theorem all eigenvalues of L have nonnegative real parts.

Passification Lemma
Problem of linear system passification is a problem of finding static linear feedback which is making initial system passive. It was solved in [8,9] for nonsquare SIMO and MIMO systems including case of complex parameters. Brief outline of SIMO systems passification is given below. Let A, B, C be real matrices of sizes n × n, n × 1, n × l accordingly. Denote by χ(s) = C T (sI − A) −1 B, s ∈ C. Let vector g ∈ R l . If numerator of function g T χ(s) is Hurwitz with degree n − 1 and has positive coefficients then function g T χ(s) is called hyper-minimum-phase.
Lemma 1 (Passification Lemma [8,9]) If there exists vector g ∈ R l such that function g T χ(s) is hyper-minimum-phase, then following is true. There exists number κ 0 > 0 such that for any κ > κ 0 there exists n × n real matrix H = H T > 0 satisfying following matrix relations

Problem statement and assumptions
Consider a network consisting of N agents modelled as linear dynamical systems: where i = 1, . . . , N, x i ∈ R n -state vector, y i ∈ R l -output or measurements vector, u i ∈ R 1 -input or control, A, B, C are real matrices of according size. By associating agents with N vertices of unit weighted digraph G and introducing set of arcs one can describe information flow in the network. For i = 1, . . . , N let us introduce notation for relative outputs where N i is a set of i-th agents neighbours. Problem is to design controllers which use relative outputs and ensure achievement of the state synchronization (consensus) of all agents: In the case of synchronization achievement asymptotical behaviour of all agents will be described by same time-dependant consensus vector which is denoted hereafter by c(t) : lim t→∞ (x i (t) − c(t)) = 0, i = 1, . . . , N.
Let us make following assumption about dynamics of a single agent. A1) There exists vector g ∈ R l such that function g T χ(s) = g T C T (sI n − A) −1 B is hyperminimum-phase. Now let us make an assumption on graph topology. A2) Digraph G has a directed spanning tree. Zero eigenvalue of Laplace matrix L has unit multiplicity iff this assumption holds [1].

Static identical control
Denote r(L) = min Re λ i where λ i are eigenvalues of L. Under assumption A2 zero eigenvalue is simple. By properties of L other eigenvalues lie in open right half of complex plane, so r(L) is positive number. Suppose that assumption A1 holds with known vector g ∈ R l . Consider following static consensus control with identical gain k s ∈ R 1 , k s > 0 : where relative output y i (t) has been defined in previous section. Denote by v(L) ∈ R N left eigenvector of L which is corresponding to zero eigenvalue and scaled such that v(L) Theorem 1 Let assumptions A1 and A2 hold. Then for all k s such that controller (4) ensures achievement of goal (3) in dynamical network (2); asymptotical behaviour is described by following consensus vector Proof. Closed loop system (2), (4) can be rewritten in a following forṁ Consider nonsingular real matrix P such that where Λ e ∈ R (N −1)×(N −1) . All eigenvalues of Λ e have positive real parts. By considering first (zero) columns of matrices P Λ = LP and (P T ) −1 Λ T = L T (P −1 ) T we can accept that first column of P is 1 N and first row of P −1 is v(L) T . Let us apply coordinate transformation z(t) = (P −1 ⊗ I n )x(t) and rewrite (6): where z 1 ∈ R n , z e ∈ R (N −1)n , z = col(z 1 , z e ). If zero solution of (8) is globally asymptotically stable, then the statements of theorem are true. For any fixed k s satisfying (5) there exists 0 < ε s < 1 such that ε s k s > κ 0 r(L) .
We can rewrite last inequality By assumption A1 there exists H = H T > 0 such that (1) is true with κ = ε s k s r(L), since κ > κ 0 . Considering following Lyapunov function V (z e (t)) = z T e (t)(Q ⊗ H)z e (t) and taking its time derivative along the nonzero trajectories of (8), we obtain Matrix relations (1) have been used here. Last inequality concludes proof.

Nonidentical control and Gain Region
Let the initial digraph G be unit weighted. Let us fix Laplace matrix L and consider static control with nonidentical gains k i > 0 : Denote Without loss of generality we can assume that network does not have a leader, since in leader case we can reduce following consideration of synchronization gain region to lower dimension N − 1.
Denote by K ′ = diag(k ′ 1 , k ′ 2 , . . . , k ′ N ). Laplace matrices L and K ′ L correspond to same digraphs which differ only in weights of arcs (recall that k i > 0, i = 1, . . . , N ). Equation for closed loop system (2),(9) can be rewritten as followṡ By repeating proof of Theorem 1 we can formulate following result.

Theorem 2 Let assumptions A1 and A2 hold. Then for all
controller (9) ensures achievement of goal (3) in dynamical network (2); asymptotical behaviour is described by following consensus vector Denote by K ⊂ O region in orthant such that for any (k 1 , . . . , k N ) ∈ K control (9) ensures achievement of the goal (3) in network (2), (9). Consider following region which is subset of K : K r ⊂ K. Let us denote Point on S ′ ε determines ray (half-line) in O with initial point at the origin. According to Theorem 2, by moving along this ray from origin, i.e. increasing k s , we will reach K r . Consider map which is continious as a composition of continious maps [11]. Image of this map is a subset of boundary ∂K r , therefore, by continuity of map h, boundary ∂K r is a hypersurface in R N . Domain S ′ ε is compact, so we can apply Weierstrass Extreme Value Theorem and arrive at following lemma.
⊂ ∂K r is continious and has minimum and maximum.
Generally, hypersurface ∂K r is not smooth in all its points.

Adaptive control
Consider the following adaptive controller: Nonrigorously, when adaptive controller (11) is applied all k i (t) will grow until they reach K.

Agents description
Suppose that each agent S i in a network is modelled as followṡ For g = 1 transfer function g T χ(s) = C T (sI 2 − A) −1 B = s+1 s 2 is hyper-minimum-phase. It can be shown that number κ 0 = 1. In the next sections different digraphs describing network topology will be considered.

Cycle digraph and scalability
Denote by L C N Laplace matrix which is corresponding to unit weighted cycle digraph which is consisting of N nodes S i with the set of arcs Eigenvalues of L C N are evenly located at circle in complex plane [6]: So, number r(L C N ) from (5) is equal to 1 − cos 2π N . For a large increasing number of agents N gain k s should grow as N 2 : inf where inf k s = κ 0 /r(L C N ) is bound given by Theorem 1. Relation (13) can be obtained using Taylor series.
Denote by ρ N ratio of gain synchronization bound k sim (N ) obtained by numerical simulations to bound given by Theorem 1: .
In the next table approximate values of ρ N for some N are given. Simulations was performed with dynamic nodes described in section 3.1 and identical gains. If a ratio of true gain synchronization bound to inf k s is near to 1 (or nondecreasing), while number of agents is large and growing, then it is possible to conclude that consensus in large cycle digraphs is hard to achieve, since an arbitrary high gains are not physically realizable, see (13).
On other hand, it is worth noting that cycle digraph is the graph with smallest number of edges which is delivering average consensus among all its nodes.

Small digraph and gain region
Consider digraph shown on Fig. 1 with dynamic nodes described in section 3.1. By Lemma 2 distance from origin to ∂K r reaches minimum. Boundary ∂K r for considering case is presented on Fig. 2. By variating eigenvalues of matrix K ′ L we can state that minimum is realized on a point for which k 2 : k 3 = 2 : 1.
Let us compare performance in two cases: • Identical gains k 2 , k . By Theorem 2 factor k is as follows Denote by e(t) = 2 i=1 x i (t) − x i+1 (t) 2 sum of error norms or disagreement measure; e (1) (t) error in the first case, e (2) (t) error in the second case. Results of 25 sec. simulations are shown on Fig. 3. From this figure one can observe following: 1. Control with nonidentical gains demonstrates better performance on initial period.

Overall synchronization time is almost same in both cases.
First observation does not hold generally for simulations with other initial conditions; second does hold. Note that consensus vector (10) does not changes for all (k 2 , k 3 ) ∈ K since subsystem S 1 is leader.

Conclusions
By means of Passification Method [9] sufficient conditions of consensus with identical and nonidentical gains are derived. Synchronous behaviour (consensus vector) is described, it can be affected by nonodentical gains in leaderless case. In [15] it is stated that there is tradeoff between performance and communication cost. Growth of gain magnitude in growing circle digraphs which have lowest communication cost for reaching average consensus is studied. For other graph topologies gain magnitude should grow slower. Three node digraph with a leader is studied: control with nonidentical gains provide performance which is not worse than identical control.