# Average Consensus Problems in Networks of Agents with Fixed and Switching Topology and Unknown Control Direction

Authors: Caixian Sun; Zhe Yang; Xiaowu Mu
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

This paper is devoted to the average consensus problems in directed networks of agents with unknown control direction. In this paper, by using Nussbaum function techniques and Laplacian matrix, novel average consensus protocols are designed for multiagent systems with unknown control direction in the cases of directed networks with fixed and switching topology. In the case of switching topology, the disagreement vector is utilized. Finally, simulation is provided to demonstrate the effectiveness of our results.


---

## Body

## 1. Introduction

An important problem that appears frequently in the context of coordination of multiagent systems is the consensus problem. It has been studied extensively during the past years. Vicsek and others put forward a model to portray MAS in 1995 [1], and since then a lot of consensus problems in different situations have been posed, such as one- and two-order MAS [2, 3], MAS with time delay [4], and switching topology [4–6]. Consensus problems have broad applications in a lot of fields, like computer science, biological science, automata theory, and so forth. Average consensus problem is one of the cases of consensus problems and has been studied a lot [7]. In this paper we consider the average consensus case.Generally, adaptive control method is useful to tackle uncertainties. In [8], an adaptive idea is used to design a robust neural network controller to deal with multiagent system with unknown nonlinear dynamics and unknown disturbances. In [9], two adaptive laws are designed to adjust the coupling weights and the neural network weights. The leader-follower synchronization problem of uncertain dynamical multiagent systems is addressed in [10], where the accurate model of each agent is not required. Moreover, in [11], a coordinated distributed adaptive control law is proposed to estimate the desired orbital velocity. But all works mentioned above cannot handle the consensus problem of multiagent systems without control direction.In fact, control direction may be unknown. Systems with unknown control direction have been studied first in the area of adaptive control in 1980s. Many results then are for linear systems. The first result was given by Nussbaum in [12], where an adaptive control law utilizing the Nussbaum-type gain was designed. Later the Nussbaum gain method was adopted in many linear and nonlinear systems to resist the lack of a priori knowledge of control direction. In [13], by Nussbaum gain, a systematic procedure is proposed to design global adaptive control of a class of nonlinear systems with unknown control direction. In [14], by incorporating Nussbaum technique, a partial-state feedback controller with unknown control direction is obtained.In this paper consensus problems are considered, but in a different situation that the control direction is unknown, we assume thatx˙i=aui, in which the sign of a is unknown. Novel control protocols are designed to reach average consensus under this consideration.Notation. Given that n∈N, we let 1(0) be the n×1 vector whose entries are 1(0).


## 2. Definitions and Preliminaries

At first, we introduce some preliminary knowledge of graph theory for the following analysis [15, 16].LetG=(V,E,A) be a weighted directed graph (digraph), consisting of a node (agent) set V={1,2,…,n}, an edge set E⊆V×V, and a weighted adjacency matrix A=[aij]∈Rn×n with nonnegative elements aij. We assume no self-loops in the graph, that is aii=0 for all i∈I. The node indexes belong to a finite index set I={1,2,…,n}. An edge of G is denoted by eij=(vi,vj). (If G is an undirected graph, eij∈E equals eji∈E.) The adjacency element is positive if the associated edge exists, that is, eji∈E⇔aij>0. The set of neighbors of node vi is denoted by
(1)Ni={vj∈V,(vj,vi)∈E}.
The indegree and outdegree of node vi are, respectively, defined as follows:
(2)deg⁡in(vi)=∑j=1n‍aij,degout(vi)=∑j=1n‍aji.A graph is said to beweight-balanced if the outdegree of each node equals its indegree.LetΔ=diag⁡(A1) be thedegree diagonal matrix of G. Let L=Δ-A be theLaplacian matrix of G. We note that L1=0.An oriented path fromvi1 to vik is a sequent list of edges (vis,vis+1)∈E, for s=1,…,k-1.A digraph is calledstrongly connected if, for any two distinct nodes vi, vj, there is a directed path from vi to vj.Lemma 1.
LetG=(V,E,A) be a weighted digraph with Laplacian L. If G is strongly connected, then rank(L)=n-1 [4].Letxi∈R denote the value of node vi. The value of a node might represent physical qualities including attitude, position, temperature, and voltage.We define theLaplacian potential of graph G as follows:
(3)ϕG(x)=xTLx.For undirected graphs, the Laplacian matrix is symmetric, and thus the Laplacian potential can be expressed as a quadratic form, that isϕG(x)=xTLx=(1/2)∑i,j‍aij(xj-xi)2, so it is nonnegative.But for digraph, when is it still true thatϕG(x) is nonnegative? We answer this by the following lemma.Lemma 2.
For weight-balanced digraphG,
(4)ϕG(x)=12∑i,j‍aij~(xj-xi)2≥0,
where aij~=(aij+aji)/2≥0.(The proof of this lemma is available in [4]).Lemma 3.
DigraphG=(V,E,A) is weight-balanced if and only if 1TL(G)=0T.Definition 4.
A smooth functionN(η) is calledNussbaum function if it is equipped with the following properties:
(5)lim⁡t→∞⁡sup⁡1t∫0t‍N(ζ)dζ=+∞,lim⁡t→∞⁡inf⁡1t∫0t‍N(ζ)dζ=-∞.Lemma 5.
LetN(η)=eη2cos⁡((π/2)η); then the following properties are satisfied.(i)
N(η)is a Nussbaum function.(ii)
Ifη(t)is a continuous, differentiable and nondecreasing function, andlimt→∞⁡η(t)=+∞, then(6)lim⁡t→∞⁡sup⁡1η(t)∫0η(t)‍ζN(ζ)dζ=+∞,lim⁡t→∞⁡inf⁡1η(t)∫0η(t)‍ζN(ζ)dζ=-∞.Proof.
(i) See [12] for the proof.
(ii) Ifη(t) is a continuous, differentiable and nondecreasing function, and lim⁡t→∞⁡η(t)=+∞, then there exist tk→+∞ such that η(tk)=k. Also we have η(t4k-1)=4k-1, η(t4k+1)=4k+1, and η(t4k+3)=4k+3.
DefineI(k)=∫0η(tk)‍ζN(ζ)dζ.
It is clear thatN(t) is positive on intervals of the form (4k-1,4k+1) and negative on intervals (4k+1,4k+3), k is a positive integer. From the nondecreasing property of η(t), we have that N(η(t)) is nonnegative on intervals of the form (t4k-1,t4k+1) and nonpositive on intervals (t4k+1,t4k+3), k is a positive integer.
It suffices to prove that(7)lim⁡k→∞1η(t4k+1)I(4k+1)=+∞,lim⁡k→∞1η(t4k+3)I(4k+3)=-∞.
To prove the former, first observe that(8)|∫0η(t4k-1)‍ζN(ζ)dζ|≤(4k-1)2e(4k-1)2,∫η(t4k-1)η(t4k+1)‍ζN(ζ)dζ≥∫η(tk¯1)η(tk¯2)‍ζN(ζ)dζ≥22(4k-12)e(4k-(1/2))2,
where η(k¯1)=4k-1/2, η(tk¯2)=4k+1/2, and tk¯1,tk¯2∈(t4k-1,t4k+1).
Combining (8) gives
(9)I(4k+1)≥e(4k-1)2[22(4k-12)e4k-3/4-(4k-1)2].
We have(10)lim⁡k→∞1η(t4k+1)I(4k+1)≥lim⁡k→∞1η(t4k+1)e(4k-1)2×[22(4k-12)e4k-3/4-(4k-1)2]≥lim⁡k→∞14k+1e(4k-1)2=+∞.
Similarly, we can prove that lim⁡k→∞⁡(1/η(t4k+3))I(4k+3)=-∞.
Thus (6) holds.Definition 6.
Letα=Ave(x)=N-11Tx, then
(11)x=α1+δ,
where δ∈Rn satisfies ∑i‍δi=0. We refer to δ as the (group)disagreement vector.Lemma 7 (Barbalat’s lemma [17]).
Letϕ:R→R be a uniformly continuous function on [0,∞). Suppose that lim⁡t→∞∫0t‍ϕ(τ)dτ exists and is finite. Then,
(12)ϕ(t)⟶0ast⟶∞.


## 3. Average Consensus Problem of MAS with Fixed Topology and Unknown Control Direction

Consider the case of fixed topologyG. Assume that G is strongly connected and weight balanced. xi∈R(i∈I) are state variables.Suppose each node of a graph is an agent with dynamics:(13)x˙i=aui,
where a∈R, |a|>0, and the sign of parameter a is unknown. Moreover, we assume a is bounded, that is, |a|≤amax⁡.We say that a state feedback(A)ui=ki(xj1,…xjmi)
is a protocol with topology G if the cluster Ji={vj1,…vjmi} of nodes with indexes j1,j2,…,jmi satisfies the property Ji⊆{vi}⋃Ni.We say protocol(A) solves the average consensus problem, if (A) satisfies that for all initial conditions, solution to (13) satisfies the average consensus condition, that is limt→∞x(t)=xave(0)1, where we let xave(t)=N-11Tx(t).This problem will be solved in the following theorem.Theorem 8.
G
=
(
V
,
E
,
A
) is a digraph which is strongly connected and weight-balanced. Consider dynamics (13), when u satisfies
(14)ui=ηN(η)∑j=1n‍aij(xj-xi),i=1,…,n,η˙=12∑ijaij(xj-xi)2,
where N(η)=eη2cos⁡((π/2)η), is a Nussbaum function and η=η(t) is a differentiable function in R. Then the average consensus problem is solved.Proof.
Together with (13) and (14), system can also be written in vector form as
(15)x˙=aηN(η)Lx,(16)η˙=xTLx.
Let us consider the positive definite function V=(1/2)xTx whose time derivative along (15) is given by
(17)V˙=xT(aηN(η))Lx=aηN(η)η.˙
Integrating from 0 intot, we arrive at
(18)12∥x(t)∥2-12∥x(0)∥2=∫0t‍aηN(η)η˙dt=∫0η(t)‍aσN(σ)dσ.
This implies that(19)∫0η(t)‍aσN(σ)dσ≥∫0η(0)‍aσN(σ)dσ-12∥x(0)∥2.
Then we can conclude thatη(t) is bounded; otherwise if η(t) is unbounded, observe that η˙=xTLx≥0 (Lemma 2) implies η(t) is nondecreasing; it follows lim⁡t→∞⁡η(t)=+∞. Furthermore, inequality (19) implies that
(20)lim⁡t→∞inf⁡1η(t)∫0η(t)‍ζN(ζ)dζ≥lim⁡t→∞1η(t)(∫0η(0)‍aσN(σ)dσ-12∥x(0)∥2).
Then from Lemma 5 we can get a contradictory inequality -∞≥0. So η(t) is bounded.
The boundedness ofη(t) together with the nondecreasing property of η(t) leads to the existence of the limit of η(t).
Moreover, from (18) we can conclude the boundedness of ∥x(t)∥2, and thus η˙ and x˙ are bounded.
From(xTLx)′=x˙TLx+xTLx˙, (xTLx)′ is bounded, which implies xTLx is uniformly continuous.
From(21)lim⁡t→+∞∫0t‍(x(τ)TLx(τ))dτ=lim⁡t→+∞η(t)-η(0)
we get that limt→+∞∫0t‍(x(τ)TLx(τ))dτ exists and is finite. By Lemma 7 (Babalat’s), we arrive at limt→+∞xTLx=0.
Owing to Lemma2, ϕG(x)=xTLx=(1/2)∑i,j‍aij~(xj-xi)2≥0 and aij~≥0, and thus limt→+∞|xj-xi|=0  (∀i,j∈I). Moreover, we denote the equilibrium of (15) as x*, and then
(22)aN(η)Lx*=0⟹x*=c1,thatis,xi*=xj*(∀i,j∈I).
Furthermore, letα(t)=Ave(x(t))=N-11Tx(t), then
(23)α˙=N-11Tx˙=N-1aηN(η)1TLx=0
(1TL=0 by Lemma 3); that is, α is a constant, which means the average of x(t) is preserved as time varies.
From above, we obtainlimt→+∞x(t)=x* and xi*=N-11Tx(0)=α, (∀i∈I).


## 4. Average Consensus Problem of MAS with Switching Topology and Unknown Control Direction

Consider a network with mobile agents. When there exists an obstacle between two agents; the communication links may fail. Then the opposite situation may arise that new links between nearby agents are created. Here we are interested to investigate whether it is still possible to reach a consensus or not in the case of a network with switching topology.Consider a switching system with statex∈Rn and switching graph G, where G belongs to a finite set
(24)Γn={G={V,E,A}:rank⁡(L(G))=n-1,1TL(G)=0T}.In fact the elements of this set are graphs of ordern that are strongly connected and weight-balanced.The switching topology of the network is modeled by using switching graphs. The switching graph isGk∈Γn, Gk=(V,Ek,Ak), k=s(t), where s(t):[0,∞)→N denotes theswitching signal, which is assumed to be a piecewise constant function continuous from the right.The protocol(A) solves the consensus problem in the way very similar to the case of the fixed topology; the only difference is that (A) is a protocol of topology Gk.This problem will be solved in the following theorem.Theorem 9.
G
k
=
(
V
,
E
k
,
A
k
)
∈
Γ
n is a switching graph, where k=s(t) is any switching signal. Lk=L(Gk). Consider (13), when u satisfies
(25)ui=ηN(η)∑j=1n‍aij(k)(xj-xi),i=1,…,n,η˙=12∑ijaij(k)(xj-xi)2,
where N(η) is a Nussbaum function and η=η(t) is a differentiable function in R. Then the average consensus problem is solved.Proof.
Whenu satisfies (25), (13) becomes
(26)x˙=aηN(η)Lkx,η˙=xTLkx.
Owing to (11) (δ is the disagreement vector), together with Lk1=0 and 1TL(G)=0T, (26) becomes
(27)δ˙=aηN(η)Lkδ,(28)η˙=δTLkδ.
Let us consider the positive definite functionV=(1/2)δTδ whose time derivative along (27) is given by V˙=δT(aηN(η))Lkδ=aηN(η)η˙.
Integrating from 0 tot, we arrive at
(29)12∥δ(t)∥2-12∥δ(0)∥2=∫0t‍aηN(η)η˙dt=∫0η(t)‍aσN(σ)dσ.
This implies
(30)∫0η(t)‍aσN(σ)dσ≥∫0η(0)‍aσN(σ)dσ-12∥δ(0)∥2.
Very similar to the proof of Theorem8, we obtain that η(t) is bounded, and thus the limit of η(t) exists. (η˙=δTLkδ≥0.)
Moreover from (29) we can conclude the boundedness of ∥δ(t)∥2, and thus (27) implies δ˙ is bounded.
Hence,(δTLkδ)′ is bounded, which can lead to the fact that δTLkδ is uniformly continuous.
What’s more, from(31)lim⁡t→+∞⁡∫0t‍(δ(τ)TLkδ(τ))dτ=lim⁡t→+∞η(t)-η(0)
we get limt→+∞∫0t‍(x(τ)TLx(τ))dτ exists and is finite; therefore applying Lemma 7 (Babalat’s) we arrive at limt→+∞δTLkδ=0.
Owing to Lemma2, ϕG(δ)=δTLkδ≥0, and thus limt→+∞|δj-δi|=0  (∀i,j∈I).
Moreover, letδ* be the equilibrium of (27); then there exist an e∈R such that δ*=e1, that is, δi*=δj*=e.
Due to the fact that∑i‍δi=0 is valid for every time, we can conclude that δi*=δj*=e=0. Therefore, limt→+∞δ(t)=δ*=0, and thus limt→+∞x(t)=α=Ave(x(0)).
Consequently, the average consensus problem is solved.


## 5. Simulation Results

In this part we use the concrete graphs to show the two results we obtained, respectively. Here the initial condition of the states is taken asx(0)=(3,5,-2,1,-7)′, and the Nussbaum function as N(η)=eη2cos⁡((π/2)η). Figure 1 shows four different digraphs each with n=5 nodes. We can observe that they are all balanced and strongly connected. To illustrate the effect of unknown control direction, in the simulation we randomly specify two constants with different signs to stand for a, respectively. The trajectories of the node values of Ga with the input coefficient a that equals 0.8147 and −1.5386 are displayed in Figure 2. We observe that they achieve consensus after some time, just as our theorem demonstrates. For the switching topology case, Figure 3(a) gives the model of switching and Figure 3(b) shows the corresponding trajectories in the cases of a=4.8219 and a=-1.3458. This result coincides with our conclusion of Theorem 9.Four balanced and strongly connected digraphs: (a)Ga, (b) Gb, (c) Gc, (d) Gd.
(a)
(b)
(c)
(d)Figure 2
Trajectories of the node values ofGa with a=0.8147 and a=-1.5386.(a)The switching model of four digraphs. The following two graphs are trajectories of the node values for the switching information flow of (a) witha=4.8219 and a=-1.3458.
(a)
(b)


## 6. Conclusion

In this paper, we present the convergence analysis of consensus protocols for networks with fixed and switching topology and unknown control direction. By Nussbaum function techniques and graph theory, two consensus protocols are constructed to tackle the difficulty caused by the unknown control direction. Finally, simulations are provided to demonstrate the effectiveness of our results.



---

*Source: 272876-2014-06-15.xml*