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

This paper is devoted to the average consensus problems in directed networks of agents with unknown control direction. In this paper, by usingNussbaum function techniques and Laplacianmatrix, novel average consensus protocols are designed formultiagent systemswith unknown control direction in the cases of directed networkswith 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.


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 twoorder MAS [2,3], MAS with time delay [4], and switching topology [4][5][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 Nussbaumtype 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 that ẋ  =   , in which the sign of  is unknown.Novel control protocols are designed to reach average consensus under this consideration.

Definitions and Preliminaries
At first, we introduce some preliminary knowledge of graph theory for the following analysis [15,16].
Let  = (, , ) be a weighted directed graph (digraph), consisting of a node (agent) set  = {1, 2, . . ., }, an edge set  ⊆  × , and a weighted adjacency matrix  = [  ] ∈ R × with nonnegative elements   .We assume no self-loops in the graph, that is   = 0 for all  ∈ .The node indexes belong to a finite index set  = {1, 2, . . ., }.An edge of  is denoted by   = (V  , V  ).(If  is an undirected graph,   ∈  equals   ∈ .)The adjacency element is positive if the associated edge exists, that is,   ∈  ⇔   > 0. The set of neighbors of node V  is denoted by The indegree and outdegree of node V  are, respectively, defined as follows: A graph is said to be weight-balanced if the outdegree of each node equals its indegree.
An oriented path from A digraph is called strongly connected if, for any two distinct nodes V  , V  , there is a directed path from V  to V  .Lemma 1.Let  = (, , ) be a weighted digraph with Laplacian .If  is strongly connected, then () =  − 1 [4].
Let   ∈  denote the value of node V  .The value of a node might represent physical qualities including attitude, position, temperature, and voltage.
We define the Laplacian potential of graph  as follows: For undirected graphs, the Laplacian matrix is symmetric, and thus the Laplacian potential can be expressed as a quadratic form, that is But for digraph, when is it still true that   () is nonnegative?We answer this by the following lemma.
(The proof of this lemma is available in [4]).
where  ∈   satisfies ∑    = 0. We refer to  as the (group) disagreement vector.

Average Consensus Problem of MAS with Fixed Topology and Unknown Control Direction
Consider the case of fixed topology .Assume that  is strongly connected and weight balanced.  ∈ R ( ∈ ) are state variables.Suppose each node of a graph is an agent with dynamics: where  ∈ R, || > 0, and the sign of parameter  is unknown.Moreover, we assume  is bounded, that is, || ≤  max .We say that a state feedback is a protocol with topology  if the cluster   = {V  1 , . . .V    } of nodes with indexes  1 ,  2 , . . .,    satisfies the property   ⊆ {V  } ⋃   .
This problem will be solved in the following theorem.
Theorem 8.  = (, , ) is a digraph which is strongly connected and weight-balanced.Consider dynamics (13), when  satisfies where () =   2 cos((/2)), is a Nussbaum function and  = () 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 Let us consider the positive definite function  = (1/2)   whose time derivative along ( 15) is given by Integrating from 0 into , we arrive at This implies that Then we can conclude that () is bounded; otherwise if () is unbounded, observe that η =    ≥ 0 (Lemma 2) implies () is nondecreasing; it follows lim  → ∞ () = +∞.Furthermore, inequality (19) implies that lim Then from Lemma 5 we can get a contradictory inequality −∞ ≥ 0. So () is bounded.The boundedness of () together with the nondecreasing property of () leads to the existence of the limit of ().

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 state  ∈   and switching graph , where  belongs to a finite set In fact the elements of this set are graphs of order  that are strongly connected and weight-balanced.
The switching topology of the network is modeled by using switching graphs.The switching graph is   ∈ Γ  ,   = (,   ,   ),  = (), where () : [0, ∞) →  denotes the switching signal, which is assumed to be a piecewise constant function continuous from the right.
The protocol () solves the consensus problem in the way very similar to the case of the fixed topology; the only difference is that () is a protocol of topology   .This problem will be solved in the following theorem.
Consequently, the average consensus problem is solved.

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 as (0) = (3, 5, −2, 1, −7)  , and the Nussbaum function as () =   2 cos((/2)).Figure 1 shows four different digraphs each with  = 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 , respectively.The trajectories of the node values of   with the input coefficient  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

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.
(a) gives the model of switching and Figure 3(b) shows the corresponding trajectories in the cases of  = 4.8219 and  = −1.3458.This result coincides with our conclusion of Theorem 9.

Figure 3 :
Figure 3: (a) The switching model of four digraphs.The following two graphs are trajectories of the node values for the switching information flow of (a) with  = 4.8219 and  = −1.3458.