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.
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 [4ā6]. 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 that , in which the sign of is unknown. Novel control protocols are designed to reach average consensus under this consideration.
Notation. Given that , we let be the vector whose entries are .
2. 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 , an edge set , and a weighted adjacency matrix with nonnegative elements . We assume no self-loops in the graph, that is for all . The node indexes belong to a finite index set . An edge of is denoted by . (If is an undirected graph, equals .) The adjacency element is positive if the associated edge exists, that is, . The set of neighbors of node is denoted by The indegree and outdegree of node are, respectively, defined as follows:
A graph is said to be weight-balanced if the outdegree of each node equals its indegree.
Let be the degree diagonal matrix of . Let be the Laplacian matrix of . We note that .
An oriented path from to is a sequent list of edges , for .
A digraph is called strongly connected if, for any two distinct nodes , , there is a directed path from to .
Lemma 1. Let be a weighted digraph with Laplacian . If is strongly connected, then [4].
Let denote the value of node . 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 , so it is nonnegative.
But for digraph, when is it still true that is nonnegative? We answer this by the following lemma.
Lemma 2. For weight-balanced digraph , where .
(The proof of this lemma is available in [4]).
Lemma 3. Digraph is weight-balanced if and only if .
Definition 4. A smooth function is called Nussbaum function if it is equipped with the following properties:
Lemma 5. Let ; then the following properties are satisfied.(i) is a Nussbaum function.(ii)If is a continuous, differentiable and nondecreasing function, and , then
Proof. (i) See [12] for the proof.
(ii) If is a continuous, differentiable and nondecreasing function, and , then there exist such that . Also we have , , and .
Define .
It is clear that is positive on intervals of the form and negative on intervals , is a positive integer. From the nondecreasing property of , we have that is nonnegative on intervals of the form and nonpositive on intervals , is a positive integer.
It suffices to prove that
To prove the former, first observe that
where , , and .
Combining (8) gives
We have
Similarly, we can prove that .
Thus (6) holds.
Definition 6. Let , then where satisfies . We refer to as the (group) disagreement vector.
Lemma 7 (Barbalatās lemma [17]). Let be a uniformly continuous function on . Suppose that exists and is finite. Then,
3. 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. are state variables.
Suppose each node of a graph is an agent with dynamics: where , , and the sign of parameter is unknown. Moreover, we assume is bounded, that is, .
We say that a state feedback is a protocol with topology if the cluster of nodes with indexes satisfies the property .
We say protocol solves the average consensus problem, if satisfies that for all initial conditions, solution to (13) satisfies the average consensus condition, that is , where we let .
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 , is a Nussbaum function and is a differentiable function in . 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 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 (Lemma 2) implies is nondecreasing; it follows . Furthermore, inequality (19) implies that
Then from Lemma 5 we can get a contradictory inequality . So is bounded.
The boundedness of together with the nondecreasing property of leads to the existence of the limit of .
Moreover, from (18) we can conclude the boundedness of , and thus and are bounded.
From , is bounded, which implies is uniformly continuous.
From
we get that exists and is finite. By Lemma 7 (Babalatās), we arrive at .
Owing to Lemma 2, and , and thus āā. Moreover, we denote the equilibrium of (15) as , and then
Furthermore, let , then
( by Lemma 3); that is, is a constant, which means the average of is preserved as time varies.
From above, we obtain and , .
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 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 , , , where 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.
Theorem 9. is a switching graph, where is any switching signal. . Consider (13), when satisfies where is a Nussbaum function and is a differentiable function in . Then the average consensus problem is solved.
Proof. When satisfies (25), (13) becomes
Owing to (11) ( is the disagreement vector), together with and , (26) becomes
Let us consider the positive definite function whose time derivative along (27) is given by .
Integrating from 0 to , we arrive at
This implies
Very similar to the proof of Theorem 8, we obtain that is bounded, and thus the limit of exists. (.)
Moreover from (29) we can conclude the boundedness of , and thus (27) implies is bounded.
Hence, is bounded, which can lead to the fact that is uniformly continuous.
Whatās more, from
we get exists and is finite; therefore applying Lemma 7 (Babalatās) we arrive at .
Owing to Lemma 2, , and thus āā.
Moreover, let be the equilibrium of (27); then there exist an such that , that is, .
Due to the fact that is valid for every time, we can conclude that . Therefore, , and thus .
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 as , and the Nussbaum function as . Figure 1 shows four different digraphs each with 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 ā1.5386 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 and . This result coincides with our conclusion of Theorem 9.
(a)
(b)
(c)
(d)
(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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is supported by the National Science Foundation of China under Grant no. 60874006. The authors highly appreciate the above financial support.