Consensus Control of Multiagent Systems with High-Order Nonlinear Inaccurate Dynamics and Dynamically Switching Undirected Topologies

This paper investigates the consensus control of a class of high-order nonlinear multiagent systems, whose topology is dynamically switching directed graph. First, the high-order nonlinear dynamics is transformed into the one-order dynamics by structuring a slidingmode plane; then, two consensus control protocols of the one-order dynamics are designed by feedback linearization, one of which is based on PD (proportion and derivative) and the other is based on PID (proportion, integral and derivative). Under these control protocols, it is proved that the consensus of new variable only requires a weaker topology condition; next, we prove that the consensus of the new variable is sufficient to the consensus of the states of multiagent systems, which implies that it only requires a weaker topology condition for the consensus of multiagent systems; finally, the study of an illustrative example with simulations shows that our results as well as designed control protocols work very well in studying the consensus of this class of multiagent systems.

The consensus of multiagent systems with a switching interaction topology has been attracting many scholars [32][33][34].In [32], the authors considered multiagent systems with first-order integrator dynamics under the switching topology and provided a control protocol under which the dynamics could achieve consensus with a rather weak topology conditions.However, they considered the second-order integrator dynamics in [33] and obtained a condition that needs the stronger topology requirement than the one in [32]; what is more, they pointed out that the condition in [32] was not sufficient to the second-order consensus algorithm with their control protocol.The authors in [34] compared the control protocol in [32] with the one in [33] and pointed out that the topology requirement was dependent on the control protocol.They transformed the high-order linear systems to first-order integrator dynamics by variable substitution and obtained the consensus conditions of high-order linear multiagent systems with a rather weak topology requirement.
The idea in [34] is interesting for high-order systems, which motivate us to consider a class of high-order nonlinear multiagent systems, whose topology is dynamically switching directed graph.We prove that the nonlinear dynamics can achieve consensus with a rather weak topology conditions similar to [32].First, the high-order dynamics can realize dimensionality reduction by structuring a sliding mode plane; then, two consensus control protocols are designed by feedback linearization, one of which is based on PD (proportion and derivative), and the other is based on PID (proportion, integral, and derivative), such that the new model can achieve the consensus under a rather weak topology condition; next, we prove that the consensus of new control protocols is sufficient to the consensus of nonlinear multiagent systems.Thus, we give a rather weak consensus conditions of high-order nonlinear multiagent systems, in which the union graph has a spanning tree frequently enough.Finally, the study of an illustrative example with simulations shows that our results as well as designed control protocols work very well in studying this class of multiagent systems.
The main contribution of this paper contains: (1) a rather weak topology condition is given for the consensus of nonlinear multiagent systems with switching structure; (2) we present two control protocols for this class of systems, under which the consensus is proved; (3) the reduce-order idea is used to transform the high-order dynamics into one-order dynamics, which simplify the difficulty of problem.
The remainder of the paper is organized as follows.Section 2 is the backgrounds and preliminaries.Section 3 is the main results of the paper.In this section, the highorder nonlinear dynamics is transformed into one-order linear dynamics, and two control protocols are designed.In Section 4, we give an illustrative example to support our new results followed by the conclusion in Section 5.

Backgrounds and Preliminaries
In [32], Ren and Beard considered a one-order linear system under the switching topology: and they designed the control protocol as follows: it was proved system (1) could achieve consensus under protocol (2) even if the switching topology satisfied a rather weak condition.Specifically to say, it could be formulated as the following lemma.Furthermore, they considered the second-order integrator in [33]: where   = [ 1 ,  2 ] was the state of Agent , and   was the control protocol.They designed the control protocol as where () was a positive scalar at time .In that book, they provided a result which needed a more stronger connectivity assumption; that is, the interaction graph needed a directed spanning tree at each time instant.Moreover, they provided some simulation examples to illustrate that, under control protocol (4), the assumption of Lemma 1 could not ensure the consensus of second-order consensus generally.Su and Lin compared the cases as above in [34] and presented a reduced-order idea.They considered the highorder linear multiagent dynamics as follows: where x i ∈   was the state vector of Agent , u i was the control input the Agent , and (A, B) was controllable.By transforming the system (5) to one-order dynamics, they presented a control protocol under which the high-order linear multiagent systems could obtain consensus with a rather weak topology.
Inspired by [34], we will consider a class of multiagent systems with  agents, which each has -order nonlinear dynamics described by where   ∈ R is the state of agent ,   ∈ R is the control input of agent , and (  ) or (  ) is inaccurate.
Denoting  1 =   , the dynamics (6) can be rewritten as The objective of this study is to design a controller such that the agents described as the system ( 6) or ( 7) can achieve consensus under the dynamically switching directed interaction topologies.
In the following, we will provide some fundamental knowledge on algebraic graph theory, which will be used in the development of this research.
Suppose G = {V, E} be a directed graph of -th order with the set of nodes V := {V 1 , V 2 , ⋅ ⋅ ⋅ , V  }, and the set of edges (i.e., ordered pairs of the agents) E ⊆ V×V.The matrix A = [  ] × is named the adjacency matrix of the graph G, where   is the weight of Agent  to Agent .For any ,  ∈ V,   > 0 if and only if  ∈ N  , where In this paper we just consider the case of simple graphs, that is,   ∉ E,  = 1, 2, ⋅ ⋅ ⋅ , .The matrix D = [  ] ∈ R × is the valency matrix of the topology G, where   is defined as Moreover, the matrix L = D − A is known as the graph's Laplacian matrix.

Main Results
This section studies the consensus of the multiagent dynamics (6).We will consider the following scenarios: (I) the dynamics (  ),  = 1, 2, ⋅ ⋅ ⋅ ,  is inaccurate, but its estimate is known; (II) the gain (  ),  = 1, 2, ⋅ ⋅ ⋅ ,  is inaccurate, but we know its upper bound and its lower bound.Before giving our main results, another lemma will be useful in our study.
Then design the control protocol as and where under the protocol ( 13), and we have the following result.
Theorem 3. Consider the multiagent system of  agents (6) under the control protocol specified by (13).If the conditions of Lemma 1 are satisfied, then the distributed control protocol (13) achieves consensus asymptotically for the multiagent systems specified by ( 6) and the consensus is reached at a constant state [, 0, 0, ⋅ ⋅ ⋅ , 0]  .
However, the terms  sgn(  ) have a strong flutter, which may need a strong control power.Next, we will consider the stable under any precision.Definition 4. The system ẋ = () is said to any precision stable if for any designated  0 , there exists , such that ‖()‖ <  0 , when  > .
Consider the set () = {|(, ) ≤ Φ}, where Φ > 0 is the precision of system state.Letting where   is the same as (12), under the control protocol (21), we have the following theorem.
Theorem 5. Consider the multiagent system of  agents (6), which is steered by the control protocol specified by (21).If the conditions of Lemma 1 are satisfied, then the protocol (21) achieves any precision stable for the multiagent systems specified by (6).

Inaccurate Gain in the
then system ( 6) can be rewritten as Letting β = √ /, then β > 1.Since  ≤ (  ) ≤ , then β−1 ≤ (  ) b−1 ≤ β The control protocol (12) with ( 11) is constructed by PD Control, which can bring in steady-state error.Thus, another control protocol is designed in order to overcome this deficiency. Letting the control protocol is designed as follows: and we have the following conclusion.

Illustrative Examples
In this section, we provide an illustrative example to show how to use the method in this research to design control protocol for the consensus of this class of multiagent systems.
Example 1.Consider the following 5-agent system running on a circle: whose topology is shown as Figure 1, {, , , } is the set of switching directed graph, it is easy to see that each of them does not obtain a spanning tree.To show the correctness of the above conclusion, we carry out the following numerical simulations.The dynamics start at  and switch to the next one after  = 0.01, and the switching rules of network topology are as follows:  →  →  →  →  →  ⋅ ⋅ ⋅ .It is easy to see that the union graph of the dynamics system (36) has a spanning tree frequently enough.Design the control protocol The simulation results are shown in Figure 2, from which we can see that the states of the 5 agents eventually converge to the same value under the protocol (37) and the final result coincides with the theoretical analysis.Simulation shows that our method is very effective in analyzing the consensus of this kind of multiagent system (36).

Mathematical Problems in Engineering
In the following, we consider Example 1 with the same switching topology and the same switching rules and design a control protocol obtaining the integration as follows: Under the same initial conditions and  = 1, the simulation results are shown in Figure 3, from which we can see that the agents converge to each other under the control protocol (38).

Conclusion
In this paper, the consensus control of a class of high-order nonlinear multiagent systems was investigated, whose topology switched dynamically, and we obtained some consensus results on this class of multiagent systems.Herein, a rather weak topology condition was given for the consensus of nonlinear multiagent systems with switching structure, and two control protocols, one of which was based on PD and the other was based on PID, were presented for this class of systems, and the stabilization was proved; furthermore, a reduced-order method was used to handle the high-order dynamics system, which simplify the difficulty of problem.Simulations showed that our results as well as designed control protocols worked very well in studying the consensus of this class of multiagent systems.However, we just take a step for this class of systems, and some questions, such as how to obtain the estimate f(), will be considered in our subsequent research.

Figure 1 :
Figure 1: The topology of switching directed graph.

Figure 2 :
Figure 2: The states of agents under the control protocol (37).

Figure 3 :
Figure 3: The states of agents under the control protocol (38).