Consensus Control of Leaderless and Leader-Following Coupled PDE-ODEs Modeled Multi-Agent Systems

: This paper discusses consensus control of nonlinear coupled parabolic PDE-ODE-based multi-agent systems (PDE-ODEMASs). First, a consensus controller of leaderless PDE-ODEMASs is designed. Based on a Lyapunov-based approach, coupling strengths are obtained for leaderless PDE-ODEMASs to achieve leaderless consensus. Furthermore, a consensus controller in the leader-following PDE-ODEMAS is designed and the corresponding coupling strengths are obtained to ensure the leader-following consensus. Two examples show the effectiveness of the proposed methods.


Introduction
Consensus in multi-agent systems (MASs) is to achieve a common group objective when agents have different initial states [1][2][3][4]. It has received great attention in the past decade as a result from its wide applications in flocking of mobile robots [5], opinion dynamics in social networks [6], formation of unmanned vehicles [7][8][9], microgrid energy management [10], traffic flow [11], etc.
More recently, there have been many important results related to PDEMASs. Ref. [32] studied a distributed adaptive controller of uncertain leader-following parabolic PDEMASs; ref. [33] studied consensus control for parabolic and second-order hyperbolic PDEMASs; ref. [34] studied distributed P-type iterative learning for PDEMASs with time delay; refs. [35,36] studied iterative learning control for PDEMASs without and with time delay; ref. [37] studied boundary control of 3-D PDEMASs with arbitrarily large boundary input delay; refs. [38,39] studied consensus and input constraint consensus of nonlinear PDE-MASs using boundary control. However, consensus control for PDE-ODEMASs has not been addressed yet, which is a new challenge. Motivated by the above, this paper studies consensus control of nonlinear coupled parabolic PDE-ODEMASs with Neumann boundary conditions. First, dealing with the leaderless case, a consensus controller of leaderless PDE-ODEMASs is designed. The leaderless consensus error system is obtained and one Lyapunov functional candidate is given. Using Wirtinger's inequality and matrix properties, coupling strengths are obtained for leaderless PDE-ODEMASs to achieve cluster consensus. Furthermore, dealing with the leader-following case, a consensus controller of leader-following PDE-ODEMASs is designed. The leader-following consensus error system is obtained and another Lyapunov functional candidate is given. The corresponding coupling strengths are obtained to ensure leader-following consensus.
The remainder of this paper is organized as follows. The problem formulation is given in Section 2. Section 3 presents a consensus control design of the leaderless PDE-ODEMAS and Section 4 gives that of the leader-following PDE-ODEMAS. An example to illustrate the effectiveness of the proposed method is presented in Sections 5 and 6 offers some concluding remarks.
Notations: λ max (·), λ 2 (·) stand for the maximum eigenvalue and smallest nonzero eigenvalue of ·, respectively. ⊗ is a Kronecker product of matrices. The identity matrix of n order is denoted by I n . ||·|| denotes the Euclidean norm for vectors in R n or the induced 2-norm for matrices in R m×n .

Consensus Control of the Leaderless PDE-ODEMAS
To achieve consensus of the leaderless PDE-ODEMAS (1), the consensus controller is designed as: where d and k are the coupling strengths to be determined, i ∈ {1, 2, · · · , N}. Assume that the topological structure A = (a ij ) N×N is defined as: The consensus error system can be obtained from (1), (2), and (7) thaṫ such that where e 0 Proof. Consider the following Lyapunov function as We haveV According to the matrix property, and where λ 2 (·) denotes the smallest nonzero eigenvalue of ·, L a,ij = −a ij when i = j, L a,ii = Therefore, L a , L b are Laplacian matrices.

Consensus Control of the Leader-Following PDE-ODEMAS
The leader agent is supposed to bė such that where x 0 0 , y 0 0 (ξ) are bounded and y 0 0 (ξ) is continuous. The leader-following consensus controller is designed as: where δ i > 0 if x i can obtain the information of x 0 ; otherwise, δ i = 0; and ρ i > 0 if y i can obtain the information of y 0 ; otherwise, ρ i = 0. Letẽ The leader-following consensus error system is obtained aṡẽ such that ∂ε i (ξ, t) ∂ξ

Definition 2. For the leader-following PDE-ODEMAS (22), (23) with any initial conditions, if
for any i ∈ {1, 2, · · · , N}, then the leader-following PDE-ODEMAS (22) Proof. Consider the Lyapunov functional candidate as One hasV Since G and H are symmetric positive definite matrices, and denotes the smallest nonzero eigenvalue and G, H are symmetric positive definite matrices.
With Theorem 1, according to (10), d > 0.50 and k > 0 are obtained. Therefore, we take d = 0.51 and k = 0.01. It can be seen in Figures 1 and 2

Example 2.
Consider a nonlinear leader-following PDE-ODEMAS composed of 1 leader agent (22) and (23) and 4 following agents (1) and (2) with coefficients the same as Example 1. In the same way, γ 1 = γ 2 = γ 3 = γ 4 = 1 are obtained. Choose δ i = ρ i = 1. With Theorem 2, according to (28), d > 2.0 and k > 0 are obtained. Therefore, we take d = 2.1 and k = 0.1. It can be seen in Figures 5 and 6 that the leader-following PDE-ODEMAS achieves consensus.  From another point of view, d = 1.9 and k = 0 do not satisfy (28). It can be seen in Figures 7 and 8 that the leader-following PDE-ODEMAS cannot achieve consensus with control gains d = 0.49 and k = 0.

Conclusions
This paper has studied consensus control of the PDE-ODEMASs. First, a consensus controller of the leaderless PDE-ODEMASs was designed. We have shown that the cluster consensus behavior can be reached for the given coupling strengths for the leaderless PDE-ODEMASs. Then, a consensus controller in the leader-following PDE-ODEMASs was designed. Leader-following consensus behavior can be arrived at for the given coupling strengths for the leader-following PDE-ODEMASs. In numerical simulations, it shows the obtained gains according to the proposed methods can ensure consensus of both leaderless and leader-following PDE-ODEMASs. On the contrary, the control with gains a little bit less than those according to the proposed methods cannot achieve consensus. There are often a great number of agents in the real world and, in future, pinning consensus, only controlling a few agents of the PDE-ODEMASs, will be studied, as well as time delays.

Conflicts of Interest:
The authors declare no conflict of interest.