Decentralized Adaptive Control for Quasi-Consensus in Heterogeneous Nonlinear Multiagent Systems

This paper proposes some novel decentralized adaptive control protocols to settle the quasi-consensus problem of multiagent systems with heterogeneous nonlinear dynamics. Based on local communication with the leader and between the followers, some innovative control protocols are put forward to adapt the control gains and coupling weights simultaneously and to steer the consensus errors to some bounded areas. In particular, two new inequalities are proposed to establish the Lyapunov-based adaptive controller design approach for quasi-consensus. Some quasi-consensus criteria are derived by utilizing the designed controllers, in which the error bound can be modulated on the basis of the adaptive controller parameters. Numerical tests are conducted to show the feasibility of the theoretical derivation. Our ﬁndings highlight quasi-consensus in heterogeneous multiagent systems without adding some additional complex nonlinear control terms to cancel the dynamical diﬀerences between agents.


Introduction
Collaborative control of multiagent systems (MASs) has become a research hotspot of distributed artificial intelligence because of its wide application in intelligent energy, multirobot formation, intelligent transportation, multiunmanned system collaboration, and other engineering systems [1][2][3][4][5].
Among them, consensus or synchronization is a key common scientific issue of cooperative control of MASs, which has aroused great concern of multidisciplinary scholars. In brief, a core issue is to design appropriate control protocols to reach an agreement between agents. So far, many significant results have been acquired, regarding the consensus patterns, models, control algorithms, etc (please see [6][7][8][9][10][11][12][13][14] and some other results).
However, the aforesaid works [1][2][3][4][5][6][7][8][9][10][11][12][13][14] on the consensus of MASs mainly concentrate upon the case in which an agent has the same dynamics as all the neighbors or the leader. In some real scenes, the mismatched parameters or dynamical differences among agents may be almost inevitable, which will thus result in heterogeneous (or nonidentical) multiagent systems (HMASs) [15][16][17]. It is remarkably that, due to heterogeneity, it is even impossible for HMASs to reach the complete consensus just by state feedback control when the coupling weights are constant. Up to now, there are few thorough research studies on complete consensus in HMASs because one has to add some additional complex nonlinear control terms or design compensators to cancel the dynamical differences [18][19][20][21][22][23][24][25][26]. Another alternative technique to deal with the heterogeneity is to transform HMASs to homogeneous ones [27,28]. Unfortunately, all of the aforementioned approaches are complex and nonintuitive, which are thus not suitable for engineering applications. Nevertheless, in many practical HMASs, the consensus error may be bounded, even small enough, which is, namely, the so-called quasi-consensus (QC) [29].
Instead, an immediate and natural question, then, is how to design a simple controller to reach QC in HMASs. Similar to it, work to date has considered the quasi-synchronization (QS) in heterogeneous complex dynamical networks (HCDN) [30][31][32][33][34]. For example, in [33,34], some QS criteria for fractional HCDN are derived via state feedback control and impulsive control, respectively. By contrast, there are few research studies about QC of HMASs [29,[35][36][37][38]. e definition of QC for MASs was first proposed in [38], and then, the definition of QC was further broadened in [29]. After that, the QC of HMASs has been further studied. e QC problem of nonlinear HMASs is studied via sampleddata control in [35]. In [36], sufficient conditions for the QC in switched HMASs are given, considering cooperation and competition interactions simultaneously. Ye and Shao [37] attempted to prove that QC in HMASs under DOS attacks can be realized by impulsive control. It should be noted, however, that, for large scale HCDN or HMASs, the computational complexity and conservativeness of QS or QC conditions in [29][30][31][32][33][34][35][36][37][38] impede their applications. In particular, it is difficult, even impossible, to check the LMIbased conditions without adaptive schemes for large-scale HCDN or HMASs. Obviously, it is an interesting and open problem to improve the QC conditions for HMASs by exploring some new control algorithms.
Motivated by the applications of decentralized adaptive control for synchronization in integer-order and fractionalorder complex dynamical networks [39][40][41][42][43], this paper aims to design some decentralized adaptive protocols to reach QC in HMASs. Different from some previous studies [37,44,45], in our controller, the coupling and feedback values between agents change adaptively. e main contributions in this paper are as follows. First, two new inequalities are proposed to establish the Lyapunov-based adaptive controller design approach for QC in HMASs. Second, some innovative control protocols are introduced to accommodate the control gains and coupling weights adaptively, to steer the consensus errors to some bounded areas.
ird, some QC criteria under the designed controllers are derived by the Lyapunov function method and the new inequalities.

Preliminaries and Problem Formation
are the separate sets of nodes and undirected edges. A � (a ij ) N×N is the weighted adjacency matrix of which the elements are nonnegative. e ith agent and the leader are modeled as the node i ∈ V, V � 0, 1, . . . , N { } and node 0, respectively. As a rule, the undirected edge (i, j) ∈ E in the weighted undirected graph G denoted by an ordered pair (V i , V j ) represents that agent i and agent j become a pair of neighbors which can get their information from each other. ere is an undirected spanning tree in an undirected graph if one agent exits an undirected path to every other distinct node.
ere are two matrices that are considered as the network topology, i.e., the weighted adjacency matrix A � (a ij ) N×N with a ij � a ji > 0 if e ij ∈ E, else a ij � 0 if e ij ∉ E, and the Laplacian matrix noted that all of them are asymmetric for the undirected graph.
Lemma 1 (see [46]). e Laplacian matrix L is constructed from the undirected network. ere are several properties in the following.

Problem Formation.
In this paper, we consider that there are N follower multiagent systems, which can be described by where ω i (t) ∈ R n can be regarded as the position vector of the agent i.f: R n × R n × R + ⟶ R n represents a continuous nonlinear vector function. A i , B i ∈ R n×n represent the system matrices of the agent i, respectively. u i (t) ∈ R n denotes the control protocol to be designed. e leader agent can be described as follows: where ω 0 (t) ∈ R n represents the leader's state vector and A, B ∈ R n×n represent the leader's system matrices, respectively. For the above heterogeneous leader-follower multiagent system, we give the following assumptions. Assumption 1. Suppose there is a normal quantity l so that the vector function f for any vector λ, ] ∈ R n satisfies Lemma 2 (see [47]). For any vector x, y ∈ R n , the following holds:

Lemma 3. Any two continuous functions satisfy
where c > 0. en, there is a t ≥ 0 so that the following holds: where * is the convolution.

Discrete Dynamics in Nature and Society
Proof.
Since v(t) + _ w(t) ≤ − cv(t), the existence of z(t) ≥ 0 makes the following equation: We calculate the Laplace transform of (8) to obtain en, we can obtain e inverse Laplace transform of (10) can be obtained as rough (10), we can obtain

Lemma 4. Any two continuous functions satisfy
where c > 0 and ε > 0. en, there is a t ≥ 0 so that the following formula holds: We can further obtain Lemma 5 (see [47]). If A, B, C, and D represent four different matrices, respectively, and the matrix products AC and B D makes sense, the Kronecker product ⊗ satisfies Definition 1 (see [29]). e leader-follower HMASs are decided to reach QC if where ξ is a nonnegative constant.

Assumption 2.
e network topology between agents is undirected and connected, and each agent can acquire the status information of the agent that has a connection relationship with it and the leader agent at any time. (2) and (3), we design the control input for all follower agents as

Our Controller. To obtain QC between HMASs
where c is a positive constant. e adaptive law for the control gains is described as where μ is a positive constant to be selected. e adaptive law for the coupling weights is described as where α ij � α ji are the positive constants to be selected.

Remark 1.
It should be noted that, in controller (19), adaptive laws (20) and (21), the coupling weights L ij (t) and the control gains r i (t) are adjusted adaptively based on local communication with the leader and between the followers. Combining adaptation of the coupling weights and control gains, adaptive law (20) ensures the QC of the follower agents, while adaptive law (21) drives the follower agents to the leader agent. Let the QC error vector be e i (t) � ω i (t) − ω 0 (t). en, the error model with the controller iscom where represents the difference between different agents. It can be obtained by Assumption 1: Since ω 0 (t) is bounded, it can be obtained that h i (ω 0 (t), t) is bounded.

Main Results
In this section, we present some theorems for achieving QC in HMASs.
Discrete Dynamics in Nature and Society
e Lyapunov function we constructed is where We take the derivative of V 1 (t) along (22) together with controller (19) and adaptive laws (20) and (21), and we can obtain rough Assumption 1 and Lemma 2, one has From (25)- (27), we can obtain We can define the Laplacian matrix Combining with (28) and (29), we have en, there is a unitary matrix U � (u 1 , . . . , u N ) so that Discrete Dynamics in Nature and Society (B 1 , B 2 , . . . , B N ), and A � diag (A 1 , A 2 , . . . , A N ).
Since I n is positive definite, by Lemma 1, we can get the following:

m T (t) Δ ⊗ I n m(t) ≥ λ 2 (Ω)m T (t) I N ⊗ I n m(t). (32)
en, we can obtain We can choose large enough k ij and d * i so that where θ > 0 is a positive constant.

Adaptive Pinning Control Protocol.
In the previous section, an adaptive controller (19) and an adaptive laws (20) and (21) for all follower agents are designed. However, it is neither realistic nor economical to control all follower agents in engineering. In view of this, the adaptive pinning control schemes are considered, where a fraction of the control gains and the coupling weights are adapted. Suppose that ε is a subset of E and HMASs (2) and (3) are connected by the undirected edge. ε is connected. e pinning adaptive protocol is written as follows: where ϑ i � 1, for i � 1, 2, . . . , N s , 0, for i � N s + 1, . . . , N .
6 Discrete Dynamics in Nature and Society e adaptive law for r i (t) is the same as (20). e adaptive law for L ij (t) is described as where (i, j) ∈ ε and L ij (0) � L ji (0) > 0. (2) and (3) can achieve QC under the pinning adaptive protocol combined with (20), (21), and (37).
Discrete Dynamics in Nature and Society After the derivative of V 3 (t) along (22), controller (19), and the decentralized adaptive pinning laws (20) and (38), the following holds: � e T (t) A + c G ⊗ I n + 1 2 I N ⊗ BB T + l 2 + 1 I n D * ⊗ I n − cλ 2 Ω * I N ⊗ I n e(t) where e following proof is similar to the previous derivation in eorem 1; thus, we will omit this part here. (20), (37), and (38).

Proof.
e Lyapunov functional is considered as follows: 8 Discrete Dynamics in Nature and Society e derivative of V 4 (t) along the error system (22), controller (37), and decentralized adaptive pinning laws (20) and (38) gives e rest of the proof is similar to eorem 1. To save space, it is thus omitted here.

Remark 2.
In eorems 2-4, some sufficient conditions are given to realize QC of HMASs (2) and (3) by using the adaptive pinning control schemes. Actually, if the follower agents (3) are connected, one can randomly choose a small fraction of coupling weights and/or the control gains to adapt. In particular, it is possible to obtain the QC by pinning one follower agent.

Numerical Examples
In this section, we will confirm the theory proposed in the paper by using digital simulation experiments.
We use five following agents and one leader agent, and the initial communication Laplacian matrix for follower agents is L � e leader's system matrices can be described as A � − 2.  Figure 1 shows the change of ‖e i (t)‖ 2 (i � 1, 2, . . . , 5) for HMASs without the controller.

Example 1.
In this case, the parameters of controller (19) and adaptive laws (20) and (21)

Example 4.
In this case, by controlling the coupling between part of the follower agent and the leader and the coupling between the follower agents, we realize the QC of the whole system. All agent system parameters are in accordance with A i and B i . We choose c � 3. e parameters in the adaptive law are μ � 0.01, α 14 � α 41 � 4.9, α 15 � α 51 � 4.2, and α 23 � α 32 � 4.9. We arbitrarily choose the 1, 2, and 3 agents, and their initial values are selected as r(0) � (1.      Discrete Dynamics in Nature and Society Figure 5 also verifies that, under the control strategy designed by eorem 4, the entire system gradually converges to a limited area.

Conclusions
e decentralized adaptive control for QC of HMASs has been studied. e combined adaptation of the coupling weights and control gains allows to drive HMASs (2) and (3) to some bounded areas. In addition, some pinning schemes have been proposed to adjust a fraction of the coupling weights and control gains. To deal with the heterogeneity, two new lemmas are proposed to derive the QC criteria. It has been shown that the QC can be obtained without requiring any global. In future works, we will extend the results to more general HMASs, such as fractional-order HMASs, HMASs with time delay, and cooperative-competitive interaction. At the same time, we will attempt to optimize control protocol and extend it to some other systems, i.e., fractional-order systems [48], memristive neural networks [49,50], and complex-valued neural networks [51,52].

Data Availability
e data used to support the findings of the study are available from the corresponding author upon request.