# Adaptive Leader-Following Consensus of Multi-Agent Systems with Unknown Nonlinear Dynamics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

_{N}(0

_{N}) is a vector in ℝ

^{N}with elements being all ones (all zeros), and I

_{n}(O

_{n}) is the n × n identity matrix (zero matrix). diag{x

_{1}, …, x

_{N}} defines a diagonal matrix with diagonal entries x

_{1}to x

_{N}. Denote by S > 0 (S < 0) that the matrix S is positive (negative) definite. ⊗ denotes the Kronecker product with well known properties (A ⊗ B)

^{T}= A

^{T}⊗ B

^{T}, A ⊗ (B+C) = A ⊗ B+ A ⊗ C, (A ⊗ B)(C ⊗ D) = AC ⊗ BD, where A, B, C and D are matrices with appropriate dimensions. If not explicitly stated, we assume that the dimensions of the matrix used in this paper are compatible.

#### 2.1. Graph Theory

_{1}, …, v

_{N}} and a set of edges $\mathcal{E}$ ⊆ × . The node indices belong to a finite index set $\mathcal{I}$ = {1, …, N}. An edge of $\mathcal{E}$ is denoted by an unordered pair (v

_{i}, v

_{j}), which means that node v

_{i}and v

_{j}can exchange information with each other. Let us define a weighted adjacency matrix = [α

_{ij}] ∈ ℝ

^{N}

^{×}

^{N}as follows: α

_{ij}= α

_{ji}> 0 if (v

_{i}, v

_{j}) ∈ $\mathcal{E}$, otherwise α

_{ij}= α

_{ji}= 0. Moreover, we assume that α

_{ii}= 0 for all i ∈ $\mathcal{I}$. The set of neighbors of node i is denoted by N

_{i}= {j ∈ $\mathcal{I}$ : (v

_{i}, v

_{j}) ∈ $\mathcal{E}$}. Define the degree matrix as D = diag{d

_{1}, …, d

_{N}} with d

_{i}= ∑

_{j}

_{∈}

_{Ni}α

_{ij}. The symmetric Laplacian matrix corresponding to the undirected graph ℊ is defined as L = D − . A path is a sequence of unordered edges of the form (v

_{i}

_{1}, v

_{i}

_{2}), (v

_{i}

_{2}, v

_{i}

_{3}), …, where v

_{ij}∈ . The graph ℊ is called connected if there is a path between any two nodes.

#### 2.2. Problem Formulation

_{i}(t) = (x

_{i}

_{1}, x

_{i}

_{2}, …, x

_{in})

^{T}∈ ℝ

^{n}is the state of agent i, u

_{i}(t) ∈ ℝ

^{m}is the control input and f

_{i}(x

_{i}(t), t) ∈ ℝ

^{m}is the matched nonlinear dynamics of agent i, which is assumed to be unknown. A ∈ ℝ

^{n}

^{×}

^{n}and B ∈ ℝ

^{n}

^{×}

^{m}are known matrices.

_{0}(t) = (x

_{01}, x

_{02}, …, x

_{0}

_{n})

^{T}∈ ℝ

^{n}is the state of the leader and f

_{0}(x

_{0}(t), t) ∈ ℝ

^{m}is an unknown continuous function.

#### Assumption 1

_{k}(x

_{k}(t), t) is parameterized as:

_{k}(x

_{k}, t) ∈ ℝ

^{r}

^{×}

^{m}is the basis function matrix and θ

_{k}∈ ℝ

^{r}is the unknown constant parameter vector to be estimated.

#### Assumption 2

_{i}, i ∈ $\mathcal{I}$). Furthermore, we define an augmented graph = ( , $\mathcal{E}$̄) to model the interaction topology between N followers and the leader (labeled as v

_{0}), where = {v

_{0}, v

_{1}, …, v

_{N}} and $\mathcal{E}$̄ ⊆ × . To depict whether the followers are connected to the leader in , we define a leader adjacency matrix = diag{α

_{10}, …, α

_{N}

_{0}} ∈ ℝ

^{N}

^{×}

^{N}, where α

_{i}

_{0}> 0 if follower v

_{i}is connected to the leader across the communication link (v

_{i}, v

_{0}), otherwise, α

_{i}

_{0}= 0. If we define a new matrix H = L+ , the following lemma plays a key role in the sequel.

#### Lemma 1

_{i}, such that all N follower agents converge to the leader, that is, lim

_{t}

_{→∞}||x

_{i}(t) − x

_{0}(t)|| = 0, ∀ i ∈ $\mathcal{I}$ for any initial conditions x

_{i}(0), i = 0, 1, …, N.

## 3. Consensus Analysis

#### 3.1. Distributed Adaptive Controller Design

_{i}of the unknown function f

_{i}(x

_{i}, t) is unavailable in the controller design. Instead, θ

_{i}has to be estimated by each follower agent during the evolution. On the other hand, for consensus protocol design, it is assumed that only relative state information can be used. More precisely, the only obtainable information for follower v

_{i}is the local neighborhood consensus error [35]:

_{ij}and α

_{i}

_{0}are defined in Section 2.

^{m}

^{×}

^{n}is the feedback gain matrix to be designed later; θ̂

_{i}and θ̂

_{0}

_{i}are the estimations of the unknown parameters θ

_{i}and θ

_{0}, respectively.

_{i}= θ̂

_{i}− θ

_{i}.

#### 3.2. Stability Analysis

_{i}and the leader v

_{0}as δ

_{i}(t) ≜ x

_{i}(t)−x

_{0}(t). Then, from the closed-loop system (7)–(8), the dynamics of δ

_{i}(t) can be obtained:

_{0}

_{i}= θ̂

_{0}

_{i}− θ

_{0}. Denote $\delta ={({\delta}_{1}^{T},\dots ,{\delta}_{N}^{T})}^{T}$; then, the error system (9) can be rewritten in a compact form as:

_{f}= diag{φ

_{1}, …, φ

_{N}} and Φ

_{0}= diag{φ

_{0}, …, φ

_{0}}.

#### Remark 1

#### Remark 2

#### Theorem 1

_{k}(k = 0, 1, …, N) are uniformly bounded. Then, the distributed control law (5)–(6) combined with the adaptive law (13) solves the leader-following consensus problem if the coupling strength c is chosen as:

_{min}(H) denotes the smallest eigenvalue of H.

#### Proof

^{N}

^{×}

^{N}, such that:

_{i}> 0 are eigenvalues of matrix H. With the state transformation δ̃ = (U ⊗ I

_{n})δ, we can continue (17) as follows:

_{f}, θ̃

_{0}∈ L

_{∞}. Hence, δ̇(t) ∈ L

_{∞}due to (10) and the assumption that φ

_{j}(j = 0, 1, …, N) are uniformly bounded. Moreover, from (18), one has:

_{2}. According to Barbalat’s lemma [37], the trajectory of error system (10) starting from any initial conditions will converge to the origin asymptotically as t → ∞, which, in turn, indicates that lim

_{t}

_{→∞}||x

_{i}(t) − x

_{0}(t)|| = 0, ∀ i ∈ $\mathcal{I}$. The proof is thus completed.

#### Remark 3

## 4. Numerical Simulation

_{i}= [x

_{i}

_{1}, x

_{i}

_{2}, x

_{i}

_{3}]

^{T}are the state variables of the i-th (i = 1, …, 5) follower agent. It is known that the Chen system depicts a chaotic behavior when α = 35, β = 3 and γ = 28. Letting:

_{i}, t) can be parameterized as $f({x}_{i},t)={\phi}_{i}^{T}({x}_{i},t){\theta}_{i}$ in Assumption 1 with:

^{T}P. Thus, according to Theorem 1, when $c>{\scriptstyle \frac{1}{2{\lambda}_{\text{min}}(H)}}=6.2$, the leader-following consensus will be achieved using the distributed control law (5)–(6) combined with the adaptive law (13). The simulation results with c = 6.5 are shown in Figures 2 and 3, from which we can see that each follower described by (19) asymptotically converges to the leader described by (20), although there exist unknown parameters.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Error trajectories δ

_{i}(t) = x

_{i}(t) − x

_{0}(t) (i = 1, …, 5) between each follower and the leader.

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**MDPI and ACS Style**

Wang, J.; Chen, K.; Ma, Q.
Adaptive Leader-Following Consensus of Multi-Agent Systems with Unknown Nonlinear Dynamics. *Entropy* **2014**, *16*, 5020-5031.
https://doi.org/10.3390/e16095020

**AMA Style**

Wang J, Chen K, Ma Q.
Adaptive Leader-Following Consensus of Multi-Agent Systems with Unknown Nonlinear Dynamics. *Entropy*. 2014; 16(9):5020-5031.
https://doi.org/10.3390/e16095020

**Chicago/Turabian Style**

Wang, Junwei, Kairui Chen, and Qinghua Ma.
2014. "Adaptive Leader-Following Consensus of Multi-Agent Systems with Unknown Nonlinear Dynamics" *Entropy* 16, no. 9: 5020-5031.
https://doi.org/10.3390/e16095020