Consensus in first-order nonlinear multi-agent systems with state time delays using adaptive fuzzy wavelet networks

Abstract

In this paper, a consensus problem is addressed for first-order multi-agent systems with unknown nonlinear dynamics under undirected graphs. Adaptive fuzzy wavelet networks are used to design two novel control algorithms for nonlinear systems without delays and nonlinear systems with state time delays. In these algorithms, adaptive fuzzy wavelet networks are employed to compensate for nonlinear dynamics of systems. Using proper Lyapunov functions, adaptive laws are obtained and the uniform ultimately bounded stability of closed-loop systems is proved. In addition, this paper uses Lyapunov–Krasovskii functions to handle unknown time delays. Three simulation examples are provided to illustrate the effectiveness of the proposed control schemes.

Keywords

Consensus multi-agent nonlinear adaptive fuzzy wavelet networks state time delay

Introduction

In recent years, many researchers have developed distributed consensus algorithms for multi-agent systems (Olfati-Saber and Murray, 2004; Ren, 2008; Ren et al., 2005; Semsar-Kazerooni and Khorasani, 2009); this significant interest is due to multi-agent systems’ broad range of applications in different areas, such as transportation, sensor networks, unmanned air vehicles, robotics and underwater vehicles (Yu et al., 2009). Consensus in multi-agent systems is defined as the arrival of all agents of a group at a common agreement, using a local distributed protocol (Meng et al., 2011). Distributed controllers have many advantages, such as less system requirement, low operational cost, high level of adaptivity, strong robustness and scalability (Cao et al., 2013; Chen and Song, 2014).

Many remarkable achievements have been acquired regarding the consensus control problem of linear multi-agent systems (Olfati-Saber and Murray, 2004; Ren, 2008; Semsar-Kazerooni and Khorasani, 2009). Olfati-Saber and Murray (2004) studied three different cases of first-order multi-agent systems, including systems with directed networks and fixed topology, systems with directed networks and switching topology and systems with undirected networks and fixed topology, considering communication time delays. Ren (2008) analysed four algorithms for the consensus problem of second-order multi-agent systems, which include systems with a bounded control input, without velocity measurement, with a leader whose information is available to all members, and with a leader whose states are available to a subset of members while the control input is bounded. A game theoretic scheme was suggested by Semsar-Kazerooni and Khorasani (2009) for achieving consensus over a common value for a group of agents with general linear dynamics.

However, to reduce the gap between theoretical models of systems and their real-world dynamics, most recent studies propose and investigate control strategies for multi-agent systems with nonlinear dynamics (Fan et al., 2014; Mei et al., 2013; Wen et al., 2014; Yu et al., 2010). Various methods, such as neural networks and adaptive control, have been used to handle unknown nonlinear dynamics of multi-agent systems to reach consensus (Chen and Song, 2014; Chen et al., 2014; Hou et al., 2009; Yu et al., 2013). Hou et al. (2009) proposed an adaptive neural-network-based control approach to solve consensus problems of first-order multi-agent systems with nonlinear dynamics; this approach was then extended to address consensus problem of higher-order systems. Yu et al. (2013) considered leaderless and leader-following cases and suggested two adaptive strategies for tuning controllers’ gains of nonlinear second-order multi-agent systems. Neural networks were used by Chen and Song (2014) to compensate for nonlinear dynamics and to reach consensus in second-order multi-agent systems with dynamic leaders and under directed graphs.

Further developments in multi-agent systems and the need for mathematical models of systems that are close to real-world applications have led researchers to consider different kinds of time delay (Chen et al., 2014; Lin and Jia, 2009, 2010; Liu et al., 2013; Meng et al., 2011; Sun and Wang, 2009). Generally speaking, there are three types of time delay in multi-agent systems: communication delay, which causes a latency in information exchange between agents; input delay, which affects the control input of systems; and state time delay, which results from the inertia that exists in the dynamics of systems (Chen et al., 2014; Meng et al., 2011; Olfati-Saber and Murray, 2004).

In this study, owing to the complexity of nonlinear systems, only state time delay is considered. The appearance of state time delay in the dynamics of almost all practical systems is an inevitable problem; therefore, it is of paramount importance to address this problem in multi-agent systems. Several notable schemes have been proposed to overcome the consensus problem of time-delayed linear multi-agent systems (Lin and Jia, 2009, 2010; Meng et al., 2011; Sun and Wang, 2009). Meng et al. (2011) studied both first- and second-order linear systems under directed networks with communication and input time delays and proper controllers, suggested to cope with the consensus problem under some conditions. Lin and Jia (2010) used linear matrix inequalities to derive sufficient conditions to reach average consensus in linear second-order multi-agent systems with time delays and jointly connected network topologies. Lin and Jia (2009) studied second-order discrete time systems with communication time delays under changing topologies and derived sufficient conditions for reaching consensus using a proposed control algorithm. However, the complexity of nonlinear multi-agent systems prevents us from directly applying control strategies of linear multi-agent systems to nonlinear ones. Chen et al. (2014) used adaptive radial basis function neural networks (RBFNNs) to compensate for nonlinear dynamics of a class of first-order nonlinear multi-agent systems and a proper Lyapunov–Krasovskii function employed to handle unknown state time delays of the system. However, applying this method leads designers to select a large control gain, which is not desired. The consensus problem of second-order nonlinear multi-agent systems with time-varying delays was investigated by Liu et al. (2013) and an adaptive strategy was proposed for tuning coupling strengths and feedback gains of the controller. However, in this work, time delays should be in linear form. A control algorithm based on adaptive fuzzy wavelet networks (AFWNs) was proposed by Taheri et al. (2017) for the consensus control problem of second-order nonlinear multi-agent systems with state time delays but, because of the differences between first- and second-order systems, the proposed algorithms in that work cannot directly be applied to first-order systems and a compatible form of AFWNs should be developed.

Many different methods have been studied for approximating nonlinear dynamics of a system, such as fuzzy logic, adaptive fuzzy approach, neural networks and wavelet networks. Combination of adaptive theory, neural networks, fuzzy logic and wavelet theory results in the appearance of AFWNs. These combine the advantages of different methods, which give them some special properties, such as multiresolution capability. This multiresolution property is useful because at a coarse resolution, wavelets can easily approximate global (low-frequency) behaviour and at a fine resolution, they can approximate local (high-frequency) behaviour of the approximated function (Ho et al., 2001b; Zekri et al., 2008). Fuzzy wavelet networks, which form the structure of AFWNs, can have a fast convergence with a small number of fuzzy rules; in comparison, neural networks need a large number of neurons and may still stick in local minima (Abiyev and Kaynak, 2008). Because of several essential properties of AFWNs, such as approximating an extensive range of nonlinear systems, fast learning ability, multiresolution capability and desired accuracy, they have been used widely in designing nonlinear control systems (Ho et al., 2001a; Lin, 2006; Shahriari-kahkeshi and Sheikholeslam, 2014; Zekri et al., 2008). Nonetheless, AFWNs have not been applied to control first-order nonlinear multi-agent systems.

This study aims to develop two control strategies to address the consensus problem of first-order nonlinear multi-agent systems with or without state time delays and proposes novel approaches based on AFWNs. In the first approach, a new consensus algorithm is provided to reach a consensus in first-order nonlinear multi-agent systems without time delays; this approach is then improved to guarantee arrival at consensus in systems with state time delays. The proposed algorithms attempt to tackle two problems: first, the problem of nonlinear dynamics of systems, which is solved using AFWNs; second, the existence of unknown state time delays of systems, which are handled using an appropriate Lyapunov–Krasovskii function. The main contributions of this paper are the use of AFWNs for approximating nonlinear dynamics of first-order nonlinear multi-agent systems, and a new algorithm to solve the consensus problem of first-order nonlinear multi-agent systems with state time delays. To mention an application, the proposed control algorithms in this paper can be applied to the consensus control of a state time-delayed multi-agent system with nonlinear dynamics, such as a combination of collaborative manipulator systems that need to reach a consensus (Hou et al., 2009), in order to achieve a specific configuration and coordination, such as picking up or landing an object.

The rest of this paper is organized as follows. In the next section, some basic concepts of graph theory and first-order multi-agent systems and some necessary lemmas, definitions and typical fuzzy wavelet networks (FWNs) are introduced. The third section proposes a FWN for approximating nonlinear dynamics of an agent, then improves it to an AFWN and suggests a control approach to reach consensus in first-order nonlinear multi-agent systems. The proposed control approach is proved in Theorem 1. A consensus control algorithm for first-order nonlinear multi-agent systems with state time delays is presented and its stability is proved in Theorem 2, in the fourth section. Two simulation examples in the fifth section illustrate the efficiency of proposed control strategies. Finally, the last section provides conclusions for this study.

Preliminaries

In this section, some basic concepts of graph theory are mentioned for first-order multi-agent systems. Then some lemmas and assumptions are presented. Finally, the structure of FWNs is studied.

Graph theory

An undirected graph $G = (V, ε, A^{g})$ of order n is defined, with a set of vertices $V = {v_{1}, \dots, v_{n}}$ , a set of edges $ε \subseteq V \times V$ and an adjacency matrix $A^{g} = [A_{ij}^{g}]$ , where $A_{ij}^{g} = A_{ji}^{g} > 0$ shows that there exists an edge between $v_{i}$ and $v_{j}$ , which means that these two nodes can exchange information between each other, and $A_{ij}^{g} = A_{ji}^{g} = 0$ denotes that there is no connection between nodes $v_{i}$ and $v_{j}$ . This paper only considers simple undirected graphs. The degree of node i is defined as the summation of the ith row of $A^{g}$ , i.e. $d_{i} = \sum_{j = 1}^{n} A_{ij}^{g}$ . The Laplacian matrix of the graph is defined as $L = D - A^{g} = [L_{ij}]$ , where $D = diag (d_{1}, d_{2}, \dots, d_{n})$ . It warrants that the property $\sum_{j = 1}^{n} L_{ij} = 0$ holds. A path between nodes in G is defined as a sequence of edges in the graph. If there exists a path between any pair of nodes, graph G is connected (Olfati-Saber and Murray, 2004; Ren, 2008; Ren et al., 2005; Semsar-Kazerooni and Khorasani, 2009).

Lemma 1. The Laplacian matrix L of undirected and connected graph $G = (V, ε, A^{g})$ is symmetric and semi-positive definite and its n real eigenvalues are given as

0 = λ_{1} (L) < λ_{2} (L) \leq λ_{3} (L) \leq \dots \leq λ_{n} (L)

Furthermore, for any $x \in R^{n}$ , satisfying $1_{n}^{T} x = 0$ , $λ_{2} (L) x^{T} x \leq x^{T} Lx \leq λ_{n} (L) x^{T} x$ holds (Godsil and Royle, 2001).

First-order multi-agent systems

The first-order nonlinear multi-agent systems are described as

{\overset{\cdot}{x}}_{i} (t) = f_{i} (x_{i} (t)) + u_{i} (t), i = 1, \dots, n

(1)

where $x_{i} (t) \in R^{m}$ denotes the position vector of agent $i, f_{i} (x_{i} (t)) : R^{m} \to R^{m}$ is an unknown continuous nonlinear function vector,and the control input vector is $u_{i} (t) \in R^{m}$ .

Assumption 1. For $x, y \in R^{m}$ , there is the nonnegative constant $ρ_{1}$ such that

∥ f (x) - f (y) ∥ \leq ρ_{1} ∥ x - y ∥

where ∥.∥ denotes the two-norm (Chen et al., 2007).

Definition 1. In equation (1), the first-order consensus is obtained if, for any initial conditions

lim_{t \to \infty} ∥ x_{i} (t) - x_{j} (t) ∥ = 0, \forall i, j = 1, 2, \dots, n

If the dynamics of agents include state time delays, then the related multi-agent system can be described as

\begin{matrix} {\overset{\cdot}{x}}_{i} = f_{i} (x_{i} (t)) + h_{i} (x_{i} (t - τ_{i})) + u_{i} (t), \\ i = 1, 2, \dots, n \end{matrix}

(2)

where $h_{i} (x_{i} (t)) : R^{m} \to R^{m}$ is the unknown continuous nonlinear function vector and $τ_{i}$ is the unknown time delay.

The following lemma, assumptions and definition are needed to obtain the main results of this paper.

Lemma 2. For appropriate matrices A, B, C and D, the Kronecker product ⊗ satisfies

$(A + B) \otimes C = A \otimes C + B \otimes C$

$(A \otimes B) (C \otimes D) = (AC) \otimes (BD)$ (Horn and Johnson, 1991).

Assumption 2. The inequality $∥ h_{i} (x_{i} (t)) ∥ \leq ρ_{i} (x_{i} (t))$ is satisfied, where $ρ_{i} (.)$ is a known positive function.

Assumption 3. The unknown time delay $τ_{i}, i = 1, 2, \dots, n$ is limited by $τ_{max}$ , where $τ_{i} \leq τ_{max}, i = 1, 2, \dots, n$ .

Definition 2. In this paper, $I_{n}$ represents the identity matrix and $1_{n}$ represents a vector with n elements equal to 1, i.e. $1_{n} = [1 \dots 1]^{T} \in R^{n}$ .

FWNs

A representative FWN with N fuzzy rules can be shown in the format (Lin, 2006; Shahriari-kahkeshi and Sheikholeslam, 2014)

\begin{matrix} R_{q} : if {\overset{'}{x}}_{1} is A_{1}^{q} \dots and {\overset{'}{x}}_{r} is A_{r}^{q} then \\ {\hat{F}}_{1}^{q} = w_{1}^{q} Π_{j = 1}^{r} g_{qj} ({\overset{'}{x}}_{j}) \\ ⋮ \\ {\hat{F}}_{m}^{q} = w_{m}^{q} Π_{j = 1}^{r} g_{qj} ({\overset{'}{x}}_{j}) \end{matrix}

(3)

where $R_{q}$ shows the qth fuzzy rule, $\overset{'}{x} = [{\overset{'}{x}}_{1} {\overset{'}{x}}_{2} \dots {\overset{'}{x}}_{r}]^{T} \in R^{r}$ and ${\hat{F}}_{k}^{q}$ ( $q = 1, 2, \dots, N, k = 1, 2, \dots, m$ ) are the fuzzy rules’ input and output, $w_{k}^{q} \in R$ denotes a real coefficient and the linguistic term $A_{j}^{q} (j = 1, 2, \dots, r)$ is specified by the Gaussian-type fuzzy membership function, $μ_{A_{j}^{q}} ({\overset{'}{x}}_{j}) = \exp (- ω_{qj}^{2} ({\overset{'}{x}}_{j} - c_{qj})^{2})$ , where dilation and translation parameters, respectively, are $ω_{qj}$ and $c_{qj}$ . In this paper, $g_{qj} ({\overset{'}{x}}_{j}) = 1 - ω_{qj}^{2} ({\overset{'}{x}}_{j} - c_{qj})^{2}$ holds in consideration of creating a fuzzy wavelet basis function. The qth fuzzy wavelet basis function $ψ_{q} = (Π_{j = 1}^{r} g_{qj} ({\overset{'}{x}}_{j})) ϕ_{q}$ is made of the firing strength of rule q, which is $ϕ_{q} = Π_{j = 1}^{r} μ_{A_{j}^{q}} ({\overset{'}{x}}_{j})$ , and the wavelet function $Π_{j = 1}^{r} g_{qj} ({\overset{'}{x}}_{j})$ . In each fuzzy rule, the multidimensional function $ψ_{q} (\overset{'}{x})$ can be described as

ψ_{q} (\overset{'}{x}) = ψ_{q} ({\overset{'}{x}}_{1}) \dots ψ_{q} ({\overset{'}{x}}_{r})

(4)

where

ψ_{q} ({\overset{'}{x}}_{j}) = (1 - ω_{qj}^{2} ({\overset{'}{x}}_{j} - c_{qj})^{2}) e^{- ω_{qj}^{2} {({\overset{'}{x}}_{j} - c_{qj})}^{2}}

The output of the FWN can be shown as

f_{k} = \sum_{q = 1}^{N} w_{k}^{q} ψ_{q} (\overset{'}{x}), k = 1, \dots, m

(5)

Consensus protocol of first-order nonlinear systems using AFWN

In this section, a FWN is described to illustrate the method of approximating the unknown nonlinear function of the ith agent of a first-order nonlinear multi-agent system. Then a proper AFWN is derived from the mentioned FWN. Finally, a control algorithm for reaching consensus using AFWN is expressed and the stability of the closed-loop system is proved in Theorem 1.

FWN for ith agent of multi-agent system

The original FWN with N fuzzy rules for approximating unknown nonlinear functions of the ith agent of equation (1) can be described in the following format

\begin{matrix} R_{q_{i}} : if {\bar{x}}_{i 1} is A_{1}^{q_{i}} \dots and {\bar{x}}_{i 2 m} is A_{2 m}^{q_{i}} then \\ F_{1}^{q_{i}} = w_{1}^{q_{i}} Π_{j = 1}^{2 m} g_{q_{i} j} ({\bar{x}}_{ij}) \\ ⋮ \\ F_{m}^{q_{i}} = w_{m}^{q_{i}} Π_{j = 1}^{2 m} g_{q_{i} j} ({\bar{x}}_{ij}) \end{matrix}

(6)

where $R_{q_{i}}$ denotes the $q_{i}$ th rule of agent i, ${\bar{x}}_{i} = [{\bar{x}}_{i 1} {\bar{x}}_{i 2} \dots {\bar{x}}_{i 2 m}]^{T} = [x_{i}^{T} u_{i}^{T}]^{T}$ and $F_{k}^{q_{i}}$ $(q_{i} = 1, 2, \dots, N, k = 1, 2, \dots, m)$ are the fuzzy rules’ input and output, $w_{k}^{q_{i}} \in R$ shows the real coefficient of the output layer and the linguistic term $A_{j}^{q_{i}} (j = 1, 2, \dots, 2 m)$ is specified by a Gaussian-type fuzzy membership function. The parameters $ω_{qj}^{i}$ and $c_{qj}^{i}$ are dilation and translation parameters, respectively. In this paper, $g_{q_{i} j} ({\bar{x}}_{ij}) = 1 - ω_{q_{i} j}^{2} ({\bar{x}}_{ij} - c_{q_{i} j})^{2}$ holds, in pursuance of the construction of the fuzzy wavelet basis function. The $q_{i}$ th fuzzy wavelet basis function $ψ_{q_{i}} = (Π_{j = 1}^{2 m} g_{q_{i} j} ({\bar{x}}_{ij})) ϕ_{q_{i}}$ is made of the wavelet function $Π_{j = 1}^{2 m} g_{q_{i} j} ({\bar{x}}_{ij})$ and the firing strength of rule $q_{i}$ where $ϕ_{q_{i}} = Π_{j = 1}^{2 m} μ_{A_{q_{i} j}} ({\bar{x}}_{ij})$ and $μ_{A_{q_{i} j}} ({\bar{x}}_{ij}) = \exp (- ω_{q_{i} j}^{2} ({\bar{x}}_{ij} - c_{q_{i} j})^{2})$ . In each fuzzy rule, the multidimensional function $ψ_{q_{i}} ({\bar{x}}_{i})$ can be described by the one-dimensional wavelet functions as

ψ_{q_{i}} ({\bar{x}}_{i}) = ψ_{q_{i}} ({\bar{x}}_{i 1}) \dots ψ_{q_{i}} ({\bar{x}}_{i 2 m})

(7)

where

ψ_{q_{i}} ({\bar{x}}_{ij}) = (1 - ω_{q_{i} j}^{2} ({\bar{x}}_{ij} - c_{q_{i} j})^{2}) e^{- ω_{q_{i} j}^{2} {({\bar{x}}_{ij} - c_{q_{i} j})}^{2}}

The output of the FWN can be shown in the following format

f_{k} = \sum_{q_{i} = 1}^{N} w_{k}^{q_{i}} ψ_{q_{i}} ({\bar{x}}_{i}), k = 1, \dots, m

(8)

To avoid complexity in the notation, the output of FWN is shown in the following matrix form

f_{i} ({\bar{x}}_{i}, c_{i}, ω_{i}, W_{i}) = W_{i} ψ_{i} ({\bar{x}}_{i}, c_{i}, ω_{i})

(9)

where the translation parameter vector is $c_{i} = [c_{11}^{i} \dots c_{(2 m) 1}^{i} \dots c_{(2 m) N}^{i}] \in R^{(2 m) N}$ , the dilation parameter vector is $ω_{i} = [ω_{11}^{i} \dots ω_{(2 m) 1}^{i} \dots ω_{(2 m) N}^{i}] \in R^{(2 m) N}$ , the weight matrix is $W_{i} = [w_{k}^{q_{i}}] \in R^{m \times N}$ and the fuzzy wavelet basis function vector is $ψ_{i} = [ψ_{1 i}, \dots, ψ_{Ni}]^{T} \in R^{N \times 1}$ . There exist ideal matrices $W_{i}^{*}$ , $c_{i}^{*}$ and $ω_{i}^{*}$ for the weight parameters matrix, translation vector and dilation vector, respectively. Using these ideal matrices, the output of the FWN for approximating the nonlinear function $f_{i} (x_{i} (t))$ can be rewritten as

f_{i} (x_{i} (t)) = W_{i}^{*} ψ_{i} ({\bar{x}}_{i}, c_{i}^{*}, ω_{i}^{*}) + ε_{i} (\bar{x})

(10)

where $ε_{i} (\bar{x})$ is an error vector and satisfies $∥ ε_{i} (\bar{x}) ∥ < \bar{ε}$ .

Assumption 4. The norms of optimal weights, $∥ W_{i}^{*} ∥$ , $∥ c_{i}^{*} ∥$ and $∥ ω_{i}^{*} ∥$ are bounded, which means that $∥ W_{i}^{*} ∥ \leq {\bar{W}}_{i}$ , $∥ c_{i}^{*} ∥ \leq {\bar{c}}_{i}$ and $∥ ω_{i}^{*} ∥ \leq {\bar{ω}}_{i}$ (Shahriari-kahkeshi and Sheikholeslam, 2014).

The estimate of $f_{i}$ using the FWN can be written as

{\hat{f}}_{i} (x_{i} (t)) = {\hat{W}}_{i} ψ_{i} (\bar{x}, {\hat{c}}_{i}, {\hat{ω}}_{i})

(11)

where ${\hat{c}}_{i} = [{\hat{c}}_{11}^{i} \dots {\hat{c}}_{(2 m) 1}^{i} \dots {\hat{c}}_{(2 m) N}^{i}] \in R^{(2 m) N}$ , ${\hat{ω}}_{i} = [{\hat{ω}}_{11}^{i} \dots {\hat{ω}}_{(2 m) 1}^{i} \dots {\hat{ω}}_{(2 m) N}^{i}] \in R^{(2 m) N}$ and ${\hat{W}}_{i} = [{\hat{w}}_{k}^{q_{i}}] \in R^{m \times N}$ are used to approximate $c_{i}^{*}$ , $ω_{i}^{*}$ and $W_{i}^{*}$ , respectively.

Since the accuracy of estimation depends directly on weights, dilation and translation matrices and their parameters, so as to attain the favourite efficiency, it is not proper to choose FWN parameters manually. Therefore, the AFWN should be employed to adjust the parameters of the FWN under appropriate adaptive laws. Adaptive learning laws are derived from a Lyapunov direct method to adapt all parameters of the FWN.

Consensus in multi-agent systems using AFWN

In this subsection, the previous FWN is developed to an AFWN for a multi-agent system with n agents, and a proper control input is proposed to attain consensus between all agents. The proposed control input is described as

u_{i} = - a_{i} \sum_{j = 1}^{n} L_{ij} x_{j} (t) - {\hat{f}}_{i} (x_{i} (t))

(12)

where $a_{i} > 0$ is the control gain. By choosing this control input, the closed-loop system is

{\overset{\cdot}{x}}_{i} = f_{i} (x_{i} (t)) - {\hat{f}}_{i} (x_{i} (t)) - a_{i} \sum_{j = 1}^{n} L_{ij} x_{j} (t)

(13)

Let us define the consensus error as $e_{i} (t) = \sum_{j = 1}^{n} L_{ij} x_{j} (t)$ and ${\tilde{f}}_{i} (x_{i} (t))$ as

{\tilde{f}}_{i} (x_{i} (t)) = f_{i} (x_{i} (t)) - {\hat{f}}_{i} (x_{i} (t))

(14)

\begin{matrix} {\tilde{f}}_{i} (x_{i} (t)) & = {W_{i}}^{*} ψ_{i} ({\bar{x}}_{i}, c_{i}^{*}, ω_{i}^{*}) + ε ({\bar{x}}_{i}) - {\hat{W}}_{i} ψ_{i} ({\bar{x}}_{i}, {\hat{c}}_{i}, {\hat{ω}}_{i}) \\ = {\hat{W}}_{i} {\tilde{ψ}}_{i} + {\tilde{W}}_{i} ψ_{i} ({\bar{x}}_{i}, {\hat{c}}_{i}, {\hat{ω}}_{i}) + {\tilde{W}}_{i} {\tilde{ψ}}_{i} + ε ({\bar{x}}_{i}) \end{matrix}

(15)

where ${\tilde{ψ}}_{i} = ψ_{i} ({\bar{x}}_{i}, c_{i}^{*}, ω_{i}^{*}) - ψ_{i} ({\bar{x}}_{i}, {\hat{c}}_{i}, {\hat{ω}}_{i})$ and ${\tilde{W}}_{i} = W_{i}^{*} - {\hat{W}}_{i}$ . To simplify the notation $x_{i} (t)$ , ${\tilde{f}}_{i} (x_{i} (t))$ , $ψ_{i} ({\bar{x}}_{i}, c_{i}^{*}, ω_{i}^{*})$ and $ψ_{i} ({\bar{x}}_{i}, {\hat{c}}_{i}, {\hat{ω}}_{i})$ are shown as $x_{i}$ , ${\tilde{f}}_{i}$ , $ψ_{i}^{*}$ and ${\hat{ψ}}_{i}$ , respectively.

To attain adaptive laws, ${\tilde{ψ}}_{i}$ should be expanded. Using the Taylor series expansion linearization method, the expansion of ${\tilde{ψ}}_{i}$ is achieved as

{\tilde{ψ}}_{i} = - {\overset{'}{ψ}}_{i {\hat{c}}_{i}}^{T} {\tilde{c}}_{i} - {\overset{'}{ψ}}_{i {\hat{ω}}_{i}}^{T} {\tilde{ω}}_{i} + o ({\bar{x}}_{i}, {\tilde{c}}_{i}, {\tilde{ω}}_{i})

(16)

where ${\tilde{ω}}_{i} = {\hat{ω}}_{i} - ω_{i}^{*}$ , ${\tilde{c}}_{i} = {\hat{c}}_{i} - c_{i}^{*}$ , ${\overset{'}{ψ}}_{i {\hat{ω}}_{i}}^{T} = \partial ψ_{i} / \partial ω_{i} |_{ω_{i} = {\hat{ω}}_{i}}$ , ${\overset{'}{ψ}}_{i {\hat{c}}_{i}}^{T} = \partial ψ_{i} / \partial c_{i} |_{c_{i} = {\hat{c}}_{i}}$ and $o ({\bar{x}}_{i}, {\tilde{c}}_{i}, {\tilde{ω}}_{i})$ is a vector containing higher-order terms. After substituting equation (16) into equation (15) and some simple manipulations, function ${\tilde{f}}_{i}$ can be written in the following format

\begin{matrix} {\tilde{f}}_{i} = {\tilde{W}}_{i} ({\hat{ψ}}_{i} - {\overset{'}{ψ}}_{i {\hat{c}}_{i}}^{T} {\hat{c}}_{i} - {\overset{'}{ψ}}_{i {\hat{ω}}_{i}}^{T} {\hat{ω}}_{i}) - {\hat{W}}_{i} {\overset{'}{ψ}}_{i {\hat{c}}_{i}}^{T} \tilde{c} - {\hat{W}}_{i} {\overset{'}{ψ}}_{i {\hat{ω}}_{i}}^{T} {\tilde{ω}}_{i} + Δ_{i} \end{matrix}

(17)

where

\begin{matrix} Δ_{i} = {\tilde{W}}_{i} ({\overset{'}{ψ}}_{i {\hat{c}}_{i}}^{T} c_{i}^{*} - {\overset{'}{ψ}}_{i {\hat{ω}}_{i}}^{T} ω_{i}^{*} + o ({\bar{x}}_{i}, {\tilde{c}}_{i}, {\tilde{ω}}_{i})) + {\hat{W}}_{i} o ({\bar{x}}_{i}, {\tilde{c}}_{i}, {\tilde{ω}}_{i}) + ε ({\bar{x}}_{i}) \end{matrix}

Theorem 1. Assume that graph G is connected and Assumptions 1 and 4 hold. Consider the following adaptive laws

{\overset{\cdot}{\hat{W}}}_{i} = γ_{w} e_{i} ({\hat{ψ}}_{i} - {\overset{'}{ψ}}_{i {\hat{c}}_{i}}^{T} {\hat{c}}_{i} - {\overset{'}{ψ}}_{i {\hat{ω}}_{i}}^{T} {\hat{ω}}_{i})^{T}

(18)

{\overset{\cdot}{\hat{c}}}_{i} = γ_{c} {\overset{'}{ψ}}_{i {\hat{c}}_{i}} {\hat{W}}_{i}^{T} e_{i}

(19)

{\overset{\cdot}{\hat{ω}}}_{i} = γ_{ω} {\overset{'}{ψ}}_{i {\hat{ω}}_{i}} {\hat{W}}_{i}^{T} e_{i}

(20)

where $γ_{w} > 0$ , $γ_{c} > 0$ and $γ_{ω} > 0$ are constant. Then, by choosing $a_{i}$ such that the two inequalities $a_{i} > ζ / 2$ , where $ζ$ is an arbitrary positive constant, and $(a_{i} - (ζ / 2)) ∥ e_{i} (t) ∥^{2} \geq (1 / 2 ζ) ∥ Δ_{i} ∥^{2}$ are satisfied, consensus in equation (13) can be reached.

Proof. Take $V (E (t), \tilde{W}, \tilde{c}, \tilde{ω}, t)$ as the Lyapunov function candidate

\begin{matrix} V (E (t), \tilde{W}, \tilde{c}, \tilde{ω}, t) = \frac{1}{2} x^{T} (t) (L \otimes I_{m}) x (t) \\ + \frac{1}{2 γ_{w}} tr ({\tilde{W}}^{T} \tilde{W}) + \frac{1}{2 γ_{c}} {\tilde{c}}^{T} \tilde{c} + \frac{1}{2 γ_{ω}} {\tilde{ω}}^{T} \tilde{ω} \end{matrix}

\begin{matrix} = \frac{1}{2} E^{T} (t) x (t) + \frac{1}{2 γ_{w}} tr ({\tilde{W}}^{T} \tilde{W}) + \frac{1}{2 γ_{c}} {\tilde{c}}^{T} \tilde{c} + \frac{1}{2 γ_{ω}} {\tilde{ω}}^{T} \tilde{ω} \\ = \frac{1}{2} \sum_{i = 1}^{n} (e_{i}^{T} (t) x_{i} (t) + \frac{1}{γ_{w}} tr ({\tilde{W}}_{i}^{T} {\tilde{W}}_{i}) + \frac{1}{γ_{c}} {\tilde{c}}_{i}^{T} {\tilde{c}}_{i} + \frac{1}{γ_{ω}} {\tilde{ω}}_{i}^{T} {\tilde{ω}}_{i}) \end{matrix}

(21)

Using algebraic graph theory, it can easily be concluded that zero is an eigenvalue of order m in matrix $(L \otimes I_{m})$ . The eigenvectors related to the eigenvalues that are equal to 0 can be written in the following format

\begin{matrix} K_{1} = [k_{1}^{T} \dots k_{1}^{T}]^{T} \in R^{nm}, \dots \\ K_{m} = [k_{m}^{T} \dots k_{m}^{T}]^{T} \in R^{nm} \end{matrix}

where $k_{i} \in R^{m}$ is a vector whose mth element is $1 / \sqrt{n}$ and other elements are equal to zero. Vectors $K_{m + 1}, \dots, K_{nm}$ show eigenvectors associated with eigenvalues $λ_{2}, \dots, λ_{n}$ of matrix $(L \otimes I_{m})$ , respectively. Based on matrix theory, the vectors $K_{1}, \dots, K_{nm}$ are considered as a set of orthogonal bases of $R^{nm}$ . Let $H = [K_{1} \dots K_{nm}] \in R^{nm \times nm}$ ; one of its features is $H^{T} = H^{- 1}$ . As a result, equation (22) gives

\begin{matrix} V_{x} & = \frac{1}{2} x^{T} (t) (L \otimes I_{m}) x (t) \\ = \frac{1}{2} x^{T} (t) (H^{T} MH) x (t) \\ = \frac{1}{2} x^{T} (t) (H^{T} M {\hat{M}}^{- 1} MH) x (t) \\ = \frac{1}{2} x^{T} (t) (H^{T} MH H^{T} {\hat{M}}^{- 1} H H^{T} MH) x (t) \\ = \frac{1}{2} x^{T} (t) (L \otimes I_{m}) H^{T} {\hat{M}}^{- 1} H (L \otimes I_{m}) x (t) \\ = \frac{1}{2} E^{T} (t) DE (t) \end{matrix}

(22)

where $M = diag (0 I_{m}, λ_{2} I_{m}, \dots, λ_{n} I_{m})$ , $\hat{M} = diag (λ_{2} I_{m}, λ_{2} I_{m}, \dots, λ_{n} I_{m})$ and $D = H^{T} {\hat{M}}^{- 1} H$ is a positive definite matrix. It is obvious from equation (22) that $V_{x}$ is a positive definite function.

The derivative of the Lyapunov function is

\begin{matrix} \overset{\cdot}{V} (E (t), \tilde{W}, \tilde{c}, \tilde{ω}, t) = \sum_{i = 1}^{n} (e_{i}^{T} (t) (f_{i} (x_{i} (t)) + u_{i} (t)) \\ - \frac{1}{γ_{w}} tr ({\tilde{W}}_{i}^{T} {\overset{\cdot}{\hat{W}}}_{i}) + \frac{1}{γ_{c}} {\tilde{c}}_{i}^{T} {\overset{\cdot}{\hat{c}}}_{i} + \frac{1}{γ_{ω}} {\tilde{ω}}_{i}^{T} {\overset{\cdot}{\hat{ω}}}_{i}) \end{matrix}

(23)

By substituting equations (12) and (18) to (20) into equation (23), and using equation (17), the derivative of $V (t)$ is

\overset{\cdot}{V} (E (t), \tilde{W}, \tilde{c}, \tilde{ω}, t) = \sum_{i = 1}^{n} (e_{i}^{T} (t) Δ_{i} - a_{i} ∥ e_{i} (t) ∥^{2})

(24)

For any positive constant $ζ$ , the following inequality holds

\overset{\cdot}{V} \leq \sum_{i = 1}^{n} (\frac{ζ}{2} ∥ e_{i} (t) ∥^{2} + \frac{1}{2 ζ} ∥ Δ_{i} ∥^{2} - a_{i} ∥ e_{i} (t) ∥^{2})

(25)

where $ζ$ is an arbitrary positive constant, chosen to reduce the effect of $∥ Δ_{i} ∥$ in equation (25). According to Assumption 4 and based on the fact that the wavelet function is bounded, it can be inferred that $Δ_{i}$ is limited, so $∥ Δ_{i} ∥ \leq δ$ is satisfied, where $δ$ is a positive constant. If $a_{i} > ζ / 2$ and inequality $(a_{i} - (ζ / 2)) ∥ e_{i} (t) ∥^{2} \geq (1 / 2 ζ) ∥ Δ_{i} ∥^{2}$ holds, then $\overset{\cdot}{V} \leq 0$ , and as a result, the variables $E (t)$ , $\tilde{W}$ , $\tilde{c}$ and $\tilde{ω}$ are globally uniformly ultimately bounded. Finally, the uniformly ultimately bounded stability implies that the error vector $E (t)$ converges to a bounded region around 0, which means that the states of the agents have a small consensus error, $e_{i} (t)$ , which satisfies the inequality $(a_{i} - (ζ / 2)) ∥ e_{i} (t) ∥^{2} \geq (1 / 2 ζ) ∥ Δ_{i} ∥^{2}$ , converge to the same number.

Remark 1. To satisfy the inequality $(a_{i} - (ζ / 2)) ∥ e_{i} (t) ∥^{2} \geq (1 / 2 ζ) ∥ Δ_{i} ∥^{2}$ , one needs to choose a large control gain $a_{i}$ ; however, choosing a larger control gain results in higher control input and greater energy consumption, which is not desirable in multi-agent systems, owing to their limited battery storage. Therefore, in choosing the proper control gain $a_{i}$ , there should be a trade-off between satisfying the aforementioned inequality and energy-storage limitations of the agent.

Consensus protocol of first-order nonlinear systems with state time delays

This section addresses the problem of consensus in first-order multi-agent systems with nonlinear dynamics and state time delays. All notation is according to that in the last section. The proposed control input and time-delayed multi-agent system are described in equations (26) and (27), respectively

u_{i} = - a_{i} \sum_{j = 1}^{n} L_{ij} x_{j} (t) - {\hat{f}}_{i} (x_{i} (t)) - \frac{1}{2} e_{i}^{- 1} ρ_{i}^{2} (x_{i} (t))

(26)

{\overset{\cdot}{x}}_{i} (t) = f_{i} (x_{i} (t)) + h_{i} (x_{i} (t - τ_{i})) + u_{i}

(27)

Substituting equation (26) into equation (27), the closed-loop system can be recast as

\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = f_{i} (x_{i} (t)) + h_{i} (x_{i} (t - τ_{i})) - {\hat{f}}_{i} (x_{i} (t)) \\ - a_{i} \sum_{j = 1}^{n} L_{ij} (x_{j} (t)) - \frac{1}{2} e_{i}^{- 1} ρ_{i}^{2} (x_{i} (t)) \end{matrix}

(28)

where $e_{i}^{- 1}$ is equal to $e_{i} / ∥ e_{i} ∥^{2}$ , and satisfies $e_{i}^{T} e_{i}^{- 1} = 1$ . The $τ_{i}$ represents the unknown time delay.

Theorem 2. Assume that graph G is connected and Assumptions 1 to 4 hold. Consider the following adaptive laws

{\overset{\cdot}{\hat{W}}}_{i} = γ_{w} e_{i} {({\hat{ψ}}_{i} - {\overset{'}{ψ}}_{i {\hat{c}}_{i}}^{T} {\hat{c}}_{i} - {\overset{'}{ψ}}_{i {\hat{ω}}_{i}}^{T} {\hat{ω}}_{i})}^{T}

(29)

{\overset{\cdot}{\hat{c}}}_{i} = γ_{c} {\overset{'}{ψ}}_{i {\hat{c}}_{i}} {\hat{W}}_{i}^{T} e_{i}

(30)

{\overset{\cdot}{\hat{ω}}}_{i} = γ_{ω} {\overset{'}{ψ}}_{i {\hat{ω}}_{i}} {\hat{W}}_{i}^{T} e_{i}

(31)

where $γ_{w} > 0$ , $γ_{c} > 0$ and $γ_{ω} > 0$ are constant. Then, by choosing $a_{i}$ such that the two inequalities $a_{i} > (ζ + 1) / 2$ , where $ζ$ is an arbitrary positive constant, and $(a_{i} - (ζ + 1) / 2) ∥ e_{i} (t) ∥^{2} \geq 1 / 2 ζ ∥ Δ_{i} ∥^{2}$ are satisfied, consensus in equation (28) can be reached.

Proof. Consider equation (32) as a Lyapunov–Krasovskii function candidate

\begin{matrix} V (E (t), \tilde{W}, \tilde{c}, \tilde{ω}, t) = \frac{1}{2} x^{T} (t) (L \otimes I_{m}) x (t) \\ + \frac{1}{2 γ_{w}} tr ({\tilde{W}}^{T} \tilde{W}) + \frac{1}{2 γ_{c}} {\tilde{c}}^{T} \tilde{c} \\ + \frac{1}{2 γ_{ω}} {\tilde{ω}}^{T} \tilde{ω} \\ + \frac{1}{2} \sum_{i = 1}^{n} \int_{t - τ_{i}}^{t} U_{i} (x_{i} (τ)) d τ \\ = \frac{1}{2} E^{T} (t) x (t) + \frac{1}{2 γ_{w}} tr ({\tilde{W}}^{T} \tilde{W}) \\ + \frac{1}{2 γ_{c}} {\tilde{c}}^{T} \tilde{c} + \frac{1}{2 γ_{ω}} {\tilde{ω}}^{T} \tilde{ω} \\ + \frac{1}{2} \sum_{i = 1}^{n} \int_{t - τ_{i}}^{t} U_{i} (x_{i} (τ)) d τ \\ = \frac{1}{2} \sum_{i = 1}^{n} (e_{i}^{T} (t) x_{i} (t) + \frac{1}{γ_{w}} tr ({\tilde{W}}_{i}^{T} {\tilde{W}}_{i}) \\ + \frac{1}{γ_{c}} {\tilde{c}}_{i}^{T} {\tilde{c}}_{i} + \frac{1}{γ_{ω}} {\tilde{ω}}_{i}^{T} {\tilde{ω}}_{i} \\ + \int_{t - τ_{i}}^{t} U_{i} (x_{i} (τ)) d τ) \end{matrix}

(32)

where $x (t) = [x_{1}^{T} (t) \dots x_{n}^{T} (t)]^{T}$ , $\tilde{W} = diag ({\tilde{W}}_{1}, \dots, {\tilde{W}}_{n})$ , $\tilde{c} = [{\tilde{c}}_{1}^{T} \dots {\tilde{c}}_{n}^{T}]^{T}$ , $\tilde{ω} = [{\tilde{ω}}_{1}^{T} \dots {\tilde{ω}}_{n}^{T}]^{T}$ , $E (t) = [e_{1}^{T} (t) \dots e_{n}^{T} (t)]^{T}$ and $U_{i} (x_{i} (t)) = ρ_{i}^{2} (x_{i} (t))$ . As for Theorem 1, it can be inferred that $V (E (t), \tilde{W}, \tilde{c}, \tilde{ω}, t)$ is a positive definite function.

The derivative of $V (t)$ can be achieved as follows

\begin{matrix} \overset{\cdot}{V} (E (t), \tilde{W}, \tilde{c}, \tilde{ω}, t) = \sum_{i = 1}^{n} (e_{i}^{T} (t) (f_{i} (x_{i} (t)) \\ + h_{i} (x_{i} (t - τ_{i})) + u_{i} (t)) \\ - \frac{1}{γ_{w}} tr ({\tilde{W}}_{i}^{T} {\overset{\cdot}{\hat{W}}}_{i}) + \frac{1}{γ_{c}} {\tilde{c}}_{i}^{T} {\overset{\cdot}{\hat{c}}}_{i} \\ + \frac{1}{γ_{ω}} {\tilde{ω}}_{i}^{T} {\overset{\cdot}{\hat{ω}}}_{i} + \frac{1}{2} (ρ_{i}^{2} (x_{i} (t)) \\ - ρ_{i}^{2} (x_{i} (t - τ_{i})))) \end{matrix}

(33)

By substituting equations (26), (29) to (31) and (17) into equation (33) and using the property $e_{i}^{T} e_{i}^{- 1} = 1$ , equation (33) can be recast in the following format

\begin{matrix} \overset{\cdot}{V} (E (t), \tilde{W}, \tilde{c}, \tilde{ω}, t) = \sum_{i = 1}^{n} (e_{i}^{T} (t) (Δ_{i} + h_{i} (x_{i} (t - τ_{i}))) \\ - \frac{1}{2} ρ_{i}^{2} (x_{i} (t - τ_{i})) - a_{i} ∥ e_{i} (t) ∥^{2}) \end{matrix}

(34)

The following inequality always holds

\begin{matrix} \overset{\cdot}{V} \leq \sum_{i = 1}^{n} (e_{i}^{T} (t) (Δ_{i}) + \frac{1}{2} {(∥ e_{i} (t) ∥^{2} + ρ_{i} (x_{i} (t - τ_{i}))}^{2}) \\ - \frac{1}{2} ρ_{i}^{2} (x_{i} (t - τ_{i})) - a_{i} ∥ e_{i} (t) ∥^{2}) \end{matrix}

(35)

Then

\overset{\cdot}{V} \leq \sum_{i - 1}^{n} (\frac{ζ}{2} ∥ e_{i} (t) ∥^{2} + \frac{1}{2 ζ} ∥ Δ ∥^{2} - (a_{i} - \frac{1}{2}) ∥ e_{i} (t) ∥^{2})

(36)

holds for any arbitrary positive constant of $ζ$ . As in Theorem 1, it can be inferred that $∥ Δ ∥ \leq δ$ , where $δ$ is a positive constant. If the inequalities $a_{i} > (ζ + 1 / 2)$ and $(a_{i} - (ζ + 1) / 2) ∥ e_{i} (t) ∥^{2} \geq (1 / 2 ζ) ∥ Δ ∥^{2}$ hold, then $\overset{\cdot}{V} \leq 0$ , and as a result $E (t)$ , $\tilde{W}$ , $\tilde{c}$ and $\tilde{ω}$ are globally uniformly ultimately bounded.

Simulations

In this section, three simulation examples are presented. The first example illustrates the effectiveness of the proposed control algorithm in the consensus control of first-order nonlinear multi-agent systems, the second example investigates the capability of the control method suggested in the fourth section for controlling time-delayed systems, and the third example shows a real-world application of the proposed control algorithm given in Theorem 2; in this example, the problem of reaching consensus in velocity of six two-link manipulators is addressed. In all of the examples, the network topologies are the same and the following Laplacian matrix is used

L = [\begin{matrix} 0.8 & - 0.3 & 0 & 0 & 0 & - 0.5 \\ - 0.3 & 0.7 & - 0.4 & 0 & 0 & 0 \\ 0 & - 0.4 & 0.5 & - 0.1 & 0 & 0 \\ 0 & 0 & - 0.1 & 0.4 & - 0.3 & 0 \\ 0 & 0 & 0 & - 0.3 & 0.9 & - 0.6 \\ - 0.5 & 0 & 0 & 0 & - 0.6 & 1.1 \end{matrix}]

Example 1

Consider the adaptive laws in Theorem 1 with five fuzzy rules for each agent, i.e. $N = 5$ , and $γ_{ω} = γ_{c} = γ_{w} = 1$ . The nonlinear multi-agent system with six agents is defined as

[\begin{matrix} {\overset{\cdot}{x}}_{i 1} (t) \\ {\overset{\cdot}{x}}_{i 2} (t) \end{matrix}] = [\begin{matrix} x_{i 2} (t) \sin (α_{i 1} x_{i 1} (t)) \\ x_{i 1} (t) \cos (α_{i 2} x_{i 2}^{2} (t)) \end{matrix}] + u_{i}

(37)

The control input $u_{i}$ is given in equation (12), where $a_{i} = 130$ . The mentioned nonlinear functions in the agents’ dynamics satisfy the Lipschitz condition and Assumption 1 holds. The constant values $α_{i 1}$ and $α_{i 2}$ are shown in Table 1. The initial values of the agents’ positions are $x_{1} (0) = [6, 2]^{T}$ , $x_{2} (0) = [3, 3 \sqrt{3}]^{T}$ , $x_{3} (0) = [- 3, 3 \sqrt{3}]^{T}$ , $x_{4} (0) = [- 6, - 2]^{T}$ , $x_{5} (0) = [- 3, - 3 \sqrt{3}]^{T}$ and $x_{6} (0) = [3, - 3 \sqrt{3}]$ . As is obvious from Figure 1, consensus is reached quickly. To examine the efficiency of the proposed control algorithm, equation (38) is employed as a performance measure (Hou et al., 2009)

e^{x} = \sum_{i = 1}^{n} e^{x_{i}}, e^{x_{i}} = | x_{i 1} - x_{A 1} | + | x_{i 2} - x_{A 2} |

(38)

where $x_{Ak} = (1 / n) \sum_{i = 1}^{n} x_{ik}$ . As shown in Figure 2, the total consensus error at the beginning is about 48 and approaches 0 gradually. Also, in Figure 3, the evolution of adaptive AFWN parameters of agent 1 is shown and it can be seen that the adaptive parameters $\hat{C}$ , $\hat{ω}$ , and $\hat{W}$ remain bounded.

Table 1.

$α_{i 1}$ and $α_{i 2}$ for ith agent.

i	1	2	3	4	5	6
$α_{i 1}$	0.6	$- 0.6$	7	$- 10$	10	5
$α_{i 2}$	0.3	0.4	$- 5$	$- 11$	11	10

Figure 1.

Position of agents in Example 1.

Figure 2.

Total consensus error $e^{x}$ of agents in Example 1.

Figure 3.

Adaptive fuzzy wavelet network parameters of agent 1 in Example 1.

As a comparison, the consensus control problem in this example is solved using adaptive RBFNNs. This method was used by Chen et al. (2014) and, with little change, we apply it to Example 1. The controller is the same as equation (12) and its parameters are equal to the parameters in Example 1, except for ${\hat{f}}_{i} (x_{i} (t))$ , which is approximated using adaptive RBFNNs as

{\hat{f}}_{i} (x_{i} (t)) = {\hat{W}}_{i}^{T} (t) S_{i} (x_{i})

where $S_{i} (x_{i}) = [s_{1} (x_{i}), \dots, s_{\hat{N}} (x_{i})], \hat{N} = 36$ (the number of neurons in the network), $s_{l} (x_{i}) = \exp [- (x_{i} - μ_{l})^{T} (x_{i} - μ_{l}) / {ϕ_{l}}^{2}], l = 1, 2, \dots, \hat{N}$ , and ${\overset{\cdot}{\hat{W}}}_{i} (t) = Γ_{i} [S_{i} (x_{i} (t)) e_{i}^{T} - σ_{i} {\hat{W}}_{i} (t)]$ is the adaptive law. As can be seen in Figures 4 and 5, in the beginning the adaptive RBFNN-based algorithm can control agents to reach consensus; however, later, owing to lack of robustness in the controller, the consensus error between agents becomes large and the multi-agent system loses its stability. Also, in this example, the proposed method of this paper used five fuzzy rules ( $N = 5$ ), by contrast, the adaptive RBFNN-based method uses 36 neurons ( $\hat{N} = 36$ ), which results in more computation.

Figure 4.

Position of agents in Example 1 using method based on adaptive radial basis function neural network.

Figure 5.

Total consensus error $e^{x}$ of agents in Example 1 using method based on adaptive radial basis function neural network.

Both the proposed methods in this paper, which are based on AFWNs, and the method used for comparison, which is based on an adaptive RBFNN, have similar structures. Therefore, the computational complexities of the proposed algorithms in this paper and the adaptive RBFNN-based approach are similar. The computational complexity of our proposed method can be shown as $O (nN)$ , where n is the number of agents and N represents the number of fuzzy rules. The adaptive RBFNN-based algorithm has a computational complexity of $O (n \hat{N})$ , where n is the number of agents and $\hat{N}$ is the number of neurons in the neural network.

Example 2

In this example, the dynamics of the agents include a state time delay. Assumptions 1 to 4 hold and the parameters of the agents are the same as in Example 1. Consider equations (27) and (26) and the adaptive laws in Theorem 2. The nonlinear functions are given as

\begin{matrix} f_{i} (x_{i} (t)) = [\begin{matrix} x_{i 2} (t) \sin (α_{i 1} x_{i 1} (t)) \\ x_{i 1} (t) \cos (α_{i 2} x_{i 2}^{2} (t)) \end{matrix}] \\ h_{i} (x_{i} (t - τ_{i})) = [\begin{matrix} β_{i 1} x_{i 1} \cos (x_{i 2} (t - τ_{i})) \\ β_{i 2} x_{i 2} \sin (x_{i 1} (t - τ_{i})) \end{matrix}] \end{matrix}

The maximum time delay is $τ_{max} = 1.9$ . The parameters of the controller and adaptive laws are $a_{i} = 150$ and $γ_{ω} = γ_{c} = γ_{w} = 10$ . Constant values $β_{i 1}$ and $β_{i 2}$ are shown in Table 2. Simulation results are obtained in Figures 6(a) and 7(a), and it can be observed that the agents reached consensus after about 0.15 s. Using the aforementioned performance measure of Example 1, it is shown in Figure 8(a) that the total consensus error decreased to a small amount. It is worth noting that using larger control gains can lead to the smaller consensus errors. To show the convergence of adaptive parameters of the proposed AFWN, the evolution of parameters $\hat{C}$ , $\hat{ω}$ , and $\hat{W}$ are shown in Figure 9(a).

Table 2.

$β_{i 1}$ and $β_{i 2}$ in ith agent.

i	1	2	3	4	5	6
$β_{i 1}$	0.9	3.5	$- 2.1$	7	4.6	5.3
$β_{i 2}$	1.2	2.6	0.6	$- 4.3$	2.8	2.4

Figure 6.

Position of agents in Example 2: (a) $a_{i} = 150$ ; (b) $a_{i} = 50$ .

Figure 7.

Position of agents in Example 2 in three-dimensional view: (a) $a_{i} = 150$ ; (b) $a_{i} = 50$ .

Figure 8.

Total consensus error $e^{x}$ of agents in Example 2: (a) $a_{i} = 150$ ; (b) $a_{i} = 50$ .

Figure 9.

Adaptive fuzzy wavelet network parameters of agent 1 in Example 2: (a) $a_{i} = 150$ ; (b) $a_{i} = 50$ .

To study the effect of varying the controller parameters, the control gains are chosen as $a_{i} = 50$ . Figures 6(b), 7(b), 8(b) and 9(b) show the position of the agents, the position of the agents in a three-dimensional view, the total consensus error and AFWN adaptive parameters, respectively. As can be seen from these figures, a consensus was reached after about 0.5 s; by contrast with the previous simulation results shown in Figures 6(a), 8(a), the agents needed more time to reach consensus. This time difference in reaching consensus is due to the decrease in control gain and, consequently, control input of the agents.

Example 3

To show the effectiveness of the proposed control algorithm in Theorem 2, a group of two-link manipulators is studied in this example. The objective of this group could be to load a workpiece or hold up an object. Six agents, with the following system dynamics, are considered (Chen et al., 2014)

\begin{matrix} {\overset{\cdot}{q}}_{i 1} = q_{i 2} \\ M_{i} (q_{i 1}) {\overset{\cdot}{q}}_{i 2} + V_{i} (q_{i 1}, q_{i 2}) q_{i 2} + G_{i} (q_{i 1}) + f_{i} (q_{i 2} (t - τ_{i})) \\ = ζ_{i} \\ i = 1, \dots, 6 \end{matrix}

(39)

where $q_{i 1}, q_{i 2} \in R^{2}$ are the vectors of position and velocity of the manipulators, respectively, $M_{i} (q_{i 1}) \in R^{2 \times 2}$ indicates the inertia matrix and it is assumed that $M_{i} (q_{i 1}) = I_{2 \times 2}$ , where $I_{2 \times 2}$ denotes the identity matrix

V_{i} (q_{i 1}, q_{i 2}) = [\begin{matrix} V_{i 11} & V_{i 12} \\ V_{i 21} & V_{i 22} \end{matrix}] \in R^{2 \times 2}

shows the centripetal Coriolis matrix of the ith agent, where $V_{i 11} = - m_{i 2} r_{i 1} r_{i 2} \sin (q_{i 12}) q_{i 22}$ , $V_{i 12} = - m_{i 2} r_{i 1} r_{i 2} \sin (q_{i 12}) q_{i 22} -$ $m_{i 2} r_{i 1} r_{i 2} \sin (q_{i 12}) q_{i 21}$ , $V_{i 21} = m_{i 2} r_{i 1} r_{i 2} \sin (q_{i 12}) q_{i 22}$ and $V_{i 22} = 0$ ; $G_{i} = [G_{i 1}, G_{i 2}]^{T} \in R^{2}$ denotes the gravitational vector of the ith agent, where $G_{i 1} = (m_{i 1} + m_{i 2}) g r_{i 1} \sin (q_{i 11}) + m_{i 2} g r_{i 2} \sin (q_{i 11} + q_{i 12})$ and $G_{i 2} = m_{i 2} g d_{i 2} \sin (q_{i 11} + q_{i 12})$ . The vector

\begin{matrix} f_{i} (q_{i 2} (t)) \\ = [α_{i 1} q_{i 21} + β_{i 1} sgn (q_{i 21}), α_{i 2} q_{i 22} + β_{i 2} sgn (q_{i 22})]^{T} \end{matrix}

indicates the friction force of the system, $τ_{i}$ shows the time delay, $ζ_{i} \in R^{2}$ denotes the torque input vector of the ith manipulator and the parameters $α_{i 1}, α_{i 2}, β_{i 1}, β_{i 2}$ are shown in Tables 3 and 4. The parameters of the manipulators are set as $g = 9.8$ m/s ², $d_{i 2} = 1$ m, $r_{i 1} = 1.6$ m, $r_{i 2} = 1.1$ m, $m_{i 1} = 1.1$ kg, and $m_{i 2} = 2.1$ kg for $i = 1, \dots, 6$ . The initial conditions of position and velocity of six manipulators are shown in Table 5.

Table 3.

$α_{i 1}$ and $α_{i 2}$ of the ith agent.

i	1	2	3	4	5	6
$α_{i 1}$	10	7	$- 11$	$- 8$	13	12
$α_{i 2}$	28	$- 15$	23	17	$- 19$	$- 17$

Table 4.

$β_{i 1}$ and $β_{i 2}$ of the ith agent.

i	1	2	3	4	5	6
$β_{i 1}$	5	2.5	$- 2.1$	7	6	$- 5.3$
$β_{i 2}$	1.1	1.6	1.6	$- 4.1$	$- 8$	3.4

Table 5.

Initial values of states in ith agent, $q_{i 11}, q_{i 12}, q_{i 21}$ and $q_{i 22}$ .

i	1	2	3	4	5	6
$q_{i 11}$	0	0	0	0	0	0
$q_{i 12}$	0	0	0	0	0	0
$q_{i 21}$	4.2	2.7	$- 3.1$	$- 4.5$	$- 1.8$	2.5
$q_{i 22}$	0.5	2.8	1.9	$- 2.2$	$- 3.1$	$- 4.3$

Equation (26), and the adoptive laws in Theorem 2 are used to design a proper controller for the six manipulators to reach consensus in their velocity. In this example, the control gain is chosen as $a_{i} = 15$ , $N = 5$ and the parameters of adaptive laws are the same as those in Example 2. As shown in Figure 10, the velocities of both joints reached a consensus and the error of consensus in Figure 11 approached zero, also Figure 12 shows the evolution of the adaptive parameters.

Figure 10.

Trajectories of velocities of first and second joints of manipulators in Example 3.

Figure 11.

Total velocity consensus error $e^{q 2}$ of both joints of manipulators in Example 3.

Figure 12.

Adaptive fuzzy wavelet network parameters of agent 1 in Example 3.

Conclusion

Two novel control algorithms were presented in this paper. The first scheme tackles the consensus control problem of first-order nonlinear multi-agent systems and uses AFWNs to compensate for nonlinear dynamics. The second control algorithm copes with consensus control problems of time-delayed multi-agent systems with uncertain nonlinearities. In the second algorithm, appropriate Lyapunov–Krasovskii functions are applied to deal with unknown state time-delayed terms. Finally, the uniformly ultimately bounded stability of both control approaches was proved. Simulation results were obtained to illustrate the effectiveness of the presented methods; then a performance measure was employed to show the total consensus error of closed-loop systems. However, there are numerous challenging problems, such as consensus control of nonlinear multi-agent systems with switching topologies and control in the presence of a fault, which we face in real-world applications. It is intended that future works will address the consensus control problem of time-delayed nonlinear multi-agent systems with switching topologies under disturbances and fault conditions. Moreover, the problem of communication time delay in multi-agent systems is another interesting research topic for future work.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Abiyev

Kaynak

(2008) Fuzzy wavelet neural networks for identification and control of dynamic plants – a novel structure and a comparative study. IEEE Transactions on Industrial Electronics 55(8): 3133–3140.

Cao

Ren

et al . (2013) An overview of recent progress in the study of distributed multi-agent coordination. IEEE Transactions on Industrial Informatics 9(1): 427–438.

Chen

Wen

Liu

et al . (2014) Adaptive consensus control for a class of nonlinear multiagent time-delay systems using neural networks. IEEE Transactions on Neural Networks and Learning Systems 25(6): 1217–1226.

Chen

Song

(2014) Cooperative tracking control of nonlinear multiagent systems using self-structuring neural networks. IEEE Transactions on Neural Networks and Learning Systems 25(8): 1496–1507.

Chen

Liu

(2007) Pinning complex networks by a single controller. IEEE Transactions on Circuits and Systems I: Regular Papers 54(6): 1317–1326.

Fan

Chen

Zhang

(2014) Semi-global consensus of nonlinear second-order multi-agent systems with measurement output feedback. IEEE Transactions on Automatic Control 59(8): 2222–2227.

Godsil

Royle

(2001) Algebraic Graph Theory. New York: Springer.

Zhang

(2001a) Fuzzy wavelet networks for function learning. IEEE Transactions on Fuzzy Systems 9(1): 200–211.

DWC

Zhang

(2001b) Fuzzy wavelet networks for function learning. IEEE Transactions on Fuzzy Systems 9(1): 200–211.

10.

Horn

Johnson

(1991) Topics in Matrix Analysis. Cambridge: Cambridge University Press.

11.

Hou

Cheng

Tan

(2009) Decentralized robust adaptive control for the multiagent system consensus problem using neural networks. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 39(3): 636–647.

12.

Lin

(2006) Nonsingular terminal sliding mode control of robot manipulators using fuzzy wavelet networks. IEEE Transactions on Fuzzy Systems 14(6): 849–859.

13.

Lin

Jia

(2009) Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies. Automatica 45(9): 2154–2158.

14.

Lin

Jia

(2010) Consensus of a class of second-order multi-agent systems with time-delay and jointly-connected topologies. IEEE Transactions on Automatic Control 55(3): 778–784.

15.

Liu

Wang

et al . (2013) Adaptive second-order consensus of multi-agent systems with heterogeneous nonlinear dynamics and time-varying delays. Neurocomputing 118: 289–300.

16.

Mei

Ren

(2013) Distributed coordination for second-order multi-agent systems with nonlinear dynamics using only relative position measurements. Automatica 49(5): 1419–1427.

17.

Meng

Ren

Cao

et al . (2011) Leaderless and leader-following consensus with communication and input delays under a directed network topology. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 41(1): 75–88.

18.

Olfati-Saber

Murray

(2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control 49(9): 1520–1533.

19.

Ren

(2008) On consensus algorithms for double-integrator dynamics. IEEE Transactions on Automatic Control 53(6): 1503–1509.

20.

Ren

Beard

Atkins

(2005) A survey of consensus problems in multi-agent coordination. In: American control conference, Portland, OR, USA, 8–10 June 2005, pp. 1859–1864. Piscataway, IEEE.

21.

Semsar-Kazerooni

Khorasani

(2009) Multi-agent team cooperation: a game theory approach. Automatica 45(10): 2205–2213.

22.

Shahriari-kahkeshi

Sheikholeslam

(2014) Adaptive fuzzy wavelet network for robust fault detection and diagnosis in non-linear systems. IET Control Theory & Applications 8(15): 1487–1498.

23.

Sun

Wang

(2009) Consensus problems in networks of agents with double-integrator dynamics and time-varying delays. International Journal of Control 82(10): 1937–1945.

24.

Taheri

Sheikholeslam

Najafi

et al . (2017) Adaptive fuzzy wavelet network control of second order multi-agent systems with unknown nonlinear dynamics. ISA Transactions 69(Suppl. C): 89–101.

25.

Wen

Duan

Chen

et al . (2014) Consensus tracking of multi-agent systems with Lipschitz-type node dynamics and switching topologies. IEEE Transactions on Circuits and Systems I: Regular Papers 61(2): 499–511.

26.

Chen

Cao

et al . (2010) Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 40(3): 881–891.

27.

Chen

Wang

et al . (2009) Distributed consensus filtering in sensor networks. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 39(6): 1568–1577.

28.

Ren

Zheng

et al . (2013) Distributed control gains design for consensus in multi-agent systems with second-order nonlinear dynamics. Automatica 49(7): 2107–2115.

29.

Zekri

Sadri

Sheikholeslam

(2008) Adaptive fuzzy wavelet network control design for nonlinear systems. Fuzzy Sets and Systems 159(20): 2668–2695.