Distributed fuzzy adaptive consensus states tracking controller for a class of nonlinear non-affine multi-agent systems with dynamic uncertainties

Abstract

This paper proposes an adaptive distributed consensus tracking protocol approach for uncertain nonlinear multi-agent systems in pure-feedback form under a directed topology where each follower is dominated by dynamic uncertainties and non-affine function of the system. By using a new intelligent variable structure controller for each follower, the provided consensus control manner solves the problem of unknown nonlinear non-affine functions of the agents. Fuzzy systems are employed both to compensate the unknown nonlinear functions obtained by recursive design procedure for followers and to use expert’s knowledge in the controller design procedure. On-line adaptation of the controller parameters, convergence of the consensus errors of the agents to zero, boundedness of all signals involved in the closed loop system, and chattering avoidance are the main characteristics of the proposed controller and the observer. Finally, to illustrate the persistency of the controller and effectiveness of this algorithm, a numerical example of a chaotic multi-agent system is investigated.

Keywords

Lyapunov stability adaptive control non-affine nonlinear system-multi-agent systems fuzzy systems

1. Introduction

In recent years, a consensus problem of multi-agent systems has received compelling attention because of its broad applications, such as UAV formation flying, wireless sensor network and smart grids. To control the multi-agent systems, one usually faces poor knowledge of the plant parameters and interconnections among agents. An adaptive control technique is then an appropriate strategy to be employed and all agents are driven to an agreement by a consensus protocol. The application of multi-agent interconnected systems can be mentioned in a variety of engineering applications such as power systems, manufacturing processes, and communication systems.

As a result of both tunable structure of the FAC and using the experts’ knowledge in controller design procedure, FAC attracted many researchers to develop appropriate controllers for nonlinear systems especially for multi-agent systems (MAS).

In the recent years, FAC is fully investigated. The Takagi-Sugeno (TS) fuzzy systems have been used to model nonlinear systems and then stable TS based controllers have been derived in [1, 2, 3]. Sliding mode fuzzy adaptive controller design for a class of multivariable TS fuzzy systems was proposed in [4]. In [5, 6], the non-affine nonlinear functions were first approximated by the TS fuzzy systems, and then stable TS fuzzy controller and observer have been designed for the obtained model. In these papers, due to the assumption that the systems should be linearizable around some operating points modeling and designing of proper controllers could be simply done.

The linguistic fuzzy systems have also been used to design controllers for nonlinear systems.

[7, 8, 9, 10, 11] have considered fuzzy stable adaptive feedback linearization controller for affine systems and furthermore in [10, 11], the zero dynamics of the mentioned system is stable. [12] proposed designing FAC sliding mode controller for a class of affine systems. [13, 14] presented a fuzzy adaptive controller procedure for a class of affine chaotic systems. Designing linear observer based fuzzy adaptive controller is derived for class of affine nonlinear systems in [15, 16, 17, 18]. [19, 20, 21] are discussed fuzzy adaptive sliding mode controllers for class of affine nonlinear time delay systems. The fuzzy adaptive output tracking controller is suggested for affine nonlinear MIMO systems in [22].

An observer based robust fuzzy adaptive controller for a class of affine nonlinear systems is derived in [23]. Both direct and indirect adaptive output-feedback fuzzy decentralized controllers are applied for affine MIMO nonlinear systems in [24]. [25] presented fuzzy based adaptive controller design for nonlinear affine systems with guaranteed ultimately boundedness of tracking error. A direct fuzzy adaptive controller design is derived for a non-minimum phase two-axis inverted pendulum servomechanism based on real-time stabilization in [26]. The restricted conditions imposed on the system dynamics and boundedness of the control gain to some known functions or constant values are the main drawback of the proposed observer based controller.

[27, 28] proposed stable FAC for non-affine nonlinear systems. The convergence of tracking errors to neighborhood of zero is the main restriction of these methods. [29, 30, 47] dealt with a decentralized fuzzy model reference controller for a class of canonical nonlinear MIMO system. Considering the interaction as a bounded disturbance and measurability of all states are the main limitation of these methods. [31] derived designing fuzzy sliding mode adaptive controller for affine nonlinear large scale systems. [32] considered fuzzy adaptive controller design for affine nonlinear time delayed systems. Considering the affine systems in designing controller procedure is the main restriction of the mentioned references.

The basic idea of consensus control is that all agents are driven to an agreement by a consensus protocol, which is designed based on local information. In consensus control, two control strategies, leaderless consensus and leader-following consensus, have been extensively developed. The most of the research results were limited to first-order or second-order multi-agent systems [33, 34, 35, 36, 37].

Recently, the high-order consensus problem has received obviously increasing attention, and several novel consensus design methods for high-order linear multi-agent systems have been developed [38]. In [38], the authors proposed a class of $l$ -order ( $l>$ 3) consensus control approaches by generalizing the first-order and second order consensus algorithms. In recent years, the challenging high-order consensus control has been extended to nonlinear multi-agent system [39]. Intelligent adaptive back-stepping technique for a class of the nonlinear strict-feedback or semi-strict feedback systems is discussed in [40, 41]. The observer-based adaptive control has been well developed for a class of the SISO the MIMO nonlinear systems in [42, 43, 44, 45]. Designing of impulsive controller is derived for consensus tracking of the nonlinear MAS in [48]. Designing leader follower controller for single integrator with time delayed communication is presented in [49].

Designing neural adaptive consensus controller is suggested for a class of affine nonlinear Multiagent systems in [50]. In [51], RBF neural adaptive consensus controller is derived for time varying affine nonlinear Multiagent systems. Fuzzy adaptive controller based on high gain observer is derived in [52] for heterogeneous second order nonlinear Multiagent system without guaranteed stability of the overall system and observer based controller. Adaptive back-stepping consensus tracking control based on classic observer is presented for nonlinear affine semi-strict-feedback Multiagent systems in [53]. Fuzzy adaptive controller based on classic sliding mode controller is developed for affine nonlinear Multiagent systems in [54]. Neural adaptive consensus tracking controller design procedure is proposed for nonlinear affine strict-feedback multi-agent systems in [55]. In [50] to [56], for simplicity it is assumed the control gains of the all agents are equal to 1.

Observer based fuzzy adaptive back-stepping controller design is discussed for a class of affine nonlinear systems in [56]. Neuro-adaptive controller is derived for a time-delay affine nonlinear system in [57]. In [58], designing observer based TS fuzzy system is proposed for a class of industrial system. Combination of the output predictive controller and fault tolerant control are derived for a class of industrial processes in [59].

Intensely motivated by present researching situation and theoretical demands of dynamics controller for multi-agent systems, so we focus on the problem of design a stable adaptive controller based on fuzzy systems for a class of multi-agent non-affine nonlinear systems. The main contributions of this paper are as follows. 1) It is considered the dynamics of the agents are unknown 2) the controller is robust against uncertainties and external disturbances, 3) boundedness of the signals in the closed loop system. Compared with the previous studies which primarily concentrated on affine SISO agents with constant control gain, the proposed method is applied on non-affine nonlinear agents.

The rest of the paper is organized as follows. Section 2 gives preliminaries. Designing fuzzy adaptive controllers is proposed in Section 3. Section 4 presents simulation results of the proposed controller and Section 5 concludes the paper.

2. Preliminaries

In this section, basic graph theory and some notations are first introduced. Then, the consensus tracking control problem of the multi-agent system is formulated. Finally, mathematical description of fuzzy system is initiated.

2.1 Basic graph theory

Consider a multi-agent system that consists of a leader and the followers. A brief survey on the graph theory is introduced here.

Let $G=(v,E)$ be a weighted graph in which $v=(v_{1},\ldots,v_{n})$ is the nonempty set of nodes, $E\subseteq v\times v$ is the set of edge, $(v_{i},v_{j})\in E$ means there is an edge from node i to node j. the topology of a weighted graph G stands for the adjacency matrix $A=[a_{ij}]\in R^{N\times N}$ and $a_{ij}>0$ if $(v_{j},v_{i})\in E$ otherwise, $a_{ij}=$ 0. It is assumed G is direct graph. The weighted in-degree of the node i is defined as $d_{i}=\sum_{j=1}^{N}{a_{ij}}$ and furthermore in-degree matrix is $D=\textit{diag}(d_{1},\ldots,d_{N})\in R^{N\times N}$ . The graph Laplacian matrix is $L=D-A\in R^{N\times N}$ . Let $\underline{1}=[1,\ldots,1]\in R^{N\times 1}$ with appropriate dimension, then $L\underline{1}=$ 0.

The set of neighbors of node i is all the nodes that the node i can get information, not necessarily vice versa for directed graph. For undirected graph, neighbor is mutual relation. A direct path from node i to node j is sequence of straight edges in form $\{(v_{i},v_{l})(v_{l},v_{k}),\ldots,(v_{m},v_{j})\}$ . The digraph has spanning tree, if the is a node (called the root) such that there is a directed path from the root to every other node in the graph. A digraph is strongly connected if for any ordered pair of nodes $[v_{i},v_{j}]$ with $i\neq j$ , there is a direct path from node i to node j.

2.2 Problem statement

Consider a nonlinear multi-agent system that consists of N interconnected following agents with uncertainty dynamic. The system model of every following agent can be described by the following non-affine nonlinear dynamics as:

$\displaystyle\left\{\begin{array}[]{l}\dot{x}_{i,l}(t)=x_{i,l+1}(t)\ \ \ \ l=1% ,2,\ldots,n-1,\\ \ \ \ i=1,2,\ldots,N\\ \dot{x}_{i,n}=f(\underline{x}_{i},u_{i})+d(t)\\ y_{i}=C_{i}^{T}\underline{x}_{i}\end{array}\right.$ (1)

where $x_{i,l}$ declares $l^{th}$ state of the $i^{th}$ agent, $n$ is number of the state in $i^{th}$ agent, $N$ is number of the agents, $\underline{x}_{i}=[x_{i,1},\ldots,x_{i,n}]^{T}\in R^{n}$ is the state vector of the system, $u_{i}\in R$ is the control input, $y_{i}\in R$ is the agent output, $f(\underline{x}_{i},u_{i})$ are a class of unknown smooth nonlinear function, and $d(t)$ considers as a bounded disturbance.

The above equation can be rewritten as:

$\displaystyle\left\{\begin{array}[]{l}\underline{\dot{x}}_{i}=A_{i}\underline{% x}_{i}+b_{s}(f(\underline{x}_{i},u_{i})+d(t))\\ y_{i}=C_{i}^{T}\underline{x}_{i}\end{array}\right.$ (2)

where $A_{i}$ and $b_{s}$ are defined below.

$\displaystyle A_{i}=\left[\begin{array}[]{ccccc}0&1&0&\ldots&0\\ 0&0&1&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&1\\ 0&0&0&\cdots&0\\ \end{array}\right]\in R^{n\times n},$ (3) $\displaystyle b_{s}=\left[\begin{array}[]{c}0\\ 0\\ \vdots\\ 1\\ \end{array}\right]\in R^{n}$

The control objective is to design adaptive fuzzy controller for system Eq. (1) such that the tracking error between the leader and the follower converge to zero while all signals in the closed-loop system of the agents remain bounded and the following agents track leader.

Consider the following assumptions concerning the agents Eq. (1) and the leader dynamics $x_{im}(t)$ as:

1) It is considered that the nonzero function $f_{u}(\underline{x}_{i},u_{i})=\partial f_{i}(\underline{x}_{i},u_{i})/% \partial u_{i}$ satisfies the following condition:

$\displaystyle f_{u}(\underline{x}_{i},u_{i})\!\geqslant\!f_{min}\!>\!0\ \ % \forall(\underline{x}_{i},u_{i})\in R^{n\times n}\!\times\!R$ (4) $\displaystyle\frac{df_{u}(\underline{x}_{i},u_{i})}{dt}\geqslant f_{dm}$

$f_{dm}$ is known constant parameter and define later.

2) The leader trajectory $x_{m}(t)$ is generated by the following desired system where $r(t)$ is reference signal.

$\displaystyle\left\{\begin{array}[]{l}\dot{\underline{x}}_{m}=A_{i}\underline{% x}_{m}+b_{s}r(t)\\ y_{m}=C_{i}^{T}\underline{x}_{m}\\ \end{array}\right.$ (5)

3) The external disturbance are bounded as:

$\displaystyle\left\|{d(t)}\right\|\leqslant d_{\max}$ (6)

Define $i^{th}$ local neighborhood synchronization error as the follow:

$\displaystyle e_{i}=\sum\nolimits_{j=1}^{N}{a_{ij}\left({\underline{x}_{j}-% \underline{x}_{i}}\right)}+b_{i}\left({x_{m}-x_{i}}\right)$ (7)

Consider the following tracking error vector for multi-agent system.

$\displaystyle\underline{e}=[e_{1},e_{2},\ldots,e_{N}]^{T}\in R^{n}_{i}$ (8)

The total error of the Multi-agent system that mentioned in Eq. (7) is sas:

$\displaystyle\underline{e}=-((L+B)\otimes I_{n})(X-X_{m})$ (9)

The total state dynamics of the Multi-agent system is as follow.

$\displaystyle\dot{X}=(I_{N}\otimes A)X+b^{\prime}_{s}(f(x_{i},u_{i})+d(t))$ (10)

Where $I_{N}$ is Identity matrix and ${b}^{\prime}_{s}$ is define below.

$\displaystyle{b}^{\prime}_{s}=[b_{s},\ldots,b_{s}]^{T}$ (11)

Taking the derivative of both sides of the Eq. (9), we have.

$\displaystyle\dot{\underline{e}}=-\left({\left({L+B}\right)\otimes I_{n}}% \right)\left({\dot{X}-\dot{X}_{m}}\right)$ (12)

Use Eqs (5) and (10) to rewrite the above equation as:

$\displaystyle\underline{\dot{e}}=-((L+B)\otimes I_{n})((I_{N}\otimes A)X+b^{% \prime}_{s}(f(x_{i},u_{i})+d(t))-(I_{N}\otimes A)X_{m}-{b}^{\prime}_{s}r)$ (13)

The above equation can be rewrite as follow

$\displaystyle\dot{\underline{e}}=-(I_{N}\otimes A)\underbrace{((L+B)\otimes I_% {n})(X-X_{m})}_{-\underline{e}}+((L+B)\otimes I_{n}){b}^{\prime}_{s}(f(x_{i},u% _{i})+d(t)-r)$ (14)

Based on properties of the Kroneker product, the consensus error dynamics can be depicted as

$\displaystyle\underline{\dot{e}}=(I_{N}\otimes A)\underline{e}-((L+B)\otimes I% _{n}){b}^{\prime}_{s}(f(x_{i},u_{i})+d(t)-r)$ (15)

To construct the controller, let $v_{i}$ be defined as follow.

$\displaystyle v=r(t)+k\underline{e}+v‘$ (16)

Consider the vector $k=[k_{1},k_{2},\ldots,k_{n}]^{T}$ chosen so that the matrix $A_{c}=(I_{N}\otimes A)-((L+B)\otimes I_{N}){b}^{\prime}_{s}k$ be Hurwitz and furthermore the $v^{\prime}$ is defined later.

Based on Eqs (14) and (16), we obtain

$\displaystyle\underline{\dot{e}}=(I_{N}\otimes A)\underline{e}-((L+B)\otimes I% _{n}){b}^{\prime}_{s}(f(x_{i},u_{i})+d(t)-v+{v}^{\prime}+k\underline{e})$ (17)

Based on assumption (1), Eq. (16) and the signal $v_{i}$ which is not explicitly dependent on the control input $u_{i}$ , the following inequality is satisfied:

$\displaystyle\frac{\partial(f({x}_{i},u_{i})-v)}{\partial u_{i}}=\frac{% \partial f({x}_{i},u_{i})}{\partial u_{i}}>0$ (18)

Using the implicit function theorem, it is obvious that the nonlinear algebraic equation $f(\underline{x}_{i},u_{i})-v=0$ is locally soluble for the input $u_{i}$ for an arbitrary $(\underline{x}_{i},v_{i})$ . Thus, there exists some ideal controller $u_{i}^{\ast}(\underline{x}_{i},v_{i})$ satisfying the following equality for a given $(\underline{x}_{i},v_{i})\in R^{n_{i}}\times R$ :

$\displaystyle f(\underline{x}_{i},u_{i}^{\ast})-v=0$ (19)

As a result of the mean value theorem, there exists a constant $\lambda$ in the range of $0<\lambda<$ 1, such that the nonlinear function $f(\underline{x}_{i},u_{i})$ can be expressed around $u_{i}^{*}$ as:

$\displaystyle f(\underline{x}_{i},u_{i})=f(\underline{x}_{i},u_{i}^{*})+(u_{i}% -u_{i}^{*})f_{u}=f(\underline{x}_{i},u_{i}^{*})+e_{u}f_{u}$ (20)

where $f_{u}=\partial f(\underline{x}_{i},u_{i})/\partial u_{i}|_{u_{i}=u_{i\lambda}}$ and $u_{i\lambda}=\lambda u_{i}+(1-\lambda)u_{i}^{*}$ .

Substituting Eq. (2.3) into the error Eq. (17) and using Eq. (19), we get

$\displaystyle\underline{\dot{e}}=\underbrace{(I_{N}\otimes A+((L+B)\otimes I_{% n})b^{\prime}k)}_{A_{c}}\underline{e}-\underbrace{((L+B)\otimes I_{n})b^{% \prime}_{s}}_{b_{c}}(e_{u}f_{u}+d(t)+v^{\prime})$ (21)

However, the implicit function theory only guarantees the existence of the ideal controller $u_{i}^{*}(\underline{x}_{i},v_{i})$ for system Eq. (19), and does not recommend a technique for constructing solution even if the dynamics of the system are well known. In the following section, a fuzzy system and classic controller will be used toobtain the unknown ideal controller.

2.3 Survey on fuzzy system

The fuzzy system contain fuzzy rule base, fuzzifier and defuzifier block in which the fuzzy rule base is consisted of a set of fuzzy IF-THEN rules. Consider the fuzzy rule base with M rules, and the $l^{th}$ rule is as follow.

$\displaystyle R^{l}(u):\ \textit{if}\ (x_{1}\ \textit{is}\ A_{1}^{l}\ldots x_{% n}\ \textit{is}\ A_{n}^{l})\ \textit{then}\ (y\ \textit{is}\ B^{l})$ $\displaystyle\quad l=1,2,\ldots,M$ (22)

where $x=[x_{1},x_{2},\ldots,x_{n}]^{T}$ and $y$ are the input and output of the fuzzy system and $A_{j}^{l}$ and $B^{l}$ are fuzzy membership function for the input and the output of the fuzzy system, respectively.

The fuzzy inference performs a mapping from fuzzy sets for input to fuzzy sets for output, based on the fuzzy IF-THEN rules.

The defuzzifier maps fuzzy sets in output to a crisp value in output. The configuration of Fig. 1 represents a general framework of fuzzy systems. The output of the system based on the sum-product inference and the center-average defuzzifier can be expressed as follow.

$\displaystyle y(x)=\frac{\sum\limits_{l=1}^{M}{y^{l}\prod\limits_{i=1}^{n}{\mu% _{A_{i}^{l}}(x_{i})}}}{\sum\limits_{l=1}^{M}{\prod\limits_{i=1}^{n}{\mu_{A_{i}% ^{l}}(x_{i})}}}$ (23)

where $\mu_{A_{i}^{l}}(x_{i})$ is the membership degree of the input $x_{i}$ to fuzzy set $A_{j}^{l}$ and $y^{l}$ is the point at which the membership function of fuzzy set $B^{l}$ achieves its maximum value. Furthermore, it is assumed that $\sum_{l=1}^{M}{\prod_{i=1}^{n}{\mu_{A_{i}^{l}}(x_{i})}}\neq$ 0 for all $x$ .

The fuzzy systems in the form of Eq. (23) are proven in [34] to be a universal approximator if their parameters are properly chosen.

The fuzzy system mentioned in Eq. (23) can be rewritten in the compact form as follow.

$\displaystyle y(x)=\theta^{T}w(x)$ (24)

where $\theta=[{y^{1},y^{2},\ldots,y^{M}}]$ is a vector grouping all consequent parameters, and $w(x)=[w_{1}(x),w_{2}(x)$ , $\ldots,w_{M}(x)]^{T}$ is a set of fuzzy basis functions defined as:

$\displaystyle w_{i}(x)=\frac{\prod\limits_{i=1}^{n}{\mu_{A_{i}^{l}}(x_{i})}}{% \sum\limits_{l=1}^{M}{\prod\limits_{i=1}^{n}{\mu_{A_{i}^{l}}(x_{i})}}}$ (25)

3. Fuzzy adaptive controller design

In last section, it has been shown that there exists an ideal control for achieving control objectives. We show how to derive a fuzzy system to adaptively approximate the unknown ideal controller.

The ideal controller can be represented as follow.

$\displaystyle u_{i}^{\ast}=\theta^{\ast}w_{(}\underline{z})+\varepsilon$ (26)

where $z_{i}=[x_{i},v_{i}]^{T}$ , and $\theta^{*}$ and $w(\underline{z}_{i})$ are consequent parameters and a set of fuzzy basis functions, respectively. $\varepsilon$ is an approximation error that satisfies $\left|\varepsilon\right|\leqslant\varepsilon_{\max}$ and $\varepsilon_{\max}>$ 0. The parameters $\theta^{*}$ are determined through the following optimization.

$\displaystyle\theta^{*}=\mathop{\arg\min}\limits_{\theta}\left[{\sup\left|{% \theta^{T}w(\underline{z})-u_{i}^{\ast}}\right|}\right]$ (27)

Denote the estimate of $\theta^{*}$ as $\theta$ and $u_{rob}$ as a robust controller to compensate approximation error, uncertainties, disturbance and interconnection term. To rewrite the controller given in Eq. (26) as:

$\displaystyle u_{i}=\theta^{T}w(\underline{z}_{i})+u_{rob}$ (28)

Consider $u_{irob}$ is defined below.

$\displaystyle u_{irob}=\textit{sign}(\underline{e}^{T}Pb_{c})\left({u_{i}+% \frac{u_{r}}{f_{\min}}+\frac{\hat{v}}{f_{\min}}}\right)$ (29)

In the above, $\theta_{i1}^{T}w_{i1}(\underline{z})$ approximates the ideal controller, $u_{r}$ is designed to compensate for bounded external disturbances, and $\hat{v}^{\prime}_{i}$ is estimation of ${v}^{\prime}_{i}$ . Define error vector $\tilde{\theta}_{i1}=\theta_{i1}-\theta_{i1}^{*}$ and use Eqs (28) and (29) to rewrite the error Eq. (21) as:

$\displaystyle\underline{\dot{e}}=A_{c}\underline{e}-b_{c}((\tilde{\theta}^{T}w% (\underline{z}_{i})+u_{rob}-\varepsilon)f_{u}+d(t)+{v}^{\prime})$ (30)

Assume that $P$ is positive definite solutions of the following matrix equation.

$\displaystyle A_{c}^{T}P+PA_{c}=-Q$ (31)

where $Q$ is positive definite matrices.

Consider the following update laws.

$\displaystyle\dot{u}_{r}=\gamma_{1}\left\|{e^{T}Pb_{c}}\right\|$ $\displaystyle\dot{\hat{v}}^{\prime}=\gamma_{2}\raise 3.01pt\hbox{${\left\|{e^{% T}Pb_{c}}\right\|}$}\!\mathord{\left/{\vphantom{{\left\|{e^{T}Pb_{c}}\right\|}% {f_{\min}}}}\right.\kern-1.2pt}\!\lower 3.01pt\hbox{${f_{\min}}$}$ $\displaystyle\dot{\theta}=\gamma_{3}wb_{c}^{T}P{e}$ (32) $\displaystyle\dot{u}_{i}=\gamma_{4}\raise 3.01pt\hbox{${\left\|{e^{T}Pb_{c}}% \right\|}$}\!\mathord{\left/{\vphantom{{\left\|{e^{T}Pb_{c}}\right\|}{f_{\min}% }}}\right.\kern-1.2pt}\!\lower 3.01pt\hbox{${f_{\min}}$}$

where $\gamma_{1}$ , $\gamma_{2}$ , $\gamma_{3}$ , $\gamma_{4}>$ 0 are constant parameters.

Q.E.D.

Theorem 1: consider the error dynamical system given in Eq. (30) for the multi-agent system (1) satisfying assumption (1), the external disturbances satisfying assumption (3), and a desired trajectory satisfying assumption (2), then the controller structure given in Eqs (28) and (29) with adaptation laws Eq. (33) makes the tracking error converges to zero and all signals in the closed loop system be bounded too.

Proof: consider the following Lyapunov function.

$\displaystyle V\!=\!\!\sum\limits_{i=1}^{N}\!{\frac{1}{2}\!\left(\!{\frac{1}{f% _{u}}\underline{e}^{T}P\underline{e}\!+\!\frac{\tilde{u}^{2}}{\gamma_{1}}\!+\!% \frac{{\tilde{v}}^{\prime 2}}{\gamma_{2}}\!+\!\frac{\tilde{\theta}^{T}\tilde{% \theta}}{\gamma_{3}}\!+\!\frac{\tilde{u}^{2}_{i}}{\gamma_{4}}}\!\right)}$ (33)

where $\tilde{\theta}=\theta-\theta^{\ast}$ , $\tilde{u}_{r}=u_{r}-d_{\max}$ , $\tilde{u}_{i}=u_{i}-\varepsilon_{\max}$ , and ${\tilde{v}}^{\prime}=\hat{v}^{\prime}-\left|{{v}^{\prime}}\right|$ . The time derivative of the Lyapunov function becomes.

$\displaystyle V=\sum\limits_{i=1}^{N}\frac{1}{2}\left(\frac{-\dot{f}_{u}}{f_{u% }^{2}}\underline{e}^{T}P\underline{e}+\frac{1}{f_{u}}\dot{\underline{e}}^{T}P% \underline{e}+\frac{1}{f_{u}}\underline{e}^{T}\right.$ $\displaystyle\ \ \ \ \left.P\dot{\underline{e}}+2\left(\frac{\tilde{u}_{r}\dot% {u}_{r}}{\gamma_{1}}+\frac{\tilde{v}^{\prime}\dot{\hat{v}}^{\prime}}{\gamma_{2% }}+\frac{\dot{\theta}^{T}\tilde{\theta}}{\gamma_{3}}+\frac{\tilde{u}^{i}\dot{u% }_{i}}{\gamma_{4}}\right)\!\!\right)$ (34)

Use Eqs (30) and (29), to rewrite above equation as:

$\displaystyle V=\sum\limits_{i=1}^{N}\frac{1}{2}\left[\frac{-\dot{f}_{u}}{f_{u% }^{2}}\underline{e}^{T}P\underline{e}+\frac{1}{f_{u}}(A_{c}\underline{e}-b_{c}% \right.((\tilde{\theta}^{T}w_{(}\underline{z}_{i})+u_{rob}-\varepsilon)f_{u}+d% (t)+{v}^{\prime}))^{T}P\underline{e}+\frac{1}{f_{u}}\underline{e}^{T}P(A_{c}{e% }-b_{c}((\tilde{\theta}^{T}w_{(}\underline{z}_{i})+u_{rob}-\varepsilon)\quad f% _{u}+d(t)+{v}^{\prime}))\left.+2\left(\frac{\tilde{u}_{r}\dot{u}_{r}}{\gamma_{% 1}}+\frac{\tilde{v}^{\prime}\dot{\hat{v}}^{\prime}}{\gamma_{2}}+\frac{\dot{% \theta}^{T}\tilde{\theta}}{\gamma_{3}}+\frac{\tilde{u}^{i}\dot{u}_{i}}{\gamma_% {4}}\right)\right]$ (35)

Using assumption (1) yields $1\mathord{\left/{\vphantom{1{f_{iu_{\lambda}}}}}\right.\kern-1.2pt}{f_{iu_{% \lambda}}}\leqslant 1\mathord{\left/{\vphantom{1{f_{\min}}}}\right.\kern-1.2pt% }{f_{\min}}$ and by assumptions (2) and (3) and Eq. (31), to rewrite Eq. (35) as follow.

$\displaystyle\dot{V}=\sum\limits_{i=1}^{N}{\frac{1}{2}}\left[\frac{1}{f_{u}}% \underline{e}^{T}\left(-\frac{\dot{f}_{u}}{f_{u}}P+A_{c}^{T}P+PA_{c}\right)% \underline{e}\right.-2\underline{e}^{T}Pb_{c}\left({\tilde{\theta}^{T}w(% \underline{z}_{i})+u_{rob}-\varepsilon}\right)\quad-\frac{2}{f_{u}}\underline{% e}^{T}Pb_{c}d(t)-\frac{2}{f_{u}}\underline{e}^{T}Pb_{c}{v}^{\prime}\left.+2% \left({\frac{\tilde{u}_{r}\dot{u}_{r}}{\gamma_{1}}+\frac{\tilde{v}\dot{\hat{v}% }^{\prime}}{\gamma_{2}}+\frac{\dot{\theta}^{T}\tilde{\theta}}{\gamma_{3}}+% \frac{\tilde{u}_{i}\dot{u}_{i}}{\gamma_{4}}}\right)\right]$ (36)

To use the sigmoid properties and the Eq. (30), $\alpha^{2}+\beta^{2}\geqslant 2\alpha\beta$ , the Eq. (37) can be rewritten as below.

$\displaystyle\dot{V}=\sum\limits_{i=1}^{N}\frac{1}{2}\left[-\frac{1}{f_{u}}% \underline{e}^{T}\left(\frac{\dot{f}_{u}}{f_{u}}P+Q\right)\underline{e}-2% \underline{e}^{T}\right.\quad Pb_{c}\tilde{\theta}^{T}w(\underline{z}_{i})-2% \underline{e}^{T}Pb_{c}\left({\textit{sign}(\underline{e}^{T}Pb_{c})\left({u_{% i}+\frac{u_{r}}{f_{\min}}+\frac{\hat{v}}{f_{\min}}}\right)}\right)+2\underline% {e}^{T}Pb_{c}\varepsilon-\frac{2}{f_{u}}{e}^{T}Pb_{c}d(t)-\frac{2}{f_{u}}{e}^{% T}Pb_{c}{v}^{\prime}\left.+2\left(\frac{\tilde{u}_{r}\dot{u}_{r}}{\gamma_{1}}+% \frac{\tilde{v}^{\prime}\dot{\hat{v}}^{\prime}}{\gamma_{2}}+\frac{\dot{\theta}% ^{T}\tilde{\theta}}{\gamma_{3}}+\frac{\tilde{u}_{i}\dot{u}_{i}}{\gamma_{4}}% \right)\right]$ (37)

To rewrite the above equation as:

$\displaystyle\dot{V}\leqslant\sum\limits_{i=1}^{N}{\frac{1}{2}}\left[-\frac{1}% {f_{u}}\underline{e}^{T}\left({\frac{\dot{f}_{u}}{f_{u}}P+Q}\right)\underline{% e}+2\underline{e}^{T}\right.Pb_{c}w^{T}(\underline{z}_{i})\tilde{\theta}-2% \left\|{\underline{e}^{T}Pb_{c}}\right\|\underbrace{\left({u_{i}-\varepsilon_{% \max}}\right)}_{\tilde{u}_{i}}-2\frac{\left\|{\underline{e}^{T}Pb_{c}}\right\|% }{f_{\min}}\underbrace{\left({u_{r}-d_{\max}}\right)}_{\tilde{u}_{r}}-2\frac{% \left\|{\underline{e}^{T}Pb_{c}}\right\|}{f_{\min}}\underbrace{\left({\hat{v}-% \left|{v}^{\prime}\right|}\right)}_{{\tilde{v}}}\left.+2\left({\frac{\tilde{u}% _{r}\dot{u}_{r}}{\gamma_{1}}+\frac{{\dot{\hat{v}}}^{\prime}}{\gamma_{2}}+\frac% {\dot{\theta}^{T}\tilde{\theta}}{\gamma_{3}}+\frac{\tilde{u}^{i}\dot{u}_{i}}{% \gamma_{4}}}\right)\right]$ (38)

Using Eq. (33), the above inequality rewrites as:

$\displaystyle\dot{V}\leqslant\sum\limits_{i=1}^{N}{\frac{1}{2}}\left({\frac{-1% }{f_{u}}\underline{e}^{T}\left({\frac{\dot{f}_{u}}{f_{u}}P+Q}\right)\underline% {e}}\right)\leqslant 0$ (39)

According to standard Lyapunov theorem, we conclude the closed loop system is stable and consequently the consensus error converges to zero. In addition, the boundedness of and the coefficient parameters is guaranteed. It completes the proof. Q.E.D.

4. Simulation results

In this section, the proposed controller is applied to two cases of the multi-agent system.

Case 1: the dynamics of the agents are Duffing chaotic system that mentioned in Eq. (41).

$\displaystyle\left\{{\begin{array}[]{l}\dot{x}_{i1}=x_{i2}\\ \dot{x}_{i2}=-x_{i1}^{3}-0.1x_{i2}+u_{i}+d(t)\\ y_{i}=x_{i1}\\ \end{array}}\right.$ (40)

The topology of the multi-agent system is proposed in Fig. 1. System has 3 agents in which one of them is leader and the others are follower.

Figure 1.

Topology of the graph G.

Figure 2.

The first state of the all agents and the desire value (Red line).

Figure 3.

The second state of all agents and the desire value (Red line).

Where the external disturbance is considered as $d(t)=4\sin(100\pi t)$ . It is assumed that the output of the leader should track the duffing system as a leader and the output of the follower should track the leader. In the mentioned controller in Eqs (29) and (30) , the state of the agents and the pseudo control input is in $[-40,40]$ , $[-10,10]$ and $[-45,45]$ and furthermore 5 Gaussian membership functions are considered for fuzzy system. The initial value of the update parameters set to zero and the constant parameters of system set as $f_{\min}=$ 1, $\gamma_{1}=$ 10, $\gamma_{2}=$ 10, $\gamma_{3}=$ 20, $\gamma_{4}=$ 10.

Figure 4.

Input of the first agent.

Figure 5.

Input of the second agent.

Based on Figs 2 and 3, it is clear that the proposed controller is robust against disturbance and has fast performance in converging to the value of leader.

Figures 6 and 7 show the inputs of the agents.

Figure 6.

Input of the third agent.

Figure 7.

The uncertainty compensator of the first agent ( $u_{i}$ ).

Figure 8.

The fuzzy parameters of the first agent ( $\theta$ ).

To show the boundedness of the closed loop system, Figs 8 and 9 show some of the parameters.

Figure 9.

The uncertainty compensator of the second agent ( $u_{i}$ ).

Figure 10.

The fuzzy parameters of the second agent ( $\theta$ ).

The membership functions of the inputs and the output of the fuzzy system are presented in Figs 12–14.

Figure 11.

The membership functions of the first state.

Figure 12.

The membership functions of the second state.

Figure 13.

The membership functions of the pseudo control.

Figure 14.

The membership functions of the output.

Figure 15.

The schematic of flexible-joint robot.

Case 2: Now we’re trying to apply the proposed method on outputs of the multi-agent flexible joint robots.

It is assumed the multi-agent system with 3 agents where $x_{i1}=q_{i1}$ , $x_{i3}=q_{i2}$ , $x_{i2}=\dot{q}_{i1}$ and $x_{i4}=\dot{q}_{i2}$ . Consider the following dynamics of the flexible-joint robot for each agent as follow:

$\displaystyle\left\{{\begin{array}[]{l}\dot{x}_{i1}=x_{i2}\\ \dot{x}_{i2}=-\frac{9.81}{I}\sin x_{i1}-\frac{4}{I}x_{i2}-\frac{100}{I}\\ \ \ \ (x_{i1}-x_{i3})+\frac{\beta}{I}\\ \dot{x}_{i3}=x_{i4}\\ \dot{x}_{i4}=-86.2(x_{i1}-x_{i3})-3.44x_{i4}\\ \ \ \ -0.86\tau_{si}-0.86\beta\\ y=x_{i4}\\ \end{array}}\right.$ (41)

The fourth states of the all agents are shown as Fig. 16.

Figure 16.

Fourth variable state of three agents.

Based on Fig. 16, it is clear that the consensus is achieved in the fourth states of the MAS. Figure 17 presents the input of the agents.

Figure 17.

The overall control input applied to the agent.

Based on the proposed applicable example, it is clear that the consensus of the forth states is reached and boundedness of the signals involved in the closed loop is guaranteed.

Based on the simulation results, boundedness of the all signals involved in the closed loop system, convergence of the states of the agents to the states of the leader and robustness of the system against uncertainties and external disturbances are investigated in this section.

5. Conclusion

In this paper, a modified adaptive distributed protocol approach has been proposed for pure-feedback nonlinear multi-agent systems where each follower is with inherent disturbances, unmodeled dynamics and non-affine dynamical system under a directed network topology. Fuzzy systems are utilized to approximate the control input derived from the distributed tracking controller design procedure. The controller structure and the derived adaptation rules guaranty the convergence of the tracking error to zero. Robustness against external disturbances and approximation errors and using knowledge of experts are the merits of the proposed method. Compared with the related results in the literature, the uncertain nonlinear non-affine systems are investigated in this paper. Finally, simulation results have been demonstrated the effectiveness of the proposed approach.

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