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Citation: J. X. Chen, J. M. Li. Global FLS-based consensus of stochastic uncertain nonlinear multi-agent systems. International Journal of Automation and Computing. http://doi.org/10.1007/s11633-021-1279-y doi:  10.1007/s11633-021-1279-y
Cite as: Citation: J. X. Chen, J. M. Li. Global FLS-based consensus of stochastic uncertain nonlinear multi-agent systems. International Journal of Automation and Computing . http://doi.org/10.1007/s11633-021-1279-y doi:  10.1007/s11633-021-1279-y

Global FLS-based Consensus of Stochastic Uncertain Nonlinear Multi-agent Systems

Author Biography:
  • Jia-Xi Chen received the M. Sc. and Ph. D. degrees in applied mathematics from Xidian University, China in 2018 and 2020, respectively. He is currently a lecturer at Department of Applied Mathematics, Xidian University, China. His research interests include adaptive control, multi-agent systems and Takagi-Sugeno (T-S) fuzzy systems.E-mail: jxchen208@163.comORCID iD: 0000-0002-9667-811X

    Jun-Min Li received the M. Sc. degree in applied mathematics from Xidian University, China in 1990, and the Ph. D. degree in systems engineering from Xi′an Jiao Tong University, China in 1997. He is currently a professor at Department of Applied Mathematics, Xidian University, China. His research interests include adaptive control, learning control of MAS, hybrid system control theory and networked control system. E-mail: jmli@mail.xidian.edu.cn (Corresponding author)ORCID iD: 0000-0001-8409-6465

  • Received: 2020-10-25
  • Accepted: 2021-01-22
  • Published Online: 2021-03-20
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

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Global FLS-based Consensus of Stochastic Uncertain Nonlinear Multi-agent Systems

Abstract: Using graph theory, matrix theory, adaptive control, fuzzy logic systems and other tools, this paper studies the leader-follower global consensus of two kinds of stochastic uncertain nonlinear multi-agent systems (MAS). Firstly, the fuzzy logic systems replaces the feedback compensator as the feedforward compensator to describe the uncertain nonlinear dynamics. Secondly, based on the network topology, all followers are divided into two categories: One is the followers who can obtain the leader signal, and the other is the follower who cannot obtain the leader signal. Thirdly, based on the adaptive control method, distributed control protocols are designed for the two types of followers. Fourthly, based on matrix theory and stochastic Lyapunov stability theory, the stability of the closed-loop systems is analyzed. Finally, three simulation examples are given to verify the effectiveness of the proposed control algorithms.

Citation: J. X. Chen, J. M. Li. Global FLS-based consensus of stochastic uncertain nonlinear multi-agent systems. International Journal of Automation and Computing. http://doi.org/10.1007/s11633-021-1279-y doi:  10.1007/s11633-021-1279-y
Citation: Citation: J. X. Chen, J. M. Li. Global FLS-based consensus of stochastic uncertain nonlinear multi-agent systems. International Journal of Automation and Computing . http://doi.org/10.1007/s11633-021-1279-y doi:  10.1007/s11633-021-1279-y
    • The research of multi-agent systems (MAS) originated from nature. There is much consensus on biological behavior in nature, such as bird migration, fish migration, firefly flicker, etc. According to the cooperation characteristics of biological groups, scholars have proposed cooperative control algorithms to study the consensus of MAS. After decades of development, a large number of results have been published on MAS research, including papers[1-10] and books[11-13]. According to different modeling subsystems, MAS are divided into two categories: deterministic MAS and stochastic MAS.

      Compared with the research results on the consensus of deterministic MAS, the consensus theoretical research results of stochastic MAS are relatively scarce, which is also the main motivation of this paper for studying the stochastic MAS. Although the theoretical research on the consensus of stochastic modeling MAS is not sufficient, the MAS modeled by stochastic models is indeed a key point in the research on the consensus of MAS, and it also has important applications in practice.

      In recent years, stochastic MAS has aroused a research boom among scholars. Researchers and engineers began to use the idea of stochastic analysis and control theory to study the consensus behavior of stochastic MAS in different probabilistic senses[14]. The content of the research is also different, mainly from four aspects, namely the consensus of linear or nonlinear stochastic MAS[15-18], the cooperative optimal consensus of stochastic MAS[19, 20], the consensus of stochastic MAS with communication white noise[21, 22], and consensus of MAS with stochastic switching topology[23, 24]. Although there are many achievements in the research on the consensus of MAS with stochastic models, there are still many problems that have not been well studied, especially the consensus problems of stochastic uncertain nonlinear MAS based on fuzzy logic system (FLS) or neural network (NN).

      For the stochastic uncertain nonlinear MAS, scholars have also done some theoretical research[25-33] based on NN/FLS. Based on the linear state observers and NN technology, Shahvali and Askari[25] studied the output tracking control problem of a multi-leader of a class of stochastic nonlinear MAS. Shahvali et al.[25-29] studied output consensus for a class of lower triangular uncertain nonlinear MAS by using NN/FLS and adaptive methods. Based on FLS and adaptive technology, Li et al.[30] investigated the output tracking control of heterogeneous stochastic nonlinear MAS in which the dynamics of leader and followers were different: followers were modeled by stochastic dynamic systems and leader was modeled as autonomous dynamic systems of ordinary differential equations. Ren et al.[31] solved the state tracking control problem of first-order stochastic nonlinear MAS based on FLS and adaptive techniques. The results of [26-29] were generalized in [32], and Wang et al.[33] considered the output tracking consensus of a class of stochastic uncertain nonlinear MAS in which each subsystem was modeled by non-strict feedback form based on FLS approximation. Based on NN/FLS and adaptive technology, Shahvali et al.[25-33] obtained semi-global uniformly ultimately bounded results in the sense of mean square or fourth moment. Whether from the aspect of theoretical research or the needs of practical applications, the results of semi-global consensus can not meet our theoretical pursuit and practical needs of the research on the consensus of stochastic uncertain nonlinear MAS. Global consensus is the final research result we need. As far as the authors know, there is no theoretical result to show the global consensus of stochastic uncertain MAS based on NN/FLS. It is still an open question, which is also the main motivation and research content of this paper.

      This paper considers the global state consensus control of first-order and second-order stochastic uncertain nonlinear MAS based on FLS, in which each subsystem is modeled by a general nonlinear system with uncertain nonlinearities and bounded input disturbances. Based on the characteristics of communication among agents, followers are divided into two categories: One group can get the information of the leader directly, the other group can not get the information of the leader directly. We design different controllers for these two kinds of followers based on the adaptive control method, respectively. Technically, the FLS is used as a feed-forward compensation term rather than a feedback function to represent some uncertain nonlinear dynamics, which overcomes the limitation of semi-global consensus and solves the global consensus problems of stochastic uncertain nonlinear MAS. Compared with the above articles, the main contributions of this paper are as follows:

      1) Compared with the consensus of deterministic MAS[1-10], the global consensus of stochastic MAS is considered in this paper.

      2) Unlike the semi-global consensus results in [25-33], the universal approximator (FLS/NN) is used as a feedback compensator to describe uncertain nonlinear dynamics. In this paper, the FLS is used as the feedforward compensator to describe the uncertain nonlinear dynamics, which avoids the restriction of the follower states in a compact set, so that the global consensus results are obtained.

      3) A time-varying gain parameter $ c_i(t) $ ($i \in \left\{1,2,\cdots, $$ n\right\}$) is introduced to replace the constant parameter $ c_0 $ in the design of a distributed controller, which avoids the limitation of knowing global information in the design of the controller′s constant gain parameter $ c_0 $, so that the fully distributed results are obtained.

    • The basic theory of graph $ \bar{G} $ is found in [34], which is omitted for simplicity. Here, we give an example of a network topology graph $ \bar{G} $ as shown in Fig. 1, where label 0 denotes the leader and labels 1−4 denote the followers. The network topology graph of followers 1−4 is undirected, and the network topology graph between leader and follower is directed. The graph $ \bar{G} $ is called a connected graph if there is a connected path from leader to any follower.

      Figure 1.  Network topology $ \bar G $

      Lemma 1.[34] If graph $ \bar{G} $ is connected, then the symmetric matrix $ H $ associated with $ \bar{G} $ is positive definite, where $H={{\left[ {{h}_{ij}} \right]}_{n\times n}}$, ${{h}_{ii}}={{b}_{i}}+\displaystyle\sum\nolimits_{j = 1}^{n}{{{a}_{ij}}}$, ${{h}_{ij}}=-{{a}_{ij}}\left(i,j\in $$ \left\{ 1,\cdots,n \right\}, i\ne j \right)$, $ a_{ij} $ denotes the amount of information transmitted between followers $ i $ and $ j $, $ b_i $ denotes the amount of information transmitted between leader and follower $ i $.

    • In this section, we consider a class of the first-order uncertain stochastic MAS that composed of one leader and $ n $ followers. The dynamics of the $ i $-th ($i \in I = $$ \left\{1,2,\cdots, n \right\}$) follower is given as

      $ {\rm{d}}{x_i} = \left( {{f_i}\left( {{x_i}} \right) + {u_i} + {d_i}\left( t \right)} \right){\rm{d}}t + g\left( {{x_i}} \right){\rm{d}}B $

      (1)

      where $x_i \in {\bf{R}}$ and $u_i \in {\bf{R}}$ represent the position and controller of the $ i $-th follower, respectively; $ B $ is a 1-D standard Brownian motion defined on a complete probability space $ (\Omega, F, P) $; $f_i(x_i): {\bf{R}} \to {\bf{R}}$ is an uncertain smooth function and $g(x_i): {\bf{R}} \to {\bf{R}}$ is uncertain smooth function; $ d_i\left( {{t}} \right) $ is an unknown but bounded input disturbance, i.e., $ \left| {{d_{i}}\left( t \right)} \right| \le a_{i} $ with $ a_{i} $ an unknown positive constant. The dynamics of the leader is presented as

      $ {\rm{d}}{x_0} = {f_0}\left( {{x_0}, t} \right){\rm{d}}t + g\left( {{x_0}} \right){\rm{d}}B $

      (2)

      where ${f_0}\left( {{x_0}, t} \right): {\bf{R}} \times \left[ 0, \infty\right) \to {\bf{R}}$ is an unknown nonlinear function to any follower $ i $ in $ G $.

      Assumption 1. 1) There exists a positive $ \bar \Omega $ such that $ \left\| x_0 \right\| \le \bar \Omega $; 2) There exists an unknown real constant $ {M_{f_0}} (>0) $ such that unknown dynamics $ \left| {{f_0}\left( {{x_0,t}} \right)} \right| \le M_{f_{0}} $.

      Assumption 2. For uncertain nonlinear functions $ {f_i}\left( \cdot \right) \in {\bf R} $ ($ i \in I $), there exists a positive constant $ {L_i} $ such that $ \left|f_i(x_i)-f_i(x_0)\right| \le {L_i} \left\|x_i- x_0 \right\| $.

      Assumption 3. There exists a positive constant $ {L_{g}} $ such that $ \left|g(x_i)-g(x_0)\right| \le {L_g} \left\|x_i- x_0 \right\|, i \in I $.

    • The consensus error for MAS (1)−(2) is designed as

      $ {\delta_i} = {x_{i}} - {x_{0}}, \;\;i \in I. $

      (3)

      The consensus objective of this section is to design controller $ u_i $ ($ i \in I $) such that consensus error $ {\delta_i} \to 0 $ as the time $ t \to \infty $.

      The local neighborhood consensus error for follower $ i $ ($ i \in I $) is defined as

      $ {e_{i}} = \sum\limits_{j \in {I}} {{a_{ij}}\left( {{x_{j}} - {x_{i}}} \right) + {b_i}\left( {{x_{0}} - {x_{i}}} \right)}. $

      (4)

      According to (4), the error vector for $ \bar G $ is given by

      $ e = -H \delta $

      (5)

      where $e = {\left[ {{e_{1}}, \cdots,{e_{n}}} \right]^{\rm{T}}}$, $\delta = {\left[ {{\delta_{1}}, \cdots ,{\delta_{n}}} \right]^{\rm{T}}} = x - {1_n}{x_{0}}$, $x = $$ {\left[ {{x_{1}}, \cdots, {x_{n}}} \right]^{\rm{T}}}$, $1_n = [1,\cdots,1]^{\rm{T}} \in {\bf{R}}^n$.

      Lemma 2.[34] If graph $ \bar G $ is connected, then it follows from (5) that

      $ \left\| {\delta} \right\| \le \frac{\left\| {e} \right\|}{\underline{\sigma }\left( H\right)} $

      (6)

      where $ \underline{\sigma }\left( H \right) (>0) $ is the minimum singular value of $ H $.

    • The It$ \hat o $ differential of $ e_i $ is

      $\begin{split} {\rm{d}}{e_i} =& \Big[ {\sum\limits_{j \in I} {{a_{ij}}\left( {{f_j}\left( {{x_j}} \right) + {u_j} + {d_j}\left( t \right)} \right)} + {b_i}\left( {{f_0}\left( {{x_0},t} \right)} \right)} -\\ & { \left( {{d_i} + {b_i}} \right)\left( {{f_i}\left( {{x_i},{v_i}} \right) + {u_i} + {d_i}\left( t \right)} \right)} \Big] {\rm{d}}t+\\ & \Big[ {\sum\limits_{j \in I} {{a_{ij}}g\left( {{x_j}} \right)} + {b_i}g\left( {{x_0}} \right) - \left( {{d_i} + {b_i}} \right)g\left( {{x_i}} \right)} \Big]{\rm{d}}B. \end{split}$

      (7)

      Since each follower $ i $ ($ i \in I $) can only communicate with its neighbor agent, and only a small part of follower agents can obtain the leader′s information in $ \bar G $, all followers are divided into two sets $ \bar I_0 $ and $ I{\backslash}{\bar I_0} $, where $ \bar I_0 $ denotes a set of followers that can obtain the leader′s signal directly and $ I{\backslash}{\bar I_0} $ represents a set of followers that cannot directly obtain the leader signal. Then, each follower $ i $ can directly use the state $ x_0 $ of the leader to design controller $ u_i $ in the set $ \bar I_0 $ and each follower $ i $ needs to estimate the leader information to design the controller $ u_i $ in the set $ I{\backslash}{\bar I_0} $. For example, in Fig. 1, the followers $ 1 $, $ 3 $ and $ 4 $ can obtain the signal of leader $ 0 $ and the follower $ 2 $ cannot directly obtain the signal of leader $ 0 $, then the sets $ \bar I_0 = \left\{1, 3, 4 \right\} $ and $ I{\backslash}{\bar I_0} = \left\{2\right\} $.

      Design the consensus control as

      $ u_i=\left\{ \begin{aligned} &c_i(t)e_i + {{\hat M}_i}\tanh \left( {\frac{{{e_i}{{\hat M}_i}}}{{\varpi_i}}} \right)- \hat W_i^{\rm{T}}\varphi_i(x_0), \; i \in {\bar I_0}\\ &c_i(t)e_i+{{\hat M}_i}\tanh \left( {\frac{{{e_i}{{\hat M}_i}}}{{\varpi_i}}} \right)+{{\hat \pi }_i}\tanh \left({\frac{{{e_i}{{\hat \pi }_i}}}{{\varpi_i}}} \right),\; i\inI\setminus{\bar I_0} \end{aligned} \right. $

      (8)

      where $ \varpi_i>0 $ is a constant, $ {c_i}\left( t \right) $ is the time-varying distributed control gain, $ \varphi_i\left(x_0\right) $ denotes a fuzzy basic function; $ \hat W_i $, $ {\hat M_i} $ and $ \hat \pi_i $ are the estimate of the unknown constants $ {W_i^*} $ (fuzzy optimal weight coefficient), $ {M_i} $ and $ \pi_i $, respectively; $\pi_i = \dfrac{1}{2}{\left\| {{W_i^*}} \right\|^2}$, $ M_i = {{M_{{{f_{0}}}}}+a_{i}}+ {{\varepsilon^*_i}} $ if $ i \in \bar I_0 $, otherwise $M_i = {{M_{{{f_{0}}}}}+a_{i}}+ {{\varepsilon^*_i}}+ \dfrac{1}{2}\phi_i^2$; $ {{\varepsilon^*_i}} $ and $ \phi_i $ denote upper bounds of $ {{\varepsilon_i}}\left(x_0\right) $ (approximation error of FLS) and $ \left\|\varphi_i\left(x_0\right)\right\| $ (fuzzy basis function vector), respectively.

      The adaptive parameters $ c_i $, $ \hat \pi_i $, $ \hat M_i $ and $ \hat W_i $ are designed as

      $ \left\{ \begin{aligned}&\dot {\hat M}_i = \zeta_i\left|e_i\right|-\sigma_{{\hat M}_i}{\hat M}_i,\; {\hat M}_i(0)>0,\; {\zeta _i}>0,\; \sigma_{{\hat M}_i}>0,\; i \in { I}\\ &\dot c_i(t) = \gamma_i e_i^2-\sigma_{c_i}c_i(t),\; \sigma_{c_i}>0,\; {\gamma _i}>0,\; i \in { I}\\ &\dot {\hat W}_i = -\eta_i\varphi_i(x_0)e_i-\sigma_{{\hat W}_i}{\hat W}_i,\; \sigma_{{\hat W}_i}>0,\; {\eta _i}>0,\; i \in {\bar I_0}\\ &\dot {\hat \pi}_i = \beta_i\left|e_i\right|-\sigma_{{\hat \pi}_i}{\hat \pi}_i,\; {\hat \pi}_i(0)>0,\; {\beta _i}>0,\; \sigma_{{\hat \pi}_i}>0,\; i \in I \setminus {\bar I_0}. \end{aligned} \right.$

      (9)

      Remark 1. It should be noted that the design of controller (8) is a typical design method of FLS as feedforward compensator. Because the communication between networks is local and not all the followers can get the leader′s signal, the follower should be divided into two categories ($ \bar I_0 $ and $ I{\backslash}{\bar I_0} $) when designing a distributed controller based on leader′s signal. In $ \bar I_0 $, the controller of follower $ i $ is designed directly by the leader′s signal. In $ I{\backslash}{\bar I_0} $, the follower′s controller is designed indirectly by the leader′s signal.

      Remark 2. Although this paper estimates the vector $ W_i^* $ in $ i \in \bar I_0 $, compared with the estimation of the vector $ W_i^* $ in $ i \in I $ in [31, 32], the designed consensus algorithms (8) and (9) also greatly reduces the computational complexity and the computational cost because this paper estimates the two norm of the vector $\dfrac{1}{2}W_i^*$ in $ i \in I{\backslash}{\bar I_0} $.

      Remark 3. Compared with the results of traditional FLS as feedback compensator to describe uncertain nonlinear dynamics[25-33], in this paper, FLS is used as a feed-forward compensator to describe uncertain nonlinear dynamics, which overcomes the limitation that the state of follower $ i $ ($ i \in I $) is in a compact set, and designs a class of global consensus controllers (8) by using an adaptive control method.

      Theorem 1. Consider the stochastic uncertain nonlinear MAS (1) and (2) under Assumptions 1−3, the distributed consensus algorithms (8) and (9), if the graph $ \bar G $ is connected, the following properties hold:

      1) The states of all followers tend to be in the neighborhood of the leader in the mean-square sense as the time $ t \to \infty $.

      2) The adaptive parameters $ c_i(t) $ ($ i \in I $), $ \hat M_i $ ($ i \in I $), $ \hat W_i $ ($ i \in \bar I_0 $) and $ \hat {\pi}_i $ ($ i \in {I}\backslash {{\bar I}_0} $) are bounded in probability.

      Proof. See Appendix. □

      Remark 4. From the proof of Theorem 1, we may see that the convergence compact set of consensus errors $ \delta $ are related to the parameters $ k $, $ {\sigma _{{{\hat M}_i}}} $ ($ i \in I $), $ {{\sigma _{{{\hat W}_i}}}} $ ($ i \in \bar I_0 $), $ {{\sigma _{{{\hat \pi }_i}}}} $ ($ i \in I{\backslash}{\bar I_0} $) and $ {{\sigma _{{c_i}}}} $ ($ i \in I $). Therefore, we can adjust these parameters to make the convergence region of consensus error $ \delta $ sufficiently small.

    • In this section, we consider a class of the second-order uncertain stochastic MAS that composed of one leader and $ n $ followers. The dynamics of the $ i $-th ($i \in I = $$ \left\{1,2,\cdots, n \right\}$) follower is given as

      $ \begin{split} &{\rm{d}}{x_i} = {v_i}{\rm{d}}t\\ &{\rm{d}}{v_i} = \left( {{f_i}\left( {{x_i},{v_i}} \right) + {u_i} + {d_i}\left( t \right)} \right){\rm{d}}t + g\left( {{x_i},{v_i}} \right){\rm{d}}B \end{split} $

      (10)

      where $x_i \in {\bf{R}}$, $v_i\in {\bf{R}}$ and $u_i \in {\bf{R}}$ represent the position, velocity, controller of the $ i $-th follower, respectively; $ B $ is a 1-D standard Brownian motion defined on a complete probability space $ (\Omega, F, P) $; $f_i(x_i, v_i): R^2 \in {\bf{R}}$ is an uncertain smooth function and $g(x_i, v_i): R^2 \in {\bf{R}}$ is an uncertain smooth function; $ d_i\left( {{t}} \right) $ is an unknown but bounded input disturbance, i.e., $ \left| {{d_{i}}\left( t \right)} \right| \le a_{i} $ with $ a_{i} $ an unknown positive constant. The dynamics of the leader is presented as

      $ \begin{split} &{\rm{d}}{x_0} = {v_0}{\rm{d}}t\\ &{\rm{d}}{v_0} = {f_0}\left( {{x_0},{v_0}, t} \right){\rm{d}}t + g\left( {{x_0},{v_0}} \right){\rm{d}}B \end{split} $

      (11)

      where ${f_0}\left( {{x_0},{v_0}, t} \right): R^2 \times \left[0, \infty\right) \to {\bf{R}}$ is unknown nonlinear function to any follower $ i $ in $ G $.

      Assumption 4. 1) There is a positive $ \bar \Omega $ such that $\left\|\left( x_0, v_0 \right)^{\rm{T}}\right\| \le \bar \Omega$; 2) There exists an unknown real constant $ {M_{f_0}} (>0) $ such that unknown dynamics $\left| {f_0}\left( x_0, $$ v_0, t \right) \right| \le M_{f_{0}}$.

      Assumption 5. For unknown nonlinear functions ${f_i}\left( \cdot \right) \in {\bf{R}}$ ($ i \in I $), there exists a positive constant $ {L_i} $ such that $\left|f_i(x_i, v_i)-f_i(x_0, v_0)\right| \le {L_i} \left\|\left(x_i, v_i\right)^{\rm{T}}- \left(x_0, v_0\right)^{\rm{T}} \right\|$.

      Assumption 6. There exists a positive constant $ {L_{g}} $ such that $\left|g(x_i, v_i)-g(x_0, v_0)\right|\le{L_g} \left\|\left(x_i, v_i\right)^{\rm{T}}-\left(x_0, v_0\right)^{\rm{T}} \right\|, $$ i \in I$.

    • The consensus error for MAS (10) and (11) is designed as

      $ {\delta_{x_i}} = {x_{i}} - {x_{0}},\; {\delta_{v_i}} = {v_{i}} - {v_{0}}, \;\;i \in I. $

      (12)

      The consensus objective of this section is to design a controller $ u_i $ ($ i \in I $) such that the consensus error between the $ i $-th follower and leader tends to zero as the time $ t \to \infty $.

      The local neighborhood consensus error for node $ i $ ($ i \in I $) is defined as

      $ \begin{split} &{e_{x_i}} = \sum\limits_{j \in {I}} {{a_{ij}}\left( {{x_{j}} - {x_{i}}} \right) + {b_i}\left( {{x_{0}} - {x_{i}}} \right)}\\ &{e_{v_i}} = \sum\limits_{j \in {I}} {{a_{ij}}\left( {{v_{j}} - {v_{i}}} \right) + {b_i}\left( {{v_{0}} - {v_{i}}} \right)}\end{split} $

      (13)

      where $ a_{ij} $ denotes the amount of information transmitted between followers $ i $ and $ j $, $ b_i $ denotes the amount of information transmitted between leader and follower $ i $.

      According to (13), the error vector for $ \bar G $ is given by

      $ e_x = -H \delta_x,\; e_v = -H \delta_v $

      (14)

      where $e_x = {\left[ {{e_{x_1}}, \cdots,{e_{x_n}}} \right]^{\rm{T}}}$, $e_v = {\left[ {{e_{v_1}}, \cdots,{e_{v_n}}} \right]^{\rm{T}}}$, $\delta_x = \left[ {\delta_{x_1}}, $$ \cdots,{\delta_{x_n}} \right]^{\rm{T}} = x - {1_n}{x_{0}}$, $x = {\left[ {{x_{1}}, \cdots, {x_{n}}} \right]^{\rm{T}}}$, $\delta_v = \left[ {\delta_{v_1}}, \cdots, $$ {\delta_{v_n}} \right]^{\rm{T}} = v - {1_n}{v_{0}}$, $v = {\left[ {{v_{1}}, \cdots, {v_{n}}} \right]^{\rm{T}}}$.

      Lemma 3.[34] If graph $ \bar G $ is connected, then it follows from (14) that

      $ \left\| {\delta_x} \right\| \le \frac{\left\| {e_x} \right\|}{\underline{\sigma }\left( H \right)},\; \frac{\left\| {\delta_v} \right\| \le \left\| {e_v} \right\|}{\underline{\sigma }\left(H \right)} $

      (15)

      where $ \underline{\sigma }\left( H \right) (>0) $ is the minimum singular value of $ H $.

    • In order to study the consensus problem of the second-order stochastic uncertain nonlinear MAS, the sliding mode error of the agent $ i $ ($ i \in I $) is defined as

      $ s_i = e_{v_i} + \sigma e_{x_i},\; \sigma>0. $

      (16)

      The It$ \hat o $ differential of $ s_i $ is

      $ \begin{split} {\rm{d}}{{{s_i}}} = &\Big[ \sum\limits_{j \in I} {{a_{ij}}\left( {{f_j}\left( {{x_j},{v_j}} \right) + {u_j} + {d_j}\left( t \right) + \sigma {v_j}} \right)} -\\ & \left( {{d_i} + {b_i}} \right)\left( {{f_i}\left( {{x_i},{v_i}} \right) + {u_i} + {d_i}\left( t \right) + \sigma {v_i}} \right) + \\ & {b_i}\left( {{f_0}\left( {{x_0},{v_0},t} \right) + \sigma {v_0}} \right) \Big] {\rm{d}}t + \Big[ \sum\limits_{j \in I} {{a_{ij}}g\left( {{x_j},{v_j}} \right)} +\\ & {b_i}g\left( {{x_0},{v_0}} \right) - \left( {{d_i} + {b_i}} \right)g\left( {{x_i},{v_i}} \right) \Big] {\rm{d}}B. \end{split}$

      (17)

      Design the consensus control as

      $ u_i = \left\{ \begin{aligned} &c_i(t)s_i + {{\hat M}_i}\tanh \left( {\frac{{{s_i}{{\hat M}_i}}}{{\varpi_i}}} \right)- \hat W_i^{\rm T}\varphi_i(x_0, v_0), \; i \in {\bar I_0}\\ &c_i(t)s_i+{{\hat M}_i}\tanh \left({\frac{{{s_i}{{\hat M}_i}}}{{\varpi_i}}}\right)+{{\hat \pi }_i}\tanh \left({\frac{{{s_i}{{\hat \pi }_i}}}{{\varpi_i}}}\right),\;i \in I\setminus{\bar I_0} \end{aligned} \right. $

      (18)

      where $ \varphi_i\left(x_0, v_0\right) $ denotes a fuzzy basic function; $ \hat W_i $, $ {\hat M_i} $ and $ \hat \pi_i $ are the estimate of the unknown constants $ {W_i^*} $ (fuzzy weight coefficient), $ {M_i} $ and $ \pi_i $, respectively; $\pi_i = \dfrac{1}{2}{\left\| {{W_i^*}} \right\|^2}$, $ M_i = {{M_{{{f_{0}}}}}+a_{i}}+ {{\varepsilon^*_i}} $ if $ i \in \bar I_0 $ and otherwise $M_i = {{M_{{{f_{0}}}}}+a_{i}}+ {{\varepsilon^*_i}}+ \dfrac{1}{2}\phi_i^2$; $ {{\varepsilon^*_i}} $ and $ \phi_i $ denote upper bounds of $ {{\varepsilon_i}}\left(x_0, v_0\right) $ (approximation error of FLS) and $ \left\|\varphi_i\left(x_0, v_0\right)\right\| $ (fuzzy basis function), respectively.

      The adaptive parameters $ c_i $, $ \hat \pi_i $, $ \hat M_i $ and $ \hat W_i $ are designed as

      $ \left\{ \begin{aligned}&\dot {\hat M}_i = \zeta_i\left|s_i\right|-\sigma_{{\hat M}_i}{\hat M}_i,\; {\hat M}_i(0)>0,\; \sigma_{{\hat M}_i}>0,\; i \in { I}\\ &\dot c_i(t) = \gamma_i s_i^2-\sigma_{c_i}c_i(t),\; \sigma_{c_i}>0, \; i \in { I}\\ &\dot {\hat W}_i = -\eta_i\varphi_i(x_0,v_0)s_i-\sigma_{{\hat W}_i}{\hat W}_i,\; \sigma_{{\hat W}_i}>0,\; i \in {\bar I_0}\\ &\dot {\hat \pi}_i = \beta_i\left|s_i\right|-\sigma_{{\hat \pi}_i}{\hat \pi}_i,\; {\hat \pi}_i(0)>0,\; \sigma_{{\hat \pi}_i}>0,\; i \in I \setminus {\bar I_0} \end{aligned} \right.$

      (19)

      where $ {\zeta _i} $, $ {\gamma _i} $, $ {\eta _i} $ and $ {\beta _i} $ are positive constants.

      Theorem 2. Consider the stochastic uncertain nonlinear MAS (10) and (11) under Assumptions 4−6, the distributed consensus algorithms (18) and (19), if the graph $ \bar G $ is connected, the following properties hold:

      1) The states of all followers tend to be in the neighborhood of the leader in the mean-square sense as the time $ t \to \infty $.

      2) The adaptive parameters $ c_i(t) $ ($ i \in I $), $ \hat M_i $ ($ i \in I $), $ \hat W_i $ ($ i \in \bar I_0 $) and $ \hat {\pi}_i $ ($ i \in {I}\backslash {{\bar I}_0} $) are bounded in probability.

      Proof. See Appendix.               □

      Remark 4. From the proof of Theorem 2, we may see that the convergence compact set of consensus errors $ \delta_x $ and $ \delta_v $ are related to the parameters $ \Delta_1 $, $ \Delta_2 $, $ {\sigma _{{{\hat M}_i}}} $ ($ i \in I $), $ {{\sigma _{{{\hat W}_i}}}} $ ($ i \in \bar I_0 $), $ {{\sigma _{{{\hat \pi }_i}}}} $ ($ i \in I{\backslash}{\bar I_0} $) and $ {{\sigma _{{c_i}}}} $ ($ i \in I $). Therefore, we can adjust these parameters to make the convergence region of consensus errors $ \delta_x $ and $ \delta_v $ sufficiently small.

    • This section gives three simulation examples to verify the above two algorithms. The communication topology between the network nodes in Examples 1−3 is shown in Fig. 1.

      Example 1. First-order MAS

      The dynamics of the follower $ i $ ($ i \in I $) are given as

      $ \quad{\rm{d}}{x_i} = (u_i + \sin{x_i} + \cos{x_i} + {d _i}(t)){{\rm{d}}t} + 0.01\cos(\pi x_i){{\rm{d}}B} . $

      (20)

      The dynamics of the leader are presented as

      $ {\rm{d}}{x_0} = (\cos{x_0} + \sin{x_0} + 0.07){\rm{d}}t + 0.01\cos(\pi x_0){\rm{d}}B .$

      (21)

      We can easily verify the system equations (20) and (21) that satisfy Assumption $ 1 $ in this paper. Based on the control protocol (8) and the parameter adaptive laws (9), we give the following conditional parameters. $ \sigma_{c_i} = 5 $ ($ i \in I $), $ \sigma_{\hat M_i} = 2 $, $ \sigma_{\pi_2} = 5 $, $ \sigma_{\hat W_i} = 2 $ ($ i \in I {\backslash}{\bar I_0} $), $ \varpi_i = 10 $ ($ i \in I $), $ {d_1}(t) = d_2(t) = d_3(t) = d_4(t) = 0.065 $, $ \zeta_1 = 0.9 $, $\eta_1 = 1$, $ \gamma_1 = 0.5 $, $ x_1(0) = -0.15 $, $ \zeta_2 = 0.2 $, $ \beta_2 = 0.2 $, $ \gamma_2 = 0.8 $, $ x_2(0) = -0.1 $, $ \zeta_3 = 0.3 $, $ \eta_3 = 0.3 $, $ \gamma_3 = 0.9 $, $ x_3(0) = 1.5 $, $ \zeta_4 = 0.4 $, $ \eta_4 = 0.4 $, $ \gamma_4 = 0.5 $, $ x_4(0) = 1.9 $, $ x_0(0) = 0.4 $. The fuzzy membership functions are given as follows: ${\mu _{0j}}\left( {{x_0}} \right) = \exp \left[ { - 10{{\left( {{x_0}-{\bar \upsilon_j}} \right)}^2}} \right]$ with $j=1, \cdots, 6$, $\bar \upsilon_1=-0.3$, $ \bar \upsilon_2 = -0.2 $, $ \bar \upsilon_3 = -0.1 $, $ \bar \upsilon_4 = 0.1 $, $ \bar \upsilon_5 = 0.2 $, $ \bar \upsilon_6 = 0.3 $. Let ${\varphi _{0j}}\left( {{x_0}} \right) = {{{\mu _{0j}}\left( {{x_0}} \right)}}/ $$ {{\displaystyle\sum\nolimits_{j = 1}^6 {{\mu _{0j}}\left( {{x_0}} \right)} }}$, $\varphi_i = [\varphi_{01}, \varphi_{02},\varphi_{03}, \varphi_{04}, $$ \varphi_{05}, \varphi_{06}]$, ($ i \in \bar I_0 $). By substituting the consensus algorithms (8) and (9) into (20), we have the simulation results shown in Figs. 23. Fig. 2 shows that tracking error $ \delta_i \to 0 $ as $ t \to \infty $. Fig. 3 shows that the control protocol $ u_i $ and the parameter adaptive laws $ c_i $, $ \hat M_i $, $ \hat W_1 $, $ \hat \pi_2 $, $ \hat W_3 $ and $ \hat W_4 $ are bounded.

      Figure 2.  Consensus errors of MAS, $ i \in I $

      Figure 3.  Time evolution of controller and adaptive parameters, $ i \in I $

      Example 2. Second-order MAS

      The network single-link robot manipulators[31] are presented to verify the effectiveness of the proposed control algorithms (18) and (19). The schematic diagram of the control single link robot manipulator is shown in Fig. 4. And the single-link robot manipulator is modeled as

      Figure 4.  Single-link robot

      $ {J_i}{\ddot \theta _i} + {m_i}g\sin \left( {{\theta _i}} \right) = {\tau _i}, \;i \in I = \left\{1,\cdots,4\right\} $

      (22)

      where $g = 10\;{\rm{m}}/{{\rm{s}}^2}$, $ {J_i} = r_i^2{J_{mi}} + {J_{li}} $, $ {J_{mi}} $ and $ {J_{li}} $ are rotational inertias of motor and link, respectively, $ r_i $ is the gear ratio, $ l_i $ is the distance from the joint axis to this link center of mass, $ m_i $ is the total mass of the link, $ {\tau _i} $ denotes external forces that are not derivable from the potential function. Here, suppose that the torque $ \tau_i $ is subject to the environmental noise and input disturbance, and is described by $ \tau_i = -y{\dot \theta_i} + u_i + d_i\left(t\right) + g\left(\theta_i, \dot \theta_i \right) \dot B $, where $ y $ is a constant, $ u_i $ denotes control input, $ d_i\left(t\right) $ denotes the input disturbance, $ \dot B $ is a 1-dimensional white noise (i.e., $ B $ is Brownian motion) and $ g\left(\theta_i, \dot \theta_i \right) $ is the intensity of the noise. Then, (22) becomes

      $ {J_i}{\ddot \theta _i} + {m_i}g\sin \left( {{\theta _i}} \right) = -y{\dot \theta_i} + u_i + d_i\left(t\right) + g\left(\theta_i, \dot \theta_i \right) \dot B . $

      (23)

      To obtain a state model for the pendulum, let us take the state variables as $ x_i = \theta_i $ and $ v_i = \dot \theta_i $. Then, (23) can be expressed as an It$ \hat o $ equation

      $ \begin{split} &{\rm{d}}{x_i} = {v_i}{\rm{d}}t\\ & {\rm{d}}{v_i} = \left( { - \frac{{{m_i}g}}{{{J_i}}}\sin {x_i} - \frac{y}{{{J_i}}}{v_i} + \frac{1}{{{J_i}}}\left( {{u_i} + {d_i}\left( t \right)} \right)} \right){\rm{d}}t+ \\ &\quad\quad\quad \frac{1}{{{J_i}}}g\left( {{x_i},{v_i}} \right){\rm{d}}B. \end{split} $

      (24)

      Here, we choose $ J_i = 1 $, $ m_i = 0.5 $, $ y = 2.5 $, $ d_i(t) = 0.2 $, $ g(x_i, v_i) = 0.1\cos\left(\pi \left(x_i+v_i\right)\right) $. Then, (24) becomes

      $ \begin{split} &{\rm{d}}{x_i} = {v_i}{\rm{d}}t\\ &{\rm{d}}{v_i} = \left( { - 5\sin {x_i} - 2.5{v_i} + {u_i} + 0.2} \right){\rm{d}}t +\\ & \quad\quad\quad0.1\cos \left( {\pi \left( {{x_i} + {v_i}} \right)} \right){\rm{d}}B. \end{split}$

      (25)

      The dynamics of the leader is presented as

      $ \begin{split} &{\rm{d}}{x_0} = {v_0}{\rm{d}}t\\ &{\rm{d}}{v_0} = \left( { -\sin {x_0} - {v_0} + 0.04} \right){\rm{d}}t +\\ &\quad\quad\quad 0.1\cos \left( {\pi \left( {{x_0} + {v_0}} \right)} \right){\rm{d}}B. \end{split} $

      (26)

      According to the designed control algorithms (18) and (19) in this paper, we select the following simulation parameters, system initial values and membership functions: $ \sigma_{c_i} = \sigma_{\hat M_i} = 0.1 $ ($ i \in I $), $ \sigma_{\pi_2} = 0.1 $, $ \sigma_{\hat W_i} = 0.1 $ ($i \in $$ I {\backslash}{\bar I_0}$), $ \varpi_i = 10 $ ($ i \in I $), $ \zeta_1 = 0.5 $, $ \eta_1 = 0.11 $, $ \gamma_1 = 0.7 $, $x_1(0) = $$ -1$, $ v_1(0) = -1.9 $, $ \zeta_2 = 0.3 $, $ \beta_2 = 0.5 $, $ \gamma_2 = 5 $, $x_2(0) = 0.5$, $ v_2(0) = 1.3 $, $ \zeta_3 = 0.1 $, $ \eta_3 = 1 $, $ \gamma_3 = 1 $, $x_3(0) = -0.5$, $v_3(0) = $$ -1.5$, $ \zeta_4 = 0.09 $, $ \eta_4 = 0.45 $, $ \gamma_4 = 0.11 $, $ x_4(0) = 1 $, $v_4(0) = $$ -2.9$, $ x_0(0) = 0.1 $, $ v_0(0) = 0.1 $; ${\mu _{0j}}( {{x_0}, {v_0}}) = \exp [ - 10 $$ ( {x_0}- {\bar \upsilon_j} )^2 - 10( {v_0}-{\bar \upsilon_j} )^2 ]$, ${\varphi _{0j}} \left( {{x_0},{v_0}} \right) = \dfrac{{{\mu _{0j}}\left( {{x_0}, {v_0}} \right)}}{{\displaystyle\sum\nolimits_{j = 1}^6 {{\mu _{0j}}\left( {{x_0}, {v_0}} \right)}}}$, $j = 1, \cdots, 6$, $ \bar \upsilon_1 = -0.3 $, $ \bar \upsilon_2 = -0.2 $, $ \bar \upsilon_3 = -0.1 $, $ \bar \upsilon_4 = 0.1 $, $\bar \upsilon_5 = 0.2$, $ \bar \upsilon_6 = 0.3 $, $ \varphi_i = [\varphi_{01}, \varphi_{02},\varphi_{03}, \varphi_{04}, \varphi_{05}, \varphi_{06}] $, ($ i \in $$ \bar I_0 $). By substituting the consensus algorithms (18) and (19) into (25), we have the simulation results shown in Figs. 56. Fig. 5 shows that consensus errors $ \delta_{x_i} $ and $ \delta_{v_i} $ asymptotically approach zero as $ t \to \infty $. Fig. 6 denotes that controller and adaptive parameters are bounded.

      Figure 5.  Consensus errors, $ i \in I $

      Figure 6.  $u_i $, $c_i $, $\hat{M}_i$ (i$\in $I), $\|\hat{W}_1\|$, $\hat{\pi}_2$, $\|\hat{W}_3\|$ and $\|\hat{W}_4\|$

      In order to show the global characteristics of the consensus algorithms (18) and (19), here we reselect enough initial values as $ x_1(0) = -227 $, $ x_2(0) = -285 $, $ x_3(0) = 218 $, $ x_4(0) = 272 $, $ v_1(0) = 80 $, $ v_2(0) = 100 $, $ v_3(0) = -80 $, $v_4(0) = $$ -100$. When these initial values are substituted into the simulation system (25), the simulation results are shown in Fig. 7. The results in Fig. 7 further verify the global characteristics of algorithms (18) and (19).

      Figure 7.  Consensus errors, $ i \in I $

      Example 3. Second-order MAS

      For the second-order MAS (10) and (11) studied, according to the design idea of feedback compensator[25-33], we can obtain the following consensus algorithms:

      $ u_i = c_i(t)e_i + {{\hat M}_i}\tanh \left( {\frac{{{e_i}{{\hat M}_i}}}{{\varpi_i}}} \right)- \hat W_i^{\rm{T}}\varphi_i(x_i, v_i) $

      (27)

      and

      $ \left\{ \begin{aligned} &\dot {\hat M}_i = \zeta_i\left|s_i\right|-\sigma_{{\hat M}_i}{\hat M}_i,\; {\hat M}_i(0)>0,\; \sigma_{{\hat M}_i}>0\\ &\dot c_i(t) = \gamma_i s_i^2-\sigma_{c_i}c_i(t),\; \sigma_{c_i}>0\\ &\dot {\hat W}_i = -\eta_i\varphi_i(x_i,v_i)s_i-\sigma_{{\hat W}_i}{\hat W}_i,\; \sigma_{{\hat W}_i}>0\end{aligned} \right. $

      (28)

      where $ i \in {I} $.

      In the consensus algorithms (27) and (28), the parameters are selected as: $ \eta_2 = 0.5 $, $ \sigma_{\hat W_2} = 0.1 $ and the values of other parameters are shown in Example 2. By substituting the consensus algorithms (27) and (28) into MAS (25) and (26), we have the simulation results shown in Fig. 8, where the initial values of the MAS (25) and (26) are set as: $x_1(0) = $$ -227$, $ x_2(0) = -285 $, $ x_3(0) = 218 $, $ x_4(0) = 272 $, $v_1(0) = $$ 80$, $ v_2(0) = 100 $, $ v_3(0) = -80 $, $ v_4(0) = -100 $, $ x_0(0) = 0.1 $ and $ v_0(0) = 0.1 $, which is the same as that selected in the second simulation verification of Example 2. From Fig. 8, we can see that when the initial selection of the distributed control system (25) is sufficiently large, the semi-global consensus algorithms (27) and (28) designed above will fail. Therefore, two comparative simulation experiments show that the global consensus algorithms (18) and (19) designed in this paper is more effective.

      Figure 8.  $\delta_{x_i}$ and $\delta_{v_i} $, $ i \in I $

    • This paper solves the global asymptotical consensus problems of the first-order and second-order stochastic uncertain MAS based on the FLS, which the FLS is used as a feed-forward compensator instead of feedback one. Based on the Lyapunov functional theory, this paper proves that all signals are bounded in the sense of probability. In future research, our interest will focus on directed graph, switching topology graph, etc.

    • This work was supported by Natural Science Foundation of China (No. 61573013).

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