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 智能系统学报  2019, Vol. 14 Issue (2): 239-245  DOI: 10.11992/tis.201709045 0

### 引用本文

YUAN Xiaolin, MO Lipo. Adaptive H synchronization of a class of fractional-order neural networks [J]. CAAI Transactions on Intelligent Systems, 2019, 14(2), 239-245. DOI: 10.11992/tis.201709045.

### 文章历史

Adaptive H synchronization of a class of fractional-order neural networks
YUAN Xiaolin , MO Lipo
School of Science, Beijing Technology and Business University, Beijing 100048, China
Abstract: This study aims to address the adaptive H synchronization problem and parameter identification problem of a class of uncertain fractional order neural networks. First, an adaptive control law is proposed to make the closed-loop system achieve H synchronization. Second, by using the robust control method and Gronwall-Bellman inequality, it is shown that the drive system and the response system can achieve synchronization under the proposed control law while satisfying the H performance. Finally, by numerical simulations, the effectiveness of the control law is verified, illustrating that the unknown parameters can also be identified using the proposed control law.
Key words: fractional-order    neural networks    adaptive    H    synchronization    unknown parameters    identification    controller

1 分数阶算子的基础理论知识

 ${}_tI_{{t_0}}^\alpha f\left( t \right) = \frac{1}{{\varGamma \left( \alpha \right)}}{\int_{{t_{\rm{0}}}}^t {\left( {t - \tau } \right)} ^{\alpha - 1}}f\left( \tau \right){\rm d}\tau$

 ${}_{{t_0}}D_t^\alpha f\left( t \right) = \frac{1}{{\varGamma \left( {n- \alpha } \right)}}\int_{{t_0}}^t {\frac{{{f^{\left( n \right)}}}}{{{{\left( {t- \tau } \right)}^{\alpha- n + 1}}}}{\rm d}\tau }$

 ${}_{{t_0}}D_t^\alpha f\left( t \right) = \frac{1}{{\varGamma \left( {{\rm{1}} - \alpha } \right)}}\int_{{t_0}}^t {\frac{{{f'}\left( \tau \right)}}{{{{\left( {t - \tau } \right)}^\alpha }}}{\rm d}\tau }$

 ${}_{{t_0}}D_t^\alpha {\left( {g\left( t \right) - h} \right)^2} \leqslant 2\left( {g\left( t \right) - h} \right){}_{{t_0}}D_t^\alpha g\left( t \right)$

 ${}_{{t_0}}I_t^\alpha {}_{{t_0}}D_t^\alpha y\left( t \right) = y\left( t \right) - \sum\limits_{k = 0}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( {{t_0}} \right)}}{{k!}}} {\left( {t - {t_0}} \right)^k}$

 $x\left( t \right) \leqslant \int_0^t {a\left( t \right)} b\left( \tau \right)\exp \left( {\int_\tau ^t {a\left( s \right){\rm d}s} } \right){\rm d}\tau + b\left( t \right)$

 $x\left( t \right)\!\! \leqslant \!\!b\left( 0 \right)\exp \left( {\int_0^t {\!\!\!a\left( \tau \right){\rm d}\tau } } \right) \!\!+\!\! \int_0^t {\!\!\dot b\!\left( \tau \right)} \exp \left( {\int_\tau ^t {\!\!\!\!a\left( s \right){\rm d}s} } \right){\rm d}\tau$

 $x\left( t \right) \leqslant b\left( 0 \right)\exp \left( {\int_0^t {a\left( \tau \right){\rm d}\tau } } \right)$

 $\frac{{\partial {z_i}\left( {w, t} \right)}}{{\partial t}} = - w{z_i}\left( {w, t} \right) + {f_i}\left( {{{ x}_i}\left( t \right)} \right)$
 ${{ x}_i}\left( t \right) = \int_0^\infty {\mu \left( w \right){z_i}} \left( {w, t} \right){\rm d}w$

2 研究问题的系统模型描述

 ${D^\alpha }{x_i}\left( t \right) \!=\! - {c_i}{x_i}\left( t \right) \!\!+\!\! \sum\limits_{j = 1}^n {{a_{ij}}{f_j}\left( {{x_j}\left( t \right)} \right)} \!+\! {I_i},\; i \!= \!1, 2, \cdots, n$ (1)

 $\left| {{f_j}\left( u \right) - {f_j}\left( v \right)} \right| \leqslant {L_j}\left| {u - v} \right|$ (2)

 ${D^\alpha }{y_i}\left( t \right) = - {c_i}{y_i}\left( t \right) + \sum\limits_{j = 1}^n {{{\hat a}_{ij}}(t)} {f_j}\left( {{y_j}\left( t \right)} \right) + {u_i}\left( t \right) + {{ w}_i}\left( t \right)$ (3)

 $D \!\!=\!\! \left\{ \!\!{{w}}\left( {{t}} \!\right) \!\! = \!\!{\left[ \!\!{{ w}_1^{\rm T}\left(\! t \!\right)\;{ w}_2^{\rm{T}}\left( \!t \!\right)\!\cdots \;{ w}_N^{\rm T}\left( \!t\! \right)}\!\! \right]^{\rm{T}}}\left| {\int_0^\infty {\sum\limits_{i \!= \!1}^N {{ w}_{{i}}^{\rm{T}}\left( t \right){{ w}_i}\left( t \right){\rm d}t \leqslant {\beta ^2}} } } \right. \!\!\right\}$

 $\mathop {\lim }\limits_{t \to \infty } \left| {{y_i}\left( t \right) - {x_i}\left( t \right)} \right| = 0, \quad i = 1, 2, \cdots, n$
3 一致性收敛分析 3.1 自适应同步

 ${u_i}\left( t \right) = - {d_i}\left( t \right){e_i}\left( t \right) + {\hat I_i}\left( t \right)$ (4)
 $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{D^\alpha }{d_i}\left( t \right) = {k_i}{e_i}{\left( t \right)^2}$ (5)
 $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{D^\alpha }{\hat I_i}\left( t \right) = - {r_i}{e_i}\left( t \right)$ (6)
 ${D^\alpha }{\hat a_{ij}}\left( t \right) = - {l_{ij}}{f_j}\left( {{y_j}\left( t \right)} \right){e_i}\left( t \right)$ (7)

 $\begin{array}{c} {D^\alpha }{e_i}\left( t \right) = - {c_i}{e_i}\left( t \right) + \displaystyle\sum\limits_{j = 1}^n {{{\hat a}_{ij}}(t)} {f_j}\left( {{y_j}\left( t \right)} \right) - \\ \displaystyle\sum\limits_{j = 1}^n {{a_{ij}}} {f_j}\left( {{x_j}\left( t \right)} \right) + {{\tilde I}_i}\left( t \right) - {d_i}\left( t \right){e_i}\left( t \right) \\ \end{array}$ (8)

 $\begin{split} V\left( t \right) &= \sum\limits_{i = 1}^n {\frac{1}{2}} {e_i}{\left( t \right)^2} + \sum\limits_{i = 1}^n {\frac{1}{{2{k_i}}}} {\left( {{d_i}\left( t \right) - {d_i}} \right)^2} + \\ & \sum\limits_{i = 1}^n {\frac{1}{{2{r_i}}}{{\tilde I}_i}{{\left( t \right)}^2} + \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{1}{{2{l_{ij}}}}} } } {{\tilde a}_{ij}}{\left( t \right)^2} \\ \end{split}$ (9)

 $\begin{array}{c} \begin{array}{*{20}{c}} {}&{} \end{array}\displaystyle\sum\limits_{j = 1}^n {{{\hat a}_{ij}}} (t){f_j}\left( {{y_j}\left( t \right)} \right) - \sum\limits_{j = 1}^n {{a_{ij}}} {f_j}\left( {{x_j}\left( t \right)} \right)= \\ \begin{array}{*{20}{c}} {}& \end{array}\displaystyle\sum\limits_{j = 1}^n {{{\tilde a}_{ij}}} (t){f_j}\left( {{y_j}\left( t \right)} \right) + \sum\limits_{j = 1}^n {{a_{ij}}} {f_j}\left( {{y_j}\left( t \right)} \right) - \sum\limits_{j = 1}^n {{a_{ij}}} {f_j}\left( {{x_j}\left( t \right)} \right)\leqslant \\ \begin{array}{*{20}{c}} {}& \end{array}\displaystyle\sum\limits_{j = 1}^n {{{\tilde a}_{ij}}} (t){f_j}\left( {{y_j}\left( t \right)} \right) + \sum\limits_{j = {\rm{1}}}^n {{a_{ij}}{L_j}{e_j}(t)}, \\ \end{array}$

 $\begin{array}{c} {D^\alpha }V\left( t \right) = \displaystyle\sum\limits_{i = 1}^n {\displaystyle\frac{1}{2}{D^\alpha }} {e_i}{\left( t \right)^2} + \sum\limits_{i = 1}^n {\frac{1}{{2{k_i}}}} {D^\alpha }{\left( {{d_i}\left( t \right) - {d_i}} \right)^2} + \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array} \displaystyle\sum\limits_{i = 1}^n {\frac{1}{{2{r_i}}}{D^\alpha }{{\tilde I}_i}{{\left( t \right)}^2} + \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{1}{{2{l_{ij}}}}} } } {D^\alpha }{{\tilde a}_{ij}}{\left( t \right)^2} \leqslant \\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{} \end{array} \displaystyle\sum\limits_{i = 1}^n {{e_i}\left( t \right){D^\alpha }} {e_i}\left( t \right) + \sum\limits_{i = 1}^n {\frac{1}{{{k_i}}}} \left( {{d_i}\left( t \right) - {d_i}} \right){D^\alpha }{d_i}\left( t \right) + \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array}\displaystyle\sum\limits_{i = 1}^n {\frac{1}{{{r_i}}}{{\tilde I}_i}\left( t \right){D^\alpha }{{\tilde I}_i}\left( t \right) + \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{1}{{{l_{ij}}}}{{\tilde a}_{ij}}\left( t \right)} } } {D^\alpha }{{\tilde a}_{ij}}\left( t \right)= \\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{} \end{array} \displaystyle\sum\limits_{i = 1}^n {{e_i}\left( t \right)} \left[{-{c_i}{e_i}\left( t \right) + \sum\limits_{j = 1}^n {{{\hat a}_{ij}}(t)} {f_j}\left( {{y_j}\left( t \right)} \right)} \right] - \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array} \displaystyle\sum\limits_{i = 1}^n {{e_i}\left( t \right)} \left[{\sum\limits_{j = 1}^n {{a_{ij}}} {f_j}\left( {{x_j}\left( t \right)} \right) + {{\tilde I}_i}\left( t \right)-{d_i}\left( t \right){e_i}\left( t \right)} \right] + \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array} \displaystyle\sum\limits_{i = 1}^n {\left( {{d_i}\left( t \right) - {d_i}} \right)} {e_i}{\left( t \right)^2} + \sum\limits_{i = 1}^n {{{\tilde I}_i}\left( t \right){D^\alpha }\left[{-{e_i}\left( t \right)} \right]} + \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array}\displaystyle\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\left[{-{{\tilde a}_{ij}}\left( t \right){f_j}\left( {{y_j}\left( t \right)} \right){e_i}\left( t \right)} \right]} } \\ \end{array}$

 $\begin{array}{c} \displaystyle {D^\alpha }V(t) \leqslant \sum\limits_{i = 1}^n {\left\{ { - {c_i}e_i^2(t) + {e_i}(t)\sum\limits_{j = 1}^n {{{\tilde a}_{ij}}} {f_j}({y_j}(t))} \right.} + \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array}\displaystyle \frac{1}{{{\varepsilon _i}}}\sum\limits_{j = 1}^n {{\mu ^2}L_{_{_{jj}}^j}^2{e_j}{{(t)}^2}} + {\varepsilon _i}{e_i}{(t)^2} + {{\tilde I}_i}(t){e_i}(t) - \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array}{d_i}(t){e_i}{(t)^2} + ({d_i}(t) - {d_i}){e_i}{(t)^2} - \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array}\displaystyle {{\tilde I}_i}(t){e_i}(t) - \left. {\sum\limits_{j = 1}^n {{{\tilde a}_{ij}}(t){f_j}({y_j}(t)){e_i}(t)} } \right\} = - \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array}\displaystyle \sum\limits_{i = 1}^n {{q_i}} {e_i}{(t)^2} \leqslant - \underline q \sum\limits_{i = 1}^n {{e_i}{{(t)}^2}} \\ \end{array}$

 $V\left( t \right) - V\left( 0 \right) \leqslant - \frac{{{\rm{2}}\underline {\rm{q}} }}{{\varGamma \left( \alpha \right)}}{\int_0^t {\left( {t - \tau } \right)} ^{\alpha - 1}}U\left( \tau \right){\rm d}\tau$

 $U\left( t \right) \leqslant V\left( t \right) \leqslant V\left( 0 \right) - \frac{{{\rm{2}}\underline q }}{{\varGamma \left( \alpha \right)}}{\int_0^t {\left( {t - \tau } \right)} ^{\alpha - 1}}U\left( \tau \right){\rm d}\tau$

 $\begin{array}{c} U\left( t \right) \leqslant V\left( 0 \right)\exp \left( { - \displaystyle\frac{{{\rm{2}}\underline q }}{{\varGamma \left( \alpha \right)}}{{\int_0^t {\left( {t - \tau } \right)} ^{\alpha - 1}}}{\rm d}\tau } \right) = \\ \begin{array}{*{20}{c}} {}&{} \end{array}V\left( 0 \right)\exp \left( { - \displaystyle\frac{{{\rm{2}}\underline q }}{{\varGamma \left( {\alpha + {\rm{1}}} \right)}}{t^\alpha }} \right) \\ \end{array}$

 ${D^\alpha }{e_i}\left( t \right) = \sum\limits_{j = 1}^n {\left( {{{\hat a}_{ij}}\left( t \right){\rm{ - }}{a_{ij}}} \right)} {f_j}\left( {{x_j}\left( t \right)} \right) = {\rm{0}}, j = 1, 2, \cdots, n$

3.2 自适应 ${H_\infty }$ 同步

 \left\{ \begin{aligned} &\! \frac{{\partial {z_i}\left( {w, t} \right)}}{{\partial t}} \!\!=\!\! - w{z_i}\left( {w, t} \right) \!- \!{c_i}{e_i}\left( t \right) \!+\! \sum\limits_{j = 1}^n {{{\hat a}_{ij}}\left( t \right){f_j}\left( {{y_j}\left( t \right)} \right)} - \\ &\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{}&{} \end{array}}\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle{\!\sum\limits_{j = 1}^n {{a_{ij}}{f_j}\left( {{x_j}\left( t \right)} \right)} \!\!+\!\! {{\tilde I}_i}\left( t \right) \!-\! {d_i}\left( t \right){e_i}\left( t \right) \!\!+\!\! {w_i}\left( t \right)} \end{array} \\ &\! {e_i}\left( t \right) = \int_0^\infty {\mu \left( w \right){z_i}\left( {w, t} \right){\rm d}w} \\ \end{aligned} \right. (10)
 $\left\{ \begin{array}{c} \displaystyle\frac{{\partial {D_i}\left( {w, t} \right)}}{{\partial t}} = - w{D_i}\left( {w, t} \right) + {k_i}{e_i}{\left( t \right)^2} \\ {{\tilde d}_i}\left( t \right) = {\displaystyle\int}_0^{\infty} {\mu \left( w \right){D_i}\left( {w, t} \right){\rm d}w} \\ \end{array} \right.$ (11)
 $\quad \left\{ \begin{gathered} \frac{{\partial {A_{ij}}\left( {w, t} \right)}}{{\partial t}} = - w{A_{ij}}\left( {w, t} \right) - {l_{ij}}{f_j}\left( {{y_j}\left( t \right)} \right){e_i}\left( t \right) \\ {{\tilde a}_{ij}}\left( t \right) = \int_0^\infty {\mu \left( w \right){A_{ij}}\left( {w, t} \right){\rm d}w} \\ \end{gathered} \right.$ (12)
 $\left\{ \begin{array}{c} \displaystyle\frac{{\partial {B_i}\left( {w, t} \right)}}{{\partial t}} = - w{B_i}\left( {w, t} \right) - {r_i}{e_i}\left( t \right) \\ {{\tilde I}_i}\left( t \right) = \displaystyle\int_0^\infty {\mu \left( w \right){B_i}\left( {w, t} \right){\rm d}w} \\ \end{array} \right.$ (13)

 $\begin{array}{c} V\left( t \right) = \displaystyle\frac{1}{2}\sum\limits_{i = 1}^n {\int_0^\infty {\mu \left( w \right){z_i}{{\left( {w, t} \right)}^2}{\rm d}w} } + \\ \begin{array}{*{20}{c}} {}&{} \end{array}\displaystyle\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{1}{{2{l_{ij}}}}} } \int_0^\infty {\mu \left( w \right){A_{ij}}{{\left( {w, t} \right)}^2}{\rm d}w} + \\ \begin{array}{*{20}{c}} {}&{} \end{array}\displaystyle\sum\limits_{i = 1}^n {\frac{1}{{2{r_i}}}} \int_0^\infty {\mu \left( w \right)} {B_i}{\left( {w, t} \right)^2}{\rm d}w + \\ \begin{array}{*{20}{c}} {}&{} \end{array}\displaystyle\sum\limits_{i = 1}^n {\frac{1}{{2{k_i}}}} \int_0^\infty {\mu \left( w \right)} {D_i}{\left( {w, t} \right)^2}{\rm d}w \\ \end{array}$

 $\begin{array}{c} \dot V\left( t \right) = \displaystyle\sum\limits_{i = 1}^n {\int_0^\infty {\mu \left( w \right){z_i}\left( {w, t} \right)\frac{{\partial {z_i}}}{{\partial t}}{\rm d}w} } + \\ \begin{array}{*{20}{c}} {}&{} \end{array}\displaystyle\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{1}{{{l_{ij}}}}} } \int_0^\infty {\mu \left( w \right){A_{ij}}\left( {w, t} \right)\frac{{\partial {A_{ij}}}}{{\partial t}}{\rm d}w} + \\ \begin{array}{*{20}{c}} {}&{} \end{array} \displaystyle\sum\limits_{i = 1}^n {\frac{1}{{{r_i}}}\int_0^\infty {\mu \left( w \right)} {B_i}\left( {w, t} \right)\frac{{\partial {B_i}}}{{\partial t}}{\rm d}w} + \\ \begin{array}{*{20}{c}} {}&{} \end{array}\displaystyle\sum\limits_{i = 1}^n {\frac{1}{{{k_i}}}} \int_0^\infty {\mu \left( w \right)} {D_i}\left( {w, t} \right)\frac{{\partial {D_i}}}{{\partial t}}{\rm d}w \\ \end{array}$

 \begin{array}{c} \dot V\left( t \right) \leqslant \displaystyle\sum\limits_{i = 1}^n {\left\{ {{\rm{ - }}\left( {{{{d}}_{\rm{i}}} + {c_i} - \left( {\sum\limits_{j = 1}^n {\frac{1}{{{\varepsilon _j}}}} } \right)L_i^2{\mu ^2} - {\varepsilon _i}} \right){e_i}{{\left( t \right)}^2}} \right\}} + \\ \begin{aligned} {}&{} \end{aligned}\displaystyle\sum\limits_{i = 1}^n {\left\{ {{e_i}\left( t \right){w_i}\left( t \right)} \right\}} \leqslant \\ \begin{aligned} {}&{} \end{aligned} \displaystyle\sum\limits_{i = 1}^n {\left\{ { - \left( {{{{d}}_{\rm{i}}} + {c_i} - \left( {\sum\limits_{j = 1}^n {\frac{1}{{{\varepsilon _j}}}} } \right)L_i^2{\mu ^2} - {\varepsilon _i} - \frac{1}{\gamma }} \right){e_i}{{\left( t \right)}^2}} \right\}} + \\ \begin{aligned} {}&{} \end{aligned}\displaystyle\sum\limits_{i = 1}^n {\left\{ {\gamma {w_i}{{\left( t \right)}^2}} \right\}} \end{array} \\
 $\begin{gathered} \dot V\left( t \right) + \sum\limits_{i = 1}^n {{e_i}{{\left( t \right)}^2}} - \gamma \sum\limits_{i = 1}^n {{w_i}{{\left( t \right)}^2}} \leqslant \\ \sum\limits_{i = 1}^n {\left\{ { - \left( {{{{d}}_{{i}}} + {c_i} - \left( {\sum\limits_{j = 1}^n {\frac{1}{{{\varepsilon _j}}}} } \right)L_i^2{\mu ^2} - {\varepsilon _i} - \frac{1}{\gamma } - 1} \right)} \right\}} {e_i}{\left( t \right)^2} \\ \end{gathered}$

 ${{{{d}}_{{i}}} + {c_i} - \left( {\sum\limits_{j = 1}^n {\frac{1}{{{\varepsilon _j}}}} } \right)L_i^2{\mu ^2} - {\varepsilon _i} - \frac{1}{\gamma } - 1} > 0$

4 仿真实验

$\alpha = 0.98,$ ${c_1} = 1, {c_2} = 1, {c_3} = 1, {I_1} = 1, {I_2} = 1,$ ${I_3} = 1$ , ${{x(t)}} = {({x_1}(t), {x_2}(t), {x_3}(t))^{\rm{T}}},$ ${{y(t)}} = {({y_1}(t), {y_2}(t), {y_3}(t))^{\rm{T}}}$ , ${f_j}\left( {{x_j}\left( t \right)} \right) = \tanh \left( {{x_j}\left( t \right)} \right), {f_j}\left( {{y_j}\left( t \right)} \right) = \tanh \left( {{y_j}\left( t \right)} \right)$ , $i, j = 1, 2, 3$

 ${{A}} = { \left( {{{{a}}_{{{ij}}}}} \right)_{{{3}} \times {{3}}}} = \left[ {\begin{array}{*{20}{c}} 2&{ - 1.2}&0 \\ {1.8}&{1.71}&{1.15} \\ { - 4.75}&0&{1.1} \end{array}} \right]$

{{{x}}^{\rm {T}}}\left( {{0}} \right) = {\left[ {\begin{aligned} {0.3}\;&{ - 0.4}&{0.9} \end{aligned}} \right]^{\rm{T}}}, {{{y}}^{\rm{T}}}\left( {{0}} \right) = {\left[ {\begin{aligned} {2.8}\;&{ - 0.4}&{4.6} \end{aligned}} \right]^{\rm{T}}}, {{\hat a}_{11}}\left( 0 \right) = 1,

${{\hat a}_{12}}\left( 0 \right) = 1, \;\;{{\hat a}_{13}}\left( 0 \right) =$ $1,\; \;\;{{\hat a}_{21}}\left( 0 \right) =$ $1, \;\;\;{{\hat a}_{22}}\left( 0 \right) =$ $1,\;\;{{\hat a}_{23}}\left( 0 \right) = 1,\;\;$

${{\hat a}_{31}}\left( 0 \right) = 1, {{\hat a}_{32}}\left( 0 \right) = 1, {{\hat a}_{33}}\left( 0 \right) = 1, {\hat I_1}\left( 0 \right) = 1, {\hat I_2}\left( 0 \right) = 1, {\hat I_3}\left( 0 \right) = 1,$

${d_1}\left( 0 \right) = 1, {d_2}\left( 0 \right) = 1, {d_2}\left( 0 \right) = 1, {k_1} = {k_2} = {k_3} = 5, {r_1} = {r_2} = {r_3} =$

$5, {l_{ij}} = 5, i, j = 1, 2, 3$

 Download: 图 1 驱动系统(1)与未加控制器的响应系统(3)状态图 Fig. 1 States of drive system (1) and response system (3) without controller

 Download: 图 2 驱动系统(1)与加控制律响应系统(3)状态图 Fig. 2 States of drive system (1) and response system (3) with controller

 Download: 图 3 ${d_1}{\text 、}{d_2}{\text 、}{d_3}$ 随时间 $t$ 的变化图 Fig. 3 ${d_1}, {d_2}, {d_3}$ varies with time
 Download: 图 4 ${\hat a_{11}}{\text 、} {\hat a_{12}}{\text 、}{\hat a_{13}}$ 的辨识 Fig. 4 Identification of ${\hat a_{11}}, {\hat a_{12}}, {\hat a_{13}}$
 Download: 图 5 ${\hat a_{{\rm{2}}1}}{\text 、}{\hat a_{{\rm{2}}2}}{\text 、}{\hat a_{{\rm{2}}3}}$ 的辨识 Fig. 5 Identification of ${\hat a_{{\rm{2}}1}}, {\hat a_{{\rm{2}}2}}, {\hat a_{{\rm{2}}3}}$
 Download: 图 6 ${\hat a_{{\rm{3}}1}}{\text 、}{\hat a_{{\rm{3}}2}}{\text 、}{\hat a_{{\rm{3}}3}}$ 的辨识 Fig. 6 Identification of ${\hat a_{{\rm{3}}1}}, {\hat a_{{\rm{3}}2}}, {\hat a_{{\rm{3}}3}}$
 $\begin{gathered} {{\hat a}_{11}} = 1.99, {{\hat a}_{12}} = - 1.2, {{\hat a}_{13}} = 0, {{\hat a}_{21}} = 1.8, {{\hat a}_{22}} = 1.7, \\ {{\hat a}_{23}} = 1.15, {{\hat a}_{31}} = - 4.75, {{\hat a}_{32}} = 0, {{\hat a}_{33}} = 1.1 \\ \end{gathered}$

 \begin{aligned} &{{{x}}^{\rm{T}}}\left( {{0}} \right) = {\left[ {\begin{array}{*{20}{c}} {0.3}&{ - 0.4}&{\!0.9} \end{array}} \right]^{\rm{T}}},\\ &{{{y}}^{\rm{T}}}\left( {{0}} \right) = {\left[ {\begin{array}{*{20}{c}} \! {0.3}&{ - 0.4}&{\!0.9} \end{array}} \right]^{\rm{T}}} \end{aligned}

 ${w_i}\left( t \right) = \left\{ {\begin{array}{*{20}{l}} 1,&{0 < t < 1} \\ 0,&{\text{其他}} \end{array}} \right.$

 Download: 图 7 $\gamma$ 随时间变化可选取值图 Fig. 7 The value of $\gamma$ varies with time

 $\int_{\rm{0}}^\infty {\sum\limits_{i = 1}^n {{e_i}{{\left( t \right)}^2}} {\rm d}t} < \int_{\rm{0}}^\infty {\gamma \sum\limits_{i = 1}^n {{w_i}{{\left( t \right)}^2}} } {\rm d}t$

 Download: 图 8 $\displaystyle\int_0^\infty {\displaystyle\sum\limits_{i = 1}^n {{e_i}{{\left( t \right)}^2}} } {\rm d}t$ 与 $\displaystyle\int_0^\infty {\displaystyle\gamma \sum\limits_{i = 1}^n {{w_i}{{\left( t \right)}^2}} } {\rm d}t$ 的能量轨迹图 Fig. 8 The energy trajectories of $\displaystyle\int_0^\infty {\displaystyle\sum\limits_{i = 1}^n {{e_i}{{\left( t \right)}^2}} } {\rm d}t$ and $\displaystyle\int_0^\infty {\displaystyle\gamma \sum\limits_{i = 1}^n {{w_i}{{\left( t \right)}^2}} } {\rm d}t$
5 结束语

 [1] ZHANG Shou, YU Yongguang, YU Junzhi. LMI conditions for global stability of fractional-order neural networks[J]. IEEE transactions on neural networks and learning systems, 2017, 28(10): 2423-2433. DOI:10.1109/TNNLS.2016.2574842 (0) [2] LI Yan, CHEN Yangquan, PODLUBNY I. Mittag-Leffler stability of fractional order nonlinear dynamic systems[J]. Automatica, 2009, 45(8): 1965-1969. DOI:10.1016/j.automatica.2009.04.003 (0) [3] HAYMAN S. The McCulloch-pitts model[C]//International Joint Conference on Neural Networks. Washington, USA, 1999: 4438–4439. (0) [4] HOPFIELD J J. Neural networks and physical systems with emergent collective computational abilities[J]. Proceedings of the national academy of sciences of the United States of America, 1982, 79(8): 2554-2558. DOI:10.1073/pnas.79.8.2554 (0) [5] YAN Zheng, WANG Jun. Robust model predictive control of nonlinear systems with unmodeled dynamics and bounded uncertainties based on neural networks[J]. IEEE transactions on neural networks and learning systems, 2014, 25(3): 457-469. DOI:10.1109/TNNLS.2013.2275948 (0) [6] PAN Yunpeng, WANG Jun. Robust model predictive control using a discrete-time recurrent neural network[C]// International Symposium on Neural Networks: advances in Neural Networks. Berlin, Heidelberg, Germany: , 2008: 883–892. (0) [7] KASLIK E, SIVASUNDARAM S. Multistability in impulsive hybrid Hopfield neural networks with distributed delays[J]. Nonlinear analysis: real world applications, 2011, 12(3): 1640-1649. DOI:10.1016/j.nonrwa.2010.10.018 (0) [8] BAO Haibo, PARK J H, CAO Jinde. Adaptive synchronization of fractional-order memristor-based neural networks with time delay[J]. Nonlinear dynamics, 2015, 82(3): 1343-1354. DOI:10.1007/s11071-015-2242-7 (0) [9] CHEN Jiejie, ZENG Zhigang, JIANG Ping. Global mittag-leffler stability and synchronization of memristor-based fractional-order neural networks[J]. Neural networks, 2014, 51: 1-8. DOI:10.1016/j.neunet.2013.11.016 (0) [10] WU Wei, CHEN Tianping. Global synchronization criteria of linearly coupled neural network systems with time-varying coupling[J]. IEEE transactions on neural networks, 2008, 19(2): 319-332. DOI:10.1109/TNN.2007.908639 (0) [11] VELMURUGAN G, RAKKIYAPPAN R, CAO Jinde. Finite-time synchronization of fractional-order memristor-based neural networks with time delays[J]. Neural networks, 2016, 73: 36-46. DOI:10.1016/j.neunet.2015.09.012 (0) [12] ABDURAHMAN A, JIANG Haijun, TENG Zhidong. Finite-time synchronization for memristor-based neural networks with time-varying delays[J]. Neural networks, 2015, 69: 20-28. DOI:10.1016/j.neunet.2015.04.015 (0) [13] YU Juan, HU Cheng, JIANG Haijun. α-stability and α-synchronization for fractional-order neural networks [J]. Neural networks, 2012, 35: 82-87. DOI:10.1016/j.neunet.2012.07.009 (0) [14] MA Weiyuan, LI Changpin, WU Yujiang, et al. Adaptive synchronization of fractional neural networks with unknown parameters and time delays[J]. Entropy, 2014, 16(12): 6286-6299. DOI:10.3390/e16126286 (0) [15] LIN Peng, JIA Yingmin, LI Lin. Distributed robust H∞ consensus control in directed networks of agents with time-delay [J]. Systems and control letters, 2008, 57(8): 643-653. DOI:10.1016/j.sysconle.2008.01.002 (0) [16] MO L, JIA Y. H∞ consensus control of a class of high-order multi-agent systems [J]. IET control theory and application, 2011, 5(1): 247-253. DOI:10.1049/iet-cta.2009.0365 (0) [17] MATHIYALAGAN K, ANBUVITHYA R, SAKTHIVEL R, et al. Non-fragile H∞ synchronization of memristor-based neural networks using passivity theory [J]. Neural networks, 2016, 74: 85-100. DOI:10.1016/j.neunet.2015.11.005 (0) [18] PODLUBNY I. Fractional differential equations: mathematics in science and engineering[M]. San Diego, Calif, USA: Academic Press, 1999. (0) [19] AGUILA-CAMACHO N, DUARTE-MERMOUD M A, GALLEGOS J A. Lyapunov functions for fractional order systems[J]. Communications in nonlinear science and numerical simulation, 2014, 19(9): 2951-2957. DOI:10.1016/j.cnsns.2014.01.022 (0) [20] KILBAS A A A, SRIVASTAVA H M, TRUJILLO J J. Theory and applications of fractional differential equations[M]. Boston: Elsevier, 2006. (0) [21] SLOTINE J J E, LI Weiping. Applied nonlinear control[M]. Beijing, China Machine Press, 2004. (0) [22] TRIGEASSOU J C, MAAMRI N, SABATIER J, et al. A lyapunov approach to the stability of fractional differential equations[J]. Signal processing, 2011, 91(3): 437-445. DOI:10.1016/j.sigpro.2010.04.024 (0) [23] TRIGEASSOU J C, MAAMRI N. State space modeling of fractional differential equations and the initial condition problem[C]//Proceedings of the 6th International Multi-Conference on Systems, Signals and Devices. Djerba, Tunisia, 2009: 1–7. (0)