0 引言

近年来混沌同步问题引起了控制界的广泛关注^[1-8]，分数阶混沌系统逐渐成为研究热点，例如：孙宁等^[9]研究了分数阶不确定混沌系统的滑模投影同步问题，实现了主从系统的投影同步.余明哲等^[10]研究了一类分数阶不确定混沌系统的自适应滑模同步问题，实现了驱动系统与响应系统的快速同步.仲启龙等^[11]基于主动滑模控制方法研究了分数阶混沌系统的同步控制问题.而多涡卷系统密匙参数更多，因而在混沌通信中得到了广泛应用.例如：孙美美等^[12]研究了一类多涡卷超混沌系统的同步控制问题, 提出了一种自适应滑模控制方案.利用滑模控制和自适应控制技术, 消除了系统不确定性和未知扰动的影响.刘恒等^[13]研究了含扰动的多涡卷系统的修正函数时滞投影同步，得到了系统取得同步的充分性条件.这些工作大多是研究分数阶系统的混沌同步，然而工程和实际应用中更需要研究分数阶混沌系统的有限时间同步问题.例如：毛北行等^[14]研究了一类分数阶复杂网络混沌系统的有限时间同步问题，并估计了系统取得混沌同步所需的时间.在设计和植入控制器时，不能忽略具有死区的非线性输入.田小敏等^[15]研究了具有死区输入的混沌系统的有限时间同步问题，证明了滑模阶段和趋近阶段均是有限时间收敛的.本文基于滑模控制并利用分数阶微积分的相关理论，研究了具有死区输入的分数阶多涡卷系统的有限时间同步问题，以得到系统取得有限时间同步的充分性条件.

定义1^[16]  Riemann-Liouville分数阶导数定义为

$ \begin{array}{l} _{{t_0}}D_t^\alpha = \frac{{{{\rm{d}}^\alpha }f\left( t \right)}}{{{\rm{d}}{t^\alpha }}} = \frac{1}{{\Gamma \left( {n - \alpha } \right)}}\frac{{{{\rm{d}}^n}}}{{{\rm{d}}{t^n}}}\int_{{t_0}}^t {\frac{{{f^{\left( n \right)}}\left( \tau \right)}}{{{{\left( {t - \tau } \right)}^{\alpha - n + 1}}}}{\rm{d}}\tau } ,\\ \;\;\;\;\;\;\;\;n - 1 < \alpha \le n \in {{\bf{Z}}^ + }. \end{array} $

1 主要结果

设计如下一类分数阶多涡卷混沌系统作为主系统：

$ \left\{ \begin{array}{l} D_t^q{x_1} = {x_2},\\ D_t^q{x_2} = {x_3},\\ D_t^q{x_3} = - \alpha {x_3} - \beta {x_2} + f\left( {{x_1}} \right), \end{array} \right. $

(1)

其中，x₁, x₂, x₃∈R³为系统的状态变量，α, β为系统参数，f(x₁)为非线性项，f(x₁)=sin(ax₁-bx₁|x₁|－(cx₁)³)，当α=0.3, β=5.1, a=11, b=1.6, c=0.4, q=0.873时出现混沌吸引子，其对应的从系统为：

$ \left\{ \begin{array}{l} D_t^q{y_1} = {y_2} + \Delta {f_1}\left( y \right) + {d_1}\left( t \right) + {h_1}\left( {{u_1}\left( t \right)} \right),\\ D_t^q{y_2} = {y_3} + \Delta {f_2}\left( y \right) + {d_2}\left( t \right) + {h_2}\left( {{u_2}\left( t \right)} \right),\\ D_t^q{y_3} = - \alpha {y_3} - \beta {y_2} + f\left( {{y_1}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\Delta {f_3}\left( y \right) + {d_3}\left( t \right) + {h_3}\left( {{u_3}\left( t \right)} \right), \end{array} \right. $

(2)

其中，h_i(u_i(t))是死区非线性输入，定义如下：

$ {h_i}({u_i}(t)) = \\ \left\{ \begin{array}{l} \left( {{u_i}\left( t \right) - {u_{ + i}}} \right){h_{ + i}}\left( {{u_i}\left( t \right)} \right),\;\;\;{u_i}\left( t \right) > {u_{ + i}}\left( t \right),\\ 0,\;\;\;\;{u_{ - i}}\left( t \right) \le {u_i}\left( t \right) \le {u_{ + i}}\left( t \right),\\ \left( {{u_i}\left( t \right) - {u_{ - i}}} \right){h_{ - i}}\left( {{u_i}\left( t \right)} \right),\;\;\;{u_i}\left( t \right) > {u_{ - i}}\left( t \right), \end{array} \right. $

其中，h_+i(t), h_－i(t)是u_i(t)的非线性函数，u_+i, u_－i是给定的常数.

$ \left\{ \begin{array}{l} \left( {{u_i}\left( t \right) - {u_{ + i}}} \right){h_i}\left( {{u_i}\left( t \right)} \right) \ge {\beta _{ + i}}{\left( {{u_i}\left( t \right) - {u_{ + i}}} \right)^2},\\ \;\;\;\;\;\;\;{u_i}\left( t \right) > {u_{ + i}}\left( t \right),\\ 0,\;\;\;\;{u_{ - i}}\left( t \right) \le {u_i}\left( t \right) \le {u_{ + i}}\left( t \right),\\ \left( {{u_i}\left( t \right) - {u_{ - i}}} \right){h_i}\left( {{u_i}\left( t \right)} \right) \ge {\beta _{ - i}}{\left( {{u_i}\left( t \right) - {u_{ - i}}} \right)^2},\\ \;\;\;\;\;\;\;\;{u_i}\left( t \right) < {u_{ - i}}\left( t \right), \end{array} \right. $

其中，β_+i, β_－i是正常数.

假设1  设不确定项Δf_i(y)和外部扰动d_i(t)有界，即存在δ_i, ρ_i > 0，使得

$ \left| {\Delta {f_i}\left( y \right)} \right| < {\delta _i},\;\;\;\;\left| {{d_i}\left( t \right)} \right| < {\rho _i}. $

定义系统误差：

$ {e_1} = {y_1} - {x_1},\;\;\;\;{e_2} = {y_2} - {x_2},\;\;\;\;{e_3} = {y_3} - {x_3}, $

很容易得到误差方程：

$ \left\{ \begin{array}{l} D_t^q{e_1} = {e_2} + \Delta {f_1}\left( y \right) + {d_1}\left( t \right) + {h_1}\left( {{u_1}\left( t \right)} \right),\\ D_t^q{e_2} = {e_3} + \Delta {f_2}\left( y \right) + {d_2}\left( t \right) + {h_2}\left( {{u_2}\left( t \right)} \right),\\ D_t^q{e_3} = - \alpha {e_3} - \beta {e_2} + f\left( {{y_1}} \right) - f\left( {{x_1}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\Delta {f_3}\left( y \right) + {d_3}\left( t \right) + {h_3}\left( {{u_3}\left( t \right)} \right). \end{array} \right. $

(3)

引理1^[17]  假设存在连续正定函数V(t)满足微分不等式$\dot V(t) \le - p{V^n}(t)$, ∀_t≥t₀, V(t₀)≥0, 其中，p > 0, 0 < η < 1是2个正常数，则对于任意给定的t₀, V(t)满足不等式：

$ {V^{1 - \eta }}(t) \le {V^{1 - \eta }}({t_0}) - p(1 - \eta )(t - {t_0}),{t_0} \le t \le T, $

并且V(t)≡0, t≥T，其中,

$ T = {t_0} + \frac{{{V^{1 - \eta }}\left( {{t_0}} \right)}}{{p\left( {1 - \eta } \right)}}. $

引理2^[18]  在Riemann-Liouville分数阶导数的定义下，如果p > q≥0，m, n是整数，0≤m－1≤p < m, 0≤n－1≤q < n，则有

$ \begin{array}{l} _\alpha D_t^p\left( {_\alpha D_t^qf\left( t \right)} \right) = D_t^{p + q}f\left( t \right) - \\ \;\;\;\;\;\;\;\sum\limits_{j = 1}^n {\left[ {_\alpha D_t^{q - j}f\left( t \right)} \right]\left| {_{t = a}} \right.\frac{{t - a}}{{\Gamma \left( {1 - q - j} \right)}}} . \end{array} $

设计滑模面

$ \begin{array}{l} {s_i}\left( t \right) = D_t^{q - 1}{e_i}\left( t \right) + D_t^{q - 2}\left[ {\frac{\lambda }{\mu }{e_i} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_i} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\left. {\left( {D_t^{q - 1}{e_i}} \right)\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right], \end{array} $

误差系统满足滑动面方程：

$ \begin{array}{l} {s_i}\left( t \right) = 0,\;\;\;{{\dot s}_i}\left( t \right) = 0 \Rightarrow {{\dot s}_i}\left( t \right) = D_t^q{e_i}\left( t \right) + \\ \;\;\;\;\;\;\;D_t^{q - 1}\left[ {\frac{\lambda }{\mu }{e_i} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_i} + \left( {D_t^{q - 1}{e_i}} \right)\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right], \end{array} $

由此易得

$ \begin{array}{l} D_t^q{e_i}\left( t \right) = - D_t^{q - 1}\left[ {\frac{\lambda }{\mu }{e_i} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_i} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left( {D_t^{q - 1}{e_i}} \right)\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right]. \end{array} $

(4)

定理1  设计控制器：

$ {u_i}\left( t \right) = \left\{ \begin{array}{l} - {\gamma _i}{\zeta _i}{\mathop{\rm sgn}} {s_i} + {u_{ - i}},\;\;\;\;{s_i} > 0,\\ 0,\;\;\;\;\;{s_i} = 0,\\ - {\gamma _i}{\zeta _i}{\mathop{\rm sgn}} {s_i} + {u_{ + i}},\;\;\;\;{s_i} < 0, \end{array} \right. $

(5)

其中，${\gamma _i} = \frac{1}{{{\beta _i}}}, {\beta _i} = \min \left\{ {{\beta _{ - i}}, {\beta _{ + i}}} \right\}, $

$ \begin{array}{l} {\zeta _1} = \left| {{e_2}} \right| + {\sigma _1} + {k_1} + \left| {D_t^{q - 1}\left[ {\frac{\lambda }{\mu }{e_1} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_1} + } \right.} \right.\\ \;\;\;\;\;\;\left. {\left. {\left( {D_t^{q - 1}{e_1}} \right)\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right]} \right| > 0,\\ {\zeta _2} = \left| {{e_3}} \right| + {\sigma _2} + {k_2} + \left| {D_t^{q - 1}\left[ {\frac{\lambda }{\mu }{e_2} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_2} + } \right.} \right.\\ \;\;\;\;\;\;\left. {\left. {\left( {D_t^{q - 1}{e_2}} \right)\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right]} \right| > 0,\\ {\zeta _3} = \left| { - \alpha {e_3} - \beta {e_2} + f\left( {{y_1}} \right) + f\left( {{x_1}} \right)} \right| + {\sigma _3} + {k_3} + \\ \;\;\;\;\;\;\left| {D_t^{q - 1}\left[ {\frac{\lambda }{\mu }{e_3} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_3} + \left( {D_t^{q - 1}{e_3}} \right)\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right]} \right| > 0.\\ 其中\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{k_i} > 0,{\sigma _i} = {\delta _i} + {\rho _i}. \end{array} $

选取滑模面：

$ \begin{array}{l} {s_i}\left( t \right) = D_t^{q - 1}{e_i}\left( t \right) + D_t^{q - 2}\left[ {\frac{\lambda }{\mu }{e_i} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_i} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\left. {\left( {D_t^{q - 1}{e_i}} \right)\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right]. \end{array} $

则系统的轨迹在有限时间T₁内收敛到原点，其中，

$ {T_1} = \frac{1}{{2\lambda }}\ln \left[ {1 + \frac{\lambda }{\mu }{{\left( {{e^{\rm{T}}}\left( 0 \right)e\left( 0 \right)} \right)}^\mu }} \right]. $

证明  选取Lyapunov函数${V_1}(t) = \sum\limits_{i = 1}^3 {{e_i}^2(t)} $

$ \begin{array}{l} {{\dot V}_1} = 2\sum\limits_{i = 1}^3 {{e_i}{{\dot e}_i}} = 2\sum\limits_{i = 1}^3 {{e_i}\left[ {D_t^{1 - q}\left( {D_t^q{e_i}} \right) + \left( {D_t^{q - 1}{e_i}} \right) \cdot } \right.} \\ \;\;\;\;\;\;\;\left. {\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right] = 2\sum\limits_{i = 1}^3 {{e_i}\left\{ { - D_t^{1 - q}\left[ {D_t^{q - 1}\left( {\frac{\lambda }{\mu }{e_i} + } \right.} \right.} \right.} \\ \;\;\;\;\;\;\;\left. {\left. {{{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_i}} \right) + \left( {D_t^{q - 1}{e_i}} \right) \cdot \frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right] + \left( {D_t^{q - 1}{e_i}} \right) \cdot \\ \;\;\;\;\;\;\;\left. {\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right\} = 2\sum\limits_{i = 1}^3 {\left( { - \frac{\lambda }{\mu }e_i^2 - {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}e_i^2} \right)} , \end{array} $

得到

$ {{\dot V}_1} = - 2\frac{\lambda }{\mu }{e^2} - {\left( {{e^{\rm{T}}}e} \right)^{1 - \mu }} = - 2\frac{\lambda }{\mu }{V_1}\left( t \right) - 2V_1^{1 - \mu }\left( t \right), $

两边同乘以μV₁^μ-1(t)，得到

$ \mu V_1^{\mu - 1}\left( t \right){{\dot V}_1}\left( t \right) + 2\lambda V_1^\mu \left( t \right) = - 2\mu , $

两边同乘以e^2λt，得到

$ \begin{array}{l} {{\rm{e}}^{2\lambda t}}\left( {\mu V_1^{\mu - 1}\left( t \right){{\dot V}_1}\left( t \right) + 2\lambda V_1^\mu \left( t \right)} \right) = \\ \;\;\;\;\;\;d\left( {{e^{2\lambda t}}V_1^\mu \left( t \right)} \right) = - 2\mu {{\rm{e}}^{2\lambda t}}, \end{array} $

两边积分，得到

$ {{\rm{e}}^{2\lambda t}}V_1^\mu \left( t \right) - V_1^\mu \left( 0 \right) = - \frac{\mu }{\lambda }{{\rm{e}}^{2\lambda t}} + \frac{\mu }{\lambda }. $

易得

$ V_1^\mu \left( t \right) = {{\rm{e}}^{ - 2\lambda t}}\left( {\frac{\mu }{\lambda } + V_1^\mu \left( 0 \right)} \right) - \frac{\mu }{\lambda }. $

$ 如果\;\;\;\;\;\;V_1^\mu \left( {{T_1}} \right) \equiv 0 \Rightarrow {{\rm{e}}^{2\lambda {T_1}}} = 1 + \frac{\lambda }{\mu }V_1^\mu \left( 0 \right), $

$ 则有\;{T_1} = \frac{1}{{2\lambda }}\left\{ {\ln 1 + \frac{\lambda }{\mu }\left[ {\sum\limits_{i = 1}^3 {{{\left( {e_i^{\rm{T}}\left( 0 \right){e_i}\left( 0 \right)} \right)}^\mu }} } \right]} \right\}. $

定理2  对误差系统(3)，具有非线性死区输入的控制器(5)，误差系统的状态轨迹能达到滑模面.

证明

$ \begin{array}{*{20}{c}} {\left( {{u_i}\left( t \right) - {u_{ + i}}} \right){h_{ + l}}\left( {{u_i}\left( t \right)} \right) = - {\gamma _i}{\zeta _i}{\mathop{\rm sgn}} {s_i}{h_i}\left( {{u_i}\left( t \right)} \right) \ge }\\ {{\beta _{ + i}}{{\left( {{u_i}\left( t \right) - {u_{ + i}}} \right)}^2} = {\beta _{ + i}}\gamma _i^2\zeta _i^{22}{s_i} \ge {\beta _i}\gamma _i^2\zeta _i^{22}{s_i},} \end{array} $

由于γ_i= $\frac{1}{{{\beta _i}}}$ > 0, ζ_i > 0, 因此上述不等式可写为

$ - {\mathop{\rm sgn}} {s_i}{h_i}\left( {{u_i}\left( t \right)} \right) \ge {\zeta _i}{{\mathop{\rm sgn}} ^2}{s_i}, $

两边同乘以|s_i|，并考虑到|s_i|sgn s_i=s_i, sgn² s_i= 1，得到

$ {s_i}{h_i}\left( {{u_i}\left( t \right)} \right) \le - {\zeta _i}\left| {{s_i}} \right|. $

选择Lyapunov函数V₂(t)=$\frac{1}{2}\sum\limits_{i = 1}^3 {{s_i}^2} $，求导得

$ \begin{array}{l} {{\dot V}_2} = \sum\limits_{i = 1}^3 {{s_i}{{\dot s}_i}} = \sum\limits_{i = 1}^3 {{s_i}\left[ {D_t^q{e_i} + D_t^{q - 1}\left( {\frac{\lambda }{\mu }{e_i} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_i} + } \right.} \right.} \\ \;\;\;\;\;\;\;\left. {\left( {D_t^{q - 1}{e_i}} \right)\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right] = {s_1}\left[ {{e_2} + \Delta {f_1}\left( y \right) + {d_1}\left( t \right) + } \right.\\ \;\;\;\;\;\;\;{h_1}\left( {{u_1}\left( t \right)} \right) + D_t^{q - 1}\left( {\frac{\lambda }{\mu }{e_1} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_1} + \left( {D_t^{q - 1}{e_1}} \right)} \right.\\ \;\;\;\;\;\;\;\left. {\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right] + {s_2}\left[ {{e_3} + \Delta {f_2}\left( y \right) + {d_2}\left( t \right) + {h_2}\left( {{u_2}\left( t \right)} \right) + } \right.\\ \;\;\;\;\;\;\;\left. {D_t^{q - 1}\left( {\frac{\lambda }{\mu }{e_2} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_2} + \left( {D_t^{q - 1}{e_2}} \right)\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right)} \right] + \\ \;\;\;\;\;\;\;{s_3}\left[ { - \alpha {e_3} - \beta {e_2} + f\left( {{y_1}} \right) - f\left( {{x_1}} \right) + \Delta {f_3}\left( y \right) + } \right.\\ \;\;\;\;\;\;\;{d_3}\left( t \right) + {h_3}\left( {{u_3}\left( t \right)} \right) + D_t^{q - 1}\left( {\frac{\lambda }{\mu }{e_3} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_3} + } \right.\\ \;\;\;\;\;\;\;\left. {\left. {\left( {D_t^{q - 1}{e_3}} \right)\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right)} \right] = \sum\limits_{i = 1}^3 {\left| {{s_i}} \right|\left( {{\zeta _i} - {k_i}} \right) - } \\ \;\;\;\;\;\;\;\left| {{s_i}} \right|{\zeta _i} \le - k\sum\limits_{i = 1}^3 {\left| {{s_i}} \right|} \le - \sqrt 2 k{\left( {\frac{1}{2}\sum\limits_{i = 1}^3 {s_i^2} } \right)^{\frac{1}{2}}} = \\ \;\;\;\;\;\;\; - \sqrt 2 kV_2^{\frac{1}{2}}\left( t \right). \end{array} $

所以系统误差能在有限时间T₂内收敛到滑模面，其中，T₂=$\frac{{\sqrt {\sum\limits_{i = 1}^3 {{s_i}^2} (0)} }}{k}$，证毕.

2 数值仿真

利用龙格-库塔法进行仿真:

$ f\left( {{y_1}} \right) = \sin \left( {a{y_1} - b{y_1}\left| {{y_1}} \right| - {{\left( {c{y_i}} \right)}^3}} \right), $

当α=0.3, β=5.1, a=11, b=1.6, c=0.4, q=0.873时出现混沌吸引子，其中,

$ \begin{array}{l} \Delta {f_1}\left( {{y_1},{y_2},{y_3}} \right) = \cos \left( {2\pi {y_2}} \right),\\ \Delta {f_2}\left( {{y_1},{y_2},{y_3}} \right) = 0.5\cos \left( {2\pi {y_3}} \right),\\ \Delta {f_3}\left( {{y_1},{y_2},{y_3}} \right) = 0.3\cos \left( {2\pi {y_2}} \right), \end{array} $

$ \begin{array}{l} {h_i}\left( {{u_i}\left( t \right)} \right) = \\ \;\;\;\;\;\;\;\left\{ \begin{array}{l} \left( {{u_i}\left( t \right) - 1} \right)\left( {0.8 - 0.1\cos \left( {{u_i}\left( t \right)} \right),\;\;\;\;{u_i}\left( t \right) > 1,} \right.\\ 0,\;\;\;\; - 1 \le {u_i}\left( t \right) \le 1,\\ \left( {{u_i}\left( t \right) + 1} \right)\left( {1 - 0.5\cos } \right)\left( {{u_i}\left( t \right)} \right),\;\;\;{u_i}\left( t \right) < - 1, \end{array} \right. \end{array} $

$ \begin{array}{l} \;\;\;\;\;\;\;{d_1}\left( t \right) = 0.2\cos \left( t \right),\;\;\;\;{d_2}\left( t \right) = 0.6{\rm{sin}}\left( t \right),\\ \;\;\;\;\;\;{d_3}\left( t \right) = \cos \left( {3t} \right),\;\;\;\;{\beta _{ + i}} = 0.4,\;\;\;\;\;{\beta _{ - i}} = 0.5, \end{array} $

$ \begin{array}{l} \;\;\;\;\;\;{\beta _i} = 0.4,\;\;\;\;{\gamma _i} = 2.5x\left( 0 \right) = {\left( {1, - 2, - 2} \right)^{\rm{T}}},\\ \;\;\;\;\;\;y\left( 0 \right) = {\left( {1,1, - 1} \right)^{\mathit{T}}},\;\;\;\lambda = 1,\;\;\;\;\mu = 0.5. \end{array} $

仿真结果如图 1~3所示，从图 1可看出，不加控制器系统无法取得同步；由图 2知，加入控制器系统可快速取得同步；由图 3知，系统的误差很快趋近于零，表明系统快速取得了同步.

3 结论

基于稳定性理论研究了分数阶多涡卷系统的有限时间同步问题，研究表明：设计非线性死区输入的控制器以及构造适当的切换函数，能够使主从系统取得有限时间同步，并给出了严格的证明.数值仿真验证了方法的有效性.