混沌系统的同步控制逐步成为研究的热门课题^[1-12], 而利用滑模控制方法研究同步逐渐成为研究不确定性系统的便捷方法, 并取得了丰富的成果.主动滑模方法, 自适应滑模方法, 积分滑模方法及比例积分滑模方法, 有限时间滑模方法, 模糊滑模等方法被相继提出.例如:文献[13]利用主动滑模方法研究了一类混沌系统的同步控制问题, 实现了驱动响应系统的主动滑模同步控制; 文献[14]研究了分数阶Van der pol振子网络的混沌同步; 文献[15]基于自适应滑模方法研究分数阶参数不确定系统的异结构混沌同步; 文献[16]研究Rössler混沌系统的自适应滑模控制问题; 实现了对Rössler混沌系统的同步控制; 文献[17]利用积分滑模滑模方法研究航天器的姿态容错控制; 文献[18]利用有限时间滑模方法研究了多涡卷混沌系统的同步.

另一方面, Victor-Carmen混沌系统的同步激起众广大学者的极大热情, 例如:文献[19]研究了Victor-Carmen混沌系统的投影同步问题; 文献[20]基于自适应滑模方法研究了分数阶Victor-Carmen混沌系统的自适应滑模控制问题.在以上研究的基础上, 研究Victor-Carmen混沌系统的有限时间滑模同步问题, 设计新型滑模面, 能够使误差动态系统有限时间内收敛到滑模面, 较大提高了滑模同步的效率.

考虑Victor-Carmen混沌系统^[20]

$ \begin{equation} \left\{ \begin{aligned} \dot {x_1}& = -x_1-\alpha x_2x_3, \\ \dot{x_2}& = -x_2+cx_3-\beta x_1x_3, \\ \dot{x_3}& = -bx_1-c x_2+ x_3+\gamma x_1x_2, \end{aligned} \right. \end{equation} $

(2.1)

其中$ x_1, x_2, x_3\in R^3 $为状态变量; $ c, b, \alpha, \beta, \gamma $为常值参数.当$ \alpha = 50, \beta = 20, \gamma = 4.1, c = 5, b = 9 $时出现怪异吸引子, 其轨相图如图 1所示.

从系统设计为

$ \begin{equation} \left\{ \begin{aligned} \dot{y_1}& = -y_1-\alpha y_2y_3+\Delta f_1(y)+d_1(t)+u_1(t), \\ \dot{y_2}& = -y_2+cy_3-\beta y_1y_3+\Delta f_2(y)+d_2(t)+u_2(t), \\ \dot {y_3}& = -by_1-c y_2+ y_3+\gamma y_1y_2+\Delta f_3(y)+d_3(t)+u_3(t), \end{aligned} \right. \end{equation} $

(2.2)

其中$ \Delta f_i(y)(i = 1, 2, 3) $为不确定项, $ y = [y_1, y_2, y_3]^T, $ $ d_i(t)(i = 1, 2, 3) $为外部扰动.

假设 1  设不确定项$ \Delta f_i(y) $和外部扰动$ d_i(t) $有界.即存在常数$ m_i, n_i>0, $使得

$ |\Delta f_i(y)|<m_i, |d_i(t)|<n_i. $

定义系统误差$ e_i = y_i-x_i\; (i = 1, 2, 3), $很容易得到误差方程

$ \begin{equation} \left\{ \begin{aligned} \dot {e_1}& = -e_1-\alpha y_2y_3+\alpha x_2x_3+\Delta f_1(y)+d_1(t)+u_1(t), \\ \dot {e_2}& = -e_2+ce_3-\beta y_1y_3+\beta x_1x_3+\Delta f_2(y)+d_2(t)+u_2(t), \\ \dot{e_3}& = -be_1-c e_2+ e_3+\gamma y_1y_2-\gamma x_1x_2+\Delta f_3(y)+d_3(t)+u_3(t). \end{aligned} \right. \end{equation} $

(2.3)

引理 1  ^[21]  假设存在连续正定函数$ V(t) $满足微分不等式

$ \dot{V}(t) \leq - pV^{\eta}(t), \forall t\geq t_0, V(t_0)\geq 0, $

式中$ p>0, 0<\eta<1 $是两个正常数.则对于任意给定的$ t_0, $ $ V(t) $满足如下不等式

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

并且$ \; V(t)\equiv 0, t\geq T, $其中

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

定理 1  在假设1条件下, 构造滑模函数$ s_i(t) = e_i+D_t^{-1}(k_1e_i+k_2|e_i|^\sigma {\hbox{sat}}(e_i)), $控制律

$ \begin{equation} \begin{array}{ll} u_1 = &e_1+\alpha y_2y_3-\alpha x_2x_3-(k_1e_1+k_2|e_1|^\sigma {\hbox{sat}}(e_1))\\ &-(k_3s_1+k_4|s_1|^\sigma {\hbox{sat}}(s_1)+k_5{\hbox{sgn}} (s_1))-( m_1+ n_1){\hbox{sgn}}(s_1) , \\ u_2 = &e_2-ce_3+\beta y_1y_3-\beta x_1x_3-(k_1e_2+k_2|e_2|^\sigma {\hbox{sat}}(e_2))\\ &-(k_3s_2+k_4|s_2|^\sigma {\hbox{sat}}(s_2)+k_5{\hbox{sgn}} (s_2))-( m_2+ n_2){\hbox{sgn}}(s_2) , \\ u_3 = &be_1+ce_2- e_3+\gamma (x_1x_2- y_1y_2)-(k_1e_3+k_2|e_3|^\sigma {\hbox{sat}}(e_3))\\ &-(k_3s_3+k_4|s_3|^\sigma {\hbox{sat}}(s_3)+k_5 {\hbox{sgn}} (s_3))-(m_3+n_3){\hbox{sgn}}(s_3), \end{array}\nonumber \end{equation} $

其中$ k_i>0\; (i = 1, 2, 3, 4, 5) $, 则系统(2.3)将在时间$ T $内收敛至切换面$ s_i = 0 $, 其中

$ T = \frac 1 r \ln(\sqrt{2}\sum \limits^3_{i = 1}|s_i(0)|+1), r = \min\{\sqrt{2}k_5, 2k_3\}. $

证  构造函数$ V(t) = \frac 1 2 \sum\limits^3_{i = 1}s_i^2, $从而得到

$ \begin{aligned} \dot{V} = &\sum \limits^3_{i = 1}s_i(\dot{e_i}+(k_1e_i+k_2|e_i|^\sigma {\hbox{sat}}(e_i)))\\ = &s_1[ -e_1-\alpha y_2y_3+\alpha x_2x_3+\Delta f_1(y)+d_1(t)+u_1(t)+(k_1e_1+k_2|e_1|^\sigma {\hbox{sat}}(e_1))]\\ &+s_2[ -e_2+ce_3-\beta y_1y_3+\beta x_1x_3+\Delta f_2(y)+d_2(t)+u_2(t)+(k_1e_2+k_2|e_2|^\sigma {\hbox{sat}}(e_2))]\\ &+s_3[-be_1-c e_2+ e_3+\gamma y_1y_2-\gamma x_1x_2+\Delta f_3(y)+d_3(t)+u_3(t)+(k_1e_3+k_2|e_3|^\sigma {\hbox{sat}}(e_3))]\\ \leq & \sum \limits^3_{i = 1}\{s_i[-(k_3s_i+k_4|s_i|^\sigma {\hbox{sat}}(s_i)+k_5 {\hbox{sgn}} (s_i))-(m_i+n_i){\hbox{sgn}}(s_i)]-(m_i+n_i)|s_i|\}\\ = &-\sum \limits^3_{i = 1}(k_3s_i^2+k_4|s_i|^\sigma {\hbox{sat}}(s_i)+k_5 |s_i|). \end{aligned} $

由于$ s_i{\hbox{sat}}(s_i) = s_i{\hbox{sgn}}(s_i/k), (|s_i|>k), s_i{\hbox{sat}}(s_i) = s_i^2/k, (|s_i|\leq k), $又$ s_i{\hbox{sgn}}(s_i) = |s_i| $, 所以$ s_i{\hbox{sat}}(s_i) = k(s_i/k){\hbox{sgn}}(s_i/k) = |s_i| (k>0), $则

$ \dot{V}\leq -\sum \limits^3_{i = 1}(k_3s_i^2+k_5 |s_i|) = -\sqrt{2}k_5\sqrt{V}-2k_3V. $

定义$ r = \min\{\sqrt{2}k_5, 2k_3\}, $所以$ \dot{V}\leq -r(\sqrt{V}+2V), $两边积分

$ dt\leq \frac {-dV}{r(\sqrt{V}+2V)}\leq \frac {-dV}{r(2\sqrt{V}+1)\sqrt{V}} = \frac {-d(2\sqrt{V}+1)}{r2\sqrt{V}+1}. $

由引理1滑模面具有可到达性.

定理 2  构造滑模函数$ s_i(t) = e_i+D_t^{-1}(k_1e_i+k_2|e_i|^\sigma {\hbox{sat}}(e_i)), $则该滑模动态系统渐近稳定并且收敛到坐标原点, 其中$ k_1, k_2>0, $ $ {\hbox{sat}}(e_i) = \left\{\begin{aligned} &{\hbox{sgn}}(e_i/k), |e_i|>k, k>0, \\ &e_i/k, |e_i|\leq k, k<0. \end{aligned}\right. $

证  滑模面上运动时$ s(t) = 0, \dot s(t) = 0\Rightarrow \dot {e_i} = -k_1e_i-k_2|e_i|^\sigma {\hbox{sat}}(e_i) $, 构造$ V(t) = \sum \limits^3_{i = 1}|e_i|, $得到

$ \begin{aligned}\dot{V}(t) & = \sum \limits^3_{i = 1}{\hbox{sgn}}(e_i)\dot e_i(t)\\ & = \sum \limits^3_{i = 1}{\hbox{sgn}}(e_i)(-k_1e_i-k_2|e_i|^\sigma {\hbox{sat}}(e_i))\\ & = \sum \limits^3_{i = 1}-k_1|e_i|-k_2|e_i|^\sigma {\hbox{sat}}(e_i){\hbox{sgn}}(e_i). \end{aligned} $

由于

$ \begin{eqnarray*} {\hbox{sat}}(e_i){\hbox{sgn}}(e_i)& = &{\hbox{sgn}}(e_i/k){\hbox{sgn}}(e_i) = 1 (|e_i|>k), \\ {\hbox{sat}}(e_i){\hbox{sgn}}(e_i)& = &e_i/k {\hbox{sgn}}(e_i) = |e_i|/k>0 (|e_i|\leq k), \end{eqnarray*} $

从而$ \dot{V}(t)\leq\sum \limits^3_{i = 1}-k_1|e_i|<0 $, 所以滑模面具有稳定性.

定理 3  若满足假设1, 构造上述滑模函数和控制律, Victor-Carmen混沌系统的主从系统(2.1)与(2.2)是有限时间滑模同步的.

证  当不在切换面上运动时, 根据定理1可知滑动模态系统可以被驱动到滑模面, 即滑模面具有可达性; 在切换面上时, 根据定理2可得系统渐近稳定, 从而$ e_i\rightarrow 0 $, 从而具有稳定性.所以(2.1)与(2.2)就取得有限时间滑模同步.

$ \alpha = 50, \beta = 20, \gamma = 4.1, c = 5, b = 9, (x_1(0), x_2(0), x_3(0)) = (2.2, 6.5, 2.5), $

$ k_1 = 8, k_2 = 2, \sigma = 0.3, k_3 = 4, k_4 = 1, k_5 = 0.1, m_i = 1.2, n_i = 1.2\; (i = 1, 2, 3), $

$ \begin{eqnarray*} \Delta f_1(y)+d_1(t)& = &0.1y_1\sin t +0.1\cos t, \\ \Delta f_2(y)+d_2(t)& = &-0.1y_2\cos t +0.1\cos t, \\ \Delta f_3(y)+d_3(t)& = &0.1y_3\sin t +0.1\cos 2t. \end{eqnarray*} $

系统参数选取如上, 定理中系统的误差曲线如图 2所示, 图中可以看到, 初始时刻系统误差距离原点较远且相差较大, 随时间推移系统误差渐趋一致, 逐渐趋近于坐标原点, 表明系统取得了同步.图中看出一段时间以后系统达到混沌同步, 当时间$ T $时刻以后, Victor-Carmen混沌系统的主从系统(2.1)与(2.2)达到滑模同步的.时间$ T $可由公式$ T = \frac 1 r \ln(\sqrt{2}\sum \limits^3_{i = 1}|s_i(0)|+1) $来推算.它给出了时刻$ T $大小估计.若选取为传统的等速趋近律$ \dot{V}\leq -\sum \limits^3_{i = 1}k_3|s_i|^2 = -2k_3V, $求得$ T = \frac 1 {2k_3} \ln( \frac 1 2\sum \limits^3_{i = 1}|s_i(0)|^2 ), $从而需要更长的时间才能趋于同步.

基于新型滑模方法研究Victor-Carmen混沌系统的有限时间同步, 取得Victor-Carmen混沌系统的主从系统达到有限时间滑模同步的充分条件, 从数学角度给出了严格证明和逻辑推理, 通过数值仿真检验方法的合理性和正确性.