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 浙江大学学报(理学版)  2017, Vol. 44 Issue (3): 302-306  DOI:10.3785/j.issn.1008-9497.2017.03.010 0

### 引用本文 [复制中英文]

[复制中文]
MAO Beixing, MENG Xiaoling. Finite-time synchronization of fractional-order multi-scroll systems with dead-zone input[J]. Journal of Zhejiang University(Science Edition), 2017, 44(3): 302-306. DOI: 10.3785/j.issn.1008-9497.2017.03.010.
[复制英文]

### 文章历史

Finite-time synchronization of fractional-order multi-scroll systems with dead-zone input
MAO Beixing , MENG Xiaoling
College of Science, Zhengzhou University of Aeronautics, Zhengzhou 450015, China
Abstract: The problem of finite-time synchronization of fractional-order multi-scroll systems with dead-zone input is studied. The sufficient conditions for the fractional order systems to get finite-time synchronization are obtained based on fractional order calculus theory. The research conclusion illustrates that fractional-order multi-scroll systems is finite-time chaos synchronization under proper conditions.
Key words: fractional order    multi-scroll systems    sliding model    chaos synchronization
0 引言

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

 $\left\{ \begin{array}{l} D_t^q{x_1} = {x_2},\\ D_t^q{x_2} = {x_3},\\ D_t^q{x_3} = - \alpha {x_3} - \beta {x_2} + f\left( {{x_1}} \right), \end{array} \right.$ (1)

 $\left\{ \begin{array}{l} D_t^q{y_1} = {y_2} + \Delta {f_1}\left( y \right) + {d_1}\left( t \right) + {h_1}\left( {{u_1}\left( t \right)} \right),\\ D_t^q{y_2} = {y_3} + \Delta {f_2}\left( y \right) + {d_2}\left( t \right) + {h_2}\left( {{u_2}\left( t \right)} \right),\\ D_t^q{y_3} = - \alpha {y_3} - \beta {y_2} + f\left( {{y_1}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\Delta {f_3}\left( y \right) + {d_3}\left( t \right) + {h_3}\left( {{u_3}\left( t \right)} \right), \end{array} \right.$ (2)

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

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

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

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

 $\left\{ \begin{array}{l} D_t^q{e_1} = {e_2} + \Delta {f_1}\left( y \right) + {d_1}\left( t \right) + {h_1}\left( {{u_1}\left( t \right)} \right),\\ D_t^q{e_2} = {e_3} + \Delta {f_2}\left( y \right) + {d_2}\left( t \right) + {h_2}\left( {{u_2}\left( t \right)} \right),\\ D_t^q{e_3} = - \alpha {e_3} - \beta {e_2} + f\left( {{y_1}} \right) - f\left( {{x_1}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\Delta {f_3}\left( y \right) + {d_3}\left( t \right) + {h_3}\left( {{u_3}\left( t \right)} \right). \end{array} \right.$ (3)

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

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

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

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

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

 $\begin{array}{l} D_t^q{e_i}\left( t \right) = - D_t^{q - 1}\left[ {\frac{\lambda }{\mu }{e_i} + {{\left( {{e^{\rm{T}}}e} \right)}^{ - \mu }}{e_i} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left( {D_t^{q - 1}{e_i}} \right)\frac{{{t^{ - \left( {1 - q} \right) - 1}}}}{{\Gamma \left( {q - 1} \right)}}} \right]. \end{array}$ (4)

 ${u_i}\left( t \right) = \left\{ \begin{array}{l} - {\gamma _i}{\zeta _i}{\mathop{\rm sgn}} {s_i} + {u_{ - i}},\;\;\;\;{s_i} > 0,\\ 0,\;\;\;\;\;{s_i} = 0,\\ - {\gamma _i}{\zeta _i}{\mathop{\rm sgn}} {s_i} + {u_{ + i}},\;\;\;\;{s_i} < 0, \end{array} \right.$ (5)

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

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

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

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

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

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

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

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

 $V_1^\mu \left( t \right) = {{\rm{e}}^{ - 2\lambda t}}\left( {\frac{\mu }{\lambda } + V_1^\mu \left( 0 \right)} \right) - \frac{\mu }{\lambda }.$
 $如果\;\;\;\;\;\;V_1^\mu \left( {{T_1}} \right) \equiv 0 \Rightarrow {{\rm{e}}^{2\lambda {T_1}}} = 1 + \frac{\lambda }{\mu }V_1^\mu \left( 0 \right),$
 $则有\;{T_1} = \frac{1}{{2\lambda }}\left\{ {\ln 1 + \frac{\lambda }{\mu }\left[ {\sum\limits_{i = 1}^3 {{{\left( {e_i^{\rm{T}}\left( 0 \right){e_i}\left( 0 \right)} \right)}^\mu }} } \right]} \right\}.$

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

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

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

 $\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}$

2 数值仿真

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

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,

 $\begin{array}{l} \Delta {f_1}\left( {{y_1},{y_2},{y_3}} \right) = \cos \left( {2\pi {y_2}} \right),\\ \Delta {f_2}\left( {{y_1},{y_2},{y_3}} \right) = 0.5\cos \left( {2\pi {y_3}} \right),\\ \Delta {f_3}\left( {{y_1},{y_2},{y_3}} \right) = 0.3\cos \left( {2\pi {y_2}} \right), \end{array}$
 $\begin{array}{l} {h_i}\left( {{u_i}\left( t \right)} \right) = \\ \;\;\;\;\;\;\;\left\{ \begin{array}{l} \left( {{u_i}\left( t \right) - 1} \right)\left( {0.8 - 0.1\cos \left( {{u_i}\left( t \right)} \right),\;\;\;\;{u_i}\left( t \right) > 1,} \right.\\ 0,\;\;\;\; - 1 \le {u_i}\left( t \right) \le 1,\\ \left( {{u_i}\left( t \right) + 1} \right)\left( {1 - 0.5\cos } \right)\left( {{u_i}\left( t \right)} \right),\;\;\;{u_i}\left( t \right) < - 1, \end{array} \right. \end{array}$
 $\begin{array}{l} \;\;\;\;\;\;\;{d_1}\left( t \right) = 0.2\cos \left( t \right),\;\;\;\;{d_2}\left( t \right) = 0.6{\rm{sin}}\left( t \right),\\ \;\;\;\;\;\;{d_3}\left( t \right) = \cos \left( {3t} \right),\;\;\;\;{\beta _{ + i}} = 0.4,\;\;\;\;\;{\beta _{ - i}} = 0.5, \end{array}$
 $\begin{array}{l} \;\;\;\;\;\;{\beta _i} = 0.4,\;\;\;\;{\gamma _i} = 2.5x\left( 0 \right) = {\left( {1, - 2, - 2} \right)^{\rm{T}}},\\ \;\;\;\;\;\;y\left( 0 \right) = {\left( {1,1, - 1} \right)^{\mathit{T}}},\;\;\;\lambda = 1,\;\;\;\;\mu = 0.5. \end{array}$

 图 1 无控制的主从系统状态 Fig. 1 State of master-slave with no control
 图 2 有控制的主从系统状态 Fig. 2 State of master-slave with control
 图 3 系统误差曲线 Fig. 3 The system errors
3 结论


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# The mean and variance of the inverse of a normal distribution [duplicate]

I would want to ask if I have a random variable $A \sim N(b,c)$ then what is the distribution of the inverse of $A$?

As stated in wikipedia and this math.stackexchange question, $1 / A$ has a bimodal distribution.
However, the mean and the variance and higher order moments of $1 / A$ are not defined (as shown here for the mean).
• (+1) Wikipedia asserts bimodality only for the case $b=0$ and $c=1$. Clearly the value of $c$ will not affect this characteristic, but in principle $b$ can. Nevertheless, $1/A$ will always be bimodal, because it has a density of $0$ at $0$ and both the positive and negative parts are unimodal (as becomes obvious when you consider the distribution of their logarithms). – whuber Nov 23 '15 at 21:46

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