﻿ 水下潜器姿态角的分数阶PID控制研究
 舰船科学技术  2016, Vol. 38 Issue (11): 129-132 PDF

Research on fractional-order PID control for underwater vehicle attitude angle
ZHAO Jian, BAI Chun-jiang, ZHANG Wen-jun
Navigation College of Dalian Maritime University, Dalian 116026, China
Abstract: Aiming at the stability problem of the control of underwater vehicle longitudinal attitude angel, a Fractional-Order Proportional Integral Derivative (FOPID) controller is presented for underwater vehicle attitude angel control system. While designing the FOPID controller, the integrated product of time and absolute error standard (ITAE) is adopted to fast optimize the parameters of FOPID controller. Finally, the transfer function of underwater vehicle is chosen as the subject investigated, and simulation experiments of FOPID and classical PID controllers are carried out. By comparing the control performance of the two different controllers, the FOPID controller presented has more satisfactory performance and stronger robustness.
Key words: autonomous underwater vehicle     fractional-order proportional integral derivative control     robustness
0 引言

PIλ Dμ 控制器的概念是由Podlubny在1997年时提出的[10]。PIλ Dμ 控制器除了兼具常规PID控制器的优点外，由于分数阶微积分自身的特性，分数阶控制器还具有许多整数阶控制器无法实现的优越性，其微分阶次μ和积分阶次λ可以进行实数范围内的任意设置，这使得PIλ Dμ 控制器具有比常规PID控制器更灵活的控制结构。近年来，一些研究者已经将PIλ Dμ 控制方法应用于航海领域，如船舶航向控制、船舶横摇控制和船舶电站柴油机调速系统控制[11 -13]，并取得了较好的控制效果。

1 分数阶微积分及分数阶PID控制器 1.1 分数阶微积分

 ${}_aD_t^\alpha = \left\{ {\begin{array}{*{20}{c}}\!\!\!\!\!\!\! \!\!\!\!\!\!{\displaystyle\frac{{{{\rm{d}}^\alpha }}}{{{\rm{d}}{t^\alpha }}},{\mathop{ Re}\nolimits} \left( \alpha \right) > 0}\text{，}\\[7pt] \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{1,{\mathop{ Re}\nolimits} \left( \alpha \right) = 0}\text{，}\\[5pt] {\int_a^t {{{\left( {{\rm{d}}\tau } \right)}^{ - \alpha }},{\mathop{ Re}\nolimits} \left( \alpha \right) < 0} }\text{。} \end{array}} \right.$ (1)

1）Cauchy的表达式为：

 ${}_aD_t^\alpha f\left( t \right) = \frac{{\Gamma \left( {\alpha + 1} \right)}}{{2{{\pi j}}}}\int_C {\frac{{f\left( \tau \right)}}{{{{\left( {\tau - t} \right)}^{\alpha + 1}}}}{\rm{d}}\tau } \text{，}$ (2)

2）Grunwald-Letnikov的表达式为：

 $_a^{GL}D_t^\alpha f\left( t \right) = \mathop {\lim }\limits_{h \to 0} \frac{1}{{{h^\alpha }}}\sum\limits_{j = 0}^{\left[ {\left( {t - a} \right)/h} \right]} {\left( { - 1} \right)^j}\left( {\begin{array}{*{20}{c}} \alpha \\ j \end{array}} \right)f\left( {t - jh} \right),$ (3)

3）Riemann-Liouville的表达式为：

 ${}_a^{RL}D_t^\alpha f\left( t \right) = \frac{1}{{\Gamma \left( {n - \alpha } \right)}}\frac{{{{\rm{d}}^n}}}{{{\rm{d}}{t^n}}}\left[ {\int_a^t {\frac{{f\left( \tau \right)}}{{{{\left( {t - \tau } \right)}^{\alpha - n + 1}}}}} {\rm{d}}\tau } \right]\text{，}$ (4)

4）Caputo的表达式为：

 ${}_0D_t^\alpha f\left( t \right) = \frac{1}{{\Gamma \left( { - \alpha } \right)}}\int_0^t {\frac{{f\left( \tau \right)}}{{{{\left( {t - \tau } \right)}^{1 + \alpha }}}}} {\rm{d}}\tau \text{，}$ (5)

1.2 分数阶PID控制器

PIλ Dμ 控制器的传递函数模型为：

 ${G_c}\left( s \right) = {k_p} + \frac{{{k_i}}}{{{s^\lambda }}} + {k_d}{s^\mu }\text{。}$ (6)

PIλ Dμ 控制器的示意图如图 1所示。其中，横轴为PIλ Dμ 控制器的积分阶次λ，纵轴为PIλ Dμ 控制器的积分阶次μ。常规的PI控制器、PD控制器和PID控制器均为PIλ Dμ 控制器平面内的一个点。

 图 1 分数阶PIλ Dμ 控制器示意图 Fig. 1 Diagram of fractional-order PIλ Dμ controller

2 水下潜器的分数阶PID控制器设计 2.1 水下潜器的传递函数模型

 $\begin{array}{l} {P_1} =\displaystyle \frac{{{{13}}{{.2 s + 73}}{{.5}}}}{{{\rm{0}}{{.015 }}{{{s}}^{\rm{4}}}{\rm{ + 1}}{\rm{.1879 }}{{{s}}^{\rm{3}}}{\rm{ + 13}}{\rm{.0786 }}{{{s}}^{\rm{2}}}{\rm{ + 36}}{{.9526 s}}}}\text{，}\\[10pt] {P_2} =\displaystyle \frac{{{{ 80}}{{.4 s + 687}}{\rm{.6}}}}{{{\rm{0}}{\rm{.015 }}{{{s}}^{\rm{4}}}{\rm{ + 1}}{\rm{.3809}}{{{s}}^{\rm{3}}}{\rm{ + 27}}{\rm{.2566 }}{{{s}}^{\rm{2}}}{\rm{ + 124}}{{.1385 s}}}}\text{，}\\[10pt] {P_3} =\displaystyle \frac{{{\rm{152}}{{.7 s + 1692}}{\rm{.8}}}}{{{\rm{0}}{\rm{.015 }}{{{s}}^{\rm{4}}}{\rm{ + 1}}{\rm{.5145}}{{{s}}^{\rm{3}}}{\rm{ + 37}}{\rm{.5676 }}{{{s}}^{\rm{2}}}{\rm{ + 218}}{{.031 s}}}}\text{。} \end{array}$ (7)
2.2 水下潜器的分数阶PID控制器

 ${J_{\rm{ITAE}}} = \int_0^\infty {t\left| {e\left( t \right)} \right|} {\rm{d}}t \text{，}$ (8)

 ${G_1} = 0.6 + \frac{{1.6}}{{{s^{0.1}}}} + 0.52{s^{0.64}} \text{，}$ (9)

 ${G_2} = \frac{{{\rm{39}}{{.23(s + 16}}{{.13)(s + 0}}{\rm{.2)}}}}{{{{s(s + 125)}}}} \text{。}$ (10)
3 仿真实验

 图 2 P2模型的仿真结果曲线 Fig. 2 Simulation result curve of modelP2

 图 3 P1模型的仿真结果曲线 Fig. 3 Simulation result curve of modelP1

 图 4 P3模型的仿真结果曲线 Fig. 4 Simulation result curve of modelP3

4 结语

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