﻿ 基于分数阶PI<sup>α</sup>D<sup>β</sup>的船用永磁同步电机控制策略研究
 舰船科学技术  2018, Vol. 40 Issue (4): 94-98 PDF

Research on control strategy of marine permanent magnet synchronous motor based on fractional order PIαDβ
PANG Ke-wang, ZHANG Ming, GUO Chang-xing
School of Electronic Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China
Abstract: Marine permanent magnet synchronous motor is a kind of nonlinear, strong coupling and multi variable complex system. It is difficult to achieve the desired effect when using conventional PID to control the speed. In order to improve the performance of the speed control system of marine permanent magnet synchronous motor, a new speed controller-radial basis function (RBF) neural network fractional PIαDβ controller is designed. The parameters of the controller are optimized online by the self-learning and self-training functions of radial basis function neural network, so that the controller can have the fast adaptability and good control performance in an unknown system. The designed controller is applied to the speed loop of marine permanent magnet synchronous motor, and the simulation experiment is carried out under the conditions of high speed and large load disturbance. The results show that the motor control system with RBF neural network fractional order PIαDβ controller has good dynamic performance and strong disturbance rejection ability.
Key words: permanent magnet synchronous motor     RBF neural network     fractional order PIαDβ controller     speed control
0 引　言

1 分数阶PIαDβ控制器 1.1 分数阶微积分的定义

 $_aD_t^\alpha = \left\{ \begin{array}{l}{{\rm d}^\alpha }/{\rm d}{t^\alpha },\;\;\;\;\;\;\;\;R(\alpha ) > 0\text{，}\\1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;R(\alpha ) = 0\text{，}\\\int_a^t {{{({\rm d}\tau )}^{ - \alpha }},\;\;\;R(\alpha ) < 0}\text{。} \end{array} \right.$

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

Riemann-Liouville分数阶积分的定义为：

 $\begin{array}{l}_aI_t^\gamma f(t) = \displaystyle\frac{1}{{\Gamma ( - \gamma )}}\int_a^t {\frac{{f(\tau )}}{{{{\left( {t - \tau } \right)}^{\gamma + 1}}}}{\rm d}\tau } \text{，}\\[12pt]\Gamma (z) = \displaystyle\int_0^\infty {{e^{ - t}}} {t^{z - 1}}{\rm d}t\text{。}\end{array}$ (2)

1.2 分数阶PIαDβ控制器及其离散

 $u(t) = {K_p}e(t) + {K_i}\int_0^t {e(\tau )} {\rm d}(\tau ) + {K_d}\frac{{{\rm d}e(t)}}{{{\rm d}t}}\text{，}$ (3)

 $u(t) = {K_p}e(t) + {K_{i0}}I_t^{ - \alpha }e(t) + {K_{d0}}D_t^\beta e(t)\text{。}$ (4)

 $u(k) = u(k - 1) + {K_p}a + {K_i}\frac{b}{{\Gamma (\alpha )}} + {K_d}\frac{c}{{\Gamma (1 - \beta )}}\text{，}$ (5)
 $a = e(k) - e(k - 1)\text{，}$ (6)
 $b = \sum\limits_{i = 1}^k {\frac{{e(i)}}{{{{(k + 1 - i)}^{1 - \alpha }}}} - \sum\limits_{i = 1}^{k - 1} {\frac{{e(i)}}{{{{(k - i)}^{1 - \alpha }}}}} } \text{，}$ (7)
 $c = \sum\limits_{i = 1}^k {\frac{{e(i)}}{{{{(k + 1 - i)}^\beta }}} - 2\sum\limits_{i = 1}^{k - 1} {\frac{{e(i)}}{{{{(k - i)}^\beta }}} + \sum\limits_{i = 1}^{k - 2} {\frac{{e(i)}}{{{{(k - 1 - i)}^\beta }}}} } } \text{。}$ (8)

2 RBF神经网络分数阶PIαDβ控制器 2.1 RBF神经网络的结构

RBF神经网络一般为包含输入层、隐层和输出层的3层网络结构，隐层输入到输出的映射是非线性的，而输出层的输入与输出为线性关系[9]。网络的结构如图1所示。

 图 1 径向基神经网络的结构 Fig. 1 Structure of radial basis function neural network

 ${h_i} = \exp \left( { - \frac{1}{{2{\sigma _i}^2}}{{\left\| {X - {C_i}} \right\|}^2}} \right)\text{。}$

${C_i} = {[{c_{i1}},{c_{i2}}, \ldots {c_{ij}} \ldots {c_{in}}]^{\rm{T}}}$ 为第i个隐层节点的中心矢量，σi为第i个隐层节点的基宽度参数。ym为神经网络的输出，与网络的输出权值向量 $\omega = [{\omega _1},{\rm{ }}{\omega _2} \ldots {\omega _i} \ldots {\omega _m}]$ hi有关，其表达式为：

 $ym = {\omega _1}{h_1} + {\omega _2}{h_2} + \cdot \cdot \cdot {\omega _m}{h_m}\text{。}$ (9)
2.2 基于RBF神经网络的分数阶PIαDβ控制器参数整定

 图 2 RBF神经网络整定控制器参数的结构 Fig. 2 Structure of RBF neural network tuning controller parameters

 $J = \frac{1}{2}{(yout(k) - ym(k))^2}\text{。}$

 $\begin{array}{l}{\omega _i}(k) = {\omega _i}(k - 1) + \eta (yout(k) - ym(k)){h_i}+\\\;\;\;\;\;\;\;\;\;\;\; a\left( {{\omega _i}(k - 1) - {\omega _i}(k - 2)} \right)\text{，}\\{\sigma _i}(k) = {\sigma _i}(k - 1) + \eta \Delta {\sigma _i} + a\left( {{\sigma _i}(k - 1) - {\sigma _i}(k - 2)} \right)\text{，}\\{c_{ij}}(k) = {c_{ij}}(k - 1) + \eta \Delta {c_{ij}} + a\left( {{c_{ij}}(k - 1) - {c_{ij}}(k - 2)} \right)\text{。}\end{array}$

 $\begin{array}{l}\Delta {\sigma _i} = (yout(k) - ym(k)){\omega _i}{h_i}\frac{{{{\left\| {X - {C_i}} \right\|}^2}}}{{{\sigma _i}^3}}\text{，}\\\Delta {c_{ij}} = (yout(k) - ym(k)){\omega _i}\frac{{{x_i} - {c_{ij}}}}{{{\sigma _i}^2}}\text{。}\end{array}$

 $\begin{array}{l}E(t) = \displaystyle\frac{1}{2}error{(k)^2}\text{，}\\error(k) = rin(k) - yout(k)\text{。}\end{array}$ (10)

KpKiKd三个参数通过采用梯度下降法进行调节，结合式（5）、式（6）、式（7）、式（8）和式（10）可得各参数变化量的表达式为：

 \begin{align}\Delta {K_p} =& - {\eta _p}\frac{{\partial E}}{{\partial {K_p}}} = - {\eta _p}\frac{{\partial E}}{{\partial yout}}\frac{{\partial yout}}{{\partial u}}\frac{{\partial u}}{{\partial {K_p}}}=\\& {\eta _p}error(t)\frac{{\partial yout}}{{\partial u}}a\text{，}\\\Delta {K_i} =& - {\eta _i}\frac{{\partial E}}{{\partial {K_i}}} = - {\eta _i}\frac{{\partial E}}{{\partial yout}}\frac{{\partial yout}}{{\partial u}}\frac{{\partial u}}{{\partial {K_i}}}=\\& {\eta _i}error(t)\frac{{\partial yout}}{{\partial u}}\frac{b}{{\Gamma (\alpha )}}\text{，}\\\Delta {K_d} = & - {\eta _d}\frac{{\partial E}}{{\partial {K_d}}} = - {\eta _d}\frac{{\partial E}}{{\partial yout}}\frac{{\partial yout}}{{\partial u}}\frac{{\partial u}}{{\partial {K_d}}}=\\&{\eta _d}error(t)\frac{{\partial yout}}{{\partial u}}\frac{c}{{\Gamma (1 - \beta )}}\text{。}\end{align}

 $\frac{{\partial yout}}{{\partial u}} \approx \sum\limits_{i = 1}^6 {{w_i}{h_i}} \frac{{{c_{i1}} - u(k)}}{{\sigma _i^2}}\text{。}$
3 永磁同步电机的数学模型

 $\begin{split}&\left[ {\begin{array}{*{20}{l}}{{U_A}}\\{{U_B}}\\{{U_C}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{R_A} + P{L_A}}&{P{M_{AB}}}&{P{M_{AC}}}\\{P{M_{BA}}}&{{R_B} + P{L_B}}&{P{M_{BC}}}\\{P{M_{CA}}}&{P{M_{CB}}}&{{R_C} + P{L_C}}\end{array}} \right]\\&\left[ {\begin{array}{*{20}{l}}{{i_A}}\\{{i_B}}\\{{i_C}}\end{array}} \right] - {\omega _r}{\varphi _r}\left[ {\begin{array}{*{20}{l}}{\sin \theta }\\{\sin \left(\theta - \frac{2}{3}\pi \right)}\\[2pt]{\sin \left(\theta + \frac{2}{3}\pi \right)}\end{array}} \right]\text{。}\end{split}$ (11)

 $\left[ \begin{array}{l}{U_d}\\{U_q}\end{array} \right] = \left[ {\begin{array}{*{20}{c}}{{R_s} + p{L_d}} & {{\rm{ - }}{{\rm{L}}_q}{\omega _r}}\\{{L_d}{\omega _r}} & {{{\rm{R}}_s} + p{L_q}}\end{array}} \right]\left[ \begin{array}{l}{I_d}\\{I_q}\end{array} \right] + {\omega _r}{\psi _r}\left[ \begin{array}{l}0\\1\end{array} \right]\text{。}$

dq0坐标系下的电磁转矩方程为：

 ${T_e} = \frac{3}{2}{n_p}\left( {{\psi _r}{I_q} + ({L_d} - {L_q}){I_d}{I_q}} \right)\text{，}$

 ${T_e} - {T_L} - B{\omega _m} = J\frac{{{\rm d}{\omega _m}}}{{{\rm d}t}}\text{。}$

4 系统仿真

 图 3 船用永磁同步电机控制结构 Fig. 3 Control structure of marine permanent magnet synchronous motor

 图 4 船用永磁同步电机速度控制仿真模型 Fig. 4 Simulation model of speed control for marine permanent magnet synchronous motor

1）电机空载启动，给定转速为1 000 r/min，在t=0.5 s时刻突加负载力矩TL=30 N·m，得到的转速变化曲线和转速误差变化曲线如图5图6所示。

 图 5 电机转速的变化曲线 Fig. 5 The curves of motor speed

 图 6 电机转速误差的变化曲线 Fig. 6 The curves of motor speed error

2）在给系统施加30 N·m负载力矩的情况下，给定速度为正弦波，周期为0.5 s，峰值为1 000 r/min。在系统运行1 s后得到的转速变化曲线和跟随误差变化曲线如图7图8所示。

 图 7 跟踪正弦波时电机转速的变化曲线 Fig. 7 The curves of the motor speed tracking sine wave

 图 8 跟踪正弦波时电机转速误差的变化曲线 Fig. 8 The error curves of the motor speed tracking sine wave

5 结　语

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