﻿ 船舶推进感应电机无速度传感器技术研究
 舰船科学技术  2017, Vol. 39 Issue (3): 69-73 PDF

Speed-sensorless research of the ship drive induction motor
HE Xin-xin, YU Meng-hong
School of Electronics and Information Jiangsu University of Science and Technology, Zhenjiang 212003, China
Abstract: In order to achieve the real-time speed of Ship drive Induction motor ,speed sensorless technology based on the full order state observers is applied to the induction motor control system . Build the full-order state observer with stator current and rotor flux as state variables, then set the full order state observer as the adjustable model, drive motor as reference model, use the model reference adaptive method to identify the motor speed in time. In view of the instability of the speed estimation system at low speed regional, try to set design criteria for feedback gain matrix that can guarantee the system stability in low speed regional. The simulation is carried out under the Matlab/Simulink simulation environment. Experimental results show that the design of the feedback gain matrix can identify the motor speed very well in the low speed regional.
Key words: full-order state observers     MRAS     ship drive motor     feedback gain matrix
0 引　言

1 推进电机全阶观测器的设计 1.1 电机空间状态方程建立

 $\left\{ {\begin{array}{*{20}{l}}{\mathop x\limits^ \bullet = Ax + Bu = \left[ {\begin{array}{*{20}{c}}{{A_{11}}} & {{A_{12}}}\\{{A_{21}}} & {{A_{22}}}\end{array}} \right]x + \left[ {\begin{array}{*{20}{c}}{{B_1}}\\0\end{array}} \right]u} \text{，}\\[10pt]{y = {i_s} = Cx = \left[ {\begin{array}{*{20}{c}}I & 0 \end{array}} \right]x} \text{。}\end{array}}\right.$ (1)

 $x = {\left[ {\begin{array}{*{20}{c}}{{i_s}} & {{\psi _r}}\end{array}} \right]^{\rm T}} = {\left[ {\begin{array}{*{20}{c}}{{i_{sa}}} & {{i_{sb}}} & {{\psi _{ra}}} & {{\psi _{rb}}}\end{array}} \right]^{\rm T}} \text{；}$ (2)

 $u = {\left[ {\begin{array}{*{20}{c}}{{u_s}} & 0\end{array}} \right]^{\rm T}} = {\left[ {\begin{array}{*{20}{c}}{{u_{sa}}} & {{u_{sb}}} & 0 & 0\end{array}} \right]^{\rm T}} \text{，}$ (3)
 $\begin{array}{l}\!\!\!\!\!\!\displaystyle{A_{11}} = - \left( {\frac{{{R_s}}}{{\sigma {L_s}}} + \frac{{1 - \sigma }}{{\sigma {\tau _r}}}} \right)I\text{，} \displaystyle {A_{12}} = \frac{{{L_m}}}{{\sigma {L_s}{L_r}}}\left( {\frac{1}{{{\tau _r}}}I - \omega J} \right)\text{，}\!\!\!\!\!\!\\\!\!\!\!\!\!\displaystyle {A_{21}} = \frac{{{L_m}}}{{{\tau _r}}}I\text{，} \displaystyle {A_{22}} = - \frac{1}{{{\tau _r}}}I + \omega J\text{，}\\\!\!\!\!\!\!\displaystyle {B_{\rm{1}}} = \frac{{\rm{1}}}{{\sigma {L_s}}}I\text{，}\\\!\!\!\!\!\!I = \left[ {\begin{array}{*{20}{c}}1 & 0\\0 & 1\end{array}} \right]\text{，} J = \left[ {\begin{array}{*{20}{c}}0 & { - 1}\\1 & 0 \end{array}} \right] \text{。}\end{array}$

1.2 电机空间状态方程的能控能观性判定

 ${P_0}\left( t \right) = B\left( t \right) = {\left[ {\begin{array}{*{20}{l}}\displaystyle {\frac{1}{{\sigma {L_s}}}} & 0 & 0 & 0\\0 & \displaystyle {\frac{1}{{\sigma {L_s}}}} & 0 & 0\end{array}} \right]^{\rm T}} \text{，}$ (4)
 \begin{aligned}\displaystyle {P_1}\left( t \right) & = - A\left( t \right)B\left( t \right) + \frac{\rm d}{{{\rm d}t}}B\left( t \right)\\& = \left[ {\begin{array}{*{20}{l}}{ \displaystyle - \frac{1}{{\sigma {L_s}}}\left( {\frac{{{R_s}}}{{\sigma {L_s}}} + \frac{{L_m^2}}{{\sigma {L_s}{L_r}{\tau _r}}}} \right) + \frac{\rm d}{{{\rm d}t}}\left( {\frac{1}{{\sigma {L_s}}}} \right)} & 0\\0 \quad \quad \displaystyle { - \frac{1}{{\sigma {L_s}}}\left( {\frac{{{R_s}}}{{\sigma {L_s}}} + \frac{{L_m^2}}{{\sigma {L_s}{L_r}{\tau _r}}}} \right) + \frac{\rm d}{{{\rm d}t}}\left( {\frac{1}{{\sigma {L_s}}}} \right)}\\\displaystyle {\frac{{{L_m}}}{{{\tau _r}}} \times \frac{1}{{\sigma {L_s}}}} \quad \quad 0\\0 \quad \quad \displaystyle {\frac{{{L_m}}}{{{\tau _r}}} \times \frac{1}{{\sigma {L_s}}}}\end{array}} \right] \text{。}\end{aligned} (5)

$\det \left[ {{P_o}\left( t \right),{P_1}\left( t \right)} \right] = L_m^2{\left( {\sigma {L_s}} \right)^4} > 0$ rank[po (t),P1(t)]T=4。

 ${Q_0}\left( t \right) = C\left( t \right) = \left[ {\begin{array}{*{20}{l}}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\end{array}} \right] \text{，}$ (6)
 $\begin{array}{l}{Q_1}\left( t \right) = - C\left( t \right){Q_0}\left( t \right) + \displaystyle \frac{\rm d}{{{\rm d}t}}{Q_0}\left( t \right) = \\\left[\!\!\!\! {\begin{array}{*{20}{l}}{\displaystyle - \left( {\frac{{{R_s}}}{{{L_s}}} + \frac{{L_m^2}}{{\sigma {L_s}{L_r}{\tau _r}}}} \right)}\!\! &\!\! 0\!\! &\!\! 0\!\!\!\! &\!\!\!\!0\\0 & { -\displaystyle \left( {\frac{{{R_s}}}{{{L_s}}} + \frac{{L_m^2}}{{\sigma {L_s}{L_r}{\tau _r}}}} \right)} & 0\!\!\! &\!\!\! 0\end{array}} \!\!\!\!\right] \text{。}\end{array}$ (7)

$\displaystyle \det \left[ {{Q_0}\left( t \right), {Q_1}\left( t \right)} \right] = \left( {\frac{{{R_s}}}{{{L_s}}} + \frac{{L_m^2}}{{\sigma {L_s}{L_r}{\tau _r}}}} \right) > 0$ ，rank [Q0(t),Q1(t) ]T = 4。

1.3 全阶状态观测器的建立以及反馈矩阵的设计

 $\left\{ {\begin{array}{*{20}{l}}{\mathop {\widehat x}\limits^ = \widehat A\widehat x + Bu + G(\widehat y - y)}\text{，}\\[6pt]{\widehat y = C\widehat x}\text{。}\end{array}} \right.$ (8)

 ${ G} = - {\left[ {\begin{array}{*{20}{c}}{{g_1}} & { - {g_2}} & {{g_3}} & { - {g_4}}\\{{g_2}} & {{g_1}} & {{g_4}} & {{g_3}}\end{array}} \right]^{\rm T}}\text{。}$ (9)

 $\begin{split}\frac{\rm d}{{{\rm d}t}}& \left[ {\begin{array}{*{20}{c}}{{e_i}}\\{{e_\psi }}\end{array}} \right] \!=\!\\& ({ A} + { GC})\left[ {\begin{array}{*{20}{c}}{{e_i}}\\{{e_\psi }}\end{array}} \right] \!\!+\! \Delta { A}\left[ {\begin{array}{*{20}{c}}{{{\widehat i}_s}}\\{{{\widehat \psi }_r}}\end{array}} \right] \!\!=\!\\& ({ A} + { GC})\left[ {\begin{array}{*{20}{c}}{{e_i}}\\{{e_\psi }}\end{array}} \right] \!\!-\! { W} \text{。}\end{split}$ (10)

 $\Delta { A} = { A} - \widehat { A} = \left[ {\begin{array}{*{20}{c}}0 & {\frac{{\sigma {L_s}{L_r}}}{{{L_m}}}J}\\0 & { - J}\end{array}} \right](\widehat \omega - \omega ) = \Delta {{ A}_1}\Delta \omega \text{。}$ (11)

 $e = {[\begin{array}{*{20}{c}} {{e_i}} & {{e_\psi }} \end{array}]^{\rm T}}$ 。 (12)

 图 1 误差方程等价结构图 Fig. 1 Error equation of the equivalent structure

 $sI \cdot {e_i} = ({A_{11}} + {G_1}){e_i} + {A_{12}}{e_\psi } - \Delta {\omega _r}\frac{{{L_m}}}{{\sigma {L_s}{L_r}}}J{\widehat \psi _r}\text{，}$ (13)
 $sI \cdot {e_\psi } = ({A_{21}} + {G_1}){e_i} + {A_{22}}{e_\psi } - \Delta {\omega _r}J{\widehat \psi _r}\text{。}$ (14)

 \begin{aligned}{e_i} = & \displaystyle s\frac{{{L_m}}}{{\sigma {L_s}{L_r}}}\left[ {sI - {A_{11}} - {G_1} - {A_{12}}{{\left( {sI - {A_{22}}} \right)}^{ - 1}}{{\left( {{G_2} + {A_{21}}} \right)}^{ - 1}}} \right]\times\\ & \displaystyle \left[ {\frac{{\sigma {L_s}{L_r}}}{{{L_m}}} \cdot \frac{{{A_{12}}}}{{{L_m}s}}{{\left( {sI - {A_{22}}} \right)}^{ - 1}} - \frac{I}{s}} \right]\Delta {\omega _r}J{\widehat \psi _r}=\\\begin{array}{*{20}{c}}\end{array} \displaystyle& s\frac{{{L_m}}}{{\sigma {L_s}{L_r}}}\left [ {{s^2}I - ({A_{11}} + {G_1} + {A_{12}})s - {A_{22}}} \right. \times\\& \displaystyle {\left. {\left( { - {G_1} \!-\! {A_{11}} \!+\! \frac{{{L_m}}}{{\sigma {L_s}{L_r}}}\frac{{ - {G_2} \!-\! {L_m}{A_{21}}}}{{\sigma {L_r}{L_s}/{L_m}}}} \right)} \right]^{ - 1}}\Delta {\omega _r}J{{\hat \psi }_r}\text{。}\end{aligned} (15)

 $\begin{array}{l}\begin{array}{*{20}{c}}{G\left( s \right) =\displaystyle \frac{{{e_i}}}{{\Delta {\omega _r}J{{\widehat \psi }_r}}}}\end{array}=\\[8pt]\begin{array}{*{20}{c}}{}\end{array} \begin{array}{l} \quad \quad \displaystyle s\frac{{{L_m}}}{{\sigma {L_s}{L_r}}}\left[ {{s^2}I \!-\! ({A_{11}} \!+\! {G_1} \!+\! {A_{12}})s \!- \!{A_{22}}} \right. \!\times\\[8pt] \quad \quad \displaystyle \left. {\left( { - {G_1} - {A_{11}} + \frac{{{L_m}}}{{\sigma {L_s}{L_r}}}\frac{{ - {G_2} - {L_m}{A_{21}}}}{{\sigma {L_r}{L_s}/{L_m}}}} \right)} \right]=\end{array}\\[8pt]\begin{array}{*{20}{c}}{}\end{array} \quad \quad \displaystyle s\frac{{{L_m}}}{{\sigma {L_s}{L_r}}}{\left[ {{s^2}I + (xI + yJ)s + mI + nJ} \right]^{ - 1}}\text{。}\end{array}$ (16)

 ${ G} = - {\left[ {\begin{array}{*{20}{c}}{{G_1}} & {{G_2}}\end{array}} \right]^{\rm T}}{\rm{ = }} - {\left[ {\begin{array}{*{20}{c}}{{g_1}} & { - {g_2}} & {{g_3}} & { - {g_4}}\\{{g_2}} & {{g_1}} & {{g_4}} & {{g_3}}\end{array}} \right]^{\rm T}}\text{，}$ (17)

 $\begin{array}{l}{G_1} = {g_1}I + {g_2}J, {G_2} = {g_3}I + {g_4}J \text{，}\\[8pt]\displaystyle m = \frac{{{R_r}}}{{{L_r}}}\left( { - {g_1} + \frac{{{R_s}}}{{\sigma {L_s}}} - \frac{{{g_3}}}{{\sigma {L_S}{L_r}/{L_m}}}} \right) + {\omega _r}( - {g_2} - \frac{{{g_4}}}{{\sigma {L_S}{L_r}/{L_m}}})\text{，}\\[10pt]\displaystyle n = \frac{{{R_r}}}{{{L_r}}}\left( { - {g_2} - \frac{{{g_4}}}{{\sigma {L_S}{L_r}/{L_m}}}} \right) - {\omega _r}( - {g_1} + \frac{{{R_s}}}{{\sigma {L_s}}} - \frac{{{g_3}}}{{\sigma {L_S}{L_r}/{L_m}}})\text{，}\\[10pt]\displaystyle x = - {g_1} + \frac{{{R_s}}}{{\sigma {L_s}}} + \frac{{{R_r}}}{{\sigma {L_r}}}\text{，}\\[8pt]y = - {g_2} - {\omega _r}\text{。}\end{array}$ (18)

 ${ G} (j\omega ) + {{ G}^*}(j\omega ) > 0 \text{，} \forall \omega > 0\text{，}$ (19)

 \begin{aligned}{ G}(j\omega ) + {{ G}^*}(j\omega ) =& \displaystyle\frac{{{L_m}}}{{\sigma {L_s}{L_r}}}{\left[ {{s^2}I + (xI + yJ)s + mI + nJ} \right]^{ - 1}} \times \\& {\left[ {{s^2}I + (xI + yJ)s + mI + nJ} \right]^*}^{ - 1} \cdot B(s) \>0 \text{。}\end{aligned} (20)

 ${ B}(s) = s{s^*}(s + {s^*})I + 2s{s^*}xI + (s + {s^*})mI + (s - {s^*})nJ \text{。}$ (21)

$s = j{\omega _e}$ 代入式 （15），ωe 为同步频率。式 （15） 展开并化简可得：

 $\begin{array}{l}{\rm B}(s) = 2\omega _e^2xI - 2j{\omega _e}nJ=\\[8pt]\begin{array}{*{20}{c}} \ \ \ \ \ \ & { \left[ {\begin{array}{*{20}{c}}{2\omega _e^2x} & {2j{\omega _e}n}\\{ - 2j{\omega _e}n} & {2\omega _e^2x}\end{array}} \right]}\end{array} > 0 \text{。}\end{array}$ (22)

 $\left\{ {\begin{array}{*{20}{l}}{2\omega _e^2x > 0}\text{，}\\{\omega _e^4{x^2} - \omega _e^2{n^2} > 0}\text{。}\end{array}} \right.$ (23)

 $\left\{ {\begin{array}{*{20}{l}}{x > 0}\text{，}\\{\omega _e^2 - \omega _c^2 > 0}\text{。}\end{array}} \right.$ (24)

 $\left\{ {\begin{array}{*{20}{l}}{{\omega _c} = \displaystyle - \frac{n}{x}}\text{，}\\{n \!=\! \displaystyle \frac{{{R_r}}}{{{L_r}}}\left( { - {g_2} - \frac{{{g_4}}}{{\sigma {L_S}{L_r}/{L_m}}}} \right) - {\omega _r}( - {g_1} \!+\! \frac{{{R_s}}}{{\sigma {L_s}}} \!-\! \frac{{{g_3}}}{{\sigma {L_S}{L_r}/{L_m}}})}\text{，}\\{x = \displaystyle - {g_1} + \frac{{{R_s}}}{{\sigma {L_s}}} + \frac{{{R_r}}}{{\sigma {L_r}}}}\text{。}\qquad\qquad\qquad\qquad\qquad\ \ (25)\end{array}} \right.$ (25)

 $\left\{ {\begin{array}{*{20}{l}}{{g_1} < \displaystyle \frac{{{R_s}}}{{\sigma {L_s}}} + \frac{{{R_r}}}{{\sigma {L_r}}}}\text{，}\qquad\qquad\qquad\qquad\qquad\qquad\ \ (26)\\[8pt]\displaystyle {\frac{{{R_r}}}{{{L_r}}}\left( { - {g_{\rm{_2}}} - \frac{{{g_{\rm{_4}}}}}{{\sigma {L_S}{L_r}/{L_m}}}} \right) = {\omega _r}( - {g_{\rm{_1}}} + \frac{{{R_s}}}{{\sigma {L_s}}} - \frac{{{g_{\rm{_3}}}}}{{\sigma {L_S}{L_r}/{L_m}}})}\text{。}\end{array}} \right.$ (26)

 $\left\{ {\begin{array}{*{20}{l}}\begin{array}{l}{g_{\rm{_1}}} = \displaystyle \frac{{{R_s}}}{{\sigma {L_s}}} + \left( {1 - \sigma } \right)\frac{{{R_r}}}{{\sigma {L_r}}} - k\frac{{{R_r}}}{{{L_r}}}\text{，}\\[6pt]{g_{\rm{_2}}} = \displaystyle - k{\widehat \omega _r}\text{，}\\[6pt]{g_{\rm{_3}}} = \displaystyle - {L_m}\frac{{{R_r}}}{{{L_r}}}\text{，}\end{array}\\[6pt] \ {{g_{\rm{_4}}} = 0}\text{。}\end{array}} \right.$ (27)
2 基于李雅普诺夫第二稳定性定理的转速自适应律推导

 $\begin{split}\frac{\rm d}{{{\rm d}t}}\left[ {\begin{array}{*{20}{c}}{{e_i}}\\{{e_\psi }}\end{array}} \right] = & ({ A} + { GC})\left[ {\begin{array}{*{20}{c}}{{e_i}}\\{{e_\psi }}\end{array}} \right] + \Delta { A}\left[ {\begin{array}{*{20}{c}}{\mathop {{i_s}}\limits^ \wedge }\\{\mathop {{\psi _r}}\limits^ \wedge }\end{array}} \right] = \\& ({ A} + { GC})\left[ {\begin{array}{*{20}{c}}{{e_i}}\\{{e_\psi }}\end{array}} \right] - { W} \text{。}\end{split}$ (28)

 $\Delta { A} = { A} - \mathop { A}\limits^ \wedge = \left[ {\begin{array}{*{20}{c}}0 & {\frac{{\sigma {L_s}{L_r}}}{{{L_m}}}J}\\0 & { - J}\end{array}} \right](\mathop \omega \limits^ \wedge - \omega ) = \Delta {{ A}_1}\Delta \omega \text{，}$ (29)

$e = {[\begin{array}{*{20}{c}} {{e_i}} & {{e_\psi }} \end{array}]^{\rm T}}$

V 函数正定性判断：e = 0， $\mathop \omega \limits^ \wedge = \omega$ 时，V（0）= 0

e ≠ 0， $\mathop \omega \limits^ \wedge \ne \omega$ 时， $V = {e_i}^2 + e_\Psi ^2 + {(\mathop \omega \limits^ \wedge - \omega )^2}/\lambda > 0$ 恒成立；

dV/dt 函数负半定判断：

 $\frac{\rm d}{{{\rm d}t}}V = \frac{\rm d}{{{\rm d}t}}({e^{\rm T}})e + {e^{rm T}}\frac{\rm d}{{{\rm d}t}}(e) + \frac{2}{\lambda }(\mathop \omega \limits^ \wedge - \omega )\frac{\rm d}{{{\rm d}t}}(\mathop \omega \limits^ \wedge - \omega )\text{，}$ (30)

 $\begin{split}\\[-12pt]\displaystyle \frac{\rm d}{{{\rm d}t}}V = & {e^{\rm T}}[(A + GC) + {(A + GC)^{\rm T}}])e + \\& 2\displaystyle \Delta {\omega _r}e_{is}^{\rm T}J\mathop {{\psi _r}}\limits^ \wedge /(\frac{{\sigma {L_s}{L_r}}}{{{L_m}}}) + \frac{2}{\lambda }\Delta {\omega _r}\frac{\rm d}{{{\rm d}t}}\mathop {{\omega _r}}\limits^ \wedge \text{。}\end{split}$ (31)

 $\mathop {{\omega _r}}\limits^ \wedge = {K_p}({e_{i\alpha }}{\mathop \psi \limits^ \wedge _{r\beta }} - {e_{i\beta }}{\mathop \psi \limits^ \wedge _{r\alpha }}) + {K_i}\int {({e_{i\alpha }}{{\mathop \psi \limits^ \wedge }_{r\beta }} - {e_{i\beta }}{{\mathop \psi \limits^ \wedge }_{r\alpha }})} {\rm d}t\text{。}$ (32)
3 系统仿真验证

${{ G}_{\rm{1}}} = {\left[ {\begin{array}{*{20}{l}} {{\rm{ - 121}}} & {{\rm{ - 120}}} & {{\rm{0}}{\rm{.6}}} & {\rm{0}}\\ {{\rm{120}}} & {{\rm{ - 121}}} & {\rm{0}} & {{\rm{0}}{\rm{.6}}} \end{array}} \right]^{\rm T}}$ 时，估算转速与实际转速仿真图，以及两者之间的局部误差放大图如图 2 和图 3 所示。

 图 2 选择增益矩阵 G 1 时，实际转速与估计转速图 Fig. 2 Speed error of actual speed and estimated speed（ G 1）

 图 3 选择增益矩阵 G 1 时，实际转速与估计转速误差局部放大图 Fig. 3 Partial enlargement of speed error （ G 1）

4 结　语

 图 4 选择增益矩阵 G 2 时，实际转速与估计转速图 Fig. 4 Speed error of actual speed and estimated speed（ G 2）

 图 5 选择增益矩阵 G 2 时，实际转速与估计转速误差局部放大图 Fig. 5 Partial enlargement of speed error （ G 2）

 [1] ﻿冯垛生, 曾月南. 无速度传感器矢量控制原理与实践[M]. 北京: 机械工业出版社 2006. [2] 李彬郎, 张斌. 全阶状态观测器在转速辨识系统中的应用改进[J]. 电气传动, 2015, 45(3): 7–11. LI Bin-lang, ZHANG Bin. Improvement of full-order state observer applied in revolution speed estimation system[J], Electric Drive, 2015 ,45(3):7-11. [3] 李永东. 交流电机数字控制系统[M]. 北京: 机械工业出版社, 2012. [4] KUBOTA H, SATO I, TAMURA Y, et al. Regenerating-mode low-speed operation of sensorless induction motor drive with adaptive observer[C]// IEEE Trans. Ind. Applicat, 2002, 38: 1081–1086. [5] YANG G, CHIN T H. Adaptive-speed identification scheme for a vector-controlled speed sensorless inverter induction motor drive[C]// IEEE Trans. Ind. Appl, 1993, 29(4): 820–825. [6] KUBOTA H, MATSUSE K, NAKANO T. DSP-based speed adaptive flux observer of induction motor[C]// IEEE Trans. Ind. Appl, 1993, 29(2): 344–348. [7] SUWANKAWIN S, SANGWONGWANICH S. Design strategy of an adaptive full-order observer for speed-sensorless induction-motor drives-track performance and stabilization[C]// IEEE Trans. Ind. Appl, 1993, 29(4): 820–825. [8] KUBOTA H, MATSUSE K. New adaptive flux observer of induction motor for wide speed range motor drives[C]// Conf. Rec. IEEE IECON’90 1990: 921–92. [9] APIRACH R, SAKORN P N. Implemantation of an adaptive full-order observer for speed-sensorless induction motor drives[C]// ICEMS 2015: 25–28.