舰船科学技术  2018, Vol. 40 Issue (7): 142-145 PDF

1. 中国船舶重工集团公司第七一三研究所，河南 郑州 450015;
2. 河南省水下智能装备重点实验室，河南 郑州 450015

Research on backstepping control algorithm of launcher-cover based on Simulink simulation
JIA Hai-jie1,2, YAN Long1,2, HOU Dong-dong1,2, JIANG Jun1,2, DING Meng-lei1,2
1. The T13 Research Institute of CSIC, Zhengzhou 450015, China;
2. Henan Key Laboratory of Underwater Intelligent Equipment, Zhengzhou 450015, China
Abstract: In this paper, the dynamic model and electro-hydraulic model of launcher-cover system are analyzed. Based on the Simulink simulation technology, the simulation model of launcher-cover system is established. Based on this, the design of the follow-up cylinder lid angular speed/time curve and angle/time is designed. The back-step controller of the curve, after simulation and experimental verification, the tracking error of the angular velocity of the launcher-cover is obviously reduced, which shows the effectiveness of the backstepping control algorithm in the launcher-cover control.
Key words: switch cover of tube     Simulink simulation     backstepping control
0 引　言

1 筒盖开盖动力学分析

 图 1 典型筒盖开盖结构原理图 Fig. 1 Typical launcher cover structure schematic

 ${({x_o} + {x_t})^2} = {L^2} + {D^2} - 2LD\cos (\beta + {\theta _t}){\text{。}}$ (1)

 $x_t=\sqrt {{L^2} + {D^2} - 2LD\cos (\beta + {\theta _t})} - {x_o}{\text{，}}$ (2)

 ${\theta _t} = arc\cos (\frac{{{L^2} + {D^2} - {{({x_o} + {x_t})}^2}}}{{2LD}}) - \beta {\text{。}}$ (3)

 $J\alpha = F \cdot D\sin \phi - mg\cos \left( {\delta + {\theta _t}} \right) {\text{，}}$ (4)

 $\cos \phi = \frac{{{{\left( {{x_o} + {x_t}} \right)}^2} + {D^2} - {L^2}}}{{2D\left( {{x_o} + {x_t}} \right)}} {\text{，}}$ (5)

 $F = \frac{{J\alpha + mg\cos (\alpha + {\theta _t})l}}{{D\sin \left( {arc\cos \left( {\displaystyle\frac{{{{\left( {{x_o} + {x_t}} \right)}^2} + {D^2} - {L^2}}}{{2D\left( {{x_o} + {x_t}} \right)}}} \right)} \right)}} {\text{。}}$ (6)
2 筒盖电液系统模型分析

 图 2 筒盖电液系统原理图 Fig. 2 Typical launcher cover electro-hydraulic system schematic

 ${Q_{dA}} \!=\! \left\{ \begin{array}{l}\!\!\!\!\!\!{C_{dd}}{\omega _{dv}}{x_{dv}}\sqrt {2\left( {{P_{ds}} - {P_{dA}}} \right)/\rho},{x_{dv}} \geqslant 0 {\text{，}}\\\!\!\!\!\!\!{C_{dd}}{\omega _{dv}}{x_{dv}}\sqrt {2{P_{dA}}/\rho},{x_{dv}} < 0{\text{。}}\end{array} \right.$ (7)
 ${Q_{dB}} \!=\! \left\{ \begin{array}{l}\!\!\!\!\!\!{C_{dd}}{\omega _{dv}}{x_{dv}}\sqrt {2{P_{dB}}/\rho}, {x_{dv}} \geqslant 0{\text{；}}\\\!\!\!\!\!\!{C_{dd}}{\omega _{dv}}{x_{dv}}\sqrt {2\left( {{P_{ds}} - {P_{dB}}} \right)/\rho}, {x_{dv}} \!< 0{\text{。}}\end{array} \right.$ (8)

 ${Q_{dL}} = {C_{dd}}{\omega _{dv}}{x_{dv}}\sqrt {\frac{{{P_{ds}} - \operatorname{sgn} \left( {{x_{dv}}} \right){P_{dL}}}}{\rho }} {\text{，}}$ (9)

 {Q_{dA}} \!\!=\!\! \left\{ \begin{aligned} & \frac{{{V_{dA}}}}{{{\beta _e}}}\frac{{{\rm d}{P_{dA}}}}{{{\rm d}t}} \!\!+\!\! {C_{dip}}{P_{dL}} \!\!+\!\! {C_{dep}}{P_{dA}} + {A_{dp}}\frac{{{\rm d}{x_{dp}}}}{{{\rm d}t}}{\text{，}} \\& \!\!-\!\! \frac{{{V_{dB}}}}{{{\beta _e}}}\frac{{{\rm d}{P_{dB}}}}{{{\rm d}t}} \!\!+\!\! {C_{dip}}{P_{dL}}{\rm{ - }}{C_{dep}}{P_{dB}} \!\!+\!\! {A_{dp}}\frac{{{\rm d}{x_{dp}}}}{{{\rm d}t}} {\text{。}} \\ \end{aligned} \right. (10)

 ${Q_{dL}} = {A_{dp}}\frac{{{\rm d}{x_{dp}}}}{{{\rm d}t}} + {C_{dtp}}{P_{dL}} + \frac{{{V_{dt}}}}{{4{\beta _e}}}\frac{{{\rm d}{P_{dL}}}}{{{\rm d}t}}{\text{，}}$ (11)

 $\frac{{J\alpha + mg\cos (\alpha + {\theta _t})l}}{{D\sin \left( {\arccos \left( {\displaystyle\frac{{{{\left( {{x_o} \!\!+\!\! {x_t}} \right)}^2} + {D^2} \!\!-\!\! {L^2}}}{{2D\left( {{x_o} \!\!+\!\! {x_t}} \right)}}} \right)} \right)}} \!\!=\!\! {P_{dA}}{A_{dP}} \!\!-\!\! {P_{dB}}{A_{dP}} \text{，}$ (12)

${P_{dL}} = {P_{dA}} - {P_{dB}}$ ，根据牛顿第二方程可得

 $m\frac{{{{\rm d}^2}{x_{dp}}}}{{{\rm d}{t^2}}} = {A_p}{P_{dL}} - {B_p}\frac{{{\rm d}{x_{dp}}}}{{{\rm d}t}} + F{\text{，}}$ (13)

3 筒盖系统反步控制器设计

 图 3 反步环控制原理图 Fig. 3 Backstepping ring control schematic

 \left\{ \begin{aligned}& {{\dot x}_1} = {x_2} {\text{，}} \\& {{\dot x}_2} = {\theta _1}{x_3} - {\theta _2}{x_2} - F/m {\text{，}} \\& {{\dot x}_3} = - {\theta _3}{x_2} - {\theta _4}{x_3} + {\theta _5}u {\text{。}} \\ \end{aligned} \right. (14)

 \left\{ \begin{aligned} &{\theta _1} = {A_p}/m {\text{，}} \\ & {\theta _2} = {B_p}/m {\text{，}} \\ &{\theta _3} = 4{\beta _e}{A_p}/{V_t} {\text{，}} \\ &{\theta _4} = 4{\beta _e}{C_{tp}}/{V_t} {\text{，}} \\ &{\theta _5} = 4{\beta _e}/{V_t} {\text{，}} \\ & u = {Q_L} {\text{。}} \\ \end{aligned} \right. (15)

 $\left\{ \begin{gathered} {e_1} = {x_1} - {x_{1d}} {\text{，}} \\ {e_2} = {x_2} - {\alpha _1} {\text{，}} \\ {e_3} = {x_3} - {\alpha _2} {\text{。}} \\ \end{gathered} \right.$ (16)

 ${V_1} = \frac{1}{2}e_1^2{\text{。}}$ (17)

 $\begin{gathered} {{\dot V}_1} = \dot e{}_1 \cdot {e_1} = e{}_1({{\dot x}_1} - {{\dot x}_{1d}}) = e{}_1({x_2} - {{\dot x}_{1d}}) \\ = e{}_1({e_2} + {\alpha _1} - {{\dot x}_{1d}}) {\text{，}} \\ \end{gathered}$ (18)

${\alpha _1} = - {k_1}{e_1} + {\dot x_{1d}}$ ，其中 ${k_1}$ 为大于0的正数

 ${V_2} = \frac{1}{2}e_2^2 + {V_1}{\text{。}}$ (19)

 $\begin{gathered} {{\dot V}_2} = e{}_2{{\dot e}_2} - {k_1}e_1^2 + e{}_1{e_2} \\ = e{}_2({\theta _1}({e_3} + {\alpha _2}) - {\theta _2}{x_2} - F/m \\ + {k_1}({x_2} - {{\dot x}_{1d}}) - {{\ddot x}_{1d}}) - {k_1}e_1^2 + e{}_1{e_2} {\text{，}}\\ \end{gathered}$ (20)

 ${\alpha _2} = - \frac{1}{{{\theta _1}}}(e{}_1 - {\theta _2}{x_2} - F/m + {k_1}({x_2} - {\dot x_{1d}}) - {\ddot x_{1d}} + {k_2}{e_2}){\text{，}}$ (21)

 ${V_3} = \frac{1}{2}e_3^2 + {V_2}{\text{。}}$ (22)

 $\begin{gathered} {{\dot V}_3} = e{}_3{{\dot e}_3} - {k_1}e_1^2 - {k_2}e_2^2 + {\theta _1}e{}_2{e_3} \\ = e{}_3( - {\theta _3}{x_2} - {\theta _4}{x_3} + {\theta _5}u - {{\dot \alpha }_2}) \\ - {k_1}e_1^2 - {k_2}e_2^2 + {\theta _1}e{}_2{e_3} {\text{，}}\\ \end{gathered}$ (23)
 $得$u = - \frac{1}{{{\theta _5}}}({\theta _1}e{}_2 - {\theta _3}{x_2} - {\theta _4}{x_3} - {\dot \alpha _2} + {k_3}{e_3})\text{，}$(24) 此时，${\dot V_3} = - {k_1}e_1^2 - {k_2}e_2^2 - {k_3}e_3^2 < 0\$ ，根据李雅普诺夫理论，系统稳定。

 图 4 电液系统Simulink仿真模型 Fig. 4 Electro-hydraulic system Simulation model
5 筒盖反步控制的仿真分析

 图 5 负载力变化曲线 Fig. 5 Load force curve

 图 6 角速度跟踪曲线 Fig. 6 Angular speed tracking curve

 图 7 角度跟踪曲线 Fig. 7 Angular tracking curve

 图 8 角速度跟踪误差曲线 Fig. 8 Angular speed tracking error curve

 [1] 田凡, 靳宝全, 程珩. 基于联合仿真的电液系统模糊PID控制研究[J]. 液压气动与密封, 2010, 6(5): 27–31. http://www.cqvip.com/QK/95681A/201006/34228965.html [2] 倪火才. 潜地导弹发射装置构造[M]. 哈尔滨:哈尔滨工程大学出版社, 1998. [3] 宋志安, 等. Matlab/simulink与液压控制系统仿真[M]. 北京:国防工业出版社, 2012. [4] 袁朝辉, 袁鸣. 电液系统中新型反步自适应控制器设计[J]. 机电工程, 2013, 07. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=jdgc201307001 [5] 石胜利, 李建雄, 方一鸣. 具有输入饱和的电液伺服系统反步位置跟踪控制[J]. 中南大学学报: 自然科学版, 2016, 47(10): 3369–3374. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=zzjsyjc201112029 [6] 胡良谋, 李景超, 曹克强. 基于MATLAB/SIMULINK的电液伺服控制系统的建模与仿真研究[J]. 机床与液压, 2003(3): 230–231. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=zbzzjs200706026