﻿ 基于新型趋近律的潜艇垂直面滑模控制仿真
 舰船科学技术  2019, Vol. 41 Issue (3): 55-61 PDF

Research and simulation of vertical plane of submarine based on sliding control based on a new reaching law
LU Bin-jie, LI Wen-kui, ZHOU Gang, CHEN Yong-bing
College of Electrical Engineering, Navy University of Engineering, Wuhan 430033, China
Abstract: In order to achieve reliable vertical plane control of the submarine, we used the submarine vertical linear model under the environment of Matlab in this paper. Considering the rudder dynamic response ,the improved sliding reaching control law was used to design the bow rudder controller and the stern rudder controller in the presence of model of nonlinearity and parameter uncertainty for submarine vertical motion. A full order state observer was designed. The bow and stern rudder controller was used to control the depth and the pitch respectively. Meanwhile, we designed the convention PID controller with best parameters, and employed Matlab simulation to realize the vertical plane control simulation with this two kinds of controllers of the submarine under three cases. The simulations were carried and results show that the sliding mode control has more excellent dynamic property and robustness compared with PID controller.
Key words: submarine     sliding mode control     robustness     reaching law     state observer
0 引　言

1 潜艇垂直面运动模型

 图 1 固定坐标系和运动坐标系 Fig. 1 Fixed coordinate system and the motion coordinate system

1）纵向方程

 $\begin{split} &m\left( {\dot u - vr + wq} \right) = \\ &\frac{1}{2}\rho {L^4}\left( {X_{qq}'{q^2} + X_{rr}'{r^2} + X_{rp}'rp} \right) + \\ & \frac{1}{2}\rho {L^3}\left( {X_{\dot u}'\dot u + X_{vr}'vr + X_{wq}'wq} \right) + \\ &\frac{1}{2}\rho {L^2}\left( {X_{uu}'{u^2} + X_{vv}'{v^2} + X_{ww}'{w^2}} \right) + \\ &\frac{1}{2}\rho {L^2}{u^2}\left( {X_{{\delta _r}{\delta _r}}'\delta _r^2 + X_{{\delta _b}{\delta _b}}'\delta _b^2 + X_{{\delta _s}{\delta _s}}'\delta _s^2} \right) + \\ &\frac{1}{2}\rho {L^2}\left( {{a_T}{u^2} + {b_T}u{u_c} + {c_T}u_c^2} \right){\text{。}} \end{split}$ (1)

2）垂向方程

 $\begin{split} &m\left( {\dot w - uq + vp} \right) = \\ &\frac{1}{2}\rho {L^4}\left( {Z_{\dot q}'\dot q + Z_{rr}'{r^2} + Z_{rp}'rp} \right) + \\ &\frac{1}{2}\rho {L^3}\left[ {Z_{\dot w}'\dot w + Z_{vr}'vr + Z_{vp}'vp + Z_{\left| q \right|{\delta _s}}'u\left| q \right|{\delta _s}} \right] + \\ &\frac{1}{2}\rho {L^3}\left[ {Z_q'uq + Z_{w\left| q \right|}'\frac{w}{{\left| w \right|}}\left| {{{\left( {{v^2} + {w^2}} \right)}^{{1 / 2}}}} \right|\left| q \right|} \right] + \\ &\frac{1}{2}\rho {L^2}\left[ {Z_0'{u^2} + Z_w'uw + Z_{vv}'{v^2} + Z_{w\left| w \right|}'w\left| {{{\left( {{v^2} + {w^2}} \right)}^{{1 / 2}}}} \right|} \right] + \\ & \frac{1}{2}\rho {L^2}\left( {Z_{{\delta _b}}'{u^2}{\delta _b} + Z_{{\delta _s}}'{u^2}{\delta _s}} \right){\text{。}} \end{split}$ (2)

3）纵倾方程

 $\begin{split} & {I_y}\dot q + \left( {{I_x} - {I_z}} \right)rp = \\ & \frac{1}{2}\rho {L^5}\left( {M_{\dot q}'\dot q + M_{rr}'{r^2} + M_{rp}'rp} \right) + \\ & \frac{1}{2}\rho {L^4}\left[ {M_q'uq + M_{\left| w \right|q}'\left| {{{\left( {{v^2} + {w^2}} \right)}^{{1 / 2}}}} \right|q} \right] + \\ & \frac{1}{2}\rho {L^4}\left[ {M_{\dot w}'\dot w + M_{vr}'vr + M_{vp}'vp + M_{\left| q \right|{\delta _s}}'u\left| q \right|{\delta _s}} \right] + \\ & \frac{1}{2}\rho {L^3}\left[ {M_0'{u^2} + M_w'uw + M_{w\left| w \right|}'w\left| {{{\left( {{v^2} + {w^2}} \right)}^{{1 / 2}}}} \right| + M_{vv}'{v^2}} \right] + \\ & \frac{1}{2}\rho {L^3}\left( {M_{{\delta _b}}'{u^2}{\delta _b} + M_{{\delta _s}}'{u^2}{\delta _s}} \right) - mgh\sin \theta {\text{，}}\quad \qquad \qquad \;\;\;(3) \end{split}$

4）运动方程

 $\left\{ \begin{array}{l} \dot \zeta = w\cos \theta - u\sin \theta {\text{，}}\\ \dot \theta = q{\text{。}} \end{array} \right.$ (4)

 $\left\{ \begin{array}{l} \dot w = {f_w} + {b_{wb}}{\delta _b} + {b_{ws}}{\delta _s}{\text{，}}\\ \dot q = {f_q} + {b_{qb}}{\delta _b} + {b_{qs}}{\delta _s}{\text{，}}\\ \dot \zeta = - u\theta + w{\text{，}}\\ \dot \theta = q{\text{。}} \end{array} \right.$ (5)

 $\left\{ \begin{gathered} \ddot \zeta = {f_\zeta } + {b_{wb}}{\delta _b} + {b_{ws}}{\delta _s} \hfill {\text{，}}\\ \ddot \theta = {f_\theta } + {b_{qb}}{\delta _b} + {b_{qs}}{\delta _s} \hfill{\text{。}} \\ \end{gathered} \right.$

2）舵机模型

 ${T_E}\dot \delta = {K_E}\left( {{\delta _E} - \delta } \right){\text{。}}$ (6)

2 基于新型趋近律的滑模控制器设计 2.1 潜艇垂直面控制策略

 图 2 首尾舵联合自动深度控制 Fig. 2 Combined automatic depth control of bow and stern rudder
2.2 传统滑模控制器设计

 $\dot \sigma = \ddot e + k\dot e = 0{\text{，}}$ (7)

 $\left\{ \begin{array}{l} {\delta _b} = \displaystyle\frac{1}{{{b_{wb}}}}\left( {{{\ddot \zeta }_d} - {k_\zeta }{{\dot e}_\zeta } - {f_\zeta } - {b_{ws}}{\delta _s}} \right){\text{，}}\\ {\delta _s} = \displaystyle\frac{1}{{{b_{qs}}}}\left( {{{\ddot \theta }_d} - {k_\theta }{{\dot e}_\theta } - {f_\theta } - {b_{qb}}{\delta _b}} \right){\text{。}} \end{array} \right.$ (8)

 ${u_{vss}} = K\operatorname{sgn} \left( \sigma \right){\text{，}}$ (9)

 $\tanh \left( {\mu \sigma } \right) = {{\left( {{e^{\mu \sigma }} - {e^{ - \mu \sigma }}} \right)} / {\left( {{e^{\mu \sigma }} + {e^{ - \mu \sigma }}} \right)}}{\text{，}}$

 \begin{aligned} & {\delta _b} = \frac{1}{{{b_{wb}}}}\left[ {{{\ddot \zeta }_d} - {k_\zeta }{{\dot e}_\zeta } - {f_\zeta } - {b_{ws}}{\delta _s}} \right] + {K_\zeta }\tanh \left( {{\mu _\zeta }{\sigma _\zeta }} \right){\text{，}}\\ & {\delta _s} = \frac{1}{{{b_{qs}}}}\left[ {{{\ddot \theta }_d} - {k_\theta }{{\dot e}_\theta } - {f_\theta } - {b_{qb}}{\delta _b}} \right] + {K_\theta }\tanh \left( {{\mu _\theta }{\sigma _\theta }} \right){\text{。}} \end{aligned} (10)

 $\left\{ \begin{array}{l} {\delta _{bc}} = {{{T_{E1}}} / {{K_{E1}}{{\dot \delta }_b}}} + {\delta _b}{\text{，}}\\ {\delta _{sc}} = {{{T_{E2}}} / {{K_{E2}}{{\dot \delta }_s}}} + {\delta _s}{\text{。}} \end{array} \right.$

 \begin{align} & {V_\zeta } = \frac{1}{2}\sigma _\zeta ^2,\;\;{\sigma _\zeta } \ne 0;\;\;{V_\theta } = \frac{1}{2}\sigma _\theta ^2,\;\;{\sigma _\theta } \ne 0{\text{；}}\\ & {\sigma _\zeta }{{\dot \sigma }_\zeta } = {\sigma _\zeta }\left( {{f_\zeta } + {b_{wb}}{\delta _b} + {b_{ws}}{\delta _s} - {{\ddot \zeta }_d} + {k_\zeta }{{\dot e}_\zeta }} \right){\rm{ = }}\\ & {K_\zeta }{\sigma _\zeta }\tanh \left( {{\mu _\zeta }{\sigma _\zeta }} \right){\text{；}}\\ &{\sigma _\theta }{{\dot \sigma }_\theta } = {\sigma _\theta }\left( {{f_\theta } + {b_{qb}}{\delta _b} + {b_{qs}}{\delta _s} - {{\ddot \theta }_d} + {k_\theta }{{\dot e}_\theta }} \right){\rm{ = }}\\ & {K_\theta }{\sigma _\theta }\tanh \left( {{\mu _\theta }{\sigma _\theta }} \right){\text{。}} \end{align}

2.3 双幂次趋近律设计

 $\dot \sigma = - \varepsilon {\left| \sigma \right|^\gamma }\tanh \left( {\mu \sigma } \right) - \lambda {\left| \sigma \right|^\chi }\tanh \left( {\mu \sigma } \right){\text{，}}$ (11)

 $\begin{array}{l} \sigma \dot \sigma = \sigma \left[ { - \varepsilon {{\left| \sigma \right|}^\gamma }\tanh \left( {\mu \sigma } \right) - \lambda {{\left| \sigma \right|}^\chi }\tanh \left( {\mu \sigma } \right)} \right]{\rm{ = }}\\ - \varepsilon {\left| \sigma \right|^{1 + \gamma }} - \lambda {\left| \sigma \right|^{1 + \chi }} < 0{\text{。}} \end{array}$

 \!\!\!\!\!\begin{aligned} & {\delta _b} \!=\! \displaystyle\frac{1}{{{b_{wb}}}}\left[\!\!\!\!\! \begin{array}{l} {{\ddot \zeta }_d} - {k_\zeta }{{\dot e}_\zeta } \!-\! {f_\zeta } \!-\! {b_{ws}}{\delta _s} \!-\! \\ {\varepsilon _\zeta }{\left| {{\sigma _\zeta }} \right|^{{\gamma _{_\zeta }}}}\!\tanh \left( {{\mu _\zeta }{\sigma _\zeta }} \right) \!-\! {\lambda _\zeta }{\left| {{\sigma _\zeta }} \right|^{{\chi _{_\zeta }}}}\!\tanh \left( {{\mu _\zeta }{\sigma _\zeta }} \right) \end{array} \!\!\!\!\!\right]{\text{，}}\\ & {\delta _s} \!=\! \displaystyle\frac{1}{{{b_{qs}}}}\left[\!\!\!\!\! \begin{array}{l} {{\ddot \theta }_d} - {k_\theta }{{\dot e}_\theta } \!-\! {f_\theta } - {b_{qb}}{\delta _b} \!-\! \\ {\varepsilon _\theta }{\left| {{\sigma _\theta }} \right|^{{\gamma _{_\theta }}}}\!\tanh \left( {{\mu _\theta }{\sigma _\theta }} \right) \!-\! {\lambda _\theta }{\left| {{\sigma _\theta }} \right|^{{\chi _{_\theta }}}}\tanh \left( {{\mu _\theta }{\sigma _\theta }} \right) \end{array}\!\!\!\!\!\right]{\text{。}} \end{aligned} (12)

 \begin{aligned} & {V_\zeta } = \frac{1}{2}\sigma _\zeta ^2,\;\;{\sigma _\zeta } \ne 0;\;\;{V_\theta } = \frac{1}{2}\sigma _\theta ^2,\;\;{\sigma _\theta } \ne 0{\text{；}}\\ & {\sigma _\zeta }{{\dot \sigma }_\zeta } = {\sigma _\zeta }\left( {{f_\zeta } + {b_{wb}}{\delta _b} + {b_{ws}}{\delta _s} - {{\ddot \zeta }_d} + {k_\zeta }{{\dot e}_\zeta }} \right){\rm{ = }}\\ & {\sigma _\zeta }\left[ { - {\varepsilon _\zeta }{{\left| {{\sigma _\zeta }} \right|}^{{\gamma _{_\zeta }}}}\tanh \left( {{\mu _\zeta }{\sigma _\zeta }} \right) - {\lambda _\zeta }{{\left| {{\sigma _\zeta }} \right|}^{{\chi _{_\zeta }}}}\tanh \left( {{\mu _\zeta }{\sigma _\zeta }} \right)} \right] < 0{\text{；}}\\ & {\sigma _\theta }{{\dot \sigma }_\theta } = \left( {{f_\theta } + {b_{qb}}{\delta _b} + {b_{qs}}{\delta _s} - {{\ddot \theta }_d} + {k_\theta }{{\dot e}_\theta }} \right){\rm{ = }}\\ & {\sigma _\theta }\left[ { - {\varepsilon _\theta }{{\left| {{\sigma _\theta }} \right|}^{{\gamma _{_\theta }}}}\tanh \left( {{\mu _\theta }{\sigma _\theta }} \right) - {\lambda _\theta }{{\left| {{\sigma _\theta }} \right|}^{{\chi _{_\theta }}}}\tanh \left( {{\mu _\zeta }{\sigma _\zeta }} \right)} \right] < 0{\text{。}} \end{aligned}

3 Luenberger状态观测器设计

$w$ $q$ 等状态量无法直接测得，需设计观测器进行估计。系统方程和量测方程：

 $\left\{ \begin{array}{l} \dot X = AX + BU + DF{\text{，}}\\ Y = CX{\text{。}} \end{array} \right.$ (13)

 $\hat X = \left( {A - LC} \right)\hat X + BU + LY{\text{，}}$ (14)

 \begin{align} & {\delta _b} = \displaystyle\frac{1}{{{b_{wb}}}}\left[ \begin{array}{l} {{\ddot \zeta }_d} - {k_\zeta }{{\dot {\hat e}}_\zeta } - {{\hat f}_\zeta } - {b_{ws}}{\delta _s} - \\ {\varepsilon _\zeta }{\left| {{{\hat \sigma }_\zeta }} \right|^{{\gamma _{_\zeta }}}}\tanh \left( {{\mu _\zeta }{{\hat \sigma }_\zeta }} \right) - {\lambda _\zeta }{\left| {{{\hat \sigma }_\zeta }} \right|^{{\chi _{_\zeta }}}}\tanh \left( {{\mu _\zeta }{{\hat \sigma }_\zeta }} \right) \end{array} \right]{\text{，}}\\ & {\delta _s} = \displaystyle\frac{1}{{{b_{qs}}}}\left[ \begin{array}{l} {{\ddot \theta }_d} - {k_\theta }{{\dot {\hat e}}_\theta } - {{\hat f}_\theta } - {b_{qb}}{\delta _b} - \\ {\varepsilon _\theta }{\left| {{{\hat \sigma }_\theta }} \right|^{{\gamma _{_\theta }}}}\tanh \left( {{\mu _\theta }{{\hat \sigma }_\theta }} \right) - {\lambda _\theta }{\left| {{{\hat \sigma }_\theta }} \right|^{{\chi _{_\theta }}}}\tanh \left( {{\mu _\theta }{{\hat \sigma }_\theta }} \right) \end{array} \right]{\text{。}} \end{align} (15)

 $\begin{array}{l} {V_\zeta } = \frac{1}{2}\hat \sigma _\zeta ^2,\;\;{{\hat \sigma }_\zeta } \ne 0;\;\;{V_\theta } = \frac{1}{2}\hat \sigma _\theta ^2,\;\;{{\hat \sigma }_\theta } \ne 0 {\text{；}}\\ {{\hat \sigma }_\zeta }{{\dot {\hat \sigma} }_\zeta } = {{\hat \sigma }_\zeta }\left[ - {\varepsilon _\zeta }{{\left| {{{\hat \sigma }_\zeta }} \right|}^{{\gamma _{_\zeta }}}}\tanh \left( {{\mu _\zeta }{{\hat \sigma }_\zeta }} \right)\right.- \\ \quad \left. {\lambda _\zeta }{{\left| {{{\hat \sigma }_\zeta }} \right|}^{{\chi _{_\zeta }}}}\tanh \left( {{\mu _\zeta }{{\hat \sigma }_\zeta }} \right) \right] < 0 {\text{；}}\\ {{\hat \sigma }_\theta }{{\dot {\hat \sigma} }_\theta } = {{\hat \sigma }_\theta }\left[ - {\varepsilon _\theta }{{\left| {{{\hat \sigma }_\theta }} \right|}^{{\gamma _{_\theta }}}}\tanh \left( {{\mu _\theta }{{\hat \sigma }_\theta }} \right) \right. -\\ \quad \left. {\lambda _\theta }{{\left| {{{\hat \sigma }_\theta }} \right|}^{{\chi _{_\theta }}}}\tanh \left( {{\mu _\theta }{{\hat \sigma }_\theta }} \right) \right] < 0{\text{。}} \end{array}$

4 仿真

1）当潜艇参数无摄动和无外界扰动时，图3图4分别给出了深度、纵倾响应曲线与舵角曲线。由图可见，SMC比PID控制使深度能更快速无超调趋近指令深度，达到指令深度后沿期望深度稳定航行，纵倾符合限制条件。滑模控制首、尾舵打舵比PID控制更为平滑。

 图 3 潜艇参数无摄动时的深度和纵倾响应 Fig. 3 Depth and pitch response without parameterperturbations of submarine

 图 4 潜艇参数无摄动时的舵角 Fig. 4 Rudder angle without parameter perturbations of submarine

2）系统部分参数摄动但无外界扰动。由于控制性能主要受垂向方程和纵倾方程有关参数影响，由式（9）知， ${f_w}$ ${f_q}$ 包含了主要系统参数，未验证滑模控制器的鲁棒性，设系统参数进行大范围摄动，令 $f_w^*$ $f_q^*$ 为系统实际参数，选取范围： $0.5{f_w} \leqslant f_w^* \leqslant 2{f_w}$ $0.5{f_q} \leqslant$ $f_q^* \leqslant 2{f_q}$ 。控制器参数不变，当系统参数分别在标称模型的50%和200%范围摄动时，SMC比PID控制使深度能更快速无超调趋近指令深度。滑模控制首、尾舵打舵比PID控制更为平滑，如图5图6所示。

 图 5 潜艇参数摄动时的深度和纵倾响应 Fig. 5 Depth and pitch response with parameterperturbations of submarine

 图 6 潜艇参数摄动时的舵角 Fig. 6 Rudder angle with parameter perturbations of submarine

3）当无摄动但有外界扰动时，为便于分析，任意选取正弦扰动信号： ${d_\zeta } = 50\;000\sin \left( {0.1{\text{π}} t} \right)$ ${d_\theta } = 50\,000$ $\cos \left( {0.2{\text{π}} t} \right)$

 图 7 正弦干扰下的深度和纵倾响应 Fig. 7 Depth and pitch response of sinusoidal interference

 图 8 正弦干扰下的的舵角 Fig. 8 Rudder angle under sinusoidal interference

5 结　语

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