﻿ 大型液体火箭姿控与跷振大回路耦合动力学建模与分析
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 力学学报  2015, Vol. 47 Issue (5): 789-798  DOI: 10.6052/0459-1879-15-137 0

### 引用本文 [复制中英文]

[复制中文]
Wang Qingwei, Tan Shujun, Wu Zhigang, Yang Yunfei, Chen Yu. DYNAMIC MODELING AND ANALYSIS OF LARGE-LOOP COUPLED BY ATTITUDE CONTROL AND POGO FOR LARGE LIQUID ROCKETS[J]. Chinese Journal of Ship Research, 2015, 47(5): 789-798. DOI: 10.6052/0459-1879-15-137.
[复制英文]

### 文章历史

2015-04-20 收稿
2015-06-15 录用
2015–06–19 网络版发表

1. 工业装备结构分析国家重点实验室, 大连理工大学工程力学系, 大连116024;
2. 大连理工大学航空航天学院, 大连116024;
3. 北京宇航系统工程研究所, 北京100076

1 传统的姿控回路模型

 图 1 传统姿控回路结构示意图 Fig. 1 Schematic diagram of traditional attitude control loop model
1.1 姿态动力学模型

(1) 俯仰通道姿态运动方程

 $\begin{array}{*{20}{l}} {\Delta \dot \theta = c_1^\varphi \Delta \alpha + c_2^\varphi \Delta \theta + c_4^\varphi \dot \varphi + c_{3x}^\varphi \Delta {\delta _{\varphi {\rm{xj}}}} + c_{3x}^{,,\varphi }\Delta {{\ddot \delta }_{\varphi {\rm{xj}}}} + }\\ {\;\;\;\;\;\;\;\;c_{3z}^\varphi \Delta {\delta _{\varphi {\rm{zt}}}} + c_{3z}^{,,\varphi }\Delta {{\ddot \delta }_{\varphi {\rm{zt}}}} + \sum\limits_{i = 1}^n {\left( {c_{1i}^\varphi {{\dot q}_i} + c_{2i}^\varphi {{\dot q}_i} + c_{3i}^\varphi {{\ddot q}_i}} \right)} + }\\ {\;\;\;\;\;\;\;\;c_1^{,\varphi }\left( {{\alpha _{wp}} + {a_{wq}}} \right) + {{\bar F}_{BY}}} \end{array}$ (1)

 $\begin{array}{*{20}{l}} {\Delta \ddot \varphi + b_1^\varphi \Delta \dot \varphi + b_2^\varphi \Delta \alpha + b_{3x}^\varphi \Delta {\delta _{\varphi {\rm{xj}}}} + b_{3z}^{,,\varphi }\Delta {{\ddot \delta }_{\varphi {\rm{xj}}}} + }\\ {\;\;\;\;\;\;\;b_{3z}^\varphi \Delta {\delta _{\varphi {\rm{zt}}}} + b_{3z}^{,,\varphi }\Delta {{\ddot \delta }_{\varphi {\rm{zt}}}} + \sum\limits_{i = 1}^n {\left( {b_{1i}^\varphi {{\dot q}_i} + b_{2i}^\varphi {q_i} + b_{3i}^\varphi {{\ddot q}_i}} \right)} + }\\ {\;\;\;\;\;\;\;b_2^{,\varphi }\left( {{\alpha _{wp}} + {\alpha _{wq}}} \right) = {{\bar M}_{BZ}}} \end{array}$ (2)

 $\Delta \varphi = \Delta \alpha + \Delta \theta$ (3)

(2) 偏航通道姿态运动方程

 $\begin{array}{*{20}{l}} {\Delta \dot \sigma = c_1^\psi \Delta \beta + c_2^\psi \Delta \sigma + c_4^\psi \dot \psi + c_{3x}^\psi \Delta {\delta _{\psi {\rm{xj}}}} + c_{3x}^{,,\psi }\Delta {{\ddot \delta }_{\psi {\rm{xj}}}} + }\\ {\;\;\;\;\;\;\;\;c_{3z}^\psi \Delta {\delta _{\psi {\rm{zt}}}} + c_{3z}^{,,\psi }\Delta {{\ddot \delta }_{\psi {\rm{zt}}}} + \sum\limits_{i = 1}^n {\left( {c_{1i}^\psi {{\dot q}_i} + c_{2i}^\psi {{\dot q}_i} + c_{3i}^\psi {{\ddot q}_i}} \right)} + }\\ {\;\;\;\;\;\;\;\;c_1^{,\psi }\left( {{\beta _{wp}} + {\beta _{wq}}} \right) + {{\bar F}_{BZ}}} \end{array}$ (4)

 $\begin{array}{*{20}{l}} {\Delta \ddot \psi + b_1^\psi \Delta \dot \psi + b_2^\psi \Delta \beta + b_{3x}^\psi \Delta {\delta _{\psi {\rm{xj}}}} + b_{3x}^{,,\psi }\Delta {{\ddot \delta }_{\psi {\rm{xj}}}} + }\\ {\;\;\;\;\;\;\;\;b_{3z}^\psi \Delta {\delta _{\psi {\rm{zt}}}} + b_{3z}^{,,\psi }\Delta {{\ddot \delta }_{\psi {\rm{zt}}}} + \sum\limits_{i = 1}^n {\left( {b_{1i}^\psi {{\dot q}_i} + b_{2i}^\psi {{\dot q}_i} + b_{3i}^\psi {{\ddot q}_i}} \right)} + }\\ {\;\;\;\;\;\;\;\;b_2^{,\psi }\left( {{\beta _{wp}} + {\beta _{wq}}} \right) = {{\bar M}_{BY}}} \end{array}$ (5)

 $\Delta \psi = \Delta \sigma + \Delta \beta$ (6)

(3) 滚转通道姿态运动方程

 $\begin{array}{l} \Delta \ddot \gamma + {d_1}\Delta \dot \gamma + {d_{3x}}\Delta {\delta _{\gamma {\rm{xj}}}} + {{d"}_{3x}}\Delta {{\ddot \delta }_{\gamma {\rm{xj}}}} + {d_{3z}}\Delta {\delta _{\gamma {\rm{zt}}}} + \\ \;\;\;\;\;\;\;{{d"}_{3z}}\Delta {{\ddot \delta }_{\gamma {\rm{zt}}}} + d_{3x}^\varphi \Delta {\delta _{\varphi {\rm{xj}}}} + \sum\limits_{i = 1}^n {\left( {{d_{1i}}{{\dot q}_i} + {d_{2i}}{q_i} + {d_{3i}}{{\ddot q}_i}} \right)} + \\ \qquad d_{3z}^\varphi \Delta {\delta _{\varphi {\rm{zt}}}} + d_{3x}^\psi \Delta {\delta _{\psi {\rm{xj}}}} + d_{3z}^\psi \Delta {\delta _{\psi {\rm{zt}}}} = {{\bar M}_{BX}} \end{array}$ (7)

(4) 弹性振动方程

 \begin{align} & {{{\ddot{q}}}_{i}}+2{{\xi }_{i}}{{\omega }_{i}}{{{\dot{q}}}_{i}}+\omega _{i}^{2}{{q}_{i}}=D_{1i}^{\varphi }\Delta \dot{\varphi }+D"_{1i}^{\varphi }\Delta \ddot{\varphi }+ \\ & \qquad D_{2i}^{\varphi }\Delta \alpha +D_{2i}^{\varphi }\left( {{\alpha }_{wp}}+{{\alpha }_{wq}} \right)+D_{3xi}^{\varphi }\Delta {{\delta }_{\varphi \text{xj}}}+ \\ & \qquad D"_{3xi}^{\varphi }\Delta {{{\ddot{\delta }}}_{\varphi \text{xj}}}+D_{3zi}^{\varphi }\Delta {{\delta }_{\varphi \text{zt}}}+D_{3zi}^{\varphi }\Delta {{{\ddot{\delta }}}_{\varphi \text{zt}}}+D_{1i}^{\psi }\dot{\psi }+ \\ & \ \ \ \ \ \ \ D"_{1i}^{\psi }\ddot{\psi }+D_{2i}^{\psi }\beta +D_{2i}^{\psi }\left( {{\beta }_{wp}}+{{\beta }_{wq}} \right)+D_{3xi}^{\psi }\Delta {{\delta }_{\psi \text{xj}}}+ \\ & \ \ \ \ \ \ \ D"_{3xi}^{\psi }\Delta {{{\ddot{\delta }}}_{\psi \text{xj}}}+D_{3zi}^{\psi }\Delta {{\delta }_{\psi \text{zt}}}+D_{3zi}^{\psi }\Delta {{{\ddot{\delta }}}_{\psi \text{zt}}}+{{D}_{4i}}\dot{\gamma }+ \\ & \qquad {{{{D}"}}_{4i}}\ddot{\gamma }+{{d}_{3xi}}{{\delta }_{\gamma \text{xj}}}+{{{{d}"}}_{3xi}}{{{\ddot{\delta }}}_{\gamma \text{xj}}}+{{d}_{3zi}}{{\delta }_{\gamma \text{zt}}}+{{{{d}"}}_{3zi}}{{{\ddot{\delta }}}_{\gamma \text{zt}}}+ \\ & \qquad \sum\limits_{j=1}^{n}{\left( {{R}_{ij}}{{q}_{j}}+{{{{R}'}}_{ij}}{{{\dot{q}}}_{j}} \right)}+{{{\bar{Q}}}_{ix}}+{{{\bar{Q}}}_{iy}}+{{{\bar{Q}}}_{iz}}+{{{\bar{Q}}}_{in}} \\ \end{align} (8)

(5) 测量方程

 $\left.\begin{array}{l} \Delta \varphi _{gz} = \Delta \varphi - \sum\limits_i {R_{zi} \left( {x_{gz} } \right)q_i } \\ \Delta \dot{\varphi }_{gz} = \Delta \dot{\varphi } - \sum\limits_i {R_{zi} \left( {x_{gz} } \right)\dot{q}_i } \\ \Delta \psi _{gz} = \Delta \psi - \sum\limits_i {R_{yi} \left( {x_{gz} } \right)q_i } \\ \Delta \dot{\psi }_{gz} = \Delta \dot{\psi } - \sum\limits_i {R_{yi} \left( {x_{gz} } \right)\dot{q}_i } \\ \Delta \gamma _{gz} = \Delta \gamma - \sum\limits_i {R_{xi} \left( {x_{gz} } \right)q_i } \\ \Delta \dot{\gamma }_{gz} = \Delta \dot{\gamma } - \sum\limits_i {R_{xi} \left( {x_{gz} } \right)\dot{q}_i } \\ \end{array}\right\}$ (9)

 ${{ x}}_{\rm a} = [\Delta \varphi ,\Delta \psi ,\Delta \gamma ,{{ q}}^{\rm T},\Delta \theta ,\Delta \sigma ,\Delta \dot{\varphi },\Delta \dot{\psi },\Delta \dot{\gamma }, {{\dot{ q}}}^{\rm T}]^{\rm T}$

 ${{ E}}_{\rm a} {{\dot{ x}}}_{\rm a} = {{ A}}_{\rm a} { x}_{\rm a} + {{ B}}_{\rm a} {{ u}}_{\rm a} + {{ w}}_{\rm a}$ (10)

 ${{ y}}_{\rm a} = {{ C}}_{\rm a} {{ x}}_{\rm a}$ (11)

 \begin{align} & {{E}_{\text{a}}}=\left[\begin{matrix} {{I}_{3}} & 0 & 0 & 0 & 0 \\ 0 & {{I}_{n+2m}} & 0 & 0 & 0 \\ 0 & 0 & {{I}_{2}} & 0 & {{E}_{35}} \\ 0 & 0 & 0 & {{I}_{3}} & {{E}_{45}} \\ 0 & 0 & {{E}_{53}} & {{E}_{54}} & {{E}_{55}} \\ \end{matrix} \right] \\ & {{A}_{\text{a}}}=\left[\begin{matrix} 0 & 0 & 0 & {{I}_{3}} & 0 \\ 0 & 0 & 0 & 0 & {{I}_{n+2m}} \\ {{A}_{31}} & {{A}_{32}} & {{A}_{33}} & {{A}_{34}} & {{A}_{35}} \\ {{A}_{41}} & {{A}_{42}} & {{A}_{43}} & {{A}_{44}} & {{A}_{45}} \\ {{A}_{51}} & {{A}_{52}} & {{A}_{53}} & {{A}_{54}} & {{A}_{55}} \\ \end{matrix} \right] \\ & {{B}_{\text{a}}}=\left[\begin{matrix} 0\ \ \ 0\ \ \ 0 \\ {{B}_{2\varphi }}{{B}_{2\psi }}{{B}_{2\gamma }} \\ 0\ \ \ 0\ \ \ 0 \\ \end{matrix} \right] \\ & {{C}_{\text{a}}}=\left[\begin{matrix} 1\ 0\ 0\ {{C}_{\varphi }}\ 0\ 0\ 0\ 0\ 0\ 0 \\ 0\ 0\ 0\ 0\ 0\ 0\ 1\ 0\ 0\ {{C}_{\varphi }} \\ 0\ 1\ 0\ {{C}_{\psi }}\ 0\ 0\ 0\ 0\ 0\ 0 \\ 0\ 0\ 0\ 0\ 0\ 0\ 0\ 1\ 0\ {{C}_{\psi }} \\ 0\ 0\ 1\ {{C}_{\gamma }}0\ 0\ 0\ 0\ 0\ 0 \\ 0\ 0\ 0\ 0\ 0\ 0\ 0\ 0\ 1\ {{C}_{\gamma }} \\ \end{matrix} \right] \\ \end{align} (12)

 \begin{align} & {{u}_{\text{a}}}=[\Delta {{\delta }_{\varphi \text{xj}}},\Delta {{{\ddot{\delta }}}_{\varphi \text{xj}}},\Delta {{\delta }_{\varphi \text{zt}}},\Delta {{{\ddot{\delta }}}_{\varphi \text{zt}}},\Delta {{\delta }_{\psi \text{xj}}},\Delta {{{\ddot{\delta }}}_{\psi \text{xj}}},\\ & \ \ \ \ \ \ \ \Delta {{\delta }_{\psi \text{zt}}},\Delta {{{\ddot{\delta }}}_{\psi \text{zt}}},\Delta {{\delta }_{\gamma \text{xj}}},\Delta {{{\ddot{\delta }}}_{\gamma \text{xj}}},\Delta {{\delta }_{\gamma \text{zt}}},\Delta {{{\ddot{\delta }}}_{\gamma \text{zt}}}{{]}^{\text{T}}} \\ \end{align} (13)

 $\begin{array}{l} amp;{w_{\rm{a}}} = [0,\;0,\;0,\;{0^{\rm{T}}},\;{0^{\rm{T}}},\;{0^{\rm{T}}},\; - {{\bar F}_{BY}} + c_1^{,\varphi }\left( {{\alpha _{wp}} + {\alpha _{wq}}} \right),\\ amp;\;\;\;\;\;\;\; - {{\bar F}_{BZ}} + c_1^{,\psi }\left( {{\beta _{wp}} + {\beta _{wq}}} \right),\\ amp;\;\;\;\;\;\;\; - {{\bar M}_{BZ}} + b_2^{,\varphi }\left( {{\alpha _{wp}} + {\alpha _{wq}}} \right),\\ amp;\;\;\;\;\;\;\; - {{\bar M}_{BY}} + b_2^{,\psi }\left( {{\beta _{wp}} + {\beta _{wq}}} \right),\\ amp;\;\;\;\;\;\;{{\bar M}_{BX}},\;{{\bar F}_Q},\;{0^{\rm{T}}},{0^{\rm{T}}}{]^{\rm{T}}} \end{array}$ (14)

 ${{ y}}_{\rm a} = \left[{\Delta \varphi _{gz} ,\Delta \dot{\varphi }_{gz} ,\Delta \psi _{gz} ,\Delta \dot{\psi }_{gz} ,\Delta \gamma _{gz} ,\Delta \dot{\gamma }_{gz} } \right]^{\rm T}$ (15)

1.2 控制系统模型

 ${{G}}\left( s \right) = \frac{b_n s^n + b_{n - 1} s^{n - 1} + ... + b_1 s + b_0 }{a_n s^n + a_{n - 1} s^{n - 1} + ... + a_1 s + a_0 }$ (16)

 图 2 姿态控制系统示意图 Fig. 2 Schematic diagram of attitude control system

 ${{ E}}_{\rm c} {{\dot{ x}}}_{\rm c} = {{ A}}_{\rm c} x_{\rm c} + {{ B}}_{\rm c} {{ u}}_{\rm c}$ (17)

 ${{ y}}_{\rm c} = {{ C}}_{\rm c} {{ x}}_{\rm c}$ (18)

 $\left. {\begin{array}{l} {{ u}}_{\rm c} = {{ y}}_{\rm a} \\ {{ y}}_{\rm c} = {{ u}}_{\rm a} \\ \end{array}} \right\}$ (19)

2 跷振回路模型

2.1 推进系统模型

 ${{ v}} = [v_1 ,v_2 ,v_3 ,...,v_{N_{\rm p} }]^{\rm T}$ (20)

 ${{ p}}_{{\rm tc}} = [p_{{\rm tco1}} ,p_{{\rm tcf1}} ,...,p_{{\rm tcoN}_{{\rm th}} } ,p_{{\rm tcfN}_{{\rm th}} }]^{\rm T}$ (21)

 ${{ M}}_{\rm p} {{\ddot{ v}}} + {{ R}}_{\rm p} {{\dot{ v}}} + {{ K}}_{\rm p} {{ v}} + {{ S p}}_{{\rm tc}} = {{ f}}_{{\rm ps}}$ (22)

 ${{\dot{ p}}}_{{\rm tc}} = {{ L p}}_{{\rm tc}} + {{ H\dot{ v}}}$ (23)

 ${{ f}}_{{\rm ps}} = {{ U}}_0 {{ r}}_{\rm p} + {{ U}}_1 {{\dot{ r}}}_{\rm p} + {{ U}}_2 {{\ddot{ r}}}_{\rm p}$ (24)

 ${{ f}}_{{\rm ps}} = {{ U}}_{\rm 0} {{ \varPhi }}_{{\rm ps}} {{ q}} + {{ U}}_{\rm 1} {{ \varPhi }}_{{\rm ps}} {{\dot{ q}}} + {{ U}}_2 {{ \varPhi }}_{{\rm ps}} {{\ddot{ q}}}$ (25)

 ${{ x}}_{\rm p}^{\rm T} = \left[{{{ v}}^{\rm T},p_{{\rm tc}}^{\rm T} ,{{\dot{ v}}}^{\rm T}} \right]$ (26)

 ${{ E}}_{\rm p} {{\dot{ x}}}_{\rm p} = {{ A}}_{\rm p} {{ x}}_{\rm p} + {{\tilde{ U}}}_{rq,0} {{ q}} + {{\tilde{ U}}}_{rq,1} {{\dot{ q}}} + {{\tilde{ U}}}_{rq,2} {{\ddot{ q}}}$ (27)

 ${{ E}}_{\rm p} = \left[{{\begin{array}{*{20}c} {{{ I}}_v } & {{ 0}} & {{ 0}} \\ {{ 0}} & {{{ I}}_{{\rm tc}} } & {{ 0}} \\ {{ 0}} & {{ 0}} & {{{ M}}_{\rm p} } \\ \end{array} }} \right],\ \ A_p = \left[{{\begin{array}{*{20}c} {{ 0}} & {{ 0}} & {{{ I}}_v } \\ {{ 0}} & {{ L}} & {{ H}} \\ { - {{ K}}_{\rm p} } & { - {{ S}}} & { - {{ R}}_{\rm p} } \\ \end{array} }} \right]\\ {{\tilde{ U}}}_{rq,0} = \left[{{\begin{array}{*{20}c} {{ 0}} \\ {{ 0}} \\ {{{ U}}_0 {{ \varPhi }}_{{\rm ps}} } \\ \end{array} }} \right],\ \ {{\tilde{ U}}}_{rq,1} = \left[ {{\begin{array}{*{20}c} {{ 0}} \\ {{ 0}} \\ {{{ U}}_1 {{ \varPhi }}_{{\rm ps}} } \\ \end{array} }} \right],\ \ {{\tilde{ U}}}_{rq,2} = \left[ {{\begin{array}{*{20}c} {{ 0}} \\ {{ 0}} \\ {{{ U}}_2 {{ \varPhi }}_{{\rm ps}} } \\ \end{array} }} \right]$
2.2 结构系统模型

 $M_{\rm s} (\ddot{ q} + Z_{\rm s} {{\dot{ q}}} + {{ \varOmega }}_{\rm s} {{ q}}) = {{ \varPhi }}^{\rm T}{{ f}}_{{\rm sp}}$ (28)

 ${{ f}}_{{\rm sp}} = {{ X}}_{{\rm sp}} {{ f}}_{\rm p} = {{ X}}_{{\rm sp}} ( {{ Q}}_0 {{ v}} + {{ Q}}_1 {{\dot{ v}}} + {{ Q}}_2 {{\ddot{ v}}}+\\\qquad {{ Q}}_{\rm p} {{ p}}_{{\rm tc}} + {{ V\ddot { r}}}_{\rm p} )$ (29)

 ${{ \varPhi }}^{\rm T}{{ f}}_{{\rm sp}} = {{ \varPhi }}^{\rm T}{{ X}}_{{\rm sp}} {{ f}}_{\rm p} = {{ \varPhi }}_{{\rm sp}}^{\rm T}( {{ Q}}_0 {{ v}} + {{ Q}}_1 {{\dot{ v}}} + {{ Q}}_2 {{\ddot{ v}}}+\\\qquad {{ Q}}_{\rm p} {{ p}}_{{\rm tc}} + {{ V \varPhi }}_{{\rm ps}} {{\ddot{ q}}})$ (30)

2.3 跷振回路模型

 ${{\tilde{ v}}} = [{{ v}}^{\rm T},{{ q}}^{\rm T},{{ p}}_{\rm c}^{\rm T} ,{{\dot{ v}}}^{\rm T},{{\dot{ q}}}^{\rm T}]^{\rm T}$ (31)

 ${{ E\dot{\tilde{ v}}}} = {{ A\tilde{ v}}}$ (32)

 ${{ E}} = \left[{{\begin{array}{*{20}c} {{{ I}}_{\rm w} } & {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} \\ 0 & {{{ I}}_{\rm q} } & {{ 0}} & {{ 0}} & {{ 0}} \\ 0 & {{ 0}} & {{{ I}}_{{\rm pc}} } & {{ 0}} & {{ 0}} \\ 0 & {{ 0}} & {{ 0}} & {{{ M}}_{\rm p} } & { - {{ U}}_{\rm 2} {{ \varPhi }}_{{\rm ps}} } \\ 0 & {{ 0}} & {{ 0}} & { - {{ \varPhi }}_{{\rm sp}}^{\rm T} {{ Q}}_{\rm 2} } & {{{ M}}_{\rm s} - {{ \varPhi }}_{{\rm sp}}^{\rm T} {{ V \varPhi }}_{{\rm ps}}^{\rm T} } \\ \end{array} }} \right] \\[3mm] {{ A}} = \left[{{\begin{array}{*{20}c} {{ 0}} & {{ 0}} & {{ 0}} & {{{ I}}_{\rm w} } & {{ 0}} \\ {{ 0}} & {{ 0}} & {{ 0}} & {{ 0}} & {{{ I}}_{\rm q} } \\ {{ 0}} & {{ 0}} & {{ L}} & {{ H}} & {{ 0}} \\ { - {{ K}}_{\rm p} } & {{{ U}}_0 {{ \varPhi }}_{{\rm ps}} } & { - {{ S}}} & { - {{ R}}_{\rm p} } & {{{ U}}_1 {{ \varPhi }}_{{\rm ps}} } \\ {{{ \varPhi }}_{{\rm sp}}^{\rm T} {{ Q}}_0 } & { - {{ M}}_{\rm s} {{ \varOmega }}_{\rm s} } & {{{ \varPhi }}_{{\rm sp}}^{\rm T} {{ Q}}_p } & {{{ \varPhi }}_{{\rm sp}}^{\rm T} {{ Q}}_1 } & { - {{ M}}_{\rm s} Z_{\rm s} } \\ \end{array} }} \right]$

3 姿控与跷振大回路耦合模型

 图 3 姿控回路与跷振回路耦合示意图 Fig. 3 Schematic diagram of coupling mechanism of attitude control loop and POGO loop
3.1 姿态动力学与跷振回路耦合模型

 ${{ w}}_{\rm a} = {{ w}}_{{\rm a},{\rm s}} + {{ w}}_{{\rm a},{\rm p}}$ (33)

 ${{ w}}_{{\rm a},{\rm s}} = \Big[0,\;0,\;0,\;{{ 0}}^{\rm T},\;{{ 0}}^{\rm T}, \;{{ 0}}^{\rm T},\\ \qquad- \bar{F}_{BY,{\rm s}} + c_1^{'\varphi } \left( {\alpha _{w{\rm p}} + \alpha _{wq} } \right),\\ \qquad- \bar{F}_{BZ,{\rm s}} + c_1^{'\psi } \left( {\beta _{w{\rm p}} + \beta _{wq} } \right),\\ \qquad- \bar{M}_{BZ,{\rm s}} + b_2^{'\varphi } \left( {\alpha _{wp} + \alpha _{wq} } \right),\\ \qquad- \bar{M}_{BY,{\rm s}} + b_2^{'\psi } \left( {\beta _{w{\rm p}} + \beta _{wq} } \right),\\ \qquad\bar{M}_{BX,{\rm s}} ,\;{{\bar{ F}}}_{Q,{\rm s}} ,\;{{ 0}}^{\rm T},{{ 0}}^{\rm T}\Big]^{\rm T}$ (34)

 $\bar{F}_{Qi,s} = D_{2i}^\varphi \left( {\alpha _{w{\rm p}} + \alpha _{wq} } \right) + D_{2i}^\psi \left( {\beta _{w{\rm p}} + \beta _{w{\rm p}} } \right)+\\ \qquad \bar{Q}_{xi,{\rm s}} - \bar{Q}_{yi,{\rm s}} - \bar{Q}_{zi,{\rm s}} - \bar{Q}_{ni,{\rm s}}$ (35)

${{ w}}_{{\rm a},{\rm p}}$为脉动项,其表达式为

 ${{ w}}_{{\rm a},{\rm p}} = \Big[0,\;0,\;0,\;{{ 0}}^{\rm T},\;{{ 0}}^{\rm T},\; {{ 0}}^{\rm T},\; - \bar{F}_{BY,{\rm p}} ,\; - \bar{F}_{BZ,p} ,\\ \qquad - \bar{M}_{BZ,{\rm p}} ,- \bar{M}_{BY,{\rm p}} ,\;\bar{M}_{BX,{\rm p}} ,\;{{\bar{ F}}}_{Q,{\rm p}}^{\rm T} ,\;{{ 0}}^{\rm T},{{ 0}}^{\rm T}\Big]^{\rm T}$ (36)

 ${{ w}}_{{\rm a},{\rm p}1}^{\rm T} = [0,\;0,\;0,\;{{ 0}}^{\rm T},\;{{ 0}}^{\rm T},\;{{ 0}}^{\rm T},\; - \bar{F}_{BY,p} ,\; - \bar{F}_{BZ,{\rm p}} ,\\ \qquad- \bar{M}_{BZ,{\rm p}} ,- \bar{M}_{BY,{\rm p}} ,\;\bar{M}_{BX,{\rm p}} ,\;{{ 0}}^{\rm T},{{ 0}}^{\rm T},{{ 0}}^{\rm T}]= \tilde{ W}_{F{\rm p}} {{ p}}_{{\rm tc}}\quad \\ {{ w}}_{{\rm a},{\rm p}2}^{\rm T} = [0,0,0,{{ 0}}^{\rm T},{{ 0}}^{\rm T},{{ 0}}^{\rm T},0,0,0,0,0,( {{ M}}_{\rm s}^{ - 1} {{ \varPhi }}_{{\rm sp}}^{\rm T}\cdot$ (37)

 $\qquad\left( {{{ Q}}_0 {{ v}} + {{ Q}}_1 {{\dot{ v}}} + {{ Q}}_2 {{\ddot{ v}}} + {{ Q}}_p {{ p}}_{{\rm tc}} + {{ V \varPhi }}_{{\rm ps}} {{\ddot{ q}}}} \right))^{\rm T},{{ 0}}^{\rm T},{{ 0}}^{\rm T}]=\\ \qquad {{\tilde{ Q}}}_{w,0} {{ v}} + {{\tilde{ Q}}}_{w,1} {{\dot{ v}}} + {{\tilde{ Q}}}_{w,2} {{\ddot{ v}}} + {{\tilde{ Q}}}_{w,p} {{ p}}_{{\rm tc}} + {{\tilde{ V}\ddot { q}}}\quad$ (38)

 ${{ w}}_{{\rm a},{\rm p}} = {{\tilde{ Q}}}_{w,0} {{ v}} + {{\tilde{ Q}}}_{w,1} {{\dot{ v}}} + {{\tilde{ Q}}}_{w,2} {{\ddot{ v}}}+ \\ \qquad (\tilde { W}_{F{\rm p}} + {{\tilde{ Q}}}_{w,{\rm p}} ){{ p}}_{{\rm tc}} + {{\tilde{ V}\ddot { q}}}$ (39)

 ${{ E}}_{\rm a} {{\dot{ x}}}_{\rm a} = {{ A}}_{\rm a} x_{\rm a} + {{ B}}_{\rm a} {{ u}} + {{ w}}_{{\rm a},s} + {{\tilde{ Q}}}_{w,0} {{ v}} + {{\tilde{ Q}}}_{w,1} {{\dot{ v}}}+\\ \qquad {{\tilde{ Q}}}_{w,2} {{\ddot{ v}}} + (\tilde { W}_{F{\rm p}} + {{\tilde{ Q}}}_{w,{\rm p}} ){{ p}}_{{\rm tc}} + {{\tilde{ V}\ddot { q}}}$ (40)

 ${{ x}}^{\rm T} = \left[{{{ x}}_{\rm a}^{\rm T} ,{{ x}}_{\rm p}^{\rm T} } \right]$ (41)

 $\left[{{\begin{array}{*{20}c} {{{\tilde{ E}}}_{\rm a} } & {{{ E}}_{{\rm ap}} } \\ {{{ E}}_{{\rm pa}} } & {{{ E}}_{\rm p} } \\ \end{array} }} \right]\left[{{\begin{array}{*{20}c} {{{\dot{ x}}}_{\rm a} } \\ {{{\dot{ x}}}_{\rm p} } \\ \end{array} }} \right] = \left[{{\begin{array}{*{20}c} {{{ A}}_{\rm a} } & {{{ A}}_{{\rm ap}} } \\ {{{ A}}_{{\rm pa}} } & {{{ A}}_{\rm p} } \\ \end{array} }} \right]\left[{{\begin{array}{*{20}c} {{{ x}}_{\rm a} } \\ {{{ x}}_{\rm p} } \\ \end{array} }} \right] + \\ \qquad \left[{{\begin{array}{*{20}c} {{{ B}}_{\rm a} } \\ {{ 0}} \\ \end{array} }} \right]{{ u}}_{\rm a}+ \left[{{\begin{array}{*{20}c} {{{ w}}_{{\rm a},s} } \\ {{ 0}} \\ \end{array} }} \right]$ (42)

 $\begin{array}{l} {{\tilde{ E}}}_{\rm a} = {{ E}}_{\rm a} + \left[{{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{\tilde{ V}}},{{ 0}},{{ 0}}} \right] \\ {{ E}}_{{\rm ap}} = \left[{{{ 0}},{{ 0}},- {{\tilde{ Q}}}_{w,2} } \right] \\ {{ E}}_{{\rm pa}} = \left[{{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{\tilde{ U}}}_{rq,2} ,{{ 0}},{{ 0}}} \right] \\ {{ A}}_{{\rm ap}} = \left[{{{\tilde{ Q}}}_{w,0} ,\tilde {W}_{F{\rm p}} + {{\tilde{ Q}}}_{w,{\rm p}} ,{{\tilde{ Q}}}_{w,1} } \right] \\ {{ A}}_{{\rm pa}} = \left[{{{ 0}},{{ 0}},{{ 0}},{{\tilde{ U}}}_{rq,0} ,{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{ 0}},{{\tilde{ U}}}_{rq,1} ,{{ 0}},{{ 0}}} \right] \\ \end{array}$
3.2 姿控与跷振大回路耦合模型

 ${{\tilde{ x}}} = \left[{{{ x}}_{\rm a}^{\rm T} ,{{ x}}_{\rm p}^{\rm T} ,{{ x}}_{\rm c}^{\rm T} } \right]^{\rm T}$ (43)

 $\left[{{\begin{array}{*{20}c} {{{\tilde{ E}}}_{\rm a} } & {{{ E}}_{{\rm ap}} } & {{ 0}} \\ {{{ E}}_{{\rm pa}} } & {{{ E}}_{\rm p} } & {{ 0}} \\ {{ 0}} & {{ 0}} & {{{ E}}_{\rm c} } \\ \end{array} }} \right]\left[{{\begin{array}{*{20}c} {{{\dot{ x}}}_{\rm a} } \\ {{{\dot{ x}}}_{\rm p} } \\ {{{\dot{ x}}}_{\rm c} } \\ \end{array} }} \right]=\\\qquad \left[{{\begin{array}{*{20}c} {{{ A}}_{\rm a} } & {{{ A}}_{{\rm ap}} } & {{{ B}}_{\rm a} {{ C}}_{\rm c} } \\ {{{ A}}_{{\rm pa}} } & {{{ A}}_{\rm p} } & {{ 0}} \\ {{{ B}}_{\rm c} {{ C}}_{\rm a} } & {{ 0}} & {{{ A}}_{\rm c} } \\ \end{array} }} \right]\left[{{\begin{array}{*{20}c} {{{ x}}_{\rm a} } \\ {{{ x}}_{\rm p} } \\ {{{ x}}_{\rm c} } \\ \end{array} }} \right] + \left[{{\begin{array}{*{20}c} {{{ w}}_{{\rm a},{\rm s}} } \\ {{ 0}} \\ {{ 0}} \\ \end{array} }} \right]$ (44)

 ${{ y}} = \left[{{\begin{array}{*{20}c} {{{ C}}_{\rm a} } & {{{ C}}_{\rm p} } & {{{ C}}_{\rm c} } \\ \end{array} }} \right]{{\tilde{ x}}}$ (45)

4 算例

 图 5 第150 秒大回路耦合模型特征值分布 Fig. 5 Eigenvalue distribution of the large-loop coupling model at 150th second

 图 6 姿控回路模型助推飞行段的滚转角仿真曲线(无量纲) Fig. 6 Simulation curve of roll angle of attitude control model during boosting time (dimensionless)

 图 7 大回路模型助推飞行段滚转角时域仿真曲线(无量纲) Fig. 7 Simulation curve of roll angle of large-loop model during boosting time (dimensionless)

5 结 论

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DYNAMIC MODELING AND ANALYSIS OF LARGE-LOOP COUPLED BY ATTITUDE CONTROL AND POGO FOR LARGE LIQUID ROCKETS
Wang Qingwei, Tan Shujun, Wu Zhigang, Yang Yunfei, Chen Yu
1. State Key Laroratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China;
2. School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, China;
3. Beijing Aerospace System Engineering Institute, Beijing 100076, China
Fund: The project was supported by the General Program of the National Natural Science Foundation of China (11072044, 11372056), Doctoral Fund of Ministry of Education of China (20110041130001) and Funds of Central Universities (DUT15LK23).
Abstract: Coupling effect always exists between structural vibration, attitude and propulsion system because of the spatial distribution characteristics of the structural vibration mode of large liquid rockets. The coupling model of attitude control-structure-propulsion system is derived to investigate the effects of the propulsion system on the stability of attitude control system. The coupling is based on mechanism that the effects of attitude control system on the attitude motion and structural vibration as well as the interaction of propulsion system and structural vibration. By including the coupling factors between the propulsion system, structural system and attitude control system, the large-loop coupling model can be used to investigate the coupling stability of the large-loop coupling system. Besides, the coupling model can be directly used for frequency-domain analysis and time-domain simulation for the non-singularity. Based on this model, the effects of the propulsion system parameters-pump gain and accumulator energy value to the stability of attitude control system are analyzed by frequency domain analysis and time domain simulation. The results show that the variations of parameters-pump gain and accumulator energy value not only result in the instability of structural vibration but also lead to the instability of attitude motion. It is concluded that it is necessary to take the effect of POGO loop into count in the design of attitude control system.
Key words: attitude control loop    POGO loop    large-loop coupling model    modeling    stability analysis