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 浙江大学学报(工学版)  2017, Vol. 51 Issue (7): 1428-1436  DOI:10.3785/j.issn.1008-973X.2017.07.022 0

### 引用本文 [复制中英文]

dx.doi.org/10.3785/j.issn.1008-973X.2017.07.022
[复制中文]
ZHENG Xue-ke, WANG Xiao-liang. Stratospheric airship control considering propeller dynamic model[J]. Journal of Zhejiang University(Engineering Science), 2017, 51(7): 1428-1436.
dx.doi.org/10.3785/j.issn.1008-973X.2017.07.022
[复制英文]

### 作者简介

orcid/org/0000-0002-0549-6434.
Email: zxk2046@sjtu.edu.cn

### 通信联系人

orcid/org/0000-0002-6343-8427.
Email: wangxiaoliang@sjtu.edu.cn

### 文章历史

Stratospheric airship control considering propeller dynamic model
ZHENG Xue-ke , WANG Xiao-liang
School of Aeronautics and Astronautics, Shanghai Jiaotong University, Shanghai 200240, China
Abstract: A novel propeller dynamic model combining with the 6-DOF airship dynamic model was proposed based on the fact that the multi-vectored thrust stratospheric airship has differences between desired thrusts and actual thrusts, high energy use, as well as the large value of the motor torque. The propeller dynamic model directly considered the command torque of the motor installed in the propeller system as the control input. A motor-propeller dynamic equation and a propeller thrust loss dynamic equation considering external disturbances and differences between flows on the propeller disk were proposed. The feedback control law based on the nonlinear state observer estimated the propeller thrust loss and the Lyapunov function analysis was introduced to guarantee "input-to-state" stabilities of the feedback system and the nonlinear observer system. A comparative study was conducted using the new model and the conventional one. The simulation results indicate that the proposed method can track three Cartesian positions and three Euler attitude angles better than the conventional one. The consumed energy is much less. The motor torque of the new dynamic model is less than that of the conventional model, which reduces the occurrence of the motor torque saturation problem.
Key words: stratospheric airship    propeller    thrust loss    nonlinear state observer    saturation

1 飞艇动力学模型 1.1 不考虑螺旋桨系统的动力学模型

Chen等[2]提出的多螺旋桨组合浮空器结构如图 1所示.

 图 1 多螺旋桨组合浮空器结构图 Fig. 1 Structure of multi-vectored airship

 $\mathit{\boldsymbol{M}}{\left( {\dot u, \dot v, \dot w, \dot p, \dot q, \dot r} \right)^{\rm{T}}} = {\mathit{\boldsymbol{F}}_{\rm{A}}} + {\mathit{\boldsymbol{F}}_{{\rm{GB}}}} + {\mathit{\boldsymbol{F}}_{\rm{I}}} + {\mathit{\boldsymbol{F}}_{\rm{T}}}.$ (1)

 ${\mathit{\boldsymbol{F}}_{\rm{A}}} = \left[{\begin{array}{*{20}{c}} {{F_{\rm{a}}} \cdot \cos \mathit{\Omega }}\\ {{F_{\rm{a}}} \cdot \sin \mathit{\Omega }}\\ {{q_\infty } \cdot {C_z} \cdot {S_{{\rm{ref}}}}}\\ {-{M_{\rm{a}}} \cdot \cos \mathit{\Omega }}\\ {{M_{\rm{a}}} \cdot \sin \mathit{\Omega }}\\ 0 \end{array}} \right].$ (2)

 ${F_{\rm{a}}} = {q_\infty } \cdot {C_x} \cdot {S_{{\rm{ref}}}},$ (3)
 ${M_{\rm{a}}} = {q_\infty } \cdot {C_{{\rm{m}}y}} \cdot {S_{{\rm{ref}}}} \cdot {L_{{\rm{ref}}}}.$ (4)

 图 2 飞艇空气动力系数 Fig. 2 Aerodynamic coefficient for airship

FT=[fx, fy, fz, mx, my, mz]T作为飞艇控制输入量.飞艇状态方程为

 $\left. \begin{array}{l} {{\mathit{\boldsymbol{\dot x}}}_1} = {\mathit{\boldsymbol{G}}_1}{\mathit{\boldsymbol{x}}_2}, \\ {{\mathit{\boldsymbol{\dot x}}}_2} = {\mathit{\boldsymbol{G}}_2}\left( {\mathit{\boldsymbol{f}} + {\mathit{\boldsymbol{F}}_{\rm{T}}}} \right). \end{array} \right\}$ (5)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{G}}_1} = \left[{\begin{array}{*{20}{c}} {\cos \psi \cos \theta }&{\cos \psi \sin \theta \sin \varphi-sin\psi cos\varphi }&{\cos \psi \sin \theta \cos \varphi + \sin \psi \sin \varphi }&0&0&0\\ {\sin \psi \cos \theta }&{\sin \psi \sin \theta \sin \varphi + \cos \psi \cos \varphi }&{\sin \psi \sin \theta \cos \varphi-\cos \psi \sin \varphi }&0&0&0\\ {-\sin \theta }&{\cos \theta \sin \varphi }&{\cos \theta \cos \varphi }&0&0&0\\ 0&0&0&1&{\sin \varphi \tan \theta }&{\cos \varphi \tan \theta }\\ 0&0&0&0&{\cos \varphi }&{ - \sin \varphi }\\ 0&0&0&0&{\frac{{\sin \varphi }}{{\cos \theta }}}&{\frac{{\cos \varphi }}{{\cos \theta }}} \end{array}} \right], }\\ {{\mathit{\boldsymbol{G}}_2} = {{\left[{\begin{array}{*{20}{c}} {m + {m_{11}}}&0&0&0&{m{x_{\rm{G}}}}&{-m{y_{\rm{G}}}}\\ 0&{m + {m_{22}}}&0&{-m{z_{\rm{G}}}}&0&{m{x_{\rm{G}}}}\\ 0&0&{m + {m_{33}}}&{m{y_{\rm{G}}}}&{-m{x_{\rm{G}}}}&0\\ 0&{ - m{z_{\rm{G}}}}&{m{y_{\rm{G}}}}&{{I_x} + {m_{44}}}&0&0\\ {m{z_{\rm{G}}}}&0&{ - m{x_{\rm{G}}}}&0&{{I_x} + {m_{55}}}&0\\ { - m{y_{\rm{G}}}}&{m{x_{\rm{G}}}}&0&0&0&{{I_x} + {m_{44}}} \end{array}} \right]}^{ -1}}.} \end{array}$

1.2 飞艇PI控制器

 ${\mathit{\boldsymbol{x}}_{1{\rm{c}}}} = {\left[{{x_{\rm{c}}}, \;\;{y_{\rm{c}}}, \;\;{z_{\rm{c}}}, \;\;{\varphi _{\rm{c}}}, \;\;{\theta _{\rm{c}}}, \;\;{\psi _{\rm{c}}}, } \right]^{\rm{T}}}.$

 ${{\mathit{\boldsymbol{\dot x}}}_{1{\rm{c}}}} = {\mathit{\boldsymbol{K}}_{\rm{p}}}\left( {{\mathit{\boldsymbol{x}}_{1{\rm{c}}}}-{\mathit{\boldsymbol{x}}_1}} \right) + {\mathit{\boldsymbol{K}}_{\rm{i}}}\int {\left( {{\mathit{\boldsymbol{x}}_{1{\rm{c}}}}-{\mathit{\boldsymbol{x}}_1}} \right){\rm{d}}t} .$ (6)

x2c=[ur, vr, wr, pr, qr, rr]T

 ${\mathit{\boldsymbol{x}}_{2{\rm{c}}}} = \mathit{\boldsymbol{G}}_1^{-1}{{\mathit{\boldsymbol{\dot x}}}_{1{\rm{c}}}}.$ (7)

 ${\mathit{\boldsymbol{F}}_{\rm{T}}} = \mathit{\boldsymbol{G}}_2^{-1}{{\mathit{\boldsymbol{\dot x}}}_{2{\rm{c}}}}-\mathit{\boldsymbol{f}}.$ (8)

 ${\mathit{\boldsymbol{F}}_{\rm{T}}} = \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{F}}_{{T_{{\rm{HV}}}}}}.$ (9)

 ${\mathit{\boldsymbol{F}}_{{T_{{\rm{HV}}}}}} = {\mathit{\boldsymbol{B}}^{-1}}{\mathit{\boldsymbol{F}}_{\rm{T}}} = {\mathit{\boldsymbol{B}}^{-1}}\left( {-\mathit{\boldsymbol{f + G}}_2^{ - 1}{{\mathit{\boldsymbol{\dot x}}}_{2{\rm{c}}}}} \right).$ (10)

 $\mathit{\boldsymbol{B = }}\left[{\begin{array}{*{20}{c}} 0&1&0&{-1}&0&0&0&0\\ {-1}&0&1&0&0&0&0&0\\ 0&0&0&0&{-1}&{ - 1}&{ - 1}&{ - 1}\\ 0&0&0&0&0&{{R_{\rm{p}}}}&0&{{R_{\rm{p}}}}\\ 0&0&0&0&{{R_{\rm{p}}}}&0&{ - {R_{\rm{p}}}}&0\\ { - {R_{\rm{p}}}}&{ - {R_{\rm{p}}}}&{ - {R_{\rm{p}}}}&{ - {R_{\rm{p}}}}&0&0&0&0 \end{array}} \right].$

 ${\mathit{\boldsymbol{F}}_{{T_{{\rm{HV}}}}}} = \left[{\begin{array}{*{20}{c}} {{f_{{\rm{1H}}}}}\\ {{f_{{\rm{2H}}}}}\\ {{f_{{\rm{3H}}}}}\\ {{f_{{\rm{4H}}}}}\\ {{f_{{\rm{1V}}}}}\\ {{f_{{\rm{2V}}}}}\\ {{f_{{\rm{3V}}}}}\\ {{f_{{\rm{4V}}}}} \end{array}} \right].$

 ${\mu _i} = \arcsin \frac{{{f_{i{\rm{H}}}}}}{{\sqrt {f_{i{\rm{H}}}^2 + f_{i{\rm{V}}}^2} }};i = 1, 2, 3, 4.$ (11)

FTHV和实际螺旋桨推力T=[T1, T2, T3, T4]T的关系为

 ${\mathit{\boldsymbol{F}}_{{T_{{\rm{HV}}}}}} = \left[{\begin{array}{*{20}{c}} {\sin {\mu _1}}&0&0&0\\ 0&{\sin {\mu _2}}&0&0\\ 0&0&{\sin {\mu _3}}&0\\ 0&0&0&{\sin {\mu _4}}\\ {-\cos {\mu _1}}&0&0&0\\ 0&{-\cos {\mu _2}}&0&0\\ 0&0&{-\cos {\mu _3}}&0\\ 0&0&0&{ - \cos {\mu _4}} \end{array}} \right]\mathit{\boldsymbol{T}}.$ (12)
2 螺旋桨动力学模型

 $2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}\dot n = {Q_{\rm{m}}}-{Q_{\rm{p}}}-2{\rm{ \mathsf{ π} }}{K_{\rm{n}}}n.$ (13)

 $J = \frac{{{u_{\rm{a}}}}}{{nD}}.$ (14)

 图 3 螺旋桨在不同轴向来流速度下的推力曲线 Fig. 3 Thrusts at different axis flow velocity

 ${T_{\rm{p}}} = {G_{{T_{\rm{p}}}}}{n^2} + {\Delta _{\rm{T}}}.$ (15)

 ${{\dot \Delta }_{\rm{T}}} =-\frac{1}{\tau }{\Delta _{\rm{T}}} + {w_{\rm{d}}}.$ (16)

ΔT作为一个随时间变化的状态量, 以一阶状态方程表示.因为螺旋桨桨盘来流速度未知、气流非定常扰动等原因, 得到ΔT的准确模型是非常困难的, 但在估计未知变量的时候经常使用该模型[9].

 ${K_{\rm{T}}} = \frac{{{T_{\rm{p}}}}}{{\rho {D^4}{n^2}}},$ (17)
 ${K_{\rm{Q}}} = \frac{{{Q_{\rm{p}}}}}{{\rho {D^5}{n^2}}}.$ (18)

KTKQJ的一次函数.KTKQ可以近似为

 ${K_{\rm{T}}} = {\alpha _1}J + {\alpha _2},$ (19)
 ${K_{\rm{Q}}} = {\beta _1}J + {\beta _2}.$ (20)

 图 4 螺旋桨推力和扭矩系数 Fig. 4 Thrust and torque coefficients for propeller

Smogeli等[10-11]用期望推力和估计进距比$\hat J$来计算KQ(J), 但是这不准确, 因为$\hat J$不等于J, 这会带来误差甚至造成系统的不稳定.

 $J = \frac{{\frac{{{G_{{T_{\rm{p}}}}}{n^2} + {\Delta _{\rm{T}}}}}{{\rho {D^4}{n^2}}}-{\alpha _2}}}{{{\alpha _1}}}.$ (21)

 ${Q_{\rm{p}}} = {k_1}{n^2} + {k_2}{\Delta _{\rm{T}}}.$ (22)

 $\begin{array}{*{20}{c}} {{k_1} = D\left[{\frac{{{\beta _1}}}{{{\alpha _1}}}{G_{{T_{\rm{p}}}}}-\left( {\frac{{{\beta _1}{\alpha _2}}}{{{\alpha _1}}}-{\beta _2}} \right)\rho {D^4}} \right], }\\ {{k_2} = \frac{{{\beta _1}}}{{{\alpha _1}}}D.} \end{array}$
3 非线性状态观测器

 $2{\rm{\pi }}{J_{\rm{m}}}\dot n = {Q_{\rm{m}}}-{k_1}{n^2}-2{\rm{\pi }}{K_{\rm{n}}}n-{k_2}{\Delta _{\rm{T}}},$ (23)
 ${{\dot \Delta }_{\rm{T}}} =-\frac{1}{\tau }{\Delta _{\rm{T}}} + {w_{\rm{d}}}.$ (24)

 $y = n.$ (25)

 $\begin{matrix} 2\text{ }\!\!\pi\!\!\text{ }{{J}_{\text{m}}}\dot{\hat{n}}={{Q}_{\text{m}}}-{{k}_{1}}{{{\hat{n}}}^{2}}-2\text{ }\!\!\pi\!\!\text{ }{{K}_{\text{n}}}\hat{n}-\\ {{k}_{2}}{{{\hat{\Delta }}}_{\text{T}}}+l\left( y-\hat{y} \right), \\ \end{matrix}$ (26)
 ${{{\dot{\hat{\Delta }}}}_{\text{T}}}=-\frac{1}{\tau }{{{\hat{\Delta }}}_{\text{T}}}+m\left( y-\hat{y} \right).$ (27)

 $2\text{ }\!\!\pi\!\!\text{ }{{J}_{\text{m}}}\dot{\tilde{n}}=-\left( {{k}_{1}}{{n}^{2}}-{{k}_{1}}{{{\hat{n}}}^{2}} \right)-\left( l+2\text{ }\!\!\pi\!\!\text{ }{{K}_{\text{n}}} \right)\tilde{n}-{{k}_{2}}{{{\tilde{\Delta }}}_{\text{T}}},$ (28)
 ${{{\dot{\tilde{\Delta }}}}_{\text{T}}}=-\frac{1}{\tau }{{{\tilde{\Delta }}}_{\text{T}}}-m\tilde{n}+w.$ (29)

 $l >-2{\rm{\pi }}{K_{\rm{n}}},$ (30)
 $\left| {\frac{{{k_2}l}}{{2{\rm{\pi }}{J_{\rm{m}}}}} + m} \right| < 2\sqrt {\frac{{2{\rm{\pi }}{K_{\rm{n}}} + l}}{{2{\rm{\pi }}\tau {J_{\rm{m}}}}}} .$ (31)

 $\begin{array}{l} {{\dot V}_{\rm{o}}}\left( {\tilde n, {{\tilde \Delta }_{\rm{T}}}} \right) =-\left( {\frac{{{K_{\rm{n}}}}}{{{J_{\rm{m}}}}}{{\tilde n}^2} + \frac{l}{{2{\rm{\pi }}{J_{\rm{m}}}}}} \right){{\tilde n}^2}-\frac{1}{\tau }\tilde \Delta _{\rm{T}}^2-\\ \;\;\;\left( {\frac{1}{{2{\rm{\pi }}{J_{\rm{m}}}}}{k_2} + m} \right)\tilde n{{\tilde \Delta }_{\rm{T}}} + w{{\tilde \Delta }_{\rm{T}}} - \frac{1}{{2{\rm{\pi }}{J_{\rm{m}}}}}{k_1}\left( {{n^2} - {{\hat n}^2}} \right)\tilde n. \end{array}$ (32)

 $\begin{array}{l} {{\dot V}_{\rm{o}}}\left( {\tilde n, {{\tilde \Delta }_{\rm{T}}}} \right) \le-\left( {\frac{{{K_{\rm{n}}}}}{{{J_{\rm{m}}}}}{{\tilde n}^2} + \frac{l}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}}} \right){{\tilde n}^2}-\frac{1}{\tau }\tilde \Delta _{\rm{T}}^2-\\ \;\;\;\;\left( {\frac{1}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}}{k_2} + m} \right)\tilde n{{\tilde \Delta }_{\rm{T}}} + w{{\tilde \Delta }_{\rm{T}}} \le - \mathit{\boldsymbol{\tilde e}}_{\rm{o}}^{\rm{T}}\mathit{\boldsymbol{Q}}{{\mathit{\boldsymbol{\tilde e}}}_{\rm{o}}} + {{\tilde \Delta }_{\rm{T}}}w. \end{array}$ (33)

 $\mathit{\boldsymbol{Q = }}\left[{\begin{array}{*{20}{c}} {\frac{{2{\rm{ \mathsf{ π} }}{K_{\rm{n}}} + l}}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}}}&{\frac{1}{2}\left( {\frac{{{k_2}l}}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}} + m} \right)}\\ {\frac{1}{2}\left( {\frac{{{k_2}l}}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}} + m} \right)}&{\frac{1}{\tau }} \end{array}} \right].$

 $\begin{array}{l} {{\dot V}_{\rm{o}}} \le-{\lambda _{\min }}{\lambda _\mathit{\boldsymbol{Q}}}{\left\| {{\mathit{\boldsymbol{e}}_{\rm{o}}}} \right\|^2} + \left| {{{\tilde \Delta }_{\rm{T}}}} \right|\left| {{w_{\rm{d}}}} \right| \le-{\lambda _{\min }}{\lambda _\mathit{\boldsymbol{Q}}}{\left\| {{\mathit{\boldsymbol{e}}_{\rm{o}}}} \right\|^2} + \\ \left\| {{\mathit{\boldsymbol{e}}_{\rm{o}}}} \right\|\left| {{w_{\rm{d}}}} \right| =-\left( {1 - \theta } \right){\lambda _{\min }}{\lambda _\mathit{\boldsymbol{Q}}}{\left\| {{\mathit{\boldsymbol{e}}_{\rm{o}}}} \right\|^2} - \theta {\lambda _{\min }}{\lambda _\mathit{\boldsymbol{Q}}}{\left\| {{\mathit{\boldsymbol{e}}_{\rm{o}}}} \right\|^2} + \\ \left\| {{\mathit{\boldsymbol{e}}_{\rm{o}}}} \right\|\left| {{w_{\rm{d}}}} \right|. \end{array}$ (34)

 $\left\| {{\mathit{\boldsymbol{e}}_{\rm{o}}}} \right\| \ge \frac{{\left| {{w_{\rm{d}}}} \right|}}{{\theta {\lambda _{\min }}{\lambda _\mathit{\boldsymbol{Q}}}}} = \rho \left( {\left| {{w_{\rm{d}}}} \right|} \right),$ (35)

 $\dot V \le-\left( {1-\theta } \right){\lambda _{\min }}{\lambda _\mathit{\boldsymbol{Q}}}{\left\| {{\mathit{\boldsymbol{e}}_{\rm{o}}}} \right\|^2}.$ (36)

4 螺旋桨动力学模型控制律设计

 ${{\bar n}_{\rm{d}}} = \sqrt {\frac{{{T_{{\rm{pd}}}}-{{\hat \Delta }_{\rm{T}}}}}{{{G_{{T_{\rm{P}}}}}}}} .$ (37)

 ${{\ddot n}_{\rm{d}}} + 2{\omega _{\rm{c}}}\xi {{\dot n}_{\rm{d}}} + \omega _{\rm{c}}^2{n_{\rm{d}}} = \omega _{\rm{c}}^2{{\bar n}_{\rm{d}}}.$ (38)

 $\begin{array}{l} {Q_{\rm{m}}} = {k_1}n\hat n + 2{\rm{\pi }}{K_{\rm{n}}}\hat n + {k_2}{{\hat \Delta }_{\rm{T}}} + \\ \;2{\rm{\pi }}{J_{\rm{m}}}{{\dot n}_{\rm{d}}}-{K_{{\rm{p1}}}}e-{K_{{\rm{p2}}}}{n^2}e. \end{array}$ (39)

 ${K_{{\rm{p1}}}} > \frac{{{\rm{\pi }}K_{\rm{n}}^2}}{{2{J_{\rm{m}}}}} + \frac{{\tau k_2^2}}{{2{\rm{\pi }}{J_{\rm{m}}}}}, {K_{{\rm{p2}}}} > \frac{{k_1^2}}{{8{\rm{\pi }}{J_{\rm{m}}}}},$

 图 5 飞艇控制系统框图 Fig. 5 Block diagram of whole control system

 $\dot e = \frac{1}{{2{\rm{\pi }}{J_{\rm{m}}}}}\left( {{Q_{\rm{m}}}-{k_1}{n^2}-2{\rm{\pi }}{K_{\rm{n}}}n-{k_2}{\Delta _{\rm{T}}} - 2{\rm{\pi }}{J_{\rm{m}}}{{\dot n}_{\rm{d}}}} \right).$ (40)

 $\begin{array}{l} \dot V \le e\dot e- \frac{{\left( {2{\rm{\pi }}{K_{\rm{n}}} + l} \right)}}{{2{\rm{\pi }}{J_{\rm{m}}}}}{{\tilde n}^2}- \left( {\frac{{{k_2}l}}{{2{\rm{\pi }}{J_{\rm{m}}}}} + m} \right)\tilde n{{\tilde \Delta }_{\rm{T}}}- \frac{1}{\tau }\tilde \Delta _{\rm{T}}^2 + \\ \;\;\;\;{{\tilde \Delta }_{\rm{T}}}{w_{\rm{d}}} \le e\left( {\dot n - {{\dot n}_{\rm{d}}}} \right) - \frac{{\left( {2{\rm{\pi }}{K_{\rm{n}}} + l} \right)}}{{2{\rm{\pi }}{J_{\rm{m}}}}}{{\tilde n}^2} - \\ \;\;\;\;\left( {\frac{{{k_2}l}}{{2{\rm{\pi }}{J_{\rm{m}}}}} + m} \right)\tilde n{{\tilde \Delta }_{\rm{T}}} - \frac{1}{\tau }\tilde \Delta _{\rm{T}}^2 + {{\tilde \Delta }_{\rm{T}}}{w_{\rm{d}}} \le \\ \;\;\;\;e\left[{\frac{1}{{2{\rm{\pi }}{J_{\rm{m}}}}}\left( {{Q_{\rm{m}}}-{k_1}{n^2}-2{\rm{\pi }}{K_{\rm{n}}}n-{k_2}{\Delta _{\rm{T}}}} \right) - {{\dot n}_{\rm{d}}}} \right] -\\ \;\;\;\;\frac{{\left( {2{\rm{\pi }}{K_{\rm{n}}} + l} \right)}}{{2{\rm{\pi }}{J_{\rm{m}}}}}{{\tilde n}^2} -\left( {\frac{{{k_2}l}}{{2{\rm{\pi }}{J_{\rm{m}}}}} + m} \right)\tilde n{{\tilde \Delta }_{\rm{T}}} -\frac{1}{\tau }\tilde \Delta _{\rm{T}}^2 + \tilde \Delta _{\rm{T}}^2{w_{\rm{d}}}. \end{array}$ (41)

 $\begin{array}{l} \dot V \le \frac{1}{{2{\rm{\pi }}{J_{\rm{m}}}}}\left( {-e{k_1}n\tilde n-2{\rm{\pi }}{K_{\rm{n}}}e\tilde n-{k_2}e{{\tilde \Delta }_{\rm{T}}}} \right) - \\ \;\;\;\;\;\frac{{\left( {2{\rm{\pi }}{K_{\rm{n}}} + l} \right)}}{{2{\rm{\pi }}{J_{\rm{m}}}}}{{\tilde n}^2} - \left( {\frac{{{k_2}l}}{{2{\rm{\pi }}{J_{\rm{m}}}}} + m} \right)\tilde n{{\tilde \Delta }_{\rm{T}}} - \frac{1}{\tau }\tilde \Delta _{\rm{T}}^2 + {{\tilde \Delta }_{\rm{T}}}{w_{\rm{d}}}. \end{array}$ (42)

 $-\frac{1}{{2{\rm{\pi }}{J_{\rm{m}}}}}e{k_1}n\tilde n =-{\left( {\frac{{{k_1}}}{{4{\rm{\pi }}{J_{\rm{m}}}}}ne + \tilde n} \right)^2} + \left( {\frac{{k_1^2}}{{16{{\rm{\pi }}^2}J_{\rm{m}}^2}}{n^2}{e^2} + {{\tilde n}^2}} \right),$ (43)
 $-\frac{{{K_{\rm{n}}}}}{{{J_{\rm{m}}}}}e\tilde n =-{\left( {\frac{{{K_{\rm{n}}}}}{{2{J_{\rm{m}}}}}e + \tilde n} \right)^2} + \left( {\frac{{K_{\rm{n}}^2}}{{4J_{\rm{m}}^2}}{e^2} + {{\tilde n}^2}} \right),$ (44)
 $\begin{array}{l} -\frac{{{k_2}e{{\tilde \Delta }_{\rm{T}}}}}{{2{\rm{\pi }}{J_{\rm{m}}}}} =-{\left( {\frac{{\sqrt \tau {k_2}}}{{2{\rm{\pi }}{J_{\rm{m}}}}}e + \frac{1}{{2\sqrt \tau }}{{\tilde \Delta }_{\rm{T}}}} \right)^2} + \\ \;\;\left( {\frac{{\tau k_2^2}}{{4{{\rm{\pi }}^2}J_{\rm{m}}^2}}{e^2} + \frac{1}{{4\tau }}\tilde \Delta _{\rm{T}}^2} \right). \end{array}$ (45)

 $\begin{array}{l} \dot V \le \left( {\frac{{k_1^2}}{{16{{\rm{ \mathsf{ π} }}^2}J_{\rm{m}}^2}}-\frac{{{K_{{\rm{p}}2}}}}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}}} \right){n^2}{e^2} + \\ \;\;\;\;\left( {\frac{{K_{\rm{n}}^2}}{{4J_{\rm{m}}^2}}-\frac{{{K_{{\rm{p}}1}}}}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}} + \frac{{\tau k_2^2}}{{4{{\rm{ \mathsf{ π} }}^2}J_{\rm{m}}^2}}} \right){\mathit{\boldsymbol{e}}^2} + \mathit{\boldsymbol{e}}_{\rm{o}}^{\rm{T}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{e}}_{\rm{o}}} + {{\tilde \Delta }_{\rm{T}}}{w_{\rm{d}}}. \end{array}$ (46)

 $\mathit{\boldsymbol{P}} = \left[{\begin{array}{*{20}{c}} {\frac{{2{\rm{ \mathsf{ π} }}{K_{\rm{n}}} + l-4{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}}}&{\frac{1}{2}\left( {\frac{{{k_2}l}}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}} + m} \right)}\\ {\frac{1}{2}\left( {\frac{{{k_2}l}}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}} + m} \right)}&{\frac{3}{{4\tau }}} \end{array}} \right].$

 $\begin{array}{l} {K_{{\rm{p}}1}} > \frac{{{\rm{ \mathsf{ π} }}K_{\rm{n}}^2}}{{2{J_{\rm{m}}}}} + \frac{{\tau k_2^2}}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}}, \\ {K_{{\rm{p}}2}} > \frac{{k_1^2}}{{8{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}}, \\ l > 4{\rm{ \mathsf{ π} }}{J_{\rm{m}}}-2{\rm{ \mathsf{ π} }}{K_{\rm{n}}}, \\ \left| {\frac{{{k_2}l}}{{2{\rm{ \mathsf{ π} }}{J_{\rm{m}}}}} + m} \right| < 2\sqrt {\frac{{3\left( {2{\rm{ \mathsf{ π} }}{K_{\rm{n}}} + l-4{\rm{ \mathsf{ π} }}{J_{\rm{m}}}} \right)}}{{8{\rm{ \mathsf{ π} }}\tau {J_{\rm{m}}}}}}, \end{array}$

5 浮空器控制仿真验证

 $P = 2{\rm{\pi }}\sum\limits_{i = 1}^4 {{Q_{{\rm{p}}i}} \cdot {n_i}} .$ (47)

 $E = \int\limits_0^t {P{\rm{d}}t} .$ (48)

 ${T_{{\rm{pd}}}} = \sqrt {f_{i{\rm{H}}}^2 + f_{i{\rm{V}}}^2} ;i = 1, 2, 3, 4.$ (49)
 ${Q_{\rm{m}}} = \frac{{D{K_{\rm{Q}}}\left| {_{J = 0}} \right.}}{{{K_{\rm{T}}}\left| {_{J = 0}} \right.}}{T_{{\rm{pd}}}}.$ (50)

5.1 飞艇“外循环”仿真

 图 6 功率随时间的变化图 Fig. 6 Time history of consumed power
 图 7 能量随时间的变化图 Fig. 7 Time history of consumed energy
 图 8 x、y、z位置轨迹随时间的变化图 Fig. 8 Time history of x, y, z tracking
 图 9 vx、vy、vz速度轨迹随时间变化图 Fig. 9 Time history of vx, vy, vz tracking
 图 10 姿态角随时间的变化图 Fig. 10 Time history of angle tracking
5.2 螺旋桨“内循环”仿真

 图 11 噪声随时间的变化图 Fig. 11 Time history of bounded input noise
 图 12 期望、实际、估计推力随时间的变化图 Fig. 12 Time history of desired thrusts, actual thrusts and estimated thrusts of propeller system
 图 13 电机命令扭矩随时间的变化图 Fig. 13 Time history of command motor torques
6 结论

(1) 本文针对飞艇的控制问题, 结合电机螺旋桨动力学模型与飞艇的六自由度动力学模型, 通过直接把电机螺旋桨扭矩信号作为控制系统的输入量, 引入电机-螺旋桨和螺旋线推力损失动力学方程, 使得飞艇模型更加准确, 避免螺旋桨性能变化所导致的控制效果变差甚至失效问题.

(2) 考虑电机螺旋桨动力学模型, 基于非线性状态观测器的反馈控制方法对螺旋桨桨推力损失进行在线估计和追踪期望推力.从理论上证明了观测器和控制器的“输入-状态”稳定性.

(3) 本文提出的方法与传统方法相比, 能够更好地跟踪位置期望值.同时, 可以大大地降低动力系统能源需求和螺旋桨电机命令扭矩, 从而延长飞艇持续巡航时间, 并避免电机由于过早出现饱和而导致的系统不稳定性.

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