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摘要: 针对一类四旋翼飞行器吊挂飞行系统的负载摆动抑制和轨迹跟踪精确控制的问题, 考虑系统存在未知外界扰动和模型动态不确定的情况, 提出一种基于扩张状态观测器(Extended state observer, ESO)的吊挂负载摆动抑制的非线性轨迹跟踪控制方法. 将四旋翼吊挂飞行系统分解为姿态、位置和负载摆动控制三个动态子系统, 分别设计非线性控制器实现欠驱动约束下的解耦控制; 设计一种扩张状态观测器, 用以估计和补偿四旋翼与吊挂负载耦合飞行的未知外界扰动与模型动态不确定性, 并验证了闭环系统的稳定性, 跟踪误差及吊挂负载摆动所有信号的一致最终有界. 最后, 利用Quanser公司的Qball2飞行器进行三维空间螺旋轨迹的跟踪控制, 仿真结果验证了未知干扰下基于扩张状态观测器的四旋翼吊挂飞行非线性控制的有效性和优越性, 实现了四旋翼吊挂系统轨迹跟踪的精确控制和飞行过程中负载摆动的快速抑制.Abstract: In order to solve the problem of load swing suppression and precise control of trajectory tracking for a class of quadrotors, considering the unknown external disturbance and model dynamic uncertainty, a nonlinear trajectory tracking control method based on extended state observer (ESO) is proposed. In this paper, the quadrotor suspension flight system is divided into three dynamic subsystems: Attitude, position and swing angle of the suspending load. Nonlinear controllers are designed to realize decoupling control under drive constraints. An extended state observer is designed to estimate and compensate the unknown external disturbance and model dynamic uncertainty in coupled flight of quadrotor and suspended load, so as to ensure the stability of the closed-loop system, tracking error and all signals in suspension system are uniformly ultimately bounded. Finally, Quanser＇s Qball2 aircraft is used to track the spiral trajectory in three-dimensional space. The simulation results show the effectiveness and superiority of the proposed control method, and realize the precise control of the trajectory tracking of the quadrotor suspension system and the rapid suppression of the load swing during the flight.
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Key words:
- Quadrotor /
- suspension flight /
- integral back-stepping /
- extended state observer (ESO)
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飞行吊挂运输因其对地面环境要求低, 无需考虑负载外形等优势, 在军用、民用领域均有广泛应用[1-3]. 传统飞行吊挂运输以单旋翼直升机为主[4-6], 近年随着四旋翼飞行器技术的发展[7-8], 四旋翼飞行器吊挂负载的控制问题逐渐成为研究热点[9].
四旋翼飞行器吊挂系统是一个多自由度、强耦合、欠驱动的复杂系统, 相比单体控制的难度增大. 越来越多学者对四旋翼吊挂系统的控制进行深入研究, 取得许多成果[10]. 文献[11]针对吊挂飞行中空气阻尼和未知载荷质量进行参数估计, 用能量法设计非线性控制器, 但动力学模型和控制器均局限于二维平面. 文献[12]用微分平滑性控制系统使系统快速稳定, 以实现吊挂飞行系统的定点控制, 但也仅研究二维平面情况. 在实际三维空间中, 系统的非线性项耦合程度加深, 适用于二维平面的设计方法不能简单扩展到三维. 因此, 在保留系统非线性的前提下将控制系统扩展到三维空间, 成为当前重点研究方向.
文献[13]考虑飞行器与吊挂负载耦合的问题, 用牛顿−欧拉方法对负载摆动效应的系统建模进行详细分析, 并设计滑模控制算法, 对系统不确定性和耦合负载摆动效应具有很强的鲁棒性. 文献[14]针对吊挂飞行系统的位置控制及负载摆动抑制问题, 采用能量分析的方法, 设计非线性控制器. 仿真结果表明, 该方法可以一定程度地抑制吊挂负载摆动. 文献[15]考虑负载离开地面的过程, 将其分为起飞、拉伸和上升三个状态. 通过设计轨迹跟踪点实现吊挂负载的最小摆动轨迹, 减小飞行器吊挂负载起飞时负载的摆动. 文献[16]考虑系统的启动问题设计混合系统表示飞行过程中的状态, 并设计非线性控制器来跟踪一系列与状态相关的路径点生成的轨迹, 可显著改善系统的跟踪性能. 文献[17]为实现吊挂摆角抑制和飞行器轨迹跟踪, 将系统模型线性化, 设计基于时间分段的非线性控制器, 在负载产生较大摆动时为快动态, 优先抑制负载摆动, 摆动消除后再进入慢动态, 进行轨迹跟踪. 文献[18]考虑吊挂系统执行紧急任务时的轨迹控制问题, 通过动态反馈将系统转化为线性能控系统, 并设计动态反馈控制律, 实现摆角大角度振荡时的负载轨迹跟踪. 文献[19]为实现有效负载摆动抑制的目标, 利用分层控制方法设计飞行器吊挂空运系统的非线性控制方法, 有效消除负载的摆动, 保证飞行器的精确定位. 文献[20]针对负载摆动引起的外力和扭矩影响, 对四旋翼的高度及姿态设计了自适应控制器, 提高了系统的稳定性. 文献[21]针对四旋翼飞行器吊挂系统的轨迹跟踪问题, 将系统分为四旋翼飞行器和吊挂负载两个子系统. 把四旋翼轨迹跟踪控制器应用于吊挂负载轨迹跟踪, 有效降低负载位置跟踪误差.
目前, 大多数文献重点研究通过四旋翼飞行器位置控制实现摆角振荡抑制, 未涉及摆角控制器的设计. 然而, 吊挂摆角自由度增大系统的耦合程度, 大幅提升了系统的控制难度. 而且吊挂摆角不能直接控制, 需通过调节飞行器位置来间接控制, 同时系统本身也易受外界甚至飞行器旋翼旋转产生的风的干扰影响, 这些复合未知干扰和动态不确定性都会影响整个四旋翼吊挂飞行系统的稳定性和吊挂飞行的安全性.
针对四旋翼飞行器吊挂飞行系统的负载摆动抑制和轨迹跟踪精确控制问题, 本文在总结前人研究成果的基础上[22-26], 结合团队在四旋翼飞行器非线性控制研究的积累[27-31], 对四旋翼吊挂飞行系统建立模型并设计控制器, 提出一种基于扩张状态观测器(Extended state observer, ESO)的四旋翼吊挂飞行非线性轨迹跟踪控制方法. 本文主要贡献有以下3点: 1)将四旋翼及吊挂负载当成整体系统进行受力分析, 建立四旋翼吊挂负载耦合系统的数学模型, 针对四旋翼吊挂飞行系统的负载摆动及飞行过程耦合严重、相互干扰的问题, 设计非线性控制器实现欠驱动约束下的解耦控制, 使飞行器轨迹跟踪效果不变的条件下抑制吊挂负载的摆动; 2)设计一种扩张状态观测器, 对未知外界干扰及模型动态不确定性进行估计和补偿, 并验证了控制器及观测器的所有信号一致最终有界; 3)利用Quanser公司的Qball2飞行器吊挂负载系统模型进行三维空间螺旋轨迹的跟踪控制仿真验证, 验证了本文方法的有效性和优越性.
1. 四旋翼吊挂系统模型
四旋翼吊挂飞行系统有4个输入、8个自由度, 是一个高度耦合、高度欠驱动的复杂系统. 不能简单将吊挂负载当作干扰进行处理, 而是要在单体四旋翼飞行器模型的基础上[32], 重新建立四旋翼吊挂负载耦合系统模型, 如图1所示.
图1飞行器为单体十字形四旋翼, 以飞行器质心点为原点$o$建立坐标系, 飞行器在惯性坐标系下的空间位置坐标为: ${\boldsymbol{I}} = \left[ {\begin{array}{*{20}{c}} X&Y&Z \end{array}} \right]$. 定义模型各个参数为: 飞行器前/后方向为$x$轴方向且后退为正, 左/右方向为$y$轴方向且向右为正, 上/下方向为$z$轴方向且向上为正; 飞行器绕$x$轴旋转角度为横滚角$\phi $且图示转向为正, 飞行器绕$y$轴旋转角度为俯仰角$\theta $且图示转向为正, 飞行器绕$z$轴旋转角度为偏航角$\psi $且图示转向为正. $\alpha $摆角为吊挂绳与飞行器$xoz$平面夹角, $\beta $摆角为吊挂绳与飞行器$yoz$平面夹角, $L$为吊挂绳索长度. 令四旋翼质心到电机轴心长度为$l$, 飞行器质量为$M$, 负载质量为$m$. 在对四旋翼吊挂系统进行建模时, 由于系统较复杂, 模型参数不确定性等因素的影响, 需对模型做出一些简化: 忽略空气阻力对飞行器吊挂系统的影响, 认为吊挂绳是刚性的且质量不计, 吊挂绳的悬挂点为飞行器质心.
单体四旋翼飞行器数学模型为:
$$\left\{ {\begin{aligned} &{\ddot x = \frac{{{a_1}{U_1}}}{M}} \\ &{\ddot y = \frac{{{a_2}{U_1}}}{M}} \\ &{\ddot {\textit{z}} = \frac{{{a_3}{U_1}}}{M} - g} \\ &{\ddot \phi = \frac{{l{U_2} + \dot \theta \dot \psi \left( {{I_y} - {I_{\textit{z}}}} \right)}}{{{I_x}}}} \\ &{\ddot \theta = \frac{{l{U_3} + \dot \phi \dot \psi \left( {{I_{\textit{z}}} - {I_x}} \right)}}{{{I_y}}}} \\ &{\ddot \psi = \frac{{{U_4} + \dot \phi \dot \theta \left( {{I_x} - {I_y}} \right)}}{{{I_{\textit{z}}}}}} \end{aligned}} \right.$$ (1) 式中
$$\left\{ {\begin{aligned} &{{a_1} = - \cos \phi \sin \theta \cos \psi - \sin \phi \sin \psi } \\ & {{a_2} = - \cos \phi \sin \theta \sin \psi + \sin \phi \cos \psi } \\ & {{a_3} = \cos \phi \cos \theta } \qquad\qquad\qquad\qquad\; \end{aligned}} \right.$$ 式中, ${U_1}$代表四个旋翼产生的总升力, ${U_2}$代表横滚力, ${U_3}$代表俯仰力, ${U_4}$代表偏航力矩. $g$代表重力加速度, ${I_x}$、${I_y}$、${I_{\textit{z}}}$代表机体绕相应坐标轴的转动惯量.
在单体四旋翼飞行器的基础上, 增加吊挂负载, 将四旋翼及吊挂负载当成整体系统进行受力分析, 建立四旋翼吊挂负载耦合系统的数学模型.
飞行器位置为$\left[ x\;\;y\;\;{\textit{z}} \right]$, 吊挂负载位置为$\left[ {{x_1}}\;\;{{y_1}}\;\;{{{\textit{z}}_1}} \right]$, 关系式为:
$$\left\{ {\begin{aligned} & {{x_1} = x + L\sin \beta } \\ &{{y_1} = y + L\cos \beta \sin \alpha } \\ & {{{\textit{z}}_1} = {\textit{z}} - L\cos \beta \cos \alpha } \end{aligned}} \right.$$ (2) 对式(2)进行二阶求导, 即可得吊挂负载加速度与四旋翼飞行器加速度之间的关系:
$$\left\{ {\begin{aligned} &{{{\ddot x}_1} = \ddot x + L{b_1}} \\ &{{{\ddot y}_1} = \ddot y + L{b_2}} \\ &{{{\ddot{\textit{z}}}_1} = \ddot {\textit{z}} + L{b_3}} \end{aligned}} \right.$$ (3) 且
$$\left\{ \begin{aligned} {b_1} =\;& \ddot \beta \cos \beta - {{\dot \beta }^2}\sin \beta \\ {b_2} =\;& - \ddot \beta \sin \beta \sin \alpha + \ddot \alpha \cos \beta \cos \alpha \;- \\ &{{\dot \beta }^2}\cos \beta \sin \alpha - {{\dot \alpha }^2}\cos \beta \sin \alpha \;- \\ & 2\dot \beta \dot \alpha \sin \beta \cos \alpha \\ {b_3} =\;& \ddot \beta \sin \beta \cos \alpha + \ddot \alpha \cos \beta \sin \alpha \;+ \\ &{{\dot \beta }^2}\cos \beta \cos \alpha + {{\dot \alpha }^2}\cos \beta \cos \alpha \;- \\ &2\dot \beta \dot \alpha \sin \beta \sin \alpha \end{aligned} \right.$$ 由加速度关系式可对其进行受力分析:
$$\left\{ {\begin{aligned} &{{F_1} = M\ddot x + m{{\ddot x}_1}} \\ &{{F_2} = M\ddot y + m{{\ddot y}_1}} \\ & {{F_3} - mg = M\ddot z + m{{\ddot {\textit{z}}}_1}} \end{aligned}} \right.$$ (4) ${F_1}$、${F_2}$、${F_3}$分别表示四旋翼吊挂系统$x$、$y$、${\textit{z}}$方向所受力. 将式(1)和式(3)代入式(4), 得:
$$\left\{ {\begin{aligned} & {\left( {M + m} \right)\ddot x + {b_1}mL = {a_1}{U_1}} \\ &{\left( {M + m} \right)\ddot y + {b_2}mL = {a_2}{U_1}} \\ &{\left( {M + m} \right)\ddot {\textit{z}} + {b_3}mL = {a_3}{U_1} - \left( {M + m} \right)g} \end{aligned} } \right.$$ (5) 采用哈密尔顿原理及拉格朗日公式计算系统总动能:
$$\frac{{\rm{d}}}{{{{\rm{d}}\tau }}}\left( {\frac{{\partial \left( {A - P} \right)}}{{\partial {{\dot q}_k}}}} \right) - \left( {\frac{{\partial \left( {A - P} \right)}}{{\partial {q_k}}}} \right) = 0$$ (6) $A$为四旋翼吊挂系统总动能, $P$为系统势能, 系统总动能$A$为:
$$\begin{split} A =\;& \frac{1}{2}M\left( {{{\dot x}^2} + {{\dot y}^2} + {{\dot {\textit{z}}}^2}} \right) + \frac{1}{2}m\left( {{{\dot x}_1}^2 + {{\dot y}_1}^2 + {{\dot {\textit{z}}}_1}^2} \right)\;= \\ & \frac{1}{2}\left( {M + m} \right)\left( {{{\dot x}^2} + {{\dot y}^2} + {{\dot {\textit{z}}}^2}} \right) \;+\frac{1}{2}m\;\times \\ & \Big[{L^2}{{\dot \beta }^2} + {L^2}{\cos ^2}\beta {{\dot \alpha }^2} + 2L\cos \beta \dot \beta \dot x \;+ \\ &2L\left( {\cos \beta \cos \alpha \dot \alpha - \sin \beta \sin \alpha \dot \beta } \right)\dot y \;+ \\ &2L\left( {\cos \beta \sin \alpha \dot \alpha + \sin \beta \cos \alpha \dot \beta } \right)\dot {\textit{z}} \Big] \end{split} $$ (7) 认为地平面是零势能, 可得系统的势能$P$为:
$$P = \left( {M + m} \right)gz - mgL\cos \beta \cos \alpha $$ (8) 得到系统总动能及势能后, 将其代入拉格朗日公式, ${q_k}$取$\alpha $和$\beta $, 计算后可得吊挂负载摆角加速度与四旋翼飞行器加速度之间的关系式:
$$\left\{ {\begin{aligned} & L\ddot \beta + \cos \beta \ddot x - \sin \beta \sin \alpha \ddot y \;+ \\ & \qquad\sin \beta \cos \alpha \ddot z + L\sin \beta \cos \beta {{\dot \alpha }^2}\; + \\ &\qquad g\sin \beta \cos \alpha = 0 \\ & \cos \beta \ddot \alpha + \cos \alpha \ddot y + \sin \alpha \ddot z\; - \\ &\qquad 2L\sin \beta \dot \beta \dot \alpha + g\sin \alpha = 0 \end{aligned}} \right.$$ (9) 由式(1)、式(5)和式 (9), 并考虑系统8个通道受到的干扰及模型不确定性$\Delta {{{d}}_{[x,y,{\textit{z}},\phi ,\theta ,\psi ,\alpha ,\beta ]}}$, 得到8个自由度的四旋翼吊挂飞行系统数学模型见式(10), 并且有:
$$\left\{ {\begin{aligned} &{{c_1} = 2ML\sin \beta \dot \alpha \dot \beta } \qquad\;\;\\ &{{c_2} = - ML\sin \beta \cos \beta {{\dot \alpha }^2}} \end{aligned}} \right.$$ $$\left\{ {\begin{aligned} &{\ddot X = \frac{{{a_1}{U_1} - mL{b_1}}}{{M + m}} + \Delta {d_x}} \\ &{\ddot Y = \frac{{{a_2}{U_1} - mL{b_2}}}{{M + m}} + \Delta {d_y}} \\ &{\ddot Z = \frac{{{a_3}{U_1} - mL{b_3}}}{{M + m}} - g + \Delta {d_z}} \\ &{\ddot \phi = \frac{{l{U_2} + \dot \theta \dot \psi \left( {{I_y} - {I_z}} \right)}}{{{I_x}}} + \Delta {d_\phi }} \\ &{\ddot \theta = \frac{{l{U_3} + \dot \phi \dot \psi \left( {{I_z} - {I_x}} \right)}}{{{I_y}}} + \Delta {d_\theta }} \\ &{\ddot \psi = \frac{{{U_4} + \dot \phi \dot \theta \left( {{I_x} - {I_y}} \right)}}{{{I_z}}} + \Delta {d_\psi }} \\ & {\ddot \alpha = \frac{{{c_1} - \cos \alpha {a_2U_1} - \sin \alpha {a_3U_1}}}{{ML\cos \beta }} + \Delta {d_\alpha }} \\ &\ddot \beta = \frac{1 }{{ML}}({c_2} -\cos \beta {a_1U_1}\; -\\ &\qquad\sin \beta \sin \alpha {a_2U_1}-\sin \beta \cos \alpha {a_3U_1}) + \Delta {d_\beta } \end{aligned}} \right.$$ (10) 由式(10)可知, 吊挂负载的摆角加速度与四旋翼飞行器的加速度相互耦合, 吊挂负载的摆动会影响飞行器的稳定, 而飞行过程的未知动态又会影响吊挂负载的状态, 因此需分别对吊挂负载摆角、飞行器位置和姿态设计控制器; 由于飞行系统多个变量均易受未知干扰, 模型也具有动态不确定性, 因此需要设计观测器对未知外界干扰及模型动态不确定性进行估计和补偿.
2. 扩张状态观测器设计
四旋翼吊挂飞行系统在实际飞行时, 机体和负载相互耦合, 未知外界干扰及模型动态不确定性相比单体四旋翼更为严重. 本文为对四旋翼吊挂飞行系统在飞行过程中所受的未知外界扰动与模型动态不确定性进行估计和补偿, 设计一种扩张状态观测器对其进行逼近.
式(10)的非线性系统可写作:
$${{\ddot{ {\eta} }}_1} = f\left( {{ {\eta}_1},{{{\dot{ {\eta} }}}_1},{{v}}(t)} \right) + {{bu}}(t)$$ (11) 式中, $f\left( {{{ {\eta}}_1},{{{\dot{ {\eta} }}}_1},{{v}}(t)} \right)$为非线性系统, ${\eta}_{1[8 \times 1]}$、${{\dot{\eta }}_{1[8 \times 1]}}$、 ${{\ddot{\eta }}_{1[8 \times 1]}}$为系统的状态变量, ${ {\eta}_1} = {{y}}{(t)_{[8 \times 1]}}$为系统输出, ${{v}}{(t)_{[8 \times 1]}}$为系统干扰及模型参数不确定性等未知函数, ${{u}}{(t)_{[8 \times 1]}}$为控制量, ${{{b}}_{[8 \times 1]}}$为已知参数. 令${\eta_2} = {{\dot{\eta }}_1}$, ${ {\eta}_3} = {{\dot{ {\eta} }}_2} - {{bu}}(t)$, ${\eta_3}$为扩张状态变量, ${{\dot{\eta }}_3} = {{ \omega}}(t)$为扰动, 则状态方程为:
$$\left\{ {\begin{aligned} &{{{{\dot{ {\eta} }}}_1} = { {\eta}_2}} \\ &{{{{\dot{ {\eta} }}}_2} = { {\eta}_3} + {{bu}}(t)} \\ &{{{{\dot{ {\eta} }}}_3} = { {\omega }}(t)} \\ & {{{y}}(t) = { {\eta}_1}} \end{aligned}} \right.$$ (12) 由式(12)设计三阶扩张状态观测器如下:
$$\left\{ {\begin{aligned} &{ {\varepsilon }(t) = {{{\hat{ {\eta} }}}_1}(t) - { {\eta}_1}(t)} \\ &{{{{\dot{\hat{ {\eta}} }}}_1}(t) = {{{\hat{ {\eta} }}}_2}(t) - {{ {\kappa }}_1}{ {\varepsilon }}(t)} \\ & {{{{\dot{\hat{ {\eta}} }}}_2}(t) = {{{\hat{ {\eta} }}}_3}(t) - {{ {\kappa }}_2}{ {\varepsilon }}(t) + {{bu}}(t)} \\ & {{{{\dot{\hat { {\eta}} }}}_3}(t) = - {{ {\kappa }}_3}{ {\varepsilon }}(t)} \end{aligned}} \right.$$ (13) 式中, ${{\hat{\eta }}_{1[8 \times 1]}}$、${{\hat{\eta }}_{2[8 \times 1]}}$、${{\hat{\eta }}_{3[8 \times 1]}}$分别为${\eta_1}$、${\eta_2}$、${\eta_3}$的观测值. ${{\kappa}_{1[8 \times 1]}}$、${{\kappa}_{2[8 \times 1]}}$、${{\kappa}_{3[8 \times 1]}}$为大于1的正实数. 下面对观测器的稳定性进行分析.
各状态变量的观测误差为:
$$\left\{ {\begin{aligned} &{{{ {\varepsilon }}_1}(t) = {{{\hat{ {\eta} }}}_1}(t) - {{ {\eta}}_1}(t)} \\ &{{{ {\varepsilon }}_2}(t) = {{{\hat{ {\eta} }}}_2}(t) - {{ {\eta}}_2}(t)} \\ & {{{ {\varepsilon }}_3}(t) = {{{\hat{ {\eta} }}}_3}(t) - {{ {\eta}}_3}(t)} \end{aligned}} \right.$$ 则
$$\left\{ {\begin{aligned} & {{{{\dot{ {\varepsilon } }}}_1}(t) = {{ {\varepsilon }}_2}(t) - {{ {\kappa }}_1}{{ {\varepsilon }}_1}(t)} \\ &{{{{\dot{ {\varepsilon } }}}_2}(t) = {{ {\varepsilon }}_3}(t) - {{ {\kappa }}_2}{{ {\varepsilon}}_2}(t)} \\ & {{{{\dot{ {\varepsilon }}}}_3}(t) = - {{ {\kappa }}_3}{{ {\varepsilon }}_3}(t) - { {\omega }}(t)} \end{aligned}} \right.$$ (14) 令
$$ \left\{ {\begin{aligned} & {{{{N}}_1} = {{ {\varepsilon}}_1}(t)}\\ & {{{{N}}_2} = {{ {\varepsilon }}_2}(t) - {{ {\kappa }}_1}{ {\varepsilon }_1}(t)}\\ & {{{{N}}_3} = {{ {\varepsilon }}_3}(t) - {{ {\kappa }}_1}{{ {\varepsilon }}_2}(t) + ({ {\kappa }}_1^2 - {{ {\kappa}}_2}){{ {\varepsilon }}_1}(t)} \end{aligned}} \right.$$ 则误差系统为:
$$\left\{ {\begin{aligned} & {{{{\dot{ N}}}_1} = {{{N}}_2}} \\ & {{{{\dot{ N}}}_2} = {{{N}}_3}} \\ &{{{{\dot{ N}}}_3} = - {{ {\kappa }}_1}{{{N}}_3} - {{ {\kappa }}_2}{{{N}}_2} - {{ {\kappa }}_3}{{{N}}_1} - { {\omega}}(t)} \end{aligned}} \right.$$ (15) 由巴尔巴辛公式[26]得误差的李雅普诺夫函数:
$$\begin{split} V =\;& \frac{{{{ {\kappa }}_1}{{ {\kappa }}_3}{{N}}_1^2 + {{ {\kappa }}_2}{{N}}_2^2 + {{({{ {\kappa }}_1}{{{N}}_2} + {{{N}}_3})}^2}}}{2} + {{ {\kappa }}_3}{{{N}}_1}{{{N}}_2} \;=\\ & \frac{1}2\Bigg[ {({{ {\kappa }}_1}{{{N}}_2} + {{{N}}_3})^2} + {{ {\kappa }}_1}{{ {\kappa }}_3}\bigg({{N}}_1^2 + \frac{2}{{{{ {\kappa }}_1}}}{{{N}}_1}{{{N}}_2} \;+ \\ & \frac{{{{{\kappa }}_2}}}{{{{ {\kappa }}_1}{{ {\kappa }}_3}}}{{N}}_2^2\bigg) \Bigg] \\[-15pt] \end{split} $$ (16) 当${{\kappa}_1}{{\kappa}_2} > {{\kappa}_3}$时, 得:
$$\begin{split} & V = \frac{1}{2}\Bigg[ ({{ {\kappa }}_1}{{{N}}_2} + {{{N}}_3})^2 + {{ {\kappa }}_1}{{ {\kappa }}_3}\bigg({{N}}_1^2 + \frac{2}{{{{ {\kappa }}_1}}}{{{N}}_1}{{{N}}_2} \;+ \\ &\;\;\;\;\frac{{{{ {\kappa }}_2}}}{{{{ {\kappa }}_1}{{ {\kappa }}_3}}}{{N}}_2^2\bigg) \Bigg] >\frac{1}{2}\bigg({{({{ {\kappa }}_1}{{{N}}_2} + {{{N}}_3})}^2} \;+\\ &\;\;\;\; {{ {\kappa }}_1}{{ {\kappa }}_3}\left({{{N}}_1} + \frac{{{{{N}}_2}}}{{{{ {\kappa }}_1}}}\right)\bigg)^2 \geq 0 \\[-15pt] \end{split} $$ (17) 得$V > 0$, 对其求导, 得:
$$\dot V = ({{ {\kappa }}_3} - {{{\kappa }}_1}{{ {\kappa }}_2}){{N}}_2^2 - ({{ {\kappa }}_1}{{{N}}_2} + {{{N}}_3}){{\omega }}(t)$$ (18) 由此可见, 当${{ {\kappa }}_1}{{ {\kappa }}_2} > {{ {\kappa }}_3}$时, $V$正定. 当扰动${\omega}(t) = 0$时, $\dot V < 0$, 误差系统在平衡点(${{\varepsilon}_1}(t) = 0$, ${{\varepsilon}_2}(t) = 0$, ${{\varepsilon}_3}(t) = 0$)大范围渐近稳定. 当扰动${\omega}(t) \ne 0$时, 令$\left| {{\omega}(t)} \right| \leq {{\omega}_0}$(${{\omega}_0}$为正常数), 系统稳定时:
$$\left\{ {\begin{aligned} & {{{{\dot{ N}}}_1} =0, \;\; {{{N}}_2} = 0} \\ &{{{{\dot{ N}}}_2} =0,\;\; {{{N}}_3} = 0} \\ &{{{{\dot{ N}}}_3} = 0} \end{aligned}} \right.$$ (19) 由此可得观测误差范围:
$$\left\{ {\begin{aligned} &\left| {{{ {\varepsilon }}_1}(t)} \right| \leq \frac{ \omega_0}{ {\kappa }_3} \\ &\left| {{{ {\varepsilon }}_2}(t)} \right| \leq \frac{\kappa _1 { \omega}_0}{ {\kappa}_3} \\ & \left| {{{ {\varepsilon }}_3}(t)} \right| \leq \frac{ \kappa _2 { \omega}_0}{ {\kappa }_3} \end{aligned}} \right.$$ (20) 3. 控制器设计
针对四旋翼吊挂飞行系统的负载摆动及飞行过程耦合严重且相互干扰的问题, 设计非线性控制器实现欠驱动约束下的解耦控制. 考虑四旋翼吊挂飞行系统在跟踪期望轨迹的同时需要降低吊挂负载的摆角, 而吊挂负载摆角的控制则会影响飞行器的位置跟踪, 此时若将吊挂负载摆角作为飞行器位置的外环来控制, 两个控制器的控制目标矛盾, 从而导致飞行器水平位置通道无法控制. 因此, 本文将吊挂负载摆角控制器的输出量和飞行器水平位置控制器的输出量转换为姿态角控制器的期望输入, 对飞行器轨迹跟踪产生的影响较小, 同时也能对吊挂摆角进行控制, 实现四旋翼吊挂飞行系统轨迹跟踪控制和吊挂负载的减摆控制. 四旋翼吊挂飞行控制系统结构如图2所示, 控制部分可分为飞行位置控制子系统、吊挂负载摆角控制子系统、飞行姿态控制子系统三个子系统.
给定期望飞行器位置${Z_d}$、${Y_d}$、${X_d}$和偏航${\psi _d}$, 期望摆角${\alpha _d}$和${\beta _d}$为0. 由高度积分反步法控制器得到电机输入量${u_{throttle}}$, 飞行器$x$、$y$方向控制器输出值${u_x}$、${u_y}$. 由摆角积分反步法控制器得到$x$、$y$方向的控制律补偿${u_\alpha }$、${u_\beta }$, 最终得到${U_x}$和${U_y}$, 经计算得到飞行器姿态期望值${\phi _d}$和${\theta _d}$. 由姿态积分反步法控制器可将期望姿态转化为电机输入量${u_{roll}}$、${u_{pitch}}$和${u_{yaw}}$, 结合高度控制器得到的${u_{throttle}}$, 经过电机模型转换可得飞行器的虚拟控制量${U_1}、$ ${U_2}、$ ${U_3}、$ ${U_4}.\;$ 经系统模型和扩张状态观测器将8个通道的输出$\hat X/{\hat{ \dot X}}、$ $\hat Y/{\hat{ \dot Y}}、$ $\hat Z/{\hat {\dot Z}}、$ $\hat \phi /{\hat{ \dot \phi}} 、$ $\hat \theta /{\hat {\dot \theta}}、$ $\hat \psi /{\hat {\dot \psi }}、$ $\hat \alpha /{\hat{ \dot \alpha}} 、$ $\hat \beta /{\hat {\dot \beta}} $和观测干扰$\Delta {{\hat{ d}}_{[8 \times 1]}}$反馈到控制器.
在位置解算时, 飞行器横滚角和俯仰角对飞行器的位置影响较小, 因此将其忽略作小角度近似, 得位置−姿态转换公式如下:
$$\left\{ {\begin{aligned} &{{\phi _d} = {U_y}\cos \psi - {U_x}\sin \psi } \\ &{{\theta _d} = - {U_y}\sin \psi - {U_x}\cos \psi } \end{aligned}} \right.$$ (21) 电机模型转换公式如下:
$$\left\{ {\begin{aligned} & {{u_{throttle}} = \frac {U_1}{4{K_t}}} \\ & {{u_{roll}} =\frac {U_2}{2{K_t}}} \\ & {{u_{pitch}} =\frac {U_3}{2{K_t}}} \\ & {{u_{yaw}} =\frac {U_4}{4{K_y}}} \end{aligned}} \right.$$ (22) 式中, ${K_t}$和${K_y}$为升力系数和反扭矩系数.
四旋翼吊挂飞行系统的位置控制器、姿态控制器和摆角控制器均采用积分反步法控制器, 可实现欠驱动约束下的解耦控制, 消除静态误差, 减小模型不确定性以及外界扰动.
3.1 位置控制器的设计
以四旋翼吊挂系统飞行器高度通道为例设计控制器.
高度期望值${Z_d}$与实际值$Z$的差为:
$${{\textit{z}}_1} = {Z_d} - Z$$ (23) 对式(23)求导, 得其跟踪误差的导数:
$${\dot {\textit{z}}_1} = {\dot Z_d} - \dot Z = {\dot Z_d} - {w_z}$$ (24) $\dot Z = {w_{\textit{z}}}$即飞行器实际的高度上升速度. 为镇定${{\textit{z}}_1}$, 令${{\textit{z}}_1}$的李雅普诺夫函数为:
$${V_1} = \frac{1}{2}{{\textit{z}}_1^2}$$ (25) 对其求导, 得:
$${\dot V_1} = {{\textit{z}}_1}{\dot {\textit{z}}_1} = {{\textit{z}}_1}\left( {{{\dot Z}_d} - {w_{\textit{z}}}} \right)$$ (26) 将飞行器高度变化速度${w_{\textit{z}}}$作为控制器的虚拟输入量, 将${w_{zd}}$作为虚拟量的期望值, 为使${\dot V_1} \leq 0$, 令:
$${w_{{\textit{z}}d}} = {k_1}{{\textit{z}}_1} + {\dot Z_d}$$ (27) 在虚拟控制量后加入积分项, 可增强控制器的鲁棒性, 消除模型不确定性的影响:
$${w_{{\textit{z}}d}} = {k_1}{{\textit{z}}_1} + {\dot Z_d} + {\lambda _1}{\chi _1}$$ (28) 式中, ${\chi _1} = \int_0^t {{{\textit{z}}_1}\left( \tau \right)} {\rm{d}}\tau ,\;{k_1}、{\lambda _1}$为大于0的常数. 虚拟控制输入量${w_{{\textit{z}}d}}$和实际高度变化速度${w_{\textit{z}}}$的差为:
$${{\textit{z}}_2} = {w_{{\textit{z}}d}} - {w_{\textit{z}}} = {k_1}{{\textit{z}}_1} + {\dot Z_d} + {\lambda _1}{\chi _1} - {w_{\textit{z}}}$$ (29) 即${\dot Z_d} = {{\textit{z}}_2} - {k_1}{{\textit{z}}_1} - {\lambda _1}{\chi _1} + {w_{\textit{z}}}$, 代入式(26), 得:
$$\begin{split} {{\dot V}_1} =\;& {{\textit{z}}_1}{{\dot {\textit{z}}}_1} = {{\textit{z}}_1}\left( { - {k_1}{{\textit{z}}_1} - {\lambda _1}{\chi _1} + {{\textit{z}}_2}} \right)= \\ & -{k_1}{{\textit{z}}_1^2} - {{\textit{z}}_1}{\lambda _1}{\chi _1} + {{\textit{z}}_1}{{\textit{z}}_2} \end{split} $$ (30) 为使${\dot V_1} \leq 0$, 需令${{\textit{z}}_2}、 {\chi _1}$趋于0, 对${{\textit{z}}_2}、 {\chi _1}$设计李雅普诺夫函数${V_2}$:
$${V_2} = \frac{1}{2}{{\textit{z}}_1^2} + \frac{1}{2}{{\textit{z}}_2^2} + \frac{1}{2}{\lambda _1}{\chi _1^2}$$ (31) ${V_2}$正定, 对式(30)求导:
$${\dot {\textit{z}}_2} = {\dot w_{{\textit{z}}d}} - {\dot w_{\textit{z}}} = {k_1}{\dot {\textit{z}}_1} + {\ddot Z_d} + {\lambda _1}{{\textit{z}}_1} - \ddot Z$$ (32) 将飞行器高度数学模型式(10)代入式(32):
$${\dot {\textit{z}}_2} = {k_1}{\dot {\textit{z}}_1} + {\ddot Z_d} + {\lambda _1}{{\textit{z}}_1} + g - \frac{{{a_3{{U_1}}} - mL{b_3}}}{{M + m}} - \Delta {\hat d_{\textit{z}}}$$ (33) $\Delta {\hat d_{\textit{z}}}$为观测器估计干扰, 则:
$$\begin{split} {{\dot V}_2} =\;& {{\dot V}_1} + {{\textit{z}}_2}{{\dot {\textit{z}}}_2} + {\lambda _1}{\chi _1}{{\dot \chi }_1}=- {k_1}{{\textit{z}}_1^2} +{{\textit{z}}_1}{{\textit{z}}_2} +{{\textit{z}}_2} \;\times \\ & \left( {k_1}{{\dot {\textit{z}}}_1} + {{\ddot Z}_d} + {\lambda _1}{{\textit{z}}_1} + g - \frac{{{a_3{{U_1}}} - mL{b_3}}}{{M + m}} - \Delta {{\hat d}_{\textit{z}}} \right) \end{split} $$ (34) 为使${\dot V_2} \leq 0$, 取控制变量:
$$\begin{split}{U_1} =\;& \frac{1}{\cos \phi \cos \theta }([ \left( {1 - k_1^2 + {\lambda _1}} \right){{\textit{z}}_1} +\left( {{k_1} + {k_2}} \right){{\textit{z}}_2} \;+ \\ & g - {k_1}{\lambda _1}{\chi _1} + {{\ddot Z}_d} - \Delta {{\hat d}_{\textit{z}}} ]\left( {M + m} \right) + mL{b_3})\end{split}$$ (35) 式中, ${k_2}$为大于0常数. 将式(35)代入式(34), 得:
$${\dot V_2} = - {k_1}{{\textit{z}}_1^2} - {k_2}{{\textit{z}}_2^2}$$ (36) ${\dot V_2}$负定, 由式(31)和式(36)可验证所设计的控制律令高度渐近稳定.
同理可得飞行器位置控制器的控制律:
$$\begin{split}{u_x} =\;& \frac{1}{U_1}([ \left( {1 - k_3^2 + {\lambda _2}} \right){{\textit{z}}_3} + \left( {{k_3} + {k_4}} \right){{\textit{z}}_4} \;+\\ & {{\ddot X}_d} - {k_3}{\lambda _2}{\chi _2} - \Delta {{\hat d}_x} ]\left( {M + m} \right) + mL{b_1})\end{split}$$ (37) $$\begin{split}{u_y} =\;& \frac{1}{U_1}([ \left( {1 - k_5^2 + {\lambda _3}} \right){{\textit{z}}_5} + \left( {{k_5} + {k_6}} \right){{\textit{z}}_6} \;+\\ &{{\ddot Y}_d} - {k_5}{\lambda _3}{\chi _3} - \Delta {{\hat d}_y} ]\left( {M + m} \right) + mL{b_2})\end{split}$$ (38) 式中, ${X_d}$和${Y_d}$为期望方向, $\Delta {\hat d_x}$和$\Delta {\hat d_y}$为估计干扰, ${k_n}\;(n = 3, \cdots, 6)$, ${\lambda _i}\;(i = 2,3)$为大于0常数, 且:
$$\left\{ {\begin{split} &{{{\textit{z}}_3} = {X_d} - X,{\chi _2} = \int_0^\tau {{{\textit{z}}_3}\left( \tau \right){\rm{d}}\tau } } \\ &{{{\textit{z}}_4} = {k_3}{{\textit{z}}_3} + {{\dot X}_d} + {\lambda _2}{\chi _2} - \dot X} \\ & {{{\textit{z}}_5} = {Y_d} - Y,{\chi _3} = \int_0^\tau {{{\textit{z}}_5}\left( \tau \right){\rm{d}}\tau } } \\ &{{{\textit{z}}_6} = {k_5}{{\textit{z}}_5} + {{\dot Y}_d} + {\lambda _3}{\chi _3} - \dot Y} \end{split}} \right.$$ (39) 3.2 摆角控制器的设计
吊挂负载摆角由于其欠驱动的特性, 无法直接控制, 需通过控制器将其转化为位置信号间接控制. 采用上文所述的积分反步法控制器, 可得:
$$\begin{split} {u_\alpha } =\;& \frac{1}{U_1}([ \left( {1 - k_7^2 + {\lambda _4}} \right){{\textit{z}}_7} + \left( {{k_7} + {k_8}} \right){{\textit{z}}_8} \;+\\ &{{\ddot \alpha }_d} - {k_7}{\lambda _4}{\chi _4} - \Delta {{\hat d}_\alpha } ]\cos \beta ML - {c_1})\end{split} $$ (40) $$\begin{split}{u_\beta } =\;& \frac{1}{U_1}([ \left( {1 - k_9^2 + {\lambda _5}} \right){{\textit{z}}_9} + \left( {{k_9} + {k_{10}}} \right){{\textit{z}}_{10}}\; + \\ &{{\ddot \beta }_d} - {k_9}{\lambda _5}{\chi _5} - \Delta {{\hat d}_\beta } ]ML - {c_2})\end{split}$$ (41) 式中, ${\alpha _d}$、${\beta _d}$为期望摆角, $\Delta {\hat d_\alpha }$、$\Delta {\hat d_\beta }$为估计干扰, ${k_n}\;(n = 7, \cdots, 10)$, ${\lambda _i}\;(i = 4,5)$为大于0常数, 且:
$$\left\{ {\begin{split} &{{{\textit{z}}_7} = {\alpha _d} - \alpha ,{\chi _4} = \int_0^\tau {{{\textit{z}}_7}\left( \tau \right){\rm{d}}\tau } } \\ &{{{\textit{z}}_8} = {k_8}{{\textit{z}}_8} + {{\dot \alpha }_d} + {\lambda _4}{\chi _4} - \dot \alpha } \\ &{{{\textit{z}}_9} = {\beta _d} - \beta ,{\chi _5} = \int_0^\tau {{{\textit{z}}_9}\left( \tau \right){\rm{d}}\tau } } \\ & {{{\textit{z}}_{10}} = {k_9}{{\textit{z}}_9} + {{\dot \beta }_d} + {\lambda _5}{\chi _5} - \dot \beta } \end{split}} \right.$$ (42) 由式(37)、式 (38)得到的飞行器位置控制器控制量和式(40)、式 (41)得到的吊挂负载摆角控制量, 可得四旋翼吊挂系统的飞行器位置控制量:
$$\left\{ {\begin{aligned} &{{U_x} = {u_x} + {u_\alpha }} \\ &{{U_y} = {u_y} + {u_\beta }} \end{aligned}} \right.$$ (43) 3.3 姿态控制器的设计
得到飞行器位置控制量后, 经式(21)将其转化为姿态角期望值. 飞行器姿态角控制为系统的内环控制, 同样采用积分反步法控制器, 结果如下:
$$\left\{ {\begin{aligned} {U_2} =\;& \frac{{{I_x}}}{l}\Bigg[ \left( {1 - k_{11}^2 + {\lambda _6}} \right){{\textit{z}}_{11}} + \left( {{k_{11}} + {k_{12}}} \right){{\textit{z}}_{12}} \;- \\ &{k_{11}}{\lambda _6}{\chi _6} + {{\ddot \phi }_d} - \Delta {{\hat d}_\phi } - \frac{{\dot \theta \dot \psi ({I_y} - {I_{\textit{z}}})}}{{{I_x}}} \Bigg] \\ {U_3}= \;& \frac{{{I_y}}}{l}\Bigg[ \left( {1 - k_{13}^2 + {\lambda _7}} \right){{\textit{z}}_{13}} + \left( {{k_{13}} + {k_{14}}} \right){{\textit{z}}_{14}} \;- \\ &{k_{13}}{\lambda _7}{\chi _7} + {{\ddot \theta }_d} - \Delta {{\hat d}_\theta } - \frac{{\dot \phi \dot \psi ({I_{\textit{z}}} - {I_x})}}{{{I_y}}} \Bigg] \\ {U_4} = \;&{I_{\textit{z}}}\Bigg[ (1 - k_{15}^2 + {\lambda _8}){{\textit{z}}_{15}} + ({k_{15}} + {k_{16}}){{\textit{z}}_{16}} \;- \\ &{k_{15}}{\lambda _8}{\chi _8} + {{\ddot \psi }_d} - \Delta {{\hat d}_\psi } - \frac{{\dot \theta \dot \phi ({I_x} - {I_y})}}{{{I_{\textit{z}}}}} \Bigg] \end{aligned}} \right.$$ (44) 式中, ${k_n}\;(n = 11, \cdots ,16)$, ${\lambda _i}\;(i = 6,7,8)$为大于0常数, ${\ddot \phi _d} 、$${\ddot \theta _d}$、${\ddot \psi _d}$为期望姿态角, $\Delta {\hat d_\phi }、$ $\Delta {\hat d_\theta }、$ $\Delta {\hat d_\psi }$取估计值, 且:
$$\left\{ {\begin{aligned} & {{{\textit{z}}_{11}} = {\phi _d} - \phi ,{\chi _6} = \int_0^t {{{\textit{z}}_{11}}\left( \tau \right){\rm{d}}\tau } } \\ &{{{\textit{z}}_{12}} = {k_{11}}{{\textit{z}}_{11}} + {{\dot \phi }_d} + {\lambda _6}{\chi _6} - \dot \phi } \\ &{{{\textit{z}}_{13}} = {\theta _d} - \theta ,{\chi _7} = \int_0^t {{{\textit{z}}_{13}}} \left( \tau \right){\rm{d}}\tau } \\ & {{{\textit{z}}_{14}} = {k_{13}}{{\textit{z}}_{13}} + {{\dot \theta }_d} + {\lambda _7}{\chi _7} - \dot \theta } \\ &{{{\textit{z}}_{15}} = {\psi _d} - \psi ,{\chi _8} = \int_0^t {{{\textit{z}}_{15}}} \left( \tau \right){\rm{d}}\tau } \\ &{{{\textit{z}}_{16}} = {k_{15}}{{\textit{z}}_{15}} + {{\dot \psi }_d} + {\lambda _8}{\chi _8} - \dot \psi } \end{aligned}} \right.$$ (45) 4. 仿真验证及结果分析
以Qball2及其吊挂系统作为仿真对象, 实验平台由Quanser公司生产的Qball2四旋翼无人飞行器、OptiTrack Flex3定位系统、 Matlab/Simulink环境的飞行控制系统和Qball2四旋翼飞行器吊挂系统四部分组成. 实验平台如图3所示:
在Matlab/Simulink仿真中, 采用Qball2四旋翼飞行器的参数, 吊挂负载为重心为中心, 质量为0.2 kg的圆球, 吊挂绳长为0.3 m, 系统参数见表1, 扩张状态观测器参数见表2.
表 1 模型参数Table 1 Model parameters参数 大小 $M$ 1.80 kg $l$ 0.20 m ${I_x}$ 0.03 kg·m2 ${I_y}$ 0.03 kg·m2 ${I_z}$ 0.04 kg·m2 ${K_t}$ 8.80 N ${K_y}$ 0.40 N $m$ 0.20 kg $L$ 0.30 m 表 2 ESO参数设置Table 2 Parameters of ESO参数 $i=X,Y,Z$ $i=\phi ,\theta ,\psi$ $i=\alpha ,\beta$ ${\kappa _{1i}}$ 55 100 100 ${\kappa _{2i}}$ 550 1000 1000 ${\kappa _{3i}}$ 5500 10000 10000 下面对四旋翼吊挂飞行系统进行轨迹跟踪控制仿真, 图4中Adp表示自适应积分反步法控制四旋翼飞行器吊挂飞行系统, ESO为本文提出的扩张状态观测器. 设置仿真时间为30 s, 在第1 s时, 飞行器启动, 起始位置为[0 0 0], 高度设定Z = 0.1 t. 在第6 s时, 设定位置 $\left[ {\begin{aligned} X \\ Y \end{aligned}} \right] = \left[ {\begin{aligned} {2\sin (\pi t/6)} \quad\\ {2\cos (\pi t/6) + 2} \end{aligned}} \right]$, 飞行器姿态加入白噪声干扰模拟飞行器飞行时的震动, 同时转动惯量分别加上0.01 kg·m2, 用来模拟参数的不确定性, 在飞行器位置加入式(46)和白噪声组成的复合干扰, 用来模拟慢时变干扰力矩:
$$0.3\left[ {\begin{array}{*{20}{l}} {\sin 0.5\pi t} \\ {\sin 0.5\pi t} \\ {\cos 0.5\pi t} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {0.1} \\ {0.1} \\ {0.05} \end{array}} \right]$$ (46) 吊挂负载摆角部分加入式(47)和白噪声组成的复合干扰力矩:
$$\left[ {\begin{array}{*{20}{l}} {\sin 0.5\pi t} \\ {\cos 0.5\pi t} \end{array}} \right]$$ (47) 图4为四旋翼吊挂飞行系统在三维空间的轨迹图, 图5 、图6为系统投影到其他平面的运动轨迹, 同时给出四旋翼及吊挂负载在仿真过程中的采样时刻. 飞行器位置跟踪误差如图7所示, ${e_x}$、${e_y}$、${e_{\textit{z}}}$为$x$、$y$、${\textit{z}}$位置跟踪误差, 可以看出, 本文设计的控制策略能很好地控制飞行器沿期望轨迹飞行, 且比自适应控制策略产生的超调更小, 跟踪误差也更小. 系统在刚运行第6 s时, $y$方向位于轨迹切线即速度最快点, 因此$y$方向会产生一定的偏差, 但能快速跟踪上.
图8为飞行过程中吊挂负载产生的摆角. 可以看出, 在没有摆角控制器的情况下, 吊挂负载产生的摆动较大且处于振荡状态, 摆角控制器的参与能将吊挂负载摆角快速稳定在较小值, 且稳定后产生的振荡更小. 图9为飞行器姿态角的期望值和实际值. 可以看出, 内环的积分反步法姿态角控制器能快速准确地跟踪期望姿态.
图10 ~ 12分别表示飞行器速度、吊挂摆角速度、飞行器角速度的实际值和估计值. 由图10 ~ 12可以看出, ESO对速度信息的估计效果较好.
图13 ~ 15为飞行器吊挂系统各个通道的估计误差曲线. 可以看出, 都能保持较小的误差, 且能快速稳定到0附近.
图16 ~ 18为系统各个通道所观测到的干扰信号. 可以看出, 本文设计的观测器能快速估计外界扰动, 同时能很好地对干扰进行补偿, 对系统的控制精度及抗干扰能力有较大的提升.
5. 结束语
考虑四旋翼吊挂飞行系统耦合严重, 外界干扰大, 模型参数不确定等约束, 针对一类四旋翼飞行器吊挂飞行系统的负载摆动抑制和轨迹跟踪精确控制的问题, 设计一种基于扩张状态观测器的吊挂负载摆动抑制的非线性轨迹跟踪控制方法. 在单体四旋翼飞行器的基础上, 增加吊挂负载, 将四旋翼及吊挂负载当成整体系统进行受力分析, 建立四旋翼吊挂负载耦合系统的数学模型, 分别设计欠驱动约束下的姿态、位置和负载摆动三个动态子系统的非线性解耦控制器; 针对四旋翼吊挂飞行系统飞行过程中的未知外界扰动和模型动态不确定性等问题, 设计一种扩张状态观测器用以估计和补偿, 同时验证了闭环系统的稳定性, 跟踪误差及吊挂负载摆动所有信号的一致最终有界. 最后, 利用Quanser公司的Qball2飞行器进行三维空间螺旋轨迹的跟踪控制, 在仿真结果验证未知干扰下, 基于扩张状态观测器的四旋翼吊挂飞行非线性控制的有效性和优越性, 是实现四旋翼吊挂飞行系统轨迹跟踪的精确控制和飞行过程中负载摆动的快速抑制的一种有效方法.
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表 1 模型参数
Table 1 Model parameters
参数 大小 $M$ 1.80 kg $l$ 0.20 m ${I_x}$ 0.03 kg·m2 ${I_y}$ 0.03 kg·m2 ${I_z}$ 0.04 kg·m2 ${K_t}$ 8.80 N ${K_y}$ 0.40 N $m$ 0.20 kg $L$ 0.30 m 
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表 2 ESO参数设置
Table 2 Parameters of ESO
参数 $i=X,Y,Z$ $i=\phi ,\theta ,\psi$ $i=\alpha ,\beta$ ${\kappa _{1i}}$ 55 100 100 ${\kappa _{2i}}$ 550 1000 1000 ${\kappa _{3i}}$ 5500 10000 10000
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