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飞行器上升段作为主动飞行段,通常选择发射坐标系作为该阶段飞行运动参考坐标系。若不考虑横向机动飞行,且认为姿态控制系统理想工作,其纵向射面内质心运动方程可表示为[7]:
$$ \left\{ \begin{gathered} \dot h = V\sin \theta \\ \dot L = \dfrac{{Rv\cos \theta }}{r} \\ \dot V = - g\sin \theta + \dfrac{{P\cos \alpha - X}}{m} \\ \dot \theta = \left( {\dfrac{V}{r} - \dfrac{g}{V}} \right)\cos \theta + \dfrac{{Y + P\sin \theta }}{{mV}} \\ \dot m = - {m_s} \\ \end{gathered} \right. $$ (1) 式中:h、L、V、
$ \theta $ 、m、$ \alpha $ 、g、r、R、$ {m_s} $ 、P、X、Y分别为飞行器纵向平面内飞行高度、航程、速度、弹道倾角、飞行器质量、攻角、重力加速度、地心距、地球半径、组合发动机燃料秒消耗量、组合发动机推力、气动阻力以及气动升力。由于航程项不影响其余状态量的计算,由此,为简化优化模型、提高计算效率,将纵向射面内质心运动方程表示为:
$$ \left\{ \begin{gathered} \dot h = V\sin \theta \\ \dot V = - g\sin \theta + \dfrac{{P\cos \alpha - X}}{m} \\ \dot \theta = \left( {\dfrac{V}{r} - \dfrac{g}{V}} \right)\cos \theta + \dfrac{{Y + P\sin \theta }}{{mV}} \\ \dot m = - {m_s} \\ \end{gathered} \right. $$ (2) 基于美国1976年公布的标准大气模型,参考文献[15]中利用拟合方法给出高度范围0~91 km间大气参数的计算公式。同时,重力加速度大小计算公式如下:
$$ g = \dfrac{{{R^2}}}{{{r^2}}}{g_0} $$ (3) 式中:R表示地球半径;r为地心距;
$ {g_0} $ 为海平面处重力加速度。文中所研究飞行器气动数据可通过数值模拟或地面风洞实验得到,并以数据表的方式进行存储。其所受气动力为:
$$ \left\{ \begin{gathered} X = qS{C_x} \\ Y = qS{C_y} \\ {C_x} = {f_x}\left( {Ma,\alpha ,h} \right) \\ {C_y} = {f_y}\left( {Ma,\alpha ,h} \right) \\ \end{gathered} \right. $$ (4) 式中:
$ q $ 为飞行动压;$ \;\rho $ 为大气密度;$ S $ 为飞行器参考面积;$ {C_y} $ 、$ {C_x} $ 分别为飞行器气动升、阻力系数,可通过气动数据插值得到。RBCC发动机推力与燃料秒流量可表示为:
$$ \begin{gathered} P = {P_R} + {P_A} \\ {{\dot m}_s} = {{\dot m}_{sr}} + {{\dot m}_{sa}} \\ \end{gathered} $$ (5) 式中:
$ {P_R} $ 和$ {P_A} $ 分别代表火箭发动机推力与冲压发动机推力,可分别由下式计算得到:$$ \begin{gathered} {P_R}{\text{ = }}{{\dot m}_{sr}}{I_{spr}} \\ {P_A} = {{\dot m}_{sa}}{E_r}{I_{spa}} \\ {{\dot m}_{sa}} = \rho SV\phi \\ \end{gathered} $$ (6) 式中:
$ {\dot m_{sr}} $ 、$ {I_{spr}} $ 分别为火箭发动机秒流量、火箭发动机比冲;$ {\dot m_{sa}} $ 、$ \rho $ 、$ S $ 、$ \phi $ 、$ {E_r} $ 、$ {I_{spa}} $ 分别为冲压发动机秒流量、当地大气密度、冲压发动机进气道横截面积、流量比、当量比、冲压发动机比冲。 -
在不考虑姿态控制的轨迹优化设计过程中,若直接选取攻角
$\alpha $ 作为控制量,优化结果中可能出现控制量变化率过大甚至一阶不连续情况,这与飞行器实际飞行情况不符[16]。针对这一问题,文中引入攻角变化率$\dot \alpha $ 作为伪控制量,攻角$\alpha $ 可视为附加的状态量,通过$\dot \alpha $ 积分得到,即$$ \alpha {\text{ = }}{\alpha _0} + \int {\dot \alpha {\rm{d}}t} $$ (7) 同时,由于文中研究对象尚处于初步设计阶段,难以针对不同模态给出准确的发动机工作模态划分准则和约束要求,所以文中暂不给定模态划分准则,而是将火箭发动机燃料秒耗量作为全程控制量,进而通过轨迹优化设计获得其取值规律,最终获得RBCC动力系统的工作模态转换准则和机理。因此,对于文中的上升段轨迹优化设计问题,最终的控制量选择为
$\dot \alpha $ 、$ {\dot m_{sr}} $ 。同理,在考虑攻角控制系统二阶滞后的情况下则选定控制量为
${\dot \alpha _c}$ 、$ {\dot m_{sr}} $ 。 -
在RBCC动力高超声速飞行器的上升段轨迹优化设计中,文中主要考虑以下约束条件对飞行轨迹的限制:
(1)过程约束
为保证上升段飞行安全性,在飞行过程中过载、动压、热流均不能超过飞行器临界值,否则将对飞行器结构造成损害。由于驻点位置是飞行器受热严重区域,为保证整体受热满足约束要求,主要对驻点热流密度加以限制。动压、过载、驻点热流密度计算公式与约束条件如下:
$$\left\{ \begin{array}{l} q = \dfrac{1}{2}\rho {V^2} < {q_{\max }}\\ n = \dfrac{{\left| {X\sin \alpha + Y\cos \alpha } \right|}}{{m{g_0}}} < {n_{\max }}\\ \dot Q = K\sqrt \rho {V^{3.15}} < {{\dot Q}_{\max }} \end{array} \right. $$ (8) 式中:
$K$ 为常数,其取值与飞行器气动外形相关。(2)状态变量约束
通常选取能够描述飞行器速度或位置状态的参数量作为状态变量,例如高度、速度等。从实际可行性以及安全飞行角度考虑,将存在状态变量约束。若用s表示状态变量,则有:
$$ {s_{\min }} \leqslant {{s}} \leqslant {s_{\max }} $$ (9) (3)控制量约束
综合考虑飞行性能、控制系统性能和发动机性能,可建立控制变量约束模型。用u表示控制变量,则有:
$$ {u_{\min }} \leqslant u \leqslant {u_{\max }} $$ (10) (4)端点约束
文中研究问题中端点约束可分为初始约束与终端约束两种。初始约束代表飞行器初始飞行状态并影响后续飞行能力;终端约束主要考虑末端高度、速度以及速度倾角等,是判断飞行器能否完成飞行任务的重要条件。用s表示状态变量,则有:
$$ \left\{ \begin{gathered} {s_{0\min }} \leqslant {\text{s}}({t_0}) \leqslant {s_{0\max }} \\ {s_{f\min }} \leqslant {\text{s}}({t_f}) \leqslant {s_{f\max }} \\ \end{gathered} \right. $$ (11) -
在助推级轨迹优化中,为挖掘助推级运载潜力文中将优化目标选取为上升段末端机械能最大,该优化目标可以表示为:
$$ J = \left( {\dfrac{1}{2}{m_f}V_f^2 + {m_f}{g_f}{h_f}} \right) \to \max $$ (12) -
凸优化问题如下式:
$$ \begin{split} &\min\;\; {f_0}\left( x \right) \\ & {\text{subject to }}{f_i}\left( x \right) \leqslant 0,i = 1, \cdots ,m \\ & \qquad \qquad a_i^Tx = {b_i},i = 1, \cdots ,p \end{split} $$ (13) 式中:目标函数
$ {f_0}\left( x \right) $ 和不等式约束函数$ {f_i}\left( x \right) $ 都是凸函数,而等式约束函数$ {h_i}\left( x \right) = a_i^Tx - {b_i} $ 是仿射函数。理论上只要能够将一个优化问题描述为凸形式,即可在多项式时间(代数运算次数为问题维度的有限次多项式函数)内得到物理可行问题的全局最优解,且不依赖任何初值。显然,上升段轨迹优化设计问题是一个高度的非线性最优控制问题,无法直接采用凸优化方法求解,因此需要先对优化问题模型进行一些处理。 -
由于上升段总飞行时间未知,为了考虑该类时间自由问题,定义新的自变量
$ \tau \in \left[ {0,1} \right] $ 和控制量$ {u_t} = {t_f} - {t_0} $ ,并且将原问题的时间区间映射到[0,1]上,表示为:$$ t = {t_0} + \left( {{t_f} - {t_0}} \right)\tau ,\tau \in \left[ {0,1} \right] $$ (14) 将原时间自变量作为新的状态变量,则有:
$$ \dfrac{{{\rm{d}}t}}{{{\rm{d}}\tau }} = {u_t},\tau \in \left[ {0,1} \right] $$ (15) 以姿态控制系统理想情况为例,设原弹道状态量为
$ x = \left[ {h,V,\theta ,m,\alpha } \right] $ ,弹道方程记为$\dfrac{{{\rm{d}}x}}{{{\rm{d}}t}} = f\left( {x,u} \right)$ ,则以$ \tau $ 为自变量的弹道运动方程表达为:$$ \dfrac{{{\rm{d}}x}}{{{\rm{d}}t}} = f\left( {x,u} \right)\dfrac{{{\rm{d}}t}}{{{\rm{d}}\tau }} = f\left( {x,u} \right){u_\tau } $$ (16) -
为了应用凸优化求解弹道优化问题,因此需要对问题进行凸化。凸化处理的基本思想是将非线性的系统方程和非线性的约束条件在给定的轨迹点处进行线性化处理。以不考虑姿态控制的上升段末端机械能最大问题为例,模型处理过程如下,设存在一个优化的初始状态和控制量序列:
$$ \begin{gathered} {{\bar x}_{(k)}} = {[{h_{(k)}},{V_{(k)}},{\theta _{(k)}},{m_{(k)}},{\alpha _{(k)}},{\beta _{(k)}},{\alpha _{c(k)}},{t_{(k)}}]^{\rm{T}}} \\ {u_{(k)}} = {\left[ {{{\dot \alpha }_{c(k)}},{m_{sr(k)}}} \right]^{\rm{T}}} \\ \end{gathered} $$ (17) 采用一阶欧拉法对状态和控制量序列下的线性状态约束进行离散处理,得到一个线性递推公式(凸约束)。设离散采用周期为T,则原连续状态约束成为:
$$ \begin{gathered} \bar x\left( {k + 1} \right) = A\left( k \right)\bar x\left( k \right) + B\left( k \right)u\left( k \right) + C\left( k \right) \\ {\text{ }}k = 1,2, \cdots ,m \\ \end{gathered} $$ (18) 其中,m为采样个数。各系数表达式为:
$$\begin{split} & {A_{\left( k \right)}} = {\left[ \begin{gathered} \dfrac{{\partial {{\bar f}_1}}}{{\partial h}}{\text{ }}\dfrac{{\partial {{\bar f}_1}}}{{\partial V}}{\text{ }}\dfrac{{\partial {{\bar f}_1}}}{{\partial \theta }}{\text{ }}\dfrac{{\partial {{\bar f}_1}}}{{\partial m}}{\text{ }}\dfrac{{\partial {{\bar f}_1}}}{{\partial \alpha }} \\ \dfrac{{\partial {{\bar f}_2}}}{{\partial h}}{\text{ }}\dfrac{{\partial {{\bar f}_2}}}{{\partial V}}{\text{ }}\dfrac{{\partial {{\bar f}_2}}}{{\partial \theta }}{\text{ }}\dfrac{{\partial {{\bar f}_2}}}{{\partial m}}{\text{ }}\dfrac{{\partial {{\bar f}_2}}}{{\partial \alpha }} \\ \dfrac{{\partial {{\bar f}_3}}}{{\partial h}}{\text{ }}\dfrac{{\partial {{\bar f}_3}}}{{\partial V}}{\text{ }}\dfrac{{\partial {{\bar f}_3}}}{{\partial \theta }}{\text{ }}\dfrac{{\partial {{\bar f}_3}}}{{\partial m}}{\text{ }}\dfrac{{\partial {{\bar f}_3}}}{{\partial \alpha }}{\text{ }}0 \\ \dfrac{{\partial {{\bar f}_4}}}{{\partial h}}{\text{ }}\dfrac{{\partial {{\bar f}_4}}}{{\partial V}}{\text{ }}\dfrac{{\partial {{\bar f}_4}}}{{\partial \theta }}{\text{ }}\dfrac{{\partial {{\bar f}_4}}}{{\partial m}}{\text{ }}\dfrac{{\partial {{\bar f}_4}}}{{\partial \alpha }} \\ \dfrac{{\partial {{\bar f}_5}}}{{\partial h}}{\text{ }}\dfrac{{\partial {{\bar f}_5}}}{{\partial V}}{\text{ }}\dfrac{{\partial {{\bar f}_5}}}{{\partial \theta }}{\text{ }}\dfrac{{\partial {{\bar f}_5}}}{{\partial m}}{\text{ }}\dfrac{{\partial {{\bar f}_5}}}{{\partial \alpha }} \\ {\text{ }}0{\text{ }}0 \\ \end{gathered} \right]_{{x_k},{u_k}}} \text{} \\ &{B_{1(k)}} = {\left[ \begin{gathered} \dfrac{{\partial {{\bar f}_1}}}{{\partial \dot \alpha }}{\text{ }}\dfrac{{\partial {{\bar f}_1}}}{{\partial {m_{sr}}}} \\ \dfrac{{\partial {{\bar f}_2}}}{{\partial \dot \alpha }}{\text{ }}\dfrac{{\partial {{\bar f}_2}}}{{\partial {m_{sr}}}} \\ \dfrac{{\partial {{\bar f}_3}}}{{\partial \dot \alpha }}{\text{ }}\dfrac{{\partial {{\bar f}_3}}}{{\partial {m_{sr}}}} \\ \dfrac{{\partial {{\bar f}_4}}}{{\partial \dot \alpha }}{\text{ }}\dfrac{{\partial {{\bar f}_4}}}{{\partial {m_{sr}}}} \\ \dfrac{{\partial {{\bar f}_5}}}{{\partial \dot \alpha }}{\text{ }}\dfrac{{\partial {{\bar f}_5}}}{{\partial {m_{sr}}}} \\ {\text{ 0 0 }} \\ \end{gathered} \right]_{{x_k},{u_k}}} \text{,}{B_{2(k)}} = {\left[ \begin{gathered} \dfrac{{\partial {{\bar f}_1}}}{{\partial {u_t}}} \\ \dfrac{{\partial {{\bar f}_2}}}{{\partial {u_t}}} \\ \dfrac{{\partial {{\bar f}_3}}}{{\partial {u_t}}} \\ \dfrac{{\partial {{\bar f}_4}}}{{\partial {u_t}}} \\ \dfrac{{\partial {{\bar f}_5}}}{{\partial {u_t}}} \\ {\text{ 1 }} \\ \end{gathered} \right]_{{x_k},{u_k}}} \text{}\\ &{C_{\left( k \right)}} = \bar f - {A_{\left( k \right)}}{\bar x_{\left( k \right)}} - {B_{1\left( k \right)}}{\alpha _{\left( k \right)}} - {B_{2\left( k \right)}}u \end{split} $$ (19) 为保证线性化的有效性,需引入信赖域约束:
$$ \left| {x - {x^k}} \right| \leqslant \delta $$ (20) 式中:
$ \delta $ 即为优化问题的信赖域,可依据具体的飞行器运动模型设定。 -
飞行过程中的动压、过载、驻点热流密度等过程约束同样需要在参考点处进行线性化处理。记
${f_c} = \left[ {{f_6}{\text{ }}{f_7}{\text{ }}{f_8}} \right] = {\left[ {\dot Q - {{\dot Q}_{\max }}{\text{ }}P - {P_{\max }}{\text{ }}n - {n_{\max }}} \right]^{\rm{T} }}$ ,过程约束可表示为$$ {f_c}\left( {x,u} \right) \leqslant 0 $$ (21) 对其进行线性化可得:
$$ {C_a}\left( k \right)x\left( k \right) + {C_b}\left( k \right) \leqslant 0 $$ (22) 式中:
$ {C_a}\left( k \right) $ 、$ {C_b}\left( k \right) $ 的求解方法与公式(19)中$ {A_{\left( k \right)}} $ 、$ {C_{\left( k \right)}} $ 的求解方法类似,此处不做赘述。 -
控制变量约束、状态变量约束的离散形式为:
$$ \begin{gathered} {u_{\min }} \leqslant u\left( n \right) \leqslant {u_{\max }} \\ {x_{\min }} \leqslant x\left( n \right) \leqslant {x_{\max }} \\ \end{gathered} $$ (23) 状态变量的端点条件表达为离散形式为:
$h\left( 1 \right) = {h_0} , V\left( 1 \right) = {V_0}$ ,$ \theta \left( 1 \right) = {\theta _0},m\left( 1 \right) = {m_0} $ ,$ \alpha \left( 1 \right) = {\alpha _0} $ ,$ t\left( 1 \right) = {t_0} $ ,$ h\left( m \right) = {h_f} $ ,$ V\left( m \right) \geqslant {V_{f\min }} $ ,$m\left( m \right) \geqslant {m_{f\min }} $ 。控制变量、状态变量约束及端点约束的离散形式已为凸约束,无需额外处理。
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机械能公式为:
$$ {f_9} = \dfrac{1}{2}m{V^2} + mgh $$ (24) 对其进行线性化可得:
$$ {f_9} = {C_A}\left( k \right)x\left( k \right) + {C_B}\left( k \right) $$ (25) 其中,
$$ \begin{split} &{C_{A(k)}} = {\left[ {\dfrac{{\partial {f_9}}}{{\partial h}}{\text{ }}\dfrac{{\partial {f_9}}}{{\partial V}}{\text{ }}\dfrac{{\partial {f_9}}}{{\partial \theta }}{\text{ }}\dfrac{{\partial {f_9}}}{{\partial m}}{\text{ }}\dfrac{{\partial {f_9}}}{{\partial \alpha }}{\text{ }}0} \right]_{{x_k},{u_k}}} \\ & {C_{B(k)}} = {f_9} - {C_{A(k)}}x\left( k \right) \end{split} $$ (26) 综上所述,原轨迹优化问题可表示为:
$$ \begin{gathered} \min {\text{ }}(25) \\ {\text{subject to }}(18),{\text{ }}(20),{\text{ }}(21),{\text{ }}(22),{\text{ }}(23) \\ \end{gathered} $$ (27) -
总结上文,基于凸优化方法的轨迹优化求解步骤如下:
(1) 设k=0时,
$ \left\{ {{{\bar x}_{\left( 0 \right)}};{u_{\left( 0 \right)}}} \right\} $ 为一条初始弹道;(2) 在第k+1次迭代中,利用式参考弹道的相关参数,求解凸优化问题
$ P_0^{\left( {k + 1} \right)} $ ,得到序列最优解$ \left\{ {{{\bar x}_{\left( {k + 1} \right)}};{u_{\left( {k + 1} \right)}}} \right\} $ ;(3) 判断收敛判据是否满足:
$$ \left| {{{\bar x}_{\left( {k + 1} \right)}}\left( n \right) - {{\bar x}_{\left( k \right)}}\left( n \right)} \right| \leqslant \varepsilon ,n = 1,2, \cdots ,m $$ (28) 其中,收敛域
$ \varepsilon $ 可以根据具体问题设定。若判据满足则进入第(4)步,否则令k=k+1,判断回溯条件并进入第(2)步;(4)
$ \left\{ {{{\bar x}_{\left( {k + 1} \right)}};{u_{\left( {k + 1} \right)}}} \right\} $ 即是全局最优状态轨迹和最优控制变量。
Trajectory optimization design of ascending stage of RBCC powered hypersonic vehicle
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摘要: 针对火箭基组合循环(Rocket Based Combined Cycle,RBCC)动力系统工作模态复杂、与飞行状态耦合程度高的特性,建立了一种适用于RBCC动力高超声速飞行器的动力段轨迹优化模型。同时,针对RBCC动力飞行器,基于凸优化理论建立了上升段轨迹优化设计框架和求解策略。在此基础上,进行了上升段末端机械能最大算例仿真。仿真结果表明,相关模型和轨迹优化方法具备良好的可行性,优化结果符合RBCC动力系统工作特点。论文提出的轨迹优化方法可有效处理复杂工作模态下RBCC助推飞行器上升段轨迹优化问题,为未来关于这一类轨迹设计与优化的工作提供了一些新的思路。Abstract: Aiming at the characteristics of complex working mode and high coupling with flight state of rocket based combined cycle (RBCC) power system, a trajectory optimization model for RBCC powered hypersonic vehicle was established. At the same time, the trajectory optimization design framework and solution strategy for RBCC powered aircraft were established based on convex optimization theory. On this basis, the maximum mechanical energy at the end of the rising section was simulated. The simulation results show that the relevant models and trajectory optimization methods are feasible, and the optimization results accord with the working characteristics of RBCC power system. The trajectory optimization method proposed in this paper can effectively deal with the trajectory optimization of RBCC assisted aircraft in the ascending phase under complex working modes, and provides some new ideas for this kind of trajectory design and optimization in the future.
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Key words:
- RBCC /
- convex optimization /
- rising section /
- mechanical energy
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