﻿ 放宽静稳定度飞机时间延迟稳定边界
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1. 北京航空航天大学 航空科学与工程学院, 北京 100083;
2. 中国航空工业成都飞机设计研究所 歼击机综合仿真航空科技重点实验室, 成都 610091

Time delay stability boundary on relaxed static stability aircraft
CHEN Xiaoming1, SUN Shaoshan2, TAO Chenggang2, TANG Yong2
1. School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China;
2. The Aviation Key Laboratory of Fighter Integrated Simulation, AVIC Chengdu Aircraft Design and Research Institute, Chengdu 610091, China
Abstract: The problem of quantitative determination between Relaxed Static Stability (RSS) and time delay boundary of the fly-by-wire control system during the preliminary design stage of aircraft is studied. Based on the longitudinal short-period equation of the combat aircraft, the time delay factor in the flight control system and the relationship between parameters in short period equation and RSS is analyzed. A characteristic equation of closed-loop system with time delay of flight control system is constructed in the form of equivalent input delay. The quantitative numerical relationship between the RSS and the time delay boundary of the flight control system is determined by the root tendency theory and the numerical calculation method. Moreover, the influence of the parameter uncertainty of control surface efficiency and dynamic derivative on the time delay boundary is also discussed. The method of this paper has certain engineering practical significance for determining the time delay boundary of the flight control system and the RSS of the aircraft in the preliminary stage of aircraft design.
Keywords: time delay stability    short period mode    relaxed static stability    controllability evaluation    preliminary design

1 战斗机电传飞控系统的时间延迟

 图 1 飞行控制系统的主要构成 Fig. 1 Main components of flight control system

1) 传感器。用于飞行控制系统的信号主要包括大气数据(气流角、动静压等)、惯性运动数据(三轴速率、三轴过载等)，当前所采用的嵌入式大气数据传感系统[6]和光学陀螺惯导方案，相比于原机械风标和机械陀螺，信号采集的稳定性和准确性都大幅提升，带来的信号测量时延极小，一般不超过2个测量周期。

2) 作动器。目前作动器的驱动方式较多，包括传统的液压、电动以及日渐成熟的电动静液作动器(Electro-Hydrostatic Actuator, EHA)[7]等，从目前的工程实践来看，这一环节的时间延迟是显著的，一般来说在50~100 ms量级。

3) 总线传输。飞控系统总线传输的时间延迟与飞行控制周期相关。参照MIL-STD-1553标准[8]，以15 ms周期为例，其对应的时延最大约22.5 ms，随着未来飞控系统总线传输技术向光传方案方向的发展[9]，这部分时延将进一步降低。

4) 飞行员操纵时延。飞行员从接收到状态信息反馈到做出操纵的时间延迟主要来自飞行员的反应时间，一般在200 ms左右[10]，加之飞行员指令输入单元(Pilot Input Unit, PIU)的40 ms左右时延，总时延约在240 ms量级。尽管飞行员操纵时延较大，但其输入是控制增稳系统(Control Augmentation System, CAS)的外部执行指令，因此在考虑CAS稳定性的时候，并不将该时延纳入系统时延。这一部分操纵时延主要涉及的问题是飞行员诱发震荡(Pilot Induced Oscillations, PIO)[11]现象。

2 静稳定度与短周期方程关系

 ${K_n} = {h_n} - h$ （1）

 ${C_{{m_\alpha }}} = {C_{{L_\alpha }}}(h - {h_n}) = - {C_{{L_\alpha }}}{K_n}$ （2）

 $\begin{array}{l} \left[ {\begin{array}{*{20}{c}} {\Delta \dot \alpha }\\ {\Delta \dot q} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - {Z_\alpha }}&1\\ {{{\bar M}_\alpha } - {{\bar M}_{\dot \alpha }}{Z_\alpha }}&{{{\bar M}_q} + {{\bar M}_{\dot \alpha }}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\Delta \alpha }\\ {\Delta q} \end{array}} \right] + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{c}} { - {Z_{{\delta _e}}}}\\ {{{\bar M}_{{\delta _e}}} - {{\bar M}_{\dot \alpha }}{Z_{{\delta _e}}}} \end{array}} \right]\Delta {\delta _e} = \mathit{\boldsymbol{Ax}} + \mathit{\boldsymbol{Bu}} \end{array}$ （3）

 $\left\{ \begin{array}{l} \begin{array}{*{20}{l}} {{Z_\alpha } = \frac{{({C_{{D^*}}} + {C_{{L_\alpha }}}){Q_*}{\kern 1pt} S}}{{m{V_*}}}}\\ {{{\bar M}_\alpha } = {C_{{m_\alpha }}}\frac{{{Q_*}Sc}}{{{I_y}}}} \end{array}\\ \begin{array}{*{20}{l}} {{{\bar M}_q} = {C_{{m_q}}}\left( {\frac{c}{{2{V_*}}}} \right)\frac{{{Q_*}{\kern 1pt} Sc}}{{{I_y}}}}\\ {{{\bar M}_{\dot \alpha }} = {C_{{m_{\dot \alpha }}}}\left( {\frac{c}{{2{V_*}}}} \right)\frac{{{Q_*}{\kern 1pt} Sc}}{{{I_y}}}} \end{array}\\ \begin{array}{*{20}{l}} {{Z_{{\delta _e}}} = {C_{{L_{{\delta _e}}}}}\frac{{{Q_*}{\kern 1pt} Sc}}{{m{V_*}}}}\\ {{{\bar M}_{{\delta _e}}} = {C_{{m_{{\delta _e}}}}}\frac{{{Q_*}{\kern 1pt} Sc}}{{{I_y}}}} \end{array} \end{array} \right.$ （4）

 图 2 配平迎角及油门位置随静稳定度变化曲线 Fig. 2 Curves of trim angle of attack and throttle position with change of static stability

 图 3 短周期方程系数随静稳定度变化曲线 Fig. 3 Curves of parameters in short period equation with change of static stability

 图 4 开环根轨迹随静稳定度变化曲线 Fig. 4 Curves of open loop root locus with change of static stability

 $\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{Ax}} + \mathit{\boldsymbol{BKu}}$ （5）

 $\mathit{\boldsymbol{sx}} = \mathit{\boldsymbol{Ax}} + \mathit{\boldsymbol{BKx}}{\rm{exp}}( - \tau s)$ （6）

 ${\rm{ det }}(s\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{A}} - \mathit{\boldsymbol{BK}}{\kern 1pt} {\rm{exp}}( - \tau s)) = 0$ （7）
3 时间延迟稳定边界

 $\begin{array}{*{20}{c}} {{s^2} - {\rm{tr}} (\mathit{\boldsymbol{A}})s - {\rm{tr}} (\mathit{\boldsymbol{BK}}) s{\rm{exp}} ( - \tau s) + }\\ {\eta {\rm{exp}}( - \tau s) + {\rm{det}} (\mathit{\boldsymbol{A}}) = 0} \end{array}$ （8）

 $\mathit{\boldsymbol{CE}}(s, \tau ) = \sum\limits_{l = 0}^n {{P_l}} ({s^\gamma }){\rm{exp}}( - l\tau s) = 0$ （9）

 $\exp ( - {\tau _{ck}}s){|_{s = {\rm{j}}{\omega _{ck}}}} = \frac{{1 - {\rm{j}}{T_{ck}}}}{{1 + {\rm{j}}{T_{ck}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {T_{ck}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \in {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathbb{R}$ （10）

 $\left\{ {\begin{array}{*{20}{l}} {{T_{ck}} = {\rm{tan}}\left( {\frac{{{\tau _{ck}}{\omega _{ck}}}}{2}} \right)}\\ {{\tau _{ck}} = \frac{2}{{{\omega _{ck}}}}({\rm{ta}}{{\rm{n}}^{ - 1}}(T) + p\pi ){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p = 0, 1, \cdots , \infty } \end{array}} \right.$ （11）

 ${\rm{exp}}( - l{\tau _{ck}}s){|_{s = {\rm{j}}{\omega _{ck}}}} = {\left( {\frac{{1 - {\rm{j}}{T_{ck}}}}{{1 + {\rm{j}}{T_{ck}}}}} \right)^l}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {T_{ck}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \in {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathbb{R}$ （12）

 $\sum\limits_{l = 0}^n {{P_l}} ({s^\gamma }){(1 - {\rm{j}}{T_{ck}})^l}{(1 + {\rm{j}}{T_{ck}})^{(n - l)}}$ （13）

 $h(s, {T_{ck}}){|_{s = {\rm{j}}{\omega _{ck}}}} = {h_R}({\omega _{ck}}, {T_{ck}}) + {\rm{j}}{h_{\rm{I}}}({\omega _{ck}}, {T_{ck}})$ （14）

 $\left\{ {\begin{array}{*{20}{l}} {{h_{\rm{R}}} = \sum\limits_{i = 0}^n {{a_i}} ({\omega _{ck}})T_{ck}^i = 0}\\ {{h_{\rm{I}}} = \sum\limits_{i = 0}^n {{b_i}} ({\omega _{ck}})T_{ck}^i = 0} \end{array}} \right.$ （15）

 $S_\tau ^s{|_{s = {\rm{j}}{\omega _{ck}}, \tau = {\tau _{ck}}}} = {\left. {\frac{{{\rm{d}}s}}{{{\rm{d}}\tau }}} \right|_{s = {\rm{j}}{\omega _{ck}}, \tau = {\tau _{ck}}}}$ （16）

 $\begin{array}{l} S_\tau ^s{|_{s = {\rm{j}}{\omega _{ck}}, \tau = {\tau _{ck}}}} = {\left. { - \frac{{\frac{{\partial C}}{{\partial \tau }}}}{{\frac{{\partial C}}{{\partial s}}}}} \right|_{s = {\rm{j}}{\omega _{ck}}, \tau = {\tau _{ck}}}} = \\ {\left. {\frac{{\sum\limits_{l = 0}^n {{P_l}} ({s^\gamma })ls{\rm{exp}}( - l\tau s)}}{{\sum\limits_{l = 0}^n {\left| {\frac{{{\rm{d}}{P_l}({s^\gamma })}}{{{\rm{d}}s}} - l{P_l}({s^\alpha })\tau } \right|{\rm{exp}}( - {\rm{j}}\tau s)} }}} \right|_{s = {\rm{j}}{\omega _{ck}}, \tau = {\tau _{ck}}}} \end{array}$ （17）

 Delay τ(s) Crossing frequency ω/(rad·s-1) Root tendency Unstable root Stability 0 0 Stable 0.280 3 5.023 8 + 2 Unstable 1.531 0 5.023 8 + 4 Unstable 2.781 6 5.023 8 + 6 Unstable 4.032 3 5.023 8 + 8 Unstable 5.283 0 5.023 8 + … … … … Unstable

 图 5 时间延迟边界随静稳定度变化曲线 Fig. 5 Curve of time delay boundary with change of static stability

 图 6 根轨迹随时间延迟变化曲线 Fig. 6 Curves of root locus with change of time delay
4.2 时延边界对舵效的敏感性

 图 7 舵效拉偏系数对时延边界的影响 Fig. 7 Influence of control efficiency departure coefficient on time delay boundary

4.3 时延边界对动导数的敏感性

 图 8 动导数拉偏系数对时延边界的影响 Fig. 8 Influence of dynamic derivative departure coefficient on time delay boundary

4.4 不确定性因素的综合影响

 图 9 舵效及动导不确定性系数对时延边界的综合影响 Fig. 9 Comprehensive influence of both control efficiency and dynamic derivative departure coefficient on static stability
5 结论

1) 分析了目前电传飞控系统中时间延迟的来源，指出在初步设计阶段进行可控性评估时可以主要考虑输入时间延迟，分析了静稳定度与短周期方程参数间的关系。

2) 根据时延系统根轨迹穿越虚轴时为纯虚数的特点，并结合根轨迹穿越时其穿越方向仅与穿越频率相关而与具体时延无关的特点，给出了基于短周期方程的闭环时延系统的稳定边界的精确求解方法。

3) 针对短周期方程参数中的主要不确定性因素，分析了其对时延边界的影响，研究表明舵效不确定性是影响时间延迟稳定边界主要因素，但在静稳定度较低甚至静不稳定的情况下，也应考虑到动导不确定性对结果的影响，实际设计中，应综合两者的影响进行设计。

4) 解决了在飞机设计初步阶段，给定静稳定度布局情况下，基于可控性考虑的精确时间延迟稳定边界求解问题，该边界对飞控系统的设计具有指导意义；另一方面，该求解方法同时提供了一种在飞控系统时间延迟已知情况下，对静稳定度布局边界的约束指标。

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http://dx.doi.org/10.7527/S1000-6893.2019.23487

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#### 文章信息

CHEN Xiaoming, SUN Shaoshan, TAO Chenggang, TANG Yong

Time delay stability boundary on relaxed static stability aircraft

Acta Aeronautica et Astronautica Sinica, 2020, 41(6): 523487.
http://dx.doi.org/10.7527/S1000-6893.2019.23487