﻿ 模式失配条件下连续时间控制系统的零控脱靶量估计误差分布
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 自动化学报  2018, Vol. 44 Issue (10): 1824-1832 PDF

1. 国防科技大学电子科学学院ATR重点实验室 长沙 410073

Distribution of Zero-effort Miss Distance Estimation Errors in Continuous-time Controlled System With Mode Mismatch
XIANG Sheng-Wen1, FAN Hong-Qi1, FU Qiang1
1. ATR Key Laboratory, College of Electronic Science, National University of Defense Technology, Changsha 410073
Manuscript received : May 10, 2017, accepted: August 17, 2017.
Corresponding author. FAN Hong-Qi  Associate professor at National University of Defense Technology. His research interest covers radar signal processing, target tracking, guidance, and control, and information fusion. Corresponding author of this paper..
Abstract: For the problem of solving the error distribution of zero-effort miss distance (ZEM) in highly maneuvering target interception, an analytical approach is proposed for the configuration of a linear estimator and a separate mode decision-maker. In the case of a fixed mode decision delay, the error of ZEM follows a biased Gaussian distribution. Finally, the correctness of the proposed method is verified by comparison with Monte Carlo simulation.
Key words: Pursuit-evasion game     highly maneuvering target     zero-effort miss distance     estimation error     mode mismatch

1 问题描述 1.1 系统运动模型

 $$${t_f} \approx \frac{{{r_0}}}{{{V_p}\cos {\phi _{p}(0)} - {V_e}\cos {\phi _{e}(0)}}}$$$ (1)

 $$$\begin{array}{*{20}{l}} {{{\dot x}_1} = {x_2}, }&{{x_1}(0) = 0}\\ {{{\dot x}_2} = {x_3} - {x_4}, }&{{x_2}(0) = {V_e}{\phi _{e}(0)} - {V_p}{\phi _{p}(0)}}\\ {{{\dot x}_3} = \dfrac{{u_e} - {x_3}}{{\tau _e}}, }&{{x_3}(0) = 0}\\[2mm] {{{\dot x}_4} = \dfrac{{u_p} - {x_4}}{{\tau _p}}, }&{{x_4}(0) = 0} \end{array}$$$ (2)

 $$$\ \begin{array}{*{20}{c}} {|{u_i}(t)| \le |a_i^{\max}|, }&{i = p, e} \end{array}$$$ (3)

 $$$\gamma = \frac{{a_p^{\max }}}{{a_e^{\max }}}$$$ (4)
 $$$\varepsilon = \frac{{{\tau _e}}}{{{\tau _p}}}$$$ (5)

 $$$\ \begin{array}{l} {\bf{\dot {\pmb x}}}(t) = {A}{\pmb x}(t) + {{\pmb B}_1}{u_p}(t) + {{\pmb B}_2}{u_e}(t)\\ {\pmb x}(0) = {(0, {x_2}(0), 0, 0)^{\rm T}} \end{array}\$$$ (6)

 \begin{align} &{{A}} = \left[{\begin{array}{*{20}{c}} 0&1&0&0\\ 0&0&1&{-1}\\ 0&0&{\dfrac{{-1}}{{{\tau _e}}}}&0\\ 0&0&0&{\dfrac{{-1}}{{{\tau _p}}}} \end{array}} \right]\nonumber\\ &{{{\pmb B}_1} = \left[{\begin{array}{*{20}{c}} 0\\ 0\\ 0\\ {\dfrac{1}{{{\tau _p}}}} \end{array}} \right]}, ~{{\pmb B}_2} = \left[{\begin{array}{*{20}{c}} 0\\ 0\\ {\dfrac{1}{{{\tau _e}}}}\\ 0 \end{array}} \right] \end{align} (7)

 $$$\ z(t) = {{\pmb D}^{\rm T}}{\Phi}({t_f}, t){\pmb x}(t)$$$ (8)

 $$${\tilde {\pmb x}}(t) = {{\Phi_F}}(t, 0){{\tilde {\pmb x}}_0} + \int\limits_0^t {{\Phi_F}(t, s){\pmb \zeta}(s){\rm d}s}$$$ (24)

 $$${\pmb \xi }(t) = {\Phi_F}(t, 0){\rm E}({{\tilde {\pmb x}}_0})$$$ (25)
 $$$\begin{split} &~~~~~~~~~~~~~~{\Sigma }(t) = {\Phi_F}(t, 0){{\tilde P}_0}{\Phi_F}^{\rm T}{(t, 0)}+ \\ &\int\limits_0^t {{\Phi_F}(t, s)[{Q} + {k}(s){R}(s){{k}^{\rm T}}(s)]{\Phi_F}^{\rm T}{{(t, s)}}{\rm d}s} \end{split}$$$ (26)

1) $t \in [0, {t_{sw}})$, 此时估计器一直保持正确的模式, 状态估计误差的均值和方差矩阵分别如式(25)和(26)所示.

2) $t \in [{t_{sw}}, {t_f}]$, 此时存在模式失配, 系统的状态方程为

 $$${\dot {\pmb x}}(t) = {A\pmb x}(t) + {{\pmb B}_1}u_P(t) + {{\pmb B}_2}{m_2} + {\pmb w}(t)$$$ (27)

 $$$\begin{array}{l} {\dot {\tilde {\pmb x}}}(t) = F(t){\tilde {\pmb x}}(t) - {{\pmb B}_2}({m_2} - {m_1}) + {\pmb \zeta}(t)\\ {\tilde {\pmb x}}({t_{sw}}) = {{{\tilde {\pmb x}}}_{sw}} \end{array}$$$ (28)

 \begin{align} {\tilde {\pmb x}}(t) =\,&{\Phi_F}(t, {t_{sw}}){\tilde {\pmb x}}({t_{sw}}) + \int\limits_{{t_{sw}}}^t {{\Phi_F}(t, s){\pmb \zeta }(s){\rm d}s}-\nonumber\\ &({m_2} - {m_1})\int\limits_{{t_{sw}}}^t {{\Phi_F}(t, s){\rm d}s} {{\pmb B}_2} \end{align} (29)

 $$${\tilde {\pmb x}}({t_{sw}}) = {\Phi_F}({t_{sw}}, 0){{\tilde {\pmb x}}_0} + \int\limits_0^{{t_{sw}}} {{\Phi_F}({t_{sw}}, s){\pmb \zeta }(s){\rm d}s}$$$ (30)

 \begin{align} {\tilde {\pmb x}}(t) =\,&{\Phi_F}(t, 0){{\tilde {\pmb x}}_0} + \int\limits_0^t {{\Phi_F}(t, s){\pmb \zeta }(s){\rm d}s} - \nonumber\\ &({m_2} - {m_1})\int\limits_{{t_{sw}}}^t {{\Phi_F}(t, s){\rm d}s{{\pmb B}_2}} \end{align} (31)

 $$${\pmb \xi }(t) = {\Phi_F}(t, 0){\rm E}\{ {{\tilde {\pmb x}}_0}\} - ({m_2} - {m_1})\int\limits_{{t_{sw}}}^t {{\Phi_F}(t, s){\rm d}s} {{\pmb B}_2}$$$ (32)

1) $t \in [0, {t_{sw}})$, 此时系统的状态方程和估计器的滤波方程分别如式(19)和(20)所示, 状态估计误差的均值和方差矩阵则分别见式(25)和(26).

2) $t \in [{t_{sw}}, {t_{sw}} + \Delta t)$, 存在模式失配, 此时系统的状态方程和估计器的滤波方程分别如式(27)和(20)所示, 状态估计误差的均值和方差则矩阵分别见式(32)和(26).

3) $t \in [{t_{sw}} + \Delta t, {t_f}]$, 估计器回到正确的目标模式上, 此时的系统状态方程为式(27), 而估计器的滤波方程为

 $$${\dot{ \hat {\pmb x}}}(t) = {A\hat {\pmb x}}(t) + {{B}_1}u_P(t) + {{B}_2}{m_2} + {k}(t)({\pmb Y}(t) - {H\hat {\pmb x}}(t))$$$ (33)

 $$$\begin{split} &{\dot {\tilde {\pmb x}}}(t) = {F}(t){\tilde {\pmb x}}(t) + {\pmb \zeta}(t) \\ &{\tilde {\pmb x}}({t_{sw}} + \Delta t) = {{{\tilde {\pmb x}}}_{{t_{sw}} + \Delta t}} \end{split}$$$ (34)

 \begin{align} {\tilde {\pmb x}}(t) =\,&{\Phi_F}(t, {t_{sw}} + \Delta t){{\tilde {\pmb x}}_{{t_{sw}} + \Delta t}} +\nonumber\\ & \int\limits_{{t_{sw}} + \Delta t}^t {{\Phi_F}(t, s){\pmb \zeta }(s){\rm d}s} \end{align} (35)

 \begin{align} {{{\tilde {\pmb x}}}_{{t_{sw}} + \Delta t}} =\,&{\Phi_F}({t_{sw}} + \Delta t, 0){{{\tilde {\pmb x}}}_0}+\nonumber\\ &\int\limits_0^{{t_{sw}} + \Delta t} {{\Phi_F}({t_{sw}} + \Delta t, s){\pmb \zeta }(s){\rm d}s}- \nonumber\\ &({m_2} - {m_1})\int\limits_{{t_{sw}}}^{{t_{sw}} + \Delta t} {{\Phi_F}({t_{sw}} + \Delta t, s){\rm d}s} {{\pmb B}_2} \end{align} (36)

 \begin{align} {\tilde {\pmb x}}(t) =\,&{\Phi_F}(t, 0){{\tilde {\pmb x}}_0} + \int\limits_0^t {{\Phi_F}(t, s){\pmb \zeta}(s){\rm d}s} -\nonumber\\ &({m_2} - {m_1})\int\limits_{{t_{sw}}}^{{t_{sw}} + \Delta t} {{\Phi_F}(t, s){\rm d}s} {{\pmb B}_2} \end{align} (37)

 \begin{align} {\pmb \xi}(t) =\,&{\Phi_F}(t, 0){\rm E}({{\tilde {\pmb x}}_0}) -\nonumber\\ & ({m_2} - {m_1})\int\limits_{{t_{sw}}}^{{t_{sw}} + \Delta t} {{\Phi_F}(t, s){\rm d}s} {{\pmb B}_2} \end{align} (38)

$\tilde z(t) = \hat z(t) - z(t)$表示ZEM的估计误差, 则

 \begin{align} \tilde z(t) =\,&{{\pmb g}^{\rm T}}(t){\hat {\pmb x}}(t) - {{\pmb x}^{\rm T}}(t){\pmb x}(t) = {{\pmb g}^{\rm T}}(t){\tilde {\pmb x}}(t)=\nonumber\\ &{\tilde x_1}(t) + {\tilde x_2}(t)({t_f} - t) + {\tilde x_3}(t) \tau _e^2\psi\left( {\frac{{{t_f} - t}}{{{\tau _e}}}}\right)-\nonumber\\ &{\tilde x_4}(t)\tau _p^2\psi\left( {\frac{{{t_f} - t}}{{{\tau _p}}}}\right) \end{align} (39)

${\mu (t) = {\rm E}\{ \tilde z(t)\} }$, ${{\sigma ^2}(t) = {\mathop{\rm var}} \{ \tilde z(t)\} }$, 则由式(39)可以得到:

 $$$\mu (t) = {{\pmb g}^{\rm T}}(t){\rm E}\{{\tilde {\pmb x}}(t)\} ={{\pmb g}^{\rm T}}(t){\pmb \xi }(t)$$$ (40)
 \begin{align} {\sigma ^2}(t) =\, &{\mathop{\rm var}} \{ \tilde z(t)\} = {\rm E}\{ {\{ \tilde z(t) - {\rm E}[\tilde z(t)]\} ^2}\}=\nonumber\\ &{\rm E}\{ {\{ {{\pmb g}^{\rm T}}(t){\tilde {\pmb x}}(t) - {{\pmb g}^{\rm T}}(t){\rm E}[{\tilde {\pmb x}}(t)]\} ^2}\}=\nonumber\\ &{\rm E}\{ {\{ {{\pmb g}^{\rm T}}(t)\{ {\tilde {\pmb x}}(t) - {\rm E}[{\tilde {\pmb x}}(t)]\} \} ^2}\}=\nonumber\\ &{{\pmb g}^{\rm T}}(t){\rm E}\{ \{ {\tilde {\pmb x}}(t) -\nonumber\\ &{\rm E}[{\tilde {\pmb x}}(t)]\} \cdot {\{ {\tilde {\pmb x}}(t) - {\rm E}[{\tilde {\pmb x}}(t)]\} ^{\rm T}}\} {\pmb g}(t)=\nonumber\\ &{{\pmb g}^{\rm T}}(t){\mathop{\rm{var}}} \{ {\tilde {\pmb x}}(t)\}{\pmb g}(t)=\nonumber\\ &{{\pmb g}^{\rm T}}(t){\Sigma }(t){\pmb g}(t) \end{align} (41)

4) ${t_{go}}$的估计精度及弹目时间常数${\tau _e}$${\tau _p}通过投影向量{{\pmb g}}(t)影响脱靶量.在具体的拦截问题中, 弹目时间常数通常可假定为确定已知的, 而雷达导引头可直接获得高精度的t_{go}测量, 因此本文分析中不考虑它们对ZEM估计误差的影响. 5) 估计器的观测精度将直接影响到Kalman增益系数k(t), 见式(21), 进而影响ZEM估计误差的均值和方差. 3 仿真实验 本节通过一个典型的TBM拦截场景验证前面理论推导的正确性, 仿真参数设置如表 1, 蒙特卡洛仿真次数设置为1 000. 表 1 仿真参数 Table 1 Simulation parameters 图 5给出了两种不同过载比\gamma=2$$\gamma=2.5$下ZEM估计误差的均值变化曲线.从实验结果可以看出, 本文推导的理论结果与蒙特卡洛仿真的曲线基本吻合.由图 5还可以看出, 当目标的运动模式改变时($t = 2$ s), ZEM的估计误差会迅速增大, 当运动模式被正确识别后($t = 2.1$ s), ZEM的估计误差将逐渐减小. 图 6给出了这两种情形下ZEM估计误差的方差分布, 可以看出本文理论推导结果与蒙特卡洛仿真的曲线同样也是吻合的. 图 5图 6的仿真结果充分说明了本文理论推导的正确性.

 图 5 ZEM估计误差的均值 Figure 5 Mean of ZEM estimation error
 图 6 ZEM估计误差的方差 Figure 6 Variance of ZEM estimation error

 图 7 各状态估计误差 Figure 7 Estimation error of every state
4 结论

 \begin{align} {\rm E}\{ {\pmb \zeta }(t){{\pmb \zeta}^{\rm T}}(t)\} = \, &{\rm E}\{ [{k}(t){\pmb v}(t)- \nonumber\\ & {\pmb w}(t)]{[{k}(t){\pmb v}(t)-{\pmb w}(t)]^{\rm T}}\} =\nonumber\\ &{k}(t){R}(t){{k}^{\rm T}}(t) + {Q} \end{align} (A3)

 \begin{align} &{\Sigma }(t) = {\rm E}\{ \tilde {\pmb x}(t) - {\rm E}\{ \tilde {\pmb x}(t)\} \} ^2={\rm E}\Bigg\{ {\Phi_F}(t, 0){{{\tilde {\pmb x}}}_0} +\nonumber\\ & \int\limits_0^t {{\Phi_F}(t, s){\pmb \zeta}(s){\rm d}s} - {\Phi_F}(t, 0){\rm E}\{ {{{\tilde {\pmb x}}}_0}\} \Bigg\}^2=\nonumber\\ & {\rm E}{\left\{ {{\Phi_F}(t, 0)[{{{\tilde {\pmb x}}}_0}- {\rm E}({{{\tilde {\pmb x}}}_0})] + \int\limits_0^t {{\Phi_F}(t, s){\pmb \zeta }(s){\rm d}s} } \right\}^2}=\nonumber\\ & {\Phi_F}(t, 0){\rm E}\left\{ {[{{{\tilde {\pmb x}}}_0}-{\rm E}({{{\tilde {\pmb x}}}_0})]{{[{{{\tilde {\pmb x}}}_0}- {\rm E}({{{\tilde {\pmb x}}}_0})]}^{\rm T}}} \right\}{\Phi_F}^{\rm T}{(t, 0)}+\nonumber\\ &\int\limits_0^t {{\Phi_F}(t, s){\rm E}\{ {\pmb \zeta } (s){{\pmb \zeta }^{\rm T}}(s)\} {\Phi_F}^{\rm T}{{(t, s)}}{\rm d}s}=\nonumber\\ & {\Phi_F}(t, 0){{{\tilde P}}_0}{\Phi_F}^{\rm T}{(t, 0)}+\nonumber\\ &\int\limits_0^t {{\Phi_F}(t, s)[{Q} + {k}(s){R}(s){{k}^{\rm T}}(s)]{\Phi_F}^{\rm T}{{(t, s)}}{\rm d}s} \end{align} (A4)

 符号说明 $P$ 导弹 $E$ 目标 $\tau_p$, $\tau_e$ 导弹和目标控制系统的时间常数 $a_p^{\max}, a_e^{\max}$ 导弹和目标最大横向加速度 ${V_p}, {V_e}$ 导弹和目标的飞行速度 ${u_p}, {u_e}$ 导弹和目标的横向加速度指令 $r$ 弹目相对距离 ${t_{sw}}$ 目标模式切换时刻 $t$ 仿真时间 ${t_f}$ 终止时刻 g 重力加速度, $9.8\rm m/{\rm{s}^2}$ $m$ 目标的运动模式 ${m_1}, {m_2}$ 目标在模式切换时刻前后的运动模式 $\Delta m$ 目标运动模式改变量, $\Delta m = {m_2} - {m_1}$ $T$ 离散采样时间间隔 ${\sigma _\theta }$ 测角精度 ${\sigma _a}$ 导弹加速度测量精度 ${s_w}$ 目标指令加速度误差的功率谱密度 $\Delta t$ 目标运动模式辨识延迟 ${\tilde {\pmb x}}$ 状态估计误差 ${\pmb \xi }, {\Sigma}$ 状态估计误差的均值和方差 $\eta (t)$ ZEM估计误差 $\mu, {\sigma ^2}$ ZEM估计误差的均值和方差

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