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复合轴跟踪系统在不同的应用场景由不同的结构组成,文中以激光通信光电跟踪系统为例,单探测器复合轴的基本结构框图如图1所示。单探测器复合轴跟踪系统主要包括主轴粗跟踪系统和子轴精跟踪系统,并且两轴共用一个探测器完成ATP的工作过程。
为了分析控制系统传递函数,单探测器复合轴控制结构图如图2所示。其中,
${G_d}\left( s \right)$ 为系统探测器的传递函数,${G_{c1}}\left( s \right)$ ,${G_{c2}}\left( s \right)$ 分别为粗、精跟踪系统控制器的传递函数,$D\left( s \right)$ 为解耦控制器,Gp1(s),Gp2(s)分别为粗、精跟踪系统被控对象的传递函数,$R\left( s \right)$ 为系统的输入引导函数,$C\left( s \right)$ 为系统的输出函数。根据梅森公式,可得单探测器复合轴跟踪系统的传递函数为:
$$ G\left( s \right) = \frac{{{G_d}{G_{c1}}{G_{p1}} + {G_d}{G_{c2}}{G_{p2}} + {G_d}D{G_{c2}}{G_{p2}}{G_{c1}}{G_{p1}}}}{{\left( {1 + {G_d}{G_{c1}}{G_{p1}}} \right)\left( {1 + {G_d}{G_{c2}}{G_{p2}}} \right) + {G_d}{G_{c2}}{G_{p2}}{G_{c1}}{G_{p1}}\left( {D - {G_d}} \right)}} $$ (1) 由公式(1)可知,解耦控制器
$D\left( s \right)$ 与探测器的传递函数${G_d}\left( s \right)$ 相等时,系统的耦合特性被消除,成为静态自主系统,系统特征方程等于粗、精跟踪系统的特征方程乘积。因此,单探测器复合轴跟踪系统稳定的必要条件是粗、精跟踪系统均为稳定系统。设复合轴跟瞄系统的误差函数为
$E\left( s \right)$ ,则误差函数可表述为:$$ E\left( s \right) = R\left( s \right) - C\left( s \right) $$ (2) 则根据公式(2)可推导出系统误差传递函数
${\phi _e}\left( s \right) = \dfrac{{E\left( s \right)}}{{R\left( s \right)}}$ 的表达式:$$ {\phi _e}\left( s \right) = \frac{1}{{\left( {1 + {G_d}{G_{c1}}{G_{p1}}} \right)\left( {1 + {G_d}{G_{c2}}{G_{p2}}} \right) + {G_d}{G_{c2}}{G_{p2}}{G_{c1}}{G_{p1}}\left( {D - {G_d}} \right)}} $$ (3) 从误差传递函数表达式(3)可以看出,当
$D\left( s \right) = {G_d}\left( s \right)$ 时,系统完全解耦,单探测器复合轴跟踪系统误差传递函数可以等效为粗、精跟踪系统各自误差传递函数的乘积,而传统复合轴系统的误差传递函数只取决于精跟踪系统的误差传递函数,因此,采用单探测器复合轴控制可以提高系统的跟踪精度。 -
单探测器复合轴跟踪系统所用探测器通常为CCD相机,由于探测器存在脱靶量滞后的现象,提升了解算解耦控制器使其逼近CCD探测器传递函数的难度,同时为了降低探测器实时处理的要求,满足粗、精跟踪系统带宽匹配的需要,文中采用卡尔曼滤波技术解决脱靶量滞后的问题;另一方面,卡尔曼滤波技术对预测脱靶量有较好的估计作用,可在短时间目标丢失无脱靶量的情况下为跟踪系统提供数字引导。基于卡尔曼滤波的单探测器控制结构如图3所示。
假设线性离散系统的过程模型以及测量模型可表示为:
$$ \begin{split} \\ \left\{ \begin{gathered} x\left( {k + 1} \right) = A\left( k \right)x\left( k \right) + w\left( k \right) \hfill \\ z\left( k \right) = C\left( k \right)x\left( k \right) + v\left( k \right) \hfill \\ \end{gathered} \right. \end{split} $$ (4) 式中:
$x\left( k \right)$ 为待估计状态变量;$ z\left( k \right) $ 为实际测量值;$ w\left( k \right) $ 为过程噪声;$ v\left( k \right) $ 为测量噪声;并假设以下条件成立:$$ \left\{ \begin{gathered} w\left( k \right) \sim \left( {0,Q\left( k \right)} \right) \hfill \\ v\left( k \right) \sim \left( {0,R\left( k \right)} \right) \hfill \\ E\left[ {w\left( k \right){w^{\rm{T}}}\left( j \right)} \right] = Q\left( k \right)\delta \left( {k - j} \right) \hfill \\ E\left[ {v\left( k \right){v^{\rm{T}}}\left( j \right)} \right] = R\left( k \right)\delta \left( {k - j} \right) \hfill \\ E\left[ {w\left( k \right){v^{\rm{T}}}\left( j \right)} \right] = 0 \hfill \\ \end{gathered} \right. $$ (5) 式中:矩阵
$ Q\left( k \right) $ 和$ R\left( k \right) $ 分别表示过程噪声协方差矩阵和测量噪声协方差矩阵;$ \delta \left( {k - j} \right) $ 为Kronecker-$ \delta $ 函数,如果$k \ne j$ ,那么$ \delta \left( {k - j} \right) $ =0;如果$k = j$ ,那么$ \delta \left( {k - j} \right) $ =1。(1)对状态变量进行预测
$$ \hat x(k + 1|k) = A(k)\hat x(k|k) $$ (6) $$ P(k + 1|k) = A(k)P(k|k){A^{\rm{T}}}(k) + Q(k) $$ (7) 式中:
$\hat x(k + 1 |k)$ 为先验估计状态值,利用k时刻及以前时刻测量值的最优估计预测(k+1)时刻的测量值,$\hat x(k|k)$ 为后验状态估计值;$P(k + 1 |k)$ 为协方差预测矩阵。(2)计算滤波增益矩阵
$$ K(k + 1) = P(k + 1|k){C^{\rm{T}}}(k){\left[ {C(k)P(k + 1|k){C^{\rm{T}}}(k) + R(k)} \right]^{ - 1}} $$ (8) (3)状态更新方程
$$ \hat x(k + 1|k + 1) = \hat x(k + 1|k) + K(k + 1)\left[ {z(k + 1) - C(k)\hat x(k + 1|k)} \right] $$ (9) $$ P(k + 1|k + 1) = \left[ {I - K(k + 1)C(k + 1)} \right]P(k + 1|k) $$ (10) 由于传统的卡尔曼滤波算法将测量噪声方差矩阵认定为固定不变的,即
$R(k)$ =K,${{K}} \in R$ 。但在实际工作过程中,由于探测器噪声以及运动平台抖动等原因,测量噪声方差矩阵是时变的。因此,文中采用自适应卡尔曼滤波的方法,对测量噪声方差矩阵实时估计,可有效抑制探测器噪声对卡尔曼滤波的影响。将公式(8)中的常数$R(k)$ 用$R(k) = \left( {1 + {\beta _k}} \right)R(k - 1)$ 代替,其中${\;\beta _k}$ 表示测量噪声方差矩阵迭代系数。定义噪声残差的方差理论值为:
$$ \varGamma \left( k \right) = C(k)P(k|k - 1){C^{\rm{T}}}(k) + R\left( {k - 1} \right) $$ (11) 定义噪声残差的方差实际值为:
$$ D\left( k \right) = \left[ {z(k) - C(k)\hat x(k|k - 1)} \right]{\left[ {z(k) - C(k)\hat x(k|k - 1)} \right]^{\text{T}}} $$ (12) 则
${\beta _k}$ 可用下式表示:$$ {\beta _k} = {K_p}\left[ {{\text{trace}}\left( {\varGamma \left( k \right)} \right) - {\text{trace}}\left( {D\left( k \right)} \right)} \right] $$ (13) 式中:trace(·)表示矩阵的迹;Kp为比例系数。通过公式(11)和公式(13)可以看出,噪声残差的方差理论值
$\varGamma \left( k \right)$ 是与测量噪声方差矩阵$R(k)$ 是单调递增的关系,根据噪声残差的方差理论值与噪声残差的方差实际值的误差,不断地调整迭代系数${\;\beta _k}$ 的大小使得误差逐渐缩小。从公式(13)进一步可以看出,假设$\varGamma \left( k \right) = D\left( k \right)$ ,此时${\;\beta _k} = 0$ ,即$R(k) = R(k - 1)$ ,测量噪声方差矩阵是固定的常数,因此,传统的卡尔曼滤波算法可看作是一种理想的假设情况,可以看作是自适应卡尔曼滤波算法的特殊情况。在实际工程中,为了尽可能提高卡尔曼滤波算法的可靠性,需要对传统的算法进行改进。 -
通过上文的分析可知,解耦矩阵的准确性直接影响复合轴系统的稳定性以及跟踪精度。单探测器复合轴跟踪系统的探测器只给精跟踪回路提供脱靶量信息,粗跟踪回路的误差信息只能通过精跟踪回路快速反射镜的偏转角度进行补偿,为了得到粗、精跟踪系统的解耦矩阵需要得到快速反射镜的偏转角度与探测器之间的变换关系。由于装调原因,相机的坐标系与快速反射镜的机械轴坐标系并不是重合的,实验忽略像旋因素,则两个坐标系之间存在角度偏转示意图如图4所示,因此进入跟踪过程之后,需要对坐标系进行旋转变换。
图4中,
$O{X_A}{Y_A}$ 表示快速反射镜的机械轴坐标系,$O{X_m}{Y_m}$ 表示相机的坐标系,则矢量${ I}$ 在$O{X_A}{Y_A}$ 坐标系下的分量${i_{xa}}$ 、${i_{xa}}$ 投影到$O{X_m}{Y_m}$ 坐标系下的分量${i_{xm}}$ 、${i_{xm}}$ 的变换公式为:$$ \left[ \begin{gathered} {i_{xm}} \hfill \\ {i_{ym}} \hfill \\ \end{gathered} \right] = H\left[ {\begin{array}{*{20}{c}} {\cos \theta }&{ - \sin \theta } \\ {\sin \theta }&{\cos \theta } \end{array}} \right]\left[ \begin{gathered} {i_{xa}} \hfill \\ {i_{ya}} \hfill \\ \end{gathered} \right] $$ (14) 式中:
$\theta $ 为两个坐标系的夹角;$H \in {R^{2 \times 2}}$ 为增益矩阵。实际标定过程中,首先保持粗跟踪在一个固定的俯仰角,分别对快速反射镜的X轴和Y轴施加等间距的电压值,记录快速反射镜的偏转角度信息和相机脱靶量信息;其次,调整粗跟踪俯仰角度,重复上述测量过程;最后,整理标定数据,辨识过程的评价方程以误差均方根值为参考,离线辨识得到增益矩阵参数。
Research on single-detector decoupling control technology based on adaptive Kalman algorithm
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摘要: 为了提高激光通信跟瞄系统的跟踪性能,增强系统的抗扰动能力,提出基于卡尔曼滤波的单探测器复合轴控制方法。首先,对单探测器复合轴系统原理进行分析,通过误差传递函数验证了解耦算法的可行性;其次,为了改善脱靶量迟滞的影响,同时降低探测器实时处理的要求,提出一种自适应卡尔曼滤波算法;最后,根据探测器坐标系与快速反射镜坐标系之间的旋转变换关系,计算出粗精系统的解耦矩阵,并搭建一套桌面实验系统进行原理验证。实验结果表明:单探测器复合轴在0.1 Hz低频扰动条件下,精跟踪系统的相对位移不会超出反射镜偏转角度的临界值,跟踪误差由2.54 μrad下降到0.86 μrad。解耦控制能够提高系统跟踪精度并增强抗扰动能力。对于以后工程中的实际应用具有一定的指导意义。Abstract: In order to improve the tracking performance of the laser communication tracking system and enhance the anti-disturbance capability of the system, a single-detector composite axis control method based on Kalman filter was proposed. Firsty, the principle of the single detector composite axis system was analyzed, and the feasibility of the decoupling algorithm was verified by the error transfer function. Then in order to improve the impact of miss distance and reduce the real-time processing requirements of the detector, an adaptive Kalman filter algorithm was proposed. Finally, according to the rotation transformation relationship between the detector coordinate system and the fast steering mirror coordinate system, the decoupling matrix of the coarse and fine system was calculated, and a desktop experimental system was built to verify the principle. With the condition of 0.1 Hz low frequency disturbance, the relative displacement of the fine tracking system will not exceed the critical value of the mirror deflection angle, and the tracking error will decrease from 2.54 μrad to 0.86 μrad. The experimental results show that the decoupling control can improve the tracking accuracy and enhance the anti disturbance ability of the system. It has a certain guiding significance for the practical application in the future engineering.
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