2. 河南科技学院 机电学院, 河南 新乡 453007
提出了改进型约束总体最小二乘多目标定位算法.首先引入辅助变量将非线性定位方程转化为伪线性方程;然后利用两步最小二乘法估计目标的初始位置,依据目标初始位置重新选择参考传感器;最后考虑伪线性方程中所有系数矩阵的噪声,采用拉格朗日乘子技术求解约束条件,利用拟牛顿算法迭代公式得到精确解.仿真结果证明了理论分析的正确性和可行性,所提算法能够达到克拉美罗下界,具有较强的鲁棒性和精确的定位性能.
2. School of Mechanical and Electrical Engineering, Henan Institute of Science and Technology, Henan Xinxiang 453003, China
An improved constrained total least squares approach was proposed for location of multiple targets. Firstly, the auxiliary variable is introduced to transform the nonlinear positioning equation into pseudo linear equation. Then the initial position of the target was estimated by two-stage weighted least squares, and the reference sensor was re-selected according to the initial estimation position. Finally, considering the noise of all the coefficients in the pseudo-linear equation and applying the Lagrangian multiplier to solve the constraint condition, an exact solution was obtained by using the quasi-Newton iterative formula. Simulations show that the proposed approach can reach the Cramér-Rao lower bound with strong robustness and precise positioning performance.
无源定位技术一直备受科研人员的关注,广泛应用于雷达、声纳、导航、监测、无线通信、传感器网络等领域[1-2].目标定位常用到达时间、到达时间差(TDOA, time difference of arrival)和到达频率差(FDOA, frequency difference of arrival)[3-4]等测量技术.定位目标时微小的传感器位置误差将导致较大的估计误差,故定位目标时应考虑传感器位置误差的影响. Ho等[5]提出了两步加权最小二乘(TSWLS, two-stage weighted least squares)算法定位多目标;Li等[6]结合传感器的位置误差和TSWLS算法定位多目标;陈少昌等[7]利用约束总体最小二乘(CTLS, constrained total least squares)算法定位目标,但CTLS算法仅对静止目标有效.笔者基于文献[5]和文献[7]提出了改进的CTLS (ICTLS, improved CTLS)算法定位多个运动目标,引入拉格朗日乘子和拟牛顿迭代公式求解带约束条件的定位方程.实验结果表明,该方法比TSWLS算法有较好的定位性能,能有效地减小定位偏差,在噪声误差较大的情况下也能达到克拉美罗下界.
1 定位场景三维空间中随机分布K个运动目标,设第i个目标的真实位置和速度分别为ui和
$ \mathit{r}_{j,i}^0 = {\left\| {{\mathit{\boldsymbol{u}}_i} - \mathit{\boldsymbol{s}}_j^0} \right\|_2} = \sqrt {{{\left( {{\mathit{\boldsymbol{u}}_i} - \mathit{\boldsymbol{s}}_j^0} \right)}^{\rm{T}}}\left( {{\mathit{\boldsymbol{u}}_i} - \mathit{\boldsymbol{s}}_j^0} \right)} $ | (1) |
其中‖·‖2表示2范数.故目标i到传感器1和j的距离差的真实值和测量值分别为
$ r_{j1,i}^0 = r_{j,i}^0 - r_{1,\mathit{i}}^0 $ | (2) |
$ {r_{j1,i}} = r_{j1,i}^0 + {\rm{\Delta }}{r_{j1,i}} $ | (3) |
其中Δrj1, i为测量误差.目标i的TDOA测量向量为
$ {\mathit{\boldsymbol{r}}_i} = {\left[ {{r_{21,i}},{r_{31,i}}, \cdots ,{r_{\mathit{M}1,i}}} \right]^{\rm{T}}} = \mathit{\boldsymbol{r}}_i^0 + {\rm{\Delta }}{\mathit{\boldsymbol{r}}_\mathit{i}} $ | (4) |
其中
当目标相对于传感器发生运动时,参照式(2) 和式(3) 得到FDOA测量值:
$ {{\dot r}_{j1,\mathit{i}}} = \dot r_{j1,i}^0 + {\rm{\Delta }}{{\dot r}_{j1,i}} = \dot r_{j,i}^0 - \dot r_{1,i}^0 + {\rm{\Delta }}{{\dot r}_{j1,\mathit{i}}} $ | (5) |
其中
为了方便表述,通常把TDOA和FDOA测量向量表示为单一的向量方程.目标i的测量向量为
为了估算运动目标的位置和速度,需要建立关于目标和传感器的TDOA和FDOA定位方程组.引入辅助变量
$ \begin{array}{l} {\left( {r_{_{j1,i}}^0} \right)^2} + 2r_{_{j1,i}}^0r_{1,i}^0 = \\ {\left( {\mathit{\boldsymbol{s}}_j^0 - \mathit{\boldsymbol{s}}_1^0} \right)^{\rm{T}}}\left( {\mathit{\boldsymbol{s}}_j^0 - \mathit{\boldsymbol{s}}_1^0} \right) - 2{\left( {\mathit{\boldsymbol{s}}_j^0 - \mathit{\boldsymbol{s}}_1^0} \right)^{\rm{T}}}\left( {{\mathit{\boldsymbol{u}}_i} - \mathit{\boldsymbol{s}}_1^0} \right) \end{array} $ | (6) |
由于
$ r_{1,i}^0 \approx {\left\| {{\mathit{\boldsymbol{u}}_i} - {\mathit{\boldsymbol{s}}_1}} \right\|_2} + \mathit{\boldsymbol{\rho }}_{{\mathit{\boldsymbol{u}}_i},{\mathit{\boldsymbol{s}}_1}}^{\rm{T}}{\rm{\Delta }}{\mathit{\boldsymbol{s}}_1} = {{\hat r}_{1,i}} + \mathit{\boldsymbol{\rho }}_{{\mathit{\boldsymbol{u}}_i},{\mathit{\boldsymbol{s}}_1}}^{\rm{T}}{\rm{\Delta }}{\mathit{\boldsymbol{s}}_1} $ | (7) |
其中
$ \begin{array}{l} 2{\left( {{\mathit{\boldsymbol{s}}_j} - {\mathit{\boldsymbol{s}}_1}} \right)^{\rm{T}}}\left( {{\mathit{\boldsymbol{u}}_i} - {\mathit{\boldsymbol{s}}_1}} \right) + 2{r_{j1,i}}{{\hat r}_{1,i}} - \\ 2{\left( {{\rm{\Delta }}{\mathit{\boldsymbol{s}}_j} - {\rm{\Delta }}{\mathit{\boldsymbol{s}}_1}} \right)^{\rm{T}}}\left( {{\mathit{\boldsymbol{u}}_i} - {\mathit{\boldsymbol{s}}_1}} \right) - 2{\rm{\Delta }}{r_{j1,i}}{{\hat r}_{1,i}} = \\ {\left( {{\mathit{\boldsymbol{s}}_j} - {\mathit{\boldsymbol{s}}_1}} \right)^{\rm{T}}}\left( {{\mathit{\boldsymbol{s}}_j} - {\mathit{\boldsymbol{s}}_1}} \right) + 2{r_{j1,i}}{\rm{\Delta }}{r_{j1,i}} - \\ 2{\left( {{\mathit{\boldsymbol{s}}_j} - {\mathit{\boldsymbol{s}}_1}} \right)^{\rm{T}}}{\rm{\Delta }}{\mathit{\boldsymbol{s}}_j} - 2{r_{j1,i}}{\mathit{\boldsymbol{d}}_{1,i}}{\rm{\Delta }}{\mathit{\boldsymbol{s}}_1} - {\left( {{r_{j1,i}}} \right)^2} \end{array} $ | (8) |
将式(8) 对时间求微分得到FDOA方程组:
$ \begin{array}{l} {\left( {{{\mathit{\boldsymbol{\dot s}}}_j} - {{\mathit{\boldsymbol{\dot s}}}_1}} \right)^{\rm{T}}}\left( {{\mathit{\boldsymbol{u}}_i} - {\mathit{\boldsymbol{s}}_1}} \right) + {{\dot r}_{j1,i}}{{\hat r}_{1,i}} + {\left( {{\mathit{\boldsymbol{s}}_j} - {\mathit{\boldsymbol{s}}_1}} \right)^{\rm{T}}}\left( {{{\mathit{\boldsymbol{\dot u}}}_i} - {{\mathit{\boldsymbol{\dot s}}}_1}} \right) + \\ {r_{j1,i}}{{\hat {\dot r}}_{1,i}} - {\left( {{\rm{\Delta }}{{\mathit{\boldsymbol{\dot s}}}_j} - {\rm{\Delta }}{{\mathit{\boldsymbol{\dot s}}}_1}} \right)^{\rm{T}}}\left( {{\mathit{\boldsymbol{u}}_i} - {\mathit{\boldsymbol{s}}_1}} \right) - \\ {\rm{\Delta }}{{\dot r}_{j1,i}}{{\hat r}_{1,i}} - {\left( {{\rm{\Delta }}{\mathit{\boldsymbol{s}}_j} - {\rm{\Delta }}{\mathit{\boldsymbol{s}}_1}} \right)^{\rm{T}}}\left( {{{\mathit{\boldsymbol{\dot u}}}_i} - {{\mathit{\boldsymbol{\dot s}}}_1}} \right) - \\ {\rm{\Delta }}{r_{j1,i}}{{\hat {\dot r}}_{1,i}} = {\left( {{{\mathit{\boldsymbol{\dot s}}}_j} - {{\mathit{\boldsymbol{\dot s}}}_1}} \right)^{\rm{T}}}\left( {{\mathit{\boldsymbol{s}}_j} - {\mathit{\boldsymbol{s}}_1}} \right) - {r_{j1,i}}{{\dot r}_{j1,i}} + \\ {{\dot r}_{j1,i}}{\rm{\Delta }}{r_{j1,i}} + {r_{j1,i}}{\rm{\Delta }}{{\dot r}_{j1,i}} - {\left( {{{\mathit{\boldsymbol{\dot s}}}_j} - {{\mathit{\boldsymbol{\dot s}}}_1}} \right)^{\rm{T}}}{\rm{\Delta }}{\mathit{\boldsymbol{s}}_j} - \\ {\left( {{\mathit{\boldsymbol{s}}_j} - {\mathit{\boldsymbol{s}}_1}} \right)^{^{\rm{T}}}}{\rm{\Delta }}{{\mathit{\boldsymbol{\dot s}}}_j} - \mathit{\boldsymbol{d}}_{2,i}^{\rm{T}}{\rm{\Delta }}{\mathit{\boldsymbol{s}}_1} - \mathit{\boldsymbol{d}}_{1,i}^{\rm{T}}{\rm{\Delta }}{{\mathit{\boldsymbol{\dot s}}}_1} \end{array} $ | (9) |
其中
$ \left( {\mathit{\boldsymbol{A + }}{\rm{\Delta }}\mathit{\boldsymbol{A}}} \right)\mathit{\boldsymbol{\theta = b + }}{\rm{\Delta }}\mathit{\boldsymbol{b}} $ | (10) |
其中:
当目标靠近参考传感器时有ui-s1≈0,在式(8)、式(9) 和式(10) 中求解位置的参量趋于零,此时目标估计位置误差较大.为此,首先利用TSWLS算法的第1步粗略估计出目标ui的位置,然后选取离目标ui最远的传感器作为新的参考传感器[8],根据新的参考传感器重新建立坐标系.在新坐标系下再次估计目标位置,由于TSWLS仅考虑了部分定位方程系数矩阵的误差,造成目标估计位置误差增大;另外,TSWLS没有考虑辅助变量
$ J\left( \mathit{\boldsymbol{\theta }} \right) = {\left( {\mathit{\boldsymbol{A\theta }} - \mathit{\boldsymbol{b}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{W}}_\theta }\left( {\mathit{\boldsymbol{A\theta }} - \mathit{\boldsymbol{b}}} \right) $ | (11) |
其中Wθ为加权矩阵,具体表达式见附录中式(37).故待求向量θ的最小二乘解为
$ \mathit{\boldsymbol{\hat \theta = }}{\rm{arg}}\;\mathop {{\rm{min}}}\limits_{\tilde \theta } {\left( {\mathit{\boldsymbol{A\tilde \theta }} - \mathit{\boldsymbol{b}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{W}}_{\tilde \theta }}\left( {\mathit{\boldsymbol{A\tilde \theta }} - \mathit{\boldsymbol{b}}} \right) $ | (12) |
其中:
求解式(12) 应考虑此约束条件.结合式(1) 和式(5) 约束条件为
$ \left. \begin{array}{l} {{\hat r}_{1,i}} = \sqrt {{\mathit{\boldsymbol{w}}^{\rm{T}}}\mathit{\boldsymbol{w}}} \\ {{\hat {\dot r}}_{j,i}} = {{\mathit{\boldsymbol{\dot w}}}^{\rm{T}}}\mathit{\boldsymbol{w}}/{{\hat r}_{1,i}} \end{array} \right\} $ | (13) |
将式(13) 变形整理代入式(11),整理得到矩阵形式的约束条件:
$ \left. \begin{array}{l} {\mathit{\boldsymbol{\theta }}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_1}\mathit{\boldsymbol{\theta }} = 0\\ {\mathit{\boldsymbol{\theta }}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_2}\mathit{\boldsymbol{\theta }} = 0 \end{array} \right\} $ | (14) |
其中:
$ \begin{array}{l} \mathit{\boldsymbol{\hat \theta }}\mathit{ = }\arg \;min{(\mathit{\boldsymbol{A\tilde \theta }} - b)^{\rm{T}}}{\mathit{\boldsymbol{W}}_{\mathit{\tilde \theta }}}(\mathit{\boldsymbol{A\tilde \theta }} - b)\\ {\begin{array}{*{20}{c}} {{\rm{s}}.}&{{\rm{t}}.}&{{\mathit{\boldsymbol{\theta }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}} \end{array}_1}\mathit{\boldsymbol{\theta }} = 0,{\mathit{\boldsymbol{\theta }}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_2}\mathit{\boldsymbol{\theta }} = 0 \end{array} $ | (15) |
求解式(15) 约束优化最小值问题时,引入拉格朗日乘子,则θ的价值函数式(11) 变为
$ \begin{array}{l} L\left( {\mathit{\boldsymbol{\theta }},{\mathit{\lambda }_1},{\mathit{\lambda }_2}} \right) = \\ {\left( {\mathit{\boldsymbol{A\theta }} - \mathit{\boldsymbol{b}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{W}}_\theta }\left( {\mathit{\boldsymbol{A\theta }} - \mathit{\boldsymbol{b}}} \right) + {\mathit{\lambda }_1}{\mathit{\boldsymbol{\theta }}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_1}\mathit{\boldsymbol{\theta + }}{\mathit{\lambda }_2}{\mathit{\boldsymbol{\theta }}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_2}\mathit{\boldsymbol{\theta }} \end{array} $ | (16) |
其中λ1和λ2为拉格朗日乘子.式(16) 变为二次约束的非凸优化问题.将式(16) 对θ求偏微分,然后令
$ \begin{array}{l} 2\left[ {{{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{D}}} \right)}^{\rm{T}}}{\mathit{\boldsymbol{W}}_\theta }\mathit{\boldsymbol{A + }}{\mathit{\lambda }_1}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_1} + {\mathit{\lambda }_2}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_2}} \right]\mathit{\boldsymbol{\theta }} - \\ 2{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{D}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{W}}_\theta }\mathit{\boldsymbol{b = 0}} \end{array} $ | (17) |
其中D=[
$ \mathit{\boldsymbol{\hat \theta = }}{\left[ {{{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{D}}} \right)}^{\rm{T}}}{\mathit{\boldsymbol{W}}_\theta }\mathit{\boldsymbol{A + }}{\mathit{\lambda }_1}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_1} + {\mathit{\lambda }_2}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_2}} \right]^{ - 1}}{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{D}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{W}}_\theta }\mathit{\boldsymbol{b}} $ | (18) |
把式(18) 代入约束条件式(14) 得到
$ \left. \begin{array}{l} {{\mathit{\boldsymbol{\hat \theta }}}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_1}\mathit{\boldsymbol{\hat \theta }} = 0\\ {{\mathit{\boldsymbol{\hat \theta }}}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_2}\mathit{\boldsymbol{\hat \theta = }}0 \end{array} \right\} $ | (19) |
式(19) 组成了关于λ1和λ2的2个高阶多项式方程组,求此方程组得到λ1和λ2并代入式(18) 得到目标的估计位置.改写式(19) 为
$ {f_k}\left( {{\mathit{\lambda }_1},{\mathit{\lambda }_2}} \right) = {\mathit{\boldsymbol{p}}^{\rm{T}}}\mathit{\boldsymbol{G}}_f^{ - {\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_k}\mathit{\boldsymbol{G}}_f^{ - {\rm{1}}}\mathit{\boldsymbol{p}} = 0,k = 1,2 $ | (20) |
其中:Gf=(A-D)TWθA+λ1Σ1+λ2Σ2,p=(A-D)TWθb.定义λ=[λ1, λ2]T,f=[f1, f2]T.利用拟牛顿迭代公式求解λ:
$ {\mathit{\boldsymbol{\lambda }}^{\left( {k + 1} \right)}} = {\mathit{\boldsymbol{\lambda }}^{\left( k \right)}} - {\alpha ^{\left( k \right)}}{\mathit{\boldsymbol{H}}^{\left( k \right)}}{g^{^{\left( k \right)}}} $ | (21) |
其中:λ(k)为第k次迭代结果,g(k)=∇f(λ(k)),α(k)为步长,H(k)为相应维数的对称矩阵且由BFGS (Broyden Fletcher Goldfarb Shanno)公式(式(22))更新.
$ \begin{array}{l} {\mathit{\boldsymbol{H}}^{^{\left( {k + 1} \right)}}} = {\mathit{\boldsymbol{H}}^{^{\left( k \right)}}} - \\ \frac{{{\mathit{\boldsymbol{H}}^{^{\left( k \right)}}}{\mathit{\boldsymbol{y}}^{^{\left( k \right)}}}{\mathit{\boldsymbol{y}}^{^{\left( k \right){\rm{T}}}}}{\mathit{\boldsymbol{H}}^{^{\left( k \right)}}}}}{{{\mathit{\boldsymbol{y}}^{^{\left( k \right){\rm{T}}}}}{\mathit{\boldsymbol{H}}^{^{\left( k \right)}}}{\mathit{\boldsymbol{y}}^{^{\left( k \right)}}}}} + \frac{{{\mathit{\boldsymbol{s}}^{\left( k \right)}}{\mathit{\boldsymbol{s}}^{\left( k \right){\rm{T}}}}}}{{{\mathit{\boldsymbol{y}}^{^{\left( k \right){\rm{T}}}}}{\mathit{\boldsymbol{s}}^{\left( k \right)}}}} + {\mathit{\boldsymbol{w}}^{\left( k \right)}}{\mathit{\boldsymbol{w}}^{\left( k \right){\rm{T}}}} \end{array} $ | (22) |
采用拟牛顿算法的BFGS公式迭代步骤如下:
1) 设定参数δ∈(0, 1),σ∈(0, 0.5),Wθ=Q-1.置精度要求ε=10-3,取初始点λ(1)=[0, 0]T,初始对称矩阵H(1)为相应的单位阵I,k=0.
2) 计算g(k)=∇f(λk),若‖g(k)‖≤ε则停止迭代,输出λ(k)作为近似解.
3) 计算搜索方向p(k)=-H(k)g(k).
4) 设mk是满足下面不等式的最小非负整数m:f(λ(k))+δmp(k)≤f(λ(k))+σδmg(k)Tp(k).
5) 由BFGS公式更新H(k+1),令k=k+1,转向步骤2).
最终通过拟牛顿算法获得λ(k)的最优值代入式(18),得到θ的最优估计值.
3 仿真实验及分析仿真实验采用文献[5]中的方案,假定有6个传感器,其位置和速度如表 1所示.
在多目标定位中以2个目标为例进行仿真实验,2个待定目标的位置和速度分别为
$ {P_{{\rm{RMSE}}}}\left| {{V_{{\rm{RMSE}}}}} \right.\left( {{\mathit{\boldsymbol{u}}_i}} \right) = \sqrt {\frac{{\sum\limits_{\iota = 1}^{10000} {\left\| {{{\mathit{\boldsymbol{\tilde u}}}_i}\left( \iota \right) - \mathit{\boldsymbol{u}}_i^0} \right\|_2^2} }}{{10000}}} $ | (23) |
其中:PRMSE和VRMSE分别为位置和速度估计值的RMSE,
图 2分别展现了TSWLS、CTLS和ICTLS算法的位置估计精度,以及位置与速度估计值的RMSE与克拉美罗下界(CRLB, Cramér-Rao lower bound)的对比结果.由图 2可知:① 位置估计,当10lg(
图 3给出了TSWLS、CTLS和ICTLS算法对2个目标定位时各自算法定位精度的RMSE与CRLB的吻合情况.在位置和速度估计结果的RMSE曲线中,上方曲线为目标
图 4描绘了TSWLS、CTLS和ICTLS算法定位2个固定目标时估计位置的RMSE与CRLB的比较结果.由图 4可知,对于目标
针对多个动态目标定位问题,提出了ICTLS的无源定位算法. ICTLS方法避免了TSWLS算法中的2个不足:① 第1步的加权最小二乘结果偏差随噪声误差的增大而增大;② 第2步引入的非线性运算降低算法精度.所提算法修正了CTLS算法中未考虑目标离参考传感器较近以及引入变量与目标之间的关系,建立了全局约束条件;同时,求解定位方程时引入了拉格朗日乘子技术,通过拟牛顿法的BFGS迭代公式进行求解,避免了Hessian矩阵的计算,加快了收敛速度.仿真实验表明传感器误差和测量误差适度的情况下,ICTLS算法比TSWLS和CTLS算法具有更低的RMSE,展现出了较强的鲁棒性和较高的定位精度.
附录
设C=[A b],ΔC=[ΔAΔb],则式(10) 变为
$ \left( {\mathit{\boldsymbol{C}} + \Delta \mathit{\boldsymbol{C}}} \right){\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{\theta }}&{ - 1} \end{array}} \right]^{\rm{T}}} = 0 $ | (24) |
方程式(24) 的CTLS解为约束最优化问题:
$ \begin{array}{l} \mathop {\min }\limits_{\Delta C,\theta } \left\| {\Delta \mathit{\boldsymbol{C}}} \right\|_{\rm{F}}^2\\ {\rm{s}}{\rm{.}}\;\;{\rm{t}}{\rm{.}}\;\;\left( {\mathit{\boldsymbol{C}} + \Delta \mathit{\boldsymbol{C}}} \right){\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{\theta }}&{ - 1} \end{array}} \right]^{\rm{T}}} \end{array} $ | (25) |
其中‖·‖F表示对矩阵求Frobenius范数,即‖H‖F=tr(HTH).设n=[ΔsT,
$ \Delta {\mathit{\boldsymbol{C}}_\iota } = {\mathit{\boldsymbol{F}}_{1n}},\iota = 1,2, \cdots ,\mathit{L + }{\rm{1}} $ | (26) |
当n不是白噪声的随机向量时,需要使向量n白化.假设Q=E(nnT)=PPT是矩阵Q的Cholesk分解,然后利用e=P-1n将n变为白噪声向量e,故Cl可改写为
$ \Delta \mathit{\boldsymbol{C}} = {\mathit{\boldsymbol{F}}_1}\mathit{\boldsymbol{Pe,}}\iota = 1,2, \cdots ,\mathit{L + }{\rm{1}} $ | (27) |
式(24) 的CTLS解变为
$ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathop {\;\;\min }\limits_{e,\theta } \left\| \mathit{\boldsymbol{e}} \right\|_{\rm{F}}^2\\ \begin{array}{*{20}{c}} {{\rm{s}}{\rm{.}}\;\;{\rm{t}}{\rm{.}}\;}&{\left( {\mathit{\boldsymbol{C + }}\left[ {{\mathit{\boldsymbol{F}}_1}\mathit{\boldsymbol{Pe,}}{\mathit{\boldsymbol{F}}_2}\mathit{\boldsymbol{Pe,}} \cdots \mathit{\boldsymbol{,}}{\mathit{\boldsymbol{F}}_{L + 1}}\mathit{\boldsymbol{Pe}}} \right]} \right)} \end{array} \times \\ \;\;\;\;\;\;\;\;\;\;{\left[ {\theta ,\;\; - 1} \right]^{\rm{T}}} = 0\; \end{array} $ | (28) |
而约束条件可变为
$ \begin{array}{l} \left[ {{\mathit{\boldsymbol{F}}_1}\mathit{\boldsymbol{Pe,}}{\mathit{\boldsymbol{F}}_2}\mathit{\boldsymbol{Pe,}} \cdots \mathit{\boldsymbol{,}}{\mathit{\boldsymbol{F}}_{L + 1}}\mathit{\boldsymbol{Pe}}} \right]{\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\theta ,}}}&{ - 1} \end{array}} \right]^{\rm{T}}} = \\ \;\;\;\left( {\sum\limits_{\iota = 1}^L {{\mathit{\boldsymbol{\theta }}_\iota }{\mathit{\boldsymbol{F}}_\iota }\mathit{\boldsymbol{P}} - {\mathit{\boldsymbol{F}}_{L + 1}}\mathit{\boldsymbol{Pe}}} } \right)\mathit{\boldsymbol{e}}\; = \mathit{\boldsymbol{EPe = }}{\mathit{\boldsymbol{G}}_\theta }\mathit{\boldsymbol{e}}\; \end{array} $ | (29) |
其中:
$ \begin{array}{l} \mathit{\boldsymbol{E}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{E}}_1}}&{{\mathit{\boldsymbol{E}}_2}} \end{array}} \right] = \sum\limits_{\iota = 1}^{L + 1} {{\mathit{\boldsymbol{\theta }}_\iota }{\mathit{\boldsymbol{F}}_\iota } - {\mathit{\boldsymbol{F}}_{L + 1}}} \\ {\mathit{\boldsymbol{E}}_1} = 2\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{B}}_1} \otimes {{\left( {{\mathit{\boldsymbol{u}}_i} - {\mathit{\boldsymbol{s}}_1}} \right)}^{\rm{T}}} - {\mathit{\boldsymbol{B}}_2}}&{{\mathit{\boldsymbol{O}}_{\left( {M - 1} \right) \times 3M}}}\\ {{\mathit{\boldsymbol{B}}_1} \otimes {{\left( {{\mathit{\boldsymbol{u}}_i} - {\mathit{\boldsymbol{s}}_1}} \right)}^{\rm{T}}} - {\mathit{\boldsymbol{B}}_3}}&{{\mathit{\boldsymbol{B}}_1} \otimes {{\left( {{\mathit{\boldsymbol{u}}_i} - {\mathit{\boldsymbol{s}}_i}} \right)}^{\rm{T}}} - {\mathit{\boldsymbol{B}}_2}} \end{array}} \right] \end{array} $ | (30) |
$ {\mathit{\boldsymbol{E}}_2} = 2\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{D}}&{{\mathit{\boldsymbol{O}}_{\left( {M - 1} \right) \times \left( {M - 1} \right)}}}\\ {\mathit{\boldsymbol{\dot D}}}&\mathit{\boldsymbol{D}} \end{array}} \right] $ | (31) |
$ \left. \begin{array}{l} \mathit{\boldsymbol{D}} = {\rm{diag}}\left( {r_2^0,r_3^0, \cdots ,r_M^0} \right)\\ \mathit{\boldsymbol{\dot D}} = {\rm{diag}}\left( {\dot {r_2^0,\dot {r_3^0, \cdots ,r_M^0}}} \right) \end{array} \right\} $ | (32) |
$ {\mathit{\boldsymbol{B}}_1} = \left[ {\begin{array}{*{20}{c}} { - 1}&1&0& \cdots &0\\ { - 1}&0&1& \cdots &0\\ { - 1}& \vdots & \vdots & \cdots & \vdots \\ { - 1}&0&0& \cdots &1 \end{array}} \right] $ | (33) |
$ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{B}}_2} = \\ \left[ {\begin{array}{*{20}{c}} {{r_{21}}{\mathit{\boldsymbol{d}}_1}}&{{{\left( {{\mathit{\boldsymbol{s}}_2} - {\mathit{\boldsymbol{s}}_1}} \right)}^{\rm{T}}}}&{{{\rm{0}}^{\rm{T}}}}& \cdots &{{0^{\rm{T}}}}\\ {{r_{31}}{\mathit{\boldsymbol{d}}_1}}&{{{\rm{0}}^{\rm{T}}}}&{{{\left( {{\mathit{\boldsymbol{s}}_3} - {\mathit{\boldsymbol{s}}_1}} \right)}^{\rm{T}}}}& \cdots &{{0^{\rm{T}}}}\\ \vdots & \vdots & \vdots & \cdots & \vdots \\ {{r_{M1}}{\mathit{\boldsymbol{d}}_1}}&{{0^{\rm{T}}}}&{{0^{\rm{T}}}}& \cdots &{{{\left( {{\mathit{\boldsymbol{s}}_M} - {\mathit{\boldsymbol{s}}_1}} \right)}^{\rm{T}}}} \end{array}} \right] \end{array} $ | (34) |
$ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{B}}_3} = \\ \left[ {\begin{array}{*{20}{c}} {{r_{21}}{\mathit{\boldsymbol{d}}_{2 + }}{{\dot r}_{21}}{\mathit{\boldsymbol{d}}_1}}&{{{\left( {{{\mathit{\boldsymbol{\dot s}}}_2} - {{\mathit{\boldsymbol{\dot s}}}_1}} \right)}^{\rm{T}}}}&{{{\rm{0}}^{\rm{T}}}}& \cdots &{{0^{\rm{T}}}}\\ {{{\dot r}_{31}}{\mathit{\boldsymbol{d}}_2} + {{\dot r}_{31}}{\mathit{\boldsymbol{d}}_1}}&{{{\rm{0}}^{\rm{T}}}}&{{{\left( {{{\mathit{\boldsymbol{\dot s}}}_3} - {{\mathit{\boldsymbol{\dot s}}}_1}} \right)}^{\rm{T}}}}& \cdots &{{0^{\rm{T}}}}\\ \vdots & \vdots & \vdots & \cdots & \vdots \\ {{{\dot r}_{M1}}{\mathit{\boldsymbol{d}}_2} + {{\dot r}_{M1}}{\mathit{\boldsymbol{d}}_1}}&{{0^{\rm{T}}}}&{{0^{\rm{T}}}}& \cdots &{{{\left( {{{\mathit{\boldsymbol{\dot s}}}_M} - {{\mathit{\boldsymbol{\dot s}}}_1}} \right)}^{\rm{T}}}} \end{array}} \right] \end{array} $ | (35) |
因此,式(28) 中的约束条件变为
$ \mathit{\boldsymbol{C}}\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{\theta }}&{ - 1} \end{array}} \right] + {\mathit{\boldsymbol{G}}_\theta }\mathit{\boldsymbol{e = }}{\rm{0}} $ | (36) |
因为Gθ是一个行满秩矩阵,Gθ的广义逆矩阵为其右伪逆矩阵
$ \begin{array}{l} {\mathit{\boldsymbol{W}}_\theta } = {\left( {{\mathit{\boldsymbol{G}}_\theta }\mathit{\boldsymbol{G}}_\theta ^{\rm{T}}} \right)^{ - 1}} = {\left( {\mathit{\boldsymbol{EQE}}} \right)^{ - 1}} = \\ {\left( {{\mathit{\boldsymbol{E}}_1}{\mathit{\boldsymbol{Q}}_\beta }\mathit{\boldsymbol{E}}_1^{\rm{T}} + {\mathit{\boldsymbol{E}}_2}{\mathit{\boldsymbol{Q}}_\alpha }\mathit{\boldsymbol{E}}_2^{\rm{T}}} \right)^{ - 1}} \end{array} $ | (37) |
由此得到CTLS的价值函数为
$ J\left( \mathit{\boldsymbol{\theta }} \right) = {\left( {\mathit{\boldsymbol{A\theta }} - \mathit{\boldsymbol{b}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{W}}_\theta }\left( {\mathit{\boldsymbol{A\theta }} - \mathit{\boldsymbol{b}}} \right) $ | (38) |
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