迭代求解最优化问题TOA定位——最小二乘问题高斯牛顿法
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迭代求解最优化问题TOA定位——最小二乘问题、高斯牛顿法
优化算法/TOA室内定位/
Target localization with range-only measurements
方法：1-非线性最小二乘估计-迭代最小二乘求解
2-极大似然估计-高斯牛顿法求解
二维仅距离测量的定位问题；
蒙特卡洛仿真
性能指标：CRLB， RMSE，**
原创不易，路过的各位大佬请点个赞
% TOA定位
%迭代最小二乘
%极大似然估计—高斯牛顿解法
%牛顿法
clc;close all;clear all;
N=50;%迭代次数
deta=0;
Runs=500;%Monte karlo
ml_runx=0;
ml_runy=0;
ls_runx=0;
ls_runy=0;
%%LS变量定义
ls_res=zeros(2,N);
ls_v=zeros(2,N);
ls_x=zeros(1,N);
ls_y=zeros(1,N);
%%ML变量定义
ml_res=zeros(2,N);
ml_v=zeros(2,N);
ml_x=zeros(1,N);
ml_y=zeros(1,N);
d0=0.5;%迭代终止条件
T=1;%采样周期
%target
xp=20000;
yp=20000;%目标位置
vp=[xp;yp];
x=zeros(1,50);
y=zeros(1,50);
%sensor
x1=0;y1=0;
x2=15000;y2=5000;
x3=30000;y3=0;%传感器位置，三个
v1=[x1;y1];
v2=[x2;y2];
v3=[x3;y3];
y=[y1,y2,y3];
x=[x1,x2,x3];
Z=zeros(3,1);%传感器量测数据
z1=0;z2=0;z3=0;
Z(1,1)=z1;
Z(2,1)=z2;
Z(3,1)=z3;
%噪声
w=zeros(3,1);
Q=(40)^2;%noise variance 40m
%非线性函数
h01=0;h02=0;h03=0;
H0=zeros(3,1);
H=zeros(3,1);
% hatv=zeros(2,1);%v=[x;y]
%%
% nonlinear funchtion h() matraix
h01=sqrt((y1-yp)^2+(x1-xp)^2);
h02=sqrt((y2-yp)^2+(x2-xp)^2);
h03=sqrt((y3-yp)^2+(x3-xp)^2);%真值
H0(1,1)=h01;
H0(2,1)=h02;
H0(3,1)=h03;
for i=1:1:Runs;
i
%白高斯噪声
w=sqrt(Q)*randn(3,1);
%measuremen z
Z=H0+w;
z1=Z(1,1);z2=Z(2,1);z3=Z(3,1);
%ML估计初值
% ml_x0=(x3*((yp-y3)/(xp-x3))-x1*((yp-y1)/(xp-x1))-y3+y1)/((yp-y3)/(xp-x3)-(yp-y1)/(xp-x1)); %20060;%初始位置
% ml_y0=((x1-x3)*((yp-y1)/(xp-x1))*((yp-y3)/(xp-x3))-y1*((yp-y3)/...
% (xp-x3))+y3*((yp-y1)/(xp-x1)))/((yp-y1)/(xp-x1)-(yp-y3)/(xp-x3)); %19770;y纵坐标初始位置
ml_x0=20060;
ml_y0=19770;
ml_v0=[ml_x0;ml_y0];
%LS估计初值
ls_x0=(x3*((yp-y3)/(xp-x3))-x1*((yp-y1)/(xp-x1))-y3+y1)/((yp-y3)/(xp-x3)-(yp-y1)/(xp-x1)); %20060;%初始位置
ls_y0=((x1-x3)*((yp-y1)/(xp-x1))*((yp-y3)/(xp-x3))-y1*((yp-y3)/...
(xp-x3))+y3*((yp-y1)/(xp-x1)))/((yp-y1)/(xp-x1)-(yp-y3)/(xp-x3)); %19770;y纵坐标初始位置
ls_x0=20060;%初始位置
ls_y0=19770;%y纵坐标初始位置
ls_v0=[ls_x0;ls_y0];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ILS开始%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%LS迭代算法
for t=1:1:N;%迭代不超过50次
if(t==1)
ls_res(:,t)=ls_v0;
ls_v(:,1)=ls_v0;
ls_x(t)=ls_x0;
ls_y(t)=ls_y0;
else
%海森矩阵
for k=1:3;
ls_H(k,1)=sqrt((ls_y(t-1)-y(k))^2+(ls_x(t-1)-x(k))^2);
end
%雅可比矩阵
for k=1:3;
ls_J(k,1)=(ls_x(t-1)-x(k))/sqrt((ls_y(t-1)-y(k))^2+(ls_x(t-1)-x(k))^2);
ls_J(k,2)=(ls_y(t-1)-y(k))/sqrt((ls_y(t-1)-y(k))^2+(ls_x(t-1)-x(k))^2);
end
%噪声方差
R=eye(3)*Q;
ls_v(:,t)=ls_v(:,t-1)+inv(ls_J'*inv(R)*ls_J)*ls_J'*inv(R)*(Z-ls_H);%最小二乘迭代位置状态拟合
end
ls_res(:,t)=ls_v(:,t);
ls_x(t)=ls_v(1,t);
ls_y(t)=ls_v(2,t);
end
for k=1:50;
ls_runx=ls_runx+(xp-ls_x(k))^2;
ls_runy=ls_runy+(yp-ls_y(k))^2;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ILS结束%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ML—牛顿法开始%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%ML迭代
for t=1:1:N;
if(t==1)
ml_res(:,t)=ml_v0;
ml_v(:,1)=ml_v0;
ml_x(t)=ml_x0;
ml_y(t)=ml_y0;
else
%海森矩阵:ml_h
for k=1:3;
ml_h(k,1)=sqrt((ml_y(t-1)-y(k))^2+(ml_x(t-1)-x(k))^2);
end
%Jml_J
for k=1:3;
ml_J(k,1)=(ml_x(t-1)-x(k))/sqrt((ml_y(t-1)-y(k))^2+(ml_x(t-1)-x(k))^2);
ml_J(k,2)=(ml_y(t-1)-y(k))/sqrt((ml_y(t-1)-y(k))^2+(ml_x(t-1)-x(k))^2);
end
%函数h的二阶梯度ml_JJ
for k=1:3
ml_JJ(k,1)=((ml_y(t-1)-y(k))^2)/(((ml_y(t-1)-y(k))^2+(ml_x(t-1)-x(k))^2)^(3/2));%横坐标的二阶导
ml_JJ(k,2)=((ml_x(t-1)-x(k))^2)/(((ml_y(t-1)-y(k))^2+(ml_x(t-1)-x(k))^2)^(3/2));%纵坐标的二阶导
ml_JJ(k,3)=-((ml_x(t-1)-x(k))*(ml_y(t-1)-y(k)))/(((ml_y(t-1)-y(k))^2+(ml_x(t-1)-x(k))^2)^(3/2));%横纵坐标的二阶导
end
%一阶梯度矩阵ml_F（函数lamuda）
ml_F(1,1)=1/Q*((z1-ml_h(1,1))*ml_J(1,1)+(z2-ml_h(2,1))*ml_J(2,1)+(z3-ml_h(3,1))*ml_J(3,1));
ml_F(2,1)=1/Q*((z1-ml_h(1,1))*ml_J(1,2)+(z2-ml_h(2,1))*ml_J(2,2)+(z3-ml_h(3,1))*ml_J(3,2));
%Hessian矩阵：ml_H
ml_H(1,1)=((z1-ml_h(1,1))*ml_JJ(1,1)-(ml_J(1,1))^2+(z2-ml_h(2,1))*ml_JJ(2,1)-(ml_J(2,1))^2+(z3-ml_h(3,1))*ml_JJ(3,1)-(ml_J(3,1))^2)/Q;
ml_H(2,2)=((z1-ml_h(1,1))*ml_JJ(1,2)-(ml_J(1,2))^2+(z2-ml_h(2,1))*ml_JJ(2,2)-(ml_J(2,2))^2+(z3-ml_h(3,1))*ml_JJ(3,2)-(ml_J(3,2))^2)/Q;
ml_H(1,2)=((z1-ml_h(1,1))*ml_JJ(1,3)-(ml_J(1,1))*(ml_J(1,2))+(z2-ml_h(2,1))*ml_JJ(2,3)-(ml_J(2,1))*(ml_J(2,2))+(z3-ml_h(3,1))*ml_JJ(3,3)-(ml_J(3,1))*(ml_J(3,2)))/Q;
ml_H(2,1)=ml_H(1,2);
%ML迭代
ml_v(:,t)=ml_v(:,t-1)-inv(ml_H)*ml_F;
end
ml_res(:,t)=ml_v(:,t);
ml_x(t)=ml_v(1,t);
ml_y(t)=ml_v(2,t);
end
for k=1:50;
ml_runx=ml_runx+(xp-ml_x(k))^2;
ml_runy=ml_runy+(yp-ml_y(k))^2;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ML—牛顿法结束%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
end
%蒙特卡洛
ml_Runsx=sqrt(ml_runx/(Runs*N));
ml_Runsy=sqrt(ml_runy/(Runs*N));
ls_Runsx=sqrt(ls_runx/(Runs*N));
ls_Runsy=sqrt(ls_runy/(Runs*N));
%ML:fisher 信息矩阵
for k=1:3;
J(k,1)=(xp-x(k))/sqrt((yp-y(k))^2+(xp-x(k))^2);
J(k,2)=(yp-y(k))/sqrt((yp-y(k))^2+(xp-x(k))^2);
end
FIM(1,1)=((J(1,1))^2+(J(2,1))^2+(J(3,1))^2)/Q;
FIM(2,2)=((J(1,2))^2+(J(2,2))^2+(J(3,2))^2)/Q;
FIM(1,2)=(J(1,1)* J(1,2)+J(2,1)* J(2,2)+ J(3,1)*J(3,2))/Q;
FIM(2,1)=(J(1,1)* J(1,2)+J(2,1)* J(2,2)+ J(3,1)*J(3,2))/Q;
ml_MSE=inv(FIM);
ml_CRLB_x=sqrt(ml_MSE(1,1))
ml_CRLB_y=sqrt(ml_MSE(2,2))
%MSE均方误差
ls_MSE=inv(J'*inv(R)*J);
ls_CRLB_x=sqrt(ls_MSE(1,1))
ls_CRLB_y=sqrt(ls_MSE(2,2))
%仿真结果
t=1:1:10;
figure(1);
plot([x1 xp],[y1 xp],'-o');hold on;
plot([x2 xp],[y2 yp],'-o');hold on;
plot([x3 xp],[y3 yp],'-o');hold on;
plot(xp,yp,'*');
axis([-10000 50000 -10000 50000]);
xlabel('East(m)'), ylabel('North(m)');
figure(2);
plot(t,ls_res(1,t),t,ml_res(1,t),'r--')
title('Estimation of East Position')
xlabel('Iteration j'), ylabel('The East Position');
legend('ILS','ML')
hold on;
figure(3);
plot(t,ls_res(2,t),t,ml_res(2,t),'r--')
title('Estimation of North Position')
xlabel('Iteration j'), ylabel('The North Position');
legend('ILS','ML');
优化算法/TOA室内定位/
Target localization with range-only measurements
方法：1-非线性最小二乘估计-迭代最小二乘求解
2-极大似然估计-高斯牛顿法求解
二维仅距离测量的定位问题；
蒙特卡洛仿真
性能指标：CRLB， RMSE，**
原创不易，路过的各位大佬请点个赞
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