UKF、EKF,CKF,STUKF.zip_EKF_ukf.stukf_卡尔曼 UKF_强跟踪_无迹滤波

% EKF UKF CKF 的三个算法 clear; x=1;% 初始状态 x_estimate = 0.5;%初始状态的估计 xpre=x_estimate ; e_x_estimate = x_estimate; %EKF的初始估计 u_x_estimate = x_estimate; %UKF的初始估计 c_x_estimate = x_estimate; %CKF的初始估计 Q = 10 % 过程状态协方差 R = 0.01 % 测量噪声协方差 P =1000;%初始估计方差 n=1; %%%%% 系统的维数 m=2*n; %%%%% CKF容积点数 w=1/m; %%%%%% 权值 kesi=[1,-1];%%%%% CKF容积点 e_P = P; %EKF方差 u_P = P;%UKF方差 Ppost=P;%CKF方差 tf = 100; % 模拟长度 x_array = [x];%真实值数组 e_x_estimate_array = [e_x_estimate];%EKF最优估计值数组 u_x_estimate_array = [u_x_estimate];%UKF最优估计值数组 c_x_estimate_array = [c_x_estimate];%CKF最优估计值数组 u_k = 1; %微调参数 u_symmetry_number = 4; % 对称的点的个数 u_total_number = 2 * u_symmetry_number + 1; %总的采样点的个数 linear = 0.5; close all; for k = 1 : tf % 模拟系统 % x = linear * x + (25 * x / (1 + x^2)) + 8 * cos(1.2*(k-1)) %状态值 % x_array = [x_array,x]; % y = (x^2 / 20) + sqrt(R) * randn; %观测值 % z=y; % x=x+ sqrt(Q) * randn; x = 0.2*x+0.01*x^2+8*cos(1.2*(k+1)) +sqrt(10)*randn %状态值 x_array = [x_array,x]; y = x^2 + 0.1*randn; %观测值 z=y; %扩展卡尔曼滤波器 %进行估计 第一阶段的估计 e_x_estimate_1 = 0.2 * e_x_estimate + 0.01* e_x_estimate^2 + 8 * cos(1.2*(k+1)); e_y_estimate = (e_x_estimate_1)^2; %相关矩阵 e_A = 0.2 + 0.01* e_x_estimate;%传递矩阵 e_H = 2*e_x_estimate_1; %观测矩阵 %估计的误差 e_p_estimate = e_A * e_P * e_A' + Q; %扩展卡尔曼增益 e_K = e_p_estimate * e_H'/(e_H * e_p_estimate * e_H' + R); %进行估计值的更新 第二阶段 e_x_estimate_2 = e_x_estimate_1 + e_K * (y - e_y_estimate); %更新后的估计值的误差 e_p_estimate_update = e_p_estimate - e_K * e_H * e_p_estimate; %进入下一次迭代的参数变化 e_P = e_p_estimate_update; e_x_estimate = e_x_estimate_2; %%%%%%无味卡尔曼滤波器UKF %采样点的选取 存在x(i) u_x_par = u_x_estimate; for i = 2 : (u_symmetry_number+1) u_x_par(i) = u_x_estimate + sqrt((u_symmetry_number+u_k) * u_P); end for i = (u_symmetry_number+2) : u_total_number u_x_par(i) = u_x_estimate - sqrt((u_symmetry_number+u_k) * u_P); end %计算权值 u_w_1 = u_k/(u_symmetry_number+u_k); u_w_N1 = 1/(2 * (u_symmetry_number+u_k)); %把这些粒子通过传递方程 得到下一个状态 for i = 1: u_total_number u_x_par(i) = 0.2 * u_x_par(i) + 0.01* u_x_par(i)^2 + 8 * cos(1.2*(k+1)); end %传递后的均值和方差 u_x_next = u_w_1 * u_x_par(1); for i = 2 : u_total_number u_x_next = u_x_next + u_w_N1 * u_x_par(i); end u_p_next = Q + u_w_1 * (u_x_par(1)-u_x_next) * (u_x_par(1)-u_x_next)'; for i = 2 : u_total_number u_p_next = u_p_next + u_w_N1 * (u_x_par(i)-u_x_next) * (u_x_par(i)-u_x_next)'; end for i = 1 :u_total_number u_y_2obser(i,:) = u_x_par(i); end %通过观测方程 得到一系列的粒子 for i = 1: u_total_number u_y_2obser(i) = u_y_2obser(i)^2; end %通过观测方程后的均值 y_obse u_y_obse = u_w_1 * u_y_2obser(1); for i = 2 : u_total_number u_y_obse = u_y_obse + u_w_N1 * u_y_2obser(i); end %Pzz测量方差矩阵 u_pzz = R + u_w_1 * (u_y_2obser(1)-u_y_obse)*(u_y_2obser(1)-u_y_obse)'; for i = 2 : u_total_number u_pzz = u_pzz + u_w_N1 * (u_y_2obser(i) - u_y_obse)*(u_y_2obser(i) - u_y_obse)'; end %Pxz状态向量与测量值的协方差矩阵 u_pxz = u_w_1 * (u_x_par(1) - u_x_next)* (u_y_2obser(1)-u_y_obse)'; for i = 2 : u_total_number u_pxz = u_pxz + u_w_N1 * (u_x_par(i) - u_x_next) * (u_y_2obser(i)- u_y_obse)'; end %卡尔曼增益 u_K = u_pxz/u_pzz; %估计量的更新 u_x_next_optimal = u_x_next + u_K * (y - u_y_obse);%第一步的估计值 + 修正值； u_x_estimate = u_x_next_optimal; %方差的更新 u_p_next_update = u_p_next - u_K * u_pzz * u_K'; u_P = u_p_next_update; %%%%%——CKF滤波——%%%%%%% %%%%%时间更新 %%%%%（1）求协方差矩阵平方根 Spost=sqrt(Ppost); %%%%%（2）计算求容积点 rjpoint(1)=Spost*kesi(1)+xpre; rjpoint(2)=Spost*kesi(2)+xpre; %%%%%（3）传播求容积点 Xminus(1)=0.2*rjpoint(1)+0.01*rjpoint(1)^2+8*cos(1.2*(k+1)); %%%%容积点经过非线性函数后的值 Xminus(2)=0.2*rjpoint(2)+0.01*rjpoint(2)^2+8*cos(1.2*(k+1)); %%%%（4）状态预测 xminus=0.5*Xminus(1)+0.5*Xminus(2); %%%%(5)状态预测协方差阵 Pminus=0.5*Xminus(1)^2+0.5*Xminus(2)^2-xminus*xminus'+Q; %%%%观测更新 %%%%%（1）矩阵分解 Sminus=sqrt(Pminus-Q); %%%%%（2）计算求容积点 rjpoint1(1)=Sminus*kesi(1)+xminus; rjpoint1(2)=Sminus*kesi(2)+xminus; %%%%%（3）传播求容积点 Z(1)=rjpoint1(1)^2; Z(2)=rjpoint1(2)^2; %%%%%%%（4）观测预测 zhat=0.5*Z(1)+0.5*Z(2); %%%%(5)观测预测协方差阵 Pzminus=0.5*(Z(1)-zhat)^2+0.5*(Z(2)-zhat)^2+R; %%%%(6)互协方差阵 Pxzminus=0.5*(rjpoint1(1)-xminus)*(Z(1)-zhat)+0.5*(rjpoint1(2)-xminus)*(Z(2)-zhat); %%%%(7)计算卡尔曼增益 K=Pxzminus*inv(Pzminus); %%%%(8)状态更新 xpre=xminus+K*(z-zhat); %%%%(9)状态协方差矩阵更新 Ppost=Pminus-K*Pzminus*K'; c_x_estimate=xpre; %进行画图程序 e_x_estimate_array = [e_x_estimate_array,e_x_estimate]; u_x_estimate_array = [u_x_estimate_array,u_x_estimate]; c_x_estimate_array = [c_x_estimate_array,c_x_estimate]; e_error(k,:) = abs(x_array(k)-e_x_estimate_array(k)); u_error(k,:) = abs(x_array(k)-u_x_estimate_array(k)); c_error(k,:)=abs(x_array(k)-c_x_estimate_array(k)); end t = 0 : tf; figure; plot(t,x_array,'k-.',t,e_x_estimate_array,'r-',t,u_x_estimate_array,'b:',t,c_x_estimate_array,'c-.'); set(gca,'FontSize',10); legend('真实值','EKF估计值','UKF估计值','CKF估计值'); xlabel('样本数目n') ylabel('信号幅度') %root mean square 平均值的平方根 e_xrms = sqrt((norm(x_array-e_x_estimate_array)^2)/tf); disp(['EKF估计误差均方值=',num2str(e_xrms)]); u_xrms = sqrt((norm(x_array-u_x_estimate_array)^2)/tf); disp(['UKF估计误差均方值=',num2str(u_xrms)]); c_xrms = sqrt((norm(x_array-c_x_estimate_array)^2)/tf); disp(['CKF估计误差均方值=',num2str(c_xrms)]); t = 1 : tf; figure; h=plot(t,e_error,'r-',t,u_error,'b:',t,c_error,'-c'); set(gca,'FontSize',10); legend('EKF估计值误差','UKF估计值误差','CKF估计值误差'); xlabel('样本数目n') ylabel('估计误差') %toc;