% EKF UKF PF 的三个算法clear;% tic;x = 0.1; % 初始状态x_estimate = 1;%状态的估计e_x_estimate = x_estimate; %EKF的初始估计u_x_estimate = x_estimate; %UKF的初始估计p_x_estimate = x_estimate; %PF的初始估计Q = 10;%input('请输入过程噪声方差Q的值: '); % 过程状态协方差R = 1;%input('请输入测量噪声方差R的值: '); % 测量噪声协方差P =5;%初始估计方差e_P = P; %UKF方差u_P = P;%UKF方差pf_P = P;%PF方差tf = 50; % 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p_x_estimate_array = [p_x_estimate];%PF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;N = 500; %粒子滤波的粒子数close all;%粒子滤波初始N 个粒子for i = 1 : Np_xpart(i) = p_x_estimate + sqrt(pf_P) * randn;endfor k = 1 : tf% 模拟系统x = linear * x + (25 * x / (1 + x^2)) + 8 * cos(1.2*(k-1)) + sqrt(Q) * randn; %状态值y = (x^2 / 20) + sqrt(R) * randn; %观测值%扩展卡尔曼滤波器%进行估计第一阶段的估计e_x_estimate_1 = linear * e_x_estimate + 25 * e_x_estimate /(1+e_x_estimate^2) + 8 * cos(1.2*(k-1));e_y_estimate = (e_x_estimate_1)^2/20; %这是根据k=1时估计值为1得到的观测值；只是这个由我估计得到的第24行的y也是观测值不过是由加了噪声的真实值得到的%相关矩阵e_A = linear + 25 * (1-e_x_estimate^2)/((1+e_x_estimate^2)^2);%传递矩阵e_H = e_x_estimate_1/10; %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;% 粒子滤波器% 粒子滤波器for i = 1 : Np_xpartminus(i) = 0.5 * p_xpart(i) + 25 * p_xpart(i) / (1 + p_xpart(i)^2) + 8 * cos(1.2*(k-1)) + sqrt(Q) * randn; %这个式子比下面一行的效果好% xpartminus(i) = 0.5 * xpart(i) + 25 * xpart(i) / (1 + xpart(i)^2) + 8 * cos(1.2*(k-1));p_ypart = p_xpartminus(i)^2 / 20; %预测值p_vhat = y - p_ypart;% 观测和预测的差p_q(i) = (1 / sqrt(R) / sqrt(2*pi)) * exp(-p_vhat^2 / 2 / R); %各个粒子的权值end% 平均每一个估计的可能性p_qsum = sum(p_q);for i = 1 : Np_q(i) = p_q(i) / p_qsum;%各个粒子进行权值归一化end% 重采样权重大的粒子多采点，权重小的粒子少采点, 相当于每一次都进行重采样；for i = 1 : Np_u = rand;p_qtempsum = 0;for j = 1 : Np_qtempsum = p_qtempsum + p_q(j);if p_qtempsum >= p_up_xpart(i) = p_xpartminus(j); %在这里xpart(i) 实现循环赋值；终于找到了这里！！！break;endendendp_x_estimate = mean(p_xpart);% p_x_estimate = 0;% for i = 1 : N% p_x_estimate =p_x_estimate + p_q(i)*p_xpart(i);% end%不敏卡尔曼滤波器%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);endfor i = (u_symmetry_number+2) : u_total_numberu_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_numberu_x_par(i) = 0.5 * u_x_par(i) + 25 * u_x_par(i)/(1+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_numberu_x_next = u_x_next + u_w_N1 * u_x_par(i);endu_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_numberu_p_next = u_p_next + u_w_N1 * (u_x_par(i)-u_x_next) * (u_x_par(i)-u_x_next)';end% %对传递后的均值和方差进行采样产生粒子存在y(i)% u_y_2obser(1) = u_x_next;% for i = 2 : (u_symmetry_number+1)% u_y_2obser(i,:) = u_x_next + sqrt((u_symmetry_number+k) * u_p_next);% end% for i = (u_symmetry_number + 2) : u_total_number% u_y_2obser(i,:) = u_x_next - sqrt((u_symmetry_number+u_k) * u_p_next); % end%另外存在y_2obser(i) 中；for i = 1 :u_total_numberu_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_numberu_y_2obser(i) = u_y_2obser(i)^2/20;end%通过观测方程后的均值y_obseu_y_obse = u_w_1 * u_y_2obser(1);for i = 2 : u_total_numberu_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_numberu_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_numberu_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;%进行画图程序x_array = [x_array,x];e_x_estimate_array = [e_x_estimate_array,e_x_estimate];p_x_estimate_array = [p_x_estimate_array,p_x_estimate];u_x_estimate_array = [u_x_estimate_array,u_x_estimate];e_error(k,:) = abs(x_array(k)-e_x_estimate_array(k));p_error(k,:) = abs(x_array(k)-p_x_estimate_array(k));u_error(k,:) = abs(x_array(k)-u_x_estimate_array(k));endt = 0 : tf;figure;plot(t,x_array,'k.',t,e_x_estimate_array,'r-',t,p_x_estimate_array,'g--',t,u_x_estimate_array,'b:');set(gca,'FontSize',10);set(gcf,'color','White');xlabel('时间步长');% lable --->label 我的神ylabel('状态');legend('真实值','EKF估计值','PF估计值','UKF估计值');figure;plot(t,x_array,'k.',t,p_x_estimate_array,'g--', t, p_x_estimate_array-1.96*sqrt(P), 'r:', t, p_x_estimate_array+1.96*sqrt(P), 'r:');set(gca,'FontSize',10);set(gcf,'color','White');xlabel('时间步长');% lable --->label 我的神ylabel('状态');legend('真实值','PF估计值', '95% 置信区间');%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)]);p_xrms = sqrt((norm(x_array-p_x_estimate_array)^2)/tf);disp(['PF估计误差均方值=',num2str(p_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)]);% plot(t,e_error,'r-',t,p_error,'g--',t,u_error,'b:');% legend('EKF估计值误差','PF估计值误差','UKF估计值误差');t = 1 : tf;figure;plot(t,e_error,'r-',t,p_error,'g--',t,u_error,'b:');set(gca,'FontSize',10);set(gcf,'color','White');xlabel('时间步长');% lable --->label 我的神ylabel('状态');legend('EKF估计值误差','PF估计值误差','UKF估计值误差');% toc;。