扩展卡尔曼滤波仿真实例 一、 问题描述 x y )0( 如图 1 所示，从空中水平抛射出的物体，初始水平速度 )0(xv ，初始位置坐标 ),0( ）；受重力 g 和阻尼力影响，阻尼力与速度平方成正比，水平和垂直 （ x kk , ；还存在不确定的零均值白噪声干扰力 xa 和 ya 。在坐标 阻尼系数分别为 y 原点处有一观测设备（不妨想象成雷达），可测得距离 r（零均值白噪声误差 r ）、 角度（零均值白噪声误差）。 y )0(xv 物体  r g 雷达 )0,0( x 图 1 雷达观测示意图 二、 建模 系统方程： f : h : 量测方程： 选状态向量  xx x    v   x  y    v    r       v y x 系统 Jacobian 矩阵 量测 Jacobian 矩阵 f  x   h  x   2 x y 2 y y g 2 a   a  a y 2 vk xx  v  x  v  y vk  y r y x    / tan( ) yx   T ,量测向量  Trz v 0 0 1 0 0 2 vk xx 0 0 0 0 0 vk y y 1 2 y  2 / yx / ) yx        1   2 x   /1 y   / (1 yx   0 0 1 20 x  (1         2 y 0 0 2 2 2 )  0 0       三、 Matlab 仿真 function test_ekf kx = .01; ky = .05; % 阻尼系数 g = 9.8; % 重力 t = 10; % 仿真时间 Ts = 0.1; % 采样周期 len = fix(t/Ts); % 仿真步数 % 真实轨迹模拟 

dax = 1.5; day = 1.5; % 系统噪声 X = zeros(len,4); X(1,:) = [0, 50, 500, 0]; % 状态模拟的初值 for k=2:len x = X(k-1,1); vx = X(k-1,2); y = X(k-1,3); vy = X(k-1,4); x = x + vx*Ts; vx = vx + (-kx*vx^2+dax*randn(1,1))*Ts; y = y + vy*Ts; vy = vy + (ky*vy^2-g+day*randn(1))*Ts; X(k,:) = [x, vx, y, vy]; end figure(1), hold off, plot(X(:,1),X(:,3),'-b'), grid on % figure(2), plot(X(:,2:2:4)) % 构造量测量 mrad = 0.001; dr = 10; dafa = 10*mrad; % 量测噪声 for k=1:len r = sqrt(X(k,1)^2+X(k,3)^2) + dr*randn(1,1); a = atan(X(k,1)/X(k,3)) + dafa*randn(1,1); Z(k,:) = [r, a]; end figure(1), hold on, plot(Z(:,1).*sin(Z(:,2)), Z(:,1).*cos(Z(:,2)),'*') % ekf 滤波 Qk = diag([0; dax; 0; day])^2; Rk = diag([dr; dafa])^2; Xk = zeros(4,1); Pk = 100*eye(4); X_est = X; for k=1:len Ft = JacobianF(X(k,:), kx, ky, g); Hk = JacobianH(X(k,:)); fX = fff(X(k,:), kx, ky, g, Ts); hfX = hhh(fX, Ts); [Xk, Pk, Kk] = ekf(eye(4)+Ft*Ts, Qk, fX, Pk, Hk, Rk, Z(k,:)'-hfX); X_est(k,:) = Xk'; end figure(1), plot(X_est(:,1),X_est(:,3), '+r') xlabel('X'); ylabel('Y'); title('ekf simulation'); legend('real', 'measurement', 'ekf estimated'); %%%%%%%%%%%%%%%%%%%%子程序%%%%%%%%%%%%%%%%%%% function F = JacobianF(X, kx, ky, g) % 系统状态雅可比函数 vx = X(2); vy = X(4); F = zeros(4,4); F(1,2) = 1; F(2,2) = -2*kx*vx; F(3,4) = 1; F(4,4) = 2*ky*vy; function H = JacobianH(X) % 量测雅可比函数 x = X(1); y = X(3); H = zeros(2,4); r = sqrt(x^2+y^2); H(1,1) = 1/r; H(1,3) = 1/r; xy2 = 1+(x/y)^2; H(2,1) = 1/xy2*1/y; H(2,3) = 1/xy2*x*(-1/y^2); 

function fX = fff(X, kx, ky, g, Ts) % 系统状态非线性函数 x = X(1); vx = X(2); y = X(3); vy = X(4); x1 = x + vx*Ts; vx1 = vx + (-kx*vx^2)*Ts; y1 = y + vy*Ts; vy1 = vy + (ky*vy^2-g)*Ts; fX = [x1; vx1; y1; vy1]; function hfX = hhh(fX, Ts) % 量测非线性函数 x = fX(1); y = fX(3); r = sqrt(x^2+y^2); a = atan(x/y); hfX = [r; a]; function [Xk, Pk, Kk] = ekf(Phikk_1, Qk, fXk_1, Pk_1, Hk, Rk, Zk_hfX) % ekf 滤波函数 Kk = Pxz*Pzz^-1; Pkk_1 = Phikk_1*Pk_1*Phikk_1' + Qk; Pxz = Pkk_1*Hk'; Xk = fXk_1 + Kk*Zk_hfX; Pk = Pkk_1 - Kk*Pzz*Kk'; Pzz = Hk*Pxz + Rk; 520 500 480 460 Y 440 420 400 380 360 0 ekf simulation real measurement ekf estimated 20 40 60 80 100 X 120 140 160 180 200 图 2 仿真结果