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1、扩展卡尔曼滤波（EKF）仿真演示（西工大 严恭敏，2012-2-4）一、 问题描述如图1所示，从空中水平抛射出的物体，初始水平速度，初始位置坐标（）；受重力和阻尼力影响，阻尼力与速度平方成正比，水平和垂直阻尼系数分别为；还存在不确定的零均值白噪声干扰力和。在坐标原点处有一观测设备（不妨想象成雷达），可测得距离（零均值白噪声误差）、角度（零均值白噪声误差）。图1 雷达观测示意图二、 建模系统方程：量测方程：选状态向量,量测向量系统Jacobian矩阵量测Jacobian矩阵三、 Matlab仿真function test_ekf kx = .01; ky = .05; % 阻尼系数 g = 9.
2、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*vx2+dax*randn(1,1)*Ts; y = y + vy*Ts; vy = vy
3、+ (ky*vy2-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,
4、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);
5、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(
6、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(x2+y2); 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/y2); function fX = fff(X, kx, ky, g, Ts) % 系统状态非线性函数 x = X(
7、1); vx = X(2); y = X(3); vy = X(4); x1 = x + vx*Ts; vx1 = vx + (-kx*vx2)*Ts; y1 = y + vy*Ts; vy1 = vy + (ky*vy2-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(x2+y2); 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 Pkk_1 = Phikk_1*Pk_1*Phikk_1 + Qk; Pxz = Pkk_1*Hk; Pzz = Hk*Pxz + Rk; Kk = Pxz*Pzz-1; Xk = fXk_1 + Kk*Zk_hfX;Pk = Pkk_1 - Kk*Pzz*Kk;图2 仿真结果
