扩展卡尔曼滤波（EKF）仿真演示
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vx(0)，初始位置坐标（x(0),y(0)）；受重力g和阻尼力影响，阻尼力与速度平方成正比，水平和垂直阻尼系数分别为kx,ky；还存在不确定的零均值白噪声干扰力?ax和?ay。在坐标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??）。
yvx(0)物体?r雷达(0,0)gx图1 雷达观测示意图
二、 建模
??vx?x?2?x??kxvxv??ax?系统方程：f: ?y??vy?2?v??kvyy?g??ay?y??r?x2?y2??r量测方程：h: ?????atan(x/y)???选状态向量x?xvxyvyT,量测向量z??r??T
??1?0??f?0?2kxvx系统Jacobian矩阵?0?x?0?0??01?22?h?x?y?量测Jacobian矩阵??x?1/y?1?(x/y)2?0?00?? 10??02kyvy??10x2?y2?x/y201?(x/y)20?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 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';
ekf simulation520realmeasurementekf estimated500480460440Y420400380360020406080100X120140160180200图2 仿真结果