基于matlab的卡尔曼滤波器设计与实现

function [x,P]=ukf(fstate,x,P,hmeas,z,Q,R) % UKF Unscented Kalman Filter for nonlinear dynamic systems % [x, P] = ukf(f,x,P,h,z,Q,R) returns state estimate, x and state covariance, P % for nonlinear dynamic system (for simplicity, noises are assumed as additive): % x_k+1 = f(x_k) + w_k % z_k = h(x_k) + v_k % where w ~ N(0,Q) meaning w is gaussian noise with covariance Q % v ~ N(0,R) meaning v is gaussian noise with covariance R % Inputs: f: function handle for f(x) % x: "a priori" state estimate % P: "a priori" estimated state covariance % h: fanction handle for h(x) % z: current measurement % Q: process noise covariance % R: measurement noise covariance % Output: x: "a posteriori" state estimate % P: "a posteriori" state covariance % % Example: %{ n=3; %number of state q=0.1; %std of process r=0.1; %std of measurement Q=q^2*eye(n); % covariance of process R=r^2; % covariance of measurement f=@(x)[x(2);x(3);0.05*x(1)*(x(2)+x(3))]; % nonlinear state equations h=@(x)x(1); % measurement equation s=[0;0;1]; % initial state x=s+q*randn(3,1); %initial state % initial state with noise P = eye(n); % initial state covraiance N=20; % total dynamic steps xV = zeros(n,N); %estmate % allocate memory sV = zeros(n,N); %actual zV = zeros(1,N); for k=1:N z = h(s) + r*randn; % measurments sV(:,k)= s; % save actual state zV(k) = z; % save measurment [x, P] = ukf(f,x,P,h,z,Q,R); % ekf xV(:,k) = x; % save estimate s = f(s) + q*randn(3,1); % update process end for k=1:3 % plot results subplot(3,1,k) plot(1:N, sV(k,:), '-', 1:N, xV(k,:), '--') end %} % Reference: Julier, SJ. and Uhlmann, J.K., Unscented Filtering and % Nonlinear Estimation, Proceedings of the IEEE, Vol. 92, No. 3, % pp.401-422, 2004. % % By Yi Cao at Cranfield University, 04/01/2008 % L=numel(x); %numer of states m=numel(z); %numer of measurements alpha=1e-3; %default, tunable ki=0; %default, tunable beta=2; %default, tunable lambda=alpha^2*(L+ki)-L; %scaling factor c=L+lambda; %scaling factor Wm=[lambda/c 0.5/c+zeros(1,2*L)]; %weights for means Wc=Wm; Wc(1)=Wc(1)+(1-alpha^2+beta); %weights for covariance c=sqrt(c); X=sigmas(x,P,c); %sigma points around x [x1,X1,P1,X2]=ut(fstate,X,Wm,Wc,L,Q); %unscented transformation of process % X1=sigmas(x1,P1,c); %sigma points around x1 % X2=X1-x1(:,ones(1,size(X1,2))); %deviation of X1 [z1,Z1,P2,Z2]=ut(hmeas,X1,Wm,Wc,m,R); %unscented transformation of measurments P12=X2*diag(Wc)*Z2'; %transformed cross-covariance K=P12*inv(P2); x=x1+K*(z-z1); %state update P=P1-K*P12'; %covariance update function [y,Y,P,Y1]=ut(f,X,Wm,Wc,n,R) %Unscented Transformation %Input: % f: nonlinear map % X: sigma points % Wm: weights for mean % Wc: weights for covraiance % n: numer of outputs of f % R: additive covariance %Output: % y: transformed mean % Y: transformed smapling points % P: transformed covariance % Y1: transformed deviations L=size(X,2); y=zeros(n,1); Y=zeros(n,L); for k=1:L Y(:,k)=f(X(:,k)); y=y+Wm(k)*Y(:,k); end Y1=Y-y(:,ones(1,L)); P=Y1*diag(Wc)*Y1'+R; function X=sigmas(x,P,c) %Sigma points around reference point %Inputs: % x: reference point % P: covariance % c: coefficient %Output: % X: Sigma points A = c*chol(P)'; Y = x(:,ones(1,numel(x))); X = [x Y+A Y-A];