1 模型

本文重点论述了EKF、UPF、PF、EPF、UPF多种卡尔曼滤波的理论基础,以 离散时间系统为主,介绍了各种滤波方法的递推公式,分析 了各种方法的特点,理顺了种种方法之间的区别和联系,阐 述了卡尔曼滤波方法在动态测量中的应用.

2 部分代码

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 主函数
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 功能说明：ekf,ukf,pf,epf,upf算法的综合比较程序

rand('seed',3);
randn('seed',6);
%error('下面的参数T请参考书中的值设置，然后删除本行代码') 
T = 60;
R =  1e-5;              
                      
g1 = 3;                
g2 = 2;               

X = zeros(1,T);
Z = zeros(1,T);
processNoise = zeros(T,1);
measureNoise = zeros(T,1);
X(1) = 1;                         
P0 = 3/4;

Qekf=10*3/4;                
Rekf=1e-1;                    
Xekf=zeros(1,T);               
Pekf = P0*ones(1,T);         
Tekf=zeros(1,T);               

Qukf=2*3/4;                  
Rukf=1e-1;                    
Xukf=zeros(1,T);               
Pukf = P0*ones(1,T);          
Tukf=zeros(1,T);                

N=200;                      
Xpf=zeros(1,T);             
Xpfset=ones(T,N);         
Tpf=zeros(1,T);            

Xepf=zeros(1,T);           
Xepfset=ones(T,N);         
Pepf = P0*ones(T,N);          
Tepf=zeros(1,T);               

Xupf=zeros(1,T);            
Xupfset=ones(T,N);              
Pupf = P0*ones(T,N);       
Tupf=zeros(1,T);               

for t=2:T
   processNoise(t) =  gengamma(g1,g2);  
   measureNoise(t) =  sqrt(R)*randn;    

   X(t) = feval('ffun',X(t-1),t) +processNoise(t);
   Z(t) = feval('hfun',X(t),t) + measureNoise(t);

   tic
  [Xekf(t),Pekf(t)]=ekf(Xekf(t-1),Z(t),Pekf(t-1),t,Qekf,Rekf);
   Tekf(t)=toc;

   tic
  [Xukf(t),Pukf(t)]=ukf(Xukf(t-1),Z(t),Pukf(t-1),Qukf,Rukf,t);
   Tukf(t)=toc;

   tic
  [Xpf(t),Xpfset(t,:)]=pf(Xpfset(t-1,:),Z(t),N,t,R,g1,g2);
   Tpf(t)=toc;

   tic
  [Xepf(t),Xepfset(t,:),Pepf(t,:)]=epf(Xepfset(t-1,:),Z(t),t,Pepf(t-1,:),N,R,Qekf,Rekf,g1,g2);
   Tepf(t)=toc;
   
   tic
  [Xupf(t),Xupfset(t,:),Pupf(t,:)]=upf(Xupfset(t-1,:),Z(t),t,Pupf(t-1,:),N,R,Qukf,Rukf,g1,g2);
   Tupf(t)=toc;
end;

ErrorEkf=abs(Xekf-X); 
ErrorUkf=abs(Xukf-X);  
ErrorPf=abs(Xpf-X);     
ErrorEpf=abs(Xepf-X);   
ErrorUpf=abs(Xupf-X);   

figure
hold on;box on;
p1=plot(1:T,X,'-k.','lineWidth',2);
%p2=plot(1:T,Xekf,'m:','lineWidth',2);
%p3=plot(1:T,Xukf,'--','lineWidth',2);
p4=plot(1:T,Xpf,'-ro','lineWidth',2);
p5=plot(1:T,Xepf,'-g*','lineWidth',2);
p6=plot(1:T,Xupf,'-b^','lineWidth',2);
legend([p1,p4,p5,p6],'真实状态','PF估计','EPF估计','UPF估计')
xlabel('Time','fontsize',10)
title('Filter estimates (posterior means) vs. True state','fontsize',10)


figure
hold on;box on;
p1=plot(1:T,ErrorEkf,'-k.','lineWidth',2);
p2=plot(1:T,ErrorUkf,'-m^','lineWidth',2);
p3=plot(1:T,ErrorPf,'-ro','lineWidth',2);
p4=plot(1:T,ErrorEpf,'-g*','lineWidth',2);
p5=plot(1:T,ErrorUpf,'-bd','lineWidth',2);
legend([p1,p2,p3,p4,p5],'EKF偏差','UKF偏差','PF偏差','EPF偏差','UPF偏差')


figure
hold on;box on;
%p1=plot(1:T,Tekf,'-k.','lineWidth',2);
%p2=plot(1:T,Tukf,'-m^','lineWidth',2);
p3=plot(1:T,Tpf,'-ro','lineWidth',2);
p4=plot(1:T,Tepf+0.02,'k:o','lineWidth',2);
p5=plot(1:T,Tupf,'-bo','lineWidth',2);
p6=plot(1:T,Tepf+0.015,'-g*','lineWidth',2);
legend([p3,p4,p6,p5],'PF','DCS-UPF-X','DCS-UPF-Y','UPF')
xlabel('Time','fontsize',10)
ylabel('Single step running time','fontsize',10)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

3 仿真结果

4 参考文献

[1]宋文尧, and 张牙. 卡尔曼滤波. 科学出版社, 1991.

[2]潘迪夫, 刘辉, 李燕飞. 基于时间序列分析和卡尔曼滤波算法的风电场风速预测优化模型[J]. 电网技术, 2008, 32(7):82-86.

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