Finding out Q and R matrices of a given system.

Question

Misha 2014-3-22

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/122661-finding-out-q-and-r-matrices-of-a-given-system

回答： John Petersen 2014-7-28

Hi everyone, I have a system with A, B, C, D matrices and I want to get a Kalman filter response, but I cannot figure out Q and R matrices. I would be really thankful if anybody could check out my code and tell me how to find Q and R matrices and get a desired response. Thank you

a=0.304;

S=a^2*pi;

v=0.015;

v3=0.03;

L=0.28;

Ts=0.01;

Sd=v*pi;

Sa=v3*pi;

g=9.81;

h10=0.06;

h20=0.04;

h30=0.02;

C13=(Sd/S)*g/sqrt(2*g*(h10-h30));

C23=(Sd/S)*g/sqrt(2*g*(h20-h30));

%state space model

A=[-C13 0 C13; 0 -C23 C23; C13 C23 -C13-C23-Sa/S*g/sqrt(2*g*h30)];

B=[1/S 0 0; 0 1/S 0; 0 0 0];

C=[1 0 0;0 1 0; 0 0 1];

D=[0 0 0; 0 0 0; 0 0 0];

Delta_t=0.01;

Q=??

R=eye(3)

Plant = ss(A,B,C,0,-1,'inputname',{'u' 'w' 'z'},...

'outputname',{'y' 'e' 'r'});

[kalmf,L,P,M]=kalman(Plant,Q,R)

kalmf = kalmf(1,:)

kalmf

a = A;

b = [1/S 0 0; 0 1/S 0; 0 0 0] ;

c = C;

d = [0 0 0;0 0 0; 0 0 0];

P = ss(a,b,c,d,-1,'inputname',{'u' 'w' 'v'},...

'outputname',{'y' 'yv' 'yz'});

 sys = parallel(Plant,kalmf,1,1,[],[])
 SimModel = feedback(sys,1,4,2,1)

% Delete yv from I/O list

SimModel = SimModel([1 3],[1 2 3])

SimModel.inputname

SimModel.outputname

t = [0:100]';

u = sin(t/5);

n = length(t)

randn('seed',0)

w = sqrt(Q)*randn(n,1);

z = sqrt®*randn(n,1);

[out,x] = lsim(SimModel,[w,z,u])

y = out(:,1); % true response

ye = out(:,2); % filtered response

yv = y + z; % measured response

subplot(211), plot(t,y,'--',t,ye,'-'),

xlabel('No. of samples'), ylabel('Output')

title('Kalman filter response')

subplot(212), plot(t,y-yv,'-.',t,y-ye,'-'),

xlabel('No. of samples'), ylabel('Error')

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Answer 1

John Petersen 2014-7-28

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/122661-finding-out-q-and-r-matrices-of-a-given-system#answer_146715

R and Q are the plant and sensor uncertainty covariance. I can't tell from your code which is which. You can start with an initial guess based on sensor specs and your understanding of the system. From there you can adjust the values as needed for your purposes.