Finding out Q and R matrices of a given system.

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')

Answer 1

[John Petersen]

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.