%Discrete Kalman filter for PID control
%Reference kalman_2rank.m
%x=Ax+B(u+w(k));
%y=Cx+D+v(k)
clear all;
close all;

ts=0.001;
%Continuous Plant
a=25;b=133;
sys=tf(b,[1,a,0]);
dsys=c2d(sys,ts,'z');
[num,den]=tfdata(dsys,'v');

A1=[0 1;0 -a];
B1=[0;b];
C1=[1 0];
D1=[0];
[A,B,C,D]=c2dm(A1,B1,C1,D1,ts,'z');

Q=1;               %Covariances of w
R=1;               %Covariances of v

P=B*Q*B';          %Initial error covariance
x=zeros(2,1);      %Initial condition on the state

u_1=0;u_2=0;
y_1=0;y_2=0;
ei=0;
error_1=0;
for k=1:1:1000
time(k)=k*ts;

rin(k)=1;
kp=8.0;ki=0.80;kd=0.20;

w(k)=0.002*rands(1);   %Process noise on u
v(k)=0.002*rands(1);   %Measurement noise on y

y(k)=-den(2)*y_1-den(3)*y_2+num(2)*u_1+num(3)*u_2;
yv(k)=y(k)+v(k);

%Measurement update
Mn=P*C'/(C*P*C'+R);
P=A*P*A'+B*Q*B'; 
P=(eye(2)-Mn*C)*P;
    
x=A*x+Mn*(yv(k)-C*A*x);
ye(k)=C*x+D;     %Filtered value
    
M=2;
if M==1          %Not using filter
 	yout(k)=yv(k);
elseif M==2      %Using filter
   yout(k)=ye(k);
end
error(k)=rin(k)-yout(k);
ei=ei+error(k)*ts;

u(k)=kp*error(k)+ki*ei+kd*(error(k)-error_1)/ts;   %PID
u(k)=u(k)+w(k);

errcov(k)=C*P*C';      %Covariance of estimation error

%Time update
x=A*x+B*u(k);

u_2=u_1;u_1=u(k);
y_2=y_1;y_1=yout(k);
error_1=error(k);
end
figure(1);
plot(time,rin,'k',time,yout,'k');
xlabel('time(s)');
ylabel('rin,yout');