echo off
% KALMDEMO demonstrates MATLAB's ability to do Kalman filtering.  
% Both a steady state filter and a time varying filter are designed
% and simulated below.

%   Copyright (c) 1986-97 by The MathWorks, Inc.
%   $Revision: 1.5 $  $Date: 1997/04/09 00:37:09 $
clc
clf
help kalmdemo
echo on
pause % Press any key to continue ...

% Given the following discrete plant
A = [1.1269   -0.4940    0.1129, 
     1.0000         0         0, 
          0    1.0000         0]

B = [-0.3832
      0.5919
      0.5191]

C = [1 0 0];

D = 0;

% Design a Kalman filter to estimate the output y based on the 
% noisy measurements  yv[n] = C x[n] + v[n]

pause % Press any key to continue ...
clc
% Let's design a steady-state Kalman filter using the 
% function KALMAN.  This function determines the optimal 
% steady-state filter gain M based on the process noise 
% covariance Q and the sensor noise covariance R.

% First specify the plant + noise model.
% CAUTION: set the sample time to -1 to mark plant 
%          as discrete
Plant = ss(A,[B B],C,0,-1,'inputname',{'u' 'w'},...
                          'outputname','y');

% Enter the process noise covariance:
Q = input('Enter a number greater than zero (e.g. 1): ');

echo off
if isempty(Q), Q = 1; end
echo on
Q

% Enter the sensor noise covariance:
R = input('Enter a number greater than zero (e.g. 1): ');

echo off
if isempty(R), R = 1; end
echo on
R

% Now design the steady-state Kalman filter with equations
%
% Time update:        x[n+1|n] = Ax[n|n-1] + Bu[n]
%
% Mesurement update:  x[n|n] = x[n|n-1] + M (yv[n] - Cx[n|n-1]) 
%
%                          where M = optimal innovation gain

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

pause % Press any key to continue ...

% The first output of the Kalman filter KALMF is the plant
% output estimate  y_e = Cx[n|n], and the remaining outputs
% are the state estimates.  Keep only the first output y_e

kalmf = kalmf(1,:)

M,   % innovation gain

pause % Press any key to continue ...
clc
% Let's see how this filter works by generating some data and 
% comparing the filtered response with the true plant response.
%
%     noise         noise
%       |             |
%       V             V
% u -+->O-->[Plant]-->O--> yv --->[      ]
%    |                            [filter]---> y_e 
%    +--------------------------->[      ]
%
% To simulate the system above, you can generate the response of 
% each part separately or you can generate both together.  To 
% simulate each separately, you would use LSIM with the plant first
% and then the filter.  Below we illustrate how to simulate both
% together.

% First, build a complete plant model with u,w,v as inputs and
% y and yv as outputs.
a = A;
b = [B B 0*B];
c = [C;C];
d = [0 0 0;0 0 1];
P = ss(a,b,c,d,-1,'inputname',{'u' 'w' 'v'},...
                  'outputname',{'y' 'yv'});

% Then connect this plant model and the kalman filter in parallel
% by specifying u as a shared input:
sys = parallel(P,kalmf,1,1,[],[]);

pause % Press any key to continue ...

% ... and connect the plant output yv to the filter input yv 
% Note:  yv is the 4th input of SYS and also its 2nd output 
SimModel = feedback(sys,1,4,2,1);
SimModel = SimModel([1 3],[1 2 3])     % Delete yv form I/O

pause % Press any key to continue ...

% The resulting simulation model has w,v,u as inputs and y,y_e as 
% outputs:
SimModel.input

SimModel.output

pause % Press any key to continue ...
clc
% You are now ready to simulate the filter behavior.
% Generate a sinusoidal input vector (known)
t = [0:100]';
u = sin(t/5);

% Generate process noise and sensor noise vectors
randn('seed',0)
w = sqrt(Q)*randn(length(t),1);
v = sqrt(R)*randn(length(t),1);

pause % Press any key to continue ...

% Now simulate the response using LSIM
[out,x] = lsim(SimModel,[w,v,u]);

y = out(:,1);   % true response
ye = out(:,2);  % filtered response
yv = y + v;     % measured response

% Compare true response with filtered response
% subplot(211), plot(t,y,'b',t,ye,'r--'), 
% xlabel('No. of samples'), ylabel('Output')
% subplot(212), plot(t,y-yv,'g',t,y-ye,'r--'),
% xlabel('No. of samples'), ylabel('Error')
echo off
clf
subplot(211), plot(t,y,'b',t,ye,'r--'), 
xlabel('No. of samples'), ylabel('Output')
title('Kalman filter response')
subplot(212), plot(t,y-yv,'g',t,y-ye,'r--'),
xlabel('No. of samples'), ylabel('Error')
echo on

pause % Press any key after the plot ...
clc
% The second plot indicates that the Kalman filter has reduced
% the error y-yv due to measurement noise, as confirmed by 
% comparing the error covariances
MeasErr = y-yv;
MeasErrCov = sum(MeasErr.*MeasErr)/length(MeasErr);
EstErr = y-ye;
EstErrCov = sum(EstErr.*EstErr)/length(EstErr);

% Covariance of error before filtering (measurement error)
MeasErrCov

% Covariance of error after filtering (estimation error)
EstErrCov

pause % Press any key to continue ...
clc
% Now let's design a time-varying Kalman filter to perform the same
% task.  A time-varying Kalman filter is able to perform well even
% when the noise covariance is not stationary.  However for this 
% demonstration, we will use stationary covariance.

% The time varying Kalman filter has the following update equations
%  
% Time update:        x[n+1|n] = Ax[n|n] + Bu[n]
%                     
%                     P[n+1|n] = AP[n|n]A' + B*Q*B'
%
%                             
% Measurement update:  
%                      x[n|n] = x[n|n-1] + M[n](yv[n] - Cx[n|n-1])
%                                                        -1
%                      M[n] = P[n|n-1] C' (CP[n|n-1]C'+R)
%
%                      P[n|n] = (I-M[n]C) P[n|n-1]

pause % Press any key to continue ...

% Generate the noisy plant response
sys = ss(A,B,C,D,-1);
y = lsim(sys,u+w);   % w = process noise
yv = y + v;          % v = meas. noise

% Now implement the filter recursions in a FOR loop:
P=B*Q*B';         % Initial error covariance
x=zeros(3,1);     % Initial condition on the state
ye = zeros(length(t),1);
ycov = zeros(length(t),1); 

%for i=1:length(t)
%  % Measurement update
%  Mn = P*C'/(C*P*C'+R);
%  x = x + Mn*(yv(i)-C*x);  % x[n|n]
%  P = (eye(3)-Mn*C)*P;
%
%  ye(i) = C*x;
%  errcov(i) = C*P*C';
%
%  % Time update
%  x = A*x + B*u(i);        % x[n+1|n]
%  P = A*P*A' + B*Q*B'; 
%end

echo off
for i=1:length(t)
  % Measurement update
  Mn = P*C'/(C*P*C'+R);
  x = x + Mn*(yv(i)-C*x);  % x[n|n]
  P = (eye(3)-Mn*C)*P;     % P[n|n]

  ye(i) = C*x;
  errcov(i) = C*P*C';

  % Time update
  x = A*x + B*u(i);        % x[n+1|n]
  P = A*P*A' + B*Q*B';     % P[n+1|n] 
end
echo on

pause % Press any key to continue ...

% Compare true response with filtered response
%subplot(211), plot(t,y,'b',t,ye,'r--'), 
%xlabel('No. of samples'), ylabel('Output')
%subplot(212), plot(t,y-yv,'g',t,y-ye,'r--'),
%xlabel('No. of samples'), ylabel('Error')
echo off
clf
subplot(211), plot(t,y,'b',t,ye,'r--'), 
xlabel('No. of samples'), ylabel('Output')
title('Response with time-varying Kalman filter')
subplot(212), plot(t,y-yv,'g',t,y-ye,'r--'),
xlabel('No. of samples'), ylabel('Error')
echo on

pause % Press any key to continue ...

% The time varying filter also estimates the output covariance
% during the estimation.  Let's plot it to see if our filter reached
% steady state (as we would expect with stationary input noise).
subplot(211)
plot(t,errcov), ylabel('Error Covar'), pause % Press any key after plot

% From the covariance plot we see that the output covariance did 
% indeed reach a steady state in about 5 samples.  From then on,
% our time varying filter has the same performance as the steady 
% state version.

pause % Press any key to continue ...
clc
% Compare covariance errors
MeasErr = y-yv;
MeasErrCov = sum(MeasErr.*MeasErr)/length(MeasErr);
EstErr = y-ye;
EstErrCov = sum(EstErr.*EstErr)/length(EstErr);

% Covariance of error before filtering (measurement error)
MeasErrCov

% Covariance of error after filtering (estimation error)
EstErrCov

pause % Press any key to continue ...

% Let's verify that the steady-state and final values of the 
% Kalman gain matrices coincide:
M,Mn


echo off
