% This is mathematically equivalent to setting the initial state estimate
% covariance to infinity.
%
% SCALAR EXAMPLE (Automobile Voltimeter):
%
 % Define the system as a constant of 12 volts:
 clear s
 s.x = 12;
 s.A = 1;
 s.dt=1
 % Define a process noise (stdev) of 2 volts as the car operates:
 s.Q = 2^2; % variance, hence stdev^2
 % Define the voltimeter to measure the voltage itself:
 s.H = 1;
 % Define a measurement error (stdev) of 2 volts:
 s.R = 2^2; % variance, hence stdev^2
 % Do not define any system input (control) functions:
 s.B = 0;
 s.u = 0;
 % Do not specify an initial state:
 s.x = nan;
 s.P = nan;
 % Generate random voltages and watch the filter operate.
 tru=[]; % truth voltage
 for t=1:20
    tru(end+1) = randn*2+12;
    s(end).z = tru(end) + randn*2; % create a measurement
    s(end+1)=kalman_filter(s(end)); % perform a Kalman filter iteration
 end
 figure
 hold on
 grid on
 % plot measurement data:
 hz=plot([s(1:end-1).z],'r.');
 % plot a-posteriori state estimates:
 hk=plot([s(2:end).x],'b-');
 ht=plot(tru,'g-');
 legend([hz hk ht],'observations','Kalman output','true voltage',0)
 title('Automobile Voltimeter Example')
 hold off