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%% Learning the Kalman Filter in Simulink Examples 
% Examples to use the Simulink model kalmanfilter.
% By Yi Cao at Cranfield University on 25 January 2008
%% Overview
% The Simulink model shows an example how the Kalman Filter can be
% implemented in Simulink. The model itself is configured with a Gaussian
% process connected with a Kalman Filter. To directly use this model, one
% only needs to provide model prarameters including parameters of the
% Gaussian process, which are state space matrices, A, B, C, and D, initial
% state, x0, and covariance matrices, Q and R; and similar parameters for
% the Kalman Filter, which can be in different values to mimic the model
% mismatch, plus the state covariance, P. The following examples show how
% this model can be used.
%
% The Kalman Filter can also be used as a standard model block to be
% connected with any other systems.


simulink图

%% Example 1, Automobile Voltmeter 
% Modified from
% <http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=5377&objectType=file Michael C. Kleder's "Learning the Kalman Filter">
%
% Define the system as a constant of 12 volts:
% 
% $$x_{k+1} = 0*x_k + 12 + w,   w \sim N(0,2^2)$$
% 
% But the Kalman filter uses different model.
%
% $$x_{k+1} = x_{k} + w, x_0=12, w \sim N(0,2^2)$$
%
% This example shows the Kalman filter has certain robustness to model
% uncertainty.
A = 0; B = 1;   % this is original definition x(k+1) = 12 + w
A1 = 1; B1 = 0; % this is original definition for Kalman filter
% Define a process noise (stdev) of 2 volts as the car operates:
Q = 2^2; % variance, hence stdev^2
Q1 = Q;
% Define the voltmeter to measure the voltage itself:
% y(k+1) = x(k+1) + v, v ~ N(0,2^2) 
C = 1; D = 0;
C1 = C; D1 = D;
% Define a measurement error (stdev) of 2 volts:
R = 2^2;
R1 = R;
% Define the system input (control) functions:
u = 12; % for the constant
% Initial state, 12 volts
x0 = 12;
x1 = x0;    %for Kalman filter
% Initial covariance as C*R*C'
P1 = 2^2;
% Simulation time span
tspan = [0 100];
[t,x,y] = sim('kalmanfilter',tspan,[],[0 u]);
figure
hold on
grid on
% plot measurement data:
hz=plot(t,y(:,2),'r.','Linewidth',2);
% plot a-posteriori state estimates:
hk=plot(t,y(:,3),'b-','Linewidth',2);
% plot true state
ht=plot(t,y(:,1),'g-','Linewidth',2);
legend([hz hk ht],'observations','Kalman output','true voltage',0)
title('Automobile Voltmeter Example')
hold off


%% Example 2, Predict the position and velocity of a moving train, 
% Modified from <http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=13479&objectType=file "An Intuitive Introduction to Kalman Filter"> by Alex Blekhman.
% 
% The train is initially located at the point x = 0 and moves
% along the X axis with velocity varying around a constant speed 10m/sec.
% The motion of the train can be described by a set of differential
% equations:
%
% $$ \dot{x}_1 = x_2,  x_1(0)=0$$
%
% $$ \dot{x}_2 = w,   x_2(0)=10, w \sim N(0,0.3^2)$$
%
% where x_1 is the position and x_2 is the velocity, 
% w the process noise due to road conditions, wind etc.
%
% Problem: using the position measurement to estimate actual velocity.
%
% Approach: We measure (sample) the position of the train every dt = 0.1
% seconds. But, because of imperfect apparature, weather etc., our
% measurements are noisy, so the instantaneous velocity, derived from 2
% consecutive position measurements (remember, we measure only position) is
% innacurate. We will use Kalman filter as we need an accurate and smooth
% estimate for the velocity in order to predict train's position in the
% future. 
%
% Model: 
% The state space equation after discretization with sampling time dt
%
% $$x_{k+1} = \Phi x_k + w_k, \Phi = e^{A\Delta t}, A=[0\,\, 1; 0\,\, 0], w_k \sim N(0,Q), Q =[0\,\, 0; 0\,\, 1]$$
%
% The measurement model is
%
% $$y_{k+1} = [1 \,\, 0] x_{k+1} + v_k, v_k \sim N(0,1)$$
%
dt = 0.1;
A = expm([0 1;0  0]*dt);
B = [0;0];
Q = diag([0;1]);
C = [1 0];
D = 0;
R = 1;
u = 0;
x0 = [0;10];


% For Kalman filter, we use the same model
A1 = A;
B1 = B;
C1 = C;
D1 = D;
Q1 = Q;
R1 = R;
x1 = [0;5];     %but with different initial state estimate
P1 = eye(2);


% Run the simulation for 100 smaples
tspan = [0 200];
[t,x,y1,y2,y3,y4] = sim('kalmanfilter',tspan,[],[0 u]);


Xtrue = y1(:,1);    %actual position
Vtrue = y1(:,2);    %actiual velocity
z = y2;             %measured position
X = y3;             %Kalman filter output
t = t*dt;           % actual time


% Position analysis
figure;
subplot(211)
plot(t,Xtrue,'g',t,z,'c',t,X(:,1),'m','linewidth',2);
title('Position estimation results');
xlabel('Time (s)');
ylabel('Position (m)');
legend('True position','Measurements','Kalman estimated displacement','Location','NorthWest');


% Velocity analysis 


% The instantaneous velocity as derived from 2 consecutive position
% measurements
InstantV = [10;diff(z)/dt];


% The instantaneous velocity as derived from running average with a window
% of 5 samples from instantaneous velocity
WindowSize = 5;
InstantVAverage = filter(ones(1,WindowSize)/WindowSize,1,InstantV);


% figure;
subplot(212)
plot(t,InstantV,'g',t,InstantVAverage,'c',t,Vtrue,'m',t,X(:,2),'k','linewidth',2);
title('Velocity estimation results');
xlabel('Time (s)');
ylabel('Velocity (m/s)');
legend('Estimated velocity by raw consecutive samples','Estimated velocity by running average','True velocity','Estimated velocity by Kalman filter','Location','NorthWest');


set(gcf,'Position', [100 100 600 800]);


%% Example 3: A 2-input 2-output 4-state system with non-zero D
dt = 0.1;
A = [0.8110   -0.0348    0.0499    0.3313
     0.0038    0.9412    0.0184    0.0399
     0.1094    0.0094    0.6319    0.1080
    -0.3186   -0.0254   -0.1446    0.8391];
B = [-0.0130    0.0024
     -0.0011    0.0100
     -0.0781    0.0009
      0.0092    0.0138];
C = [0.1685   -0.9595   -0.0755   -0.3771
     0.6664    0.0835    0.6260    0.6609];  
D = [0.2 0;0 0.1];
% process noise variance
Q=diag([0.5^2 0.2^2 0.3^2 0.5^2]);
% measurement noise variance
R=eye(2);
% random initial state
x0 = randn(4,1);
% Kalman filter set up
% The same model
A1 =A;
B1 = B;
C1 = C;
D1 = D;
Q1 = Q;
R1 = R;
% However, zeros initial state
x1 = zeros(4,1);
% Initial state covariance
P1 = 10*eye(4);
% Simulation set up
% time span 100 samples
tspan = [0 1000];
% random input change every 100 samples
u = [(0:100:1000)' randn(11,2)];
% simulation
[t,x,y1,y2,y3,y4] = sim('kalmanfilter',tspan,[],u);
% plot results
% Display results
t = t*dt;
figure
set(gcf,'Position',[100 100 600 800])
for k=1:4
    subplot(4,1,k)
    plot(t,y1(:,k),'b',t,y3(:,k),'r','linewidth',2);
    legend('Actual state','Estimated state','Location','best')
    title(sprintf('state %i',k))
end
xlabel('time, s')