1. Jean Walrand – EECS – UC Berkeley
Probability in EECS
Jean Walrand – EECS – UC Berkeley
Kalman Filter

2. Kalman Filter: Overview
X(n+1) = AX(n) + V(n); Y(n) = CX(n) + W(n); noise ⊥
KF computes L[X(n) | Yn]
Linear recursive filter, innovation gain Kn, error covariance Σn
If (A, C) observable and Σ0 = 0, then Σn → Σ finite, Kn → K
If, in addition, (A, Q = (ΣV)1/2) reachable, then filter with Kn = K is asymptotically optimal (i.e., Σn → Σ)

3. Rudolf (Rudy) Emil Kálmán
Kalman Filter 1. Kalman
He is most noted for his co-invention
and development of the Kalman Filter that is widely used in control systems, avionics, and outer space manned and unmanned vehicles.
For this work, President Obama awarded him with the National Medal of Science on October 7, 2009.
Rudolf (Rudy) Emil Kálmán
b. 5/19/1930 in Budapest
Research Institute for Advanced Studies Baltimore, Maryland from 1958 until 1964.
Stanford 64-71
Gainesville, Florida 71-92

4. Kalman Filter 2. Applications
Tracking and Positioning:
Apollo Program, GPS, Inertial Navigation, …

5. Kalman Filter 2. Applications
Communications:
MIMO

6. Kalman Filter 2. Applications
Communications:
Video Motion Estimation

7. Kalman Filter 2. Applications
Bio-Medical:

8. Kalman Filter 3. Big Picture

9. Kalman Filter 3. Big Picture (continued)

10. Kalman Filter 4. Formulas

11. Kalman Filter 5. MATLAB for n = 1:N-1 V = normrnd(0,Q);
W = normrnd(0,R);
X(n+1,:) = A*X(n,:) + V;
Y = C*X(n+1,:) + W;
S = A*H*A' + Q*Q’;
K = S*C'*(C*S*C'+R*R’)^(-1);
H = (1 - K*C)*S;
Xh(n+1,:) = (A - K*C*A)*Xh(n,:) + K*Y;
end
System KF

12. Kalman Filter 5. MATLAB (complete)
% KF: System X+ = AX + V, Y = CX + W (by X+ we mean X(n+1))
% \Sigma_V = Q^2, \Sigma_W = R^2
% we construct a gaussian noise V = normrnd(0, Q), W = normrnd(0, R)
% where Z is iid N(0, 1)
% The filter is Xh+ = (A - KCA)Xh + KY (Here Xh is the estimate)
% K+ = SC'[CSC' + R^2)
% S = AHA' + Q^ (Here H = \Sigma = est. error cov.)
% H+ = (1 - KC)S
% CONSTANTS
A = [1, 1; 0, 1];
C = [1, 0];
Q = [1; 0];
R = 0.5;
SV = Q*Q';
SW = R*R';
N = 20; %time steps
M = length(A);
%SYSTEM
X = zeros(M, N);
Xh = X;
H = zeros(M, M);
K = H;
X(:,1) = [0; 3]; %initial state
for n = 1:N-1 % These are the system equations
V = normrnd(0,Q)
W = normrnd(0,R;);
X(:,n+1) = A*X(:,n) + V;
Y = C*X(:,n+1) + W;
%FILTER
S = A*H*A' + SV; %Gain calculation
K = S*C'*(C*S*C'+SW)^(-1);
H = (1 - K*C)*S;
%Estimate update
Xh(:,n+1) = (A - K*C*A)*Xh(:,n) + K*Y;
end
%PLOT
P = [X(2,:); Xh(2,:)]';
plot(P)

13. Kalman Filter 6. Example 1: Random Walk
C = 1 ;
Q = 0.2;
R = 0.3;
A = 1 ;
C = 1 ;
Q = 0.2;
R = 1;

14. Kalman Filter 6. Example 1: Random Walk
C = 1 ;
Q = 0.2;
R = 0.3;
A = 1 ;
C = 1 ;
Q = 0.2;
R = 1;

15. Kalman Filter 6. Example 1: Random Walk
C = 1 ;
Q = 10;
R = 10;

16. Kalman Filter 7. Example 2: RW with unknown drift
C = [1, 0];
Q = [1; 0];
R = 0.5;

17. Kalman Filter 8. Example 3: 1-D tracking, changing drift
Q = [1; 0.1];
R = 0.5;

18. Kalman Filter 9. Example 4: Falling body
C = [1, 0];
Q = [10; 0];
R = 40;

19. Kalman Filter 10. Simulink