Probability in EECS

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

2. Kalman Filter: Overview 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. Kalman Filter1. 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 Filter2. Applications Tracking and Positioning: Apollo Program, GPS, Inertial Navigation, …

5. Kalman Filter2. Applications Communications: MIMO

6. Kalman Filter2. Applications Communications: Video Motion Estimation

7. Kalman Filter2. Applications Bio-Medical:

8. Kalman Filter3. Big Picture

9. Kalman Filter3. Big Picture (continued)

10. Kalman Filter4. Formulas

11. Kalman Filter5. 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 Filter5. 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^2 (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); %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) %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;

13. Kalman Filter6. Example 1: Random Walk A = 1 ; C = 1 ; Q = 0.2; R = 0.3; A = 1 ; C = 1 ; Q = 0.2; R = 1;

14. Kalman Filter6. Example 1: Random Walk A = 1 ; C = 1 ; Q = 0.2; R = 0.3; A = 1 ; C = 1 ; Q = 0.2; R = 1;

15. Kalman Filter6. Example 1: Random Walk A = 1 ; C = 1 ; Q = 10; R = 10;

16. Kalman Filter7. Example 2: RW with unknown drift A = [1, 1; 0, 1]; C = [1, 0]; Q = [1; 0]; R = 0.5;

17. Kalman Filter8. Example 3: 1-D tracking, changing drift A = [1, 1; 0, 1]; C = [1, 0]; Q = [1; 0.1]; R = 0.5;

18. Kalman Filter9. Example 4: Falling body A = [1, 1; 0, 1]; C = [1, 0]; Q = [10; 0]; R = 40;

19. Kalman Filter10. Simulink