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# Probability in EECS - PowerPoint PPT Presentation

Probability in EECS. Jean Walrand – EECS – UC Berkeley. Kalman Filter. Kalman Filter: Overview. Overview X(n+1) = AX(n) + V(n); Y(n) = CX(n) + W(n); noise ⊥ KF computes L[X(n ) | Y n ] Linear recursive filter, innovation gain K n , error covariance Σ n

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### Probability in EECS

Jean Walrand – EECS – UC Berkeley

Kalman Filter

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)

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

Kalman Filter2. Applications

Tracking and Positioning:

Apollo Program, GPS, Inertial Navigation, …

Kalman Filter2. Applications

Communications:

MIMO

Kalman Filter2. Applications

Communications:

Video Motion Estimation

Kalman Filter2. Applications

Bio-Medical:

Kalman Filter3. Big Picture

Kalman Filter3. Big Picture (continued)

Kalman Filter4. Formulas

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

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;

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;

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;

Kalman Filter6. Example 1: Random Walk

A = 1 ;

C = 1 ;

Q = 10;

R = 10;

Kalman Filter7. Example 2: RW with unknown drift

A = [1, 1; 0, 1];

C = [1, 0];

Q = [1; 0];

R = 0.5;

Kalman Filter8. Example 3: 1-D tracking, changing drift

A = [1, 1; 0, 1];

C = [1, 0];

Q = [1; 0.1];

R = 0.5;

Kalman Filter9. Example 4: Falling body

A = [1, 1; 0, 1];

C = [1, 0];

Q = [10; 0];

R = 40;