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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

Bio-Medical:

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

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

C = [1, 0];

Q = [1; 0.1];

R = 0.5;

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