Probability in eecs
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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 ) | Y n ] Linear recursive filter, innovation gain K n , error covariance Σ n

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Probability in EECS

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Probability in eecs

Probability in EECS

Jean Walrand – EECS – UC Berkeley

Kalman Filter


Kalman filter overview

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 filter 1 kalman

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 filter 2 applications

Kalman Filter2. Applications

Tracking and Positioning:

Apollo Program, GPS, Inertial Navigation, …


Kalman filter 2 applications1

Kalman Filter2. Applications

Communications:

MIMO


Kalman filter 2 applications2

Kalman Filter2. Applications

Communications:

Video Motion Estimation


Kalman filter 2 applications3

Kalman Filter2. Applications

Bio-Medical:


Kalman filter 3 big picture

Kalman Filter3. Big Picture


Kalman filter 3 big picture continued

Kalman Filter3. Big Picture (continued)


Kalman filter 4 formulas

Kalman Filter4. Formulas


Kalman filter 5 matlab

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 filter 5 matlab complete

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 filter 6 example 1 random walk

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 filter 6 example 1 random walk1

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 filter 6 example 1 random walk2

Kalman Filter6. Example 1: Random Walk

A = 1 ;

C = 1 ;

Q = 10;

R = 10;


Kalman filter 7 example 2 rw with unknown drift

Kalman Filter7. Example 2: RW with unknown drift

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

C = [1, 0];

Q = [1; 0];

R = 0.5;


Kalman filter 8 example 3 1 d tracking changing drift

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 filter 9 example 4 falling body

Kalman Filter9. Example 4: Falling body

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

C = [1, 0];

Q = [10; 0];

R = 40;


Kalman filter 10 simulink

Kalman Filter10. Simulink


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