Probability in eecs
Download
1 / 19

Probability in EECS - PowerPoint PPT Presentation


  • 107 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Probability in EECS' - tara


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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


ad