PROJECT REPORT

(Project Semester May – August 2012)

ENHANCING RESULTS ON KALMAN FILTER

Submitted by

Vijit Dubey

Department of Electrical Engineering

Punjab Engineering College, Chandigarh

India

Under the Guidance of

Dr. Jeffrey Burl

Associate Professor, Electrical and Computer Engineering

Michigan Technological University

Houghton, Michigan

USA

ACKNOWLEDGEMENT

It has been an enriching experience for me to undertake a Project at Michigan Technological

University, Houghton. At the outset I would like to express my gratitude to my mentor Dr. Jeffrey

B. Burl, who assigned me this project after assessing my aptitude in this particular field and spent

his valuable time in giving inputs related to the project and clearing my doubts on the same.

There are some people without whom the completion of this Project was not possible. I would like

to express my deep regards to Dr. Dennis for wise inputs and the time he spared for discussions. I

would also like to thank the entire faculty of Electrical Engineering Department for giving their

valuable time to my Project.

Introduction

Some basic topics were covered during the internship project. Initially weeks we had a run though

of basic concepts of MATLAB. Plotting basic functions was learnt. Many other examples were

covered to get a basic idea of functionality of topics to be covered in the reviewing of the thesis.

Some of the examples covered are discussed below.

Plotting a sine wave

Plotting a cosine wave with amplitude at specific points

Plotting a composite wave

Plotting a Magnitude vs Frequency graph

MATLAB code:

clear % Equation for filter: y(k) =.65*sin(k)+.65*sin(k-1)+.65*sin(k-

2)+..+.65*sin(k-15) s=16; kt=zeros(s); fr=60; T=1/s/fr; % step is of 0.01 so T=1 would be fine y=zeros(1,s); tt=0:1/s/fr:(s-1)/s/fr; kt(1)=1; for k=1:s,

for n=1:s, p(n) = cos(2*pi*fr*(s+1-k)*kt(s+1-n)/(fr*s))+j*sin(2*pi*fr*(s+1-

k)*kt(s+1-n)/(fr*s)); y(k)=p(n); end

end % finding implulse at 16 points in a period figure(1) clf stem(tt,y); % finding frequency response % H(z) = .65+.65*z^(-1)+.65*z^(-2)+...+.65*z^(-15) f=0:0.1:300; z=exp(-j*2*pi*f*T); F=0; for v=1:s Hofz= y(s+1-v)*z.^(v-1); F=F+Hofz; end figure(2) clf subplot(311) plot(F); subplot(312) plot(f,abs(F)); subplot(313) plot(f,angle(F));

Plotting the results on graph

Work on KALMAN Gain

The results of thesis written by an alumni of MTU were quite unexpected. The model was working

perfectly if sampling was done at 10. However, if sampling was done at any other rate. The results were

coming out to be much unexpected.

The old MATLAB code and results are discussed below.

OLD MATLAB Code:

% 11 state clear sample_count=10; cycles=6; f=60; t=0:1/sample_count/60:cycles/60; % not using 15/16/60 to get a full wave

till 2*pi y=sin(2*f*pi*t); points=sample_count; dt=1/600; w=2*pi*60; lambda=0.5; P=[cos(w*dt) -sin(w*dt) 0 0 0 0 0 0 0 0 0; sin(w*dt) cos(w*dt) 0 0 0 0 0 0 0 0 0; 0 0 cos(2*w*dt) -sin(2*w*dt) 0 0 0 0 0 0 0; 0 0 sin(2*w*dt) cos(2*w*dt) 0 0 0 0 0 0 0; 0 0 0 0 cos(3*w*dt) -sin(3*w*dt) 0 0 0 0 0; 0 0 0 0 sin(3*w*dt) cos(3*w*dt) 0 0 0 0 0; % 11*11 0 0 0 0 0 0 cos(4*w*dt) -sin(4*w*dt) 0 0 0; 0 0 0 0 0 0 sin(4*w*dt) cos(4*w*dt) 0 0 0; 0 0 0 0 0 0 0 0 cos(5*w*dt) -sin(5*w*dt) 0; 0 0 0 0 0 0 0 0 sin(5*w*dt) cos(5*w*dt) 0; 0 0 0 0 0 0 0 0 0 0 exp(-dt/lambda)]; Q=P; C=[1 0 1 0 1 0 1 0 1 0 1]; U=1*diag([0.005^2 0.005^2 0.005^2 0.005^2 0.005^2 0.005^2 0.005^2 0.005^2

0.005^2 0.005^2 0.005^2]); % 11*11 B=0.01^2; M=1*diag([(100)/6 (100)/6 (5^2)/6 (5^2)/6 (1^2)/6 (1^2)/6 (1^2)/6 (1^2)/6

(2^2)/6 (2^2)/6 (10^2)/12]); % 11*11 vhat0=[0;0;0;0;0;0;0;0;0;0;0]; % 11*1 err=zeros(1,length(y)); for n=1:points K(:,n)=M*C'/(C*M*C'+B); % 11*1 Z=(eye(11)-K(:,n)*C)*M; % 11*11 M=P*Z*P'+Q*U*Q'; % 11*11 end

flag=0;

for n=1:length(y)

if n<=points if n==1 vhat(:,n)=P*vhat0+K(:,n)*(y(n)-C*P*vhat0); else vhat(:,n)=P*vhat(:,n-1)+K(:,n)*(y(n)-C*P*vhat(:,n-1)); end elseif flag==1 && k<=length(K(1,:)) vhat(:,n)=P*vhat(:,n-1)+K(:,k)*(y(n)-C*P*vhat(:,n-1)); k=k+1; if k==length(k(1,:)) flag=0; end else vhat(:,n)=P*vhat(:,n-1)+K(:,end)*(y(n)-C*P*vhat(:,n-1)); bla(:,n)=P*vhat(:,n-1); err(n)=abs(y(n)-y(n-points)); if err(n)>0.1 flag=1; k=1; end end

end

est=abs(vhat(1,:)-1i*vhat(2,:)); % fundamental est2=abs(vhat(3,:)-1i*vhat(4,:)); % 2 harmonic est3=abs(vhat(5,:)-1i*vhat(6,:)); % 3 harmonic est4=abs(vhat(7,:)-1i*vhat(8,:)); % 4 harmonic est5=abs(vhat(9,:)-1i*vhat(10,:)); % 5 harmonic est_dc=abs(vhat(11,:)); % dc offset

figure(1) plot(60.*t,y,'k',60.*t,est,'b:h',60.*t,est2,'g:+',60.*t,est3,'r-

s',60.*t,est4,'b:x',60.*t,est5,'b:o',60.*t,est_dc,'k-->') % figure(2) % plot(60.*t,est,'b') % figure(3) % plot(60.*t,est2,'g') grid on xlabel('number of 60hz cycles') ylabel('currrent A') Title('11 state model') legend('signal input','fundamental wave','2nd harmonic','3rd harmonic','4th

harmonic','5th harmonic','DC offset');

Results

for sample count = 10, cycles=6, frequency=60:

for sample count = 16, cycles=6, frequency=60:

for sample count = 16, cycles=6, frequency=60:

New MATLAB code after correction:

% 11 state clear sample_count=16; cycles=6; f=60; t=0:1/sample_count/60:cycles/60; % not using 15/16/60 to get a full wave

till 2*pi y=sin(2*f*pi*t); points=sample_count; fs=points*60; dt=1/fs; w=2*pi*60; lambda=0.5; P=[cos(w*dt) -sin(w*dt) 0 0 0 0 0 0 0 0 0; sin(w*dt) cos(w*dt) 0 0 0 0 0 0 0 0 0; 0 0 cos(2*w*dt) -sin(2*w*dt) 0 0 0 0 0 0 0; 0 0 sin(2*w*dt) cos(2*w*dt) 0 0 0 0 0 0 0; 0 0 0 0 cos(3*w*dt) -sin(3*w*dt) 0 0 0 0 0; 0 0 0 0 sin(3*w*dt) cos(3*w*dt) 0 0 0 0 0; % 11*11 0 0 0 0 0 0 cos(4*w*dt) -sin(4*w*dt) 0 0 0; 0 0 0 0 0 0 sin(4*w*dt) cos(4*w*dt) 0 0 0; 0 0 0 0 0 0 0 0 cos(5*w*dt) -sin(5*w*dt) 0; 0 0 0 0 0 0 0 0 sin(5*w*dt) cos(5*w*dt) 0; 0 0 0 0 0 0 0 0 0 0 exp(-dt/lambda)]; Q=P; C=[1 0 1 0 1 0 1 0 1 0 1]; U=1*diag([0.005^2 0.005^2 0.005^2 0.005^2 0.005^2 0.005^2 0.005^2 0.005^2

0.005^2 0.005^2 0.005^2]); % 11*11 B=0.01^2; M=1*diag([(100)/6 (100)/6 (5^2)/6 (5^2)/6 (1^2)/6 (1^2)/6 (1^2)/6 (1^2)/6

(2^2)/6 (2^2)/6 (10^2)/12]); % 11*11 vhat0=[0;0;0;0;0;0;0;0;0;0;0]; % 11*1 err=zeros(1,length(y)); for n=1:points K(:,n)=M*C'/(C*M*C'+B); % 11*1 Z=(eye(11)-K(:,n)*C)*M; % 11*11 M=P*Z*P'+Q*U*Q'; % 11*11 end

flag=0;

for n=1:length(y) if n<=points if n==1 vhat(:,n)=P*vhat0+K(:,n)*(y(n)-C*P*vhat0); else vhat(:,n)=P*vhat(:,n-1)+K(:,n)*(y(n)-C*P*vhat(:,n-1)); end elseif flag==1 && k<=length(K(1,:)) vhat(:,n)=P*vhat(:,n-1)+K(:,k)*(y(n)-C*P*vhat(:,n-1)); k=k+1; if k==length(k(1,:))

flag=0; end else vhat(:,n)=P*vhat(:,n-1)+K(:,end)*(y(n)-C*P*vhat(:,n-1)); bla(:,n)=P*vhat(:,n-1); err(n)=abs(y(n)-y(n-points)); if err(n)>0.1 flag=1; k=1; end end

end

est=abs(vhat(1,:)-1i*vhat(2,:)); % fundamental est2=abs(vhat(3,:)-1i*vhat(4,:)); % 2 harmonic est3=abs(vhat(5,:)-1i*vhat(6,:)); % 3 harmonic est4=abs(vhat(7,:)-1i*vhat(8,:)); % 4 harmonic est5=abs(vhat(9,:)-1i*vhat(10,:)); % 5 harmonic est_dc=abs(vhat(11,:)); % dc offset

figure(1) plot(60.*t,y,'k',60.*t,est,'b:h',60.*t,est2,'g:+',60.*t,est3,'r-

s',60.*t,est4,'b:x',60.*t,est5,'b:o',60.*t,est_dc,'k-->') % figure(2) % plot(60.*t,est,'b') % figure(3) % plot(60.*t,est2,'g') grid on xlabel('number of 60hz cycles') ylabel('currrent A') Title('11 state model') legend('signal input','fundamental wave','2nd harmonic','3rd harmonic','4th

harmonic','5th harmonic','DC offset');

Results

for sample count = 10, cycles=6, frequency=60:

for sample count = 16, cycles=6, frequency=60:

for sample count = 32, cycles=6, frequency=60:

Conclusion

The results received were due to a minor mistake in the MATLAB code. The changes were

rectified and accurate results were received.