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%%% ------ ***** Tanzeela Shakeel****----------%%%%%%%------- **** FA13-MSEE-019 *****------------%%%%%%%------- **** Assignment #1 *****------------%%%%%%%------- ****Date: 13-03-2014 ****------%%%%%%%------- **** Non Linear Control System ****------%%%%%%%------- **** Phase portrait ****------%%%%

Question #1

functiondy=myfunc(t,y)clcclose alldy(1)=y(2);dy(2)=-3*y(1)-y(2)-y(1)*y(1);

dy=dy';return

Calling Function file code Q# 1

clcclose all

options=odeset('Maxstep',0.01);[t,y]=ode45('myfunc',[0,20],[-2;4],options);[t1,y1]=ode45('myfunc',[0,20],[-2;6],options);[t2,y2]=ode45('myfunc',[0,20],[2;6],options);

figureplot(y2(:,1),y2(:,2),'g',y1(:,1),y1(:,2),'r', y(:,1),y(:,2),'b')

title('Phase Potrait of Q1') % titlelegend('initial condition 2', 'initial condition 1','initial condition ')legend('Location','NorthWest') % move legend to upper left

figure

plot(t, y(:,1),'r')hold on

plot(t,y(:,2),'b')title('time potrait of Q1') % title

legend('initial condition1 y1','initial condition1 y2')

legend('Location','NorthWest') % move legend to upper leftfigure

plot(t1, y1(:,1),'r')hold onplot(t1,y1(:,2),'b')title('time potrait of Q1') % title

legend('initial condition1 y1','initial condition1 y2')legend('Location','NorthWest') % move legend to upper left

figure

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plot(t2, y2(:,1),'r')hold onplot(t2,y2(:,2),'b')

title('time potrait of Q2') % titlelegend('initial condition1 y1','initial condition1 y2')legend('Location','NorthWest') % move legend to upper left

Figure

Question #2

functiondx=myfu(t,x)clcclose alldx(1)=x(2);%e=0.1;e=1.0%e=5.0;dx(2)=-x(1)+e*(1-x(1)*x(1))*x(2);dx=dx';

-3 -2 -1 0 1 2 3 4-6

-4

-2

0

2

4

6

Phase Potrait of Q1

initial condition 2

initial condition 1

initial condition

0 2 4 6 8 10 12 14 16 18 20-2

-1

0

1

2

3

4

time potrait of Q1

initial condition1 y1

initial condition1 y2

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return

Calling Function file code Q# 2

options=odeset('Maxstep',0.01);[t,x]=ode45('myfu',[0,20],[-2;4],options);% figure

% plot(x(:,1),x(:,2));

[t1,x1]=ode45('myfu',[0,20],[-2;6],options);

% figure%plot(x1(:,1),x1(:,2),'r', x(:,1),x(:,2))

[t2,x2]=ode45('myfu',[0,20],[2;6],options);

figureplot(x2(:,1),x2(:,2),'g',x1(:,1),x1(:,2),'r', x(:,1),x(:,2),'b')title('Phase Potrait of Q2') % title

legend('initial condition 2', 'initial condition 1','initial condition ')legend('Location','NorthWest') % move legend to upper left

figure

plot(t, x(:,1),'r')hold on

plot(t,x(:,2),'b')title('time potrait of Q2') % title

legend('initial condition1 x1','initial condition1 x2')legend('Location','NorthWest') % move legend to upper left

figure

plot(t1, x1(:,1),'r')hold onplot(t1,x1(:,2),'b')

title('time potrait of Q2') % titlelegend('initial condition1 x1','initial condition1 x2')legend('Location','NorthWest') % move legend to upper left

figure

plot(t2, x2(:,1),'r')hold onplot(t2,x2(:,2),'b')

title('time potrait of Q2') % title

legend('initial condition1 x1','initial condition1 x2')legend('Location','NorthWest') % move legend to upper left

Figure

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Question #3

functiondy =funct(t,y)

clcclose all%y(1)=y%y(2)=dydy(1)=y(2);

ify(1)*y(2)>0

dy(2)= -(2*y(1))

else

dy(2)=-(1/2*(y(1)))

enddy=dy'

return

Calling Function file code Q# 3

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

4

5

6

7

Phase Potrait of Q2

initial condition 2

initial condition 1

initial condition

0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4

5

time potrait of Q2

initial condition1 x1

initial condition1 x2

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clcclose all

options=odeset('Maxstep',0.01);[t,y]=ode45('funct',[0,20],[-2;4],options);%figure

% plot(y(:,1),y(:,2));

% [t1,y1]=ode45('funct',[0,20],[-2;6],options);

%figure%plot(y1(:,1),y1(:,2),'r', y(:,1),y(:,2))

[t2,y2]=ode45('funct',[0,20],[2;6],options);

figureplot(y2(:,1),y2(:,2),'g',y1(:,1),y1(:,2),'r', y(:,1),y(:,2),'b')title('Phase Potrait of Q3') % title

legend('initial condition 2', 'initial condition 1','initial condition ')legend('Location','NorthWest') % move legend to upper left

figure

plot(t, y(:,1),'r')hold on

plot(t,y(:,2),'b')

title('time potrait of Q3') % titlelegend('initial condition y1','initial condition y2')legend('Location','NorthWest') % move legend to upper left

figure

plot(t1, y1(:,1),'r')hold onplot(t1,y1(:,2),'b')

title('time potrait of Q3') % titlelegend('initial condition1 y1','initial condition1 y2')legend('Location','NorthWest') % move legend to upper left

figure

plot(t2, y2(:,1),'r')hold onplot(t2,y2(:,2),'b')

title('time potrait of Q3') % titlelegend('initial condition1 y1','initial condition1 y2')legend('Location','NorthWest') % move legend to upper left

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Figure

Question #4functiondy =func(t,y)clcclose all%y(1)=y%y(2)=dydy(1)=y(2);

ify(1)+y(2)>0

dy(2)= -1 ;

elsey(1)+y(2)

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Calling Function file code Q# 3

clcclose all

options=odeset('Maxstep',0.01);[t,y]=ode45('func',[0,20],[-2;4],options);%figure%plot(y(:,1),y(:,2));

[t1,y1]=ode45('func',[0,20],[-2;6],options);

%figure%plot(y1(:,1),y1(:,2),'r', y(:,1),y(:,2))

[t2,y2]=ode45('func',[0,20],[2;6],options);

figureplot(y2(:,1),y2(:,2),'g',y1(:,1),y1(:,2),'r', y(:,1),y(:,2),'b')

title('Phase Potrait of Q4') % titlelegend('initial condition 2', 'initial condition 1','initial condition ')legend('Location','NorthWest') % move legend to upper left

% time potrait

figure

plot(t, y(:,1),'r')hold on

plot(t,y(:,2),'b')title('time potrait of Q4') % titlelegend('initial condition y1','initial condition y2')legend('Location','NorthWest') % move legend to upper left

figure

plot(t1, y1(:,1),'r')hold onplot(t1,y1(:,2),'b')

title('time potrait of Q4') % titlelegend('initial condition1 y1','initial condition1 y2')legend('Location','NorthWest') % move legend to upper left

figure

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plot(t2, y2(:,1),'r')hold onplot(t2,y2(:,2),'b')title('time potrait of Q4') % title

legend('initial condition2 y1','initial condition2 y2')legend('Location','NorthWest') % move legend to upper left

Figure

-10 -5 0 5 10 15 20 25-6

-4

-2

0

2

4

6

Phase Potrait of Q4

initial condition 2

initial condition 1

initial condition

0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4

5

6

7

time potrait of Q4

initial condition y1

initial condition y2

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Question #5

Figure: