Modelarea si Simularea unui sistem electric

Modelarea si Simularea unui sistem electric

Se considera un circuit RLC:

intr: U(t)

ies: UL

R1=5

R2=100

C=150uF

L=5H

Formule:

Legea lui Ohm-> UR(t)=iR(t)*R

;

Aceasta schema simuleaza un proces electric in care avem o intrare, respectiv o iesire.

In aceasta schema sunt 3 tipuri fundamentale de comportament din punct de vedere energetic :

Element de tip I (Inductiv):

Aceste elemente modeleaza un comportament de tip acumulativ de energie similar comportamentului bobinelor, sau similar comportamentului maselor in miscare cu energie cinetica.

Element de tip C (Capacitiv):

Modeleaza elemente fizice care acumuleaza energie similar condensatoarelor.

Element de tip R (rezistiv):

Modeleaza elemente similare ca si comportament cu rezistente.

Aceste elemente sunt denumite:

I,C -> acumulative

R  -> disipatoare

Din punct de vedere energetic rationamentele sunt similare pentru orice domeniu al fizici.Fiecare domeniu utilizeaza variabile specifice pentru descrierea fenomenelor.

Modelarea sistemului:

In acest process electric am folosit doua legi ale fizicii:

Legea lui Ohm

Legea II a lui Kirchoff

Calculele pentru modelarea matematica:

Modelul I/O

i1=i2+i3

U(t)=i1*R1+i2*R2+UC

0=UL-UC-i2*R2=>+i2R2=UL

i3=

UC=

U(t)=i1*R1+UL=>i1=

i2=i1-i3=

U(t)=U-UL+

=>

Modelul ISO

ISO:

U(t)=R1+R2+C

y=Cx+Du

Pentru simularea sistemului continuu si discret am ales testarea sistemului la semnalul treapta, impuls si rampa.

Model I/O

Treapta

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

clear all;

close all;

clc;

num=[0.0418 44.44 ];

den=[1 47.77 14814814.81 0];

figure(1);

step(num,den)

xlabel('Treapta cu semnal continuu')

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

num=[0.0418 44.44 ];

den=[1 47.77 14814814.81 0];

[num,den]=c2dm(num,den,0.01,'zoh')

figure(1)

dstep(num,den)

xlabel('Treapta cu semnal discret')

Impuls

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

num=[0.0418 44.44 ];

den=[1 47.77 14814814.81 0];

[num,den]=c2dm(num,den,0.01,'zoh')

t=0:0.1:10;

u=t;

impulse(num,den)

xlabel('Impuls cu semnal continuu');

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

num=[0.0418 44.44 ];

den=[1 47.77 14814814.81 0];

[num,den]=c2dm(num,den,0.01,'zoh')

t=0:0.1:10;

u=t;

dimpulse(num,den,u)

xlabel('Impuls cu semnal discret');

Rampa

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

num=[0.0418 44.44 ];

den=[1 47.77 14814814.81 0];

t=0:0.1:10;

u=t;

dlsim(num,den,u)

xlabel('Rampa cu semnal continuu');

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

num=[0.0418 44.44 ];

den=[1 47.77 14814814.81 ];

[num,den]=c2dm(num,den,0.01,'zoh')

t=0:0.1:10;

u=t;

dlsim(num,den,u)

xlabel('Rampa cu semnal discret');

Model ISO

Treapta

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

R1=50;

R2=100;

C=0.00015;

L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0];

D=0;

ts=0.0000001;

[num,den]=ss2tf(A,B,C,D);

figure(1)

step(num,den)

xlabel('Treapta cu semnal continuu')

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

R1=50;

R2=100;

C=0.00015;

L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0];

D=0;

ts=0.0000001;

[num,den]=ss2tf(A,B,C,D);

[num,den]=c2dm(num,den,ts,'zoh')

figure(1)

dstep(num,den)

xlabel('Treapta cu semnal discret')

Impuls

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

R1=50;

R2=100;

C=0.00015;

L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0];

D=0;

ts=0.0000001;

[num,den]=ss2tf(A,B,C,D);

figure(1)

impulse(num,den)

xlabel('Impuls cu semnal continuu')

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

R1=50;

R2=100;

C=0.00015;

L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0];

D=0;

ts=0.0000001;

[num,den]=ss2tf(A,B,C,D);

[num,den]=c2dm(num,den,ts,'zoh')

figure(1)

dimpulse(num,den)

xlabel('Impulse cu semnal discret')

Rampa

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

R1=50;

R2=100;

C=0.00015;

L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0];

D=0;

t=0:0.1:10;

u=t;

[num,den]=ss2tf(A,B,C,D);

figure(1)

dlsim(num,den,u)

xlabel('Rampa cu semnal continuu')

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

R1=50;

R2=100;

C=0.00015;

L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0];

D=0;

t=0:0.1:10;

u=t;

ts=0.0000001;

[num,den]=ss2tf(A,B,C,D);

[num,den]=c2dm(num,den,ts,'zoh')

figure(1)

dlsim(num,den,u)

xlabel('Rampa cu semnal discret')

Pentru analiza stabilitatii am ales diagrama Nyquist si Bode in frecventa.

I/O

Bode

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

clear all;

close all;

clc;

num=[0.0418 44.44 ];

den=[1 47.77 14814814.81 0];

figure(1);

bode(num,den)

xlabel('Diagrama Bode cu semnal continuu')

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

num=[0.0418 44.44 ];

den=[1 47.77 14814814.81 0];

[num,den]=c2dm(num,den,0.01,'zoh')

figure(1)

bode(num,den)

xlabel('Diagrama Bode cu semnal discret')

Nyquist

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

clear all;

close all;

clc;

num=[0.0418 44.44 ];

den=[1 47.77 14814814.81 0];

figure(1);

nyquist(num,den)

xlabel('Diagrama Nyquist cu semnal continuu')

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

num=[0.0418 44.44 ];

den=[1 47.77 14814814.81 0];

[num,den]=c2dm(num,den,0.001,'zoh')

figure(1)

nyquist(num,den)

xlabel('Diagrama Nyquist cu semnal discret')

ISO

Bode

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

R1=50;

R2=100;

C=0.00015;

L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0];

D=0;

[num,den]=ss2tf(A,B,C,D);

figure(1)

bode(num,den)

xlabel('Diagrama Bode cu semnal continuu')

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

R1=50;

R2=100;

C=0.00015;

L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0];

D=0;

ts=0.0000001;

[num,den]=ss2tf(A,B,C,D);

[num,den]=c2dm(num,den,ts,'zoh')

figure(1)

Bode(num,den)

xlabel('Diagrama Bode cu semnal discret')

Nyquist

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

R1=50;

R2=100;

C=0.00015;

L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0];

D=0;

[num,den]=ss2tf(A,B,C,D);

figure(1)

nyquist(num,den)

xlabel('Diagrama Nyquist cu semnal continuu')

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

R1=50;

R2=100;

C=0.00015;

L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0];

D=0;

ts=0.0000001;

[num,den]=ss2tf(A,B,C,D);

[num,den]=c2dm(num,den,ts,'tustin')

figure(1)

nyquist(num,den)

xlabel('Diagrama Nyquist cu semnal discret')

Observabilitatea si Controlabilitatea

Model I/O

clear all

close all

num=[0.0418 44.44 ];

den=[1 47.77 14814814.81]

[A,B,C]=tf2ss(num,den);

n=length(A);

Mc=Ctrb(A,B);

Mo=Obsv(A,C);

Rank(Mc);

Rank(Mo);

disp('Sistemul este:')

if n==Rank(Mc)

disp('Controlabil,')

else

disp('Necontrolabil,')

end

disp('si')

if n==Rank(Mo)

disp('Observabil.')

else

disp('Neobservabil.')

end

[ABAR,BBAR,CBAR,T,K] = CTRBF(A,B,C)

Model ISO

clc

clear all

close all

R1=50;

R2=100;

C=0.00015;

L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0];

n=length(A);

Mc=Ctrb(A,B);

Mo=Obsv(A,C);

Rank(Mc);

Rank(Mo);

disp('Sistemul este:')

if n==Rank(Mc)

disp('Controlabil,')

else

disp('Necontrolabil,')

end

disp('si')

if n==Rank(Mo)

disp('Observabil.')

else

disp('Neobservabil.')

end

[ABAR,BBAR,CBAR,T,K] = CTRBF(A,B,C)

Identificarea experimentala

Pentru identificarea sistemului am ales functia ARX nerecursiva

Model I/O

Implementarea programului :

num=[0.0418 44.44 0];

den=[1 47.77 14814814.81];

Ts=0.1;

t=0:Ts:10;

sysc=tf(num,den);

sysd = c2d(sysc,Ts,'tustin');

u=ones(size(t));

u=square(t);u=u';

%bucla deschisa fara zg

y=lsim(sysd,u,t);

stairs(t,y);

save date_bd_fz u y Ts;

%bucla deschisa cu zg

zg=(2*rand(size(y))-1)*0.01;

y=y+zg

figure(2);

stairs(t,y);

save date_bd_cz u y Ts;

%bucla inchisa fara zg

ut=u

sysi=feedback(sysd,1,-1);

y=lsim(sysi,ut,t);

u=ut-y;

y1=y

figure(3);

stairs(t,y)

save date_bi_fz u y Ts;

%bucla inchisa cu zg var1

y=y+zg

save date_bi_cz1 u y Ts;

%bucla inchisa cu zg var2

sys2=feedback(1,sysd,-1);

y2=lsim(sys2,zg,t);

y=y1+y2;

u=ut-y;

save date_bi_cz2 u y Ts;

%----- ----- --------------identificare----- ----- ----------------

clear all;

close all;

clc;

m=menu('alegeti datele dorite','bucla deschisa fara zg','bucla deschisa cu zg','bucla inchisa fara zg','bucla inchisa cu zgomot var1','bucla inchisa cu zgomot var2')

if m==1 ;

load date_bd_fz;

end

if m==2 ;

load date_bd_cz;

end

if m==3;

load date_bi_fz;

end

if m==4;

load date_bi_cz1;

end

if m==5

load date_bi_cz2

end

dat=iddata(y,u,Ts);

nn=[3 3 0];

M=arx(dat,nn);

yest=idsim(u,M);

eest=yest-y;

plot(yest,'r');hold on;

plot(y,'.b');

figure

plot(eest)

Model ISO

R1=50; R2=100; C=0.00015; L=5;

A=[-(R1+R2)/C -1/C ; 1/L 0 ];

B=[1/C ; 0];

C=[1 0]; D=0;

[num,den]=ss2tf(A,B,C,D);

Ts=0.1;

t=0:Ts:10;

sysc=tf(num,den);

sysd = c2d(sysc,Ts,'tustin');

u=ones(size(t));

u=square(t);u=u';

%bucla deschisa fara zg

y=lsim(sysd,u,t);

stairs(t,y);

save date_bd_fz u y Ts;

%bucla deschisa cu zg

zg=(2*rand(size(y))-1)*0.01;

y=y+zg

figure(2);

stairs(t,y);

save date_bd_cz u y Ts;

%bucla inchisa fara zg

ut=u

sysi=feedback(sysd,1,-1);

y=lsim(sysi,ut,t);

u=ut-y;

y1=y

figure(3);

stairs(t,y)

save date_bi_fz u y Ts;

%bucla inchisa cu zg var1

y=y+zg

save date_bi_cz1 u y Ts;

%bucla inchisa cu zg var2

sys2=feedback(1,sysd,-1);

y2=lsim(sys2,zg,t);

y=y1+y2;

u=ut-y;

save date_bi_cz2 u y Ts;

%----- ----- --------------identificare----- ----- ----------------

clear all;

close all;

clc;

m=menu('alegeti datele dorite','bucla deschisa fara zg','bucla deschisa cu zg','bucla inchisa fara zg','bucla inchisa cu zgomot var1','bucla inchisa cu zgomot var2')

if m==1 ;

load date_bd_fz;

end

if m==2 ;

load date_bd_cz;

end

if m==3;

load date_bi_fz;

end

if m==4;

load date_bi_cz1;

end

if m==5

load date_bi_cz2

end

dat=iddata(y,u,Ts);

nn=[3 3 0];

M=arx(dat,nn);

yest=idsim(u,M);

eest=yest-y;

plot(yest,'r');hold on;

plot(y,'.b');

figure

plot(eest)

Afisarea grafica :

Model I/O

Treapta

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

Impuls

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

Rampa

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

Model ISO

Treapta

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

Impuls

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

Rampa

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

I/O

Bode

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

Nyquist

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

ISO

Bode

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

Nyquist

-------- ----- ------ ---Semnal continuu-------- ----- ------ --

-------- ----- ------ ---Semnal discret-------- ----- ------ -----

Observabilitatea si Controlabilitatea

Model I/O

den =

1.0e+007 *

0.0000 0.0000 1.4815

Sistemul este:

Controlabil,

si

Observabil.

ABAR =

1.0e+007 *

0 0.0000

-1.4815 -0.0000

BBAR =

0

1

CBAR =

1.0e+005 *

-6.1926 0.0004

T =

0 1

1 0

K =

1 1

Model ISO

Sistemul este:

Controlabil,

si

Observabil.

ABAR =

1.0e+006 *

0 0.0000

-0.0067 -1.0000

BBAR =

1.0e+003 *

CBAR =

1

T =

1

0

K =

1

Identificarea experimentala

Model I/O

Bucla deschisa fara zgomot

Bucla deschisa cu zgomot

Bucla inchisa fara zgomot

Bucla inchisa cu zgomot var1

Bucla inchisa cu zgomot var2

Model ISO

Bucla deschisa fara zgomot

Bucla deschisa cu zgomot

Bucla inchisa fara zgomot

Bucla inchisa cu zgomot var1

Bucla inchisa cu zgomot var2