Teoria Sistemelor si Reglajelor Automate

Teoria Sistemelor si Reglajelor Automate

Lucrare Personala

1.Studiul elementului aperiodic de ordin 1

2.Studiul SLN invariant de ordinul 2

I.Studiul elementului aperiodic de ordinul 1

1.1Modele matematice de tipul intrare-iesire:ecuatia diferentiala,functia de transfer,caracteristici de frecventa.

a)=>T=K∙r

b)

c)W(jω)=A(ω)∙

1.2.Scheme de modelare,in Simulink,pentru calculul raspunsului indicial(K=1,T=5(s))

1.2.1.Schema de modelare in baza ecuatiei diferentiale;

1.2.2.Schema de modelare in baza functiei de transfer

1.3.Calculul raspunsului indicial si a functiei pondere in MATLAB.

figure(1)

%Raspunsul indicial al EIO1

t=0:0.01:20;

num=[1];den=[5 1];

ys=step(num,den,t);

v=t;df=diff(v)./diff(t);df1=0.95*df;td=t(2:length(t));

plot(t,ys,'-r',td,df,'-k',td,df1,'-k'),grid

figure(2)

%Calculul functiei pondere pt EIO1

t=0:0.01:30;

num=[1];den=[5 1];yi=impulse(num,den,t);

plot(t,yi,'-r'),grid

title('Functia pondere a EIO1')

1.4.Determinarea performantelor in raport cu referinta treapta unitara pentru K=1 si T=5(s),utilizand una din variantele 1.2.1,1.2.2. sau 1.3.

t=0:0.01:40;

T=5;

y=1-exp(-t/T);

v=t;

df=diff(v)./diff(t);

df1=0.95*df;df2=1.05*df;

td=t(2:length(t));

plot(t,y,'-r',td,df,'-k',td,df1,'-b',td,df2,'-k'),grid

xlabel('t(s)')

ylabel('y(t)')

[X,Y]=ginput

1.5.Calculul caracteristicilor de frecventa U(ω),V(ω),A(ω),φ(ω),pentru K=3 si T=5(s).

k=3;T=5;

%omega=w

w=-10:0.01:10;

figure(1)

%caract.U(omega)

subplot(221)

u=k./(1+(w*T).^2);plot(w,u,'-k'),grid

title('Caract. U(omega)')

xlabel('omega'),ylabel('U(w)')

%Caract.V(omega)

subplot(222)

v=-(k*w*T)./(1+(w*T).^2);plot(w,v,'-k'),grid

title('Caract. V(omega)')

xlabel('omega'),ylabel('V(w)')

%Caract.A(omega)

subplot(223)

a=sqrt(u.^2+v.^2);plot(w,a,'-k'),grid

title('Caract. A(omega)')

xlabel('omega'),ylabel('A(w)')

%Caract.Fi(omega)

subplot(224)

Fi=-atan(w*T);plot(w,Fi,'-k'),grid

title('Caract. Fi(omega)')

xlabel('omega'),ylabel('Fi(w)')

figure (2)

%Locul de transfer

w=-60:0.01:60;

u=k./(1+(w*T).^2);v=-(k*w*T)./(1+(w*T).^2);

plot(k./(1+(w*T).^2),-(k*w*T)./(1+(w*T).^2),'-k'),grid

title('Locul de transfer')

xlabel('U(omega)'),ylabel('V(omega)')

figure(3)

%Caract. lg. de pulsatie A(omega),Fi(omega)

w=logspace(-1,1,200);

num=[3];den=[5 1];[mag,phase,w]=bode(num,den);

subplot(211)

semilogx(w,20*log10(mag)),grid

title('Caract.lg.A(omega)')

xlabel('omega'),ylabel('Adb(omega)')

subplot(212)

semilogx(w,phase),grid

title('Caract.lg.Fi(omega)')

xlabel('omega'),ylabel('Fi(omega)')

1.6.Calculul caracteristicilor logaritmice de frecventa A(ω) si φ(ω),cu programe in MATLAB(K=10,T=5(s)).

%Caracteristici logaritmice de pulsatie A(omega),Fi(omega)

w=logspace(-1,1,200);

num=[3];den=[5 1];[mag,phase,w]=bode(num,den);

subplot(211)

semilogx(w,20*log10(mag)),grid

title('Caracteristica logaritmica amplitudine-pulsatie')

xlabel('omega(rad.sec)'),ylabel('Adb(omega)')

subplot(212)

semilogx(w,phase),grid

title('Caracteristica logaritmica faza-pulsatie')

xlabel('omega(rad.sec)'),ylabel('fi(grade)')

II.Studiul SLN invariant de ordinul 2

2.1.Modele matematice de tipul intrare-iesire si respectiv intrare-stare-iesire(T=1(s));

;;

;;

;

Y(t)=C∙X(t).

2.2.Scheme de modelare in Simulink(T=1(s))

2.2.1.Schema de modelare in baza ecuatiei diferentiale;

2.2.2.Schema de modelare in baza functiei de transfer;

2.2.3.Intocmirea schemei de modelareutilizand variabilele de faza

2.3.Calculul functiei pondere,pentru ξ=0.25 si ω=2π cu program in MATLAB.

t=0:0.01:4;

csi=0.25;w=2*pi;%w=omega

a=exp(-csi*w*t);

b=sqrt(1-csi^2);fi=acos(csi);

c=w*b;d=sin(c*t+fi);

y=1-(a/b).*d;

yst=1;ym=max(y)

sigma=ym-yst

tr1=log(0.05*b)/(-csi*w)

tr2=4/(csi*w)

v=t;df=diff(v)./diff(t);

df1=1.05*df;df2=0.95*df

td=t(2:length(t))

plot(t,y,'-r',td,df,'-b',td,df1,'-k',td,df2,'-k'),grid

xlabel('t(s)')

ylabel('y(t)')

gtext('Raspunsul SLN2 cu csi=0.25')

2.4.Calculul raspunsului indicial pentru ξ=0.5 si ω=2π utilizand functia de transfer si variabilele de stare,cu program in MATLAB.

num=[3 2];num=conv(num,[1 1]);num=conv(num,[2 3]);num=conv(num,[1 1 1])

den=[1 2 3 4 3 1]

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

iu=1;%Nr. de intrari

[num1,den1]=ss2tf(A,B,C,D,iu)

[Z,P,K]=ss2zp(A,B,C,D,iu)

[A1,B1,C1,D1]=zp2ss(Z,P,K)

[num2,den2]=zp2tf(Z,P,K)

w=2*pi;csi=0.25;A=[0 1;-w^2 -2*csi*w];B=[0;w^2];C=[1 0];D=[0];

t=0:0.001:4;

figure(1)

% calc rasp imdicial

x0=[0;0];%vect. starii initiale

v=t;u=diff(v)./diff(t);df1=1.05*u;df2=0.95*u;

td=t(2:length(t));

[y,x]=lsim(A,B,C,D,u,td,x0);

plot(td,y,'-r',td,df1,'-k',td,df2,'-k'),grid

2.5.Determinarea performantelor in raport cu referinta treapta unitara pentru T=1(s);ξ=0.25;0.707;1.0;3.0.,utilizand una din variantele 2.2.1.,2.2.2.,2.2.3.

figure(1)

t=0:0.01:7;

csi1=0.25;csi2=0.707;w=2*pi;

num1=[w^2];num2=num1;

den1=[1 2*csi1*w w^2];

den2=[1 2*csi2*w w^2];

ys1=step(num1,den1,t);

ys2=step(num2,den2,t);

ym1=max(ys1);yst=1;sigma1=ym1-yst

ym2=max(ys2);sigma2=ym2-yst

v=t;df=diff(v)./diff(t);

df1=0.95*df;df2=1.05*df;td=t(2:length(t));

plot(t,ys1,'-r',t,ys2,'-b',td,df,'-k',td,df1,'-k',td,df2,'-k'),grid

figure(2)

%SLN cu csi=1.0 si csi=3.0

t=0:0.01:7;

csi1=1.0;csi2=3.0;w=2*pi;

y1=1-(1+w*t).*exp(-w*t);

r21=-62.1979;r22=-0.6347;

y2=1-(r22/(r22-r21))*exp(r21*t)+(r21/(r22-r21))*exp(r22*t);

v=t;df=diff(v)./diff(t);df1=0.95*df;

td=t(2:length(t));

plot(t,y1,'-r',t,y2,'-b',td,df1,'-k'),grid