Modele :
       dT2
Theta*-----+T2 = K*T1(t-Tho)
        dt
avec :
  Theta : cst de tps
  Tho : retard pur

1) Solution homogene (sans second
membre)

         dT2
Theta * ----- + T2 = 0
          dt

dy(t)/y(t) = -1/Theta * dt
=> ln(y)=1/Theta*int(0,T,1*dt)

ln(y)=-(1/Theta)*t + cste 

y=e^(-t/Theta + A)
 =e^A * e^-t/Theta
 
y(t)=C1*e^-t/Theta

T2(t)=C1*e^-t/Theta

T2(t=0)=C1=T20

T2L(t)=T20*e^-t/Theta

T1(t-2)=echelon=1    Theta*t>0

(Theta*d*T2F(t))/dt + T2F(t) =K
Condition initiale : T2F(0)=0

T2F(t)=C2*e^-t/Theta + C3

(d*T2F(t))/dt=C2/Theta *e^-t/Theta

-Theta*(C2/Theta)*e^-t/Theta
+ C2*e^-t/Theta + C3 = K

C3=K

T2F(t)=C2*e^-t/Theta + K

T2F(0)=C2+k=0 => C2=-k

T2F(t)=-k*e^-t/Theta +k
      =k(1-e^-t/Theta)
      
2.3)Equation numerique

Theta*p*T2(p)+T2(p)=k*T1(p).e^-Tho*p

T2(p)/T1(p) =
(K*e^-Tho*p)/(1+Theta*p)
-> Broida

T*z(F(p))=(1-z^-1)z(F(p)/p)

T2(z)/T1(z) = K(1-a)/z-a
avec a=e^-Delta/Theta

(k(1-a)z^-1)/(1-az^-1)

T2(z)(1-a*z^-1)=k(1-a)z^-1 *T1(z)

T2(z)-a*z^-1 *T2(z)
=k(1-a)z^-1 *T1(z)

T2(n)-a*T2(n-1)=k(1-a)T1(n-1-r)
=> Modele

2.5) Solutions libre/forcee

T2(k)-a*T2(k-1)=K(1-a)T1(k-1-r)

T2L(K+1)=a*T2m(k)
T2L(K+2)=a*T2m(K+1)=a^2 *T2m(k)
T2L(K+H)=a^H * T2m(k)

T2F(K)=K(1-a)
T2F(K+1)=T2F(K+1)+K(1-a)
        =K(1-a)a+K(1-a)
        =K(a-a^2 +1-a)
        =K(1-a^2)
T2F(K+2)=aT2F(K+2)+K(1-a)
        =aK(1-a^2)a+K(1-a)
        =K(a-a^3 +1-a)
        =K(1-a^3)
T2F(K+H)=K(1-a^H)

2.7)Trajectoire de reference
(1-Lamda^H)(c-T2p(n))=T2m(n+h)-T2m(n)
E(n+H)=Lamda^H *E(n)

2.9) Regulateur predictif

E(n)=c-T2p(n)

T1(z)=(1-Lamda^H)/(K(1-a^H)) 
*E(z) + T2m(z)/K

T2m(z)/T1(z)= (z^-r *(1-a)Kz^-1)
              ------------------
                   1-az^-1
                   
Schema bloc cours