Esercizio sul calcolo della risposta forzata

Esercizio pag 245 cambiamento

X(k+1) = (1/2 1; 0 1/4) X(k) + (0; 1) u(k)

u(k) = 1, k > 0

Evoluzione forzata = ?

A = (1/2 1; 0 1/4)

B = (0; 1)

X(z) = [x₁(z); x₂(z)] = (zI - A)^-1 B U(z) = H(z)U(z)

ovv X(z) = H(z)U(z) = (zI - A)^-1 B U(z)

u(k) = 1, k > 0

U(z) = z/(z - 1)

(zI - A)^-1 = 1/det(zI - A) (zI - A)^c

zI - A = (z 0; 0 z) - (1/2 1; 0 1/4) = (z - 1/2 -1; 0 z - 1/4)

(zI - A)^c = (z - 1/2 -1; 0 z - 1/4)^c = (z - 1/4 1; 0 z - 1/2)

det(zI - A) = (z - 1/2)(z - 1/4)

H(z) = (zI - A)^-1 B = 1/((z - 1/2)(z - 1/4)) (z - 1/4 1; 0 z - 1/2) (0; 1)

= 1/((z - 1/2)(z - 1/4)) (1)

= [H₁(z); H₂(z)]

X(z) = [ 1/((z - 1/2)(z - 1/4)) ; (z - 1/2)/((z - 1/2)(z - 1/4)) ] U(z)

= [ 1/((z - 1/2)(z - 1/4)) ; (z - 1/2)/((z - 1/2)(z - 1/4)) ] (z/(z - 1)) = [x₁(z); x₂(z)]

X(k) = Z^-1[X(z)] = [ Z^-1[x₁(z)]; Z^-1[x₂(z)] ]

x₁(z) = z/((z - 1/2)(z - 1/4)(z - 1)) = R₁/(z - 1/2) + R₂/(z - 1/4) + R₃/(z - 1)

x₂(z) = (z - 1/2)z/((z - 1/2)(z - 1/4)(z - 1)) = R_1'/(z - 1/2) + R_2'/(z - 1/4) + R_3'/(z - 1)

X_f(k) = ?

si domanda propria

Esercizio pag 245 (Antonia?)

X(k+1) = [ 1/2 1/4 ] X(k) + [ 0 ] u(k)

[ 0 1/4 ] [ 1 ]

u(k) = 1, k > 0

Evoluzione forzata = ?

A = [ 1/2 1/4 ]

[ 0 1/4 ]

B = [ 0 ]

[ 1 ]

X(z) = [ x1(z) ] = (zI - A)^-1 B U(z) = H(z)U(z)

[ x2(z) ]

ovr X(z) = H(z)U(z) = (zI - A)^-1 B U(z)

U(k) = 1, k > 0

U(z) = z / (z - 1)

(zI - A)^-1 = 1/det(zI - A) (zI - A)^c

zI - A = [ z 0 ] - [ 1/2 1/4 ] = [ z-1/2 -1/4 ]

[ 0 z ] [ 0 1/4 ] [ 0 z-1/4 ]

(zI - A)^c = [ z-1/4 1/4 ]

[ 0 z-1/2 ]

det(zI - A) = (z-1/2)(z-1/4)

H(z) = (zI - A)^-1B = 1/((z-1/2)(z-1/4)) [ z-1/4 1/4 ] [ 0 ][ 1 ]

= 1/((z-1/2)(z-1/4)) [ 1/4 ][ (z-1/2)/((z-1/2)(z-1/4)) ]

= [ H1(z) ][ H2(z) ]

X(z) = [ 1/((z-1/2)(z-1/4)) ] U(z)

[ (z-1/2)/((z-1/2)(z-1/4)) ]

= [ 1/((z-1/2)(z-1/4)) ] (z/(z-1)) = [ X1(z) ]

[ (z-1/2)/((z-1/2)(z-1/4)) ] [ X2(z) ]

X(k) = Z^-1[X(z)] = [ Z^-1[X1(z)] ][ Z^-1[X2(z)] ]

X1(z) = z / ((z-1/2)(z-1/4)(z-1)) = R1/(z-1/2) + R2/(z-1/4) + R3/(z-1)

X2(z) = (z-1/2)z / ((z-1/2)(z-1/4)(z-1)) = R1'/(z-1/2) + R2'/(z-1/4) + R3'/(z-1)

X4(k) = ?

δ dx1(z) > δ mx1(z) (strettamente propria)