1 Predittore di Kalman

1
Predittore di Kalman
Sistema:
(
x(t + 1) = F (t) x(t) + v1 (t)
•
y(t) = H(t) x(t) + v2 (t)
Ipotesi:
• E [v1 (t)] = 0
E [v2 (t)] = 0
∀t
• E [v1 (t1 ) v1 (t2 )0 ] = V1 (t1 ) δ(t2 − t1 )
∀t1 , t2
• E [v2 (t1 ) v2 (t2 )0 ] = V2 (t1 ) δ(t2 − t1 )
∀t1 , t2
• E [v1 (t1 ) v2 (t2 )0 ] = V12 (t1 ) δ(t2 − t1 )
(
1 t=0
• dove δ(t) =
0 t 6= 0
∀t1 , t2
• E [x(1)] = x1
• Var [x(1)] = P1
• E [vi (t) x(1)0 ] = 0
i = 1, 2 ∀t
Predittore:
• x̂(t + 1|t) = F (t) x̂(t|t − 1) + K(t) e(t)
• e(t) = y(t) − H(t) x̂(t|t − 1)
• P (t+1) = F (t) P (t) F (t)0 +V1 (t)−K(t)·[H(t) P (t) H(t)0 + V2 (t)]·K(t)0
• K(t) = [F (t) P (t) H(t)0 + V12 (t)] · [H(t) P (t) H(t)0 + V2 (t)]−1
• x̂(1|0) = x1
• P (1) = P1
1
2
Controllo a minima varianza generalizzato
w(t)-
y ◦ (t)-
H(z)
+
-f −
6
1
G(z)
C(z)
u(t)- −k
z B(z)
+
?
-f
+
F (z)
6
Sistema:
• A(z) y(t) = B(z) u(t − k) + C(z) w(t)
Cifra di merito:
• J = E (P (z) y(t + k) + Q(z) u(t) − y ◦ (t))2
Controllore:
• G(z) u(z) + F (z) y(t) − H(z) y ◦ (t) = 0
• G(z) = PD (z) · [B(z) QD (z) E(z) + C(z) QN (z)]
• F (z) = F̃ (z) QD (z)
• H(z) = C(z) PD (z) QD (z)
Equazione diofantina:
• C̃(z) = Ã(z) E(z) + F̃ (z) z −k
• C̃(z) = C(z) PN (z)
• Ã(z) = A(z) PD (z)
2
1
A(z)
y(t)-