Formulario, Sistemi e modelli

Continui e discreti

Formule continue

CONTINUOX(t) = e^Ft X(0)e^Ft = T e^F_qt T^-1

V(x) = x^TP xṼ(x) = x^T(F^TP + PF) x

Q = - (F^TP + PF)

ω(t) = H e^Ft G̅ + Jδ(t)

W(s) = H(sI - F)^-1G + J

Ψ(s) = W(s) U(s) + H(sI - F)^-1 X(0)

Formule discrete

DISCRETOX(t) = F^t X(0)F^t = T F_q^t T^-1

ΔV(x) = + x^T[F_rPF - P] xQ = P - FPF

ω(t) = H F^(t+1) G̅ + Jδ(t)

W(z) = H(zI - F)^-1G̅ + J

Ψ(z) = W(z)U(z) + H(zI - F)^-1 z^j(0)

Stima e varianze

Θ̂_MV = (F^TR^-1F)^-1 F^TR^-1y

Var(Θ̂_MV) = (F^TR^-1F)^-1 = σⁱ₂

Θ̂_MQ = (G^TG)^-1 G^T z̅ = σⁱ₂[Θ̂ - kG, Θ̂ + kG]

k = 2 → 95%
k = 3 → 99%

Θ̂(y) = argmin_Θ̂ (- log (P(y; Θ)))

MAX VERI(Θ) = E [ (d/dt(-log(p₁, p₂)))² ]

MSE_Θ̂ = Var + BIAS² = (E(x²) - E(x)²) + ‖Θ - E[Θ̂(y)]‖²

Distribuzioni e probabilità

P(x̅) = 1/√(2π²G)e^{-1/2(x - μ)2/G2}

Formule alternative continue

ContinuoX(t) = e^FtX(0)e^Ft = T·e^βtT^-1

V(x) = x^TPx

V̇(x) = x^T(F^TP+PF)x

Q̇ = -(F^TP+PF)

ω(t) = H·e^FtḠ + Jδ(t)

W(s) = H·(sI-F)^-1G + J

Ψ(s) = W(s)·U(s) + H(sI-F)^-1X(0)

Formule alternative discrete

DiscretoX(t) = F^tX(0)F^t = T·F_β^tT^-1

ΔV(x) = x^T[FPF-p]x

Q₂ = p-FPF

ω(t) = H·F^(t₊₁)Ḡ + Jδ(t)

W(z̄) = H(zI-F)^-1G + J

Ψ(z) = W(z)U(z) + H(zI-F)^-1z^X(0)

Identificazioni e stime

Θ̂_MV = (F^TRF)^-1F^TRy

Var(Θ̂_MV) = (F^TRF)^-1 = σ_i²

Θ̂_MQ = (G^TG)^-1G^Tz̄ = σ_i²[Θ̂ - kG, Θ̂ + kG]

k = 2 → 95%
k = 3 → 99%

Θ̂(y) = argmin_Θ̂(-log(P(y;Θ)))

I(Θ) = E [ ( d/dt(-log(P₁,P₂)) )² ]

MSE_Θ̂ = Var + Bias² = (E(x²)-E(x)²) + ||Θ - E[Θ̂(y)]||²

P̂(x) = ¹/_√2πσe^{-1/2(x-μ)²/σ²}

=0 ⇒ = ° classi minori.= _{Σ (0)}/_Σ = _{Σ (∞)}/_Σ (∞)=-·=0 ⇒ vett lineare e autonomo·=- ➝ in presenza di ingressi(-)=0 ➝ discreti